Handbook of Thermal Plasmas 3030849341, 9783030849344

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Maher I. Boulos Pierre L. Fauchais Emil Pfender Editors

Handbook of Thermal Plasmas

Handbook of Thermal Plasmas

Maher I. Boulos • Pierre L. Fauchais • Emil Pfender Editors

Handbook of Thermal Plasmas With 1236 Figures and 147 Tables

Editors Maher I. Boulos Department of Chemical and Biotechnology Engineering University of Sherbrooke Sherbrooke, QC, Canada

Pierre L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS) University of Limoges Limoges, France

Emil Pfender Department of Mechanical Engineering University of Minnesota Minneapolis, MN, USA

ISBN 978-3-030-84934-4 ISBN 978-3-030-84936-8 (eBook) https://doi.org/10.1007/978-3-030-84936-8 © Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

In memoriam to: Alice Boulos Kilchenmann (1942–2021) My Life Partner, the Love of my Life, for her Encouragement and Unwavering Support Emil Pfender (1925–2016) For his Leadership and Pioneering Contribution to the Field of Thermal Plasma Maher I. Boulos

Foreword

Anyone who has witnessed a lightning strike has seen a thermal plasma. Anyone who has travelled in a car, train, or airplane, or watched a construction welder working on a steel bridge or a new high-rise building has benefitted from thermal plasma applications. Yet few people have heard of thermal plasmas despite their immense value to industry and society. Indeed, thermal plasmas were already in common use decades before the term “plasma” was coined by Irving Langmuir in the 1920s. Thermal plasmas are characterized by high temperatures, large energy fluxes, intense radiative emission, and high densities of electrons and reactive species. These properties give thermal plasmas immense value. The first widespread implementation of thermal plasmas (or any type of plasma, for that matter), carbon arc lamps, occurred towards the end of the nineteenth century. Ever since, thermal plasmas have been used in an ever-increasing range of applications. An incomplete list includes high-power plasma jets to test materials for re-entry in the earth’s atmosphere; low- and high-voltage circuit breakers for electrical supply protection; electric arc furnaces for steelmaking and other metallurgical and mineral processing applications; plasma waste treatment and gasification; plasma cutting of metals and arc welding to join metals; plasma spraying of wear-resistant and thermal-barrier coatings; plasma spheroidization of metal and ceramic powders, including the micron-sized powders used in many 3D-printing methods; plasma nanoparticle production for electronic applications; wire-arc additive manufacturing for 3D printing of large metal components; and inductively coupled plasmas used with mass or optical spectroscopy for elemental analysis. As well as being practical, thermal plasmas are scientifically interesting. Thermal plasma systems come in wide varieties – they range from a few watts to hundreds of megawatts and include DC and AC electric arcs and DC, RF, and microwave plasma jets. Even the simplest examples, such as stable DC arcs in pure argon, hold many secrets – a welder who uses such an arc every day is unlikely to realize that its temperature may exceed 20,000 K and that the plasma may flow at 200 m/s or more. Once the interactions of the plasma with electrodes and other materials are considered, the complexity increases dramatically. A detailed understanding requires expertise in plasma physics, chemical kinetics, thermodynamics, statistical physics, materials science and metallurgy, electromagnetism, fluid dynamics, radiative vii

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transfer, and more. Development of applications requires electrical, mechanical, and chemical engineering skills. Countless scientists and engineers have devoted their careers to studying these phenomena, bringing new insights and developments every year. The three most celebrated names in thermal plasma research and development are Maher Boulos, Pierre Fauchais, and Emil Pfender. Ever since Vol. 1 of their landmark Thermal Plasmas – Fundamentals and Applications appeared in 1994, plasma researchers have asked for Vol. 2. At last, “Vol. 2,” and much more, has arrived in the form of the new Handbook of Thermal Plasmas, which contains an updated version of Vol. 1 (10 chapters presenting the fundamentals of thermal plasmas) and 22 new chapters covering thermal plasma generation, plasma, and particle dynamics, including diagnostics and modelling, industrial applications, and thermophysical properties. The handbook was worth the long wait. It is, in a word, a masterwork. Seldom does a reference work present so much information so cohesively and accessibly. The authors’ expertise, accumulated over their long and productive careers in academia and industry, has allowed them to assemble the diverse knowledge required to cover the expansive landscape of thermal plasma science and technology. Unlike many handbooks, the authors have cooperated closely together, and with guest coauthors in the last few chapters, ensuring that the chapters complement each other to provide a connected and comprehensive guide. The list of those who would benefit from reading parts, or all of the handbook, is long. The work contains essential information for students and researchers new to the field, and even the most experienced thermal plasma researchers and engineers will find a wealth of valuable material. But a much wider audience, those for whom thermal plasmas are a tool rather than a focus, should also use the resource. The group includes researchers who study the products of thermal plasma processes, such as plasma-sprayed coatings and welded metal components, and the many technologists who routinely apply thermal plasmas. The Handbook of Thermal Plasmas will undoubtedly be accepted as the standard reference in the field. The information and insights contained therein will be instrumental to the education of current and future generations of researchers and engineers and will inevitably contribute to the development of new and improved products and processes. The authors have provided a work of immense value, for which anyone using or interested in thermal plasmas owes them a debt of gratitude. CSIRO Manufacturing Lindfield, NSW, Australia

Anthony B. Murphy

Preface

Thermal plasma technology has evolved over the past five decades into an advanced interdisciplinary science that is attracting increasing attention. In the early 1960s and 1970s, its initial applications were mostly triggered by the aerospace program and the need for the development of high-power arc gas heaters for high-temperature supersonic wind tunnels. Several commercial entities led by Chemische Werke Hüls AG in Germany, Tioxide and Tetronics in the UK, Union Carbide, Westinghouse, and Plasma Energy Corporation (PEC) in the USA, Aerospatiale in France, and SKF in Sweden captured this opportunity for the development of plasma-based technologies for the Metallurgical and Chemical Process Industries (CPI). Industrial-scale applications of thermal plasmas also included plasma cutting and welding; plasma spray coating and the deposition of near net-shaped parts; plasma cladding of preforms for the fiber-optics industry; and more recently the plasma synthesis of advanced materials including micron-sized, ultrafine, and nanopowders and plasma waste treatment. As engineers or research scientists undertake a design or an R&D project along any of these areas, they are faced with the major challenge of having to be acquainted with a wide range of scientific disciplines varying from plasma physics, statistical thermodynamics, high-temperature chemical kinetics, advanced transport phenomena, and materials science. Publications dealing with thermal plasma technology tend to be dispersed over a wide range of scientific journals, which makes it rather difficult to access for the expert as well as the newcomer. Having maintained an active R&D program in this field for more than four decades and taught regular graduate level and continuing education courses covering different aspects of thermal plasma technology, we have decided a few years ago to put our collective knowledge, diverse experiences, and backgrounds together in a single reference textbook devoted to the fundamentals and applications of thermal plasmas. We choose to address the book to practicing engineers and research scientists looking for a simple and clear review of the principal fundamental concepts involved rather than an exhaustive survey of the subject. This book may also serve as an introductory text for graduate students entering the field of thermal plasma technology. Ample references are given in each chapter for a more in-depth study of the topics covered.

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Preface

In order to make the handbook more “reader friendly,” the material is presented in five distinct parts each dealing with a well-defined aspect of the science and technology as follows. Part I, Fundamentals of Thermal Plasmas, Chaps. ▶ 1–▶ 10, is essentially an updated version of our earlier book Thermal Plasmas, Fundamentals and Applications, published in 1994 by Plenum Press, now part of the Springer group. It is devoted to the more fundamental concepts of plasma physics and gaseous electronics, thermodynamics, and transport properties of plasmas. It is complemented by a large volume of thermodynamic, transport, and radiation data presented in graphical and tabulated form in Part V of this book. Part II, Generation of Thermal Plasmas, Chaps. ▶ 11–▶ 19, is dedicated to a review of the basic concepts used for the generation of plasmas with Chaps. ▶ 11– ▶ 17 covering DC plasma torch designs and electrode phenomena for such applications such as plasma cutting, welding, plasma transferred arc (PTA) deposition, wirearc and plasma spraying, torches. Chapters ▶ 18 and ▶ 19 cover respectively RF inductively coupled plasma ((ICP) and high-power plasma torches and transferred arc furnaces. Special attention is given to present fundamental aspects of torch design and performance without infringing on technological proprietary information. Part III, Plasma and Particle Dynamics, Chaps. ▶ 20–▶ 24, covers plasma and in-flight particle diagnostic techniques as well as mathematical modelling of plasma and particle dynamics. Chaps. ▶ 20 and ▶ 21 are dedicated to plasma diagnostics including optical emission and absorption spectroscopy, laser techniques, flow visualization, and probe techniques. Chapter ▶ 22 provides a brief introduction of standard powder characterization techniques followed by in-flight particle velocity and temperature measurements. These include laser Doppler anemometry and two-wavelength pyrometry. The remaining two chapters of this part, Chaps. ▶ 23 and ▶ 24, are dedicated respectively to momentum, heat, and mass transfer phenomena under plasma conditions, as well as particle trajectory and temperature history calculations in plasma flows, including plasma-particle interactions under dense particle loading conditions. Part IV, Industrial Applications of Thermal Plasmas, Chaps. ▶ 25–▶ 30, covers a wide range of industrial-scale applications of thermal plasma technology with Chap. ▶ 25 providing an example of plasma process integration in an industrial scale environment for the thermal spray industry, including auxiliary equipment’s, automation, environmental, and health concerns. Subsequent chapters are dedicated to different industrial sectors, with Chap. ▶ 26 covering the role of plasma in the thermal spray-coating industry; Chap. ▶ 27 plasma in the aerospace industry; and Chaps. ▶ 28, ▶ 29, and ▶ 30 covering, respectively, applications of plasma technology in the metallurgical, chemical process, and the waste treatment industries. It should be pointed out that as coverage made in these chapters goes back a few decades, a number of examples provided for industrial-scale applications might not have survived technoeconomic pressures and have since been abandoned. They are, nevertheless, maintained because of the intrinsic value of the technology and the fact that in our rapidly changing technoeconomic environment and available energy

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resources, interest might be revived in some of these technologies for the same product or alternate applications. Part V, Thermodynamic, Transport, and Radiation Properties of Thermal Plasmas, Chaps. ▶ 31 and ▶ 32, presents a compilation of relevant thermodynamic, transport, and radiation property data in graphical and tabulated form for a wide range of gases, including the effect of the presence of metal vapors on the properties of pure gases. Such data is indispensable for plasma equipment and process design calculations. They were based on established data basis currently used in a number of leading research centers around the world, as well as the large volume of data available in the open literature. This book has grown out of work done by the authors and their co-workers at their respective universities. Thanks are due to many former students and co-workers of the authors who contributed directly or indirectly to this book. The loss of our dear colleague Professor Emil Pfender, who passed away on January 28, 2016, sadly deprived him from seeing this Handbook completed in spite of the time and effort he had devoted to this project. His memory is cherished by us and many of his friends and colleagues for his valuable contribution to the field of thermal plasmas. We also want to thank all guest coauthors who joined us for the completion of the last eight chapters of this handbook, Prof. Dr. M. Auweter-Kurtz, Dr. Y. Cressault, Dr. R. Henne, Prof. J.W. Jurewicz, and Dr. A.B. Murphy, whose contributions are gratefully acknowledged. Special thanks are due to our life partners, Alice Boulos, Paulette Fauchais, and Maja Pfender, for their patience, understanding, and encouragement in the many years of sustained effort that led to the realization of this Handbook. Sherbrooke, QC, Canada Limoges, France Minneapolis, MN, USA February 2023

Maher I. Boulos Pierre L. Fauchais Emil Pfender

Contents

Volume 1 Part I

Fundamentals of Thermal Plasmas . . . . . . . . . . . . . . . . . . . . .

1

1

The Plasma State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

3

2

Basic Atomic and Molecular Theory . . . . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

57

3

Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

103

4

Fundamental Concepts in Gaseous Electronics . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

139

5

The Plasma Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

181

6

Thermodynamic Properties of Plasmas . . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

211

7

Transport Properties of Gases Under Plasma Conditions . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

257

8

Plasma Radiation Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

311

9

Thermodynamic Properties of Non-equilibrium Plasmas Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

.......

385

Transport Properties of Non-Equilibrium Plasmas . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

427

10

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Contents

Part II

Generation of Thermal Plasmas . . . . . . . . . . . . . . . . . . . . . .

471

11

Basic Concepts of Plasma Generation . . . . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

473

12

Thermal Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

507

13

Electrode Phenomena in Plasma Sources . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

553

14

DC Plasma Torch Design and Performance . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

595

15

Plasma Torches for Cutting, Welding, and PTA Coating . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

659

16

Wire-Arc Spray Torches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

743

17

Plasma Spray Torches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

795

18

RF Inductively Coupled Plasma Torches . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

849

19

High-Power Plasma Torches and Transferred Arcs . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

905

Volume 2 Part III 20

Plasma and Particle Dynamics . . . . . . . . . . . . . . . . . . . . . . .

Plasma Diagnostics, Optical Emission and Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

961

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21

Plasma Diagnostics, Lasers, Flow Visualization and Probe Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025 Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

22

Powders and In-Flight Particle Diagnostics . . . . . . . . . . . . . . . . . . 1093 Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

23

Plasma-Particle Momentum, Heat and Mass Transfer . . . . . . . . . . 1153 Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

24

Plasma–Particle Interactions in Thermal Plasma Processing . . . . . 1217 Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

Contents

Part IV

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Industrial Applications of Thermal Plasmas . . . . . . . . . . . .

1311

25

Plasma Spray Process Integration . . . . . . . . . . . . . . . . . . . . . . . . . 1313 Maher I. Boulos, Pierre L. Fauchais, Rudolf H. Henne, and Emil Pfender

26

Plasma in the Thermal Spray Coating Industry . . . . . . . . . . . . . . 1423 Maher I. Boulos, Pierre L. Fauchais, Rudolf H. Henne, and Emil Pfender

27

Plasma in the Aerospace Industry . . . . . . . . . . . . . . . . . . . . . . . . . 1509 Maher I. Boulos, Monika Auweter-Kurtz, Pierre L. Fauchais, and Emil Pfender

28

Plasma in the Metallurgical Industry . . . . . . . . . . . . . . . . . . . . . . . 1581 Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

29

Plasma in the Chemical Process Industry . . . . . . . . . . . . . . . . . . . . 1661 Maher I. Boulos, Jerzy W. Jurewicz, Pierre L. Fauchais, and Emil Pfender

30

Plasma in the Waste Treatment Industry . . . . . . . . . . . . . . . . . . . . 1739 Maher I. Boulos, Jerzy W. Jurewicz, Pierre L. Fauchais, and Emil Pfender

Part V Thermodynamic, Transport, and Radiation Properties of Thermal Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1805

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Thermodynamic and Transport Properties of Gases over the Temperature Range 300–30,000 K . . . . . . . . . . . . . 1807 Maher I. Boulos, Yann Cressault, Pierre L. Fauchais, Anthony B. Murphy, and Emil Pfender

32

Radiation Properties of Gases over the Temperature Range 300–30,000 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1923 Maher I. Boulos, Yann Cressault, Pierre L. Fauchais, and Emil Pfender

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1943

About the Editors

Maher I. Boulos University of Sherbrooke QC, Canada Professor Maher I. Boulos is an Emeritus Professor at the Chemical and Biotechnology Engineering Department of the University of Sherbrooke, Sherbrooke, Québec, Canada. He obtained his bachelor’s degree in Chemical Engineering from Cairo University, Egypt, in 1963 and his Master’s and Ph.D. degrees from the University of Waterloo, Ontario, Canada, in 1968 and 1972. He has a long experience in the areas of fluid dynamics and heat and mass transfer under plasma conditions. Following a post-doctoral year at McGill university, in Montreal, Québec, he joined the University of Sherbrooke in 1973 as an assistant professor, was promoted to associate professor in 1976, and professor in 1981. He founded the Plasma Technology Research Center in 1985. In 1990, he founded Tekna Plasma Systems Inc., a spin-off of the University of Sherbrooke, with the mandate of developing and promoting industrial applications of thermal plasma technology. He retired from his regular faculty position at the university in January 2007 to focus on running the Tekna Group, which has grown to be a world leader in induction plasma technology. In 2013, he transitioned from his CEO position to Chief Technology Officer of the Tekna Group to focus on technology development, a position he held until he retired from Tekna in 2018. Over his 35-year career at the University of Sherbrooke, Professor Boulos supervised or co-supervised 57 M.Sc. and Ph.D. students, published more than 150 papers in refereed scientific journals and more xvii

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About the Editors

than 200 in international conference proceedings, and contributed to 35 patents. He is the co-author of the textbooks Thermal Plasmas: Fundamentals and Applications (Plenum, 1994) and Thermal Spray Fundamentals: From Powder to Part (Springer, 2014 and 2021). Professor Boulos is a Fellow of the Canadian Academy of Engineering and has been the recipient of numerous distinctions, including being inducted into The Thermal Spray (TSS) Hall of Fame by the Thermal Spray Society of ASM International in 2003 and receiving the LionelBoulet Prix du Québec in 2007 from the Government of Québec for his contribution to the scientific and economic development of Québec. He was also the recipient of the Plasma Chemistry Award in 2015, the highest recognition awarded by the International Plasma Chemistry Society (IPCS) for lifetime achievements in the field of Plasma Chemistry. Pierre L. Fauchais Université de Limoges Limoges, France Professor Pierre Fauchais is an Emeritus Professor at the University of Limoges, France. He received Diplomas in both Engineering: Aeronautics and Related Thermal Problems and Physics (1961), his Ph.D. in Mechanics and Aeronautics (1963), and his State Doctoral Thesis in Physics (1968) from the University of Poitiers, France. From 1961 to 1968, he was a Researcher at National Research Centre (CNRS) at ENSMA Poitiers. He joined the University of Limoges as Assistant Professor in 1968, was promoted to Professor in 1973, and Exceptional Professor in 1988. His work was devoted to thermodynamic and transport properties of thermal plasmas, arc technology, plasma diagnostics (temperatures and velocities), sensors for particles in flight and at impact, laser anemometry, fast pyrometers (50 ns), and imaging, plasma spraying, and production of nanostructured coatings. During his long career at the University of Limoges, Professor Fauchais was a prolific writer with close to 400 scientific papers published in leading journals, 9 book chapters, 280 peer-reviewed international conference papers, 430 international conference papers, and 8 patents. He delivered 52 plenary lectures and 82 topical

About the Editors

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lectures at international meetings. He is the co-author of the textbooks Thermal Plasmas: Fundamentals and Applications (Plenum, 1994) and Thermal Spray Fundamentals: From Powder to Part (Springer, 2014 and 2021). Professor Fauchais was inducted into The Thermal Spray (TSS) Hall of Fame by the Thermal Spray Society of ASM International in 1998 and received The Plasma Chemistry Award (PCA) in 2001, the highest recognition awarded by the International Plasma Chemistry Society (IPCS) for lifetime achievements in the field of plasma chemistry. Professor Fauchais has received 34 best paper awards and has been a Fellow of the ASM since 2002. He directed or co-directed 92 Ph.D. and 15 State theses or equivalent (HDR). Besides regular teaching, he organized and taught numerous continuing education short courses in conjunction with international conferences and specialized courses adapted to industrial needs. Emil Pfender University of Minnesota Minneapolis, MN, USA Professor Emil Pfender was an Emeritus Professor at the Department of Mechanical Engineering, University of Minnesota. He earned his Diploma in Physics in 1953 and Dr. Ing. in Electrical Engineering in 1959, both from the Technical University of Stuttgart, Germany, where he became Chief Assistant and Lecturer at the Institute for Gaseous Electronics. In 1961, he spent a year as Visiting Scientist at the Plasma Physics Branch of the Air Force Research Laboratories at Wright Patterson Air Force Base. In 1964, Professor Pfender was recruited to direct the High Temperature Laboratory (now the High Temperature and Plasma Laboratory) in the Department of Mechanical Engineering at the University of Minnesota. He joined the Department as an Associate Professor and became Professor in 1967. Under his leadership, the HTL (HTPL) grew to become one of the world’s leading centers in the field of plasma science and technology, with 4 faculty members and over 20 graduate students and research assistants as well as several postdocs and visiting scientists. The University of Minnesota awarded Professor Pfender a Distinguished Alumni Professorship in 1989 and the Ernst Eckert

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Professorship in 1994. He formally retired from the University in 2000, but as an Emeritus Professor remained an active member of the Graduate Faculty in Mechanical Engineering, co-advising several Ph.D. students up to the time of his death in January 2016. Professor Pfender was a Fellow of the American Society of Mechanical Engineers since 1981 and a member of the US National Academy of Engineering since 1986. He was a recipient of many well-deserved honors from leading scientific societies, including the Alexander von Humboldt Award of the German Government (1978), the Gold Honorary F. Krizik Medal for Merit in the Field of Technical Sciences of the Czech Academy of Sciences, and an Honorary Doctorate from the Technical University of Ilmenau, Germany. In 1995, he was the first recipient of the Plasma Chemistry Award, then given by the International Union of Pure and Applied Chemistry, and subsequently awarded by the International Plasma Chemistry Society for lifetime achievement in plasma chemistry.

Contributors

Monika Auweter-Kurtz Steinbeis-Transferzentrum Aerospace Technology and Application, Stuttgart, Germany Maher I. Boulos Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada Yann Cressault LAPLACE, Université de Toulouse (UPS), Toulouse, France Pierre L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France Rudolf H. Henne Institute of Engineering Thermodynamiques, German Aerospace Center (DLR), Stuttgart, Germany Jerzy W. Jurewicz Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada Anthony B. Murphy CSIRO Manufacturing, Lindfield, NSW, Australia Emil Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA

Emil Pfender: deceased. xxi

Part I Fundamentals of Thermal Plasmas

1

The Plasma State Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Plasma State, Fourth State of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 What Is a Plasma? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Plasma Temperature(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Different Types of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Natural Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Man-Made Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Nonequilibrium, Man-Made Cold Plasmas (Te
Th) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Glow Discharges and Some of Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Corona Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Dielectric-Barrier Discharges (DBD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Thermal, Man-Made Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Basic Concepts Used for the Generation of Thermal Plasmas . . . . . . . . . . . . . . . . . . . . . . . 4.6 Thermal Plasma Sources and Their Fields of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Latin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 5 6 7 10 11 14 16 17 21 23 27 28 34 52 52 53 53

M. I. Boulos (*) Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada e-mail: [email protected] P. L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France e-mail: [email protected] E. Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] © Springer Nature Switzerland AG 2023 M. I. Boulos et al. (eds.), Handbook of Thermal Plasmas, https://doi.org/10.1007/978-3-030-84936-8_1

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List of Abbreviations

AC CLTE CVD DBD DC DLC EAF EB-PVD EM ER GM HID HSW ICP ICP-MS ICP-OES LLRW LTE LUX MIG MIPs MSW NASA PA-CVD PA-PVD PLTE PS-PVD PVD RF SSW TIG TPH TWD UV VOCs

1

Alternating current Complete local thermodynamic equilibrium Chemical vapor deposition Dielectric barrier discharges Direct current Diamond-like carbon Electric arc furnace Electron beam-physical vapor deposition Electromagnetic Erosion rate General motors High intensity discharge Hospital solid waste Inductively coupled plasma Inductively coupled plasma mass spectrometry Inductively coupled plasma optical emission spectroscopy Low-level radioactive waste Local thermodynamic equilibrium Lux per Watt Metal inert gas Microwave-induced plasmas Municipal solid waste National Aeronautics and Space Administration Plasma-assisted chemical vapor deposition Plasma-assisted physical vapor deposition Partial local thermodynamic equilibrium Plasma sprayed-physical vapor deposition Plasma vapor deposition Radio frequency Sewage sludge waste Tungsten inert gas Ton per hour Traveling wave discharges Ultraviolet Volatile organic compounds

Introduction

This chapter serves as an introduction to the vast and rapidly growing field of plasma science and technology. A brief introduction defining what is a plasma and the different types of plasmas, whether natural or man-made, is presented. This is

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The Plasma State

5

followed by a review of the main characteristics and applications of man-made plasmas. Separate sections deal with nonequilibrium, man-made plasmas, whether glow discharges, corona discharges, dielectric barrier discharges (DBD), or microwave plasmas and their respective industrial applications. Thermal, man-made plasmas are discussed next, identifying the basic concepts for the generation of such plasmas followed by a description of principal thermal plasma sources and their respective industrial applications. Whenever applicable, forward referencing is made to subsequent chapters in the handbook where further details are given on the different subjects.

2

The Plasma State, Fourth State of Matter

The plasma state is frequently referred to as the fourth state of matter. The definition is derived from the concept that the solid state is a ground state or first state of matter, which exists at the lowest end of the temperature and specific enthalpy scale. As illustrated in Fig. 1, the increase of the specific enthalpy of matter results in corresponding increase of its temperature, until its melting point is reached, at which it passes to the liquid state, the second state of matter. The further increase of the specific enthalpy of matter results in the corresponding increase of its temperature until its passage, at the vaporization temperature, to the vapor phase, the third state of matter. With the subsequent increase of the specific enthalpy of matter, it eventually reaches the plasma state, the fourth state of matter, at which molecules start to dissociate and atoms to ionize forming a mixture of molecules, atoms, and ions resulting from ionization of molecules and/or atoms and electrons all in local electrical neutrality. This implies that plasmas are electrically neutral with negative electrons and negatively charged ions, if any, balanced by positively charged ions. Molecules, atoms, and ions are either in fundamental or excited states. The lifetime of excited species is low (varying from a few ns to the μs range). Plasmas also

Atoms

Specific enthalpy (MJ/kg)

Fig. 1 Schematic representation of the four states of matter as function of temperature and the specific enthalpy of matter

Ions

THE PLASMA STATE Molecules Vapor

Electrons

Liquid Solid Tm

Tv

Temperature (K)

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M. I. Boulos et al.

contain photons emitted as a result of transition of molecules, atoms, and ions from an excited state to lower energy levels or their corresponding ground states. Photons can also be emitted by electromagnetic radiation produced by the deceleration (bremsstrahlung) of a charged particle when deflected by another charged one, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into a photon because energy is conserved. These processes are at least partially responsible for the luminosity of plasmas. Of course if excited atoms or molecules emit photons when de-exiting, reciprocally photons can excite atoms or molecules in lower excitation states or in ground state and plasmas can also be produced by photoionization. This classification of plasma as a “state of matter” is justified by the fact that more than 99% of the known universe is in the plasma state. A typical example is the Sun, whose interior temperatures exceed 107 K. The high energy content of a plasma compared to that of solids, liquids, or ordinary gases lends itself to a number of important applications. Many textbooks have been devoted to plasma physics and engineering such as Cobine (1958), Brown (1959), Cambel (1963), Mitchner and Kruger (1973), Peratt (1992), Bittencourt (2004), and Fridman (2004, 2008).

2.1

What Is a Plasma?

As mentioned earlier, plasmas consist of a mixture of electrons, ions, and neutral particles. Since the masses of ions and neutrals are much higher than the electron mass (mH =me ¼ 1836, where mH = mass of the hydrogen atom, lowest atom mass, and me = electron mass), neutrals and ions are classified as the heavy particles or the heavy component in plasma. The mixture of electrons, ions, and neutrals in the ground state, excited species, and photons can be designated as plasma only if the negative and positive charges balance each other, i.e., overall plasma must be electrically neutral. This property is known as quasi neutrality. In contrast to gas at ambient temperature and pressures, plasmas are electrically conducting due to the presence of free charge carriers. In fact, plasmas may reach electrical conductivities exceeding those of metals at room temperature. For example, hydrogen plasma at atmospheric pressure (100 kPa) and a temperature of 106 K has approximately the same electrical conductivity as that of copper at room temperature. Plasmas generally have a low enough density to obey Maxwell–Boltzmann statistics rather than Fermi–Dirac or Bose–Einstein ones. However, the dynamical behavior of plasmas is generally more complex than that of gases and fluids, due to: (i) Weak Coulomb scattering, resulting in mean free paths of the electrons and ions often larger than the plasma’s macroscopic length scales. This allows the particles’ momentum distribution functions to deviate significantly from their equilibrium Maxwellian forms and, in particular, to be highly anisotropic.

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(ii) The long-range electromagnetic fields allowing charged particles coupling to each other electromagnetically and acting in concert as modes of excitation (plasma waves) that behave like single dynamical entities. A more rigorous definition of the plasma state, taking second-order effects into account, will be given in ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics,” of this part.

2.2

Plasma Temperature(s)

The temperatures of a plasma, as in any gaseous medium, are defined by the average kinetic energy of its constituents or particles (molecule, atom, ion, or electron), which according to Maxwell–Boltzmann statistics is given by the equation: 1 2 3 mv ¼ kT (1) 2 2 pffiffiffiffiffi where m is the mass of the particle, v2 is its root mean square (rms) or effective velocity, k is the Boltzmann constant, and T represents the absolute temperature (K). Equation 1 results from Maxwell–Boltzmann distribution, which can be expressed by dnv ¼ n f ðvÞ dv

(2)

with the distribution function f(v) defined as f ðvÞ ¼ ðdnv =dvÞ and  3   4 2kT 2 2 mv2 f ðvÞ ¼ pffiffiffi v exp  (3) 2kT π m qffiffiffiffiffiffi As shown in Fig. 2, f(v) reaches a maximum at vm ¼ 2kT m . The number density of molecules with velocities between v and v þ dv is given by dnv. From this distribution it follows that the average velocity is v¼

ð1 0

rffiffiffiffiffiffiffiffi 8kT vf ðvÞdv ¼ πm

(4)

And, the mean square velocity is v2

¼

ð1 0

rffiffiffiffiffiffiffiffi 3kT v f ðvÞdv ¼ m 2

(5)

The establishment of a Maxwell–Boltzmann distribution among the particles in plasma or in an ordinary gas depends strongly on the interaction between the particles, i.e., on the collisional frequency and on the energy exchange during each

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Fig. 2 Maxwellian distribution function evolution with velocity (Boulos et al. 1994)

collision. By applying the conservation equations to an elastic binary collision of particles with mass m and m0 , one can show that on the average kinetic energy exchanged is given according to Howatson (1965) and Chang and Pfender (1990) by ΔEkin ¼

2mm0 ðm þ m 0 Þ2

(6)

This result implies that for particles of the same mass ðm ¼ m0 Þ, ΔEkin ¼ 1=2 , and therefore any distortion of the Maxwell–Boltzmann distribution among particles of the same mass will be eliminated by fewer than 10 successive collisions. These considerations demonstrate that in a collision-dominated plasma, we can assume that heavy species and electrons among themselves will have a Maxwell–Boltzmann distribution that permits the definition of a corresponding temperature for these species. If the subscript, r, designates the various components (such as electrons, ions, and neutrals) in a plasma, then the Maxwell–Boltzmann distribution for each of these components can be written in terms of their kinetic energy Er ¼ 1=2 mv2r (see ▶ Chap. 3, “Kinetic Theory of Gases,” for details) as   2nr Er exp  dEr dnEr ¼ pffiffiffi ðkTr Þ3=2 E1=2 r kTr π

(7)

where Tr represents the temperature of the component r. As the following discussion will show, the temperatures of the various components of plasma may or may not be the same.

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Let’s consider the energy exchange between electrons and heavy species. With m0 ¼ me (electron mass) and m ¼ mh (mass of the heavy species), we find from Eq. 6 that ΔEkin ¼

2me mh

(8)

Since me  mh, a large number of collisions (>103) are required to equilibrate the temperatures of the electrons and heavy particles and eliminate energy (or temperature) differences between them. In an electric discharge, which is one of the most common ways to generate and maintain plasma, the high-mobility electrons pick up energy from the applied electric field and then transfer part of this energy to the heavy particles through elastic collisions. Even with an excellent collisional coupling (high collision frequency) between electrons and heavy particles, there will always be a difference between the electron temperature and the temperature of the heavy species in the plasma. The energy transferred from an electron to a heavy particle in a single e elastic collision may be expressed by 32 kðTe  Th Þ 2m mh , where Te and Th represent, respectively, the electron and the heavy particle temperatures. The energy that an electron acquires from the electric field (E) between collisions is given by e E vd τe , where vd is the average drift velocity (see ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics,” for definition) and τe the average time of flight between collisions. With ‘e ; ve rffiffiffiffiffiffiffiffiffiffi 8kTe ve ¼ πme τe ¼

(9)

(10)

and ‘e ¼ mean free path (mfp) of the electrons (see ▶ Chap. 3, “Kinetic Theory of Gases,” for definition), it follows that for steady-state situation, Te  Th 3πmh ¼ Te 32me

e‘e E 3 2 kTe

!2 (11)

According to Eq. 11, kinetic equilibrium ðTe ¼ Th Þ requires that the energy that the electrons acquire in an electric field between collisions must be very small compared to the average kinetic energy of the electrons. Another interpretation of Eq. 11 considers the fact that, ‘e  1p, with p as the absolute pressure; thus Te  Th ΔT ¼  Te Te

 2 E p

(12)

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This relation shows that the parameter (E/p) plays a governing role for determining the kinetic equilibrium situation in plasma. For small values of (E/p), the electron temperature approaches the heavy particle temperature, which is one of the basic requirements for the existence of local thermodynamic equilibrium (LTE) in plasma. Additional conditions for LTE include excitation and chemical equilibrium as well as certain limitations on the gradients in the plasma. Details of LTE requirements for plasma will be discussed in ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics.” The (E/n) ratio between the electric field, E, and the concentration of neutral particles, n, is often used instead of (E/p), because the mean energy of electrons (and therefore many other properties of discharge) is a function of (E/n). Increasing the electric filed intensity, E, by some factor, q, has the same consequences as lowering gas density, n, by factor, q. The SI unit for (E/n) is V m2, but the Townsend unit (Td = 1021 V . m2) is frequently used. Plasma that is in kinetic equilibrium and simultaneously meets all other LTE requirements is classified as thermal plasma. In contrast, plasmas with strong deviations from kinetic equilibrium ðTe
Th Þ are classified as nonequilibrium plasmas, which are often, though not necessarily, nonthermal. Both types of plasmas will be further discussed in the following section.

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Different Types of Plasmas

Solar core and Thermonuclear fusion plasmas

104 103 102

Nebula and Solar corona

106 Lightning 105

10 Glow discharges

Flames

High pressure arcs 104 and RF discharges microwave 103 and DBD

1016

1020

1 10−1 10−2 108

107

Ionosphere

1012

Number density of charged particles

Fig. 3 Classification of plasmas

1024 (m−3)

Electron Temperature (K)

Electron Temperature (eV)

Plasmas can be classified based on a wide range of criteria. One of the simplest is their classification based on their origin or nature that is as natural or man-made plasmas. The main difference is in the way the plasma manifests itself in nature, whether it is the result of a natural phenomenon such as lightening and aurora borealis or that of a human activity such as in arcs and different types of electrical discharges. One of such classifications is illustrated in Fig. 3 in which the different

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The Plasma State

11

types of plasmas are shown as the number density of charged particles (m3) versus the electron temperature (in eV or K) diagram. The range of electron densities indicated varies from 108 to 1024 (m3) which corresponds to pressure variations from high vacuum to pressures above atmospheric. The electron temperatures are expressed in terms of either units of eV (l eV corresponds to 7,740 K for a Maxwell–Boltzmann plasma) or in terms of degrees K. At the low end of the pressure or number densities of charged particles, one identifies two naturally existing plasmas, the ionosphere with relatively low energy density or temperature levels, less than 0.1 eV or 103 K, and the nebula and solar corona at energy levels of 102 up to 103 eV (around 106 K). At pressures close to atmospheric and above (charged particle densities of 1024), we have lightning, which is one of the most frequently observed manifestations of natural plasmas. Most man-made plasmas also fall in this range such as atmospheric and highpressure arcs and RF discharges with energy levels around 1 or 2 eV and temperatures in the range of 10,000–20,000 K. At considerably higher electron energies and temperatures, we also have solar core and thermonuclear fusion plasmas. In between we find a number of other man-made plasmas with different energy levels such as in flames, glow discharges, microwave plasmas, and dielectric barrier e discharges (DBD). The degree of ionization defined as ξ ¼ nenþn varies widely 10 in flames to a few percentages in the range of 5–10% in from as low as 10 high-pressure arcs, up to full and multiple ionization levels in thermonuclear fusion plasmas. In the following specific examples, the principal characteristics of the different types of natural and man-made plasmas are briefly discussed.

3.1

Natural Plasmas

As mentioned previously, natural plasmas comprise more than 99% of the universe known today. At sea level, in air three-body recombination (see ▶ Chap. 3, “Kinetic Theory of Gases,” for definition) of free electrons with molecular oxygen limits their lifetimes to only about 16 ns, and thus no natural plasma can exist under these conditions (Becker et al. 2005). On the other hand, at high altitudes the pressure decreases, with a corresponding rise of the lifetime of free electrons. At an altitude of 30,000 ft, the lifetime of a free electron is 119 ns, while at 60,000 ft, it reaches 1.83 μs. When reaching 60 km above the sea level (beginning of ionosphere), high fluxes of extreme ultraviolet (solar photons) and also collisions with free electrons maintain free electron densities in the range of 105–1010 m3 which is characteristic of the charged particle densities in the ionosphere and supports the notion that 99% of the universe is in a plasma state (Suplee 2009). In fact all of the observed stars in interstellar and interplanetary media as well as the outer atmospheres of planets are essentially in a plasma state. The Sun is the closest star to the Earth and is the center of our solar system. In a star the plasma is bound together by gravitational forces, and the enormous energy it emits originates in thermonuclear fusion reactions within the interior. The temperature is more than 106–107 K at the core: fully ionized plasma. Heat is transferred from the interior to the exterior by radiation in

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Fig. 4 Photograph of the sun, June 9th. 2002, courtesy of NASA

the outer layers. In the Sun’s outer layer, the temperature is lower than in the radiative zone and heavier atoms are not fully ionized. As shown in Fig. 4, the density of gases is low enough to allow convective currents to develop a giant, spinning ball of a temperature of about 6,000 K on the surface. A transition layer, the tachocline, separates the radiative zone and the convective zone. The light from the Sun heats our planet and makes life possible on it. The Sun is also an active star that displays sunspots, solar flares, erupting prominences, and coronal mass ejections. These phenomena, which are all related to the Sun’s magnetic field, impact our near-Earth space environment and determine our “space weather.” In the vicinity of a hot star, the interstellar medium consists almost entirely of completely ionized hydrogen, ionized by the star’s ultraviolet radiation. A corona surrounds the Sun and other celestial bodies (Aschwanden 2004). The Sun’s corona extends millions of kilometers into space and is most easily seen during a total solar eclipse. Associated to the Sun, the solar wind is a stream of charged particles (plasma) released from its upper atmosphere. It mostly consists of electrons and protons with energies usually between 1,500 and 104 eV. The stream of particles varies in density, temperature, and speed over time and over solar longitude. These particles can escape the Sun’s gravity because of their high kinetic energy and the high temperature of the corona. Two of the earliest known natural plasma phenomena are lightning and the aurora borealis. The aurora borealis, shown in Fig. 5, is a natural light display in the sky particularly in the Arctic and Antarctic regions, caused by the collision of energetic charged particles with atoms in the high-altitude atmosphere (thermosphere). The charged particles are directed by the Earth’s magnetic field into the atmosphere. The aurora borealis appears as a diffuse, widespread (of astronomic dimensions),

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Fig. 5 Aurora borealis

Fig. 6 Lightning

low-luminosity event. Lightning, on the other hand, as shown in Fig. 6, is a massive electrostatic discharge between the electrically charged regions within an agglomeration of clouds or between a cloud and the surface of the Earth. The charged regions within the atmosphere temporarily equalize themselves through a lightning flash. In the case of lightning, narrow, high-luminosity channels are observed with numerous streamers branching out of the main core of the lightning channel.

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M. I. Boulos et al. Discharge tube E=V/d Cathode (–)

Anode (+)

d

Fig. 7 Discharge tube

3.2

Man-Made Plasmas

Plasmas started to be studied in the middle of the nineteenth century. They were essentially produced using a discharge tube at low pressure as shown in Fig. 7. A direct-current, DC, voltage was applied between parallel electrodes creating an electric field, E, inside the tube E = V/d, where V is the applied voltage and d being the distance between both electrodes. As the applied voltage is increased over a certain value, depending on the nature of the gas and its pressure, breakdown occurs between the two electrodes and the gas becomes conducting. With the increase of the applied voltage and consequently of the current passing through the discharge, the electric discharge passes through the following three regimes, with distinct current–voltage characteristics as shown in Fig. 8 after Chang and Pfender (1990) and Pfender (1978).

3.2.1 Townsend Discharge Townsend discharge occurs below the breakdown voltage (A to E in Fig. 8). At low voltages, the only current is that due to the generation of charge carriers by external sources such as cosmic rays or other sources of ionizing radiation (A to B). When increasing the applied voltage, the free electrons carrying the current gain enough energy to cause further ionization, resulting in an electron avalanche. In this regime, the current increases from below 1010 to over 106 A, for very little further increase in voltage C to D in Fig. 8. In this area of V–I characteristics, except for corona discharges D to E in Fig. 8, and the breakdown itself at E in Fig. 8, the discharge remains invisible to the eye. However, if one electrode is a very thin wire or a point, extremely nonuniform electric fields prevail close to it. With sufficiently high potential V applied to such an electrode, breakdown of the gas near the wire surface occurs at potentials below the spark breakdown potential. The resulting discharge is also known as the corona discharge if operated at high (atmospheric) pressures. Of course nonequilibrium conditions prevail in the Townsend discharge regime. 3.2.2 Glow Discharge Glow discharge takes place in the current region between E and H in Fig. 8. The breakdown avalanche is a cascade reaction involving electrons in a region with a

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dark discharge

Glow discharge

arc discharge

Townsend regime

D

Corona

E Breakdown voltage

Voltage (V)

H C

Glow-to-arc transition

F Saturation regime

G

I

Normal glow Abnormal glow

A 10−10

Nonthermal

B Background ionization

10−8

10−6

10−4

10−2

1

J Thermal arc K 102

104

Current (A)

Fig. 8 Evolution of discharge characteristics as function of the discharge current in a tube discharge

sufficiently high electric field in a gaseous medium that can be ionized. Following an original ionization event, caused by such events as ionizing radiation, the positive ion drifts toward the cathode, while the free electron drifts toward the anode of the discharge tube. If the electric field is strong enough, the free electron gains sufficient energy to liberate a further electron when it collides with another molecule or atom. The two free electrons then travel toward the anode and gain sufficient energy from the electric field to cause further ionization as a consequence of subsequent collisions. This chain reaction depends on the free electrons gaining sufficient energy between collisions to sustain the avalanche. As the current further increases, space charges develop, leading to a sudden drop of the voltage across the electrodes (subnormal glow discharge) shown in Fig. 8 as the range E to F. Between F and G in Fig. 8, the voltage across the tube is almost current independent as well as the electrode current density, j. When the current I increases (F to G in Fig. 8), the fraction of cathode surface occupied by plasma increases until plasma covers the entire cathode surface at G. For the formation of glow discharge, the mean free path of the electrons has to be reasonably long but shorter than the distance between the electrodes; glow discharges therefore do not readily occur at both too low and too high gas pressures. At higher currents, the normal glow turns into abnormal glow (G to H in Fig. 8), the voltage across the tube gradually increases, and the glow discharge covers more and more of the surface of the electrodes. Electrons striking the gas atoms and ionizing them facilitate glow discharge. The breakdown voltage for the glow discharge depends nonlinearly on the product of gas pressure and electrode distance according to Paschen’s law:

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M. I. Boulos et al.

V¼

aðpdÞ lnðpdÞ þ b

(13)

where p is the absolute pressure in the discharge tube and d the gap distance (m), while a and b are constants linked to gas composition. For a certain pressure  distance value (p . d), there is a lowest breakdown voltage. The increase of the breakdown voltage with decreasing electrode gap is a consequence of the reduced ionization efficiency of electrons, because the mfp of electrons becomes larger than the electrode gap. All plasmas produced under these conditions are out of equilibrium.

3.2.3 Arc Discharges Arc discharges occur at higher currents, over the range from H to K in Fig. 8. There is an important voltage drop in the transition region due to the much lower cathode fall in the arc region. The cathode is heated to temperatures sufficient for thermionic electron emission, which is a much more efficient electron emission mechanism compared to that in the glow discharge regime (gamma mechanism). The cathode fall in the arc region is typically around 10 V, whereas in the glow discharge, cathode falls are typically around 100 V. The transition region from I to J in Fig. 8 represents a hybrid between the glow and arc discharge with strong deviations from equilibrium. As the current further increases (J to K in Fig. 8), the arc plasma approaches LTE. A more detailed discussion of the principal characteristics and major areas of industrial applications of man-made plasmas will follow. This is divided into two sections depending on the nature of the plasma state, one dealing with nonequilibrium man-made cold plasmas and the other on man-made thermal plasmas.

4

Nonequilibrium, Man-Made Cold Plasmas (Te
Th)

Nonequilibrium plasmas are frequently classified as “cold” plasmas, because of the low temperature of the heavy species ðTe
Th Þ. The plasmas or discharges can be classified according to their time dependence (transient or steady state), importance of space charge effects or of heating of the neutral gas species, and presence of a surface close to the discharge (Nijdam et al. 2012). Nonequilibrium plasma systems can be generated at pressures ranging from 10 kPa or less up to atmospheric pressure. According to the previously discussed (E/p) criterion, substantial deviations from kinetic equilibrium are expected for large values of (E/p). Typical values of (E/p) or (E/n) are several orders of magnitude higher for nonthermal plasmas, compared to thermal plasmas. A typical value, of a nonequilibrium glow discharge operated at a pressure of 0.1 Pa, is on the order of E/p = 107 V/m.kPa.

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The Plasma State

4.1

17

Glow Discharges and Some of Their Applications

Figure 9 shows the basic physical structure of a nonequilibrium discharge, generally run in Ar or other noble gases with Te = 104 K, while Th ’ 300 K. The different regions of the discharge shown in Fig. 9 are not always found in any given condition. These depend on the dimensions, pressure and voltage, and nature of the exciting field. Glow discharges can be generated in direct-current (DC) mode, which requires that the cathode must be conductive. The pressure, voltage, and current are interrelated, but only two parameters can be controlled, the third being dependent on the two variable parameters. Glow discharges may also be generated by alternating current (AC), radio frequency (RF), and other source applied in order to establish a negative bias voltage on the electrode surface. Capacitively coupled RF discharges are still the most common plasmas used in dry etching in the electronic industry. In this case nonelectrical conductive materials can be used. Both DC and RF glow discharges can be operated in pulsed mode allowing higher instantaneous powers to be applied without excessively heating the electrodes. As shown earlier in Fig. 8, in the case of a DC glow discharge, current can increase by several orders of magnitude at constant voltage. It is controlled by resistive ballast. Most of the voltage fall takes place close to the cathode. In the cathode glow, electrons are energetic enough to excite the neutral atoms with which they collide. The negative glow, shown on the RHS of Fig. 9, the brightest intensity of the entire discharge, has a relatively low electric field and is long compared to the cathode glow. Electrons that have been accelerated in the cathode region to high speed produce ionization, and slower ones produce atom excitations. These slower electrons are responsible for the negative glow. The electron number density in the negative glow discharge is typically in the range of 1015 to 1016 e/m3 with energies varying between 1 and 2 eV. The positive column has a small positive electric field (about 100 V/m). Glow discharges can be run at very low pressure, as low as 10 Pa, with voltages of about 100 V. At atmospheric pressure, voltages of a few kV are needed combined with considerably shorter distance between the electrodes (Chapman 1980; Marcus and Broekaert 2003; Hippler et al. 2008).

Anode dark space

Positive column

Faraday dark space

Cathode dark space Aston dark space

Anode (+)

Cathode (−)

Cathode glow Anode glow Fig. 9 Basic physical structure of a glow discharge

Negative glow

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Applications of glow discharges include elemental analysis of solids, liquids, and gases. In the case of elemental analysis of solids, which is the most common, the sample is used as the cathode (Marcus and Broekaert 2003). Gas ions striking the sample surface sputter it. The sputtered atoms, in the gas phase, are detected by atomic absorption, atomic emission, and mass spectrometry. When sample atoms are ionized, the ions can then be detected by mass spectrometry. Both bulk and profile analysis of solids may be performed with glow discharge (Marcus and Broekaert 2003). Glow discharge is used for surface cleaning and modification of metallic biomaterials (Aronsson et al. 1997). Low-pressure glow discharges are used in lamps as well as in plasma-assisted physical or chemical vapor deposition processes, while atmospheric pressure glow discharges are also used for decontamination, cleaning, preparation, and modification of biomaterial and implant surfaces.

4.1.1 Lamps Lamps represent an important application of glow discharges. Among the different types of lamps shown in Fig. 10, florescent lamps are probably the best-known gas-discharge lamps. They mostly use a noble gas such as argon, neon, krypton, and xenon or a mixture of these gases with additional doping materials, such as

Fig. 10 Range of lamps commonly used in industrial, residential, and commercial applications

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19

mercury, sodium, and metal halides. Gas-discharge lamps offer higher energy conversion efficiency, lux per watt (LPW), more than most conventional incandescent lamps.

4.1.2 Plasma-Assisted Physical Vapor Deposition (PA-PVD) Plasma-assisted physical vapor deposition (PA-PVD) corresponds to a group of vacuum processes for the deposition of layers composed of (primarily) metals, alloys, nitrides, oxides, carbides, borides, sulfides, silicides, fluorides, and mixtures of these. Layer thickness ranges from a few 10 nm to a few tens μm (Frey and Khan 2013). Evaporated or sputtered particles can reach the substrate only if their mean free paths are in the few meters range, which requires vacuum levels lower than 1 Pa (typical of glow discharge). The process is a line of sight one, and the substrate must be moved accordingly to achieve a coating with a uniform thickness. The deposited layer growth rate is influenced by evaporation or sputter process with the assistance of excited atoms and molecules and/or ions generated in plasmas (Erkens et al. 2011; Bunshah 2001). At least one part of the particles involved in layer growth possesses larger energy than the thermal energy of evaporation. The simplest system represents cathode sputtering: an electric field is generated between the material to be deposited (the target), necessarily a metal or an alloy (cathode) and the substrate (anode). Voltages up to 1,000 V are used. The surrounding atmosphere is argon; the mass of its ions, accelerated toward the cathode, should be high enough for sputtering (mechanical process). For ceramic materials, the corresponding metal is sputtered and the reactive gas, which is responsible for the formation of the ceramic, is injected in the chamber (process close to chemical vapor deposition). To increase the process efficiency, “magnetron sputtering” is used with a magnetic field applied behind the target, and parallel to its surface, to intensify the sputtering process. Due to the magnetic field, the electrons from the glow discharge no longer move parallel to the electric field lines, but instead along a spiral track. They are thus able to ionize more gas molecules on their longer path to the target. Electron and ion densities are highest in this zone. However, because of their high mass, ions are hardly deflected by the magnetic field, and the greatest erosion of the target occurs below this zone. Arc PVD is also used, with an arc initiated between the cathode (the material to be evaporated) and an anode (the chamber wall in most cases). The plasma ignition is generated by laser evaporation or a high-voltage and high-frequency spark. With arcs of 10–100 A, the cathode is melted locally at different points, which are in continuous motion over its surface. The evaporated cathode material is present in the form of highly ionized plasma. The ions are accelerated (to velocities ranging from a few 104 to a few 105 m/s) and frequently manifest multiple ionizations (Erkens et al. 2011). This process is generally used for the evaporation of metals such as Ti, Al, or Cr and their alloys such as AlTi but also carbon, for deposition of extremely hard amorphous carbon layers. Reactive gases can also be added, for example, to deposit nitrides. According to Erkens et al. (2011), arc PVD is used for all types of tools (cutting, primary shaping, shaping and forming, and machining of plastics), for car and engine components (including piston rings, bucket tappets, and

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Fig. 11 TiN-PVD-coated tools (Erkens et al. 2011)

power train components), for hydraulic components (such as pistons), for medical tools and instruments (e.g., bone punches), and for turbine blades, in machine parts (e.g., collet chucks) and in decorative coatings (bathroom fittings, for instance). Glow discharge-based techniques are extensively used for coating tools, for example, with TiN and diamond-like carbon (DLC), using PVD and PA-PVD, respectively, as shown in Figs. 11 and 12.

4.1.3 Plasma-Assisted Chemical Vapor Deposition (PA-CVD) The CVD process is used for the deposition of a layer on a substrate from the gas phase by means of reactions of gas phase constituents. Compared to PVD process, they are not a line of sight process allowing for a more uniform layer deposition on three-dimensional surfaces of complex shape with undercuts or hollow parts. In the conventional CVD process, the reaction occurs in the high-temperature zone (900–1,400  C) where the substrate is typically located, which provides the energy necessary to activate the reaction. In the PA-CVD process, the energy necessary for the activation of the gases is supplied by means of high-energy electrons in the plasma (Frey and Khan 2013). Plasma excitation (the generation of effectively free electrons, ions, radicals, and excited species) is accomplished by means of glow discharges (DC voltage, pulsed DC voltage, medium frequency, and radio frequency) or by means of microwaves (Erkens et al. 2011; Bunshah 2001). Conventional hard coatings such as TiN, TiCN, or Al2O3 are achieved using a combination of plasma-assisted and thermal CVD in the 400–600  C temperature range. Hard

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Fig. 12 DLC steel gear components for the aviation industry coated by PA-PVD (Erkens et al. 2011).

amorphous carbon layers, known as DLC layers, are achieved at temperatures below 200  C with pulsed glow discharges or high-frequency discharges using precursors such as C2H2, hexamethyldisiloxane (HMDS), or tetramethylsilane (TMS) (Erkens et al. 2011).

4.2

Corona Discharges

Corona discharges use nonsymmetric pair of electrodes, for example, a flat or a slightly curved surface for the cathode and a pointed surface or a wire as anode. This arrangement results in an increased intensity of the electric field near the pointed element or wire, with a corresponding increase in the gas ionization level. In Fig. 13a two such electrode configurations are illustrated. Figure 13a shows the case for a point electrode used as the anode and the flat electrode as the cathode, while in Fig. 13b the electrodes are coaxial with the central wire being anode and the outer cylindrical shell cathode. The discharge develops in the high field region near the sharp electrode and it spreads out toward the cathode (Jogi 2011; Nijdam et al. 2012). The main risk is the transition of the discharge to an arc, which can be avoided if: • The voltage is low enough to stop the spreading of the discharge before the cathode is reached. • The voltage is lowered when the cathode is reached, which requires a complex power supply.

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a

b

High electric field ionization zone

Ion drift to the other electrode

Fig. 13 Corona discharges principle. (a) Positive point, (b) positive wire (Jogi 2011)

a

b

Ionization

• • •

smaller volume, lower electron concentration Higher energy electrons

• • •

Large volume, Higher electron concentration Lower energy electrons

Fig. 14 Comparison of positive and negative corona discharges (Jogi 2011)

The corona is said to be positive when the highly curved electrode is connected to the positive output of the power supply and a negative corona when it is connected to the negative output of the power supply. When voltages are relatively low, the space charge near the sharp electrode disappears due to diffusion and recombination and the discharge dies out. The process is self-repetitive. Discharge properties depend strongly on the polarity of the sharp electrode, as illustrated in Fig. 14. With DC power sources, the power input in continuous coronas is rather limited by the voltage range that could be used while still avoiding the development of sparks. The use of pulsed power supplies offers means of avoiding the

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Fig. 15 Discharges in a 40 mm gap in atmospheric air with a 54 kV pulse, 30 ns rise time, and half-width of about 70 ns. The images are acquired with (a) short (50 ns) and (b) long (1,800 ns) exposure times (Nijdam et al. 2012)

development of streamers by limiting the duration of voltage pulses to less than the streamer development and propagation time, which is typically in the 100–300 ns ranges for a 10–30 mm gap. A small ball of light around the point electrode characterizes the corona discharge, the ball expands and forms a shell, and then eventually, the expanding shell breaks up into multiple streamer channels, as shown in Fig. 15. Corona discharges are used in many applications, the most important of which is the manufacture of ozone and in electrostatic precipitators. Air ionizers are commonly used for sanitization of pool water, removal of unwanted volatile organics, such as chemical pesticides, solvents, or chemical weapons agents, from the atmosphere. Corona discharges are also used for improvement of wettability of polymer films to improve compatibility with adhesives or printing inks, inactivation of bacteria, removal of unwanted electric charges from the surface of aircraft in flight and thus avoiding the detrimental effect of uncontrolled electrical discharge pulses on the performance of avionic systems, etc. (Parvulescu et al. 2012; Fridman 2004; Dobrynin et al. 2011).

4.3

Dielectric-Barrier Discharges (DBD)

They are somewhat similar to corona discharges, but a dielectric layer covers one or two of the electrodes in the discharge gap. At a sufficiently high voltage between the electrodes, the discharge starts in the gas volume. It spreads out until it reaches the electrodes, but at the dielectric it builds up a space charge that cancels the applied electric field. At that moment the discharge stops. With dielectric barrier (electrical insulator), these discharges require alternating voltages for their operation. Typical planar DBD configurations are schematically shown in Fig. 16. According to Kogelschatz (2003): “The dielectric constant and thickness, in combination with the time derivative of the applied voltage, dV/dt, determine the amount of displacement current that can be passed through the dielectric(s). To

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High voltage AC generator

M. I. Boulos et al.

High voltage electrode Dielectric barrier Discharge Ground electrode

Fig. 16 Common planar and cylindrical dielectric barrier discharge configurations (Kogelschatz et al. 1999)

transport current (other than capacitive) in the discharge gap the electric field has to be high enough to cause breakdown in the gas. In most applications the dielectric limits the average current density in the gas space. It thus acts as a ballast which, in the ideal case, does not consume energy.” Dielectric barrier materials are glass or silica glass and in special cases also ceramic materials, thin enamel, or polymer layers. At very high frequencies, the current limitation by the dielectric becomes less effective. For this reason DBDs are normally operated between 50–60 Hz to about 10 MHz. When the electric field in the discharge gap is high enough to cause breakdown, a large number of micro-discharges are observed in most gases when the pressure is of the order of 105 Pa. The number of micro-discharges per unit of electrode area and time depends on the power density and their strength (energy density, transferred charge) and is determined by the gap spacing, pressure, and dielectric properties. Figure 17 shows micro-discharges in a 1-mm gap containing atmospheric pressure air, photographed through a transparent electrode. According to Kogelschatz (2003), the ionic and excited atomic and molecular species initiate chemical reactions that finally result in the synthesis of a desired species (e.g., ozone, excimers) or the destruction of pollutants (e.g., volatile organic compounds (VOCs), nerve gases, odors, NH3, H2S, NOx, SO2, etc.). DBDs can be used to: • Generate optical radiation by the relaxation of excited species in the plasma, mainly the generation of UV or vacuum UV radiation. Those excimer ultraviolet lamps can produce light with short wavelengths. Many excimers have been produced that way (Kogelschatz 2003). UV photons can also be utilized to excite phosphors that convert UV radiation to visible light. This principle is used on a large scale in fluorescent lamps and energy-saving lamps.

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25

Fig. 17 End-on view of micro-discharges in atmospheric pressure dry air in a plate ozonizer (original size, 6  6 cm; exposure time, 20 ms) (Kogelschatz 2003)

Fig. 18 Configuration of discharge tubes in a technical ozone generator (not to scale) (Kogelschatz 2003)

HV fuse

Discharge gap

Outer steel tube

Metal coating

Glass tube

Gas flow Gas flow

∼ Cooling water flow

• Industrial ozone generation is performed by DBD using oxygen or air, at 0.1 and 0.3 MPa. Generally ozone is produced with cylindrical discharge tubes of about 20–50 mm diameter and 1–3 m length. A typical configuration is presented in Fig. 18. The Pyrex glass tubes are closed at one side, mounted inside slightly wider stainless steel tubes to form annular discharge gaps of about 0.5–1 mm radial width. Large ozone installations reach input powers of several MW, the ozone production reaching 100 kg/h. • Based on the mature ozone generation technology, large atmospheric pressure gas flows, with negligible pressure drop, can be treated for pollution control. DBC discharges are used to treat diesel exhaust gases.

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• Plastic and other polymer materials have nonpolar chemically inert surfaces making them non-receptive to bonding, printing inks, coatings, and adhesives. DBC discharges substantially increase the surface energy of different substrates that can be treated at speeds over 10 m/s. These discharges are also used for treatment of textiles at atmospheric pressure and room temperature. The treatment can be used to modify the surface properties of textile to improve wettability, absorption of dyes, and adhesion and for sterilization.

4.3.1 Microwave Plasmas In microwave-induced plasmas (MIPs), the plasma receives its sustaining energy from electromagnetic (EM) microwaves called surface waves or traveling wave discharges (TWD) for sustaining the plasma at frequencies in the range 300 MHz–300 GHz. The plasma can be created at pressures between 10–3 Pa to several 100 kPa, in discharge dielectric tubes with diameters from 0.5 to 150 mm, without the presence of electrodes, which prevents its contamination by electrode material. The microwave systems that are used to sustain MIPs are comparable to those found in domestic microwave ovens. A typical reactor is presented schematically in Fig. 19. The wave generated by the magnetron travels to a coupler system, which drives it into a cavity where the heated target is placed; some shielding protects the user from the microwave radiation. The MIP is created if the target is a dielectric container filled with a gas. The energy is coupled to the charged particles of the plasma. Electrons play a key role in the energy transfer because they oscillate with the frequency of the EM field and collide with heavy particle to which they transfer energy. The electrons with higher energies are responsible for ionization processes, which maintain the discharge. Many different ways allow creating and operating MIPs. The size and shape of the plasma depend on the gas chemical composition and Fig. 19 Schematic diagram showing the microwave system components and the plasma torch (Jogi 2011)

Sliding short resonator Waveguide

Gas flow

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27

the pressure in the container. Many applicators exist: cavity microwave plasma generators, waveguide microwave plasma generators, surface wave plasma generators, slow wave plasma generators, wave beam microwave plasma generators, etc. For more details see Moisan and Pelletier (1992) and Lebedev (2010). MIPs can be used for (Uhm et al. 2006; Fridman 2008): • Production of carbon nanotubes with C2H2 (carbon source) and using iron pentacarbonyl as the source of metal catalyst particles. • Diamond synthesis: crystalline diamond predominantly composed of {100} and {111} faces was grown on a non-diamond substrate from a gaseous mixture of hydrogen and methane under microwave glow discharge conditions. • Deposition of hydrophobic or hydrophilic layers on surfaces via plasma polymerization. • Abatement of fluorinated compound gases (hydrofluorocarbons, perfluorocarbons, SF6) resulting from semiconductor fabrication systems. • Decontamination of chemical and biological warfare agents. • Plasma Assisted Chemical Vapor deposition (PA-CVD).

4.4

Thermal, Man-Made Plasmas

Thermal, man-made plasmas are also classified occasionally in the North American and European literature as hot or equilibrium plasmas. In contrast, they have been classified in the Russian literature as low-temperature plasmas in order to distinguish them from thermonuclear plasmas. While, by definition, thermal plasmas are in, or close to, LTE, it has become increasingly clear that the existence of LTE in thermal plasmas is the exception rather than the rule. Many plasmas that are classified as thermal plasmas do not meet all requirements for LTE, i.e., they are not in complete local thermodynamic equilibrium (CLTE). As will be discussed later in more detail, one of the main reasons for deviations from CLTE is the lack of the excitation equilibrium (Boltzmann distribution). In particular, the lower-lying energy levels of atoms may be under populated due to the high radiative transition probabilities of these levels, resulting in a corresponding overpopulation of the ground state. Because of the small contribution of excited species to the enthalpy of plasma, this type of deviation from CLTE is immaterial for most engineering applications. For this reason, such plasmas are still treated as thermal plasmas or, more accurately, as plasmas in partial local thermodynamic equilibrium (PLTE). Caution must be exercised, however, if emission spectroscopy is used for diagnostics in such plasmas. Substantial errors may be incurred if energy levels that deviate from excitation equilibrium are used. More serious deviations from LTE may be expected in the fringes of plasma or in the vicinity of walls or electrodes. Deviations from both kinetic ðTe 6¼ Th Þ and chemical (composition) equilibrium may be found in such regimes. In high-speed plasma flows, deviations from chemical equilibrium are likely because chemical reactions cannot follow the

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rapid macroscopic motion of the species; a chemically “frozen” situation results. In this case electron densities may be substantially higher than one would expect from the prevailing temperatures. A more detailed discussion of such deviations from LTE will follow in ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics.”

4.5

Basic Concepts Used for the Generation of Thermal Plasmas

The generation of man-made, thermal plasmas is possible using two broad ranges of physical concepts. These are: • Electric arcs which result from the passing of an electric current (DC or AC) through a gas between two electrodes. • Capacitive or inductive coupling of the energy in the plasma. These are often also referred to as electrodeless plasma generation.

4.5.1 Thermal Arcs Passing an electric current through a gas between two electrodes may generate thermal arcs. Since gases at room temperature are excellent insulators, a sufficient number of charge carriers must be generated to make the gas electrically conducting. This process is known as electrical breakdown, and there are many possible ways to accomplish this breakdown. Breakdown of the originally nonconducting gas establishes a conducting path between a pair of electrodes. The passage of an electrical current through the ionized gas leads to an array of phenomena known as gaseous discharges. In most cases such discharges are produced by direct current between the cathode and the anode. Alternating current sources working at low frequency (less than 100 Hz) produce arcs that resemble direct-current ones: on each half cycle, the arc is initiated by breakdown, and the electrodes interchange roles as anode and cathode as current reverses. As the frequency of the current increases, there is not enough time for all ionization to disperse on each half cycle, and the periodic breakdown of the gas is no longer needed to sustain the arc; the voltage vs. current characteristic becomes nearly ohmic. Such gaseous discharges are the most common though they are not the only means for producing thermal plasma. Finally, heating gases (vapors) in high-temperature furnace can also produce plasmas. However, because of inherent temperature limitations, this method is restricted to metal vapors with low ionization potentials. Because of space limitations, this section will be limited to a presentation of the most common arc-generated, steady-state thermal plasmas (including those produced with low-frequency alternative current). Transient plasmas and plasmas that are not produced by electrical discharges such as those produced by laser beams, high-energy particle beams, shock waves, or heating in a furnace will not be included in this book. The potential distribution in an arc shows a peculiar behavior, as indicated in Fig. 20. Steep potential drops in front of the electrodes and relatively small potential gradients in the arc column suggest dividing the arc into three parts:

1

The Plasma State

Fig. 20 Typical potential distribution along an arc (Boulos et al. 1994)

29 Cathode

Anode

Voltage

Arc column

Va’

V c’

d c’

da’

• The cathode region • The anode region • The arc column The arc column is a true plasma that will approach a state of LTE in a high-intensity arc. High-intensity arcs, with which we are mainly concerned in the following, are defined as a discharges operated at current levels, I > 50 A, and pressures p > 10 kPa (0.1 atm). These are characterized by strong macroscopic flows induced by the arc itself. Any variation of the current-carrying cross section of the arc gives rise to a pumping action of the type shown in Fig. 21. The effect is a consequence of the interaction of the arc current with its own magnetic field. At sufficiently high currents (I > 100 A) and axial current density variations, flow velocities of the order of 100 m/s are produced. The cathode jet phenomenon, also known as the Maecker effect (Finkelnburg and Maecker 1956; Boulos et al. 1994; Pfender 1999), is a typical example. Temperatures and charged particle densities, which are the most important properties of arc plasma, can vary over a wide range. These properties are determined by the arc parameters, including the arc geometry. For arc applications, it is useful to classify arc columns in terms of their methods of stabilization. There is a direct link between the method of stabilizing the arc column and the options available for the design of arc devices. For stable operation, most electric arcs require some kind of stabilizing mechanism that must be either provided externally or produced by the arc itself. Here the term stabilization refers to a particular mechanism that keeps the arc column in a given stable position, i.e., any accidental excursion of the arc from its equilibrium position causes an interaction with

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Fig. 21 Induced flow resulting from arc constriction using a watercooled diaphragm (pumping action) (Boulos et al. 1994)

Diaphragm

Induced flow

Induced flow

the stabilizing mechanism such that the arc column is forced to return to its equilibrium position. This stable position is not necessarily a stationary one; the arc may, for example, rotate or move along rail electrodes with a certain velocity. Stabilization implies in this case that the arc column can only move in a well-defined pattern controlled by the stabilizing mechanism (Finkelnburg and Maecker 1956). Free-burning arcs, as the name implies, have no external arc stabilization mechanism imposed on the arc. This does not exclude the possibility that this arc generates its own stabilizing mechanism. Although high-intensity arcs may be operated in the free-burning arc mode, they are frequently also classified as selfstabilized arcs if the induced gas flow due to the interaction of the self-magnetic field with the arc current is the dominant stabilization mechanism. Arcs operated at extremely high currents (up to 100 kA) known as ultrahigh-current arcs should also be mentioned in this category. Although most experiments in this current range utilize pulsed discharges, the relatively long duration ð 10 msÞ of the discharge justifies classifying them as arcs. There is considerable interest in such arcs for applications involving melting and steelmaking, chemical arc furnaces, and highpower switchgear. Visual observations of ultrahigh-current arcs in arc furnaces reveal a rather complex picture of large, grossly turbulent plasma volumes, vapor jets emanating from the electrodes, and parallel current paths with multiple, highly mobile electrode spots. In this situation there is no evidence for any dominating stabilizing mechanism. Induced gas flows and vapor jets exist simultaneously, interacting with each other in a complicated way. For certain polarities of the arc and certain electrode materials, stable vapor jets that are able to stabilize the arc

The Plasma State

a

31

b

d

Plasma gas

e Plasma gas

Plasma gas

Plasma gas

c

Plasma gas

Plasma gas

1

Fig. 22 Arc configurations commonly used for the integration of an arc in a plasma-generating device

column have been observed. Thus, the generation of vapor jets by the arc represents another possible mechanism for self-stabilization of arcs (Edels 1973). The integration of an arc in a plasma-generating device is possible using any of the arc configurations illustrated schematically in Fig. 22. These can be grouped as one of the following two setups: • Blown arc configuration (plasma torch) represented on Fig. 22a, c for a hot or cold cathode, respectively. In each of these two cases, the arc is struck between an internal cathode and anode integrated in the plasma torch, with an adequate flow of plasma-forming gas in the torch in order to insure the stabilization mechanism of the arc and the continuous motion of the arc root on the surface of the cold electrodes (anode in the case of Fig. 22a and both cathode and anodes in the case illustrated in Fig. 22c). • Transferred arc configuration represented in Fig. 22b, d, e. In cases b and d, the plasma torch comprises only the cathodic electrode, while the anode is external to the plasma torch and can be the workpiece as in the case of plasma welding and cutting, or a molten metal bath, as in the case of metallurgical applications of thermal plasmas. Alternately the arc polarity can be inversed, as shown in Fig. 22e, with the internal electrode in the plasma torch acting as anode, while the external electrode being the cathode. Such a configuration is often identified as a reversed polarity mode of operation. Photographs of laboratory scale transferred arcs are shown in Figs. 23 and 24. The transferred arc shown in Fig. 23 is formed between a hot cathode and a molten copper anode. The arc length is 100 mm, with a 750 A arc current. A photograph

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Fig. 23 Photograph of an Ar/H2 transferred arc, 100 mm long, 750 A

Fig. 24 Photograph of a transferred arc struck between two plasmas torched over a molten metal bath

of a transferred arc between two plasma torches is shown in Fig. 24. In this case one of the two torches is acting as cathode (right-hand side), while the second one (left-hand side) is acting as anode. It may be noted that in this case the current flows from the first torch to the molten metal bath and back from the molten metal bath to the second torch.

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4.5.2 Inductively Coupled Discharges While the concept of electrodeless induction heating of gases was recognized as early as 1945, by Babat (1947), the first demonstration of the continuous operation of an inductively coupled radio frequency (RF) discharges can be traced back to Reed (1961a, b). The basic concept used is illustrated on the left-hand side of Fig. 25. General reviews on the subject are given by Eckert (1974) and Boulos (1997). The plasma is generated through the electromagnetic coupling of the energy from an induction coil into the plasma discharge maintained in a coaxial plasma confinement tube. A high frequency current is applied to the coil generating an alternating magnetic field in the discharge region, in which it induces an alternating annular electric field, which sustains the discharge through ohmic heating. A photograph of pure argon, inductively coupled discharge, confined in an openended, air-cooled, quartz tube is shown on the right-hand side of Fig. 25. Typical radial profiles of the induced electric field, electrical current density, and electrical conductivity at the middle section of the induction coil are given in Fig. 26 after Eckert (1971) and Eckert (1972). These show the radial variation of the induced electric field, E (V/cm) and the radial profile of the electrical conductivity of the plasma, σ (mho/cm), and of the induced electric current density, j (A/cm2). The radial distribution of the local power generation can be calculated as (j  E) which gives rise to an off-axis maximum in the annular region known as the skin depth δ. The value of the skin depth is typically in the tens of mm range. It is function of the frequency of the applied magnetic field, the electrical conductivity

Powder Central gas

Sheath gas

RF Electrical Supply (MHz)

Fig. 25 Basic concept for energy transfer in an inductively coupled plasma torch (left) and photograph of a 12 kW open-air, inductively coupled argon discharge (right)

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Fig. 26 Radial distribution of the magnitude of the induced electric field (E), current density (J), and electrical conductivity (σ) in an argon induction discharge, f = 2.6 MHz, Po = 25 kW (After Eckert (1972))

of the plasma, and the permeability of the medium as given by Eq. 14. For more details see ▶ Chap. 18, “RF Inductively Coupled Plasma Torches”: 1 δ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi πξo σ f

4.6

(14)

Thermal Plasma Sources and Their Fields of Applications

A summary of the range of the principal, man-made, thermal plasma sources considered is presented in Fig. 27 as function of their nominal power rating (kW) and the plasma velocity range (m/s). With the exception of plasma sources for ballistic and aerospace reentry testing, most of these sources operate at essentially atmospheric pressure or soft vacuum conditions.

4.6.1 Plasma Torches with Hot Cathodes These are among the most commonly used thermal plasma sources for a wide range of applications including plasma cutting and welding and plasma spraying of protective coatings and near-net-shaped parts. In these devices, an electric arc generates the high-temperature plasma through resistive energy dissipation by the current flowing through the partially ionized gas within the torch. In order to allow the current to flow through the gas, the gas temperatures must be sufficiently high to have an appreciable degree of ionization resulting in sufficiently high electrical conductivity. For most plasma-forming gases, the required temperatures are around 8,000 K and above at atmospheric pressure. A schematic of a typical dc plasma

The Plasma State

Plasma velocity (m/s)

1

Segmented dc plasma torches for ballistic and aerospace reentry testing Hot cathode DC plasma torches for plasma spraying, cutting and welding Cold cathode DC plasma torches for plasma R.F. induction plasma metallurgical applications torches for materials and waste treatment synthesis and processing

104 103 102 10 1 10−1

35

1

10

100

1000

10 000

Plasma power rating (kW)

Fig. 27 Classification of man-made thermal plasma sources according to their nominal power rating and plasma velocity range

Arc column

Anode attachment

Cooling Water out Plasma gas Hot Cathode

Cooling Water in

Plasma jet Anode nozzle Cold gas Boundary layer

Fig. 28 Schematic representation of a DC plasma torch with a hot cathode design

torch is shown in Fig. 28. The arc is struck between a central cathode (usually a rodor button-type design) and a cylindrical annular anode nozzle. The plasma gas is injected (axially or as a vortex) at the base of the cathode, heated by the arc, and exits the nozzle as a high-temperature, high-velocity plasma jet. In plasma torch designs used for plasma spraying, electrons are supplied through thermionic emission from the hot cathode material. A mixture of tungsten doped with 1% or 2% by weight of a material with a low work function value, such as ThO2, La2O3, or LaB6, is widely used as torch cathode. The consequence is a significantly reduced cathode operating temperature and longer cathode life. The arc column characteristics are determined by the energy dissipation per unit length, i.e., by the arc current, the plasma gas flow and composition, and the arc channel diameter. The torch anode usually consists of a water-cooled copper channel, sometimes lined with a tungsten or tungsten copper sleeve. Numerous nozzle designs are used

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to maintain the anode arc root attachment in continuous motion over the annular anode surface and to optimize the stability and characteristics of the plasma jet emmerging from the plasma torch. This jet is typically highly turbulent with continuous motion as shown by the results of three-dimensional arc modeling studies shown in Fig. 29. The turbulent nature of the flow is also observed in optical photographs of the emerging plasma jet shown on the right-hand side of the same figure. Alternately, arc rotation on the surface of the anode could also be enhanced through the addition of a field coil surrounding the anode as shown in Fig. 30. Through this arrangement, the interaction of the generated axial magnetic field, with the radially oriented arc current near the arc root attachment on the anode  ! point ! surface, gives rise to a tangential electromagnetic force J  B which further Fig. 29 Results of threedimensional simulation of arc inside plasma torch (left) (Reproduced with kind permission of Juan Pablo Trelles) and photographs of the emanating plasma jet (right) (Fauchais et al. 2014)

Fig. 30 Schematic of a plasma torch with stick-type hot tungsten cathode and a water-cooled cold anode, with magnetic field rotation of the arc root on the anode surface

1

The Plasma State

37

enhances the velocity of rotation of the arc root, provided, of course, that the direction of the force is in the same direction as that of the aerodynamic vortex motion in the torch. It should be pointed out that the use of a strong vortex motion, with or without magnetic field enhancement, has been the subject of extensive studies in the former Soviet union (Zhukov 1979), applied to reduce electrode erosion in a wide range of cold-electrode, plasma torch designs. Using hot cathode, arc-generated thermal plasmas for metal cutting and welding is a well-established technology (Finch 2007). Both TIG (tungsten inert gas) and MIG (metal inert gas) welding processes are in wide use today (Lucas 1990). In TIG welding, a transferred arc configuration is used where the non-consumable tungsten electrode serves as the cathode and the workpiece is the external anode. Inert gas or gas mixtures (Ar, He) are blown along the cathode to prevent contamination from the surroundings. In MIG welding, the arc is maintained, also in the transferred arc mode, between the workpiece and a consumable wire electrode that is fed continuously through the torch at controlled speeds. Inert gas is fed simultaneously through the torch into the weld zone, protecting the weld from the contaminating effects of the atmosphere (Wilden et al. 2006). Other welding methods make use of submerged arcs. Plasma cutting is also a well-established technology Thomas (2012). An example of a typical plasma cutting torch design is shown in Fig. 31. Torches with stickor button-type cathodes (made of tungsten or hafnium) with laminar or vortex injection of plasma-forming gas (oxygen) are used (Nemchinsky and Severance 2006; Zhou et al. 2008). In plasma cutting, the specific heat fluxes at the workpiece are at least one order of magnitude higher (typically in the range from 100 to 200 kW/cm2) than for arc welding. This difference implies that the arc for plasma cutting must be extremely constricted, resulting in high current densities and correspondingly high temperatures in the axis of the arc, even at relatively low currents (20 A). New developments in this area were reported over the past few years, including air-operated and air-cooled low-amperage cutting torches that are extensively used in automotive repair shops. Another more recent development considers highcurrent cutting torches for underwater cutting. This technology may become important in the dismantling of nuclear power plants that are beyond their useful life span. Carbon-arc cutting, also known as air arc cutting, is an arc cutting process where metal is cut and melted by the heat of a graphite electrode. A compressed air jet removes molten metal. This process is most often used for cutting and gouging aluminum, copper, iron, magnesium, and carbon and stainless steels. Because the air jet blows the molten metal away, oxidation is rather low. The main purpose for air carbon arc is gouging and removal of old or defective welds so that they can be redone or the equipment dismantled. It allows removing the minimum possible amount of material so that the joint can be re-welded. For more information see ▶ Chap. 15, “Plasma Torches for Cutting, Welding, and PTA Coating.” In plasma spraying, which is one of the most important applications of these torches, the material to be sprayed is in the form of a fine powder (typically with a mean particle diameter between 20 and 100 μm), suspended in a carrier gas, and is

38

M. I. Boulos et al.

a

b

Axial gas injection

Vortex gas injection

Hot cathode holder

Anode-nozzle

Button type cathode

Tungsten stick type cathode

Plasma gas

Hot cathode Exit nozzle

Fig. 31 Typical design of a plasma cutting torch (Eliot 1991)

injected into the plasma jet, where the powder particles are accelerated, heated, and melted. A schematic representation of the plasma spraying process is shown in Fig. 32. As the molten powder particles impinge at high velocities (from 100 to 300 m/s) on the surface of a substrate, they form splats, which superimpose on each other resulting in a more or less dense coating. Any material, provided its melting and vaporization or decomposition temperatures are separated at least by 300 K, can be sprayed. Plasmas are mainly used to spray ceramic materials either in air (oxides) or controlled atmosphere (carbides, borides, etc.) but also metals or alloys in air, if oxidation is not a problem, or under soft vacuum, especially, for example, to achieve diffusion adhesion for bond coats (Fauchais et al. 2014). To coat big parts, such as big rolls in the paper industry, plasma torches up to 250 kW have been developed with particle feed rates up to 20 kg/h (Morishita et al. 1991). Recently, Oerlikon Metco has developed a new process called PS-PVD using a high-power plasma spray torch (180 kW–3000 A, gas flow rate up to 200 slm) working at a pressure as low as 0.05–0.2 kPa (0.5–2 mbar). Under such low-pressure conditions,

1

The Plasma State

39

Plasma jet core

Coating Air engulfment

Plasma plume Plasma torch Particles injector

Splats Unmelted particles

Details of coating

Substrate

Pores

Fig. 32 Schematic representation of plasma spray process: the plasma gas entering the torch at the base of the cathode is heated by the arc between the cathode and the cylindrical anode. Spray particles are injected into the plasma jet and transported to the substrate

the plasma jet reaches more than 2 m in length and up to 0.4 m in diameter. The coating obtained exhibits a microstructure similar to that of EB-PVD coatings (von Niessen.and Gindrat 2011). These coatings are used against wear, corrosion and oxidation, thermal protection, clearance control and bonding. Other applications include freestanding spray-formed parts, medical applications, replacement of hard chromium, and electrical and electronic applications. Industries using them are aerospace, land-based turbines, automotive, electrical and electronic, land-based and marine applications, medical applications, ceramic and glass manufacturing, printing, pulp and paper, metal processing, petroleum and chemical, electrical utilities, textile and plastic, polymers, reclamation, and nuclear. For example, Fig. 33 presents ceramic coating plasma sprayed on a large drum for manufacturing paper. Plasma spray coating has also been traditionally used extensively in the aerospace industry. Figure 34 shows components in a typical jet engine, which are plasma coated with different thermal barrier coatings, wear resistance, and abradable coatings. Globally, plasma spraying represents about 50% of the thermal spray market which was evaluated to be about $ 4.6 billion in 2012. For more information see ▶ Chap. 26, “Plasma in the Thermal Spray Coating Industry.”

4.6.2 Plasma Torches with Cold Electrodes The cold electrodes used in these torches usually consist of a water-cooled metal sleeve, mostly copper, steel, silver, or an alloy of these materials (Heberlein 1999). Such electrodes can easily work with oxidizing or reducing plasma-forming gases. For the cathode, electron emission is the result of evaporation and ionization of the

40

M. I. Boulos et al.

Fig. 33 Plasma spray applications in the paper and printing industry (Reproduced with kind permission of Sulzer-Metco AG, Switzerland)

Fig. 34 Plasma sprayed components in a typical jet engine (Reproduced with kind permission of Oerlikon-Metco AG, Switzerland)

1

The Plasma State

41

metal in very small cathode spots with the consequence that it is necessarily associated with materials loss. Studies of cathode spots under plasma conditions show spot diameters of 1–2 mm for currents ranging from 800 to 1,400 A, with current densities of 108–109 A/m2. The erosion rate (ER) in (μg/C) has been found to follow the general equation: ER ¼ A  Im

(15)

where I is the arc current. The constants A and m depend on the plasma-forming gas and the operating conditions. A critical design feature of such torches is the introduction of a strong vortex motion in the flow pattern of the gases in the discharge cavity driving the arc roots over the electrode surfaces leading to a more uniform distribution of the electrode erosion over as large an area as possible. The motion of the arc root can be enhanced by the superposition of a magnetic field coil around the electrodes, cathode, and/or anode, The interaction of the generated axial magnetic field with the arc at its point of attachment to the electrode gives rise to a tangential electromagnetic force acting on the arc which is responsible for the rotation of the arc root over the inner surface of the electrode. An example of such an arrangement was given in Fig. 30 for the arc root rotation on the surface of a cold anode using a superimposed magnetic field. Similar designs using electromagnetic fields for the rotation of the arc root attachment on the surface of a cold electrode were adopted by Aerospatiale in France in their complete line of high-power, cold-electrode plasma torches (Fig. 35) where only the back electrode, normally the anode, had a field coil for the rotation of the arc root attachment. The need for the generation of a strong vortex motion in the discharge cavity required the use of relatively high plasma gas flow rates (about 200–300 m3/MW) resulting in relatively low enthalpies of the plasma jets emerging from these torches ( 1 are defined as excited states of the atom. A transition from a higher quantum orbit (nu) to a lower quantum orbit (n‘) leads, according to one of Bohr’s postulates, to the emission of a photon of frequency ν, i.e.,   2 E u
E1 m e ð Z0 Þ e 4 1 1 ν¼ ¼

h ð4πε0 Þ2 4πℏ3 n2u n21

(12)

where nu > n‘ and both nu and n‘ are integers. In terms of the wave number σ = 1/λ = ν/c, it follows: 

1 1 σ ¼
R1 ðZ Þ 2
2 nu n1 0 2

 (13)

where R1 is the Rydberg constant, R1 ¼

me e4 ð4πε0 Þ2 4πcℏ3

(14)

The essential predictions of Bohr’s theory can be summarized as follows: (a) The lowest energy state that an atom or ion (particle) can assume is the state for which n = 1, known as the ground state.

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63

(b) Excitation of a particle to a level where (n > 1) requires energy, the particle can acquire in an electrical discharge. (c) The excess energy of a particle is generally emitted in the form of radiation. An excited particle returns to the ground state by a single transition or by a series of transitions in which the electron assumes states of successively lower energies. (d) A very large number of such energy transitions comprise an atomic and/or ionic spectrum, i.e., all possible transitions will occur during observation of such a spectrum. The wave number, σ, of the emitted spectral lines is given by Eq. 13 for all possible combinations of nu and n‘, where nu > n‘. For the sake of simplicity, the hydrogen atom (Z0 = 1) will be considered in the following discussion. For a fixed value of n‘ = 2, Eq. 13 becomes  σ ¼ R1

1 1
22 n2u

 with

nu ¼ 3, 4, 5, 6, . . .

(15)

This expression describes the Balmer spectrum of the hydrogen atom, provided that R1 is identified as RH, the Rydberg constant of the hydrogen atom. RH is obtained by replacing the electron mass in Eq. 14 by the reduced mass μ = me/(1 + me/M). This extremely small correction of R1 (about 0.05%) accounts for the finite mass of the hydrogen nucleus compared to that of the electron and brings predictions and experiments into perfect agreement. In addition to the Balmer spectrum, Bohr’s theory predicted other spectral series for the hydrogen atom: n‘ n‘ n‘ n‘ n‘

¼ 1, nu ¼ 2, nu ¼ 3, nu ¼ 4, nu ¼ 5, nu

¼ 2, 3, 4, ¼ 3, 4, 5, ¼ 4, 5, 6, ¼ 5, 6, 7, ¼ 6, 7, 8,

... ... ... ... ...

Lyman series ðultravioletÞ Balmar series ðvisibleÞ Paschen series ðnear infraredÞ Brackett series ðinfraredÞ Pfund series ðfar infraredÞ

The success of Bohr’s theory was particularly impressive because none of the spectral series other than the Balmer and Paschen series were known at the time Bohr developed his theory. The existence of the other spectral series was experimentally confirmed afterward. The predictions of Bohr’s theory work equally well for other one-electron systems such as He+, Li++, etc., which have spectra similar to that of the hydrogen atom but with larger wave numbers (as given by Eq. 13).

2.3

Line Absorption

The previous discussion referred to spectra produced by emission of radiation (emission spectra). In absorption spectra, absorption of radiation is considered. Such spectra are also easy to understand in terms of Bohr’s theory. Atoms or ions

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M. I. Boulos et al.

can only absorb radiation with appropriate wavelengths or energies. The absorption of radiation of frequency υu‘ will increase the energy level of the atom from its lower excited level E‘ to an upper level Eu according to Eq. 2. Equation 2 can be rewritten in terms of the wave number, σ, as σ¼

1 E u
E‘ ¼ λ hc

(16)

At room temperature, almost all atoms are in the ground state, so absorption can occur only from this state. This implies that for hydrogen, only the Lyman series will appear in absorption at room temperature. Since absorption occurs from one level only (the ground state), absorption spectra show many fewer lines than emission spectra.

3

The Hydrogen Atom and Its Eigenfunctions

3.1

The Schro¨dinger Equation

The general equation describing the propagation of a wave can be written as ∇2 Ψ


1 ∂2 Ψ ¼0 u2 ∂t2

(17)

where ∇2 represents the Laplace operator, Ψ the wave function, and u the phase velocity of the wave. The last term in Eq. 17 is the second derivative of the wave function with respect to time. In Cartesian coordinates the operator ∇2 can be expressed by ∇2 ¼

∂2 ∂2 ∂2 þ þ ∂x2 ∂y2 ∂z2

(18)

Equation 17, which is valid for macroscopic systems, will be transformed into an equation (the Schro¨dinger equation) that holds for microscopic (atomic scale) systems. For this transformation, quantum mechanical concepts such as the wavelength of matter (de Broglie wavelength) must be introduced. In terms of the de Broglie wavelength λ¼

h mv

(19)

where h is Planck’s constant, m is the mass of a particle, and v is its velocity; the phase velocity of a wave can be expressed by u ¼ λν ¼

hν mv

(20)

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Basic Atomic and Molecular Theory

65

The numerator in Eq. 20 represents not only the well-known energy of a photon but also, in terms of de Broglie’s postulate, the energy, E, of any particle. Since the total energy of a particle is comprised of kinetic and potential energy, the kinetic energy can be expressed as 1 2 mv ¼ E
VP 2

(21)

m2 v2 ¼ 2mðE
VP Þ

(22)

or

Substituting Eq. 22 into Eq. 20 gives u¼

hν ½2mðE
VP Þ1=2

(23)

and substituting Eq. 24 into the general wave equation (Eq. 17) leads to ∇2 Ψ


2mðE
VP Þ ∂2 Ψ ¼0 ∂t2 h2 ν2

(24)

In general, the wave function Ψ is a function of location and time, i.e., Ψ = Ψ(x, y, z, t). The wave function can also be expressed as a product of an amplitude ψ and a periodic function in time: Ψðx, y, z, tÞ ¼ ψðx, y, zÞexpð
2πiνtÞ (25) pffiffiffiffiffiffiffi where i ¼
1 is the imaginary unit. Substituting Eq. 25 into Eq. 24 results in the famous Schro¨dinger equation: ∇2 ψ þ

8π2 m ðE
VP Þψ ¼ 0 h2

(26)

This equation is a homogeneous second-order partial differential equation that describes the steady-state properties of atomic systems.

3.2

Solution of the Schro¨dinger Equation

If the potential energy of an atomic system VP(x, y, z) is specified, solutions of the Schro¨dinger equation can be obtained in terms of the so-called eigenfunctions ψ(x, y, z) that are a consequence of the boundary conditions of the system. The parameters that appear in these eigenfunctions are the quantum numbers. Solutions of the Schro¨dinger equation also define the possible energy states of atomic systems, known as the eigenvalues. For an atomic system the eigenvalues can assume only quantized values, as discussed in the previous paragraph.

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The mathematical formalism required for the solution of the Schro¨dinger equation, even for the simplest atomic system, is rather extensive. In general, the choice of an appropriate coordinate system depends on the geometry of the problem. For example, in the case of the H atom, a spherical coordinate system (r, θ, ϕ) is used since the potential energy depends only on r. Substituting the value of VP (Eq. 7) in Eq. 26, the Schro¨dinger equation for the H atom can be written in spherical coordinates as follows:       ∂2 ψ 2∂ψ 1 1 ∂ ∂ψ 1 ∂2 ψ 8π2 me e2 þ sin θ þ E þ þ þ ψ¼0 ∂r2 r∂r r2 sin θ ∂θ ∂θ 4πε0 r sin2 θ ∂ϕ2 h2 (27) It is customary to solve this type of partial differential equation by separation of variables. In this process it is assumed that the solution of Eq. 27 can be expressed as a product of three functions (R, θ, Φ), where each function depends on only a single variable, i.e., ψðr, θ, ϕÞ ¼ RðrÞ  ΘðθÞ  ΨðϕÞ

(28)

By introducing Eq. 28 into Eq. 27, the partial differential equation with the variables (r, θ, and ϕ) transforms into three ordinary differential equations, i.e.,  2    d2 R 2 dR 8π me e2 A þ þ E þ R¼0
dr2 r dr r2 4πε0 r h2 1 d sin θ dθ



   dΘ m2 sin θ þ A
Θ¼0 dθ sin2 θ d2 Ψ þ m2 Ψ ¼ 0 dϕ2

(29)

(30)

(31)

with A and m2 as separation constants. Mathematical considerations show that: – The differential equation for R(r) has only discrete solutions in which the quantum number ‘ takes on integer values between 0 and n – 1, i.e., 0‘n
1

(32)

– The solution of the differential equation for Θ(θ) is valid for discrete values of the quantum number m over the range between 0 and  ‘, i.e., m ¼ 0,  1,  2, . . . ,  ‘

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Basic Atomic and Molecular Theory

67

– The solution of the equation for Φ(ϕ) is valid only for the n and ‘ quantum numbers as defined above. The solutions ψ(r, θ, ϕ), which depend on n, ‘, m, are known as the eigenfunctions of the hydrogen atom. The complete solution in non-normalized form can be expressed as ðmÞ

ψðρ, θ, ϕÞ ¼ ρ‘  un
‘
1 ðρÞexpð
ρ=2ÞP‘ ð cos θÞexpðimϕÞ

(33)

ðmÞ

With ρ ¼ 2r=r1 n  P‘ ðcos θÞ is a polynomial of degree ‘ calculated according to Rodrigues formula, and un
‘
1 is a polynomial of degree n
‘
1. This equation provides a complete description of the behavior of the electron in the stable energy states of the H atom.

3.3

Quantum Numbers

As shown in the last section, the eigenfunctions and energy states of the H atom are characterized by the three quantum numbers n, ‘, m‘ with n ‘ þ 1,

and 0  jm‘ j  ‘

(34)

It is customary to add a subscript ‘ to the quantum number m to avoid confusing it with the particle mass. A fourth quantum number for the electron spin, s, does not follow from this calculation. According to the Pauli exclusion principle, the states of two electrons in an atom cannot be identical. If n, ‘, and m‘ are identical for two electrons, then s has to be different. The spin quantum number, s, can assume only two values: s¼

1 2

(35)

The equation for the energy eigenvalues of the H atom is governed by the quantum numbers m‘ and ‘. For every ‘, there are 2‘ + 1 different eigenfunctions. With the restriction that ‘  n – 1, one finds that the total number of eigenfunctions for a given energy state is n
1 X

ð2‘ þ 1Þ ¼ n2

(36)

‘¼0

This result shows that there can be more than one eigenfunction for one energy state, a property known as degeneracy. Disregarding the electron spin, the energy eigenvalues of the H atom are (n2
l)-fold degenerate. Including the electron spin adds a factor of two to the number of eigenfunctions per energy state because of the two possible spin orientations of the electron.

68

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M. I. Boulos et al.

Probability Distribution

Before considering the probability distribution for various combinations of quantum numbers, a physical interpretation of the quantum numbers is useful: n: Principal quantum number, describes the energy state of an atomic system. ‘: Quantum number for the angular momentum of the electron, usually called the azimuthal quantum number, describes the shape of electron “orbits.” m‘: Magnetic quantum number, describes the orientation of the electron “orbits.” s: Spin quantum number, assumes the values  12, depending on the orientation of the electron spin. Considering only the r-dependence of the wave function, a probability distribution function P can be established: dP ¼ jψ2 jdτ ¼ R2  4πr2 dr

(37)

and p = 4πr2R2 is the probability density. Figure 3 shows examples of such probability density distributions on a relative scale for different combinations of n and ‘.

Fig. 3 Examples of probability density distributions for finding the electron in a hydrogen atom

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Basic Atomic and Molecular Theory

69

As previously discussed, the possible electronic states in an H atom are characterized by a set of four quantum numbers (n, ‘, m‘, s). The first two quantum numbers in this set are of particular interest for describing electronic states. It is customary to assign letters to the values of the second quantum number (‘), i.e., ‘¼0!s ‘¼1!p ‘¼2!d ‘¼3!f ‘¼4!g ‘¼5!h Starting with ‘ = 3 (f), the letters follow the alphabet. The first three letters are derived from the appearance of the spectra of the more complicated alkaline atoms, which show four characteristic spectral series referred to as the sharp series (s), the principal series (p), the diffuse series (d), and the Bergmann series (f). The alkaline atoms show an important similarity to the H atom. As will be shown later in this chapter, the inner electron shells of the alkaline atoms are completely filled and the outermost shell has only one electron. As a first approximation, such atoms can be treated as one-electron systems. Some examples of the notations used for various combinations of the first and second quantum numbers are listed below:  n¼1 1s ‘¼0 n¼2 2s ‘ ¼ 0 n¼2 2p ‘¼1 n¼3 3s ‘ ¼ 0 n¼3 3p ‘¼1 n¼3 3d ‘¼2

electron electron electron electron electron

 n¼4 4s ‘ ¼ 0 n¼4 4p ‘¼1 n¼4 4d ‘¼2 n¼4 4f ‘¼3

electron electron electron electron

⋮

electron

4

The Structure of More Complex Atoms

4.1

Atomic Structure

In general, an atom can have Z0 protons in the nucleus, resulting in a total nuclear charge of +Z0 e. In addition, a nucleus contains a certain number of neutrons, which, together with the protons, make up the mass of the nucleus. Since an atom is

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M. I. Boulos et al.

electrically neutral, the total charge of the electrons surrounding the nucleus must be -Z0 e. The nuclear mass of an atom in atomic mass units is indicated by a superscript after the atomic symbol, and the nuclear charge is indicated by a subscript in front of the atomic symbol, for example, 1H

1

, 2 He4 , 3 Li6 , 18 Ar40 , 22 Ti48 , 82 Pb207

Isotopes have the same nuclear charge, but the number of neutrons in the nucleus differs. Typical examples of pairs of isotopes are 3Li6 and 3Li7 and 92U235 and 238 . Such isotopes can not be distinguished chemically since the chemical 92U properties of elements are determined by the arrangement of the electrons around their nucleus and not by the mass of the nucleus. In atomic models of more complex atoms, it is assumed that the electrons are arranged in shells. The first two quantum numbers of an electronic state define the electron shells, as shown below: n¼ main shells : ‘¼ subshells :

1 K 0 s

2 L 1 p

3 M 2 d

4 N 3 f

These definitions provide a schematic outline of atomic structure, which is shown in Table 1 for the first four electron shells. The number of electrons in each subshell is indicated by superscripts following the designation of the electron state (last column of Table 1). Using this schematic, a table of the electronic structures of the elements can be established (Table 2). Table 2 outlines the electronic structures of the elements up to completion of the 4 s shell. The horizontal double lines in Table 2 indicate completed main shells, and the single lines correspond to completed subshells. Table 1 Schematic outline of atomic structure

Shell K L M

N

n 1 2 2 3 3 3 4 4 4 4

‘ 0 0 1 0 1 2 0 1 2 3

Designation of electrons 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f

Number of electrons in subshell 2 2 6 2 6 10 2 6 10 14

Total number of electrons in shell 2 8

Configuration of completed shell 1 s2 2 s22p6

18

3s23p63d10

32

4s24p64d104f14

2

Basic Atomic and Molecular Theory

71

Potassium (K) reveals an irregularity. A new shell is started even though the 3d subshell is not yet completed. This irregular behavior has been confirmed by spectroscopic observations as well as by energetic considerations. For Cu the opposite effect is observed: two electrons move into the 3d subshell leaving only one electron in the 4s subshell. This irregular behavior occurs several times throughout the schematic electron arrangement of the elements. From an extended Table 2, the electron configuration for every element can be derived. Copper, for example, has the electronic configuration: 1s22s22p63s23p63d104s1. Knowledge of the electron configuration of atoms is very important because the chemical properties of atoms are determined mainly by their electron configurations. For example, elements such as the alkali metals

Table 2 Electronic structure of the elements Z0 1 2 3 4 5 6 • • • 10 11 12 13

18 19 20 21 • • • 28 29 30 • • •

Element H He Li Be B C • • • Ne Na Mg Al • • • Ar K Ca Sc • • • Ni Cu Zn • • •

Number of electrons in subshells 1s 2s 2p 3s 3p 1 2 2 1 2 2 2 2 1 2 2 2 • • • • • • • • • 2 2 6 2 2 6 1 2 2 6 2 2 2 6 2 1 • • • • • • • • • • • • • • • 2 2 6 2 6 2 2 6 2 6 2 2 6 2 6 2 2 6 2 6 • • • • • • • • • • • • • • • 2 2 6 2 6 2 2 6 2 6 2 2 6 2 6 • • • • • • • • • • • • • • •

3d

4s

1 • • • 8 10 10 • • •

1 2 2 • • • 2 1 2 • • •

4p

4d

4f

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M. I. Boulos et al.

(Li, Na, K, Ru, and Cs), all of which have one outer electron, show strong similarities in their chemical behavior.

4.2

Electronic States of Atoms

Like atomic energy levels, the angular momentum of an atom, which is character! ized by ‘ in wave mechanics, can assume only discrete values of energy levels. For pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! a bound electron, the magnitude of ‘ is ‘ð‘ þp1ffiffiÞffi  h, where ‘ is the azimuthal quantum number. Thus, a d electron (‘ = 2) has 6 units of angular momentum, where the unit of the angular pffiffiffi momentum is h. Similarly, a 5p electron (an electron in the fifth (0) shell) has 2 units of angular momentum. Electrons with identical values of n and ‘ are called equivalent electrons. The number of equivalent electrons (r) in a subshell is written as a superscript to the r 2 right of the subshell: pffiffiffi n‘ . Thus, 6d denotes 2 electrons in the 6th (P) shell, each electron having 6 units of angular momentum. !

According to classical field!theory, the angular momentum vector ‘ of an atom in a magnetic or electric field F will prescribe a cone with the field direction as axis and will have a constant component m as shown in Fig. 4a. Quantum theory allows only discrete values of m‘ between
‘ and + ‘ as shown in Fig. 4b. The quantum number ‘ can assume all integer values from 0 to n
1.

4.2.1 Momentum The remaining quantum number is the quantum number s (not to be confused with the subshells designated by ‘ = 0), which is associated with the electron spin. The Fig. 4 (a) Precession of the ⇀

angular momentum ‘ of an electron in a magnetic or an electric field. (b) Space ⇀

quantization of ‘ in a field for ⇀

‘ ¼ 3 (after Hertzberg)

2

Basic Atomic and Molecular Theory

73

quantum number s can only assume two values, s = 1/2. Therefore, there can be only two s electrons (‘ = 0) in the first shell (n = 1, ‘ = 0, m‘ = 0, s = 1/2). The second shell (n = 2) can contain two s electrons (n = 2, ‘ = 0, m‘ =0, s = 1/2) and six p electrons (n = 2; ‘ = 1; m‘ =
1, 0, 1; s = 1/2) and so on. Since each electron is characterized by an angular momentum quantum number, ‘, and a spin quantum number, s, these individual quantum numbers can be added to ! ! ! describe the total angular momentum j ¼ ‘ þ s . The total angular momentum quantum pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinumber j = ‘  1/2 is associated with the total angular momentum jðj þ 1Þ  h. When a magnetic field is applied, the total angular momentum vector of the electron orbit can assume (2j + 1) different orientations in the magnetic field, i.e., mj = j, (j
1), . . .
(j
1),
j.

4.2.2 Energy Transitions As shown in Sect. 2.1, the transition of an electron from one energy level to another is associated with the emission or absorption of electromagnetic radiation, i.e., hνu‘ ¼ Eu
E‘ Only transitions between certain energy levels are permitted. The permitted transitions are determined by selection rules that are based on the observed spectra of atoms. The lowest energy state of the atom is the ground state. The other energy states (in which the electron is still bound to the particle) are the electronically excited states. Their lifetime is generally short (10
8 to 10
6 s). The resonance excited state is a special type of excited state in which the probability of radiative transition to the ground state is very high (the lifetime of electrons in this state is extremely small ~10
8 s). The resonance state is the first excited state, and thus, it can appear in both emission and absorption. Photons with the corresponding energy for transition to the resonance excited state will be absorbed by particles in the ground state with very high efficiency. Radiative de-excitation or electronic excitation can cause the transition of an electron to an energy level from which the transition rules forbid a radiative transition to a lower energy level. The lifetime of such a state, called a metastable state, can be orders of magnitude higher than for spontaneous transitions. Such states can be de-excited only by collisions with other particles or by absorption of radiative energy. Thus, such particles act as an energy reservoir. When one or more electrons are completely removed from a particle, the particle is said to be in an ionized state. Electrons removed by ionization processes are crucial for maintaining an electrical discharge. The ionization energy depends on the atomic number Z0 . For the noble gases (He, Ne, Ar, etc.), in which each shell contains the maximum number of electrons allowed by the Pauli principle (completed shells), higher energies are required to remove a bound electron than for other elements. The higher the atomic number Z0 (and principal quantum number n), the greater the number of completed shells shielding the positive charges of the

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nucleus. It is, therefore, easier to remove one electron from Xe than from Kr. Ionization energies vary from 24.5 eV for He down to 14 for Kr and 13 for Xe. The alkali metals (Li to Fr) have the lowest ionization energies, because the binding energy acting on their single outermost electron is low due to shielding by the electrons in the inner, completed shells. The chemical activity of the alkali metals is, correspondingly, very high and they are known as electropositive elements. The increase of the ionization potential is gradual from the alkali metals toward elements with the next completed shell. In general, the energy level of the first excited state (resonance state) follows a pattern similar to the variation of the ionization energies with Z0 or n (Table 3). Negative ions are formed by the attachment of an additional electron to a neutral atom. Of course, elements that need only one electron to form a completed shell (e.g., F, which has only seven electrons in the L shell compared with 8 for Ne) will very easily attach an extra electron due to the field resulting from the nucleus and the electronic charges. Thus, the halogens (from F to At) are the most strongly electronegative elements. Elements with two electrons missing from a completed shell (e.g., O) also tend to favor electron attachment. Table 4 lists the electron affinity of various atoms.

4.3

Designation of Electron Configurations

The nomenclature adopted for describing the transitions between atomic energy levels generally refers to the nature of the coupling between the electrons that are effective in producing the spectrum. Because the core of completed atomic shells has zero angular momentum, only electrons in partially filled outer shells need to be considered. We do, however, need to include those situations in which an atom with completed shells, such as argon, is excited so that an electron from a completed shell is moved to an outer, incomplete shell. Electrons interact with each other through coupling forces arising from: – Electrostatic repulsions between electrons – Magnetic fields resulting from both the orbital motions and the spins of the electrons – Exchange forces between electron spins (these forces can be understood only on a quantum mechanical basis) Both the energy levels of the atom and the probability of transitions between electronic levels depend on the nature and magnitude of these interactions. The potential energy of an atom is determined by the energies of its electrons in their various orbits. Each individual electron can be assigned an orbital quantum number, ‘i, and a spin quantum number, si, (si = 1/2). Most of the various interactions are accounted for by either L–S coupling or j–j coupling (Herzberg, 1944). j–j coupling occurs when the magnetic interaction between the orbital

Li

N

O – – – – Ne –

3

7

8

11

Na

He

2

10

Element H

Z 1

22.99

15.999 – – – – 20.183 –

14.007

6.941

4.004

Atomic weight (10
3 kg) 1.008

Table 3 Electronic data for selected atoms

5.139

13.618 – – – – 21.564 –

14.534

5.391

24.481

Ionization energy (eV) 13.659

2.102 2.104 3.191

9.521 – – – – 16.847 18.380

1.848 1.848 10.335 10.329 10.325

3p2P01/2 3p 2P03/2 4s2S1/2

3s3S01 – – – – 3s1P01 3p3S1

2p2P01/2 2p2P03/2 3s4P3/2 3s4P3/2 3s4P5/2

Energy and designation of the first excited states (eV) (
) 10.198 2p2P01/2 10.198 2p2P03/2 21.216 2p1P01

2.384 2.384 3.575 3.575 0.020 0.028 1.967 4.189 9.146 16.618 16.670 16.714 – – –

19.818 20.614 – 2p3 2D3/2 2p3 2D5/2 2p32P01/2 2p32P03/2 2p43P1 2p43P0 2p41D2 2p41S0 3s5S02 4s3P2 4s3P1 4s3P0 – – –

2s 3S1 2s 1S0 –

Energy and designation of the metastable states (eV) (
) 10.198 2s2S1/2

61,200 144,000 40 166 – – 110–147 0.76–0.90 – 0.8–430 – 0.8
24.8 – – –

0.0197 9,000 –

(continued)

Radiative lifetime of metastable states (s) 0.12

2 Basic Atomic and Molecular Theory 75

Element Cl

At

K

Kr

Z 17

18

19

36

83.80

–

39.944

Atomic weight (10
3 kg) 35.453

Table 3 (continued)

13.999

4.341

15.755

Ionization energy (eV) 12.967

11.623 11.821 1.609 1.616 10.03 10.64 11.30

4s2P01 4s2P03 4p2P01/2 4p2P03/2 5s3P01 5s1P01 5p3S1

Energy and designation of the first excited states (eV) (
) 9.202 4s2P3/2 9.281 4s2P1/2

Energy and designation of the metastable states (eV) (
) 0.109 3p5 2P01/2 8.421 4s4P5/2 8.987 4s4P3/2 9.029 4s4P1/2 11.548 4s2P00 11.723 4s2P02 – – – – 9.91 5s3P2 10.56 5s3P0

Radiative lifetime of metastable states (s) 81 – – – 1.3
55.9 1.3
44.9 – – 1.85 0.49
1.0

76 M. I. Boulos et al.

2

Basic Atomic and Molecular Theory

Table 4 Electronic data for selected atoms

77

Ion formed H
(1 s2) H
(2 s2) from 2 s H
(2 s2p) from 2 s C
O
F
Cl
NO
O
2

(eV) 0.746 0.434 0.460 1.249 1.466 3.448 3.612 0.911 0.438

OH


1.830

angular momentum and the spin angular momentum of each electron becomes dominant compared to the electrostatic and exchange interactions between different electrons. It can be used for intermediate coupling (as in the case of noble gases).

4.3.1 L–S Coupling In L–S coupling, the orbital angular momenta of the individual electrons are strongly coupled among themselves. Therefore, the total ! orbital angular momentum ! L is formed by combining the orbital angular momenta ‘ i of the various electrons. ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The magnitude of the orbital angular momentum L is LðL þ 1Þ  h, where L is the associated quantum number  (Herzberg, 1944). ! Because the spins s i of the individual electrons can be regarded as strongly i

!

!

coupled, the total spin momentum S is formed by combining the spin momenta s ! i of the separate electrons. The magnitude of the spin angular momentum S is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SðS þ 1Þh, where S is the associated quantum number (Herzberg, 1944). ! In L–S coupling the total orbital angular momentum L and the total spin ! momentum S are coupled by weak magnetic forces. A total angular momentum ! J , characterized by the quantum number J, can be expressed by J¼LþS

(38)

!

The magnitude of the total momentum J corresponding to the quantum number J is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi JðJ þ 1Þℏ, where J assumes values between J ¼ L þ S, L þ S
1, L þ S
2, . . . , L
S when S < L and J ¼ L þ S, L þ S
1, L þ S
2, . . . , S
L when S > L

(39)

In the first case (S < L), there are 2S + 1 possible values of J, while in the second case, there are 2L + 1 possible values of J. 2S + 1 is called the multiplicity. A given L-value together with the corresponding multiplicity defines a spectral term. A given J for a given term defines a spectral level, and spectral lines arise from

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M. I. Boulos et al.

transitions between spectral levels. A multiplet consists of all possible levels of a given term. Designation of L-Values The designations used for the total angular momentum quantum numbers are analogous to the designations used for orbital angular momenta of single electrons, except that capital letters are used instead of small letters. The total orbital angular momentum designations are L = 0, 1, 2, 3, 4, . . ., corresponding to the S, P, D, F, G,. . ., levels, respectively. Designation of Terms A term is defined by the L-value and the multiplicity. It is formed by placing the number denoting the multiplicity (2S + 1) as a left-hand superscript on the L-value designation. For example, if S = 1/2 and L = 2, the multiplicity is given by 2S + 1 = 2, and the designation appropriate to L = 2 is D. Therefore, the term is designated by 2D. Designation of Levels A level corresponding to a given term and a given J-value is designated by adding the J-value as a right-hand subscript to the term designation. In the example above where S = 1/2 and L = 2, there are two possible J-values, J = 5/2 and J = 3/2, according to Eq. 39. The corresponding two levels of the D term are designated by 2 D5/2 and 2D3/2 and shown schematically in Fig. 5. Splitting into 2 J + 1 states may occur, for example, in the presence of an electric or a magnetic field. 2 J + 1 is often referred to as the statistical weight of the level and is important for deriving the radiative power of a spectral line or for evaluating partition functions, as will be discussed in chpater 1.6 on thermodynamic propoerties. Designation of Parity X The parity is either odd or even depending on whether ‘i has an odd or even i X ‘i denotes the sum of the angular momentum quantum value. The expression i

numbers for the electrons in the atom. For completed shells, the total X number of electrons is even and so is the parity. Thus, we need to evaluate ‘i only for i

electrons in partially filled shells. Fig. 5 2D term split into levels by electron spin

Energy

Levels Term

2D

5/2

2D

3/2

2D

The term is split by the action of the electron spin

2

Basic Atomic and Molecular Theory

79

Odd parity is designated by a superscript “o” to the right of the term symbol, e.g., P , while an even parity is designated by omitting the superscript “o.”

2 o

Example According to the previous definitions, the ground state of a copper atom will be designated by

 1s2 2s2 2p6 3s2 3p6 3d10 4s 2 S1=2 Since the shells up to 3p do not play any significant role in the formation of the spectrum, this designation is abbreviated by

 3d10 4s 2 S1=2 Because all the shells up to and including 3d are complete, only the outer s electron contributes to the total angular momentum and to the resultant spin. L = 0 and S = 1/2 result in multiplicity of 2S + 1 = 2, leading to the term designation 2S1/2. Excitation of the 4 s electron to other orbits such as 4p or 5d leads, according to the previous arguments, to 2Po and 2D terms, respectively. The 2Po term is split into two levels, 2Po1/2 and 2Po3/2, and the 2D term into the levels 2D5/2 and 2D3/2. On the other hand, if one of the electrons of the copper 3d shell is excited, one obtains the configuration 3d9 4 s nx where n > 4 and x corresponds to s, p, d, f, . . .,. The core now has a residual spin of 1/2, which can couple with the 1/2 spin of both the 4 s and the nx electrons to give a total resultant spin of S ¼ 1=2 þ 1=2 þ 1=2 ¼ 3=2; 2S þ 1 ¼ 4 or S ¼ 1=2 þ 1=2
1=2 ¼ 1=2; 2S þ 1 ¼ 2: The corresponding multiplicities are 4 and 2, known as quartets and doublets, respectively. Energy of the Spectral Levels The energy of a bound electron increases with the increase of the principal quantum number n and, in general, with the value of the angular momentum quantum number, ‘. The following relatively simple rules are applicable for ground state terms, though they do not necessarily apply to higher configurations: – Triplet terms lie below singlet terms in an energy diagram (the energy is smaller for higher S). – The difference in term values between two levels J + 1 and J of a multiplet with the same L and S is proportional to J + 1, e.g., in the case of a 3P2,1,0 multiplet, the term difference between the 3P2 and 3P1 is twice that between the 3P1 and the 3P0.

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The tables of Moore (1949, 1952, 1958), for example, list the energies of the different excited states corresponding to the spectral terms for most of the atoms and their ions. Selection Rules for Dipole Radiation in the Case of L–S Coupling The transition probability for the emission or absorption of radiation between any two term values is governed by the square of the appropriate matrix element (see Cohen-Tannoudiji et al. (1977)). It is possible to provide a set of simple rules (selection rules) that indicate whether electric dipole transitions are allowed or forbidden. In strict L–S coupling, these rules are: (a) The parity must change. (b) The multiplicity must remain unchanged, i.e., intercombination lines are forbidden. (c) J must change by 1 or 0, however, the transition J = 0 to J = 0 is not allowed. (d) L must change by 1 or 0; however, the transition L = 0 to L = 0 is not allowed. As previously shown, the excitation of the outer 4s electron of copper to 4p and 5d orbits leads to 2P0 and 2D terms, respectively. The 2P0 term is then split into the two levels 2 Po3=2 and 2 Po1=2 , and the 2D term is split into the two levels 2 D5=2 and 2 D3=2 . Applying the selection rule gives rise to three possible transitions, as shown in Fig. 6a. If doublet transitions involve an S term, only two transitions are possible, as j–j coupling shown in Fig. 6b. The maximum number of transitions between two quartet terms is 9, as shown in Fig. 6c.

4.3.2 j–j Coupling With increasing Z0 and the corresponding increase in the distance between the outermost electrons, their coupling among themselves (L–S coupling) decreases. The spin–orbit interaction becomes predominant, resulting in (j–j) coupling for the ⇀ ⇀ heavy species and noble gases. The coupling of S i and ‘ i results in a total angular  ! ! ! ! momentum j i , for the ith bound electron j i ¼ ‘i þ Si . The total angular !

momenta j i of the individual electrons combine to form a total angular momentum J. The correspondence between the two designations ((j–j) and (L–S)) is rather simple and will be illustrated for the He atom with the terms related to an s, p configuration: ‘ ¼ 0 s, p 1 ‘2 ¼ 0

s1 ¼ 1=2 s2 ¼ 1=2

Basic Atomic and Molecular Theory

81

a

b 2D 5/2 ∆J=–1 402.2 nm

Forbidden ∆J=–2

∆J=0 406.3 nm

406.2 nm ∆J=–1

Terms 5 d2D

4p2Po

Levels

Terms 6 s2S 453.1 nm ∆J=+1

Levels

2D 3/2

2P0 3/2

6s 2S 1/2 448.0 nm ∆J=0

2

4p2Po

4p2Po3/

2

4p2Po1/

2

2P0 1/2

Levels 4D 1/ 4D 2 4D3/2 4D5/2 7/

c

' ' ' '

Terms 4D

469.7 nm 454.0 nm 484.2 nm 467.5 nm 458.7 nm 479.7 nm 470.5 nm 465.1 nm

∆J=0 ∆J=+1 ∆J=–1 ∆J=0 ∆J=+1 ∆J=–1 ∆J=0 ∆J=+1

'

450.9 nm ∆J=+1

2

Z4Fo

Z 4F03/ 2 Z 4F05/ 2 4 0 Z F 7/ 2 Z 4F09/ 2

Fig. 6 Examples of transitions in the copper spectrum: (a) three lines in a doublet system, (b) two lines in a triplet system, (c) nine lines in a quartet system

In L–S coupling, L = 1 and S = 1 or S = 0 give 3P2,1,0 or 1P1. With j–j coupling ð‘1 , s1 Þ ¼ j1 ¼ 1=2 ð‘2 , s1 Þ ¼ j2 ¼ 3=2, 1=2 corresponding to two terms ð1=2, 1=2Þ or J ¼ 1, 0 ð1=2, 3=2Þ or J ¼ 2, 1 The four terms obtained from the two types of coupling correspond to each other in the following way: P1 . . . J ¼ 1 ð1=2, 3=2Þ 3 P2 . . . J ¼ 2 3 P1 . . . J ¼ 1 ð1=2, 3=2Þ 3 P0 . . . J ¼ 0 1

82

5

M. I. Boulos et al.

Excited States of Diatomic Molecules

When a particle consists of more than one atom, the determination of its energy states becomes a rather complex problem compared to the situation encountered with atoms. The discussion in this section will be restricted to diatomic molecules.

5.1

Energy States

Figure 7 gives a schematic representation of a diatomic molecule in which each of the two atoms (X and Y) is surrounded by a cloud of electrons and the two atoms are joined by outer, shared electron orbits. For example, in the H2 molecule the two H atoms are held together by forces that arise from the two shared electrons, which move around the two nuclei in an orbit resembling a three-dimensional 8. The balance of the electrostatic forces involved results in a finite separation r between the two atomic centers. The atoms can, however, vibrate in an r-dependent force field about an equilibrium internuclear distance. The nuclear “dumbbell” can also rotate about two axes orthogonal to the internuclear axis. Therefore, vibrationally and rotationally excited energy states are possible in addition to the electronically excited states of X and Y. The solution of the Schro¨dinger equation shows, as it did for atoms, that only discrete energy levels are allowed, thus defining quantum numbers for each of the electronically, the vibrationally, and the rotationally excited states. This solution is much more complex than the solution for atoms because the electrostatic forces vary with r and because potential energies must be determined for each electronic orbit. The results given by this solution are very sensitive to the choice of potential energies in the ground state as well as in the excited states. Figure 8, for example, is a typical energy level diagram used for diatomic molecules, approximated by a Morse potential: VðrÞ ¼ De ð1
expð
aðr
re ÞÞÞ2

Fig. 7 Schematic representation of a diatomic molecule

Molecular electron bonding orbital Atomic inner shell Electron orbitals r Nuclei

Internuclear axis

(40)

Basic Atomic and Molecular Theory

Fig. 8 Potential level diagram of a diatomic molecule XY in the ground state and in an electronically excited state

83 ( XY ) X +Y

Potential energy, Vp

2

( XY ) 4 3 2 1 V=0

X +Y De r = f(v)

re Atomic distance, r

where a is a constant for the particular molecular state, re is the equilibrium distance between the nuclei, and De is the energy difference between the equilibrium position of the nuclei (r = re) and the free atoms (r = 1); it is often referred to as the dissociation energy with respect to the minimum energy (this energy is not the real dissociation energy, which is given by D0 = De
1/2 hν0; at rest, when the vibrational quantum number v = 0, the vibrational energy of the molecule is not zero but 1/2 hν0, where ν0 is the oscillation frequency for the classical vibration of two masses separated by a distance re). The part of the potential energy curve for r > re corresponds to an attractive potential and the part for r < re to a repulsive potential. It should be noted that, except for low values of the vibrational quantum number v, the vibrations are asymmetrical, as shown by the line r ¼ f ðvÞ passing through the middle of the classical vibrational amplitudes (see Fig. 9). Taking into account the different quantum numbers involved (electronic, vibrational (v), and rotational (J)), the solution of the wave equation in the simplest case when the various excitation modes (electronic, vibrational, rotational) may be considered as independent [Herzberg 1959] (the Born–Oppenheimer approximation) leads to energy eigenvalues of the molecule (with the Morse potential) described by ha E ¼ Ee þ π

sffiffiffiffiffiffi  De 1 h2 þ 2 2 J ð J þ 1Þ vþ 2 2μ 8π μre

(41)

where the first term represents the noninteracting electronic energy. The second term contains the vibrational energy, which is assumed to be simply harmonic (this is true for vibrational levels close to the bottom of the Morse potential, which can be approximated in this region by a simple parabola (Fig. 9)). The third term accounts for the rotational energy of the molecule (using the rigid rotator approximation without coupling). In this expression μ is the reduced mass of the molecule.

