Physics and Mechanics of New Materials and Their Applications: Proceedings of the International Conference PHENMA 2023 (Springer Proceedings in Materials, 41) 3031522389, 9783031522383
This book presents 60 selected peer-reviewed contributions from the international conference Physics and Mechanics of Ne
145 72 71MB
English Pages 673 [667] Year 2024
Preface
Contents
Contributors
Processing Techniques of Advanced Materials
Control Over Morphological Characteristics of the Pt/C Catalysts Obtained by the Liquid-Phase Synthesis
1 Introduction
2 Experimental Section
2.1 Preparation of the Pt/C Catalysts
2.2 Study of the Composition and the Structural and Morphological Characteristics of the Pt/C Catalysts
2.3 Study of the Electrochemical Behavior of the Catalysts
3 Results and Discussion
3.1 Effect of the Temperature Mode on the Average Size of Pt NPs and the ESA
3.2 Effect of the Presence of a Carbon Support During the Pt/C Synthesis on the Average Size of Pt NPs and the ESA
3.3 Effect of the Alkali Content in the Reaction Medium (OH:Pt Ratio) on the Average Size of Pt NPs and the ESA
3.4 Comparison of the Obtained Catalyst with Its Commercial Analog
4 Conclusions
References
Study of Various Approaches to the Synthesis of PtCo/C Electrocatalysts for Fuel Cells
1 Introduction
2 Experimental Part
2.1 Preparation of the Catalysts
2.2 Studying of the Catalysts Structure
3 Results and Discussions
3.1 Structural Characteristics of Pt3Co/C Materials, Synthesized by the Borohydride Method
3.2 Structural Characteristics of PtCo/C Materials, Synthesized on a Cobalt-Containing Substrate
3.3 Electrochemical Characteristics of PtCo/C Materials Synthesized on a Cobalt-Containing Substrate
3.4 Structural Characteristics of Pt3Co/C Materials, Synthesized by the Polyol Method
3.5 Structural Characteristics of Pt3Co/C Materials, Synthesized by the Modified Polyol Method
4 Conclusions
References
Advantages of Bimetallic Electrocatalysts for Cathodes in a Proton Exchange Membrane Fuel Cell
1 Introduction
2 Materials and Methods
2.1 Materials
2.2 Synthesis Method
2.3 Methods to Study the Composition and the Structure of the Catalysts
2.4 Electrochemical Methods
3 Results and Discussion
4 Conclusions
References
Carbon Nanoparticles from Thermally Expanded Cointercalates of Graphite Nitrate with Organic Substances
1 Introduction
2 Experimental
2.1 Synthesis of Graphite Nitrate Cointercalation Compounds and Thermally Expanded Graphite Preparation
2.2 GNCCs and TEG Sample Characterization Methods
2.3 Dispersions of Carbon Nanoparticles
3 Results and Discussions
3.1 Graphite Nitrate Cointercalation Compounds as Precursors of Thermally Expanded Graphite
3.2 Pre-organized TEGs for Carbon Nanoparticles
3.3 Morphology of Carbon Nanoparticles from Expanded Graphite Nitrate Cointercalants
4 Conclusions
References
Comparative Study of Photocatalytic Activities of Sn- or F-doped and Sn-F Co-doped TiO2 Nanomaterials
1 Introduction
2 Research Method
2.1 Materials
2.2 Preparation of Materials
2.3 Characterization
3 Results and Discussion
4 Conclusion
References
Silicon-on-Silicon Oxide Metalens: Design and Fabrication Aspects
1 Introduction
2 Design Principles
3 Metalens Design and Numerical Verification
3.1 Metalens Design
3.2 Numerical Simulation Verification
4 Fabrication Aspects
4.1 Silicon on Isolator Structure Fabrication
4.2 Silicon Microstructure Formation by Plasma Chemical Etching
5 Conclusion
References
Preparation, Phase Composition, Nanostructure, Dielectric and Ferroelectric Characteristics of Sr0.6Ba0.4Nb2O6 Thin Films Grown on a Silicon Substrate in an Oxygen Atmosphere
1 Introduction
2 Materials and Methods
3 Results and Discussion
4 Conclusion
References
Influence of the Elemental Composition and Thermodynamic Background on the Processes of Phase Formation in Solid Solutions of the Triple System NaNbO3 - KNbO3 - BiFeO3
1 Introduction
2 Objects, Methods of Obtaining and Studying Samples
3 Results and Discussions
4 Conclusion
References
Deposition and Characterization of Magnetron Sputtered TiN Coatings with Variable Stoichiometry
1 Introduction
2 Materials and Methods
3 Results and Discussion
4 Conclusion
References
Influence of Synthesis Conditions on the Characteristics of Antimony Sulfoiodide
1 Introduction
2 Materials, Synthesis and Research Methods
2.1 Materials
2.2 Synthesis
2.3 Research Methods
3 Results and Discussion
4 Conclusion
References
Study of Nanosecond Repetitively Pulsed Discharges Effects on the Technological Properties of Sulfide Minerals from Low-Grade Copper-Nickel Ores
1 Introduction
2 Experimental
2.1 Minerals
2.2 Research Technique
3 Results and Discussion
3.1 Effect of Nanosecond Electromagnetic Pulses and Dielectric Barrier Discharge on Morphology, Chemical Composition and Microhardness of Minerals Surface
3.2 Effect of HPEMP and DBD on Physicochemical and Flotation Properties of Sulfide Minerals
4 Conclusions
References
Modelling of Phase Transitions in the Process of Cryopreservation of Biological Material
1 Introduction
2 Research Method
3 Finite Element Modelling
4 Results and Discussion
5 Conclusion
References
Physics of Advanced Materials
Using Quantum Dot Structure and Suitable Material for Increasing Propagation Length of Surface Plasmon Polariton
1 Introduction
2 Simulation and Results
3 Conclusion
References
Specific Heat Capacity of Light Rare-Gas Crystals in the Model of Deformable Atoms
1 Introduction
2 Specific Heat Capacity with Inclusion of the Three-Body Interaction and Deformation of Electron Shells