84

M. I. Boulos et al.

a

b S

J

N

J N K Λ

M

Λ L

S

L

Ω = Λ + S

Fig. 9 Coupling of angular momentum vectors for a homonuclear molecule in Hund’s coupling in case (a) and case (b)

As already mentioned, spectral energies are proportional to 1/λ, and it is therefore customary to express them in units of cm
1(E/h.c). With the classical notation (see Herzberg (1959)) Eq. 41 becomes

with and where

T ¼ TðeÞ þ GðvÞ þ FðJÞ GðvÞ ¼ ωe  ðv þ 1=2Þ Ee TðeÞ ¼ hc JðffiJ þ 1Þ FðJÞ ¼ Bs e  ffiffiffiffiffi a De h ωe ¼ and Be ¼ 2 2 cπ 2μ 8π μcre

(42)

ωe corresponds to the vibrational term when the vibration is assumed to be harmonic, and Be is the rotational constant of the molecule at rest (the separation distance re between the two nuclei is the equilibrium distance; the moment of inertia is I ¼ μr2e ). The differences between the various energy modes (see Fig. 9) are very significant: a few eV between electronically excited states, approximately 0.1 eV between vibrational levels and approximately 0.01 eV between rotational levels. Of course expression (41) is oversimplified because in solving the Schro¨dinger equation, one has to take into account: – The degree of anharmonicity of the actual molecule (Morse potential instead of a simple parabola) – The change in the internuclear distance in the higher vibrational state resulting in modification of the rotation – The centrifugal correction arising from the increase in the internuclear distance during higher-energy rotation – The spin effect

2

Basic Atomic and Molecular Theory

85

Taking these effects into account results in more complex expressions for the energy. These expressions will be explained through the momentum description of the molecule.

5.2

Classification of the Electronic States of Diatomic Molecules

The motion of the electrons and nuclei in the molecule, the resultant spin, and even the spins of the individual electrons are not independent of each other. This is rather evident, because the motions of the nuclei and the electrons create electric currents, which produce magnetic fields. An interaction is therefore to be expected, because an intrinsic property of the spin is its magnetic momentum. The different relevant vectors in the momentum description of the molecule are: !

– The orbital electronic angular momentum L !

– The spin electronic angular momentum S

!

– The angular momentum of rotation of the nuclei N (The latter is orthogonal to the internuclear axis.)

5.2.1 Orbital Angular Momentum In a diatomic molecule the charges of the nuclei create a strong electric field in the ! direction of the internuclear axis, around which the angular momentum L processes with a constant component ML(h/(2π)), where ML can assume only the values L, L
1, . . .,-L. In an electric field, unlike in a magnetic field, reversing the direction of motion of all the electrons does not change the energy of the system but changes ML into
ML. Therefore, the energy is a function only of ML, and states with different ML have, in general, widely differing energies, since the electric field that causes this splitting is very strong. Therefore, the electronic states are classified ! according to ML, written as Λ. The corresponding angular momentum vector Λ represents the component of the electronic angular momentum along the internuclear axis (see Fig. 9). The state of a molecule is designated according to its value of L, as shown in the following table: Λ STATE

0 Σ

1 Π

2 Δ

3 θ

When different electronic states of the same molecule have the same value of Λ, the Greek symbols (Σ, Π, Δ, θ, . . .) are proceeded by a Roman capital letter. The ground state for electronic levels is designated by X, and the others by A, B, C, . . ., in order of increasing energy. Π, Δ, θ, . . . states are twofold degenerate since ML can have two values, +Λ and
Λ, with the same energy.

86

M. I. Boulos et al.

5.2.2 Spin ! The spins of individual electrons combine to form a total spin momentum S . The corresponding quantum number S can be an integer or a fraction,!depending on whether the total number of electrons is even or odd. In Σ states, S , which is not affected by an electric held, is fixed in space as long as the molecule does not rotate. On the other hand, if Λ 6¼ 0, there will be an internal magnetic field in the direction of the internuclear axis as a result of the orbital motion of the electrons. This ! magnetic field causes a precession of S about the internuclear axis with a constant component Ms  (h) (see Fig. 9). For molecules, Ms is denoted by Σ (this is not the same as the one Σ corresponding to Λ = 0) with quantum theory allowing the values Σ = S, S
1, . . .,
S, i.e., 2S + 1 values. The multiplicity of the electronic states is indicated by a superscript (corresponding to 2S + 1) to the left of the Greek letter denoting the value of Λ. For example, B3π denotes the electronic excited state B with Λ = 1 and S = 1. In contrast, for Σ states (Λ = 0) Ms = 0 even if the spin quantum number S 6¼ 0. In this case one gets a singlet state even if the value of 2S + 1 is given, as, for example, in the case of the 2Σ state of Nþ 2. 5.2.3 Total Angular Momentum of the Electrons The total electronic angular momentum about the internuclear axis, designated by ! ! ! ! ! Ω, is obtained by adding Λ and Σ (see Fig. 9a). Because Λ and Σ are colinear (along the line joining the nuclei), an algebraic addition is sufficient; thus, the quantum number Ω is given by Ω ¼ jΛ þ Σj

(43)

For Λ 6¼ 0 there are 2S + 1 different values of Ω, corresponding to somewhat different energies of the resulting molecular states. For a multiplet, the electronic energy can be written as !

!

Te ¼ T0 þ A Λ  Σ

(44)

where T0 is the term value when the spin is neglected and A is a constant for a given multiplet.

5.2.4 Angular Momenta for the Rotation of the Molecule According to classical mechanics, the rotation of a diatomic molecule surrounded by a “rigid” cloud of electrons leads to identical moments of inertia IB for both axes perpendicular to the intermolecular axis and passing through the center of gravity. The moment of inertia IA about the internuclear axis is much smaller than I , but the ! B corresponding angular momenta are of the same order of magnitude J , since the electrons rotate much faster than the heavy particles. The total angular momentum is the sum of the angular momentum taken at right angles to the internuclear axis ! ! (designated by N) and the total electronic angular momentum along the axis Ω (see Fig. 9a).

2

Basic Atomic and Molecular Theory

87 !

Quantum mechanics indicates that J is quantized, i.e., !

jJ j ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ð J þ 1Þ h

(45)

The rotational quantum number J can only be an integer (0, 1, 2, . . .). Thus, the energy levels for rotation will be given by FðJÞ ¼ Be  J  ðJ þ 1Þ þ ðA
Be ÞΛ2

(46)

with Be ¼

h h h ¼ and A ¼ 2 8π2 c IB 8π2 μc r2e 8π c IA

(47)

where c is the velocity of light. Note that, because of the small values of IA, A > Be.

5.2.5 Coupling of Rotation and Electronic Motion In an actual molecule, rotational, vibrational, and electronic transitions occur simultaneously, and the influence of the corresponding interactions is very important for the energy states of the molecule. The interaction of vibrational and electronic transitions is accounted for by the interaction potential V(r). Therefore, the interactions of rotational and electronic transitions need closer examination. The simplest case is, of course, when S and Λ are zero (these are the Σ states for ! which the angular momentum of nuclear rotation N is identical with the total angular momentum J). This case corresponds to a simple rigid rotator. The other cases have been classified by Hund as cases a, b, and c and intermediate cases. Only cases a and b will be briefly described here. Hund’s Case (a) ! ! ! For this case, the interaction of the vectors S and Λ is strong. S is coupled very strongly with the internuclear axis, precessing around it at a constant angle in such a ! ! way that the axial component of S is quantized (Σ takes 2S + 1 values). Ω is defined ! and combined vectorially with the angular momentum of rotation of the nuclei N (not nuclear spin) to form a resultant angular momentum for the molecule (see Fig. 9a), i.e., !

!

!

J ¼Ω þ N

!

(48) !

!

J , which is of constant amplitude, has a fixed direction around which N and Ω ! ! (as well as the internuclear axis itself) precess much more slowly than do L and S about the internuclear axis. For a given Ω, J takes the values

88

M. I. Boulos et al.

J ¼ Ω, Ω þ 1, Ω þ 2, . . . Levels for J < Ω do not exist.

Hund’s Case (b) ! As in case (a), L processes rapidly around the internuclear axis of the molecule and ! ! Λ is quantized, but in this case the magnetic field associated with Λ is so weak that ! ! the interaction between Λ and S is small compared with the effect of the rotation of ! ! the molecule on the spin. S is no longer coupled with the axis; therefore, Σ does not ! exist and Ω is not defined. ! ! Λ and the orbital nuclear moment N, which are parallel and perpendicular to the ! internuclear axis, respectively, form a resultant K : !

!

!

K ¼Λ þ N !

(49)

!

around which Λ and N precess (see Fig. 9b). The corresponding quantum number K is an integer that can assume the values K ¼ Λ, Λ þ 1, Λ þ 2, . . . !

!

!

!

K and the resultant spin S form the total angular momentum J  J , which has a !

!

fixed direction around which K and S precess slowly compared with the molecular rotation. !

!

!

J ¼K þ S

(50)

For given values of K, J can assume the values J ¼ K þ S, K þ S
l, . . . , ðK
SÞ Thus, each level K is composed of 2S + 1 sublevels. Figure 9b indicates the position of these vectors. The coupling case depends on the excitation of the molecule, particularly on its rotational excitation. As soon as J increases sufficiently, the rotational speed of the molecule, which is usually small compared with the speed of precession ! ! of S around Λ (case(a)), becomes comparable with it; in this situation the influence of rotation becomes predominant (case (b)). For example, Shemansky and Jones (1968) assert that a state belongs to Hund’s case (a) as long as A/Be J.

2

Basic Atomic and Molecular Theory

5.3

89

General Remarks About Molecular Spectra

As already mentioned in the Born–Oppenheimer approximation (see Eq. 41), the total energy can be expressed by the sum of the electronic, vibrational, and rotational energy (T = Te + Ge(v) + FV(J)). The electronic energy, including the coupling of the spin and the angular momentum of the electrons, is given by Eq. 44. The vibrational energy depends on the interaction potential (see Fig. 8), which in the simple case of the harmonic oscillator is reduced to a parabola (V(r) = f  (r
re)2). In the actual situation a cubic term or even a fourth-order term has to be added to this expression. In this case the solution of the Schro¨dinger equation leads to       1 1 2 1 3 G ð vÞ ¼ ω e v þ þ ω e ye v þ þ ...
ωe xe v þ 2 2 2

(51)

with ωe ye  ωe xe  ωe . The values of xe, ye (constants for each electronic state) can be found, for example, in Herzberg (1959). The rotational energy, as we have already seen, has to be corrected to take into account not only the interaction between the angular momentum of the nuclei (at a right angle to the internuclear axis) and the electronic angular momentum (see Fig. 9) but also the interaction between the rotation and the vibration. During vibration, the internuclear distance and consequently the moment of inertia and the rotational constant Be are changing. Since the period of vibration is very small compared to the period of rotation, one uses a mean value for Be in the vibrational state considered, and Bv is written as   1 Bv ¼ Be
αe v þ 2

(52)

where αe is a constant (for a given electronic state) that is small compared to Be (see Herzberg (1959)). The detailed quantum mechanical equations lead finally to FðJÞ ¼ Bv JðJ þ 1Þ
Dv J2 ðJ þ 1Þ2 þ ðA
Bv ÞΛ2

(53)

  1 Dv ¼ De þ βe v þ 2

(54)

with

and

90

M. I. Boulos et al.

De ¼

4B3e ω2e

(55)

Table 5 gives some energy values for various excited states of selected molecules. The spectral emissions of excited molecules consist of spectral bands due to electronic, vibrational, and rotational levels. Each line within a band corresponds to a transition between rotational levels, transitions between vibrational levels determine the band structure, and the location of possible potential curves in the energy level diagram is determined by electronic excitations. The details of molecular transition rules are rather complex; therefore, only the main rules will be discussed here, and the complexity of molecular spectra will be demonstrated with a relatively simple example. The wave numbers of spectral lines corresponding to transitions between two electronic states (in emission or absorption) are given by

 σ ¼ T0e
T00e þ ðG0
G00 Þ þ ðF0
F00 Þ

(56)

where the single-primed letters refer to the upper state and the double-primed letters to the lower state. The upper and lower states are determined by the transition rules. The selection rules for transitions between electronic terms ΔΛ = Λ0 – Λ00 and between electronic levels are ΔΛ = 0, 1, ΔΩ = 0, 1, and ΔΣ = 0 for the spin; there are also some rules for symmetry properties (see Herzberg (1959)). There are no selection rules for vibrational transitions. The selection rules for rotational transitions ΔJ = J0 – J00 are given by ΔJ = 0, 1, with the restriction that transitions from J = 0 to J = 0 or from Ω = 0 to Ω = 0 are not allowed. The line series for which ΔJ = 0 is called the Q branch, the series with ΔJ = +1 is the positive or R branch, and the series with ΔJ =
1 is the negative or P branch. A band represents the collection of all lines (of the Q, R, P branches) that belong to one-electron transition and one specific vibrational transition v.

5.4

The N+2 (
1) Spectra

As an example consider the simple case of the (1
) system of Nþ 2 , which is often seen in thermal plasmas where nitrogen is the plasma gas or when nitrogen is entrained during the operation of a plasma jet in atmospheric air (Baronnet and Fauchais 1971). In this system transition occurs between the electronically excited þ 2 þ state B2 Σþ u and the fundamental state X Σg of N2 .

5.4.1 Rotational Structure Since both the upper and lower states are Σ states, Hund’s case (b) always applies; ! with Λ = 0 and Σ = 0, the spin vector S is not coupled with the internuclear axis at ! all and Ω is not defined. In this case S = 1/2, and therefore,

Molecular weight 10
3kg 2.02

202

18.02

28.01

Molecule H2

Hþ 2

H2O

N2

9.756

9.509

2.648

Thermal dissociation energy (eV) 4.588

0.198 0.293

1Σþ g

0.285

Ev (eV) 0.545

–

2Σþ g

Designation 1Σþ g

Ground states

Table 5 Electronic data for selected molecules

Er (eV) 1.50E2 7.40E3 1.00E3 4.98E4 8.400 8.590

a1 Σþ u a Πg 1

6.224

–

–

A3 Σþ u

–

–

Metastable states Ee Designation (eV) C3Πg 11.87

–

0.017–0.5

1.36

–

–

τm (s) 1.02–1.76

15.58

13.0

–

23.4

18.7

H2O+ Nþ 2

–

N+

OH+

–

Dissociation ionization Energy (eV) State 18.0 H+ –

First ionization Energy (eV) State 15.426 Hþ 2

2 Basic Atomic and Molecular Theory 91

92

M. I. Boulos et al.

J ¼ K  1=2 The transition rule for K requires that ΔK = 1 ΔK = 0 being forbidden for Σ levels. The separation of the two sublevels J, for a given K, will in general be very small compared to the separation of successive rotational levels; if we neglect the sublevels, each band consists of a P branch (ΔK =
1) and an R branch (ΔK = +1). Using the simplified formula (53) in Eq. 56 and writing K0 as a function of K00 (denoted as K in the following) result in



 σP ðKÞ ¼ σ00
B0v0 þ B00v00 K þ B0v0
B00v00
D0v0 þ D00v00 K2 þ . . .

(57)



 σR ðKÞ ¼ σ 00 þ 2B0v0
4D0v0 þ 3B0v0
B00v00
12D0v0 K þ B0v0
B00v00
13D0v0 þ D00v00 K2 þ . . .

(58)

and

where σ00 is the wave number for J0 = J00 = 0. The selection rule ΔK = 1 gives K = 1 for the smallest value of the P branch (ΔK =
1) and K = 0 for the R branch (ΔK =
1) (see Fig. 10). In such a case the proceeding Eqs. 57 and 58 show that there is no line at the position σ = σo. In fact, if the fine structure of the band (the separation of the J lines) is considered, the structure of the band is more complex: each band (P and R) is divided into three branches because each line is separated into three components according to the values ΔJ (see Fig. 11). However, the differences in wavelength between the three J lines in the P branch and the three J lines in the R branch are ¨ , and unless a very high-resolution monochromator (resolution smaller than 0.1 A power greater than 200,000) is used, only one line is seen in the spectra. In the following discussion, only lines will be considered from the P and R branches, corresponding to ΔK =
1 and +1, respectively, neglecting ΔJ. Table 6 presents the values of the corresponding coefficients for the system. If each branch is represented by plotting the wave number of each line along the abscissa and the corresponding K value along the ordinate, the Fortrat parabola (see Fig. 12 for the N2 (1
) transition) is obtained; it can take two forms according to the values of B and B (the Dv terms in Eq. 53 are negligible). Fig. 10 Rotational transitions for the P and R branches of the transition B2 þ 2 þ Σþ u ! X Σg of N2 (1
) transition

K' 4

K' 3

2

1

0 ΔK = +1

ΔK = –1 K" 4

K" 3

2

1

0 P

R

Basic Atomic and Molecular Theory P2

∆J = ∆K = –1

∆J = 0 ∆K = –1

∆J = ∆K = –1

K´

1 2

1

R1

R2 RQ

21

∆J = 0 ∆K = 1

12

∆J = ∆K = 1

P1 PQ

93

∆J = ∆K = 1

2

K˝ = K´ + 1 2 1

K˝ = K´

2 1

K˝ = K´ – 1

2

J´ = K´ + 1/2 J´ = K´ – 1/2

J˝ = K˝ + 1/2 = K´ + 3/2 J˝ = K˝ – 1/2 = K´ + 1/2 J˝ = K˝ + 1/2 = K´ + 1/2 J˝ = K˝ – 1/2 = K´ + 1/2 J˝ = K˝ + 1/2 = K´ – 1/2 J˝ = K˝ – 1/2 = K´ – 3/2

Fig. 11 Transitions for the principal ðΔK ¼ ΔJÞ and satellite branches ðΔK 6¼ ΔJÞ for the Nþ 2 ð1
Þ

Because of the quadratic term in K2, some of the lines of the same branch run together to form a “band head” where only lines of the same branch are found. In the case of the Nþ 2 0
0(1
) transition, the first 26 lines of the P branch form the band head shown in Fig. 13. The first peak of the band head is the integral of the first 17 P lines. Of course the R lines are found only for wave numbers higher than S0 (see Fig. 12), and the R branch corresponds to lines of the P branch with > 26. Because of the dispersion of the monochromator used to register the spectra shown in Fig. 13 and the small wavelength difference between two adjacent lines of P and ¨ ), the line corresponding to Kp = KR + 27 almost R branches (less than 0.2 A overlaps with the line R corresponding to KR, and both lines merge in a unique line, sometimes referred to as a global or total line. This is why the lines corresponding to wavelengths smaller than 391.0 nm in Fig. 13 are referred to by two K numbers, one for the line from the P branch and one for the line from the R branch. Figure 13 (which is based on experimental results) also shows that the line intensities alternate. This alternation occurs because the rotational statistical weight includes the nuclear spin (Fauchais et al. 1980), which is different for the symmetrical (index s) and for the antisymmetric (index a) levels (see Herzberg 1959 for symmetry properties): gS ¼

I Iþ1 ga ¼ 2I þ 1 2I þ 1

where I is the spin vector quantum number. In the (1
) spectra of N2, I =1, the symmetrical levels corresponding to odd values of the initial state of K0 will have their intensity multiplied by 2/3, while those corresponding to even values will have their intensity multiplied by 1/3. Thus, in the line notation we have chosen (K = K00 ), the lines with Kp or KR even (corresponding to K0 = K  1, i.e., K0

X2 Σþ g

B 2 Σþ u

State

0

Te cm
1 25461.5

2507.19

ωe cm
1 2419.84

16.14

ωexe cm
1 23.19

Be cm
1 2.083 1.9328

ωeye cm
1
0.5375
0.0400 0.0208

αe – 0.0195

βe – – 0.2910
6

De cm
1 – 5.7510
6

0.1118

re nm 0.1075

–

σoo cm
1 B X 25566.0

2 þ þ þ Table 6 Electronic, vibrational, and rotational constants for the transition B2Sþ u ! X Sg of N 2 of the states of N 2 values to be used in equs (51 – 58)

94 M. I. Boulos et al.

2

Basic Atomic and Molecular Theory

95

Rotational quantum number, K (–)

80 70 60 50

P- Branch

40 R- Branch 30 20 10

σ0 25.6

25.8

26.0

26.2

Wave number, σ (10

26.4

26.6

–1cm)

þ 2 þ Fig. 12 Fortrat parabola of the 0–0 band of the transition B2 Σþ u ! X Σg of N2

391.5

KR = 7 KP = 34

KR = 5 KP = 32

391.0

KR = 6 KP = 33

KR = 3 KP = 30 KR = 4 KP = 31

KR = 1 KP = 28 KR = 2 KP = 29

KR = 0 KP = 27

KP = 26

KP = 24

KP = 23

KP = 25

KP = 21

KP = 18 KP = 20 KP = 22

Fig. 13 (0-0) Band head and first rotational lines of the þ 2 þ system B2 Σþ u ! X Σg of N2

390.5

Wave length λ (nm)

odd) will have twice the intensities of the lines for which K is odd (K = K  1 then being even). Thus, the lines with even Kp are about twice the intensity of the lines with odd Kp; this can be seen very clearly from Fig. 13 for 49 of this band overlap with the 1–1 band and so on (see Fig. 15 for a detailed spectrum of the 0–0 band). Thus, the interpretation of the molecular spectra is not straightforward, and a lot of care must be taken in evaluating rotational and vibrational temperatures from these spectra.

6

Nomenclature

Units are indicated in parentheses; whenever no units are given, the parameter is dimensionless.

6.1 a A

Latin Alphabet Constant Separation constant

2

Basic Atomic and Molecular Theory

ap Be c d D0 De e E E‘ Eu En0 E1 f Fv(J) g G(v) h ℏ I i !

j

!

99

Coefficients Rotational constant of a diatomic molecule at rest (cm
1) Velocity of light (c = 299,792,458 m/s) Electron subshell

 Dissociation energy D0 ¼ De
12 hν0 Energy difference between the equilibrium position of the nuclei of a diatomic molecule and the free atom (cm
1) Elementary charge (electron charge = 1.6  10
19 A s) Energy (eV or cm
1) Energy of the excited lower level ‘ (eV or cm
1) Energy of the excited upper level u (eV or cm
1) Energy distribution of the hydrogen atom (eV or cm
1) Energy of the ionized state between the ground state ‘ and the state n (eV or cm
1) Electron subshell Rotational energy of a diatomic molecule Statistical weight Vibrational energy of a diatomic molecule Planck’s constant (h = 6.626  10
34 J.s) Reduced Planck’s constant (ℏ = h/2π) Moment of inertia or spin vector quantum number pffiffiffiffiffiffiffi Imaginary unit (i=
1) ! ! ! Total angular momentum: J ¼ ‘ þ s (kg.m/s)

J

! J1

Total angular momentum for the ith bound electron (kg.m/s) ⇀ ! Total angular momentum J ¼ Σi J1 ðkg:m=sÞ

K

Main electron shell

K k⇀ L L! ‘ ‘ m me m‘ M n⇀ N P(m) ‘ p(r) r R(r) R1

K ¼Λ þ N (see Fig. 9b) Boltzmann constant (k = 1.38  10
23 J/K) Orbital electronic angular momentum (diatomic molecule) Main electron shell Quantum number of the angular momentum of an atom Quantum number for hydrogen (‘=0, 1, 2, 3, n
1) Mass of a particle (or separation constant) Electron mass (me=9.10938356  10
31 kg) Magnetic quantum number Mass of an atom (kg) Principal quantum number for hydrogen (n=1,2,3…) Angular momentum of rotation of the nuclei of a diatomic molecule Polynomial of degree 1 and of order m Probability density function Radial coordinate Radial part of the wave function 

⇀

⇀

⇀

⇀

Rydberg constant for hydrogen

4

R1 ¼ ð4πεmÞe2e4πcℏ3 0

100

re rn r1 ⇀ S S s ⇀ s t T(e) Te Th Tk u u(ρ) v ve Vp X X* XY XY* x y z Z0 Z

6.2

M. I. Boulos et al.

Equilibrium distance between the nuclei (diatomic molecule) Radius of nth Bohr orbit (proportional to n2) Radius of first Bohr orbit (ground state) (r1=5.29177  10
11 m) Total spin momentum Associated quantum number Spin quantum number Spin vector Time (s) Equivalent electronic energy of a diatomic molecule (cm
1) Electron temperature (K, eV) Temperature of the heavy particles (K, eV) Kinetic energy (J) Phase velocity of a wave (ratio wavelength period) Polynomial Vibrational quantum number Electron velocity (m/s) Potential energy (J) Symbol for an atomic chemical species in the ground state Symbol for an atomic chemical species in an excited state Diatomic molecule in its ground state Diatomic molecule in an excited state Coordinate Coordinate Coordinate Number of protons in a nucleus Number of protons and neutrons in a nucleus

Greek Alphabet n X ∂2 f ) ∂x2i i¼1

Δ

Laplace operator (divergence of the gradient: Δf ¼

Δ

Laplace operator r2 ¼ ∂∂2 x þ ∂∂2 y þ ∂∂2 z

ε0 θ λ λn λul

Dielectric constants (ε0= 8.86  10
12 A.s/V.m) Cylindrical coordinate wavelength Wavelength (nm) Parameter (λn= E1/En) Wavelength corresponding to a transition from an upper level u to a lower level 1 Associated quantum number Angular momentum Reduced mass Frequency (Hz)

Λ ! Λ μ ν

2

2

2

2

Basic Atomic and Molecular Theory

ρ σ σe Φ Ψ ψ* Ψ ωe

Reduced coordinate Wave number (m
1) Electrical conductivity (Ohm
1 m
1) Cylindrical coordinate Amplitude of wave function Conjugate complex wave function Wave function Vibrational term when vibration is assumed to be harmonic

Ω

Ω ¼Λ þ S

⇀

6.3 u ‘ n 1

⇀

⇀

101

⇀

Subscripts Upper energy level Lower energy level Refers to principal quantum number Infinite mass (or infinite distance)

References Baronnet JM, Fauchais P (1971) Measuring the rotational-vibrational temperature of Ne molecule produced in a nitrogen DC plasma torch. J Phys 32:50–58 (in French) Cohen-Tannoudiji C, Diu B, Laloe¨ F (1977) Quantum mechanics, vol 1. Wiley, New York Condon EU, Shortley GH (1951) The theory of atomic spectra. Cambridge University Press, Cambridge, 441 pp Delcroix JL (1966) Physique Des Plasmas. Monographic Dunod, Paris Fauchais P, Lapworth K, Baronnet JM (1980) First report on measurement of temperature and concentration of excited species in optically thin plasmas. In: Fauchais P (ed) IUPAC subcommittee on plasma chemistry. Limoges University, Limoges Herzberg G (1944) Atomic spectra and atomic structure. Dover, New York Herzberg G (1959) Spectra of diatomic molecules. D. van Nostrand Company, New York Lide DR (2003) CRC handbook of chemistry and physics, 84th edn. CRC Press, Boca Raton, 2616 pp Moore CE (1949) Atomic energy levels. NBS Circ 467:1 Moore CE (1952) Atomic energy levels. NBS Circ 467:2 Moore CE (1958) Atomic energy levels. NBS Circ 467:3 Morrison MN, Estle TL, Lane NF (1976) Quantum states of atoms, molecules and solids. PrenticeHall, Englewood Cliffs, 575 p Shemansky DE, Jones AV (1968) Type-B red aurora; The O2+ first negative system and the N2 first positive system. Planet Space Sci 16:1115–1130 Svanberg S (2004) Atomic and molecular spectroscopy: basic aspects and practical applications. Springer, New York

3

Kinetic Theory of Gases Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Particles and Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Cross Sections and Collision Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Collision Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Collision Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Collision Frequencies and Scattering Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Mean Free Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Total Effective Cross Section Qi(v) for Collision Processes . . . . . . . . . . . . . . . . . . . . . . . . 4 Elementary Processes for Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Elementary Processes for Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Inelastic Collisions of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Particle Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Maxwellian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Collision Probabilities and Mean Free Paths in a Particle Ensemble . . . . . . . . . . . . . . .

104 104 106 106 108 110 111 112 113 114 114 116 118 119 119 120 122 125 127

M. I. Boulos (*) Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada e-mail: [email protected] P. L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France e-mail: [email protected] E. Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] © Springer Nature Switzerland AG 2023 M. I. Boulos et al. (eds.), Handbook of Thermal Plasmas, https://doi.org/10.1007/978-3-030-84936-8_3

103

104

M. I. Boulos et al.

7 Reaction Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Binary Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Three-Body Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Latin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

128 128 131 132 133 133 136 136

Introduction

This chapter deals with the fundamentals of the kinetic theory of gases and the basic mechanism behind transport phenomena under plasma conditions. A brief introduction of particles and collisions is presented introducing the concepts of collision cross section and frequencies. This is followed by elementary processes for elastic and inelastic collisions, including excitation and ionization mechanisms. The concept of distribution functions and reaction rates for binary and three-body recombination is also briefly discussed.

2

Particles and Collisions

In a plasma, six basic types of particles can be identified: – Free electrons (designated as e). The mass of an electron me is very low 1 mH , with mH being the mass of compared to atoms and molecules (me ¼ 1856 the hydrogen atom, the lightest one) – Atoms and molecules in their fundamental or ground state (designated as X) – Excited atoms and molecules (generally designated as X*) – Positive ions (atomic ions Xþ , Xþþ , Xþþþ , . . . or molecular ions such as Xþ 2) – Negative ions; certain atoms and molecules, particularly those with an almostcompleted outer electron shell, form negative ions when an electron attaches itself to the neutral particle – Photons; they have no mass, and their velocity c is equal to the velocity of light When two particles initially separated by a large distance, d, approach one another, they start to interact; if after the interaction some measurable (in principle) change has occurred, we say that a collision has taken place. According to the classical mechanical model in which actual particles (excluding photons) are replaced by rigid spheres without any electronic structure, a collision would occur only if such spheres made physical contact. In reality, particles “sense” each other as a result of their electronic structure long before they come into physical contact. The initial interaction as two particles approach each other is due to a mutual deformation of their electronic shells (e.g., polarization), resulting in an attractive force (for the case of the Lennard-Jones potential, this

3

Kinetic Theory of Gases

Fig. 1 Typical interactions between two neutral particles

105

m1

Impact parameter m2

force is  d7 ). As the outer electron shells of the two particles begin to penetrate each other, a strong repulsive force is experienced due to the positively charged nuclei (for the Lennard-Jones potential, this force is  d13 ). The interaction potential between two particles usually consists of a longer-range attractive part and a very short-range repulsive part. Interactions of neutral particles following, for example, the Lennard-Jones potential description are classified as short-range interactions (see Fig. 1). In contrast, interactions between charged particles
result in long-range interactions because of the far-reaching Coulomb force  d2 between charged particles. The result of this mutual influence or collision is that the particles will sensibly deflect each other’s paths. A collision is defined as a resulting deflection greater than a certain minimum value. The kinetic and/or potential energy of the participating particles changes as a consequence of a collision, and the distinction between elastic and inelastic collisions is based on this energy change. It might be appropriate at this point to mention that for the description of elementary processes in plasma physics, energy is often defined in units of electron-Volts (eV) or photon energy. The unit of electron-Volt (eV) is defined as the energy gained by an electron when passing through a potential difference of 1 V. Energy = e. E. L, where e: electron charge (e = 1.602  1019 C) E: electric field (V/m) L: distance between electrodes (m) Thus, 1 eV = 1.602  1019 (J) In spectroscopic tables, energy units are often expressed in terms of photons of frequency ν, emitted by the de-excitation of the state of energy E2 to the lower energy E1. Thus, E2  E1 ¼ h:υ ¼ h:c=λ ¼ h:c:σ where h: Planck’s constant (h = 6.626  1034 J. s) c: Velocity of light (c = 2.998  108 m/s) υ = frequency of the associated wave (s1)

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λ: wavelength (m) σ: wave number (independent of c) Ei in most tables are expressed in σ units = cm1, with 1 cm1 ¼ 1:24  104 eV ¼ 1:99  1023 J Elastic collisions are collisions in which the total kinetic energy is conserved. Practically all collisions in neutral gases at ambient temperatures are elastic. In the elastic collision process, the fraction of energy, K, transferred from one particle of mass m to another of mass M averaged over all angles is K¼

2mM ðm þ MÞ

2

¼

2m M

when, m 0 indicates a transformation of part of the internal energy of one of the particles into kinetic energy of the other particle (a superelastic collision).

3

Cross Sections and Collision Frequencies

For more details, the reader can refer to the books Bittencourt (2004), Gould and Tobochnik (2010), Delcroix (1966), Fowler (1956), Landau and Lifshitz (1967), and Reif (1988).

3.1

Collision Probabilities

Let P(t) be the probability that a particle with relative velocity

!  ! ! V ¼ v1  v2

survives a time t without suffering a collision (of course, P(0) = 1 and P(1) = 0). To describe the collision, let w.dt be the probability that a particle suffers a collision between t and t + dt. w is thus the probability per unit time that a particle suffers a collision; w is also known as the collision rate.

3

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107

The customary assumption is that w is independent of the past history of the particle, i.e., it does not matter when the particle suffered its last collision. Assum! ing that V does change appreciably in times of the order of w1, it follows that Pðt þ dtÞ ¼ PðtÞ ð1  w:dtÞ

(2)

1 dP ¼ w P dt

(3)

Hence

or

0

1

ðt

P ¼ C  exp@ wðt0 Þ  dt0 A

(4)

0

!

If w does not vary with V , it follows that P ¼ C  expðw  tÞ

(5)

and because P(0) = 1, C must be equal to one, and hence, P ¼ expðw  tÞ

(6)

If P*(t)dt is defined as the probability that a particle, after surviving without collisions for a time t, suffers a collision in the time interval from t to t + dt P ðtÞ  dt ¼ expðwtÞ  w  dt

(7)

P*(t)  dt is also normalized (asserting that one particle collides at some time): 1 ð



1 ð

P ðtÞ  dt ¼ 0

1 ð

expðwtÞwdt ¼ 0

expðyÞdy ¼ 1

(8)

0

The mean time between two collisions (referred to as the “collision time” or “relaxation time”) is defined as 1 ð

τ¼t¼

t  P ðtÞ  dt

(9)

0 1 ð

τ¼

expðwtÞ  t  wdt 0

τ¼

1 w

(10) (11)

This expression will be used for deriving an expression for the mean free path (mfp).

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3.2

Collision Cross Sections

Consider two particles of respective masses mi and mj, with position vectors ! ri vj . From a frame of reference that is fixed with and ! rj and velocities ! vi and ! ! ri respect to particle j, their motion is described by their relative position Rij ¼ ! ! ! ! !  rj and their relative velocity Vij ¼ vi  vj . In this frame of reference, it can be assumed that particle j is the target (see Fig. 2a) at rest and that a uniform flux (Fi) of type i particles per unit area and unit time impinges with a relative ! velocity Vij on the target. A number dNi of particles of type i will be scattered !

by the target particle per unit time and will have final velocities in the range V0ij !

!

to V0ij þdV0ij . At large distances from the target particle, we can define a small solid angle dΩ0 about the direction θ, φ of the scattered beam (see Fig.  2b).  If the  !0   ! collision process is elastic so that energy is conserved, then  V  ¼  V . dNi is proportional to the incident flux Fi and to the solid angle: !  dNi ¼ Fi  σ V i, j , θ, φ  dΩ0

Fig. 2 (a) Scattering process viewed from a frame of reference in which the target particle j is at rest, (b) Collision process between two hard spheres of radii ri and rj, (c) Incident flux of particles i scattered by particles of type j

a

(12)

Fi

j

Vij

b i

j

Vij

F

q dq

dW

c

dx

dy

dz V ij

dA

3

Kinetic Theory of Gases

109

!  σ V i, j , θ, φ is called the differential scattering cross section. It has the dimensions of area per unit solid angle since Fi is expressed per unit area and unit time. The total number Ni of particles of type i scattered per unit time in all directions is obtained by integrating (Eq. 13) over the full solid angle: ð Ni ¼ Ω


 Fi  σij  dΩ0 ¼ Fi  σ0 Vij

(13)

0

where
 σ0 Vij ¼

ð   ! σ V ij , θ, φ  dΩ0 Ω

(14)

0

is called the total scattering coss section; it has units of area (m2) and depends on the magnitude of ð Ni ¼ Ω


 Fi  σij  dΩ0 ¼ Fi  σ0 Vij

(13)

0

where
 σ0 Vij ¼

ð   ! σ V ij , θ, φ  dΩ0

(14)

Ω0

is called the total scattering cross section; it has units of area (m2) and depends on the magnitude of the relative velocity of the two particles. The differential cross section is often written as !  ! 
 σij V ij , θ, φ  dΩ0 ¼ Pij V ij , θ, φ σij Vij dΩ0

(15)

where Pij(Vij, θ, φ) is a proportionality factor. Since
 σ0 Vij ¼

ðπ 2ðπ

!  σij V ij , θ sin θdθdφ

(16)

0 0

we have ðπ  
 ! σ0 Vij ¼ 2π σij V ij , θ sin θdθ

(17)

0

if there is no dependence on the azimuthal angle φ. The differential cross section is primarily of importance for transport phenomena, where the average scattering angle is an important parameter.

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The total collision cross section σ0(Vij) can be interpreted as the effective geometrical blocking area that the field particles present to the beam of incident particles (see Fig. 2a) and thus for hard, spherical particles.
2 σ 0 ¼ π ri þ rj

(18)

which reduces for two identical particles to σ0 ¼ 4π r2i

(19)

Although atoms do not have any well-defined dimension but interact with other atoms over a certain range of distances, it is still useful to associate a radius r with an atom. In the simple model of Bohr, r ¼ n2 r1 =Z0, where n is the principal quantum number, r1 the radius of the first Bohr orbit, and Z0 the charge of the nucleus. For hydrogen, r1 = 0.529  1010 m, and if we use the area presented by a sphere of radius r1 as an estimate of the order of magnitude of the total cross section, we obtain
2 σ0 ¼ 4π  5:3  1011 ¼ 3:53  1020 m2

(20)

Thus, depending on the atoms considered, a first approximation indicates a range from 1020 to 1019 m2 for the total collision cross section σ0. The Bohr radius gives generally the right order of magnitude for σ0. The actual value of σ0 is usually greater than (4π r21). This definition of the total cross section makes no distinction between the types of collisions that may occur. In fact, it represents an averaging over the elastic and all other relevant nonelastic collision cross sections. Of course, σ0 also depends on ! the relative velocity V ij of the target particle j and of the colliding particle i. It is also clear that for inelastic collisions, σ0 will be zero if the relative kinetic energy of the colliding particles does not exceed a threshold energy, that is, the energy of the corresponding excited state. For example, for the first excited state of the H atom, this value is 10.2 eV; for the N atom, 10.34 eV; for the O atom, 9.15 eV; and for the Ar atom, 11.5 eV. On the other hand, if the relative velocity is very high, the time for interactions becomes short and the probability of the collision occuring is reduced.

3.3

Collision Frequencies and Scattering Cross Sections

The collision frequency or collision rate is defined as τ1, i.e., the inverse of the mean time between collisions, which in turn can be related to the total cross section. If ni is the density of particles of type i, their relative flux incident on any one particle j situated in a volume d3r is given by (see Fig. 2c)

3

Kinetic Theory of Gases

111


 ni Vij  dt  dA ¼ ni Vij Fi ¼ dt  dA

(21)

where Vij is the mean relative velocity, which will be defined in Sect. 3.5.5. A fraction ni Vij σ0 of these incident particles is scattered per unit time in all possible directions by one target particle. The total number of particles i scattered by all particles j in volume d3r is then given by


ni Vij σ0




nj d3 r



(22)

Dividing this by the number nid3r of type i particles in d3r gives the collision probability or collision frequency υ = τ1 per unit time for one particle of this type: 0 τ1 ij ¼ υij ¼ Vij σij nj

(23)

This collision probability may be enhanced by a large density of target particles j, a large relative particle speed (e.g., electrons vs. heavy particles), and a large total scattering cross section.

3.4

Mean Free Paths

The mean free path (mfp) ‘ij is the average path length traversed by a particle of type i between two successive collisions with particles of type j. In time τij, the particle i, covers a distance (τijvi ), where vi is the mean velocity of particle i, ‘ij ¼

vi 1  Vij nj σ0

(24)

For example, if one considers collisions between electrons i and heavy species j, Vij ¼ vi  vj  vi , and thus, ‘e ¼

1 n  σ0

(25)

where n is the number density of neutral particles (neglecting ion density). In these calculations, τij, ‘ij, and σ0 have been assumed to be independent pffiffiffi of Vij. Would this dependence be taken into account, the ratio vi =Vij ¼ 1= 2 . For a mixture of different chemical species j, the mean free path for particles of type i (e.g., electrons) is defined as 1 ‘i ¼ pffiffiffiX (26) nj σ0 Vij 2 j

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It is useful to evaluate the mean free path ‘ in a cold gas at atmospheric pressure. Since the density of the molecules in a gas at 0  C and 105 Pa is n = 2.69  1025 m3, with σ = 1019 m2, it results that ‘¼

1 ¼ 3:7  107 m nσ

(27)

In an ideal gas (see Eq. 2) for 100 < p < 10 MPa, it results generally in ‘  T=p

(28)

For T < 600 K, an equation of the form ‘ðT Þ ¼

‘300 1 þ C=T

(29)

can be used. But in gases at very high temperatures such as in thermal plasmas, the relationship has the form ‘ðTÞ ¼ const  T5=4

(30)

To obtain the distribution of mean free paths, consider a group of N0 particles, initially moving through the gas with a velocity v. Let N(x) be the number of particles reaching x without undergoing any collisions. The number of particles that will undergo collisions between x and x + dx and will leave the initial group of particles will be dN = N(x)  dw. Hence, with dw = n  σ  dx (see Eqs. 12 and 24), P ð xÞ ¼

NðxÞ ¼ expðnσxÞ ¼ expðx=‘Þ N0

(31)

where P(x) is the probability of a free path exceeding a distance x, i.e., 37% of the free path lengths are shorter than ‘ and 63% exceed ‘.

3.5

Total Effective Cross Section Qi(v) for Collision Processes

The result of an encounter is then defined as the number of collisions per unit path length for test particles i with particles of type j: 1
1  m (32) ‘ij Notice that due to the presence of particles of density nj in Eq. 32, the pressure and temperature at which Qi is calculated must be specified. Usually Qi, also called the effective cross section per unit volume, is given at 133 Pa (1 Torr). The total effective cross section for all collisions is the sum of the particular cross sections for particular types of collisions. Qi ¼ nj σij ¼

3

Kinetic Theory of Gases

113

Q¼

X

Qi

(33)

i

4

Elementary Processes for Elastic Collisions

40

Cross section for scattering of electrons, σ (10−20 m2)

Cross section for scattering of electrons, σ (10−20 m2)

As will be shown later, electron interactions are the most frequent processes in plasmas. Therefore, special attention will be given to interactions with electrons. The concept of differential cross section is very important, because at energies above 1 eV, electrons are not scattered isotropically by gas molecules. Instead, they show preferred scattering directions with pronounced forward scattering. In light gases, these extremes are most noticeable over a relatively small range of electron energies (up to 6 eV in H2 and 15 eV in He), but in heavier gases, they are observed at energies up to 800 eV. The total cross section σ0 generally varies strongly with the incident electron energy (see Fig. 3a from Brown (1959) for noble gases, see also Gould and Tobochnik (2010)) and reveals the same general behavior for atoms in the same column of the periodic table of elements. For noble gases below 1 eV, the cross section is very small due to diffraction effects (the Ramsauer effect, see Present (1958)). For electron energies exceeding 20 eV, the cross section decreases monotonically. Electrons with such energies may gain more and more energy from the electric field in a discharge and simultaneously lose less and less energy by collision (runaway electrons). Figure 3b Brown (1959) shows the behavior of the total cross section for some selected diatomic molecules.

a Xe

30

Kr

20

Ar 10 He 0 0

2

4

6

8

Electron energy, E½ (eV)½

10

40

b CO

30

20

N2

10 O2 0 0

2 4 6 8 Electron energy, E½ (eV)½

10

Fig. 3 Cross sections, σO for total scattering of electrons (a) in various noble gases, (b) in diatomic gases. Brown (1959)

114

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Elementary Processes for Inelastic Collisions

Although inelastic collisions belong to the next chapter on gaseous electronics, some of the basic features of inelastic collisions will be discussed in this section. During every collision, a substantial fraction of the kinetic and/or internal energy of the colliding particles will be exchanged or converted into other forms of energy (internal energy, chemical energy). There are numerous types of inelastic collision processes (more than 200). In this book, the discussion will be restricted, however, to those that are of particular importance for thermal plasmas. For more details on other types of inelastic collisions, the reader is referred to the following publications of the subject: Massey (1969), Massey and Burhop (1969), Massey (1971), Massey and Gilbody (1974).

5.1

Excitation

As already mentioned, when an atom or a molecule of type X in the ground state absorbs sufficient energy during a collision (with an electron or a heavy particle) or through absorption of a photon, one of its bound electrons reaches a higher energy level; this atom or molecule is said to be in an excited state, denoted by X*. In the following discussion, the subscript i will refer to the chemical species and the index s to the excited states (of chemical species i).

5.1.1 Excitation by Photons Schematically, the process of excitation by a photon can be written as X þ hυ ) X Photons can produce excited states if the energy of the incident photons, hυ, is at least equal to the energy difference between the upper (Eu) and lower (E‘) states < Eu  E‘ ¼ E ). The probability of the process depends on the selection rules governing the reverse process. Thus, a metastable state (from which an electron cannot revert to the ground state by emission of radiation) cannot be excited from the ground state by absorption of radiation. The cross sections for photoexcitation are usually very small ( 0.1 eV) are generally much larger than the rotational ones.

5 Excitation cross section, Qex (cm−1)

Fig. 4 Calculated effective cross sections, Qex, for the excitation of the mercury atom for the initial state 73S1 by electron impact (133.32 Pa or 1 Torr) Francis (1960)

Total 4

p=1 Torr

3 63 P2

2

63 P1

1

63 P0 4

5

6 7 Electron energy, E (eV)

8

9

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M. I. Boulos et al.

5.1.3 Excitation by Impact of Atoms or Ions Atoms can be excited by collisions with fast atoms or ions, but the energies required to reach σmax are two orders of magnitude higher than the corresponding energies for electrons. Therefore, this process is of little importance in thermal plasmas Francis (1960), and McDaniel (1964)).

5.2

Ionization

When an atom or a molecule of type r has absorbed sufficient energy (Eion) to release one of its outer electrons, the particle is said to be ionized. Ionization can occur through absorption of a photon, electron impact, or impact of a heavy particle.

5.2.1 Ionization by Photons In the following discussion, the photon energies will be restricted to the optical regime. For the ionization of an atom or molecule in the ground state, the photon energy hυ must be equal to or greater than Eion (hυ > Eion for the process hυ þ X ) Xþ þ e). This transforms in terms of the wavelength of the incident photon into 1240 (36) Eion where Eion is in eV. For the alkali elements, λi must be between 200 and 300 nm and for noble gases smaller than 50 nm. The corresponding cross section rises sharply from zero at the threshold (Eion) to a value of about 1021 m2 as the photon energy is increased and then passes through successive steps as other electrons are removed. Ionization of an already excited atom by photons can also occur: λi
Eion

Figure 5 from von Engel (1965) shows the effective electron collision cross sections for ionization of various atoms and molecules. The cross section increases rapidly once the threshold energy has been surpassed. The maximum values are on the order of 1020 m2 for σmax (Qi  10 to 20 cm1 for p = 133 Pa) and correspond, in general, to electron energies around 100 eV for most gases (except for alkaline metals, for which they are about 20 eV).

Kinetic Theory of Gases

Fig. 5 Effective cross section, Qion, for ionization by impact in selected gases (p = 133.32 Pa) Von Engel (1965)

117

Ionization cross section, Q ion(cm−1)

3

20 10 5

Hg

Cs

N2

2

H

1

Ar

Na

0.5 0.2 0.1 10

He

p=1 Torr 2

5 102

2

Ne H2

5

103

2

5

104

Electron energy, E (eV)

High-energy electrons can give rise to double ionization of atoms. For example, electrons with E > 80 eV can lead to double ionization of helium: e þ He ) Heþþ þ 3e Since electron energies of this magnitude are practically nonexistent in the thermal plasmas considered here, such processes will be neglected. Collisions between electrons and molecules may give rise to molecular ions, for example, H2 þ e ) Hþ 2 þ 2e; and also to dissociative ionization: H2 þ e ) Hþ þ H þ 2e For the hydrogen molecule, σmax occurs at approximately the same energy for both processes, but the value of σmax for dissociative ionization is one order of magnitude smaller than the value for molecular ion formation.

5.2.3 Ionization by Impact of Atoms or Molecules If the target particle of mass m2 is assumed to be at rest, then the incident particle (mass m1) must have at least a kinetic energy Es (threshold) Es ¼

m1 þ m2 Eion m2

(37)

derived from momentum and energy conservation. In general, Es is substantially higher than Eion. Usually, the increase of the cross section after passing the threshold energy is lower than that for electrons, and σmax occurs for energies in

118

M. I. Boulos et al.

the keV range. Again, such particle energies are beyond the range considered in the context of this book. For metastable atoms, ionization may occur by collisions between excited atoms, i.e., Xm þ Xm ) Xþ 2 þe or Xm þ Xm ) Xþ þ Xþ þ e The probability of the first process is supposed to be much smaller than that of the second process McDaniel (1964).

5.3

Inelastic Collisions of the Second Kind

Collisions of the second kind are those in which internal energy (excitation energy) from one particle is transferred to another particle. The excitation energy of the colliding particle may be transferred as kinetic energy to the resultant particles or it may cause excitation or ionization of the receiving particle. In the case of ionization, the electron removed from the particle carries off most of the excess energy. As a rule, the cross section for this process is small when the energy difference (ΔE) is large and the velocity of the impacting particle is small.

5.3.1 Associative Ionization The production of molecular ions can occur in the following way: X þ X ) Xþ 2 þe provided that the energy condition E Eb is met, where Eb is the binding energy of Xþ 2.

5.3.2

Ionization of Already Excited Atoms by Electron Impact eðE1 Þ þ X ) Xþ þ e þ eðE2 Þ

This process can occur when E1 Eion E. The excited atom involved in this process is usually a metastable atom, because of the short lifetime of regular excited states.

5.3.3 Charge Exchange Processes In this type of process, there are ionized states both before and after the encounter: Xþ þ Y ) X þ Yþ

3

Kinetic Theory of Gases

119

but there is no production of free electrons. If X = Y, one has a so-called resonance phenomenon and the collision probability assumes a maximum. In this case, one has a charge exchange process, especially when one of the particles is fast (an ion accelerated in an electric field (subscript f)) and the other relatively slow (subscript s). Schematically, this process can be written  þ Xþ f þ Xs ) Xf þ Xs

5.3.4 The Penning Effect Of special interest for low-pressure electrical discharges is the process in which ionization of one particle occurs by impact with a metastable particle with excitation energy Em Eion (the Penning effect). A typical example is ionization of argon (Eion = 15.76 eV) by the metastable atom of Ne 3P2 (Em = 16.53 eV). The probability for ionization in this case is large (almost unity per collision) because ΔE ¼ 0:77 eV is small, and the effect is further enhanced by the long lifetime of the metastable state. For example, Ne-Ar and He-Ar mixtures show the Penning effect, whereas He-Ne and Ar-Kr do not have the necessary favorable energy level combinations as shown in the following table: Atom Helium Neon Argon Krypton

6

First metastable state Em(eV) 19.8 16.6 11.5 9.9

Ionization energy Eion (eV) 24.6 21.6 15.7 14

Distribution Functions

The quantities we have just discussed are related to the relative velocities (energies) of individual particles on impact. But to describe plasma on a macroscopic scale, microscopic properties have to be averaged, and for this purpose, the velocity distribution of the particles is needed. For more details, see Fowler (1956), Hirschfelder (1954), Landau and Lifshitz (1967), Mayer and Mayer (1940), Munster (1969), Reif (1988).

6.1

Definition

  ! ! The velocity distribution function f r , v , t depends, in general, on the velocity !

(components of the velocity vector v in velocity space), on time, and on the space ⇀ coordinate (components of the position vector r ). The number of particles dwelling

120

M. I. Boulos et al. ⇀

at time t in the volume element dx.dy.dz, (which, for convenience, is denoted as d r) ! ! ! and having velocities in the range from v to v þ dv is given by   ! ! ! ⇀ (38) d6 N ¼ f r , v , t :dv :dr   ! The total particle number density n r , t is then given by   ð   ! ! ! ! n r , t ¼ f r , v , t :dv

(39)

and the integration is carried out over  the total  velocity space (according to the ! ! presence of exponential terms in f r , v , t ; in most cases, integration may be !

performed for velocity vectors v varying between 1 and þ1). The distribution function is normalized by the requirement that þ1 ð

  ! ! ! f n v , r , t dv ¼ 1

(40)

1   ! ! A steady state is characterized by a distribution function f r , v independent of t;   ! a uniform distribution function is characterized by a function f v , t independent     ! ! ! !  ! of r ; an isotropic distribution exists when f r , v , t depends on r , t, and  v  ¼ v

(the absolute value of v has no directional dependence). For an isotropic distribution, the fraction of particles with velocities between v and v + dv can be expressed by   ! 4πv2 f v, r , t dv

(41)

4πv2. dv is the volume between the spheres of radii v and v + dv in velocity space.

6.2

Particle Fluxes

The fundamental kinetic description of a partially ionized gas is provided by the velocity distribution function for each species i. Particle fluxes in such a gas may result from the random motion of the particles. There is, however, no net flux of particles of any type, unless there are gradients that are able to give rise to net particle fluxes.   ! ! Any function χ r , v , t that describes particle properties in terms of position !

!

r and velocity v at time t has a mean value defined by þ1 ð      1 ! ! ! ! !   < χ r ,t > ¼ f r , v , t χ r , v , t  dv ! n r , t 1



!

(42)

3

Kinetic Theory of Gases

121

where n is the mean number density of particles per unit volume as defined by ! Eq. 40. The mean velocity of the particles in volume d r is given by !

¼  !  v :f r , v , t dv n r , t 1

!

(43)

!

This mean velocity, denoted below by v g , is referred to as the species fluid (or average) velocity relative to some laboratory frame of reference. For collision-dominated gases, it is more convenient to consider particle velocities with respect to a local frame of reference  that  moves with the mean mass ! ! velocity (center of gravity) of the fluid v g r , t , rather than with respect to a laboratory frame of reference. Therefore, the so-called peculiar velocity is introduced:   ! ! ! ! (44) U¼ v  v g r , t and by definition

D !E D E ! ! U ¼ v  vg

(45)

which holds only if the gas consists of a single chemical species (see ▶ Chap. 7, “Transport Properties of Gases under Plasma Conditions”). The concept of fluxes is very important for transport phenomena, which are determined by the calculation of fluxes of various quantities. Let us consider a ! surface element dA (with its normal to n ) that divides the whole space in two ! regions: (+) on the side of n , (-) on the other  side  (see Fig. 6). This surface may, for ! ! example, move with the mean velocity v g r , t . In general, the peculiar velocities !

!

U are much higher than v g , and thusparticleswill cross dA in both directions ! ! carrying different physical properties χ r , v , t with them. The normal compo  ! nent of the fluxFn r , t of χ through dA is then defined as the total amount of χ carried per unit time and unit surface from the (-) side to the (+) side of the specified surface. ! ! If Un ¼ n  U > 0, particles will cross dA from (-) to (+), and the number of particles crossing dA during time dt corresponds to the particles contained  in a ! ! !  cylinder of base dA and length U dt and a corresponding height of  n U dt. This cylinder contains

    ! !  ! ! ! f r , v , t dv  n  U dtdA

(46)

particles, and each of these particles transports the property χ. The total amount of χ transported through dA from (-) to (+) is then obtained by integrating over all ! ! velocities for which n  U > 0:

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Fig. 6 Particles crossing the area element dA moving with the gas velocity vg in time dt from (a) the (-) to the (+) side, and (b) from (+) to () side

(+) (−)

(+) (−)

n

n

dA Udt

Udt

vj vj

Fþ n ¼

ð

      ! ! ! !  ! ! ! f r , v, t   n  U   χ r , v, t dv

(47)

! !

n  U >0

!

!

Similarly, for the particles passing from (+) to (-) and corresponding to n  U < 0, one finds ð       ! ! !  ! ! χ ! ! F ¼ f r , v , t  n  U , v , t dv (48)  r   n ! !

n  U n1  n2

(87)

For example, if f10 and f20 are Maxwellian distributions for the same temperature (or for the same mean kinetic energy), the reaction coefficient can be written as

k12 ¼ n1 n2

ðm1 m2 Þ3=2 ð2kTÞ3

þ1 ð þ1 ð

1 1




m1 v21 þ m2 v22 exp  v12 σðv12 Þd3 v1 d3 v2 2kT

(88)

where d3v stands for dvx.dvy.dvz. ! Using the coordinates of the position vector of the center of mass r i with the corresponding velocity   ! ! m v þ m v 1 2 1 2 ! c¼ (89) m1 þ m2 and the relative velocity and expressing d3vl2 in spherical coordinates d3 v12 ¼ v212 dv12 dΩv

(90)

where dΩv denotes an infinitesimal solid angle defined by the direction of v12 and integrating over the whole space (dΩv ¼ 4π if isotropic) results in

130

M. I. Boulos et al.

k12 ¼

1 ð


 ! c2 exp ðm1 þ m2 Þc2 =ð2kTÞ dc

ð4πÞ2 ðm1 m2 Þ3=2 ð2πkTÞ

3 0

1 ð


 !  v312 exp m12 v212 =ð2kTÞ σ12 ðv12 Þdv 12 0

(91)

with m12 ¼

m1 m2 ðm1  m2 Þ

Using the expression f 02

ð1


 pffiffiffi y2 exp y2 dy ¼ π=4

0

results finally in k12 ¼

4  m12 3=2 π 2kT

ð1

xexpðxÞσ12

0

rffiffiffiffiffiffiffiffiffiffiffi

2kT x dx m12

(92)

where x ¼ m12  v212 =2kT. This expression shows explicitly that the reaction rate is essentially a macroscopic quantity that depends on the thermodynamic state of the gas. In the expression for k12 , the integration has been performed over all velocities (from zero to infinity). Most processes, however, have threshold energies, and therefore integration has to be performed only from the corresponding threshold velocity to infinity. The probability of collisions per unit time corresponds to the rate of production of species A and B. Hence, the mass action law for this reaction can be written as dnA dnB ¼ dt dt ¼ kd for the forward reaction

k12 nAB ne ¼ In general, the rate coefficient k12

(93)

AB þ e ) A þ B is different from the coefficient k34 ¼ kr for the reverse reaction A þ B ) AB þ e !

because σ12 is different from σ34 and the relative velocity v 12 of the electrons with ! respect to the molecules AB is higher than the relative velocity v 34 of particles A  with respect to particles B . However, if thermodynamic equilibrium prevails, a microreversibility relationship exists between kd and kr.

3

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131

Usually, the reaction rate coefficient k is expressed in m3/s. For collisions between atoms (e.g., Ar at 300 K with a Maxwellian distribution), we have σ12  5:52  1019 m2 v12  4:0  102 m=s and thus k  2:2  1016 m3 =s As an order of magnitude, k  1017 m3 =s, for the probability that a reaction will occur (to a first approximation, values below 1017 m3/s can be ignored).

7.2

Three-Body Reactions

Even at relatively low pressures, three-body reactions may become important. Schematically, a typical three-body reaction can be written as A þ B þ M ) AB þ M0 The third
body does  not participate in the reaction; it merely absorbs the excess energy E0M > EM . If f1, f2, and f3 are the distribution functions of the particles A, B, and M, with respective velocities of v1, v2, and v3, dnAB ¼ dt

ððð   ! ! ! ! ! ! k v 1 , v 2 , v 3 f 1 f 2 f 3 dv 1 dv 2 dv 3

(94)

where k is the differential reaction rate expressed in m6/s, i.e.,   ! ! ! ! k v 1 , v 2 , v 3 ¼ σ12 v 12 V12, 3

(95)

In this expression, V12,3 is the interaction volume of the intermediate compound with the third body. The reaction rate constant is given by   ! ! ! k ¼ < k v 1, v 2, v 3 > Typical mean values are in the ranges ! 17 σ12 v 12  10

m3 =s


3 V12, 3  109 m3

k  1044 m6 =s

(96)

132

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Recombination

Recombination is defined as the mutual attachment of particles in the course of an encounter, for example, between a positive ion and an electron or between a positive ion and a negative ion. As previously shown, the process of recombination is generally described by a coefficient k, as given by the rate equation, which describes the recombination of charges during binary collisions. The three most important recombination processes for thermal plasmas will be considered here.

7.3.1

Radiative Recombination Xþ þ e ) X þ hυ0 ) X þ hυ1 þ hυ0

where 1 hυ0 ¼ Ei  E þ me v2e 2 The reverse process is photoionization, which cannot be neglected in thermal plasmas. Under thermodynamic equilibrium conditions, there would be a microbalance between photoionization and photorecombination, i.e., a high probability for photoionization would also entail a high probability for photorecombination.

7.3.2 Dissociative Recombination In this process, part of the neutralization energy is used for the dissociation process. The electron is captured into a nonradiative repulsive state (XY)*i , which then dissociates: ðXYÞþ þ e ) ðXYÞi ) X þ Y ) X þ Y þ hυ1 þ hυ2 This process is one of the most effective recombination processes in the presence of molecular ions. The recombination coefficient increases as the density of the gas molecules increases, passes through a maximum, and then decreases for pressures exceeding one atmosphere. The maximum value is on the order of 1013 m3/s.

7.3.3 Three-Body Recombination At higher pressures, a third body, Y, which can be either a neutral particle or an electron, removes part of the neutralization energy: Xþ þ e þ Y ) X þ Y The third particle, Y, is frequently a slow electron, because the reverse process, ionization by electron impact Bittencourt (2004), efast þ X ) Xþ þ e þ eslow

3

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133

has a high probability. For these recombination reactions, the reaction rate coefficient k is denoted by α, which is defined by dnþ dne ¼ ¼ α nþ ne  αn2e dt dt

(97)

for a quasineutral plasma (note that α is defined for a two-body reaction and is expressed in m3/s). In the case of a Maxwellian distribution among the particles, the velocity ve of the electrons is much higher than that of the ions, which can be considered stationary with respect to the electrons. With this assumption, α becomes 1 ð h  me 3=2 α¼ σ12 ðuÞf o ðuÞu1=2 du π 2kT

(98)

0

where u = E/kT and where f (u).du is given by Eq. 66. Electron-ion recombination coefficients are generally lower than those for ion-ion recombination because of the higher velocities of the electrons. Radiative recombination, which results from trapping free electrons into atomic levels with emission of radiation, has a coefficient between 1018 and 1019 m3/s. Nevertheless, this process is sufficiently important to be partially responsible for the high luminosity of thermal plasmas. If there is a high concentration of molecular ions in a plasma, dissociative recombination becomes very important because of its large recombination coefficient, which is in the order of 1013 m3/s. 0

8

Nomenclature

Units are indicated in parentheses; whenever no units are given, the parameter is dimensionless.

8.1 A c Cij !

c dA dN(v) e E Eb

Latin Alphabet Surface (m2) Velocity of light (c = 2.998  108 m/s) Net rate of increase of particles in the control volume due to collisions Center-of-mass velocity of two particles (m/s) Elementary surface (m2) Number of particles with velocities between v and v+dv Electron charge (e = 1.602  1019C) Electric field (V/m) Binding energy of the molecular ion

134

Ei Eion Em Es E*  f(v) ! ! f r , v ,t   ! ! fn r , v , t Fi Fþ n F n

h J! J k   ! k12 v 12   ! ! ! k v 1, v 2, v 3 K ‘ ‘i ‘ij ‘ L m mH mi M n ! n ni p ! p

M. I. Boulos et al.

Energy of the excited level i (EV or cm1) Ionization energy of an atom (eV) Energy of a metastable state (eV) Threshold energy (eV) Energy of an excited state (eV) Maxwellian distribution function Velocity distribution function at time t at the tip of the position ! vector r Normalized distribution function Incident flux of particles i (m2s1) Flux per unit time and unit surface carried to the (+) side of a ! surface defined by its normal n and moving with the gas velocity ! vy Flux per unit time and unit surface carried to the (-) side of a ! surface defined by its normal n and moving with the gas velocity ! vy Planck’s constant (h = 6.626  1034 Js, or kgm2/s) Associated quantum number  ! P ! Total angular momentum J ¼ i Ji Boltzmann’s constant (k = 1.38  1023 J/K)

3

Mean direct binary collision rate or reaction rate (m /s) 6

Mean direct three-body collision rate (m /s) Fraction of energy transferred by elastic collision from one particle of mass m to another of mass M averaged over all angles Mean free path (m) Mean free path of particles of type i in a mixture of different chemical species (m) Mean free path traversed by a particle of type i between two successive collisions with particles of type j (m) As index, lower energy level Distance between electrodes (m) Mass of a particle (kg)   Mass of hydrogen atom

mH me

¼ 1856

Mass of the species i (kg) Mass of a particle (kg) Principal quantum number Normal to a surface dA Number density of particles of chemical species i (m3) Absolute value of momentum Momentum vector

3

Kinetic Theory of Gases

135

pij(Vij, θ, φ) P(x) P(t)

Proportionality factor defining the differential cross section Probability of a free path to exceed a distance x (m) Probability that a particle with relative velocity !  ! ! V ¼ v 1  v 2 survives a time t without suffering a collision

P*(t).dt

Q

Probability that a particle, after surviving without collisions for a time t, suffers a collision in the time interval between t and t + dt Total effective cross section for collision processes of particle i with particles j (m1) X 
Qi Total effective cross section Q ¼

u

As index, upper energy level

U

Peculiar velocity of a particle

Un

particle velocity and v g the gas velocity (m/s) !  ! ! Normal component of the peculiar velocity U ¼ v  v g (m/s)

Qi

i !

!  ! ! ! U ¼ v  v g where v is the

!

! ri ! rj ! Rij

T v vm v v2 ! vi ! vg V V0 !

V !ij V 0ij Vij w.dt X X* Xm X+ X++ X2

Position vector of particle i (m) Position vector of particle j (m) !  ! ! Relative position of particles i and j Rij ¼ ri  rj Temperature (K) Absolute value of the velocity (m/s) Most probable velocity (m/s) Mean velocity (m/s) Mean square of the effective velocity (m/s) Velocity vector of a particle of chemical species i (m/s) Mean velocity referred to as the species fluid (or average) velocity relative to some laboratory frame of reference (m/s) Relative velocity of particles of chemical species i and j before collision (m/s) Relative velocity of particles of chemical species i and j after collision (m/s) Relative velocity vector of particles i and j before collision (m/s) Relative velocity vector of particles i and j after collision (m/s) Mean relative velocity of particles i and j (m/s) Probability that a particle suffers a collision between t and t + dt, also called collision rate Designation for the chemical species in its ground state Designation for the chemical species in an excited state Designation for the chemical species in a metastable state Designation for the chemical species in its first ionized state Designation for the chemical species in its second ionized state Designation for the molecule made of two X atoms in its ground state

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M. I. Boulos et al.

Xþ 2 Y Z’

8.2

Designation for the ionized molecule Designation for the chemical species in its ground state Number of protons in the nucleus

Greek Alphabet

α ΔE ΔEtot ΔJ ΔP θ λi Λ σ σ0 σij σimax σ0(Vij) !  σ V ij , θ, φ !  σij V ij , θ, φ τ τij υ υij φ  ! ! χ r , v,t Ω

Recombination reaction rate coefficient (m3/s) Energy difference between two excited states (eV or cm1) Total kinetic energy exchange during a collision (eV, J, cm1) Difference in the angular momentum of the atom between its initial and final states Change in angular momentum during collisions (ΔP = hΔJ) Angle in spherical coordinates Maximum incident wavelength for ionization by photons (nm) Wavelength (nm) Wave number (m1) Total scattering cross section (m2) Differential scattering cross section (m2) Maximum value of σ(m1) Total scattering cross section (m2) 2

Differential scattering cross section (m ) 2

Differential scattering cross section (m ) Mean time between collisions also called relaxation time (s) Mean time between collisions of two particles of types i and j Frequency of the associated wave (s1 or Hz) Collision frequency between two particles of types i and j (s1) Azimuthal angle in spherical coordinates ! ! Particle properties in terms of position r and velocity v at time t Solid angle (ster.)

References Bittencourt JA (2004) Fundamentals of plasma physics. Springer, p 678 Brown SC (1959) Basic data of plasma physics. MIT Press, Cambridge, MA Delcroix JL (1966) Physique des Plasmas, vol 1 and 2. Monographies, Dunod, Paris Fowler G (1956) Statistical thermodynamics. U.K. Press, Cambridge Francis G (1960) Ionization phenomena in gases. Butterworth, London Gould H, Tobochnik J (2010) Statistical and thermal physics. Princeton Univeristy Press, Princeton Hirschfelder JD (1954) Molecular theory of gases and liquids. Wiley, New York Landau L, Lifshitz E (1967) Physique statistique. MIR, Moscow

3

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Massey HSW (1971) Electronic and ionic impact phenomena. Slow collisions of heavy particles. Clarendon, Oxford Massey HSW, Burhop EHS (1969) Electronic and ionic impact phenomena. Collisions of electrons with atoms. Clarendon, Oxford Massey HSW, Gilbody HB (1974) Electronic and ionie impact phenomena. Recombination and fast collisions of heavy particles. Clarendon, Oxford Mayer JE, Mayer GM (1940) Statistical mechanics. Wiley, New York McDaniel EW (1964) Collision phenomena in ionized gases. Wiley, New York Mitchner M, Kruger CH Jr (1973) Partially ionized gases. Wiley, New York Munster A (1969) Statistical thermodynamics, vol 1. Springer/Academic, New York/Berlin Present RD (1958) Kinetic theory of gases. McGraw-Hill, New York Reif F (1988) Fundamentals of statistical and thermal physics. McGraw-Hill, New York Von Engel A (1965) Ionized gases. Clarendon, Oxford

4

Fundamental Concepts in Gaseous Electronics Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Generation of Charge Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Direct Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Indirect Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Loss of Charge Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Motion of Charge Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Drift in Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Diffusion of Charge Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Motion of Charge Carriers in Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Thermal Excitation and Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Boltzmann Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Saha Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Complete Thermal Equilibrium (CTE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Concept of Local Thermodynamic Equilibrium (LTE) . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Deviations from LTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Rigorous Definition of the Plasma State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Debye Length in a Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Characteristic Lengths in Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

140 140 141 141 141 142 142 148 152 153 154 157 161 163 166 167 168 170

M. I. Boulos (*) Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada e-mail: [email protected] P. L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France e-mail: [email protected] E. Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] © Springer Nature Switzerland AG 2023 M. I. Boulos et al. (eds.), Handbook of Thermal Plasmas, https://doi.org/10.1007/978-3-030-84936-8_4

139

140

M. I. Boulos et al.

7 Quasi-neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Charge Separation by Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Charge Carrier Separation by Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Plasma Sheaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Latin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 172 173 175 177 177 179 180

Abbreviations

AC CTE CLTE DC LTE PLTE RF

1

Alternating current Complete thermodynamic equilibrium Complete local thermodynamic equilibrium Direct current Local thermodynamic equilibrium Partial local thermodynamic equilibrium Radio frequency

Introduction

It is beyond the scope of this book to even attempt a comprehensive treatment of fundamental concepts of gaseous electronics, which experienced immense growth during and after the Second World War. In this chapter the fundamentals behind the generation, loss, and motion of charge carriers are discussed. This is followed by a review of thermal excitation and ionization, definition of the plasma state, quasineutrality, and plasma sheaths. For a comprehensive treatment of the subject, the reader is referred to a number of books (Capitelli et al. 2012; Finkelnburg and Maecker 1956; Griem 1964; Gupta 2007; Huddlestone and Leonard 1965; Lee et al. 1973; Lochte-Holtgreven 1995; Loeb (1961); Massey et al. 1969; Mitchner and Kruger 1973; M€uller and Weiss 2005; Reif 2009) that may be considered classics in this field.

2

Generation of Charge Carriers

As described in ▶ Chap. 1, “The Plasma State,” electrical discharges are the most common method of producing gaseous plasmas. In order to maintain a steady-state electrical discharge, charge carriers must be produced at the same rate as they vanish. In principle, charge carriers are comprised of negative and positive particles. The former includes electrons and negative ions, and the latter, positive ions only. In the context of this book, negative ions play a minor role, and they will therefore be neglected in this section. Charge carriers can be produced by ionization processes in a gas volume or by liberation of charged particles from confining walls and, in particular, from

4

Fundamental Concepts in Gaseous Electronics

141

electrodes. Liberation of electrons from the cathode, for example, is vital for maintaining DC or AC discharges. The following discussion will be restricted to ionization in the gas volume. Interactions among the plasma constituents (electrons, ions, neutrals, and photons) can lead to an array of different ionization processes. Among all conceivable interaction processes in plasma, binary interactions have a much higher probability, and therefore, they are particularly important. In general, ionization is defined as the removal of one or more electrons from a neutral particle or from an ion (single or multiple ionization, respectively). Such ionization processes always produce free electrons and positive ions. These processes can occur through a simple encounter (direct ionization) or through a sequence of encounters and energy exchange processes (indirect ionization). The latter processes are characterized by intermediate energy states.

2.1

Direct Ionization

Direct ionization by photons, by electron impact, and by collisions among heavy particles was discussed in Sect. 5.2 in ▶ Chap. 3, “Kinetic Theory of Gases.” It should be emphasized that these ionization processes refer to particles initially in the ground state. Regardless of the nature of an ionization process or any other encounter between plasma constituents, such encounters are governed by the conservation laws for mass, charge, momentum, and energy. By applying the conservation laws, it is possible to single out the most important ionization process in the given plasma. In this context, for example, ionization by collisions among heavy particles is for practical purposes negligible because the required particle energies are beyond those encountered in the plasmas covered in this book.

2.2

Indirect Ionization

As mentioned previously, indirect ionization implies that intermediate energy states precede the actual ionization process. The processes can produce such energy states (e.g., excited states). Typical examples of indirect ionization requiring at least two steps for producing charge carriers (associative ionization; ionization of already excited atoms by electron impact, by photons, and by excited atoms; and ionization by the Penning effect) have already been discussed. Again, the conservation laws govern all of these various processes.

3

Loss of Charge Carriers

There are a large number of conceivable processes by which charge carriers vanish from plasma. Only those processes that are important for the plasmas considered in this book will be discussed here.

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In general, charge carrier losses can result from: • Drift to the electrodes resulting from electric fields in the case of DC or AC discharges • Diffusion to the surrounding walls with subsequent recombination • Volume recombination Electron attachment (formation of negative ions) will not be considered here. Drift of positive ions to the cathode results in neutralization of the ions upon impact on the cathode. Electrons drifting to the anode are absorbed into the metal lattice. Drift of charge carriers and diffusion of charge carriers due to particle density gradients will be discussed in more detail. In the volume recombination process, an electron is captured by a positive ion; in this process both the ionization energy (Eion) and the kinetic energy of the colliding electron (12 m:v2 ) are released:
 1 þ 2 e þ X ) X þ Eion þ me ve 2 The possible recombination mechanisms depend on the disposal of this energy within the restraints of the conservation laws. This energy, for example, cannot be assumed by the neutral particle after recombination, because this would violate momentum conservation. There are essentially three recombination mechanisms that are of interest in the context of this book: • Radiative recombination • Three-body recombination • Dissociative recombination

4

Motion of Charge Carriers

4.1

Drift in Electric Fields

This section begins with a microscopic description of the motion of a charged particle in a uniform electric field in the absence of any other forces. By first establishing the average behavior of a charged particle in an ensemble of particles, macroscopic relations (e.g., mobilities and electrical conductivity, which will be discussed in the second part of this section) can be derived. The drift velocity of an electron is due to the fact that it is accelerated by the electric field towards the anode, but before reaching it, it collides with many heavy species and its trajectory is thus continuously deflected as illustrated in Fig 1. The final electron trajectory is therefore considerably longer than it would have been without collisions, giving rise to a considerably lower effective mean electron drift ! velocity, v ed , between the cathode and the anode.

4

Fundamental Concepts in Gaseous Electronics

143

Fig. 1 Movement of an electron in an electric field with collisions with heavy particles

The electric force acting on charge carriers exposed to an electric field accelerates them according to Newton’s second law of motion, which, in the case of electrons, assumes the form !

me

! du e ¼ eE dt

(1) !

!

where u e is the electron drift velocity, me is the electron mass, and E is the electric field strength. A similar equation holds for positive ions in an electric field: !

M

! du i ¼ eE dt

(2)

!

where u i is the ion drift velocity and M is the ion mass. For simplicity it is assumed that the ion is singly charged. Because of the similarity of Eqs. 1 and 2, only electrons will be considered in the following derivations. The derivation of the final expressions for ions is left to the reader. Integration of Eq. 1 assuming constant electric field strength results in ! ue !

!

¼ u e0 

!

eE t me

(3)

where u e0 is the initial velocity of the electron at the moment when the electric field is applied. Equation 3 indicates that the drift velocity of the electron increases linearly with time. This increase is limited, however, by collisions with other particles in the plasma. (Only elastic collisions will be considered in the context of this analysis. This assumption is justified by the fact that elastic collisions dominate in plasma.) If we assume that the free flight time of an arbitrarily chosen electron is τe, then ! integration of Eq. 3 determines the distance s e that this electron will travel during the time interval τe: !! eE ! ! (4) s e ¼ u e0  τe  :τ2e 2me

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In the following discussion, the average behavior of an electron will be considered, rather than the behavior of an arbitrarily chosen electron. The laws of statistics may be applied to determine this average behavior, because of the large number of electrons per unit volume in typical plasma (ne > 1020 m3). Since the initial velocities of the electrons are randomly distributed, the first term on the right! hand side of Eq. 4 will vanish, i.e., u e0 :τe ¼ 0. As previously shown, the mean free path length or the mean free flight time follows a statistical distribution according to 1 ð

τe ¼ 0

τe expðτe =τe Þdτe τe

(5)

By applying Eq. 5 to Eq. 4 for averaging, one finds ! se

!

eE ¼  2me

1 ð

0

!

τ2e eE expðτe =τe Þdτe ¼  τ2e τe me

(6)

!

The corresponding mean drift velocity (often denoted v ed ) becomes !e vd

! ue

¼

!

eE ¼  τe me

(7)

Substituting τe ¼

‘e ! ve

(8) !

(where ‘e is the mean free path length of the electrons and v their mean thermal velocity) into Eq. 7 results in ! ue

¼

! e‘e ! E ¼ μeE me ve

(9)

where μe is the electron mobility: μe ¼

e‘e me ve

(10)

An analogous expression can be derived for the average ion drift velocity: ! ui

!

¼ μ iE

(11)

e‘i mvi

(12)

with μi ¼

4

Fundamental Concepts in Gaseous Electronics

145

where ‘i is the mean free path of the ions and vi is the mean thermal velocity of the ions. At this point it is interesting to make an order of magnitude comparison between the drift velocity of charged particles and their random (thermal) velocity. For this comparison, the region close to the anode will be considered for an atmospheric pressure, high-intensity argon arc (welding arc) and specify the following parameters (order of magnitude): T = 104 K, ‘e  ‘i  106 m, E = 5  102 V/m, me = 9.1  1031 kg, M = 6.8  1026 kg. The drift and thermal velocities for ions and electrons are: Electrons:


 8kT 1=2 ve ¼ ’ 106 m=s πme  
e‘  ! e E ’ 102 m=s  ue  ¼ m e ve Ions:


 8kT 1=2 vi ¼ ’ 103 m=s πM  
e‘  ! i E ’ 1m=s  ui  ¼ Mvi

The drift velocities are several orders of magnitude smaller than the corresponding thermal velocities. Based on these findings, we can see that the motion of the charged particles is analogous to that of a swarm of flies on a hot summer day drifting in a slight wind. In plasma, electrons and ions drift in opposite directions under the influence of an applied electric field, and this drift gives rise to an electric current of density:   ! ! ! ! ! ! (13) j ¼ j i þ j e ¼ e ni u i  ne u e ¼ e ð ni μ i þ ne μ e Þ E where ni and ne are the ion and electron densities, respectively. Since the electric field is the only driving force for electron and ion currents, Ohm’s law can be written in simple form as !

!

j ¼ σe E

(14)

A comparison of Eqs. 13 and 14 provides a simple expression for the electrical conductivity σ e ¼ e ð ni μ i þ ne μ e Þ

(15)

This expression can be further simplified because ni = ne (for singly charged ions) and μi  μe . With this simplification, Eq. 15 becomes

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M. I. Boulos et al.

σe ¼ ene μe

(16)

The justification for neglecting μi compared to μe follows from Eqs. 10 and 12. The ratio μi ‘i me ve ¼ μ e ‘ e M vi

(17)

transforms with ve ¼ vi




M me

1=2 (18)

into μi ‘i me 1=2 ¼ (19) μe ‘e M qffiffiffiffi Since ‘i < ‘e in a plasma and the ratio mMe already exceeds 40 for hydrogen, neglecting μi compared to μe is indeed justified. Equation 18 in this simple form is valid only when kinetic equilibrium (Te = Th) prevails in the plasma. In deriving the expression for the electron mobility, Eq. 10, a number of implicit, simplifying assumptions have been made: The degree of ionization in the plasma is assumed to be small ðξ  1Þ, which implies that Coulomb interactions are negligible, i.e., only collisions between electrons and neutral particles have been taken into account. Averaging over the velocity distribution functions has been omitted. Local imbalances of charge neutrality have been neglected. By averaging over the velocity distribution function, Eq. 10 becomes μe ¼

2 e‘e π m e ve

(20)

Removing all the simplifying assumptions leads to the more complex expression derived by Gvosdover (1937): μe ¼

e pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ne e 4 πkme T þ ‘k γe ðkTÞ2

where γe ¼

2 π 3kT ln 1=3 2 2e2 ne

!

!

(21)

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Fundamental Concepts in Gaseous Electronics

147

and ‘k is a mean free path length associated with the Ramsauer cross section. For plasmas with a low degree of ionization ðξ  1Þ, Eq. 21 reduces to μe ¼

e‘e ðπkTme Þ1=2

pffiffiffi 2 2 e‘e ¼ π me ve

(22)

pffiffiffi Except for the 2 factor, Eq. 22 is identical to Eq. 20. In general, the mobility of a charged particle is a function of its kinetic energy. The electric field and the mean free path, in turn, determine the kinetic energy that a particle can acquire. The maximum kinetic energy that an electron, for example, can acquire between two collisions is given by Tk ¼ e‘e E

(23)

where ‘e and E are parallel to each other. Since ‘e  1p , the electron mobility becomes a function of E/p: μe ¼ f ðE=pÞ Figure 2 shows the electron mobility in different gases as function of E/p. These are order of magnitudes higher than the corresponding ion mobilities values given in Fig 3. For different ions. The latter, according to Eq.12, are function of the ion mass, mi, in addition to the E/p value, and therefor;
 E ,M μi ¼ f p

Fig. 2 Electron mobility in hydrogen versus E/p

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Fig. 3 Ion mobility versus molecular weight Ion mobility, μi (m2/ V.s)

4.8 4.4 Li

4.0 3.6 Na Al

2.8

K

N2

2.4 2.0

4.2

NH3

3.2

0

40

Na(NH3) Ca Rb Kr

In Xe Ba Cs

80 120 160 Molecular weight of ions, M (–)

Ti Hg

200

Diffusion of Charge Carriers

In this section diffusion of plasma charge carriers due to concentration gradients as well as ambipolar diffusion associated with the large difference between electron and ion mobilities will be considered. Fick’s law can describe ordinary diffusion: ! jk

¼ Dk grad nk

(24)

where jk represents the flux of charged particles (m2.s1) of type k, Dk is their diffusion coefficient, and nk is their number density. The motion of charge carriers due to diffusion gives rise to an electric current of density ! je

¼ eDe grad ne

(25)

for the electrons, and the corresponding ion current density can be written as ! ji

¼ eDi grad ni

(26)

where De and Di are the electron and ion self-diffusion coefficients, respectively. These diffusion coefficients are related to gas kinetic parameters by Dk ¼

‘k vk 3

(27)

This expression follows from kinetic gas theory (see ▶ Chap. 7, “Transport Properties of Gases Under Plasma Conditions”). At kinetic equilibrium, the ratio of the electron and ion diffusion coefficients is given by

4

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Fig. 4 One-dimensional situation in which a plasma borders a wall

Sheath

Plasma

Wall

Grad (n) Ex

ne= ni = n

X

De ‘e ve ‘e ¼ ¼ Di ‘ i vi ‘i

rffiffiffiffiffi M μe ¼ m μi

(28)

This relationship will be used later on in this section. Charged particle density gradients are particularly steep close to plasmaconfining walls. The onedimensional situation in which a plasma borders a wall is shown in Fig. 4. The gradients of electron and ion density in the vicinity of the wall drive electron and ion fluxes towards the wall, but the electron flux initially exceeds the ion flux because of the higher electron mobility. Since the wall is assumed to be isolated (no net current flow), it will acquire a negative potential, producing an electric field, Ex, that points towards the wall (see Fig. 4). This field subsequently balances electron and ion fluxes (electrons are retarded and ions are accelerated); thus, in a steady-state situation, electrons and ions reach the wall at the same rate and recombine on impact with the wall. This process is known as ambipolar diffusion. In this case the wall serves as the third collision partner for three-body recombination. The combined electron and ion current in this situation can be written as j ¼ je þ ji ¼ eðni μi þ ne μe ÞEx þ eðDe  Di Þ

dn ¼ 0 dx

(29)

Since there is no net current flow to an electrically isolated wall, Eq. 29 allows to determine the electric field responsible for ambipolar diffusion: Ex ¼ 

De  Di dn ni μi þ ne μe dx

(30)

For simplicity, it will be assumed that the plasma contains singly ionized species only (ne = ni = n), so Eq. 30 reduces to Ex ¼ 

De  Di 1 dn μi þ μe n dx

(31)

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M. I. Boulos et al.

Using Eq. 31 and applying Eqs. 25 and 26 to the specified one-dimensional situation, the following expression is obtained for the number of charge carriers arriving at the wall per unit area and unit time: je j De μi þ Di μe dn dn ¼ Da ¼ i¼ dx e e μi þ μe dx

(32)

where Da is the ambipolar diffusion coefficient. With μi  μe , the ambipolar diffusion coefficient reduces to Da ¼ Di þ

μi De μe

(33)

and with Eq. 28, which is valid for kinetic equilibrium, results in Da ¼ 2:Di

(34)

This relation indicates that in plasma in which kinetic equilibrium prevails, the ions diffuse at twice the rate they would in the absence of electrons. This finding has important consequences for situations in which heat transfer by diffusion of charged particles becomes significant. As we have just pointed out, ambipolar diffusion plays an important role in regimes of steep particle density gradients as, for example, in the vicinity of confining walls. In such regimes, the ambipolar diffusion process may govern the density distribution of charged particles. In low-pressure discharges (glow discharges), where the ambipolar diffusion coefficient assumes large values, charge carrier losses are frequently dominated by ambipolar diffusion to the walls of the plasma container. In general, the charge carrier continuity equation can be written as ! @nk þ div I k ¼ Sk @t

(35)

where Sk represents the source term describing the net volume production of charge carriers of type k. Rewriting Eq. 35 for electrons assuming that the source term for electron production is proportional to the electron density, i.e., Se ¼ υi :ne, where υi is the net ionization coefficient, results in ! @ne þ div I e ¼ υi ne @t

(36)

For steady-state conditions and considering only electron flux resulting from ⇀  ambipolar diffusion I e ¼ Da :grad ne , Eq. 36 reduces to Δne þ

υi ne ¼ 0 Da

(37)

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where Δ is the Laplace operator. As an example Eq. 37 will be solved for a discharge vessel consisting of a pair of infinitely long, parallel plates separated by a distance L. For this one-dimensional case, Eq. 37 becomes d2 ne υ i þ ne ¼ 0 dx2 Da

(38)

With the boundary conditions ne ¼ 0

L 2

(39)

for x ¼ 0

(40)

rffiffiffiffiffiffi υi x Da

(41)

for x ¼ 

dne ¼ 0 dx Equation 38 has the solution

ne ¼ ne ð0Þ cos

where ne(0) is the electron density for x = 0. The first boundary condition requires that ne ð0Þ cos

rffiffiffiffiffiffi υi L  ¼0 Da 2

or rffiffiffiffiffiffi υi L π  ¼ ð2k þ 1Þ , k ¼ 0, 1, 2, . . . 2 Da 2

(42)

Equation 38 has solutions only in terms of eigenvalues characterized by the integer k. Only the fundamental mode will be considered, or the fundamental eigenvalue for k = 0:
2   υi L π 2  ¼ 2 2 Da or  π 2 υi 1 ¼ ¼ 2 L Da Λ

(43)

where Λ is defined as the characteristic diffusion length, which is a function only of the geometry.

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For an infinitely long cylinder of radius R, the corresponding characteristic diffusion length becomes 1 ¼ Λ2


 2:405 2 R

(44)

and the corresponding solutions of Eq. 37 are Bessel functions. For a cylinder of radius R and finite length L, which can be produced by the intersection of a pair of infinitely long parallel plates with an infinitely long cylinder, the corresponding characteristic diffusion length becomes 1 ¼ Λ2


 2:405 2  π 2 þ R L

(45)

The characteristic diffusion length of more complex configurations can be similarly determined by this intersection method.

4.3

Motion of Charge Carriers in Magnetic Fields

The effects that uniform, constant magnetic fields exert on charged particle motion will be briefly presented. For the more complex situations involving no uniform and/or time-varying magnetic fields, the reader should consult the pertinent textbooks on this subject (Allis 1956; Cambel 1963; Lee et al. 1973; Mitchner and Kruger 1973; Schmidt 1979; Uman 1964). ! Without collisions if a charged particle of mass m, charge q, and velocity v is ! exposed to a uniform and constant magnetic field of induction B , then the particle will experience a force !   dv ! ! F ¼ m ¼ q vB dt

!

!

(46)

Multiplying both side of Eq. 46 by v (dot product) it can be demonstrated that the kinetic energy of the particle remains constant. The movement of a charged particle around a guiding center with a linear motion at constant velocity superimposed in the direction of the applied magnetic field, will give rise to a helical motion as shown in Fig. 5. However, thermal plasmas are collision-dominated, and collisions among charge carriers and with other plasma constituents cause severe disturbances in the previously discussed trajectories of such particles. Between collisions the drift imposed ! ⇀ ! ⇀ on charge carriers by the E and B fields is still present. As soon as E and B fields are applied to a plasma, electric currents will be induced, and these currents, in turn, interact with the applied fields.

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153

Fig. 5 Motion of charged particle in a magnetic induction field

Fig. 6 Creation of a self-

!

j

magnetic field of induction B in a rotationally symmetric

Fr = j x B

!

Fr

arc of current density j and resulting magnetic force

B

For example, in a rotationally symmetric steady arc as shown schematically in Fig. 6, the applied electric field gives rise to a current of density j, which induces a ⇀ self-magnetic field of induction B . The interaction of this field with the current produces a magnetic body force !

F ¼

!

!

j B

(47)

!

The force F r accelerates charged particles towards the center of the arc, giving rise to a pressure gradient and an increase of the pressure in the arc center. For current densities < 106 A/m2, this pressure increase is negligible (typical for wall-stabilized arcs with currents in the order of 100 A). For current densities >108 A/m2, the situation becomes quite different. In confined, pulsed arcs, operated ! with currents I in the order of 105 A, F r gives rise to the “pinch” effect, a substantial reduction of the arc diameter. Another example of extremely high current densities is the constriction of an arc in front of a sharp-tipped cathode (TIG welding arc) leading to the formation of a cathodic plasma jet. This effect will be discussed in detail in Part II, ▶ Chap. 12, “Thermal Arcs.”

5

Thermal Excitation and Ionization

In this section, it is assumed that only the thermal energy of the particles is responsible for excitation and ionization or, in general, for any type of energy exchange among the particles.

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Assigning a temperature to an ensemble of particles (a large number of particles) implies that the particle speeds or particle energies follow a particular distribution, namely, the Maxwell-Boltzmann distribution. Among all conceivable distributions, the Maxwell-Boltzmann distribution represents the most probable distribution. An important feature of the Maxwell-Boltzmann distribution appears in the speed as well as in the kinetic energy distribution function. The kinetic energy of the particles appears in the exponential term, i.e.,
 kinetic energy exp  kT Boltzmann demonstrated that other forms of energy (potential energy, chemical energy, etc.) can replace the kinetic energy in this expression (this form is known as the general Boltzmann distribution). The following section will show how to arrive at the Boltzmann distribution.

5.1

Boltzmann Distribution

Only the basics of this derivation will be described in this section. For more details the reader is referred to Lee et al. (1973) and Laurandeau (2005). Let’s assume a system has N indistinguishable particles (as, e.g., in a uniform gas) with a total energy E. The thermodynamic probability, W, which is usually a very large number, represents the different ways the energy E can be distributed among the N particles: W ¼

N! N! ¼ N1 !N2 ! . . . Ni ! ∏i Ni !

(48)

As shown in ▶ Chap. 12, “Thermal Arcs” for the case of the H-atom, more than one eigenfunction belongs to a single energy state and this degeneracy must be taken into account. Using the principles of quantum statistics (Bose-Einstein statistics), this degeneracy results in a modified thermodynamic probability W ¼ ∏k

ðgk þ Nk  1Þ! ðgk  1Þ!Nk !

(49)

where gk represents the multiplicity (degeneracy) of energy state Ek. Considering for simplicity a monatomic gas, the position, as well as the momentum of every atom in a system of N atoms can be described by a set of six coordinates (Cartesian coordinates) x, y, z, px , py , pz

4

Fundamental Concepts in Gaseous Electronics

155

where the first three describe the position and the other three the momentum of an atom (p = mv, where m is the mass of an atom). By specifying a six-dimensional space (x, y, z, px, py, pz) known as the phase space, the state of every atom in terms of its location and momentum can be represented by a point (a phase point) in this phase space. This phase space can be subdivided into small volume elements H ¼ dx  dy  dz  dpx  dpy  dpz

(50)

denoted as cells and numbered from 1 to k. These cells, however, must remain sufficiently large that the number of phase points per cell remains large (this is necessary for the application of statistical laws). The distribution of the atoms with respect to location and momentum at a given instant can be translated into a distribution of phase points in phase space such that N1 phase points fall into cell 1, N2 phase points into cell 2. . . and Nk phase points into cell k, where ΣNk ¼ N

(51)

The distribution just described is called a macrostate of the system. Only the numbers of phase points per cell are specified, not their individual coordinates within a cell. In contrast, a microstate is defined by a complete specification of all six coordinates of the phase points in a cell. In quantum theory, according to Heisenberg’s uncertainty principle, any quantity in physics that has the dimension of an action (energy x time) can change only in multiples of h (Planck’s constant), i.e., dx  dy  dz  dpx  dpy  dpz  h3

(52)

As a consequence, a volume element in phase space can never be smaller than h3, i.e., H will have a minimum size Hmin = h3 called a “compartment” (g). Each cell must contain a large number of phase points; a cell will, in general, consist of many compartments: g ¼

H 1 h3

(53)

A state for which the thermodynamic probability reaches a maximum (maximum number of microstates for this particular macrostate) is also a state of maximum entropy [S = k.ln(W)] and thus represents the equilibrium state. To determine the Boltzmann distribution (the most probable distribution), we have to find the conditions under which W or ln(W) reaches a maximum (Eq. 49). In the statistical interpretation, gk in Eq. 49 represents the number of compartments in cell k. Using the approximation of Stirling for factorials of large numbers and Lagrange’s method of undetermined multipliers, one obtains the Bose-Einstein distribution function:

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M. I. Boulos et al.

N0k 1 ¼ gk B expðβEk Þ  1

(54)

The Maxwell-Boltzmann statistics can be considered a special case of the more general quantum statistics. The Maxwell-Boltzmann statistics is characterized by N0k  gk, i.e., with the Maxwell-Boltzmann statistics, many compartments in phase space will be empty. With this assumption, Eq. 54 reduces to N0k 1 ¼ gk B expðβEk Þ

(55)

where B  1 and N0k represents the most probable (equilibrium) distribution of phase points in phase space (a Maxwell-Boltzmann distribution). Since only equilibrium distributions will be considered here, the upper index (0) will be dropped. The constant B follows from N¼

X

X Nk ¼ B gk expðβEk Þ

k

k

or B¼

N Σgk expðβEk Þ

(56)

The denominator of this equation is known as the partition function or sum over all states X Q¼ gk expðβEk Þ (57) Introducing Eqs. 56 and 57 into Eq. 55 results in the final form of the MaxwellBoltzmann distribution N k gk ¼ expðEk =kTÞ N Q

(58)

where β ¼ 1=kT can be derived from the second law of thermodynamics. The energy Ek has not been specified in deriving the Maxwell-Boltzmann distribution, i.e., Ek can represent any type of energy (kinetic, potential, chemical, etc. including combinations of such energies). Ek has the units of cm1, it could also be expressed in other units knowing that; 1 cm1 ¼ 1:24  104 eV ¼ 1:99  1023 J The special case in which Ek represents the sum of chemical and kinetic energy will be considered in the next section. If Ek stands for only the kinetic energy of the particles, Eq. 58 transforms into the Maxwell distribution of molecular energies or velocities.

4

Fundamental Concepts in Gaseous Electronics

157

Equation 58 can be written in terms of particle densities by dividing with the volume of the system: nk gk ¼ expðEk =kTÞ n Q

(59)

This equation represents the Boltzmann distribution of excited states if nk is interpreted as the number density of excited atoms in the quantum state k, gk as their statistical weight, Ek as the energy of this excited states (with E1 = 0 X as the energy of the ground state), Q as the partition function of the atoms, and n ¼ nk k

as the total number density of atoms regardless of their excitation state. Equation 59 can be generalized to r-times ionized atoms by adding an additional index r:  n r, k g r, k ¼ exp Er, k =kT nr Qr

(60)

where nr,k is the number density of r-times ionized atoms in the quantum state k, gr,k is their X corresponding  statistical weight, and Er,k their energy. Qr ¼ gr, k expX Er, k =kT represents the partition function of r-times ionized atoms and nr ¼ nr, k die the number density of r-times ionized atoms regardless k

of their excitation state. In a state of thermodynamic equilibrium, the population of excited states as described by Eq. 59 follows a Boltzmann distribution. This fact is one of the cornerstones for the treatment of thermal plasmas in which the particles approach a state of thermodynamic equilibrium or of local thermodynamic equilibrium. Such Boltzmann distributions will therefore be extensively used throughout this book.

5.2

Saha Equilibrium

Eggert, based on thermodynamic principles, has also derived the equation that describes thermal ionization, known today as the Saha equation. Since ionization by collision with high-energy electrons is the most important ionization process in thermal plasma, we will consider such a collision with a neutral atom A: efast þ A ¼ Aþ þ eslow þ e The energy of the electron liberated in this process can be written as Ei þ

 1  2 px þ p2y þ p2z 2me

(61)

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The first term (Ei) represents the potential energy (ionization energy) of the electron with respect to the positive ion, and the second term represents the kinetic energy of the liberated electron. If d6ne is the number density of electrons liberated by ionization in cell H = dx.dy.dz.dpx.dpy.dpz, then the number of electrons per compartment will be d6 ne d6 ne  h3 ¼ gk H



  d6 ne  h3 1 1  2 EI þ px þ p2y þ p2z =n0 exp  kT 2me H

(62)

(63)

with n0 as the number density of neutral atoms in the ground state. According to the Boltzmann principle Equation 63 does not consider the effects of degeneracy. Integrating over the volume and all possible momenta results in


ð þ1 ð ð  ne ΔV Ei 1  2 2 2 exp p þ py þ pz dpx dpy dpz 3 exp 2me kT x n0 kT h -1

(64)

The integral in Eq. 64, over px, gives rise to;


þ1 ð

exp -1

 1 2 px dpx ¼ ð2πme kTÞ1=2 2me kT

(65)

The same results are obtained for integration over py and pz, which when substituted back in Eq. 64, results in; ne ΔVð2πme kTÞ3=2 expðEi =kTÞ n0 h3

(66)

The electrons are derived from ionization processes that produce an equal number of positive ions. Therefore, ΔV will be chosen such that the ion from which an electron has been liberated falls into this volume element, ΔV. ni = 1, where ni is the ion density. Without considering degeneracy, the expression becomes ne ni n0




2πme kT h2

3=2 expðEi =kTÞ

(67)

Generalizing this expression for r-times ionized atoms and considering degeneracy results in

4

Fundamental Concepts in Gaseous Electronics

159


 ne nrþ1, 0 ge grþ1, 0 2πme kT 3=2 ¼ expðErþ1 =kTÞ n r, 0 g r, 0 h2

(68)

where Er+1 represents the energy required for transforming an r-times ionized atom into an (r + l)-times ionized atom. The statistical weight ge of the electrons is 2 (due to 2 possible spin orientations). Equation 68 is known as the Saha equation. Considering the total number of r-times and (r + l)-times ionized atoms (not only those in the ground state), one finds nr ¼

X k

n r, k

and

nrþ1 ¼

X

nrþ1, k

k

resulting in an alternative form of the Saha equation
 ne nrþ1 2Qrþ1 2πme kT 3=2 ¼ expðErþ1 =kTÞ nr Qr h2

(69)

X  where Qr ¼ gr, k exp Er, k =kT . A similar expression holds for Qr+1. Rewriting the Saha equation for the first ionization, one finds
 ne ni 2Qi 2πme kT 3=2 ¼ expðEi =kTÞ n Qo h2

(70)

The term n denotes the density of all neutral particles regardless of their level of excitation. With pe = nekT as the partial pressure of the electron gas, Eqs. 68 and 69 can be expressed by 2 grþ1, 0 ð2πme Þ3=2 ðkTÞ5=2 nrþ1, 0 pe ¼ expðErþ1 =kTÞ n r, 0 g r, 0 h3

(71)

nrþ1 2Qrþ1 ð2πme Þ3=2 ðkTÞ5=2 pe ¼ expðErþ1 =kTÞ nr Qr h3

(72)

or

Equations 71 and 72 apply regardless of other ionized species present in the plasma. Considering the first ionization, i.e.,
 ne ni 2Qi 2πme kT3=2 ¼ expðEi kTÞ n Q0 h2

(73)

letting n ¼ ne þ n ¼ ni þ n and with ξ = fraction of ionized atoms in the gas, one obtains

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M. I. Boulos et al.

ξ ¼

ne ni ¼ since ni ¼ ne n e þ n ni þ n

(74)

and ne ¼ ξn ni ¼ ξn n ¼ ð1  ξÞn Applying Dalton’s law p = (ne + n + ni).kT, the Saha equation becomes ξ2 2Qi ð2πme Þ3=2 ðkTÞ5=2 ¼ expðEi =kTÞ Qo 1  ξ2 h3 p

(75)

where ξ ¼ ξðT, p, Ei Þ and X  gr, s exp Er, s =kT Qr ¼ s

The Saha equations as just described require a correction, because the ionization level in plasma is slightly lowered by the effects of electric or magnetic fields. Here we will consider only electric field effects. The charged particles in the plasma establish electric micro fields, which broaden the energy levels, particularly the upper levels, thus removing the degeneracy. Therefore, the upper levels overlap so that the ionization appears to be lowered by δΕi, as represented in Fig. 7. The summation for the partition function is also affected by the lowering of the ionization energy, because only discrete levels are included in this summation. For moderate temperature levels, it is usually sufficient to consider the first two or three terms in the sum Qr ¼

X

 gr, k exp Er, k =kT

(76)

k

For hydrogen and elements similar to hydrogen, Unso¨ld (1927) found that the lowering of the ionization potential could be expressed by Fig. 7 Schematic representation of the lowering of δEi

E Ei

0

δEi

Fundamental Concepts in Gaseous Electronics

Fig. 8 Degree of ionization as a function of temperature for selected gases and vapors

161

1.00 Degree of ionization, ξ (–)

4

P=100kPa 0.75 Cu

0.50

H

Ar

12

16

He

0.25

0.00

0

4

8

20

24

28

Temperature, T (103K)

δEi ¼ 7  109 n1=3 e

(77)

where δΕi has units of electron volts for n in [m3]. As an example let’s consider hydrogen under the following conditions: T ¼ 20, 000 K ne ¼ 2  1023 m-3 p ¼ 1 atm The lowering of the ionization potential in this case is δΕi ¼ 0:4 eV. The degree of ionization as a function of temperature is shown in Fig. 8 for a number of selected gases and vapors. Equation 69 can be rewritten with the reduced ionization energy E rþ1 as
  nrþ1 2Qrþ1 2πme 3=2 pe ¼ ðkTÞ5=2 exp E rþ1 =kT 3 nr Qr h

(78)

Substituting the appropriate values for the constants yields the working form of the Saha equation:

 nrþ1 5 Qrþ1

5040 þ log T þ log10 p ¼ Erþ1  0:48 log10 T 2 nr e Qr

(79)

for E rþ1 in eV, pe in J/cm3, and n in cm3.

5.3

Complete Thermal Equilibrium (CTE)

It is useful to consider first plasma in a state of complete thermodynamic equilibrium (CTE) even though this state cannot be realized under laboratory conditions. For the sake of simplicity, it will be assumed that the plasma is generated from a monatomic gas or a mixture of monatomic gases.

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CTE prevails in a uniform, homogeneous plasma volume if kinetic and chemical equilibriums as well as every conceivable plasma property are unambiguous functions of the temperature. The temperature, in turn, has to be the same for all plasma constituents and their possible reactions. More specifically, the following conditions must be met: (a) The velocity distribution functions for particles of every species r that exists in the plasma, including the electrons, must follow a Maxwell-Boltzmann distribution
 4v2r mr v2r f ðvr Þ ¼ pffiffiffi exp  kT πð2kT=mr Þ3=2

(80)

where vr is the velocity of particles of species r, mr is their mass, and T is their temperature, which is the same for every species r and, in particular, is identical to the plasma temperature. (b) The population density of the excited states of every species r must follow a Boltzmann distribution n r, k ¼ n r

 g r, k exp Er, k =kT Qr

(81)

where nr is the total number density of ions of species r, Qr is their partition function, Er,k is the energy of the kth quantum state, and gr,k is the statistical weight of this state. The excitation temperature T, which appears explicitly in the exponential term and implicitly in the partition function Qr, is identical to the plasma temperature. (c) The particle densities (for neutrals, electrons, and ions) are described by the Saha equations, which can be considered a mass action law: nrþ1 ne 2Qrþ1 ð2πme kTÞ3=2 ¼ expðErþ1 =kTÞ nr Qr h3

(82)

where Er+1 represents the energy required to produce an (r + l)-times ionized atom from an r-times ionized atom (ionization energy). The ionization temperature T in this equation is identical to the plasma temperature. Lowering of the ionization potential has been disregarded in the Saha equation. (d) The electromagnetic radiation field is that of blackbody radiation of the intensity Bov as described by Planck’s function
Boυ

¼

2h3ν c2




1 ehν=kT  1

 (83)

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The symbol ν stands for the frequency, h represents Planck’s constant, and c is the velocity of light. The temperature of this blackbody radiation is again identical to the plasma temperature. The plasma that follows the ideal model described by Eqs. 80, 81, 82, and 83 would have to dwell in a hypothetical cavity whose walls were kept at the plasma temperature, or else the plasma volume would have to be so large that the central part of this volume, in which CTE prevails, would not sense the plasma boundaries. In this way the plasma would be penetrated by blackbody radiation of its own temperature. Actual plasma, of course, deviates from these ideal conditions. The observed plasma radiation, for example, is much less than the blackbody radiation, because most plasmas are optically thin over a wide wavelength range. Therefore, the radiation temperature of a gaseous radiator deviates appreciably from the kinetic temperature of the plasma constituents or the already mentioned excitation and ionization temperatures. In addition to radiation losses, plasmas suffer irreversible energy losses by conduction, convection, and diffusion, which also disturb the thermodynamic equilibrium. Thus, laboratory plasmas as well as some of the natural plasmas cannot be in a CTE state. The following sections will discuss deviations from CTE and the associated concept of LTE.

5.4

The Concept of Local Thermodynamic Equilibrium (LTE)

The following discussion will be limited to optically thin plasmas, a situation that is frequently met in laboratory scale arc or RF inductively coupled plasmas. In contrast to CTE situation, LTE in optically thin plasmas does not require a radiation field that corresponds to the blackbody radiation intensity of the respective LTE temperature. It does require, however, that collision processes (not radiative processes) govern transitions and reactions in the plasma and that there be a microreversibility among the collision processes. In other words, a detailed equilibrium between each collision process and its reverse process is necessary. Steady-state solutions of the respective collision rate equations will then yield the same energy distribution as that of a system in complete thermal equilibrium, with the exception of the rarefied radiation field. LTE further requires that local gradients of the plasma properties (temperature, density, heat conductivity, etc.) be sufficiently small that a given particle that diffuses from one location to another in the plasma finds sufficient time to equilibrate, i.e., the diffusion time should be of the same order of magnitude or larger than the equilibration time. From the equilibration time and the particle velocities, an equilibration length that is smaller in regions of small plasma property gradients (e.g., in the center of an electric arc) can be derived. Therefore, with regard to spatial variations, LTE is more probable in such regions. Heavy particle diffusion and resonance radiation from the center of a nonuniform plasma source help to reduce the effective equilibrium distance on the outskirts of the source.

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The following sections will systematically discuss the important assumptions for LTE, based on plasmas that approach LTE.

5.4.1 Kinetic Equilibrium It can be safely assumed that each species (electron gas, ion gas, or neutral gas) in a dense, collision-dominated, high-temperature plasma will assume a Maxwellian distribution (excluding regions close to walls and electrodes). However, the temperatures defined by these Maxwellian distributions may be different from species to species. Such a situation, which leads to a two-temperature description, will be discussed in the following paragraphs. The electric energy fed into an arc, for example, is dissipated in the following way: the electrons, because of their high mobility, pick up energy from the electric field and partially transfer it to the heavy plasma constituents through collisions. Because of this continuous energy flux from the electrons to the heavy particles, there must be a “temperature gradient” between these two species, that is, Te > Th, where Te is the electron temperature and Th the temperature of the heavy species, assuming that ion and neutral gas temperatures are the same. In the two-fluid plasma model defined in this manner, two distinct temperatures Te and Th may exist. The degree to which Te and Th deviate from each other will depend on the thermal coupling between the two species. The difference between these two temperatures can be derived from an energy balance, assuming that the kinetic energy exchanges by elastic collisions are equal to the energy gained by electrons from the electric field E, neglecting inelastic collisions: ð3=2ÞkðTe  Th Þð2me =mh Þ ¼ eEved τe

(84)

The left-hand side represents the fraction of energy transferred from one electron of mass me to a heavy particle of mass mh. On the right-hand side, ved is the drift velocity of the electrons, defined as !

!

ved ¼ μe  E

(85)

and τe is the mean time between two collisions of electrons with heavy species. Using Eq. 22 and writing that τe ¼

‘e ve

(86)

where ve is the mean thermal velocity of the electrons, it follows that Te  Th πmh ð‘e eEÞ2 ¼ Te 24me ðkTe Þ2

(87)

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165

Since the term (π mh/24 me) is already 243 for hydrogen, the amount of (directed) energy (‘e. e. E) that the electrons pick up along one mean free path length has to be very small compared with the average thermal energy (k.Te) of the electrons. Low field strengths, high pressures, ‘e  1=p, and high temperature levels are favorable for kinetic equilibrium among the plasma constituents. The field strength and pressure requirements are usually summarized in the parameter E/p. In glow discharges, which are characterized by high electron temperatures and low temperatures for the heavy species, E/p assumes values on the order of 105 V/m.Pa, whereas in typical thermal arcs, E/p is on the order of 102 V/m.Pa. At low pressures, for example, appreciable deviations from kinetic equilibrium may occur. The semi-schematic diagram in Fig. 21 in ▶ Chap. 1, “The Plasma State,” shows how electron and gas temperatures separate in an electric arc with decreasing pressure. For an atmospheric argon high-intensity arc with E = 1300 V/m, ‘e ¼ 3  106 m, mAr =me = 7  104, and Te = 30  103K, the deviation between Te and Th is on the order of 1% (Finkelnburg and Maecker 1956).

5.4.2 Excitation Equilibrium In order to establish the criteria for excitation equilibrium, every conceivable process that may lead to excitation or de-excitation has to be taken into account. For simplicity, only the most prominent mechanisms (collisional and radiative excitation and de-excitation) will be considered: Excitation 1. Electron collisions 2. Photo-absorptions

De-excitation 1. Collisions of the second kind 2. Photoemissions

In CTE, micro-reversibilities would have to exist for all processes, i.e., in the above scheme, excitation by electron collisions would have to be balanced by the reverse process, namely, collisions of the second kind, and excitation by the photoabsorption process would have to be balanced by photoemission processes, which include spontaneous and induced emissions. Furthermore, the populations of excited states would have to follow a Boltzmann distribution. Micro-reversibility for the radiative processes holds only if the radiation field in the plasma reaches the intensity B0υ of blackbody radiation. However, actual plasmas are frequently optically thin over most of the spectral range, so that the situation for excitation equilibrium seems to be hopeless. Fortunately, if collisional processes dominate, photo-absorption and photoemission processes do not have to balance: only the sums on the left-hand side and the right-hand side of the scheme above have to be equal. Since the contribution of the photon processes to the number of excited atoms is almost negligible when collisional processes dominate, the excitation process is still close to LTE.

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5.4.3 Ionization Equilibrium For ionization equilibrium, again only the most prominent mechanisms leading to ionization and recombination will be considered Ionization 1. Electron collisions 2. Photo-absorptions

Recombination 1. Three-body recombinations 2. Photo-recombinations

In a perfect thermodynamic equilibrium state (CTE) with cavity radiation, a micro-reversibility would exist among the collisional and radiative processes, and the particle densities would be described by the Saha equation. Without cavity radiation, the number of photo-ionizations is almost negligible, requiring a total balance of all processes involved (instead of micro-reversibility). Photo-recombinations, especially at lower electron densities, are not negligible. The frequency of the three remaining elementary processes is a function of the electron density only, leading (for a certain electron density) to the same order of magnitude frequency for these elementary processes. The result is an appreciable deviation between actual and predicted (from Eq. 82) electron densities. Only for sufficiently large electron densities does the Saha equation predict correct values. For smaller electron densities, the corona formula, which considers ionization by electron impact and photo-recombination only, must be used. The particle concentrations in low-intensity arcs at atmospheric pressure, for example, must be calculated using this formula. Significant deviations of the electron density predicted by the Saha equation from the true electron density may also occur in the fringes of highintensity arcs, RF discharges, and plasma jets. In summary, LTE exists in a steady-state, optically thin plasma when the following conditions are simultaneously fulfilled: (a) The different species that form the plasma have a Maxwellian distribution. (b) E/p is small enough and the temperature is sufficiently high that Te = Th. (c) Collisions are the dominating mechanism for excitation (Boltzmann distribution) and ionization (Saha equilibrium). (d) Spatial variations of the plasma properties are sufficiently small.

5.5

Deviations from LTE

In addition to the two extreme cases, namely, LTE (based on Saha ionization equilibrium) and corona equilibrium, conditions between these two limiting cases are also of interest. In this range three-body recombinations as well as radiative recombination and de-excitation are significant. Over the past 40 years, a large number of investigations have been reported on the subject of radiative-collisional processes, LTE, and deviations from LTE. The results of these studies up to 1966 are summarized in three books on plasma diagnostics (Huddlestone and Leonard 1965; Lochte-Holtgreven 1968; Hutchinson 2002). Drawin (1970) presents a comprehensive review of the validity conditions for LTE, including a discussion of

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complete local thermodynamic equilibrium (CLTE) and partial local thermodynamic equilibrium (PLTE). This distinction is associated with the population of excited levels that may deviate from ideal Boltzmann distributions. In the case of optically thin plasmas, the lower-lying excited energy levels tend to be underpopulated with respect to the ground state. This situation is referred to as PLTE provided that all the other conditions for LTE are met. The electron densities, ne, required for CLTE in optically thin plasma are substantially higher than those needed for the less stringent requirement of PLTE. Griem (1964) established the following criterion for the existence of CLTE in optically thin homogeneous plasma
ne  9  10

23

E21 EH þ

3


kT EH þ

 (88)

where E21 represents the energy gap between the ground state and the first excited level, EHþ ¼ 13:58 eV is the ionization energy of the hydrogen atom, and T is the plasma temperature. This criterion shows the sensitivity of the required electron density for CLTE to the energy of the most critical, first excited state. It is obvious that deviations from CLTE or even from PLTE will occur in regions of low electron densities as, for example, in plasma regions adjacent to walls or in arc fringes and in all types of low-density plasmas of laboratory dimensions. For many years the existence of CLTE in atmospheric pressure, high-current arcs has not been questioned. Only recently have deviations from CLTE been found in such arcs. It has been shown that a pressure of approximately 300 kPa is necessary to reach a state of CLTE in the central portion of a free-burning argon arc at currents of 300–400 A. These conditions correspond to an electron density of approximately 1024 m3 in the center. Deviations from CLTE still persist in the outer regions of such arcs, where the electron density drops substantially below 10 24 m3. Numerous analytical as well as experimental studies over the past years demonstrate that LTE (CLTE or PLTE) in high-intensity arcs is the exception rather than the rule. Several studies have shown that besides the underpopulation of lowerlying energy levels, deviations from LTE can frequently be attributed to strong gradients in the plasma and the associated diffusion effects. More details on LTE and deviations from LTE are discussed in reference books by Huddlestone and Leonard (1965), Griem (1964), Lochte-Holtgreven (1995), Mitchner and Kruger (1973), and Hutchinson (2002).

6

Rigorous Definition of the Plasma State

Plasma consisting of a mixture of electrons, ions, and neutral particles in the gaseous state is overall electrically neutral, as pointed out in the first chapter. This electrical neutrality of plasma, however, applies only for sufficiently large

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plasma volumes, i.e., V > λ3D, where λD, the Debye length, is a characteristic length in a plasma; it will be discussed in the following section. Although a plasma can be treated in a first approximation as electrically neutral, deviations from neutrality must be considered in a second approximation; such deviations, however, are restricted to distances on the order of a Debye length.

6.1

The Debye Length in a Plasma

In 1923, Debye and H€uckel were the first to develop the concept of electric shielding of a positive ion by negative ions (and vice versa) for strong electrolytes. This concept, however, also applies to gaseous plasmas. Because of Coulomb forces acting between charged particles, a positive ion is, on the average, surrounded by more than one electron; this surrounding electron cloud provides an effective shielding of the positive ion charge. The accumulation of negative charges in the vicinity of a positive ion represents a net negative space charge, i.e., a deviation from charge neutrality occurs over the dimension of the electron cloud, which is known as a Debye sphere. These Debye spheres are dynamic in nature and overlap each other. In the following paragraph, an expression will be derived for the dimension of a Debye sphere (or Debye length) in a uniform plasma of electron density ne = ni (singly ionized species only). In this derivation, it will be assumed that the equilibrium ion density distribution ni ¼ ni, 0 is not affected by the electron clouds forming around positive ions; the electrons establishing a dynamic equilibrium in the cloud retain their Maxwell-Boltzmann distribution, and no recombination with the positive ions occurs (the potential energy eV > eV. Based on this assumption, the space charge density can be expressed by

ρe‘ ¼ eneo


 eV eV eV  1þ 1 ¼  2neo kT kT kT

(119)

Introducing Eq. 119 into Eq. 118 results in d2 V 1  V ¼ 0 dy2 λ2D

(120)

 1=2 with λD ¼ εo kT=e2 ne, o The solution of Eq. 120 with the boundary condition V = V0 for y = 0 is given by  pffiffiffi  V ¼ V0 exp  2 y=λD (121) where V0 is either the floating potential that the wall assumes or the negative biasing potential of the wall. Figure 12 shows a sketch of this result.

4

Fundamental Concepts in Gaseous Electronics

– Sheath Electric potential, V

Fig. 12 Evolution of the potential in the sheath

177

λD/

Plasma

2

y

Vo

The thickness of the sheath is on the order of a Debye length. Because of the exponential potential variation in the sheath, the definition of the sheath edge is somewhat arbitrary. Specification of the sheath edge varies in the literature from 1 to 10 Debye lengths. Since the space charge always assumes the opposite sign of the wall potential, this potential is effectively “screened off” from the plasma, i.e., any electrical disturbance imposed on the plasma is felt only over the sheath thickness ðλD Þ. This fact is extremely important in the application of electric probes as plasma diagnostic tools.

9

Nomenclature

Units are indicated in parentheses; whenever no units are given, the parameter is dimensionless.

9.1 !

B B0υ c Da De Di Dk e efast eslow Ex Er+1

Latin Alphabet Magnetic induction (V.s/m2) Intensity of blackbody radiation (J/ster.m2) Light velocity (c = 3xl08 m/s) Ambipolar diffusion coefficient (m2/s) Electron diffusion coefficient (m2/s) Ion diffusion coefficient (m2/s) Diffusion of plasma charge carriers (m2/s) Electron charge (e = 1.6  l019 A.s) Fast electrons Slow electrons Electric field responsible for ambipolar diffusion (V/m) Energy required for transforming an r-times ionized atom into an (r + 1)times ionized atom

178 !

E ! Er E rþ1 Eion EH þ Er,k eV f(v) ! F ! Fr g gk gr,k h H I! I !e Ik j je j! i Jk k! J ‘e ‘i L me M ne ni nr,k Nk Nok p pe px py pz q Qr rL rmin

M. I. Boulos et al.

Electric field (V/m) Electric field in radial direction (V/m) Reduced ionization energy E rþ1 = Erþ1  δErþ1 ðeVÞ Ionization energy (eV) Hydrogen atom ionization energy (EH+=13.6 eV) Energy of chemical species r in the excited state k (cm1) Electron volt (1 eV = 1.6*1019 J) Maxwellian distribution function Force vector (N) Force in the radial direction (N) Number of compartments (h3) in the phase space volume dx.dy.dz.dpx.dpy. dpz Statistical weight of excited state k Statistical weight of chemical species r in excited state k Planck’s constant (h = 6.626  1034 J.s) Elementary volume in phase space: dx.dy.dz.dpx.dpy.dpz Arc current (A) Electrons flux (s1m2) Species k flux (s1m2) Electric current density (A/m2) Electron current density (A/m2) Ion current density (A/m2) Flux of charged particles (m2s1) Boltzmann constant (k = 1.38  1023 J/K) Current density vector (Am2) Electron mean free path (m) Ion mean free path (m) length (m) Electron mass (me = 9.11  1031 kg) Ion mass (kg) electrons density (m2) electrons density (m2) Particle number density of chemical species r in excited state k (m3) Number of particles in excited state k equilibrium distribution of phase points in phase space (MaxwellBoltzmann distribution) Total pressure (Pa) Partial pressure of an electron gas (Pa) Component of the momentum (px = m.vx) (kg.m/s) Component of the momentum (py = m.vy) (kg.m/s) Component of the momentum (pz = m.vz) (kg.m/s) Electrical charge (A.s) Partition function of chemical species r Larmor radius for the circular motion (rL = m.v/q.B (m)) Landau parameter (m)

4

Fundamental Concepts in Gaseous Electronics

179

R ! se

Arc radius (m) Distance that an electron travels during time interval τe (m)

Sk t T Te Th Tk ! ue ue ! u eo

Mean distance travelled by the electron (m) Source term (see Eq. 35) (m3.s1) Time (s) Temperature (K) Electron temperature (K) Heavy species temperature (K) Maximum kinetic energy acquired by an electron between two collisions (J) Electron drift velocity (m/s) Mean electron drift velocity (m/s) Initial velocity of the electron when the electric field is applied (m/s)

! se

! ue ! ue ! ui ! ui !

v

! ve

ve vi !e vd ! vD vk v⊥ V W x X X+ y z

Mean drift electron velocity (m/s) Mean drift ion velocity (m/s) Ion drift velocity (m/s) Mean ion drift velocity (m/s) Particle velocity (m/s) Electron velocity (m/s) Mean electron velocity (m/s) Mean ion velocity (m/s) Mean electron drift velocity (m/s) Electron drift velocity in magnetic and electric fields (m/s) Charged particle velocity component parallel to the magnetic field (m/s) Charged particle velocity component perpendicular to the magnetic field (m/s) Electrical potential (V) Thermodynamic probability Position coordinate (m) Chemical species Singly ionized chemical species Position coordinate (m) Position coordinate (m)

9.2

Greek Alphabet

β δEi δN Δp(r) ε0 γe λD Λ

Lagrange multiplier [β = 1/(k.T)] Ionization potential lowering (eV) Particle number variation Pressure variation along the plasma radius (Pa)  Dielectric constant ε0 ¼ 8:86  1012 A:s=V:m Gvosdover parameter Debye length (m) Diffusion length (m)

180

μe μi μ0 υi ξ ρ ρel σe τe ωe ∇

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Electron mobility (m2/V.s) Ion mobility (m2/V.s)  Magnetic permeability constant μ0 ¼ 1:26  106 Hy=m Net ionization coefficient Fraction of ionized atoms in the gas Specific mass (kg/m3) Electric space charge (C/m3) Electrical conductivity (ohm1.m1) Mean free flight time for electrons (s) Larmor frequency (s1) Laplace operator

References Allis WP (1956) Motion of ions and electrons. In: Encyclopedia of physics, vol 21. Springer, Berlin Cambel AB (1963) Plasma physics and magnetofluid mechanics. McGraw-Hill, New York Capitelli M, Colonna G, D’Angola A (2012) Fundamental aspects of plasma chemical physics thermodynamics, vol 66, Springer series on atomic, optical, and plasma physics. Springer, New York Delalondre C (1990) Mode´lisation ae´rothermodynamique d’arcs e´lectroniques a` forte intensite´ avec prise en compte du de´se´quilibre thermodynamique local et du transfert thermique a` la cathode. Ph.D. thesis, University of Rouen Drawin HW (1970) Spectroscopic measurement of high temperatures (a review). High Temp High Pressures 2:359 Finkelnburg W, Maecker H (1956) Elektrische Boˆgen und thermisches Plasma. In: Flu¨gge S (ed) Encyclopedia of physics, vol 23. Springer, Berlin Griem HR (1964) Plasma spectroscopy. McGraw-Hill, New York Gupta MC (2007) Statistical thermodynamics. Amazon, p 528 Gvosdover SD (1937) Phys Z Sov 12:164 Huddlestone RH, Leonard SL (eds) (1965) Plasma diagnostic techniques. Academic, New York, p 627 Hutchinson IH (2002) Principles of plasma diagnostics, 2nd edn. University of Cambridge Press, Cambridge Laurandeau NM (2005) Statistical thermodynamics: fundamentals and applications. Cambridge University Press, New York Lee JF, Sears FW, Turcotte DL (1973) Statistical thermodynamics, 2nd edn. Addison-Wesley, Reading Lochte-Holtgreven W (ed) (1995) Plasma diagnostics. AIP Press, New York, p 928 Loeb LB (1961) Basic processes of gaseous electronics. University of California Press, Berkeley/ Los Angeles Massey HSW, Burhop EHS, Gilbody HB (1969) Electronic and ionic impact phenomena, vol 4, 2nd edn. Oxford University Press, New York Mitchner M, Kruger CH Jr (1973) Partially ionized gases. Wiley, New York Mu¨ller I, Weiss W (2005) Entropy and energy: a universal competition. Springer, Berlin Reif F (2009) Fundamentals of statistical and thermal physics. Waveland Press, Long Grove, p 651 Schmidt G (1979) Physics of high temperature plasmas, 2nd edn. Academic, New York Uman MA (1964) Introduction to plasma physics. McGraw-Hill, New York

5

The Plasma Equations Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Entropy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Onsager’s Reciprocity Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Heat of Transition and Energy Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Diffusion and Energy Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Example of Mass and Energy Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Transport Equations for a Fully Ionized Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Plasma Exposed to an Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Plasma Exposed to Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Determination of Transport Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Latin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

182 182 183 183 183 184 185 188 193 196 198 200 202 202 206 207 207 208 209

M. I. Boulos (*) Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada e-mail: [email protected] P. L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France e-mail: [email protected] E. Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] © Springer Nature Switzerland AG 2023 M. I. Boulos et al. (eds.), Handbook of Thermal Plasmas, https://doi.org/10.1007/978-3-030-84936-8_5

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1

Introduction

In contrast to the preceding chapters, which considered the microscopic behavior of particles, this chapter will deal with the motion of the various plasma components (electrons, ions, neutrals) on a macroscopic scale under the influence of external as well as internal forces. Since this treatment will be restricted to thermal plasmas, which can be conveniently generated by electric arcs or high-power RF discharges, temperature is the most important parameter. Thus, the derivation of the plasma equations (for current, momentum, mass, and heat flow) will be based on thermodynamics and irreversible thermodynamics. This particular approach provides a straightforward survey of the various effects that govern the behavior of such plasmas, although it leaves the absolute values of the coefficients (e.g., the friction coefficient of the diffusion coefficient) open. These values will be derived from kinetic theory. The application of the equations derived in this chapter is restricted to a two-component mixture which implies fully ionized plasmas consisting of electrons and positive ions. This may appear rather restrictive, but these equations can still (with a reasonable approximation) be applied to highly ionized plasmas which are rather common in the field of thermal plasma technology. On the other hand, these equations can be extended to three components (electrons, positive ions and neutral particles) as shown by W. Finkelnburg and H. Maecker (1950).

2

Definitions

Plasma may consist of a mixture of k components. Chemical reactions in the plasma, particle density and temperature gradients, and macroscopic velocities for the plasma components are all permitted. The velocity of the center of mass, vg , follows from ! ρvg ¼

X

!

ρk v k

ρ¼

with

k

X

ρk

(1)

k

where ρk and ! vk represent the mass density and velocity of an individual component, respectively. The mass concentration is given by ck ¼

ρk ρ

where

P

ck ¼ 1

(2)

The mass flux of component k relative to the center of mass is ! Ik

and

X k

!

  ! ! ¼ ρk  v k  v g

I k ¼ 0 (from Eqs. 1 and 3)

(3)

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183

3

Conservation Equations

3.1

Conservation of Mass

There are four alternative forms of the conservation of mass equation. Two refer to a single component and the other two refer to the entire mixture:   @ρk ! þ div ρk v k ¼ Γk @t

(4)

! dck þ div I k ¼ Γk dt   @ρ ! þ div ρ v g ¼ 0 @t   dρ ! þ ρ  div v g ¼ 0 dt

ρ

(5) (6) (7)

Γk represents the generation rate of particles of component k (units: kg/m3. s). Equation 6 follows from Eq. 4 by summation over all components, considering that X Γ ¼ 0. Eq. 5 reduces to Eq.4 if Eqs. 2, 3, and 6 are applied in addition to the k k substantive derivative  d @ ! ¼ þ v g  grad dt @t Finally, Eq. 7 can be verified by introducing the substantive derivative.

3.2

Conservation of Momentum

Only external forces are able to move the center of gravity. Therefore, the momentum equations for the total mixture contain only external forces, for example: Force Gravity Electric forces Magnetic forces Pressure forces

!

mk g !

Force/unit mass !

g

!

ek E  ! ! ek v k x B

ek E =mk  ! ! ek =mk v k x B

mk =ρk grad pk

1=ρk grad pk

In these examples, mk represents the mass, ek the charge, and pk the pressure of ! ! an individual component; g is the acceleration due to gravity; E is the electric field ! strength; and B is the magnetic field strength.

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The overall momentum equation, which is identical to the basic equation of hydrodynamics, assumes the form !

ρ

X ! dv g ¼ grad p þ ρk  F k dt k

(8)

The corresponding equations for the individual components cannot be derived from Eq. 8 because they contain, in addition to external forces, internal forces due, for example, to friction among the various components.

3.3

Conservation of Energy

It is assumed that the total energy of a given system is composed of kinetic and thermal energy. By multiplying Eq. 8 by ! vg , the kinetic energy is obtained: X ! 1 d 2 ! ! ρ vg ¼ v g  grad p þ v g  ρk F k 2 dt k

(9)

From the first law of thermodynamics, it follows that dq ¼ dug þ pdv0

(10)

where q and ug represent heat and internal energy per unit mass, respectively, and v0 ¼ 1=ρ is the specific volume. Equation 10 can be written as ρ

dug dq ! ¼ρ þ p div v g dt dt

ρ

dv0 1 dρ ! ¼ div v g ¼ ρ dt dt

(11)

because

According to Eq. 7, the right-hand side of Eq. 11 represents the change in the internal energy plus the change in compression work. In general, such changes are caused by net heat fluxes and internal heat generation. Therefore, Eq. 11 can be written in the form of an energy balance as ρ

X! ! dug ! ! þ p div v g ¼ div q þ Fk I k dt k !

(12)

The heat flow vector q contains heat conduction without mass flow as well as heat carried by the mass flows. The potential energy converted into heat by friction is

5

The Plasma Equations

185

represented by the last term in Eq. 12. By adding Eqs. 9 and 12, the total energy equation is obtained: 2 d vg þ ug ρ dt 2

!

  X! ! ! ! F k ρk v k ¼ div p v g þ q þ

(13)

k

This equation has the typical form of a balance equation: ðChange of total energyÞ ¼ ðdivergence of energy flowÞ þ ðenergy source termÞ

3.4

Entropy Balance

A similar balance equation can also be established for the entropy. This derivation will be useful for obtaining information on mass and heat fluxes. A combination of the first and second law of thermodynamics yields dsg X dck dq dug dv0 ¼ þp ¼T þ μk dt dt dt dt dt k

(14)

where sg is the entropy per unit per mass and μk is the chemical potential of component k. Combining Eqs. 11, 12, and 14 results in

T

X dck X! ! dsg ! þρ ¼ div q þ μk Fk I k dt dt k k

(15)

Using Eq. 5, the second term in Eq. 15 can be transformed into ρ

X k

μk

X X ! dck ¼ div I k þ μk Γk dt k k

(16)

or ρ

! dsg 1 X 1X 1 1X ! ! ! μk div I k  μk Γk  div q þ Fk I k þ T T T dt T

Equation 16 can be rearranged to form a typical balance equation !

ρ

! ! Σμk I k

dsg q ¼ div dt T

! grad

þ

 q

T

T

! ! μ  þ Σ I k F k  Tgrad k  Σμk Γk T T

(17) of the form

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ðChange of entropyÞ ¼ ðdivergence of entropy flowÞ þ ðentropy source termÞ The various terms in the previous equations require further explanation. For this purpose, some simple examples will be examined. For an interpretation of the terms in the energy and entropy balance equation, the term ! ! I k Fk

  ! ! ! ¼ F k ρk v k  v g

will be first considered. !

!

!

F k ρk v k ¼ work done by force F k ðper unit volume and unit timeÞ !

!

F k ρk v g ¼ work necessary for accelerating the center of mass ðper unit volume and unit timeÞ

The difference of these two terms   ! ! ! ! ! F k ρk v k  v g ¼ F k I k is the work that is transformed into heat (per unit volume and unit time). For further clarification, an example will be considered: !

e ! E ¼ electric field force ðper unit massÞ me ! me ! j ¼ mass flux associated with the flow of electrons Ik¼ e ðper unit area and unit timeÞ

Fk ¼ 

! !

!

!

F k I k ¼ j e E ¼ dissipated Joule heat ðper unit volume and unit timeÞ The term T grad ðμk =TÞ in Eq. 17 must be also a mechanical force that when ! multiplied by the flow friction work that is transformed into  vector Ik describes  ! heat. The next term q  grad T=T does not lend itself to a separation into mass flow and force. Nevertheless, the term ðgrad  T=TÞ is, ina generalized sense, also ! considered a driving “force.” The product q  grad T=T therefore also represents a heat generation term. All the terms that appear in the numerator of the last term of Eq. 17 are called “energy dissipation” terms. When divided by T, these terms represent the entropy production (per unit volume and unit time), which must be 0 according to the second law of thermodynamics. The term μkΓk contains the chemical affinity of component k. At chemical equilibrium, this term vanishes. But this term will exist if deviations from chemical

5

The Plasma Equations

Fig. 1 Deviation from chemical equilibrium in a plasma-wall boundary layer

187

Wall

Plasma ne (frozen)

T

Tw

ne plasma

T plasma

ne (equil.) 0

equilibrium occur (e.g., if the chemical composition of plasma does not follow its temperature, as in “frozen flow”). This type of deviation from chemical equilibrium may occur within the thermal boundary layer separating the plasma from an adjacent wall, as sketched in Fig. 1. In this situation, electron diffusion due to the steep gradients of density and/or temperature is so fast that relaxation processes (recombination) cannot follow – the chemical processes appear to be “frozen.” Since μk Γk ! 0 as chemical equilibrium is approached, this term is useful for describing relaxation phenomena. ! As stated earlier, the heat flow vector, q , comprises pure heat conduction (without mass flow) and the heat flow carried by the mass flow. This heat flow is X ! reduced by the term μk I k which represents the part of the energy flow that gives rise to chemical reactions. The chemical potential μk is frequently interpreted as a measure of the driving force tending to cause a chemical reaction to take place. In the last part of this section, the momentum equations for individual components will be considered. These equations will be written for a two-component mixture, assuming that there are two internal forces (friction and thermal diffusion); this leads to two additional terms in each of the two momentum equations:   ! dv 1 grad T ! ! þ n1 n2 ε12 v 1  v 2 ¼ F 1 ρ1  grad p1  ρ1 y1 T dt

(18)

  ! dv 2 grad T ! ! ρ2 þ n1 n2 ε21 v 2  v 1 ¼ F 2 ρ2  grad p2  ρ2 y2 T dt

(19)

!

ρ1

!

In these equations, friction is assumed to be proportional to the relative velocity between the two components. The particle densities for components 1 and 2 are designated as n1 and n2, respectively, and both ε12 and ε21 represent friction coefficients. The last terms on the right-hand sides of Eqs. 18 and 19 are the forces responsible for thermal diffusion, with the coefficients y1 and y2 determining the magnitude of these forces.

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The sum of Eqs. 18 and 19 should be identical to Eq. 8 if the last term of this equation is written for two components only. Since !

ρ1

!

!

dv g dv 1 dv 2 þ ρ2 ¼ρ dt dt dt

according to Eq. 1, this identity is indeed fulfilled if ε12 ¼ ε21

and ρ1 y1 ¼ ρ2 y2

Friction forces act only between the two components, and therefore they are not able to move the center of mass. Thus, ε12 ¼ ε21 must be valid. A similar argument holds for the forces responsible for thermal diffusion, leading to ρ1 y1 ¼ ρ2 y2 . !

!

I1¼I2¼

 ρ1 ρ2  ! ! v1  v2 ρ

(20)

Subtracting Eq. 19 from Eq. 18 gives ! v1



! v2

" ! ! ! ! ρ1 ρ2 1 dv 1 dv 2 1 þ  grad p1 ¼ F1  F2  ρ1 ρ n1 n2 ε12 dt dt  1 grad T þ grad p2  ðy1  y2 Þ ρ2 T

(21)

Finally, introducing Eq. 21 into Eq. 20 results in ! I1

" ! ! ! ! ρ1 ρ2 1 dv 1 dv 2 1 þ  grad p1 ¼ F1  F2  ρ1 ρ n1 n2 ε12 dt dt  1 grad T þ grad p2  ðy1  y2 Þ ρ2 T

(22)

In the following an expression will be derived for individual mass flows relative to the center of mass.

4

Onsager’s Reciprocity Relations

As shown in the previous section, the entropy production, Es can be expressed in the following way: Es ¼

Σ “fluxes” x “forces” T

5

The Plasma Equations

189

The relation between the fluxes and the “driving forces” that give rise to these fluxes can, in a first approximation, be assumed to be linear. This is a common approach that is applied to electrical and thermal conduction, for example, as well as to diffusion: Ohm’s law :

!

j ¼ σe grad V

!

Fourier’s law : q ¼ κ grad T !

Fick’s law : S ¼ D grad n The justification for this approach will be discussed later. In the relations given above, the gradients of the potential V, the temperature T, and the number density n are regarded as “driving forces.” The coefficients σe, κ, and D represent electrical conductivity, thermal conductivity, and the diffusion coefficient, respectively. The same approach can be applied to more complicated situations in which a flux depends on several driving forces. In the following, a two-component mixture will be considered in which each flux is assumed to depend on three different types of driving forces. This leads to the following set of phenomenological equations: !

!

!

!

!

!

!

!

I 1 ¼ L11 X 1 þ L12 X 2 þ L1u X u I 2 ¼ L21 X 1 þ L22 X 2 þ L2u X u !

!

!

!

Q ¼ Lu1 X 1 þ Lu2 X 2 þ Luu X u

(23) (24) (25)

The Lk‘, Lku, and Lu‘ in these equations represent coefficients that will be deter! mined in later paragraphs, and the X ‘ stands for the “driving forces.” These “driving forces” will be identified with the same type of “forces” that appear in the entropy production term of Eq. 17, i.e., ! μ1 ! X 1 ¼ F 1  T grad T ! ! μ X 2 ¼ F 2  T grad 2 T ! 1 X u ¼  grad T T

These linear approximations require, as shown by the gas kinetic calculations of Enskog (1917), that the parameters of state do not vary appreciably over a mean free path length. Choosing the temperature Ti as the parameter of state for particles of type i, this requirement can be expressed by λi  jgrad Ti j  Ti

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M. I. Boulos et al.

where λi represents the mean free path length of particles of type i. Similar relations hold for other parameters of state. From

X

!

Ik¼0

k¼1

it follows that

X

Lk‘ ¼ 0 and

X

k

Lku ¼ 0

(26)

k

because the forces can be arbitrarily chosen. In addition, Onsager’s reciprocity theorems derived from kinetic statistics Onsager (1931a, b) require that the following relations hold: ðLk‘ Þ ðLku Þ

¼ ðL‘k Þ ¼ ðLuk Þ

(27)

i.e., the matrices of the coefficients are symmetric. This relationship, however, is ! ! only valid if the forces Xk and the fluxes Ik are chosen in such a way that their product divided by T represents an entropy production term. Thus, the forces were chosen according to the last term of Eq. 17. From Eqs. 26 and 27, it follows that X

Lk‘ ¼ 0

and

‘

X

Lu‘ ¼ 0

‘

With this result one may write in general ! Ik

¼

n1 X

! !  ! Lk‘ X ‘  X u þ Lku X u

(28)

  Lu‘ X‘  Xu þ Luu Xu

(29)

‘¼1

and !

q¼

n-1 X ‘¼1

If this result is applied to a two-component mixture, Eqs. 28 and 29 reduce to ! I1

! ! !  ! ¼  I 2 ¼ L11 X 1  X 2 þ Llu X u ! !  ! q ¼ Llu X 1  X 2 þ Luu X u

!

Returning to the previously introduced identification of forces results in !

!

!

!

X 1  X 2 ¼ F 1  F 2  T grad

μ1  μ2 T

(30) (31)

5

The Plasma Equations

191

where μk represents the chemical or Gibbs potential, which can be expressed by μk ¼ ug, k þ pv0, k  Tsg, k ¼ hg, k  Tsg, k

(32)

(u, s, v are expressed per unit mass). The specific volume v0,k of component k can be expressed as v0 , k ¼

pk 1 p kT and pv0, k ¼ k ¼ p ρk ρk m k

(33)

where k is the Boltzmann constant. Statistical mechanics provides an expression for the entropy of component k;

sg , k ¼

ug, k k þ lnQk mk T

(34)

where Qk is the partition function of component k. Considering only the translational energy of particles (continuous energy distribution) results in Qk ¼

 en 2πmk kT 3=2 : nk h2

(35)

In this equation, en ¼ 2:718 is the base of natural logarithms, and h is Planck’s constant: 5 ln Qk ¼ 1  ln pk þ ln T þ ln C 2

(36)

In Eq. 36, all constants are summarized in the single term, ln C. Introducing Eqs. 33, 34, and 36 into Eq. 32 results in  μk k 5 ¼ ln pk  ln T  ln C 2 T mk And grad and finally

 μk k 1 5 grad T ¼ grad pk  2 T T m k pk

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M. I. Boulos et al.

T grad

μk grad T 1  grad pk ¼ hg, k T ρk T

(37)

Introducing the mole fraction xk ¼ pk =p results in kT kT kT gradðln pk Þ ¼ ½gradðln pÞ þ gradðln xk Þ ¼ v0, k grad p þ grad xk (38) mk mk m k xk

For the case of a binary mixture, the right-hand side of Eq. 38 becomes v0, 1 grad p þ

kT grad c1 mc1

where x1 ¼ c1 m2 =m, and m ¼ c1 m2 þ c2 m1 ; dx1 =dc1 ¼ m1 m2 =m2 Thus, two alternative forms of Eq. 37 can be given for a two-component mixture: μ1 grad T 1  grad p1 ¼ hg , 1 T ρ1 T

(39)

μ1 grad T kT  v0, 1 grad p  ¼ hg , 1 grad c1 T T mc1

(40)

T grad or T grad

Introducing Eq. 39 into Eqs. 23 and 24 results in ! I1

!

¼I2

 ! ! 1 1 grad T ¼ L11 F 1  F 2  grad p1 þ grad p2 þ L11 ðh1  h2 Þ ρ1 ρ2 T  Llu

grad T T

(41)

or in the other form (using Eq. 40) results in ! I1

!

¼I2

   ! ! kT ¼ L11 F 1  F 2  v0, 1  v0, 2 grad p þ grad c1 mc1 c2 þ L11 ðh1  h2 Þ

and

grad T grad T  Llu T T

(42)

5

The Plasma Equations

193



! F1

! F2

1 1   grad p1 þ grad p2 q ¼ Llu ρ1 ρ2   grad T grad T  Luu þLlu hg, 1  hg, 2 T T

!

 (43)

!

An alternative expression for q can be obtained by replacing the terms in brackets in Eq. 43 by those in Eq. 42. Equations 41 and 42 already show a similarity to Eqs. 21 and 22, respectively.

5

Heat of Transition and Energy Fluxes

Assuming that there is no temperature gradient but that the other forces still exist, ! ! ! there will be a mass flow I1 ¼  I2 6¼ 0 and also a heat flow q 6¼ 0. Since the two mass fluxes are opposite and equal, they must carry different amounts of energy with them. ! Let’s assume that I1 carries the overbalance of energy defined as q*(per unit ! mass of the flow I1 ), which is called the beat of transition. According to this definition, !

!

q I 1 ¼ q , q < 0

(44)

and q ¼

!

q L1u ¼ I1 L11

(45)

Eq. 45 is a consequence of the assumption that grad T = 0. Equation 45 permits elimination of one coefficient so that Eq. 42 can be rewritten as "

! I1

¼

! I2

  ! ! kT ¼ L11 F 1  F 2  v0, 1  v0, 2 grad p  grad c1 mc1 c2 #    grad T  q  hg, 1 þ hg, 2 T

(46)

Or in the alternative form !

!

I1 ¼I2 

  grad T ! ! 1 1 ¼ L11 F 1  F 2  grad p1 þ grad p2  q  hg, 1 þ hg, 2 ρ1 ρ2 T (47)

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M. I. Boulos et al.

Comparing Eq.47 with Eq.22, it is noted that the two equations will be identical provided that  L11 ¼

ρ1 ρ2 1 n1 n2 ε12 ρ

y1 -y2 ¼ q  hg, 1 þ hg, 2

(48) (49)

and !

dv 1 dv2 ¼ dt dt

(50)

Elimination of L1u from Eq. 43 using Eq. 45 and applying Eq. 41 result in an interesting expression for the heat flux:  grad T ! q ¼ q I 1  Luu  L11 q2 T

!

(51)

The first term on the right-hand side represents the energy carried by the mass flow. The second term describes the energy flow due to conduction only and may be aptly termed as ! q I¼0

 grad T ¼ Luu  L11 q2 T

(52)

Assuming that we have a homogeneous mixture of two gases exposed to a temperature gradient (all other “forces” shall vanish), Eq. 43 reduces to ! qT

   grad T ¼  Luu  q L11 hg, 1  hg, 2 T

(53)

!

The heat flux q T is driven solely by the temperature gradient, which, however, also gives rise to a mass flow   grad T I TD ¼ L11 q  hg, 1 þ hg, 2 T

!

(54)

Equation 54 follows from either Eq. 41 or 42 with the previously stated assumption. ! The mass flux I TD is due to thermal diffusion. According to previous arguments, this mass flux carries a heat flux ! q TD

  grad T ! ¼ I TD q ¼ L11 q q  hg, 1 þ hg, 2 T

(55)

5

The Plasma Equations

195

! q TD

!

represents the heat flux due to thermal diffusion. Since the heat flux q T contains the contribution due to thermal diffusion as well as that caused by pure conduction, the following relation must hold ! qT

 grad T ! !  q TD ¼ q I¼0 ¼  Luu  L11 q2 T

(56)

which is identical to Eq. 52. Under the specified conditions, the temperature gradient initially gives rise to diffusional mass fluxes, which, however, build up concentration gradients that in turn lead to ordinary diffusion fluxes opposite to those caused by thermal diffusion. After ! steady-state conditions are established, both ITD and q! TD are substantially reduced, and heat transfer is essentially by conduction only. This situation is sketched in Fig. 2 where one-dimensional conditions are assumed. For this case it can be shown that ! qT

!

 q I¼0 ¼ κI¼0 grad T

Measurements of the heat conductivity under steady-state conditions yield essentially the value of κI¼0 , whereas calculations lead to a value κ defined by ! qT

¼ κ grad T

The difference between these two values is usually rather small except for very light particles, particularly for electrons. Fig. 2 Thermal diffusion and ordinary diffusion in a binary mixture. Concentration Ci0 is due to thermal diffusion, and concentration Ci is due to ordinary diffusion

196

6

M. I. Boulos et al.

Diffusion and Energy Fluxes

The results from the previous section will now be compared with the equations derived by Enskog (1917) and Waldmann (1949, 1950) from kinetic theory. A binary mixture will be considered assuming that there are no external forces and no pressure gradients. For this particular case, the equations for mass flux and heat flux have the following form: !

grad T I 1 ¼ ρD grad c1  αρDc1 c2 T  αkT ! q ¼ ρD þ hg, 1  hg, 2 grad c1  κ grad T m

(57) (58)

or in alternate form !

q¼



αkT þ hg, 1  hg, 2 m



! I1

 κI¼0 grad T

(59)

In these equations, D represents the binary diffusion coefficient, α the thermal diffusion factor, κ the heat conductivity (which includes the contribution of thermal diffusion), and κI¼0 the ordinary heat conductivity without mass fluxes. Comparison of the first terms in Eqs. 46 and 57 results in the following correlation between the coefficients L11

kT ¼ ρD mc1 c2

(60)

or L11 ¼

ρDmc1 c2 kT

Using the previously derived expression for L11 (Eq. 48), an expression for the friction coefficient becomes ε11 ¼

kT Dðn1 þ n2 Þ

(61)

Comparing the second terms in Eqs. 46 and 57 results in   L11 q  hg, 1 þ hg, 2 ¼ αρDc1 c2 or q  hg, 1 þ hg, 2 ¼

αkT ¼ y1  y2 m

(62)

5

The Plasma Equations

197

The last equality is identical to Eq. 49. Introducing Eq. 57 into Eq. 59 results in !

q ¼ ρDq grad c1  q αρDc1 c2

grad T  κI¼0 grad T T

(63)

The last two terms may be combined to form the term (κ grad T) where κ ¼ κI¼0 þ q αρDc1 c2

1 T

(64)

Introducing the previously evaluated coefficients into Eq. 46 and into the alternative form of Eq, 43 provides the final equations !

!

I1

¼I2¼



  ! ρDmc1 c2 ! kT αkT grad T F 1  F 2  v0, 1  v0, 2 grad p  grad c1  T kT mc1 c2 m

(65) and  h   ! ! m αkT þ hg, 1  hg, 2 F 1  F 2  v0, 1  v0, 2 grad p q ¼ ρD c1 c2 kT m  kT  grad c1  κ grad T (66) mc1 c2

!

The last equation can be written in alternative form as  ! αkT q¼ þ hg, 1  hg, 2 I 1  κI¼0 grad T m

!

(67)

In Eqs. 65 and 66, the following “force” terms can be exchanged:   kT 1 1 grad c1 ¼ grad p1  grad p2 v0, 1  v0, 2 grad p þ ρ1 ρ2 mc1 c2

(68)

Finally, the coefficients y1 and y2 can be calculated from Eqs. 49 and 62: y1 ¼

αkT αkT c2 and y2 ¼ c1 m m

(69)

Now that the various coefficients have been determined, the resulting equations will be applied to a chemically reacting gas in the following section.

198

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M. I. Boulos et al.

Example of Mass and Energy Fluxes

@y Assuming that steady state @t ¼ 0 in a binary mixture and that there are no ! prevails,  external forces so that vg ¼ 0 . The assumption of a binary mixture requires that ! ! div I1 ¼ Γ1 and div I2 ¼ Γ2 which means that the sources for the mass fluxes are the corresponding mass ! ! generation terms. Since I1 ¼  I2 , it follows that Γ1 þ Γ2 ¼ 0 assuming that the binary mixture consists of nitrogen molecules and nitrogen atoms. Such a mixture can be maintained if the temperature is sufficiently high; this can be accomplished by heating nitrogen in a plasma torch. It is also assumed that the plasma jet emanating from the torch is confined in a circular tube so that a rotationally symmetric temperature profile will exist with (grad T)r 6¼ 0, with exception of the axis (Fig. 3). Since the concentration of atoms in the plasma jet is a function of the temperature, there will be a continuous diffusional flux of N atoms to colder regions (on the periphery) where the atoms will recombine into molecules, and because of the higher concentration of N2 molecules in the colder regions, there is a molecular flux in the opposite direction so that !

!

div I a ¼ div I m and Γa ¼ Γm Here index “a” refers to the atoms and “m” to the molecules. The atoms carry an enthalpy

Fig. 3 Temperature distribution and gradients in a nitrogen cylindrical plasma jet (a = atoms, m = molecules)

Nitrogen plasma jet

grad (na) Ia

T

grad (nm) Im

Tw Radius, r

5

The Plasma Equations

199

ha ¼

5 kT 1 D0 þ Ea þ 2 ma 2 ma

(70)

With them, where ma is the mass of an N atom, Ea is the electronic excitation energy per unit mass, and D0 represents the dissociation energy. It is assumed that each atom carries half of the dissociation energy as potential energy. The corresponding enthalpy carried by the molecules is hm ¼

5 kT þ Em 2 mm

(71)

where mm = 2ma represents the mass of an N2 molecule and Em is the excitation energy per unit mass. Em includes electronic excitation as well as excitation of vibrational and rotational energy states. Combining Eq. 70 and 71 gives  1 5 kT þ D0 þ Ea  Em ha  hm ¼ 2ma 2 The corresponding mass flux according to Eq. 57 is then !

I a ¼ Im ¼ ρD grad ca  αρDca cm

grad T T

(72)

or (assuming chemical equilibrium) with grad ca ¼

@ca grad T @T

(73)

we have ! Ia

 @ca 1 þ αρDca cm ¼  ρD grad T T @T

The heat flux is given by !

q¼



  ! αkT 1 5 kT þ D0 þ Ea  Em I a  κI¼0 grad T þ 2ma 2 m

(74)

or in alternative form

   αkT 5kT D0 @ca þ þ þ Ea  Em ρD þ αca cm =T þκI¼0 grad T q ¼ 4ma 2ma m @T (75)

!

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M. I. Boulos et al.

The expression in brackets in the last equation can be referred to as e κ so that !

κ grad T q ¼ e

(76)

The value of e κ contains contributions due to ordinary diffusion, thermal diffusion, enthalpy interdiffusion (or heat of transition), and pure heat conduction. It is obvious from Eq. 75 that diffusion and enthalpy interdiffusion are interrelated, as are thermal diffusion and enthalpy interdiffusion, i.e., the corresponding terms in Eq. 75 cannot be separated. An interesting demonstration of the phenomena described in this section is observed during the reentry of space vehicles into the Earth atmosphere. Limiting the analysis to nitrogen, which is the predominant component (79% vol.) in the atmosphere, there will be a binary mixture of nitrogen atoms and molecules close to the surface of the space vehicle during the reentry phase. A mass flux of atoms is driven to the surface due to concentration gradients and thermal diffusion as shown in Eq. 74. This mass flux carries different forms of energy as described in the bracket of Eq. 74. This energy will be released upon impact of the atoms on the surface of the space vehicle. An important contribution is the dissociation energy. In addition, there will be heat transfer by conduction (last term in Eq. 74) which is usually smaller than the heat released by the mass flux of the atoms. In the next section, the equations that have been derived for a binary mixture will be applied to a fully ionized plasma representing a typical case of a two-component mixture.

8

Transport Equations for a Fully Ionized Plasma

We will assume that the plasma consists of electrons and positive ions only. This assumption excludes recombination, at least for the time being. For the sake of simplicity, it will be assumed that the ions of mass mi are singly charged (+e). The electron mass and charge will be denoted by me and -e, respectively. The mass fluxes of ions and electrons follow from Eq. 3: ! Ii

  ! ! ¼ ni m i v i  v g

or   e ! ! ! I i ¼ eni v i  v g mi

(77)

  e ! ! ! I e ¼ ene v e  v g me

(78)

and correspondingly

5

The Plasma Equations

201

The total current density can be written as !

!

!

j¼ jiþ je

or !

!

!

j ¼ eni v i  ene v e !

(79) !

Introducing Eqs. 77 and 78 into Eq. 79 and using I i ¼  I e result in !



1 1 þ j ¼ e mi me



! Ie

!

þ eðni  ne Þ v g

(80)

The last term of this equation contains the net electrical space charge eðni  ne Þ ¼ ρe1

(81)

Applying conservation of mass in the form of Eq. 4 to ions and electrons yields   @ni ! ¼ div ni v i @t

(82)

  @ne ! ¼ div ne v e @t

(83)

and

Combination of these two equations after multiplying Eq. 82 by e and Eq. 83 by  e leads to the current conservation equation: @ρe1 þ div @t

!

j ¼ 0:

(84)

In order to obtain an explicit expression for the current density, a previously derived expression for the mass flux (Eq. 47) will be applied to the flow of electrons. Introduction of this expression into Eq. 78 yields  

! ! 1 1 1 1 grad T  (85) j ¼ e m þ m L11 F e  F i  ρ grad pe þ ρ grad pi ðq  he þ hi Þ T i e e i

!

For the plasma densities of interest in this section, net space charges cannot exist and plasma sheaths will be excluded ðρel ¼ 0Þ. It will be shown later that net space charges may become important for tenuous plasmas for which the density of charge carriers falls below 1014 cm3. Equation 85 will be further modified by introducing, step by step, external forces that play an important role in the behavior of plasma.

202

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M. I. Boulos et al.

Plasma Exposed to an Electric Field

Here the electric field is the only external force acting on the charge carriers, i.e., grad pe = grad pi = grad T = 0. Then !

Fe ¼ 

e ! E me

and !

Fi ¼

e ! E mi

(86)

Applying Eq. 86 to Eq. 85 results in   1 1 1 1 ! j¼e þ þ L11 e E mi me mi me

!



or  ! mi þ me 2 j¼e L11 E mi me

!

2

(87)

If plasma is exposed to an electric field only, Ohm’s law can be written as !

!

j ¼ σe E

Therefore,  mi þ mc 2 σe ¼ e L11 mi me 2

8.2

(88)

Plasma Exposed to Electric and Magnetic Fields

In this situation the corresponding forces acting on electrons and ions can be written as ! e ! e ! ! F e ¼  m E  m v e B e e ! e ! e  ! ! Eþ v i B Fi ¼ mi mi

Introducing Eq. 89 into Eq. 85 results in

(89)

5

The Plasma Equations



!

j ¼ σe

203

 ! ! mi me mi me ! ! v e B þ v i B Eþ me ðmi þ me Þ mi ðmi þ me Þ !

The last expression for the current density can be simplified because me  mi, i.e.,

 ! ! me  ! ! vi B j ¼ σe E þ v e B þ mi

!

(90)

The drift velocities of electrons and ions can be eliminated from Eq. 90 by using Eq. 79 and the definition of the center of mass velocity (Eq. 1): !

!

!

ρi v i þ ρe v e ¼ ρL v g

(91)

where ρL þ ρi þ ρe Multiplying Eq. 77 by mi =e and adding it to Eq. 89 yield ! v e ðne me

!

þ ne m i Þ ¼ ρ L v g 

mi ! j e

(92)

or !

mi ! j eρL

!

me ! j eρL

! ve

¼ vg 

! vi

¼ vg þ

and (93)

where the fact that ne = ni has been used. Elimination of electron and ion velocities from Eq. 90 by using Eqs. 91 and 93 yields !



!

j ¼ σe E þ



! ! v g B

  ! mi ! ! me ! me ! ! j B þ j B  v g B þ ρL e mi ρL e

(94)

The last term in Eq. 94 can be neglected compared with the previous term, resulting in

 ! ! 1 ! ! ! j ¼ σe E þ v g B  j B enL

!

(95)

where nL ¼ ne ¼ ni . Due to the presence of a magnetic field in addition to the electric field, two  ! ! additional terms appear in Ohm’s law. The term vg B represents the induced

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field caused by the movement of the center of mass. Only for the special case in which ! ! vg and B are parallel will this term vanish. The last term represents a counteracting field due to the interaction of the current with the applied magnetic field. A bulk motion of the plasma can be caused by an applied magnetic field, as can readily be seen from the momentum equation: !

ρ

! ! dv g ¼ grad p þ j B dt

(96)

The interaction of the magnetic field with the current gives rise to an acceleration of the plasma and/or establishes a pressure gradient. Two extreme cases are conceivable: a free plasma flow and a completely obstructed flow. In the first case no pressure gradient can develop, i.e., the first term on the right-hand side of Eq. 96 vanishes. In the second case, the initial acceleration of the plasma comes to a complete stop; in this situation the Lorentz force is balanced by a pressure gradient. An acceleration of the plasma can be initiated with the application of a magnetic field. There is, however, a counteracting force represented by the last term in Eq. 95. The plasma reaches its final velocity as soon as this term, which tends to reduce the current density, stops the current flow entirely. In this situation, which is analogous to an induction motor running without load (zero current), the magnitude of the final velocity of the plasma is determined by the equation; ! ! ! B E ¼ v g

(97)

If the acceleration of the plasma is impeded, a pressure gradient will form, requiring a modification of Eq. 95: !



!

j ¼ σe E þ

! ! v g B

  1 ! ! me mi 1 1  j Bþ grad pe þ grad pi enL ρi eðmi þ me Þ ρe (98)

Since mi me, the last term in this equation reduces to ð1=enL Þ grad pe. With grad p = grad pe + grad pi = 2 grad pe and the momentum Eq. 96, the last two terms in Eq. 98 can be replaced to give !

"

!

j ¼ σe E þ

! ! v g B

1  enL

!

dv g 1 grad p þ ρ 2 dt

!# (99)

Besides the “forces” included in this equation, there may be a temperature gradient that is also able to drive an electric current. This current will be denoted by JT (see Eq. 85):

5

The Plasma Equations ! jT

205



1 1 grad T ¼e þ L11 ðq  he þ hi Þ mi me T

(100)

Where, q  he þ hi ¼ αkT m and m ¼ 2me or ! jT

¼ σe

αk grad T 2me

(101)

The electric current driven by thermal diffusion is frequently small compared with the other contributions. In the neighborhood of electrodes, however, grad T may assume such large values that the corresponding “driving force” reaches a value of several volts. In addition to the previously discussed forces, other forces (e.g., gravity, centrifugal forces, or inertia) may be responsible for the current flow. For laboratory plasmas, gravity does not play a significant role; the same is true for inertia. It should be pointed out that the electrical conductivity, σe, which appears in the current equation, is assumed to be a scalar. This is, however, no longer true if strong magnetic fields are present. In this situation the electrical conductivity changes to a tensor, which assumes different values parallel and normal to the orientation of the magnetic field. Finally, the energy flux in fully ionized plasma will be considered. Applying Eq. 67 to a fully ionized plasma leads to !

q¼ 

 ! αkT me mi þ he  hi j κI¼0 grad T m e ðm e þ m i Þ

(102)

Alternatively, if me  mi with he ¼

5kT 5kT Ei and hi ¼ þ 2me 2mi mi

where Ei is the ionization energy, resulting in

  ! kT me 2Ei αþ5 q¼  5þ j κI¼0 grad T 2e mi kT

!

(103)

The first term of this equation represents the energy carried by the electric current, and the second term describes pure beat conduction. The term in parentheses can ! be neglected compared with (α + 5). The current density j can be taken from Eq. 99.

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M. I. Boulos et al. !

An interesting conclusion can be drawn from Eq. 103 by forming div q : !

!

div q ¼  j

@f ðTÞ grad T  divðκI¼0 grad TÞ @T

(104)

where

  kT me 2Ei αþ5 þ5 f ð TÞ ¼ 2e mi kT !

!

The scalar product j grad T 6¼ 0 if the grad T is not normal to j , i.e., in this case ! j grad T represents an energy source term. This situation prevails, for example, in the boundary layers close to electrodes in an electric discharge (Dinulescu et al. 1980; Hsu et al. 1983; Sanders et al. 1984). In the previously derived equations for current and energy fluxes, there are coefficients (σe, κI¼0 , and α.) that must be determined. The following section contains expressions for these coefficients.

9

Determination of Transport Coefficients

From solutions of the Boltzmann transport equation (see ▶ Chap. 3, Kinetic Theory of Gases), Spitzer (1950) derived expressions for the transport properties of a fully ionized plasma:  32π2 ε2o me v3e 2 3=2 σe ¼ 2 γE 3π e Z lnðqÞ γT γE

(106)

  320 π2 ε2o m2e kv5e 2 3=2 3 δE γT ¼ δT 1  3π 3e4 ZlnðqÞ 5 δT γE

(107)

α¼3 κI¼0

(105)

where v2e ¼

3 kT me

q¼

4 π εo k T 1=3

e2 Zni

and

Z¼

X nz z 2 ne

represent the average ion charge. In addition, there are four transport coefficients (δT, δE, ΥT, ΥE) that are functions of the average ion charge. Values for these coefficients can be found in the following table:

5

The Plasma Equations

207 Z=1 0.5816 0.2727 0.4652 0.2252

γE γT δE δT

Z=2 0.6833 0.4131 0.5787 0.3563

Disregarding the minor influence of the logarithmic term in Eqs. 102 and 103, it is important to emphasize that σe  v3e  T3=2 κI¼0  v5e  T5=2 Both electrical and thermal conductivity are governed by the free electrons, a result that is not surprising considering the high mobility of free electrons in plasma. For Z = 1, one finds that κI¼0 k2 ¼2 2T σe e a relation that has been derived for pure metals. There is obviously a close relationship between a metal (a “solid-state plasma”) and fully ionized gaseous plasma. After considering a typical two-component mixture (a fully ionized plasma), the treatise should be extended to a three-component mixture, which can be applied in general to any plasma consisting of a mixture of neutral particles, ions, and electrons. The resulting equations are more complicated but similar to those derived for a two-component mixture. The derivation of these equations can be found in Finkelnburg and Maecker (1950).

10

Nomenclature

Units are indicated in parentheses; whenever no units are given, the parameter is dimensionless.

10.1 !

B C ck Ck D Do en ! E

Latin Alphabet Magnetic induction (V.s/m2) Constant, Eq.36 Mass concentration Mass fraction of component k Diffusion coefficient (m2/s) Dissociation energy (J, eV) Base of naturel logarithmes (en = 2.718) Electric field strength (V/m)

208

Ea Es ek ! Fk ! g h hg ! Ik ! je k 9 Lk‘ = Lku ; Lu‘ me mk n p(k) Qk q* q ! q ! S sg t T ug V ! vg ! vk vo ! X‘ xk yk Z

M. I. Boulos et al.

Electronic excitation energy (J, eV) Entropy production (W/m3.K) Electric charge of component k (C) Body force acting on component k (N/kg) 2 Acceleration due to gravity (m/s ) 34 Planck’s constant (h = 6.626 10 J.s) Specific enthalpy (kJ/kg) 2 Mass flux of component k, relative to the center of mass (kg/m .s) 2 Electron current density (A/m ) 23 J/K) Boltzmann constant (k = 1.38x10 Onsager coefficients in Eqs. 23, 24, and25 Electron mass (me = 9.11x1031kg) Mass of component k (kg) Particle number density (m3) Partial pressure of component k (Pa) Partition function of component k Heat of transition (J/kg) Heat exchange with surroundings (J/kg) Heat flux (W/m2) 2 1 Diffusion flux (m .s ) Entropy per unit mass (J/kg. K) Time (s) Absolute temperature (K) Internal energy (J/kg) Electrical potential (V) Velocity of the center of mass (m/s) Velocity of component k (m/s) Specific volume (m3/kg) Driving forces (N) Mole fraction of component k Thermal diffusion coefficient of component k (J/kg) Mean ion charge (C)

10.2

Greek Alphabet

α 9 δE > > = δT Υε > > ; ΥT εij Γk

Thermal diffusion factor Spitzer coefficients in Eqs. 102, 103, and 104 Friction coefficient between components i and j Generation rate of particles of component k (kg/m3.s)

5

κ λi μk ρ ρk σe τ

The Plasma Equations

209

Thermal conductivity (W/m.K) Mean free path of particle of type I (m) Chemical potential of component k (J/kg) Mass density of the mixture (kg/m3) Mass density of component k (kg/m3) Electrical conductivity (mho/m) Time (s)

References Dinulescu HA, Pfender E (1980) Analysis of the anode boundary layer of high intensity arcs. J Appl Phys 51(6):3149 Elenbaas W (1935) Physica 2:169 Enskog D (1917) Dissertation, University of Uppsala, Sweden Finkelnburg W, Maecker H (1950) Electric arcs and thermal plasmas. In: Fl€ ugge S (ed) Handbook of physics, vol 22. Springer, Berlin, p 254 Helier G (1935) Physica 6:389 Hsu KC, Etemadi K, Pfender E (1983) Study of the free-burning high-intensity argon arc. J Appl Phys 54(3):1293 Onsager L (1931a) Phys Rev 37:405 Onsager L (1931b) Phys Rev 38:2265 Sanders NA, Pfender E (1984) Measurements of anode falls and anode heat transfer in atmospheric pressure, high-intensity arcs. J Appl Phys 55(3):714 Spitzer L Jr (1950) Physics of fully ionized gases, NY interscience. 2nd edn. Dover books on Physics, USA Waldmann L (1949) Zs f Naturforschung 4a:105 Waldmann L (1950) Zs f Naturforschung Sa:322

6

Thermodynamic Properties of Plasmas Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Thermodynamic Functions for CTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Partition Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Thermodynamic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Computation of Partition Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Composition of a Plasma at Constant Pressure in CTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Equilibrium Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Calculation of the Plasma Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Thermodynamic Properties of Plasmas in CTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Specific Heat at Constant Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Enthalpy and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Complementary Issues in Plasma Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Presence of Solid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Calculations at Constant Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Sound Velocity and Adiabatic Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212 212 213 213 214 215 221 224 225 226 229 238 238 241 246 246 247 248

M. I. Boulos (*) Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada e-mail: [email protected] P. L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France e-mail: [email protected] E. Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] © Springer Nature Switzerland AG 2023 M. I. Boulos et al. (eds.), Handbook of Thermal Plasmas, https://doi.org/10.1007/978-3-030-84936-8_6

211

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7 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Latin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

250 250 252 253

List of Abbreviations

LCE LTE CTE

1

Local chemical equilibrium Local thermodynamic equilibrium Complete thermodynamic equilibrium

Introduction

Physical and chemical processes occurring in thermal plasmas, and their interactions with gases, particulate matter, or liquids, are highly complex and require knowledge of composition, thermodynamic, and transport properties of the plasma. In this chapter, the relationships between thermodynamic functions and partition functions for plasmas in local thermodynamic equilibrium (LTE) and local chemical equilibrium (LCE) are discussed. The methods of calculation of these partition functions are given, followed by a review of the different ways to calculate the composition of a plasma, and of the corresponding thermodynamic properties. As shown in Part II, “▶ Generation of Thermal Plasmas,” and Part III, “▶ Plasma and Particle Dynamics,” even if the plasma core can be assumed to be in complete thermodynamic equilibrium (CTE), this may not be the case in the fringes or in the plume of thermal plasmas, where two different temperatures can be defined (one for the electrons and the other for the heavy species). Separate discussions on the effects of such deviations from LTE or LCE on the plasma properties, are presented in ▶ Chaps. 9, “Thermodynamic Properties of Non-equilibrium Plasmas,” and ▶ 10, “Transport Properties of Non-equilibrium Plasmas.”

2

General Remarks

A plasma process frequently involves complex interactions between heat, mass, and momentum transfer under steady-state and transient conditions, turbulent phenomena, thermal radiation, chemical reactions, and phase changes when dealing with heating and accelerating of particulate matter dispersed in the plasma flow. Modeling of such processes (see, e.g., Chang and Ramshaw 1993, 1996; Lee and Pfender 1987; McKelliget et al. 1982; Westhoff et al. 1992) requires in the first place the knowledge of composition, thermodynamic, and transport properties of the plasma. In this chapter, the bases of these calculations are considered assuming that local thermodynamic equilibrium (LTE) and local chemical equilibrium (LCE) are achieved. LTE implies that the characteristic time of the slowest reaction is small compared to that of the flow and of the diffusion time along the temperature and

6

Thermodynamic Properties of Plasmas

213

composition gradients (Rat et al. 2008). In fact, complete thermodynamic equilibrium (CTE) can be considered instead of LTE if collision processes (not radiative processes) govern transitions and reactions the plasma, and if there is a microreversibility among the collision processes. Chemical equilibrium is the state in which both reactant and product concentrations do not change with time, which occurs when the forward reactions proceed at the same rate as that of the reverse reactions. Such a state is known as dynamic equilibrium (Hill 1987). Under LTE conditions, the plasma composition can be calculated by either of the following two equivalent methods (André et al. 1997): • The mass action law involving the equilibrium constants Kp(T) of different reactions (Saha and Guldberg–Waage laws) with the Dalton and species conservation laws as the neutrality condition • Gibbs free energy minimization The composition of the gas (e.g., in a nitrogen plasma: the molar fractions of N2, + ++ Nþ 2 , N, N , N , and e) depends strongly on its temperature, which is a result of the energy balance between the electrical energy dissipated and the heat losses that occur mainly at the fringes of the plasma. In such a complex mixture, where a dynamic equilibrium exists between dissociation, ionization, and recombination, the total energy content depends on the energy of the various particles (frozen energy) and on the chemical reactions among them (reactive energy). Thus, the thermodynamic properties (enthalpy, entropy, and specific heat) depend strongly on the composition of the plasma. The latter is calculated through the minimization of the Gibbs free energy (for a given temperature and pressure), which in turn depends on the chemical potentials of the different chemical species present in the plasma. At high temperatures (T >6,000 K), these chemical potentials can be calculated only from statistical thermodynamic considerations (see ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics”) through quantities called partition functions which are related to the internal energy levels of the different chemical species of the plasma (energies can be determined spectroscopically).

3

Thermodynamic Functions for CTE

3.1

Notation

To identify the thermodynamic function of any chemical species, the simplest notation is to use a chemical symbol as the subscript for the thermodynamic function considered. However, as soon as the number of chemical species in the plasma increases (approximately 20–50) the composition must then be obtained computationally, and numerical subscripts need to be used. For example Ei,s corresponds to the energy of the i th species (e.g., nitrogen) in the excited states s, while Ei,0 would correspond to the ground state. Tables of energy levels for the excited states (e.g., Moore 1949, 1952, 1958) are usually arranged by increasing values, thus a second

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M. I. Boulos et al.

index for the energy of the excited state is generally sufficient. The total number of the species considered is thus split among the different energy levels Ni ¼

X

Ni, s

(1)

s

where the summation is performed over all possible excited states, including the ground state. Similar considerations hold for the diatomic molecular energy levels. With these indexes, we can write the chemical reactions in a matrix notation suitable for computer use. However, to acquaint the reader who is not familiar with plasmas, most of the plasma properties will be presented using nitrogen as an example; the pertinent chemical species will be classified by the chemical index, followed, if necessary, by the additional index s for the excited states. When only the chemical index is given, it means that a property of the entire chemical species is considered, regardless of excitation state.

3.2

Partition Functions

The number of particles of chemical species i that will be found in a quantum state s of energy Ei,s is given by Boltzmann’s relationship (see ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics”)   gi, s  exp Ei, s =kT Ni, s ¼X   Ni gi, s  exp Ei, s =kT

(2)

s

where gi,s is the statistical weight of the state s (i.e., the number of wave functions related to the energy level s). The denominator of Eq. 2 is called the atomic or molecular internal partition function Qint i (T) for particles of species i: Qint i ð TÞ ¼

s X

  gi, s  exp Ei, s =kT

(3)

s

This quantity is dimensionless, and the summation is over all possible states of s, limited to a value s*, which will be specified for the cases of atoms and diatomic molecules. In general (Mayer and Mayer 1940), for a given particle of species i, two types of energies have to be considered: the energy of translation (kinetic energy) and the energy related to the internal degrees of freedom. For an atom, these degrees of freedom include the excitation energy of the bound electrons while for a molecule they include the rotational energy of the molecule and the vibrational energy of the nuclei (see Sect. 5 in ▶ Chap. 2, “Basic Atomic and Molecular Theory”), in addition to the energy related to the excitation of the bound shared electrons.

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Thermodynamic Properties of Plasmas

215

If the translational and internal   energies of the particles are supposed to be int independent Ei, s ¼ Etrans þ E i, k i, j , the partition function Qi, which is then the product of the translational and internal partition functions, can be written Qi ¼

Qtri



Qint i

  2πmi kT 3=2 V int ¼  V  Qint i ¼ 3 :Qi 2 Λi h

(4)

where mi is the mass of the chemical species i, V is the plasma volume, and Λi is the thermal de Broglie length:  Λi ¼

h2 2πmi kTi

1=2 (5)

All these equations hold provided the hypothesis of Boltzmann is fulfilled (for more details, see see ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics,” and Landau and Lifschitz 1967). This means the number of accessible quantum states must be much higher than the number of particles, that is, there should be a very low probability of finding two particles simultaneously in the same quantum state. In other words, the condition 

2  π  mi  k  T h2

3=2  V=N  1

(6)

must be fulfilled. Notice that the Maxwell-Boltzmann distribution is a limiting case of the BoseEinstein and Fermi-Dirac quantum distributions.

3.3

Thermodynamic Functions

Boltzmann’s statistical treatment allows to express the thermodynamic functions through the partition functions of the system under consideration. For example, according to Mayer and Mayer (1940), the Helmholtz free energy, F, related to a reference energy F0 is given by F  F0 ¼ k  T  lnQtot

(7)

where Qtot is the partition function of all the particles of the thermodynamic system under consideration. The energy of the system is then assumed to be the sum of the energies of the different particles (non-interacting particles), resulting in; Qtot ¼

∏i QNi i ∏i N i !

(8)

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Qi is the partition function of a single particle of the chemical component i, Ni is the total number of particles of species i in the system, and Πi ¼ Πi¼K i¼1 . Using Eqs. 7 and 8, and the Stirling formula (lnN!=N.lnNN), F can be calculated from X F  F0 ¼  i Ni  k  T  lnðen  Qi =Ni Þ (9) where en is the base of natural logarithms (en = 2.7182818) and Σi ¼ ΣK i¼1 . However, for Eq. 8 to be valid it is essential that all interconnected energy levels be referred to the same reference, for example, to free atoms in their lowest quantum states (ground states).

3.3.1 Perfect Gases The right-hand side of Eq. 9 (valid when Eq. 6 is fulfilled, i.e., no quantum effects) is a function of the temperature T and either the volume or the pressure. The righthand side can be determined easily provided the energies and degeneracies of the excited states (as well as the plasma composition) are known. For example, for nitrogen at temperature T and pressure p, the following equation is obtained in which only the species N2, N, N+ and e are considered. For the initial mixture one mole of N2 at T = 0 K, and p = 1 atm (with free energy F (0)) is postulated.          en  QN2 en  Q N en  Q N þ en  Q e F  F0 ¼ kT NN2 ln þ NN ln þ ðNNþ Þln þ Ne ln NN2 NN NNþ Ne

(10) It is thus possible to calculate all other thermodynamic functions through the partition functions for the various types of particles present in the plasma. For example, the pressure p is given by  p¼

   @F @lnQi ¼ Σi Ni  k  T  @V T, Ni @V T, Ni

(11)

and the internal energy by " #   F   int  @ T 3 2 lnQi ¼ Σi Ni kT þ kT E  E0 ¼ T 2 @T V, Ni @V T, Ni 2

(12)

3.3.2 Debye Correction As soon as the temperature increases, the densities of the charged species also increase, and the long-range Coulomb interactions result in an interaction energy that must be added to the thermodynamic functions calculated under the perfect gas assumption.

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Thermodynamic Properties of Plasmas

217

With the Debye model (Drawin 1972), (see also Mayer and Mayer 1966), Helmoltz free energy, F, becomes F  F0 ¼ 

X i

Ni kT  lnðen Qi =Ni Þ 

kTV 12πλ3D

(13)

with 0

11=2 E0 kTV B K C λD ¼ @ 2 X 2 A e Zi N i

(14)

i¼1

where E0 is the vacuum permittivity and Zi is the number of ionic charges of species i (e.g., Zi = 1 for N+, Zi = 1 for e, Zi = 1 for Nþ 2 , Zi ¼ 0 for N and N2). þ In a nitrogen plasma containing the species N2 , Nþ 2 , N, N , and e, K X

Z2i Ni ¼ NNþ þ Ne þ NNþ2

i¼1

For higher temperatures with a gas containing mainly N+, N++, and e, K X

Z2i Ni ¼ NNþ þ 4NNþþ þ Ne

i¼1

The Coulomb field also modifies the pressure: p¼

Ni kT kT  V 24πλ3D

(15)

while the Gibbs free energy becomes G  G0 ¼ pV þ F  F0 ¼ 

X

Ni kT  lnðQi =Ni Þ 

kTV 8πλ3D

(16)

The Debye correction will be greatest for the highest values of charged particle densities. For N2, Ar, H2, and O2, which all have similar ionization potentials, the maximum electron density at atmospheric pressure occurs around 14,000–15,000 K, (see, e.g., Figs. 1, 2, 3, 4, and 5) and remains almost constant for higher values of T. Thus, the Debye correction is the highest at these temperatures. In most cases, these corrections are rather small (less than 2 or 3%). Thus, it is generally sufficient to calculate the composition with no Debye correction and then evaluate the corresponding corrections (λD depends on Ni) to the thermodynamic functions. The Debye correction is illustrated in Table 1, where corrections to the enthalpy of air plasmas are given at pressures of 20, 100 and 500 kPa.

218 1026 Ar Number density, ni (m−3)

Fig. 1 Temperature dependence of the composition (species number densities) of argon plasma at atmospheric pressure (starting from one mole of Ar at room temperature) (Pateyron et al. 1986; Boulos et al. 1994)

M. I. Boulos et al.

1024 Argon, 100kPa 1022 e 1020 Ar+ 1018

Ar++

1016 0

2

4

6

8

10

12

14

Temperature, T (103 K) 1026 He Number density, ni (m−3)

Fig. 2 Temperature dependence of the composition (species number densities) of a helium plasma at atmospheric pressure (starting from one mole of He at room temperature) (Pateyron et al. 1986; Boulos et al. 1994)

1024 Helium, 100kPa 1022 e

He+

1020 1018 1016 0

2

4

6

8

10 12 14 16 18 20 22 24

Temperature, T (103 K)

1026 N2 Number density, ni (m−3)

Fig. 3 Temperature dependence of the composition (species number densities) of nitrogen plasma at atmospheric pressure (starting from one mole of N2 at room temperature) (Pateyron et al. 1986; Boulos et al. 1994)

Nitrogen, 100kPa

N

1024 e N

1022

N2 1020

e N+2 N–

1018 N+

e 1016

N++

N+2 0

2

4

6

8

10

Temperature, T (103 K)

12

14

Thermodynamic Properties of Plasmas

Fig. 4 Temperature dependence of the composition (species number densities) of oxygen plasma at atmospheric pressure (starting from one mole of O2 at room temperature) (Pateyron et al. 1986; Boulos et al. 1994)

1026

219

O2

Oxygen, 100kPa

24

Number density, ni (m−3)

6

10

e

O O2

1022 O

1020

O–

O+2

e O+

1018 e O–2

1016 0

2

4

O2

O–

O++

6

8

10

12

14

Temperature, T (103 K)

1026

Number density, ni (m−3)

Fig. 5 Temperature dependence of the composition (species number densities) of hydrogen plasma at atmospheric pressure (starting from one mole of H2 at room temperature) (Pateyron et al. 1986; Boulos et al. 1994)

10

H2

Hydrogen, 100kPa

24

e

H 1022

H+

1020

H2

H e H+

1018

H–

1016 0

2

4

6

8

Temperature, T

10

12

14

(103 K)

3.3.3 Virial Correction When two atoms approach each other, an interaction potential (e.g., of the Morse type, see ▶ Chap. 2, “Basic Atomic and Molecular Theory”) governs their behavior because the electronic shells surrounding the nuclei interact with each other (see, e.g., Fig. 1 in ▶ Chap. 3, “Kinetic Theory of Gases”). This interaction potential becomes repulsive only when the two atoms come very close together (a few tens of angstro¨ms). If the mean free path is large compared to the distance at which the interaction starts, the deflection of the particles will result in mean trajectories quite similar to those predicted by the classical model. If ‘ij is the mean free path of particle i approaching a particle j, then an interaction potential Vij is active for distances smaller than dij. No virial corrections will be necessary as long as

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Table 1 Debye and virial corrections for the enthalpy of air (calculated at 20 kPa, 100 kPa and 500 KPa) Pressure (kPa) 20

100

500

T (K) 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,000 14,000 15,000 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,000 14,000 15,000 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000

Perfect gas values (J/kg) .7540E + 06 .1980E + 07 .4212E + 07 .7995E + 07 .1073E + 08 .1892E + 08 .3355E + 08 .4089E + 08 .4555E + 08 .5333E + 08 .6904E + 08 .9705E + 08 .1310E + 09 .1542E + 09 .1653E + 09 .7540E + 06 .1977E + 07 .3772E + 07 .7386E + 07 .9965E + 07 .1466E + 08 .2575E + 08 .3751E + 08 .4342E + 08 .4861E + 08 .5705E + 08 .7246E + 08 .9790E + 08 .1291E + 09 .1532E + 09 .7540E + 06 .1975E + 07 .3555E +07 .6492E + 07 .9386E + 07 .1249E + 08 .1908E + 08 .3028E + 08 .3992E + 08 .4568E + 08

Debye corrections 0.00% +0.00% +0.00% 0.00% +0.00% +0.00% +0.00% +0.01% +0.06% +0.24% +0.60% +0.85% +0.54% +0.09% 0.12% +0.00% +0.00% +0.00% 0.00% +0.00% +0.00% +0.00% +0.01% +0.05% +0.19% +0.55% +1.17% +1.58% +1.11% +0.28% 0.00% +0.00% 0.00% 0.00% 0.00% +0.00% +0.00% +0.01% +0.04% +0.14%

Virial corrections +0.00% +0.00% +0.00% +0.00% +0.00% 0.00% +0.00% +0.00% +0.00% +0.00% +0.00% 0.00% +0.00% +0.00% 0.00% +0.01% +0.00% +0.00% +0.00% +0.00% +0.00% 0.00% 0.00% +0.00% +0.00% +0.00% 0.00% +0.00% +0.00% +0.00% +0.03% +0.02% +0.01% +0.00% +0.00% +0.00% +0.00% 0.00% +0.00% +0.00%

Real gas values (J/kg) .7540E + 06 .1980E + 07 .4212E + 07 .7995E + 07 .1073E + 08 .1892E + 08 .3355E + 08 .4090E + 08 .4558E + 08 .5346E + 08 .6945E + 08 .9788E + 08 .1317E + 09 .1544E + 09 .1651E + 09 .7541E + 06 .1977E + 07 .3772E + 07 .7386E + 07 .9965E + 07 .1466E + 08 .2575E + 08 .3751E + 08 .4344E + 08 .4870E + 08 .5736E + 08 .7331E + 08 .9944E + 08 .1305E + 09 .1537E + 09 .7542E + 06 .1976E + 07 .3556E + 07 .6492E + 07 .9386E + 07 .1249E + 08 .1909E + 08 .3029E + 08 .3993E + 08 .4574E + 08 (continued)

6

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221

Table 1 (continued) Pressure (kPa)

T (K) 11,000 12,000 13,000 14,000 15,000

Perfect gas values (J/kg) .5125E + 08 .5966E + 08 .7375E + 08 .9592E + 08 .1242E + 09

Debye corrections +0.44% +1.11% +2.14% +2.90% +2.37%

Virial corrections +0.00% +0.00% 0.00% 0.00% +0.00%

‘ij  dij

Real gas values (J/kg) .5148E + 08 .6032E + 08 .7533E + 08 .9870E + 08 .1271E + 09

(17)

where dij is the distance at which the particles start to repulse each other. If Eq. 17 does not hold for all colliding species, the perfect gas law has to be modified to pV N ¼ 1 þ B ð TÞ NkT V where N ¼

K X

(18)

Ni and B(T) is the second virial coefficient, which can be calculated

i¼1

using the interaction potentials Vij as follows:

BðTÞ ¼

K, K X Ni Nj Bij ðTÞ N2 i, j

(19)

where Bij ðTÞ ¼ 2π

 ð1   Vij ðrÞ exp  1 r2 dr kT 0

(20)

Note that the calculation is somewhat similar to that of the collision integrals (see ▶ Chap. 7, “Transport Properties of Gases Under Plasma Conditions”). In most cases, the virial corrections are negligible for the pressures generally used in thermal plasmas (0.2–5 atm), as illustrated for air in Table 1. They are, of course, highest for the highest densities, i.e., for the lowest temperatures.

3.4

Computation of Partition Functions

3.4.1 Translational Partition Functions To calculate a thermodynamic function, the ratio Qi/Ni is needed (see, e.g., Eq. 9). For the translational degree of freedom, the desired quantity is

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Qtri ¼ Ni

  2π mj  kT 3=2 V V  ¼ Λ3 i  Ni Ni h2

(21)

where Λi is the thermal de Broglie length. Qtri /Ni depends on the result of the calculation of the composition Ni, but as will be shown (see Sect. 6.3), the composition depends only on the quantity Qi 3=2 66 ¼ Λ3 i ¼ 2:77721  10  ðmi  TÞ V

ðwhere mi is in kg and T in KÞ (22)

which is a straightforward calculation.

3.4.2 Limitations of the Internal Partition Functions The second term involved in calculating the composition is the internal partition function. Internal partition function for atoms. For atoms, this function includes only the electronically excited states and is given by Eq. 3. Though this formula seems simple, its evaluation poses considerable difficulties (Drawin 1972): (a) The statistical weights and the energies must be obtained from solutions of the Schro¨dinger equation. For systems of a complicated level structure, it is very difficult to obtain exact solutions. (b) The summation in Eq. 3 has to be performed over all possible discrete energy states which exist only for energies Ei,s, lower than EI δEI, where δEI is the lowering of the ionization potential EI. The bound electron is no longer bound when its energy reaches EI  δEI (see Sect. 4.2 and Fig. 7 in ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics”). From a practical point of view, one of the main problems is that δEI depends on the composition of the plasma, and it becomes necessary to use an iterative procedure for rigorous calculations. However, as demonstrated by Fauchais (1962) and Fauchais et al. (1969 and 1975), the various theories about the breakoff s* of the partition functions give about the same values for the thermodynamic properties, and it is thus generally sufficient to use the theory of Gurvich and Kvlividze (1961) as a first approximation to calculate δEI. Although this theory is not rigorous, it does provide s*, which is correlated with the principal quantum number n, as a function of T and p: n2 r1 ¼ Z1 eff 



3V 4πN

1=3

¼ Z1 eff



3kT 4πp

1=3

¼ 1:48848  108 Z1 eff

 1=3 T p

(23)

6

Thermodynamic Properties of Plasmas

223

p is in Pa, T is in K, and Zeff is the effective charge, taken as the stage of ionization plus 1 for the particle being considered: 1 for a neutral atom, 2 for an ion such as N+, 3 for N++, etc. ln this case, the calculation of n* is straightforward for a given T and p. Note, however (Fauchais 1962), that even for differences up to 120% in the value of n* for the various theories, the composition of the main species remains the same within 5%. This is true only if the same limitation theory is used for the calculation of the partition functions for all the species of the plasma. The differences for the minor species (for which molar fractions are less than 103, for example) may be much higher (up to 40%) depending on the limitation theory used, but such differences do not change the general behavior of the plasma. Internal partition function for molecules. For molecules, the internal partition function can be written as

Qint ¼

" X

 # "X  #  # "X Ee Ev ðeÞ Er ðe, vÞ g  exp gv exp gr exp   e e kT kT kT v r (24)

where ge, gv, and gr denote the electronic, vibrational, and rotational degeneracies, respectively. For an electronic state designated by 2S + 1 (see ▶ Chap. 2, “Basic Atomic and Molecular Theory,” and Herzberg (1950)), ge ¼ 2S þ 1 for Σ states ð^ ¼ oÞ

(25)

ge ¼ 2ð2S þ 1Þ for π, Δ, ∅ states ð^ 6¼ 0Þ

(26)

gv ¼ 1

(27)

However,

for vibrational states of quantum number v, and gr ¼ 2J þ 1

(28)

for rotational states described by the quantum number J. In Eq. 24, the summations have to be limited (e.g., when the dissociation limit for v is reached, Mc Bride and Gordon (1967, 1992) and Beyer (1982)). The different methods all yield similar results (within 3% for nitrogen, for example) except when the fundamental state is a multiplet (e.g., Oþ 2 , or NO), in which case a method with a limitation must be used. The main problem in calculating the molecular partition function often arises when the interactions between the various excited states complicate the calculation (see Mayer and Mayer 1960).

224

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3.4.3 Data Base Data for atoms are available in spectroscopic tables such as those of C.E. Moore (1949, 1952, 1958) and Sansonetti and Martin (2009). These tables first give the denomination of the spectral level according to the L-S or the j-j coupling, (see ▶ Chap. 2, “Basic Atomic and Molecular Theory”), followed by its energy expressed in cm1 (in Eq. 3. Boltzmann’s constant must be modified to k=hc ¼ 0:695). The degeneracy g is given by (2 J + 1), where J is the angular momentum of the spectral level under consideration. The total number of levels to be accounted for depends on both the temperature and energy of each level. For example, for T 13,000 K for an argon atom), the density of the considered species is already low (see Fig. 1). It should be pointed out, however, that the use of different limitation theories for the different plasma components could introduce rather significant errors (up to 30%) in the composition and thermodynamic properties. It is therefore important when using tables at high temperatures (T >8,000 K) to use data from only one source, if possible.

4.3.3 Composition of Simple Plasma Gases First, we will discuss the gases most commonly used in plasma generators: argon, nitrogen, hydrogen, helium, and oxygen (see Fauchais et al. 1969, 1975; Drellishak 1963; Capitelli et al. 1969; Pateyron et al. 1992; Murphy 1995; Murphy and Arundell 1994). Argon and helium are the simplest. For argon at atmospheric pressure and temperatures below 35,000 K, the following species must be taken into account: 1 mole of Ar ðT0 , p0 Þ ) n0Ar  Ar þ n0Arþ  Arþ þ n0Arþþ Arþþ þ n0Arþþþ  Arþþþ þ n0e  e ðT, pÞ

Three equilibrium constants corresponding to the three successive ionization steps Ar⇄Arþ þ e, Arþ ⇄Arþþ þ e and Arþþ ⇄Arþþþ þ e are necessary. Figure 1 shows the temperature dependence of the argon plasma equilibrium composition at atmospheric pressure, starting from room temperature. As the temperature increases, the particle density (NAr/V) of argon atoms decreases monotonically due to progressive ionization, which is completed at about 15,000 K. At still higher temperatures, the density of Ar+ decreases steadily with the appearance of Ar++, while the electron density remains almost constant. The total particle

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M. I. Boulos et al.

p density nT decreases with temperature for a given pressure (nT ¼ kT in the ideal case). If the calculations are performed at higher temperatures, they show that the density of Ar+++ starts to rise over 1015 at 19,000 K. Figure 1 also illustrates a very important point: the steep variation in the particle densities with temperature. For example, ne varies over five orders of magnitude between 4,000 and 10,000 K while remaining almost constant for T >15,000 K. It is also important to note that for T 20,000 K), He+ density is only three to four times higher than Ar++ density. Water Plasma Aubreton et al. (2009a) have calculated the composition of water plasma (e/H/O mixtures) in the temperature ranges of 400–30 000 K (p = 0.05 and 0.1 MPa) and 500–30,000 K (p = 0.5 and 1 MPa). The ten following species were considered: e, H, O, H+, O+, O++, H2, O2, OH, and H2O. Figure 12 presents the plasma composition in molar fractions at 1 MPa. It can be seen that the high pressure promotes the minor species. As an example, xHO2 ¼ 2  104 at p = 1 MPa and T = 3,900 K, while for p = 0.05 MPa xHO2 ¼ 4:5  105 at 3,300 K. In fact, three types of species can be observed: negligible ones xi 6¼ 0 and K X

!

ni  mi  ¼ 0

(32)

i¼1

This allows defining the diffusion velocity of chemical species i, which is the flow rate of particles of type i with respect to the mass average velocity of the gas: !

!

!

U i ðr, tÞ ¼ < v i  v o >

(33)

Clearly, it is also the average of the peculiar velocity: ð ! ! ! < U i > ¼ 1=ni U i  f i  dv i

(34)

All these quantities can be calculated only if the distribution functions fi are known through the solution of the Boltzmann equation. As was shown in ▶ Chap. 3, “Kinetic Theory of Gases,” Sect. 6.3, the Boltzmann equation for a single chemical species i can be written as

Df / i =D / cf

ð35Þ

where D / is the operator:

F ∂ + vi ⋅∇r + i ⋅∇v ð36Þ ∂t mi ! ! ∇v is the gradient of the velocity components i.e., @=@vx, @=@vy, @=@vz , while ∇r X / c f is the collision term (written as corresponds to @=@x, @=@y, @=@z, and D cij ). / = D

j

Since collisions are the key point allowing calculation of the distribution function and thus the transport properties, deviation angles and cross sections for collisions must first be defined (Child (1974)). The first assumption is that the internal degrees of freedom of the particles are unaffected by elastic collisions. The interactions between particles depend only on their relative positions and velocities.

7

Transport Properties of Gases Under Plasma Conditions

269 !

!

The two colliding particles, 1 and 2, have initial velocities v 1 and v 2 , so they ! ! ! move with a relative velocity V ¼ v 1  v 2. After the collision, their velocities are ! !0 ! ! ! v 1 and v 02 and they move with a relative velocity V 0 ¼ v 01  v 02 . Conservation of the total momentum allows a change of coordinates because the ! change implies that the velocity of the center of mass v c is time independent, and thus calculations can be developed in a frame of reference that moves with the center of mass. Assuming elastic collisions implies that the total kinetic energy of the two that V2 = V0 2. Thus, ! colliding ! particles remains unchanged in a collision, so ! 0 V ¼ V , and the only effect of the collision is to change V in direction but not in magnitude. ! ! In the frame relative to the center of mass, the position vectors r 1 and r 2 have opposite directions, and their magnitude has a fixed ratio. It is then easy to demonstrate that the motion of the two particles can be reduced to the solution of a simple one-particle problem in which particle 2 is thought of as the target. The scattering process appears (as shown in Fig. 5) to be the motion of a particle of ! reduced mass μm ¼ mm1þmm2 and velocity V . In Fig. 5, the impact parameter b is 1 2 defined as the distance of closest approach of the particles if there is no interaction between them. The collision process can thus be described by merely specifying the polar angle ! 0 θ and azimuthal angle φ0 of the final relative velocity V 0 with respect to the relative ! velocity V before the collision. ! In Fig. 5, the motion of particle 1, whose mass is μm and velocity is V , can be described in polar coordinates (r, θ) with the origin at the position of particle 2   ! ! ! r ¼ r 1  r 2 . From the equations of motion, it can be demonstrated that !

!

!

r  V ¼K !

(37)

!

which means that the plane defined by V and r moves parallel to itself (orthogonal ! to the fixed vector K ).

Fig. 5 Schematic of the scattering process V⬘ 1⬘ 1

V b

q

r

q⬘ 2

b

270

M. I. Boulos et al.

The force acting on the particles is a function of the distance between them. It is more convenient to use the potential energy of interaction V(r) rather than the force of interaction !

F ðr Þ ¼  0

The deviation angle θ is then defined as 1 ð

0

θ ¼π2 rm

dVðrÞ dr

(38)

dr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 2 r4 =b  r2  2r4 VðrÞ= μm g212 b2

(39) !

dr ¼ 0, and g12 ¼ j V j where rm is the distance of closest approach, defined by dθ ! ! ¼ j v 1  v 2 j. The quantity θ0 is not an observable quantity because it is impossible to single out a particular impact parameter on a molecular scale. The observable quantity is σ0 , the scattering cross section, which can be measured from molecular beam experiments or calculated through θ0 once the interaction potential V(r) is known. ! In the classical mechanical approach, particles having initial velocities v 1 and ! v 2 can be scattered in their relative motion through various angles θ0 and φ0 . The angles are functions of b, and the scattering process must be described in statistical terms in order to take into account the particle distributions f1 and f2 before the collision and f0 1 and f0 2 after the This is  done in terms of the quantity σ0 ,  collision.! !

! ! ! ! which is defined so that σ0 v 1 , v 2 ! v0 1 , v0 2 dv0 1 dv0 2 is the number of

particles per unit time (per unit flux of type 1 particles incident with relative ! velocity V upon a type 2 particle) emerging after scattering with respective final ! !

! ! !

! velocities between v0 1 and v0 1 þ dv0 1 and between v0 2 and v0 2 þ dv0 2. According !0 !0 to conservation of momentum ! !and energy, v 1 and v 2 must have a constant center of mass velocity and V ¼ V 0 . Thus, σ0 = 0 if these conditions are not fullfilled. Some symmetry properties of σ0 are important in deriving the Boltzmann equation: σ0 is invariant • For reversed collisions  (reversed  velocities) • For space inversion

!

!

r !  r , especially for the “inverse” collision (original

collision with initial and final states interchanged). A simpler but less symmetrical quantity, the differential scattering cross section, ! can also be introduced (see Sect. 2.2): σ V  dΩ0 is the number of particles !

scattered per unit time and emerging with a final relative velocity V 0 in a direction given by the solid angle dΩ0 about θ0 and φ0 . From these definitions, it follows:

7

Transport Properties of Gases Under Plasma Conditions

271

ðð  !  

! ! ! ! ! ! ! σ V 0 dΩ0 ¼ σ0 v 1 , v 2 ! v 01 , v 02 dv0 1  dv 0 2

(40)

! ! vc0 V 0

! vc0

is the velocity of the center of mass after collision. The integration is over where ! ! 0 all values  !  of vc0 and V . 0 σ V , often defined as σ(θ0 , φ0 ), can be easily related to the impact parameter b when considering all the particles of the incident beam whose impact parameter is between b and b + db. The number of collisions, if ϕ is the flux of incident particles with a unique ! relative velocity V, is given by ϕ.2.π.b.db, while the number of scattered particles is: !  ϕσ V , θ 2π sin θ0 dθ0 (the scattering is supposed to be symmetric for the φ0 angles, whose integration gives 2π), and thus ! σðV ÞdΩ is the collision cross section for particles emerging after collision with a ! relative velocity V 0 into a solid angle range dΩ0 about θ0 and φ0  !  b db (41) σ V , θ0 ¼  sin θ0 dθ0 The absolute value is taken because in experiments σ is measured in crossed beams of atomic or molecular jets, and since diffusion occurs in all directions, it is not possible to distinguish between positive and negative deflection angles. The total elastic collision cross section is defined as 2ðπð π

QðVÞ ¼ 0 0

σ0 ðθ0 , φ0 , VÞ sin θ0 dθ0 dφ0

ðπ

(42) 0

0

0

¼ 2π σ ðθ , VÞ sin θ dθ

0

0

With these definitions, it is not too difficult, for a single chemical species, to show / c f is given by the following triple integral (integrated over that the collision term D ! ! all the possible velocities v 2 of the target particles and the resulting velocities v 1 ! and v 2 of particles 1 and 2 after collision):

/ cf = D

∫ ∫ ∫ (f2′f1′ − f1f 2) ⋅ V ⋅ σ′(v1, v2 → v1′, v ′2) ⋅ dv′1 ⋅ dv′2 ⋅ dv 2

v2 v2′ v1′

ð43Þ

The distribution functions f1 and f2 are related to the colliding particles before collision when f0 1 and f0 2 are the distributions after collision. If σ(Ω0 ) (see Eq. 40) is used instead of σ0 , we have

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M. I. Boulos et al.

/ cf = D

∫ ∫ (f 2′f1′ − f 2f1) ⋅ v ⋅ σ′ ⋅ d Ω′dv 2

ð44Þ

v2 Ω′

  ! ! ! ! ! ! In these Eqs. 43 and 44, σ0 ¼ σ0 v 1 , v 2 ! v 01 , v 02 dv 01  dv 02 is the collision cross !

!

!

!

section for particles with initial velocities between v 1 and v 1 þ dv 1 and v 2 and ! ! ! ! ! ! v 2 þ dv 2 and final velocities (after collision) between v 01 and v 01 þ dv 01 and v 02 and !0 !0 v 2 þ dv 2 / c f term causes all the difficulties in solving the Boltzmann Of course, the D integrodifferential equation. It is important to note that the expression of the collision term has been established (see ▶ Chap. 3, “Kinetic Theory of Gases,” Sect. 6.3) assuming that f does not vary appreciably during a time interval on the order of the free flight time, i.e., over a spatial distance on the order of the range of intermolecular forces. Thus a shielded Coulomb’s potential is used to keep this assumption not too far from reality for charged particles, with their long-range interaction potential. When K different chemical species are considered (electrons and heavy species), K equations must be written for the K species. For example, given a particle of chemical species i, it follows: K

Df / i = ∑ ∫∫ (fi′ ⋅ f j′− fi .f j ) ⋅ Vij ⋅ σij ⋅ dv j ⋅ dΩ = D / c fi

ð45Þ

j=1

The solution of these K Boltzmann equations gives K distribution functions, allowing for the calculation of the mean values of the various properties (i.e., macroscopic properties) and fluxes.

4.2

Fluxes

As defined in ▶ Chap. 3, “Kinetic Theory of Gases,” Sect. 6.2, the total flux of a quantity χi (related to a chemical species i) across an elementary surface oriented by ! ! its normal n and moving in the flow direction with the mean flow velocity v 0 (see Eq. 29) is given by ð ! ! ! ! J i ¼ χi f i U i n dU i (46) The different fluxes outlined in Sect. 1 are related to the transport of mass, momentum, and energy. With χi ¼ mi , we have ð

! ! ! J i ¼ mi f i  U i  dU i ¼ ni  mi  < U i >

!

(47)

7

Transport Properties of Gases Under Plasma Conditions

273

!

where J i is the mass flux vector and < Ui > the mean peculiar velocity (see Eq. 34). ! With χi ¼ mi Uix where Uix is the component of peculiar velocity U i in the x direction, we can write ð ! !

! Jix ¼ mi Uix  U i  f i  dU i ¼ ni  mi  < Uix  U i > (48) This is the flux vector associated with the transport of the χ component of ! momemtum (relative to v 0 ). This vector has components proportional to Uix  Uix, Uix  Uiy, and Uix  Uiz. Similar flux vectors are obtained for the y and z components of the momentum, making a total of three flux vectors associated with momentum transfer. The components of the three vectors form a symmetrical second-order tensor pi corresponding to a partial pressure and given by !

!

pi ¼ ni mi < U i  U i >

(49)

representing the flux of momentum through the gas. With χi ¼ 12 mi U2i , it follows that ð  ! ! 1 1

! 2 ¼ mi Ui  U i  f i  dU i ¼ mi ni hU2i  U i i (50) 2 2 ! q i is the flux vector associated with the transport of kinetic energy of particles of species i. The sum of these vectors over all K components gives the heat flux vector ! q , whose components qx, qy, and qz represent the flux of kinetic energy in the x, y, and z directions, respectively. Calculation of the mass flux vector, the pressure tensor, and the heat flux vector ! q will allow the determination of the transport coefficients, provided f is known. Thus, the main problem is to determine f. The fundamental hydrodynamic equations of continuity, motion, and energy balance can be derived from the Boltzmann equation without determining the form of the distribution function fi. To do this (see Hirschfelder et al. (1964)), the Boltzmann equation (45) for the ith component is multiplied by the quantity χi associated with the ith species and ! integrated over v i . This new set of equations is called the transfer equations. It can then be demonstrated that the integrals of the right-hand side, i.e., the ones ! related to collisions, vanish if χi is mi, mi U i , or 12 miU2i , yielding the well-known Navier–Stokes equations relative to mean values. ! qi

4.3

Calculation of Distribution Functions

As in the preceding section, only the outline of the calculation procedure will be presented for equilibrium plasmas.

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The theoretical study of transport coefficients for equilibrium plasmas was first developed for monatomic gases through the Boltzmann equation. The results were then modified by Hirschfelder et al. (1964), Chapman and Cowling (1952), Devoto (1966). Devoto (1967), Delcroix and Bers (1994) to account for the high temperatures and the presence of excited particles and ions. A series solution of the Boltzmann equation can be obtained by introducing a perturbation parameter ξ in such a manner that the frequency of collisions can be varied in an arbitrary manner, without affecting the relative number of collisions of a particular kind. The Boltzmann equation is then written as;

Df / i=

1 D / cf i ξ

ð51Þ

l/ξ measures the frequency of collisions (and is proportional to σ). (If ξ is very small, collisions would be very frequent and the gas would behave like a continuum in which local equilibrium is maintained everywhere.) The distribution function is expanded in a series as follows f i ¼ f i ð 0Þ þ ξ  f i ð 1Þ þ ξ 2  f i ð 2Þ þ . . .

(52)

fi(0) is obtained by assuming that the right-hand side of the Boltzmann equation, which depends only on collisions, must be zero when equilibrium is reached. With the help of the H-theorem of Boltzmann, it can be shown (Hirschfelder et al. (1964)) that f i ð 0Þ ¼ ni 



  mi 3=2 mi  U2i  exp  2kT 2kT

(53) !

which corresponds to Maxwellian distributions for the peculiar velocity U i at a unique temperature T (fi(0) is usually denoted fi(0)). In Enskog’s first-order approximation, fi(1) is written as f i ð 1Þ ¼ f i ð 0Þ  ϕi

(54)

where ϕi is a perturbation function (ϕi  1). The perturbation function ϕi depends only on space and time via the species densities, the average flow velocity, the temperature, and their space derivatives (see Eqs. 56 and 57). ϕi is linear in the derivatives. The coefficients appearing in the expression of ϕi are then expanded in a finite series of Sonine’s polynomials Athye (1965), Devoto (1965), Devoto (1967), Devoto (1968), Devoto and Li (1968), Devoto (1973). The final results, i.e., the transport coefficients, are expressed as function of complex quantities called bracket integral, which are themselves functions (through collision integrals) of the interaction potentials characterizing the different collisions.

7

Transport Properties of Gases Under Plasma Conditions

5

Transport Properties of Equilibrium Plasmas

5.1

Main Parameters

275

In thermal plasmas, equilibrium is assumed. According to the high electron number density, ne, elastic collision frequencies are very high, so energy transfer is important and leads to an equal distribution of the energy between the mean heavy particles energy and that of electron energy. In principle, tracking all collisions between all particles in the thermal plasma flow would allow accurately calculating transport properties, but even with supercomputers the task is not yet possible. Thus, as presented summarily before, one uses statistical means representing the state of the gas by distribution functions based on the most probable mechanical states of the system (Bottin et al. (2006)). The theory of the transport properties of gas mixtures is based on solving the Boltzmann integral differential equation describing the evolution of particles’ energy distribution by the Chapman–Enskog method Chapman and Cowling (1952) applied to complex mixtures. The calculation has been developed by Hirschfelder et al. (1964). The Boltzmann equation is solved by expressing the distribution function as a series of the type shown in Eq. 52, the first-order expression, as shown in Eq. 54, being generally used. In principle, such calculations were established for weakly ionized gases, but Butler and Brokaw (1957) demonstrated they were valid for thermal plasmas (see also the reviews of Murphy and Arundell 1994; Gleizes et al. 2005; Bottin et al. 2006). It is necessary to know the plasma composition (particle number densities) and the collision integrals Ω‘,s i,j , as defined by Hirschfelder et al. (1964). The particle number densities for minority atomic species at high temperatures are not very precise, and the uncertainties may affect the viscosity calculation (Capitelli 1972). Ω‘,s ij depends on the potential characterizing the interaction between particles i and j: !1=2 ð     1 kT ‘, s Ωij ¼ exp γ2ij  γ2sþ1  Qlij gij  dγij (55) ij 2πμij 0 with γij ¼

μij  g2ij

and μij ¼

mi  mj mi þ mj

(56) ! ! gij ¼ v i  v j is the absolute value of the relative velocity of the colliding 2kT

particles, and Ω‘ij is the quantum collision cross section. The two parameters Ω(1,1) and s define the order of the collision integrals, and their values depend on i,j the number of terms in the Sonine polynomials used to calculate the perturbation function (see Eq. 54). This expansion is best carried out with Laguerre–Sonine polynomials since they possess certain orthogonality properties, which simplify the mathematical analysis (Bottin et al. 2006). The expansion is limited to a certain number because the transport properties computed with the Sonine

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expansion asymptotically tend to the properties computed with the exact Chapman–Enskog procedure. The second problem is the calculation of the quantum collision cross section. The fundamental defect in the classical theory is that no particle trajectory can ever be precisely defined because of the inherent uncertainties associated with it. This uncertainty problem is solved by employing a wave function and its associated interference pattern in place of the limiting deflection for a well-defined classical trajectory (see Sect. 3.1). The precise connection between the scattering amplitude of this wave function and the differential cross section is obtained by comparing the incident flux with the rate of scattering into a given solid angle dΩ. It can be shown that for the case of elastic scattering with a potential V(r) the wave function can be expanded with Legendre functions. Depending on the degree ‘ of the development, the weighted corresponding cross sections are given by ðπ   !  !
Q V ¼ 2π σ θ0 , V  1  cos ‘ θ0  sin θ0  dθ0 ‘

(57)

0

or, in terms of impact parameter, ðπ  !
 Q V ¼ 2π 1  cos ‘ ðθ0 Þ  b  db ‘

(58)

0

As already emphasized, these collision cross sections depend on the interaction potential V(r). An important difficulty is the choice of the best-adapted potential, because many of them exist (Bottin et al. 2006) and the choice also depends on the interacting particles. Bottin et al. (2006) have presented the most used potentials according to the interaction types: Neutral-neutral: different simplifying assumptions are used where the simplest is that of rigid spheres of diameter d:  φ¼

1, 0,

r < d, r > d:

(59)

which is convenient for first-hand calculations. The Sutherland potential is also rather simple:  φ¼

1,
 6 ε dr ,

r < d, r > d:

(60)

The Lennard-Jones potential when long-range interactions dominate φ ¼ 4ε

 

r0 12 r0 6  r r

(61)

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277

Repulsive potential, such as φ ¼ A expðαrÞ:

(62)

A and α being obtained from experiments Charged-charged: Very often, Coulomb potential shielded at the Debye length λD is used: φ ¼ φ0

  λD r exp  λD r

(63)

with φ0 ¼

e 4πε0 λD

(64)

A negative sign in Eq. 63 stands for attractive (ion–electron) interactions, whereas a positive sign holds for repulsive (ion–ion and electron–electron) interactions. Ion-neutral: Often treated as a neutral–neutral interaction, except if the exchange is resonant, such as a transfer of charges. From a practical point of view, collision integrals are calculated as follows: • If the transfer cross section Q‘ is known (from experience or theory), Eq. 55 is integrated numerically. • If the interaction potential of the collision is known (Q‘ depends directly on it through the scattering angle θ0 ), the determination of Ω‘,s ij is performed with the help of tables (such as those of Mason et al. 1967; Monchick 1959; Smith and Munn 1964; Levin et al. 1989). When the interaction involves different excited states, each with its own potential, collision integrals are calculated successively for each potential and then a mean collision integral is calculated such as P Ω ‘, s ¼

 Ωk‘, s k gk

gk kP

(65)

where g is the statistical weight of the potential k. The transport coefficients are calculated using the well-known Chapman-Enskog method (Chapman and Cowling 1952) with different levels of approximation (number of terms used in the finite Sonine polynomial expansion), see, for example, Murphy and Arundell (1994) Viscosity, μ: is calculated, according to Hirschfelder et al. (1964), to a first approximation Electrical conductivity, σ: is calculated to a third-order approximation, neglecting the contribution of the ions, as described by Devoto (1968).

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Thermal conductivity, κ: is written as the sum of three components, respectively the translational, internal, and reaction thermal conductivities: κ ¼ κtr þ κint þ κreac

(66)

κtr can be broken down into contributions from ions, atoms, and molecules (κh) and from electrons (κe) κtr ¼ κh þ κe

(67)

κh is calculated to a second-order approximation (Devoto 1968); higher-order approximations are necessary for electron transport. If, at temperatures at which electron-neutral collisions dominate, the third-order approximation may still be insufficient, at these temperatures κh κe second-order approximation is still accurate (Murphy and Arundell 1994). κint must be calculated to the second order, but its value is generally small compared to κtr and κreac. The reaction thermal conductivity corresponds to ionized or dissociated species releasing the reaction enthalpy when the reverse reaction occurs after diffusion into a lower temperature region. Calculations are performed according to expressions derived by Butler and Brokaw (1957) and Brokaw (1960). Very often the electronically excited states are not included in such calculations, in spite of the fact (Capitelli and Devoto 1975) that low-lying excited states of nitrogen atoms and ions can influence the charge-exchange reactions, when calculating the thermal conductivity of nitrogen plasma. A large volume of data has been published for the thermodynamic and transport properties of gases under thermal plasma conditions. One has to be very careful when using such data, especially transport properties, because the collision integrals on which the calculations are based are not always well known, and thus large uncertainties may result. To illustrate this evolution, Aubreton et al. (2013) used the interaction potentials given by three authors, among several, Capitelli and Devoto (1975), Murphy and Arundell (1994), and Aubreton and Elchinger (2004), to calculate the thermal conductivity of nitrogen plasma at atmospheric pressure. The corresponding results are presented in Fig. 6. With potentials of Capitelli and Devoto (1975) very divergent results were obtained. This figure illustrates the importance to have good description of interaction (i.e., accurate potential interactions). For example, Fig. 7 shows the influence of such uncertainties on the H-H interaction potentials for the thermal conductivity of hydrogen. Similarly, Murphy (2000) has shown the large discrepancies appearing at the peak of thermal conductivity at around 3800 K, as illustrated in Fig. 8. Diffusion coefficients: In the case of mixtures, multicomponent diffusion coefficients must be computed by means of linear systems (Rat et al. 2001). The combined diffusion coefficients are defined by an expression for the mass flux of gas A relative to the mass-averaged velocity (Murphy 1993, 1994, 1996, 2000). The combined diffusion coefficients have the advantages that only one “gas” conservation equation needs be solved, rather than N-1 species conservation equations for a plasma containing N species, and that at most four diffusion coefficients need be calculated, rather than

Transport Properties of Gases Under Plasma Conditions

Fig. 6 Dependence on temperature of thermal conductivities of a nitrogen plasma at p = 100 kPa calculated by (Aubreton et al. 2013) using interaction potentials of Capitelli (1975), Murphy and Arundell (1994) and Aubreton and Elchinger (2004), (Aubreton et al. 2013)

279

6 Total thermal conductivity, κ (W/m.K)

7

Murphy and Arundell (1994) Capitelli and Devoto (1973) 4

2 Aubreton and Elchinger (2004). 0

10

5

15

20

Temperature , T (103 K)

Hydrogen, Pressure =100 kPa

(W/m.K)

30

Belayev et al (1967) 20

Thermal conductivity,

Fig. 7 Temperature dependence of the thermal conductivity of the hydrogen plasma for two different values of the H-H interaction potential at atmospheric pressure (Gorse 1975; Boulos et al. 1994)

Vanderslice et al (1962)

10

0 0

5 10 15 Temperature, T (103 K)

20

one ordinary diffusion coefficient for each pair of species and one thermal diffusion coefficient for each species Murphy (2012). The corresponding mass flux JA of gas A relative to the mass–averaged velocity is given by JA ¼

  n2 mA mB DxAB ∇xB þ DPAB ∇lnP þ DEAB E  DTAB ∇lnT ρ

(68)

DxAB , DPAB , DTAB , and DEAB are, respectively, the combined ordinary, pressure, temperature, and electric field diffusion coefficients, mA and mB are the number–density–weighted average mass of the species present in gas A and gas B respectively, xB is the sum of the mole fractions of the species of gas B, n is the

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Murphy (2000) Thermal conductivity, κ (W/m.K)

20

Baronnet et al. (1985)

Aubreton and Fauchais (1983)

18

Boulos et al. (1994)

16 Behringer et al. (1968) 14

12

Capitelli et al. (1976)

10 3.0

3.5

4.0

4.5

5.0

Temperature, T (103 K)

Fig. 8 Detail of the peak centered on 3800 K of the thermal conductivity of hydrogen (Aubreton and Fauchais 1983; Behringer et al. 1968; Murphy 2000)

number density, ρ is the mass density, P is the total pressure, T is the temperature, and E is the externally applied electric field. Murphy (2000) gives the different expressions of DxAB , DPAB , DTAB , and DEAB and also presents the relations between these coefficients: p q @xj 1 X X si mj Daij mB i¼2 j¼1 @xB   p q ρj @xj 1 X X ¼ si mj Daij xj  þ P mB i¼2 j¼1 ρ @P ! p q Ta 2 X D n X a @xj i ¼ mA si  mj Dij T mi ρ j¼1 @P i¼2 p q X X e 1 ¼ si mj xj Zj Dij kB T mB i¼1 j¼1

DxAB ¼ DPAB DTAB DEAB

(69)

These coefficients obey the following relationship DPAB ¼ DPBA , DTAB ¼ DTBA , and DEAB ¼ DEBA :DxAB ¼ DxBA ;

(70)

nj, ρj, xj, Zj, and mj are, respectively, the number density, mass density, mole fraction, charge number, and mass of the jth species. The electronic charge is e, and Boltzmann’s constant is kB. Species 1 is the electron, species 2 to p belong to gas A, and species p + 1 to q belong to gas B. The si are stoichiometric coefficients, equal to the number of atoms contained in a molecule of species i, normalized to the

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Transport Properties of Gases Under Plasma Conditions

281

number–density–weighted average number of atoms contained in all the species of the gas of which species i is a component.

5.2

Second Parameters

The calculation of partition functions plays a key role on the one hand on plasma composition and thermodynamic properties and on the other on transport properties (Aubreton et al. 2013). The databases to determine the partition functions for diatomic and polyatomic molecules have been described in Aubreton et al. (2009b) with data from Gurvich et al. (1989) and NIST (2006). In thermal plasma, the partition function is the product of two independent terms: the translational partition function and the internal partition function neglecting nuclear partition function Aubreton et al. (2013). Unfortunately, the evaluation of the internal partition functions at high temperature is extremely complex: missing energy levels in tables, use of cutoff criteria, etc. Concerning the cutoff criteria, one can use only the reduction of ionization energy using the Debye-H€ uckel approach (A), introduce a cutoff criterion using the principal quantum number n depending only on the gas or plasma densities (B), or simplify partition functions for which we take into account only the low electronic levels (En < EI/2) (C) as did Aubreton et al. (2013). These authors choose Ar–Cu mixtures (the ionization 1 1 energies are quite different): EAr and ECu I ¼ 127, 000 cm I ¼ 62, 000 cm ) to show the influence of plasma composition determination on transport properties. For example Fig. 9 presents the total thermal conductivity and the four contributions versus temperature, for p = 0.1 MPa and 25% Ar–75% Cu mixture in molar percentage. The temperature range was limited to 3000 to 13,000 K to emphasize the discrepancy between the results obtained with methods A, B, and C. Below 3000 K population of excited levels is low, and at T > 13,000 K, the total thermal conductivity is controlled by the translational thermal conductivity of electrons,

(W/m.K)

1.2

Thermal conductivity,

Fig. 9 Total thermal conductivity and its four contributions calculated with methods A, B and C (see text) total partition function for p = 100 kPa, Ar/Cu 75% molar (Aubreton et al. 2013)

Method C Method A & B

0.8 Ar/Cu 75% molar, p=100 kPa 0.4

0.0 3.0

5.0

7.0 9.0 11.0 Temperature, T (103 K)

13.0

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M. I. Boulos et al.

which is the same for the three methods. The discrepancy is essentially observed between the results obtained with the three theories, only for inelastic processes: Kint and Kreac. We will not comment other results of this type, our main objective being to point out that this problem, very often neglected in transport properties’ calculations, is not necessarily negligible.

6

Transport Coefficients of Gases in CTE

As in Sect. 4, when the distribution functions are known, transport properties can be calculated if either the interaction potentials or cross sections of the different elementary collisions involved are known. However, according to the uncertainties in the determination of the distribution functions, the accuracy of these calculations is far below that of the thermodynamic properties.

6.1

Examples for Simple Gases

The results obtained for the properties of the simple gases most commonly used in plasmas (H2, N2, Ar, and He) are of primary importance in understanding heat and momentum transfer between plasmas and condensed particles. In the following examples, all the results will be presented at atmospheric pressure.

6.1.1 Electrical Conductivity In the presence of an electric field (in arcs or RF discharges, for example) or under the action of other driving forces such as temperature, pressure, or concentration gradients, transport within the plasma occurs by electric charges that constitute an electric current. As shown in Sect. 3.4, using oversimplified expressions, the electrical conductivity depends mainly on the electrons’ density. The collision integrals are calculated to a third-order approximation, neglecting the contribution of the ions. As for viscosity, the choice of potentials is important (Murphy and Arundell 1994). Figure 3, shows the evolution of the electrical conductivity σe (logarithmic scale) versus temperature at atmospheric pressure for He, Ar, N2, and H2. The electrical conductivity increases sharply with temperature and reaches a limiting value of 10, 000ðΩ:mÞ1 or (S/m) before second ionizations occur. The influence of ionization on the electrical conductivity is obvious: the plasma is electrically conducting when the gas is sufficiently ionized (simplified Eq. 25) shows that σe is proportional to the electron density ne, i.e., the degree of ionization of the plasma). Ar, N2, and H2, which have comparable ionization energies, consequently have about the same electrical conductivity (ne) at the same temper1 ature. For example, at 10,000 K, σe is about 2000 ðΩ:mÞ for these three gases, 1 while He has a very low value of σe ¼ 500 ðΩ:mÞ due to its higher ionization energy. Oxygen’s electrical conductivity is quite similar to those of Ar, N2, and H2

Transport Properties of Gases Under Plasma Conditions

Fig. 10 Temperature dependence of the electrical conductivity of O2 by (Murphy and Arundell 1994; Neumann and Sacklowski 1968)

283

14 Electrical conductivity, se (kA / V.m)

7

12 10 8 6 4 Neumann and Sacklowski (1968)

2 0 0

10

20

30

40

3

Temperature, T (10 K)

(Murphy and Arundell 1994). For these gases, σe is almost negligible below 7000 K. It is easier to see it when σe is represented in a linear scale as for oxygen in Fig. 10.

6.1.2 Viscosity The viscosity coefficient establishes the proportionality between the friction force in the direction of the flow and the velocity gradient in the orthogonal direction. Calculation of this coefficient requires the first approximation of the pressure tensor according to Chapman-Enskog (see Chapman and Cowling 1952). As a first approximation, μ is proportional to the square root of the product of mass and temperature and inversely proportional to the Ω(2,2) collision integral, at i,j least for charged particle molar fractions below 1%. This is illustrated by comparing Ar and He values presented in Fig. 1: in spite of the fact that mAr =mHe ffi 10, the 2, 2 2, 2 viscosity ofpHe ffiffiffiffiffi up to 10,000 K is almost equal to that of Ar because ΩArAr =ΩHeHe is close to 10. As the temperature reaches 10,000 K, where ionization is pertinent (see Fig. 2), the viscosity of Ar starts to decrease due to the reduction of the charged particle mobility as the long-range Coulomb forces induce a lowering of the transport of momentum. For pure He this decrease is not observed until T = 17,000 K due to the higher ionization potential of He (see Fig. 11 of ▶ Chap. 6, “Thermodynamic Properties of Plasmas”). Up to these two maxima the variation of pffiffiffi μ with T is a good first approximation because the collision integral decreases slowly with temperature. For diatomic gases, dissociation results in a change of the slope of μ (T), as is illustrated by the behavior of H2 (see Figs. 5 and 1 of ▶ Chap. 6, “Thermodynamic Properties of Plasmas”), where the change in the slope at 3500 K is due to the change in collision integrals (from H2– H2 to H– H) and the pffiffiffi corresponding change from 2 for mH2 to 1 for mH. As it does for Ar, μ for H2 starts to decrease at 11,000 K due to the presence of charged species. Similar results were obtained with oxygen: maximum value at about 10,000 K, slight slope change around 4300 K when dissociation is achieved (Murphy and Arundell 1994).

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Capitelli et al (1976) Gorse (1975)

Viscosity, µ (10–5 kg/m.s)

30

Amdur & Mason (1958) Kulik et al (1963) Kulik (1971) Schreiber et al (1972) Vargaftik (1975) Aeschliman & Cambell (1970) Kanazawa & Kimura (1967) Bonilla et al (1956) Asinouski et al (1967) Kannappan & Bose (1973) Bahadori & Soo (1966)

20

10 Argon Pressure = 100 kPa 0 0

5

10

15

20

25

Temperature, T (103 K)

Fig. 11 Temperature dependence of the viscosity of argon at atmospheric pressure (Capitelli et al. 1976; Lesinski and Boulos 1990)

The viscosity of plasmas at 10,000 K lies typically in the range from 0.0310 to 0.3310 kg/m.s. The maximum values are approximately ten times higher than those of the same gases at room temperature. This difference partly explains the difficulty of mixing cold gases with plasmas or introducing solid particulates into a thermal plasma stream in the temperature range from 5000 K to 10,000 K. Note that the choice of the interaction potentials to be used in the calculation of the collision integrals plays a very important role in calculating the viscosity, as illustrated in Fig. 11 (Lesinski and Boulos 1990) for argon. The agreement is better for temperatures above 12,000 K because most authors use the same calculations for the interactions between charged particles. Murphy and Arundell (1994) showed for argon viscosity that at low temperature, the agreement with experimental measurements was within 1.5%, while deviations up to 3% occurred around 10,000 K, differences due to the potential used. The viscosity of Ar has been calculated by Chen et al. (1996) for different interaction potentials and values recommended. Polynomial expressions have been developed for calculating the argon viscosities, which will be useful for numerical work and other applications in thermal argon plasmas.

6.1.3 Thermal Conductivity The thermal conductivity is of primary importance for thermal plasmas. It controls the energy losses in the arc or RF discharge fringes and thus the discharge behavior as well as the heat transfer to solid materials or particulates in flight. The translational thermal conductivity is written as the sum of two terms, one due to electrons and the other due

7

Transport Properties of Gases Under Plasma Conditions

285

to heavy particles (Eq. 67). It was shown in Sect. 3.3 in the discussion on equilibrium composition that at high temperatures, the plasma gas is dissociated and ionized. The dissociation and ionization phenomena make a large contribution to energy transport, and the corresponding term has a high value at the temperatures where these phenomena occur. Since each species has an internal energy component (due to vibrational, rotational, and electronic excitation) at high temperature, energy is transferred through inelastic collisions of the second kind. This effect is taken into account by the internal thermal conductivity. The total thermal conductivity is considered to be the sum of these three contributions (Eq. 66). The contributions of these three terms will now be considered using nitrogen as an example, since it has been studied in detail from an equilibrium point of view. Figure 2 shows that for nitrogen, the translational and internal conductivities of neutral particles make a small contribution to the total thermal conductivity. The main contribution is clearly the reactional one, which shows two maxima corresponding to the dissociation and ionization energies. At temperatures higher than 10,000 K, the translational thermal conductivity of electrons becomes significant. It is interesting to compare Fig. 2 with Fig. 12, which shows the different components of thermal conductivity of argon as a function of temperature. No dissociation occurs for argon at low temperatures (T < 10,000 K), where the translational contribution of neutral species is the most significant. The reactional contribution, along with the contribution of the electrons, becomes important at 10,000 K because Ar is significantly ionized (see Fig. 11 of ▶ Chap. 6, “Thermodynamic Properties of Plasmas”). At very high temperatures, when the first ionization is completed, the translational thermal conductivity of the electrons is mainly responsible for the energy transfer until second ionization occurs. Errors of a factor of 2 can occur in the translational thermal conductivity due to uncertainties in the interaction potentials. The same uncertainties may occur in the other transport properties. The data in Figs. 13 and 14 represent the results of both experimental and theoretical studies compiled by Lesinski and Boulos (1990) for argon and nitrogen at atmospheric pressure. Due to significant experimental 3.0 Thermal conductivity, κ (W/m.K)

Fig. 12 Temperature dependence of the contribution of the different components to the thermal conductivity of argon plasma at atmospheric pressure (Gorse 1975; IUPAC 1982; Boulos et al. 1994)

2.5 2.0 1.5 1.0 0.5 0.0

0

5

10

15

Temperature, T (103 K)

20

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M. I. Boulos et al.

Capitelli et al (1976)

Thermal conductivity, κ (W/m.K)

3.0

Gorse (1975) Morris et al (1970) Emmons (1967) Kulik et al (1963) Kulik (1971)

2.0

Vargaftik (1975) Kanazawa & Kimura (1967) Kannappan & Bose (1973)

1.0

Knopp & Cambel (1966) Jordan & Swift (1973)

0.0

0

Argon Pressure = 100 kPa

5

10

15

20

25

3

Temperature, T (10 K)

Fig. 13 Temperature dependence of the thermal conductivity of argon at atmospheric pressure (Capitelli et al. 1976; Lesinski and Boulos 1990)

Thermal conductivity, κ (W/ m.K)

8.0

Schreiber et al (1972)

Plantikow (1970)

Morris et al (1970)

Yun et al (1962)

Hermann & Schade (1970)

Burhorn (1959)

Capitelli et al (1976)

Vargaftik (1975)

Gorse (1975)

6.0

4.0

2.0 Nitrogen Pressure = 100 kPa 0 0

5

10

15

20

25

3

Temperature, T (10 K)

Fig. 14 Temperature dependence of the thermal conductivity of nitrogen at atmospheric pressure (Lesinski and Boulos 1990)

7

Transport Properties of Gases Under Plasma Conditions

287

difficulties and uncertainties in the theoretical databases used, important differences between the calculated and measured transport properties can be observed.

6.2

Examples for Complex Gas Mixtures

For complex gas mixtures, the calculations become more difficult, first because the number of interaction potentials to be taken into account increases drastically (C2K for K species, including electrons) and second because of the lack of data about interaction potentials (when they are unknown, the species are supposed to be interacting as hard spheres). In the following paragraphs, the transport properties of mixtures such as air, Ar-H2, N2-H2 and Ar-He will be described. Because of their important influence on certain transport properties, mixture-containing metal vapors (e.g., Ar-Cu and air-Cu) will also be included. These combinations of gases have been chosen because air is commonly used in cold electrode vortex plasma torches, Ar-H2 and Ar-He are mainly used in plasma spraying, and finally, Ar-Cu and air-Cu mixtures show the influence of electrode erosion on the plasma properties, metal vapor being also important in welding and cutting.

Fig. 15 Temperature dependence of the electrical conductivity of air at four different pressures (50, 100, 200 and 500 kPa) (Pateyron et al. 1990; Boulos et al. 1994)

Electrical conductivity, σe (kA / V.m)

6.2.1 Electrical Conductivity The electrical conductivity of air is almost independent of the molar ratios of nitrogen or oxygen, since the electrical conductivities of the two components are almost the same. The influence of pressure is illustrated in Fig. 15, which shows the variation of σe versus temperature for air plasma at four different pressures. When the pressure increases, ionization is delayed, and σe decreases slightly at temperatures lower than 12,000 K. At higher temperatures the electron density increases with pressure, and thus σe increases, too. Because σe is proportional to ne, σe is almost the same for the Ar-H2mixture (see Fig. 16) whatever the percentage of hydrogen may be (ionization starts at a temperature 1000 K higher for H2 than for Ar). For T > 22,000 K the increase of 20 500 kPa

Dry Air

200 kPa

15

100 kPa 50 kPa

10

5

0 0

5

10 Temperature, T

15 (103 K)

20

25

Fig. 16 Temperature dependence of the electrical conductivity of Ar-H2 plasmas at atmospheric pressure (Pateyron et al. 1992; Boulos et al. 1994)

M. I. Boulos et al.

Electrical conductivity, se (kA / V.m)

288 12 10 8 6 Pure Ar 4 Ar / 30% H2 molar

Pure H2

2 0 0

5

10

15

20

3

Fig. 17 Temperature dependence of the electrical conductivity of Ar-He plasmas at atmospheric pressure (Pateyron et al. 1992; Boulos et al. 1994)

Electrical conductivity, σe (kA / V.m)

Temperature, T (10 K)

12 Ar / He mixtures, p =100 kPa 10 Pure Ar 50%

8

60% 70% 80%

6

90% He

4 2 0

Pure He 0

5

10 15 Temperature, T (103 K)

20

the Ar density modifies the collision cross sections and thus σe. For the Ar-He mixture (see Fig. 17), σe is mostly controlled by the argon present in the mixture (up to 80 vol.% He). Even for 90 vol.% He and temperatures up to 13,000 K, the electrical conductivity is close to that of pure argon. When copper vapor is introduced into the plasma, the electrical conductivity may be drastically changed, as shown by Mostaghimi and Pfender (1984) for an argon plasma with contamination levels of copper ranging from 0.01% to 5% by volume. Figure 4 shows the corresponding results. The electrical conductivity of the argon plasma at T = 5000 K increases by a factor of 28 when 1% Cu is added. Similar results are obtained for air at atmospheric pressure (see Fig. 18 from Pateyron et al. (1990)). At T > 17,000 K the Cu ions have only a small influence on the electrical conductivity of the plasma. Murphy et al. (2009) showed that for Ar-Fe plasmas the presence of just 1 mol% iron vapor has a very large effect on the electrical conductivity for temperatures up to around 10,000 K, as shown in Fig. 19.

Transport Properties of Gases Under Plasma Conditions

Fig. 18 Temperature dependence of the electrical conductivity of air plasmas at atmospheric pressure in the presence of a few mole percent of copper (Pateyron et al. 1990; Boulos et al. 1994)

Electrical conductivity, σe (A / V.m)

7

289

104 Dry Air-Cu Pressure = 100kPa 103 10% Cu 102

1.0% Cu 0.1% Cu

10

1

Dry Air

0

2

4

6

8

10

12

14

Fig. 19 Temperature dependence of the electrical conductivity of pure Ar and pure iron vapor plasmas at atmospheric pressure, and Ar/iron vapor mixtures with 1 and 10% molar fractions of Fe. (Murphy et al. 2009)

Electrical conductivity, se (kA / V.m)

Temperature, T (103 K)

12 Ar / Fe mixtures, p =100 kPa

10 8

100% Fe Ar / 10% Fe

6

Ar / 1% Fe

4

100% Ar 2 0

0

5

10

15

20

25

Temperature, T (103 K)

These results are typical for metallic impurities in plasmas, since metallic species ionize at lower temperatures than the common plasma gases. Figure 20 presents the evolution with temperature of σe for water plasma at four different pressures Aubreton et al. (2009a). As the ionization degree increases with temperature (for a given pressure), so does the electron density ne, σe increases too. For T = 13,500 K, the different σe are independent of pressure. At lower temperatures, the electrical conductivity decreases as pressure increases. For T = 10,000 K the ionization reaction is delayed when pressure increases. At higher temperatures (around 24,000 K), the first ionization reaction is completed and the electron density increases with pressure. Mixtures of water and carbon at high temperatures were studied by Wang Weizong et al. (2012). In such mixtures, a large number of chemical species (69 heavy species in all) as well as electrons, including relevant atoms, ions, and

Fig. 20 Temperature dependence of the electrical conductivity of water vapor plasma at four different pressures (50, 100, 500 and 1000 kPa) (Aubreton et al. 2009a)

M. I. Boulos et al.

Electrical conductivity, σe (kA / V.m)

290 25

1000 kPa

20

500 kPa

15

100 kPa

10 50 kPa

5 0 0

5

10

15

Fig. 21 Temperature dependence of the electrical conductivity of carbon and water plasmas (Cq (H2O)1-q) at different molar amounts of carbon q from 0 to 1 in steps of 0.1 (Wang Weizong et al. 2012)

Electrical conductivity, σe (kA / V.m)

Temperature, T

20

25

30

(103 K)

16 q =0

q=0.0 q=0.1 q=0.2 q=0.3 q=0.4 q=0.5 q=0.6 q=0.7 q=0.8 q=0.9 q=1.0

12

8

q =1

4

0

0

5

10

15

20

25

30

Temperature, T (103 K)

molecules, were taken into account. In the C–H–O system, the species composition and thus thermodynamic properties strongly depend on the ratios of carbon and water vapor as well as the phase transition of carbon. As the proportion of carbon increases, a greater number of species appear in appreciable concentrations in the reacting mixture, leading to a complex reaction field due to the high reactivity of C/H/O species. The equilibrium composition of the plasma has, of course, a great influence on mass, momentum and heat transfer. Figure 21 presents the electrical conductivity of the mixture Cq (H2O)1-q. At temperatures lower than 18,000 K, the ionization temperature decreases as the carbon concentration increases, due to the lower ionization energy of carbon atoms relative to those of hydrogen and oxygen, and hence the electrical conductivity increases. Above 22,000 K, as the mole fraction of carbon increases, the

7

Transport Properties of Gases Under Plasma Conditions

291

electrical conductivity of the mixture decreases even though the electron number density rises. As more carbon is added to the water vapor, the increasing electron number density is counteracted by the influence of the formation of multiply charged ions, for which the Coulomb collision integrals are larger. Including the carbon phase transition has negligible influence on the electrical conductivity, since significant ionization does not occur at the relevant temperatures. Calculations for other mixtures have been performed, for example, water and ethanol used when spraying suspensions Pateyron et al. (2013), equimolar CO-H2 plasmas, typical of biomass, from 500 K to 30,000 K for p = 0.1, 0.2, 0.5, and 1 MPa (Aubreton et al. 2009b), for Mars and Titan atmospheres (André et al. 2010; Colonna et al. 2013), CO2–CF3I mixture for circuit breakers (Yokomizu et al. 2009), etc.

6.2.2 Viscosity Figure 22 from Pateyron et al. (1986, 1990, 1992) gives the viscosity of air at different pressures (50 to 500 kPa). The evolution of μ with temperature is quite similar to that of nitrogen (see Fig. 1). The small variations in the slope are due to the dissociation of oxygen between 3000 and 5000 K and to N0+ formation and destruction between 5000 and 8000 K. The rapid reduction in the slope at T > 7000 K corresponds to nitrogen dissociation. At temperatures higher than 10,000 K, the viscosity decreases more rapidly when the pressure is reduced, in agreement with the charged particle densities, which increase when the pressure decreases. Between 3000 and 10,000 K, the pressure effects, which are not very significant, are linked successively in the O2 dissociation, NO+ formation and destruction, and N2 dissociation. For T > 20,000 K when the densities of the charged particles are almost constant the viscosity reaches a constant value which is highest for the highest densities of charged particles i.e., for the highest pressure. Bacri and Raffanel (1989) and A. D’Angola et al. (2008, 2012) have also calculated air viscosity at different pressures and Murphy (1995) and Murphy and Arundell (1994) mixtures of air with Ar, N2, and O2. 28 Viscosity, μ (10–5 kg/m.s)

Fig. 22 Temperature dependence of the viscosity of air at four different pressures (50, 100, 200 and 500 kPa) (Boulos et al. 1994)

24

Dry air

20 16 12

p = 500 kPa 200 kPa

8

100 kPa 50 kPa

4 0

0

2

4

6

8

10

Temperature, T

(103 K)

12

14

292 30 Ar –H2 mixtures p =100kPa

25 Viscosity, μ (10–5 kg/m.s)

Fig. 23 Temperature dependence of the viscosity of Ar-H2 mixture at atmospheric pressure (Pateyron et al. 1992; Boulos et al. 1994)

M. I. Boulos et al.

0.5 0.4 0.3

20

Pure Ar Wilke (1950)

0.2 15

0.1

10 Pure H2

5

0

0

5

10

15

20

3

Temperature, T (10 K)

In an Ar-H2 mixture at atmospheric pressure, the viscosity is mainly dominated by Ar for H2 vol.% up to 40–50% (see Fig. 23). Intuitively, this can be explained by using the first approximation for viscosity (μ is proportional to the square root of the product of mass and temperature and inversely proportional to the Ω2,2 i,j collision 2, 2 2, 2 at 8000 K: Ω ¼ 4:8, Ω integral) with the following values for Ω2,2 i,j HH ¼ 2:16, ArAr 2, 2 ΩArH ¼ 3:4. Assuming that the simplified expression can be used for the diffusion of H in Ar with m ¼ μ (reduced mass), it q canffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi be seen ffi that the contributions of H-H 2, 2 and Ar-H to the viscosity (proportional to m=Ωi, j ) are low compared with that of Ar (as long as the Ar density is high). At temperatures above 22,000 K, the density of Ar++ increases (see Fig. 11 of ▶ Chap. 6, “Thermodynamic Properties of Plasmas”) and the viscosity decreases. Similar results were obtained by Murphy (2000). In Ar-N2 mixtures differences between pure Ar, pure N2 and 50 mol% of each differ by less than 3%. In N2-H2 mixtures the viscosity of the mixture is the highest with the maximum % of N2 and the lowest for the highest H2 %, as illustrated in Fig. 24. , 2 ¼ 1:2 and Ω2, 2 In contrast, for the Ar-He mixture at 8000 K (with Ω2HeHe ArHe ¼ 2:0), the contributions of these interactions, especially that of Ar-He, can no longer be neglected compared with that of Ar-Ar. Thus the viscosity of this mixture, up to 11,000 or 12,000 K, is higher than that of the pure initial components (see Fig. 25). For the case of an Ar-Cu mixture, Mostaghimi and Pfender (1984) used the first approximation of Sonine for the heavy particles and the sixth approximation for the electrons. The unknown potentials were assumed to be hard sphere potentials. The plasma was at atmospheric pressure with temperatures ranging from 1000 K to 24,000 K, and the contamination level of copper ranged from 0.01% to 5% by volume. Both the first and second ions of copper and argon were considered, see

Transport Properties of Gases Under Plasma Conditions

Fig. 24 Temperature dependence of the viscosity of N2-H2 mixture at atmospheric pressure (Murphy 2012)

Viscosity, μ (10–5 kg/m.s)

7

293

50% H2

25

N2 /H2 Mixtures

75% H2 20 15 25% H2

10 5 0

0

5

10

15

20

25

30

Temperature, T (103 K)

50 Pure He

Calculated Viscosity, μ (10–5 kg/m.s)

Fig. 25 Temperature dependence of the viscosity of Ar-He mixtures at atmospheric pressure (Pateyron et al. 1992; Boulos et al. 1994)

40 Wilke’s (1950) interpolation 30 Linear interpolation 20 Pure Ar 10

0

Ar –He mixtures (20 – 80% vol.) p = 100kPa 0

5

10

15

20

Temperature, T (103 K)

Fig. 26. The main conclusion was that the effect of Cu vapor on the viscosity was, for all practical purposes, negligible (see Fig. 26). A quite similar result was obtained for Ar-Fe plasma (1% and 10% Fe) by Murphy et al. (2009). Aubreton et al. (2009a) calculated the viscosity of a water vapor plasma as function of temperature at pressures ranging from 50 to 1000 kPa. As shown in Fig. 27, the ionization reaction is delayed by pressure for a Tmax  10,000 K, for p = 50 kPa up to a Tmax  12,500 K for p = 1 MPa. At low temperatures, the values of viscosity principally depend on neutral–neutral interactions for which the average collision integrals vary smoothly with temperature: hence in this region, the viscosity increases with T1/2. The slight changes of slopes between 3000 and 7000 K are due to dissociation phenomena. At higher temperatures (T > Tmax), ionization occurs and the ion–ion collision integrals are very large leading to a decrease in viscosity.