3 Conclusions
References
Advanced Piezo-Active 2–1–2 Composites with Large Parameters for Hydroacoustic and Energy-Harvesting Applications
1 Introduction
2 Structure of the Composite, Its Effective Properties, Figures of Merit, and Components
3 Results and Discussion
3.1 Hydrostatic Figures of Merit ( Qh* )2 of Composites Based on PZN–xPT
3.2 Volume-Fraction (m) Dependence of Hydrostatic Parameters
3.3 Links Between Piezoelectric Properties and Figures of Merit
3.4 Maxima of Figures of Merit at Variations of the Aspect Ratio ηc
3.5 Comparison to Data on Some 2–2-Type Composites
4 Conclusion
References
Novel Lead-Free 2–2-Type Composites with High Piezoelectric Sensitivity and Strong Hydrostatic Response: Examples of 2–1–2 Connectivity
1 Introduction
2 Model of the Piezo-Active 2–1–2 Composite, Its Effective Properties, Hydrostatic Parameters and Components
2.1 Model Concepts, Effective Electromechanical Properties and Related Hydrostatic Parameters
2.2 Components of 2–1–2 Composites
3 Piezoelectric Properties and Hydrostatic Parameters of Composites
4 Conclusion
References
Experimental Investigation of Band Gaps in Two-Layered Elastic Metamaterials with Arrays of Strip-Like Voids
1 Introduction
2 Design and Manufacturing
2.1 Design of Elastic Metamaterial
2.2 Manufacturing of Specimen
3 Experimental Setup
4 Experimental Results
5 Conclusion
References
Pyroelectric and Piezoresonant Responses in the Region of the AFE-FE Phase Transition of the Pb(Zr1−xTix)O3 Solid Solutions
1 Introduction
2 Research Method
2.1 Materials and Research Methods
3 Results and Discussion
4 Conclusion
References
Dielectric, Piezoelectric and Pyroelectric Properties of Ceramic Solid Solutions Based on PZT System
1 Introduction
2 Research Method
2.1 Materials and Research Methods
3 Results and Discussion
4 Conclusion
References
Structural Order/Disorder Phenomena Investigation in Pb-Containing Complex Perovskites with Relaxor Ferroelectric Properties
1 Introduction
2 Materials and Methods
2.1 Fabrication of Samples and X-ray Structural Studies
2.2 Investigation of Dielectric Properties of Samples
2.3 Determination of Phase Composition of the Samples with Using the Retrieve Crystallographic Database
3 Experimental Results and Discussion
4 Conclusion
References
Switching Processes and Ferroelectric Hysteresis in Porous PZT Type Piezoceramics
1 Introduction
2 Experimental
3 Results and Discussions
4 Conclusion
References
Microstructure, Complex Electromechanical Parameters and Dispersion in Porous Piezoceramics
1 Introduction
2 Objects of the Study and Methods of Measurements
3 Results and Discussions
3.1 Microstructure Study
3.2 Study of Electromechanical Properties
4 Conclusion
References
The Behavior of Dielectric Properties in the Solid Solutions Based on Na0.5Bi0.5TiO3 and Na0.5K0.5NbO3
1 Introduction
2 Research Method
2.1 Sample Preparation
2.2 Experimental Methods
3 Results and Discussion
4 Conclusion
References
Ferroelectric and Dielectric Properties of Solid Solution Ceramics Based on Bismuth Ferrite and Lead Ferroniobate Multiferroics with Germanium Dioxide Additive
1 Introduction
2 Obtaining and Research Methods
3 Experimental Results and Discussion
4 Conclusions
References
Resonant Microwave Response in Strontium Titanate Single Crystals
1 Introduction
2 Obtaining and Research Methods
2.1 Microstrip Measuring Cells
2.2 Method for Calculating the Dissipation Coefficient of a Sample on an MSL
2.3 Waveguide Measuring Cell
2.4 Method for Calculating the Dissipation Coefficient of a Sample in a Waveguide
3 Experimental Results and Discussion
4 Conclusions
References
Crystal Structure, Microstructure, Piezoelectric and Dielectric Properties of Piezoceramic Bi7Sr2TiNb5O27
1 Introduction
2 Experiment
3 Results and Discussion
4 Conclusions
References
The Relationship of Electrical Conductivity, Morphology and Protective Properties of Organic Films with the Type of Inhibitor and Its Concentration
1 Introduction
2 Experimental Results and Discussion
2.1 Inhibition of the Steel Corrosion in Sulfuric Acid
2.2 Microstructure of the Investigated Films
2.3 Electrical Conductivity of Protective Films
3 Conclusion
References
Negative Electrical Capacitance of the Organic Anticorrosion Films
1 Introduction
2 Problem Formulation
3 Experimental Results and Discussion
3.1 Film Samples
3.2 Negative Electrical Capacitance of the Organic Films
3.3 Analysis of the Equations of Frequency Dependences C′(ω) and G″(ω) Using the Impedance Representation of the Relaxation Polarization of the Films with Negative Capacitance
3.4 Comparison of Experimental Results and Theoretical Calculations
4 Conclusion
References
Broadband Optical Limiting of Single-Walled Carbon Nanotubes with Tetracarboxy-Substituted Phthalocyanine Ligand Composite
1 Introduction
2 Experimental Part
2.1 Materials
2.2 Spectral Measurement of the Materials Prepared
2.3 Spectral Measurement of the Materials Prepared
3 Results and Discussion
4 Conclusion
References
Mechanics of Advanced Materials
Identification of the Effective Properties of PZT-Ni and PZT-Air Composites Considering a Non-uniform Partly Polarized Field
1 Introduction
2 Mathematical Model
2.1 Representative Volume Element (RVE) Model
2.2 Boundary Value Homogenization Problems
2.3 Modeling the Non-uniform Polarization Field
2.4 Computing the Effective Moduli
3 Results and Discussion
4 Conclusions
Appendix
References
Finite Element Investigation of Mechanical Properties of Highly Porous Nanoscale Materials Composed of Regular Lattices from Gibson-Ashby Cells of Variable Geometry
1 Introduction
2 Homogenization Approaches
3 Construction of Representative Volume Element (RVE)
4 Computational Experiments and Results
5 Conclusions
References
Methodology of Determination of Porosity Parameters in the Theory of Microdilation
1 Introduction
2 Statement and Solving Methods of the Contact Problems
3 The Scheme for Finding Microdilatation Parameters
4 Conclusions
References
Molecular Dynamic Study of Dependency on Mechanical Characteristic of Nanocrystalline Copper over Various Temperature and Strain Rate
1 Introduction
2 Simulation Model and Methods
3 Results and Discussion
3.1 Effect of Temperature on Mechanical Properties and Elastic Constant at a Fixed Strain Rate of 5.0 × 109 s−1
3.2 Effect of Strain Rates on Mechanical Properties and Elastic Constant at a Fixed Temperature of 300 K
3.3 Potential Energy Dependence of Single Crystal Nano Copper on Strain Rate and Temperature
4 Conclusion
References
Applied Theory of Vibrations of a Composite Electromagnetoelastic Bimorph with Damping
1 Introduction
2 Problem Statement
3 Solution
4 Numerical Experiment
5 Conclusions
References
Features of Backscattering of Short Longitudinal Waves on Spatial Defects of Canonical Form Located in Elastic Bodies
1 Introduction
2 Problem Statement
3 Solution Method
4 The Case of p-p Reflection
5 Numerical Results
6 Conclusion
References
Delamination Detection in a Multilayer Carbon Fiber Reinforced Plate Based on Acoustic Methods: Numerical and Experimental Study
1 Introduction
2 Experiment
3 Finite Element Model of the System Under Study
4 Numerical and Experimental Results
5 Conclusions
References
Theoretical Approaches for the Damage Identification in the Timoshenko Beam Based on Solving a Coefficient Inverse Problem
1 Introduction
2 Problem Statement and Methods
2.1 Research Aim
2.2 Statement of the Modal Analysis Problem
2.3 Optimization Problem Formulation
2.4 Minimization of the Misfit Functional
2.5 Obtaining the Fréchet Derivative
3 Algorithm for Solving the Coefficient Inverse Problem for the Moment of Inertia
4 Defect Identification Based on Recovery of Two Coefficients
5 Conclusion
References
Calculation of the Stress State of a Three-Layer Spherical Shell Based on Exact, Asymptotic Solutions and Solutions According to Some Applied Theories
1 Problem Statement
2 Solution Method
2.1 Exact Solution
2.2 Asymptotic Solution
2.3 Equations of Shell Theory, Obtained Under the Assumption that the Kirchhoff-Love Hypotheses are Satisfied for the Entire Package
2.4 Equations of Broken Element Theory
2.5 Calculation Based on Theory that Consider Transverse Shear Deformations
2.6 Comparative Analysis of Numerical Results
3 Conclusions
References
Determination of Stress-Strain State in Layered Structures Made of Isotropic and Transversally Isotropic Materials for the Case of Source of the Elastic Waves Inside a Layer Package
1 Introduction
2 Problem Statement
3 Matrix Representation of Fields in a Layer with Sources
4 Conclusions
References
Multi-parameter Assessment of Wear Resistance of Antifriction Ion-Plasma Coatings Deposited on a Cemented Steel Substrate
1 Introduction
2 Research Method
2.1 Materials and Research Methods
2.2 Multiparametric Assessment Methodology
3 Results and Discussion
4 Conclusion
References
Mathematical Model of the Strength of Modified Cement Stone
1 Introduction
2 Research Method
3 Results and Discussion
4 Conclusion
References
Influence of Man-Made Raw Materials on the Physical and Mechanical Properties of Organomineral Compositions
1 Introduction
1.1 Purpose of the Study
1.2 Research Area
2 Research Methods
3 Results and Discussion
4 Conclusion
References
The Effect of Variations in Coconut Fiber Ash Waste as Added Material in Mortar
1 Introduction
2 Research Method
2.1 Materials
2.2 Mixture Composition of Mortar Specimens
2.3 Preparing Process of Coconut Fiber Ash Material
2.4 Making Specimen of Mortar
2.5 Testing of Specimens
3 Results and Discussions
3.1 Density (Unit-Weight)
3.2 Water Absorption
3.3 Compression Strength
4 Conclusion
References
Applications of Advanced Materials
Embedding Epitaxial VO2 Film to Switchable Two-Band Filter on the Surface Acoustic Waves
1 Introduction
2 Theoretical Study of Switchable/Tunable Filter on SAW
3 Calculated Switchable SAW Filter with a Built-in Vanadium Dioxide Film
4 Conclusion
References
Structural Scheme of an Electromagnetoelastic Actuator for Nanotechnology Research
1 Introduction
2 Research Method
2.1 Method of Mathematical Physics
2.2 Structural Scheme
3 Results and Discussion
4 Conclusion
References
Design and Analysis of Piezoaeroelastic Energy Harvester for Mid-Range Wind Velocity Applications
1 Introduction
2 Development of Novel Bluff Body
3 Mathematical Modelling
4 Results
5 Conclusion
References
A Novel Design of Motorbike Suspension System for Energy Harvesting
1 Introduction
2 Modeling, Design, and Method
3 Parameters Study and Results Analysis
3.1 Effect of Scissor Angle
3.2 Effect of the Width of the PZT Patch
3.3 Effect of Thickness of PZT Patch
3.4 Effect of the Velocity of the Motorbike in Different Road Classes
4 Conclusion
References
Energy Harvesting Devices Based on Solid Solutions of Barium Titanate-Zirconate and Silver Niobate
1 Introduction
2 Objects and Methods of Study
2.1 SS BaTi1−xZrxO3
2.2 SS (1 – x)BaTi0.85Zr0.15O3 + xAgNbO3
3 Results and Discussions
4 Conclusions
References