294 3.0 Viscosity, μ (10–5 kg/m.s)

Fig. 26 Temperature dependence of the viscosity of Ar plasma containing a few mole percent of Cu vapor at atmospheric pressure (Mostaghimi and Pfender 1984; Boulos et al. 1994)

M. I. Boulos et al.

Pure Ar

2.0 1.5 1.0 0.5 0.0

5

0

10 15 Temperature, T (103 K)

20

25

3.0 Viscosity, μ (10–5 kg/m.s)

Fig. 27 Temperature dependence of the viscosity of water vapor plasma at different pressures (50, 100, 500, 1000 kPa) (Aubreton et al. 2009)

1 % Cu 5% Cu

2.5

1000 kPa

2.5

500 kPa

2.0

100 kPa

1.5

50 kPa

1.0 0.5 0.0

0

5

10

15

20

25

Temperature, T (103 K)

Viscosity calculations for other mixtures have been performed, for example, water and ethanol used when spraying suspensions (Pateyron et al. (2013)), equimolar CO-H2 plasmas, typical of biomass, from 500 K to 30,000 K for p = 0.1, 0.2, 0.5, and 1 MPa Aubreton et al. (2009a), for Mars and Titan atmospheres André et al. (2010), Colonna et al. (2013), etc.

6.2.3 Thermal Conductivity Fig. 28 shows the evolution of the thermal conductivity of dry air with temperature at different pressures.The double peaks of the thermal conductivity observed at temperature between 3000 and 9000 K are due to the dissociation of oxygen at around 3500 K, and that of nitrogen at 7000 K. The ionization peaks of O and N occur at almost the same temperature, around 14,000 K. The peaks are somewhat broadened compared to those of pure gases, but their amplitudes are in almost the same ratio as that of N2 and in air at room temperature. At higher pressure (above atmospheric), the 02 and N2 dissociation peaks are shifted to higher temperatures

Transport Properties of Gases Under Plasma Conditions

295

6.0 (W/m.K)

Fig. 28 Temperature dependence of the thermal conductivity of air at four different pressures (50, 100, 200 and 500 kPa) (Pateyron et al. 1990; Boulos et al. 1994)

Thermal conductivity,

7

5.0

Dry air p = 50 kPa 100 kPa 200 kPa 500 kPa

4.0 3.0 2.0 1.0 0.0

0

2

4

6

8

10

12

14

Temperature, T (103 K)

100 vol. % H2

(W/m.K)

15

Thermal conductivity,

Fig. 29 Temperature dependence of the thermal conductivity of Ar-H2 plasmas at atmospheric pressure with different molar fractions, x, of H2 (Pateyron et al. 1990; Boulos et al. 1994)

Ar –H2, p =100 kPa

90 vol. % H2 10

50 vol. % H2 30 vol. % H2

5

0

0

5

10

15

20

Temperature, T (103 K)

and lower maximum values are obtained. The ionization peaks are also shifted to higher temperatures and their values increase slightly with pressure. When H2 is added to Ar (recall the high dissociation and ionization peaks of H2), the thermal conductivity of the mixture increases with the H2 vol.%, especially near the dissociation and ionization temperatures (see Fig. 29). For T > 20,000 K the curves merge together due to the fact that the hydrogen atom/ion composition remains unchanged while the maximum ionization of Ar is not yet reached. In the Ar-He mixture, it can also be seen that an increase in the He content results in a steady increase in the thermal conductivity of the mixture, the increase being almost proportional to the He vol.% (see Fig. 30). For T > 17,000 K the ionization peak for He is noted. In both mixtures (Ar-H2 and Ar-He), the values of κ are higher close to the Ar ionization temperature for vol.% above 70% than the highest value of κ for the pure component (H2 for Ar-H2 and He for Ar-He).

296

(W/m.K)

8.0 Pressure =100kPa Ar –He mixtures

Pure He 90% He

6.0

70%

Thermal conductivity,

Fig. 30 Temperature dependence of the thermal conductivity of Ar-He plasmas at atmospheric pressure with different molar fractions, x, of He (Mostaghimi and Pfender 1984; Boulos et al. 1994)

M. I. Boulos et al.

50%

4.0 30% 10%

2.0

Pure Ar 0.0

0

5

10

15

20

Temperature, T (103 K)

In the presence of steep temperature gradients, careful attention must be given to the way in which the mean effective thermal conductivity is calculated. Bourdin et al. (1983) have suggested using the mean integrated thermal conductivity defined as  ð Tp 1 κ¼ κðtÞ dt (71) Tp  Ts Ts where Ts and Tp are the temperature limits across which the energy transfer is taking place. Figure 31 represents κ for the Ar-H2 mixture. Note that the important advantage of adding H2 to the mixture is that the mean integrated thermal conductivity is drastically increased as soon as the temperature exceeds 4000 K. In the Ar-He mixture (see Fig. 32), the increase with temperature is more regular than for Ar-H2. He ionization increases slightly for T > 17,000 K. A comparison between the thermal conductivity of these two gas mixtures typically used in plasma spraying is given in Fig. 33. These are Ar + 30 vol.% H2 and Ar + 60 vol. % He. For T > 24,000 K, due to He ionization, [κ (Ar-He) > κ (Ar-H2)]. The contribution of the various species to the thermal conductivity of a plasma contaminated by metal vapor depends on temperature (see Fig. 34, showing data for an Ar-Cu plasma). For T 10,000 K the total thermal conductivity is governed by the contributions of chemical reactions (ionization) and free electrons. For the mixture Ar-iron vapor the addition of 10 mol% iron vapor approximately doubles the thermal conductivity for temperatures between 5000 K and 10,000 K, as shown in Fig. 35. For water vapor plasma, as observed in Fig. 36 from Aubreton et al. (2009a), the increase in thermal conductivity at high temperatures (T > 25,000 K) is controlled

Fig. 33 Temperature dependence of the mean integrated thermal conductivity, κ, of plasmas respectively of Ar-30 vol.% H2 and Ar-60 vol.% He, at atmospheric pressure (Pateyron et al. 1992; Boulos et al. 1994)

Mean Integ. thermal conductivity,

Pressure =100kPa Ar –H2 mixtures

Pure H2 4.0

90% H2 70%

2.0

50% 30% 10%

Pure Ar

0.0 0

(W/m.K)

Fig. 32 Temperature dependence of the mean integrated thermal conductivity, κ, of Ar-He plasmas at atmospheric pressure (Pateyron et al. 1992; Boulos et al. 1994)

297

6.0

4.0

Mean Integ. thermal conductivity,

Fig. 31 Temperature dependence of the mean integrated thermal conductivity, κ, of Ar-H2 plasmas at atmospheric pressure (Pateyron et al. 1992; Boulos et al. 1994)

(W/m.K)

Transport Properties of Gases Under Plasma Conditions

3.0

5

10 15 Temperature, T (103 K)

20

Pressure =100kPa Ar –He mixtures

2.0

1.0

Pure He

90% He 70% 50% 30% 10%

Pure Ar 0.0 0

5

10 15 Temperature, T (103 K)

20

3.0 Mean Integral thermal conductivity, (W/m.K)

7

Pressure =100kPa Ar –H2 mixtures 2.0

Ar + 30% H2

1.0 Ar + 60% He

0

0

5

10 15 Temperature, T (103 K)

20

298

(W/m.K)

4.0

Thermal conductivity,

Fig. 34 Temperature dependence of the thermal conductivity of Ar plasma containing a few mol percentage of Cu vapor at atmospheric pressure (Mostaghimi and Pfender 1984; Boulos et al. 1994)

M. I. Boulos et al.

Cu Cu Cu Cu Cu

3.0

/ / / / /

Ar Ar Ar Ar Ar

0.0 0.01 0.0 1. 5.

% % % % %

2.0 5% Cu 1.0 5% Cu 0.0

5

0

10

15

20

25

Thermal conductivity,

Fig. 35 Temperature dependence of the thermal conductivity of Ar, iron vapor and mixtures of argon with 1 mol% and 10 mol% of iron at atmospheric pressure (Murphy et al. 2009)

(W/m.K)

Temperature, T (103 K)

4.0 Ar / Fe mixtures, p =100 kPa 3.0 100% Ar 2.0

Ar / 10% Fe

0.0

100% Fe

Ar / 1% Fe

1.0

5

0

10

15

20

25

Temperature, T (103 K)

(W/m.K)

12

Thermal conductivity,

Fig. 36 Temperature dependence of the thermal conductivity of water vapor plasma at different pressures (50, 100, 500, 1000 kPa) (Aubreton et al. 2009)

10 8

1000 kPa

6

500 100

4 50 2 0

0

5

10

15

20

Temperature, T (103 K)

25

30

7

Transport Properties of Gases Under Plasma Conditions

299

by the translational thermal conductivity of electrons. At low temperatures, up to 15,000 K, the translational contribution of heavy species is important and is dominant in the temperature range from 6000 to 8000 K for 100 kPa of pressure. The shape of thermal conductivity is disturbed by the two strong maxima of the reaction component, (a) at 4000 K for dissociation reactions and (b) at 15,000 K for ionization reactions (p = 100 kPa). These peaks are shifted to higher temperatures due to the pressure. Finally, the contribution of the internal conductivity is always negligible for temperatures above 2500 K since the energy stored by atomic species, neutral and ions is relatively small. At low temperatures this contribution is important because of the H2O molecule Aubreton et al. (2009a). Thermal conductivity calculations for other mixtures have been performed, for example, water and ethanol used when spraying suspensions Pateyron et al. (2013), showing the increase of κ with the ethanol percentage, equimolar CO-H2 plasmas, typical of biomass, from 500 K to 30,000 K for p = 100, 200, 500 and 1000 kPa (Aubreton et al. 2009a, b), for Mars and Titan atmospheres André et al. (2010), Colonna et al. (2013).

6.2.4 Diffusion Coefficient Different diffusion coefficients are considered, see Eqs. 68, 69, and 70. For example Murphy (2000) has calculated them for Ar-H2. Figure 37 shows the temperature dependence of the combined ordinary diffusion coefficient for different mixtures of argon and hydrogen. Ordinary diffusion coefficients for a mixture of two species i and j vary as function of, 

1 1 þ mi mj

12

3 ð1, 1Þ T2 =Ωi, j

where mi and mj are the masses of the two species. According to Murphy (2000), similar trends are found for the combined ordinary diffusion coefficient, whose magnitude is much larger for argon–hydrogen mixtures than for mixtures of argon with nitrogen, oxygen, and air. The combined ordinary diffusion coefficient is also 8 Ordinary diffusion coefficient, (m2/s)

Fig. 37 Combined ordinary x diffusion coefficient DArH2 of mixtures of argon and hydrogen. Percentages refer to mole fractions (Murphy 2000)

25% H2 6

Ar /H2 mixtures

1% H2 99% H2

4 50% H2 75% H2

2

0

0

5

10 15 20 Temperature, T (103 K)

25

30

300

M. I. Boulos et al.

larger for argon–hydrogen mixtures than for mixtures of argon and helium for temperatures below around 12,000 K. At higher temperatures, the diffusion coefficient for argon-helium mixtures containing significant amounts of helium continues to increase, reaching a peak of around 0.09 m2/s at 13,000 K for a mixture of equal parts of argon and helium. This is because helium ionizes at a higher temperature and, hence, Coulomb interactions, for which Ω(1,1) is much larger, i,j are not important at intermediate temperatures. The combined pressure and combined temperature diffusion coefficients, shown in Figs. 38 and 39, respectively, exhibit a strong dependence on the relative concentrations of argon and hydrogen at all temperatures. The combined temperature coefficient, referred to as the combined thermal diffusion coefficient, is, in fact, the sum of terms involving both the ordinary diffusion coefficients and the thermal diffusion coefficients (Murphy 1997). Unlike some other gas mixtures, the combined pressure diffusion coefficient for argon and hydrogen is always positive. Its value is large compared to mixtures of argon with nitrogen, oxygen, and air

0.05 Pressure diffusion coefficient, (m2/s)

Fig. 38 Combined pressure p diffusion coefficient DArH2 of mixtures of argon and hydrogen. Percentages refer to mole fractions (Murphy 2000)

Ar / H2 mixtures

75% H2 0.04

50% H2

0.03

25% H2

0.02

99% H2 1% H2

0.01 0

0

5

10

15

20

25

30

Temperature, T (103 K)

0.06 Pressure diffusion coefficient, (g/m.s)

Fig. 39 Combined temperature diffusion T coefficient DArH2 of mixtures of argon and hydrogen. Percentages refer to mole fractions (Murphy 2000)

25% H2

50% H2

0.04 75% H2

0.02

1% H2

0.00 –0.02

99% H2

Ar /H2 mixtures

–0.04 0

5

10

15

Temperature, T

20 (103

25 K)

30

7

Transport Properties of Gases Under Plasma Conditions

301

Murphy and Arundell (1994) and Murphy (1995) and similar to values for argon–helium mixtures (Murphy 1997). The combined temperature diffusion coefficient is small compared to values found for mixtures of argon with other gases. Figures 40 and 41 show the evolution of the combined ordinary diffusion coefficient (CODC) as a function of temperature for various concentrations of metal, for both iron and copper metals (for air–metal mixtures the percentages are given using mass proportions) (Cressault and Gleizes 2010). The maxima are observed for temperatures between 8500 K and 12,000 K. They depend significantly on the concentration of the metal in the mixture. This dependence is indirect and is due to the influence of gas proportions on the ionization degree of the plasma. In the low temperature range, the coefficient increases with temperature because of the diffusion between neutral particles (atoms and molecules), whereas when the plasma is dominated by Coulomb’s interactions at high temperature, the mobility of electrons and ions strongly decreases diffusion.

14 Air / Fe, p= 1 bar

12

with 10% Fe with 30% Fe with 50% Fe with 70% Fe with 90% Fe

10 (m2/s)

Ordinary diffusion coefficient,

Fig. 40 Combined ordinary diffusion coefficient of iron in air at atmospheric pressure, for different concentrations of iron vapor in the mixture (mass proportions) (Cressault and Gleizes 2010)

8 6 4

90% Fe

2 0

10% Fe 0

5

10

15

20

25

30

Temperature, T (103 K)

14 Air / Cu, p= 1 bar

with 10% Cu with 30% Cu with 50% Cu with 70% Cu with 90% Cu

12 (m2/s)

Ordinary diffusion coefficient,

Fig. 41 Combined ordinary diffusion coefficient of copper in air at atmospheric pressure, for different concentrations of copper vapor in the mixture (mass proportions) (Cressault and Gleizes 2010)

10 8

90% Cu

6 4 2

10% Cu

0 0

5

10

15

20

Temperature, T (103 K)

25

30

302

M. I. Boulos et al.

According to Figs. 40 and 41, the CODC of air–Fe and air–Cu plasmas are very similar. Cressault and Gleizes (2010) explain this behavior by the combination of two properties: the atomic masses and the ionization energies of the two gases (air and vapor). Indeed, this maximum value increases when the gas having the lowest mass is increasingly present in the plasma. The maximum of the CODC evolution with temperature moves towards higher temperatures when the concentration of metal vapor increases.

6.3

Precisions of Such Calculations

Results of calculations depend on the plasma composition of the gas mixture and the Debye–H€uckel correction must be applied to the chemical potential of charged species, and to the total number density of the plasma (Murphy 2012). The second problem is the collision integrals for the different possible interactions, whose number can increase very fast. For example with the simple mixture N2-H2 the species considered by Murphy (2012) in the calculations were H, H+, H, H2, Hþ 2, + þ þ 2þ 3þ 4þ , N, N , N , N , N , N , N, N , N , NH , H , NH , HNNH and the electron. H 2 2 3 2 2 2 The choice of the interaction potentials is very important for the transport property calculated. This is illustrated in Fig. 8, where the maximum value of thermal conductivity varies between 20.2 W/m.K and 15.0 W/m.K depending on the interaction potential chosen. In his paper Murphy (2012) discusses in details the choice of the different interaction potentials for the nitrogen and hydrogen gases using the simple mixture N2-H2. The author points out that “the viscosity, thermal conductivity and electrical conductivity at atmospheric pressure when compared to recently-published data, present only small discrepancies, which can be explained in terms of the intermolecular potentials from which the collision integrals for two important interactions were calculated. Larger discrepancies exist with older published values, since less accurate collision integrals were used in those studies.” Aubreton et al. (2013) in their paper also point out the importance of the choice of the interaction potentials, especially for complex mixtures (with a lot of species) where numerous interaction potentials are unknown. They also show that the calculation of internal partition functions of monatomic species (neutral and ions) plays an important role in the plasma composition and thus in the transport properties. At last, charge (e.g., H(n)-H+) and excitation (e.g., H(n)-H(m)) transfers can lead to non-negligible differences between transport properties (Sourd et al. 2007).

6.4

Mixing Rules and Their Limitations

Due to the complexity of the calculation procedure for the accurate determination of the transport properties of gaseous mixtures, there is a strong need for simplified mixing rules that can be used to calculate the properties of a given gaseous mixture

7

Transport Properties of Gases Under Plasma Conditions

303

as a function of its composition and the corresponding properties of the pure gases. By neglecting all interactions between the components of the two gases and using only a first-order development of Chapman-Enkog expressions, Wilke (1950) has developed a semiempirical expression giving the viscosity of the mixture: 1 0 μmix ¼

C K B X B xi μ i C C B C BX K A i¼1 @ xj Zij

(72)

j¼1

where 2   1 1 M i 2 4 Zi,j ¼ pffiffiffi 1 þ 1þ Mj 8

μi μj

!12  1 32 Mj 4 5 Mi

(73)

K is the number of chemical species in the mixture, xi and xj are the mole fractions of species i and j, μi and μj are the viscosities of species i and j at the mixture temperature and pressure, and Mi and Mj are the corresponding molecular weights (Z is dimensionless, and for i = j, Z = 1). Figure 23 shows that the agreement between the exact calculation and Wilke’s is rather good for Ar-H2 mixtures. For Ar-He mixtures (e.g., 20 vol.% Ar–80 vol.% He), the exact calculation gives quite a different result from those calculated using either Wilke’s expression or a linear interpolation (see Fig. 25). These differences can be explained easily by considering the relative importance of the neglected values for Ar-He compared to those of Ar-H2 or Ar-H for Ar-H2 mixtures. In this last case, of course, Wilke’s expression works well. When dealing with calculation of the thermal conductivity of mixtures κ, it is important to keep in mind that Wilke’s formula (Eq. 72) also neglects all reactive contributions to κ. It is therefore not surprising that differences of up to 60–70% were observed between the exact calculation and those obtained using the mixing rule, especially close to dissociation or ionization temperatures. Gleizes et al. (2010) have made a careful study of mixing rules applied to mixtures of gas and metal vapors. They showed that For mass density, no satisfactory mixing law at low temperatures exists in molecular gas plasma, while at high temperatures a very good approximation is obtained when using the electron number density. For specific heat, mixing rules are not very precise, but a linear interpolation with mass proportions can be satisfactory. For electrical conductivity, any simple interpolation is valid at high temperatures (T > 10,000 K). At lower temperatures and rather low metal proportions (

305

Equilibrium distribution function Component of the
force in the x-direction (N) !! Relative velocity g12 ¼ jv v2 ðm=sÞÞ 1 Planck’s constant (h = 6.626  1034 J.s) Molar enthalpy of species i (kJ/mol) Molar dissociation enthalpy of nitrogen molecule (kJ/mol) Molar ionization enthalpy of nitrogen atom (kJ/mol) Energy flux (W/m2) Particle flux (part/m2.s) Momentum flux in x-direction Flux density of χ or net quantity of χ (particles number, transverse momentum, energy, charge) transported across per unit area and unit time Flux of the quantity x Boltzmann constant (k = 1.38  1023 J/K) Mean free path (m) Mean free path in z-direction (m) Number–density–weighted average mass of the species present in gas A Number–density–weighted average mass of the species present in gas B Mass of the particle of chemical species i (kg) Atomic mass of chemical species i (kg) Number density (m3) Number density of chemical species i (m3) Mole number of chemical species i Pressure (Pa) Heat flux due to temperature gradients in x direction (W/m2) Pressure tensor Flux vector for the transport of kinetic energy of particles of species i (J/m2.s) Reactional heat flux vector (J/m2.s) Heat flux vector in z-direction (J/m2.s) Bracket integral Product of the reduced collision integral by the cross-sectional area (m2) Quantum collision cross section Elastic collision cross section at the degree 1 of the Legendre functions Position vector !  Relative position vector r ¼ r1  r2 (m) Distance of closest approach (m) Perfect gas constant (R = 8.32 J/K.mole) Absolute temperature (K) Local macroscopic velocity (m/s) ! Peculiar velocity Uι ¼ ! v0 of particles of species i (m/s). vi  ! Peculiar velocity in the x-direction Diffusion velocity of particles of species i

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! vi !

v ! v0 V(r) ! Vij

Velocity of particle of species i (m/s) Mean velocity of particles (m/s) Mean flow velocity {m/s) Interaction potential ! ! Relative velocity before collision ðV ij ¼ v i  vj Þ (m/s)

Vij0

Relative velocity after collision ðV 0ij ¼ v 0i  v 0j Þ (m/s)

xi Zij Zj

Mole fraction of species i Dimensionless coefficient to calculate the viscosity of a mixture Charge number

!

7.2

!

!

!

Greek Alphabet

ε ζ θ θ0 χ hχii χi κ κ κint κR κtr μ μe μm μmix ξ ρ ρ0

Mean kinetic energy of particles (J) Inverse of the frequency of collision (s) Angle in spherical coordinates Deviation angle Variable Mean value of χi Mean value of χi Thermal conductivity (W/m.K) Mean integrated thermal conductivity, see Eq. 71 (W/m.K) Internal thermal conductivity (W/m.K) Reactional thermal conductivity (W/m.K) Translational thermal conductivity (W/m.K) Gas viscosity (kg/m.s) 2 Electron mobility
(m /V.s)
 Reduced mass μm ¼ mi mj = mi þ mj (kg) Molecular viscosity of a mixture, Eq. (72) (kg/m.s) Perturbation parameter: 1/ξ measures the frequency of collisions Specific mass (kg/m3) ! Collision cross section for particles with initial velocities between (v 1 ! ! ! ! ! and v1 þ dv1 ) and (v2 and v2 þ dv2 ) and velocities after collision !

!

!

!

!

σe σ0 σen σij ! σð v Þ:dΩ0

between (V 01 and V 01 þ dV 01 ) and (V 02 þ dV 02 ) Electrical conductivity (mhos/m) Total collision cross section (m2) Total collision cross section between electrons and neutral particles (m2) Total collision cross section between particles of type i and j (m2) Collision cross section for particles emerging after collision with a

τzx φ ϕi

relative velocity V 0 into a solid angle range dΩ0 about θ0 and φ0 Stress in a plane at z in direction x (N/m2) Angle in spherical coordinates Perturbation function for calculating the distribution function

!

7

Ωlsij ls Ωij !

∇r ! ∇v

Transport Properties of Gases Under Plasma Conditions

307

Hirschfelder’s collision integral for particles of species i and j (m2) 2 Reduced collision integral for particles  of species i and j (m ) Position gradient vector, Velocity gradient vector,

@ @ @ @x , @y , @z



@ @vx

,

@ @vy

,

@ @vz



References André P, Aubreton J, Clain S, Dudeck M, Duffour E, Elchinger M.F, Izrar B, Rochette D, Touzani R, Vacher D (2010) Transport coefficients in thermal plasma. Applications to Mars and Titan atmospheres. Eur Phys J D 57:227–234 Athye W.F (1965) A critical evaluation of methods for calculating transport coefficients of partially and fully ionized gases. NASA TN, ND-2611 Aubreton J, Fauchais P (1983) Influence des potentiels d’intéraction sur les propriétés de transport des plasmas thermiques: exemple d’application le plasma argon hydroge`ne à la pression atmosphérique. Revue de Physique Appliquée 18(1):51–66 Aubreton J, Elchinger M.F, Vinson J.M (2009a) Transport coefficients in water plasma: part I: equilibrium plasma. Plasma Chem Plasma Process 29:149–171 Aubreton J, Elchinger M.-F, Hacala A, Michon U (2009b) Transport coefficients of typical biomass equimolar CO–H2 plasma. J Phys D Appl Phys 42:095206 (13 pp) Aubreton J, Elchinger M.F, Andre P (2013) Influence of partition function and interaction potential on transport properties of thermal plasmas. Plasma Chem Plasma Process 33:367–399 Bacri J, Raffanel S (1989) Calculation of transport coefficients of air plasmas. Plasma Chem Plasma Process 9(1):133–154 Balescu R (1988) Transport processes in plasmas, vols 1 and 2. North-Holland, Amsterdam Behringer K, Kollmar W, Mentel J (1968) Messung der Wa¨rmeleitfa¨higkeit von Wasserstoff zwischen 2000 und 7000 K, Z. Physics 215(2):127–151 Bottin B, Vanden Abeele D, Magin T.E, Rini P (2006) Transport properties of collision-dominated dilute perfect gas mixtures at low pressures and high temperatures. Prog Aerosp Sci 42:38–83 Boulos M, Fauchais P, Pfender E (1994) Thermal plasmas: fundamentals and applications, vol 1. Plenum Press, New York/London Bourdin E, Boulos M, Fauchais P (1983) Transient heat conduction to a single sphere under plasma conditions. Int J Heat Mass Transf 26:567 Brokaw RS (1960) Thermal conductivity of gas mixtures in chemical equilibrium. J Chem Phys 32:1005 Butler J.N, Brokaw R.S (1957) Thermal conductivity of gas mixtures in chemical equilibrium. J Chem Phys 26:1636 Capitelli M (1972) Cut-off criteria of electronic partition functions and transport properties of thermal plasmas. J Plasma Phys 7(1):99–106 Capitelli M.J (1975) Charge transfer from low-lying excited states: effects on reactive thermal conductivity. Plasma Phys 14:365–371 Capitelli M, Gorse C, Fauchais P (1976) Transport coefficients of Ar-H2 high temperature. J Chem Phys 73:755 Chapman S, Cowling T (1952) The mathematical theory of non-unifonn gases. Cambridge University Press, New York Chen W.L.T, Heberlein J, Pfender E (1996) Critical analysis of viscosity data of thermal argon plasmas at atmospheric pressure. Plasma Chem Plasma Process 16(4):635–650 Child M.S (1974) Molecular collision theory (Pub). Academic, New York Colonna G, D’Angola A, Laricchiuta A, Bruno D, Capitelli M (2013) Analytical expressions of thermodynamic and transport properties of the Martian atmosphere in a wide temperature and pressure range. Plasma Chem Plasma Process 33:401–431

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Cressault Y, Gleizes A (2010) Calculation of diffusion coefficients in air–metal thermal plasmas. J Phys D Appl Phys 43:434006 (6 pp) D’Angola A, Colonna G, Gorse C, Capitelli M (2008) Thermodynamic and transport properties in equilibrium air plasmas in a wide pressure and temperature range. Eur Phys J D 46:129–150 D’Angola A, Colonna G, Bonomo A, Bruno D, Laricchiuta A, Capitelli M (2012) Phenomenological approach for the transport properties of air plasmas. Eur Phys J D 66:205 Delcroix J.L, Bers A (1994) Physique des plasmas, vols 1 and 2. Inter Editions/CNRS Editions, Paris in French Devoto R.S (1965) The transport properties of a partially ionized monoatomic gas. PhD thesis, Stanford University Devoto R.S (1966) Transport properties of ionized monoatomic gases. Phys Fluids 9 (6):1230–1240 Devoto R.S (1967) Simplified expressions for the transport properties of ionized monoatomic gases. Phys Fluids 10(10):2105–2112 Devoto R.S (1968) Transport coefficients of partially ionized hydrogen. J Plasma Phys 2 (4):617–631 Devoto R.S (1973) Transport coefficients of ionized argon. Phys Fluids 16:616 Devoto R.S, Li C.P (1968) Transport coefficients of partially ionized helium. J Plasma Phys 2 (1):17–32 Gleizes A, Gonzalez J.J, Freton P (2005) Topical review, thermal plasma modelling. J Phys D Appl Phys 38:R153–R183 Gleizes A, Cressault Y, Teulet P (2010) Mixing rules for thermal plasma properties in mixtures of argon, air and metallic vapors. Plasma Sources Sci Technol 19:055013 (13 pp) Gorse C (1975) Contribution au calcul des propriétés de transport des plasmas des mélanges Ar-H2 et Ar-N2, PhD, University of Limoges (in French) Hirschfelder J.O, Curtiss C.F, Bird R.B (1964) Molecular theory of gases and liquids, 2nd edn. Wiley, New York IUPAC Subcommission on Plasma Chemistry (1982) Thermodynamic and transport properties of pure and mixed thermal plasmas at LTE. Pure Appl Chem 54(6):1221–1238 Lesinski J, Boulos M (1990) Thermodynamic and transport properties of argon, nitrogen and oxygen at atmospheric pressure over the temperature range 300–30,000 K. Internal report, University of Sherbrooke, Quebec Levin E, Partridge H, Stallcop J.R (1989) Collision integrals and high temperature transport properties. RIACS (NASA) Tech Rep 89:43 Mason EA, Monchick L (1962) Heat conductivity of polyatomic and polar gases. J Chem Phys 36:1622 Mason E.A, Munn R.J, Smith F.J (1967) Transport coefficients of ionized gases. Phys Fluids 10 (8):1827–1832 Mitchner M, Kruger C.H (1973) Partially ionized gases. Wiley, New York Monchick L (1959) Collision integrals for the exponential repulsive potential. Phys Fluids 2 (6):695–700 Mostaghimi J, Pfender E (1984) Effects of metallic vapor on the properties of an argon arc plasma. Plasma Chem Plasma Process 4(2):129–139 Murphy A.B (1993) Diffusion in equilibrium mixtures of ionized gases. Phys Rev E 48:3594 Murphy A.B (1994) Erratum: diffusion in equilibrium mixtures of ionized gases (Phys. Rev. E 48, 3594 (1993)). Phys Rev E 50:5145 Murphy A.B (1995) Transport coefficients of air, argon-air, nitrogen-air, and oxygen-air plasmas. Plasma Chem Plasma Process 15(2):279–307 Murphy A.B (1996) A comparison of treatments of diffusion in thermal plasmas. J Phys D Appl Phys 29:1922 Murphy A.B (1997) Transport coefficients of helium and argon-helium plasmas. IEEE Trans Plasma Sci 25(5):809–814

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Murphy A.B (2000) Transport coefficients of hydrogen and argon–hydrogen plasmas. Plasma Chem Plasma Process 20(3):279–297 Murphy A.B (2012) Transport coefficients of plasmas in mixtures of nitrogen and hydrogen. Chem Phys 398:64–72 Murphy A.B, Arundell CJ (1994) Transport coefficients of argon, nitrogen, oxygen, argonnitrogen, and argon-oxygen plasmas. Plasma Chem Plasma Process 14(4):451–490 Murphy A.B, Tanaka M, Yamamoto K, Tashiro S, Sato T, Lowke J.J (2009) Review article, modelling of thermal plasmas for arc welding: the role of the shielding gas properties and of metal vapour. J Phys D Appl Phys 42:194006 (20 pp) Neumann W, Sacklowski U (1968) Beitr Plasma Phys 8:57 Pateyron B, Aubreton J, Elchinger M.F, Delluc G, Fauchais P (1986) Thermodynamic and transport properties of N2, 02, H2, Ar, He and their mixtures, Internal report LMCTS. University of Limoges, France Pateyron B, Elchinger M.F, Delluc G, Fauchais P (1990) Thermodynamic and transport properties of air and air – Cu at atmospheric pressure. Internal report, LMCTS, University of Limoges Pateyron B, Elchinger M.F, Delluc G, Fauchais P (1992) Thermodynamic and transport properties of Ar-H2 and Ar-He plasma gases for spraying at atmospheric pressure – part 1: properties of the mixtures. Plasma Chem Plasma Process 12(4):421–448 Pateyron B, Calve N, Pawłowski L (2013) Influence of water and ethanol on transport properties of the jets used in suspension plasma spraying. Surf Coat Technol 220:257–260 Rat V, Aubreton J, Elchinger M.F, Fauchais P (2001) Calculation of combined diffusion coefficients from the simplified theory of transport properties. Plasma Chem Plasma Process 21 (3):355–369 Reif F (1988) Fundamentals of statistical and thermal physics. McGraw Hill, New York Smith F.J, Munn R.J (1964) Automatic calculation of the transport collision integrals with tables for the Morse potential. J Chem Phys 41(11):3560–3568 Sourd B, André P, Aubreton J, Elchinger M.-F (2007) Influence of the excited states of atomic nitrogen N(2D), N(2P) and N(R) on the transport properties of nitrogen. Part II: nitrogen plasma properties. Plasma Chem Plasma Process 27:225–240 Wang W.Z, Yan JD, Rong M.Z, Murphy A.B, Spencer J.W (2012) Thermophysical properties of high temperature reacting mixtures of carbon and water in the range 400–30,000 K and 0.1–10 atm. Part 2: transport coefficients. Plasma Chem Plasma Process 32:495–518 Wilke C.R (1950) A viscosity equation for gas mixtures. J Chem Phys 18:517–519 Yokomizu Y, Ochiai R, Matsumura T (2009) Electrical and thermal conductivities of hightemperature CO2–CF3I mixture and transient conductance of residual arc during its extinction process. J Phys D Appl Phys 42:215204 (14 pp)

8

Plasma Radiation Transport Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 General Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Gaseous Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Radiation Mechanisms in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Spontaneous Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Induced Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Microreversibility Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Effective Radiative Lifetime of an Excited State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Radiation Emission and Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Classification of Emitted Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Line Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Continuum Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Total Effective Radiation of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Thermal Plasma Radiation Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

312 313 313 314 317 319 319 320 320 320 322 324 324 328 334 343 352

M. I. Boulos (*) Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada e-mail: [email protected] P. L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France e-mail: [email protected] E. Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] © Springer Nature Switzerland AG 2023 M. I. Boulos et al. (eds.), Handbook of Thermal Plasmas, https://doi.org/10.1007/978-3-030-84936-8_8

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5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Classical Plasma Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Plasma Seeded with Metallic Vapors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Blackbody Radiation of High-Temperature Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Two-Temperature Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Emission and Calculation of Net Emission Coefficient (NEC) . . . . . . . . . . . . . . . . . . . . . 7.2 Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Latin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Superscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357 357 365 372 374 374 376 377 377 379 380 380 380

Abbreviations

2T GMAW LTE NEC Non-LCE Non-LTE

1

Two-temperature, also NLTE Gas-metal arc welding Local thermodynamic equilibrium Net emission coefficient Non-local chemical equilibrium Non-local thermodynamic equilibrium

Introduction

Radiative energy transfer is one of the principal properties of gases under plasma conditions. It is a direct consequence of the excitation to higher energy states of the elementary particles in a plasma and their return to lower energy states, or the ground state, by emission of radiation over a wide range of the spectrum. In this chapter, following a general definition of general concepts of blackbody and gaseous radiation, a review is presented of the radiation emission and absorption in plasmas. This includes line and continuum radiation, total effective radiation of plasmas, and thermal plasma radiation modeling. Examples are given of the contribution of line and continuum emission to the total volumetric emission of argon and nitrogen at atmospheric pressure as function of temperature. This is followed by an introduction to the concept of effective or net emission coefficient (NEC) as a means of taking into account self-absorption in plasmas. This is followed by a discussion of mixing rules for complex plasma gas mixtures. Examples are given of the total volumetric emission coefficients of gases such as argon, nitrogen, hydrogen, helium, air, water vapor, and their mixtures at atmospheric pressure over the temperature range from 5000 to 25,000 K. The effect of the presence of metal vapors such as copper and iron in the plasma gases is discussed. Data are provided for different metal vapor concentrations ranging from a few percentage points up to pure metal vapor plasmas. A brief discussion is presented of blackbody radiation of high-temperature, high pressure plasmas, and of two-temperature nonequilibrium plasmas.

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313

2

General Concepts

2.1

Definitions

The radiation energy that passes through a cross section dS within the solid angle element dΩ (measured in steradian (ster)) in the direction θ with respect to the ! surface normal n (see Fig. 1) during a time interval dt, at frequencies between υ and υ + dυ, contains an amount of energy (see General Literature) given by dEυ ðθ, φÞ ¼ Iυ ðθ, φÞ  dυ  dS  cos θ  dΩ  dt:

(1)

The quantity Iυ(θ, φ), which refers to unit surface, unit time, and unit frequency, is called the monochromatic radiation intensity and is usually expressed in J/ster.m2 (ster is the unit of the solid angle). The total intensity is obtained by integrating over all frequencies 1 ð

Iðθ, φÞ ¼

Iυ ðθ, φÞ  dυ:

(2)

0

Here I(θ, φ) is expressed in W/m ster. It should be noted that the dimensions of Iυ(θ, φ) and I(θ, φ) are different. By integrating over all possible angles θ (from 0 to π/2) and φ (from 0 to 2π) and all frequencies, the total radiation intensity is obtained as 2

ððð I¼

Fig. 1 Definition of the monochromatic radiation intensity of the radiation field

Iυ ðθ, φÞ  dυ  dθ  dφ:

(3)

dΩ = sinθ dθ dϕ

ϕ

θ n

ds

I(v, θ, ϕ)

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The intensity can also be expressed as a function of the wavelength λ instead of the frequency υ. In this case, since Iυ  dυ ¼ Iλ  dλ and that dυ ¼ c  dλ , it follows that λ2 Iυ ¼

λ2 Iλ c

(4)

and (units of Iυ) = (units of Iλ) ms. Thus, Iλ is expressed in W/m3ster. The total radiation flux passing through dS from left to right, with dΩ ¼ sin θ  dθ  dφ, is 2π ð

þ

H ¼þ

π=2 ð

dφ φ¼0

1 ð

dθ

Iυ ðθ, φÞ  dυ  cos θ  sin θ;

(5)

Iυ ðθ, φÞ  dυ  cos θ  sin θ:

(6)

υ¼0

θ¼0

and the radiation from right to left is 

2ðπ

H ¼

ð0 dφ

φ¼0

1 ð

dθ

π=2

υ¼0

The quantities H+ and H are called radiation fluxes and are expressed in W/m2. For isotropic radiation (which is independent of θ and φ), Iυ ðθ, φÞ ¼ Iυ, and since the radiation is the same in all directions, Hþ ¼ H ¼ π  I:

(7)

Since radiation propagates with the velocity of light c, photons will travel in time dt, a distance l = c.dt. It is then possible to define a corresponding volume dV = dS.l. For the special case of an isotropic radiation field, the photons pass through dV under all solid angles from 0 to 4π, and hence uυ ¼

4π  Iυ : c

(8)

uυ is expressed in Js/m3. The integration of uυ over all frequencies gives the total radiation density, expressed in J/m3. In the volume that contains matter of refractive index nr instead of a vacuum, the radiation density must be multiplied by n3r . In plasmas, however, it is generally assumed that nr = 1.

2.2

Blackbody Radiation

2.2.1 Planck’s Law The spectral dependence of the blackbody radiation field in a vacuum is a function only of T and is given by Planck’s radiation law. The density of the radiation field in the frequency interval υ to υ + dυ follows from the formula

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uoυ ðTÞ  dυ ¼

8πhυ3 dυ :  3 expðhυ=ðkTÞÞ  1 c

(9)

The superscript o indicates that uo is related to blackbody radiation. In a vacuum, the radiation field is isotropic, and the blackbody radiation intensity Ioυ often written as Bυ, is obtained from Eq. 8: Bυ  dυ ¼

2h  υ3 dυ :  expðhυ=ðkTÞÞ  1 c2

(10)

By using the relationship Iυ  dυ ¼ Iλ  dλ, Eqs. 9 and 10 can also be expressed as functions of wavelength: uoλ ðTÞ  dλ ¼ Bλ  dλ ¼

8πhc dλ  5 expðhc=kTλÞ  1 λ

2hc2 dλ :  λ5 expðhc=kTλÞ  1

(11)

(12)

uoλ is expressed in J/m4 and Bλ in W/m3  ster Since 2hc2 ¼ 2c1 ¼ 1:1909  1016 W m2 and hc=k ¼ c2 ¼ 0:014386 m  deg (all spectroscopic tables are given in cm1, and c2 is given in cmdeg), the numerical form of Eq. 12 is Bλ ðTÞ ¼ 1:1909  1016  λ5

1 expð0:014386=λTÞ  1

(13)

where λ is in m and T in K. Figure 2 represents Bλ(T) versus λ for various temperatures.

2.2.2 Wien’s Law For a fixed temperature T, Planck’s formula gives an intensity maximum Imax at a distinct wavelength, λmax. The important relationship λmax  T ¼ const: ¼ 2886 μm  K

(14)

follows from ðdIλ ðTÞ=dλ; ¼ 0Þ and is known as Wien’s law. The corresponding position of maximum values of Bλ(T) is represented by the dotted curve in Fig. 2.

2.2.3 Stefan–Boltzmann Law The total flux (intensity emitted per unit surface per unit time into the half sphere) is obtained by integration of dH0 given by, 1 ð

dH ¼

Bυcos θdΩdυ ¼

0

υ¼0

2π4 k4 4 T cos θdΩ 15c2 h3

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M. I. Boulos et al.

Fig. 2 Blackbody radiation density versus wavelength Radiation inensity, Bλ (1011W/m3. ster)

1.4

1.2

1.0

0.8

2000K

0.6

1800K 1500K

0.4

1200K 1000K

0.2 0.0

0

2

4 6 Wave length, λ (μm)

8

10

It follows that 2π4 k4 T4 H0 ¼ 15c2 h3

π=2 ð

cos θdΩ ¼

2π5 k4 4 T ¼ σ s T4 15c2 h3

(15)

0

This is the Stefan–Boltzmann radiation law, and the value of σs is σ s ¼ 5:671  108

W=m2  K4 :

(16)

The radiation intensity (see Eq. 7) is B ð TÞ ¼

σs 4 T ¼ 1:80513  108 T4 : π

(17)

Finally, for the case of an isotropic blackbody radiation field, the total radiation density is given by u o ð TÞ ¼

4σs 4 T J=m3 : c

(18)

The total flux emitted by a blackbody per unit surface area and unit time into the solid angle element dΩ and in the direction θ is obtained by integrating Planck’s radiation law over all frequencies υ and multiplying the result by cos θ  dΩ:

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1 ð

dH ¼

Bυ ðTÞ  cos θ  dΩ  dυ ¼

0

ν¼0

2.3

2π4 k4  T4  cos θ  dΩ: 15c2 h3

(19)

Gaseous Radiation

It will be assumed in the following calculations that the radiation field is isotropic and thus that Iυ ðθ, φ, TÞ ¼ Iυ ðTÞ ¼ Iυ :

(20)

2.3.1 Volumetric Emission Coefficient The volumetric monochromatic emission coefficient is the radiant energy dE emitted by a volume element dV in a frequency interval between v and v + dv into a solid angle dΩ per unit time dt (see Griem (1964)). ευ ¼

dE dυ  dV  dΩ  dt

(21)

ελ ¼

dE : dλ  dV  dΩ  dt

(22)

or

As already shown, the difference in units (ευ in J/m3ster and ελ in W/m4ster) comes from Eq. 4. The volume element dV should be so small that absorption and induced emission processes occurring within it are negligible. The integrated volumetric emission coefficient εL(λo) of a spectral line is ð εL ð υ o Þ ¼

ð ευ  dυ

line

or

ε L ð λo Þ ¼

ελ  dλ

(23)

line

where υo indicates that εL(λo) is related to the line centered at the frequency υo ¼ υu‘. Technically this integral is to be taken over all frequencies since the line is theoretically infinite in extent. In practice, some judgment has to be made to decide how far out on the line the integral must be taken. The practical evaluation of the integral can be complicated by interferences from neighboring lines or when the background continuum is not constant over the range of integration. The units of εL(υo) (W/m3ster) are the same as those of εL(λo).

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2.3.2 Absorption Coefficient If Iυ is the specific monochromatic intensity of radiation passing through an absorbing medium of thickness dx, then κυ0 , the absorption coefficient per unit length at frequency υ, is given by (see Pecker-Wimel (1967)): dIυ ¼ κ0υ  Iυ  dx:

(24)

Note that the upper index 0 is introduced here because induced emission is not considered, and κυ0 is generally expressed in cm1. It is in general a function of the wavelength, the gas properties at the frequency υ, and the direction in which the radiation propagates. If we consider dIλ instead of dIυ, the relationship would be the same: dIλ ¼ κ0λ  Iλ  dx;

(25)

thus κ0λ ¼ κ0υ . In general, after integrating over the thickness L of a gaseous layer, 0

ðL

1

Iυ ¼ Iυ, 0 exp@ κ0υ  dxA

(26)

0

¼ Iυ, 0 expðτυ Þ

(27)

where Iυ,0 is the radiation intensity entering the gaseous layer and τυ represents the optical depth of the layer.

2.3.3 Relationship Between Emission and Absorption For a parallel beam of radiation intensity Iυ, the increase in intensity across an element of thickness dx is given by the difference between emission and absorption within the element: dIυ ¼ ευ ðxÞ  dx  Iυ ðxÞ  κ0υ ðxÞ  dx:

(28)

With the boundary condition Iυ(0) = 0, this differential equation has the solution ðL

0

ðL

1

Iυ ðLÞ ¼ ευ ðxÞ  exp@ κ0υ ðtÞ  dtA  dx: 0

(29)

x

If κυ0 and ευ are constant throughout the gas, Eq. 29 reduces to I υ ð LÞ ¼

  ευ  1  exp κ0υ  L : 0 κυ

(30)

The ratio ευ/κυ0 is called the source function, Sυ, and under complete equilibrium it can be shown to be equal to the Planck function Bυ(T) (see Eq. 10). In the general case of a medium whose refractive index is nr, the source function is given by

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Plasma Radiation Transport

319

Sυ ¼ ευ =nr  κ0υ :

(31)

There are two limiting forms for the expression for Iυ(L). The first is the optically thin approximation in which κ0υ  L  1. This leads, after expanding the exponential into a series, to ðL Iυ ðLÞ ¼ ευ ðxÞ  dx:

(32)

0

The second occurs when κ0υ  L is large compared with 1. For uniform conditions, I υ ð LÞ ¼

εy ¼ Bυ ðTÞ κ0υ

(33)

where Bυ(T) is Planck’s monochromatic radiation intensity. This situation corresponds to the behavior of an optically thick plasma.

3

Radiation Mechanisms in Plasmas

3.1

Spontaneous Emission

An excited atom in a higher quantum state u may return to a lower energy state ‘ by emitting a photon of energy h  υu‘ (see Fig. 3). In the absence of incident radiation, the number of atoms leaving u in a time interval dt is proportional to Nu(t), the population of the state at time t, or dNu ðtÞ ¼ Au‘  Nu ðtÞ  dt:

(34)

Fig. 3 Spontaneous emission, induced emission, and absorption Absorption

Spontaneous emission

Induced emission +0

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M. I. Boulos et al.

Integration yields Nu ðtÞ ¼ Nu ð0ÞexpðAu‘  tÞ

(35)

where Nu(0) is the population of state u at initial time t = 0. The constant Au‘ is called the spontaneous transition probability and is defined as the probability per second that an atom in the state u will spontaneously decay to state ‘ by emitting a photon at frequency υu‘. In fact, Eq. 34 can also be written in terms of the lifetime τu‘, defined as the time necessary to reduce the population of state u by a factor e. Note that τu‘ ¼ 1=Au‘ :

(36)

And Au‘ is expressed in s1. Typical values of Au‘ are between 106 and 108 s1; the highest values are obtained for resonance states of atoms (i.e., transition from the first excited state to the ground state).

3.2

Induced Emission

Induced or forced emission plays a key role for the population inversion in lasers. When a plasma is imbedded in a radiation field of density u0v, then radiation from the plasma may occur not only by spontaneous emission but also by induced emission. In the elementary process of induced emission, a photon of frequency v interacts with an atom or ion in a higher quantum state and thus will force this atom or ion to emit a photon of the same frequency and in the same direction as the oncoming photon. The number of such transitions per second and m3 is proportional to a transition probability Bu, to the density of the radiation field uv, and to the number density of atoms or ions in the appropriate excited state.

3.3

Absorption

Consider a volume with monochromatic radiation of density u0υ given by Eq. 9 and atoms capable of being raised from state ‘ to state u by absorbing radiation of energy h.υ‘u. The probability per second that an atom in state ‘ will absorb a quantum h.υ‘u and pass to state u when exposed to isotropic radiation u0υ dυ in the frequency range between υ‘u and υ‘u þ dυ is proportional to u0υ and is written as B‘uu0υ (see Fig. 3). Notice that the dimensions of transition probability for absorption, B‘u, are m3/J.s2.

3.4

Microreversibility Principle

In any assembly of atoms, the equilibrium must be dynamic. At any given instant, some atoms are absorbing radiation while others are emitting radiation, and it is found that equilibrium is maintained only if each individual interaction process is balanced by its own reverse process (microbalance).

8

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321

Based on the principle of dynamic balance, it is then possible to consider the details of an individual process and to derive relations that connect absorption and emission processes. If absorption and emission are the only processes effective between states ‘ and u, then uoυ  B‘u  N‘ ¼ Au‘  Nu ;

(37)

and if the atomic populations are governed by Boltzmann’s law (see Eq. 58 and 59 of Part I, ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics”), uoυ

  Au‘ gu h  υ‘u ¼   exp B‘u g‘ kT

(38)

where gu and g‘ are the statistical weights of the states u and ‘. Equation 38, however, is not the final balance equation, because induced emission must be included in this balance. As previously mentioned these induced emissions are proportional to the density of the radiation field uoυ, the transition probability Bu‘, and to the number of atoms in the upper quantum state u(Nu). Thus, applying the principle of detailed balance for an equilibrium situation, it follows that uoυ  B‘u  N‘ ¼ Au‘  Nu þ Bu‘ uoυ Nu

(39)

which leads to uoυ ¼

Au‘ =Bu‘ : g‘ B‘u hυ 1   exp kT gu Bu‘

(40)

By comparing this expression with Planck’s law (Eq. 9), it follows that g‘  B‘u ¼ gu  Bu‘

(41)

and Au‘ ¼

8  π  h  υ3  Bu‘ c3

(42)

or Au‘ ¼

1:66543  1032  Bu‘ λ3

with λ in meters. These transition probabilities (Au‘, Bu‘, B‘u) are called Einstein coefficients. Once one is known (calculated or measured), the two others can be deduced from Eqs. 41 and 42. They are related to the volumetric emission and absorption coefficients (see

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M. I. Boulos et al.

Eqs. 21 and 24). If ni,u is the number density of atoms of species i excited to state u, the number of quantum transitions becomes Aiu‘  ni, u and thus in an isotropic plasma ευ ¼

1  Ai  ni, u  h  υu‘ : 4π u‘

(43)

l/4π refers to the unit solid angle. Equation 28 can then be written taking into account the relationship between Bυ and uoυ (see Eq. 8): dIυ ¼

Au‘ Io hυu‘ nu dx þ υ hυ‘u dxðBu‘  nu  B‘u  n‘ Þ: 4π c

(44)

Of course, υu‘ ¼ υ‘u : Assuming a Boltzmann distribution corresponding to a temperature T and using Eq. 41, it follows that    d0 Iυ Au‘ B‘u hυu‘ o ¼ nu hυu‘  n‘ Iυ hυu‘ 1  exp  : dx 4π c kT

(45)

The second term on the right-hand side of Eq. 45 is the effective absorption term. Without the spontaneous emission term, Eq. 45 becomes equivalent to the differential equation that defines the absorption coefficient (see Eq. 24), and thus κυ ¼

   hυu‘ hυu‘ B‘u n‘ 1  exp  : c kT

(46)

  In the case exp hυkTu‘  1 (induced emission is negligible), Eq. 46 reduces to the familiar relation κ0υ ¼

3.5

B‘u n‘ hυu‘ : c

(47)

Effective Radiative Lifetime of an Excited State

In Sect. 3.1 the radiative lifetime of an excited state was defined as the reciprocal of the spontaneous transition probability Au. However, an excited state u can be depopulated by emission of radiation to a family of lower states ‘ or by absorption of radiation to a set of higher states u0 (see Fig. 4). The sum of the transition probabilities must be equal to the reciprocal of the lifetime of the state, assuming superelastic collisions to be negligible. Thus

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Plasma Radiation Transport

323

Fig. 4 Atomic transitions from a state u

ℓ

+0

because

X X X 1 ¼ Au‘ þ Bu‘  uo ðυu‘ Þ þ Buu0 uo ðυuu0 Þ τu u0 ‘ ‘

(48)

  Au‘ hυu‘ ¼ exp 1 Bu‘ uo ðυu‘ Þ kT

(49)

it follows that    1 ! X    1 X 1 hυu‘ g0 hυuu0 þ 1 1 ¼ Au‘ 1 þ exp Au0 u u exp : kT gu kT τu u0 ‘

 hυ 

(50)

The influence of the stimulated radiation appears kT . For T = 300 K  hυ  through exp  5  1036 , and the effect of the and for a photon of wavelength 600 nm, exp kT stimulated emission is insignificant in the visible spectral region where hυ=kT  1. Therefore X 1 ¼ Au‘ : (51) τu ‘ However, at 9000 K exp significant.

 hυ  kT

 16 , and departures from Eq. 51 can become

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M. I. Boulos et al.

Fig. 5 Excited levels of an atom or an ion after Cabannes and Chapelle (1971) and Boulos et al. (1994). En,l i,z being the energy of the excited state and Ei, zþ1 being the ionization energy of the atom of species i with electrical charge z.e

Energy

Free Electrons

Free - Free

Free - Bound Bound Electrons

Ionization Excitation

Bound - Bound

0

4

Radiation Emission and Absorption

4.1

Classification of Emitted Radiation

The various types of spectra observed in plasmas are classified according to the particles that emit them and the degrees of freedom they set into action. Figure 5 (from Cabannes and Chapelle (1971)) shows a schematic of the various excited states of an atom or an ion and the corresponding bound–bound, bound–free, and free–free transitions.

4.1.1 Bound–Bound Transitions Due to their electronic excitation, atoms and ions (except H+) emit a spectrum of lines such that hc Eu  E‘ ¼ h  υu‘ ¼ (52) λu‘ (where u and ‘ are, respectively, the upper and lower excited levels between which the transition takes place). The radiation frequency υ (or wavelength λ) is characteristic of both the atom or ion and the emitting levels. Electrons changing their orbits remain bound to the nucleus, and this type of transition is called a “bound–bound” transition. The corresponding wavelengths extend from the infrared to the far ultraviolet. The spectra of diatomic molecules are by far more complex than those of single atoms, depending on the energy levels of the molecules (see Sect. 5 of Part I, ▶ Chap. 2, “Basic Atomic and Molecular Theory”), and the following excited states have to be considered: – Electronic excitation at energies of a few eV – Vibrational excitation at energies in the order of 0.1 eV – Rotational excitation at energies from 104 to 102 eV

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325

Dissociation

C(3P)+N(2D) 8

6 C(3P)+N(4S)

Vibrational excitation

4

Electronic excitation 2

B2Σ +

Rotational level

A2∏l

6

Energy, E(eV)

Wave number, s (104 cm–1)

CN Molecule

4

2

X2Σ + 0 1 2 3 Nuclear distance, r (10–10 m)

Fig. 6 Energy curves for diatomic molecule (CN) (Cabannes and Chapelle 1971; Boulos et al. 1994)

Figure 6 (from Cabannes and Chapelle (1971)) gives the energy diagram for a CN diatomic molecule with various electronic excited states like the fundamental X2Σ+ state, the first excited A2Π1 state, and second excited B2Σ+ state (see Sect. 5 of Part I, ▶ Chap. 2, “Basic Atomic and Molecular Theory,” for the denomination of the states). Electronic transitions are modified by the presence of vibrational and rotational effects that give rise to groups of bands in band systems. For atoms, each band system corresponds to a single electronic transition. If the energy is written using spectroscopic notation, E ¼ Te þ G þ F

(53)

where Te, G, and F are, respectively, the electronic term, the vibrational term, and the rotational term expressed in cm1. The wave numbers of the spectral lines are given by    σ ¼ E0  E00 ¼ T0e  T00e þ ðG0  G00 Þ þ F0  F00

(54)

where the single-primed letters refer to the upper state and the double-primed letters to the lower state.

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M. I. Boulos et al.

A2 p

X2 Σ

(Red system)

3883 Å 0-0 1-1 2-2

3510 Å

4216 Å 0-1 0-2

B2 Σ

1-0 2-1

4606 Å Sequences

0-2 7-3 4-6

7-3 6-2 5-1

5-2 4-1

CN Molecular spectra

A2 p

(Violet system)

Fig. 7 Schematic spectrum of violet and red systems of the CN molecular spectra (Cabannes and Chapelle 1971; Boulos et al. 1994)

For a given electronic transition, the expression σE ¼ T0e  T00e is a constant. Neglecting the rotational terms, which are very small, the variable part of the frequency is ðG0  G00 Þ , which gives the electronic transition a coarse structure called the vibrational structure. Since there is no selection rule for the vibrational quantum number (v), a large number of bands are expected for electronic transitions. Each band is designated by the two vibrational quantum numbers v0 and v00 . Some characteristic groups are observed in the band system. The groups are called sequences, and for each one, Δv = v0  v00 is a constant. The separation of bands is small for a sequence because the vibrational constants ωe and xe do not change much from one electronic state to another, and thus G0  G00 varies slowly with v0 when Δv is a constant. Figure 7 shows the sequences for the violet and the red systems of the diatomic molecule CN. Notice that due to the significant overlapping in sequences, only a few bands are observed in each sequence. Each band has a rotational structure due to the rotational  motion  of the nuclei. The corresponding frequency for a given electronic band ( T0e  T00e and (G0  G00 ) are then constants) is obtained from the difference, F0  F00 . These rotational spectra are themselves very complex and are classified into branches according to the value of ΔJ = J0  J00 = 0  1. The electronic excitation of diatomic molecules gives band systems in the UV, visible, and IR regions. If the electronic state does not change during a transition, infrared bands called the vibration–rotation bands may be observed (mainly for the fundamental X state). The wave number of the spectral lines is then σ ¼ ðG0  G00 Þ þ ðF0  F00 Þ:

(55)

Generally, one or only a few bands corresponding to the lowest values of the vibrational quantum number are observed.

8

Plasma Radiation Transport

4.1.2

327

Free–Bound and Free–Free Transitions

Free–Bound Transitions Because a free electron may assume non-quantized kinetic energies, its recombination with an ion will result in continuum radiation. For example, the recombination of free electrons and N+ ions gives rise to “free–bound” transitions resulting in a continuous spectrum observable in the visible and the ultraviolet range. This same phenomenon also occurs for the recombination of free electrons with N++, N+++, Nþ 2 , etc. The excess energy of an electron with velocity ve is converted into radiation according to the relation me  v2e =2 þ EINþ  δENþ  EN, j ¼ hυ

(56)

where EINþ is the ionization energy corresponding to the reaction N ! N+ + e and δENþ is its lowering; ENj is the energy of the excited state j to which the electron is trapped. A threshold value exists for the wavelength resulting from the trapping of electrons with zero velocity in the ENj excited state: Δ0 EN ¼ h  υmin λmax ðmÞ ¼

or

λmax ¼

hc Δ0 EN

1:98647  1025 1:23944  106 1:00048  102 ¼ ¼ 0 Δ0 EN ðJÞ Δ0 EN ðeV Þ Δ EN ðcm1 Þ where

Δ0 EN ¼ EINþ  δENþ  En , j :

(57)

(58) (59)

Because the electrons are distributed over a large spectrum of energies, their recombination into energy level j will give rise to a spectrum between υmin and large values of υ Because electrons can be trapped into every available energy state of an atom, the number of continuous spectra coincides with the number of available energy levels. Free–Free Transitions Continuous spectra occur when free electrons loose part of their kinetic energy in the Coulomb field of positive ions or neutral particles. This loss in kinetic energy is converted into radiation reflecting the Maxwellian distribution of the electrons. The emitted energy is a continuum called “bremsstrahlung,” which is typically in the infrared. Because radiation is emitted by free electrons whose kinetic energy is 1 2 2 me ve hυ; the spectral wavelength depends on the electron temperature. Free–free transition may also occur during electron–atom or electron–molecule encounters. Energetic electrons may penetrate the outermost electron shell or shells

328

M. I. Boulos et al.

of a neutral particle. As the electron “sees” the positive nucleus, the situation is similar to that in the encounter with a positive ion.

4.2

Line Radiation

4.2.1 Line Broadening The absorption or emission of discrete radiation quanta implies that monochromatic radiation of frequency υu‘ should be observed. The examination of such radiation by a spectrometer shows, however, that the spectral line has a definite width δv or δλ around υu‘ or λu‘. When the resolving power of the spectrometer used for line analysis is high enough (see Part III, ▶ Chap. 20, “Plasma Diagnostics, Optical Emission and Absorption Spectroscopy”), it is observed that the line width depends on the properties of the emitting source. The spectral lines emitted by a gas discharge are found to be rather narrow at low gas pressure (Griem 1974). At higher pressures and correspondingly higher densities of atoms, ions, and electrons in the source, there will be more frequent collisions among the radiating atoms, and the resulting effects are described as pressure broadening. When charged particles are present in significant quantities, they generate electric micro-fields that perturb the normal energy levels via Stark broadening of the spectral lines. Note that magnetic fields may also result in the broadening of discrete energy levels through the Zeeman effect. In addition, these effects may result in a displacement of the maximum of the lines called line shift. When the pressure is progressively reduced, collision and Stark broadening are also reduced, and the line width asymptotically approaches a temperaturedependent finite value. The remaining width of the line is due to random motion of the radiating atoms and is described as the Doppler effect. When the Doppler width is reduced to negligible values (by reducing the thermal velocities of the radiating atoms), the remaining width is known as the natural width and is associated with the interactions of the atoms. This natural width is generally much narrower than the Doppler broadening. In general, the frequency or wavelength distributions of emission or absorption can be described by a shape factor Pðv  v0 Þ such that ð þ1

Pðυ  υo Þ  dυ ¼ 1

with υo ¼ υu‘

Pðλ  λ0 Þ  dλ ¼ 1

with λo ¼ λu‘ :

1

or ð þ1 1

These relationships mean that Eq. 43 can be written as

(60)

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Plasma Radiation Transport

329

Fig. 8 Line profile and line shift

P(λ)

2δ

Δλ λo

ελ ¼

λ λmax

hc  Ai  ni, u  Pðλu‘ Þ: 4πλu‘ u‘

(61)

As the integration is performed over the line contour or over the line wavelength range δλ, εL(λ) results from the integration of Eq. 61 are expressed in W/m3.ster (not in W/m4.ster, as already indicated in Sect. 2.3.1). Equation 61 is then written as ð εL ðλu‘ Þ ¼

ελ  dλ ¼ line

1 hc ni, u Aiu‘ 4π λu‘

(62)

which is the line emission coefficient. Figure 8 shows the characteristic parameters of a line profile: 2δ is the width at half of the maximum intensity, λo is the central wavelength of the line, and Δλ is the shift. In the following discussion, we will restrict our description to the natural, Doppler, and Stark broadenings and to the resulting profiles. It must be emphasized that when LTE prevails, the emitted line profile is identical to its absorption profile according to Kirchhoff’s law. Since the self-absorption coefficient varies across the line width, the emission line profile reflecting this absorption will be more distorted at the line center compared to its wings. Natural Line Width The fundamental broadening of a spectral line stems from the hypothetical situation in which an atom is motionless and is shielded from electric and magnetic fields. Under these conditions the line profile is Lorentzian as defined by

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M. I. Boulos et al.

δ i: PN ðλÞ ¼ h π ð λ  λo Þ 2 þ δ 2

(63)

The half width is given by δ ¼ δm þ δn

with

δm ¼

m 1 X

Amn =4π

and

δn ¼

n¼1

n1 X Anm =4π:

(64)

m¼1

In most practical cases for thermal plasmas, this broadening is negligible compared to Stark and Doppler broadening and may be neglected in the calculations. Doppler Line Width If a monochromatic light source is moving with a velocity, v, in the direction of the line of sight, the emitted light frequency appears to be shifted by an amount (v.υ0/c) compared to the frequency υ0 of the source at rest. The random motion of the atoms in a source causes the spectral line emitted or absorbed by these atoms to be broadened. If the velocity distribution is Maxwellian, the resulting line profile is   ! 1 λ  λo 2 PD ðλÞ ¼ exp  Δλp Δλp π1=2

(65)

where  Δλp ¼ λo

2kT mc2

1=2 (66)

and m is the mass of the emitting atom. The width at half the maximum intensity is given by 2δD ¼ 2Δλp ðln2Þ1=2 ¼ 2λo

  pffiffiffiffiffiffiffiffiffiffi 2kT ln2 1=2 ¼ 7:16  107 λo T=M 2 mc

(67)

where M is the atomic mass (in g). δD is expressed in the units of λo. In thermal plasmas the Doppler effect usually results in a small contribution to the line broadening. For example, at T = 10,000 K and p = 100 kPa, 2δD ¼ 3:5  102 nm for the Hβ line at 486.1 nm, while 2δD = 9  103 nm for the NI nitrogen atomic line at 493.5 nm, and 2δD = 2.5 103 nm for the Ar-I atomic line at 696.5 nm. Equation 65 can be rewritten in terms of δD as follows: "   # ðln2Þ1=2 λ  λo 2 exp ln2  P D ð λÞ ¼ : δD δD  π1=2

(68)

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Plasma Radiation Transport

331

Stark Broadening The Stark effect becomes predominant as soon as the rate of ionization exceeds 1%. The perturbation time resulting from a charged particle moving with a relative mean velocity v with respect to the emitting particle is given by τs ¼ b=v

(69)

where b is the impact parameter. Because of the large difference between the velocities of electrons and/or atoms, the collision time τs for electrons can be regarded as short compared with the time between successive collisions, and the broadening is determined on the basis of the impact approximation. In this model the emitting system is virtually unperturbed most of the time, and broadening is described in terms of impacts that are well separated in time. For the slowly moving ions, the perturbation is practically constant over the time of interest (1/Δυ), and thus their motion can be neglected completely. Their effect is calculated using the quasi-static approximation in which a static Stark effect is assumed, taking into account the statistical distribution of the electric fields. Griem (1974) and Traving (1968) have given expressions for the half-line Stark width δe,i and the Stark shift Δe,i for which the electron density ne is the most important parameter. For example, Griem (1974) has expressed δe,i through the electron density and the parameters w, d, and α which are tabulated for different lines, temperatures, and electron densities in forms such as n

o 1=6 1=2 2δe, i ¼ 2 1 þ 1:75  104 : n1=4 : 1017 : ne : w e : α 1  0:068 ne : T

(70)

where δe,i is expressed in nm and ne in cm3; the relationship holds for 0:05 104 1=4 ne and α 0.5. The resulting line profile is Lorentzian, for example, PðλÞ ¼

1 δe 

: π λ  λo  Δe, i 2 þ δ2 e, i

(71)

For the nitrogen lines at T = 13,200 K, ne = 1.05.1017 cm3, and 2δe,i = 0.269 nm for the NI line at 493.5 nm and 0.124 nm for the NI line at 746.8 nm. At such temperatures, the Doppler broadening, which is 2δD = 0.011 nm for NI at 493.5 nm and 0.0164 nm for NI at 746.8 nm, is less than 13% of the Stark broadening. Hydrogen atomic lines are often used to determine the electron density (δe,i is by far more sensitive to ne than to Te). Figure 9 represents, for example, the Hα line broadening at T = 10,000 K versus ne; 2δe,i is calculated with the parameters determined by Hill (1964). The experimental values of Weise (1975) and Ehrich (1986) are in rather good agreement with the calculated values. The coefficients used to calculate 2δe,i should be checked carefully; the relative error between the calculated and measured values, especially for species such as Fe, Cu, F, and S, can reach values up to 30% (Rahmani 1989).

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Hill (1963) Wiese (1975) Ehrich(1986)

1.0 Stark width, 2δe,i (nm)

Fig. 9 The Stark width versus electron density at half maximum of the Hα line at 10,000 K, calculated by Baronnet (1978)

Ha at T=10 4 K

0.5

0.0 0

5

10

Electron density, ne (10

15 22

–3

m )

Resulting Profiles In thermal plasmas, the line profile represents the convolution of the Doppler (Gaussian) and Stark (Lorentzian) profiles. Assuming that the two processes are independent, the result is a Voigt profile defined as

Pða, bÞ ¼

a expðy2 Þ dy π ðb  yÞ2 þ a2

(72)

δL ðln2Þ1=2 δD

(73)

ðλ  λo  ΔλÞ ðln2Þ δD

(74)

2π1=2 δD

with and

þ1 ð

1

b¼

1

a¼

Δλ being the line shift.

4.2.2 Volumetric Spectral Emission Coefficient Neglecting Absorption When absorption is neglected, the line intensity is calculated according to Eq. 62. Assuming a Boltzmann distribution for ni,u, it follows that   ð hc 1 Ei, u 1 εL ðλu‘ Þ ¼ ελ  dλ ¼ ni  gi, u exp  Ai  i 4π kT Qe‘ ðTÞ u‘ λu‘

(75)

where index i stands for the chemical species i and Qie‘(T) is the corresponding electronic partition function. With the densities ni in m3, the transition probabilities in s1, and the wavehc lengths in m, the coefficient 4π ¼ 1:58078  1026 . In Eq. 75, the temperature

Plasma Radiation Transport

Fig. 10 Volumetric emission coefficient versus temperature at atmospheric pressure (W/m3  ster) of the NI lines at 493.503 and 746.831 nm and the OI line 777.196 nm (Baronnet 1978)

333 Line emission coefficient, e L (W/m3.ster)

8

106

O I = 777.196 nm

10 N I = 493.503 nm

10–4

N I = 746.831 nm 10–9 10–14 4

6 8 10 12 Temperature, T (103 K)

14

appears explicitly only in the exponential term, but the density of the species ni is also strongly dependent on temperature, whereas Qie‘, the electronic partition function, is rather weakly temperature dependent. Thus, for a given pressure, εL depends strongly on T, mainly through the exponential term and to a lesser degree (except when dissociation is not completed or ionization important) through ni. For example, in Fig. 10 (from Baronnet (1978)), which shows the volumetric emission coefficient εL(λ) as a function of temperature at p = 100 kPa for NI and OI lines, it is clear that the emission coefficient of NI at 493.5 nm is negligible as long as dissociation is not completed. The same is true for OI at 777.2 nm. This figure illustrates a very important experimental fact: most detectors (photomultipliers, photodiodes, etc.) can detect variations in a range of three to four orders of magnitude but not more. Thus, according to Fig. 10, temperature measurements through the absolute values of εL(λ) will be restricted to temperature ranges between about 8000 and 16,000 K. In Fig. 11 (from Pfender (1981)), which illustrates the emission coefficients (normalized to the maximum intensity) of two argon lines, Ar-I (neutral atom) and Ar-II (first ion), it is clear that the neutral line will show up in the spectrum only in the temperature interval from 10,000 K to 25,000 K, since the density of argon atoms is very low beyond 25,000 K (see Part I, ▶ Chap. 5, “The Plasma Equations,” Fig. 5.4). The normalized emission coefficient is the emission coefficient divided by its maximum value (which occurs at approximately 15,500 K for Ar-I and 25,500 K for Ar-II). According to Eq. 75, the emission coefficient of an atomic or ionic line of the species i will increase with pressure for a given temperature (about two orders of magnitude if the pressure is raised from 100 kPa to 10 MPa) due to the change in density, ni. The maximum line emission coefficient is also shifted to higher temperatures because dissociation occurs at a higher temperature when the pressure is raised, thus shifting the position of the maximum density of atoms.

334

M. I. Boulos et al. 1.0 Normalized line emission coefficient, e L/e max (-)

Fig. 11 Relative emission coefficient of two argon lines: Ar-I 763.5 nm and Ar-II 480.6 nm, as function of temperature at atmospheric pressure (Pfender 1981)

0.8

0.6 Ar II 480.6 nm Ar I 763.5 nm

0.4

0.2

0.0

0

5

10 15 20 Temperature, T (103 K)

25

30

Similar results are obtained for diatomic molecules, whose emission coefficients can be given (Fauchais et al. 1974) by ð εL ðλo Þ ¼

ελ  dλ ¼ line

0 0 0 0 hc  Ann00,,vv00,,KK,00J, J00  ni ðn0 , v0 , K0 , J0 Þ 4πλ

(76)

where ni(n0 , v0 , K0 , J0 ) is the density of the particles of species i in the emitting state; their quantum numbers are n0 for the electronic term, v0 for the vibrational term, and K0 and J0 for the rotational term. ni(n0 , v0 , K0 , J0 ) is given by the Boltzmann equation. Figure 12 represents the emission coefficient of the 0–0, 0–1, and 0–2 band heads of the B2 Σuþ ) X2 Σgþ transitions for Nþ 2 . Here also the maximum corresponds to the maximum density of Ni (see ▶ Chap. 6, “Thermodynamic Properties of Plasmas,” Sect. 3, Fig. 3).

4.3

Continuum Radiation

4.3.1 General Relationships According to Griem (1964), the absorption coefficient for particles of species i and electrical charge z.e (where z = 0 for an atom, z = 1 for its first ion, etc.) is given by κi, zþ1 ðυ, TÞ ¼

X n, ‘

nni,,z‘ σni,,z‘

(77)

Plasma Radiation Transport

Fig. 12 Absolute volumetric emission coefficients (W/m3  ster) of the band heads O–0, O–1, and O–2 of the system B2 Σu ! X2 Σg of Nþ 2 , at atmospheric pressure, as a function of temperature (Fauchais et al. 1974)

335 104 Line emission coefficient, e L (W/m3 .ster)

8

102 1 10–2 10–4

O-O Band O-1 Band O-2 Band

10–6 10–8 0–10 0

5 10 Temperature, T (103 K)

15

where nn,‘ i,z is the density of the chemical species i, z.e is the electrical charge of the excited state defined by the principal quantum number n and the azimuthal quantum number ‘, and σn,‘ i,z is the cross section for photoionization by a photon hν as calculated by Bates (1962). The emission or absorption coefficient of the atom of electrical charge z.e is denoted by the index z + 1 because of its correspondence to the recombination of electrons with ions. For an optically thin plasma, we can substitute Eq. 77 into Eq. 33, and then, using Eq. 10, it can be shown that at equilibrium: ‘ εni,,zþ1 ¼

   1 2hυ3 hυ exp nni,,z‘ σni,,z‘ :  1 kT c2

(78)

Using the Saha equation (Eq. 67 and 68, Part I, ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics”), ni,z can be expressed in terms of the electron and parent ion densities as follows: !   e‘ EIi, zþ1  δEi, zþ1 ne  ni, zþ1 2πme kT 3=2 Qi, zþ1 ¼2 exp  ni, z kT h2 Qe‘ i, z

(79)

where EIi, zþ1 is the first ionization energy for the reaction Xi, z ⇄Xi, zþ1 þ e, δEi,z + 1 is its lowering (see Sect. 5, Part I, ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics”), and Qe‘ i,z is the electronic partition function of species i, whose electrical charge is z.e. As in most cases, the ion is in its ground state (denoted by the upper index 1); using the Boltzmann equation, Eq. 2, Part I, ▶ Chap. 6,

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“Thermodynamic Properties of Plasmas,” to express ni, zþ1 versus n1i, zþ1, it follows that ‘ εni,,zþ1 ðυ Þ

¼

h4 υ3 c2 ð2πme kTÞ3=2



σni,,z‘  gin,,z‘  ne  n1i, zþ1 g1i, z

:

! (80) EIi, zþ1  δEi, zþ1  Eni,,z‘   hυ  1 exp exp kT  1 kT As is the case for line emission, one must be careful with the units of Eq. 80, which ‘ ‘ ðυÞ in J/m3ster. If e is expressed in terms of the wavelength, εni,,zþ1 ðλÞ is in gives εni,,zþ1 4 W/m ster, since   c ε λ ¼ 2 εðυÞ: λ

(81)

4.3.2 Free–Bound Transitions In this case, the emission for the different atoms of species i and their parent ions of charge z + 1 is given by εfb ¼

zmax X zmax XX ‘ εni,,zþ1 : i z¼0 n, ‘

In the following paragraphs the notation

zmax X

will be abbreviated to

(82) X : z

z¼0

The summation is performed for all atoms and ions involved and for each of the levels n, ‘ compatible with frequencies in excess of ν0 (corresponding to electrons trapped with close to zero velocity (see Sect. 3.1)). Therefore, hυo ¼ EIi, zþ1  δEi, zþ1  Eni,,z‘ :

(83)

If the resulting ion is in an excited state n0 , ‘0 and not in its fundamental ground 0 0 state, the energy of this excited state Eni, ,z‘ has to be added to the ionization energy Ei, zþ1 . Equation 83 shows clearly that the quantum states to be taken into account depend on the frequency or the wavelength at which the calculation is performed. Moreover, a limit nmax has to be imposed on the principal quantum number. According to Griem (1964), the levels with high values of n can be considered to be hydrogenic, i.e.,

nmax i, z ’

ð z þ 1 Þ 2 EH þ δEi, zþ1

!1=2 (84)

8

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337

where EHþ is the ionization energy of hydrogen. Equation 84 implies that the lowering of the ionization potential is identical to the limit of the discrete energy levels (i.e., the point at which they overlap into a continuum close to the ionization limit). Two different approaches will be used for the calculation of the photoionization cross sections: Hydrogenic Levels For hydrogenic levels (levels with high n values), the classical photoionization cross section defined by Kramers (1923) is used:

σni,,z‘

class

  σhα 2 EHþ 3 ðz þ 1Þ4 n ¼ 3=2 πr1 Gi, z ðυÞ: hυ n5 3

(85)

Gni,z is the Gaunt factor, which depends weakly on υ and which is generally close to 2 unity, α ¼ 2πe hc is the constant of fine structure, and r1 is the Bohr radius (see Eq. 5). The relative cross section defined by Eq. 86 is frequently used instead of σn,‘ i,z (see Griem (1964)). n, ‘ Gi, z

¼

σni,,z‘



σni,,z‘ gni,,z‘

class

 g1i, zþ1  2  n2

:

(86)

From Eq. 78, it can be shown that "

‘ εin,,zþ1

# !   1 n, ‘  4 Ein,,z‘ 16α0 3 EHþ ðz þ 1Þ g1i, zþ1 ni, z Gi, z hυ 1  : ¼  3  exp exp  kT Qi, z n kT 33=2 π

(87) The constant in brackets is the right-hand side of Eq. 87 and is equal to 3.0077  1026 J/ster. εfb(υ) is calculated in J/m3ster. For high quantum numbers n, the levels are very close to each other, and the summation in Eq. 82 can be replaced by an integral. At frequencies, υ < ðz þ 1Þ2

EHþ ‘ ¼ 1. According to Cabannes and Chapelle (1971), εfb can be rewritten , Gni,,zþ1 hn0 2 in the form: " εfb ðυÞ ¼

16πe6 3c3 ð6πm3 kÞ1=2 ð4πεo Þ3

#

  X X zmax ne hυ 1  exp  z2 niz : kT T1=2 i z¼1

(88)

The value of the constants in square brackets on the right-hand side of Eq. 88 can be replaced by C1 which is equal to 5.44692  1052 J. m3 K1/2/ster. For a single ion εfb is proportional to n2e . Other calculations are also possible using the Gaunt factors given by Menzel and Pekeris (1935) and Peach and Seaton (1962).

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Nonhydrogenic Atoms and Ions For nonhydrogenic atoms and ions, the photoionization cross section in Eq. 80 can be evaluated for each level, n, using the quantum defect method of Burgess and Seaton (1960): σni,,z‘ ¼

X   2 4α0 2 hυ n‘ πr1 C‘0 g n‘ , ‘0  4 EHþ ðz þ 1Þ 0 3 ‘ ¼‘1

(89)

where g is a complex expression as given by Burgess and Seaton (1960) and where n*‘ is the effective quantum number, defined as n‘ ¼

ð z þ 1 Þ 2 EH þ Ei, zþ1  Eni,,z‘

! (90)

with ‘þ1 2‘ þ 1 ‘ C ‘0 ¼ 2‘ þ 1 C ‘0 ¼

if

‘0 ¼ ‘ þ 1

if

‘0 ¼ ‘  1:

(91)

An expression similar to Eq. 88 has also been proposed by Cabannes and Chapelle (1971):   X X g1i, zþ1 ne hυ εfb ¼ C1 1=2 1  exp  ðz þ 1Þ2 ni, zþ1 e, ‘ ξi, z ðυ, TÞ kT T Qi, zþ1 z i

(92)

where ξ is the Biberman factor, which takes into account the electronic structure of the atoms. ξ depends strongly on the frequency, υ, and weakly on the temperature. For example, Fig. 13 shows the Biberman factor for an argon plasma at 8000 K at a pressure of 105 kPa (Krey and Morris 1970). The first peak is observed at λ = 87.6 nm and corresponds to the photoionization of the fundamental state of argon.

4.3.3 Free–Free Transitions Free–free transitions can be calculated for thermal plasmas using a hydrogenic approximation (Kramers 1923). Equation 80 is used with

 Giff, z ðυ, TÞ σni,,z‘ ¼ σni,,z‘ class

(93)

where Gi,z ff is the free–free Gaunt factor, and assuming hydrogenic behavior, gni,,z‘ ¼ 2n2 g1i, zþ1 and

(94)

8

Plasma Radiation Transport

339

Fig. 13 Biberman factor ξ (T, λ) for an argon plasma at p = 100 kPa and T = 8000 K (Essoltani 1991)

ðz þ 1ÞEHþ EIi, zþ1  Eni,,z‘ ¼ : (95) n2   hυ  1  hυ  1 . The emission For high frequencies, Gffi, z  1 and exp kT ffi exp  kT e, i coefficient, εff is then given by the simplified equation; εeff, i

  ne hυ X ¼ C1 1=2 exp  ðz þ 1Þ2  ni, zþ1 Giff, z : kT i, z T

(96)

The Gaunt factors Gi,z ff have been tabulated by Karsas and Letter (1961). Equation 95 shows clearly that εff is proportional to n2e (assuming a singly ionized plasma) and the εff will have significant values at atmospheric pressure only for ne > 1021 m3 (about T > 9000 K for Ar, H2, N2, and O2). For low frequencies Eq. 95 can be written as (see Cabannes and Chapelle (1971)) εeff, i ¼ C1

ne X ni, zþ1 ðz þ 1Þ2  Gi, z ðυ, TÞ T1=2 i, z

(97)

with pffiffiffi     3 T3=2 5 ln 2:1  108 e Gi, z ðυ, TÞ ¼  γ π 2 2υ where γ = 0.577 (Euler’s constant).

(98)

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M. I. Boulos et al.

When the electron density is low, the free–free radiation due to collisions with neutral atoms of density na must also be taken into account (see Cabannes and Chapelle (1971)). Assuming only elastic collisions, it follows that εea ff

" #   X 32e2 k 3=2 1 ¼ na Ga ðυ, TÞ ne T3=2 3 2πm 4πεo 3c a

(99)

where the constant C2 (in brackets on the right-hand side of Eq. 99) is 3.4213  1043 J.m.K3/2ster. The number densities na and ne are expressed in m3. The Gaunt factor for the neutrals, Ga(υ, T), is given by 1 ð

Ga ðυ, TÞ ¼

σea ðxÞx2 expðxÞdx

(100)

x0

with x = mv2/2kTe and x0 = hυ/kTe  σea(x) is the electron-neutral impact elastic cross section (m2), which is a function of the electron velocity ve (neglecting the neutral atoms’ velocity compared with that of the electrons). Equation 100 can be simplified by choosing a constant mean (average) value for σea (see Cabannes and Chapelle (1971)): Ga ðυ, Te Þ ¼ σea

  !   hυ 2 hυ 1þ 1þ : exp  kT kT

(101)

At high frequencies ðhυ=kTe  1Þ,  εea ff ðυ, Te Þ

¼

C2 ne T3=2 e

hυ kT

2



 hυ X exp  na σea : kT a

(102)

At low frequencies ðhυ=kTe  1Þ, 3=2 εea ff ðυ, Te Þ ¼ 2C2 ne T

X

na σea :

(103)

a ei In general εea ff is more important than εff at low temperatures. For example, in the ei case of an argon plasma at atmospheric pressure, εea ff ¼ εff at λ = 300 nm for ne/na  3 3  10 which corresponds to a temperature Te  8500 K.

4.3.4 Total Continuum Radiation The total continuum radiation includes the free–free (Eq. 96 or Eq. 99) and the free–bound (Eq. 86 or Eq. 91) radiation. For high frequencies (hυ/(kT)>>1), the continuum is reduced to the free–bound radiation, while for low frequencies (hυ/ (kT) ν0 with ν0 ¼ 3:52  1015 Hz), κv0 has a rather high value, varying from about 1 to 10 cm1. In this high-frequency zone, the continuum is mainly due to free–bound transitions; in fact the frequency υ0 corresponds to the ionization energy of the fundamental state of the nitrogen atom: EINþ ¼ hυ0N with EINþ ¼ 14:55 eV. Any photon whose frequency is higher than ν0 can induce photoionization of the ground state according to the reaction Xg þ hυ ! Xþ þ e : Because the ground state is highly populated and the photoionization cross section of the reaction is high, κv0 is also high. In fact, for the nitrogen atom, the ground state can be considered to correspond to three levels with, respectively, values of 0 eV, 0.238 eV, and 3.76 eV. The ionization energies of the last two levels are, respectively, 12.17 eV and 10.79 eV, corresponding to the threshold frequencies v1N ¼ 2:94  1015 Hz and v2N ¼ 2:65  1015 Hz. • In the second zone (1014 < ν < 2:65  1015 Hz, corresponding to the near IR, visible, and near-UV spectra), κV0 varies very little. Its value is between 104 and 105 cm1. • In the far IR (zone 3, υ < 1014 Hz), the continuum is mainly due to free–free radiation, and κν0 increases when the frequency decreases down to the plasma frequency.

352

M. I. Boulos et al. 102 101 Absorption coefficient, κ ν⬘ (cm–1)

Fig. 19 The nitrogen continuum absorption coefficient κν0 as a function of frequency at T = 10,000 K and p = 100 kPa (Rahmani 1989)

100 10−1 10−2 10−3 10−4 10−5 Zone

10−6

Zone II

III 0

1

2

Zone I u 0N 4 u 1N 3 u 2N

5

Frequency, n (1015 Hz)

When temperature increases, the behavior of κv0 varies in each zone depending on the phenomena controlling κv0 . When temperature increases, κv0 decreases in zone 1 (fewer atoms in the ground state), and it is almost constant in zone 2 and increases in zone 3 (with electron density). Variations with pressure are closely linked to the plasma composition, especially the concentration of electrons and ionized species.

4.5

Thermal Plasma Radiation Modeling

The aim of a thermal plasma model is to provide a self-consistent description of the plasma starting from macroscopic parameters such as the geometry, the current intensity, the nature of the gas and materials used, the mass flow rate, and/or some boundary conditions (Gleizes et al. 2005). Due to the lack of knowledge of certain phenomena and the complexity of the calculations, simplifying assumptions are needed. This is the case of the plasma radiation. Thermal plasmas emit strong radiation, particularly in the UV and visible range of the spectrum, which is far from that of a blackbody. Moreover plasma arcs and jets present strong radial temperature or density gradients which also gives rise to diffusion effects. The first simplification that can be used is local thermodynamic equilibrium (LTE), which is justified by the high collision frequencies of the different species in the plasma. Unfortunately LTE does not apply to radiation, which is not in equilibrium with particle distribution. The radiation transfer depends both on spatial and spectral parameters (Gleizes

8

Plasma Radiation Transport

353

et al. 1991; Naghizadeh-Kashani et al. 2002; Modest 2003; Gleizes et al. 2005). Moreover, for any spatial coordinate, all mechanisms of emission and absorption of radiation must be considered. These depend in turn on frequency (or wavelength), continuum and line and molecular band profiles, and intensities which makes calculations very heavy. Gleizes et al. (2005) point out that a good detailed representation of the overall spectrum for air needs three million interval ranges for each temperature value and is a three-dimensional phenomenon. These authors also pointed out that rigorous treatment of radiation transfer in plasma modeling would require a large number of iterations. Results of radiation modeling also depend on (Cressault and Gleizes (2013)): – The number of the considered atomic levels, atomic lines, and transition probabilities and their sources – The radiative mechanisms and the corresponding cross sections used to calculate the continuum – The approaches to calculate the line broadening – The assumptions used in the calculation of the net line radiation – The geometry considered for the final calculation Thus it is not surprising that important differences can be observed between different authors for the same plasma radiation calculation. The radiation emitted by all the plasma volumes, which are generally at different temperatures and have different compositions, must be taken into account. The transfer between the emission volume element and the considered one must be calculated accounting for local absorption and emission (Cressault and Gleizes 2013).

4.5.1 Approximate Solutions The simplest model (Gleizes et al. 2005), called the net emission coefficient (NEC), eN, consists in calculating the net radiation (net means the difference between emission and absorption, i.e., the divergence of the radiation intensity) in the center of an isothermal sphere. This assumption of constant temperature seems very restrictive, but it has been demonstrated that it is valid for the central regions of arcs and thermal plasmas (Gleizes et al. 1992); also, it is easy to use, εN, which gives information on the radiation of gases or mixtures of gases. In this case, we find: εN ¼

ð1 0

  Bv κ0v exp κ0v Rp dv

(131)

where εN (W/m3  ster) is the power radiated by a unit volume of the plasma per unit solid angle and the absorption coefficient κv (m1) is the inverse of a length, κν is replaced by κv0 taking induced emission into account, Rp is the radius of the sphere, and Bυ is the blackbody radiation ðεv =κv ¼ Bv ðT, υÞÞ . Figure 20 (Gleizes et al. 2005) shows the influence of plasma radius on the NEC, the fictitious curve at Rp = 0 corresponding to the assumption of an optically thin case: about 90% of

M. I. Boulos et al. Net emission coefficient, eN (W/m3.ster)

354 1012 Rp = 0 mm

1010

0.1

1.0

10

100

108

106 104 0

5

10

15 20 25 30 Temperature, T (10 3 K)

35

40

Fig. 20 NEC in air for various plasma thicknesses (Gleizes et al. 2005)

the radiation is absorbed in the first mm around the emission point, which is mainly due to the strong self-absorption of resonance lines. Figure 21 (Gleizes et al. 2005) shows the effect of the nature of the species on the NEC; the metallic vapor having a low ionization potential presents high values of εN at intermediate temperatures. According to Gleizes et al. (2005), to calculate the proportion of radiation reaching the wall or the electrodes, the NEC properties can be combined with the view factor method. But to take into account the absorption at the plasma edges and the energy deposited on a wall by radiation, the P1 model is preferred (Deron et al. 2004). This method corresponds to a first-order development of the radiative transfer equation (Eby et al. 1998). In this approach, the mean absorption coefficient is assumed to be constant over the frequency band, for a given temperature. This method requires long calculation times, as one more equation is needed for each band. As the number of bands for the gas description is around 5 (Gleizes et al. 2005), an alternative is to obtain a first solution using the NEC and then use the P1 model to achieve the convergence.

4.5.2 Pressure Effect As it has been shown in Chapter 6, increasing the pressure results in increasing the temperatures at which dissociations and ionizations occur. Thus pressure increase will also modify the radiation properties. Figure 22 presents the results obtained by Peyrou et al. (2012) for three pressures. To keep all results on the same figure, they have represented, for different Rp, the reduced NEC, i.e., the ratio εN/p, the pressure p being given in bar. For an optically thin plasma (Rp = 0), the reduced NEC generally increases with p due to the effects on the absorption spectrum. When Rp increases, the increase in the reduced absorption coefficient with pressure is counterbalanced by the escape factor exp(κv(T, p).Rp), and the reduced NEC may decrease when the pressure is increased (Peyrou et al. 2012).

Plasma Radiation Transport Net emission coefficient, εN (W/m3.ster)

8

355

1010

109

Xi = 100% 10% Ar – Cu mixture R = 5 mm

108 1% 0.1% 107

0

5

Xi = 0%

10 15 Temperature, T (103 K)

20

25

Fig. 21 Influence of copper vapor in the NEC of argon. Xi is the molar fraction of copper vapor (Gleizes et al. 2005)

1012

Net emission coefficient, εN /p (W/m3.ster.bar)

1011 1010 109 108 107

Rp (mm) Pressure 0 0.1 1.0 0.1 MPa (1 bar) 10.0

106 105 104

5

10

Rp (mm) Pressure 0 0.1 1 MPa 1.0 (10 bar) 10.0

15 20 Temperature, T (10 3 K)

Rp (mm) Pressure 0 0.1 10 MPa 1.0 (100 bar) 10.0

25

30

Fig. 22 Reduced NEC of air plasmas for different values of Rp at 0.1 MPa or 1 bar (solid lines), 1 MPa or 10 bar (dashed lines), and 10 MPa or 100 bar (dotted–dashed lines) (Peyrou et al. 2012)

4.5.3 Comparison of Methods of Calculation Dixon et al. (2004) have compared three radiation semiempirical models based on net emission coefficients (Zhang et al. 1987), the five-band P1 model (Eby et al. 1998), and the method of partial characteristics (Aubrecht and Lowke 1994; Sevast’yanenko 1979, 1980) to calculate the radiation transfer in an SF6 nozzle arc. (i) For net emission calculation, Zhang et al. (1987) proposed, for a nitrogen nozzle arc, a one-dimensional model and assumed that the arc was

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axisymmetric and the radiation transport occurred only in the radial direction. Such a model in principle performed well if axial variation in temperature was slow compared with radial variations, reasonable assumption in DC plasma flows. (ii) The method of Sevast’yanenko (1979, 1980) where the basic assumption is that in the integration of the equation of radiation transfer the temperature varies linearly between the point of interest and the current point on the integration path. This allows the integration over frequency to be done in advance and the results tabulated. (iii) The five-band P1 method, which started in 1966 (Cheng 1966). It was assumed that the absorption coefficient was constant, or an effective value was found, which is a function of composition and temperature, by some averaging technique such as the weighted-sum-of-gray gases (Becker et al. 1998). For example, Gleizes et al. (2005) have used five bands to calculate the absorption coefficients of SF6. Dixon et al. (2004) concluded that the five-band PI model proposed by Gleizes can be used to accurately predict the temperature in SF6 arcs, though the simpler model of Zhang et al. (1987) gave equally good results.

4.5.4 Mixing Rules In industrial applications of plasmas, many gases are used: Ar–H2, Ar–He, Ar–H2–He, water in plasma spraying, SF6 and CO2 in circuit breakers, Ar or air, and metal vapors in welding, cutting, etc. According to the complexity of radiation calculations for a single gas, as for compositions and thermodynamic properties (see ▶ Chap. 6, “Thermodynamic Properties of Plasmas”) or transport properties (see ▶ Chap. 7, “Transport Properties of Gases Under Plasma Conditions”), mixing rules have been developed and tested by comparing the obtained results with those resulting from rigorous calculations. Gleizes et al. (2010) have tested some mixing rules for the evolution of the NEC in the case of air–copper plasma: linear interpolation of the NEC following both mole and mass proportions and linear interpolation of the function NEC/ne. Figure 23 presents the results obtained for the NEC air–Cu plasma (99% air, 1% Cu). Gleizes et al. (2010) have shown that there are not very good mixing rules for the air–Cu and also for air–Fe and argon–metal mixtures. They explain why results are not as good as it could have been expected. At high temperatures (T > 12,000 K), a linear interpolation with molar proportions seems satisfactory. At lower temperatures they have studied several ways for mixing rules. Some tendencies have been detected but the mixing rules are then rather complicated and not correct in all cases. Finally simple linear interpolations may represent an acceptable compromise.

8

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357

Net emission coefficient, εN (W/m3.ster)

1010

109

exact mole mass

108

107

106

With no mass With no mole 5

10

15 20 Temperature, T (103 K)

25

30

Fig. 23 Simple interpolation laws for the NEC of air (90 vol.%) and Cu (10 vol.%), p = 0.1 MPa, Rp = 10 mm (Gleizes et al. 2010)

5

Examples

5.1

Classical Plasma Gases

5.1.1 Argon Plasma The argon plasma is probably the most intensively studied plasma. Figure 24 shows the evolution with temperature of the total volumetric emission coefficient εT ð1 ð0 ¼ ελ dλ ¼ εv dν expressed in W/m3. The data, compiled by Boulos (1984), 0

1

are for a temperature range of 8000–24,000 K and optically thin plasma; the resonance lines are neglected. The data from different sources Krey and Morris (1970), Emmons (1967), and Mensing and Boedeker (1969) are relatively dispersed, especially at high temperatures, where there is up to an order of magnitude difference between the extremes. A more recent compilation of the data of Krey and Morris (1970), Emmons (1967), Evans and Tankin (1967), Mensing and Boedeker (1969), Miller and Ayen (1969), Owano et al. (1990), Yabukov (1965), and Wilbers et al. (1991) shows less widely dispersed results (see Fig. 25). It should be pointed out that the results of Fig. 25 are expressed in W/m3  ster, while those of Fig. 24 are in W/m3. Correspondence between the two can be obtained by multiplying the results in W/m3  ster by 4π. Only Yabukov (1965), Owano et al. (1990), Emmons (1967), and Wilbers (1991) have extended their calculations to lower temperatures (down to 5000 K). For example, the more recent results of Wilbers et al. (1991) show higher values than those of Yabukov (1965).

M. I. Boulos et al.

Total emission coefficient, εT (W/m3)

358 1011 1010

Pure Argon Pressure = 100 kPa

109 108 107

Emmons (1967) Evans and Tankin (1967) Moskvin (1968) Mensingand Boedeker (1969) Krey and Morris (1970)

106 105 104

0

5

10 Temperature, T

15

20

(103

25

K)

Fig. 24 Total volumetric emission coefficient of argon plasma as function of temperature. The plasma is assumed to be optically thin and the resonance lines neglected (Data compiled by Boulos (1984))

Total emission coefficient, eT (W/m3.ster)

109 108

Pure Argon Pressure = 100 kPa

Evans and Tankin (1967) Krey and Morris (1970)

107 106

Emmons (1967)

105

Owano et al. (1990)

104

Miller and Ayen (1969)

103

Yabukov (1965)

102 Wilbers (1991)

10 4

6

8 10 Temperature, T (103 K)

12

14

Fig. 25 Total volumetric emission coefficient of an argon plasma as function of temperature. The plasma is assumed to be optically thin and the resonance lines neglected (Data compiled by Boulos (1984))

According to Wilbers et al. (1991), the difference occurs because their spectral range (100–105 nm) is larger than that of Yabukov (1965). Similar considerations hold when comparing the results of Wilbers et al. (1991) to those of Owano et al. (1990) who made measurements from 250 to 2500 nm. The latter did not account for the resonance lines below 200 nm. Also the continuum emissivity between 100 and 250 nm and above 2500 nm was not taken into account. For their measurements, the deviations between the measured values and the upper limits increase with increasing temperature. This result is probably due to the increasing influence of the continuum emissivity in the omitted wavelength regions.

Plasma Radiation Transport Effective emission coefficient, eE (W/m3)

8

1010

359

R = 0 mm R = 1 mm

108

106 Pure Argon Pressure = 100 kPa

104

102 5

6

7

8 9 10 Temperature, T (103 K)

11

12

Fig. 26 Evolution of the effective volumetric emission coefficient, εE, versus temperature for argon plasma at 100 kPa. The plasma is assumed to be optically thin (R = 0) or partially absorbing (R = 1 mm) (Essoltani 1991)

Of course, very different results are obtained when considering resonance lines and absorption, as illustrated in Fig. 26 from Essoltani (1991) (note that the effective emission coefficient is expressed in the same units as the total volumetric emission coefficient). The evolution of the resonance and excited state lines at atmospheric pressure as a function of temperature is shown for two cases, without (R = 0) and with (R = 1 mm) absorption. It is clear from this figure that in optically thin plasmas the resonance lines are responsible for the main contribution (almost two orders of magnitude higher than that of the other lines). However, as soon as absorption is taken into account, the absorption of the resonance lines over a radius of 1 mm significantly reduces the net emission (by almost three orders of magnitude), while that of the other excited lines remains essentially unaffected. The same tendencies with εE expressed in W/m3  ster instead of W/m3 are shown in Fig. 27 from Essoltani (1990). Here again, very strong absorption of the resonance line is observed: for a plasma radius of 2 mm, the escape factor is lower than 103 when T 10,000 K and is equal to about 0.016 at T = 15,000 K. The escape factor increases with temperature because of the decrease in the density of the groundstate atoms and the broadening of the lines due to the Stark effect. As shown in Fig. 26, the absorption of the other lines is rather small.

5.1.2 Nitrogen Plasma Figure 28 from Boulos et al. (1994) shows the results for a nitrogen plasma in W/m3 obtained by various authors (Hermann and Schade 1972; Barfield 1977; Krey and Morris 1970), assuming optically thin plasmas. Here again a certain dispersion of the data can be observed. Figure 29 compares the results in W/m3ster of the calculations of Rahmani (1989) with those of Hermann and Schade (1972) and

360

M. I. Boulos et al.

Effective emission coefficient, eE (W/m3.ster)

1011 1010

R = 0 mm

109 R = 2 mm

108 107

Pure Argon Pressure = 100 kPa

106 105 104 103

5

10

15 20 Temperature, T (103 K)

25

Total emission coefficient, eT (W/m3)

Fig. 27 Evolution of the effective volumetric emission coefficient, εE, versus temperature for an argon plasma at 100 kPa. The plasma is assumed to be optically thin (R = 0) or partially absorbing (R = 2 mm) (Essoltani 1990)

1012 Pure Nitrogen Pressure = 100 kPa

1011 1010 109 108

Hermann and Shade (1970) Krey and Morris (1970) Schreiber et al. (1972) Neuberger (1973) Barfield (1977)

107 106 105

0

5

10 15 Temperature, T (10 3 K)

20

25

Fig. 28 Total volumetric emission coefficient of nitrogen plasma assumed to be optically thin as a function of temperature at 100 kPa (Boulos 1984)

with the experimental results of Ernst et al. (1973). The experimental values are in rather good agreement with predictions up to 20,000 K. When considering absorption and the effective emission coefficient, as we did for argon, the results are rather different, as illustrated in Fig. 30 for a plasma radius of 2 mm. It is interesting to note that for T 17,000 K, the continuum radiation is almost equivalent to that of the lines in εE. Figure 31 shows the evolution of the effective emission coefficient for nitrogen plasma for different plasma radii at atmospheric pressure. As we saw for argon, the radiation is absorbed (almost 95%) in the first millimeter of the plasma. For T > 18,000 K the absorption

Plasma Radiation Transport

Fig. 29 Total volumetric emission coefficient of nitrogen plasma assumed to be optically thin as a function of temperature at 100 kPa (Rahmani 1989)

361 1011 Total emission coefficient, eT (W/m3.ster)

8

Ernst et al. (1973) 1010 Hermann and Schade (1970) 109

Rahmani (1989) Pure Nitrogen Pressure = 100 kPa

108

10

15 20 Temperature, T (103 K)

25

becomes more important. It is also important to note that εE becomes almost constant for T > 16,000 K because the electron density becomes almost constant for higher temperatures (see Fig. 3, ▶ Chap. 6, “Thermodynamic Properties of Plasmas”). Thus, the Stark line width δv also reaches a limiting value, and because the optical depth is inversely proportional to δv (see Eq. 129), absorption does not vary anymore. Moreover, the Stark width of the ionic lines is also less than that of the atomic lines; thus, the atomic lines (for the same absorption coefficient) will be more absorbed. These two facts explain the relative increase of absorption for T > 18,000 K.

5.1.3 Other Plasmas Used in Industry In plasma spraying and in cutting and welding argon, hydrogen, helium, and their mixtures are commonly used. Figure 32 from Cressault et al. (2010a) presents the NEC of pure Ar, H2, and He. As it can be seen results are rather close for Ar and H2. Some differences occur for low and high temperatures: they are due at low temperature to the molecular phenomena with H2 resulting in slightly higher εN, while at high temperature, ionic lines do not exist for hydrogen, and εN decreases slightly while it increases for Ar with the formation of Ar++ and the corresponding electrons. For He, as could be expected with its high ionization energy (24.6 eV), the NEC is the weakest for temperatures lower than 20,000 K: 300 times weaker at 10,000 K, 52 times (weaker) at 15,000 K, and 1.35 times at 20,000 K (Cressault et al. 2010). For higher temperatures, the NEC is similar to that of Ar. When considering mixtures, those of Ar and H2 are not very different from NEC results obtained with pure Ar and H2 (Cressault et al. 2010a). For Ar–He Fig. 33 from Cressault et al. (2010a) shows that helium only plays a role at very high temperatures (center of plasma) and for important proportions (>70 vol.%) in good agreement with results obtained with pure gases.

362 1012

Effective emission coefficient, eE (W/m3.ster)

Fig. 30 Evolution of the effective volumetric emission coefficient of nitrogen plasma with a radius of 2 mm, as a function of temperature at 100 kPa. Total coefficient with no absorption, effective total emission coefficient, effective emission coefficient for λ > 200 nm (no resonance lines) (Rahmani 1989)

M. I. Boulos et al.

1011

1010

Pure Nitrogen Pressure = 100 kPa Total emission eT eN, R=2 mm ec, R=2 mm

109

108 eT, λ>200 nm, R=2 mm 107

106

10

25

1012

Effective emission coefficient, εE (W/m3.ster)

Fig. 31 Effective volumetric emission coefficient of nitrogen plasma for different plasma radii R (R = 0, corresponds typically to an optically thin plasma) as a function of temperature at 100 kPa (Rahmani 1989)

15 20 Temperature, T (10 3 K)

1011 R=0 mm 1010

1 mm 5 mm 10 mm

109

15 mm

108

20 mm 107 Pure Nitrogen Pressure = 100 kPa

106

10

15 20 Temperature, T (10 3 K)

25

Plasma Radiation Transport Net emission coefficient, εN (W/m3.ster)

8

363

1011 109 Pure H2 Riad (1986)

107

Pure Ar Erraki (1999)

105

Pure He

103

Pressure =100 kPa Rp = 1 mm

10 10–1

5

10

15 20 Temperature, T (103 K)

25

30

Net emission coefficient, εN (W/m3.ster)

Fig. 32 Net emission coefficient of pure Ar, H2, and He for Rp = 1 mm (results are also compared with those of Riad (1986) and Erraki (1999)) (Cressault et al. 2010a)

1012 1010

Pure Ar

108

25% He 50% He 75% He

106 104

Pure He Pressure = 100 kPa Rp = 1 mm

102 1

5

10

15 20 Temperature, T (10 3 K)

25

30

Fig. 33 Net emission coefficient of Ar–He mixtures at atmospheric pressure for Rp = 1 mm (Cressault et al. 2010a)

The presence of pollutants in the environment, particularly in water, is one of the main concerns of our society. Many fields of industrial activities are concerned by the detection of pollutants and the decontamination of wastewater (Hannachi et al. 2008). For that elimination of toxic organic molecules or the detection of hazardous metallic contaminants of soiled water, several plasma processes are involved, and contaminant detection is often achieved by laser-induced breakdown. That is why Hannachi et al. (2008) have studied the net emission coefficient (NEC) for water–air–MgCl2/CaCl2/NaCl mixtures. Radiation of air plasma, already presented in Fig. 20, is not very different from that of water as shown in Fig. 34 from Hannachi et al. (2008). When discrepancies occur with other authors, they are

M. I. Boulos et al.

Net emission coefficient, εN (W/m3.ster)

364

1010 108 Air

106 104

H2O Pressure =100 kPa Rp =1 mm

102 1

0

5

10 15 20 Temperature, T (103 K)

25

30

Fig. 34 Net emission coefficient of air and water plasmas at atmospheric pressure and Rp = 1 mm (solid and dashed lines are due to Hannachi et al. (2008), while the data for H2O are due to Riad et al. (1995, 1998) and that of air due to Naghizadeh-Kashani et al. (2002)) (Hannachi et al. 2008)

often linked to the fact that they have neglected molecular bands or continuum. According to Hannachi et al. (2008), the NEC of air is slightly higher than that of water for temperatures up to 7000–8000 K, due to the nitrogen molecular bands and the existence of resonance lines of neutral atomic nitrogen. Above 20,000 K, radiative processes are dominated by ionic lines (O+ and N+). The NEC of water is consequently lower than that of air because O+ is the only contributor. Hannachi et al. (2008) have calculated NECs for 100% air, water, CaCl2, MgCl2, and NaCl, and results are presented in Fig. 35 for a plasma radius Rp = 1 mm. For pure alkaline salts, the NEC is strongly increased below 12,000 K compared with those of air and H2O (see Fig. 34). Species such as Mg, Ca, or Na have low ionization potentials. The population of excited electronic levels is then efficient even at low temperature leading to a strong increase in the NEC mainly due to resonance lines (Hannachi et al. 2008). For the three salts and temperature below 7500 K, the main contributions to the NEC are radiative processes involving Mg, Ca, and Na (mainly through resonance lines) rather than chlorine, which is negligible because of its high ionization potential. Between 7500 and 30,000 K, NEC values for MgCl2 and CaCl2 are quasi-identical. At high temperature, Ca+ or Mg+ lines are predominant (ionization energies of magnesium, Mg and Mg+, and calcium, Ca and Ca+, are close together with those of chlorine (Cl and Cl+ lines)). However the NEC of NaCl exhibits different behaviors with a decrease in εN between 7000 and 11,000 K where the ionic species Na+ is the majority in the plasma and Na+ lines have weak intensities. Above 11,000 K, the NEC of NaCl is again increasing because of chlorine species. Hannachi et al. (2008) have confirmed the key role of the self-absorbed lines upon the total NEC value. Other gases or gas mixtures have been studied, for example, for circuit breakers: SF6 by Randrianandraina et al. (2011), SF6–N2 (Gleizes et al. 1991), SF6–C2F4 (Jan et al. 2014), and CF3I–CO2 to replace SF6 (Cressault et al. 2011).

Plasma Radiation Transport Net emission coefficient, εN (W/m3.ster)

8

365

CaCl2

1010

108 NaCl 106

MgCl2 H2O

104

102

Air

0

5

Pressure = 100 kPa Rp = 1 mm 10 20 15 Temperature, T (10 3 K)

25

30

Fig. 35 NEC for pure gases (air, water, CaCl2, MgCl2, and NaCl) at atmospheric pressure with Rp = 1 mm (Hannachi et al. 2008)

5.2

Plasma Seeded with Metallic Vapors

When spraying metals or during cutting or welding, metal vapors interact with plasma gases, and radiation properties can be drastically modified, especially during welding and cutting (Boselli et al. 2013; Murphy 2013). Thus radiative properties of plasma gases (Ar, Ar–H2, He, air, SF6, air) with metal vapors (Al, Cu, Fe, Si, W) have been studied by Essoltani et al. (1990), Gleizes et al. (1991, 1993), Essoltani et al. (1994a, b), Raynal et al. (1995), Menart and Malik (2002), Menart et al. (2002), Cressault et al. (2008, 2010b), Moscicki et al. (2008), and Aubrecht et al. (2010). Compared to conventional gases used in thermal plasmas, the absorption length on the net emission coefficient of metal vapor is generally very important. This is illustrated in Fig. 36 for pure iron vapor. The absorption is very strong in the first 1 mm, but beyond this radius, there is little further absorption. That is why in welding arcs, the NEC is mainly calculated with an absorption radius of 1 mm, with the use of a larger radius resulting in small differences to the calculations. The calculations for pure Ar and Ar + 1 mol% Fe, presented here, have been developed by Essoltani (1991). Figure 37 shows the total volumetric emission coefficients for iron and argon in optically thin plasma at 100 kPa. In spite of the low percentage of iron (1 mol%), iron radiates much more than argon does, especially at low temperatures (T 10,000 K), due to two factors: (1) the large number of iron lines (3226 lines in the iron calculation, compared with less  than 500 for argon) and (2) the low ionization potential of iron EIFeþ ¼ 7:9 eV and the low energy of the first excited level of iron (E1 = 0.85 eV). The corresponding value for Ar-I is 15.75 and 11.54 eV, respectively. For T > 12,000 K the argon emissivity exceeds that of iron.

M. I. Boulos et al.

Net emission coefficient, e N (W/m3.ster)

366 1012 Rp = 0

1011

Rp = 1 mm

1010 109 Rp = 15 mm

108 107 106 105 104

0

5

10 15 20 Temperature, T (103 K)

25

30

Total emission coefficient, eT (W/m3)

Fig. 36 Comparison of net emission coefficients for 100% iron vapor, for different plasma radii (Murphy 2010)

1010 108 Ar + 1% Fe 106 104

Pure Argon

102

Pressure = 100 kPa

10 10–2 10–3

3

4

5

6 7 8 9 Temperature, T (103 K)

10

11

12

Fig. 37 Total volumetric emission coefficient of Ar–Fe (1 mol%) plasma assumed to be optically thin, as a function of temperature at 100 kPa (Essoltani 1991)

The importance of iron for the total volumetric emissivity εT is illustrated in Fig. 38, where the evolution of εT versus temperature for the Ar–Fe mixture has been plotted for various iron mole percentages. Below 7000 K even 0.001% of iron results in a dramatic increase in εT (eight orders of magnitude at 3000 K). When considering the effective volumetric emission coefficient εEFe of 1 mol% of Fe in Ar at different plasma radii (see Fig. 39), it can be seen that:

8

Plasma Radiation Transport

367

Total emission coefficient, eT (W/m3)

1014 1012 1010 108

90% Fe 10% Fe 5% Fe 1% Fe 0.1% Fe 0.01% .Fe 0.001% Fe

106 104 102 10

Pure Argon

10–2 10–3

3

4

5

Pressure = 100 kPa

6 7 8 9 Temperature, T (103 K)

10

11

12

Effective emission coefficient, eE (W/m3)

Fig. 38 Total volumetric emission coefficient of Ar–Fe plasmas assumed to be optically thin, for different mole percent of iron as a function of temperature at 100 kPa (Essoltani 1991)

1011 Ar + 1% Fe Pressure = 100 kPa

1010 109 108

R=0 mm R=1 R=5 R=10

107 106 105

3

4

5

6 7 8 9 Temperature, T (103 K)

10

11

12

Fig. 39 Effective volumetric emission coefficient of Ar–Fe (1 mol%) plasmas for different plasma radii as a function of temperature at 100 kPa (Essoltani 1991)

• With no absorption (R = 0) εEFe increases dramatically with temperature from 3000 to 7000 K in connection with the high densities of the excited levels in this range. temperature  Due to the low value of the iron ionization potential EIFeþ ¼ 7:9 eV for T > 7000 K, ions are produced and the neutral atoms’ densities decrease, slightly reducing εEFe

M. I. Boulos et al.

Effective emission coefficient, eT (W/m3)

368 1014 90% Fe

1012

10% Fe 5% Fe

1010 108 106 104

1% Fe

102

0.1% Fe 0.01% Fe 0.001% Fe

1 10–2

Pure Argon

10–4 10–6

3

4

5

Pressure = 100 kPa R=10 mm

6 7 8 9 Temperature, T (103 K)

10

11

12

Fig. 40 Effective volumetric emission coefficient of Ar–Fe plasmas as a function of temperature at 100 kPa for different mole percent of iron and a plasma radius of 10 mm (Essoltani 1991)

• Due to the important contribution of the resonance lines to the total emissivity, εEFe decreases very rapidly within the first millimeter of plasma and much more slowly at radii greater than 1 mm. When comparing Fig. 40, which gives εE for various iron percentages and a plasma radius of 10 mm, to Fig. 38, which gives εT for an optically thin plasma, the importance of the absorption of the argon resonance lines in the whole temperature range is obvious, while the resonance lines of iron mainly absorb at temperatures below 9000 K. In both cases, however, it is clear that the presence of iron vapor, even at low percentages, significantly increases the plasma emission below 10,000 K. The calculations for argon and copper plasmas have been performed by Gleizes et al. (1990). Figure 41 compares the effective volumetric emission coefficients for argon plasma containing 1 mol% iron and one containing 1 mol% copper; the arc radius is 10 mm. Because of the lower number of Cu lines compared to those of Fe, εEFe is almost one order of magnitude higher than εECu. Notice that (as already emphasized for Ar–Fe plasma) the absorption of argon in the Ar–Cu plasma is higher than that of Cu because of the strong absorption of Ar resonance lines, especially at high temperatures (see Fig. 42 from Gleizes et al. (1990)). For a given percentage of Cu (1 mol%), the influence of the thickness of the plasma is illustrated in Fig. 43 (from Gleizes et al. (1990)). As we saw for iron, overall there is a strong absorption at low temperatures corresponding to high densities of absorbing atoms. Also note that the values of εE are hardly dependent on the plasma radius as soon as it is greater than 1–2 mm. Finally, the influence of the copper mole percentage is illustrated in Fig. 44, which shows the evolution of εE with temperature (at 100 kPa) for a plasma radius of 2 mm.

8

Plasma Radiation Transport

369

Effective emission coeff., eT (W/m3. ster)

1010 Ar + 1% Fe 109

108 Ar + 1% Cu 107

106

Ar/Cu and Ar /Fe plasmas Pressure = 100 kPa R=1 mm 5

6

7

8

9

10

Temperature, T

11

12

(103

K)

13

14

15

Fig. 41 Effective volumetric emission coefficient of Ar–Fe and Ar–Cu plasmas (both with 1 mol% of metallic vapor) for a plasma with a radius of 1 mm as a function of temperature at 100 kPa (Gleizes et al. 1990)

100 R=5 mm 10% Cu Relative proportion of Cu-emission (%)

Fig. 42 Relative proportion of copper emission in Ar–Cu plasma, at 100 kPa; solid line R = 0 (no absorption); dotted line R = 5 mm (Gleizes et al. 1990)

75

R= 0

R=5 mm 50

1% Cu R= 0 R=5 mm

25 R= 0 0.1% Cu

0

5

10

15 20 Temperature, T (103 K)

25

370

M. I. Boulos et al.

Fig. 43 Influence of plasma thickness on the effective emission coefficient of Ar–Cu (1 mol%) plasma at 100 kPa (Gleizes et al. 1990)

1011

Effective emission coefficient, eE (W/m3 . ster)

Fig. 44 Influence of the copper concentration (mole %) on the effective emission coefficient of Ar–Cu plasma at 100 kPa and a plasma radius of 2 mm (Gleizes et al. 1990)

1010 10% Cu 1%

109

108 0.1% 107 Pure Argon 106 Ar /Cu plasma Pressure = 100 kPa 105

104 5

R=2 mm

10

15 20 Temperature, T (103 K)

25

8

Plasma Radiation Transport

371

Net emission coefficient, e N (W/m3.ster)

1011 1010 109

Pressure =100 kPa Rp =1 mm

Pure Cu 108 107

10% Cu 1% Cu

106

0.1% Cu Pure Air

105

5

10

15 20 Temperature, T (103 K)

25

30

Fig. 45 NEC in air–copper mixtures at atmospheric pressure (Gleizes et al. 2010)

The presence of metal vapor in conventional gases used in thermal plasma generation has in general a significant influence on their NEC because of the low values of excitation and ionization potentials for the metal (Gleizes et al. 2010). This influence is particularly important at low and intermediate temperatures as it can be seen in Fig. 45 showing the NEC in air–Cu mixtures. As a first approximation, the evolution of the NEC with temperature is rather similar to that of the electron number density. For example, Fig. 46 (Murphy (2013)) shows that the net emission coefficients for argon are much lower than those for aluminum, which are in turn much lower than those for iron. Metal vapor has a strong influence on the arc temperature distribution as illustrated by Murphy (2013) for a gas–metal arc welding (GMAW) of thin sheets of aluminum. The arc current was 95 A, the wire electrode radius was 0.6 mm and fed at a rate of 72 mm/s, the wire electrode–workpiece distance is 5 mm, and the wire electrode and workpiece composition was aluminum alloy AA5754. Murphy (2013) has calculated that the temperature in the presence of aluminum vapor was 9800 K and without aluminum vapor 14,000 K. In addition to the much lower temperature, a local temperature minimum was apparent in the aluminum vapor case, extending on the axis for about 1 mm below the wire tip. The net NEC depends on the temperature, the plasma size, and the concentration of the metallic vapors in the mixture. Cressault and Gleizes (2013) have studied the NEC of Ar–Cu, Ar–Fe, and Ar–Al: it increases with the temperature with fast variations at low and intermediate T, not only due to the fast evolution of the electron number density but also due to the excited-level number density; it decreases when the plasma size increases. Its values dramatically diminish within the first millimeter due to a strong absorption around the emitting point and more

M. I. Boulos et al.

Net emission coefficient, εN (W/m3.ster)

372 105 Iron 104 Aluminum 103

Argon

102 10

1 0

5

10 15 Temperature, T (103 K)

20

25

Fig. 46 Net emission coefficients for plasmas in argon, aluminum, and iron. The absorption length is 1 mm (Murphy 2013)

particularly due to the resonance lines that are very emissive but strongly absorbed when Rp > 0 mm (Cressault and Gleizes 2013). Figure 47 shows that the absorption is more pronounced with aluminum than with iron or copper in the first millimeter for temperatures higher than 10,000 K, results confirmed by Fig. 48 which shows the relative absorption coefficient as function of temperature for Ar/Cu, Ar/Fe, and Ar/Al mixtures. The relative absorption coefficient is defined in this case as the ratio of the net emission coefficient εN for (Rp = 1) to the net emission coefficient in the absence of absorption (Rp = 0).

6

Blackbody Radiation of High-Temperature Gases

Some of the emission coefficients described in the previous paragraphs are based on the assumption that the plasma is optically thin, with all the radiation escaping from the plasma. This assumption may fail for the following two situations: (i) In the first case very strong absorption occurs for resonance lines (transition between the first excited state and the fundamental level) for which the absorption coefficient κv0 (see Eq. 47) is so high that a layer of thickness L of a fraction of a millimeter is already sufficient for complete absorption (Finkelnburg and Peters 1957) (see also the results presented in Sect. 4). In the immediate neighborhood of such a resonance line, the absorption coefficient is usually many orders of magnitude smaller. For lines corresponding to lower-level transitions, the absorption coefficient may be such that in a layer of a few millimeters self-absorption will be responsible for a reduction of 20–50% of the line intensity. This reduction may lead to significant errors when such lines are used for spectroscopic temperature measurements with no corrections for this absorption.

Plasma Radiation Transport

Net emission coefficient, εN (W/m3.ster)

8

373

1012 90% Ar + 10% metal vapor 1011 1010 109 108 Rp = 0 mm Rp = 1 mm

107 106

0

Fe

20 10 15 Temperature, T (103 K)

5

Al

Cu

25

30

Fig. 47 Comparison of the NEC for 90% Ar–10% metal vapor mixtures (mass proportions) at atmospheric pressure and Rp = 0 and 1 mm (Cressault and Gleizes 2013)

Relative absorption coefficient (%)

100

80 60

Pure Argon

40 10% metal

Pure 20

0

Cu Fe Al 5

10

15

Temperature, T

20 (103

25

30

K)

Fig. 48 Absorption phenomena for pure gases and Ar–10% metal vapor mixture (mass) at atmospheric pressure (Cressault and Gleizes 2013)

(ii) The optically thin assumption may also fail for high-pressure plasmas (p > 100 kPa). Finkelnburg and Peters (1957) have calculated the conditions for which the continuous radiation of a laboratory plasma approaches that of blackbody radiation εv 0:9 ε0v in the visible range of the spectrum (500.0 nm). Figure 49 shows the results of their calculation for different gases. By considering only the singly ionized species of these gases, all curves merge into a common curve that corresponds to an ionization degree of 100%.

374

M. I. Boulos et al.

He (24.46 eV)

100

Pressure p, (MPa)

H (13.59 eV) Hg (10.38 eV)

10

Xe (12.0 eV)

Cs (3.87 eV)

1.0 R = 2 mm 0.1 0

10

20 30 Temperature, T (103 K)

40

50

Fig. 49 Regime in which a plasma control volume with a radius of 2 mm can be considered to be a blackbody (After Finkelnburg and Peters (1957))

Above this common curve, a laboratory plasma with a layer thickness of 2 mm or larger would become a blackbody radiator at the plasma temperature. Argon, nitrogen, and oxygen, which have ionization energies between 14 and 16 eV, would fall between the curves for helium and hydrogen, i.e., become blackbody radiators for temperatures T > 2.104 K and pressures p > 20 MPa. Cesium, with the lowest ionization potential, would require a minimum pressure of 0.5 MPa to become a blackbody radiator at 5000 K.

7

Two-Temperature Plasma

Relatively few papers have been published for plasma radiative emission calculations under nonequilibrium, two-temperature (2T) conditions. The short analyses given below are those of Gleizes (2014) who made valuable contributions to this problem. In the following it is assumed that differences between the temperatures of heavy particles and electrons are not too important and that both have Maxwell’s distributions and Boltzmann equilibrium is achieved.

7.1

Emission and Calculation of Net Emission Coefficient (NEC)

7.1.1

Atomic Gas

Emission The main assumption in this case is that the degree of ionization should be higher than 104, giving rise to sufficiently high electron densities to control the excitation-de-excitation processes. The populations of excited states are calculated using the Boltmann equation at Tex = Te. The emitted radiation implies that:

8

Plasma Radiation Transport

375

– Free electrons loosing part of their kinetic energy in the Coulomb field of positive ions: corresponding to bremsstrahlung. This type of radiation is proportional to n2e and depends on Te. – Radiative recombination is proportional to n2e and also linked to Te. – The line emission is the main component of radiation. According to Saha’s law, the population of excited states is proportional to n2e . However for temperatures in the 12,000–14,000 K (depending on considered species), LTE can be assumed. Accordingly a first approximation could be that the emission intensity of the 2T plasma is given by "

# n2e I ðTe , Th Þ ¼ I :Te  2 : ne 

(132)

Variables with star as exponent are those calculated at LTE with Th = Te. The real value of ne must be calculated with 2T Saha equation coupled to Dalton’s law. Of course if nonlocal chemical equilibrium (non-LCE) is also considered, reaction kinetics must be taken into account. Emission and Absorption: Net Emission Coefficient (NEC) The problem becomes more complex because NEC considers both emission and absorption. If the emission is proportional to the density of emitting species, absorption is defined through a term exp(K.x) where, K, the absorption coefficient, is proportional to the density of absorbing species. For atomic species, absorption depends on resonance radiation lines and radiative recombination. Calculating NEC in 2T conditions is accordingly rather complex. For species resulting from dissociated, or almost dissociated, molecular gases, in hot plasma regions, resonance emission is strongly absorbed. Nevertheless, the radiation that escapes (net radiation), which is part of the resonance emission, is still not negligible, and the calculation of the NEC using Eq. 132 will be overestimated. This is due the proportionality with n2e , which increases the emission while the absorption in 2T plasma is higher than it is in LTE case. Thus maybe the error would be lower with no proportionality NECðTh , Te Þ ffi NEC ðTh ¼ Te Þ

(133)

star index corresponding to values calculated in LTE. However this has not been checked yet. Metal Vapors In the presence of metal vapors, generally resonance lines are strongly reabsorbed and Eq. 133 is probably applicable. However if the metal vapor concentration is low (1023 m
3), resulting in sufficiently high frequency of elastic collisions between electrons and heavy species to maintain thermal equilibrium. However, when the pressure is lowered or the electric field increased (e.g., near the electrodes of arcs at atmospheric pressure) and large gradients exist (near cold walls or when a cold gas is injected into the plasma), the equilibrium between the temperatures of the electrons and heavy species cannot be attained. In ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics,” Sect. 5.4.1., Eq. 84 shows that the electron temperature Te differs from that of the heavy species Th when the energy gained by the electrons between two collisions
e  E  ved  τe is no longer negligible compared to the thermal energy (kTe). At atmospheric pressure in the high-temperature core of the DC plasma jet, the difference between Te and Th is less than 2% and equilibrium prevails. At lower pressures (kPa), the mean free path ‘e increases significantly, and Te will be almost twice that of the value of Th, even in the hot core of the jet. It must also be pointed out that in regions close to walls and electrodes the boundary layer thickness corresponds to about 102 mean free paths. The heavy particles reach the wall with almost the same temperature as that of the wall, while the electrons lose only a few percent of their kinetic energy and maintain almost the same temperature as in the equilibrium regions. Furthermore equilibrium is not readily attained when a cold gas is injected into the plasma. The energy exchange among heavy particles is very fast, but the thermalization of the electrons with the heavy particles requires rather long distances (10 times the mean free path) (Hsu and Pfender 1981). An excellent overview of the problems induced by calculations presented below was presented in the paper of Rat et al. (2008).

3.1

Calculation Basis

Assuming constant pressure p, the total pressure can be expressed by Dalton’s law: p þ δp ¼ p1 þ

K X

pi

(1)

i¼2

Electrons are characterized by the index one, while other species have indexes >1. δp is the pressure correction term resulting from the electrostatic interactions

390

M. I. Boulos et al.

(see Sect. 2.3.2 and ▶ Chap. 6, “Thermodynamic Properties of Plasmas,” Eq. 15); this term cannot be neglected for nonequilibrium regions. Equation 1 can be rewritten as p þ δp ¼ n1 kTe þ

K X

ni kTh

(2)

i¼2

Introducing the parameter θ ¼ Te =Th , it becomes K X θðp þ δpÞ ¼ θ  n1 þ ni kTe 1¼2

(3)

with   kTe
24 Te ¼ 2:147  10 ðPaÞ 24πλ3D λ3D

δp ¼

(4)

λD is the Debye length. Equation 14 in chapter 6 for the Debye length has to be modified by introducing the electron temperature Te and the electron density along with the heavy particle temperature for all the heavy species. For example, for nitrogen plasma containing only N2, N, N+, and e, it follows that λ
2 D

 ¼

e2 ε0 k



ne nN þ þ Te Th

 or

λ2D

  εo kTe 1 ¼  2 1 þ Te =Th e ne

(5)

since nNþ ¼ ne . To illustrate the calculation nitrogen will be considered. At equilibrium, the following species have been considered: e, N, N+, N2, and Nþ 2 . Figure 1 shows that 1026 Nitrogen, 100kPa Number density, ni (m–3)

Fig. 1 Evolution with temperature of the number density (m
3) of the different species for a nitrogen plasma at p = 100 kPa in LTE (André et al. 1999)

N2

1024

1022 N 1020

1018

e

1

2

3

4

5 3

Temperature, T (10 K)

N+ 6

N2+ 7

9

Thermodynamic Properties of Non-equilibrium Plasmas

391


4 for T > 5000 K, Nþ 2 ions are present but with a low molar fraction (between 10
6 + and 10 ), the main ion being N (even when N is at its maximum and the N+ density is more than one order of magnitude larger). When considering higher temperatures (up to 14,000 K) (see Fig. 3 in ▶ Chap. 6, “Thermodynamic Prop20 erties of Plasmas,” the density of Nþ the maximum being about at 2 is below 10 8000 K. However, for the same electron temperature at 15,000 K, for example, if the ratio is θ = 3, then Th = 5000 K. The dissociation of N2 molecules, which is mainly controlled by collisions between heavy particles, is low, while the electron energy is high. Since the main component of the plasma is N2 at Th = 5000 K, the highenergy electrons will have a high probability of removing an electron from N2 but a low probability of taking an electron off N, whose population is very low. Thus under these conditions, the main ion will be Nþ 2 . The composition at equilibrium is obtained by satisfying dGp,T = 0 together with the other conditions of nitrogen conservation and electrical neutrality (see respectively Eqs. 34, and 35 in ▶ Chap. 6, “Thermodynamic Properties of Plasmas”). Proceeding as in Sect. 3.1 in for the mass action law in CTE,

 
 dG ¼ 2μN
μN2 dηN þ ðμNþ þ μe
μN ÞdηNþ þ μNþ2 þ μe
μN2 dηNþ2

(6)

where dηN, dηNþ , and dηNþ2 are, respectively, the production rates of NðN2 , 2NÞ, Nþ ðNþ þ e , NÞ,

and


þ  Nþ 2 N2 þ e , N2 :

The chemical potentials (see Eqs. 48, and 49 in ▶ Chap. 6, “Thermodynamic Properties of Plasmas”) are then written for electrons and heavy particles as   p þ kTe ln 1 p

(7)

  p μi ¼ μ0i þ kTh ln i p

(8)

μ1 ¼

μ01

and

Thus, replacing p with the corresponding values of the partition functions for electrons results in "

  #   kTe 2π me k Te 3=2 p 
kTe lnQe þ kTe ln 1 þ E001 μ1 ¼
kTe  ln p p h2

Qe ¼ 2 for electrons. For atoms and ions, one obtains

(9)

392

M. I. Boulos et al.

"

    # kTh 2π mi k Th 3=2 pi i
kT μi ¼
kTh  ln  lnQ ð T Þ þ kT ln þ E00i h h h int P p h2

(10) In this equation, the temperature at which Qiint must be calculated (Th in Eq. 10) is still a matter of debate.

3.2

Partition Function Calculation in Two-Temperature Model

In all composition calculations, partition functions are the key factors; see, for example, the chemical potentials μi in Eqs. 10 and 9, and the main question is at what temperature they must be calculated: Te, Th, or Tex (excitation temperature). Most authors agree on the fact that elastic collisions between heavy species depend on Th, while for inelastic collisions (excitation and ionization of atoms) they depend on Te. For molecules it can be supposed that the quantum motions of rotation and vibration are uncoupled, and energies of molecules are translation, excitation, rotation, and vibration (Herzberg 1959). Thus three temperatures should be considered: excitation Tex, vibration Tv, and rotation Tr (André 1995). If most authors agree on the fact that rotational energies must be calculated at Th, it is not the case for vibrational energies. Some authors, such as Gleizes et al. (1999), suggested using Th only, while others consider that they depend on both. Gleizes et al. (1999) and Tanaka and Yokomizu (1997) suggested that the dissociation and ionization reactions should be governed by an excitation temperature, Tex, which controls the internal energy states of heavy species. When ne is high, the inelastic electron collisions are mainly responsible for the transitions between excited levels, and Tex  Te. At low vales of ne, these transitions are mainly due to inelastic collisions of atoms or molecules, and Tex  Th. Tanaka and Yokomizu 1997 and Cliteur et al. 1999 have defined an excitation temperature Tex by assuming a Maxwellian energy distribution for both electrons and heavy species, Tex also depending on elastic collision frequencies, electron and heavy species temperatures, and collision loss factors:

Tex A ¼

υ e, A T e þ υ e, A þ

X Xjðj6¼eÞ

υ j , A Th

υ jðj6¼eÞ j, A

(11)

where νe,A and νj,A are the elastic collision frequencies given by υj, A ¼ nj σj, A ðTh Þ υh

(12)

υe, A ¼ ne σe, A ðTe Þ υe

(13)

9

Thermodynamic Properties of Non-equilibrium Plasmas

393

where σe,A and σj,A are the elastic cross sections and ve and vh the mean thermal speeds of electrons and heavy species, respectively. However Gleizes et al. (1999) pointed out that the expressions of Tanaka and Yokomizu (1997) and Cliteur et al. (1999) are not correct because Tex should depend on inelastic collisions and not on elastic ones. Consequently, they have defined an excitation temperature from a simplified kinetic approach depending on ne, Te, and Th. For that they introduced the following excitation temperature: 

    ΔE ΔEðTe
Th Þ 4 exp
ffi 10 ne exp þ nA kTex kTe Th

    o
1 ΔEðTe
Th Þ ΔE ΔE  104 ne exp ; exp þ nA exp kT h kTe Th kTe

(14)

where ΔE is the energy gap between two consecutive excited levels and nA the atom’s number density. The excitation temperatures tend to Th at low ne and Te at high ne. André et al. (2001) have also defined an intermediate temperature T* between Te and Th, which tends to Te when ne is high (>10
23 m
3) and to Th when ne is low ( 5.10
22–10
23 m
3

394

M. I. Boulos et al.

Fig. 2 Evolution of the intermediate temperature T* (as defined by Eq. 15) and Te, with the heavy species temperature for a hydrogen plasma (André et al. 2001)

LTE exists and below these values θ increases when ne decreases. That is why André et al. (2001) have proposed the following relationship:
 Θ ¼ 1 þ A:ln ne =nmax e

(18)

where nmax ¼ 10
23 m
3 and A =
0.2 have been selected in order to obtain θ = e 2 at 5000 K, θ = 1.2 at 9000 K, and θ = 1 at 13,500 K, in agreement with experimental results (Fauchais et al. 1989; Gordon and Kruger 1993). For example, for a hydrogen thermal plasma, the evolutions of T* (Eq. 15) and Te as function of Th are presented in the Fig. 2 from André et al. 2001. T* is the same as Th below 6000 K, and between 9000 and 13,500 the same as Te, which
K, it is almost  becomes equal to Th over 13,500 K ne  1023 m
3 . For small deviation from equilibrium, Te is about twice as high as T* at 5000 K and about five times at 1500 K. However, for practical purposes, below 4500–5000 K, the electron density is negligible. Finally, T* damps the strong variations of Te between 7000 and 9000 K, where the electron density is still negligible In all cases results depend strongly on the partition functions (Aubreton et al. 2013). The partition function has to be calculated carefully as illustrated by Capitelli et al. (2008) who studied the role of Debye–H€ uckel electronic energy levels on the thermodynamic properties of hydrogen plasmas including isentropic coefficients. Aubreton et al. (2013) have studied the influence of partition functions on thermodynamic and transport properties of Ar–Cu mixtures. They point out that up to the principal quantum number n = 5, the tables of NIST tabulate the energy levels of the different electronic states but not beyond. The solution is then to assume that the energy levels have hydrogenic structure. The choice of the cutoff criteria to determine the maximum value of the principal quantum number is also important. With this mixture they used three methods of calculation of partition functions and compared results for plasma thermodynamic and transport properties. The first method works on nmax (maximum allowed principal quantum number) and the second on energy depending

9

Thermodynamic Properties of Non-equilibrium Plasmas

395

on the species as in the first method. In the third method they tested simplified partition functions. Two methods gave similar results, which was not the case with the third. They concluded that it is necessary to use the most complete data, and for temperatures higher than 12,000 K, it is better to complete the data and use one of the two first methods. Capitelli et al. (2012) developed a state-to-state approach for the calculation of the partition function of monatomic species in thermal plasmas. They showed that the consideration of excited states is very important. The state-to-state plasma kinetics is now passing from a qualitative understanding to a quantitative predictive tool especially for molecular H2, N2, and O2 plasmas and their mixtures.

3.3

Calculations of the Plasma Composition

3.3.1 Dissociation Reaction Since it can be assumed that dissociation is mainly due to collisions between heavy particles, according to Eq. 10, it can be found that for the dissociation of N2:  Kp ðNÞ ¼

p2N ¼ pN2

QN int ðTe Þ



2 QN int ðTh Þ 



3=2

2πmN kT h2



2  kTh

 2πmN2 kT 3=2 h2

 kTh

  2Eo
EooN2 exp
oN kT

(19)

E00i being energy of the chemical species i in its ground state related to an absolute reference state and 2E00N
E00N2 being the dissociation energy.

3.3.2 Ionization Reactions When considering the reaction Nþ þ e , N with Eqs. 9 and 10, μNþ þ μe
μN ¼ 0 results in     kTh tr p þ int QNþ ðTh Þ
kTe lnQNþ ðTe Þ þ kTh ln N þ EooNþ
kTh ln p p     kTe tr pe Qe ðTe Þ
kTe ln2 þ kTe ln
kTe ln þ Eooe p  p    kTh tr p int QN ðTh Þ þ kTh lnQN ðTe Þ þ kTh ln N þ EooN
dENþ ¼ 0 þkTh ln p p and thus finally  pNþ 1=θ pN       EINþ
δENþ 2πme kTe 3=2 Qint Nþ ðTe Þ ¼ kTe 2 exp
kTe Qint h2 N ðTe Þ

KP ðNþ Þ ¼ pe



(20)

396

M. I. Boulos et al.

where EINþ is the ionization potential for the reaction ðNþ þ e , NÞ and δENþ its lowering. with the ionization of molecules, for example,
Similarþ calculations  N2 , N2 þ e result in


Kp Nþ 2



 ¼ pe

pNþ2 pN2

 ¼ kTe

1=θ

2πme kTe h2

3=2 2

Qint Nþ ðTh Þ 2

Qint N2 ðTh Þ

!1=θ exp


EINþ
δENþ2

!

2

kTh

(21)

where EINþ is the ionization energy of Nþ its lowering. It has been assumed 2 and δENþ 2 2

that mN2 ¼ mNþ2 and the main difference for the molecular ions compared to dissociation is the exponent 1/θ for the partition function ratio. Of course Tex can replace the temperature Th. Instead of considering partial pressures, one can use number densities and the Eq. 20 becomes KnNþ

 1=θ  3     EINþ
δENþ nN þ 2πme kTe 2 Qint Nþ ðTe Þ ¼ ne ¼ 2 exp
nN kTe Qint h2 N ðT e Þ

(22)

This approach, generally called that of Potapov who was the first to propose it, has been criticized by van de Sanden et al. (1989). They proposed a generalized Saha law differing from that of Potapov by the lack of the exponent (1/θ) on the left hand side of Eq. 22, i.e., KnNþ ¼ ne

   3     EINþ
δENþ nN þ 2πme kTe 2 Qint N þ ð Te Þ 2 ¼ exp
nN kTe Qint h2 N ð Te Þ

(23)

This equation was similar to that proposed by Chen and Han (1999). At last we will cite the method of André (1995), André et al. (1996) who generalized the Gibbs free energy minimization to obtain the multi-temperature plasma composition. This minimization was achieved using the Lagrange multipliers, considering the neutrality condition, mass conservation, and Dalton’s law. The Gibbs free energy was minimized to obtain the plasma composition and was written as 2 0 13   N X p n T i tri A5 G¼ (24) ni 4μ0i þ RTtri ln 0 þ RTtri ln@XN p n T i¼1 k trk k¼1

ni, μ0i ,

0

where Ttri, p, p , and R are, respectively, the number density, the chemical potential at pressure p0, the translational temperature of species i, the total pressure, the pressure of 105 Pa, and the ideal gas constant.

9

Thermodynamic Properties of Non-equilibrium Plasmas

397

The potentials μ0i are given by μ0i ¼
RTtri ln

 tr  Qi ðTtri Þ
RTex lnQei ðTex Þ
RTv lnQvi ðTv Þ
RTr lnQri ðTr Þ þ e0i N

(25) where Qtri , Qei , Qvi , Qri , and e0i are, respectively, the translational, electronic, vibrational, and rotational partition functions, with the corresponding temperatures and the enthalpy of formation of species i at 298.15 K and 105 Pa. By definition all levels of the same types (excitation, vibration, rotation) follow a Boltzmann distribution with their own temperature. The results will strongly depend on the choices of Ttri, Tex, Tv, and Tr and the way partition functions will be calculated.

3.4

Results Obtained with Different Methods

In the following the results obtained by André et al. (1999) using the Gibbs free energy minimization method are presented (see Eqs. 29 and 31 in ▶ Chap. 6, “Thermodynamic Properties of Plasmas”) to calculate the compositions of N2 plasma, at atmospheric pressure, under LTE and 2T conditions. These are given in Figs. 1 and 3, respectively, at LTE and NLTE with θ ¼ 2 . Since LCE was assumed in both cases, the concentration of N2 increases, following the perfect gas law nN2 ¼ kTp h for θ ¼ 0 and 1 (Fig. 1), while for θ ¼ 2 (Fig. 3), the dissociation of NðnN2 ¼ nN Þ occurs, respectively, at a heavy particle temperature between 5750 and 6600 K. A comparison of both Figs. 1 and 3 shows that at 2T (θ = 2), due to the electrical neutrality, the electronic concentration is higher, while concentrations of þ Nþ 2 and N are close to that of electrons. Aubreton et al. (1998) as André et al. (1999) used the Gibbs free energy minimization method to calculate the composition of N2 plasma at atmospheric 1026 Nitrogen, 100kPa Number density, ni (m–3)

Fig. 3 Evolution with temperature of the density (m
3) of different nitrogen species in a nitrogen plasma at p = 100 kPa in NLTE, θ = 2 (André et al. 1999)

N2 1024 e N

1022

N+2 1020

1018

1

N+

2

3

4

5

6

Heavy particle temperature, T (103K)

7

398

M. I. Boulos et al.

pressure at either LTE or 2T with θ ¼ 2 , LCE being assumed. As it could be int expected the choice of the temperature at which partition functions Qint N2 and QNþ are 2

calculated plays a role in compositions. Figure 4 for N2 and Fig. 5 for Nþ 2 show the influence of the temperature at which the vibrational partition function is calculated: Tv = Th or Tv = 1.5Th or Tv = Te = 2Th in both figures. As it can be seen for N2 (Fig. 4) higher values are obtained at 2T compared to LTE. For Nþ 2 (Fig. 5) the effect is more pronounced than with N2 because the energies of the first excited electronic states are much lower for Nþ 2 than for N2. Aubreton et al. (1998) also studied the influence of the vibrational temperature on the densities of various species of the nitrogen plasma. First Fig. 6 presents

Fig. 4 Evolution with temperature of the internal partition function of N2+ at p = 100 kPa at LTE (—) and at NLTE: θ = 2 (Aubreton et al. 1998)

80 +

+

Partition function, (1000) N2

Fig. 5 Evolution with temperature of the internal partition function of Nþ 2 at p = 100 kPa, at LTE (—), and at NLTE: θ = 2 (Aubreton et al. 1998)

N2 p =100 kPa

Tv = 2Th

60 Tv = 1.5Th 40

Tv = Th

Equilibrium Te = 2Th

20

0

0

2

4

6

8

10

12

Translation temp. of heavy species

14

(103K)

9

Thermodynamic Properties of Non-equilibrium Plasmas

399

results obtained at LTE up to 9000 K, which are identical to those of André et al. (1999). 2T results are presented in Fig. 7 (θ ¼ 2, Tv = Th) and 8 (θ ¼ 2, Tv = Te = 2Th) and can be compared with those obtained at LTE, presented in Fig. 6. θ ¼ 2 results in easier ionization of N: the temperature at which the nN and nþ N curves cross is now only 7500 K against 14,700 K at equilibrium. Of course, the higher energy of the electrons induces a higher level of ionization of N2: the Nþ 2 peak value is shifted to a lower temperature (about 6000 K against 7500 K at equilibrium) and its peak value is almost 2 orders of magnitude higher. Up to 5500 K nothing is changed for nNþ and nNþ2 . When Tv = Te = 2Th (compare Figs. 7 and 8), as would be expected, no variation occurs for nNþ compared to the previous case, but nN2 increases (its maximum is 2  1022 m
3 against 8  1021 m
3) and its

1026

Number density, ni (m–3)

Fig. 6 Evolution with temperature of the density (m
3) of the different species of nitrogen plasma at p = 100 kPa, 2T: Tv = Th, Te = 2xTh (Aubreton et al. 1998)

Pure N2

Tv = Th

p =100 kPa

Te = 2 x Th

N2

1024

N+ N 1022

e

+

N2

1020 0

1026 Pure N2 p =100 kPa Tv = Te = 2 x Th Number density, ni (m–3)

Fig. 7 Evolution with temperature of the density (m
3) of the different species of nitrogen plasma at p = 100 kPa, 2T (Aubreton et al. 1998)

1 2 3 4 5 6 7 8 9 Trans. Temp. of heavy species, T (103K)

N2

1024

e

N

1022

+

N2

N+ 1020 0

1 2 3 4 5 6 7 8 9 Trans. Temp. of heavy species, T (103K)

400 1026

Number density, ni (m–3)

Fig. 8 Evolution with temperature of the density (m
3) of the different species of nitrogen plasma at p = 100 kPa, LTE (Aubreton et al. (1998))

M. I. Boulos et al.

Pure N2 Te = Tv = Th

N2

p=100 kPa

1024 N

1022

N+ 1020

+

N2

e 1018

0

2

4

6

8

10

12

14

Temperature, T (103K)

Fig. 9 Evolution of the composition of an argon plasma with electron temperature, Te, with different values of θ = 1.0, 2.0, and 3.0 (Rat et al. 2002a)

position is shifted to higher temperatures: from 5800 K to about 6500 K. Higher vibrational excitation allows a slightly easier ionization of N2, Nþ 2 being a more stable species, which survives at slightly higher temperatures. Rat et al. (2002a) calculated the plasma composition of argon plasma for three values of θ = 1, 2, and 3, and the results are represented in Fig. 9 as function of Te. The composition was calculated according to the modified Saha equation of van de Sanden. Below Te = 10,000 K, the electron number density increases as the electron temperature increases while still being considerably lower than the number density of argon atoms. The latter increases as θ increases for a given value of

9

Thermodynamic Properties of Non-equilibrium Plasmas

401

Electron number density, ni (m–3)

1023 4

1022 1021 1020 1019

1

1018

p = 100 kPa θ=2

1017 2

1016 1015 3

3

4 4

5 6 7 8 9 10 Electron temperature, Te (103K)

11

12

Fig. 10 Influences of the excitation temperature and of the choice of Potapov and van de Sanden laws on the electron number density in SF6 plasma at atmospheric pressure (see the text for curves numbering) (Gleizes et al. 1999)

Te. Above Te = 15,000 K, the electron number density increases slightly with the increase of θ, while the number density of argon atoms decreases as θ decreases. Gleizes et al. (1999) have studied, under LCE conditions, a 2T SF6 plasma at atmospheric pressure and the influence on its composition both of the excitation temperature and of the choice of the 2T calculation law (Potapov and van de Sanden). Results for the electron number density are illustrated in Fig. 10: • Curve (1) was obtained with Eqs. 19 and 20 or 22 (Potapov method), partition functions of Eq. 20 or 22 being calculated with Te. • Curve (2) was obtained with the same equations but with Tex = Te for atoms and Tex = Th for molecules. • Curve (3) was obtained with the same equations where equilibrium was calculated with Eq. 23 for ionized molecule and Eq. 22 for atoms. • Curve (4) was obtained with Eq. 23 (van de Sanden method). The following comments by Gleizes et al. (1999) were made about these curves. Differences between curves 1 and 2 are important at low temperatures as they show the large influence of the choice of Tex. The choice of the Saha law only (compare curves 3 and 4) is not so important, but has a rather strong influence at moderate temperatures, when diatomic molecules are present. Rat et al. (2002b) studied the dissociation reaction of H2, in the mixture Ar–H2 (50% mole %), by using the excitation temperature Tex proposed by Tanaka and Yokomizu (1997). Ghorui et al. (2007) used the method of van de Sanden et al. (1989) to calculate the composition, thermodynamic, and transport properties of O2 plasma under NLTE conditions at pressures between 100 kPa and 7 MPa. Calculations were

402

M. I. Boulos et al.

performed taking into account deviation from composition equilibrium due to diffusion or delayed recombination effects. Ghorui et al. (2008) also calculated the properties of N2–O2 plasmas under NLTE conditions. Figure 11 presents variation of the composition with T for a N2–O2 (20 vol%) at a pressure of 0.5 MPa under LTE conditions. As the temperature increases beyond 3000 K, dissociation of oxygen begins. Since the total pressure is constant, a sharp rise in the partial pressure or number density of the oxygen atoms causes a decrease in the partial pressure or number density of the N2 molecules. Similar observations are made under NLTE conditions, when θ = 2 (see Fig. 12), with dissociation, which depends essentially on Th, starting at about Te = 6000 K against at 3000 K at equilibrium. Initially, only a small fraction of the N atoms resulting from the dissociation of the N2 molecules combines with O to form NO as indicated by the appearance of small NO peaks. As the temperature increases further, dissociation of

Fig. 11 Variation of composition with Te, at pressure 0.5 MPa for N2–O2 (20 vol%) with θ = 1: LTE (Ghorui et al. 2008)

Fig. 12 Variation of composition with Te, at pressure 0.5 MPa for N2–O2 (20 vol%) with θ = 2 (Ghorui et al. 2008)

9

Thermodynamic Properties of Non-equilibrium Plasmas

403

O2 becomes complete and strong dissociation of N2 starts above a certain temperature Th. At a constant total pressure, the increase in the partial pressure from N species causes a decrease in the partial pressures of O. The kink in the mole fraction curve of O near this region (Fig. 11) bears this signature. As temperature increases further, the first and second ionizations of the atoms start gradually. While the molecular dissociation peaks of N2 and O2 are widely separated, their first ionization peaks are seen to be close to each other due to nearly equal first ionization energies. It is observed that in general an increase in θ causes a shifting of the peaks toward higher electron temperatures. A remarkable effect of the thermal nonequilibrium is noticed in the formation of NO and NO+. As θ increases, more pronounced peaks of NO+ appear. Relatively lower ionization energy of NO (compared to N and O) allows them to easily form NO+ through ionization. While at lower θ, NO and NO+ coexist together, at higher θ, formation of substantially less NO may be noted. At higher values of θ, NO formation starts at significantly higher electron temperatures, Te, with the consequence that as soon as the NO molecules are formed, they ionize forming NO+. A significant NO+ concentration and almost no NO peak are observed in the figures (see, e.g., Fig. 13, θ = 10), accordingly. The formation of NO+ at higher values of θ has the important consequence of important enhancement in the electrical conductivity at relatively low gas temperatures. Similar results were obtained by Colombo et al. (2009) using a 2T model according to van de Sanden et al. (1989), for Ar–H2 (50% mole fraction) (see Fig. 14) and Ar–N2 (50% mole fraction) (see Fig. 15). The results are presented in terms of the mole fractions of different species for Ar–H2 (50% mole fraction) as function of the electron temperature, for different values of the nonequilibrium parameter: θ = 1 (LTE) or 2 (2T). Composition of argon–hydrogen plasma has been obtained considering the species Ar, Ar+, Ar2+, Ar3+, Ar4+, H2, H, H+, and e. For nitrogen–hydrogen plasma H2, H, H+, N2, N, N+, N2+, N3+, and e have been considered. For θ = 2, a shift of the nitrogen and hydrogen dissociation toward higher temperatures is observed resulting in a lower concentration of nitrogen and hydrogen atoms. For mixtures including nitrogen, thermal nonequilibrium Fig. 13 Variation of composition with Te, at pressure 0.5 MPa for N2–O2 (20 vol%) with θ = 10 (Ghorui et al. 2008)

404 0.7 H

0.6 Mole fraction (–)

Fig. 14 Electron temperature dependence of the mole fractions of different species for Ar–H2 (50% mole fraction) for different values of the nonequilibrium parameter: —— θ = 1,    θ = 2 (Colombo et al. 2009)

M. I. Boulos et al.

H2

0.5

e

0.4

Ar

H+

0.3 Ar+

0.2

Ar2+

Ar3+

0.1 0.0

10 20 30 Electron temperature, Te (103K)

40

0.7 H

0.6

e Mole fraction (–)

Fig. 15 Electron temperature dependence of the mole fractions of different species for N2–H2 (50% mole fraction) for different values of the nonequilibrium parameter: —— θ = 1,    θ = 2 (Colombo et al. 2009)

0

0.5 0.4 H2

N2

0.3

N+ N

0.2

H N2+

0.1 0.0 0

10 20 30 Electron temperature, Te (103K)

40

ionization takes place in a narrower range of temperature. Electron mole fraction decreases with increasing value of θ, while electron number density increases. Colombo et al. (2011) have also calculated, using the same method, the composition of CO2 plasmas in LTE and NLTE. Results for LTE are presented in Fig. 16. At room temperature the gas is composed of CO2 molecules, which dissociate into CO and O2 at temperatures around 4000 K. With the further increase of temperatures, CO and O2 dissociate around 4500 K into atomic oxygen, whereas single carbon atoms appear at 6000 K from the dissociation of CO. Carbon ionization starts at 7000 K, whereas oxygen ions become relevant around 10,000 K. Double ionized particles appear over 20,000 K. Aubreton et al. (2004) studied the effect of the nonequilibrium parameter, θ = Te/Th, on the plasma composition and thermodynamic and transport properties of ternary mixtures Ar–H2–He, which are commonly used in DC plasma spraying. Calculations were performed at atmospheric pressure, over the

9

Thermodynamic Properties of Non-equilibrium Plasmas

Mole fraction (–)

Fig. 16 Temperature dependence of the composition for a CO2 mixture at atmospheric pressure (Colombo et al. 2011)

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

405

CO2 O e CO O+

C C+

O2 0

5

10

15

C2+

20

25

O2+ 30

Electron temperature, Te (103K)

Number density, ni (m–3)

1026 He

1024 H2

1022

e

H+

H

He+

1020

Ar++

Ar+ ArH+

1018 1016

0

5

10

H2+ 15

Ar He++

HeH+ 20

25

30

Electron temperature, Te (103K)

Fig. 17 Evolution with temperature of the equilibrium composition of an Ar–H2–He (30–10–60 mol%) mixture at atmospheric pressure (Aubreton et al. 2004)

temperature range 300–30,000 K. The nonequilibrium composition was calculated using two techniques: the equilibrium constant (van de Sanden et al. (1989) method) and pseudo-kinetic methods (NLCE), results of the latter being presented in Sect. 3.2. For the method of van de Sanden, they used the excitation temperature of molecules as proposed by Tanaka and Yokomizu (1997) and Cliteur et al. (1999). They also calculated the specific heat at constant pressure obtained by a five-point numerical differentiation of the specific mass enthalpy; results are presented in Sect. 2.5. Figure 17 shows the equilibrium composition of the Ar–H2–He (30–10–60 mol%) plasma as a function of temperature, at atmospheric pressure. It is observed that the dissociation of H2 is maximum ðnH2 ¼ nH Þ at about 3200 K, the number density of hydrogen atoms increasing by nine orders of magnitude between 900 and 4200 K.
7 The ions He–H+ and Hþ 2 are in very small quantities (molar fractions below 10 ). At about 15,000 K, the electron number density reaches a plateau, as does the number density of H+ and He+ (at least up to 25,000 K).

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Number density, ni (m–3)

1026 He 1024

e

H2 H+

H

1022

Ar+

He+ 1020

Ar++ H2+

Ar +

HeH

1018

He++

ArH+ 1016

0

5

10

15

20

25

30

Electron temperature, Te (103K)

Fig. 18 Evolution with the electron temperature of the nonequilibrium composition of an Ar–H2–He (30–10–60 mol%) mixture at atmospheric pressure for θ = 3 (Aubreton et al. 2004)

Figure 18 presents the nonequilibrium composition obtained using the equilibrium constant method for the same mixture at atmospheric pressure, for values of θ = Te/Th = 3. It is noted that the dissociation maximum is at the same heavy species temperature, Th, as before, with a shift to a higher electron temperature through the factor θ. Under nonequilibrium conditions, there is a shift of the composition curves, this displacement being directly linked to temperatures used in equilibrium constants. Moreover, for θ = 3, the majority of ions at all temperatures are atomic ions, the density of which is shifted to lower temperatures compared with that obtained at equilibrium. It should also be noted that Hþ 2 increases by almost four orders of magnitude due to the ionization of H2 long before dissociation of H2 starts.

3.5

Thermodynamic Properties

Enthalpy, entropy, and specific heat of species i at constant pressure depend on partition functions and their temperature derivatives. For example, the enthalpy of species i can be written as hg ¼

1 X xi hi mg i

(26)

where mg is the mass of the plasma (hg in J/kg), xi is the molar fraction of species i, and hi is linked to the internal partition function Qint i of species i: hi ¼ kT2

@lnQint 5 i þ T þ e0i 2 @T

(27)

9

Thermodynamic Properties of Non-equilibrium Plasmas

407

e0i being the reference energy. The heat capacity at constant pressure is obtained as the derivative of Eq. 27 relatively to T.   int 2 @lnQint 5 2 @ lnQi i þT þ cpi ¼ k 2T 2 @T @2T

(28)

According to what has been presented previously (Sect. 3.2 of this Chapter), the temperatures (Te, Th, or Tex), at which dissociation and ionization of molecules occur, will have a significant influence on the results through the derivation of the molar fractions and partition functions. Trelles et al. (2007) calculated Ar plasma properties at different values of θ. Figures 19 and 20 present, respectively, the gas enthalpy and the specific mass of

Enthalpy, h= hh+he (J/kg)

108 Enthalpy 107

θ=1 θ=3

106

θ=5 θ=7

105

θ = 10 0

5

10

15

20

25

30

Electron temperature, Te (103K)

Fig. 19 Enthalpy of two-temperature three-component argon plasma (Trelles et al. 2007)

Mass density, ρ(kg/m3)

10 θ=1

Mass Density

θ=3

1

θ=5 θ=7 θ = 10

10–1

10–2 0

5

10

15

20

Electron temperature, Te

25

30

(103K)

Fig. 20 Specific mass of two-temperature three-component argon plasma (Trelles et al. 2007)

408 200 Pure N2 p=100 kPa Enthalpy, (106 J/kg)

Fig. 21 Evolution with temperature of the enthalpy (J/kg) of the different species of nitrogen plasma at p = 100 kPa, at LTE, Te = Th (—), and at NLTE, Te = 2Th, with Tv =1.5 Th (- - -) and Tv =2Th (-. -.) (Aubreton et al. 1998)

M. I. Boulos et al.

150 Non-equilibrium Te = 2Th

100

Equilibrium Te = Th

Tv = 2Th 50

0

Tv = 1.5Th

0

2

4

6

8

10

12

14

Translation temp. of heavy species (103K)

the plasma. Calculations were performed for different values of θ using only three species (i.e., Ar, Ar+, e). The marked dependence of the properties on θ for higher values is evident. However as Ar++ species are neglected, enthalpies are significantly underestimated, while the total mass density is overestimated, at temperatures above  13,000–15,000 K. For example, Fig. 21 presents the evolution with temperature of the enthalpy (J/kg) of the nitrogen plasma for which compositions under different conditions have been calculated (see Figs. 6, 7, and 8). As could be expected from the results obtained for composition when θ ¼ 2, ionization occurs almost at the same heavy species temperature as that of dissociation (see Figs. 7 and 8). Thus, instead of two important successive significant variations for dissociation and ionization at equilibrium, only a single major increase of the enthalpy shows up between 5000 and 8000 K (see Fig. 21). Of course the increase of vibrational temperature (or energy) easing the ionization of Nþ 2 and delaying the disappearance of molecular species shifts the variations of enthalpy with dissociation toward slightly higher temperatures. These results are confirmed by the evolution of the specific heat with temperature (see Fig. 22). The ionization peak for θ = 2 is almost independent of Tv, while the dissociation peak occurs at higher temperature, closer and closer to the ionization temperature, with the increase of θ from 1 to 2. The size of the dissociation peak increases too with vibration temperature. Gohrui et al. (2008) performed similar 2T calculations for the enthalpy of N2–O2 mixture with 20 vol% of O2. The result presented in Fig. 23 took into account 17 components including different molecules, atoms, and ions of oxygen, nitrogen, and nitrogen oxides NO and NO+. The slope changes correspond to O2 and then N2 dissociations followed by ionizations almost simultaneously of N and O (see Figs. 11 and 12). The decrease of the enthalpy with the increase of θ is similar to that observed in Fig. 19 for Ar. Colombo et al. (2009) have calculated the specific heat, cp, for Ar–H2 (50% mole fraction) (see Fig. 24) and Ar–N2 (50% mole fraction) (see Fig. 25). The specific heat of argon–hydrogen shows the characteristic peak due to the dissociation of H2

Thermodynamic Properties of Non-equilibrium Plasmas

Fig. 22 Evolution with temperature of the specific heat (kJ/kg.K) of the different species of nitrogen plasma at p = 100 kPa, at LTE, Te = Th (—), and at NLTE, Te = 2Th, with Tv = Th (...), Tv =1.5 Th (- - -), and Tv =2Th (-. -.) (Aubreton et al. 1998)

70 Specific heat, (kJ/kg.K)

9

409

Pure N2 p=100 kPa

60 Non-equilibrium Te = 2Th

50 40

Tv = 2Th

30

Tv = 1.5Th

20

Tv = Th

Equilibrium Te = Th

10 0

0

2

4

6

8

10

12

14

Translation temp. of heavy species (103K)

109

Enthalpy, (J/kg)

Fig. 23 Variation of enthalpy with Te, at pressure 0.5 MPa for N2–O2 (20 vol%) for different values of θ: NLTE (Ghorui et al. 2008)

108

10 1

2

3

15 20 Te/Th

5

107 106 5

103 0

10

20

30

40

Electron temperature, Te (103K)

30 Total specific heat (kJ/kg.K)

Fig. 24 Electron temperature dependence of the total specific heat for Ar–H2 (50% mole fraction) for different values of the nonequilibrium parameter —— θ = 1, —— θ = 2,    θ = 3 (Colombo et al. 2009)

Ar/H2 50 vol%

25 20 15

θ=1

10 θ=2

5

θ=3 0

0

10

20

30

Electron temperature, Te (103K)

40

410 80 Total specific heat (kJ/kg.K)

Fig. 25 Electron temperature dependence of the total specific heat for N2–H2 (50% mole fraction) for different values of the nonequilibrium parameter —— θ = 1, —— θ = 2,    θ = 3 (Colombo et al. 2009)

M. I. Boulos et al.

70 θ=3

60 50

N2/H2 50 vol%

θ=2

40 30

θ=1

20 10 0

0

10

20

Electron temperature, Te

30

40

(103K)

(Th  3700 K) and the peaks due to the ionization of argon and hydrogen atoms and ions (Te  15,000, 25,000, and 37,500 K); in the case of nitrogen–hydrogen mixtures, the total specific heat shows peaks due to the dissociation of nitrogen and hydrogen molecules (Th  6800 and 3700 K, respectively) and those due to the ionization of nitrogen and hydrogen atoms and ions (Te  15,000 and 29,500 K). Since the heavy particle temperature governs dissociation peaks, as θ increases they shift toward higher electron temperature: for θ=3 the dissociation peak of hydrogen molecules in argon–hydrogen mixtures gives a contribution to the peak due to first ionization of argon and hydrogen atoms; in nitrogen–hydrogen mixtures, for θ = 3 the peak due to dissociation of nitrogen reaches the temperature range where first ionization takes place (Te  15,000 K), resulting in a pronounced shift of the atom ionization toward higher electron temperature. Moreover, in this case the first ionization takes place in a narrower range of electron temperature resulting in a higher peak of total specific heat. In both argon–hydrogen and nitrogen–hydrogen mixtures, the ionization peaks shift slightly toward higher Te as θ increases. Colombo et al. (2011) calculated the evolution with the electron temperature of the specific heat at atmospheric pressure of CO2/H2mixture (50–50 vol.%) at three values of θ =1 (LTE), 2 and 3 (2T) assuming LCE; see Fig. 26. In thermal equilibrium, the total specific heat shows the typical peaks due to dissociation of CO2 and O2 molecules around 4000 K, dissociation of CO around 7000 K, and ionization of carbon and oxygen atoms at 15,000 K. Since heavy particle temperature has a dominant effect on the localization of dissociation peaks on the electron temperature axis, they shift toward higher electron temperature as θ increases, causing also a pronounced shift of the atom ionization in the same direction. For θ > 1 the first ionization takes place in a narrower range of electron temperature, resulting in a higher peak of total specific heat. The results obtained at LTE are in good agreement with those of Sourd et al. (2006). Aubreton et al. (2004) calculated the enthalpy of the Ar–H2–He (30–10–60 vol%) plasma at atmospheric pressure, under LTE and NLTE conditions. The corresponding compositions for such mixtures were presented in Figs. 17 and 18.

9

Thermodynamic Properties of Non-equilibrium Plasmas

411

Fig. 26 Electron temperature dependence of the total specific heat for a CO2 mixture for different values of the nonequilibrium parameter θ (Colombo et al. 2011)

450 Enthalpy (106 J/kg)

Fig. 27 Evolution with the electron temperature of the specific mass enthalpy for the Ar–H2–He (30–20–50 mol%) mixture at equilibrium and for different values of θ = Te/Th (Aubreton et al. 2004)

400 350

θ=3

Ar / H2 / He

θ =1.6

300 250

θ =1

200 150 100 50 0

0

5

10

15

20

25

30

Electron temperature, Te (103K)

Figure 27 presents the evolution with the electron temperature of the specific enthalpy, calculated at LTE and for different values of (θ = Te/Th). The enthalpy increases linearly with temperature up to the onset of dissociation at around 2000 K. Once dissociation is completed the same linear trend is observed up to the point where ionization levels are significant around 14,500 K for H and Ar, 22,000 K for He, and 25,000 for Ar++. The trend is independent of the value of θ at temperatures below 14,500 K, above which an increase of θ gives rise to a further increase of the specific enthalpy. The effect seems to be associated with the presence of Ar++ and He+ ions in the mixture, since as observed earlier in Fig. 19, it had the opposite effect for a pure Ar plasma at values of θ varying between 1 and 10. Figure 28 represents the evolution with the electron temperature of the specific heat at LTE and 2T for different values of θ = Te/Th. It should be pointed out that, when θ increases, the dissociation of H2 (the characteristic temperature being Th = Te/θ  3800 K) is shifted to higher electron temperatures. Above 11,000 K, the maximum of the ionization peaks strongly depends on θ.

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M. I. Boulos et al.

Fig. 28 Evolution with the electron temperature of the specific heat for the Ar–H2–He (30–20–50 mol%) mixture at equilibrium and for different values of θ = Te/Th (Aubreton et al. 2004)

Specific heat (103 J/kg.K)

120 100

Ar / H2 / He θ=3

80 60

θ=1.6

40

θ=1

20 0

0

5 10 15 20 25 Electron temperature, Te (103K)

4

Deviations from Local Chemical Equilibrium

4.1

Calculation Basis

30

The method consists in determining kinetic coefficients of forward and reverse reactions, respectively, kf and kr. The equilibrium chemical reactions are generally written as N X

vji Ai $

i¼1

N X v0ji Ai

j ¼ 1, 2, . . . , L;

(29)

i¼1

where Ai denotes the chemical symbol of species i while vji and vji0 are, respectively, the forward and the reverse stoichiometric coefficients, N is the total number of species, and L is the number of reactions.

4.1.1 Stationary Kinetic Calculation Richley and Tuma (1982) have proposed a stationary kinetic calculation, neglecting diffusion and convection. At steady state one has the following rate equation for the jth reaction: N

v

N

v0

kjf ∏ ni ij
kjr ∏ ni ij ¼ 0 i¼1

(30)

i¼1

The forward reaction coefficient kf is written as kf ¼ ATBi expð
C=Ti Þ

(31)

where Ti is the temperature associated with the forward reaction, A and B are constants, and C is the activation energy. In principle these parameters are found in

9

Thermodynamic Properties of Non-equilibrium Plasmas

413

the literature. If the coefficient of the reverse reaction kr is missing, it can be calculated according to Eq. 30 as kf =kr ¼ Kn

(32)

This relationship assumes that the equilibrium constant is independent of the colliding third body. Calculations for diatomic species can be performed at Te or Th or Tex as already described in Sect. 2.3, ▶ Chap. 7, “Transport Properties of Gases Under Plasma Conditions.” When steady state is not achieved, the net generation rate, Ri, of a species is dependent on the rates of the forward and reverse reactions of the chemical reactions considered (Ye et al. 2007): Ri ¼ mi

X j

  N ν N ν0 νij kjf ∏ ni ij
kjr ∏ ni ij

(33)

νij ¼ νrij
νfij

(34)

i¼1

i¼1

with

kjf being the forward reaction rate coefficient Eq. 30. Of course if the plasma is spatially homogeneous, Ri = 0.

4.1.2 State-to-State Approach As previously mentioned, to calculate kr from kf, the detailed balance principle (the equilibrium constant), or the use of the Maxwellian distribution function in the integration of the reaction cross section, implies the assumption of equilibrium. In reality the reverse reaction rate should be obtained by integrating the reaction cross section, considered over a distribution function that is not necessarily Maxwellian. Especially under nonequilibrium conditions, for which the electron number density is low, collisional–radiative models have to be coupled with the Boltzmann equation to deduce the distribution function (Rat et al. 2008). That is why Capitelli et al. (2007) proposed a state-to-state approach. The advantage of the state-to-state approach is the characterization of nonequilibrium vibrational distributions and non-Maxwellian distribution under plasma conditions, a characterization which is hidden by macroscopic approaches. The difficulty occurs in the preparation of a database of cross sections, which includes their dependence on the internal energy (actually authors have only considered H2, N2, and O2). The second difficulty is the introduction of the state-to-state approach in robust fluid dynamic codes. The description of Capitelli et al. (2007) is limited to the insertion of the state-to-state approach in 1D fluid dynamics. 4.1.3 Pseudo-equilibrium Calculation Another problem with the stationary kinetic calculation is the computing time, which is much longer than calculations in LCE. André et al. (2001) have proposed a

414

M. I. Boulos et al.

modified pseudo-equilibrium calculation, which gives almost the same results as those of kinetic calculations for determining the composition of hydrogen and nitrogen plasmas at atmospheric pressure but with a computing time two to three orders of magnitude shorter. Pseudo-equilibrium calculation also takes into account the reactions with low activation energies instead of ionization reactions while keeping all the species present in the kinetic calculation. This means that the plasma composition is calculated with the equilibrium constants corresponding to low
 þ þ 2 activation energies X2 ) 2X, e þ Xþ where X is 2 ) 2X, X2 þ X ) X þ X hydrogen (X = H) or nitrogen (X = N). Such calculations are justified by the fact that under the conditions prevailing in nonequilibrium thermal plasmas, the most important species are molecules and their ions. Based on experimental results, the electron temperature, Te, seems to be directly related to the temperature of the heavy species, Th. The ratio Te/Th varies as a function of the logarithm of the ratio max (ne/nmax e ), ne
being the electron  density in the plasma core for which equilibrium 23
3 . The kinetic calculations have been performed is achieved nmax 10 m e assuming micro-reversibility where the reverse kinetic rate coefficient kr is calculated by Eq. 32. When electrons are involved in both forward and reverse reactions, kf and Kn are expressed as functions of Te. However, when the direct reaction involves electrons while the backward one is due to collisions between heavy species (or the reverse), a temperature T* between Te and Th is introduced. T* is determined as a function of the ratio of the electron flux to that of neutral species in such a way that T* = Te for ne = 1023 and T* = Th for low values of ne (ne < 1015 m
3).

4.2

Example of Results for Compositions

4.2.1 Stationary Kinetic Calculation When considering LCE and two-temperature Ar–H2 plasma, Rat et al. (2002b) took into account elastic processes. The corresponding reactions are summarized in Table 1. Temperatures given in Table 1 are those at which partition functions were calculated. Figure 29 presents results obtained in LCE at 2T with θ = 1.6. The heavy particle temperature, Th, at which dissociation is a maximum is the same as that under LTE conditions, with a shift to a higher electron temperature by the Table 1 Reactions considered by Rat et al. (2002b) elastic processes to calculate the composition of two-temperature Ar–H2 plasma

Number 1

Reactions H2 þ M $ 2H þ M

Temperatures Tex

2

e þ H $ Hþ þ 2e

Te

3

e þ H2 $ Hþ 2 þ 2e

Te

4

þ

e þ Ar $ Ar þ 2e þ

Te

5

e þ ArH $ Ar þ H

Tex

6

e þ Arþ $ Ar2þ þ 2e

Te

Thermodynamic Properties of Non-equilibrium Plasmas

Fig. 29 Evolution with the electron temperature of the nonequilibrium composition of an Ar–H2 (50 mol%) mixture at atmospheric pressure for θ = 1.6 obtained using the equilibrium constant method (Rat et al. 2002b)

415

1026 Number density (m–3)

9

1024 H+ 1022

e H2

H+

Ar Ar+

1020

Ar2+

H

1018

ArH+ H2+

1016 0

5

10

15

20

25

Electron temperature, Te (103K)

factor θ. Under these conditions, there is a shift in composition curves, directly linked to temperatures used in equilibrium constants. Moreover, for θ = 1.6, the majority of ions at all temperatures are the atomic ones. When considering NLCE the problem becomes more complex (Rat et al. 2002b). + þ In LCE, the species H
, Hþ 2 , H3 , and ArH , in the temperature range where they exist, are minor species compared to the species Ar, Ar+, Ar2+, H2, H, and H+. However in NLCE, minor species can no longer be neglected because they play a role in the dissociation regime. The following reaction must be considered under NLCE conditions; Dissociation; H2 þ M , 2H þ M first ionization of H, H2 and Ar e þ H , Hþ þ 2e e þ H2 , Hþ 2 þ 2e e þ Ar , Arþ þ 2e reactive charge transfer Hþ þ H2 , H þ Hþ 2 þ Hþ 2 þ H2 , H3 þ H Arþ þ H2 , Ar Hþ þ H þ Ar þ Hþ 2 , ArH þ H and dissociative recombination: Hþ 3 þ e , H2 þ H ArHþ þ e , Ar þ H Hþ 2 þ e , 2H Once reactions have been chosen, the main problem is to find the values of the different parameters of kf (Eq. 31). For example, such values have been gathered by Rat et al. (2002b) for the Ar–H2 plasma. The kr coefficients were calculated according to Eq. 32. Figure 30 presents results obtained and can be compared with Fig. 29 obtained in LCE. In Fig. 30 a strong discontinuity is observed at Te

416 1026 Number density (m–3)

Fig. 30 Evolution with the electron temperature of the nonequilibrium composition of an Ar–H2 (50 mol%) mixture at atmospheric pressure for θ = 1.6 obtained using the stationary kinetic calculation to represent NLCE (Rat et al. 2002b)

M. I. Boulos et al.

1024

e H2

1022

Ar

H++

Ar+ 1020

Ar2+

H

1018

ArH+

1016

H2+ 0

5

10

15

20

25

Electron temperature, Te (103K)

= 11,000 K which is typical of multi-temperature stationary kinetic calculations. Dissociation follows the same evolution as at equilibrium, that is, for Te < 5000 K, the electron number density ne is negligible and Tex  Th. This induces a shift of dissociation to the higher electron temperature when θ = Te/Th increases. Dissociation follows the same evolution as at equilibrium: for Te < 5000 K the electron number density ne is negligible and Tex  Th. This induces a shift of dissociation to the higher electron temperature when θ = Te/Th increases. The heavy particle + temperature Te/θ at which electrons, Ar+, H+, Hþ 2 , and ArH appear at a number 15
3 density of about 10 m corresponds to that at equilibrium. Below the discontinuity temperature Te = 11,000 K, ionization is controlled by charge transfer and dissociative recombination reactions, which prevent hydrogen and argon atoms from ionizing. When Ar+ and H+ are created, they transfer their charge through H2 to form, respectively, ArH+ and Hþ 2 . Then, the latter are destroyed through dissociative recombination reactions. Moreover, for Te < 11,000 K, nHþ nArþ , results are different from those obtained at equilibrium; see Fig. 29. Consequently, it seems that the ArH+ ions favor the coupling gases: Ar and H2.
þ between both  þ However, the destruction reaction of Hþ H þ H , H þ H has not been 2 2 2 3 considered (Rat et al. 2002b). Ye et al. (2007) have also calculated Ar–H2 (5.77 vol% H2) plasma in NLCE with the reactions summarized in Table 2, where no dissociative recombination was considered. Figures 31 and 32 present results they obtained up to Te = 10,000 K. NLCE was assumed and kr was deduced from kf assuming micro-reversibility. Figure 31 was calculated for Te = Th, while Fig. 32 was obtained for Te = 3Th. As could be expected at Te = Th results are similar to those calculated by Rat et al. (2002b). With Te = 3Th, as previously demonstrated, the plasma composition is strongly affected by the degree of thermal nonequilibrium. The hydrogen atom density peaks at Th = 3800 K for θ ¼ 1 and at Th = 6400 K for θ ¼ 3, indicating that increasing the electron energy leads to more dissociations induced by electron impact. There is also a noticeable increase in the Hþ 2 density at θ ¼ 3 due to the greater H2 density at higher electron temperatures. In addition, the electrons are mainly generated by the

Thermodynamic Properties of Non-equilibrium Plasmas

Table 2 Chemical reactions considered for the Ar–H2 plasma (Ye et al. 2007)

Fig. 31 NLCE plasma composition for an Ar–H2 mixture (5.77% vol. H2) at 27 kPa and with Te/Th = 1 (Ye et al. 2007)

417

I. Dissociation of H2 1. 2H þ H2 $ H2 þ H2 2. 3H $ H2 þ H 3. e þ H2 $ e þ 2H 4. Ar þ 2H $ Ar þ H2 II. Ionization of H 5. e þ H $ e þ e þ Hþ III. Ionization of H2 6. e þ H2 $ e þ e þ Hþ 2 IV. Ionization of Ar 7. e þ Ar $ e þ e þ Arþ

1024

Number density (m–3)

9

θ=1

Ar

1022 H2

1020

e

H+

H

1018 Ar+ 1016

0

2

4

6

8

10

Electron temperature, Te(103K)

θ=1

Ar

1024

Number density (m–3)

Fig. 32 NLCE plasma composition for an Ar–H2 mixture (5.77% vol. H2) at 27 kPa and with Te/Th = 3 (Ye et al. 2007)

H2

1022

e H+

Ar+

1020

H H2+

1018 1016 0

2

4

6

8

10

Electron temperature, Te(103K)

ionization of Ar for the whole range of temperatures at θ ¼ 3, while they are mostly produced by the ionization of H for temperatures below 5500 K at θ ¼ 1. However no discontinuity is observed.

418 1026 1025 Number density (m–3)

Fig. 33 Evolution with the electron temperature of the nonequilibrium composition of an Ar–H2–He (30–10–60 mol%) mixture at atmospheric pressure for θ = 3 obtained using the stationary kinetic calculations (Aubreton et al. 2004)

M. I. Boulos et al.

θ=3

He

1024

e

H2

1023 1022

H

1021

He+

H

1020

He

Ar+

Ar

Ar++

1019 1018 1017

He++

0

5

10

15

20

25

30

Electron temperature, Te(103K)

Aubreton et al. (2004) also calculated the Ar–H2–He (30–10–60 mol%) plasma composition, as they did in LCE (see Figs. 17 and 18), using the stationary kinetic calculation, neglecting diffusion and convection. For that they determined kinetic coefficients of direct and reverse reactions kd and kr. In Fig. 33, for the same plasma composition and θ = 3, a strong discontinuity is observed at Te = 9800 K, which is denoted in the following as TD, the discontinuity temperature (compare with Fig. 18 in LCE and θ =3). This kind of discontinuity is typical of multi-temperature stationary kinetic calculations (Gleizes et al. 1999; Rat et al. 2001a). Dissociation follows the same evolution as at equilibrium, that is, for Te < 5000 K, the electron number density is negligible and Tex  Th (see Eq. 11). This induces a shift of dissociation to the higher electron temperature when θ = Te/Th increases. Also the heavy particle temperature, Te/θ, at which electrons, Ar+, H+, He+, Ar++, and He++ appear at a number density of 1017 m
3, corresponds to that at equilibrium. This means that below TD, charge transfer and dissociative recombination reactions, which prevent hydrogen, argon, and helium atoms from ionizing, control ionization. Below TD one has the following reaction routes: 1. Ionization e þ X ) Xþ þ 2e (X = H, Ar, He, and H2) 2. Charge transfer Xþ þ H2 ) XHþ þ H 3. Dissociative recombination e þ XHþ ) X þ H When nH2 is too small (dissociation of H2 is negligible), the efficiency of step 2 drops and ionization occurs sharply, leading to the discontinuity. Thus according to the method chosen to calculate the plasma composition, LTE and LCE, 2T and LCE, and 2T and NLCE, results can be rather different. Yu et al. (2001) developed a two-temperature chemical kinetic model to study the effect of vibrational nonequilibrium on the chemical composition of atmospheric pressure nonthermal nitrogen plasmas. By comparison with a vibrationally specific collisional–radiative model, it was shown that the vibrational levels in two-temperature, atmospheric pressure nonequilibrium nitrogen plasmas tended to

9

Thermodynamic Properties of Non-equilibrium Plasmas

419

follow a Boltzmann distribution and that the vibrational temperature was approximately equal to the gas temperature at low electron number densities and to the electron temperature at high electron number densities. The kinetic model failed to predict this behavior because of the assumptions made on the temperature dependence of several key rates.

4.2.2 Pseudo-Equilibrium Calculation André et al. (2001) have compared calculations performed with the pseudoequilibrium method (see Sect. 3.1) with those of stationary kinetic ones (see Sect. 3.1.1) for H2 and N2. In this method θ is calculated according to Eq. 18 and the intermediate temperature T* according to Eqs. 15 and 17. In the following we will present results obtained for N2 with the reactions considered in Table 3. Figure 34 presents results obtained by using the kinetic coefficients. The kf and Kx are calculated at Te and T*. Compared to what occurred with hydrogen, a discontinuity occurs at about 6500 K, for which no clear explanation has been found. It might result from a shift of the dissociation regime to the ionization one. Maximum dissociation is observed at 7500 K. At 6500 K dissociation of N2 is beginning. The dissociative recombination reaction for nitrogen, which occurs at a much lower energy (about 50% lower) than that of H2, might result in a strong Table 3 Reactions N þ M ! 2N þ M 1. 2 considered for the M ¼ N2 calculation of pseudo2. M ¼ N equilibrium for N2 plasma 3. M ¼ e (André et al. (2001)) 4. e þ N ! Nþ þ 2e 5. e þ N2 ! Nþ 2 þ 2e ! 2N 6. e þ Nþ 2 7. Nþ þ N2 ! Nþ 2 þN

1026

Number density (m–3)

Fig. 34 Evolution with the heavy species temperature T of the number density of the different species of a nitrogen plasma at p = 100 kPa calculated either according to the kinetic reaction rates depicted in Table 2 using the temperatures Te and T* for kf and kr (thick lines) or at equilibrium (thin lines) (André et al. 2001)

1025

N2

N

1024 1023 1022 N+

1021 1020 1019

e

N+ N2 N2+

N2

e

0

N2+ 3 6 9 12 Heavy species temperature, T (103K)

15

420

M. I. Boulos et al.

enhancement of the electron production and probably plays an important role in the abrupt variation of the density curves of N2, N, N+, and Nþ 2 . Cliteur et al. (1999) obtained a similar result at about 7000 K for SF6 plasma at atmospheric pressure. The composition presented in Fig. 35 represents the same calculations as for Fig. 34 but neglecting the charge transfer. The Nþ 2 density is about the same with and without charge transfer. The shift from a behavior dominated by dissociation to one dominated by ionization is much smoother without the charge transfer: the ioniza+ þ
tionþ process is classical  ðN , N þ eÞ because the process of N destruction þ N þ N2 , N2 þ N no longer exists. The pseudo-equilibrium calculation was performed using T* (varying between Th and Te), instead of T, with the three reactions: 1 of Table 3 (dissociation) and 4 and 5 of Table 3 (ionization). Results are presented in Fig. 36 together with equilibrium calculations. With the modified pseudo-equilibrium calculation, the ionizations of N and, more importantly, of N2 at T < 5000 K are not damped by the l/θ exponent in the equilibrium constants of the Saha–Potapov method. As well, the 1026

Number density (m–3)

Fig. 35 Evolution with the heavy species temperature T of the number density of the different species of a nitrogen plasma at p = 100 kPa calculated either according to the kinetic reaction rates using Te and T*, but neglecting the charge transfer reaction (thick lines), or at equilibrium (thin lines) (André et al. 2001)

1025

N

1023

N 1022 N+

1021

e

e

1020 1019

N+ N2+

0

3 6 9 12 Heavy species temperature, T (103K)

15

1026

Number density (m–3)

Fig. 36 Evolution with the heavy species temperature T of the number density of the different species of nitrogen plasma at p = 105 Pa, calculated either at pseudoequilibrium with T* and the three classical reactions (1, 4, 5 in Table 3) (thick lines) or at equilibrium (thin lines) (André et al. 2001)

N2

1024

1025

N2

1024 N

1023 1022

e e

1021 N2+

1020 1019

N+ 0

N+ N2+

3 6 9 12 Heavy species temperature, T (103K)

15

9

Thermodynamic Properties of Non-equilibrium Plasmas

421

ionization of N starts earlier. Over 5000 K the decrease of Nþ 2 is faster which is in good agreement with the faster dissociation of N2. Finally (Andre et al. 2001) when comparing the multi-temperature mass action law calculation at T* (based on the Saha–Potapov equation) with the kinetic calculations, the agreement is rather good with the results obtained by neglecting charge transfer. The only difference for nitrogen is a higher (by one order of magnitude) density of Nþ 2 , which, however, is below 1021 m
3. Rat et al. (2001) have compared the compositions of SF6 plasma at atmospheric pressure calculated either by kinetic or pseudo-equilibrium. Only slight discrepancies for species below 1022 m
3 were observed, for example, F
below 5000 K where the electrons are the main negative species in the pseudo-equilibrium calculations. However, the main advantage of pseudo-equilibrium is that the computing time is about two orders of magnitude shorter than that of kinetic calculations. Over 6000 K there is not much difference between both equilibrium and multi-temperature calculations. Below 6000 K the most important differences, up to a few orders of magnitude, between kinetic and multi-temperature calculations, are

+ noticed mainly for e, S+, Sþ 2 , F , S , and SF species. The multi-temperature methods neglecting the exponent 1/θ in the equilibrium constant do not really improve the agreement with kinetic calculation results. This is illustrated in Fig. 37 showing the evolution with the heavy species temperature of the electron density ne and fluorine negative ion density nF
, calculated using different methods. A In this figure, the curves identified as nA e and nF
are those reported by André et al. C C (1996). The corresponding curves ne and nF
are due to Cliteur et al. (1999), while K P P the curves identified as nK e and nF
and ne and nF
were obtained André et al. (2001) using respectively the kinetic theory and assuming pseudo-equilibrium. Rat et al. (2001) make the following comments of the presented results. Except for F
calculated with the method of André et al. (2001), probably due to the exponent

Number density (m–3)

1024

4.0 3.5

θe

1023 1022 1021

ne

1020

nF –

C

3.0

K

2.5

ne nP e

A

K

ne

2.0

C

nF –

1.5

P

nF –

1019

1.0

A

nF–

1018 1017

0

3

5

7

0.5

9

11

13

0.0 15

Heavy species temperature, T (103K)

Fig. 37 Evolution with the heavy species temperature of the electron density ne and fluorine negative ion density nF
, calculated using different methods (After Rat et al. (2001))

422

M. I. Boulos et al.

1/θ, over 6000 K all methods give the same results for ne and nF
, in good agreement with the variations of Te and T*. Below 5000 K, drastic differences show up between, on the one hand, kinetic or pseudo-equilibrium calculations and, on the other hand, the multi-temperature calculations with the methods of André et al. (2001) or Cliteur et al. (1999). This is largely due to the low ionization energy 19
3 þ of Sþ at 3000 K 2 , (EI = 8.3 eV) which gives rise to S2 ion densities below 10 m using the kinetic or pseudo-equilibrium calculation, while according to Cliteur et al. (1999) the corresponding situation occurs at T below 2200 K. Of course, F
production is linked to the electron production in the (Cliteur et al. 1999) multitemperature calculation, while in the kinetic or pseudo-equilibrium calculation, it is compensated at the beginning by Sþ 2. It should be pointed out that attention has been recently given by a number of researchers to the modelling the two-temperature chemically non-equilibrium modelling plasmas under a wide range of conditions. That includes the work of Baeva et al. (2012) on the modelling of argon transferred arcs. The chemically non-equilibrium modelling of the inductively coupled nitrogen plasma at atmospheric pressure reported by Tanaka and Sakuta (2002) and that of inductively coupled argon-nitrogen plasma by Tanaka (2004). Wang et al. (2014) also reported the two-temperature chemical-non-equilibrium modeling of a high-velocity argon plasma flow in a lowpower arc-jet thrusters. In all of these cases the results, while rather interesting, needs further validation against experimental data for general conclusions to be drawn.

5

Nomenclature

Units are indicated in parentheses; whenever no units are given, the parameter is dimensionless.

5.1 Ai cp cpi dij e dηi E en Ei,s E00,i ED X EIXþ

Latin Alphabet Chemical symbol of species i Specific heat at constant pressure (kJ/kg.K) Specific heat of species i at constant pressure (kJ/kg.K) Distance at which interaction between particles of types i and j has to be taken into account (m) Electron charge (e = 1.602,176,565  10
19C) Production rates of i Electric field (V/m) Base of natural logarithm: (e = 2.7182818) Energy of the chemical species i in the excited state s (eV or cm
1) Energy of the chemical species i in its ground state related to an absolute reference state (eV or cm
1) Dissociation energy of the diatomic molecule X2 (eV) Ionization energy of the atom X (eV)

9

Thermodynamic Properties of Non-equilibrium Plasmas

F ge G h hg hi Hi HD X HIX i k kfij krij Ke(X) Kni Kp(X+) Kp(X) ‘e ‘ij mi Mi ne nmax nmax e ni ni,s nT Ni Ni,s p pi p0 Qi Qtri Qiint Qtot R Ri T TD Th Te Tex

423

Helmholtz free energy (J) Electronic statistical weight Gibbs free enthalpy (J) Plank’s constant (h = 6.626  10
34 J.s) Specific enthalpy of the gas (kJ/kg) Specific enthalpy of the species i (kJ/kg) Molar enthalpy of the chemical species i (J) Molar dissociation enthalpy of the molecule X2 (J) Molar ionization enthalpy of the molecule X (J) Index of the chemical species Boltzmann’s constant (k = 1.38  I0
23 J/K) Kinetic rate of the forward reaction in NLCE (m3/s) Kinetic rate of the reverse reaction in NLCE (m3/s) Equilibrium constant to produce the species X Equilibrium constant to produce the species i density Partial pressure equilibrium constant for ionization of X Partial pressure equilibrium constant for dissociation of X2 Electron mean free path (collisions with heavy particles) (m) Mean free path between collisions of particles of type i and j (m) Mass of a particle of chemical species i (kg) Atomic mass of chemical species i (kg) Number density of electrons (m
3) Maximum allowed principal quantum number
 Electron number density over which LTE exists nmax ¼ 10
23 m
3 e Number density of chemical species i, independent of their excitation state (m
3) Number density of chemical species i in the excited state (eV) Total number density (nT = p/kT) (m
3) Number of particles of chemical species i, independent of their excitation state (m
3) Number of particles of chemical species i in the excited state, s (m
3) Pressure of the gas (Pa) Partial pressure of the chemical species i (Pa) Reference pressure (Pa) Partition function of the chemical species i Translational partition function of the chemical species i Internal partition function of the chemical species i
 Total partition function of the gas Qtot ¼ ∏i QNi i =∏i Ni Ratio of the electron flux to that of neutral species Net generation rate of a species through forward and reverse reactions Equilibrium temperature (K) Discontinuity temperature (K) Heavy species temperature (K) Electron temperature (K) Excitation temperature (K)

424

Tr Tv T* Ttri vi vi V Ved xi

M. I. Boulos et al.

Rotation temperature (K) Vibration temperature (K) Intermediate temperature T* between Te and Th (K) Translational temperature of species i (K) Velocity of particle i (m/s) Mean velocity of particle i (m/s) Volume of the gas or of the plasma (m3) Mean drift velocity of electrons (m/s) Molar fraction of chemical species i

5.2

Greek Alphabet

δp Δp ΔE δENþ ΔH

Pressure correction term resulting from the electrostatic interactions (Pa) Lowering of the pressure (Pa) Energy gap between two consecutive excited levels (eV) Ionization potential lowering for N+ ion production (eV) Enthalpy change of a thermodynamic state at T and p with respect to a reference state (J) Vacuum permittivity (εo = 8.854,187,817  10
12 F/m) Specific heat ratio (γ=cp/cv) Debye length (m) Chemical potential for the chemical species i (J) Chemical potential for the electrons (J) Part of the chemical potential of chemical species i that depends only on temperature or at reference pressure p0 (J) Ratio of electron to heavy particle temperature (θ = Te/Th) Elastic collision frequencies between electron and species A (s
1) Elastic collision frequencies between species j and species A (s
1) Elastic collision frequency between electron and species A (m2) Elastic collision frequency between species j and species A (m2) Mean free flight time for electrons (s) Forward stoichiometric coefficient Reverse stoichiometric coefficient

εo γ λD μi μe μoi θ νe,A νj,A σe,A σj,A τe υij υij0

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André P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (1999) In: Fauchais P, Amouroux J (eds) Plasma concentrations out of equilibrium: N2 (kinetic method and mass action law), Ar–CCl4 and Ar–H2CCl4 (mass action law). The Annals of the New York Academy of Sciences, New York, pp 85–94 André P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2001) A new modified pseudoequilibrium calculation to determine the composition of hydrogen and nitrogen plasmas at atmospheric pressure. Plasma Chem Plasma Process 21(1):83–105 Aubreton J, Elchinger MF, Fauchais P (1998) New method to calculate thermodynamic and transport properties of a multi-temperature plasma: application to N2 plasma. Plasma Chem Plasma Process 18(1):1–27 Aubreton J, Elchinger MF, Fauchais P, Rat V, André P (2004) Thermodynamic and transport properties of a ternary Ar–H2–He mixture out of equilibrium up to 30,000K at atmospheric pressure. J Phys D Appl Phys 37:2232–2246 Aubreton J, Elchinger MF, Andre P (2013) Influence of partition function and interaction potential on transport properties of thermal plasmas. Plasma Chem Plasma Process 33:367–399 Baeva M, Kozakov R, Gorchakov S, Uhrlandt D (2012) Two-temperature chemically non-equilibrium modeling of transferred arcs. Plasma Sources Sci Technol 21:055027, 13 pp Boselli M, Colombo V, Ghedini E, Gherardi M, Sanibondi P (2013) Two-temperature modeling and optical emission spectroscopy of a constant current plasma arc welding process. J Phys D Appl Phys 46(22):224009 Capitelli M, Giordano D, Colonna G (2008) The role of Debye-H€ uckel electronic energy levels on the thermodynamic properties of hydrogen plasmas including isentropic coefficients. Phys Plasmas 15:082115 Capitelli M, Armenise I, Bruno D, Cacciatore M, Celiberto R, Colonna G, De Pascale O, Diomede P, Esposito F, Gorse C, Hassouni K, Laricchiuta A, Longo S, Pagano D, Pietanza D, Rutigliano M (2007) Non-equilibrium plasma kinetics: a state-to-state approach. Plasma Sources Sci Technol 16:S30–S44 Capitelli M, Armenise I, Bisceglie E, Bruno D, Celiberto R, Colonna G, D’Ammando G, De Pascale O, Esposito F, Gorse C, Laporta V, Laricchiuta A (2012) Thermodynamics, transport and kinetics of equilibrium and non-equilibrium plasmas: a state-to-state approach. Plasma Chem Plasma Process 32:427–450 Chen X, Han P (1999) On the thermodynamic derivation of the Saha equation modified to a twotemperature plasma. J Phys D Appl Phys 32:1711–1718 Chen DM, Hsu KC, Pfender E (1981) Two-temperature modeling of an arc plasma reactor. Plasma Chem Plasma Process 1:295–314 Cliteur GJ, Suzuki K, Tanaka Y, Sakuta T, Matsubara T, Yokomizu Y, Matsumura T (1999) On the determination of the multi-temperature SF6 plasma composition. J Phys D Appl Phys 32:1851–1856 Colombo V, Ghedini E, Sanibondi P (2009) Two-temperature thermodynamic and transport properties of argon–hydrogen and nitrogen–hydrogen plasmas. J Phys D Appl Phys 42:055213 Colombo V, Ghedini E, Sanibondi P (2011) Two-temperature thermodynamic and transport properties of carbon–oxygen plasmas. Plasma Sources Sci Technol 20:035003 Fauchais P, Coudert JF, Vardelle M (1989) In: Flamm D, Aucellio F (eds) Plasma diagnostics, vol 1. Academic, New York, pp 349–446 Ghorui S, Heberlein JVR, Pfender E (2007) Thermodynamic and transport properties of twotemperature oxygen plasmas. Plasma Chem Plasma Process 27:267–291 Ghorui S, Heberlein JVR, Pfender E (2008) Thermodynamic and transport properties of twotemperature nitrogen-oxygen. Plasma Chem Plasma Process 28:553–582 Gleizes A, Chervy B, Gonzalez JJ (1999) Calculation of a two-temperature plasma composition: bases and application to SF6. J Phys D Appl Phys 32:2060–2067 Gordon MH, Kruger CH (1993) Temperature and density measurements in a recombining argon plasma with diluent. Plasma Chem Plasma Process 13:365–378 Herzberg G (1959) Spectra of diatomic molecules. D. van Nostrand Company, New York

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Hsu KC, Pfender E. Calculation of thermodynamic and transport properties of a two-temperature argon plasma. In: Proceedings of the fifth international symposium on plasma chemistry, vol. 1144. Heriot- Watt University Edinburgh, Scotland Park J, Heberlein J, Pfender E, Candler G, Chang CH (2008) Two-dimensional numerical modeling of direct- current electric arcs in nonequilibrium. Plasma Chem Plasma Process 28:213–231 Potapov AV (1966) High Temp 4:48 Rat V, André P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2001) A modified pseudoequilibrium model competing with kinetic models to determine the composition of a twotemperature SF6 atmosphere plasma. J Phys D Appl Phys 34:2191–2204 Rat V, André P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2002a) Transport coefficients including diffusion in a two-temperature argon plasma. J Phys D Appl Phys 35:981–991 Rat V, André P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2002b) Two-temperature transport coefficients in argon–hydrogen plasmas – II: inelastic processes and influence of composition. Plasma Chem Plasma Process 22(4):475–493 Rat V, André P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2002c) Two-temperature transport coefficients in argon–hydrogen plasmas – I: elastic processes and collision integrals. Plasma Chem Plasma Process 22(4):453–474 Rat V, Murphy AB, Aubreton J, Elchinger MF, Fauchais P (2008) Treatment of non-equilibrium phenom- ena in thermal plasma flows. J Phys D Appl Phys 41:183001, 28 pp Richley E, Tuma DT (1982) On the determination of particle concentrations in multitemperature plasmas. J Appl Phys 53:8537–8542 Sourd B, Aubreton J, Elchinger MF, Labrot M, Michon U (2006) High temperature transport coefficients in e/C/H/N/O mixtures. J Phys D Appl Phys 39:1105 Tanaka Y (2004) Two-temperature chemically non-equilibrium modelling of high-power Ar–N2 inductively coupled plasmas at atmospheric pressure. J Phys D Appl Phys 37:1190–1205 Tanaka Y, Sakuta T (2002) Chemically non-equilibrium modelling of N2 thermal ICP at atmospheric pressure using reaction kinetics. J Phys D Appl Phys 35:468–476 Tanaka Y, Yokomizu Y (1997) Particle composition of high-pressure SF6 plasma with electron temperature greater than gas temperature. IEEE Trans Plasma Sci 25(5):991–995 Trelles JP, Heberlein JVR, Pfender E (2007) Non-equilibrium modeling of arc plasma torches. J Phys D Appl Phys 40:5937–5952 Trelles JP, Chazelas C, Vardelle A, Heberlein JVR (2009) Arc plasma torch modeling. J Therm Spray Technol 18(5–6):728–752 van de Sanden MCM, Schram PPJM, Peeters AG, van der Mullen JAM, Kroesen GMW (1989) Thermodynamic generalization of the Saha equation for a two-temperature plasma. Phys Rev A 40:5273–5276 Wang H-X, Sun W-P, Sun S-R, Murphy AB, Ju Y (2014) Two-temperature chemicalnonequilibrium modelling of a high-velocity argon plasma flow in a low-power arcjet thruster. Plasma Chem Plasma Process 34:559–577 Ye R, Murphy AB, Ishigaki T (2007) Numerical modeling of an Ar–H2 radio-frequency plasma reactor under thermal and chemical nonequilibrium conditions. Plasma Chem Plasma Process 27(2007):189–204 Yu L, Pierrot L, Laux CO, Kruger CH (2001) Effects of vibrational nonequilibrium on the chemistry of two-temperature nitrogen plasmas. Plasma Chem Plasma Process 21(4):483–503

Transport Properties of Non-Equilibrium Plasmas

10

Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Simplified Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Chapman–Enskog Method and Stefan–Maxwell Relations . . . . . . . . . . . . . . . . . . . . . . . . . 3 Non-equilibrium Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Solution of Rat et al. (2001b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Solution of Zhang et al. (2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Examples of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Monoatomic Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Diatomic Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Complex gases and mixtures in NLTE and NLCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Latin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

428 428 428 429 431 432 439 441 441 448 455 464 464 466 467

M. I. Boulos (*) Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada e-mail: [email protected] P. L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France e-mail: [email protected] E. Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] © Springer Nature Switzerland AG 2023 M. I. Boulos et al. (eds.), Handbook of Thermal Plasmas, https://doi.org/10.1007/978-3-030-84936-8_10

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Abbreviations

ES GS LCE LTE NLCE NLTE RTC 2T

1

Excited states Ground state Local chemical equilibrium Local thermodynamic equilibrium Non-local chemical equilibrium Non-local thermodynamic equilibrium Reactive thermal conductivity Two-temperature plasma (Non-LTE)

Introduction

Simulation of thermal plasma processes requires transport coefficients such as electrical and thermal conductivities, viscosity, and diffusion coefficients. As already shown in ▶ Chap. 7, “Transport Properties of Gases Under Plasma Conditions,” calculation of such coefficients, even under equilibrium conditions (both LTE and LCE), is rather involved. In this chapter, derivations of such coefficients under non-equilibrium conditions will be described, considering: (a) The situation of two-temperature plasmas in local chemical equilibrium (LCE) (b) Two-temperature plasmas with deviations from local chemical equilibrium (NLCE) Work on these complex problems is still in progress. After introducing simplified models, detailed analyses of recent approaches for calculating transport coefficients are presented, based on solutions of the Boltzmann equation, using a modified Chapman–Enskog method which includes coupling of electrons with heavy species. Finally, selected examples of such coefficients are presented for monatomic gases (Ar, He) and diatomic gases (H2, N2, O2, CO2) used for the generation of thermal plasmas. Some examples of more complex mixtures will also be shown (Ar/H2, N2/H2, Ar/H2/He).

2

General Remarks

2.1

Simplified Models

In ▶ Chap. 7, “Transport Properties of Gases Under Plasma Conditions,” a simplified derivation of the transport properties of gases was presented based on two important assumptions: • Gas is sufficiently diluted that only two-particle collisions need to be taken into account. • Between two collisions, the particles travel with their mean random velocity v.

10

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429

However, the averaging does not take into account the distribution functions of species, and if these expressions are easy to use and allow physical interpretation of results, the transport coefficients can be relatively inaccurate. As pointed out by Rat et al. (2008), they can be applied to two-temperature plasma if the correct temperature is chosen: • Momentum transfer is dominated by heavy species, and therefore, the viscosity depends on the heavy species temperature. • The translational thermal conductivity independently contains the contribution of electrons at Te and heavy species at Th. • The electrical conductivity is linked to the electron density and thus to Te or θ = Te ⁄ Te (see Fig.1 for the electrical conductivity of a non-equilibrium argon plasma). • The self-diffusion coefficient is often expressed in terms of ambipolar diffusion, for example, Da = D(1 + Te/Th), where D is the self-diffusion coefficient of heavy species, Da being driven by the mass of heavy species and weighted by the ratio Te/Th. Moreover, this is only valid if the departure from equilibrium is not too large. For more details see Rat et al. (2008).

2.2

Chapman–Enskog Method and Stefan–Maxwell Relations

As already discussed in ▶ Chap. 7, “Transport Properties of Gases Under Plasma Conditions,” a more exact formulation requires (taking into account the velocity distributions) calculating fluxes of all species through a reference surface dA moving with the mean fluid velocity. The non-equilibrium transport properties will differ from those calculated at equilibrium (Rat et al. 2008): – The necessity to calculate the concentrations or molar fractions of the different species, which are used in the expressions of transport properties. Thus, a transport property will depend on the way it has been calculated: 2T and LCE or 2T and NLCE (Potapov (1966)) as described in ▶ Chap. 9, “Thermodynamic Properties of Non-equilibrium Plasmas.” – Departures from equilibrium depend on the different relaxation processes to analyze, such as the self-relaxation time, τc, corresponding to the time required for self-collisions to bring a species toward the Maxwellian distribution function at its own kinetic temperature (Spitzer 1956). – At equilibrium the transport coefficients are calculated from the Boltzmann equation, which is solved by assuming for species, i, a Maxwellian distribution ð0Þ with a first-order perturbation f i ð1 þ ϕi Þ. The Chapman–Enskog (CE) method then consists in assuming that Φi is a linear combination of particle concentration, total pressure, external forces, velocity, and temperature gradients, which cause the transport phenomena. As described in ▶ Chap. 7, “Transport Properties of Gases Under Plasma Conditions,” for each

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species i and each gradient, the linear coefficients are expanded into a series of Sonine polynomials, which fix the order of approximation ξ. It is assumed that the composition, the mass-average velocity, and the temperature are not affected by the perturbation function. The derivation gives rise to the definition of collision integrals, taking into account the interaction potential between colliding species. This approach, due to the work of Devoto (1965, 1967), was modified by Bonnefoi (1983) and Aubreton et al. (1986), but their approaches lead to the decoupling of the electrons and heavy species, i.e., neglecting the electron–heavy species collision terms. Such decoupling does not allow mass conservation to be satisfied. More recently, Rat et al. (2001b) suggested maintaining the coupling between electrons and heavy species in solving the systems of linear equations involved in the Chapman–Enskog method. Figure 1 from Rat et al. (2008) shows a comparison between the electrical conductivities of an atmospheric-pressure argon plasma as a function of the electron temperature, for different values of the ratio θ = Te/Th, using the approach of Devoto (1965, 1967) modified by Bonnefoi (1983) and Aubreton et al. (1986) and that obtained using the approach of Rat et al. (2001b, 2002a). At equilibrium both approaches converge, but the differences increase with θ. Devoto’s approach does not satisfy mass conservation, and his simplified expressions provide results (electrical conductivity and translational electron thermal conductivity) that can be quite different from those obtained with a full calculation, i.e., retaining the coupling between electrons and heavy species in the Boltzmann equation.

Fig. 1 Dependence on electron temperature of the electrical conductivity of atmospheric-pressure argon plasma, calculated by Rat et al. (2008) for different values of the ratio Te/Th using Devoto’s approach (Devoto 1965, 1967) and modified by Bonnefoi (1983) and Aubreton et al. (1986) and the approach of Rat et al. (2001b, 2002a)

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431

– The combined diffusion coefficients of Murphy (1993) as described in ▶ Chap. 7, “Transport Properties of Gases Under Plasma Conditions,” Sect. 4 lead to unphysical results even when Te tends toward Th. The numerical treatment of diffusion is particularly complex even at equilibrium because a large number of coefficients exist: N(N1)/2 for ordinary diffusion coefficient with N species and N1 thermal diffusion coefficient. That is why Murphy (1997) has introduced the treatment of diffusion in term of gases such as Ar and H2 instead of Ar, Ar+, H2, H, H+. When using the simplified theory of transport coefficients of Devoto to calculate them, the symmetry relationships DxAB ¼ DxBA and DTAB ¼ DTBA are not fulfilled when the degree of ionization exceeds 10%. This simplified theory is therefore not adapted to account for diffusion in a multi-temperature plasma [Rat et al. (2001b)]. – At last at equilibrium collision integrals between species, i and j are calculated at temperature T, but when one species is at Th, while the other at Te (with θ ¼ Te =Th ), an effective collision temperature T* must be defined, as demonstrated by Rat et al. (2001b): T ¼



1 mi þ mj




mi mj þ Ti Tj

1

:

(1)

To conclude this section, it can be stated that the calculation of transport coefficients in non-equilibrium conditions is more complex than at equilibrium. However in all thermal spray processes non-equilibrium phenomena are often present, as shown for example by Baeva et al. (2012) for transferred arcs, Boselli et al. (2013) for welding arc, and for arc plasma torches used in plasma spraying Trelles et al. (2007) and Trelles et al. (2009).

3

Non-equilibrium Transport Properties

As pointed out in the previous Sect. 2.2, the theories of Devoto (1965, 1967) modified by Bonnefoi (1983) and Aubreton et al. (1986) neglecting the electron–heavy species collision terms do not allow mass conservation to be satisfied. Caesar et al. (2000) developed a computation method of thermodynamic and transport properties of multi-temperature plasmas. An alternate approach by Capitelli et al. (2012) used a state-to–state model offering detailed information about internal distributions affecting thermodynamics, transport coefficients and kinetics, properly accounting for the presence of excited states. However it is rather complex and still at its beginning In the following, highlights are presented of the approaches of Rat et al. (2001b) and the most recent one of Zhang et al. (2013). Both are based on the solution of the Boltzmann equation using a modified Chapman–Enskog method taking into account the coupling between electrons and heavy species.

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3.1

M. I. Boulos et al.

Solution of Rat et al. (2001b)

3.1.1 Equations According to their model, transport properties are derived, without Bonnefoi’s assumptions, in nonreactive two-temperature plasma, assuming chemical equilibrium is achieved. The electron kinetic temperature Te is supposed to be different from that of heavy species Th. Only elastic processes are considered in a collisiondominated plasma. The Boltzmann equation, according to the Chapman–Enskog method, is used to calculate transport coefficients. The solution of these systems allows writing transport coefficients as linear combinations of collision integrals, which take into account the interaction potential for a collision between two particles. Extending the definition and calculation of bracket integrals introduced by Chapman and Cowling (1970) allows obtaining these linear combinations in the thermal non-equilibrium case. The obtained results are rigorously the same as those of Hirschfelder et al. (1964) at thermal equilibrium. The distribution function fi of the ith species is the solution of the integrodifferential equation of Boltzmann (Hirschfelder et al. 1964): N Df i X ¼ Dt j¼1

ððð   ! f 0i f 0j  f i f j gb db dε d v j :

(2)

where fi0 is the distribution function after collision of the ith species, g is the relative velocity of the species i and j, and b and ε are, respectively, the impact parameter and the incidence azimuthal angle. The mixture can be characterized by the ! knowledge of the unknowns ni, v 0 the mass-averaged velocity, Te, and Th, by solving the equations of continuity, momentum, and energy obtained from the Boltzmann equation. This equation according to Chapman and Cowling (1970) can also be written as: N Df i X ¼ Dt j¼1

ðð 

 ! f 0i f 0j  f i f j gσij dΩdv j :

(3)

!

In this equation v j is the velocity of species j, σij is the differential collision cross section, and Ω is the solid angle. It is assumed that the zero-order approximation function is Maxwellian at Te for electrons and Th for heavy species. The distribution function of the ith species, solution of Eq. 2, is approximated by a Maxwellian distribution function, f(0) i ð0Þ perturbed by Φi such as Φi  1, f i ð1 þ ϕi Þ which is the first-order perturbation function of the ith species with the temperature, Ti. Equation 2 becomes: N X Df i ð0Þ ¼ Ii þ Dt j¼1 ð 0Þ

ððð

ð0Þ ð0Þ

fi fj

h  i Φ0i þ Φ0j Ki  Φi  Φj gb db dε dvj :

(4)

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433

i By definition Df Dt corresponds to

ð0Þ

!

ð0Þ

Df i @f Fi ! ! ! ð0Þ ¼ i þ v i ∇ fi þ ∇ Dt @t mi !

ð 0Þ ! v i fi

(5)

!

where v i , F i , and mi are, respectively, the velocity, external force, and mass of species i. I(0) i represents the zero-order approximation of Chapman–Enskog expansion and does not vanish for two colliding particles with different temperatures: ð0Þ

Ii

¼

N ððð   X ! ð0Þ ð0Þ ð0Þ ð0Þ f 0 i f 0 j  f i f j gb db dε dV j :

(6)

j¼1

Rat et al. (2001b) pointed out that the Maxwellian f(0) and the unknowns Φi are i assumed to vary slowly in space and time over a distance of a mean free path and over the time of a mean free flight. This assumption justifies, in a first approximation, neglecting the derivatives of f(0) i as well as the products of Φi with derivatives of f(0) i on the left-hand side of Eq. 4. Non-equilibrium is defined by the parameter θ; θij ¼

Ti ; Tj

(7)

and Rat et al. (2001b) introduced a term Ki(Wi, θij), taking into account the thermal non-equilibrium when electrons and heavy species collide: ð0Þ ð0Þ

f0i f0j

ð0Þ ð0Þ  ¼ f i f j Ki Wi , θij

(8)

where      2 Ki Wi , θij ¼ exp  W0 i  W2i 1  θij

(9)

!

and Wi0 is the reduced velocity after collision. Wi is defined as !

Wi ¼




mi 2kTi

1=2

!

Vi:

(10)

However, it has to be noted that Ki = Kj and, of course, when two species with the same temperature collide, Ki = 1. The introduction of Ki allows defining of bracket Df 0 integrals to be generalized out of thermal equilibrium. Thus Dti can be calculated for i = 1 (electrons) and heavy species ði 6¼ 1Þ. In these expressions there appears the quantity Q(0) 1 corresponding to the exchanged kinetic energy between electrons Df 0 and heavy species during collisions. Equations Dti are also modified to take into

434

M. I. Boulos et al.

account the coupling between electrons and heavy species. It is assumed that the ! ! thermal non-equilibrium defined by θij depends on space vector r , that is, θð r Þ. Hence, !

!

!

∇ lnTe ¼∇ lnTh þ ∇ lnθ:

(11) Df 0

Df 0

Taking into account all these definitions, expressions of Dt1 (electrons) and Dti (heavy species) have been obtained by Rat et al. (2001b) from which a general form of Φi in Eq. 4 has been established: !

!

$

! !

Φi ¼ A i  ∇ lnTh  B i :∇ υ 0 þ

X

!

!

ð0Þ

C ji  d j þ Di Q1 þ

N X ! ! ! ! E ji  gj ∇ lnθ  F i  ∇ lnθ

j¼1

j¼1

(12) !

!

!

!

The previous unknowns Φi are replaced by new ones: A i, B i, C ji, E ji, and Fi. It has to !

!

!

!

be noted that A i , C ji, E ji, and Fi are vectors, B i is a second-order tensor, and Di are scalars. This approach permits to define new unknowns that are expressed as linear combination of Sonine polynomial expansion coefficients. The expansion coefficients are complex combinations of bracket integrals, which are written as linear combinations of collision integrals Ω‘,s, as shown by Chapman and Cowling (1970). After few calculations, Rat et al. (2001b) obtained expressions similar to those of Chapman and Cowling (1970) at equilibrium with bracket integrals. However, the expressions of bracket integrals corresponding to the collision electrons–electrons or heavy species–heavy species are the same as at thermal equilibrium, but with the correct corresponding temperatures. For collisions between molecules of type i and type j, according to Rat et al. (2008), these integrals are defined as: ð ‘, s Þ

Ωij

¼ π1=2

ð1 0

ð ‘Þ

γ2sþ2 eγ ϕij dγ 2

(13)

with the transport cross section ð‘Þ

ϕij ¼

ð1



1  cos ‘ χ gb db:

(14)

0

In these equations, γ2 is a reduced energy, g is the relative velocity of the encounter, and b is the impact parameter. γ2 is defined by   γ2 ¼ μij =2kT g2

(15)

where T is the effective temperature of collisions defined by Eq. 2 and μij the reduced mass:

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Transport Properties of Non-Equilibrium Plasmas

μij ¼

435

mi mj : mi þ mj

(16)

Other collision integrals can be defined following the approach of Hirschfelder et al. (1964). In both cases their values, as at equilibrium, will strongly depend on the interaction potential chosen for the calculation.

3.1.2

Transport Coefficients

Diffusion The diffusion velocity is given by !

Vi ¼

ð ! 1 ! V i f i dV i : ni

(17)

where !

!

!

V i ¼ v i  v 0;

(18)

! nimi! v1 being the velocity of the ith species and v 0 the mass-average velocity

! υ0

¼

N 1X ! ni m i υ i : ρ i¼1

(19)

Compared to equilibrium equations, with two temperatures electrons and heavy particles must be considered separately. Written in terms of Th and θ, with species 1 the electron and species 2 to N the heavy species, according to Rat et al. (2001b, 2008), the diffusion mass flux, using Eq. 12, becomes: Ji ¼

N    mi n X mj Dij dj þ Dθij gj ∇lnθ  DTi ∇lnTh  Dθi ∇lnθ: ρkTi j¼1

(20)

In Eq. 20 there appear two-temperature ordinary diffusion coefficients Dij and a thermal diffusion coefficient:
 ni ρkTi kTi 1=2 ji Dij ¼ ci0 nmj 2mj


DTi

kTi ¼ ni mi 2mi

cjii0 and ajii0 being expansion coefficients.

(21)

1=2 ai0

(22)

436

M. I. Boulos et al. 

New thermal non-equilibrium diffusion coefficients Dθij and Dθi also appear (Rat et al. 2001b) taking into account the temperature difference between electrons and heavy species and corresponding to diffusion due to the gradient of the temperature ratio θ ¼ TThe . It has to be noted that the diffusion velocity of electrons depends on the heavy species gradient. In fact, Th can be considered as a reference temperature, the temperature difference Te and Th being established by the  introduction of Dθi . The diffusion coefficient Dθij introduces mass transfer in the mixture, which tends to eliminate the temperature difference between electrons and heavy species: Dθij


 ni ρkTi kTi 1=2 ji ¼ ei0 : nmj 2mi

 Dθi




kTi ¼ ni m i 2mi

(23)

1=2 f i0 :

(24)

These two-temperature diffusion coefficients satisfy the symmetry relationships resulting from the fact that the sum of mass fluxes (relative to the mass-average velocity) in the mixtures vanishes, which are similar to those defined at equilibrium (Rat et al. 2002a): N X mi ðmk Dih  mk Dik Þ ¼ 0 Ti i¼1 N X

DTi ¼ 0

(25)

(26)

i¼1 N X mi  mh Dθih  mk Dθik ¼ 0 Ti i¼1 N X



Dθi ¼ 0

(27)

(28)

i¼1

For a binary mixture (e.g., electrons diffusing in argon at low temperature), the constraint dealing with the vanishing sum of mass fluxes (Eq. 25) allows one to write an asymmetric relationship Rat et al. (2002b) between two-temperature ordinary diffusion coefficients: Dij ¼ θij Dji :

Electrical Conductivity Rat (2001) wrote the current density vector as

(29)

10

Transport Properties of Non-Equilibrium Plasmas

j¼

N X

437

eni Zi Vi ;

(30)

i¼1

and accounting for Ohm’s law, he has shown that the electrical conductivity was ( ) N N  e2 n X Zi X σei ¼  mj nj Zj Dij : ρk i¼1 Ti j¼1

(31)

However, the contribution of heavy species can be neglected (the ratio of the part due to ions to that due to electrons is below 103 Rat (2001), and finally σel ¼

N e2 n X mj nj Zj D1j : ρkT j¼1

(32)

Viscosity The pressure tensor is defined as $

p¼

N  X

  $ ð0Þ $ ni kTi 1  di1 Q1 U 2μi S

(33)

i¼1 $

$

where U is unity tensor, and S is the stress tensor; see Rat et al. (2001b) for its definition, which is the same as at equilibrium. Q(0) 1 corresponds to a kinetic-energy transfer between electrons and heavy species, and di1 is a perturbative term. Finally, μi is the viscosity coefficient of the ith species, the corresponding relationships being defined in Rat et al. (2001a, b). Thermal Conductivity At equilibrium, the thermal conductivity, κ, comprises 4 terms: heavy species translational thermal conductivity, κh; electron translational thermal conductivity, κe; reactional thermal conductivity, κR; and the internal thermal conductivity, κint, being generally negligible. Translational Thermal Conductivity

For translational thermal conductivities the heat flux is given by !

q¼

1X mj 2 N

ð

!

!

V2j V j f j dV j :

(34)

j¼1

After introducing Eq. 1, Rat et al. (2001a, b) point out that the calculation is quite different from that performed at equilibrium because of the asymmetry of

438

M. I. Boulos et al.

calculations introduced by the presence of different temperatures. Rat et al. (2001a, b), inserting Eqs. 1 and 12 in Eq. 34, obtain: !

q ¼

XN i¼1

! 5 kTi ni Vi 2



κ0i

!

∇ Th 

θ κ0 i T i

!

∇ lnθ 

N X κD ij j¼1

nj m j

! dj

! (35)

where 5 4

κ0i ¼  k θ κ0 i

¼

5  kni 4

κD ji

¼


 Ti 2kTi 1=2 ni ai1 Th mi

(36)

!
 N X 2kTi 1=2 ji f i1  ei1 gj mi j¼1

5 n n m kT 4 i j j i

(37)


 2kTi 1=2 ji ci1 : mi

(38)

The translational thermal conductivity of the species, i in a two-temperature plasma, and that resulting from the temperature difference is, respectively, κi0 and κ0 θi (see Rat et al. 2001a, b). It has to be noted that, in the expression of thermal conductivity of electrons κi0 , the term θ ¼ Te =Th (when i = 1) takes into account the fact that their heat flux is !

calculated with respect to the temperature gradient of heavy species ∇ Th . The thermal conductivity κ0 θi due to the non-equilibrium effect highlights the energy transfer between the two subsystems consisting of electrons and heavy species. Rat et al. (2001a, b) point out that the translational thermal conductivity can also be defined by introducing the specific enthalpy hi of the ith species such that ρi hi ¼ 52 ni kTi . Reactional Thermal Conductivity An expression for the reactional thermal conductivity in a two-temperature, nonequilibrium Ar plasma using the basic concept of Butler and Brokaw and neglecting external forces was developed by Chen and Li (2003). Rat et al. (2008) presented a review of the problems linked to the reactive thermal conductivity accounting for inelastic collisions, which mainly give rise to dissociation and ionization reactions. If forward and backward reaction rates are sufficiently fast compared with diffusion processes, the chemical composition reaches equilibrium with the local temperature. Since plasma composition depends on local temperature, a temperature gradient will lead to greater concentrations of molecules (in the case of dissociation) in lower-temperature regions and atoms in higher-temperature regions. The associated concentration gradients result in the diffusion of atoms toward the low-temperature regions where they recombine, releasing the energy of dissociation. Thus, dissociation, ionization, and other chemical reactions contribute to the energy transport

10

Transport Properties of Non-Equilibrium Plasmas

439

along the temperature gradient. Defining the total heat flux as a function of the gradients of T(e) and T(h) only does not take into account the way energy is transferred during inelastic collisions between electrons and heavy species, and does not consider either the role of the thermal conductivities of electrons and heavy species in the reactional component of the thermal conductivity. In plasmas formed exclusively from monatomic gases, no dissociation occurs and the only reaction is ionization. At low temperatures, electrons are produced by heavy particle collisions, but as temperature increases, the ionization process is primarily due to collisions with electrons. The inverse three-body recombination reaction also occurs in the presence of electrons. Thus κR essentially depends on Te. In molecular gases, ionization is predominantly due to electrons. However, for a majority of polyatomic and diatomic molecules, dissociation occurs when the temperature is lower than 5,000 K, and the plasma is only weakly ionized. As a first approximation, dissociation is due to heavy-particle collisions, and thus the corresponding part of κR should be attributed to the heavy particles. The problem is more complex for a few stable molecules (N2 in particular) for which dissociation occurs at higher temperatures; it is then difficult to separate the electron and the heavy-particle contributions. The influence of electronically excited atoms on transport processes, especially at high temperature, must also be considered (Rat et al. 2008). Capitelli and his coworkers (2002, 2004) and Bruno et al. (2006) investigated this subject for many years.

3.2

Solution of Zhang et al. (2013)

According to Zhang et al. (2013), the calculation proposed by Rat et al. (2001a, b, 2008) overcome the problems of approaches of Devoto (1967) and Bonnefoi (1983), but the complete theory is extremely complex and time-consuming to implement and, as a consequence, has not been widely accepted. In addition, the recent studies of Colombo et al. (2008, 2009, 2011) showed that the results of the simplified theory of Devoto (1967) and Bonnefoi (1983), with the exception of diffusion coefficients, presented only slight differences with those obtained from the complete theory, after certain corrections were made. In their study, in contrast to the complete theory of Rat et al. (2001a, b, 2008), reasonable simplifications are made on the basis of the fact that the mass of electrons, me, is much smaller than those of heavy species, mh (ions and neutral species). For instance, the change of the first-order perturbation function of heavy species is neglected compared with that of electrons in the electron–heavyparticle collisions. Second, the coupling between the subsystems of electrons and heavy species is considered in this study, instead of regarding them as two isolated systems, as was assumed in the simplified theory of Devoto (1967). Although the subsystems of electrons and heavy species can be assumed to be mutually adiabatic due to the extreme weakness of the translational kinetic-energy coupling between electrons and heavy species, work or particle transfer may take place between these

440

M. I. Boulos et al.

two subsystems due to the occurrence of chemical reactions (e.g., ionization, excitation, and their corresponding reverse processes). Zhang et al. (2013) start from the solution of the Boltzmann equation according to Chapman and Cowling (1970) Eq. 3 and used the first-order perturbation ϕi but they neglected the change in the first-order perturbation function of heavy species compared with that of electrons:  0  ϕi  ϕi  ϕ01  ϕ1 : (39) Colombo et al. (2008, 2009) comparing this simplified theory with the complete one of Rat et al. (2001a, b, 2008) found only minor differences. According to this simplification, Zhang et al. (2013) obtained for electrons Eq. 40 and for heavy species Eq. 41: ð 0Þ

Df 1 ¼ Dt

ðð

N X  ! ð0Þ f 1 f ð0Þ ϕ01 þ ϕ0  ϕ1  ϕ gσ11 dΩ dv þ

ðð

ð0Þ ð0Þ  0 ϕ1

f1 fj

!  ϕ1 gσ1j dΩ dv j ;

j¼2

(40) ð0Þ

Df i ¼ Dt

ðð

ð0Þ ð0Þ  0 ϕ1

fi f1

N X !  ϕ1 gσi1 dΩ dv 1 þ

ðð

ð0Þ ð0Þ

fi fj



 ! ϕ0i  ϕ0j  ϕ0  ϕj gσij dΩ dv j :

j¼2

(41) Contrary to the simplified theory of Devoto (1967), the first term on the right hand side of Eq. 41 was not neglected. ! They used the definition of the diffusion driving force d i introduced by Rat et al. (2001a, b) for electrons, ! d1

¼


 N ! ! ρ1 X x 1 θ ρ1 ! θp !  nj X j  n1 X 1 þ ∇ p þ 2 ∇ x1 ; D ρ j¼1 ρ D

(42)

and heavy species, ! di


 N ! ! ρi X x i ρi ! p! xi ðθ  1Þp !  ¼ nj X j  ni X i þ ∇ x1 ∇ p þ ∇ xi  D ρ j¼1 D ρ D2

ð i  2Þ : (43)

In these equations ðXi ¼ ni =nÞ is the mass fraction of species i, D ¼ 1 þ x1 ðθ  1Þ and θ ¼ Te =Th . The perturbation function is that given by Rat et al. (2001a, b) in Eq. 12. Then the transport coefficients are calculated on the bases defined by Rat et al. (2001a, b). Finally the simplified theory of the transport properties of two-temperature plasma proposed by Zhang et al. (2013) relies on the solution of the Boltzmann equations using a modified Chapman–Enskog method. In this study they used the physical fact me =mh  1 and the inclusion of the coupling between the electron and heavy

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441

species subsystems. While they are treated as separate systems, mass, momentum, and energy exchanges between these two subsystems are considered, which are particularly important for the treatment of the diffusion processes. Based on these two newly modified assumptions, simplified yet complete transport coefficients are obtained for 2T plasmas. Compared to the calculations of Rat et al. (2001a, b), those of Zhang et al. (2013) present no increase in the complexity of the expressions for the transport coefficients, especially in the number of collision integrals required.

4

Examples of Results

4.1

Monoatomic Gases

4.1.1 Argon Calculations of transport coefficients of two-temperature partially ionized argon plasma have been performed at atmospheric pressure using the expressions given in Rat et al. (2001a, b). The 2T plasma composition has been obtained at chemical equilibrium using the method of van de Sanden et al. (1989) to compare the results of transport coefficients. Transport coefficients due to ∇ln θ are not displayed, θ = Te/ Th being a fixed parameter. It has to be underlined that the presented results at equilibrium (θ = 1.0) are in good agreement with those of Murphy (2000). Rat et al. (2002a) calculated the plasma composition for three values of θ: 1, 2, and 3, as presented in Fig. 9 in ▶ Chap. 9, “Thermodynamic Properties of Non-equilibrium Plasmas.” Viscosity. The viscosity of the argon plasma (Rat et al. 2002a) is presented in Fig. 2 as a function of the electron temperature, Te, for different values of θ. As viscosity mainly depends on the mass of species, the electron contribution is small, leading to the same results between theories of Bonnefoi  (1983) and Rat 1=2 et al. (2001b). First viscosity increases with temperature  Th until the longrange Coulomb interactions overcome the interactions between neutral species. The drastic change of slope at 10,000 K highlights the ionization regime and corresponds to an electron number density equal to about 1022 m3. At fixed electron temperature, the viscosity decreases when θ increases. As the viscosity directly depends on the square root of the mass, it is therefore linked to heavy species whose temperature is Th = Te/θ. Thus, at fixed Te, the collision integrals are calculated at Te/θ, which decreases as θ increases. As the collision integrals generally decrease as temperature increases, their values increase as θ increases and the viscosity decreases (Rat et al. 2002a). It is interesting to compare these results to those of Colombo et al. (2008) who used the decoupling method of Devoto (1967) instead of the non-simplified theory of Rat et al. (2001b). They also used the assumption of local chemical equilibrium. Figure 3 presents the argon plasma viscosity evolution with Te as calculated by Colombo et al. (2008) and compared with the results of Murphy and Arundell (1994) at equilibrium ðθ ¼ 1Þ and those of Rat et al. (2002a).

442

M. I. Boulos et al. 30

Viscosity, μ (10−5 kg/m.s)

25

θ =1.0 θ =1.1

20

θ =1.3 θ =1.6

15

θ =2.0 θ =3.0

10

5 0 0

5

10

15

20

Electron temperature, Te

25

(103 K)

Fig. 2 Evolution of the viscosity with the electron temperature for an argon plasma at atmospheric pressure for different values of θ (Rat et al. 2002a)

Viscosity, μ (10−5 kg/m.s)

30 25 θ =1.0 20

Murphy et al (1994) + θ =1.0 Rat et al (2002a)

θ =2.0 15

X θ =2.0 θ =3.0

θ =3.0 10 5 0 0

10

20 30 40 Electron temperature, Te (103 K)

50

Fig. 3 Evolution of the viscosity with the electron temperature of an argon plasma at atmospheric pressure for different values of θ. Comparison with results of Rat et al. (2002a) in LTE and NLTE and those of Murphy and Arundell (1994) in LTE (Colombo et al. 2008)

Results of argon viscosity in 2T compared with those of Rat et al. (2002a) are slightly different for electron temperatures higher than 12,000 K because in the computation of composition, different methods have been used.

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Electrical conductivity. As presented in Fig. 1, the electrical conductivity, σel, increases with the electron temperature, i.e., with the ionization degree. It decreases weakly around 21,000 K. The presence of multi-charged ions at high temperature (21,000 K) involves an increase of collision integrals. At fixed Te, the electrical conductivity slowly increases with θ, following the weak increase of electron number density. Moreover, the Arþþ number density reaches 1022 m3 at lower temperature when θ increases, implying a shift of the maximum of σel due to the presence of Arþþ ions (Rat et al. 2002a). The results of Colombo et al. (2008) (see Fig. 4) are in good agreement with results reported by Rat et al. (2002a). Slight differences still remain because of the different theories used: that of Devoto (1967) and that of Rat et al. (2002a). Finally, 2T electrical conductivity of argon plasma calculated by Colombo et al. (2008) is in good agreement with that reported by Rat et al. (2002a) when the same definition for Debye length is used. In LTE the agreement is good with results reported by Murphy and Arundell (1994). Thermal conductivity. Rat et al. (2002a) also point out that the values of the electron thermal conductivity, κe, are rather different when calculated by the theories of Rat et al. (2002b) and Bonnefoi (1983) because the simplified theory of Bonnefoi underestimates the electron thermal conductivity when non-equilibrium conditions are applied. The discrepancy reaches more than 40% for θ = 3.0. The reactive thermal conductivity (RTC) accounts for inelastic collisions, which mainly give rise to dissociation and ionization reactions (only ionization for argon). Since plasma composition depends on local temperature, a temperature gradient results in higher atom concentrations in higher-temperature regions. These concentration gradients result in the diffusion of atoms toward the low-temperature regions where they recombine, releasing the energy of dissociation. It leads to addition of a

Electrical conductivity, σe (103 S/m)

16

Colombo et al (2008) θ =1.0 θ =2.0 θ =2.0

14 12 10 8 6

Rat et al (2002a) X θ =2.0 θ =3.0

4 2 0

0

10

20 30 Electron temperature, Te (103 K)

40

50

Fig. 4 Evolution of electrical conductivity of argon with electron temperature for different values of θ. Comparison of results of Colombo et al. (2008) with those of Rat et al. (2002a) in LTE and 2T and those of Murphy and Arundell (1994) in LTE (Colombo et al. 2008)

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Fig. 5 Comparison of the calculated values of the reactive thermal conductivity of two-temperature argon plasma at atmospheric pressure for four different cases. Case 1; Case 2; Case 3; Case 4 (Chen and Li 2003)

reactive contribution to the total heat flux along the temperature gradient, namely, qR ¼ κR  ∇T, where κR is the reactional thermal conductivity. Chen and Li (2003) have calculated κR for argon using different methods: Case 1, Saha law of van de Sanden et al. (1989) and expression of κR proposed by Hsu (1982), Case 2 Saha law of van de Sanden et al. (1989) and expression of κR proposed by Chen and Li (2003), Case 3 Saha law of van de Sanden et al. (1989) and expression of κR proposed by Hsu (1982), and at last Case 4 Saha law of Chen and Han (1999) with expression of κR proposed by Hsu (1982). Figure 5 shows the large differences between results, which are consequences of both the method used to calculate plasma composition and the method used to calculate the RTC for large deviations from kinetic equilibrium, θ = 5. Diffusion coefficients. Calculations of diffusion coefficients were also developed by Rat et al. (2002a). Figure 6 presents the ratio of two-temperature ordinary diffusion coefficients De–Ar and DAr–e calculated for different values of the nonequilibrium parameter θ at low temperatures (Te < 10,000 K). Up to around 5,000 K, the binary mixture approximation is valid, and Eq. 29 holds. The asymmetric relationship between De–Ar and DAr–e means that non-equilibrium conditions influence the diffusion of electrons more than that of heavy species. It should be pointed out that Eq. 29 is satisfied, independently of the approximation order ξ, as long as the binary approximation is valid. But values of diffusion coefficients also strongly depend on the approximation order ξ: at Te = 3,000 K, De–Ar (ξ = 3) = 9.16 m2 s1 and DAr–e (ξ = 1) = De–Ar = 4.17 m2 s1 for θ = 1.0 (it is worth noting that the equality (DAr–e (ξ = 1) = De–Ar is only true at low temperature). Figures 7 and 8 from Rat et al. (2002a) illustrate the two-temperature thermal diffusion coefficients as functions of the electron temperature and θ. The electron thermal diffusion coefficient weakly increases with θ (see Fig. 7) and represents a small contribution to diffusion. For example, at about 14,000 K, where there are no multiply charged ions at atmospheric pressure, DTArþ  5 106 kg=m:s and DTe

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Fig. 6 Evolution of the ratio De–Ar/DAr–e with electron temperature for an argon plasma at atmospheric pressure for different values of θ (Rat et al. 2002a)

445

Ratio of the diffusion coefficients (−)

3.5 θ=3.0

3.0 2.5

θ=2.0

2.0

θ=1.6 θ=1.3 θ=1.1 θ=1.0

1.5 1.0 0.5 0.0

0

2

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6

Electron temperature, Te

8

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(103 K)

Electron thermal diffusion coeff. DTe (10−8 kg/m.s)

11 θ =1.0 θ =1.3 θ =1.6 θ =2.0 θ =3.0

9 x 7 5 3 1 −1 0

5

10

15

20

25

Electron temperature, Te (103 K) Fig. 7 Evolution of the two-temperature thermal diffusion coefficient DTe with electron temperature for different values of θ (Rat et al. 2002a)

 4 108 kg=m:s for θ = 1.0. The heavy species thermal diffusion coefficients (see Fig. 8) exhibit almost symmetry because of the very small value of the electron thermal diffusion coefficients. It has to be noted that Eq. 26 must be satisfied whatever the value of the non-equilibrium parameter θ. Consequently, for Te < 20,000 K, DTAr  DTArþ . Moreover, the dependence of thermal diffusion coefficients of heavy species on θ can be attributed to collision integrals as shown for viscosity. Non-equilibrium Ar plasma have also been studied by Al Mamun et al. (2010), Burm et al. (2002) and Paik and Pfender (1990).

446 6 Thermal diffusion coeffs. (10−6 kg/m.s)

Fig. 8 Evolution of the two-temperature thermal diffusion coefficients DTAr, DTArþ , and DTArþþ with electron temperature for an argon plasma at atmospheric pressure for different values of θ (Rat et al. 2002a)

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Ar+, θ =1.0 Ar+, θ =1.3

4

Ar+, θ =2.0 2 0 25

Ar++, θ =1.0

−2 −4 −6

Ar++, θ =1.3

Ar, θ =2.0

Ar++, θ =2.0

Ar, θ =1.3 Ar, θ =1.0 5

10

15

20

25

3

Electron temperature, Te (10 K)

4.1.2 Argon–Helium Argon–helium mixtures, both monoatomic gases, are widely used because on the one hand the mixture exhibits a higher viscosity over 10,000 K at equilibrium due to its higher ionization temperature and also because helium increases the plasma enthalpy and the thermal conductivity of the mixture compared to the case of pure argon. The study of the plasma composition shows that ionization is especially favored as θ increases Aubreton et al. (2004b). Viscosity. Figure 9 from Aubreton et al. (2004b) presents the viscosity of argon–helium mixture (25 mol% of argon) as a function of the heavy species temperature, for different values of the non-equilibrium parameter θ, at atmospheric pressure. It shows the significant change in the slope corresponding to ionization, i.e., the collision integrals of charged species are dominant which drastically reduces the viscosity because the collision integrals of charged species are three orders of magnitude higher than those of neutral species. The dependence of the viscosity on θ is much more pronounced. The part of the non-equilibrium curves, which overlap the equilibrium one, corresponds to the neutral–neutral interaction regime, i.e., before efficient ionization. Plasma composition shows that ionization is especially favored as θ increases (see Figs. 17 and 18 of ▶ Chap. 9, “Thermodynamic Properties of Non-equilibrium Plasmas”). The significant change in the slope corresponds to ionization, i.e., the collision integrals of charged species are dominant which drastically reduces the viscosity because the collision integrals of charged species are three orders of magnitude higher than those of neutral species. Thermal conductivity. Figure 10 shows the electron temperature dependence of the thermal conductivity of an argon–helium mixture (25 mol% of argon), for different values of the non-equilibrium parameter θ, at atmospheric pressure. The main contributions to the total thermal conductivity are the electron translation thermal conductivity as well as the reactional thermal conductivity, which depends on ionization reactions. That is why the total thermal conductivity is plotted as a

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30 Ar (25%) + He (75%)

Viscosity, μ (10−5 kg/m.s)

25

θ =1.0

20

θ =1.1

15

θ =1.3 θ =1.6

10 θ =2.0 θ =3.0

5 0

0

5

10

15

20

Heavy species temperature,Th

25

30

(103 K)

Fig. 9 Heavy species temperature dependence of the viscosity of an argon–helium mixture (25 mol% of argon), for different values of the non-equilibrium parameter θ, at atmospheric pressure (Aubreton et al. 2004b)

Thermal conductivity, K (W/m.k)

20 18 Ar (25%) + He (75%)

16 14

θ =3.0

12

θ =2.0

10

θ =1.6

8 6

θ =1.3 θ =1.1 θ =1.0

4 2 0

0

5

10

15

20

Electron temperature, Te

25

30

(103 K)

Fig. 10 Electron temperature dependence of the thermal conductivity of an argon–helium mixture (25 mol% of argon), for different values of the non-equilibrium parameter θ, at atmospheric pressure (Aubreton et al. 2004b)

function of Te. Below 10,000 K, it can be seen that, at fixed Te, the thermal conductivity, depending mainly on the heavy species temperature, decreases as θ increases. At low temperatures, the translational contribution of heavy species is dominant. However, as soon as ionization occurs, above 10,000 K, the first peak corresponds to the ionization of argon atoms and the second one to the ionization of

M. I. Boulos et al.

Electrical conductivity, σe(3) (103 S/m)

448 25 GS ES

20

θ =3.0 θ =2.0 θ =1.0

15

10 Ar (25%) + He (75%)

5

0

5

10

15

20

25

Electron temperature, Te

30

35

40

(103 K)

Fig. 11 Third-order electrical conductivity σel vs. electron temperature in ground-state GS and excited-state ES argon–helium (25–75 mol%) plasma mixture p = 1 atm for different values of the non-equilibrium parameter θ (Sharma et al. 2011a)

helium atoms and Ar+ ions. At high temperatures, the translational contribution of electrons seems to dominate the behavior of the thermal conductivity, especially as θ increases. However, it should be kept in mind that the translational contribution of electrons is θ times higher with respect to the heavy species temperature gradient than with respect to the electron temperature gradient. Electrical conductivity. Sharma et al. (2011a) calculated the transport properties of an Ar–He 25–75 mol% mixture using the Chapman–Enskog method and the simplified approach of Devoto (1967) and Bonnefoi (1983). The plasma composition was calculated using the modified Saha equation of van de Sanden et al. (1983). Figure 11 presents their results for the electrical conductivity of this mixture, calculations being performed at the three-order approximation of Sonine polynomials. Moreover, they made calculations by assuming that all atoms are in the ground state (GS) or in excited states (ES). For the electrical conductivity, σel, differences between ES and GS are negligible and for temperatures higher than 14,000 K, σel, increase with θ. Capriati et al. (1992) have studied non-equilibrium pure helium.

4.2

Diatomic Gases

4.2.1 Hydrogen Sharma et al. (2011b) calculated also the transport properties of hydrogen (H2). For the composition they used the Saha law of Potapov (Eq. 22 of ▶ Chap. 9, “Thermodynamic Properties of Non-equilibrium Plasmas”), Qint H being calculated at Te Qint and referred to as S1; the same with the ratio of partition functions QHintþ at a power H

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1/θ, Qint H being calculated at Th, referred to as S2; the equation of van de Sanden (Eq. 23 of ▶ Chap. 9, “Thermodynamic Properties of Non-equilibrium Plasmas”), Qint H being calculated at Te, referred to as S3; and finally the equation of van de int Sanden, Qint H being calculated at Th, referred to a S4. In all cases QHþ was assumed to + be 1 (degeneracy of H ), and in S1 to S3 cases, the exponential term with the ionization potential was calculated at Th, while in S4 it was calculated at Te. The partition function of H was calculated at the ground state (GS) or with excited states (ES). Results are illustrated in Fig. 12a–b for the electrical conductivity of H2 up to 40,000 K at atmospheric pressure, the third order of Sonine polynomials being used.

Electrical conductivity, σe(3) (103 S/m)

a 30 GS ES

25 20

θ =3.0 θ =2.0 θ =1.0

15 10 5 0

5

10

15

20

25

Electron temperature, Te Electrical conductivity, σe (3) (103 S/m)

b

30

35

40

35

40

(103 K)

30 GS ES

25

Ar (25%) + He (75%)

20 15

θ =1.0 θ =3.0

10 θ =2.0 5 0

5

10

15

20

25

30

Electron temperature, Te (103 K)

Fig. 12 Variation of H2 plasma third-order electrical conductivity σel vs. electron temperature using different Saha equations for ground-state GS and excited-state ES cases at p = 1 atm (a) S1 (b) S4; see text above for the meaning of S1 and S4 and for GS and ES (Sharma et al. 2011b)

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Figure 12a shows for S1 that σel increases with θ above Te = 15,000 K (for S2 it is the same and for S3 the increase occurs above Te = 12,000 K). Below these temperatures this trend is reversed. Figure 12b shows that S4 behaves very differently due to the presence of Th in the exponential factor. This illustrates that the plasma composition calculation is very important for the transport properties of gases at two-temperature. Results of Colombo et al. (2008) for argon are rather close to those of Rat et al. (2002a).