Aspect Ratio Optimization of Piezoceramic Piezoelements for Maximizing Energy Conversion in Energy Harvesting Applications
1 Introduction
2 Objects of the Study and Methods of Measurements
3 Results and Discussions
4 Conclusion
References
Renewable Energy: Charcoal Briquettes from Coconut Shells
1 Introduction
2 Methods
2.1 Renewable Energy
2.2 Biomass Energy – Environmentally Friendly Energy
2.3 Basic Concepts of Biomass
2.4 Research Methodology
3 Results and Discussion
3.1 Benefits and Waste of Coconut Trees
3.2 Coconut Shell Briquettes
3.3 The Process of Making Coconut Shell Briquettes
3.4 Advantages of Coconut Shell Briquettes
3.5 Advantages of Briquettes as an Alternative Energy
3.6 Ecological Impact of Using Charcoal Briquettes
3.7 Advantages of Briquettes Compared to Fuel Oil
4 Conclusion
References
Behavior of Magnetic Nanoparticles in the Phantom of the Biological Medium
1 Introduction
2 Materials and Measurement Techniques
3 Results and Discussion
4 Conclusion
References
Simulation of the Interaction of the Keratoprosthesis with the Cornea
1 Introduction
2 Geometric Model and Mathematical Formulation of the Problem
3 Numerical Analysis
4 Conclusion
References
Image Reconstruction Algorithm of Optoacoustic Signal Transformation Based on Neural Networks
1 Introduction
2 Research Method
3 Results and Discussion
4 Conclusion
References
Effective Method of Pipeline Transport Monitoring
1 Introduction
2 Research Method
2.1 Aim of the Study
2.2 Research Proposal
2.3 Description of the Experimental Setup
3 Results of the Experiment
4 Results and Discussion
5 Conclusion
References
Modeling of the Temperature Profile When Heating Fuel Oil with Water Vapor in a Single-Pass Tubular Heat Exchanger
1 Introduction
2 Basics of Thermal Calculation of Heat Exchangers
3 Mathematical Modeling of Temperature Profile During Heating of Fuel Oil with Water Steam in a Single-pass Tubular Heat Exchanger
4 Conclusion
References
Optimization of Parameters of a Hydraulic Mobile Pumping Unit for Hydraulic Fracturing
1 Introduction
2 Research Method
3 Results and Discussion
References
Designing a Mobile High-Pressure Well Service Pump Unit on a Vehicle Chassis
1 Introduction
2 Research Method
3 Results and Discussion
4 Conclusion
References
Temporal Mapping of Coastal Areas Using Landsat Satellite Imagery
1 Introduction
2 Research Data
3 Research Methodology
4 Results and Discussion of Research
4.1 Coastal Land Cover in Surabaya 1994–2023
4.2 The Trend of Changes in Land Cover Area
4.3 Comparison of Land Cover Data Using Landsat and Sentinel Satellite Images
4.4 Evaluation of Accuracy Testing
5 Conclusion
References
Teyeng Batik: A New Innovation for Making Motifs on Batik Using Iron Rust Stains
1 Introduction
2 Methods
2.1 Conceptual Batik
2.2 Batik Characteristics
2.3 Types of Batik According to the Manufacturing Technique
2.4 Research Methodology
3 Results and Discussion
3.1 Iron Rusting Process
3.2 Rusting Process in Teyeng Batik
3.3 Steps for Making Teyeng Batik
3.4 Teyeng Batik Motifs
4 Conclusion
References
Analysis of Balanced Furniture Product Requirements with Fuzzy Goal Programming Model Development
1 Introduction
2 Problem Formulation
2.1 Experimental Details
2.2 Objective Function
2.3 Constraint Function
3 Results and Discussion
3.1 Objective Function Model
3.2 Form of Constraints
3.3 Fuzzy Membership
3.4 Fuzzy Goal Programming
4 Conclusion
5 Suggestions
References
Correction to: Molecular Dynamic Study of Dependency on Mechanical Characteristic of Nanocrystalline Copper over Various Temperature and Strain Rate
Correction to: Chapter 33 in: I. A. Parinov et al. (Eds.): Physics and Mechanics of New Materials and Their Applications, SPM 41, https://doi.org/10.1007/978-3-031-52239-0_33
Author Index
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Springer Proceedings in Materials
Ivan A. Parinov Shun-Hsyung Chang Erni Puspanantasari Putri Editors
Physics and Mechanics of New Materials and Their Applications Proceedings of the International Conference PHENMA 2023
Springer Proceedings in Materials
41
Series Editors Arindam Ghosh, Department of Physics, Indian Institute of Science, Bengaluru, India Daniel Chua, Department of Materials Science and Engineering, National University of Singapore, Singapore, Singapore Flavio Leandro de Souza, Universidade Federal do ABC, Sao Paulo, São Paulo, Brazil Oral Cenk Aktas, Institute of Material Science, Christian-Albrechts-Universität zu Kiel, Kiel, Schleswig-Holstein, Germany Yafang Han, Beijing Institute of Aeronautical Materials, Beijing, Beijing, China Jianghong Gong, School of Materials Science and Engineering, Tsinghua University, Beijing, Beijing, China Mohammad Jawaid , Laboratory of Biocomposite Technology, INTROP, Universiti Putra Malaysia, Serdang, Selangor, Malaysia
Springer Proceedings in Materials publishes the latest research in Materials Science and Engineering presented at high standard academic conferences and scientific meetings. It provides a platform for researchers, professionals and students to present their scientific findings and stay up-to-date with the development in Materials Science and Engineering. The scope is multidisciplinary and ranges from fundamental to applied research, including, but not limited to: • • • • • • • • •
Structural Materials Metallic Materials Magnetic, Optical and Electronic Materials Ceramics, Glass, Composites, Natural Materials Biomaterials Nanotechnology Characterization and Evaluation of Materials Energy Materials Materials Processing
To submit a proposal or request further information, please contact one of our Springer Publishing Editors according to your affiliation: European countries: Mayra Castro ([email protected]) India, South Asia and Middle East: Priya Vyas ([email protected]) South Korea: Smith Chae ([email protected]) Southeast Asia, Australia and New Zealand: Ramesh Nath Premnath (ramesh. [email protected]) The Americas: Michael Luby ([email protected]) China and all the other countries or regions: Mengchu Huang (mengchu.huang@ springer.com) This book series is indexed in SCOPUS database.
Ivan A. Parinov · Shun-Hsyung Chang · Erni Puspanantasari Putri Editors
Physics and Mechanics of New Materials and Their Applications Proceedings of the International Conference PHENMA 2023
Editors Ivan A. Parinov I. I. Vorovich Institute of Mathematics, Mechanics and Computer Science Southern Federal University Rostov-on-Don, Russia
Shun-Hsyung Chang Department of Microelectronic Engineering National Kaohsiung University of Science and Technology Kaohsiung, Taiwan
Erni Puspanantasari Putri Institute for Research and Community Service Universitas 17 Agustus 1945 Surabaya Surabaya, Indonesia
ISSN 2662-3161 ISSN 2662-317X (electronic) Springer Proceedings in Materials ISBN 978-3-031-52238-3 ISBN 978-3-031-52239-0 (eBook) https://doi.org/10.1007/978-3-031-52239-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024, corrected publication 2024 This work is subject to copyright. All rights are solely and exclusively licensed 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 Paper in this product is recyclable.
Preface
The quick acceleration of scientific and technological progress requires the design and improvement of advanced materials and composites, together with the development of modern devices for broad applications. The high-level results, obtained for designed materials and technologies, rapidly determine their consumers, aimed to industrial production and its improvement. Achievement of optimal solutions is directly connected with effective chemical, physical, and mechanical studies, modern computational methods, and approaches, as well as the results of a full-scale experiment. The current level of science and technology requires the devices and equipment with high measurement accuracy, reliability, and operational durability. Moreover, scientific and technical developments should be aimed to solution of the environmental and biomedical problems of modern life. This collection of 60 papers presents selected reports of the 2023 International Conference on “Physics and Mechanics of New Materials and Their Applications” (PHENMA 2023), Surabaya, Indonesia, October 3–8, 2023, https://phenma2023.sfe du.ru/. The conference was sponsored by the Ministry of Education and Science of Russian Federation, Russian Science Foundation, South Scientific Center of Russian Academy of Science, Ministry of Science and Technology of Taiwan, The Society of Materials Science, Kyushu Branch, (Japan), Kitakyushu Convention & Visitors Association (Japan), The Korean Society for Composite Materials (South Korea), National Foundation for Science & Technology Development (Vietnam), Vietnam Association of Science Editing (VASE), Vietnam Union of Science and Technology Associations (VUSTA), New Century Education Foundation (Taiwan), Ocean & Underwater Technology Association (Taiwan), Fair Well Fishery Co. (Taiwan), Woen Jinn Harbor Engineering Co. (Taiwan), Lorom Group (Taiwan), Longwell Co. (Taiwan), Don State Technical University (Russia), Kyushu Institute of Technology (Japan), Korea Maritime and Ocean University (South Korea), Hanoi University of Science and Technology (Vietnam), Vietnam Maritime University, Vinh Long University of Technology Education (Vietnam), Hanoi University of Industry (Vietnam), Vietnamese-German University (Vietnam), Ho Chi Minh City University of Agriculture and Forestry (Vietnam), Research Institute of Agriculture Machinery (Vietnam), PDPM Indian Institute of Information Technology, and South Russian Regional Centre for Preparation and Implementation of International Projects (Russia). The topic of the PHENMA 2023 continued ideas of previous international symposia and conferences: PMNM-2012 (Rostov-on-Don, Russia), PHENMA 2013 (Kaohsiung, Taiwan), PHENMA 2014 (Khon-Kaen, Thailand), PHENMA 2015 (Azov, Russia), PHENMA 2016 (Surabaya, Indonesia), PHENMA 2017 (Jabalpur, India), PHENMA 2018 (Busan, South Korea), PHENMA 2019 (Hanoi, Vietnam), PHENMA 2020 (Kytakyushu, Japan), PHENMA 2021–2022 (Divnomorsk, Russia), whose results have been published in the following edited books: “Physics and Mechanics of New Materials and Their Applications”, Ivan A. Parinov, Shun-Hsyung Chang (Eds.), Nova Science
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Preface