4.2.2 Nitrogen Results presented below, except diffusion coefficients, are those of Colombo et al. (2008). They used the decoupling method of Devoto (1967) instead of the non-simplified theory of Rat et al. (2001b) and assumed local chemical equilibrium. (a) Thermal conductivity. Different components of the total thermal conductivity, κ of nitrogen in LTE are presented in Fig. 13. When dissociation and ionization take place, the main contribution to κ is that of the reactive term, whereas at higher temperatures the translational electron contribution is the most important. Results of Colombo et al. (2008) are in good agreement with those of Murphy and Arundell (1994) as it can be seen in Fig. 13. In 2T Fig. 14 presents the translational thermal conductivities of nitrogen (those of electrons, κetr, and heavy species, κhtr) for different values of θ. Once the dissociation temperature has been reached, the non-equilibrium thermal conductivity of electrons ðθ > 1Þ swiftly approaches the equilibrium one. κetr presents a very low increment as θ increases for fixed electron temperature, because it depends mainly on electrons. Colombo et al. (2008) point out that results depend on formulations used for Debye length in the computation of Coulomb collision integrals.

Thermal conductivity, K (W/m.k)

9 8

Translational Reactive Internal Total

7 6 5

Murphy and Arundell (1994)

4 3 2 1 0 0

5

10

15

20

Temperature, T

25

30

35

40

(103 K)

Fig. 13 Components of the thermal conductivity of nitrogen in LTE (Colombo et al. 2008)

Transport Properties of Non-Equilibrium Plasmas

451 0.8

Electron translational thermal conductivity, K etr (W/m.k)

2 θ =1.0 θ =2.0 θ =3.0

0

0.7 0.6

8

0.5 0.4

6

0.3

4

0

0.2

Murphy and Arundell (1994), θ=1

2

0.1 5

0

10

15

20

30

25

35

Heavy particle translational thermal conductivity, (W/m.K)

10

0.0 45

40

Electron temperature, Te (103 K)

Fig. 14 Translational thermal conductivity of nitrogen in NLTE (Colombo et al. 2008)

Viscosity, μ (10−5 kg/m.s)

30 Murphy and Arundell (1994), θ=1

25 20

θ =1.0 θ =2.0 θ =3.0

15 10 5 0

0

5

10

15

20

25

30

35

40

45

Electron temperature, Te (103 K)

Fig. 15 Viscosity of nitrogen in NLTE (Colombo et al. 2008)

(b) Viscosity. As already pointed out, viscosity is primarily a property of the heavy particles, for a given Te. If θ increases, Th decreases, resulting in a drop in the momentum transport and hence in a drop in the viscosity value. As shown in Fig. 15, when increasing θ the peak values for viscosity shift toward higher Te values, and its maximum value diminishes. (c) Electrical conductivity. Electrical conductivity depends only on collision integrals for interactions involving electrons: with higher values of θ, the electrical conductivity starts rising fast once ionization temperature has been reached and then quickly stabilizes near the values observed for lower values of

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Electrical conductivity, σe (103 S/m)

20 18 θ =1.0 θ =2.0 θ =3.0

16 14 12 10 8

Murphy and Arundell (1994), θ=1

6 4 2 0

0

5

10

15

20

25

Electron temperature, Te

30

35

40

45

(103 K)

Fig. 16 Electrical conductivity of nitrogen in LTE and NLTE (Colombo et al. 2008)

θ. For nitrogen, Fig. 16, the electrical conductivities are lower than the respective LTE conductivity, for θ > 1 and Te < 15, 000K , while for electron temperatures higher than 15,000 K, conductivities increase as θ increases. For argon, as seen earlier in Fig. 4, the electrical conductivity decreases weakly around 27,000 K because multi-charged ions involve an increase of collision integrals. (d) Diffusion coefficients. Meher et al. (2014) have calculated multicomponent diffusion coefficients for 2T nitrogen plasma following the first perturbation technique of Chapman and Enskog. Binary, thermal, thermal ambipolar, and general ambipolar diffusion coefficients are presented for electron temperature ranging from 300 to 50,000 K and θ from 1 to 5 at atmospheric pressure. For example, Fig. 17 presents results of calculations of the thermal diffusion coefficient of electrons together with those obtained by Wang et al. (2011), which are in good agreement. Non-equilibrium N2 plasma has also been calculated by Sourd et al. (2007a, b).

4.2.3 Oxygen According to calculations of Colombo et al. (2008), the electrical conductivity evolution is rather close to that of nitrogen with less dispersion between the different values of θ below 15,000 K. This is probably due to the fact that the dissociation of O2 occurs around 3,500 K against 7,000 K for N2. Roughly the evolution of O2 viscosity with θ is similar to that of N2, with slightly higher maxima values for O2. The evolution of thermal conductivity is similar for O2 and N2, except that the reactional conductivity of N2 occurs at about 7,000 K against

Transport Properties of Non-Equilibrium Plasmas

Thermal diffusion coefficient (10−8 kg/m.s)

10

453

35 θ =1 θ =5 θ =10 θ =15

30 25 20 15 10 5

θ =1.0

5

10

15

0 0

10

20

30

40

50

Electron temperature, Te (103 K)

Fig. 17 Thermal diffusion coefficient of electrons for a nitrogen plasma at atmospheric pressure and different values of θ (Meher et al. 2014), results being compared with those of Wang et al. (2011)

3,500 K for O2 also with a peak twice smaller for O2 compared to that of N2. Nonequilibrium calculations were also performed by Ghorui et al. (2007) for N2 and the same authors Ghorui et al. (2008) for N2 and O2.

4.2.4 Carbon Dioxide Colombo et al. (2011) calculated transport properties of CO2 and different mixtures of CO2. As indicated previously for N2 calculations, they used the Chapman–Enskog method up to the third order. Properties for 2T plasmas were obtained using both the two-temperature theory developed by Rat et al. (2001b) and the simplified theory by Devoto (1967) neglecting the coupling between electrons and heavy particles. They have shown that for carbon–oxygen mixtures results relative to thermal and electrical conductivities are very close with these two theories. However, some discrepancies were found for ordinary diffusion coefficients of the type electron–heavy particle: (a) Viscosity. Figure 18 presents the viscosity of CO2 plasma at atmospheric pressure for different values of θ. As seen before, with growing values of the non-equilibrium parameter θ, the viscosity peaks around Te  10,000 K decreases due to ionization increase for higher electron temperature. (b) Thermal conductivity. Figure 19 presents the thermal conductivity, κ, for different values of θ. At LTE the first two peaks correspond to dissociation and ionization reactions depending on Th. When θ increases, peaks shift toward higher electron temperatures because reactions depend on heavy-particle

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Viscosity, μ (10−5 kg/m.s)

35

θ =1 θ =2 Rat (2008) θ =3 Rat (2008) θ =2 Devoto (1967) θ =3 Devoto (1967)

30 25 20 15 10 5 0

0

5

10

15

20

25

30

Electron temperature, Te (103 K)

Fig. 18 Electron temperature dependence of the viscosity for a CO2 mixture for different values of the non-equilibrium parameter θ at atmospheric pressure, calculations performed according to Rat et al. (2008) and Devoto (1997) (Colombo et al. 2011)

Total thermal conductivity, K (W/m.k)

18 θ =1 θ =2 Rat (2008) θ =3 Rat (2008) θ =2 Devoto (1967) θ =3 Devoto (1967)

16 14 12 10 8 6 4 2 0

0

5

10

15

Electron temperature, Te

20

25

30

(103 K)

Fig. 19 Electron temperature dependence of the thermal conductivity for a CO2 mixture for different values of the non-equilibrium parameter θ at atmospheric pressure, calculations performed according to Rat et al. (2008) and Devoto (1997) (Colombo et al. 2011)

temperature. Three peaks can be seen at θ = 1. For higher values of θ, the dissociation peaks reach the ionization peak. For temperatures higher than 15,000 K, the translational contribution is the most important. (c) Electrical conductivity. Figure 20 presents the electrical conductivity, σel, for different values of θ. At low temperature, when increasing θ, σel decreases

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Electrical conductivity, σe (103 S/m)

20 18

θ =1 θ =2 Rat (2008) θ =3 Rat (2008) θ =2 Devoto (1967) θ =3 Devoto (1967)

16 14 12 10 8 6 4 2 0

0

5

10

15

20

25

30

Electron temperature, Te (103 K)

Fig. 20 Electron temperature dependence of the electrical conductivity σel for a CO2 plasma at atmospheric pressure for different values of θ, calculations performed according to Rat et al. (2008) and Devoto (1997) (Colombo et al. 2011)

because ionization shifts toward higher electron temperatures. For electron temperatures higher than 17,000 K, σel increases with growing values of θ.

4.3

Complex gases and mixtures in NLTE and NLCE

4.3.1 Ar–H2 Mixtures Under Non-equilibrium Conditions Rat et al. (2002b), taking into account the coupling between electrons and heavy species in thermal plasmas and considering the nonlocal chemical equilibrium (NLCE), have calculated the composition (see in ▶ Chap. 9, “Thermodynamic Properties of Non-equilibrium Plasmas,” Fig. 30 for NLCE and Fig. 29 at LTE for comparison) and transport properties of argon–hydrogen mixtures at atmospheric pressure. When comparing, for θ = 1.6, the different plasma compositions obtained either through an equilibrium constant (LTE Fig. 29 of ▶ Chap. 9, “Thermodynamic Properties of Non-equilibrium Plasmas”) or a stationary kinetic method (NLCE Fig. 30 of ▶ Chap. 9, “Thermodynamic Properties of Non-equilibrium Plasmas”) discontinuity at Te = 11,000 K and an ionization delay are observed in stationary kinetic calculations, compared to the equilibrium constant method. Of course this important difference will modify transport properties as illustrated below. (a) Electrical conductivity. The evolution of the electrical conductivity σel of Ar–H2 (50 mol%) mixture with the electron kinetic temperature is shown in Fig. 21 using, for θ = 1.6 and θ = 2.0, the composition calculated either by the stationary kinetic calculation or by the equilibrium constant method. The

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Electrical conductivity, σe (S/m)

105 104 103 Kp (θ =2.0)

102

Kp (θ =1.6) Kinetic (θ =1.6)

10

Kinetic (θ =2.0) 1.0

0

5

10

15

Electron temperature, Te

20

25

(103 K)

Fig. 21 Evolution at atmospheric pressure of the electrical conductivity σel of an Ar–H2 (50 mol%) mixture with the electron kinetic temperature using compositions calculated by the stationary kinetic calculation (kinetic) and the equilibrium constant method (Kp) for θ = 1.6 and θ = 2.0 (Rat et al. 2002b)

evolution of σel follows closely the evolution (Rat et al. 2002b) of the electron number density. The discontinuity found in composition curves obtained with kinetic method occurs in σel at about 11,000 K. A shift to the higher electron temperature of σel calculated with the kinetic method is observed when θ increases. At low temperature, which is before the discontinuity, plasma composition is governed by reactions between heavy species. (b) Viscosity. The evolution with Te of the viscosity of Ar–H2 (50 mol%) mixture, using a composition calculated by the stationary kinetic calculation and the equilibrium constant method, is presented in Fig. 22. for θ = 1.6 and θ = 2.0. Large discrepancies between both calculations are observed between 8,000 and 14,000 K. Moreover, a discontinuity, at around 11,000 K, occurs with the kinetic viscosity. In this case the maximum is shifted to a higher electron temperature with respect to the viscosity calculated at NLTE. This is due to the fact that ionization delay occurs in the kinetic viscosity, which consequently increases with temperature until the discontinuity is reached (Rat et al. 2002b). (c) Thermal conductivity. In non-equilibrium thermal plasmas, the total thermal conductivity cannot be written as at equilibrium (Rat et al. 2002b). Indeed, the translational and internal contributions have been defined with respect to the heavy species temperature gradient, whereas the reactional contribution has been defined with respect to the electron temperature gradient. As a result, considering θ as a given parameter in calculations, the total thermal conductivity is arbitrarily defined with respect to the heavy species temperature and is written as κtot ¼ κe þ κh þ κint þ θ  κr . Figure 23 presents, for θ = 1.6, the electron temperature dependence of thermal conductivities, namely, κtot, κe, κh, κint, and κr, calculated by the steady-state kinetic method. κint is too small to

10

Transport Properties of Non-Equilibrium Plasmas

457

20

Viscosity, μ (10−5 kg/m.s)

18 16

Kinetic (θ =2.0)

14 Kinetic (θ =1.6)

12 10

Kp (θ =1.6)

8 6

Kp (θ =2.0)

4 2 0

0

5

10

15

20

25

Electron temperature, Te (103 K)

Fig. 22 Evolution at atmospheric pressure with the electron temperature of viscosity of an Ar–H2 (50 mol%) mixture using compositions calculated by the steady-state kinetic calculation (kinetic) and the equilibrium constant method (Kp) for θ = 1.6 and θ = 2.0 (Rat et al. 2002b)

Thermal conductivity, K (W/m.K)

10 9

Ar(50%) +H2 (50%) θ =1.6

8

Ktot

7 6

Kr

5

Ke

4 3 2 Kh

1 0

0

5

10

15

Electron temperature, Te

20

25

(103 K)

Fig. 23 Evolution at atmospheric pressure with the electron temperature of thermal conductivities of an Ar–H2 (50 mol%) mixture using compositions calculated by the steady-state kinetic method where κtot, κe, κh, κint, and κr are, respectively, the total, electron translational, heavy species translational, internal, and reactional conductivities for θ = 1.6 (Rat et al. 2002b)

appear in the figure. Rat et al. (2002b) give the following explanations for the behavior of the different terms of κtot: • κh is dominant at low-electron temperature (Te < 3,500 K) and up to Te  10,000 K where dissociation is completed and ionization has not begun.

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Thermal conductivity, K (W/m.K)

14 12

Ar(50%) +H2 (50%)

Kinetic (θ =2.0)

10

Kp (θ =2.0)

Kinetic (θ =1.6)

8 6 4

Kp (θ =1.6)

2 0

0

5

10

15

Electron temperature, Te

20

25

(103 K)

Fig. 24 Evolution at atmospheric pressure with the electron temperature of the total thermal conductivity of an Ar–H2 (50 mol%) mixture using compositions calculated by the steady-state kinetic calculation (NLCE) and the equilibrium constant method (NLTE) for θ = 1.6 and θ = 2.0 (Rat et al. 2002b)

• κe increases strongly above the discontinuity temperature (11,000 K) for which the ionization avalanche occurs and becomes dominant in κtot for Te >18,000 K; • The contribution of κr to κtot reaches its maximum value around 6000 K (Th = 3750 K). At higher temperatures a second peak is observed around 13,000 K which is due secondary ionization giving rise to the important increase of the number density of the Ar+ ion between 11,000 and 13,000 K. Figure 24 presents the comparison of 2T and NLCE calculations on the one hand and those performed at 2T and LCE on the other for κ for θ = 1.0, 1.6, and 2.0. For θ = 1.6, a good agreement is observed between the dissociation peaks for both the NLCE and 2T on the one hand and those of 2T and LCE on the other. However, for θ = 2.0, the maximum value of the 2T and NLCE dissociation peak is higher than that of the 2T and LCE method. Furthermore, for a temperature range of 8,000–11,000 K, the NLCE method leads to an ionization delay of argon atoms, κtot(NLCE) < κtot(LCE), but at temperatures above the avalanche phenomena, nAr+ increases faster, giving κtot(NLCE) > κtot(LCE). The results of Colombo et al. (2009) who calculated the NLTE transport properties of Ar–H2 (50 mol%) are quite similar to those of Rat et al. (2002b) at NLTE.

4.3.2 N2–H2 Under Non-equilibrium Conditions For details about the method used by Colombo et al. (2009), see Sect. 3.2. The plasma composition has been presented in Fig. 14 of ▶ Chap. 9, “Thermodynamic Properties of Non-equilibrium Plasmas” for θ ¼ 1 and 2.

Transport Properties of Non-Equilibrium Plasmas

Electrical conductivity, σe (103 S/m)

10

459

18 16

N2(50%) +H2 (50%)

14 12

θ =1

10

θ =2

8 6 4

θ =3

2 0

0

5

10

15

20

25

30

35

40

Temperature, T (103 K)

Fig. 25 Electron temperature dependence of the electrical conductivity at atmospheric pressure for 50 mol% nitrogen and 50 mol% hydrogen for different values of the non-equilibrium parameter (Colombo et al. 2009)

(a) Electrical conductivity. Electrical conductivities, for different values of θ, for 50 mol% nitrogen and 50 mol% hydrogen, are presented in Fig. 25. At low temperatures, for increasing values of θ, conductivities decrease because dissociation shifts toward higher electron temperatures. For electron temperatures higher than 15,000 K, conductivity increases with growing values of θ: even though in Fig. 15 of ▶ Chap. 9, “Thermodynamic Properties of Non-equilibrium Plasmas,” electron mole fraction decreases as θ increases, electron number density increases with growing values of non-equilibrium parameter justifying the increase in σel. (b) Viscosity. Viscosities, for different values of θ, are presented in Fig. 26. With growing values of θ, the peak around Te  10,000 K diminishes and ionization shifts toward higher Te. This effect is more pronounced than with argon–hydrogen since nitrogen dissociation occurs at higher heavy-particle temperature (Th  7,000 K) than hydrogen dissociation (Th  3,700 K), resulting in a proportionally greater shift toward higher Te as θ increases. (c) Thermal conductivity. Figure 27 presents the thermal conductivity for a 50 mol% nitrogen and 50 mol% hydrogen mixture for various values of θ; three peaks are clearly distinguished for θ = 1, whereas for higher θ, the dissociation peaks reach the ionization peak. In the temperature range where dissociation and ionization take place, a large contribution to total thermal conductivity is due to the reactive term, whereas for temperatures higher than 15,000 K, the translational contribution is the most important.

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Viscosity, μ (10−5 kg/m.s)

25 N2(50%) +H2 (50%)

20 θ=1

15

θ=2 10 θ=3 5

0

0

5

10

15

20

25

30

35

40

Temperature, T (103 K)

Fig. 26 Electron temperature dependence of the viscosity at atmospheric pressure for 50 mol% nitrogen and 50 mol% hydrogen mixture for different values of the non-equilibrium parameter (Colombo et al. 2009)

Thermal conductivity, K (W/m.K)

25 N2(50%) +H2 (50%) 20 θ=3

15

θ=2

10

θ=1 5

0

0

5

10

15

20

25

Electron temperature, Te

30

35

40

(103 K)

Fig. 27 Electron temperature dependence of the thermal conductivity at atmospheric pressure for 50 mol% nitrogen and 50 mol% hydrogen mixture for different values of the non-equilibrium parameter (Colombo et al. 2009)

4.3.3 Ar–H2–He (30–10–60 mol%) Under Non-equilibrium Conditions Aubreton et al. (2004a) have calculated the composition of the mixture Ar–H2–He (30–10–60 mol%) at atmospheric pressure both at 2T (method of van de Sanden et al. (1989)) and at NLCE using the kinetic method. The corresponding

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461

compositions are presented in ▶ Chap. 9, “Thermodynamic Properties of Non-equilibrium Plasmas,” in Fig. 17 for LTE and in Fig. 18 for NLCE where a strong discontinuity is observed at Te = 9,800 K. Only thermal conductivity will be considered among the different transport properties. Figures 28 and 29 show the electron temperature dependence of the thermal conductivity of the argon–hydrogen–helium (30–10–60 mol%) mixture for different values of θ at atmospheric pressure, assuming LCE. Figure 28 presents results up to 30,000 K, while Fig. 29 is limited to 10,000 K in order to better detail results below this temperature. The reaction thermal conductivity depends on ∇Te and ∇Th, but as ∇Te = θ.∇Th (θ is fixed in calculations), the total thermal conductivity has been arbitrarily defined with respect to the temperature gradient of heavy species. The main contributions to the total thermal conductivity are the electron translation thermal conductivity for temperatures over 10,000 K as well as the reactional thermal conductivity, which mostly depends on ionization reactions. As a result, the total thermal conductivity is plotted as a function of Te. (a) At high temperatures, κe seems to dominate the behavior of κtot, especially as θ increases, and it reduces the effect of κr and screens it especially when the ionization of helium takes place i.e., in the temperature range 20,000–25,000 K. (b) For Te < 15,000 K, κr seems to dominate κtot; the ionization peak of κr reaches its maximum value around 14,000 K, which can be attributed to the ionization of Ar and H. (c) At low temperatures, κh is dominant. The dissociation of H2 is shifted to higher values of Te (corresponding in fact roughly to Th = 3,400 K). At LTE, the

Thermal conductivity, K (W/m.K)

20 18

Ar(30%) +H2 (10%) + He(60%)

16

θ =3.0 θ =2.0

14

θ =1.6

12 10 8 6 4

θ =1.1

2 0

θ =1.3

θ =1.0 0

5

10

15

Electron temperature, Te

20

25

30

(103 K)

Fig. 28 Evolution at atmospheric pressure with the electron temperature of the thermal conductivity of Ar–H2–He (30–10–60 mol%) mixture for different values of the non-equilibrium parameter θ: 300 K < Te < 30,000 K (Aubreton et al. 2004a)

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Thermal conductivity, K (W/m.K)

7 Ar(30%) +H2 (10%) + He(60%)

6 5

θ =3.0

θ =1.3

4

θ =1.6

θ =1.1 θ =1.0

3

θ =2.0

2 1 0

0

2

4

6

8

10

Electron temperature, Te (103 K)

Fig. 29 Evolution at atmospheric pressure with the electron temperature of the thermal conductivity of Ar–H2–He (30–10–60 mol%) mixture for different values of the non-equilibrium parameter θ: 300 K < Te < 10,000 K (Aubreton et al. 2004a)

dissociation peak of H2 reaches its maximum value around 3,400 K, and for non-equilibrium plasma, for example, θ = 2, the maximum value of this peak is attained at Te = θ.Th = 7,000 K. Moreover, for θ = 3, this hypothesis is no longer valid because the dissociation is due mainly to the following reaction þ routes: e þ H2 ! Hþ 2 þ 2e and H2 þ e ! 2H. When an atomic ion appears, an ion–molecule charge transfer must be added to the previous process, which explains the maximum value of the dissociation peak around 7,800 K for θ = 3 instead of 10,500 K. To conclude, the thermal conductivity is mainly controlled by the percentage of H2 up to 5,000 K and by the percentages of both H2 and He at temperatures above this level. Figure 30 presents the NLCE case with the evolution with Te of the total thermal conductivity of the same Ar–H2–He (30–10–60 mol%) mixture at atmospheric pressure for different values of θ. For more details Fig. 31 presents the same evolution but only up to 14,000 K. As with the composition derived from the Saha modified law (2T but LCE), it can be pointed out that when θ increases, the dissociation of H2 is shifted to higher values of Te; the maximum of the dissociation peak is almost invariant with the heavy species temperature for θ = 1, 1.1, 1.3, 1.6, 2, and 3. Above 11,000 K the maximum of the ionization peaks strongly depends on θ. Furthermore, for a temperature range of 8,000–11,000 K, the stationary kinetic method leads to an ionization delay of

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463

Thermal conductivity (W/m.K)

20 18

Ar(30%) +H2 (10%) + He(60%)

16 14

θ =3.0

12

θ =2.0 θ =1.6

10 8 6 4

θ =1.3 θ =1.0 θ =1.1

2 0

0

5

10

15

20

25

30

Electron temperature, Te (103 K)

Fig. 30 Evolution at atmospheric pressure with the electron temperature of the thermal conductivity of an Ar–H2–He (30–10–60 mol%) mixture for different values of the non-equilibrium parameter θ: 300 K < Te < 30,000 K (Aubreton et al. 2004a)

Thermal conductivity, K (W/m.K)

14 Ar(30%) +H2 (10%) + He(60%)

12

θ =3.0

10 8

θ =2.0

6 θ =1.1 θ =1.3 θ =1.6

4 θ =1.0

2 0

0

2

4

6

8

10

12

14

Electron temperature, Te (103 K)

Fig. 31 Evolution at atmospheric pressure with the electron temperature of the thermal conductivity of an Ar–H2–He (30–10–60 mol%) mixture for different values of the non-equilibrium parameter θ: 300 K < Te < 14,000 K (Aubreton et al. 2004a)

argon atoms, κtot (2T and NLCE) < κtot (2T and LCE) except for θ = 3, but at temperatures above the avalanche phenomena, nArþ increases faster, giving κtot (2T and NLCE) > κtot (2T and LCE).

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Nomenclature

Units are indicated in parentheses; whenever no units are given, the parameter is dimensionless.

5.1 !

Latin Alphabet Vectorial coefficients for calculating the perturbation function Φi Impact parameter defined as the perpendicular distance between the path of a projectile and the center of the field created by the object that the projectile is approaching (m) Number of atoms in a molecule of species i Second-order tensor allowing calculation of the perturbation function Φi Center of mass velocity for two particles of type i and j (m/s) Vectorial coefficients for calculating the perturbation function Φi Diffusion driving force of species i

Aι b

bi

$

Bi ! vc !j ci ! di

dA ! dr ! dv

Elementary surface (m2) Elementary volume in ordinary space (m3)   ! Elementary volume in velocity space: d v ¼ dvx ; : dvy ; : dvz ðm3 =s3 Þ

Da Di Dij

Ambipolar diffusion coefficient (m2/s) Scalar coefficients for calculating the perturbation function Φi Ordinary diffusion coefficient (m2/s)

DTi

Thermal diffusion coefficient (kg/m.s)

Dθij

Diffusion coefficient (m2/s)

DX AB

Mean value of the diffusion coefficient between species A and B (m2/s)

DX BA

Mean value of the diffusion coefficient between species B and A (m2/s)

DTAB

Mean value of the thermal diffusion coefficient between species A and B (kg/m.s)

DTBA

Mean value of the thermal diffusion coefficient between species B and A (kg/m.s)



Dθi e !

Non-equilibrium diffusion coefficient Elementary charge (C) Electric field (V/m)

!

Externally applied field (V/m) Distribution function of species i after collisions of the i species Distribution function of species i Maxwellian distribution function External force (N)

E E fi’ fi f0i Fi Fx

Component of the force in the x-direction (N)

10

Transport Properties of Non-Equilibrium Plasmas

g h Hi I0i !

j

! JE ! Jn ! J px ! Jx

k Ki(Wi, θij) ‘ ‘z mA mB mi Mi n ni p $ p ! q ! qi ! qR ! qz qlsij ls Qij !

r r rm R $ S T Te Ti Th

!

T* $

U

465

Relative velocity of the species i and j (m/s) Planck’s constant (h = 6.626 1034 J.s) Molar enthalpy of species i (kJ/mol) Zero-order approximation of Chapman–Enskog expansion Current density vector Energy flux (W/m2) Particle flux (part/m2.s) Momentum flux in x-direction Flux of the quantity x Boltzmann constant (k = 1.38 1023 J/K) Term taking into account the thermal non-equilibrium when electrons and heavy species collide Mean free path (m) Mean free path in z-direction (m) Number–density–weighted average mass of the species present in gas A Number–density–weighted average mass of the species present in gas B Mass of the particle of chemical species i (kg) Atomic mass of chemical species i (kg) Number density (m3) Number density of chemical species i (m3) Pressure (Pa) Pressure tensor (Pa) Heat flux (J/m2.s) Flux vector associated with the transport of kinetic energy of particles of species i (J/m2.s) Reactional heat flux vector (J/m2.s) Heat flux vector in z-direction (J/m2.s) Bracket integral Product of the reduced collision integral by the cross-sectional area presented by particles considered as hard spheres (m2) Position vector ! Relative position vector ð r  ¼ r1  r2 Þ (m) Distance of closest approach (m) Perfect gas constant (R = 8.32 J/K.mole) Stress tensor Absolute temperature (K) Electron temperature (K) Absolute temperature of species i (K) Heavy species temperature (K) h  i1 mj mi 1 Effective collision temperature T ¼ mi þm þ Ti Tj j Unity tensor

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! vi

! !  v 0 ¼ U i of particles of species i (m/s)

Ui

Peculiar velocity

!

Peculiar velocity in the x-direction

U ix vi vi0 ! vi v ! v0 V(r) !

V ij !

Velocity of particle i before collision (m/s) Velocity of particle i after collision (m/s) Velocity vector of particle of species i (m/s) Mean velocity of particles (m/s) Mass-average velocity (m/s) Interaction potential !  ! ! Relative velocity before collision V ij ¼ v i  v j (m/s) !

!

!

!

Relative velocity after collision V 0ij ¼ v 0i  v 0j (m/s)  1=2 ! ! mi Reduced velocity before collision Wi ¼ 2kT Vi i

!

Reduced velocity after collision

V 0ij Wi W0i xi Zi

5.2 ΔEI ε φi or Φi Φ γ2 κ κe κh κr κint κi0 μ μe μj μij θ θij σel σei σii

Molar fraction of species i Charge number of the ith species

Greek Alphabet Ionization potential (eV) Incidence azimuthal angle First-order perturbation of the Maxwellian distribution of the Boltzmann equation First-order perturbation function of the ith species with the temperature, Ti   Reduced energy γ2 ¼ μij =2kT g2

Thermal conductivity (W/m.K) Translational thermal conductivity of electrons (W/m.K) Translational thermal conductivity of heavy species (W/m.K) Reactional thermal conductivity (W/m.K) Internal thermal conductivity (W/m.K) translational thermal conductivity of species i (W/m.K) Molecular viscosity (kg/m.s) Electron mobility (m2/V.s) Viscosity coefficient of the ith species (kg/m.s) mi mj Reduced mass μij ¼ mi þm (kg) j Ratio of electron to heavy-particle temperatures  ðθ ¼ Te =T hÞ Ratio of particle i to particle j temperatures θij ¼ Te =Th Electrical conductivity (S/m) Electrical conductivity (S/m) Differential collision cross section

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Transport Properties of Non-Equilibrium Plasmas

σj,A νj,A ξ χ Ω Ω(ls) ij

467

elastic collision cross-section (m2) elastic collision frequency (s1) order of approximation (m2) Angle of deflection Solid angle Collision integral

References Al-Mamun SA, Tanaka Y, Uesugi Y (2010) Two-temperature two-dimensional non chemical equilibrium modeling of Ar–CO2–H2 induction thermal plasmas at atmospheric pressure. Plasma Chem Plasma Process 30:141–172 Aubreton J, Bonnefoi C, Mexmain J.M. (1986) Calculation of thermodynamic and transport properties in Ar-O2 plasma in non-local thermodynamic equilibrium at atmospheric pressure (in French). Rev Phys Appl 21(6):365–376 Aubreton J, Elchinger MF, Fauchais P, Rat V, André P (2004a) Thermodynamic and transport properties of a ternary Ar–H2–He mixture out of equilibrium up to 30 000K at atmospheric pressure. J Phys D Appl Phys 37:2232–2246 Aubreton J, Elchinger MF, Rat V, Fauchais P (2004b) Two-temperature transport coefficients in argon–helium thermal plasmas. J Phys D Appl Phys 37:34–41 Baeva M, Kozakov R, Gorchakov S, Uhrlandt D (2012) Two-temperature chemically non-equilibrium modeling of transferred arcs. Plasma Sources Sci Technol 21:055027 (13 pp) Bonnefoi C (1983) Contribution to the study of solving methods of Boltzmann equation in a twotemperature plasma: example Ar-H2 mixture. State thesis, University of Limoges, France (in French) Boselli M, Colombo V, Ghedini E, Gherardi M, Sanibondi P (2013) Two-temperature modelling and optical emission spectroscopy of a constant current plasma arc welding process. J Phys D Appl Phys 46:224009 (11 pp) Bruno D, Catalfamo C, Laricchiuta A, Giordano D, Capitelli M (2006) Convergence of ChapmanEnskog calculation of transport coefficients of magnetized argon plasma. Phys Plasmas 13:072307 Burm K.T.A.L., Goedheer W.J., Schram D.C. (2002) An alternative quantitative analysis of nonequilibrium transport coefficients in argon plasmas. Plasma Chem Plasma Process 22(3):413–435 Caesar T, Bollmann H, Steinmetz E, Wilhelmi H (2000) Efficient and accurate methods for the computation of thermodynamic and transport properties of multitemperature thermal plasma part I: thermodynamic and heavy particle transport properties. Plasma Chem Plasma Process 20(1):13–22 Capitelli M, Celiberto R, Gorse C, Laricchiuta A, Minelli P, Pagano D (2002) Electronically excited states and transport properties of thermal plasmas: the reactive thermal conductivity. Phys Rev E 66:016403 Capitelli M, Celiberto R, Gorse C, Laricchiuta A, Pagano D, Traversa P (2004) Transport properties of local thermodynamic equilibrium hydrogen plasmas including electronically excited states. Phys Rev E 69:026412 Capitelli M, Armenise I, Bisceglie E, Bruno D, Celiberto R, Colonna G, D’Ammando G, De Pascale O, Esposito F, Gorse C, Laporta V, Laricchiut A (2012) Thermodynamics transport and kinetics of equilibrium and non-equilibrium plasmas: a state-to-state approach. Plasma Chem Plasma Process 32:427–450 Capriati G, Colonna G, Gorse C, Capitelli M (1992) A parametric study of electron energy distribution functions and rate and transport coefficients in non-equilibrium helium plasmas. Plasma Chem Plasma Process 12(3):237–260

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Chapman S, Cowling TG (1970) The mathematical theory of non-uniform gases, 3rd edn. Cambridge University Press, Cambridge, UK Chen X, Han P (1999) On the thermodynamic derivation of the Saha equation modified to a twotemperature plasma. J Phys D Appl Phys 32:1711–1718 Chen X, Li HP (2003) The reactive thermal conductivity for a two-temperature plasma. Heat Mass Transf 46:1443–1454 Colombo V, Ghedini E, Sanibondi P (2008) Thermodynamic and transport properties in nonequilibrium argon, oxygen and nitrogen thermal plasmas. Prog Nucl Energy 50:921–933 Colombo V, Ghedini E, Sanibondi P (2009) Two-temperature thermodynamic and transport properties of argon–hydrogen and nitrogen–hydrogen plasmas. J Phys D Appl Phys 42:055213 (12 pp) Colombo V, Ghedini E, Sanibondi P (2011) Two-temperature thermodynamic and transport properties of carbon–oxygen plasmas. Plasma Sources Sci Technol 20:035003 Devoto RS (1965) The transport properties of a partially ionized monoatomic gas. PhD thesis, Stanford University, USA Devoto RS (1967) Simplified expressions for the transport properties of ionized monoatomic gases. Phys Fluids 10(10):2105–2112 Freton P, Gonzalez JJ, Ranarijaona Z, Mougenot J (2012) Energy equation formulations for twotemperature modeling of ‘thermal’ plasmas. J Phys D Appl Phys 45:465206 Ghorui S, Heberlein JVR, Pfender E (2007) Thermodynamic and transport properties of twotemperature oxygen plasmas. Plasma Chem Plasma Process 27:267–291 Ghorui S, Heberlein JVR, Pfender E (2008) Thermodynamic and transport properties of twotemperature nitrogen-oxygen. Plasma Chem Plasma Process 28:553–582 Hirschfelder JO, Curtiss CF, Bird RB (1964) Molecular theory of gases and liquids, 2nd edn. Wiley, New York Hsu KC (1982) PhD thesis, University of Minnesota Meher KC, Tiwari N, Ghorui S, Das AK (2014) Multi-component diffusion coefficients in nitrogen plasma under thermal equilibrium and non-equilibrium conditions. Plasma Chem Plasma Process 34:949–974 Murphy AB (1993) Erratum: diffusion in equilibrium mixtures of ionized gases. Phys Rev E 50:5145. Phys Rev E 48:3594 Murphy AB (1997) Demixing in free burning arcs. Phys Rev E 55:7473–7494 Murphy AB (2000) Transport coefficients of hydrogen and argon–hydrogen plasmas. Plasma Chem Plasma Process 20:279–297 Murphy AB, Arundell CJ (1994) Transport coefficients of argon, nitrogen, oxygen, argon-nitrogen, and argon-oxygen plasmas. Plasma Chem Plasma Process 14(4):451–490 Paik S, Pfender E (1990) Argon plasma transport properties at reduced pressures. Plasma Chem Plasma Process 10(2):291–304 Potapov AV (1966) Chemical equilibrium of multitemperature systems. High Temp 4:48–51 Rat V (2001) Contribution to the calculation of transport properties in non-equilibrium thermal plasmas taking into account a coupling between electrons and heavy species: applications to argon and argon–hydrogen mixture. PhD thesis, University of Limoges, France (in French) Rat V, André P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2001a) A modified pseudoequilibrium model competing with kinetic models to determine the composition of a twotemperature SF6 atmosphere plasma. J Phys D Appl Phys 34:2191–2204 Rat V, André P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2001b) Transport properties in a two-temperature plasma: theory and application. Phys Rev E 64:26409–26428 Rat V, André P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2002a) Transport coefficients including diffusion in a two-temperature argon plasma. J Phys D Appl Phys 35:981–991 Rat V, André P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2002b) Two-temperature transport coefficients in argon–hydrogen plasmas – II: inelastic processes and influence of composition. Plasma Chem Plasma Process 22(4):475–493

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Rat V, Murphy AB, Aubreton J, Elchinger MF, Fauchais P (2008) Treatment of non-equilibrium phenomena in thermal plasma flows, topical review. J Phys D Appl Phys 41:183001 (28 pp) Sharma R, Singh G, Singh K (2011a) Higher-order contributions to transport coefficients in twotemperature hydrogen thermal plasma. Phys Plasmas 18:063504 Sharma R, Singh G, Singh K (2011b) Theoretical investigation of thermo-physical properties in two-temperature argon-helium thermal plasma. Phys Plasmas 18:083510 Sourd B, André P, Aubreton J, Elchinger M-F (2007a) Influence of the excited states of atomic nitrogen N(2D) and N(2P) on the transport properties of nitrogen, part I: atomic nitrogen properties. Plasma Chem Plasma Process 27:35–50 Sourd B, André P, Aubreton J, Elchinger M-F (2007b) Influence of the excited states of atomic nitrogen N(2D), N(2P) and N(R) on the transport properties of nitrogen part II: nitrogen plasma properties. Plasma Chem Plasma Process 27:225–240 Spitzer L (1956) Physics of fully ionized gases. Inter Science, New York Trelles JP, Heberlein JVR, Pfender E (2007) Non-equilibrium modeling of arc plasma torches. J Phys D Appl Phys 40:5937–5952 Trelles JP, Chazelas C, Vardelle A, Heberlein JVR (2009) Arc plasma torch modeling. J Therm Spray Technol 18(5–6):728–752 van de Sanden MCM, Schram PPJM, Peeters AG, van der Mullen JAM, Kroesen GMW (1989) Thermodynamic generalization of the Saha equation for a two-temperature plasma. Phys Rev A 40:5273–5276 Wang W, Rong M, Yan JD, Murphy AB, Spencer JW (2011) Thermophysical properties of nitrogen plasmas under thermal equilibrium and non-equilibrium conditions. Phys Plasmas 18(113502):1–18 Zhang X-N, Li H-P, Murphy AB, Xia W-D (2013) A numerical model of non-equilibrium thermal plasmas. I. Transport properties. Phys Plasmas 20:033508

Part II Generation of Thermal Plasmas

Basic Concepts of Plasma Generation

11

Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Electrical Discharge Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Direct Current (DC) Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Dark Current and Townsend Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Townsend Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Townsend Breakdown Criterion and Paschen Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Glow Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Transition from Glow to Arc Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Corona Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Spark Breakdown and Streamer Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Alternating Current (AC), Radio Frequency (RF), and Microwave (MW) Discharges . . . 4.1 Alternating Current (AC) Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Radio Frequency (RF) Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Microwave (MW) Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Latin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

474 474 475 477 479 482 485 488 490 491 492 492 494 502 503 503 504 505

Emil Pfender: deceased. M. I. Boulos (*) Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada e-mail: [email protected] P. L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France e-mail: [email protected] E. Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA © Springer Nature Switzerland AG 2023 M. I. Boulos et al. (eds.), Handbook of Thermal Plasmas, https://doi.org/10.1007/978-3-030-84936-8_11

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Abbreviations

AC CVD DC MW mfp RF TWD UV

1

Alternating current Chemical vapor deposition Direct current Microwave mean free path Radio frequency Traveling wave discharge Ultra violet

Introduction

As described in Part I of this book dealing with fundamental concepts of the plasma state, passing an electric current through a gas gives rise to the formation of plasmas. Since gases at room temperature are excellent insulators, a sufficient number of charge carriers must be generated to make the gas electrically conducting. This process is known as electrical breakdown, and there are a number of ways by which this may be accomplished. The application of a strong electric field between a pair of electrodes can lead to a breakdown of the originally nonconducting gas leading to the establishment of a conducting path. The passage of the electrical current through the gap between the electrodes leads in turn to an array of phenomena known as gaseous discharges. These are the most common, but not the only means for producing a plasma. For certain applications, plasmas are produced by electrode-less radio frequency (RF), capacitive or inductively coupled discharges, microwaves, shock waves, lasers, or high-energy particle beams. Finally, heating gases (vapors) in a high-temperature furnace may also lead to the generation of plasma. Because of inherent temperature limitations, this method is restricted to metal vapors with low ionization potentials. This chapter is devoted to a brief review of the different types of discharges, which are commonly used for the generation of plasmas. Only plasmas produced by electrical means will be considered. This includes discharges produced by DC, AC, RF, or microwave fields. Common features and basic differences among such discharges are discussed with emphasis on discharges, which allow the production of thermal plasmas. Since this is an introductory chapter, the treatment of the various types of discharges will be restricted to their basic features.

2

Electrical Discharge Systems

Electrical discharge systems consist in general of three elements schematically represented in Fig. 1. (Brown 1959, 1966; Chanin 1971; Fl€ ugge 1956a, b; Fridman 2008; Fridman and Kennedy 2004; Goldman and Goldman 1978; Hirsh and Oskam 1978; Loeb 1955; McDonald and Tetenbaum 1978; Mitchner and Kruger 1973). The discharge is sustained by a power source, and there are different ways in which the

11

Basic Concepts of Plasma Generation

Fig. 1 Electric discharge systems (Chanin 1971)

475

Coupling Mechanism

Power Supply DC AC RF Microwave

Resistive Capacitive Inductive Microwave antenna

Electrical Discharge (Plasma) Pressure Power levels Electric Fields Magnetic Fields

power from this source can be coupled into the discharge (coupling mechanisms). The discharge itself is governed by a number of basic parameters, including power level, pressure, and electric and magnetic fields. Although the simple principle shown in Fig. 1 is always valid, actual discharge systems may be much more complex due to a wide variety and combination of power sources, coupling mechanisms, discharge geometries, and plasma confinements. Resistive coupling generally refers to systems having electrodes in direct contact with the ionized gas or plasma. The electric field required to sustain the plasma is modified by positive and negative space charges within and at the boundaries of the plasma. A finite potential difference at the gas/electrode boundary exists due to the accumulated charges. This potential then usually supports collision processes, which are vital for sustaining the discharge. Capacitive or inductively coupled discharges do not have electrodes in direct contact with the plasma and hence are often referred to as electrode-less discharges. The electric field is induced in the inductively coupled discharges by a time-varying magnetic field. The situation in inductively coupled discharges is analogous to a transformer where the plasma acts as a one-turn (short circuit) secondary coil. Capacitive coupled discharges, often referred to as polarization discharges, are characterized by the fact that the electrodes, through which the power is delivered to the discharge, are separated from the discharge by a dielectric barrier, such as the walls of the plasma container. Oscillating electrostatic charges at the dielectric barrier surfaces produce the electric field of the capacitive coupled discharge. The properties of the plasma produced by an electric discharge can be readily altered by externally controllable variables, such as gas pressure, particle concentration, temperature, gas flow rate, power supply characteristics, frequency, voltage, and magnetic field intensity. Chemical reactions, which occur within the discharge, are affected by changes of these parameters since they may result in altering of the electron and gas temperatures, of the gas excitation, dissociation, and ionization, and of other collision processes important for plasma chemical processes.

3

Direct Current (DC) Discharges

In this section, a continuous sequence of DC discharges will be considered, starting with extremely small currents close to the detection limit, up to currents of hundreds or even thousands of amperes which are typical for arc discharges. A simple discharge vessel will be considered consisting of a cylindrical glass tube with a

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pair of plane electrodes at each of its ends, with a distance, d, between the two electrodes. The tube is filled with a noble gas at a pressure, p, of approximately 1 kPa. A DC power source, with variable voltage output, is connected through a series resistance to the electrodes (Fig. 2). A schematic representation of the voltageto-current relationship is presented in Fig. 3 for the discharge regimes that will be discussed. Roughly the voltage–current evolution comprises three types of discharges: dark, glow, and arc discharges. The transition between dark and glow discharges takes place at currents between 10
5 and 10
4 A, while the transition between glow and arc discharges takes place at currents above 1 A (see Part I, ▶ Chap. 1, “The Plasma State,” Fig. 1.8). A special type of DC discharge, known as the corona discharge, will be discussed in Sect. 3.6 of this chapter.

E

Fig. 2 Schematic of discharge tube circuit

-

+ I

- +

10-20

10-16

10-10

10-6

10-3

10-1

Arc discharge

Abnormal glow discharge

Transition to the arc

S

Normal glow discharge

Subnormal glow discharge

Breakdown region

Self-sustained current

Current sustained by external agents

Voltage, V (V)

Townsend discharge

R

J 1

K 10

100

Current, I (A) Fig. 3 Schematic representation of the different regimes for a DC discharge and their corresponding voltage–current characteristics

11

3.1

Basic Concepts of Plasma Generation

477

Dark Current and Townsend Discharge

For a discharge tube with a value of p.d < 200 Pa.m (> ‘e; volume recombination and diffusion losses have been neglected.

non-self-sustained dark current

Basic Concepts of Plasma Generation

Voltage, V (V)

11

479

semi-self-sustained dark current

1

2

Log (current, I (A))

Fig. 6 Characteristic of non-self-sustained and semi-self-sustained dark current

In Eq. 4 or 5, the simultaneous generation of positive ions has been ignored. As shown by Townsend, positive ions travel towards the cathode and by impinging on the cathode secondary electrons are liberated (γ – effect). These secondary electrons represent starting points for new electron avalanches. By considering the various generations of electron avalanches, Townsend (1915) found the following expression: ne ¼

ne, o expðαdÞ 1
γ½expðαdÞ
1

(6)

je ¼

jo expðαdÞ 1
γ½expðαdÞ
1

(7)

or

where jo is the electron current density at the cathode. The coefficient γ represents the number of secondary electrons released at the cathode by one impinging positive ion. In general, γ γ) based on available experimental data. By applying the secondary emission coefficient Γ to Eq. 7, one obtains je ¼

j0 expðαdÞ 1
ΓðexpðαdÞ
1Þ

(13)

Table 2 Electrons emitted by the g effect, after Knoll et al. (1935) Element Ba K Mg Al Cu Hg W C Fe Ni Pt

Work function (eV) 2.1 2.25 3.3 4.25 4.29 4.53 4.54 4.55 4.77 4.99 5.36

Ar 0.14 0.22 0.077 0.12 0.058 – – – 0.058 0.058 0.058

H2 – 0.22 0.125 0.095 0.050 0.008 – 0.014 0.061 0.053 0.020

He 0.100 0.17 0.031 0.021 – 0.020 – – 0.015 0.019 0.010

Air – 0.077 0.038 0.035 0.025 – – – 0.020 0.036 0.017

N2 0.14 0.12 0.089 0.10 0.066 – – – 0.059 0.077 0.059

Ne – 0.22 0.11 0.053 – – – – 0.022 0.023 0.023

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This relation describes the current behavior in the semi-self-sustained region of the discharge as shown together with other parts of the characteristic in Fig. 6.

3.3

Townsend Breakdown Criterion and Paschen Law

The specified limitation for Townsend breakdown applies to laboratory systems with electrode gaps in the order of cm, and values of p.d < 200 Pa.m. There is no general agreement in the literature about this limit (see for example (Fridman and Kennedy 2004)). Breakdown separates the regimes of the non-self-sustained or semi-self-sustained dark current from the self-sustained current. If the discharge circuit does not impose a severe limitation on the magnitude of the current, experimental evidence shows a drastic increase of the current at the transition point between these two regions. Postulating je ! 1 mathematically approximates this fact. This implies, according to Eq. 13 that ΓðexpðαdÞ
1Þ! 1, which is known as the Townsend’s breakdown criterion; ΓðexpðαdÞ
1Þ ¼ 1

(14)

Equation 14 implies that a single electron starting from the cathode will form an electron avalanche of magnitude, exp(αd), close to the anode. At the same time (exp (αd)-1), positive ions are formed. By assuming that all of these ions impinge on the cathode, and by taking all effects which may contribute to electron emission from the cathode into account, the number of secondary electrons emitted from the cathode will be Γ(exp(αd)-1). If this number replaces the initial electron starting at the cathode, then the discharge will be self-sustaining, i.e., one finds ΓðexpðαdÞ
1Þ ¼ 1 which is the Townsend breakdown criterion. This criterion is valid for values of p.d < 200 Pa.m or (p.d < 150 Torr. cm). The magnitude of the breakdown potential for an operating pressure, p, and a given gas with Townsend constants A and B (Eq. 10 and Table 1) can be calculated as follows: Using Eq. 14 results in expðαdÞ ¼

1 þ1 Γ

or   1 1 þ1 α ¼ ln d Γ

and

  α 1 1 ¼ ln þ1 p p:d Γ

Introducing Eq. 10 into this expression results in     B 1 1 A :exp
ln þ1 ¼ E=p p:d Γ

(15)

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Basic Concepts of Plasma Generation

483

or lnA


   B 1 ¼ ln ln þ 1
lnðp :dÞ E=p Γ

(16)

At the point of breakdown, the breakdown field strength is ðE ¼ VB =dÞ where VB is the breakdown potential, and d, the distance between the electrodes. Rearranging Eq. 16 results in VB ¼

B : ð p : dÞ    1 þ1 lnðp:dÞ þ lnA
ln ln Γ

(17)

which is known as the Paschen law for breakdown. It is obvious from Eq. 17 that minor variations of Γ have little effect on the breakdown potential. Since A and B are constants which depend on the type of gas employed, Eq. 17 demonstrates that VB is only a function of the product (p.d). Figure 7 shows schematically the behavior of the Paschen law. The curve passes through a minimum, which separates the regime of “narrow gap” breakdown from the regime of “wide gap” breakdown.

  The curve has an asymptote for ðp:dÞas , V ¼ ln Γ1 þ 1 =A , indicating that the breakdown potential approaches infinity. This situation prevails if the pressure, p, and the electrode gap, d, are such that ‘e > d (i.e., no ionizing collisions even at extremely high potentials!). As an example, Fig. 8 shows Paschen curves for a number of different gases. Most data in the literature refer to (p.d) in units of (Torr.cm = 1.33 Pa.m). It should be pointed out that Eq. 17 does not provide any information about the current, which separates the non-self-sustained from the self-sustained discharge

( p.d) as

‘Wide ’ gap breakdown

Breakdown potential, V (V)

‘Narrow ’ gap breakdown

Fig. 7 Schematic of Paschen’s law

( p.d ) min

Pressure x electrode gap, p.d (Pa. m)

484 Fig. 8 Paschen curves for different gases and uniform electric fields (After V.P. Raizer, see General Literature)

M. I. Boulos et al.

10 000

H2

1 000

Ar Ne Ne + 0.1%Ar

100 0.1

1.0

10

100

Pressure x electrode gap, p.d (Torr.cm)

regime. It was assumed for deriving Eq. 17 that the current shows a rapid increase at breakdown (je ! 1) and the question arises whether or not experiments provide definite values for the current, in other words a well-defined breakdown point on the discharge characteristic. This does not seem to be the case, because electrical stability requires a rather high resistance in series with the discharge around breakdown. Observations show current bursts immediately before breakdown, and these bursts are associated with the previously mentioned statistics. For this reason, it is more meaningful to define a breakdown range in terms of the discharge current, and this range may bridge several orders of magnitude. Care has to be exercised in using Eq. 17, which applies well at and around the Paschen minimum, but fails for smaller and larger values of (p.d). For the majority of pure gases, the minimum values of VB fall between 100 and 500 V corresponding to values of (p.d) in the range from 0.1 to 10 Pa.m, and as shown in Fig. 8 this minimum is rather shallow. Impurities in the gas will somewhat affect VB, especially in case of impurities with low ionization potential, or if the system allows ionization through the Penning effect. The net result of those effects is a lowering of VB, below the prediction of Paschen’s law. In contrast, the pressure of electro-negative gases such as oxygen or halogens in the main gas will lead to an increase of the breakdown potential due to electron attachment to electro-negative gases and formation of negative ions. Such ions have a relatively low mobility compared to electrons, i.e., they will reduce the prebreakdown electron current flow. As the current increases after breakdown, the discharge characteristic shows a falling trend (a few volts), indicating that breakdown leads to one or several effects, which favor charge carrier production. It seems that one of the most important effects is the formation of a minor ion space charge in front of the cathode which distorts the electric field distribution, leading to enhanced field strengths in front of the cathode which, in turn, favor charge carrier production.

11

Basic Concepts of Plasma Generation

485

This regime of slightly falling characteristic after breakdown, together with the regime of the semi-self-sustained dark current, is defined as the Townsend discharge as indicated in Fig. 5. As the current is further increased, space charge effects in front of the cathode determine the behavior and characteristic of the discharge which is discussed in the following section.

3.4

Glow Discharges

3.4.1 Subnormal Glow Discharge As the current increases further to values of approximately 10
5 to 10
4A in a typical laboratory scale discharge vessel (current densities of 10
6 to 10
4 A/cm2), see Fig. 3, pronounced positive space charges form in front of the cathode, leading to strong distortions of the potential distribution as indicated schematically in Fig. 9. This figure shows already an indication of a “cathode fall” (sudden potential drop) in front of the cathode with correspondingly high electric fields, which, in turn, favor charge carrier generation in this area by electron impact. It is not surprising that the voltage required for sustaining the discharge shows a substantial drop in this regime (negative slope of the characteristic), which frequently imposes severe problems in terms of electrical stability. In fact, electrical stability may not be possible at all, i.e., the discharge may become intermittent (Pfender 1953). In this subnormal glow regime, the discharge becomes faintly visible with an indication of bright and dark regions. Further increase of the current leads to the regime of the normal glow discharge. 3.4.2 Normal Glow Discharge A constant voltage drop over a relatively large current range characterize the normal glow discharge with current densities in the range from 10 to 100 A/m2 (depending Fig. 9 Change of the potential distribution as space charges in front of the cathode develop (schematically)

E je

No space charges

Cathode

Anode

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M. I. Boulos et al.

on the pressure), and at the same time the current density remains constant. This implies that in a configuration as shown in Fig. 2, only part of the available cathode surface carries electric current in the lower current regime of the normal glow discharge. As the current increases, more and more of the cathode surface carries current, i.e., the current-carrying area of the cathode is directly proportional to the current. One of the most distinctive features of a normal glow discharge is the characteristic arrangement of luminous and “dark” areas along the tube as indicated in Fig. 10. These bright and dark areas are directly linked to the existence of space charges and the corresponding potential distribution. Since there are excellent descriptions and interpretations of these observations in the literature (Engel 1965, 1983), they will not be discussed here. Two important similarity laws for the normal glow discharge should be mentioned. The thickness, dcn, of the cathode

1st,2nd … cathode layers positive column negative glow

anode glow

+

− Aston dark space

Cathode dark space

Faraday dark space

Anode dark space Light intensity Electric field

Potential Positive space charge density

Total J+

J-

Negative space charge density Current density Gas temperature

Fig. 10 Schematic of a normal glow discharge (Francis 1956)

11

Basic Concepts of Plasma Generation

487

dark space (cathode fall) is directly proportional to the mfp of electrons, ‘e, and since ‘e  ð1=pÞ one finds ðp:dÞcn ¼ const:

(18)

The thickness of the normal cathode fall spacing will always adjust according to the pressure. For pressures in excess of 1.3 kPa, the cathode fall spacing for a plane electrode configuration (Fig. 2) is typically in the order of 1 mm so that the negative glow (Fig. 10) appears to be “attached” to the cathode. Another important similarity law may be derived from Eq. 18 and the space charge law (there is a strong positive space charge in front of the cathode) resulting in


 je =p2 ¼ const:

(19)

where the constant depends on the cathode material and the type of gas in the discharge vessel. Although the positive column (Fig. 10) in a glow discharge is not essential for the existence of the discharge (it may vanish for a sufficiently short discharge tube), the positive column represents true plasma since there are no net space charges. Glow discharges, also known as “cold” plasmas, are of technical importance for a number of applications, because the heavy particle temperature, Th, remains close to room temperature, whereas the electron temperature, Te, may reach values of 2  104 K or even more. The main reason for Te> > Th is the poor collisional coupling between electrons and heavy particles for conditions typical for glow discharges. Plasma processing of temperature sensitive materials is feasible with such nonequilibrium plasmas.

3.4.3 Abnormal Glow Discharge By further increasing the current beyond the normal glow discharge regime, the potential required for sustaining the discharge increases. Simultaneously, the thickness of the cathode fall spacing, the extent of the negative glow, and the Faraday dark space (Fig. 10) decrease, whereas the current density and the potential drop across the cathode fall spacing increase. The substantially higher field strength in the abnormal cathode fall region combined with the higher current density leads to a strong increase of the power dissipation (je.E) associated with an increase of the gas temperature. At the same time, the higher ion current density at the cathode causes appreciable heating of the cathode. If the cathode consists of a refractory material (for example, W, Mo, Ta, or C), its temperature will rise as the current increases, reaching temperatures in excess of 103 K. In the case of nonrefractory cathode materials, the cathodes will start to melt. For a liquid cathode (for example, Hg or molten copper), there will be extensive evaporation of cathode material. At the end of the abnormal glow regime, the voltage required for sustaining the discharge reaches a maximum. Further increase of the current leads to a transition to the arc discharge, which is discussed in the following section.

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3.5

M. I. Boulos et al.

Transition from Glow to Arc Discharges

3.5.1 Transition to the Arc In the case of a cathode consisting of a refractory metal, the transition from the abnormal glow to the arc occurs gradually with a rapid drop in the voltage required for sustaining the discharge as shown in Fig. 11a. Thermionic emission which, compared to the γ – effect, is a much more effective emission mechanism, becomes important in this regime and as a consequence the required discharge voltage drops substantially. For a liquid cathode, the transition to the arc discharge region occurs abruptly and, depending on the direction of the current change, there is a pronounced hysteresis as indicated in Fig. 11b. The level at which the transition to the arc occurs can vary widely depending on the surface conditions of the electrodes. By adjusting the discharge parameters to the critical point, the discharge may continuously jump between the abnormal glow and the arc regimes. 3.5.2 The Arc Discharge In an arc discharge, the current typically exceeds 1A and may be as high as 100 or 1000 A (Finkelnburg and Maecker 1956; Pfender 1978), as shown in the schematic representation (Fig. 3) of the entire range of DC discharges. Compared to glow discharges, arcs reveal at least three distinctive features, which may be used for defining an arc discharge (Boulos et al. 1994; Pfender 1978). (a) The current density in the column of an arc may be as high as 106 A/m2, whereas the positive column of a glow discharge carries current densities in the order of 10 A/m2. (b) The cathode fall in an arc is typically 15 V, whereas glow discharges require cathode falls close to, or exceeding, 100 V. The main reason for the relatively Refractory cathode

b. Liquid (Hg) cathode Voltage, V (V)

Voltage, V (V)

a.

Current, I (A)

Current, I (A)

Fig. 11 Schematic of the transition from the abnormal glow discharge to the arc

11

Basic Concepts of Plasma Generation

489

low cathode fall in arcs is associated with the more efficient electron emission mechanism (thermionic emission for refractory metal cathodes). (c) The luminosity of an arc, which is a function of pressure and temperature, is orders of magnitude higher than in a glow discharge. For this reason, electric arcs found widespread application in the lighting industry. In fact, the first application of arcs for illumination dates back to 1820. In addition, in arcs one frequently finds more or less constricted arc roots at the electrodes (“arc spots”), in particular at the cathode. Current densities in such arc spots on thermionically emitting cathodes are in the range from 107 to 108 A/m2, whereas on field emission cathodes (“cold” cathodes) current densities in the tiny arc spots may exceed 1011 A/m2. Arcs frequently reveal a falling current–voltage characteristic. This, however, is not always the case as will be shown in Part II, ▶ Chap. 12, “Thermal Arcs.” Although the overall arc voltage in a given discharge vessel is lower than that of a glow discharge in the same vessel and at the same pressure, the overall voltage drop over the discharge does not provide a useful criterion for distinguishing an arc from other types of discharges. Depending on the arc length and the energy balance of the arc column, the overall voltage drop of an arc may be very high. In vortex-stabilized, high-pressure arc gas heaters, for example, arc voltages may reach 15 kV (see Part II, ▶ Chap. 13, “Electrode Phenomena in Plasma Sources”). Finally, it should be pointed out that there is a basic difference between low-pressure (p < 1 kPa) and high-pressure (p >> 1 kPa) arcs. As it has been discussed in Part I, ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics,” the kinetic energy exchange is given by: Δ Ekin ¼

2me mh ðme þ mh Þ2

(20)

Of course if the collision is that of an electron of mass me with a heavy particle of mass mh, Eq. 20 becomes Δ Ekin ¼

2me mh

(21)

Thus, the energy transferred from an electron to a heavy species in a single elastic collision may be expressed by: 2kðTe
Th Þ

2me mh

(22)

where Te and Th are, respectively, the temperatures of electrons and heavy species. The energy that an electron acquires from the electric field E between collisions is: e E vd τ e

(23)

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M. I. Boulos et al.

105 Temperature (K)

Fig. 12 Behavior of electron temperature (Te) and heavy particles temperature (Th) in an arc plasma

Te

104

103

102 10-4

Th

10-3

10-2 10-1 1.0 10 Pressure, (kPa)

102

103

where vd is the average drift velocity and τe the average time of flight between collisions, for more details see Part I, ▶ Chap. 4, “Fundamental Concepts in Gaseous Electronics.” Finally, according to the expressions of vd and τe it results: Te
Th π:mh ¼ Te 24 me

  ‘e e E 2 kTe

(24)

Since ‘e  1p, with, p being the pressure, Eq. 24 results in; Te
Th  Te

 2 E p

(25)

and the parameter (E/p) plays a governing role for determining the kinetic equilibrium. In low-pressure arcs, the heavy particle temperature, Th, is relatively low and much lower than the electron temperature, Te, because of the poor collisional coupling between electrons and heavy particles. In contrast, in high-pressure arcs, the heavy particle temperatures are substantially higher and the plasma may attain a state of local thermodynamic equilibrium (LTE) (Boulos et al. 1994). This state has been extensively discussed in chapter 4 of Vol. 1. Typical values of Te and Th for an arc plasma as function of pressures are given in Fig. 12. These show a gradual decrease of the difference between Te and Th with the increase of pressure, with the two converging into a single (LTE) value at pressures above 10 kPa.

3.6

Corona Discharges

In an electrode configuration in which one electrode consists of a very thin wire or of a point, extremely nonuniform electric fields prevail close to the wire surface (Goldman and Goldman 1978; Loeb 1965). By applying a sufficiently high potential to this electrode configuration, breakdown of the gas near the wire surface will occur

11

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491

at potentials below the spark breakdown potential. The resulting discharge belongs in the realm of glow discharges and is known as the corona discharge if operated at high (atmospheric) pressures. For potentials close to the onset of the corona discharge, the discharge is intermittent, consisting of current bursts induced by external ionization (natural radioactivity, cosmic radiation, etc.). This mode of operation is comparable to that in a Geiger counter. The corona discharge is well known in connection with highvoltage transmission lines, causing a continuous power loss. The corona discharge is also of technical importance as, for example, electrostatic filters, precipitators, and ozonizers. A typical corona discharge arrangement consists of a central wire, of radius r, surrounded by an external metallic cylindrical electrode of radius R where R/r must surpass a certain limiting value. If the wire (“active” electrode) is the cathode, the resulting discharge is known as the “negative corona,” whereas for reversed polarity, results in the “positive corona.” Processes in the high electric field region close to the wire surface govern the discharge in both cases. For further information on corona discharges, the reader may consult some of the more general references (Cobine 1958; Goldman and Goldman 1978; LlewellynJones 1967; Loeb 1965; Lowke and Morrow 1994; Kogelschatz 2003; Sigmond 1978).

3.7

Spark Breakdown and Streamer Mechanism

3.7.1 Spark Breakdown The limitation of spark breakdown to values of p.d > 200 Pa.m applies to laboratory systems with electrode gaps, d, in the order of cm. Townsend breakdown, discussed in Sect. 1.2.3, requires typically breakdown time intervals in the order of 10
5s, due to the relatively low mobility of the ions and the associated γ-effect (liberation of secondary electrons from the cathode by ion impact) which is crucial for the Townsend mechanism. Since γ is a strong function of the cathode material and of the type of gas used for the discharge (see Table 2), Townsend breakdown will also be a function of these parameters. In contrast, spark breakdown does not depend on the cathode material and the breakdown time is in the range from 10
7 to 10
6 s, i.e., up to two orders of magnitude shorter than Townsend breakdown as shown for the first time by Rogowski (1926). In addition, the breakdown voltages for spark breakdown are orders of magnitude higher than breakdown voltages required for Townsend breakdown. According to Paschen’s law, these voltages are typically in the order of 100 V, whereas spark breakdown requires voltages in the order of 10 kV. For example, spark breakdown of a 1 cm gap in atmospheric air requires 30 kV. In the case of Townsend breakdown, the ionized region appears rather diffuse. In contrast, spark breakdown reveals a relatively thin, highly luminous channel designated by Raether as “kanal” (Raether 1964).

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3.7.2 Streamer Mechanism Similar as in the case of the initial phase of the Townsend breakdown, a starting electron close to the cathode will give rise to the development of an electron avalanche. If the electron multiplication factor in the head of this avalanche αd  20, the strong space charge field, due to a slight separation of electrons and positive ions (caused by the applied electric field), will give rise to the emission of energetic photons from this region. In this situation, the electric field induced by space charges reaches approximately the same magnitude as the externally applied field. The high-energy photons will be absorbed over small distances, causing photo ionization, giving rise to the formation of new electron avalanches. As these “daughter” avalanches grow, the heads of these avalanches may again produce energetic photons which, in turn, will start a third generation of avalanches and so on. As the avalanches, propagating in field direction, merge, they will form a conducting channel to the anode (negative streamer). At the same time, a similar mechanism of avalanche formation towards the cathode will lead to a positive streamer. As both negative and positive streamers which grow with velocities in the order of 106 m/s (the positive streamer grows slightly slower) reach the electrodes, a highly conducting channel is established allowing the flow of high currents resulting in a spark or, depending on the power source, in an electric arc. Making electron avalanches visible in cloud chambers, as pioneered by Raether 1964, has substantially enhanced understanding of the streamer mechanism. For more details on streamers, including pertinent references, the reader may consult a recent book (Fridman and Kennedy 2004).

4

Alternating Current (AC), Radio Frequency (RF), and Microwave (MW) Discharges

4.1

Alternating Current (AC) Discharges

Both glow and arc discharges may be generated and maintained by an AC power source. This approach is preferred in a number of applications, because of the simplicity and convenience of the power sources, but the specific features of cathode and anode are no longer maintained in this mode of operation (Brown 1956). At low frequencies (e.g., 50 or 60 Hz), the discharge characteristics are similar to those of DC discharges, although the direction of the current changes from half-cycle to half-cycle, lagging somewhat behind the applied voltage (phase shift). This lag is due to ionization and recombination processes during the AC cycle. For consecutive cycles, the potential for restarting the discharge is lower due to the presence of residual ionization from the previous cycle, which results in the gas breakdown at lower potentials than its normal breakdown potential. This effect becomes more pronounced at higher currents. Since arcs are frequently operated with alternating currents, the dynamic characteristic of such arcs, which refers to rapid changes of the current, will be briefly discussed. The changes are so fast that the equilibrium state of the arc is no longer

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a A B

Voltage

B

C Voltage, V (V)

Current

Time

A

b

C

Positive half-cycles 0 Negative half-cycles

0

Current, I (A)

Fig. 13 Characteristics of an AC low current carbon arc at 50 Hz (a) Current and voltage as a function of time, (b) Corresponding dynamic characteristic (Finkelnburg and Maecker (1956))

feasible. Figure 13a shows current and voltage of a typical AC arc as a function of time (for clarity shown with opposite signs), and Fig. 13b shows the corresponding dynamic characteristic which is a strong function of the AC frequency, f. For the limiting case of, f ! 1 the state of the arc can no longer follow this rapid change of the arc parameters, i.e., the arc behaves like an ohmic resistor and the dynamic characteristic shrinks to a straight line with positive slope. Figure 13a shows that after the arc has been established, the voltage in a consecutive half-cycle reaches a peak while the arc current remains practically zero. This point corresponds to the ignition of the discharge. After ignition, the voltage drops rapidly (falling arc characteristic), reaching a plateau and, at the same time, the current increases, following the sinusoidal AC characteristic of the power source. As the current approaches zero, the arc voltage increases following the voltage imposed by the power supply, passing through zero. During the negative half-cycle, arc current and voltage follow the same pattern as described for the positive half-cycle. Since the process is periodic, the cross plots of a full cycle result in the dynamic characteristic shown in Fig. 13b. The hysteresis of the dynamic characteristic is a consequence of thermal effects in the arc. After arc ignition (point A), the increasing current, following the falling characteristic (B), heats the plasma so that the voltage required for sustaining the arc is reduced in part C of the characteristic (decreasing current). As the current drops to a critical value, the arc extinguishes, since the available voltage can no longer sustain the arc. In the negative half-cycle, the process repeats itself with reversed polarity. By increasing the length of the arc, the effect of the hot electrodes on the plasma is diminished and the maximum voltage required for arc ignition increases, leading to a more pronounced hysteresis of the dynamic characteristic.

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Radio Frequency (RF) Discharges

4.2.1 General Considerations Radio frequency (RF) discharges offer another possibility to generate thermal as well as nonthermal plasmas (Eckert 1974; Boulos 1997). In this case, none of the electrodes has to be in contact with the plasma, which may be advantageous if electrode contamination poses a problem. This holds for both inductively and capacitive coupled plasmas. Considering a capacitive coupled discharge with two parallel electrodes and a neutral gas in the gap between them subjected to a sinusoidal varying electric field, the ionization processes can become markedly different from those in steady fields. At low frequencies of the applied electric field, the polarity of the electrodes will change slowly and the breakdown mechanism is essentially the same as under steady fields. The only difference will be that ionization in the gas will be subject to a slowly varying field. It is evident that charge carriers remaining from a preceding half-cycle in the electrode gap can significantly alter the breakdown mechanism in the case of high frequency fields. For an alternating field, Eo cos ωt, with positive ions having a mobility, μi, a critical frequency, fc, can be defined corresponding to the ability of the ions to traverse the electrode gap, d, during a half-cycle. The distance, x, a positive ion will travel from the instant of the voltage peak, is ω ðt

x ¼ μi E0 cos ωt dt

(26)

0

x¼

μi E0 cos ωt ω

(27)

The critical frequency for positive ion clearance of the gap, d, is f¼

μi E0 πd

(28)

The same analysis as for the ionic motion can be applied to the motion of the electrons that are concentrated at the tip of the avalanche. This will lead to another critical frequency that depends on the electrode separation, electron mobility, and magnitude of the electric field. Since electron drift velocities are generally two orders of magnitude higher than those of ions under the same conditions, the critical frequency for electrons should also be about two orders of magnitude higher. If electrons cannot reach either electrode, electron losses which only occur by recombination, attachment, and/or diffusion, breakdown will be governed by these loss mechanisms. Another breakdown mechanism will become effective at a definite frequency, fco, called the cutoff frequency. When μe, the electron mobility, replaces μi in Eq. 28 results in f co ¼

μ e E0 πd

(29)

11

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495

This cutoff frequency specifies a distinct limit between two different breakdown mechanisms. With (f < fco), electrons are lost by virtue of their mobility towards the electrodes and breakdown is governed by the mobility-controlled mechanism. When (f > fco), electron losses are mainly due to diffusion and the breakdown is governed by the diffusion-controlled breakdown mechanism. At high frequencies of the electric field, not even the electrons produced by electron impact in the gas will have time to reach the opposite electrode. These electrons will oscillate in the gap, undergoing collisions with the gas particles. When the field is adequately high, they will produce more and more electrons until breakdown is achieved. This occurs without the participation of the electrodes, which have become superfluous and may be placed, if desired, outside of the insulating chamber containing the gas under study. Restricting the attention to medium- and high-pressure gases, it is possible to classify the conditions according to the relation between the frequency of the power source and the ionization frequency, νi, of the electrons. Low-frequency field with (f fc) the ions do not reach the electrodes at all, and their accumulation enhances the field and consequently α; hence, less applied field will be required to produce breakdown. The motion of the electrons due to their mobility governs breakdown. At still higher frequencies neither the ions, nor the electrons will reach the electrodes. Electrons will be lost mainly by diffusion, and breakdown occurs when the rate of production of electrons compensates for the rate of their loss by diffusion. With fields of very high frequency and/or very low pressures, i.e., (f > νi), the electrons are now under the influence of a standing wave with oscillatory electric and magnetic components. The gap acts as a cavity as in the case of a microwave guide system, according to the frequency of the field, the geometry of the cavity, and the mode of excitation. In general, RF discharges allow for larger plasma volumes and smaller gradients compared to DC discharges, especially at higher pressures. Atmospheric pressure RF plasma torches in which the plasma approaches LTE may operate at substantially lower velocities than comparable DC plasma torches which is important for certain applications. Table 3 shows a compilation of distinctive parameters of thermal and nonequilibrium RF plasmas. In this table, Eeff refers to an effective electric field, Table 3 Thermal and nonequilibrium RF plasmas

Thermal plasmas p > 10 kPa

Nonequilibrium plasmas p  10 kPa

P > 109 W=m3

P < 106 W=m3

Te Th

Te Th

Approach LTE Eeff =p < 1 V=m:kPa

Strong non- LTE Eeff =p > 105 V=m:kPa

496

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which will be defined later on in this section. Thermal RF plasmas are finding increasing applications for plasma spraying, spheroidization of powders chemical vapor deposition (CVD), plasma chemical synthesis, plasma sintering, and plasma waste destruction. Nonequilibrium RF plasmas are indispensable for the microelectronics industry (plasma etching), for surface modification, and plasma CVD, in particular of organic materials.

4.2.2 High Frequency Breakdown As mentioned earlier, the breakdown mechanism at high frequencies is controlled by diffusion phenomena. The configuration to consider in this case is shown in Fig. 14. Although reference is made to a capacitive coupled discharge, the principle of breakdown applies to an inductively coupled situation as well. For ‘Ee > L the equation of motion for the electrons can be written as ⇀

me

⇀ d ue ¼
e E dt

(30) ⇀

⇀

where me is the electron mass, u e the electron drift velocity and E the electric field which may be expressed by ⇀

E ¼ Eo expðiωtÞ

(31)

withpEffiffiffiffiffiffi o as ffi the amplitude, (ω = 2πf) as the radian frequency of the power source and i ¼
1 as the imaginary unit [exp(iωt) = cos ωt + i sin ωt]. Integrating Eq. 30 results in; ⇀

⇀

u e ¼ u eo þ

eEo i expðiω tÞ me ω

(32)

⇀

where u eo refers to the initial drift velocity of an electron. From this equation an ⇀ ⇀ electron mobility, μe, may be derived u e ¼ μe E , i.e., μe ¼ Fig. 14 Schematic arrangement for a capacitive coupled RF discharge

e i me ω

(33)

11

Basic Concepts of Plasma Generation

497

The average power dissipation per unit volume in the discharge vessel may be expressed by, ⇀

⇀

⇀

P¼j :E

(34)

P ¼ ene Reðμe ÞE2

(35)

⇀

With j ¼ ene u e one obtains

and since the real part of μe expressed as Reðμe Þ ¼ 0, P ¼ 0. This result shows that the electrons are not able to pick up energy from the electric field for ‘Ee > L, because drift velocity of the electrons and accelerating electric field are 90 out of phase. Electron multiplication, which is a necessary condition for breakdown, is not feasible in this situation. Electron multiplication becomes only possible if there are ionizing collisions in the discharge vessel, i.e., ‘Ee < L. For ‘Ee < L Eq. 30 is modified to include a collision term me

! du e ! ¼
eE expðiωtÞ þ ðme νÞu e 0 dt

(36)

 ! In this equation, ν refers to the collision frequency of electrons and ðme νÞu e represents a friction term in this differential equation. Equation 36, which is a linear, nonhomogeneous differential equation, has the solution ⇀

ue ¼


eEo expðiωtÞ þ C1 expð
ν tÞ me ðν þ iωÞ

(37)

Only the periodic part of the solution is of interest, because as t ! 1 the second term on the RHS of Eq. 37 vanishes. From Eq. 37, an electron mobility may be derived as μe ¼

e e ðν
iωÞ ¼ me ðν þ iωÞ me ðν2 þ ω2 Þ

(38)

with Re ðμe Þ ¼

me

eν þ ω2 Þ

ðν2

(39)

and ⇀

⇀

P¼j :E¼

e 2 ne ν E2 me ðν2 þ ω2 Þ

(40)

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M. I. Boulos et al.

By defining E2eff ¼

ν2 E2 ν 2 þ ω2

(41)

e 2 ne 2 E me ν eff

(42)

results in; P¼

as the average power dissipation per unit volume. This power dissipation leads to the multiplication of the original electrons present in the discharge vessel. Breakdown will occur when electron production exceeds or, in the limit, matches electron losses. Electrons may be lost by volume recombination, electron attachment, and/or by diffusion to the surrounding walls, followed by recombination. Volume recombination is proportional to ne2, i.e., it is not important for breakdown because electron densities are relatively small. Electron attachment is important in electro-negative gases, which will be excluded for the following considerations. Electron losses by diffusion are important and may be described by Fick’s law ⇀

Γ e ¼
De grad ne

(43)

⇀

where Γ e describes the electron flux per unit area and unit time. Natural effects continuously produce electrons (see Sect. 2.1), and this quantity will be denoted by nA m
3s
1. In addition, electrons are produced by ionizing collisions, νine m
3s
1, where νi is the ionization collision frequency. With the previously defined quantities, a charge carrier balance equation (continuity equation) may be established ⇀ @ne ¼ νi ne þ n_ A
div Γ e @t

The breakdown point may be approached in a quasi-steady fashion, i.e., and, therefore, De Δne þ νi ne þ n_ A ¼ 0

(44) @ne 0 @t (45)

where Δ is the Laplace operator. In the following breakdown will be considered between a pair of long parallel plates c separated by a distance L. With this assumption and L λ, λ ¼ where c is the f velocity of light and f the frequency of the power source, Eq. 45 transforms into De

d2 ne þ νi ne þ n_ A ¼ 0 dx2

(46)

11

Basic Concepts of Plasma Generation

499

De, νi, and n_ A are assumed to be uniform in the entire volume between the pair of plates.
 By specifying boundary conditions dne =dx ¼ 0 for x ¼ 0, and ne ¼ 0 for x ¼ L2 , Eq. 46 can be solved 2

rffiffiffiffiffiffi vi cos 6 D n_ A 6  rffiffiffiffiffieffi ne ¼ 6 vi vi 4 cos De



3

x

7 
17 7 L 5 2

(47)

This equation describes the electron density distribution between the two plates (Fig. 15) for the competing electron production processes and the electron loss process. Similar as in the case of DC breakdown, RF breakdown will occur when ne reaches very high values (mathematically ne ! 1). This is the case when rffiffiffiffiffiffi  νi L cos !0 De 2

(48)

rffiffiffiffiffiffi νi L π ¼ ð2k þ 1Þ 2 De 2

(49)

or

where k = 0, 1, 2, etc. For the following, only the first mode (k = 0) will be considered, i.e., rffiffiffiffiffiffi νi L π ¼ De 2 L

(50)

π 2 νi 1 ¼ ¼ 2 L De Λ

(51)

or

where ðΛ ¼ L=πÞ is the characteristic diffusion length for this configuration. Equation 51 represents the RF breakdown criterion for a pair of parallel plates. Fig. 15 Schematic arrangement for RF breakdown between parallel plates

x L

500

M. I. Boulos et al.

The previously defined ionization coefficient ðη ¼ α=EÞ (see Eq. 11) may be written as η¼

νi νi ¼ ue E μ e E2

(52)

since the first Townsend coefficient α may be expressed by α¼

νi ue

(53)

The DC ionization coefficient η according to Eq. 52 may now be compared to an analogous ionization coefficient for RF ionization ζ¼

νi De E2eff

(54)

The electric DC field, E, is analogous to the effective RF field, Eeff, and in the case of DC discharges the lifetime of the electrons is governed by the mobility, μe, whereas in RF discharges the lifetime of electrons is governed by the diffusion coefficient, De, consistent with the assumption that diffusion is the primary electron loss mechanism. The RF breakdown criterion (Eq. 51) may also be expressed in terms of the RF ionization coefficient ζ¼

νi 1 ¼ De E2eff , b Λ2 E2eff , b

(55)

where Eeff,b represents the RF breakdown field strength. From Eq. 55 it follows that E2eff , b ¼

1 Λ2 ζ

(56)

or 1 Eb ¼ Λ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ν 2 þ ω2 ζν2

(57)

Considering that the collision frequency, ν, is a function of pressure, Eq. 57 implies that Eb ¼ f ðΛ, pÞ

(58)

A schematic representation of the variation of Eb as function of pressure, for a discharge between parallel plate electrodes with the inter-electrode spacing, L as

11

Basic Concepts of Plasma Generation

501

Fig. 16 Schematic of RF breakdown curves for parallel plates of separation L1 and L2

parameter is given in Fig. 16. These show that at low pressures, because of the low collision frequency, the electron multiplication is impaired requiring high voltages for a breakdown of the gas. With the gradual increase of pressure, the required breakdown field strength decreases. On the other hand, at high pressures the situation is similar to that for DC breakdown. Since ‘e  1/p the requirement of e‘Ee E > EI gives rise to an increase of the Eb value with increasing pressure. The transition between these two modes represent a minimum value for Eb at ω = 2πνI, known as the “resonance case” which is the point of most efficient energy transfer from the electric field to the electrons. Larger separation of the electrodes reduces the breakdown field. This finding is directly related to the assumption that diffusion governs electron losses are function of the (surface/volume) ratio of the discharge vessel and this decreases with increasing L resulting in reduced electron losses. A comparison of the magnitude of DC and RF breakdown fields reveals that RF breakdown fields may be appreciably lower than DC fields. This fact is associated with the electron oscillation in RF fields, which enhances the lifetime of electrons and, therefore, their efficiency for ionization. Similar as in the case of DC breakdown, experimental results may be presented by an RF breakdown voltage as a function of the product, p.Λ, i.e., Eeff , b :Λ ¼ f ðp:ΛÞ

(59)

In this way, both geometry and the power source frequency are included in a simple relationship. The characteristic diffusion length can be easily determined for simple geometries and combinations of such geometries.
2 1 Parallel infinite plates: ¼ Lπ Λ2
2 1 Infinite cylinder of radius R: ¼ 2:4 R Λ2
2
2 For a cylinder of radius (R) and length (L) Λ12 ¼ Lπ þ 2:4 R

502

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Breakdown voltage, Ee. Λ (V)

1000

100

10 0.1

1.0 10 Pressure x diffusion length, p. Λ (Torr.cm)

100

Fig. 17 RF breakdown in hydrogen. The solid line refers to theoretical predictions; the points are experimental results (Brown 1966)

As an example, Fig. 17 shows RF break down for Hydrogen as function of (p Λ). For further information the reader is referred to Fulcheri et al. (2015) for AC, and Freeman, M. P. and Chase, J. D., (1968) for RF discharges.

4.3

Microwave (MW) Discharges

According to Lebedev (2010), microwave discharges (MD) are the electrical discharges generated by the electromagnetic waves with frequencies exceeding 300 MHz. The used wavelengths of microwaves are in the range from millimeters up to several tens of centimeters and should correspond to allowed microwave frequencies for industrial, medical, and scientific applications. The frequency 2.45 GHz is the most commonly used. Microwave discharges are widely used for generation of quasi-equilibrium and nonequilibrium plasma for different applications. Microwave plasma can be generated at pressures as low as 10
3 Pa up atmospheric pressure in the pulse and continuum wave regimes at incident powers ranged between several Watts and hundreds of kW. Breakdown of a microwave discharge is similar to that of RF discharges, Fig. 18 shows microwave breakdown for a frequency of 2.8 GHz and differently sized resonant cavities for a pressure range from 13 Pa to 39.5 kPa (0.1–300 Torr) for helium with a small amount of mercury. There is an excellent agreement between theoretical predictions and experiments (McDonald and Tetenbaum 1978). Over the past years, microwave discharges attracted increasing attention as a viable plasma source for a number of applications, including deposition of highquality diamond films. Conventional microwave discharges require that the discharge be an integral part of the microwave circuit. This requirement imposes inherent limitations on the flexibility of the discharge parameters, in particular on the configuration and size of the plasma volume.

Basic Concepts of Plasma Generation

Electric field, E (V/cm)

11

503

1000 Λ=0.0505 cm Λ=0.101 cm

100 Λ=0.152 cm Λ=0.809 cm

1.0

10

Theoretical Experimental

100

Pressure, p (Torr) Fig. 18 Microwave breakdown (f = 2.8 GHz) in (He/Hg mixture) (McDonald and Tetenbaum 1978)

Recent developments of more flexible microwave devices make use of electromagnetic surface waves, also known as Traveling Wave Discharges (TWD), for sustaining the plasma. Microwave discharges are operated with frequencies ranging from 1 MHz to more than 10 GHz and pressures from 10
3 Pa up to several hundred kPa, and discharge tube diameters from 0.5 to 150 mm. For more information on such discharges, the reader is referred to the pertinent literature (McDonald and Tetenbaum 1978; Moisan and Zakrzewski 1991).

5

Nomenclature

Units are indicated in parentheses; whenever no units are given, the parameter is dimensionless.

5.1

Latin Alphabet

A B C d dcn Da De Di e

Constant (m
1 Pa
1) Constant (V/m.Pa) Constant (
) Distance between cathode and anode or electrode gap (m) Thickness of the cathode fall for a normal glow discharge (m) Ambipolar diffusion coefficient (m2/s) Electron diffusion coefficient (m2/s) Ion diffusion coefficient (m2/s) Electron charge (e = 1.6  10
19 A.s) Electric field (V/m)

⇀

E

⇀

E eff Eeff,b

Effective field producing the same energy as a steady electric field (V/m) Effective electric field at breakdown (V/m)

504

EI Eo f fc fco i I j je je(n) je,o ji jo L ‘e

M. I. Boulos et al.

‘Ee mh mi n n_ A ne ne,o P P(‘x) p Re(μe) T T Te Th ⇀ ue ⇀ u eo vk vd ved V VB x

Ionization energy (eV) Amplitude of the electric field (V/m) Frequency of the discharge (s
1) Critical frequency for ions to reach the electrodes (Hz) Critical frequency for ions to reach the electrodes (cutoff frequency) (Hz) pffiffiffiffiffiffiffi Imaginary unit (i ¼
1) Discharge current (A) Electric current density (A/m2) Electron current density (A/m2) Electron current density at plane n (A/m2) Electron current density at the cathode (A/m2) Ion current density (A/m2) Saturation current density (A/m2) Separation of two parallel plates (m) Mean free path (mfp) of the electrons between two successive collisions with neutrals (m) Component of the electron mfp in field direction (m) Mass of the heavy species (kg) Mass of the ion (kg) Neutral particle number density (m
3) Electron production rate by natural effects (m
3s
1) Electron number density (m
3) Electron number density at the cathode (m
3) Average power density (W/m3) Probability of a mean free path > ‘x (
) Pressure (Pa) Real part of the electron mobility (m2/V.s) Time (s) Temperature (K) Electron temperature (K) Heavy species temperature (K) Electron drift velocity (m/s) Initial electron drift velocity (m/s) Velocity of particles of species k (m/s) Mean drift velocity (m/s) Drift velocity of the electrons (m/s) Voltage (V) Breakdown voltage (V) Coordinate (m)

5.2

Greek Alphabet

α β

First Townsend coefficient (m
1) Second Townsend coefficient (m
1)

11

γ Γ

⇀

Γe Δ εo ζ Λ η μe μi νe νi τe ω

Basic Concepts of Plasma Generation

505

Secondary emission coefficient (-) Secondary emission coefficient (-) (generalization of γ) Electron flux (m
2s
1) Laplace operator Vacuum permittivity (εo = 8.86  10
12 A.s/V.m) RF ionization coefficient (V
2) Characteristic diffusion length for electron (m) First Townsend’s coefficient defined relative to the voltage drop (V
1) Electron mobility (m2/V.s) Ion mobility (m2/V.s) Collision frequency of electrons (s
1) Nonelastic collision frequency for neutral ionization by electrons (s
1) Average time of flight between collisions Angular frequency (ω = 2πf) (s
1)

References Boulos MI (1997) The inductively coupled radio-frequency plasma. J High Temp Mater Process 1:17–39 Boulos MI, Fauchais P, Pfender E (1994) Thermal plasmas: fundamentals and applications, vol 1. Plenum Press, New York Brown SC (1956) Breakdown in gases: alternating and high-frequency fields. In: Fl€ ugge S (ed) Encyclopedia of physics. Gas discharges II, vol XXII. Springer, Berlin Brown SC Jr (1959) Basic data of plasma physics. Wiley, New York Brown SC (1966) Introduction to electrical discharges in gases. Wiley, New York Chanin LM (1971) Fundamental concepts of plasma chemistry, In: Chapter 2, Glow discharges, continuing education course. University of Minnesota, St Paul MN Cobine JD (1958) Gaseous conductors. Dover, New York Eckert HU (1974) The induction arc: a state-of-the-art review. High Temp 6:99–134 Finkelnburg W, Maecker H (1956) Electric arcs and thermal plasmas. In: Fl€ ugge S (ed) Encyclopedia of physics. Gas discharges II, vol XXII. Springer, Berlin Fl€ ugge S (ed) (1956a) Encyclopedia of physics. Gas discharges I, vol XXII. Springer, Berlin Fl€ ugge S (ed) (1956b) Encyclopedia of physics. Gas discharges II, vol XXII. Springer, Berlin Francis G (1956) The glow discharge at low pressures. In: Fl€ ugge S (ed) Encyclopedia of physics. Gas discharges II, vol XXII. Springer, Berlin Freeman MP, Chase JD (1968) Energy-transfer mechanism and typical operating characteristics for the thermal RF plasma generator. J Appl Phys 39:180–190 Fridman A (2008) Plasma chemistry. Cambridge University Press, Cambridge Fridman A, Kennedy LA (2004) Plasma physics and engineering. Taylor & Francis, New York Fulcheri L, Fabry F, Takali S, Rohani V (2015) Three-phase AC arc plasma systems: a review. J Plasma Chem Plasma Process 35:565–585 Goldman M, Goldman A (1978) Gaseous electronics. In: Hirsh MN, Oskam HJ (eds), Chapter 4, Corona discharges, pp 219–290. Science Direct Elsevier Inc Hirsh MN, Oskam HJ (1978) Gaseous electronics. Electrical discharges, vol 1. Academic, New York Knoll M, Ollendorff F, Rompe R (1935) Gasentladungstabellen. Springer, Berlin, p 79 Kogelschatz U (2003) Dielectric-barrier discharges: their history, discharge physics, and industrial applications, invited review. Plasma Chem Plasma Process 23(1):1–46

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Lebedev YA (2010) Microwave discharges: generation and diagnostics. J Phys Conf Ser 257:012016 Llewellyn-Jones F (1967) Ionization avalanches and breakdown. Methuen, London Loeb LB (1955) Basic processes of gaseous electronics, 2nd edn. University of California Press, Berkeley Loeb LB (1965) Electrical coronas. University of California Press, Berkeley Lowke JJ, Morrow R (1994) Theory of electric corona including the role of plasma chemistry. Pure Appl Chem 66(6):1287–1994 McDonald AD, Tetenbaum SJ (1978) Gaseous electronics, Chapter 3. In: Hirsh MN, Oskam HJ (eds) High frequency and microwave discharges, vol 1. Academic, New York Mitchner M, Kruger CH (1973) Partially ionized gases. Wiley, New York Moisan M, Zakrzewski Z (1991) Plasma sources based in the propagation of electromagnetic surface waves. J Phys D Appl Phys 24:1025–1048 Pfender E (1953) Beitrag zum quantitativen Verlauf der Entladungsgenetik. Z Angew Phys 5:450 Pfender E (1978) Electric arcs and arc gas heaters, Chapter 5. In: Hirsh MH, Oskam HJ (eds) Gaseous electronics, vol 1. Academic, New York Raether H (1964) Electron avalanches and breakdown in gases. Butherworths, Washington Rogowski W (1926) Townsend’s Theorie und der Durchschlag der Luft bei Stosspannung. Arch Elektrotech 16:496 Sigmond RS (1978) Electrical breakdown in gases. In: Meek J, Graggs JD (eds) Chapter 4, Corona Discharges. Wiley & Sons, New York Townsend JS (1915) Electricity in gases. Clarendon Press, Oxford von Engel A (1965) Ionized gases, 2nd edn. Clarendon, Oxford von Engel A (1983) Electric plasmas, their nature and uses. Taylor and Francis, London von Engel A, Steenbeck M (1934) Elektrische Gasentladungen, ihre Physik und Technik, vol 1. Springer, Berlin, Germany, p 98

Thermal Arcs

12

Maher I. Boulos, Pierre L. Fauchais, and Emil Pfender

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Arc Column and Electrode Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Arc Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Electrode Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Arc Characteristics and Electrical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Current-Voltage Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Electrical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Classification of Arcs According to Their Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Latin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

508 508 508 510 518 531 531 534 536 547 547 549 549

Abbreviations

AC AJD CJD

Alternative current Anode jet dominated Cathode jet dominated

Emil Pfender: deceased. M. I. Boulos (*) Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada e-mail: [email protected] P. L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France e-mail: [email protected] E. Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA © Springer Nature Switzerland AG 2023 M. I. Boulos et al. (eds.), Handbook of Thermal Plasmas, https://doi.org/10.1007/978-3-030-84936-8_12

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DC i.d. ISPC LTE mfp MW OFHC RF TIG TWD 1-D 2-D

1

Direct current Internal diameter International Symposium on Plasma Chemistry Local Thermodynamic Equilibrium Mean free path Microwave Oxygen-free high-purity copper Radio frequency Tungsten Inert Gas Traveling wave discharge One-dimensional Two-dimensional

Introduction

DC and AC arcs are governed essentially by the same basic features. In the case of AC arcs, the fact that the electrode polarity changes continuously at the line frequency obscures the basic phenomena involved. Emphasis will therefore be placed in this chapter on DC arcs which is divided into two principal section: The first deals with the arc column and the electrode regions. This includes the classical Elenbaas-Heller model, which provides means for obtaining basic trends of arc column behavior such as the maximum temperature, which is feasible in an arc as function of the power input. This is followed by the Watson model which represents a simple, single-fluid description of an arc column. This section also covers a description of the electrode regions (cathode and anode) and different current attachment mechanisms. The second part of this chapter is devoted to the current-voltage characteristics of arcs and their electrical stability. A classification of the different methods used for stabilizing the arc column is presented. This includes free-burning arcs, selfstabilized arcs, gas-stabilized arcs, wall-stabilized arcs, vortex-stabilized arcs, electrode-stabilized arcs, and finally magnetically stabilized arcs. The integration of the arc into the design of a plasma generating device, often referred to as arc gas heaters, or plasma torches or transferred arcs is dealt with in Part II, ▶ Chap. 14, “DC Plasma Torch Design and Performance,”.

2

The Arc Column and Electrode Regions

2.1

General Considerations

By definition, in a thermal arc the thermodynamic state of the plasma generally approaches LTE, which includes kinetic as well as chemical equilibrium, except for the arc fringes or close to a cooled wall. Such arcs, known as high-intensity arcs,

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typically require currents above 100 A and pressures in excess of 10 kPa. At lower pressures electron and heavy particle temperatures are significantly different. A large number of references related to thermal arcs are listed in a survey on “Electric Arcs and Thermal Plasmas” (Finkelnburg and Maecker 1956) and in a more recent review on “Electric Arcs and Arc Gas Heaters” (Pfender 1978). There are three principal features, which distinguish an arc from other discharge modes. For the sake of simplicity, the following discussion will be restricted to steady-state DC arcs.

2.1.1 Relatively High Current Densities The current density in the arc column may reach values in excess of 106 A/m2 which is considerably higher than the corresponding values of 10–100 A/m2 typical for the positive column of a glow discharge. The situation is even more pronounced at the electrodes. Arcs may attach to the electrodes, and in particular to the cathode, in the form of tiny spots in which current densities can be as high as 1010 A/m2. The associated heat flux densities are of the order of 109–1011 W/m2, values requiring special precautions to protect the mechanical integrity of the electrodes. 2.1.2 Low Cathode Fall As indicated in Fig. 1 the potential distribution in an electric arc changes rapidly in front of the electrodes forming the so-called cathode and anode fall. The cathode fall is of particular interest; it assumes values of around 10 V in contrast to the typical cathode falls in glow discharges, which usually exceed 100 V. This relatively low cathode fall is a consequence of the more efficient electron emission mechanisms at the cathode compared with those prevailing in glow discharges. It should be pointed out that the thicknesses dc0 and da0 of the electrode regions in Fig. 1 are enlarged, particularly if high-pressure arcs are considered. Also, Vc0 does not, in general, represent the true cathode fall. The actual cathode fall Vc corresponding to the electron space charge region (see Fig. 1) may be several volts less than Vc0 . Although the overall arc voltage in a given discharge vessel is lower than that of a glow discharge in the same vessel and at the same pressure the overall voltage drop over the discharge does not provide a useful criterion for distinguishing an arc from Fig. 1 Typical potential distributions along an arc

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other types of discharges. Depending on the arc length, and the energy balance for the arc column, the overall voltage drop of an arc may be very high.

2.1.3 High Luminosity of the Column This criterion provides a useful distinction between an arc and other discharge modes, provided the pressure is sufficiently high (p  1 kPa). The extremely high luminosity of the column of high-pressure (p  100 kPa) thermal arcs finds many applications in the illumination field.

2.2

The Arc Column

2.2.1 Definition Since cathode and anode regions may be considered as thin “boundary layers” overlying the electrodes, the column with its comparatively small potential gradient (see Fig. 1) represents the main body of the arc. In contrast to the regions immediately in front of the electrodes in which space charges exist, the arc column represents true plasma in which quasi-neutrality prevails. The pressure in the arc column is uniform and equal to the pressure in the surrounding fluid with the exception of arcs operated at extremely high current levels (I > 5000 A). In such arcs, the interaction of the current with the self-induced magnetic field produces a pressure gradient in radial direction (pinch effect) so that the pressure becomes elevated on the axis of the arc column. For a given arc current the conditions in the column (temperature distribution and associated distribution of thermodynamic and transport properties) adjust themselves in such a way that the field strength required for driving this current is minimized (Steenbeck minimum principle) (Peters 1956). The relatively small field strength prevailing in the arc column may also be interpreted as a consequence of the favorable energy balance, which is, to a large degree, determined by the charge carrier balance.

Je

A

C

B

Ji Control volume Fig. 2 Control volume within arc column

Arc column

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In the following a small cylindrical control volume will be considered in a rotationally symmetric, fully developed arc column (Fig. 2). The applied electric field imposes a drift velocity on electrons and ions in the column which gives rise to a certain current flow. The contribution of the ions to this current flow is almost negligible due to the imbalance of ion and electron mobilities. Steady state requires that the same number of electrons entering the control volume through surface A per unit time must leave through surface B. The same argument holds for the ions traveling in the opposite direction. There is, however, a continuous loss of charge carriers by ambipolar diffusion across surface C, accompanied by recombination outside of the control volume. Neutral particles diffuse in the opposite direction maintaining the mass balance within the control volume. Neutral particles must be ionized in the control volume at the same rate, as charge carriers are lost in order to maintain steady-state conditions. Enhanced cooling of the arc fringes by convection, for example, increases not only the temperature gradient but also the charged particles density gradient, resulting in a corresponding increase of diffusion losses of charged particles. These losses must be compensated by a correspondingly higher rate of ionization in the control volume, i.e., by higher field strength. In summary, the arc column responds to increased cooling of its fringes by an increase of the field strength, E, and, therefore, of the energy dissipation, I.E, per unit length of the arc column. This higher energy dissipation leads to higher temperatures in the core of the arc column. The net effect of cooling of the arc fringes is, therefore, an increase of the core temperature, provided that the current is kept constant. Although the previous arguments indicate the proper trend of the described effects, the picture is not quite complete because the charge carrier losses account only for part of the total energy losses in the arc column. The entire energy balance, including losses by heat conduction, convection, and radiation, determines the actual field strength or power dissipation in the arc column. The dominating process responsible for ionization in the arc column is due to electron impact. The field strength in the arc column in the case of high-pressure arcs (p > 10 kPa) is by far insufficient for an electron to accumulate enough kinetic energy over a mean free path to make an ionizing collision, i.e., e:‘e :E 3500

Current density (A/m2) 107–108

Pressure =Ambient

Ambient

Cathode attachment Fixed or slowly moving Rapidly moving

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sections are met. If an arc, for example, is forced through a diaphragm, which reduces its cross section, plasma jets are induced with the flow directed away from the location of the most severe constriction of the arc column (Maecker effect, Maecker 1955). Such an arrangement (Fig. 8) acts as pump; gas is continuously moving towards the opening of the diaphragm by suction and the gas ingested, and heated by the arc, is then accelerated away from the orifice in the diaphragm. For an analytical description of this phenomenon, momentum and continuity equation are required which, in vector notation, may be written for a steady arc neglecting viscous effects as: !

ρ

dv þ grad p ¼ dt

!

!

J  B

(15)

! div ðρv ¼ 0

(16) !

!

where ρ is the plasma density, v the plasma velocityvector, p the pressure, and B the ! ! self-induced magnetic field vector. The J  B force, which, in general, is responsible for the pinch effect in current-carrying plasma columns, may build up a pressure gradient and/or accelerate the plasma. Equation 16 determines which fraction of the magnetic body force is used for plasma acceleration. For a rotationally symmetric arc, the radial pressure gradient and the resulting overpressure in the arc may be expressed by Zr0 jðsÞ:BðsÞ ds

ΔpðrÞ ¼

(17)

r

where r0 is the radius of the arc column beyond which the electrical conductivity is negligible (T < 7000 K for most plasma gases). Fig. 8 Schematic of the pumping action induced by arc constriction

B I Diaphragm

Induced flow Induced flow

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With !

!

rot B ¼ μ0 B

(18)

one obtains BðrÞ ¼

μ0 r

Zr j:s ds

(19)

0

where μo = 1.26  106 Hy/m is the permeability of vacuum. If the current density distribution j (r) is known, Δp (r) can be calculated. Assuming a uniform current density distribution j (one step model) over the cross section of the arc, j¼

I π r20

(20)

where I represents the total arc current, Δp(r) can be determined. Combining Eqs. 17, 19, and 20 one finds  
 μ0 I: j r2 1 2 ΔpðrÞ ¼ 4π r0

(21)

According to this model, Eq. 21, the overpressure on the arc axis at the point of constriction is proportional to the product of total arc current, I, and current ! density j . With the increase of the constriction of the arc channel in the cathode region, the current density as well as the self-induced magnetic field will increase which, according to Eq. 21, will also increase the overpressure on the arc axis. The axial pressure gradient pointing towards the cathode will induce a flow in the opposite direction away from the cathode with a maximum velocity, vmax, given by

vmax

sffiffiffiffiffiffiffiffiffiffi μ0 I j ¼ 2π ρ

(22)

where ρ is the average plasma mass density. For a free-burning 200 A carbon arc, maximum velocities of the order of 100 m/s have been found by Finkelnburg and Maecker (1956). The maximum velocity depends critically on the arc constriction in the cathode region, which may be influenced by the cathode shape in the case of thermionically emitting cathodes. The cathode jets listed under (ii), (iii), and (iv) in this section originate at the cathode surface (see Fig. 9). These jets contain cathode material and/or impurities either in vapor form or as particulate matter, including gases stemming from

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Arc column

Metal droplets ejected

Vortex

Arc column in which metal vapor diffuses

Current stream lines

Solid electrode

Liquid electrode

Fig. 9 Schematic representation of the arc attachment at the cathode (Teste 1994)

chemical reactions on the cathode surface. Oxidation of carbon steel, for example, will produce CO and CO2. Since metal vapors as well as many impurities have a lower ionization potential than permanent gases, these materials will have a strong influence on the plasma properties including the transport coefficients of the plasma in the cathode region. The cathode jets may also enhance the “stiffness” of the cathode region and the adjacent arc column. In fact, the electromagnetically induced cathode jet may serve as a stabilizing mechanism for a free-burning arc. The interaction of externally applied magnetic fields with arcs is of continuing interest for arc applications as well as for basic research. If, for example, a transverse magnetic field is applied to an arc, which may move freely, one finds that the arc including the electrode attachments move in a direction mutually perpendicular to the current flow and the magnetic field. Arcsat pressures p > 100 kPa move in the ! ! direction given by the magnetic body force j  B . At reduced pressures, some arcs move in the reverse direction (retrograde motion), which is one of the most puzzling problems associated with the cathode region in arc physics. Although many theories have been advanced and much experimental work has been done to explain this phenomenon, there is still no generally accepted explanation. It seems that no general explanation can be expected as long as the emission mechanism and other elementary processes in the cathode region cannot be clearly defined.

2.3.2 Anode Region There are a number of similarities and common features of cathode and anode region in an arc. In contrast to the cathode region, the anode region plays a more passive role, which is reflected in a comparatively small number of investigations and available data. The anode region, as any other part of an electrical discharge, is governed by the conservation equations including the current equation. Unfortunately, any attempt to

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solve these equations for the anode region faces three major problems. First of all, the conventional conservation equations apply only as long as the continuum approach is valid. Since the anode fall spacing is in the order of one mean free path length of the electrons, the continuum approach is no longer valid for that part of the anode region. Secondly, the application of the conservation equations requires that the plasma is in LTE or at least that its thermodynamic state is known. There is serious doubt that LTE exists for the entire anode region. Close to and in the anode fall region deviations from LTE occur due to differences in electron and heavy particle temperatures and due to deviations from chemical composition equilibrium. Finally, the specification of realistic boundary condition faces similar problems as in the cathode region. The principal task of the anode fall, namely, to provide electrical continuity between the high-temperature plasma of the arc and the low-temperature anode surface embraces several effects imposed by the conservation equations. Conservation of energy requires, for example, that field strength and current density adjust themselves so that the ohmic heating in the same volume element compensates for the total net energy losses. Because of the steep gradients of the plasma parameters in front of and normal to the anode surface, losses in axial direction may be substantially larger than in radial direction. Conservation of charge carriers demands ion production in the anode fall which, however, amounts to only a small fraction (approximately 1%) of the total current. At the anode surface, the electrons (ion emitting anodes shall be excluded) exclusively carry the current, which gives rise to a net negative space charge. This space charge tapers off with increasing distance in normal direction from the anode due to ion production mentioned above until the unperturbed state in the plasma column is reached. The potential drop in the anode fall zone is a consequence of the net space charge adjacent to the anode surface (Poisson equation). The specific characteristics of the anode fall may be summarized as follows: • Production of ions in order to maintain the ion flux passing through any cross section of the plasma column towards the cathode. Because of the low ion mobility, the required ion production is rather small. • The directed velocity of the ions due to the high field strength in the anode fall has to be transformed into random motion as the plasma column is approached to match the boundary conditions at the interface plasma column anode fall region. • The opposite is true for the electrons, which receive a large directional velocity component as they approach the anode. In this way, the anode fall provides the necessary electron collection. • Charge carrier generation in the anode fall zone may occur by two basically different ionization mechanisms, namely, field ionization (F-ionization) and thermal ionization (T-ionization). • Field ionization seems to play an important role in low intensity arcs where anode falls in the order of the ionization potential of the working fluid have been observed. For high-intensity arcs, which are of interest in the context of this book, the anode falls are substantially lower (a few volts or even negative). For

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such arcs thermal ionization is the governing ionization mechanism. Electrons in the high-energy tail of the Maxwellian distribution are responsible for ionization. (i) Arc attachment The attachment of the arc at the anode surface may occur diffusely as well as severely constricted (spot), and it is still not entirely clear under which conditions constriction will occur. Chemical reactions on the anode surface as, for example, encountered in arcs operated in atmospheric air or in other oxidizing fluids seem to favor a constricted anode root which, at the same time, may move more or less randomly over the anode surface with appreciable velocities. Anode evaporation is another mechanism, which leads, in general, to spot formation. In high-current arcs with relatively small electrode separation (a few centimeters), the diffuse anode attachment is directly associated with the cathode jet. The wellknown bell shape of a free-burning high-intensity arc is a typical example. The intense cathode jet impinging on the anode surface pushes hot plasma against the anode eliminating the need for ionization in the anode fall zone. By increasing the electrode gap under otherwise identical conditions, the influence of the cathode jet at the anode is diminished and, finally, at sufficiently large gaps the arc forms a single or several spots at the anode surface. Further evidence that anode spot formation and cathode jet are intimately related is illustrated in Fig. 10. In this configuration, the axis of the cathode is parallel to the surface of a plane anode so that the cathode jet does not impinge on the anode. The deflected cathode jet provides evidence that there must be an appreciable constriction of the anode attachment. Any constriction of the current path leads to the previously described pumping action which results in this case in an anode jet which causes the observed deflection of the cathode jet from the anode surface. The relative strength of the two jets determines the angle of deflection. These observations suggest that the anode attachment is governed by the thermal conditions at and adjacent to the anode surface. Any effect which has a favorable influence on the energy balance in the anode region, in the sense that internal heat generation by the arc may be decreased, seems to favor a diffuse arc attachment. In a suitable arc configuration the cathode jet is able to provide a continuous flow of hot plasma into the anode region reducing in this way the necessity of heat generation by the arc itself. Sanders et al. (1982) studied the influence on the anodic arc Fig. 10 Interaction of cathode and anode jet in an arc

Anode Cathode

Anode jet Plasma column

Cathode jet

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a Cathode pumping

Cathode

Anode

b Cathode pumping

Cathode

Anode

Fig. 11 Anode attachment of a segmented wall-stabilized Ar arc. (a) Cathode jet dominated mode (bell shape) arc. (b) Anode jet dominated mode

attachment by using a transferred arc stabilized by a water-cooled segmented wall. For an atmospheric pressure argon arc, with an i.d. of the stabilized channel of 10 mm, and an arc current of approximately 300 A, a diffuse arc attachment was obtained as shown in Fig. 11a. Under such conditions, known as the cathode jet dominated (CJD) anode region, the current densities at the anode were in the range of 106–107 A/m2 (Smith and Pfender 1976). The mechanism of constricted anode arc attachment called anode jet dominated mode (AJD) (see Fig. 11b) is related to the cooling of the anode region. As discussed earlier, the arc column constricts more and more as the radial heat losses increase. Simultaneously, the heat dissipation in the arc column increases or, at a given arc current, the field strength in the arc column rises. The described effect in the anode region occurs if the influence of the cathode jet on this region is strongly reduced or entirely eliminated. The relatively low temperature in the vicinity of the anode induces, as a primary effect, a certain constriction, which, however, is always accompanied by the already mentioned pumping effect. The cold gas adjacent to the anode surface is accelerated towards the center of the arc and to a certain degree ingested into the arc reducing its diameter further according to this additional heat removal mechanism. A stronger constriction leads to higher velocities of the cold gas, which in turn constricts the arc even more. While there seems to be no bound for this process which may be termed a “flow induced thermal pinch,” the increase of heat dissipation in conjunction with heat conduction in radial direction counterbalances the constriction due to the induced gas flow, establishing a steady-state situation. Due to the arc constriction in front of the anode, the current densities at the anode end of the transition region (constriction region) are substantially higher than in the arc column. This is accompanied by a corresponding increase of

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the field strength and the plasma temperature in the constriction region. There is evidence that the steep gradients of the plasma parameters in the constriction zone are responsible for strong deviations from LTE. Since the arc attachment at the anode may be sharply constricted, the corresponding current densities may be as high as 107–109 A/m2. When the heat fluxes imposed on the anode result in anode evaporation, the behavior of the arc may markedly change, depending on the type of attachment. In the case of CJD attachment, the anode vapor (in most cases copper) does not readily mix with the plasma gas, with the gases close to the anode surface composed mostly of pure copper (Kaddani et al. 1994), at temperatures significantly different from those of the argon plasma column (Degout and Catherinot 1986). In the case of AJD attachment the copper vapor from the anode mixes readily with the plasma forming gas (Gleizes et al. 1995), resulting in a modification of the arc voltage. The increase of the electrical conductivity of the plasma by the metal vapor results in a voltage decrease across the arc, which is partially compensated by the increase of the electric field due to the added cooling of the arc resulting from the significant radiation losses from the metal vapor. (ii) Boundary layer in front of an anode The anode region may be divided into several zones (Pfender 1978, 1993). A flow-affected zone, as mentioned before, followed towards the anode by a layer in which the presence of the relatively cold anode is felt and which is characterized by steep gradients of temperature and particle densities. In the usual terminology, this layer may be designated as a boundary layer. In the immediate vicinity of the anode surface is the sheath as sketched in Fig. 12, with a thickness of the order of the Debye length (see Part II, ▶ Chap. 11, “Basic Concepts of Plasma Generation,” Sect. 4.7). A one-dimensional analysis of this anode boundary layer reveals substantial deviations from LTE (Dinulescu and Pfender 1980). The temperature of the heavy species approaches the temperature of the anode in the immediate vicinity of the Fig. 12 Scale lengths in the plasma/anode boundary layer. L anode boundary layer thickness, ‘e electron mean free path, λD Debye length

Constricted anode attachment ℓe L λD Anode

Anode boundary layer

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Fig. 13 Electron and heavy particle temperature in the anode boundary layer of a high-intensity arc (Dinulescu and Pfender 1980)

anode surface, whereas the electron temperature remains sufficiently high to ensure the required electrical conductivity as shown for a typical example in Fig. 13. Temperature and density gradients in the anode boundary layer contribute substantially to the electric current flow so that the potential drop across the boundary layer may become negative. Equation 23 (generalized Ohm’s law) describes the current flow for this situation:
 1 dpe dTe j ¼ σe E þ þϕ e : ne dx dx

(23)

In this equation, j is the current density; Te, the electron temperature; σe, the electrical conductivity; E, the electric field strength; ne, the electron density; pe, the partial pressure of the electron gas; ϕ, the thermal diffusion coefficient; and x, the distance from the anode surface for the one-dimensional model. Electric probe measurements at and close to a plane water-cooled anode surface in atmospheric pressure, high-intensity argon arcs, for different arc configurations which, in turn, result in two distinctly different anode arc roots (diffuse and constricted), are in qualitative agreement with the previous analysis (Sanders and Pfender 1984). In the case of a constricted anode arc root, the potential drop across the anode surface is positive, whereas in the case of a diffuse anode arc root, a negative anode fall has been found (Sanders and Pfender 1984). This fact has a strong bearing on the heat flux carried by the electric current. In the case of a positive anode fall, the conventional model may describe the heat flux due to the electron current

5 kTe qel ¼ j þ ϕa þ Va 2 e

(24)

and in the case of a negative anode fall (Dinulescu and Pfender 1980) results in

Thermal Arcs

Fig. 14 Electron temperatures at the anode (Leveroni and Pfender 1991)

529 15 Electron Temperature, Te (kK)

12

14 13 12 0.7 cm 1.0 cm 1.2 cm 1.5 cm

11 10 50


qel ¼ j

100

5 eϕ 2 kσe



150 Arc current , I (A)

dTe þ ϕa þ e

200

250

(25)

where ϕa, is the work function of the anode material; Va, the anode fall; and k, the Boltzmann constant. Measurements of electron temperatures and electron and ion densities in a freeburning argon arc at atmospheric pressure have been performed with a Langmuir probe flush with the anode surface (Leveroni-Calvi 1985; Leveroni and Pfender 1991). The electron temperature follows directly from the slope of the probe characteristic. Electron temperatures as a function of the arc current for various arc gaps are shown in Fig. 14. It should be pointed out that both increase of the arc current and decrease of the electrode gap enhance the velocity of the cathode jet in front of the anode, i.e., they reduce the thickness of the anode boundary layer resulting, as expected, in a steeper electron temperature gradient at the anode. The electron temperature throughout the boundary layer up to the anode surface stays above 104 K. The same probe can be used to determine current densities in the arc axis at the anode (maximum current density). Typical results are shown in Fig. 15, which indicates a clear nonmonotonic dependence on the overall arc current I, displaying a minimum which, for the shorter gaps, is very pronounced. The increase in je for arc currents falling below the value at which the minimum occurs can be interpreted in terms of a constriction of the currentcarrying cross section associated with the need for the arc to maintain sufficiently high temperatures and electrical conductivity in the core. Larger values of je compensate by ohmic heating for the losses by conduction due to the steeper radial, and possibly axial, gradients induced by the constriction typical of low current regimes. This current constriction at low arc currents appears more severe at narrower gaps. With increasing I, je eventually increases also due to the growing influence of the cathode flow and the resulting higher temperatures in front of the anode, which enhances the electrical conductivity as well as the gradients in front of the anode.

Fig. 15 Electron current densities at the anode (Leveroni and Pfender 1991)

M. I. Boulos et al.

Electron current density, j e (kA/cm2)

530 2.5 2.0 1.5 1.0

0.7 cm 1.0 cm 1.2 cm 1.5 cm

0.5 0.0 50

100

150 Arc current, I (A)

200

250

These two effects combined produce an effective boundary layer structure at the anode, where large gradients exist over a distance much shorter than the characteristic length of the arc and where the resulting diffusion forces play a governing role as the current driving mechanism. The position of the current density minimum shifts to higher arc currents for larger gaps, indicating again that the cathode-induced flow governs the establishment of diffusion dominated anode attachment. The heat flux carried by the electrons according to Eqs. 24 or 25 shows a similar behavior as the electron current density. This finding is of practical importance for the design of arc plasma devices. By keeping the anode boundary layer thin, the specific heat flux at the anode arc root may be reduced. As previously mentioned, the gradients become extremely steep in the case of a very thin boundary layer, so that the electric current is predominantly driven by these gradients. In the case of electron currents, this may result in negative anode falls as previously discussed. But not only electron currents, also ion currents are driven towards the anode, establishing an electron retarding positive sheath in front of the anode. Since this sheath is more positive than the anode, it explains the negative anode fall and the associated reduction in the electron current reaching the anode. Some of the gradient driven positive ions will actually reach the anode resulting in an ion current. Although this ion current is approximately two orders of magnitude less than the electron current, it plays an important role for the interpretation of the interaction between boundary layer and anode (Leveroni-Calvi 1985). Besides heat transfer qel to the anode due to the current flow, heat transfer by conduction and convection qc and by radiation from the plasma qr have to be also considered. The local heat flux at the anode arc root, qa may be expressed as; qa ¼ qc þ qr þ qel

(26)

where, qc is the local heat flux by conduction and convection, qr local heat flux by radiation, and, qel is the heat flux due to the electrons given by Eqs. 24 and 25.

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The convective/conductive part will depend strongly on the composition of the plasma forming gas (see Part I, ▶ Chap. 7, “Transport Properties of Gases Under Plasma Conditions”) and the convective part on the attachment mode (CJD or AJD). Based on studies in a free-burning argon arc for currents between 50 and 350 A, Sanders and Pfender (1984) concluded that approximately 50% of the anode heat flux is due to the current flow, 45% is due to conduction and convection, and the remainder (5%) represents radiative heat transfer. This distribution, however, depends strongly on the arc configuration, the arc gas composition, the power input, and on the pressure. In configurations in which the cathode jet impinges on the anode, convective heat transfer is enhanced and it may even dominate anode heat transfer at higher current (power) levels (Eberhart and Seban 1966). Radiative heat transfer which is relatively small in the case of atmospheric pressure argon arcs may contribute substantially to anode heat transfer if the plasma contains metallic or other vapor components with low lying energy levels (arc lamps). And at high pressures (p > 50 MPa), radiation may become the dominating heat transfer mechanism.

3

Arc Characteristics and Electrical Stability

One of the most common and useful representations of the overall arc behavior is the current-voltage characteristics or in modified form the current-field strength characteristics. The latter, of course, apply only for the arc column.

3.1

Current-Voltage Characteristics

In this section, only characteristics of steady DC arcs will be considered. The measurement of current-voltage characteristics is a straightforward procedure as long as the arc is stable. The characteristic consists of pairs of current-voltage readings, which, by definition, represent steady values, although it may take, in some cases, rather long time to establish a steady-state situation. There is a wealth of information on measured arc characteristics (low intensity arcs) with numerous attempts to establish empirical correlations of the form Varc = f (I). The relation given by Eq. 27, developed by Ayrton in 1902, applies only to arcs with falling characteristics since the current intensity, I, appears in the denominator. Varc ¼ a þ b  ‘a þ

c þ d  ‘a I

(27)

where ‘a represents the arc gap length, and the constants a, b, c, and d must be determined empirically. Many modifications of the Ayrton equation have been proposed for various types of arcs. Without the possibility of attaching physical significance to the various constants, such equations are of little value.

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M. I. Boulos et al.

The overall arc voltage is, according to Fig. 7, given by Varc ¼ V0c þ V0a þ

Z

‘a d0a d0c

Edx

(28)

where E represents the field strength in the arc column. Since dc0 and da0 0 dI

(31)

or   dV Rs >   dI

(32)

According to this inequality the load line must intercept the characteristic from above for stable operation. In a more detailed electrical stability analysis the arc

Fig. 19 Electric stability of an arc with falling characteristic

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inductance and capacitance must be taken into account. A calculation based on small disturbances from the equilibrium state leads to two conditions for arc stability dV þ Rs > 0 dI

(33)

1 dV 1 þ >0 L dI Rs C

(34)

and

The first condition is identical with the Kaufman criterion whereas the second criterion establishes an upper limit for the load resistance L Rs <   dV C  dI

(35)

L, represents the arc inductance which is assumed to be in series with an inductionfree arc and C the capacitance of the arc configuration which is assumed to be in parallel with a capacity-free arc configuration. The second criterion is not critical for high-current arcs because L is an increasing function of the current, C is relatively small, and |dV/dI| does not assume extremely large values in high-current arcs. When fluctuations are superimposed to the DC current, and if |dI/dt| is high, the corresponding voltage differential would be less negative, mainly due to the thermal inertia of the gas in the arc channel (Morgensen 1987). The associated time constant is in the order of 1 ms, and the effect is noticeable at 50 or 60 Hz. The arc resistance is generally positive in the kHz range. The thermal inertia will also cause hysteresis effects at higher frequencies as illustrated in Fig. 20 (Morgensen 1987).

Voltage, V (V)

Fig. 20 Dynamic behavior of the arc when a small AC current is superimposed (Morgensen 1987)

High frequency AC Low frequency AC

Current, I (A)

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3.3

M. I. Boulos et al.

Classification of Arcs According to Their Stabilization

This classification is particularly useful because of the direct link between the method of stabilizing the arc column and the options available for the design of arc gas heaters. Most electric arcs require for their stable operation some kind of stabilizing mechanism which must be either provided externally or which may be produced by the arc itself. The term stabilization, as applied in this paragraph, refers to a particular mechanism which keeps the arc column in a given, stable position, i.e., any accidental excursion of the arc from its equilibrium position causes an interaction with the stabilizing mechanism such that the arc column is forced to return to its equilibrium position. This stable position is not necessarily a stationary one; the arc may, for example, rotate or move along rail electrodes with a certain velocity. Stabilization implies in this situation that the arc column can only move in a welldefined pattern, controlled by the stabilizing mechanism.

3.3.1 Free-Burning Arcs As the name implies, no external stabilizing mechanism is imposed on the arc in this case. This does not exclude, however, that the arc generates its own stabilizing mechanism. Considering a low-intensity arc established between two horizontal carbon electrodes in atmospheric air, natural convection effects will bend the arc column upwards in the form an “arc.” Historically, the arc derives its name from this peculiar shape, which is referred to as convection-dominated or convection-stabilized arc. If gravity is eliminated (von Engel 1965), the arc column becomes rotationally symmetric and wider and, at the same time, the arc voltage drop is somewhat reduced as a consequence of the more favorable energy balance. Free convection effects are not quite as obvious if the arc is operated vertically. The induced free convection flow velocities of laboratory-scale arcs are in the order of 1 m/s. Thus, additional flow fields can easily eliminate free convection effects, a situation which arises, for example, in high-intensity arcs. 3.3.2 Self-Stabilized Arc Although high-intensity arcs may be operated in the free-burning arc mode, they are frequently classified as self-stabilized arcs if the induced gas flow, due to the interaction of the self-induced magnetic field and the arc current, is the dominating stabilizing mechanism. The transition from a low-intensity to a high-intensity arc, which occurs at atmospheric pressure above currents of 50 A, manifests itself in a drastic change of the stability of the arc column. Below 50 A the arc column is subject to irregular motion induced by free convection effects. For arc currents in the range from 50 to 100 A, the column becomes suddenly motionless and stiff with a visually well-defined boundary. The cathode jet phenomenon, which has been described previously, gives rise to this transition. According to Eq. 22 the

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maximum velocity in the jet, vmax, is related to total current and current density at the cathode by, vmax  ðI:jÞ1=2

(36)

As soon as this velocity substantially exceeds those induced by free convection effects (in the order of 1 m/s), the described transition will occur. The current at which this transition takes place depends on the conditions in the cathode region (current density and variation of the current density in axial direction). The phenomenon is usually reversible, i.e., by lowering of the current, a transition to the free convection-dominated low-intensity arc occurs. It should be emphasized that the arc temperature at which the transition from lowto high-intensity arc occurs is substantially lower for arcs in molecular gases (for example, nitrogen or air) than for monatomic gases (for example, argon). In molecular gases a hot core is formed in the arc as soon as the dissociation temperature in the axis is surpassed which changes the current density distribution and, therefore, influences the jet formation. As pointed out earlier in Sect. 2.3, for a self-stabilized arc, the heat flux at the anode depends strongly on the Cathode jet (CJD) which for an arc in a controlled (inert) atmosphere depends, in-turn, on; • The arc current I (Maecker 1955; Sanders et al. 1982): the cathode jet velocity increases with I, according to Eq. 22. • The distance between the cathode tip and the anode, dAK: the cathode jet velocity close to the anode decreases when dAK increases (Maecker 1955; Sanders et al. 1982). • The shape of the cathode tip: with a more pointed cathode tip, the cathode jet velocity increases (Gauvin 1989; Maecker 1955; Lin 1985; Petrie and Pfender 1970; Troi 1983; Young et al. 1983). For example, in argon atmosphere with dAK = 30 mm, a conical cathode tip (cone angle of 60 ) with I  300 A leads to the CJD mode, while under the same conditions the CJD mode is obtained with I > 250 A for a cone angle of 40 . • The nature of the plasma gas: the constriction of the plasma column and thus the cathode jet velocity. The constriction increases when the heat flux potential (see Eq. 5) increases. For the following plasma gases it increases in the sequence: Ar, He, N2, H2. Arcs operated at extremely high currents (up to 100 kA) known as ultrahighcurrent arcs should also be mentioned in this category. Although most experiments in this current range utilize pulsed discharges, the relatively long duration (~10 ms) of the discharge justifies classifying them as arcs. There is considerable interest in ultrahigh-current arcs in connection with melting and steel making, utilization in chemical arc furnaces and high-power switchgear. Visual observations of ultrahighcurrent arcs in arc furnaces reveal a rather complex picture of large, grossly turbulent plasma volumes; vapor jets emanating from the electrodes; and parallel current paths

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with multiple, highly mobile electrode spots. In this situation, there is no evidence for any dominating stabilizing mechanism. Induced gas flows and vapor jets exist simultaneously, interacting with each other in a complicated way. Depending on the polarity of the arc and the electrode materials, stable vapor jets have been observed which are able to stabilize the arc column. Thus, the generation of vapor jets by the arc represents another possible mechanism for self-stabilization of arcs. A survey (Edels 1973) gives a comprehensive description of the characteristic features and properties of ultrahigh-current arcs, including 120 pertinent references up to 1970. Continuing studies in this area are mainly concerned with radiation properties and flow fields in such arcs.

3.3.3 Gas Stabilization Gas stabilized arcs rely on forced convection as the stabilizing mechanism. Hot cathode transferred arcs in air require shrouding by a protecting gas (Young et al. 1983) to avoid the contact of the cathode tip with air. Besides its protecting effect the shrouding gas modifies the arc stabilization in the neighborhood of the cathode tip up to a distance of 10–20 cm in the case of efficient shrouds. This is the case for tungsten inert gas (TIG) welding applications (Maecker 1955; Petrie and Pfender 1970; Troi 1983; Schoeck and Eckert 1961; Nestor 1962) where the shrouding gas (in general, argon) is injected through an annular space surrounding the tungsten stick cathode (see Fig. 21a). In this application dAK is short (roughly between 5 and 30 mm) and the plasma forming gas protects cathode and anode surface rather well from oxidation provided its flow rate is sufficient. Unfortunately, this simple shrouding concept, as shown by Coudert et al. (1993), causes gas recirculation close to the cathode tip. The recirculation velocity depends both on the cathode tip cone angle, the position of the injector with respect to the cathode tip, and the plasma forming gas flow rate m_ pl. This recirculation modifies the cathode tip heating and reduces the plasma jet constriction. In summary: • In most cases lower velocities, temperatures, and arc current densities are obtained compared to those with a free-burning arc for the same cathode cone Fig. 21 Gas injection nozzle around a stick-type cathode with a conical shape tip. (a) Straight nozzle. (b) Converging nozzle (Coudert et al. 1993)

a

b

Straight nozzle

Converging nozzle

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angle and arc current. Thus, for given I and dAK, a high value of m_ pl can transform a CJD into an AJD mode in a controlled atmosphere of argon (the AJD mode may lead to anode surface oxidation) • The lowest values of velocities, temperatures, and current densities are obtained for the highest recirculation velocities Therefore in many applications, especially when powder is injected close to the cathode tip, the gas injection nozzle has to have a conical shape with the cone angle identical to that of the cathode tip and the same axis, see Fig. 21b. It is even possible to inject a shrouding gas around the plasma gas, see Fig. 22 (Maecker 1960; Young et al. 1983; Young and Pfender 1989; Choi and Gauvin 1982; Sheer et al. 1974). In this case, the cathode jet velocity can be increased drastically with the plasma forming gas velocity (Coudert et al. 1993). A shrouding gas can be injected close to the plasma gas injection as shown in Fig. 22, in order to protect the anode surface from oxidation which is feasible with a reasonable shroud gas flow rate if dAK is relatively small. For example, the efficiency of the conical injection, with a gap of 1 mm between the 40  conical cathode tip and the nozzle wall (see Fig. 22), with dAK = 30 mm, and an argon flow rate of 30 slm, the CJD mode is obtained at 150 A instead of 250 A with a cylindrical gas injection around the cathode.

3.3.4 Wall-Stabilized Arcs The principle of wall stabilization of arcs has been known for more than 60 years, being introduced in connection with arc lamps. A long arc enclosed in a narrow water-cooled tube with circular cross section will assume a rotationally symmetric, coaxial position within the tube. Increasing heat conduction to the wall, which reduces the temperature and, therefore, the electrical conductivity at this location, will compensate any accidental excursion of the arc column towards the wall. In short, the arc will be forced to return to its equilibrium position. In this situation, Fig. 22 Conical gas injection close to the cathode tip with a shrouding gas

540

M. I. Boulos et al.

increased thermal conduction and the associated secondary effects represent the stabilizing mechanism. Arcs enclosed in glass or quartz tubes are widely used as light sources. Waymouth J.R. (1971) published a comprehensive review on arc light sources. In order to cope with the extremely high wall heat fluxes experienced with highintensity arcs enclosed in small diameter tubes, Maecker introduced metal tubes as arc vessel, consisting of a stack of electrically insulated, water-cooled disks (usually Cu) often referred to as neutrodes. This arrangement is known as the wall-stabilized, cascaded arc, which has been extensively used as a basic research tool. The same principle found widespread application in the design of high-power arc gas heaters with considerably higher arc voltages, which will be discussed later on. Figure 23 shows a cutaway view of a typical wall-stabilized arc. Due to the much higher electrical conductivity of metals compared with that of the arc column, segmentation of the tube enclosing the arc is necessary because a continuous metal tube would cause a double arc through the metal wall (arcing from the cathode to the metal tube and from the metal tube to the anode) seeking the path of least resistance. If E is the field strength in the arc column and d the thickness (in field direction) of an individual segment, the following condition has to be met for avoiding double arcing: Zd E : dx < ðVc þ Va Þ

(37)

0

Plasma jet Anode Coolant channels DC source Arc Gas flow Cathode Conducting material Electrical insulator Fig. 23 Cutaway view of a wall-stabilized, cascaded arc

12

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The minimum voltage required for establishing and maintaining an arc is the sum of the cathode fall, Vc, and the anode fall, Va. A reduction of the inside diameter of the segments (arc constriction) leads to a marked increase of the field strength, keeping the other arc parameters the same, and accordingly to higher energy losses by heat conduction. In this situation the maximum thickness of a segment may be only 1.2–2.0 mm. It is essential for retaining the wall-stabilizing effect that the diameter of the tube containing the arc is smaller than the diameter of a free-burning arc would be operated under the same conditions. If the tube diameter is too large, the stabilizing effect due to heat conduction is lost and, at the same time, free convection effects may cause serious distortions of the arc column. The mechanism of wall stabilization as such does not require any flow in the constrictor tube, although a certain axial flow may be desirable or, under certain conditions, inevitable. The latter is particularly true if cathode jet phenomena are involved which have been discussed in a previous section. A small axial flow component in wall-stabilized arcs is frequently desirable for removing impurities in the plasma released by the electrodes or the confining walls. Wall-stabilized arcs, which operate with strong superimposed axial flow, belong in the category of arc gas heaters and they will be discussed in the corresponding section. Since arc constrictor tubes with circular cross section are able to produce rotationally symmetric arcs, this configuration is usually preferred although more complex geometries are also possible. The maximum possible temperature or enthalpy attainable in a constricted, wallstabilized arc is limited by the highest permissible heat flux, which the wall is able to withstand. Sophisticated water-cooling arrangements permit wall heat fluxes up to 200 MW/m2. In order to overcome this limitation a modified wall-stabilized arc configuration has been suggested. By replacing the water-cooled wall with a transpirationcooled constrictor, convective/conductive heat fluxes to the wall may be reduced by the thicker boundary layer, while radiated heat losses are captured by the radially oriented gas flow through the porous wall as indicated schematically in Fig. 24. The temperature gradient at the wall is drastically reduced by this cooling mechanism and, therefore, appreciably higher power inputs per unit length of the arc and correspondingly higher temperatures on the arc axis become feasible.

Fig. 24 Schematic of a transpiration-cooled arc

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M. I. Boulos et al.

Theoretical considerations by Anderson and Eckert (1967) indicate that temperatures up to 60,000 K on the axis of a hydrogen arc at atmospheric pressure would be possible if limitations due to electrode problems are disregarded. With other gases, such as argon or nitrogen, the possible peak temperatures are lower owing to the high volumetric radiation of these gases at high-temperature levels. As long as heat transfer from the hot plasma to the wall of the porous constrictor is mainly caused by conduction and convection, transpiration cooling is a very efficient cooling mechanism. If radiation becomes predominant, transpiration cooling loses its effectiveness. Other problems associated with transpiration cooling are its requirement for relatively high transpiration gas flow rates and its inherently unstable behavior leading to strong temperature variations along the constrictor wall. A more uniform temperature distribution has been obtained with a segmented plenum chamber as shown in Fig. 25.

3.3.5 Vortex-Stabilized Arcs Similar as in the case of gas stabilization, vortex stabilization relies also on forced convection as the stabilizing mechanism. The principle of vortex stabilization of arcs has been already reported around the turn of the century (Schoenherr 1909). In the case of vortex- or swirl-stabilization the arc is confined to the center of a tube in which an intense vortex of a gas or liquid is maintained. Centrifugal forces drive the cold fluid towards the walls of the arc chamber, which, in this way, is thermally well

Electrical connection

Copper anode

Water-cooled end segment

Water inlet

Chamber cover Water inlet Separator plates Porous tungsten Electrical insulation Tungsten disc Auxiliary anode

Water outlet Transpirant gas inlet Support bolts W-cathode Cathode gas inlet

Water inlet Cathode flow evener

Auxiliary anode Electrical connection Cathode support

Electrical connection

Water outlet

Water inlet

Fig. 25 Transpiration-cooled arc with multiple plenum chambers

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protected. In addition to the circumferential component of the vortex flow, there is also an axial component superimposed which supplies continuously cold fluid. A well-known example of a vortex-stabilized arc is the so-called Gerdien arc from Gerdien and Lotz (1923), which is schematically illustrated in Fig. 26. In this case the stabilizing fluid is water, and the arc plasma is generated from water vapor in the core of the vortex. Because of the extreme cooling of the arc fringes, the power dissipation per unit volume and the associated arc temperatures reach much higher values than feasible in wall-stabilized arcs. Arc temperatures in excess of 50,000 K have been found using an arc current of 1450 A and confining the arc to a diameter 107 V/cm) and low cathode temperatures, i.e., for cathode materials with low boiling point. From Eqs. 10 and 11 follows that field emission is possible for current densities j > 1011 A/m2, even if unrealistically low work functions are postulated. For intermediate temperatures and substantial field strengths, both thermionic emission (Eq. 10) and field emission (Eq. 11) may be simultaneously involved. The corresponding emission process is known as thermionic field emission or TF – emission. In the case of a nonthermionic cathode, the arc attachment at the cathode never consists of a single emitting area. The arc root is usually composed of a number of small emitting sites which are close together at higher pressures and which seem to move randomly, with sites disappearing and reappearing simultaneously in spite of the steady-state conditions in the arc. The number of emitting sites increases with current, which may be an indication that the current per site is fairly constant. It seems that the surface state (oxide layers, impurities, mechanical imperfections, etc.)

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is an important if not the governing parameter for the behavior of the cathode roots on nonthermionic cathodes. Even under extremely clean and macroscopically welldefined conditions, the definition of the surface state on a microscopic scale remains a formidable problem. Analytical approaches are faced with this dilemma, and it appears quite natural that there are so many seemingly conflicting theories because there may be no single theory, which applies to all possible conditions. Guile (1971, 1981) summarizes in his reviews many observations and measurements in the cathode region of nonthermionic arcs as, for example, spot size, current density and cathode fall, spot splitting, movement and evaporation with the resulting cathode tracks and cathode erosion, surface cleaning and oxidation, influence of electrode material including grain size and boundaries, surface state and chemisorbed gases, and the influence of the pressure (vacuum arcs and high pressure arcs). In addition to the previously discussed electron emission processes at the cathode, other possible contributions by other mechanisms cannot be ruled out. Electrons may be liberated from the cathode by individual field components due to the statistical variation of the space charge, by the Auger effect, by the external photo effect, by excited (metastable) atoms, by positively charged oxide layers on the cathode surface, by lowering of the work function produced by monatomic surface layers (e.g., Th on W), or by negative space charges inside the cathode. The various possibilities have been summarized by Guile (1971, 1981) in conjunction with many pertinent references.

3.2

Cathode Materials and Erosion

In most cases due to the necessity of coping with high heat fluxes, the cathodes are made of copper, which provides a high thermal conductivity (κ = 360 W/m.K) sometimes doped with a few wt% of chromium. The constriction of the arc in the vicinity of the cathode, as illustrated in Fig. 9 (Part II, ▶ Chap. 12, “Thermal Arcs”), has been observed for different cathode materials including tungsten and carbon (Herring and Nichols 1949), copper and brass (Harris 1980), and operating conditions as emphasized by Szente et al. (1992). Some researchers suggested that the cathode arc attachment of both vacuum arcs (Guile 1981; Harris 1980; Mitterauer 1975; Emtage 1975) and atmospheric pressure arcs (Guile 1981; Drouet and Grabu 1976) consists of multiple independent cells or cathode spots as confirmed by a paper of J€uttner (1995), while the existence of these cells has been disputed by others (Daalder 1974; Djakov and Holmes 1971). Electrode erosion depends (Smith and Pfender 1976) on the instantaneous local heat flux to the electrode surface. The arc heats the cathode in four different ways: by radiation, convection (from the plasma), Joule heating, and ion bombardment. The radiation and convection terms affect a large area; although they can account for a large fraction of the energy transferred to the cathode, they are too diffuse to cause erosion and can, in general, be neglected for erosion studies (Prock 1986). The Joule heating and the ion bombardment terms are therefore the main heat sources for

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cathode erosion. The relative importance of these two terms is still questionable. Some researchers propose that Joule heating is the most important term (Daalder 1978), while others suggest that ion bombardment is the dominant heat source for erosion (J€uttner 1981). It has been reported (Szente et al. 1987b) that the arc velocity affected cathode erosion rates, higher for lower arc velocities, i.e., the erosion is lower for shorter arc residence time on a particular cathode area. No precise measurements have been published for either the overall area of the arc attachment or the area of the individual cathode spots forming the attachment (Szente et al. 1992). However, it has been reported that different surface conditions could affect the arc attachment dimensions and that these in turn affect the erosion rate (Porto et al. 1982; Hantzsche 1981). Since rapid movement of the arc roots will reduce electrode erosion, many theoretical studies were devoted to the movement of the arc in a magnetic field (Guile and Naylor 1968; Guile and Hitchcock 1975; Roman and Meyers 1966). These studies express the fact that the Lorentz force acting on the arc due to an external magnetic induction, B, is balanced by a force equivalent to drag forces (aerodynamic drag and “surface drag”) which retards the motion (Szente 1987b; Munz and Habelrich 1992): BI‘a þ 0:5 ‘a CD de ρ v2 þ Fs ¼ 0

(12)

Where; B magnetic field strength (Vs/m2) I arc current (A) ‘a arc length in the direction perpendicular to the electrodes (m) CD aerodynamic drag coefficient (dimensionless) de arc diameter perpendicular to its motion (m) v arc velocity (m/s) Fs surface drag force (N) In a more general form, the arc velocity, v, is expressed by: v ¼ K B1=n I1=m

(13)

with K as a constant. Depending on the authors, the exponents l/n and l/m may vary as 0.4 < 1/n < 0.75 and 0.3 < 1/m < 0.6. However, the experimental results are often not in good agreement with Eq. 13. This is probably due to the fact that such calculations cannot account for the actual surface conditions and possible surface contamination. It seems that no explanation has yet been found about the ability of a metallurgical structure to control the motion of the cathode spot, i.e., to generate its own existence conditions (spot temperature, electric field at the surface, power density at the interface, ionization rate of the gas close to the cathode, etc.). Available information is restricted to the studies of Guile (1971, 1981) and Szente et al. (1987a, b, 1989, 1990a, b, 1991, 1992). Their results are sometimes contradictory proposing,

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575

for example, a reduction of the erosion with an increase of the velocity of the cathode spot, while Guile and Hitchcock (1975) report the opposite. With magnetic fields different parameters have been studied for copper cathodes: the plasma-forming gas (steam, hydrogen, and oxygen). Results obtained up to 1994 and reported by Szente et al. (1987a, b, 1989, 1990a, b, 1991, 1992), Munz and Habelrich (1992), Teste (1994), Gleizes (1995), Meunier et al. (1992), and Meunier and Desaulniers-Soucy (1994) were obtained at atmospheric pressure for different plasma gases (Ar, He, CO, N2, O2, Cl2, H2S, CH4, Air) gas flow rates and magnetic flux densities from 0 to 0.15 T, with either cylindrical electrodes (Szente et al. 1992) or rotating disk electrodes (Teste 1994). The most important conclusions are the following: • Without magnetic field, the arc moves continuously at the cathode surface, sometimes interrupted by a jump to a new track where it moves again continuously (Teste et al. 1994). The erosion rates decrease with increasing arc velocity. • With magnetic fields exceeding 0.025 T (Szente et al. 1992), the cathode spots, which make up the arc attachment, are aligned and closely spaced. For magnetic fields below 0.025 T, the cathode spots separate from each other, causing a weaker dependence of erosion rate on arc velocity. Szente et al. (1988) showed that the arc velocity was function of B0.60 throughout the range investigated. The erosion rates dropped from 9.0 to 1.0 μg/C as the arc velocity was increased from 15 to 135 m/s. • Contaminated cathode surfaces (Szente et al. 1992) caused by either using polyatomic gases or adding less than 1 vol% of polyatomic gases to noble gases resulted in the lowest erosion rates. This was attributed to a combination of lower residence time of the arc attachment on a particular cathode area, lower current densities, and a broad distribution of cathode spots. • The motion of the cathode spots was controlled by the creation of favorable conditions for electron emission at the cathode surface. Teste et al. (1994) developed a thermal 3-D model to account for the arc spot motion. Assuming a surface velocity of 100 m/s, a power dissipation at the cathode of, P = 7.5 kW, and a surface power density, Q(r, t) = 5.0  109 W/m2, values which are typical for those found in the literature, the maximum surface temperature at the cathode spot was estimated to be 600  C. This result suggests that no erosion occurs either by vaporization or droplet ejection. However, when looking at the surface, many tiny craters can be observed. Moreover, with jc ~ 108 A/m2, the electric field at the cathode for electron emission should be around, 8  109 V/m. Teste et al. (1994) have accordingly modified their model assumptions, postulating that Q (r, t) = Veq jc, where Veq is equivalent to Vc. They further assumed that Veq was constant (thus jc varies with the surface area of the cathode spot), and the motion of the arc root should consist of successive jumps and successive anchorage points equidistant and independent of the mean value of the arc velocity v. They have also considered the simultaneous existence of two arc roots, with the same total current intensity as that of the arc. The existence of the second arc root was assumed to start at ts/2. Under

576

M. I. Boulos et al.

such conditions, Q(r,t) can reach values of, 5.0  1010 W/m2, which is a heat flux sufficiently high for cathode spot vaporization. An experimental device allowing delivering a current pulse of up to 1200 A in 5 ms was built by Teste et al. (1994) to test the model with two types of cathodes: Cu and CuCr (2 wt% Cr) with a thermal conductivity, κ, which is 80% of that of highpurity oxygen-free (HPOF) copper. For pulse duration of 5 ms, the total erosion in the form of copper vapor and ejection of liquid copper droplets was higher with CuCr than with pure Cu (97 μg/A.s against 8.0 μg/A.s) in spite of the fact that the viscosity of CuCr is higher than that of Cu. With shorter pulse duration (0.7 ms, high dI/dt), the erosion of CuCr (1.1 μg/A.s) is less than that of copper (2.15 μg/A.s). If the vaporization is about the same at similar temperatures, the higher viscosity of CuCr reduces the droplet ejection and thus the electrode erosion. As an example, Fig. 13 represents the evolution of the surface power density Q (x, t) with the stagnation time ts, for Pmax = 7.5 kW. It is clear that the power density at the arc root on the cathode surface increases, with the decrease of the dwell time ts (or the arc velocity increases). The overall erosion rate of copper cathodes as a function of the arc current was the subject of numerous studies by Guile (1971) who proposed the following correlation. Gc ¼ const: Ia

(14)

where Gc is the erosion rate in (kg/A.s) and I the arc current in A. According to Guile (1971), the exponent a = 2.0 for ð45  I  800AÞ. Since 1995 numerous studies were devoted to cold cathode erosion with:

Surface power density, (W/m2)

1010

Copper Tevap = 2595°C 109

108

0

1

2

3

4

5

Stagnation time, ts

6

7

8

9

10

(10-4 s)

Fig. 13 Surface power density Q(r,t) as a function of the dwell time, ts (Pmax = 7.5 kW) (Teste et al. 1994)

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– Desaulniers-Soucy and Meunier (1995) studied the influence of Ar contamination with 1 vol% CO or N2. The Cu density increases with both, more with N2 than with CO. – A copper ion recombination zone extending 2 mm from the cathode was observed in the nitrogen contamination case. – Kwak and Munz (1996) worked the erosion of cathodes made of titanium, stainless steel 314, copper-nickel 10% and 30%, and copper 122, for magnetically rotated arcs operating in argon, nitrogen, and argon/hydrogen mixtures at a constant magnetic flux density of 0.1 T. Titanium, and stainless steel, gave very low erosion rates in argon (0.2 and 0.3 μg/C, respectively). Cupronickels were shown to be suitable for nitrogen and hydrogen plasmas. The slope of hydrogen solubility versus temperature in the cathode material was found to be important in determining hydrogen plasma erosion characteristics. – Essiptchouk et al. (2007), using air as plasma-forming gas, showed the existence of a critical temperature of about 500–600 K for the transition from micro- to macroerosion for a magnetic field intensities in the range 0.01–0.35 T. When the macroerosion regime is reached, the cathode erosion rate strongly increases with temperature. The experimental data obtained can be represented as an exponential function of the time, for the electrode surface underneath the arc root to reach the fusion temperature of the electrode material. – Chau et al. (2007) worked on nontransferred plasma torches operating in air with an external induction coil to generate a magnetically driven arc and a circular swirling action to produce a vortex flow structure. Experimental results suggest that the axial magnetic field is the most important parameter to operate the welltype cathode-plasma torches at high output power with a limited cathode erosion rate. The work emphasizes the importance of the external magnetic field for reducing the cathode erosion. – Rao and Munz (2007, 2008) performed measurements on copper cathodes with an external magnetic field of 0.04 T in 1.33 mPa vacuum with different surface roughness and surface patterns. Results obtained indicate that both surface roughness and surface patterns affect the erosion rate. Patterns perpendicular to the direction of cathode spot movement give lower erosion rates, and isotropic surfaces give lower erosion rates than patterned surfaces at the same roughness. – The same authors but in 2008 studied thermally sprayed copper cathodes having grain size distributions from submicrometers to a few micrometers. The different microstructures were formed using high velocity oxygen fuel (HVOF) spraying and vacuum plasma spraying (VPS) methods with different annealing times. Annealing HVOF coatings at 600  C for 8 h produced near equiaxed grains with a 2–3 μm average size. These coatings showed 60% higher steady-state arc velocities and up to 68% lower erosion rates compared with massive copper cathodes having a 20–23 μm average grain size. Similar electrode erosion studies were carried out on high-power plasma torches such as those of Aerospatiale in France, Tioxide in Great Britain, and Westinghouse

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in USA. The results are, however, difficult to compare with those obtained under laboratory conditions. The results of Eschenbach et al. (1964) and Fey and McDonald (1976) for industrial scale plasma torches with OFHC copper cathodes, running in air or oxygen seem, better represented by the following correlation: Gc ¼ const: Ia =A

(15)

where A is the electrode surface area on which the arc root moves. No general consensus has been found in the literature regarding the numerical value of the exponent, a. George A.P. (1985) found a = 2.0 for I < 1000 A, while Fey and McDonald (1976) have shown that for I  1000 A a = 3.2. It is believed (George 1985) that there is a transition at about 1000 A in the dependence of the erosion rate on arc current. More recently Hare et al. (1992) have proposed some sort of Arrhenius law to express the erosion rate m_ c in kg/s. lnðmc Þ ¼ A  B=T

(16)

where T is a mean temperature at the cathode surface (K), A, the frequency factor and B an activation energy. Table 2 lists the values of A and B obtained by Hare et al. (1992) from the results of Tioxide (Guile 1971; Fey and McDonald 1976). The values of “B” are reasonably constant while large variations in “A” are observed. As noted by George (1985), it might be due to the geometry sensitivity. The scatter of the experimental results compared to the values obtained by Eq. 16 is very high (up to 40%). In conclusion, it is clear from these results that understanding of the phenomena occurring at the surface of cold copper cathodes is still poor and much more work has to be done to achieve a better understanding. The copper cathode erosion is rather high compared to the hot cathode erosion as shown in Figs. 3 and 12. For currents less than 1000–1200 A, typical erosion rates are in the range from 109 to 107 kg/A.s. Following the copper emission at the torch exit allows monitoring continuously the cathode erosion (Meunier and Desaulniers-Soucy 1994), in order to relate it to the operating conditions and avoid those detrimental for the cathode lifetime. This lifetime can be estimated based on the maximum weight of material (M), which can be eroded: τ‘ ¼

M ðGc :IÞ

(17)

with

Table 2 Coefficients of Eq. 16 (Hare et al. 1992)

Reference Hare et al. 1992 Guile 1971 Fey et al. 1976

A () 11.6 12.9 7.0

B (K) 2100 1600 2200

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Electrode Phenomena in Plasma Sources

M ¼ ρCu : A et

579

(18)

where Gc is the erosion rate, ρCu is the specific mass of copper, A is the electrode surface area, and et is the wall thickness which is allowed to erode. It is determined by heat conduction and mechanical strength considerations, which are virtually independent of the scale of operation.

4

Anodes

As mentioned earlier (Sect. 2.3.2, ▶ Chap. 12, “Thermal Arcs”), the anode plays a more passive role in the arc circuit. It is usually made of a water-cooled conductor such as copper. In certain cases, the material treated in an arc can act itself as the anode as in the case of transferred arc furnaces and a variety of plasma welding and cutting applications. The behavior of the arc at the anode surface and accordingly that of the anode depends strongly on the heat flux distribution at the anode surface, its surface roughness, and oxidation state. Since arc stability on the surface of the anode is one of the most critical aspects affecting anode lifetime, this section will deal separately with both static and dynamic aspects of arc behavior on the anode surface. Heberlein et al. (2010) have presented a survey of the actual knowledge of the anode region of electric arcs and Shkol’nik (2011) the development of concepts of anode phenomena in arc discharges in a gaseous medium over a wide pressure range, as well as in vacuum arcs.

4.1

Static Behavior

In this section, a simple electrode configuration (“free-burning” arc) will be considered. As underlined in Part II, ▶ Chap. 12, “Thermal Arcs,” Sect. 2.3.2, the heat flux at the anode (see Eqs. 24, 25, and 26 of ▶ Chap. 12, “Thermal Arcs”) comprises electron energy transfer, qel, the conduction heat transfer which depends on the nature of the gas through its thermal conductivity, the convective heat transfer which is strongly related to the cathode jet, and radiative heat transfer which is similar for different plasma gases, but increases by a few orders of magnitudes as soon as metal vapors are mixed with the plasma gas (anode evaporation). All of these phenomena, whether or not dominated by the cathode jet, are coupled, which makes the prediction of the heat flux qA (W/m2) and the power, QA (W), dissipated at the anode rather difficult. Figure 14 gives a schematic representation of the arc in (a) cathode jet dominated (CJD) mode, (b) anode jet dominated (AJD) mode, and (c) an erratic attachment mode. In the CJD mode, the arc assumes a bell shape and no severe constriction is observed at the anode. In the AJD mode, the arc is severely constricted at the anode with an anode jet pumping surrounding gas and anode vapors away from the anode, provided the anode material is molten and starts to vaporize. Anode and cathode jet

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Fig. 14 Transferred arc configuration (a) CJD, (b) AJD, (c) EA modes

meet somewhere between cathode and anode (generally closer to the anode than to the cathode) forming a stagnation area forcing gas flow in radial direction. When the cathode jet is too weak, there is no more a well-defined anode jet and the arc strikes at the anode erratically forming tiny plasma columns (see Fig. 14c), resulting in a rather unstable configuration called Erratic Attachment (EA) mode (Lerrol 1987–1988). A separate discussion of the conditions under which an arc becomes CJD or AJD will be presented in Sects. 3.2.3a and 3.2.3b. The effect of heat transfer to the anode caused by the cathode jet has been well demonstrated by Nagashima et al. (1988) who measured the heat flux distribution on a water-cooled copper anode as a function of the arc current, I, the argon plasma gas flow rate, m_ pl , the anode cathode distance dAK. This heat transfer was characterized by the maximum heat flux measured at the cathode axis qAm divided by the half width rw of the heat flux profile q*w (W/m3), a quantity, which represents the sharpness of the heat transfer distribution. In summary, – q*w increases with I for a given dAK; this increase is due to higher values of qAm when rw increases more slowly with I. – q*w decreases with m_ pl for given I and dAK: qAm increases little but rw increases drastically with m_ pl . With a conical injection (see Fig. 21b of ▶ Chap. 12, “Thermal Arcs”) for a given m_ pl the power dissipated in the anode QA as well as rw increase with the plasma gas velocity, i.e., when the gap between the two cones is reduced (Coudert et al. 1993). – Any gas blown close to the anode surface induces a transition from the CJD to the AJD mode (depending on its thermal conductivity) and an increase of the q*w value (Nagashima et al. 1988). – q*w increases with dAK for given I and :m pl which is mainly due to a decrease of rw and a slight increase of qA (at least for dAK < 12.5 mm). Measurements (Coudert et al. 1993) of the power dissipated in the anode QA show that when dAK reaches a value where the CJD mode is shifted to the AJD mode, QA remains almost constant. Thus, the mean heat flux dissipated at the anode increases drastically.

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The effect of the nature of the plasma gas has been studied at Limoges (unpublished results) and by Gauvin (1989) by comparing transferred arcs working with the same gas injection geometry, the same plasma gas flow rate, arc current, and anode cathode distance. The power dissipated at the anode when shifting from argon to nitrogen plasma gas is increased due to the constriction of the arc, resulting in a higher velocity of the cathode jet, and the transfer efficiency (ηA ¼ QA =P where P is the power dissipated in the arc) is increased by about 15%. The mean heat fluxes dissipated at the anode are in the range from 107 to 108 W/m2 in the CDJ mode, and they can reach 1010–1011 W/m2 in the AJD mode. In the first case, the anode can be cooled by standard techniques while in the latter, it will be impossible to provide such high cooling rates and local anode melting and vaporization will occur, unless the arc root is kept in continuous motion over the surface of the anode. The anode evaporation has different effects on anode heat transfer depending on the arc attachment. In the CDJ mode, the heat flux is distributed over a rather wide area, and with the cathode jet impinging on the anode, almost no metal vapor diffuses into the arc which comprises two parts: the plasma column consisting of plasma gas at temperatures similar to that observed with a cold anode and a thin layer where metal vapor dominates at a lower temperatures (Menart and Lin 1999; Degout and Catherinot 1986). In extractive metallurgy, such a configuration results in low losses by evaporation as shown by Maske (1985) (less than 3 wt% of Cr in FeCr extractive metallurgy). According to measurements by Gauvin (1989), the heat transferred to the anode with and without evaporation is about the same. When the ADJ mode is established, the metal vapor penetrates the arc increasing its electrical conductivity σe. As a consequence, as measured by Gleizes et al. (1995), the plasma temperature decreases, reducing the enhanced radiation due to the presence of copper vapor (Gleizes et al. 1995). The voltage decrease is partially compensated by the constriction of the arc due to the enhanced radiative losses. In the ADJ mode, evaporation increases significantly (13–14% of evaporated Cr according to Maske (1985)). Parisi and Gauvin (1991) measured the heat-transfer rates from an axially enclosed transferred arc on a molten metal bath to a surrounding water-cooled cylindrical sleeve, 150 mm high. They showed that the energy radiated by the plasma column within the enclosure, and that radiated from the arc below the enclosure were responsible for most the heat load to the surrounding sleeve.

4.2

Dynamic Behavior

4.2.1 Anode Perpendicular to the Arc Axis In this section, the dynamic behavior of a low-intensity arc (I = 40 A) but with a strong gas constriction (conical injection along the cathode cone, see Fig. 21b in Part II, ▶ Chap. 12, “Thermal Arcs”) will be discussed. The problem is rather complex and the observed phenomena depend on the relative velocity between the arc (the cathode is moved parallel to the anode) and the anode surface, the nature of the anode surface (mainly through its thermal diffusivity), its oxidation state, and its roughness. Sobrino et al. (1995) have initiated such studies by using video camera

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Fig. 15 Transferred arc with the cathode moving with respect to the anode at vAK. (a) Photos of the arc (top), Arc Voltage fluctuation (middle), and arc trace (bottom), for vAK = 60 mm/s, (b) Photos of the arc (top), Arc Voltage fluctuation (middle), and arc trace (bottom), for vAK = 240 mm/s

pictures of the arc synchronized with the voltage fluctuation and the arc velocity. Figure 15a, b show typical results obtained with a transferred arc between a thoriated tungsten cathode with conical gas injection and a copper anode (mean surface roughness Ra ~ 0.7 μm). Figure 15a refers to a relative velocity between the arc and the anode surface of vAK = 60 mm/s, while Fig. 15b refers to vAK = 240 mm/s. For both cases, the arc current was I = 40 A, the interelectrode spacing 5.0 mm, and the cathode gas flow rate 2.0 slm (Ar). The time elapsed between two successive frames is 0.04 s. In Fig. 15a, the first three and the last three frames are associated with a voltage ramp. During one voltage ramp, the arc root remains attached at the same location while the arc column moves and remains “rigid.” The location of the anode attachment becomes molten. After two successive restrikes, the voltage fluctuation tends to fade out leaving an almost continuous molten track of the arc. At the end when the arc is stationary, one can see the central molten zone with the redeposited copper in its fringes. When vAK increases (see Fig. 15b), the same phenomenon is observed with steep voltage ramps and only a few continuous molten tracks. With very smooth surfaces (Ra = 0.15 0.05 μm) at low velocity (vAK = 60 mm/s), the arc root moves at the same velocity as the cathode (no saw tooth shape voltage), but as soon as the velocity reaches 240 mm/s a few restrikes are observed. When the surface roughness increases, both the frequency and length of the voltage jumps increase systematically. All these observations indicate that the time, ts, necessary to establish an anodic steady state is not negligible (of the order of a few ms) and the characteristic time of any movement of the arc has to be compared with ts. It is worth noting that similar observations were obtained with a 200 A arc with the same type of constriction and a cathode tip-anode gap of 25 mm. In both cases (60 and 200 A), the arc attachment at the anode was of the CJD type.

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4.2.2 Anode Parallel to the Arc Axis This is the case of most plasma torches using either stick or button-type cathodes or working with cold cathodes, and this configuration creates the most important voltage instability. The problem is complex because a cold gas boundary layer exists between the plasma and the anode where nonequilibrium conditions exist with Te > Th (Trelles et al. 2006, 2007), which is the condition necessary for the current to flow across this boundary layer (Paik et al. 1993; Dinulescu and Pfender 1980). In this case, as shown in Fig. 10, ▶ Chap. 12, “Thermal Arcs,” the arc attachment at the anode is in the AJD mode with high local heat fluxes up to 109 W/m2. The first study of Wutzke et al. (1968) with a superimposed flow of argon and a flat anode has shown that the arc at the anode is characterized by a motion of the anode arc root driven by the flow followed by a sudden restrike near the cathode between the arc column and the anode as illustrated in Fig. 16. This behavior results in saw tooth-shaped voltage traces because there is a linear relationship between the arc voltage and arc length (Wutzke et al. 1968).

Fig. 16 Frames from high-speed movie (5000 frames/s) of arc (see configuration Fig. 10 Part 2, ▶ Chap. 12, “Thermal Arcs”) in restrike mode (argon gas) (Wutzke et al. 1968)

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Measurements performed with a segmented anode have shown that the arc column root seems to split at the attachment point and moves from one segment to the next. In this case, part of the current flows through one segment and the rest flows through the other. Due to this restrike mode, the heat flux is more uniformly distributed along the anode (time integrated heat flux measured with water-cooled segments which have a response time on the order of 1 s while the frequency of the restrike mode is between 500 and 2500 Hz). With two parallel anode plates at the same distance from the cathode axis, the breakdown on either plate seems to be nearly random in some cases, whereas in others the restrike occurs preferentially on the opposite anode plate. This seems to be associated with the deflection of the arc column by the anode jet which pushes the arc column closer to the opposite anode plate. With cylindrical nozzles used, for example, in plasma spraying (stick type cathode and generally cylindrical anode nozzle), the connecting arc leg between the main arc column and the anode surface (see Fig. 16) is exposed to several forces. The dominant one is the drag force exerted by the cold gas in the boundary layer. Since the gas density in the connecting arc leg is much lower than that in the cold boundary layer, the cold gas in this boundary layer will mostly flow around the connecting arc leg, exerting a drag in the downstream direction. The drag force intensity is linked to the boundary layer gas velocity, as well as on the mass density of the gas. In general, the arc-anode attachment movement has been divided into three different categories as shown in Fig. 17 (Duan and Heberlein 2000), the steady mode of operations with very low fluctuations of the voltage trace, the takeover mode of operation with random fluctuations, and the restrike mode where the arc

100 90

restrike

Voltage (v)

80 70 60

50

take-over

40 30 20

steady 0

1.0

2.0 Time (ms)

3.0

4.0

Fig. 17 Typical voltage traces for the restrike mode, the takeover mode, and the steady mode of operation of a plasma torch (Duan and Heberlein 2000)

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585

attachment moves downstream inside the anode nozzle until an upstream restrike occurs with a new connection between the arc column and the nozzle (Coudert et al. 1994, 1995a, b; Planche et al. 1997; Duan and Heberlein 2000, 2002; Dorier et al. 2000). A clear correlation exists between the mode value and the thickness of the boundary layer. A thicker boundary layer is associated to restrike mode, which is typical for higher gas flow rates, higher hydrogen or helium percentages in the plasma gas, nitrogen plasma gas, or lower currents. When operating with pure Ar or Ar/He mixtures in the takeover mode, addition of hydrogen or nitrogen beyond a certain amount constricts the arc and increases the boundary layer thickness with a shift to restrike-like behavior. Increasing the arc current reduces the anode boundary layer thickness and the arc root fluctuations, but the effect of increasing the total mass flow rate and the fraction of the secondary gas hydrogen can be different for different nozzle designs and different if the primary gas is argon or nitrogen (Nogues et al. 2007). For the steady mode, only minimal fluctuations are observed, but unfortunately the lifetime of the anode is very limited. In the restrike mode, the total amplitude of fluctuations of the voltage is between 40% and 100% of the average voltage value Vm, with frequencies between 2 and 6 kHz. In the takeover mode, this fluctuations amplitude is in the order of 20–50% of Vm, with a much wider Fourier spectrum of their frequencies. For plasma spray torches, an arc motion to distribute the heat input to the anode is needed to limit the anode erosion, but strong variations of the voltage and the power must be avoided, which, for example, limits the vol% of H2 to 5–6 vol%. In the restrike mode, the situation varies as soon as the anode surface starts to erode (Ra > 0.5 0.1 μm) as shown by Zhukov (1979) and Coudert et al. (1995a, b). It seems that the arc root at the anode surface does not move continuously; it rather jumps to a new position. It is the arc junction (Fig. 18a), which is pushed downstream increasing the voltage (saw tooth shape) up to a value where a short circuit creates a new arc root either upstream (Fig. 18c) or downstream (Fig. 18b) of the preceding arc root. Figure 19 shows the corresponding time evolution of the voltage. With a new nozzle, the time intervals, τ1, between two successive voltage jumps are rather dispersed (Planche et al. 1995), and they may correspond to an arc root displacement (and not arc shunting) as observed by Wutzke et al. (1968). Ten to twenty minutes later the dispersion of the values of τ1 becomes much smaller and it seems reasonable, according to what has been described in the previous section, to assume that the arc root does not move anymore but rather jumps between asperities created by arc erosion. The amplitude of the voltage fluctuations becomes also lower and more regular as soon as the anode surface is eroded. For given working conditions, the arc erosion is distributed along the anode nozzle corresponding roughly to 1.5–2 times the nozzle diameter. Under restrike mode conditions, the anode erosion corresponds to between 5.0  1011 and 5.0  1010 kg/A.s. Lawton (1971) and Hu et al. (2013) studied the effect of a magnetic flux at the cylindrical anode of a torch with a stick type cathode of doped tungsten. The measured voltage waveforms indicated that the arc root attachment mode would be controllable by an external magnetic field. However, only the appropriate external

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Fig. 18 Schematic of the restrike mode within a cylindrical nozzle: successively at t = 0, at t = δt (downstream short circuit), at t = δt + ε (upstream short circuit)

90

Voltage, V (V)

70 60 Um(ti) Um(ti+1)

50 ti

40 0.0

0.1

0.2

σi

τi 0.3

ti+1

0.4

0.5

0.6

Time, t (ms)

Fig. 19 Representation of the corresponding voltage fluctuations

magnetic field could inhibit and reduce the instability of the plasma torch and prolong the life of the anodic electrode. The constricted arc attachment between the arc column and the anode surface with strong heat fluxes enhances the anode erosion at the arc root. Yang and Heberlein (2007) show that at the arc root instability exists leading to an electron

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587

temperature run-away and localized evaporation of anode material. After a certain working time, the erosion patterns can result in preferred arc attachment spots leading to increased erosion and establishment of preferred tracks for the arc attachment movements. Rigot et al. (2003), studying the erosion patterns and the associated arc voltage traces, have shown that the preferred anode attachment slightly moves upstream and becomes slightly more constricted. The amount of erosion correlates well with the residence time of the arc at a specific location. The residence time varies from 30 to 150 μs, with the lower values occurring much more frequently. For a new anode, the attachment times remains smaller than 150 μs, while for a worn anode shows an increasing fraction of residence times at a specific attachment location exceeding this value. Of course the anode erosion can also be strongly influenced by the anode design and the gas injection method (Sun and Heberlein 2005). The long-term stability of a spray torch shows that over the period between 10 and 40 h of operation (depending on the working conditions and the number of restarts) the voltage and torch power decrease steadily, resulting in a decreasing power level, which must be compensated by an increase of the arc current. After this steadily power decrease, electrodes must be replaced. Coudert et al. (2007; Coudert and Rat 2008) and Rat and Coudert (2010, 2011) showed that in fact the mean arc spot lifetime did not exclusively depend to the cold boundary layer (CBL), and they showed that the restrike mode is superimposed to the main fluctuation of arc voltage, observed around 4–5 kHz. This main oscillation is essentially due to compressibility effects of the plasma-forming gas in the rear part of the plasma torch, that is, in the cathode cavity. The plasma torch behaves like a Helmholtz resonator. The system is best described in terms of a mass-spring model. The cold gas in the cathode cavity is equivalent to the spring and the plasma in the anode nozzle channel to the oscillating mass (Coudert et al. 2007) the movements of which are superimposed on the continuous flow. Rat and Coudert (2010) used an acoustic stub mounted on the torch body and connected to the cathode cavity. It has permitted to significantly reduce the mean relative amplitude of arc voltage. Meanwhile, the previous beating phenomenon in the envelope of the arc voltage signal was altered by the use of the acoustic stub, showing it was indeed coupled with the plasma torch. The restrike mode can be almost completely suppressed with a diffuse attachment of the arc at the anode when the anode surface is at elevated temperatures (~3000 K at the point of the arc attachment) (Malmberg et al. 1991; Neumann 1975). It should be pointed out that among the observed modes of arc operation in a plasma torch, the takeover mode is the most benign mode giving rise to the lowest anode erosion rates.

4.3

Anode in a Molten State

When the anode is molten, which is the case when welding, cutting, and melting for metallurgical operations, the problem is different from that with solid anodes. The vapors generated can penetrate the plasma and modify its properties (thermal conductivity, radiation, and electrical conductivity) and the surface tension of the liquid, varying with its temperatures distribution within the molten zone can

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substantially modify the direction of recirculatory flow in this part of the anode (Lowke 2003). Tanaka et al. (2003) have modeled free-burning arcs and their electrodes for clarifying the heat transfer phenomena in the welding process and the effects controlling the weld penetration. They showed that the drag force of the cathode jet and the Marangoni force mainly dominated the calculated convective flow in the molten anode. Different surface tension properties can change the direction of recirculatory flow in the molten anode and dramatically vary the weld penetration geometry. Strong convective flow outward at the surface of the weldpool can lead to a shallower weld than for heat transfer by conduction alone. According to Osterhouse et al. (2013), in plasma arc cutting the arc is surrounded by a high velocity vortex flow, melting and blowing away the molten metal from the kerf. The arc attaches inside the kerf, providing a large heat flux and determining the flow dynamics of the plasma. The location of the arc attachment is very important because it strongly affects the fluid dynamics inside the kerf and the heat flux at the arc attachment. Moreover, the Joule heating in the arc provides a huge energy source above the anode attachment. Thus, it is very important to control the arc attachment, which can be – to a certain degree – controlled by the cutting speed and the stand-off distance (distance between the torch exit and the work piece).

5

Nomenclature

Units are indicated in parentheses; whenever no units are given, the parameter is dimensionless.

5.1

Latin Alphabet

A A’

Cathode surface on which the arc moves (m2) Constant for Richardson-Dushman equation (A’ = 1.2  106 A/m2.K2)

B B B’ CD dAK da0 dc0 de dK dKS e et

Self-magnetic induction (Vs/m ) Magnetic field strength (Vs/m2) Coefficient depending on plasma gas (m/A0.5) Aerodynamic drag coefficient (dimensionless) Cathode tip-anode distance (m) Thickness of the anode region (m) Thickness of the cathode region (m) Current-carrying arc diameter (m) Diameter of imbedded tungsten cathode (m) Diameter of the molten cathode spot (m) Electron charge (e = 1.6  1019 A.s) Maximum wall thickness which can be eroded (m) Electric field (V/m)

!

!

E E

2

Electric field strength (V/m)

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Electrode Phenomena in Plasma Sources

589

EI FS Gc h I

Ionization energy (eV) Surface drag force (N) Erosion rate (kg/C) Plasma enthalpy (J/kg) Arc current (A)

j

Current density vector (A/m ) Current density at the cathode (A/m2) Electron current density (A/m2) Ion current density (A/m2) Current density at the cathode spot (A/m2) Boltzmann constant (k = 1.38  1023 J/K) Arc length (m) Arc length in the direction perpendicular to the electrodes (m) Mean free path (m) Maximum weight of cathode material, which can be eroded (kg) Equation the molecular weight of the ions (kg) Erosion rate at the cathode (kg/s) Plasma gas flowrate (slm or kg/s) Density of electrons (m3) Density of species r (m3) Power dissipated in an arc (P = V  I) (W) Pressure (Pa) Heat flux at the cathode (W/m2) Heat removed from the anode by the cooling water (W) Heat removed from the cathode by the cooling water (W) Local heat flux at the anode (W/m2) Maximum heat flux at the anode (W/m2) Maximum heat flux divided by the half width rw of the heat flux profile (W/m3) Mean integrated surface roughness (μm) Radial coordinate (m) Half width at mid height of the heat flux profile (m) Electron temperature (K) Heavy species temperature (K) Ion temperature (K) Arc root dwell time at the cathode (s) Arc velocity (m/s) Voltage (V) Anode fall (V) Potential drop at the anode (V) Cathode fall (V) Potential drop at the cathode (V) Ionization potential (V) Mean fluctuating voltage (V) Relative velocity between the arc and the anode surface (m/s) Arc velocity or radial component of velocity (m/s)

!

jc je ji js k ‘ ‘a ‘e M Mi m_ c m_ pl ne nr P p Q(r,t) QA QK qA qAm q*w Ra r rw Te Th Ti ts v V Va Va' Vc V'c VI Vm vAK v

2

590

5.2 εo η !

ϕ ∅ Φa ϕc Φc κ μ μ0 ρ ρcu σe τ1

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Greek Alphabet Vacuum dielectric constant (εo = 8.86  1012 As/V  m) Transfer efficiency (%) 2 Heat flux vector (W/m ) Thermal diffusion coefficient Work function of the anode material (V) Thermionic work function (V) Cathode work function (V) Thermal conductivity (W/m K) Molecular viscosity (kg/m.s) Magnetic permeability of vacuum (μ0 = Hy/m) Mass density of the plasma (kg/m3) Mass density of copper cathodes (kg/m3) Electrical conductivity (A/V.m) Life time of the cathode (s)

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Arc Stabilization in DC Plasma Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Free Arc-Length Constrictor Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fixed Arc-Length Constrictor Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Gas Flow Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Magnetic Rotation of the Arc Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Electrode Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Hot Cathode Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Cold Cathode Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Anodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Cooling of the Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Erosion of the Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Ignition of the Arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Arc Ignition by Electrode Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Arc Ignition Using a Wire or Rod Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Arc Ignition by Pre-ionization with a High-Voltage, High-Frequency Discharge . . . 5.4 Starting Circuitry and Power Supply Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

596 597 599 599 600 603 606 612 612 621 624 629 631 634 634 635 635 636

Emil Pfender: deceased. M. I. Boulos (*) Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada e-mail: [email protected] P. L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France e-mail: [email protected] E. Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA © Springer Nature Switzerland AG 2023 M. I. Boulos et al. (eds.), Handbook of Thermal Plasmas, https://doi.org/10.1007/978-3-030-84936-8_15

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6 Performance Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Plasma Torches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Interaction of the Plasma Jet and the Surrounding Atmosphere . . . . . . . . . . . . . . . . . . . . . 6.3 Transferred Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Latin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

636 636 641 645 649 649 652 652

Abbreviations

AC AJD AS BOF BTC CJD DC EAF i.d. ISPC ITSC MV OFHP OMA PEC PI PID scmh slm SVC STATCOM TIG USGPM UTSC WTC

1

Alternative current Anode jet dominated AeroSpatial Blast oxygen furnace Button-type cathode Cathode jet dominated Direct current Electric arc furnace Internal diameter International Symposium on Plasma Chemistry International Thermal Spray Conference Mode value Oxygen-free high-purity copper Optical multichannel analyzer Plasma energy corporation Proportional integral Proportional integral differential Standard cubic meter per hour Standard liter per minute Static VAR compensator Static synchronous compensator Tungsten inert gas US Gallon per minute United Thermal Spray Conference Well-type cathode UTSC

Introduction

Because of the broad range of present and potential industrial applications of thermal plasma technology, there is no universal plasma torch design that fits all. These can vary widely depending on whether they are to be used for cutting and welding, plasma spraying, metallurgical applications, treatment of waste materials, or the synthesis and processing of high-purity materials. In each of these cases, the design of the plasma torch has to aim for the following objectives;

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• High-energy efficiency, since the principal role of a plasma torch is to convert electrical energy into thermal energy at the highest level of energy availability • Heating of a wide range of gases from inert (Ar, He, etc.) to chemically active (N2, H2, O2, air, CH4, CO2, etc.) • Long electrode life, which has a direct impact on process reliability and economics • Safety and reliability of operation, which is a prime requirement for acceptance of the technology on an industrial scale In this chapter, the basic design features of DC plasma torches and transferred arcs are reviewed. This covers arc stabilization techniques, which was discussed in some details in Part II, ▶ Chaps. 12, “Thermal Arcs,” and ▶ 13, “Electrode Phenomena in Plasma Sources.” Attention is given to torch design based on either the concepts of using a free or fixed arc length and aerodynamic control of the flow pattern in the discharge region surrounding the arc. This is followed by details of electrode designs and their impact on electrode life. A review is presented of typical plasma torch current-voltage characteristics, its energy efficiency, and the interaction of plasma jets with the surrounding atmosphere. Detailed discussion of the design of plasma torches for specific applications is given in subsequent chapters of this part: Part II, ▶ Chaps. 18, “RF Inductively Coupled Plasma Torches.” and ▶ 19, “HighPower Plasma Torches and Transferred Arcs.”

2

Basic Concepts

The design and performance of DC plasma sources has evolved considerably over the past five decades. As schematically represented in Fig. 1, a broad range of plasma torches and transferred arc furnaces were developed over this period depending on their ultimate use and the process needs. These can be grouped, based on the nature of the electrodes, into two broad categories: hot-cathode sources (Fig. 1a, c) and cold-cathode sources (Fig. 1b, d, e). Alternately, they can also be grouped, based on the electrode configuration, into blown-arc sources (Fig. 1a, b) or transferred arc sources (Fig. 1c, d, e). In the latter case, we can also identify the configuration given in Fig. 1c, d as being of a straight-polarity and (Fig. 1e) in “reverse-polarity.” In all of these cases, the choice of the electrodes as well as of any other component of an arc heater depends on the desired performance for the specific application in which the plasma torch is to be used (Arc Plasma Processes 1988; Barcza 1987; Camacho 1988; Fauchais and Vardelle 1997; Hamblyn 1977; Sololenko and Zhukov 1994; Pfender 1978; Sololenko 2000; Zhukov 1979; Zhukov and Zasypkin 2007). In certain applications, consumable electrodes (for example, carbon, graphite or metal) may be acceptable or even desirable, whereas in other applications extreme care must be exercised to avoid contamination of the processed material by the electrodes. In such cases, the use of nonconsumable electrodes is to be considered. In general, electrode erosion can be minimized or, at least, spread

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Plasma gas

c

Plasma gas

d

Plasma gas

e

Plasma gas

Plasma gas

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Plasma gas

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Fig. 1 Arc configurations commonly used for the integration of an arc in a plasma generating device

uniformly over the whole surface of the electrode, through the use of rapid movement of the arc root by means of flow and/or magnetic fields, which avoids localized electrode erosion, causing premature electrode failure and costly process shutdowns. In dealing with the design of arc plasma sources, whether gas heaters (blown-arc plasma torches) or transferred arc furnaces, it is important to recognize the gradual shift of the plasma technology from AC to DC electric sources because of the improved arc stability and the generally reduced electrode erosion in DC sources compared to AC devices. This is a direct consequence of the ability to optimize the choice of the electrode material to its polarity in DC sources, which is obviously not possible in AC devices. It may also be noted that DC power supply technology has evolved considerably through the availability of high-power, fast switching thyristors, which makes most DC power supplies costing only 5–10% more than comparable AC sources at the same power rating. It should be pointed out, however, because of the phase control strategy on the rectifiers, the arc voltage swings induce large reactive power variations on the power network and a static VAR compensator (SVC) or a static synchronous compensator (STATCOM) is always added to avoid flicker effect (Ladoux et al. 2005). Compared with AC furnace, the power grid disturbance with DC equipped with SVC and shift control is considerably lower than that of AC furnace (about 7% of the power grid disturbance of AC furnaces) (Jones et al. 2011). As an indication of this gradual technology shift, especially in high-power sources, it is noted that in 1988, 125 DC furnaces working at 100 MW, with graphite cathode and bottom anode below the molten bath, had replaced AC ones. Presently 75% of the existing AC furnaces are to be replaced by DC ones due to fewer flickers on the lines, less noise, lower graphite consumption (1–1.5 kg/t against 1.8–3 kg/t with AC), less wear of refractory liners, and a better power control. This is not to

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DC Plasma Torch Design and Performance

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imply, however, the gradual disappearance of AC plasma sources, since in specific applications, such as the metallurgical industry, three-phase, AC electric arc furnaces, using graphite electrodes, have also regained interest for heating molten metal baths with no bottom electrode in the crucible (Neusch€ utz 2000). The production of steel by DC arcs, which represented about 10% of the total production in the sixties, was up to 35% around the turn of the century.

3

Arc Stabilization in DC Plasma Sources

3.1

Free Arc-Length Constrictor Design

In the simplest system, the arc is forced through a constrictor tube located between the cathode and anode. The power input per unit length of a constricted arc and the associated enthalpy level increase significantly. This principle is found in many conventional plasma torches, such as those used in plasma spraying, for example, which are equipped with nozzle-shaped anodes with the nozzle serving as well as an arc constrictor (see Fig. 2 which represents a schematic of a torch with a hot sticktype cathode, and axial plasma-forming gas injection). However, power levels are generally below 50–60 kW. Such anodes are also used in DC plasma torches with a button-type hot cathode or a well-type cold cathode, associated with a vortex injection of the plasma-forming gas. A configuration where the arc is allowed to strike at any point along the anode nozzle is known as a “free arc length” torch. As it will be shown later, the arc strikes at the anode when the cold boundary layer overlying the anode/nozzle wall has been heated sufficiently (T > 3000 K) by the plasma column. Thus, for a given arc Main arc root attachment

Plasma forming gas

Upstream arc short circuit

downstream arc short circuit Entrained cold gas

Cathode

Cold gas engulfment Vortex ring

Insulator Cold boundary layer Hot boundary layer

Plasma column Anode

Fig. 2 Schematic representation of an atmospheric pressure DC plasma-spraying torch (Fauchais and Vardelle 1997)

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current, the arc voltage will depend on the arc constriction, i.e., on the nozzle internal diameter (i.d.), the composition, and flow pattern of the plasma-forming gas.

3.2

Fixed Arc-Length Constrictor Design

In the transferred arc operation, the length of the arc is fixed and the nozzle, if any, is kept on floating potential serving only as an arc constrictor. However, the arc length has to be adapted to avoid double arcing (between the cathode and the nozzle, and/or between the nozzle and the anode). The criterion associated with double arcing is given in Eq. 36 of ▶ Chap. 12, “Thermal Arcs.” Four different configurations are commonly used in DC plasma torches (see Part II, ▶ Chap. 12, “Thermal Arcs,” Sect. 3).

3.2.1 Cascaded Arc The wall-stabilized cascaded arc, shown schematically in Part II, ▶ Chap. 12, “Thermal Arcs,” Fig. 23, served primarily as a basic research tool (Kroesen et al. 1990) at that time. As already pointed out of Part II, ▶ Chap. 12, “Thermal Arcs,” to achieve long arcs the use of segmented, water-cooled, and electrically insulated disks (usually Cu) was necessary to avoid double arcing through the nonsegmented metal wall (arcing from the cathode to the metal tube and from the metal tube to the anode). Equation 37 in Part II, ▶ Chap. 12, “Thermal Arcs,” defines the maximum thickness of each insulated disk to avoid double arcing. At the end of the nineties, however, the spray industry realized the interest of such cascaded arcs to increase the torch power level without increasing the arc current (Zierhut et al. 1998). Indeed, the heat losses in the cooling system of the torch vary almost linearly with the arc current, but the variation with the voltage is about V0.2. Thus, doubling the arc voltage for the same arc current does not double the heat losses in the cooling system. For high-power arcs, Zhukov (Zhukov 1994; Zhukov and Zasypkin 2007) and other Russian workers have demonstrated the need for additional gas injection. This is illustrated in Fig. 3 for a torch with a button-type cathode, a vortex gas injection, and a grounded anode separated from the cathode by a sleeve at floating potential (often called neutrode). The structure of the arc exhibits three zones: • In the first zone, A-B, the arc is stabilized on the channel axis and a cold layer is formed at the wall limiting the heat transfer between the relatively cold medium surrounding the arc and the arc itself. The zone is characterized by relatively low electric field strength and low heat losses (mainly by radiation). At the end of this zone, the cold boundary layer is heated and becomes turbulent. • In the second zone, B-C, the turbulent boundary layer becomes fully developed and a large amount of hot gas is dispersed in radial direction. This zone is often called transient zone. The arc column becomes less stabilized revealing magnetohydrodynamic instabilities. As the losses to the wall increase, the arc constricts

14

DC Plasma Torch Design and Performance

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itself and the electric field starts to increase. Carefully designed cold gas injection at the beginning of the second zone may lengthen the first zone. • The third zone, C-D, corresponds to the fully developed turbulent flow, the arc being often divided into a multitude of conducting channels. In this area, the arc usually strikes to the wall. • The electric fields as well as the heat losses (see Fig. 3) increase sharply in the fourth zone D-E reaching a maximum in this zone accompanied by a significant reduction of the torch lifetime.

3.2.2 Segmented Constrictor with Gas Injection As demonstrated by Zhukov (1994) and his coworkers, the need for additional gas injection arises from the behavior of long arcs stabilized by a vortex flow such as shown schematically in Fig. 3. Figure 4 shows the principle of a torch with a segmented-nozzle with cold gas injection between each pair of segments for regeneration of the cold boundary layer by cooling the fully developed hot turbulent boundary layer (see Fig. 3, zone D-E). Anode

Separation sleeve

1 2 3

Button Cathode

E, Q

A

B

C

D

E

ET.I ηi

0

Q∑

Es.I Qb

Qb

Fig. 3 Structure of interaction of the arc with the surrounding gas: (1) the boundary of the jet core; (2) electrical arc; and (3) the boundary of the thermal layer of the arc (Zhukov and Zasypkin 2007)

Sheath gas

Magnetic coil

Anode

Cathode

Fig. 4 Principle of an arc constrictor with insulated segments and intersegment gas injection (Zhukov 1979)

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It should also be noted that a magnetic field may help the anode root displacement at the anode end of the torch. Additional sheath gas has to be injected between the segments in order to insure their thermal protection (Zhukov 1994). The fraction of the total mass flow rate of the sheath gas, which is injected in the ith segment, is defined as, xi = (ρivi/ρovo), whose value can vary between 0.1 and 1). The index “o” corresponds to the mean gas flow rate in the torch. High values of xi result in high injection velocities vi (0.5–0.7 of the sound velocity of the cold plasma-forming gas) corresponding to high Reynolds numbers (Re = 104–105). For these conditions (Zhukov 1994), established a semiempirical relationship between the mean ratio of the segment length ZH and its diameter d by ZH ¼

ZH 1:3  Re0:25

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ¼
d 1 þ 1:85  103 I= d  μ0 h0 σ0

(1)

With I, μo, ho, and σo being, respectively, the current (A), the viscosity (kg/m. s), the enthalpy (J/kg), and the electrical conductivity (A/V. m) of the plasma at the temperature To corresponding to 1 vol% of electrons. The Reynolds number, Re, is calculated using the following expression: Re ¼

Gd μ0

(2)

where G is the mass flux of the plasma gas (G = ρ.vi)(kg/m2.s), d the internal diameter (i.d.) of the cylindrical nozzle, and μ0 is the viscosity of the plasma (kg/s.m) at the temperature To corresponding to 1% of electrons. Equation 1 has been checked for (I/d) values varying from
1 to 104 A/m  and Re from 104 to 105. For these conditions, Eq. 1 predicts values of ZH :Re0:25 between 1.3 and 0.8. High electric fields and correspondingly high voltages (up to 6–8 kV) are feasible with such torches, which are commercially available (e.g., Acurex [Aerotherm], SKF [Santen et al. 1986; Thörnblom 1989], and Aerospatiale (Van den Brook et al. 1987), using high flows of plasma forming gases (from 200 to 300 m3/h). Extensive developments in this area have also been reported in the former USSR (Pustogarov 1994; Zhukov 1989).

3.2.3 Transpiration-Cooled Constrictor A torch using a transpiration-cooled constrictor is shown in, Part II, ▶ Chap. 12, “Thermal Arcs,” Fig. 24. In this case, porous tungsten, molybdenum, chromium, or W-Cu may be used. The gas injection through the porous interelectrode insert increases the electric field drastically (values between 5000 and 20,000 V/m), resulting in torches working in the kV range, provided high specific gas flow rates are used (between 1 and 100 kg/m2.s) (Pustogarov 1994). Many torches of this type have been developed in the former USSR, but they are rarely found in Western countries.

14

DC Plasma Torch Design and Performance

Vortex gas injection

603

Water cooling Magnetic coil

Button-type cathode

d1

l1

d2

l2

Fig. 5 Principle of a plasma torch with a step change in the anode internal diameter (quasi-fixed arc length) (Zhukov and Zasypkin 2007)

3.2.4 Cylindrical Torch Nozzle with a Step Change in Its Diameter The fourth configuration uses a nozzle with a step change in its diameter (see Fig. 5). The length ‘1 of the smaller diameter (d1) part has to be shorter than the length ‘ca that the arc would assume without the step change in the anode diameter d2. Under such conditions due to the recirculation, which occurs downstream of the step change, the arc strikes close to this step change.

3.3

Gas Flow Pattern

The main purpose of an arc gas heater is the well-controlled, efficient heating of gases to a certain enthalpy level. An important prerequisite for achieving this goal is the proper choice of the gas flow pattern and the resulting interaction of the arc with the superimposed flow, including stabilization of the arc column. There are a number of options available to the designer: • • • •

Coaxial flow Cross flow Radial flow Vortex flow These options refer to the flow pattern with respect to the arc column.

3.3.1 Coaxial Flow In the case of coaxial flow, the gas is usually introduced from the cathode end of the arc heater flowing parallel to the axis of the arc and forming a shroud of cold gas around the arc column. Some of the cold gas from the shroud permeates gradually

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into the arc is heated to the prevailing temperature level and accelerated downstream as shown schematically in Fig. 2 for a constricted arc. The coaxial flow pattern is particularly useful in conjunction with arc constrictors. It is widely used in high enthalpy arc heaters as well as in plasma torches. Sometimes a vortex component is superimposed to the coaxial flow. As already pointed earlier, the coaxial flow may result in recirculation along the wall of a conical tip cathode, and in this case, it is preferable to inject the gas parallel along the cathode surface (conical injection). With cold plasma-forming gas injection velocities up to 50 m/s (generally corresponding to Re < 1000–2000) are required, the distance between the cathode wall and the surrounding injection tube may be as small as 0.2–0.5 mm and centering of the flow becomes a serious problem. In order to avoid too high cold gas velocities close to the molten cathode tip, the concentric conical injection tube is kept just long enough to ensure laminar flow along the cathode. The end of the injection cone may be flared to reduce the problem of concentricity of the flow.

3.3.2 Cross Flow Arcs in cross-flow have been extensively studied in the 1960s and 1970s, and a few applications have been also reported, which use this principle for the entire length of the arc column, for example, in coaxial arc heaters (Konotop 1994). However, almost in every conceivable arc heater configuration, some parts of the arc (sometimes only the cathodic or anodic parts) are exposed to cross-flow, which allows arc root displacement at the electrode surface. In many arc heater designs, this displacement is a desirable feature for reducing electrode erosion 3.3.3 Radial Flow In a few instances, arcs have been exposed to radial flow as, for example, in the case of transpiration-cooled constrictors. In arc constrictors with insulated segments and intersegment gas injection, the injection is more in tangential rather than in radial direction. 3.3.4 Vortex Flow Besides coaxial flow, vortex flow is the most common flow pattern found in arc gas heaters. This is largely due to the fact that a vortex flow provides an excellent stabilizing mechanism for the arc, even in rather wide tubes. A vortex flow also leads inherently to an efficient thermal protection of the wall of the constriction tube, since the cold gas layer at the tube wall remains intact over longer distances than in the case of coaxial injection. This is achieved, however, at the expense of flow rates which are much higher (up to one order of magnitude) compared to those used with coaxial injection for the same power level. The length, z, of the zone where the cold boundary layer thickness is still sufficient to prevent turbulent heat transfer from the plasma column can be estimated by a semiempirical equation similar to Eq. 1 (Zhukov 1989).

14

DC Plasma Torch Design and Performance

Z¼

605

z 1:435  Re0:27  ¼
d 1 þ 1:3  103 K1:1

(3)

d pffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ0 h0 σ0 I

(4)

With, K¼

Usually the vortex is generated with a vortex chamber with a diameter D, three to five times larger than the anode diameter, d (see Fig. 6). The gas in injected through tangential injection ports with high velocities. According to Zhukov (1979), the length of the tangential ports (Fig. 6) for gas injection must be three to four times larger than their diameter and the number of injection ports must be at least four and they have to be equidistant along the injection chamber perimeter. As in most cases, a high velocity is required for the injected plasma-forming gases, resulting in high-pressure drops in the injection ports (Morgensen and Thörnblom 1987). If the velocity in the narrowest section of the injection port approaches the velocity of sound, the gas flow will become choked and proportional to the inlet pressure, independent of the pressure inside the plasma generator. For most gases, the critical pressure ratio (chamber to inlet pressure) is close to 0.5 (Zhukov 1989), which requires an inlet pressure more than twice the plasma generator pressure. Usually there has to be a compromise between the desired high injection velocities (and high vortex swirl numbers) and the compression cost. A reasonable compromise is to limit the injection velocity of the swirl gas to 0.4–0.7 of the sound velocity. Another compromise has to be found for the ratio, R ¼ ‘v =D, where ‘v is the height and D the diameter of the cold plasma-forming gas injection chamber. Small values of R reduce the swirl by friction, but large values of Fig. 6 Principle of a DC plasma torch with gas vortex injection and a well-type cathode

Copper cathode

d

Lc

Ceramic insulator

lv Copper anode

C

Gas injection ports

d

AA’

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R may lead to recirculation problems in the central part of the vortex chamber. Another role of the vortex injection is to provide the arc root with a helicoidally motion on the anode and well-type cathode walls. For the anode, there is a strong effect of the vortex even in the zone where the arc strikes the anode wall. The vortex induces a helicoidally motion of the arc root, giving rise to a uniform distribution of the electrode erosion over a large fraction of the anode. The level of swirl in the flow is generally characterized by the swirl number, S defined as; S¼

1 ð 1  y3 Þ tan θ 3 ð1  y2 Þ1:5

(5)

where y = D/d with d as the anode internal diameter, D the diameter of the vortex chamber, and tan θ is the ratio of the azimuthal velocity in the gas injection chamber to that of the axial velocity in the nozzle. For the well-type cathode, where the erosion is rather important, the vortex effect is not as pronounced as it is at the anode. This was shown by the cold flow calculations of Brilhac J.F. (1993) and Brilhac et al. (1995). The results are directly linked to the electrode erosion area and depend strongly on the torch internal design, especially the vortex injection chamber, the diameters and lengths of the well-type cathode and anode, and the cold gas flow rate G (kg/s). The arc root/cold gas interaction depends on the gas momentum in the “cold” layer between the arc column and the anode/nozzle wall which, in turn, is linked to the plasma-forming gas mass flow rate (kg/s), but not to its volumetric flow rate (slm) (Marotta et al. 1993; Erin et al. 1995). The results of Brilhac et al. (1995) have shown (for an air plasma) that the erosion area in the well-type cathode depends on the axial velocity, vz, of the cold gas close to the wall. This velocity increases with G and depends on the ratio of the diameter of the cathode to that of the anode as well as on the length (or depth) of the well-type cathode. Almost independent of these parameters, the azimuthal velocity, va, in the cathode erosion area is between 10 and 50 m/s while it is higher in the anode erosion area. Since the erosion at the cathode is more severe, an auxiliary magnetic field is often used to assist the arc root motion on the cathode surface. However, it must be pointed out that the drag force of the vortex acts on the gas (neutral and charged particles) in the relatively thin column connecting the main arc column to the wall, while the magnetic field acts only on the charged species of these connecting columns.

3.4

Magnetic Rotation of the Arc Root

Many studies have been devoted to the interaction of an arc with a magnetic field (Pfender 1978; Fauchais and Vardelle 1997; Guile and Sloot 1975). The most comprehensive surveys, which includes comparisons between simulations and experiments, have been published by Minoo et al. (1994, 1995) and Arsaoui et al. (1994). These authors have studied the influence of a magnetic field in the presence of vortex flow, considering the torch depicted in Fig. 7a.

14

DC Plasma Torch Design and Performance

607

a r

z

Anode region

Cathode region Gas injection chamber

b

Magnetic coil 2R0

c

ea

2a0

Cathode spot Cathode cavity

Z2

Z3 Zb

r

Uθ Arc column

Z1

y

ez 0

ez x erx ea=ez

Z4 Zbmax

θ

er

Zc

Fig. 7 DC vortex plasma torch with its magnetic coil around the well-type cathode (a) schematic of the torch. (b) Position of coil, arc root and the location of the maximum radial component Br of the magnetic field. (c) System of coordinates (Arsaoui et al. 1994)

The torch has been designed for high power (2 MW) and installed at the “Les Renardières” research center of EDF in France. The arc current ranges from 400 to ! 1000 A for air flow rates between 0.05 and 0.23 kg/s. The magnetic field B is produced by a coil with its own power supply, allowing to vary the coil current Ib independently of the arc parameters. With Bmax = 1 T, the magnetic field orientation may be in  z-direction (see Fig. 7b). In general, the magnetic field has two components one in the z-direction, Bz, and the other in the r-direction, Br (see Fig. 7c), while the vortex flow close to the wall has an azimuthal component, Ua, assumed to be counterclockwise. The magnetic field configuration consists of ten coils with each coil having 20 windings. Minoo et al. (1994, 1995) calculated the different components of the field. Considering one coil of radius am made of a single wire and defining the coordinate system as shown in Fig. 7c, d, the radial component of the magnetic field Br becomes: Br ¼

μ0 I K z 4π r ðam :rÞ1=2

" J1 þ

a2m þ z2 þ r2 ðam  rÞ2 þ z2

# J2

(6)

And the axial component Bz: Bz ¼

μ0 I K z 4π r ðam :rÞ1=2

" J1 þ

a2m  z2  r2 ða m  r Þ2 þ z 2

# J2

(7)

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The azimuthal component vanishes (Ba = 0) With 4am :r

K2 ¼

ðam þ rÞ2 þ z2

π=2 ð

J1 ¼


0

π=2 ð

J2 ¼

dΨ 1  K2 sin2 Ψ

1=2

(8)

(9)


1=2 1  K2 sin2 Ψ dΨ

(10)

Br ¼ BΨ ¼ 0

(11)

0

For, r = 0 (z-axis)

and, Bz ¼

μ0 I a2m 3=2 2 z2 þ a2m


(12)

For the entire magnetic field coil consisting of ten individual coils with 20 windings per coil Br ¼

10 X 20 X

Bijr

(13)

Bijx

(14)

j1 i1

and Bz ¼

10 X 20 X j1 i1

Applying in Eqs. 6 and 7, general indices i and j to am and z, respectively, K, J1, and J2 transform into Kij, Jij1, and Jij2 in the corresponding Eqs. 8, 9, and 10. The radius aim of the ith winding may be expressed as aim ¼ R1 þ

Δa þ ði  1ÞΔa 2

(15)

and zj ¼ z 

Δa  ðj  1ÞΔa 2

(16)

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DC Plasma Torch Design and Performance

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with Δa ¼

R2  R1 L ¼e ¼ 10 20

(17)

where R1 and R2 are, respectively, the internal and external radii of the magnetic coil and L is its length. Figure 8a represents the evolution of Br (in Tesla) as a function of z at the cathode wall (r = R0) for different coil currents (from 500 to 1000 A). Br becomes zero at zb (see Fig. 7b) and reaches a maximum at (zb)max regardless of coil current or radius at which it is calculated (see Fig. 8b calculated at Ib = 750 A for different values of r). For affecting the current connection to the cathode, the values of Br and Bz close to the cathode wall will be the most important. Figure 9a shows Bz as a function of z, for r = Ro, at different values of the magnetic coil current, Ib. Bz reaches a maximum at z = Zb, which corresponds to the center of the magnetic coil. In Fig. 9b, the physical position corresponding to the different z-values are those given in Fig. 7b. Z1 and Z4 correspond to the cathode a Magnetic flux density, Br(T)

0.10

Ib=1000A 900 800

0.05

700 600 500

0.00

-0.05

-0.10

r = Ro

0.0

0.1

0.2

0.3

0.4

Distance, Z(m)

b Magnetic flux density, Br(T)

Fig. 8 Magnetic flux density, Br as a function of Z. (a) At r = Ro, for different values of magnetic coil current Ib and (b) at Ib = 750 A, at different radial positions (Arsaoui et al. 1994)

0.08 3r1

2.5r1

0.04

3.5r1=R0

2r1 r1

0.00 Ib= 750A

-0.04 Zb

-0.08 0.0

0.1

Zbmax

0.2 Distance, Z(m)

0.3

0.4

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a 1.0 Ib=1000A Magnetic flux density, Bz(T)

Fig. 9 Magnetic flux density, Bz, (a) at r = R0, as a function of z, for different values of magnetic coil current Ib and (b) at Ib = 750 A, as function of (r/rm), for different values of z. See Fig. 7 (b) for the corresponding values of z (Arsaoui et al. 1994)

r = Ro

900

0.8

800

0.6

700 600

0.4

500

0.2 0.0 0.0

0.1

b 0.7 Magnetic flux density, Bz(T)

0.2

0.3

0.4

Distance, Z(m)

Z=Zb

0.6

Z3

0.5

Z2

0.4 Z1

0.3

Ib = 750A

0.2 0.1

Z4

0.0 0.0

0.01

0.02

0.03

0.04

Distance, r/rm(-)

cavity’s two extremities, Z2 and Z3 correspond to the position of the two ends of the magnetic coil, and Zb corresponds to the symmetry plane (center) of the magnetic coil. The result shows that for Ib = 750 A, Bz is almost independent of r. If Zc is the position of the arc root on the surface of the electrode, it can be demonstrated experimentally that regardless of the vortex and/or the magnetic field and Bz sign, Zc > ðZb Þmax > Zb

(18)

This inequality is found by establishing a force balance for the z-component of all forces acting on the cathode region (Minoo et al. 1994), i.e.,
2 a0 Δ‘ρ0 CD v0z þ

ð ΔV

!

! !

j  B  e z dv ¼ 0

(19)

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DC Plasma Torch Design and Performance

611

where ao is the diameter of the connecting arc column  to the cathode surface, Δ‘ is its length, ΔV is the corresponding volume (Δ‘:π a2o =4 , CD is the drag coefficient, j the ! arc current density, and e z the unit vector in z-direction. After some simplifications, Eq. 19 reduces to ρv2z CD ¼ πa0 ja Br

(20)

An analysis of all possible cases according to the sign of Bz, Br, and ja and according to the force equilibrium condition Fzg þ Fzb ¼ 0 and Steenbeck’s minimum principle shows that Zc > Zbmax as observed experimentally where Zc corresponds to the stable position of the arc. The motion of the arc in the cathode region can be determined by considering the force balance in azimuthal direction (index a), neglecting the contribution of ! surface forces (Minoo et al. 1994, 1995). The drag force F ag acting on the gas ! contained in the connecting arc column varies with v a the azimuthal component ! of the gas velocity. At the torch axis, v a ¼ 0 and it reaches a maximum close to ! the cathode wall, but it decreases in the boundary layer. The magnetic force F ab ! behaves as B z and its variation with r is very small. Thus, two cases may occur: (a) Bz > 0 !

!

!

In this case, Fab is negative and F a ¼ F ab þ F ag is negative from r = 0 to r = rI and positive for rI < r  R  eBL where R is the well-type cathode internal radius ! and eBL the thickness of the flow boundary layer. As shown in Fig. 10a, F a induces a deformation of the arc column with time. As time increases from t1 to t3, the arc column path changes from Ωt1M1 to Ωt3M1. At time t3, the deformation brings the arc column close to the cathode surface, resulting in arc shunting (breakdown) and the arc root jumps from M1 to M2. The same process occurs at t5 where the arc root jumps from M2 to M3. However, at t7 the arc root will jump from M3 to N due to arc shunting. By this process, the arc ! ! root may jump either in þ e Θ or  e Θ direction. But since the integral contribution of ! ! the drag force F ab is stronger than the surface force F ag , the overall effect (after a ! sufficiently long time) is a rotation in þ e Θ direction. (b) Bz < 0 !

!

!

In this case, F ag and F ab are positive and act in þ e Θ direction, causing the deformation of the arc column ΩLP’ or ΩMQ’ shown in Fig. 9b. Near the ! cathode surface, shunting will occur and the arc root jumps always in the þ e Θ direction.

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Fig. 10 Evolution of the arc shape with time and jumping process of the cathode arc spot (a) Bz > 0 Fa ¼ Fag þ Fab , (b) Bz < 0 Fa ¼ Fag þ Fab , Fag > 0 and Fab > 0 (Zhou et al. 1994)

This behavior has been confirmed by high-speed frames taken through an optical fiber disposed at the bottom of the well-type cathode (Minoo et al. 1994).

4

Electrode Designs

4.1

Hot Cathode Designs

4.1.1

Tungsten with or Without Doping (Arc Currents Below 1000–1200 A) As already shown in Part II, ▶ Chap. 13, “Electrode Phenomena in Plasma Sources,” for arc currents exceeding 50–100 A, hot cathodes are cooled mainly by their thermionic emission (90 to 95% of their thermal load) (Zhou et al. 1994; Heberlein 2000). The erosion is essentially controlled by thermal loads, which may overheat the cathode spot. These thermal loads are associated with physical processes in the near electrode region of the arc discharge, at the electrode surface and in the crystal lattice of the metal (Cobine and Burger 1995; Eberhart and Seban 1966; Heberlein 2000; Lowke 1997; Zhou et al. 1994; Zhou and Heberlein 1996; Benilov and Marotta 1994). In the case of Arc currents below 1000–1200 A, either stick, rod, or button-type cathodes are used. The rod-type cathodes are characterized by the length-to-diameter ratio, defined as (4‘K/πd), as well as by the cone angle of the conical tip. A number of simple rules have to be followed in order to keep the cathode erosion as low as possible: (i) Even if only about 5% of the total power input is dissipated at the cathode, the thermal contact between the hot cathode and the water-cooled copper holder must be as perfect as possible.

DC Plasma Torch Design and Performance

Fig. 11 Evaporation and oxidation rates evolution with tungsten cathode temperature under different conditions (Zhukov 1979)

613

100 O2 , p=133 Pa Specific energy requirement, qu(C/μg)

14

102 O2 , p=1.3 Pa

104 106

Vacuum, p 4) for stabilizing the arc root. These results in mass flow rates

M. I. Boulos et al.

Specific erosion rate, G(kg/C)

616

10-11 8 6 4 2 10-12 8 6 4

Ar N2 H2 He

2 10-13

0

200

400 600 Arc current, I(A)

800

1000

Fig. 13 Evolution with arc current of the specific erosion of tungsten stick-type cathodes (top) and button-type cathodes (bottom) with different plasma gases (Zhukov 1989)

10-9

Specific erosion rate, G(kg/C)

Fig. 14 Specific erosion of tungsten rod cathodes with different electrode dimensions, in inert plasma gas (Argon) (Zhukov 1994)

Ar, d=5mm, l=50mm

10-10

Ar, d=4mm, l=15mm

ℓk

ℓk> 20 mm ℓk = 5 - 10 mm

10-11

Ar N2 H2 He

10-12

10-13

0

200

400

600

800

1000

Arc current, I(A)

exceeding (three to five times) those necessary with stick-type cathodes for the same power level, and the average gas temperatures are substantially lower (4–5000 K lower).

14

DC Plasma Torch Design and Performance

a

617

b

+

+

-

c

e

-

+

d

+

+

Fig. 15 Schematics of different hot cathode designs (a) axial injection, (b) vortex injection with a low swirl, (c) conical injection, (d) conical injection with a constant surface, (e) radial injection

(v) Stick-type cathodes may be used with many different gas injection configurations: parallel to the cathode jet axis (see Fig. 15a) with a vortex having a swirl number below 2 obtained with some sort of screw injector around the cathode rod (see Fig. 15b), with an injection along the conical tip (see Fig. 15c) which drastically constricts the plasma column; the surface between both cones can also be kept constant (see Fig. 15d) and finally radial injection (see Fig. 15e). (vi) Cylindrical cathodes are often of the well type made of copper with five to eight tungsten buttons imbedded in the Cu tube. This results in splitting of the arc roots especially when the arc current is sufficiently high for the arc column to fill the available channel (see Fig. 16 (Zhukov 1994)).

4.1.2

Tungsten with or Without Doping (Arc Currents Between 1000 and 6000 A) Few data have been published on such electrodes (Camacho 1988; Barcza 1987; Hackmann and Bebber 1992; Moore et al. 1989). Their shape is of the button-type, but the button is not completely imbedded in copper. Their diameters depend on the current to be carried by the electrode and its material of construction (its thermal and electrical conductivities, its melting point, and work function). Water cooling is required. A thoriated tungsten cathode capable to carry currents in this range requires a tip size of 10–20 mm in diameter. The current density for continuous thermionic emission is approximately 30 MA/m2 (3 kA/cm2), i.e., for a current of 10 kA a tip size of 20 mm diameter would be required (Camacho 1988). The Voest-Alpine torch cathode (M€uller et al. 1989) consists of a short, stubby tungsten electrode of close to 2.5 cm in diameter. It is slightly protruding from a

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Fig. 16 (a) Cylindrical copper cathode with imbedded tungsten buttons (b) arc current splitting between the imbedded tungsten anodes (Zhukov 1994)

Fig. 17 Voest-Alpine transferred arc plasma torch: (a) conventional (b) with a shroud gas (M€ uller et al. 1989)

a Ar

b Cooling water

Cooling water

Ar

O2

O2

copper or tungsten nozzle, and its position is adjustable from the back end of the torch to make up for electrode erosion (see Fig. 17a). The plasma gas is argon and the torch can operate up to 6000 A. This torch can be equipped with another watercooled jacket (see Fig. 17b) so that any shroud gas can be added according to the process requirements. The Tetronics (Moore et al. 1989) high-current cathodes have a truncated conical shape (see Fig. 18) capable to operate up to 5000 A with a tip diameter of 15 mm. The tungsten electrode with the surrounding nozzle is protected by the flow of an inert gas through a narrow annular slot between the electrode and the nozzle. For all these cathodes, the protective argon gas flow rate is relatively low usually below 200 slm. Cathode erosion is mainly due to the formation of high-pressure

14

DC Plasma Torch Design and Performance

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Fig. 18 Tetronics transferred arc plasma torch (Arc Plasma Processes 1988)

−

+

bubbles in the molten cathode material and ejection of small metal droplets (Heberlein 2002). The erosion rates are in the range from 1010 to 109 kg/C (Zhukov 1977).

4.1.3 Zirconium, Hafnium, Hafnium Carbide Cathodes In contrast to W, these cathode materials can tolerate oxygen-containing plasma gases, but as already mentioned in Part II, ▶ Chap. 13, “Electrode Phenomena in Plasma Sources,” all these cathodes cannot sustain currents in excess of 300 A due to their rather poor thermal conductivity. The only possible design option is the buttontype cathode. For higher currents, Russian researchers have tested cylindrical hollow copper cathodes with six buttons of cathode material imbedded (Timoshevskii 1994) (see Fig. 15) and currents up to 1000 A were feasible. Yin et al. (1999) studied a Hf cathode working with air as plasma gas and showed that during arc initiation very high current densities (>109 A/m2), three to eight times exceeding steady state conditions, were observed creating high pressures at the point of arc cathode attachment, resulting in rapid expansion of the arc root. And it seems that arc ignition contributes primarily to cathode erosion. Moreover, after interrupting the arc current, the solidified oxide (HfO2, ZrO2) becomes a perfect dielectric and arc reignition can take place only on the copper holder/cathode interface contributing to its wear. Typical erosion rates are in the range from 1011 to 1010 kg/C for arc currents below 300 A. These rates increase sharply with the increase of the arc current (4  1010 kg/C per 100 A). With the multiarc arrangement (inserts in a tubular electrode), this erosion drops down to 1011 kg/C. 4.1.4 Graphite Electrodes Graphite cathodes are mostly used at current levels above 1000 A with the exception of very special torches such as water-stabilized plasma torches (Hrabovsky et al. 1997). The electrodes are rod-shaped, sometimes with a hole drilled through its center for injection of the plasma gas and/or the particles to be treated. It should be noted that their erosion rate is much higher than that of copper for arc currents below 15 kA, which implies that during operation these cathodes have to be continuously advanced to keep the cathode tip at the same location. These electrodes are usually not cooled except in some cases the base of the electrode may be water-cooled. The cathode size has to be adapted to the arc current and to the grade

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of the graphite used. Their erosion rates can reach 300 μg/C, i.e., close to 100 kg/h at 100 kA, which is almost double of that for tubular electrodes at the same current rating (Barcza 1987). A unique, vortex-stabilized, plasma source was developed in the 1990s at the Institute of Plasma Physics (IPP) of the Czech Academy of Sciences in Prague (Hrabovsky et al. 1997). A schematic and photograph of the plasma torch is presented in Fig. 19. Water is injected tangentially into the discharge cavity and creates the vortex in the chamber (Fig. 19a). The arc is struck between a consumable cathode, made of a graphite rod continuously advanced, or an inert-gas shielded thoriated tungsten cathode, and a water-cooled copper anode in the form of a rotating disk external to the torch (Fig. 19b). The arc currents vary between 300 and 600 A with relatively stable voltages varying between 267 and 293 V. Temperatures of the water-vapor plasma vary between 19,000 and 28,000 K and velocities between 2500 and 7000 m/s. Of course at such temperatures the centerline density of the plasma is very low: between 0.9 and 2 g/m3.

Fig. 19 Schematic representation and a photograph of the water-vapor plasma torch with its external rotating, water-cooled, copper anode developed at IPP in Prague (Hrabovsky et al. 1997)

14

DC Plasma Torch Design and Performance

4.2

621

Cold Cathode Designs

4.2.1 General Remarks In most cases, cold cathodes are made of copper or copper alloys operated with arc currents limited to 1200–1500 A. Beyond these values, copper erosion becomes excessive. Typical design includes tubular electrodes usually with a well-type cathode. For very high-power arcs (10 MW and beyond), coaxial electrodes are used (Konotop 1994) (see Fig. 20). It has to be kept in mind that compared to hot stick-type cathodes, the use of cold cathodes with high vortex flows (swirl number more than 5 calculated for cold plasma-forming gas flow) requires plasma-forming gas flow rates at least three to five times higher than that for hot cathodes with the same power rating. The arc root motion at the cathode surface has been the subject of intensive studies (J€uttner 1997). It is generally accepted that erosion rate decreases with the increase of the velocity of the cathode spot motion. The latter is observed to increase significantly with the addition of diatomic gases in a monatomic plasma-forming gas (even as low as 1 vol%). The cathode spot motion also depends on the vortex efficiency and magnetic field arc rotation depending on the specific torch design. Generally, the erosion rate of cold cathodes depends on • • • • •

Cathode material Plasma forming gas composition Efficiency of the gas vortex Magnetic field configuration Arc current

Fig. 20 Coaxial plasma torch with magnetic stabilization of the arc root motion: (a) axial discharge, (b) discharge through a mixing chamber, and (c) lateral discharge (Zhukov and Zasypkin 2007)

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Numerous correlations of the cathode erosion as a function of the arc current and the magnetic field intensity have been published. These seem to be specific for the particular plasma torch design for which it has been established. Compared to hot cathodes, the erosion rates of cold cathodes are higher (109 to 107 kg/C). Longer electrode lifetimes can be achieved through the uniform distribution of the erosion over a large surface of the electrode: up to 1000 h for well-type cold cathodes at a power level of 1.5 MW with air against 100 h for stick-type cathodes at 50 kW with Ar-H2 (plasma spray torch). This is due to the lower specific enthalpies of the plasma (five to ten times), which are typical for cold cathode torches, and to the fact that the cathode surface area available for arc attachment is one to two orders of magnitude larger than corresponding hot cathodes.

4.2.2 Cathode Materials In most cases, oxygen-free high-purity (OFHP) copper is used with nitrogen, air, CO–CO2, and steam as plasma-forming gases. Some electrodes made of copper are doped with chromium (2 wt%) which gives rise to slightly lower erosion rates with air than that of pure copper. According to Testé et al. (1994), the effect could be a consequence of the difference between the lifetimes of the cathode spot. With pure oxygen as plasma gas, the erosion rate seems to be slightly lower when using silvercopper alloys. With argon-hydrogen and pure hydrogen, the erosion rates can reach 2  108 kg/C compared to 5  109 kg/C with oxygen or steam for 100 A arc currents (Szente et al. 1989). At this point, while the reason for this finding are not clear, the influence of the vortex motion of the plasma-forming gas on the arc root motion will exert a strong impact on the cathode erosion rate. Low carbon steel as cathode material has also been used with success by H€ uls for decades in their 8.5 MW plasma torches working with H2 and CH4 as plasma gases (Gladish 1969). Titanium, stainless steel, and copper-nickel alloys have also been used as cathode materials for arcs operating in argon, argon-hydrogen, and nitrogen (Kwak and Munz 1996). Titanium and stainless steel showed the lowest erosion rates with argon (0.2 and 0.3  109 μg/C at 100 A compared with those of OFHP copper of 15  109 μg/C). Copper-nickel alloy (10 and 30 wt% Ni) has been shown to be suitable for nitrogen and hydrogen plasma gases. For hydrogen, its solubility into the cathode material as function of the cathode temperature was found to be important for the cathode erosion. High gas solubility into the cathode material can result in mechanical erosion due to microexplosions near the cathode surface. Tungsten cold cathodes have also been used with certain chemical products such as H2S. In general, they are made of a tungsten insert surrounded by a copper holder. The arc current is, however, limited due to the poor thermal conductivity of the cathode compared to that of copper. 4.2.3 Magnetic Field Configuration The effect of the magnetic field on the arc has been described earlier in Sect. 3.4. The position of the plane in which the arc cathode spot rotates is determined by the arc

14

DC Plasma Torch Design and Performance

623

current, the components of the magnetic field Br and Bz, the gas velocities νz and νa, the velocity of the arc spot rotation Vaarc, and the radius of the arc cathode spot a0 (Minoo et al. 1995). The cathode spot’s motion over the cathode surface is neither continuous nor by successive jumps. The motion is the result of a repetition process of birth of new spots and disappearance of old spots provoked by the magnetic field and the gas flow. The calculations presented in Minoo et al. (1995) allow determining the arc cathode spot rotation velocity Vaarc and the radius of the arc spot a0 which is in the mm range (radius which is, in fact, the location where many tiny arc cathode spots coexist). It is clear that the magnetic field has to be adjusted to the torch working conditions (see next section). Thus, the magnetic coil should be positioned according to the mass flow rate of the plasma-forming gases as well as the arc current and the current in the coils. This implies that for high-power torches, the coil cannot simply be in series with the arc, but requires its own power source. For spreading the erosion over a wider cathode area, one solution is to let some parameters such as Br undergo continuous changes with time. For low-power torches (below 100 kW), magnetic fields below 0.1 T are sufficient, while for high-power torches (up to 2 MW) fields of 0.3–0.7 T are required.

4.2.4 Vortex Flow Due to its configuration, the vortex is by far more efficient at the anode than at the well-type cathode. In the latter case, the gas flow has to pass along the wall and return axially. Experiments have shown that the erosion area at the cathode is strongly linked to the gas flow close to the cathode surface (Jestin et al. 1989; Brilhac et al. 1995). For a low-power DC torch (~50–80 kW) with no magnetic field, calculations neglecting the plasma influence have shown that the arc will strike in regions where the azimuthal gas velocity is between 10 and 50 m/s corresponding to axial velocities vz between 2 and 8 m/s. Similar results were observed by Marotta et al. (1993) for low-power DC torches and by Minoo et al. (1995). The position where this velocity range is obtained depends on the mass flow rate of the plasma-forming gas, the gas injection chamber design, the depth of the cathode-well, lc, and the diameter ratio (dc/da) where dc is the cathode diameter and da the anode diameter (Brilhac et al. 1995). If (dc/da) < 1, the “good” velocity distribution is close to lc or even extends beyond the end of the well, which results in lower erosion rates. For (dc/da) > 1, higher erosion rates are found, with the erosion mostly distributed inside the well. However, it moves towards the end of the well when dc increases, and thus, ‘c has to be increased to ensure arc attachment in the middle of the well. As emphasized in the previous section, the magnetic field plays an important role in determining the arc root position, which depends also on the gas velocities. A compromise has to be found. The lowest erosion rates are achieved, however, when the gas and magnetic field rotations are in the same direction.

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Anodes

4.3.1 General Remarks In most cases (98%), anodes are tubular. One of the very few exceptions is the waterstabilized torch, which uses an external anode that is water-cooled and rotating (Hrabovsky et al. 1997). Sometimes the anodes are nozzle-shaped with a convergent throat and a divergent section especially when the torch is designed to work in a surrounding soft vacuum or to produce supersonic jets at atmospheric pressure. In most cases, however, anodes have a cylindrical shape with a convergent section that is a simple cone. To achieve quasi-fixed arc length with a vortex injection, they have a step change in their diameter as already illustrated in Fig. 5 for a torch with a button-type cathode or the anode is separated from the cathode by insulated rings without or with gas injection between them (Fig. 5 again with a button-type cathode). Generally, the anode material is made of oxygen-free high-purity (OFHP) copper, but copper-silver alloys are also used with pure oxygen as plasma gas. Copper anodes with tungsten inserts have been also in use (Neumann 1976). Torches with thick (6 mm) tungsten inserts give rise to tungsten wall temperature close to 2000 K. This temperature enlarges the cross-section of the anode arc root and the anode root velocity increases, and therefore, the erosion rate is reduced and the temperature distribution within the plasma jet becomes more uniform. In industrial plasma torches used for spraying, the tungsten inserts are approximately 3 mm thick. Their erosion is reduced compared to that of copper anodes used under the same conditions. However, due to high thermal stresses in the near-surface layer, grains of tungsten are pulled out of the anode randomly and they can be detrimental for the application (e.g., when spraying thermal barrier coatings (Wigren et al. 1997). The same problem can also be observed with copper if the initial grain size is not minimized. The copper anode erosion rate is usually lower than that of cathodes (see Fig. 21) for DC plasma torches with high swirl number vortex flows. For high arc currents (1000–4000 A), the shaded region in Fig. 21, with a 1 T magnetic field, an air plasma-forming gas, and an anode internal diameter of 0.1 m, shows an erosion rate of approximately 109 kg/C, a value typically obtained with low arc currents. However, when moving the arc root in axial direction, lower erosion rates can be obtained. The dynamic behavior of the arc root at the anode plays also an important role for anode erosion similar to that for the cathode arc roots. With increasing pressure in the arc chamber, anode erosion also increases. This is due to both the arc column and the arc root constriction and the resulting increase of the radiative losses as shown in Fig. 22. Wigren et al. (1997) have shown that for a DC plasma spray torch the location of the electric power connection to the anode nozzle has a nonnegligible effect on the mean voltage, the amplitude of the voltage fluctuations, and the torch thermal efficiency. The current connection affects the total length of the arc column, and the highest efficiency is obtained for the longest arc. This corresponds to an electrical connection to both the downstream and the upstream parts of the anode. However,

14

DC Plasma Torch Design and Performance

625

Specific erosion rate, G(kg/C)

1

B=0

10-9

Air

100 150 H2 410h

10-10

Air

d=0.1m p=1-40x105Pa

84h

Water vapor

10-11

25%Ar

8 4 2

10-12

0

200

N2 Ar

400

600

800

1000

4000

Arc current, I(A)

Radiated power ratio, φ (%)

Fig. 21 Specific erosion of copper tubular anodes (Zhukov 1994) 30

20 Air, I=500A

10

0

2 4 6 Arc chamber pressure, p (105Pa)

8

Fig. 22 Evolution with pressure of the ratio of the radiated density Qr on the product ExI inside an anode 2  102 m in internal diameter, working with air at 500 A (Zhukov 1994)

the arc root lifetime is the shortest with an upstream connection, resulting in a longer lifetime of the anode.

4.3.2 Dynamic Behavior As already discussed in Part II, ▶ Chap. 13, “Electrode Phenomena in Plasma Sources,” Sect. 4.2, many papers have been devoted to the study of the dynamic behavior of the arc in the anode nozzle of DC plasma spray torches and for torches with strong vortex flows (Brilhac 1993; Brilhac et al. 1995; Heberlein 2000; Collares and Pfender 1997; Coudert et al. 1995, 1996; Planche 1995; Planche et al. 1997, 1998; Duan et al. 1999). Heberlein (2000) and Malmberg et al. (1991) have

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identified the following types of voltage waveforms, reflecting three types of operating modes; • The restrike mode characterized by a saw-tooth voltage-time behavior and large voltage fluctuation amplitude • The takeover mode which has an approximately sinusoidal or triangular shape of the waveform with a relative low fluctuating amplitude • Steady mode with an almost flat voltage profile Mixed modes of operation involving restrike and takeover were also identified as possible. These different operating conditions depend mainly on the thickness of the relatively cold boundary layer between the arc column and the anode surface. Its thickness which, in turn, is linked to the nozzle internal diameter, the arc current, the mass flow rate of the plasma-forming gas, and the vol% of diatomic gases. A typical example is shown in Fig. 23 (Heberlein 2000) obtained with a plasma torch with an 8 mm anode internal diameter (straight bore), working with an Ar-He mixture as plasma-forming gas and different arc currents. Depending on the working conditions, different operating modes are obtained. The restrike mode corresponds to a highly constricted arc column: low arc current (100 A) and high mass flow rate (120–40 slm). The takeover mode is observed as soon as the arc column radius rc approaches that of the anode-nozzle radius R (100 A, 40–20 slm), and finally the steady mode corresponds to a situation where the gas dynamic drag exerted on the connecting arc column to the anode is balanced by the magnetic body force of the self-magnetic field (900 A, Ar 60 slm). It is to be noted that the steady mode does not 100 90

restrike

Voltage (v)

80 70 60 50 take-over

40 30 20

steady 0

1.0

2.0

3.0

4.0

Time (ms)

Fig. 23 Arc operating modes with 8 mm i.d. nozzle, Ar-He mixtures of plasma-forming gas and different arc currents (Duan and Heberlein 2002)

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correspond necessarily to the lowest anode erosion, especially if there is no swirl. Moreover, it does not seem to correspond either to a diffuse anode arc root attachment. The work of Duan and Heberlein (2002) has also clearly shown that whatever the plasma-forming gas species, restrike and takeover modes could take place simultaneously to form a mixed mode. In fact, the mode depends upon the characteristics of the cold boundary layer, which develops between the anode-nozzle and the arc column. Figure 24 for Ar-He mixtures displays, for example, the evolution of the mode value (MV) versus the cold gas boundary layer thickness (measured by end-on imaging of the arc), where MV = 2 corresponds to the restrike mode, MV = 1 to the takeover mode, and MV = 0 to the steady mode. When the boundary layer thickness decreases (for example by increasing the arc current intensity), the arc column characteristic diameter increases leading to a decrease of the cold gas mass flow rate and hence to the He ratio. In this case, the mode value tends to 1 (takeover mode). On the opposite, when the boundary layer thickness increases (for example by decreasing the arc current intensity), the arc column characteristic diameter decreases leading to increases of both the cold gas mass flow rate and the He ratio. In this case, the mode value tends to 2 (restrike mode). As already underlined, operating a plasma spray torch under the takeover mode is preferable regarding the coating overall characteristics. As soon as there is more than 5 vol% of H2 or 10 vol% of N2 in an Ar plasmaforming gas, a pure restrike mode has been found as shown (Part II, ▶ Chap. 13, “Electrode Phenomena in Plasma Sources,” Fig. 19). It is only when the current reaches high values (e.g., 900 A with an 8 mm i.d. anode nozzle) that a mixed mode 2.0

Mode value

1.5

1.0

Ar = 60 slm Ar = 100 slm

0.5

Ar/He = 58/20 slm Ar/He = 98/20 slm

0.0 0.6

0.8

1.0

1.2

1.4

1.6

Boundary layer thickness, δ[mm] Fig. 24 Evolution of the mode value versus the cold boundary layer thickness for Ar and Ar-He operated DC plasma torches (Duan and Heberlein 2002)

628

Straight

90

60

70

40

50

20

30

Swirl 0

0.2

0.4 0.6 Time, t(ms)

0.8

1.0

Voltage, U(V)

a Voltage, U(V)

Fig. 25 (a) Arc voltage in a restrike mode obtained with N2-H2 mixture with straight and vortex injection of the plasma-forming gas. (b) Fast Fourier transforms of arc voltage signals (Dorier et al. 2000)

M. I. Boulos et al.

0

b 0.8

Spectral power (a.u.)

Swirl

Straight

0.4

0

0

2

4 6 8 Frequency, f(kHz)

10

12

with a low percentage of takeover mode can be observed with diatomic gases. The gas injection mode (straight or vortex) can also modify the shape of the arc voltage signal as shown in Fig. 25a for N2-H2 plasma (Dorier et al. 2000). As it can be seen in Fig. 25b, the corresponding power spectrum is also different for these two modes of injection of the plasma-forming gas. In the restrike mode, the arc voltage V exhibits large fluctuations (ΔV/Vm > 0.8) and a high mean value Vm (60–90 V). In this mode, the frequency of arc root fluctuation ranges between 2 and 8 kHz (Heberlein 2002). As DC torches are supplied with constant current sources (i.e., the arc current remains almost constant, and close to the set value whatever the voltage), this signifies that the dissipated power varies linearly with the arc voltage, while the losses in cooling water remains relatively constant since it depends mostly upon the arc current (Roumilhac 1990; Roumilhac et al. 1990; Betoule 1994). As an example, considering Vm = 60 V, ΔV = 60 V and I = 600 A, the dissipated power in the torch varies between 18 and 54 kW. In such conditions, it is not surprising that the plasma jet fluctuates in length and position at these characteristic frequencies as schematically shown in Fig. 26. Hence, the plasma flow can be considered as successions of warm and cold puffs (Coudert et al. 1996).

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Fig. 26 Typical fluctuations of a d.c. plasma spray jet working in the restrike mode (aperture time for each view: 104 s): torch nozzle i.d. 7 mm, conical injection, I = 600 A, 45 slm, 15 slm H2 (Coudert et al. 1996; Planche et al. 1997)

Finally, it should be noted that using a hot tungsten anode with temperatures exceeding 1150 K, the amplitude of the fluctuations is reduced by a factor of 2 (Neumann 1976). An important question is how measuring techniques, such as emission spectroscopy, can monitor continuously fluctuating jets such as those presented in Fig. 26. The answer is that there are two possibilities for spectroscopic measurements: either integrated measurements with an integration time larger than the voltage fluctuations (3 < f < 10 kHz) or instantaneous values taken at a much faster rate than the voltage fluctuations. In the latter case, it would be possible to obtain mean temperature values, as well as their standard deviations, σT, as shown in Fig. 27. The presented temperature distributions were determined by measuring the absolute value of the Ar-I 727.2 nm line using an OMA detector with an integration time of 0.7 s including Abel inversion. This measurement results in a population temperature, which is not necessarily that of the heavy species of the flow. These distributions, however, provide a good estimate of the trend of the temperature behavior with different working parameters. When measurements are performed with an integration time larger than the voltage fluctuations (3 < f < 10 kHz), isotherms obtained would be between those presented in Fig. 27 at T+σT and T-σT.

4.4

Cooling of the Electrodes

The arc generates high heat fluxes due to conduction, convection, and radiation mainly at the arc roots. Current densities at the arc attachment can reach at hot cathode surfaces 107–108 A/m2 and at cold cathode surfaces 1010–1012 A/m2. At

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Fig. 27 Isotherms of a DC plasma jet  standard deviation due to arc root fluctuations: nozzle i.d. 10 mm, arc current 632 A, voltage 61 V, Ar 45 slm, H2 15 slm, thermal efficiency 56% (Coudert et al. 1995; Planche 1995)

anode surfaces, current densities from 108 to 109 A/m2 have been observed. These high current densities lead to concentrated ohmic losses further enhanced by heat dissipation from the processes at the electrodes (plasma solid interface). There are no structural materials available that can withstand the corresponding heat fluxes and except for hot cathodes, the electrodes survive because the arc roots are not stationary with spot lifetimes in the order of 100 μs. At any rate, the resulting mean thermal loads demand efficient heat transfer from the surfaces to be cooled to the cooling medium, which, in most cases, is water. Cooling of hot cathodes with maximum heat dissipation of 5% of the total energy input to the arc is not a problem except in the case of DC plasma spray torches where the cooling circuit of the cathode is in series with that of the anode. In that case, the design of the cathode holder must provide sufficiently large cross-sections of the water-cooling tubes to avoid reduction of the water flow rate required for anode cooling which is by far more critical than that of the cathode. Most electrode failures are associated with local film boiling, even if the mean outlet water temperature remains rather low (40–50  C). Boiling crisis and burnout occurs locally when the bulk liquid temperature close to a hot surface spot approaches the boiling point. The vapor bubbles formed at the hot surface do no longer collapse, but rather form a vapor film and the corresponding heat transfer coefficient drops drastically, resulting in melting and local failure of the electrode. Thus, for tubular electrodes, a few simple rules have to be followed to avoid this problem (Yask’o 1969; Zhukov 1979; Morgensen and Thörnblom 1987): • The water pressure has to be sufficiently high to shift the boiling point to higher temperatures. For small torches (P < 100 kW), typical values for the water pressure are in the 1.2–1.8 MPa range, while for large torches (a few MW) they

14

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631

are in the 2.5–3 MPa range (at such pressures, deformation of too thin electrode may occur). • The flow velocity has to be increased to ensure an intensively turbulent water flow along the cooled surface. The Nusselt number (Nu) is related to the Prandtl (Pr) and Reynolds (Re) numbers by f # rffiffiffi f 8 1 þ 8:7 ðPr  1Þ 8

Nu ¼ Re  Pr "

(21)

where f is the Fanning friction factor (flow resistance coefficient). From this relationship follows the heat transfer coefficient h (W/m2.K) as v w : f : ρw : c w p # rffiffiffi f 8 1 þ 8:7 ðPr  1Þ 8

h¼ "

(22)

where vw is the water velocity and ρw and cw p are, respectively, the water density and specific heat at constant pressure. Equation 22 shows that h increases with the water velocity and with f, which, in turn, increases with the surface roughness. The flow velocity depends on the pressure drop Δp along the channel ‘ v2 Δp ¼ f   ρw w δ 2

(23)

where ‘ is the cooling channel length and δ its thickness. Thin channels are used to increase the pressure drop (laminated flows), and usually an iterative calculation has to be performed to choose δ. • De-ionized water has to be used exclusively to avoid the formation of calcium oxide on the hot walls since such films reduce the heat transfer to the water. This implies that the cooling circuit must be a closed loop, using a heat exchanger for water cooling. To limit the possibility of local film boiling, the water inlet temperature has to be kept below 10–15 C and the outlet temperature limited to 40 –50 C. The cooling circuit has to be designed according to the heat flux, which has to be removed including a safety factor. For example, for a 1 MW torch with a thermal efficiency of 70%, the cooling system must have a capacity of approximately 400 kW.

4.5

Erosion of the Electrodes

Most of the published work is related to plasma spray torches with stick or buttontype cathodes. The wear of the electrodes resulting from the arc attachment spot erosion is not negligible and drives their mean lifetime (which can evolve between a

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few to 100 h depending on the working conditions and the number of restarts) (Coudert et al. 2005). When plasma spraying, for example, this wear affects ultimately in a drastic manner the heat and momentum transfers to particles (Leblanc and Moreau 2002) and thus the thermo-mechanical properties of the coating. This electrode wear has to be compensated, especially during the spraying of large parts (lasting longer) in order to keep as constant as possible the operating conditions and hence as homogeneous as possible the resulting coating microstructure. Indeed, erosion wear of both electrodes is quite different. The major erosion wear of the cathode occurs during the very first working hours. The cathode erodes due to the diffusion and evaporation of thoria (Heberlein 2002) and this erosion leads to a lower plasma flow velocity (Coudert et al. 1995). The erosion wear mechanisms of the anode are by far more complex than those of the cathode, and many works, often contradictory, have been devoted to them (Heberlein 2000, 2002; Leblanc and Moreau 2002; Duan et al. 1997; Rigot et al. 2003; Pfender et al. 1991). The general trend however is an almost regular voltage drop during a few tens of hours working time. Then, this drop increases drastically and it leads to the ejection of tungsten (anode with tungsten insert) or copper (anode with no insert) particles issuing from the anode. This phenomenon is of course totally detrimental to the coating characteristics due to metal particle embedding. This is why in industrial spray booths the electrodes are usually systematically replaced long before (i.e., only a few tens of hours working time) this event occurs. Rigot et al. (2003) have shown that the erosion is due, for given operating conditions, to the shortening of the arc column together with a smaller area of the arc root attachment. This results in longer arc root lifetimes (i.e., for new electrodes, stagnation time is below 160 μs, while for worn ones, stagnation times can reach 200 μs) leading to the melting and evaporation of the anode. Another predominant factor is also the way the plasma-forming gas mixture is injected within the anode (i.e., axially, radially or in vortex) and the number of gas injectors (Duan et al. 2000). Attempts have been made to characterize the time history of the anode erosion through time-resolved measurements of the spectrum of sound emitted by the torch, establishing a criterion for replacing the anode before its erosion becomes detrimental for the process (Duan et al. 1997). Recently Rigot et al. (2003) have developed an electronic device based on three signals averaged on long time periods (hours) allowing, for given spray conditions as those used in spray booths for production, determining three stages of wear: no problem – watch out – dangerous. When the anode wear develops, the plasma flow becomes unstable, the fluctuation frequencies increase, and the jet average length shortens. To compensate the lower arc average voltage resulting from the anode erosion, one way is to increase either the arc current intensity or the secondary plasma gas flow rate. Nevertheless, it has to be kept in mind that when increasing the secondary plasma gas flow rate, the cold boundary layer thickness rises amplifying hence the arc instabilities. Indeed, as it will be shown in the section devoted to the plasma–particle interactions the best way to keep constant the power dissipated in the plasma torch is to compensate the anode erosion by increasing the arc current intensity only, reducing in such a way the instabilities and their consequences on the resulting coating structure

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At this time, systematic studies of the erosion have been reported mostly for the restrike mode. When the cold boundary layer thickness between the arc column and the anode surface increases (e.g., by decreasing the arc current keeping all other controllable parameters the same), the mean lifetime of the anode spot at a particular location increases or the voltage fluctuation frequency decreases (Coudert et al. 1996; Planche 1995). It is, however, difficult to compare different torches, because their working conditions may be quite different. For example, for an 8 mm i.d. anode, a power level of 35 kW (500 A, 70 V) may be realized either with a DC spray torch using 50 slm N2 and no swirl, with a button-type cathode torch (swirl number ~ 4) 100 slm N2, (290 A, 120 V), or with a well-type cathode (swirl number ~ 6 cathode i.d. – 10 mm) 160 slm N2, (100 A, 350 V). For the same type of torches and the same operating conditions, but for a fixed arc current, for example of 100 A, the anode spot mean lifetimes are, respectively, 160 μs, 200 μs, and 130 μs for rather different power levels (16, 30, and 35 kW, respectively). These results clearly show that the anode spot mean lifetime depends also strongly on the swirl and the type of cathode used. For the well-type cathode, it has not been possible to isolate voltage fluctuation frequencies related to the cathode. The total voltage frequency depends upon the ratio dc/da and the magnetic field acting on the cathode arc root (Brilhac 1993; Brilhac et al. 1995; Duan and Heberlein 2002). Figure 28 represents the mean lifetime of the anode spot of a DC plasma spray torch with a conical injection and no swirl. The torch is working with 60 slm of an Ar-H2 (45 to 15 slm) mixture. When the anode spot lifetime falls below 80 μs, the anode erosion becomes rather low. However, the anode lifetime (as well as that of the cathode) depends also on the number and frequency of arc restarts as well as on the working conditions. This 160

Arc root life time, (μs)

d = 8mm Ar-H2

120

d=7

80 d=6 40

0

0

200

400

600

Arc current, I(A)

Fig. 28 Evolution with arc current and different anode diameters of the anode arc root lifetime τ for an Ar-H2 plasma (45 slm Ar-15 slm H2) (Coudert et al. 1995; Planche et al. 1997)

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implies that a torch will have the highest electrode lifetime for a relatively narrow window of working parameters. Considering an Ar-H2 mixture with a flow rate of 60 slm and a 7 mm i.d. anode, the longest lifetimes are achieved with 400 < I < 650 A. Below 400 A, the relatively long anode spot lifetime increases copper vaporization. Over 700 A, the arc column almost fills the anode bore and radiation loss from the arc increase and simultaneously the arc length decreases, reducing the anode area over which the arc restrikes. Similar remarks can be made for torches operated with high swirl number vortex flows. In the restrike mode, the voltage fluctuation frequency varies also with anode wear (Brilhac 1993; Coudert et al. 1993; Planche 1995). With a new anode, the internal surface is smooth and the arc root tends to be distributed over a wide area. After some tens of minutes of operation, the electrode starts to wear and the arc attachment area shrinks. After some tens of hours, the anode surface reveals rather deep erosion marks enhancing the arc root lifetime up to a level where the anode wear becomes excessive.

5

Ignition of the Arc

Three basically different ways can be used to initiate the arc. The choice of the method for ignition depends mainly on the arc arrangement and, in particular, on the electrode configuration.

5.1

Arc Ignition by Electrode Contact

If one or both electrodes are movable, electrode contact may be established after an electric potential is applied to the electrodes. The short circuit current passing through the contact bridge between the electrodes heats the contact point to temperature levels sufficient for thermionic emission of electrons. At the same time, electrode material is evaporated and ionized at the contact point providing the required charge carriers for developing an arc as soon as the electrodes are separated. There is, however, a lower current limit for drawing an arc that is mainly a function of the electrode material. If the discharge circuit allows at least this minimum current to be drawn, an arc may be established in the described way. However, this method leads to sever electrode erosion (craters) where the arc is initiated, and such craters are preferred starting points for further electrode erosion. For high-power vortex torches, when the gap between the electrodes is too large for a high-frequency, high-voltage spark, a metal or graphite auxiliary rod may be used to establish the contact between cathode and anode. In most cases, the primary contact is with the anode, i.e., the rod is at the potential of the anode. As the rod touches the cathode, small volume plasma develops initiating an arc between the rod and the cathode. As the rod is withdrawn, the arc is transferred to the anode. To limit rod and cathode erosion, a resistance is often placed in series between the rod and the

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anode with a value in the range of 10 and 100 Ω, depending on the open circuit voltage, in order to limit the arc current dissipated during the arc initiation step.

5.2

Arc Ignition Using a Wire or Rod Explosion

A frequently used method for initiating the arc employs a thin wire or a graphite rod between 0.5 and a few mm in diameter stretched across the electrode gap making contact with both electrodes. The applied open circuit voltage leads to explosive heating of the wire or rod supplying the necessary charge carriers for establishing an arc.

5.3

Arc Ignition by Pre-ionization with a High-Voltage, HighFrequency Discharge

In this case, the arc gap is momentarily supplied with a high-voltage (4000–8000 V) pulse from a spark inductor (see Fig. 29), leading to a high-frequency (a few MHz) spark. The power supply and monitoring instruments have to be protected from the highvoltage, high-frequency pulse, which is not as easy as it sounds. If possible, the starting plasma-forming gas should be pure argon which lends itself to arc initiation. The argon flow rate has to be sufficiently small to avoid blowing out the created low-current arc (~100 A). With long segmented anodes (Morgensen and Thörnblom 1987), it is necessary to initiate a short arc upstream, e.g., between the cathode and the first segment. This arc will provide sufficient ionization for successively connecting the segments to the anode potential and successively transfer the arc from segment to segment and finally Fig. 29 Schematic of a spark inductor 200V

L1

R1

7000V

C1 C2 R2 L2

dc source

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Segmented plasma generator

Power supply

Timer

Fig. 30 Step wise ignition of a segmented arc plasma generator (Morgensen and Thörnblom 1987)

to the anode (see Fig. 30). Here again the whole starting process may advantageously be performed in argon atmosphere. In general, any auxiliary plasma source (plasma jet, DC spark, etc.) may also serve as a means for preionizing the arc gap.

5.4

Starting Circuitry and Power Supply Matching

The starting circuitry and the main power supply have to be carefully matched in order to secure a smooth transition to the running mode. For example, with a sticktype cathode, the use of argon as plasma-forming gas allows limiting the open circuit voltage to 40–60 V and, thus, reducing the transient arc current peak which may reach five times or even more that of the steady state current, leading to damage to the electrodes. Once the arc has been started, a gradual increase of the arc current in combination with a gradual shift from argon to the final plasma working gas should establish the desired operating conditions.

6

Performance Characteristics

6.1

Plasma Torches

6.1.1 Current Voltage Characteristics As already outlined in Part II, ▶ Chap. 12, “Thermal Arcs,” voltage-current (V-I) characteristics are falling in both wall- and vortex-stabilized arcs. The arc connects to the tubular anode wall at the location where the boundary layer between the arc column and the anode wall becomes turbulent. An increase of the arc current gives rise to a corresponding increase of the plasma column diameter and of the electric field in the arc column increase (see Fig. 31).