Publishers, New York, 2013, 444 p. ISBN: 978-1-62618-535-7; “Advanced Materials – Physics, Mechanics and Applications”, Springer Proceedings in Physics. Vol. 152. Shun-Hsyung Chang, Ivan A. Parinov, Vitaly Yu. Topolov (Eds.), Springer, Heidelberg, New York, Dordrecht, London, 2014, 380 p. ISBN: 978-3319037486; “Advanced Materials – Studies and Applications”, Ivan A. Parinov, Shun-Hsyung Chang, Somnuk Theerakulpisut (Eds.), Nova Science Publishers, New York, 2015, 527 p. ISBN: 978-163463-749-7; “Advanced Materials – Manufacturing, Physics, Mechanics and Applications”, Springer Proceedings in Physics, Vol. 175, Ivan A. Parinov, Shun-Hsyung Chang, Vitaly Yu. Topolov (Eds.). Heidelberg, New York, Dordrecht, London: Springer Cham. 2016. – 707 p., ISBN: 978-3319263229; Advanced Materials – Techniques, Physics, Mechanics and Applications, Springer Proceedings in Physics, Vol. 193, Ivan A. Parinov, Shun-Hsyung Chang, Muaffaq A. Jani (Eds.). Springer Nature, Cham, Switzerland. 2017. – 627 p., ISBN: 978-3-319-56062-5; Advanced Materials – Proceedings of the International Conference on “Physics and Mechanics of New Materials and Their Applications”, PHENMA 2017, Springer Proceedings in Physics, Vol. 207, Ivan A. Parinov, Shun-Hsyung Chang, Vijay K. Gupta (Eds.). Springer Nature, Cham, Switzerland. 2018. – 640 p. ISBN: 978-3319789187; Advanced Materials – Proceedings of the International Conference on “Physics and Mechanics of New Materials and Their Applications”, PHENMA 2018, Springer Proceedings in Physics, Vol. 224, Ivan A. Parinov, Shun-Hsyung Chang, Yun-Hae Kim (Eds.). Springer Nature, Cham, Switzerland. 2019. – 666 p. ISBN: 978-3-030-19893-0; Advanced Materials - Proceedings of the International Conference on “Physics and Mechanics of New Materials and Their Applications”, PHENMA 2019, Springer Proceedings in Materials, Vol. 6, Ivan A. Parinov, Shun-Hsyung Chang, Banh Tien Long (Eds.). Springer Nature, Cham, Switzerland, 2020. – 621 p. ISBN: 978-3-030-45119-6; Physics and Mechanics of New Materials and Their Applications – Proceedings of the International Conference PHENMA 2020, Springer Proceedings in Materials, Vol. 10, Ivan A. Parinov, Shun-Hsyung Chang, YunHae Kim, Nao-Aki Noda (Eds.). Springer Nature, Cham, Switzerland, 2021. – 601 p. ISBN: 978-3-030-76480-7; Physics and Mechanics of New Materials and Their Applications – Proceedings of the International Conference PHENMA 2021-2022, Springer Proceedings in Materials, Vol. 20, Ivan A. Parinov, Shun-Hsyung Chang, Arkady N. Soloviev (Eds.). Springer Nature, Cham, Switzerland, 2023. – 577 p. ISBN: 978-3-031-21571-1, respectively. The papers of the PHENMA 2023 are divided into four scientific directions: (i) processing techniques of advanced materials, (ii) physics of advanced materials, (iii) mechanics of advanced materials, and (iv) applications of advanced materials. Into framework of the first direction, control of morphological characteristics of the Pt/C catalysts, obtained by the liquid phase synthesis, is performed with the study of various approaches to the synthesis of PtCo/C electrocatalysts for fuel cells. It discussed an advantage of bimetallic electrocatalysts for cathodes in a proton exchange membrane fuel cell. Then carbon nanoparticles from thermally expanded cointercalates of graphite nitrate with organic substances are processed and studied. Comparative study of photocatalytic activities is present for Sn- or F-doped and Sn-F co-doped TiO2 nanomaterials. Design and fabrication aspects are considered for silicon-on-silicon oxide metalens with the following discussion of different characteristics of Sr0.6 Ba0.4 Nb2 O6 thin films grown
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on a silicon substrate. Thermodynamic background of the phase formation in solid solutions of the triple system NaNbO3 – KNbO3 – BiFeO3 is studied in necessary detail. Then the deposition and characterization of magnetron-sputtered TiN coatings with variable stoichiometry are investigated. It estimated the influence of synthesis conditions on the characteristics of antimony sulfoiodide with study of nanosecond repetitively pulsed discharges effects on the technological properties of sulfide. The first section is finished by modeling of phase transitions in the process of cryopreservation of biological material. The physical topic is opened by the study of using quantum dot structure for increasing propagation length of surface plasmon polariton. Then specific heat capacity of light rare-gas crystals is considered in the model of deformable atoms. Advanced piezoactive 2–1–2 composites with large parameters for hydroacoustic and energy harvesting applications are modeled, too. Band gaps in two-layered acoustic metamaterials with arrays of strip-like voids are investigated experimentally. Dielectric, piezoelectric, and pyroelectric properties of ceramic solid solutions, based on PMN-PT and PZT, are studied in detail with the consideration of structural order/disorder phenomena. Microstructure, switching processes, and ferroelectric hysteresis are discussed for porous piezoceramics. Then specific considerations are devoted to ceramic, based on bismuth ferrite, lead ferroniobate multiferroics with germanium dioxide additive, and strontium titanate single crystals. Promising Aurivillius phases are presented by study of the microstructure, piezoelectric, and dielectric properties of piezoceramic Bi7Sr2TiNb5O27. Organic films are investigated in respect of their electric and physical properties, morphology, and protective characteristics. The second section is finished by the discussion of the broadband optical limiting of single-walled carbon nanotubes with tetracarboxy-substituted phthalocyanine ligand composite. The section of mechanics is opened by identification of the effective properties of PZT-Ni and PZT-Air composites considering a non-uniform partly polarized field. Investigation of mechanical properties of highly porous nanoscale materials, composed of regular lattices from Gibson-Ashby cells of variable geometry, is present with developed a methodology for determination of porosity parameters in the theory of microdilation. Molecular dynamic is applied to study of nanocrystalline copper in dependence of mechanical characteristic on various values of temperature and strain rate. Then applied theory of vibrations of a composite electromagnetoelastic bimorph with damping is developed. Features of backscattering short longitudinal waves are considered on spatial defects of canonical form, located in elastic bodies. Numerical and experimental studies are devoted to delamination detection in a multilayer carbon fiber reinforced plate, based on acoustic methods. Theoretical approaches for the damage identification in the Timoshenko beam are based on solving a coefficient inverse problem. Then it calculated the stress state of a three-layer spherical shell by using exact, asymptotic solutions, and solutions according to some applied theories. This topic is continued by determination of stress-strain state in layered structures made of isotropic and transversally isotropic materials for the case of source of the elastic waves inside a layer package. Multi-parameter assessment is obtained for wear resistance of antifriction ion-plasma coatings, deposited on a cemented steel substrate. Finally, influence of man-made raw materials on the physical and mechanical properties of organomineral compositions with effect of variations in coconut fiber ash waste as added material in mortar is considered.
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Discussion of embedding epitaxial VO2 film to switchable two-band filter on the surface acoustic waves opens in the fourth section. Then structural scheme of an electromagnetoelastic actuator is proposed for nanotechnology research. Design and analysis of piezoaeroelastic energy harvester for mid-range wind velocity applications and motorbike suspension system for energy harvesting are developed. Energy harvesting devices, based on solid solutions of barium titanate-zirconate and silver niobate, are discussed, too, together with aspect ratio optimization of piezoceramic elements for maximizing energy conversion. Charcoal briquettes from coconut shells are proposed for renewable energy. Medical and biological applications are presented by consideration of the behavior of magnetic nanoparticles in the phantom of the biological medium, simulation of the interaction of the keratoprosthesis with the cornea, and image reconstruction algorithm of optoacoustic signal transformation, based on neural networks. Then effective method of pipeline transport monitoring is proposed with modeling of the temperature profile by heating fuel oil with water vapor in a single-pass tubular heat exchanger. Moreover, it studied an optimization of parameters of a hydraulic mobile pumping unit for hydraulic fracturing with designing a mobile high-pressure well service pump unit on a vehicle chassis. Finally, an innovation for making motifs on batik using iron rust stains and balanced furniture product requirements with fuzzy goal programming model development are discussed. This book will be interesting for students of different levels of education, scientists, and engineers, which R&D of nano-, piezomaterials and structures, other promising materials, and composites with optimal structure-sensitive and electrophysical properties. The book also presents experimental and theoretical problems and computer simulation methods, devoted to designing the promise devices, based on advanced materials, which are suitable for wide scientific, technical, and technological directions. The book discusses new research results in the materials science, chemistry, condensed matter physics, mechanics, computational and physical modeling and experiment, fabrication of modern materials and composites, mathematical processing test results, simulation methods, and various applications. This edited book was partially supported by the Ministry of Science and Higher Education of the Russian Federation, state task in the field of scientific activity, scientific project no. FENW-2023–0012. November 2023
Ivan A. Parinov Shun-Hsyung Chang Erni Puspanantasari Putri
Contents
Processing Techniques of Advanced Materials Control Over Morphological Characteristics of the Pt/C Catalysts Obtained by the Liquid-Phase Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yu. Bayan, K. Paperzh, M. Danilenko, D. Alekseenko, Yu. Pankova, I. Pankov, and A. Alekseenko Study of Various Approaches to the Synthesis of PtCo/C Electrocatalysts for Fuel Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. R. Beskopylny, S. V. Belenov, D. K. Mauer, and V. S. Menshchikov Advantages of Bimetallic Electrocatalysts for Cathodes in a Proton Exchange Membrane Fuel Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angelina Pavlets, Ekaterina Kozhokar, Yana Astravukh, Ilya Pankov, Alexey Nikulin, and Anastasia Alekseenko Carbon Nanoparticles from Thermally Expanded Cointercalates of Graphite Nitrate with Organic Substances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. V. Raksha, A. A. Davydova, V. A. Glazunova, Y. V. Berestneva, A. B. Eresko, O. N. Oskolkova, P. V. Sukhov, V. V. Gnatovskaya, G. K. Volkova, V. V. Burkhovetskij, A. S. Doroshkevich, and M. V. Savoskin Comparative Study of Photocatalytic Activities of Sn- or F-doped and Sn-F Co-doped TiO2 Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. G. Volkova and E. M. Bayan Silicon-on-Silicon Oxide Metalens: Design and Fabrication Aspects . . . . . . . . . . E. Yu. Gusev, S. P. Avdeev, S. V. Malokhatko, V. S. Klimin, V. V. Polyakov, S. Wang, X. Ren, D. Chen, L. Han, Z. Wang, W. Zhang, and O. A. Ageev Preparation, Phase Composition, Nanostructure, Dielectric and Ferroelectric Characteristics of Sr0.6 Ba0.4 Nb2 O6 Thin Films Grown on a Silicon Substrate in an Oxygen Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . A. V. Pavlenko, Ya. Yu. Matyash, D. V. Stryukov, and N. V. Makinyan