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d = 6mm

Electric field, E(V/m)

d=7 2000

1500

d = 10 d=8

1000

500 100

200

300

400

500

600

Current, I(A)

Fig. 31 Evolution with the arc current for different anode-nozzle i.d. of the electric field in a DC plasma-spraying torch working with an Ar-H2 mixture (45 slm Ar – 15 slm H2) and an axial plasmaforming gas injection (Coudert et al. 1995)

But simultaneously the arc length decreases (i.e., the arc strikes closer to the cathode tip), because the boundary layer between the arc and the anode is heated more when its thickness decreases. Thus, according to Eq. 28, Part II, ▶ Chap. 12, “Thermal Arcs,” when the arc length decreases faster as E increases, the voltage drops with increasing arc current. However, when the arc column diameter approaches that of the anode bore, the losses in the arc fringes increase drastically and the arc compensates for these losses by increasing its electric field. Thus, the V-I characteristic may become slightly rising. (a) Free arc length torches Excluding the voltage drop in the cathode and anode region, the arc voltage will reach a maximum either when the arc length and/or its electric field assumes a maximum. The arc length depends on two parameters: • The thickness of the boundary layer between arc column and anode surface • The momentum of the gas in this boundary layer acting on the connecting arc leg to the anode surface (this leg between the anode arc root and the main arc column may be continuously moving, see Sect. 2.3 of Part II, ▶ Chap. 12, “Thermal Arcs”) The momentum of the gas in the anode boundary layer is directly linked to the mass flow rate of the plasma-forming gas. The lower the mass flow rate, the shorter the arc will be, and the lower will be the frequency of the restrike (high anode spot

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lifetime), resulting in severe anode erosion. For example, a plasma torch with a 7-mm nozzle i.d. will perform satisfactorily with pure argon at 45 slm, while it will require a somewhat higher plasma gas flow rate with nitrogen, up to 120–150 slm with pure He, and close to 200 slm with pure H2. It is to be noted that when shifting successively from Ar to He, N2, and H2, the arc column radius (for the same arc current and anode i.d.) will decrease. This results in a thicker and colder boundary layer, which will have a higher inertia, due to the increase of the specific weight of the gas with the decrease of its temperature. In summary, • For given working conditions (arc current, torch anode i.d., gas injector design), the voltage will increase when shifting successively from argon to helium, nitrogen, air, and hydrogen. As soon as a diatomic gas is added to form a mixture with a monatomic gas, the voltage will increase. This is due to the higher thermal conductivity of diatomic gases, resulting in increasing cooling of the arc fringes and corresponding shrinking of the arc diameter. The voltage tends to increase slightly when the plasma-forming gas mass flow rate is increased. • When comparing a wall-stabilized (even with a swirl of a low swirl number < 1.5–2) with a vortex-stabilized torch (with a high swirl number >3–4) for the same operating conditions, the voltage will be higher and increase more drastically with the plasma-forming gas mass flow rate in the latter case. The boundary layer becomes thicker with high vortex flow, and the cold gas momentum acting on the connecting arc leg becomes more efficient resulting in much longer arcs. The voltage drop observed when the arc current increases for button-type hot cathodes (vortex-stabilized torches) is higher than that obtained with wall-stabilized torches using stick-type hot cathodes. When using a cold well-type cathode, due to the longer arc, the voltage is higher than that obtained with a button-type cathode. The curve 1 in Fig. 32 from Zhukov (1994) shows a typical V-I characteristic for a vortex-stabilized torch with a hot cathode. Notice on the schematic of the torch that operation with air, as the main plasma gas, requires a chamber with an argon vortex around the tungsten cathode to protect the latter from oxidation. When all the other conditions are kept the same (mass flow rate and gas composition, arc current, gas injector, etc.), the arc voltage drops when the nozzle diameter increases. As the arc length increases with increasing anode length, the electric field reduction becomes more important, resulting in a falling characteristic (see curve 1 Fig. 32). (b) Fixed arc length torches DC torches with fixed arc length are typically achieved using segmented anodes, as shown on the top of Fig. 32, with or without inter-segment gas injection, in order to maintain a fixed length of the arc. At low arc currents, the arc voltage shows slightly decreasing characteristics, followed by a gradual increase of the voltage at high currents (see curve 3 in Fig. 32).

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3

N=const

Voltage, V(V)

l2 > lcy d2 2

α

d2

1

d3

lcy

l2< lcy l2 Current, I(A) Fig. 32 Electrical characteristics with the three designs most commonly used for vortex-stabilized plasma torches (Zhukov 1994)

For torches with a step change in the anode internal diameter, as shown at the bottom of Fig. 32, the arc voltage characteristics are more complex depending on the anode dimensions and torch operating conditions. Considering first the simple case of a torch with a constant cylindrical anode diameter d2 and a length ‘2. The torch characteristics will be decreasing as shown by curve 1 in Fig. 32. On the other hand, if the same torch is designed with a step increase of its internal diameter (from d2 to d3 with d3 > d2, at ‘s, see Fig. 32) the torch characteristics will depend on the length of the arc, which in turn depends on the plasma gas flow rate. At low plasma gas flow rates, when the arc length is shorter than ‘s, the characteristic will be falling. As the plasma gas flow rate is increased and the arc strikes beyond the step, ‘s the arc V-I characteristic becomes rising. For more details see [Zhukov 1994)]. It is worth noting that the arcs with a rising characteristic (R = dV/dI) are much easier to stabilize than those with a falling characteristic.

6.1.2 Thermal Efficiencies The thermal efficiency of a torch is defined as ηth ¼

V  I  Pth VI

(24)

where Pth accounts for the losses in the cooling circuits. As a general rule Pth  a  I þ b

(25)

i.e., the losses depend mainly on the arc current I and very little on the arc voltage (a ~ Vx with 0.2    0.3). Thus, for a given arc current and a given torch, the general trend is that shown in Fig. 33 for a stick-type cathode plasma spray torch.

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Thermal efficiency, η (%)

0.56

0.54 d=6mm 0.52

d=7 d=8

0.50

0.48 300

Ar - H2

d = 10

400

500

600

Current, I(A)

Fig. 33 Evolution with the arc current of the thermal efficiency of a DC plasma torch with a sticktype cathode for different anode-nozzle i.ds. The torch is working with 45 slm Ar and 15 slm H2 (Betoule 1994)

For a given arc current, all phenomena, including a voltage increase, will result in an improvement of the thermal efficiency. As, for example, by increasing the percentage of diatomic gases in the plasma-forming gas, by increasing the mass flow rate of the plasma-forming gas, and by decreasing the anode internal diameter (as shown in Fig. 33), the thermal efficiency will increase. Values of ηth vary between 0.2 and 0.95. Low values are obtained with wallstabilized torches working with pure argon and high currents (~1000 A), whereas very high efficiencies are found in the case of high-power vortex plasma torches (1–2 MW). Although the thermal efficiency of the torch is important, the environment in which the torch is operated may modify this efficiency. For example, high-power torches operated in high-temperature furnaces may require additional external cooling, which may substantially affect the overall thermal efficiency of the torch system.

6.1.3 Enthalpies The mean enthalpy of the heated gas is defined as h ¼ ηth  V  I=G

(11:26)

where G is the plasma gas mass flow rate. This implies that the kinetic energy of the flow is negligible. In terms of enthalpy levels, two types of plasma torches may be distinguished: • Wall-stabilized torches where G is rather low • Vortex-stabilized torches with high vortex flow (G is five to ten times higher for the same power level)

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Torches of the first type have enthalpies ranging from 100 to 1000 MJ/kg (30–300 kWh/kg), while those of the second type fall between 14 and 100 MJ/kg (4 and 30 kWh/kg).

6.2

Interaction of the Plasma Jet and the Surrounding Atmosphere

6.2.1 Turbulent Mixing and Its Effect on Plasma Jets In most cases, DC plasma jets exit the anode nozzle at velocities generally higher than 700–800 m/s. The steep velocity and temperature gradients associated with such flows gives rise to the generation of high turbulence in the flow at the exit level of torch nozzle. These can have a significant influence on the individual trajectories of solid parties injected into the plasma flows. In most cases, this turbulence is treated using either a K-ε or a Reynolds Stress approach. Among the approaches that attempt description of the large-scale turbulence in the jet are the assumption of a time varying radial profile for the plasma temperatures and velocities at the nozzle exit (Dussoubs et al. 2001; Park et al. 1997, 1999) and the use of a two-fluid model (Huang et al. 1995a, b, c). However, most calculations are performed assuming LTE. In their model, Huang et al. (1995c) considered steady turbulent argon plasma issuing into a stagnant argon environment at atmospheric pressure, as shown in Fig. 34. One of the fluids is the cold gas moving towards the jet axis, while the other fluid is the hot plasma moving in the outward direction in addition to the movement in the axial direction (see Fig. 34). The surrounding atmosphere is entrained as fragments, which intermingle with the plasma. Whatever may be the type of calculation for the turbulent mixing, the net result is a strong cooling of the jet. When the plasma jet flows in an argon atmosphere at atmospheric pressure, the plasma jet loses energy as a result of heating the cold surrounding argon gas entrained within the jet. Typical temperature distribution, Free stream boundary Inlet plane

Thermal fringe Momentum fringe x

Ux

Inlet

In-moving fragment

Out-moving fragment

Axis

x

Fig. 34 Illustration of two-fluid approach for simulating cold gas entrainment; the plasma jet consists of hot argon plasma with a radial velocity component away from the jet axis (400 A, 23.9 V, thermal efficiency 0.422, and Ar 35.4 slm, anode nozzle i.d. 8.5 mm (Huang et al. 1995c)

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measured by emission spectroscopy, is shown in Fig. 35a (see Part III, ▶ Chap. 20, “Plasma Diagnostics, Optical Emission and Absorption Spectroscopy”). However, if the surrounding atmosphere is nitrogen (N2), the cooling is more important, as shown in Fig. 35b, because of the dissociation of N2 around 7000 K. With air, one

Fig. 35 Effect of the environment on plasma jet temperature profiles, with air showing the strongest quench effect (a) argon ambient gas, (b) nitrogen, and (c) air (Roumilhac et al. 1990)

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must also consider the dissociation of oxygen around 3500 K and the jet cooling is more important as illustrated in Fig. 35c.

6.2.2 Nozzle Extensions The use of nozzle extensions has been studied essentially for plasma spray torches. As shown previously, the interaction of the high velocity plasma jet with the surrounding atmosphere has a significant impact on the cooling rate of the jet at the exit of the nozzle. Betoule (1994) showed that the fast cooling of the plasma jet due to the mixing of the plasma with the surrounding atmosphere can be delayed through the use of nozzle extensions. For stick-type cathode torches, the use of a nozzle extension resulted in a reduction of the rate of drop of the temperatures and velocities of the jet at the exit of the nozzle, despite the power lost in the extension cooling circuit. For example, Betoule (1994) used a torch with an anode-nozzle i.d. of 7 mm, with a plasma-forming gas flow rate of [45 slm (Ar) + 15 slm (H2)] and arc current of 600 A corresponding to a power input to the plasma torch of 40 kW. At a distance of 2 mm downstream of the torch nozzle exit, without extension, the gas axial temperature and velocity were 13,000 K and 2000 m/s, respectively. At 52 mm downstream of the nozzle exit, the axial temperature and velocity values drop down to 5200 K and 500 m/s due to the mixing with the ambient air. This implies that particles injected radially at the torch nozzle exit would travel in the plasma jet with rapidly dropping temperatures and velocities. On the other hand, in the presence of a 50-mm-long, water-cooled copper extension, the corresponding axial temperature and velocity, at 52 mm from the torch nozzle exit, were 11,000 K and 1700 m/s, respectively, in spite of 5.7 kW lost in cooling circuit of the extension. This implies that with a nozzle extension, particles injected at the extension entrance would travel in plasma where temperatures and velocities much higher than those with no extension. Unfortunately such a scheme, with radial injection of particles between the torch and its extension, offers an increasing risk of particle deposition on the inner wall of the extension leading eventually to its clogging. This is, however, not the case with axial injection of the particles in the plasma flow, such as in the Mettech Axial III plasma torch, which has an extension and is capable of attaining particle velocities which are two to four times those obtained with conventional plasma torches with stick type cathode (Moreau et al. 1995). This torch also allows for the spraying of suspensions such as YSZ particles with mean sizes around 100 nm, which reach velocities up to 800 m/s (Oberste-Berghaus et al. 2008). Conventional plasma spray torches, with radial powder injection, are limited to maximum particle velocities below 300 m/s under comparable spraying conditions (Fauchais et al. 2013). 6.2.3 Laminar Jets Many experiments were performed, mostly with stick-type cathode torches, to produce long, 40–50 times the jet diameter, laminar plasma jets of pure argon, pure nitrogen, and mixtures of argon and nitrogen flowing in air or controlled atmosphere, see for example Bauchire et al. (1997), Krejci et al. (1993), Pan

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et al. (2001), Cheng et al. (2006), Pan et al. (2006). Of course in parallel models were developed for analyzing the plasma jet behavior. See for example Bauchire et al. (1997), Chang and Pfender (1990a, b), Cheng et al. (2006). Compared to turbulent short plasma jets, the long laminar plasma jets are very stable with low noise-emission intensity. As the entrainment of ambient gas into the plasma jet is significantly reduced, the axial gradients of plasma jet parameters (temperature, velocity, and species concentration) are much smaller than those observed in short turbulent jets. The torch structure, gas feeding mass flow rate, power dissipated, anode nozzle internal diameter, and characteristics of power supply all affect the plasma jet length and stability of an atmospheric DC arc jet. Models have shown that only the molecular diffusion mechanism is involved in the laminar plasma jet, the mass flow rate of ambient air entrained into the laminar plasma jet being comparatively small and less dependent on the jet inlet velocity. For the turbulent plasma jet where the turbulent transport mechanism is dominant, the entrainment rate of ambient air into the turbulent plasma jet is more than one order of magnitude larger and almost directly proportional to the jet inlet velocity (Cheng et al. 2006). The length of the high-temperature region of the laminar plasma jet increases with increasing jet inlet velocity or inlet temperature, within certain limits, while the jet inlet velocity or temperature shows a reduced effect on the short turbulent plasma jet. These results are illustrated in Fig. 36 for an anode nozzle i.d of 8 mm, an arc current of 200 A and different argon flow rates.

Fig. 36 Variation of the visual plasma jet length with the argon mass flow rate. (a) 0.17, (b) 0.21, (c) 0.24, (d) 0.27, (e) 0.29, and (f) 0.37 (g/s). Arc current 200 (A) (Cheng et al. 2006)

(a) 0.17 g/s (Ar)

(b) 0.21 g/s (Ar)

(c) 0.24 g/s (Ar)

(d) 0.27 g/s (Ar)

(e) 0.29 g/s (Ar)

(f) 0.37 g/s (Ar)

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Transferred Arcs

6.3.1 Current Voltage Characteristics (a) Hot cathodes As discussed in Part II, ▶ Chap. 12, “Thermal Arcs,” transferred arcs are often gas stabilized (Coudert et al. 1993; Tsantrizos and Gauvin 1982). In contrast, free burning arcs are operated without superimposed flow, even if the surrounding atmosphere, entrained by magneto-hydrodynamic forces acting close to the cathode, becomes plasma gas. The important working parameters of a transferred arc are: the plasma-forming gas momentum close to the arc root at the cathode, the arc current, and the arc length. The latter is particularly important since beyond a critical value ‘c (depending on the two other parameters) the arc attachment at the anode shifts from the cathode jet dominated (CJD) mode with a rather diffuse attachment at the anode, to the anode jet dominated (AJD) mode, with a more or less severely constricted arc root (see ▶ Chap. 12, “Thermal Arcs,”). In the first case, even if the anode material is in a molten state, and some vapor is present, the metal vapor remains close to the anode and does not modify the general arc behavior. Typical shapes of the V-‘AK characteristics for a nitrogen plasma at an arc current of 250 A are shown in Fig. 37a (Tsantrizos and Gauvin 1982). The plasma shroud gas was varied between 7.1 and 21.3 slm. It is noted in both cases that the arc voltage increases monotonously with

Fig. 37 Voltage evolution of a DC transferred arc with a hot cathode as function of the anodecathode distance, ‘AK for a given arc current and different plasma-forming gas flow rates (a) with no anode material vaporization and (b) with anode material vaporization (Tsantrizos and Gauvin 1982)

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the increase of the arc length. In the absence of anode material vaporization, no particular change in the slope is observed when ‘AK > ‘c (i.e., when shifting from the CJD mode to the AJD mode). In the presence of anode material vaporization, on the other hand, the metal vapor is entrained into the arc, by the anode jet in the AJD mode. Due to the high electrical conductivity of the easily ionized metal vapor, the arc voltage drops significantly. However, as the arc radiation due to the neutral vapor increases, it will cool the arc, which may somewhat constrict itself opposing the drop of the electric field (Gleizes et al. 1994) giving rise to a more gradual change of the voltage drop as shown in Fig. 37b. Coudert et al. (1993) studied the effect of the electrode configurations on the arc voltage characteristics. The different configurations studied are schematically represented in Fig. 38. The results presented in Fig. 39a showed that plasmaforming gas velocity, or more specifically its momentum, close to the cathode tip, has an important influence on the arc voltage. The effect is a result of local constriction of the arc and the increase of the electric field strength (over some distance from the cathode depending on the design when forced convection plays a role). When injecting a diatomic gas into the plasma-forming gas, the arc column becomes constricted and the voltage increases. For example, with the same mass flow rate, arc length, and arc current, the voltage may increase by a factor of 3 to 4 when shifting from pure argon to pure nitrogen. The voltage evolution with arc current for a given value of the arc length, ‘AK, is shown in Fig. 39b when using the same electrode configurations defined in Fig. 38. For given plasma-forming gas momentum, the voltage decreases first when I increases. At low values of I, the arc is constricted and thus E and V are relatively high. For I > 100–200 A, with its G1

G3

G2 10 mm

10 mm

10 mm

5.6 mm

8.6 mm 60°

40° G4

40° 10 mm

G5

10 mm

1 mm 40°

0.5 mm 40°

Fig. 38 Different configurations of the cathode arrangement with its nozzle for gas flow control close to the cathode tip (all dimensions in mm) (Coudert et al. 1993)

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Fig. 39 Voltage evolution for an Argon DC transferred arc with a hot cathode and water-cooled copper anode with different injector configurations as given in Fig. 37, plasma gas flow rate = 30 slm (a) effect of arc length for an arc current I = 290 A and (b) effect of arc current for a given arc length ‘AK = 26 mm, (Coudert et al. 1993)

electrical conductivity almost constant, the arc has to increase E slightly to compensate for the enhanced losses due to the arc broadening (Coudert et al. 1993). Figure 39b also shows the important influence of the plasma-forming gas momentum close to the cathode tip. The momentum depends on the way the gas is injected. Configurations G1–G3 are quite inefficient, and moreover, G2 and G3 induce recirculation along the conical tip of the cathode (Gourlaouen et al. 1998). The G4 and G5 configurations will increase the gas velocity close to the cathode tip which is, by far, more efficient, provided the gas velocity close to the cathode tip is not too high to blow out the molten tungsten cathode tip. (b) Cold electrodes Vortex-stabilized plasma torches, such as those depicted in Fig. 7a, can be used for transferred arc operation. The outlet tubular electrode is shortened to avoid double arcing, and this electrode is kept at floating potential. The well-type electrode may be either cathode or anode, and the metallic material to which the arc is transferred has then the opposite polarity. The configuration with the well-type electrode as anode is generally preferred because the cathode erosion is more severe with such torches. Moreover, it allows performing electrochemistry in the molten bath by extracting the cations. Under both operating conditions, the transferred arc presents characteristics similar to those shown in Fig. 39. However, the voltage behavior with increasing ‘AK is by far steeper and the V-I characteristic is almost flat for a given cathode-anode distance. On the other hand, the voltage increases significantly with the increase of the plasma-forming gas flow rate, keeping the other parameters constant. The length of the part of the arc column, which is well established by the vortex flow in the case of cold cathodes, is shorter than that obtained with gas stabilized transferred arcs with hot cathodes. With hot cathodes, the same power level requires arc currents two to four times higher than those typical

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for cold electrodes. Thus, for the same power level of the transferred arc, the length ‘s of the wall-stabilized part of the arc (almost straight arc column) is about two to three times shorter than that with the cold cathode vortex transferred arc. Moreover, when the material to which the arc is transferred is the cathode, and if ‘AK > ‘c, cathode spots with the corresponding cathode jets will form on the surface of the material, moving erratically over a surface area with a diameter close to that of ‘AK (Moore et al. 1989).

6.3.2 Heat Transferred to the Anode There are no published results on heat transferred to the material from a DC vortex transferred arc. Thus, the following comments will be restricted to results published for hot cathode transferred arcs. The processes occurring at an anode are quite complex and have already been described. The heat flux qa received by the anode may be written as (Gauvin 1989; Parisi and Gauvin 1990, 1991). qa ¼ qc þ qr þ qe‘

(27)

where qc is the heat flux due to conduction and convection, qr is the heat flux due to absorbed radiation from the plasma column, and qe‘ is the heat flux due to electrons collected by the anode. The latter depends on the arc attachment mode (CJD or AJD). The heat fluxes qr and qc will depend strongly on the constriction of the arc column by the cold flow injected close to the cathode tip. Part of the energy dissipated within the arc will be lost by radiation and convection to the arc surroundings. The more severely the arc is constricted along its path the higher the arc radiation losses per unit volume to the surroundings. Part of this energy has to be removed or used in the process, especially for high-power (~MW) arcs if melting of the refractory lining of the furnace has to be avoided (Parisi and Gauvin 1991). Parisi and Gauvin (1990) made systematic studies of the energy radiated to a 15 cm long sleeve surrounding a transferred arc (22 cm long) where 12–24 kW were dissipated. Compared to the energy dissipated in the arc along the length of the sleeve (Vg  I, where Vg is the voltage drop along the arc within the sleeve), the radiated energy to the sleeve can reach 70% with Ar and 50% with N2. Similar results were obtained by Pateyron (1987) and Mehmetoglu and Gauvin (1983) who have shown that the radiated power increases with increasing arc constriction. The power dissipated at the anode increases with increasing arc length (resulting in dissipated power increase), increasing arc current, increasing percentage of diatomic gases in the plasma-forming gas (increasing the arc voltage), and with increasing constriction of the plasma column (increasing also the arc voltage) (Coudert et al. 1993; Parisi and Gauvin 1991) as shown in Fig. 40. The plateau obtained for increasing ‘AK (Fig. 40a) is due to the fact that for a given arc constriction (G  vi), and arc current, the arc shifts over a critical distance progressively from a CJD mode to an AJD mode. The increase of the heat flux with G  vi is mainly due to the convective contribution (20–30% of qa at most) (Nagashima et al. 1988).

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649

Fig. 40 Evolution of the energy dissipated in the copper water-cooled anode, Qi of a hot cathode transferred arc for different cathode arrangements (see Fig. 38) with argon (30 slm) as forming gas (a) with the anode-cathode distance ‘AK for an arc current I = 290 A and (b) with the arc current for an arc length, ‘AK = 26 mm (Coudert et al. 1993)

The increase of qa with arc current is mainly linked to the size of the arc root and the current density j both increasing with I (see Eqs. 24 and 25 of ▶ Chap. 12, “Thermal Arcs”). The percentage of the arc energy transferred to the anode varies between 50% and 75%. The smallest values are obtained with pure argon due to its higher radiation losses. The heat losses in the cathode cooling circuit are usually between 4% and 8% of the total power dissipated in the arc. It is interesting also to mention that Japanese researchers have shown that when constricting the anodic arc root in the CJD mode with a cold gas flow directed towards the arc root, the heat flux dissipated at the anode increases (Nagashima et al. 1988). This is probably due to the drastic increase of the current density with this forced cooling.

7

Nomenclature

Units are indicated in parentheses; whenever no units are given, the parameter is dimensionless.

7.1

Latin Alphabet

a ao am aφ a0 B

Sound velocity (m/s) Diameter of the connecting arc column (m) Radius of magnetic field coil (m) Coefficient relating the heat potential to the enthalpy (kg/m.s) Slope of the transient voltage signal (V/s) Magnetic induction (Tesla)

650 !

B Br Bz CD cv cp cw p d d1 d2 D da dAK dc ea eBL Ed0 ez f Fab Fag Fzg Fzb G h h ho h I Ib Id j k k ‘ ‘1 ‘K ‘AK ‘c ‘v L

M. I. Boulos et al.

Magnetic field (Tesla) Radial component magnetic induction (Tesla) Axial component magnetic induction (Tesla) Drag coefficient Specific heat at constant volume (J/kg.K) Specific heat at constant pressure (J/kg.K) Specific heat of water at constant pressure (J/kg.K) Electrode or neutrode diameter (m) Smaller diameter of cylindrical torch nozzle with a step change in its diameter (m) Larger diameter of cylindrical torch nozzle with a step change in its diameter (m) Internal diameter of the vortex chamber (m) Anode internal diameter (m) Anode-cathode distance (m) Internal diameter of cold well-type cathode (m) Unit vector in azimuthal direction Flow boundary layer thickness close to well-type cathode wall (m) Open circuit voltage of the power source (V) Unit vector in axial direction Fanning friction factor Azimuthal component of magnetic force (N) Azimuthal component of gas drag force (N) Axial component of drag force (N) Axial component of magnetic force (N) Mass flux of the gas (G = ρ.v)(kg/m2.s) Harmonic rank Heat transfer coefficient (W/m2.K) Specific enthalpy (MJ/kg)
Mean specific enthalpy: h ¼ ðP  Pth Þ =m_ g Þ (MJ/kg) Arc current (A) Coil current (A) Maximum arc current achievable in a torch (A) Current density (A/m2) Boltzmann constant (k = 1.38  1023 J/K) Integer Cooling channel length (m) Length of the smaller diameter (d1) of cylindrical torch nozzle with a step change in its diameter (m) Length of a stick-type tungsten cathode (m) Anode – cathode distance in a transferred arc (m) Critical anode – cathode distance (m) Length of the vortex chamber (m) Length of nozzle extension or characteristic length (m)

14

DC Plasma Torch Design and Performance

n0 P pa p Pr Pth q qa qc qr qe‘ QK R R1 R2 R’ Re ReZ S Sc sσi SV SVmin T0 tr ts U Ua v va v’a vc vi vmax vw vz V Va VK Vmin,i y z zb zbmax

Flow rate of the air entrained by the plasma jet (kg/s) Power (kW) or (MW) Ambient pressure (Pa) Pulse index (6 for example for a Graetz bridge) () Prandtl number (Pr = cp μ/k) Power loss in the cooling circuits (W) Heat flux (W/m2) Heat flux received by the anode (W/m2) Heat flux due to conduction and convection (W/m2) Heat flux due to absorbed radiation (W/m2) Heat flux due to electrons (W/m2) Heat flux removed at the cathode by the cooling water (W/m2) Negative resistance of the arc (dV/dI) (Ω) Internal radius of the magnetic coil (m) External radius of the magnetic coil (m) Length to diameter ratio of the vortex chamber, (R’ = ‘v/D) Reynolds number (Re = ρ.v.d/μ) also see Eq. 2 Reynolds number in a torch with a vortex flow (-) Swirl number Electrode cross-sectional surface (m2) Criterion for the voltage jump [σi/Sσi = (σi + Ua)I/G h0] Voltage criterion (SV = σe0.V.d/I) Dimensionless voltage [SVmin = (Vmin,i  UA  UK).d.σ0/I] Temperature at which the electrons molar fraction is 1% or 3% (K) Response time of a control unit (s) Time between two successive triggers in a thyristor bridge (s) Plasma gas velocity (m/s) Anode voltage drop (V) local gas velocity (m/s) Azimuthal component of the gas velocity (m/s) Magnetic induction azimuthal component (Tesla) Elongation velocity, vc, of the arc connecting column (m/s) Cold gas velocity (m/s) Maximum gas velocity, or center line velocity at the nozzle exit (m/s) Water velocity (m/s) Axial component of the gas velocity Arc voltage (V) Anode voltage drop (V) Cathode voltage drop (V) Minimum voltage corresponding to the ith voltage jump Diameter ratio (D/d) Axial distance along the plasma jet axis (m) Axial distance where Br = 0 (m) Axial distance where Br is maximum (m)

651

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Zi Zc ZH Zs

Different positions inside the plasma torch (Fig. 7b) (m) Axial position where the arc strikes (m) Segment length (m) Distance between torch nozzle exit and substrate or stand-off distance in spraying (m) Ratio of the length of the of the zone where the cold boundary layer thickness is still sufficient to prevent turbulent heat transfer to the anode

Z

7.2

Greek Alphabet

α δ Δ‘ Δp ΔVol ε εt γ ηth κ λ μ μ0 ρi ρw σe σi σo σT τ φ

Angle between two torches or triggering angle of a thyristor ( ) Thickness of the water channel (m) Length of the arc connecting column (m) Water pressure drop along the channel (MPa) Volume of the arc connection column (m3) Total emissivity of the plasma (W/m3.ster) Total emission of the plasma (J/m3.s) Isentropic coefficient (γ = cp/cv) Thermal efficiency of the torch Thermal conductivity (W/m.K) Mean free path (m) Viscosity (kg/m.s) or (Pa.s) Viscosity at temperature To (kg/m.s) or (Pa.s) Cold gas volumetric mass (kg/m3) Water volumetric mass (kg/m3) Electrical conductivity (A/V.m) or (mho/m) Voltage jump in the restrike mode (V) Electrical conductivity at temperature To (A/V.m) Standard deviations of temperature (K) Lifetime of the arc roots (s) ðT Heat flux potential: ½φ  φref ¼ κðTÞ:dT (W/m)

ωc

Cut-off frequency (s1)

Tref

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Ushio M, Tanaka K, Tanaka M (1995) Electrode heat transfer and behavior of tungsten cathode in arc discharge. In: Fauchais P, Boulos M, van der Mullen J (eds) Proceedings of the international symposium on heat and mass transfer under plasma conditions. Begell House, New York/ Wallingford, pp 265–272 Van den Brook J, Labrot M, Pineau D (1987) Revue Générale de Thermique (in French) 310:527–541 Wigren J, Pezjyd L, Karlsson H (1997) The effect of tungsten contamination from the spray gun on the performance of a thermal barrier coating. In: Thermal spray: proceedings of the 1st UTSC Yask’o OI (1969) Correlation of the characteristics of electric arcs. J Phys D Appl Phys 2(5):733 Yin F, Schein J, Hackett C, Heberlein J (1999) Investigation of the cathode behavior in a plasma cutting torch. In: Hrabovsky M, Konrád M, Kopecky V (eds) Proceedings of ISPC-14, vol I. Institute of Plasma Physics AS CR, Czech Republic, pp 49–55 Zhou X, Heberlein J (1995) Emission spectroscopic diagnostics and result comparison of theoretical and experimental investigations in the cathode region. JV Heberlein R (ed) Proceedings ISPC-12, St. Paul., MN, USA Zhou X, Heberlein J (1996) Characterization of the arc cathode attachment by emission spectroscopy and comparison to theoretical predictions. Plasma Chem Plasma Process 16 (1S):229S–244S Zhou X, Heberlein J, Pfender E (1994) Theoretical study of factors influencing arc erosion of cathode. IEEE Trans Compon Packag Manuf Technol A 17(1):107–112 Zhou X, Ding B, Heberlein JVR (1996) Temperature measurement and metallurgical study of cathodes in DC arcs. IEEE Trans Compon Packag Manuf Technol A 19(3):320–328 Zhukov MF (ed) (1977) Electric arc plasmatrons. The USSR Academy of Science, Siberian Chapter, Institute of Thermal Physics, Novosibirsk/USSR Zhukov MF (1979) Basical calculations of plasmatrons (in Russian). Nauka/USSR, Novosibirsk Zhukov MF (1989) Electric arc generators of low temperature plasma. In: Solonenko OP, Fedorchencko AI (eds) High temperature dust laden jets. VPS, NL, pp 5–19 (in Russian) Zhukov MF (1994) Linear direct current plasma torches. In: Sololenko OP, Zhukov MF (eds) Investigations and design of thermal plasma generators. Cambridge Interscience, Cambridge/ England Zhukov MF, Zasypkin IM (2007) Thermal plasma torches: design, characteristics, applications. Cambridge Interscience, Cambridge/England, 596 pp Zierhut J, Haslbeck P, Landes K, Barbezat G, M€uller M, Sch€ utz M (1998) TRIPLEX – an innovative three-cathode plasma torch. In: Proceedings. ITSC-98, Nice, vol 2. ASM International, Materials Part, OH/USA

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Plasma arc Cutting (PAC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Evolution of Different Cutting Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Basic Concepts of the Plasma arc Cutting (PAC) Torch Design . . . . . . . . . . . . . . . . . . . . . 3.3 PAC Torch Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Optimization of the PAC Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Plasma ARC Welding (PAW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Overview of Plasma Arc Welding Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Gas Tungsten Arc Welding (GTAW)/Tungsten Inert Gas (TIG) Welding . . . . . . . . . . . 4.3 Wire Electrode Plasma Arc Welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Laser-Assisted Arc Welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Basic Phenomena in Plasma Arc Welding Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Plasma Transferred Arc (PTA) Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Equipment and Operating Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Process Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Effect of Process Parameter Changes on Coating Properties . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Process Modifications and Adaptations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Emil Pfender: deceased. M. I. Boulos (*) Department of Chemical and Biotechnology Engineering, University of Sherbrooke, Sherbrooke, QC, Canada e-mail: [email protected] P. L. Fauchais Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS), University of Limoges, Limoges, France e-mail: [email protected] E. Pfender Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA © Springer Nature Switzerland AG 2023 M. I. Boulos et al. (eds.), Handbook of Thermal Plasmas, https://doi.org/10.1007/978-3-030-84936-8_47

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6 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Latin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

733 733 734 734

Abbreviations

AC-GMAW CA-TIG CMT DE-GMAW ESW FCAW GMAW GTAW ISPC ITSC LCHW LHS LSHW MAG MIG NTSC PAW PCTIG PTA PWHT RHS SAW TIG UTSC

1

Alternating current–gas metal arc welding Cap active flux tungsten inert gas Cold metal transfer Double-electrode gas metal arc welding Electroslag welding Flux-cored arc welding Gas metal arc welding Gas tungsten arc welding International Symposium on Plasma Chemistry International Thermal Spray Conference Laser-CMT hybrid welding Left-hand side Laser-standard short-circuiting arc welding Metal active gas Metal inert gas welding National Thermal Spray Conference Plasma arc welding Pulsed current tungsten inert gas welding Plasma transferred arc Post-weld heat treatment Right-hand side Submerged arc welding Tungsten inert gas for welding United Thermal Spray Conference

Introduction

Plasma cutting, plasma welding, and plasma transferred arc coating represents one of the most established groups of thermal plasma technologies which are used worldwide on an industrial scale. Their main advantages are in the precision and speed of the cut over relatively thick plates and the cleanliness and quality of the welds and coatings. Plasma torches used for cutting and welding are generally of relatively low power, 10 to 40 kW. Their design is based on the use of a hot cathode, operating in a transferred arc mode, with the part to be cut or welded acting as the anode to make use of the high heat fluxes associated with anode root arc attachments. Cutting and welding torches differ, however, in their performance

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objectives and accordingly in the nature of the plasma gas used. Cutting torches tend to use essentially oxidizing gases at high flow rates in order to enhance the cutting efficiency and get rid of metal dross, while welding torches use inert or reducing atmosphere in order to avoid contamination of the weld spot with debris of metal oxide. Plasma torches used for PTA coating also operate in the transferred arc mode to create a pool of molten metal at the coating location in which the coating materials in the form of a fine powder are uniformly dispersed. Once the molten metal cools and freezes, it creates a metal matrix composite coating, which is often used in hard facing or rebuilding of worn-off parts. This chapter is devoted exclusively to processes using plasmas, the materials to be welded, cut or coated being necessarily electrically conducting. The chapter is divided into three parts. The first dealing with plasma cutting torches, giving highlights of different torch designs and the fundamental science behind them. This is followed by a brief review of typical performance characteristics and fundamental research activities in this area. The second part of this chapter deals in a similar way with welding torches presenting corresponding design and performance characteristics. The third and final part of this chapter is devoted to PTA torches used for coatings and hard facing, describing their basic design features and typical performance characteristics.

2

General Remarks

According to Renault et al. (2011), the world market for cutting and welding equipments was estimated in 2009 at 12.8 billion US$ distributed between different industrial sectors as shown in Fig. 1, with the maintenance and repair activity

Fig. 1 World distribution of plasma arc cutting and welding markets between different industrial sectors (Renault et al. 2011)

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representing 33.5%, followed by construction and infrastructure 18%, automotive and machinery 14%, oil and gas industry 9.5%, ship buildings 9%, pipelines and mills 6%, petrochemical 5.8%, and power 4.2%. Cutting is the separation of a physical object, into two or more parts, through the application of an intense and localized force. Any sharp object is capable of cutting if it is sufficiently harder than the object being cut, provided it is applied with sufficient force. Since the end of the nineteenth century, cutting has been achieved by turning, drilling, and grinding, processes that generate heat which must be controlled to avoid overheating of the part and of the cutting tool used. The first nonmechanical cutting technique developed was based on heating the metal with a flame up to a critical temperature followed by its localized oxidation at the cut zone. The process was limited to steel and other ferrous metals easily oxidized by an oxygen-rich gaseous stream. It is only in the 1950s that plasma cutting was developed for cutting metals that could not be flame cut, such as stainless steels, aluminum, and its alloys and copper. Plasma cutting, however, does not rely on oxidation to work, and thus it can cut aluminum, stainless steel, and any other conductive material. The process started with the use of conventional plasma torches with stick-type tungsten cathode and a relatively small anode nozzle, working with nonoxidizing gases. This was followed by operating the plasma torch in the transferred arc mode with the workpiece acting as anode. With the further development of DC plasma torches, new types of electrodes with the possibility to work with oxidizing gases were used for cutting. According to Renault et al. (2011) from 1976 to 2006, the design of plasma cutting torches has rapidly evolved, working with lower currents, using smaller nozzle diameters, less plasma-forming gas flow rates for an equivalent or faster cutting speed and superior cut quality with less dross and oxides removal after cutting and minimal part deformation. Highlights of some of these developments are summarized in Table 1. Presently, cutting can be achieved by: • Mechanical means such as turning, drilling, and grinding • High-pressure water jet to which oxide particles are frequently added for enhanced performance • Flames, such as oxyacetylene burners used for heating and melting the metal at the location of the cut

Table 1 Advances in plasma arc cutting technology over the period 1976 and 2006 (Renault et al. 2011)

Thickness (mm) 6.0 12.0 25.0

1976 N2 plasma/H2O injection Nozzle orifice Current (mm) (A) 4.0 300 4.0 350 4.0 400

Cutting speed (m/s) 3.35 2.29 0.76

2006 O2 plasma/air injection Nozzle orifice Current (mm) (A) 1.1 100 1.5 150 2.1 300

Cutting speed (m/s) 3.81 2.29 1.77

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• Transferred arc plasmas with the metal generally acting as anode • Lasers with highly focused beams resulting in a localized heating and melting of the material with an excellent spatial resolution. Welding is a process that joins materials, mostly metals or thermoplastics, by local melting and coalescence. In most welding processes, a filler material is added to the pool of molten metal (the weld pool) which on cooling and freezing forms a strong joint. Pressure can be used in conjunction with heat to produce the weld. Welding is different from soldering and brazing, where a lower-melting point material is used to form a bond between the workpieces, without melting them. Up to the end of the nineteenth century, only forge welding was practiced in which blacksmiths joined iron and steel by heating and hammering. Arc welding and oxy-fuel welding were among the first processes to be developed in the beginning of the twentieth century and advanced rapidly when World War I and II drove the demand for reliable and inexpensive metal joining methods. Welding is presently the basic technology, universally accepted for permanently joining parts and manufacturing structures in a wide range of industries. Different plasma welding processes are commonly used. These include: • Tungsten inert gas (TIG), also called gas tungsten arc welding (GTAW) • Plasma arc welding (PAW) • Gas metal arc welding (GMAW) commonly referred to as metal inert gas (MIG) or metal active gas (MAG) welding • Submerged arc welding (SAW) that uses an automatically fed consumable cathode and a blanket of granular fusible flux • Electroslag welding (ESW) which is a single-pass welding process for thicker materials between 25 and 300 mm, in a vertical or close to vertical position The plasma transferred arc (PTA) coating process was developed for the rebuilding and repair of worn parts and the hard facing of parts exposed to extreme abrasion and corrosion. Regular steel parts with an appropriate PTA coating can exhibit superior corrosion and wear behavior even compared to specialty alloys. The process is significantly different from other coating processes, including plasma spraying, as the surface of substrate is locally molten in the process creating a strong metallurgical bond between the coating and the substrate if they are metallurgical compatibles. The substrate, in most of the cases, serves as the anode for the transferred arc and only exceptionally as cathode to significantly limit the heat flux to the substrate and clean its surface. The powder to be deposited is introduced into the arc plasma though two or more orifices located on a ring surrounding the exit of the plasma torch nozzle. A further annular slot surrounding the powder injection ring provides the shield gas flow, necessary to avoid reaction of the molten metal with the environmental air. However, in the heat penetration region and in particular the region where the coating material is mixed with the substrate material, the mechanical and metallurgical properties of the coating can differ from those of the substrate or the coating material. The PTA process combines good control over the process with high deposition rates (typically 3 to 10 kg/h up to 20 kg/h) and good coating quality with practically no porosity and

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dilution of less than 7% (better than 5% have been reported). Feeding of multiple powders allows for a range of coating compositions to be obtained. Compared to other thermal spray processes, PTA has the major advantage that alloy formation can take place in the metal pool during the process, allowing for a wider range of combinations of metal compounds or metal matrix composite compositions than with any other spray deposition process. In a variance of PTA coating technology, tungsten inert gas (TIG) welding, which is a free burning, un-constricted arc welding process, has also been used for coating applications with the filler material introduced in form of a wire. Because of the less stable arc attachment, the coating obtained has higher dilution values and larger heat affected zones compared to the standard PTA process. Deposition rates are of the order of 3 kg/h or less. Metal inert gas (MIG) welding using a wire as anode and the workpiece as the cathode is also a possible alternative to PTA. This has the advantage that oxide layers on light metals can be cleaned off before deposition. Deposition rates are in the order of 6 to 9 kg/h (Gebert and Bouaifi 2005), and the dilution is somewhat less than with the TIG welding-coating process. Compared to other weld overlay processes, the coating quality is less sensitive to the torch–substrate distance.

3

Plasma arc Cutting (PAC)

3.1

Evolution of Different Cutting Processes

As already mentioned earlier, apart from mechanical machining, four processes are commonly used for cutting. These are oxy-fuel, arc plasma, water jet, and laser cutting. Oxy-fuel cutting is one of the first nonmechanical cutting techniques developed. It is based on burning, or oxidizing the metal, and thus necessarily limited to steel and other ferrous metals supporting oxidation. Plasma arc cutting (PAC) was developed in the early 1950s for metals that could not be flame cut, such as stainless steels, aluminum, and its alloys and copper. Plasma cutting, however, does not rely on oxidation to work, and thus it can cut aluminum, stainless steel, and any other conductive materials. According to Renault et al. (2011), the design of plasma arc cutting torches have evolved rapidly since the 1950s with the following milestones: • 1955 – use of conventional plasma torches with stick-type tungsten cathode and an anode nozzle with a relatively small diameter. Oxidizing gases being not compatible with tungsten electrodes, nitrogen was used as plasma and shielding gas. • 1957 – the arc was transferred to the workpiece which acts as anode. A stick-type tungsten cathode was still used with either oxygen or water shielding injected downstream the cathode.

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• 1958 – broader use of other gases for shielding depending on the nature of the workpiece: air or oxygen for mild steel, carbon dioxide for stainless steel, and an argon/hydrogen mixture for aluminum. • 1961 – introduction of button-type, water-cooled cathode combined with the use of swirling shield gas. • 1965 – water shielding of plasma cutting torches, similar to dual flow except that water was substituted for the downstream shield gas to provide a better cooling effect. • 1975 – new design of the arc chamber and arc constriction. • 1976 – development of arc stabilization, complex gas management, oxidizing gas control. • 1977 – underwater cutting to reduce the noise level of the plasma arc and eliminate fume development (for arc currents higher than 100 A). • 1989 – introduction of hafnium (Hf) button-type electrode allowing for the use of oxygen as plasma gas combined with the developments of superior electronic process controller. • 1990 – deep water plasma cutting for the dismantling of nuclear reactors and auxiliary components. Water cutting: Carl Johnson of Durox International in Luxembourg used for the first time in 1956 water jet for the cutting of plastic shapes using a thin stream of high-pressure water. In 1962, Philip Union Carbide explored the same concept using pulsating water jets with water pressures up to 340 MPa to cut metals, stones, and other materials. In 1970, systems working with water pressures up to 480 MPa were used with further development devoted to the reduction of nozzle erosion. While cutting with water was also possible for softer materials, the addition in the 1980s of an abrasive to the water turned the water jet technology into a modern machining tool for all materials. The process has been called “abrasive water jet” (Momber and Kovacevic 1997). Laser cutting was developed in the 1960s with the first laser-cutting production machine used in 1965 to drill holes in diamond dies. This machine was made by the Western Electric Engineering Research Center. In 1967, the British pioneered laser-assisted oxygen jet cutting for metals. In the early 1970s, this technology was put into production to cut titanium for aerospace applications. At that time CO2 lasers were limited to cutting non-metals, such as textiles, due to their relatively low power rating which could not provide the necessary energy to compensate for heat losses by conduction to the bulk metal. The choice of solidstate lasers is limited by the wavelengths where they are better absorbed. Laser cutting is not suitable either for highly reflective materials such as aluminum (Petković et al. 2016). Each of the above-described cutting techniques has obviously its advantages and limitations. Besides the standard techniques for cutting by machining with tools and oxy-fuel flame, the most advanced cutting techniques are plasma, water jet, and laser which, according to Krajcarz et al. (2014), are characterized by:

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• Plasma arc cutting is limited to the cutting of metals or electrically conductive materials that can be relatively large but simple. The thickness treated is medium and thick (up to 50 mm and more). It can also treat metals covered with a medium thickness of rust. It is fast with a good cutting precision and the relatively low operating costs compared to water jet or lasers. Plasma arc cutting, however, cannot cut multilayers but can give rise to material hardening and thermal deformation over relatively wide areas. It can also produce hazardous fumes. • Abrasive water jet is the best for thick and thin parts with small and large size, even with complicated shapes. However it is relatively slow. Most of solids are suitable, covered or not by rust, or composites. It induces no thermal deformation and produces no hazardous fumes. Multilayers can be cut. However it has the highest operating costs. • Laser beam can cut small and large parts with relatively complex shapes. The treated material can be thin to medium in thickness (up to 6–10 mm) and would tolerate a small degree of dust cover. The cutting precision is high and the operating costs are relatively low. However, the solids must be homogeneous with no reflections for the laser wavelength. Multilayer cutting is impossible. Laser cutting also gives rise to hardening and local deformation over small area surrounding the cut. They can produce hazardous fumes.

3.2

Basic Concepts of the Plasma arc Cutting (PAC) Torch Design

3.2.1 Introduction Plasma arc cutting (PAC) is based on the use of highly controlled plasma energy sources for the localized heating and melting of the workpiece to be cut. PAC torches are mostly operated in the transferred arc mode with the workpiece to be cut as the anode. It is only when cutting dielectric materials, such as glasses and ceramics, that standard blown arc torches are used. The transferred arc configuration takes full advantage of the enhanced energy transferred to the anode arc root (see Sect. 3.4.2.1 of Part II, ▶ Chap. 13, “Electrode Phenomena in Plasma Sources”) (Eliot 1991; Sung Je Kim 2009; Eddie Paul 2011). Depending on the nature of the material to be cut, the process can also make use of the energy generated through the oxidation (combustion) of the material to be cut for the acceleration of the cut speed as in the case of using oxygen plasma to cut a steel part. It is also necessary to have a highvelocity plasma gas flow with a sufficient inertia in order to blow the molten metal away from the cut region. Cutting torches are therefore generally equipped with nozzle with i.d. as small as possible in order to accelerate the plasma flow. 3.2.2 Arc Stabilization Mechanism A wide range of plasma cutting torch designs and arc stabilization techniques have been developed and commercially used in industry. Most of these are covered by a large number of patents and proprietary information, which make a detailed discussion of their design and performance rather limited (Colombo et al. 2011). Schematic representations of two typical PAC torch designs are given in Fig. 2. The design

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b

Hot cathode holder Anode-nozzle Part to be cut (anode)

Fig. 2 Plasma cutting torches with (a) stick-type cathode and axial plasma-forming gas injection and (b) button-type cathode and vortex plasma-forming gas. (Eliot 1991)

Fig. 3 Plasma cutting torch with button-type cathode, vortex plasma-forming gas, and external water vortex (Eliot 1991)

Plasma forming gas with vortex injection

Button type cathode Water cooling Water vortex inlet Nozzle Water jacket

given in Fig. 2a shows a torch with a stick-type thoriated tungsten cathode with axial injection of the plasma-forming gas. Because of the sensitivity of thoriated tungsten to the presence of oxygen in the plasma gas, these torches can only be used with nonoxidizing gases such as N2 or Ar-H2. The torch design given in Fig. 2b uses, on the other hand, a button-type cathode with vortex injection of the plasma-forming gas. The cathode is made in this case of either hafnium (Hf) or zirconium (Zr) for operation with air or oxygen or of thoriated tungsten for operation with nonoxidizing gases. In each of these torch designs, with axial or vortex arc stabilization, the arc is struck between the central electrode which acts as cathode and the part to be cut which acts as anode, with currents up to 200 A or more. In order to increase the arc constriction, decrease the fume emission, and reduce the thermally affected zone of the workpiece, a water vortex can also be added at the nozzle exit as shown schematically in Fig. 3. Due to additional energy losses in the water, this torch is operated at higher power levels. The arc current can vary between 15 and 640 A but remains limited to 200 A with Hf or Zr cathodes. In order to keep the current density high, the nozzle orifice is kept as small as possible with its internal diameter directly dependent on the arc current. For example, with nitrogen as plasma gas, the i.d. of the cutting torch nozzle can be as small as

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0.9 mm for arc currents up to 30 A, 1.2 mm for currents up to 60 A, and 1.5 mm for currents up to 120 A. The gas flow rates are adjusted with the arc chamber pressure reaching up to sonic velocities in some cases.

3.2.3 Choice of Plasma Gas The choice of the proper plasma gas for the operation of plasma arc cutting torches has a critical influence on the torch performance and the process economics. Gases best adapted for plasma cutting must have a high specific enthalpy and a high thermal conductivity, which is generally the case with diatomic gases such as H2, N2, and O2. They should also have a high molecular mass in order to compensate for the significant drop in the specific mass of the gas at high temperatures. Pure argon is only used to start the arc but never for cutting due to its relatively low specific enthalpy and thermal conductivity. Ar/H2 mixtures are typically used with 20 vol.% H2. The presence of hydrogen gives rise to a significant increase of the specific enthalpy and thermal conductivity of the mixture compared to that of pure argon. Hydrogen also helps to trap, at least partially, oxygen and provides neat and clean cutting grooves. N2 is a good alternative plasma gas due to its high specific enthalpy and high thermal conductivity, which allows fast cutting without excessive oxide and slag formation. The thickness of the workpieces in this case is limited to 25 to 30 mm. The addition of H2 to N2 contributes to achieving better performance. It is to be noted, however, that nitrogen is susceptible to partial desolation in the molten metal in the cut area, which could make it more difficult to weld the part in the same cut area. O2 is also a good alternative plasma gas for plasma arc cutting since it is useful for energy transport. At the top of the cut where temperatures can exceed 2200
C, there is no exothermic oxidation for iron, while at the bottom of the cut, oxidation adds to the heating of the cut area and gives rise to a reduction of the surface tension and the viscosity of the molten metal which helps for the drainage of the formed slag at the bottom of the cut. The proper choice of the plasma gas has to be made taking also into account the electrode material since it will have an impact on the electrode lifetime. Zirconium or hafnium cathodes exhibit much higher wear, and consequently shorter lifetime, than that of the tungsten electrodes when working with nonoxidizing gases. Moreover, as the arc current is limited, by electrode design, to a 200 to 300 A, the maximum power of the plasma torch will depend on the nature of the plasma gas used, which has a direct effect on the torch voltage. The torch power will in turn affect the maximum thickness of the workpiece to be cut. While dry air is frequently used to replace oxygen for economic reasons, its use gives rise to the formation of nitrogen oxides, which requires an efficient ventilation of the environment around the welding area. Recently a new plasma arc cutting torch was proposed (Maslani et al. 2013; Sember et al. 2011) working with water vapor and presented in Fig. 4. The liquid water is vaporized with a heating element and introduced into the arc chamber where the cathode is made of hafnium insert in a copper holder. The nozzle i.d. is 1 mm working typically with a current of 60 A.

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Fig. 4 Schematic view of a steam plasma arc cutting torch (Maslani et al. 2013)

Liquid water Heating element

Cathode

Vaporizer

Steam

Cutting nozzle Plasma

Convective heat transfer between the arc and the workpiece which can vary between 20% and 50% of the total heat flux impacting on the workpiece and increases, almost linearly, with increasing torch to part distance and with the arc current (Valetoux 1998). With an O2 plasma and a current of 15 A, steel sheets with a thickness up to 1.5 mm can be cut. With a current of 120 A, the corresponding thickness limit of the steel plates, which can be cut, is between 8 and 20 mm.

3.3

PAC Torch Characteristics

3.3.1 General Remarks Because of their relatively tight design, PAC torches are virtually inaccessible by most diagnostic techniques. Studies of the characteristics and performance of PAC torches have to rely mainly on a combination of energy balance, optical diagnostic techniques, and mathematical modeling of the flow, temperature, and electromagnetic fields in the discharge. An excellent review of studies of the arc characteristics in a PAC torch based on the use of Schlieren techniques combined with high-speed imaging was reported by Colombo et al. (2009). The review gives valuable insight on the following topics: • • • •

Visualization of the cutting front through high-speed imaging Position and geometry of the shock wave in the arc/plasma jet at the nozzle exit Studies of electrode erosion phenomena Pilot arcing and piercing phases in plasma arc cutting of mild steel and stainless steel • Cathode region phenomena during the different phases of pilot arcing • Phenomena taking place during double arcing • Particle ejection from the cathode during pilot arcing

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Electrode with Hf insert

Sapphire window Shield gas cap

Viewing port

Nozzle

Fig. 5 Schematic representation of (a) standard PAC nozzle and (b) modified nozzle for cathode visualization (Peters et al. 2005)

Most of these studies were carried out on the external region of the arc/plasma jet downstream of the torch nozzle or through end view observation of the arc in the discharge cavity. Observation of the arc root attachment on the cathode needed for studies of cathode erosion had to rely on the use of specially modified torches as schematically represented in Fig. 5 (Peters et al. 2005). Such observations and associated data helped in optimizing the design of torch components, determining process characteristics and plasma operative conditions (e.g., cathode erosion mechanism, double arcing, effect of anode attachment location, dross formation, effects of non-perfectly aligned torch head components). Mathematical modeling, on the other hand, is the most valuable and complimentary tool to diagnostics by offering means of calculating the flow, temperature, and electromagnetic fields in the discharge under steady-state and transient condition which are essential for the understanding of the basic phenomena involved and their interactions. Modeling, however, is not a simple task either because of the need to take into account a wide range of complex phenomena such as: • The interaction between the thermal plasma and molten cathode surface. • The current distribution close to the cathode. • The gas flow pattern in the discharge cavity, which can have a swirl velocity component. • The possibility of having sonic, compressible flow, at the exit nozzle of the torch. • The arc attachment at the anode (generally parallel to the torch axis) controls the heating and melting process of the part being cut and consequently the cutting speed. • Chemical nonequilibrium and deviations from LTE.

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• Metal vapors resulting from the local evaporation of the workpiece being cut will diffuse and mix with the plasma flow affecting their thermodynamic and transport properties and, most significantly, the radiation properties of the plasma. Models imply the solution of the mass fraction, the momentum, and the energy conservation equations to which are added the equations of the vector potential. Most calculations are performed at equilibrium (LTE) except those of Ghorui et al. (2007) who assumed chemical nonequilibrium and two temperatures (Te for electrons and Th for heavy species). The arc radiation is taken into account using the net emission coefficient or the P1 model; see for example Freton et al. (2002, 2003). Calculations are often 2D (Freton et al. 2002, 2003) with the swirl injection of the gas into the arc cavity being simulated by an equivalent surface corona, and calculations are performed with the jet assumed to have both laminar and turbulent regions in the flow (k-ε or Prandtl mixing length) (Zhou et al. 2009a, b; Chau et al. 2011; Colombo et al. 2011; Phi Long 2013). Whatever may be the model, the thermodynamic, transport, and radiation properties of the plasma gas must be known either at equilibrium or under nonequilibrium two-temperature conditions. For more details, see Part I of this Handbook, Chapters 4 through 10. Colombo et al. (2008) have also reported extensive studies of the thermodynamic and transport properties of Ar, O2, and N2 under nonequilibrium conditions. Cressault and Gleizes (2010) reported data for diffusion coefficients in air–metal thermal plasmas.

3.3.2 Flow and Temperature Fields in a PAC Torch The model proposed by Ghorui et al. (2007) considers nonequilibrium effects using a two-temperature description of the plasma and chemical nonequilibrium due to diffusion. A schematic of the torch simulated and the computational domain are given in Fig. 6. The torch had a water-cooled copper nozzle with an internal diameter of 2 mm and length of 2.6 mm (distance JF on Fig. 6). The cathode was a buttontype, with a water-cooled copper holder and a 2 mm diameter Hf insert flush to the holder surface placed at a distance of 5 mm behind the nozzle entrance level (distance EJ on Fig. 6). Oxygen was used as the plasma gas injected in a vortex mood upstream of the cathode. The arc current was 200 A, and the internal pressure Fig. 6 Schematic of PAC torch simulated and the computational domain (Ghorui et al. 2007)

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in the torch was 478 kPa. Isocontours of the electron temperature, Te, and heavy particle temperature, Th, in the discharge cavity are given in Fig. 7. With the exception of the core region of the arc close to its centerline, a wide difference is observed in the distribution of the electron and heavy particle temperatures. On the torch axis, the temperature of the electrons and heavy particles are rather close, showing an immediate increase close to the cathode surface dropping with distance from the cathode, only to regain its maximum value of 28,000 K near the nozzle exit. In the radial direction, the heavy particle temperature drops sharply near the nozzle wall, leaving a thin layer of cold gas attached to the wall and thereby preventing it from melting, while the electron temperature continues to be rather high even in the close vicinity of the nozzle wall, r = 1.0 mm as shown in Fig. 8. Computed distributions of velocities inside the torch are presented in the vector plot of Fig. 9. The gas velocity is low before entering the zone near the tip of the cathode, where intense heating by the arc current causes a rapid expansion of the gas. Due to the nozzle constriction, it results in a significant pressure buildup in the torch, as seen in Fig. 10, which limits the gas velocity increase in the immediate vicinity of the cathode. As the high-pressure gas moves toward the torch nozzle, combined with the significant heating of the plasma by the arc current, the plasma velocity increases significantly reaching up to 8000 m/s at the nozzle exit. Figure 11 shows the axial velocity profile along the centerline of the torch and its radial distribution at the torch exit level, z = 21 mm. Figure 11a shows a strong acceleration of the gas along the torch axis especially in the exit nozzle channel. The corresponding radial velocity distribution at the torch exit is shown in Fig. 11b. It is to be noted that the pressure

Plasma Torches for Cutting, Welding, and PTA Coating

Fig. 8 Comparison of the computed electron and heavy particle temperatures with spectroscopic measurements carried out at 0.5 mm downstream of the torch nozzle exit level, oxygen plasma, arc current = 200 A (Ghorui et al. 2007)

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Numerous other studies have been reported in literature dealing with the modeling of the flow and temperature fields inside a commercial PAC torch such as that of Zhou et al. (2009a, b) on the effect of plasma–gas swirl on the performance of a highly constricted plasma cutting arc and comparative study of turbulence models. Interesting results are presented by Chau et al. (2011) with modeling steam plasma flow in a nontransferred torch, Colombo et al. (2011) about design-oriented simulation of PAC consumables and experimental validation of results, and Phi Long (2012)

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Fig. 12 Distribution of the current density inside a PAC torch operating with pure oxygen, arc current = 200 A (Ghorui et al. 2007)

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related to numerical investigation of the effect of cathode holder shape on Hf cathode evaporation for an oxygen PAC torch. Of course the temperature distribution in the workpiece (anode) and the geometry of the cutting front are very important parameters, which have been the subject of numerous studies. Nemchinsky (2005) reported a study of the anode layer in a high-current arc in atmospheric pressure nitrogen; Teulet et al. (2006) reported an experimental study of an oxygen plasma cutting torch, arc–material interaction, energy transfer, and anode attachment; and Martin et al. (2013) reported measurements of the energy released due to the oxidation reaction during plasma arc cutting of steel with an oxygen plasma.

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Swirl velocity,

Fig. 14 Radial profiles of swirl velocity for different swirl numbers 4 mm downstream from the nozzle exit, nozzle i.d. 1 mm, arc current = 200 A (Zhou et al. 2009)

(m/s)

3.3.3 Arc Dynamics In PAC torches, the plasma-forming gas is often injected with a swirl orientation in the discharge cavity to improve plasma column confinement and provide a more stable constricting arc (Zhou et al. 2009a). The swirl number of the flow, Sw, defined as the ratio between the axial flux of angular momentum and the axial flux of axial momentum normalized with a characteristic radius, is of critical importance for the proper control of the flow pattern around the arc. Due to centrifugal forces, increasing the swirl number moves the location of maximum pressure to the plenum chamber edge while reducing the swirl number moves it closer to the axis. The large pressure difference in the radial direction constricts the arc and leads to large current density and Ohm heating close to the axis. This has the consequence of increasing the arc temperature near the cathode and accelerates the cathode erosion rate, which is a limiting factor in the arc process, especially with oxygen as cutting gas. Changes in the swirl number also affect significantly the velocity distributions in the discharge cavity and in the plasma jet at the exit of the PAC torch. Figure 14 presents radial profiles of the tangential velocity at a distance of 4 mm downstream from the nozzle exit for different swirl numbers (Zhou et al. 2009a). Considering that the PAC torch is normally operated with a distance of 3 to 6 mm between the torch nozzle exit and the surface of the workpiece, it is not surprising that a high swirl number can affect the bevel angles for the left and right edges of the cutting kerf. Zhou et al. (2008) reported a study of the effect of the nozzle length, arc current, and plasma gas flow rate, on the performance of a PAC torch. The results show that a long nozzle gives rise to a high-velocity plasma jet with high heat flux. Both arc voltage and backup chamber pressure increase with the nozzle length. High arc current also increases plasma velocity and temperature, enhances heat flux, and augments chamber pressure. Strong plasma gas mass flow produces a pinch effect on plasma arc inside the torch, increases the arc voltage and power, and accordingly increases the plasma velocity, temperature, and heat flux to the workpiece. Figure 15 shows the effect of increasing the arc current over the range of 100 to 200 A on the axial temperature profile along the centerline of the torch. The plasma temperature decreases sharply downstream of the cathode region followed by significant increase in the nozzle channel to temperatures in the 24,000 to 28,000 K range before immerging from the torch. Calculations were performed with no shield gas, 200 Sw =40

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the radius of the nozzle bore being 1.0 mm, and the length from cathode tip to nozzle inlet is 5.2 mm; the length of the cylindrical nozzle is 3.8 mm. The temperature profile At 200 A shows a subsequent peak in the temperature profile at a distance of 3 mm downstream of the nozzle exit which is associated with the a shock wave, where the kinetic energy converts to inner energy. No such shock waves are observed at lower arc currents of 100 or 150 A. Unfortunately, one of the phenomena that put a limit to increasing capabilities of the PAC process is double arcing (Nemchinsky 1998; Nemchinsky and Severance 2006). As schematically illustrated in Fig. 16, double arcing occurs when the voltage drop across the cold gas envelope surrounding the central arc channel reaches its electrical breakdown voltage giving rise to arcing between the central plasma core and the inner cold wall of the torch nozzle. As a consequence of double arcing, the arc, which normally connects the cathode to the workpiece, is split into two parts, one connecting the cathode and the nozzle and the other connecting the nozzle and the workpiece. This event leads to catastrophic damage of the nozzle and electrode. Guo

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Fig. 17 Schemes for (a) dual gas structure (referring to Peters et al. 2007) and (b) double-nozzle structure introduced by Guo et al. (2010)

et al. (2010) proposed the use of a double-nozzle structure, as shown in Fig. 17, which is less vulnerable to the double arcing phenomenon compared to single-nozzle structure, even with a shield cap. However with longer nozzle shock wave in the plasma, jet may be stronger. Figure 18a, b shows the axial profiles of temperature and velocity for different nozzle lengths. There are slight variations for both temperature and velocity in the plasma torch chamber, which are located between the cathode tip and the nozzle inlet. In the nozzle channel, the longer is the nozzle length, the lower the temperature and velocity are. Downstream of the nozzle exit, the axial variation of the temperature and velocity is more pronounced for the longer nozzle length. The longer nozzle gives rise to a better quality of cut kerf (Guo et al. 2010). Alternate approaches to overcome double arcing have also been proposed with varying successes such as the use of a dual-swirling PAC torch design proposed by Wang J. et al. (2011).

3.3.4 Cathode Erosion Cathode erosion varies widely depending on the cathode design and the operating conditions of the PAC torch. It is of critical importance since it has a direct effect on the cathode lifetime and accordingly on the process reliability and economics. The principal operating conditions that have an impact on the cathode erosion are: • • • • •

Cathode design Nature of the plasma gas and its flow rate Torch current in transient and steady-state operation Torch startup and shutdown procedures Frequency of startups and shutdowns

In the case of stick type, refractory metal cathode erosion results from the evaporation of the hot cathode material at the tip of the electrode into the near-cathode plasma.

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According to Nemchinsky (2012), as the metal vapor is heated by the plasma, it is ionized almost instantly. The ions created, under the influence of the local electric field, have a high probability of returning to the cathode on which they deposit. A small fraction of the evaporated metal diffuses into the bulk of the plasma stream and accordingly becomes lost. The fraction of the total evaporated metal that is not redeposited on the cathode, called the escape factor, is of the order of a few percentage. Nemchinsky (2012) estimates an erosion rate close to 1 μg/s for a free-burning argon arc with a thoriated tungsten cathode and an arc current of 200 A. Nemchinsky and Showalter (2003) have also shown how the flow pattern in the vicinity of the cathode can influence the cathode erosion. Zhou et al. (2009a, b) reported that for Sw < 3, the plasma moves axially toward the cathode near the axis, which according to Nemchinsky (2002) makes the evaporated atom’s return to the cathode more probable thus reducing cathode erosion. In the case of button-type cathodes with a hafnium insert, erosion is the principal mechanism responsible for electrode wears which is in the form of particulates that

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Fig. 19 Illustration of cathode and nozzle wear (a) new cathode and nozzle, (b) normal cathode wear and pit formation and (c) cratered cathode pit (Peters et al. 2008)

are ejected from the molten hafnium pool. This phenomenon occurs during steadystate operation due to fluid dynamic effects near the cathode surface (Nemchinsky 2005) and under the transient conditions during arc ignition and extinction between cuts (Peters et al. 2005. Peters et al. (2008) performed measurements without the shield gas. Figure 19 illustrates the cathode and nozzle condition for new cathode, Fig. 19a, worn cathode, Fig. 19b, and cratered cathode, Fig. 19c. The amount of deposits on the internal surfaces of the nozzle increases as the cathode erosion increases. There is also the possibility of the arc damaging to the nozzle channel during operation. The formation of the crater can be catastrophic for torch operation if copper is eroded too quickly. In this case, the arc cannot be sustained or restarted. Boselli et al. (2013) showed that there was no influence of the recess depth on the time for the stabilization of the pilot arc attachment on the cathode insert. They also showed that droplets of molten cathode material can be expelled from the emissive surface during pilot arc shutdown and, in general, during the arc-off transient contributing to the cyclic erosion of the electrode. They also pointed out the influence of the swirl component of the plasma gas on the cut quality, which is influenced by cathode instabilities induced by a very high swirl flow. Colombo et al. (2009) used the modified PAC torch design given in Fig. 5 to follow the erosion of a button-type cathode with Hf insert. Droplet ejection during different phases of the cutting process was observed in agreement with observations reported by Heberlein (2007). Based on the results of their mathematical modeling work, Phi Long et al. (2012) showed that the total amount of mass loss due to hafnium cathode evaporation was enhanced with increasing gas flow rate, gas pressure, and cathode diameter. The results were in agreement with experimental data in terms of the average surface temperature of the cathode. Phi Long et al. (2013) reported, on the other hand, that the shape of the cathode holder could have a significant effect on the Hf erosion rate, through its influence on the flow pattern of the plasma in the vicinity of the cathode. Comparing the three cathode holder designs given in Fig. 20, they showed that the convex cathode holder not only reduced the gas flow velocity away from the cathode surface but also gave rise to the formation of a vortex in the vicinity of the cathode moving in the opposite direction. The cathode surface temperature decreased

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markedly with higher protrusion of the convex cathode holder, Fig. 20c, resulting in a significant decrease of mass loss of from the Hf cathode. Rotundo et al. (2012) experimentally studied the microstructural modifications of the Hf insert in plasma arc cutting electrodes operating at 250 A during first cycles. Colombo et al. (2011) studied the behavior of Hf cathodes at the beginning of their service life when operating at high current levels (250 A). They have drawn conclusions concerning the optimization of the dimensions of the initial recess of the Hf insert fit to avoid the massive deposition of material on the nozzle inner surface that would cause a strong reduction of electrode and nozzle service life and a rapid degrading of cut quality. Chau et al. (2011) have modeled steam plasma in a DC nontransferred torch equipped with well-type cathode with a strong vortex flow.

3.4

Optimization of the PAC Process

3.4.1 Independent Parameters of the PAC Process The optimization of plasma arc cutting, as much as any other technology, requires a clear definition of the quality criteria required in the final product and a fundamental understanding of the basic phenomena involved and the role played by different independent variables. According to Colombo et al. (2009, 2010); and Chamarthi et al. (2013), the main independent parameters influencing the plasma cutting process are: – – – – – –

Arc voltage or arc current intensity Nature of the plasma-forming gas and its flow rate Torch nozzle orifice diameter and length Orientation of the torch with respect to the workpiece and standoff distance Thickness of the workpiece Cutting speed

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Fig. 21 Scheme of typical arc cutting industrial problems (Narimanyan 2009) Non-square Cut edges Top edge rounding Dross

The quality of the cut is evaluated in terms of a number of parameters illustrated in Fig. 21 after Narimanyan (2009). Two types of defects are observed, the non-square (or rounded) cut edges at the top of the plate and the formation of dross at the bottom edge of the kerf. The former is generally a result of overheating of the metal at the upper surface of the plate being cut. The dross formation on the lower surface of the plate depends in a complex way on the thickness of the plate and the gas flow in the cut region, which is needed to remove the molten metal as it is formed. Other cut defects occasionally observed according to Kavka et al. (2011) include splats of molten metal, edge inclination, and draglines.

3.4.2 Energy Balance A first step in evaluating the performance of a PAC torch, or for that matter any plasma device, is to carry out an energy balance in order to access the energy distribution into the system and how much of it is being used for the process performance. Girard et al. (2006) and Teulet et al. (2006) performed an extensive study of the PAC torch under different operating conditions. The work of Girard et al. (2006) was devoted to spectroscopic temperature field measurements in the plasma jet at the exit of the torch nozzle. This was followed by the study reported by Teulet et al. (2006), where they addressed the arc–material interaction, energy transfer, and anode attachment for oxygen as plasma gas under different operating conditions. The different energy components distributed into the system are schematically presented in Fig. 22. These include the following: Qtot total energy dissipated in the arc (Qtot = U  I) Qlost-up losses above the plate mainly due to radiation and convection Qkerf available energy brought by the arc inside the kerf (Qkerf = Qtot  Qlost-up) Qoxy energy generated through the oxidation of iron QHCL conduction losses inside the workpiece Qmelt energy utilized in melting steel Qlost-down losses beneath the workpiece in the extinguishing arc Typical energy balance data for PAC torch operated with oxygen at plasma gas as function of arc voltage, U, cutting speed, vt, and oxygen inlet pressure, PO2, are given in Fig. 23a, b, and c, respectively, after Teulet et al. (2006).

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Fig. 22 Energy balance in a PAC process (Teulet et al. 2006)

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While the energy Qmelt used in melting/cutting the steel is weakly affected by changes in the plasma gas mass flow rate, it increases significantly by the increase of the arc voltage, U, and cutting speed, vt. Strong behavior similarities were clearly identified between heat conduction losses, QHCL, and melting power, Qmelt, because both are part of a single term corresponding to energy deposited inside the plate. Teulet et al. (2006) also visualized the arc attachment at the anode. They showed a slight motion of the arc root toward the bottom of the cut-front surface when the inlet pressure (i.e., the mass flow rate of oxygen) increased. They found also that when the cutting speed, vt, increased, a second luminous spot was located immediately below the top of the plate, as indicated in Fig. 24. Bini et al. (2008) also found that cutting speed and arc voltage affect the kerf formation mechanism and their interaction is also important in defining the inclination of the cut. Kavka et al. (2013) studied the effect of the nature of the plasma gas (air or water vapor) and its flow rate on the properties of transferred arc and cutting phenomena (analysis of cutting performance and energy balance of the cutting process). Despite significant difference in gas and arc properties, oxygen and steam cutting exhibited essentially similar features. Colombo et al. (2009), using high-speed imaging, studied arc root attachment in the kerf region during plasma arc piercing and cutting in various operating conditions. They confirmed that for the cutting speed over 0.9 m/min, a second luminous spot can be observed close to the top of the cut-front surface, probably due to anode spot oscillation or a double attachment of the arc root. Osterhouse et al. (2013) presented a new method of measuring the location of the arc attachment through measurement of the arc voltage. The validity of this method was supported by the observation of kerf shape, dross particle size, and dross adhesion to the workpiece.

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Plasma Torches for Cutting, Welding, and PTA Coating

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3.4.3 Cut Profile Wang et al. (2013) in their study of the dual swirling plasma cutting parameters formed longitudinally a cut of 80 mm length in the middle of the plate. After cutting, an observed section was obtained at 20 mm apart from the end of the cut to evaluate kerf shape. The shape factors of the cut usually include top kerf width (Kt), bottom kerf width (Kb), bevel angle (θ), straightness or flatness (St), and the adherent dross to the cut, as shown in Fig. 25. They studied the influence of different working parameters on the cutting profile. The experimental conditions were as follows: arc current 45–75 A; cutting speed 5–50 mm/s; thickness of workpiece 6 mm; torch standoff distance 2 and 3 mm; plasma gas flow rate 31 slm, at a gas feed pressure of 0.81 MPa; shield gas flow 12 slm; nozzle internal diameter 1.2 mm; and Hf cathode insert diameter 1.1 mm. Figure 26 shows photographs of several examples of cut cross-section. It is noted that the lower is the arc currents and the slower cutting speeds, the more dross attaches to the bottom of the cut, particularly on the right side (i.e., low-speed side) of the cut. It is also demonstrated that the difference between the left and right bevel

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3 mm Fig. 26 Photographs of kerf shape at different cutting energy parameters (h = 3 mm) (Wang J et al. 2011)

angles obviously becomes great with increasing cutting speed at 75 A. As a result, the dross-free and nearly perpendicular cut surfaces of acceptable flatness are exhibit on the low-speed side of the cut. The operating gas is a combination of the plasma and shielding gases for the dual plasma arc cutting. If the gas contains oxygen, the heat produced from the oxidation reaction of oxygen with metal, along with the electric heat of the arc, will contribute to the formation of kerf. Figure 27 indicates the kerf shape for different cut speeds at 30 and 40 mm/s. It is noted that dross attached to the bottom of the cut increases when operating with air in the plasma or shield gas. For the “O2 + O2” process, an indentation and plenty of oxide dross readily occur on the internal sides of the cut. Colombo et al. (2012) used high-speed imaging, characterized by a very small time scale (less than 1 ms), to describe phenomena taking place in plasma cutting and follow the arc attachment in the kerf. They also investigated fluid dynamic instabilities using statistical analysis of high-speed Schlieren image. The presented results were only an initial stage of a study aiming at transforming information achievable with the Schlieren technique, which are typically qualitative, into quantitative data. Chamarthi et al. (2013) studied the influence of voltage, cutting speed, and plasma gas flow rate on the geometry of the cut profile using the design of experiment (DoE) techniques. They found that the arc voltage is the main parameter and it influences all the aspects related with the cut quality rather than the effect on the arc power. Besides the arc voltage, the cutting speed showed a noticeable effect. Salonitis and Vatousianos (2012) monitored the quality of the cut by measuring the kerf taper angle (taper), the edge roughness, and the size of the heat-affected zone (HAZ). They used a statistical analysis of the results, considering the cutting speed, the cutting current, the plasma gas pressure, and the distance of the plasma torch

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Fig. 27 Photographs of kerf shape with different operating gases (I = 60 A and h = 2 mm) (Wang J et al. 2011)

Plasma + Shield

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from the workpiece surface (cutting height). They found that the surface roughness and the taper were mainly affected by the cutting height, whereas the cutting current mainly influenced the heat-affected zone. Deli and Bo (2011) used a variable interval fuzzy-PI quantification algorithm with a self-adjustable factor in the full domain to define a control strategy of the process. They found that this approach was significant for optimizing cutting control and improving the process effect. Gariboldi and Previtali (2005) studied the quality of cuts performed on titanium sheets using two cutting gases N2 and O2 with N2 shield gas in both cases. The microstructural features of the titanium sheet were severely affected by the thermal cycle brought about by the plasma torch pass-by and by interaction with the cutting atmosphere in a wide layer. The thickness of this layer, related to the position along the cutting edge, overcame 1 mm for cuts performed using nitrogen and for the slowest cut with oxygen. All cutting edges displayed an external layer of oxide, resulting from interaction with oxygen even when both cutting and shield gases were nitrogen.

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4

Plasma ARC Welding (PAW)

4.1

Overview of Plasma Arc Welding Processes

Welding is one of the most widely used manufacturing steps in an industry. It is the basic technology for joining metals with annual sales of welding equipment and supplies in the world estimated in 2010 at 6 trillion US$ (Hirata 2011). Because of the wide range of industrial applications requiring metal welding, a large number of welding processes were developed to meet different working conditions and performance requirements. These can be grouped into the following three broad techniques: • Tungsten electrode plasma arc welding • Wire electrode plasma arc welding • Hybrid techniques combining lasers with one of the above plasma arc welding techniques The great majority of the PAW techniques are based on the use of DC current though in a few cases, pulsed DC or AC currents have been used. Tungsten electrode plasma arc welding is one of the most widely used techniques in the transferred arc mode though it could also be used in a nontransferred arc configuration. The technique in this later case is closer to oxy-fuel flame welding, though with a higher-energy density than flames while being significantly less than transferred arc torches. If necessary, filler rod is introduced continuously into the plasma jet immerging from a typical hot-cathode plasma torch. In the case of operation in a transferred arc mode, the arc is struck between a hot-doped (often thoriated) tungsten cathode and the workpiece to be welded which acts as anode. Depending on the details of the torch design and the use of sheath gas, the technology is often reoffered as gas tungsten arc welding (GTAW) or tungsten inert gas (TIG) welding. Wire electrode plasma arc welding is operated in the transferred arc mode, with the transferred arc struck between a consumable wire feed centrally into the torch acting as cathode and the workpiece to be welded acting as anode. A number of variations have been developed based on this concept including: • • • •

Gas metal arc welding (GMAW)/metal inert gas (MIG) welding Flux-cored arc welding (FCAW) Submerged arc welding (SAW) Electroslag welding (ESW)

A brief description of the main design features of different arc welding devices, whether based on “tungsten” or “wire” electrodes, is presented followed by a more detailed analyses of some of the fundamental phenomena involved in the arc welding process.

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Gas Tungsten Arc Welding (GTAW)/Tungsten Inert Gas (TIG) Welding

This process uses a nonconsumable doped tungsten cathode protected from surrounding air by an inert shielding gas (often Ar or He) with the parts to be welded acting as anode (Finch 1997). As shown in Fig. 28, the arc is struck between the tungsten cathode tip and the parts to be welded which acts as anode. A wire or a rod of the filler metal (filler rod) is continuously advanced into the arc region where it is molten forming a weld pool which is kept clean through the use of a shielding gas at a flow rate high enough to cover the weld pool and block off oxygen and impurities from contaminating the pool. The cathode-to-anode distance is set in such a way that the arc attachment at the anode is anode jet dominated (AJD) (see Part II, ▶ Chap. 13, “Electrode Phenomena in Plasma Sources,” Fig. 14a). The power source normally used in such cases is of the constant current type. The torch is cooled by air for currents below 400 A and by water for higher currents. The initial striking of the arc is achieved using a high-frequency generator. If necessary, secondary shielding gas can also be used to cool the back of the part to be welded and avoid its excessive heating. When using Ar-H2 as the plasma gas, attention should be given to the possibility of desolation of the hydrogen in the molten metal bath. Since transferred arc mode of operation results in high-energy density to the arc root attachment, the technology can be used with relatively thick parts. Depending on the process parameters, the plasma arc can even pierce through the parts to be welded and push the weld pool to the sides. It is even possible to form a weld pool, which holds itself by its own surface tension. The molten metal behind the keyhole flows and solidifies on the lower side of the part. Generally, TIG or GTAW can be used for: • Micro-plasma welding with currents between 0.1 < I < 20 A • Medium plasma welding 20 < I < 100 A • Keyhole welding I > 100 A

– Plasma gas

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Hot wirePower source

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Fig. 28 Schematic of TIG/GTAW setup

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Fig. 29 Photograph of a typical TIG/GTAW setup for steel pipe welding (curtesy of Air Liquide)

The equipment is often operated manually requiring skilled welders for a wide range of applications in the aerospace industry for the welding of thin workpiece such as small-diameter, thin-wall tubing, of nonferrous metals. They are also used for making first-pass welds for piping of various sizes, maintenance and repair work, and repairing tools and dies, especially components made of aluminum and magnesium. A photograph of a typical TIG/GTAW setup used for the welding of stainless steel pipes is shown in Fig. 29.

4.3

Wire Electrode Plasma Arc Welding

These processes comprise:

4.3.1 Gas Metal Arc Welding (GMAW)/Metal Inert Gas (MIG) Welding Gas metal arc welding (GMAW), also called metal inert gas (MIG) welding or metal active gas (MAG) welding was, developed in the early 1940s for the welding of aluminum and other nonferrous materials and shortly after applied to steels because of its relatively higher welding speed compared to alternate welding processes. It is a welding process in which the electric arc is struck between the continuously advancing welding wire which acts as a cathode and the metallic workpiece as anode (American Welding Society 2004; Davies 2003). The molten metal falling from the wire tip accumulates into the weld grove heating the workpieces and causing them to melt and join. As shown in Fig. 30, an appropriate shielding gas is fed through the nozzle surrounding the wire in order to protect it. Ar and He were initially used as shielding gases only to be replaced later on by less expensive gases such as carbon dioxide (CO2) which is presently more commonly used. The power is generally supplied using a constant voltage source, though constant current or alternating current systems can also be used. Air-cooled torches, such as the Thermal Dynamics ESAB “Classic-6” shown in Fig. 31, are used for currents up to 600 A with a 60% duty cycle. Water-cooled torches, on the other hand, can be operated with currents up

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Oxide film Shielding gas

Wire electrode +

Molten pool

Shielding gas

Wire electrode

Arc Workpiece

Gas metal arc welding Fig. 30 Principle of gas metal arc welding (GMAW), also called metal inert gas (MIG) welding or metal active gas (MAG) welding Fig. 31 Photographs of the “Classic-6” air-cooled welding torch by (Thermal Dynamics ESAB)

to 600 A at 100% duty cycle, using a welding wire diameter of 1.6 mm and Ar or CO2 as shielding gas. Depending on the working conditions, the method of the transfer of the molten metal to the weld region can be either globular, shortcircuiting, spray, or pulsed spray. Each of these has distinct properties and corresponding advantages and limitations in terms of the quality of the weld achieved. Pulsed spray method uses a pulsing current, at a frequency between 30 and 400 pulses per second, to melt the filler wire and allow one small molten droplet to fall into the weld pool with each pulse. The pulse provides a stable arc and no spatter since no short-circuiting takes place. It is suitable for nearly all metals, and thicker electrode wire can be used as well. The relatively small weld pool allows for welding in all positions. The process can be semiautomatic or fully automated, with maximum welding speeds of 85 mm/s.

4.3.2 Flux-Cored Arc Welding (FCAW) As illustrated in Fig. 32, flux-cored arc welding (FCAW) is almost identical to MIG welding except for the use of a special tubular wire filled with flux, which is continuously fed into the arc. Generally FCAW can be operated using either a

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Contact tip Solidified slag

Nozzle Flux core wire Arc

Solidified weld metal Base material Gas shielding

Fig. 32 Principle of flux-cored arc welding (FCAW)

constant voltage or a constant current welding power supply. The shielding gas is not necessarily required in this case since the flux itself is sufficient to protect the weld pool from the surrounding atmosphere through the generation of gaseous protection and the formation of a molten slag layer, which covers the weld. This process is widely used in construction because of its high welding speed and portability.

4.3.3 Submerged Arc Welding (SAW) In submerged arc welding (SAW), the molten weld and the arc zone are protected from atmospheric contamination by being “submerged” under a blanket of granular fusible flux consisting of lime, silica, manganese oxide, calcium fluoride, and other compounds (see Figs. 33 and 34). When molten, the flux becomes conductive and provides a current path between the electrode and the workpieces. This thick layer of flux completely covers the molten metal thus preventing spatter and sparks as well as suppressing the intense ultraviolet radiation and fumes that are generated by the welding process. The process works with flat or horizontal-fillet welding positions. Currents in the range of 300 to 2000 A are commonly used, with relatively high deposition rates approaching 45 kg/h compared to ~5 kg/h (max) for MAW. The process is mostly operated in the automatic or mechanized mode though semiautomatic (handheld) SAW devices are also available. 4.3.4 Electroslag Welding (ESW) Electroslag welding (ESW) is a single-pass welding process used for thick (greater than 25 mm up to 300 mm) materials in a vertical or close to vertical position. A wire that is fed into the desired weld location initially strikes an electric arc. Flux is added until the molten slag, reaching the tip of the electrode, extinguishes the arc. The wire is then continually fed through a consumable guide tube (that can oscillate if desired) toward the surfaces of the metal workpieces, and the filler metal is melted using the electrical resistance of the molten slag (see Fig. 35). The wire and tube are pulled upward, while a copper starting plate that was placed at the bottom, which can be water-cooled if desired, is used to keep the weld material in place between the plates being welded. ESW is mainly used to join thick carbon steel plates and/or sections. The DC current used is typically around 600 A, with a corresponding voltage in the 40–50 V range.

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Fig. 33 Principle of submerged arc welding (SAW)

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Electrode wire Drive rolls

Flux hopper

Drive motor Servo control

DC power supply Weld pool Submerged arc Flux Workpiece

Fig. 34 Photograph of submerged arc welding (SAW), Courtesy of Air Liquide

4.4

Laser-Assisted Arc Welding

With the rapid development of CO2 lasers to the 5–10 kW power range, and of Nd: YAG lasers to 0.3–3 kW, industrial applications of laser technology for metal cutting and welding became accessible. When a high-power laser beam is focused on the joint between two workpieces, heat flux intensities can reach up to 104 MW/m2 (Hügel and Schinzel 2004). With such high localized and intense heating, a weld with a high depth-to-width ratio and minimal thermal distortion of the workpiece can be achieved. Good-quality weld requires, however, careful and accurate positioning of the laser beam and proper preparation of the workpiece prior to welding. However, laser and plasma arc welding are not mutually exclusive since the two techniques can be combined as shown in Fig. 36 to produce what is known as hybrid welds. In the laser–arc hybrid welding, the arc power (generally MIG or TIG) controls the weld width and the root opening bridging ability, while the laser

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Fig. 35 Principle of electroslag welding (ESW)

Electrode wire feed Guide tube

Power supply

Molten slag Weld pool

Starting plate

Weld

Workpieces to be Welded Electroslag welding

Fig. 36 Geometric configuration of the laserMIG welding system (Casalino et al. 2014)

Welding direction

Laser beam

MIG torch 40°

d=2.5 mm

power controls the penetration. The laser is generally found to stabilize the arc voltage and current and to reduce the column resistance, which increases the depthto-width ratio of the resulting weld (Roepke et al. 2010; Casalino et al. 2014). The microstructure of three types of welds, MIG, laser, and MIG–laser hybrid, is given in Fig. 37 after Ono et al. (2002). These show that the penetration depth in hybrid welding is comparable to that of laser welding (Fig. 37b, c). It may be noted, however, that while in the hybrid case (Fig. 37c) the arc overheats and melts the base metal near its surface, it does not deepen the penetration compared to that of the laser weld (Fig. 37b). In the MIG case (Fig. 37a), the addition of metal through the use of a wire gives rise to the formation of a weld bead on the surface of the weld which is also observed in the hybrid case as shown in Fig. 37c. The larger amount of weld metal observed in the weld bead in this case is advantageous in terms of the gap tolerance and bead width in lap welding (Ono et al. 2002). Gao et al. (2007) showed that shielding gas is very important for the process stability and the efficient synergetic effects between the CO2 laser and TIG arc,

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Fig. 37 Cross sections of (a) MIG weld, (b) laser weld, and (c) MIG–Laser hybrid weld (Laser output; 3 kW, Arc current; 100 A, Welding speed; 1 m/min) (Ono et al. 2002)

resulting in deeper weld penetration depth. They carried out a detailed experiment on the welding of 316 L stainless steel plates using CO2 laser–TIG hybrid, with different shielding gas compositions: Ar, He, and Ar-He. The results demonstrated that fully penetrated welds could only be obtained using the hybrid protecting method with the appropriate shielding gas parameters. Liu and Hao (2009) studied the interactions between low-power laser pulse and arc plasma (TIG) in the hybrid welding process of magnesium alloy. They also found that with the assistance of arc plasma, the threshold value of average power density to form keyhole welding for YAG laser is only 3.3  108W/m2. Moreover, the distribution of energy density during laser pulse was modulated to improve the formation and stability of laser keyholes. Liu and Chen (2011) also found that the laser affects the arc plasma through the keyhole formed on the magnesium alloy workpiece. Depending on the welding parameters, they showed the existence of three kinds of interactions taking place between laser and the arc plasma. Wu and Xiao (2015) investigated the effects of laser radiation on the characteristics of the (TIG) arc by applying a high-power (1 kW) lab CO2 laser and a Yb:YAG laser. The arc characteristics were affected by the CO2 laser radiation, but not by that of YAG laser up to 5.66  1010 W/m2. Arcs generated with Ar were affected by CO2 laser but not those generated using He as the plasma gas. The arc voltage–current curve shifted downward when CO2 laser interacted with the TIG arc in argon. It seems that the different behaviors of the arc when interacting with CO2 and YAG lasers at atmospheric pressure with argon and helium result from the different inverse Bremsstrahlung absorption coefficient. Li et al. (2014) investigated the interaction between the arc and the laser in laser–metal active gas (MAG) hybrid welding of mild steel using Ar + 20 vol.% CO2 as shielding gas. They employed three arc modes: cold metal transfer, pulsed spray arc, and standard short-circuiting arc. It was found that both the stability and mechanical properties of laser-cold metal transfer hybrid welding (LCHW) was the best, while those of laser-standard short-circuiting arc welding were the worst. Wahba et al. (2015) performed similar studies, using CO2 as shielding gas. They observed that the use of 100% CO2 gave rise to the increase of the depth of the weld,

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and decrease of its porosity, at the expense of the formation of a large number of spatters. The latter could be decreased though the use of a buried arc. The welding parameters were optimized, thanks to observations using high-speed video photography. A significant reduction in spatter generation could then be obtained by optimizing the relative distances between the two heat sources in the X and Y directions to produce a proper profile of the arc cavity that could trap any generated spatters.

4.5

Basic Phenomena in Plasma Arc Welding Processes

The plasma arc welding processes depend on a large number of independent variables, which can be grouped as follows: • Thermophysical properties of the workpiece and filler material, such as thermal conductivity, thermal diffusivity, specific heat, melting and vaporization temperatures, viscosity and surface tension of the molten metal pool, and radiative properties of any generated metal vapor • Plasma gas and sheath gas composition and flow rates • Plasma power (arc current and voltage) and possible modulation of the power supply, which controls the energy dissipated in the weld and consequently the weld penetration • Geometric considerations including the relative position of the parts to be welded: joint gap dimensions and the positioning of the welding torch, its angle relatively to the workpiece as well as torch-to-workpiece distance and the welding speed. • Workpiece preparation including edges and surface finish Numerous studies have been reported in the open literature dealing with different aspects of plasma arc welding technologies. These mostly aimed at the understanding of the basic phenomena involved and process optimization through a combination of advanced diagnostic studies and mathematical modeling. Diagnostic measurements of the plasma were performed mainly using emission spectroscopy (Mirapeix et al. 2006; Liu and Hao 2009). For example, Mirapeix et al. (2006) have developed an optimized technique for real-time spectral analysis of thermal plasmas, with application in the monitoring and defect detection of industrial welding processes, particularly arc welding. Direct observation of the weld pool and the keyhole during welding process is also possible using CCD camera provided an appropriately designed narrow band filter is used to block the intense radiation from the arc. Plasma modeling also provides a valuable tool to predict the plasma characteristics in regions of the flow, which are not accessible for measurements and predict the process performance necessary. Modeling necessitates, however, a good knowledge of the thermodynamic and transport properties of the plasma and shield gases and the influence of the presence of varying concentrations of metal vapors, generated

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during welding process, on those properties (Cressault and Gleizes 2004, 2007, 2013; Cressault et al. 2012, 2013; Dassanayake and Etemadi 1989). Considering that the principal objective of welding is to reconstitute the welded part in a mechanically solid and structurally sound part, it is important that the optimization of the welding process be also accompanied by a systematic and detailed characterization of the microstructure and mechanical properties, including residual stress, of the welded parts. In the following, a review of fundamental studies and development work carried out on different aspects of the welding process is presented.

4.5.1 Arc Constriction and Weld Pool Profile Tungsten inert gas (TIG) or gas tungsten arc welding (GTAW) are mostly used to weld stainless steel and nonferrous materials, such as aluminum and magnesium. They can also be applied to nearly all metals, with a notable exception being zinc and its alloys. It is not recommended to use them manually because of the need to maintain a short arc length, while avoiding any contact between the electrode and the workpiece, which is rather difficult without proper instrumentation and automation. As pointed out by Toropchin et al. (2014), the design of the ceramic nozzle surrounding the tungsten electrode in TIG or GTAW is of critical importance for the stability of the arc. Schematics of some of the configurations studied are given in Fig. 38, with Fig. 38a, b showing a straight ceramic nozzle and Fig. 38c a converging nozzle. The corresponding shape of the arcs obtained with pure argon with each of these nozzle designs is superposed on the figure. It may be noted that the convergent nozzle design gives rise to a more constricted arc compared to that of the other two design options. While it is relatively easy to calculate the temperature and velocity distributions for the arc with pure argon as plasma gas, the problem is considerably more complex once the workpiece starts melting and vaporizing. The metal vapors evolving from the molten metal pool diffuse into the arc modifying significantly the thermodynamic, transport, and radiative properties of the plasma. Moreover, in order to model the weld pool, one must also consider the different forces acting on the melt, which gives rise to internal recirculation of the molten metal. Among the forces that need to be considered are: • The Marangoni effect by which the presence of a gradient in surface tension of the melt induces the liquid to flow away from regions of low surface tension • The drag force acting opposite to the direction of the flow on the surface of the pool • The electromagnetic forces due to the arc current • The buoyancy effects which gives rise to an upward force exerted by the molten metal in an opposition to the weight of an immersed object The conservation equations of fluid dynamics and heat transfer are used to derive the flow and temperature fields in the arc and weld profile of the molten metal pool. Typical results obtained for each of the geometries given in Fig. 38 are shown in

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c 35

b

a Ceramic nozzle

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Fig. 38 Geometric configurations of the ceramic arc confinement nozzle with different internal diameters of the nozzle, locations of the cathode tip (Δz is the distances between the cathode tip and the anode) and the nozzle exit, (a) id = 11 mm, Δz = 6 mm, (b) id = 11 mm, Δz = 10 mm, (c) id = 8.1 mm, Δz = 16 mm (Toropchin et al. 2014)

Fig. 39 for an argon plasma with arc currents of 200 A (Fig. 39a) and 150 A (Fig. 39b, c). These show temperatures as high as 18,000 to 20,000 K near the cathode tip, with strong temperature gradients in the radial direction specially in the presence of the constricted ceramic nozzle shown in Fig. 39c. The lower part of Fig. 39 shows the corresponding profile of the molten metal pool in the workpiece. The results are blown out for clarity in Fig. 40, for the case of 150 A arc using the ceramic nozzle of Fig. 38c. The results given in Fig. 40a, b also show, respectively, the corresponding flow vectors in the weld pool “with” and “without” inclusion of the Marangoni force in the calculation model. The markedly different weld depths in each of these two cases are related to basic material properties such as specific heat, electrical and thermal conductivity, and model formulation. According to Toropchin et al. (2014), the Marangoni effect gives rise to a dependence of the direction of melt recirculation in the molten metal pool on the variation of surface tension of the melt with temperature. Comparing a material with falling surface tension characteristic, versus a material, such as mild steel, with a rising characteristic, reviles a significantly different weld pool profiles and circulation patterns of in the molten metal pool in each of these two cases as shown in Fig. 41. It is important to point out that the weld pool profile formed in plasma arc welding depends on a large number of parameters, which includes the chemistry of the materials to be welded and that of the plasma and shield gases. The effect is clearly demonstrated experimentally by Tanaka and Lowke (2007) for the welding of stainless steel plates (SUS 304) under essentially the same physical conditions

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699 Plasma Temperature (K)

Plasma Temperature (K)

Plasma Temperature (K)

23355 22141 20928 19714 18501 17288 16074 14861 13647 12434 11221 10007 8794 7580 6367 5154 3940 2727 1513 300

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Fig. 39 Corresponding weld pool profiles and temperature fields in an argon arc plasma for the same geometrical parameters as in Fig. 38, for an arc current (a) I = 200 A, (b) and (c) I = 150 A (Toropchin et al. 2014)

Fig. 40 Flow vectors in the weld pool for mild steel S235 at a current of 150 A. (a) Computation including the Marangoni forces and (b) computation neglecting the Marangoni forces (Toropchin et al. 2014)

with tungsten electrode 3.2 mm in diameter, a distance of 5 mm between the electrode tip and the steel plate, an arc current of 150 A, at atmospheric pressure with argon as plasma gas. Photographs of the cross section of two stainless steel plates with sulfur contents of 40 and 220 ppm are given in Fig. 42. These show as well a significant difference in the corresponding weld pool profiles for each of these

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Surface tension

Surface tension

700

Temperature

Temperature

Fig. 41 Schematic illustration of convection in the weld pool driven by Marangoni force resulting from the temperature dependence of the surface tension (Toropchin et al. 2014)

Fig. 42 Cross sections showing weld regions for stainless steel with different sulfur content, 150 A arcs with a 5 mm electrode separation at atmospheric pressure (Tanaka and Lowke 2007)

two plates depending on their sulfur content. The weld profile for the plates with 40 ppm sulfur being wide and shallow in sharp contrast to the profile obtained with the plates containing 220 ppm of sulfur in which the weld pool is narrow and deep. Further support for the influence of the composition of the plasma and shield gases on the weld metal pool profile was provided by Tanaka and Lowke (2007). Micrographs given in Fig. 43 show typical cross sections of welds for stainless steel, conducted with an arc current of 150 A and arc gap of 3 mm, using Ar and He as plasma gases and a mixture of Ar-CO2 or Ar-10% H2-CO2 as sheath gas. The corresponding arc voltages varied with the plasma and shield gas composition being at 11.8 V for pure Ar, while for an Ar arc shielded with CO2 15 V, and 18.9 V for Ar-10 vol.% H2-CO2 shielded. These show the penetration depth for the argon plasma gas CO2-shielded arc to be 4.9 mm (Fig. 43b), 6.9 mm for an Ar + 10 vol.% H2 mixture as the inner nozzle gas CO2-shielded (Fig. 43c) while the depth for the Argon arc is only 2.0 mm (Fig. 43a). The effect of CO2 shielding is due to the better arc constriction, as predicted by calculations. Tanaka and Lowke (2007) also point out that mathematical modeling is a powerful tool for the understanding and control of the arc welding process provided that the model incorporates both the arc plasma as the heat source and the workpieces. The need for such an integration of these two distinct fields is due to “the

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Fig. 43 Measured weld shape in stainless steel (SUS-304, 30 ppm sulfur) for an arc current = 120 A, arc gap = 3 mm, welding speed = 90 mm/min and different welding gases (a) Ar-TIG, (b) Ar + CO2 and (c) Ar + 10% H2 + CO2 (Tanaka and Lowke 2007)

close interaction between the electrode, the arc plasma and the weld pool, which constitute the welding process, and must be considered as a unified system.” Murphy et al. (2009, 2010) further confirmed the necessity to include the arc plasma in the computational domain taking into account the influence of the metal vapors released from the molten metal pool on the transport and radiative properties of the plasma. The most important effect of the metal vapor is the increased electrical conductivity at low temperatures, and the significant increase of radiation heat losses from the arc column, which leads to lower heat flux density and current density at the weld pool, implying a shallower weld pool. They pointed out that the product of specific heat and mass density is particularly important in determining the arc constriction and the weld pool depth. A modeling study of the influence of the nature of the plasma gas, Ar or He, on the arc and the weld pool was reported by Yamamoto et al. (2008). Figure 44 shows the temperatures, velocities, and iron vapor molar fractions (other components were not considered) in 150 A arc in argon and helium, with a 304 stainless steel workpiece as anode. The temperature interval in the iso-contours for the plasma is 2000 K, while it is 200 K in the cathode and 250 K in the anode. The concentration of the iron vapor released from the surface of the melt pool increases sharply with time following the initiation of the arc, only to stabilize shortly after only 5 s. The maximum concentration of the iron vapor in the arc depends on the arc current and the nature of the plasma gas, reaching 0.2 vol.%, for an argon arc compared to 7.0 vol.% for the helium arc at the same arc current of 150. The effect is due to the higher melt pool temperature with a helium arc compared to that of an argon arc. Such high metallic vapor concentration levels can be detrimental to the welding process not only because of their influence on the plasma properties and the process energy efficiency but also because of the eventual homogenous condensation of the vapor in the colder regions of the flow forming a fine nanometer sized fume which represent a serious health hazard to the operator (Tashiro et al. 2010). The use of active flux for deeper penetration can also modify the weld pool characteristics. Morisada et al. (2014) have proposed what they called cap active flux tungsten inert gas (CA–TIG) welding using atmospheric oxygen, to increase the penetration depth of a weld. An especially designed nozzle cap entrained air in the molten pool. The

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1700 K Interval 200 K

Ar, t=20.0

–6 150 A, 10.4 V –4 Axial distance z, (mm)

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Max 7 cm/s 500 K Interval 250K

SUS 304 (LS)

300 K

–10 –8 –6 –4 –2

0

2

4

Radial distance r (mm)

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10 500 K Interval 250K

–2

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6 2000 K

Max 49 cm/s

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6 8

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–6 –4

–4

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He, t=20.0

–6 150 A, 18.5 V

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Radial distance r (mm)

Fig. 44 Temperature fields (RHS) and iron vapor mole fraction fields and velocity vectors (LHS) in the arc and electrodes for 150 A arc in argon and helium at 1 atm., with a 304 stainless steel anode, after 20 s of operation (Yamamoto et al. 2008)

penetration depth for SUS304 stainless steel was increased by the reversal of the Marangoni convection due to the entrained oxygen, and it reached three times deeper than that of the conventional TIG welding. Authors say that the effect of the entrained air on the mechanical properties of the joint formed by the CA–TIG welding is negligible, and the joint efficiency is high enough for conventional applications. According to Karanunakarani and Balasubramanian (2011), in conventional welding, fusion zones typically exhibit coarse columnar grains because of the prevailing thermal conditions during weld metal solidification. It often results in inferior weld mechanical properties and poor resistance to hot cracking. It is thus necessary to control solidification structure in welds, and thus temperatures and thermal gradients in welds. This is especially the case when welding thin sheet materials with high thermal conductivity and diffusivity such as aluminum and magnesium alloys. Using pulsed current welding can solve the problem. When welding AA6351-T6 aluminum alloy joints, pulsed current resulted in lower peak temperatures and lower magnitude of residual stresses compared with constant current welding. The joint obtained with this technique exhibited superior tensile properties compared with constant current welding technique with the formation of finer grains. Razal Rose et al. (2012) studied the influence of pulsed current, base current, pulse frequency, and pulse on time on tensile strength of the PCTIG welded AZ61 A magnesium. They observed that the effect of peak current and pulsing frequency initially increase the tensile properties and then decrease the tensile properties irrespective of changes in base current and pulse on time. A maximum tensile strength of 199.5 MPa was obtained with the optimum welding parameters. Pulsed current was more sensitive than the other parameters followed by pulse on time, pulse frequency, and base current.

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4.5.2 Keyhole Formation in Plasma Arc Welding The arc used in plasma arc welding is constricted by a small nozzle (see Fig. 38c) and has a much higher velocity (300–2000 m/s) and heat input intensity (109–1010 W/m2) than that in conventional gas tungsten arc welding. According to Wu et al. (2014) the process which has the best penetration and permits welding in keyhole mode in relatively thick material in a single pass, is the most effective for many applications, minimizes electrode contamination, and has large tolerance to joint gaps and misalignment; the high depth-to-width ratio of the plasma weld also can greatly reduce angular distortion and residual stress. As the process is nonlinear, is dynamic, and involves complex systems with multiple inputs and outputs and multiple large number of process parameters, it requires multiple sensors be integrated to monitor multiple variables during the welding processes resulting in higher investment than with other welding processes. Wu et al. (2014) have listed the different parameters on which the process depends: • Plasma torch nozzle shape and size • Tungsten electrode dopant material and diameter • Tungsten electrode distance between the tip of the electrode and the nozzle exit, parameter, which has a significant effect on weld bead formation and arc penetration • Composition and flow rate of the plasma gas • Welding current, which controls the arc pressure and temperature • Welding speed, which is strongly connected to the arc current and plasmaforming gas • Distance from the nozzle exit to the workpiece • Composition and flow rate of the shielding gas According to the importance of the keyhole, it is necessary to define it. It depends on velocity and temperature levels of the plasma arc when it impinges on the area where the two workpieces are to be joined. First the material is melted and a molten liquid pool (weld pool) is created. Then because of its high velocity and the associated momentum, the plasma arc penetrates through the molten pool and forms a keyhole in the weld pool (Fig. 45). Moving the welding torch, the associated keyhole will cause the flow of the molten metal surrounding it to the rear region where it re-solidifies to form a weld bead (Wu et al. 2013, 2014), as shown in Fig. 45. During welding, the keyhole can be observed from the backside of the workpiece either with an ultrahigh shutter speed vision system including a strobe-illumination unit (pulse laser) or with a camera (Wu et al. 2014). But once the keyhole is established, its width does not change with an increasing welding current and a decreasing welding speed (Zhang and Zhang 1999). Thus authors point out that the width of the keyhole gives no adequate information about the stable state of the keyhole and should not be used as an indication of the robustness of the keyhole process in plasma arc welding. Zhang et al. (2015) have also developed a single costeffective vision sensor to observe both the keyhole and the weld pool simultaneously from the underside of workpiece in PAW. From the plasma arc ignition to its

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b

a

Plasma torch

Electrode Welding direction

Torch Plasma jet

Weld pool

Weld bead

Plasma jet Workpiece

Weld pool Keyhole

Keyhole Front surface

Workpiece

Solid-liquid interface

Back surface

Liquid-plasma interface

Fig. 45 Schematic diagram of keyhole formation in plasma arc welding: (a) keyhole inside the weld pool (Wu et al. 2013) after its formation, (b) evolution of the weld and keyhole with the torch moving (Wu et al. 2014)

extinguishment, the dynamic evolution of the keyhole and the weld pool experienced six stages. They found that there was a time lag between the time of achieving the quasi-steady state taken by the keyhole and the weld pool. Thus they determined the influence of welding parameters on keyhole and weld pool. To spray thin sheets of stainless steel plate of 0.1 and 0.3 mm in thickness, PAW torch with nozzle i.d. of 0.1 to 0.3 mm was developed (Yoshioka et al. 1993). Of course arc current was limited to a few A with Ar as plasma gas and voltage around 45 V. Appropriate welding conditions were obtained, and it was demonstrated that there was large position tolerance of torch for the working distance and the deviation from the welding line. Of course in parallel to these measurements, complex models were developed (Wu et al. 2013, 2014; Jian and Wu 2015). Jian and Wu (2015) surveyed the modeling and simulation progress on the keyhole evolution and its interaction with the weld pool. They also presented some examples of the predicted heat transfer and fluid flow in PAW. Figure 46 presents the transient evolution process of the plasma arc–weld pool–keyhole. From the plasma arc ignition to the full penetration of the workpiece, the plasma arc shape varies as the weld pool surface deformation and the keyhole expansion take place. The keyhole shape is like an upside-down frustum of a cone. For this study case, the top and bottom diameters of the keyhole begin to stabilize after 0.5 s, while the keyhole depth increases with time. Comparisons of the measured and calculated weld profiles at transverse cross section were in good agreement.

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Plasma Torches for Cutting, Welding, and PTA Coating

a

0

b 103 K 26 22 18 14 10 1.7 1.3 0.9 0.5

Z (mm)

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800 m/s

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0 2 r (mm)

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4 Z (mm)

103 K 26 22 18 14 10 1.7 1.3 0.9 0.5

4 Z (mm)

103 K 22 18 14 10 1.7 1.3 0.9 0.5

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Z (mm)

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6 800 m/s

8 10 12 14

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0 2 r (mm)

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Fig. 46 Evolution process of plasma arc–weld pool–keyhole at different times. (a) t = 0.001 s, (b) t = 0.5 s, (c) t = 1.0 s and (d) t = 1.48 s (material SUS304, plasma gas Ar) (Jian and Wu 2015)

During PAW process, keyhole formation is very sensitive to any changes of the welding parameters. Jia et al. (2009) showed that the linear relationship among the welding current, efflux plasma voltage, and backside weld width exists so that the efflux plasma voltage signals can be used to control the penetration extent of PAW process. With this control, the PAW process is stable, and satisfactory weld formation is obtained. Chen and Pan (2011) developed a dynamically controlled PAW process to further reduce the heat input. For that, the keyhole is deliberately allowed to close for a specified period of time to reduce the heat input and the mass of the weld pool. Results with Ti-6Al-4 V welding show that it reduces prior-beta (β) grain size and prohibits formation of hard martensite phases in the fusion zone. Thus the ductility and hardness of the welded joint are improved. Sun et al. (2011) developed pulse cycle where the welding current changes from peak current to background current, resulting in periodic variation of expanding and contracting the keyhole, corresponding to “one keyhole in each pulse.”

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4.5.3 Metal Transfer in Wire Electrode Arc Welding GMAW process is popular because of its high productivity and its robotic development. As shown in Fig. 30, the electrode wire is the anode and receives the highest heat transfer, while the part to be welded is the cathode; the continuous movement of the cathodic spots at the surface metal cleans its surface and heats it. The wire anode metal can be transferred according to three modes (American Welding Society 2004): • The short-circuit transfer or short arc obtained with the lower current and the lower wire feed rate. A molten drop forms at the tip of the wire electrode, but instead of dropping to the weld pool, it bridges the gap between the electrode and the weld pool. This induces a short circuit and extinguishes the arc, but it is quickly reignited after the surface tension of the weld pool pulls the molten metal bead off the electrode tip. This process occurs about 100 times per second! Fig. 47a represents the phenomenon. This type of metal transfer provides better weld quality and fewer spatters than the globular variation and allows for welding in all positions. • The globular one where a ball of molten metal from the electrode tends to build up on its end, often in irregular shapes with a larger diameter than the electrode itself (Fig. 45b) (Miller Electric 2012). When the droplet finally detaches either by gravity or short-circuiting, it falls to the workpiece, leaving an uneven surface and often causing spatter. • Spray transfer achieved when the current and voltage increases beyond the range of short-circuit globular transfers (see Fig. 47c). This process eliminates spatter and results in a high-quality weld finish (Cary and Helzer 2005). However according to the higher heat input and larger weld pool area (for a given weld electrode diameter), it is generally used on workpieces of thicknesses above about 6.4 mm and mainly when welding horizontal parts. Besides the arc current, other parameters can modify the transition spray globular, as shown by Zielinska et al. (2009): transition between the two transfer

Fig. 47 Main classification of metal transfer mode: (a) short-circuit or short arc transfer, (b) globular transfer, (c) spray transfer (Hirata 2011)

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modes is linked to the existence and disappearance of a rather insulating oxide “gangue” at the wire extremity whose nature depends of the shielding gas. Chemical reactions at high temperature such as oxidation–reduction reactions between shielding gas and melted metal govern the transition of the spray arc to globular transfer mode. Finally the welding parameters influencing the quality, productivity, and cost of welding joint are the arc welding current and voltage, welding speed, torch angle, free wire length, nozzle distance, welding position and direction, and lastly the flow rate of gas (Ibrahim et al. 2012). Welding current is the factor that determines the depth of weld penetration, which is also influenced by the welding speed and arc voltage. However the weld hardness decreases when the arc current increases, and it also depends on the welding speed. Karadeniz et al. (2007) also show that in steel, the depth of penetration increases linearly with increasing welding current up to a limit, the voltage increases also the penetration but less than welding current. Feng et al. (2012) studied influence of welding velocity on the impact behavior of the globular metal transfers and showed the existence of a critical welding velocity. With too low welding velocity, the droplet impacted inside the weld pool, while over a certain value the droplet impacted outside the weld pool. According to Murphy (2011a), there are still many unresolved physics questions about GMAW or MIG welding: • Can arc welding be modeled accurately without including the arc in the computational domain? • What effect cools the arc center in MIG welding? • What is the metal vapor concentration in MIG welding? • Does an electric current pass through the droplets? • Where is welding fume formed, and how can it be reduced? • How do the arcs interact in tandem welding? The same author (Murphy 2011b) has developed a three-dimensional computational model of GMAW that takes into account all important effects, except for the influence of metal vapor. The model treats the wire electrode, arc, and workpiece self-consistently and allows the shape of the weld pool, including the surface deformation, to be predicted. Results are presented for welding of an aluminum alloy. The current density distribution at the interface between the arc and the weld pool is strongly dependent on the surface profile of the weld pool. This in turn affects the temperature distribution in the weld pool. The momentum transferred by the droplet affects the direction of flow in the weld pool and, together with the energy transfer, increases the weld pool depth. Schnick et al. (2010) developed a computational model of the argon arc plasma in gas metal arc welding (GMAW) that includes the influence of metal vapor from the electrode. The main source of iron vapor is the wire extremity. The occurrence of a central minimum in the radial distributions of temperature and current density is demonstrated, contradicting other measurements and also the predictions of previous

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models, which do not take metal vapor into account. This results in a shallower weld pool, and reduces weld quality and efficiency of the welding process. Typical results are shown in Fig. 48. Murphy (2010) in his review studied the formation of the metal vapor and its diffusion. In his later paper on the topic with other researchers, Murphy (2013) showed that higher arc currents, and the use of iron or steel electrodes, favor the formation of a local temperature minimum due to the strong radiative emission from Velocity (m/s) Fe fraction (g/g) 1e+000 5e−z001

a

2e−001 1e−001 5e−002 2e−002 1e−002 5e−003 2e−003 1e−003 5e−004 2e−004 1e−004

0

50 100 150 200

T Fluid (k) 16000 15000 14000 13000 12000 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0

b

c

Fig. 48 Fields of iron vapor mass fraction (left) and temperature (right) and flow velocity (vectors) in (a) 250 A, (b) 400 A and (c) 100 A arcs with an input flow of iron vapor at the tip of the wire anode. The dimensions are 15 mm horizontally by 10 mm vertically (Schnick et al. 2010)

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metal vapor. They also showed that wire shapes participate to the iron vapor production: 0.0175 g/s with a wire diameter of 1 mm which end is rounded against 0.099 g/s with a large droplet before its detachment. To weld materials which are lighter in weight and possess good thermal and electrical conductivity, such as aluminum, pulse gas metal arc welding (GMAW-P) allows avoiding excessive heating of base material and producing spray transfer at lower mean currents (Praveen et al. 2005, Praveen and Yarlagadda 2005). GMAW-P was invented during the mid-1960s. It overcomes the drawbacks of globular mode while achieving the benefits of spray transfer. It is characterized by pulsing of current between low-level background current and high-level peak current in such a way that mean current is always below the threshold level of spray transfer. The background current is to maintain arc, whereas peak currents are long enough to make sure detachment of the molten droplet. An alternating current–gas metal arc welding (AC-GMAW) process with an improved current waveform was proposed by Arif and Chung (2015). In this process, an increase in the welding current and electrode-negative (EN) ratio resulted in fingerlike penetration. Concentrated heat input was observed at the center of the weld pool at high current, and this input increased with the EN ratio. A typical AC-GMAW pulse is presented in Fig. 49. Results can be summarized as follows: • Increasing the EN ratio increases the penetration depth. • Fingerlike penetration is observed in AC-GMAW at welding currents above 200 A. • Adopting the AC-GMAW process reduces arc interruptions and spatter generation under the effect of external magnetic fields. • Controlled heat input through an induction process can provide a solution to the lack of fusion at the joint sides when welding steel plates with thickness above 5 mm. (Lu et al. 2014) also developed a double-electrode gas metal arc welding (DE-GMAW), which is a novel welding process in which a second electrode, Fig. 49 Conventional AC-GMAW pulse (EP electrode positive, EN electrode negative) (Arif and Chung 2015)

EP

0 Time

EN

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nonconsumable or consumable, is added to bypass part of the wire current. The bypass current reduces the heat input in nonconsumable DE-GMAW or increases the deposition rate in consumable DE-GMAW.

5

Plasma Transferred Arc (PTA) Coating

5.1

Basic Concepts

The process is closer to plasma arc welding than other plasma spray coating processes since the coating is fused to the substrate which is part of the electrical circuit serving in most of the cases as the anode to the transferred arc and only in a few cases as the cathode. The technique is comparable to tungsten inert gas (TIG) and metal inert gas (MIG) welding except for the feeding of the material to be deposited in the form of powder. It is also generally operated at higher power levels than TIG and MIG welding which gives rise to higher deposition rates of the order of tens of kg/h compared to 3 kg/h or less for TIG and 6 to 9 kg/h for MIG (Gebert and Bouaifi 2005). In contrast, in laser weld deposition, the substrate material is melted by a laser beam, and the deposition material is introduced either in the form of a powder or in the form of a wire (Lewis and Schlienger 2000). This process, while limited to low deposition rates comparable with TIG and MIG, is used advantageously for high precision coating of small areas because of the ability to precisely control the movement of the laser beam. The equipment is rather expensive. The same is true for electron beam overlaying, which is used for some special cases. A schematic of a typical PTA torch setup is illustrated in Fig. 50. The PTA torch consists of a hot, thoriated tungsten cathode surrounded by a water-cooled copper nozzle constricting the arc between the cathode and the workpiece. The plasma gas is introduced at the base of the cathode tip. The nozzle also serves as a “pilot arc” anode, i.e., there is a current supplied by a separate power supply flowing from the nozzle/pilot arc anode to the cathode. This pilot arc pre-ionizes the gap between the cathode and the workpiece and allows easy establishment of the transferred arc and a good control of its arc current. The pilot arc further stabilizes the transferred arc when the plasma is cooled by the powder injection and assures the correct positioning of the anode attachment and the powder deposition area. The powder is introduced into the arc plasma though two or more orifices located on a ring surrounding the exit of the nozzle. A further annular slot surrounding the powder injection ring provides the shield gas flow, necessary to protect the molten metal from contact with the ambient air. Because of the local melting of the substrate material at the anodic arc root attachment, the coating is different from those obtained using alternate standard thermal spray coating techniques. The powder feed into the arc is also mostly melted, in-flight, and recovered into the molten pool of the substrate material

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Plasma Torches for Cutting, Welding, and PTA Coating W cathode Plasma gas Cooling water Powder feed

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–

–

+

+

Pilot arc power

Transferred arc power

Shield gas

Coating Shield gas Powder injection Substrate Substrate movement

Fig. 50 Schematic of PTA torch

Fig. 51 Defining the dilution of a PTA coating

Coating A B Substrate

with which it is mixed and alloyed. The consequence is that a good metallurgical bond or fusion bond is achieved between the coating and the substrate with a very low porosity. However, the heat penetration region and in particular the region where the coating material is mixed with the substrate material can change the properties of both the substrate and the coating. This mixing region is expressed in terms of the “dilution.” As illustrated in Fig. 51, representing a cross section though a bead of deposited material mixed with the substrate material, the dilution (D), is defined as the ratio of the area where the substrate and coating materials are mixed (B), to the total area in which the coating material appears (A + B). That is, D = B/(A + B) %. The dilution is a major consideration when the functionality of the coating is evaluated. It is typically less than 5% for coatings of a thickness larger than 1 mm. It is also considerably lower than corresponding values obtained in TIG and MIG welding as well as submerged arc welding (SAW) in which dilution values of 15 to 40% are reached, requiring the deposition of several layers (Hallen et al. 1991).

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The PTA process combines good control over the process with high deposition rates (typically 3 to 10 kg/h, with up to 20 kg/h possible) and good coating quality with practically no porosity and dilution of less than 7%. Compared to other thermal spray processes, PTA has the major advantage that alloy formation takes place in the molten metal pool during the process, provided the two metals are compatible, allowing for a wider range of combinations of metal compounds or metal matrix composites compared to spray deposition process. According to DuMola and Heath (1997), PTA has the following advantages: • High powder feed rates and deposition efficiencies (>95%) • Tolerance to variations in feed powder quality, allowing for the use of low-cost powders • Low porosity coatings (almost theoretical density) • Thicker coatings in one pass can be achieved requiring fewer passes for part building • Metallurgical bond with strong adhesion to the substrate • Fine microstructure coating with smooth surfaces finish of the coating • Molten metal pool protected by a gas shield preventing oxidation during the deposition process PTA technology has, on the other hand, the following limitations: • • • • • • • •

The requirement of a metallic substrate. Limitations on the substrate shape complexity. Dilution changing some of the coating and the substrate properties. High heat input into the substrate, requiring process modifications or active substrate cooling when smaller size substrates are coated. The process is not adapted to vertical surfaces due to the influence of gravity on the fluid flow and the coating geometry (Wilden et al. 2006). Coatings are poorly adapted to complex substrate shapes including corners and grooves (Gatto et al. 2004). Residual stresses can be rather high. Metallurgical compatibility between coating and substrate must be considered.

5.2

Equipment and Operating Parameters

As illustrated in Fig. 52, a typical PTA installation is constituted essentially of the following seven principal components: • The PTA torch, as shown in Fig. 50, consists of a central thoriated tungsten (2% ThO2) cathode, typically 6 to 8 mm diameter (although smaller diameters have been used). The plasma gas is introduced into the torch at the base of the cathode and usually high-purity argon or an argon–helium mixture. Downstream of the

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Plasma Torches for Cutting, Welding, and PTA Coating

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Gas supply

HF starter pilot arc power supply

+

Powder feeder

– +

Transferred arc power supply Torch oscillator

Cooling water supply

Substrate movement

Fig. 52 The various components of a PTA system

cathode tip is a water-cooled copper nozzle, typically 2 to 4 mm in diameter, depending on the operating power of the torch. This nozzle serves as the pilot arc anode, and it constricts and stabilizes the arc. Alignment of the cathode with the nozzle axis is of critical importance for obtaining quality coatings because a slight misalignment can lead to unstable or strongly asymmetric arcs, uneven powder heating, uncontrolled substrate heating, and reduced electrode life. Surrounding this nozzle is a ring with powder injection orifices. The powder is injected, depending on the process and material, either into the arc where it is fully melted or close to be melted or into the fringes of the arc to keep it only partially melted. In the first case, there is a secondary nozzle (sometimes called focusing nozzle) with a diameter of about double that of the constricting nozzle surrounding the powder injection ring. Torches that do not have this secondary nozzle have the ring surrounding the powder injection orifices at the same level as the constriction nozzle exit, and this arrangement results in powder injection into the fringes of the arc with less heating of the powders. The powder can also be fed into the molten metal pool by an extension of the tube that transports the powder beyond the secondary nozzle to a location close to the substrate surface. Surrounding this secondary nozzle is the ring introducing the shield gas. The shield gas consists typically of argon with 5–7% hydrogen added, and it provides a curtain that protects the molten metal from reacting with the surrounding atmosphere. Shield gas flow rates are multiples of the plasma gas flow rates. For deposition of metals that will react with hydrogen, e.g., Ti or Al, pure argon can be used as shield gas. • Two independent power supplies, one for the pilot arc and another for the transferred arc. Both power supplies are controlled in the constant current mode with relatively low open-circuit voltages (