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Influence of the Elemental Composition and Thermodynamic Background on the Processes of Phase Formation in Solid Solutions of the Triple System NaNbO3 − KNbO3 − BiFeO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. O. Moysa, L. A. Shilkina, K. P. Andryushin, A. S. Pavlenko, and L. A. Reznichenko Deposition and Characterization of Magnetron Sputtered TiN Coatings with Variable Stoichiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrey L. Nikolaev, Vasilina A. Lapitskaya, Evgeniy V. Sadyrin, Pavel E. Antipov, Ivan O. Kharchevnikov, and Sergey S. Volkov Influence of Synthesis Conditions on the Characteristics of Antimony Sulfoiodide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. E. Oparina, E. S. Medvedeva, and T. G. Lupeiko
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Study of Nanosecond Repetitively Pulsed Discharges Effects on the Technological Properties of Sulfide Minerals from Low-Grade Copper-Nickel Ores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Igor Zh. Bunin and Irina A. Khabarova Modelling of Phase Transitions in the Process of Cryopreservation of Biological Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Andrey Matrosov, Arkady Soloviev, Irina Serebryanaya, Olga Pustovalova, and Daria Nizhnik Physics of Advanced Materials Using Quantum Dot Structure and Suitable Material for Increasing Propagation Length of Surface Plasmon Polariton . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Watheq F. Shneen and Sabah M. M. Ameen Specific Heat Capacity of Light Rare-Gas Crystals in the Model of Deformable Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Ie. Ie. Gorbenko, E. A. Pilipenko, and I. A. Verbenko Advanced Piezo-Active 2–1–2 Composites with Large Parameters for Hydroacoustic and Energy-Harvesting Applications . . . . . . . . . . . . . . . . . . . . . 151 V. Yu. Topolov Novel Lead-Free 2–2-Type Composites with High Piezoelectric Sensitivity and Strong Hydrostatic Response: Examples of 2–1–2 Connectivity . . . . . . . . . . 167 V. Yu. Topolov and S. A. Kovrigina
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Experimental Investigation of Band Gaps in Two-Layered Elastic Metamaterials with Arrays of Strip-Like Voids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Mikhail V. Golub, Artur D. Khanazaryan, Kirill K. Kanishchev, Ilya A. Moroz, Olga V. Doroshenko, and Sergey I. Fomenko Pyroelectric and Piezoresonant Responses in the Region of the AFE-FE Phase Transition of the Pb(Zr1−x Tix )O3 Solid Solutions . . . . . . . . . . . . . . . . . . . . . 188 A. A. Pavelko, A. A. Martynenko, and L. A. Reznichenko Dielectric, Piezoelectric and Pyroelectric Properties of Ceramic Solid Solutions Based on PZT System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 A. A. Pavelko, A. A. Martynenko, K. P. Andryushin, E. V. Glazunova, A. V. Nagaenko, L. A. Shilkina, and L. A. Reznichenko Structural Order/Disorder Phenomena Investigation in Pb-Containing Complex Perovskites with Relaxor Ferroelectric Properties . . . . . . . . . . . . . . . . . . 209 Alla Lebedinskaya and Angela Rudskaya Switching Processes and Ferroelectric Hysteresis in Porous PZT Type Piezoceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 I. A. Shvetsov, N. A. Shvetsova, E. I. Petrova, D. I. Makarev, and A. N. Rybyanets Microstructure, Complex Electromechanical Parameters and Dispersion in Porous Piezoceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 M. G. Konstantinova, P. A. Abramov, N. A. Shvetsova, I. A. Shvetsov, M. A. Lugovaya, and A. N. Rybyanets The Behavior of Dielectric Properties in the Solid Solutions Based on Na0.5 Bi0.5 TiO3 and Na0.5 K0.5 NbO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 E. V. Glazunova, A. S. Chekhova, L. A. Shilkina, I. A. Verbenko, A. V. Nazarenko, L. A. Reznichenko, and V. A. Isaev Ferroelectric and Dielectric Properties of Solid Solution Ceramics Based on Bismuth Ferrite and Lead Ferroniobate Multiferroics with Germanium Dioxide Additive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 K. M. Zhidel, A. V. Pavlenko, E. V. Glazunova, and L. A. Reznichenko Resonant Microwave Response in Strontium Titanate Single Crystals . . . . . . . . . 252 P. A. Astafev, A. A. Pavelko, K. P. Andryushin, A. R. Borzykh, A. M. Lerer, Y. A. Reizenkind, I. V. Donets, A. V. Pavlenko, and L. A. Reznichenko
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Crystal Structure, Microstructure, Piezoelectric and Dielectric Properties of Piezoceramic Bi7 Sr2 TiNb5 O27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 Sergei V. Zubkov, Ivan A. Parinov, Alexander V. Nazarenko, and Yuliya A. Kuprina The Relationship of Electrical Conductivity, Morphology and Protective Properties of Organic Films with the Type of Inhibitor and Its Concentration . . . 275 E. N. Sidorenko, S. P. Shpanko, A. V. Nazarenko, M. A. Bunin, and A. V. Shloma Negative Electrical Capacitance of the Organic Anticorrosion Films . . . . . . . . . . 285 E. N. Sidorenko, S. P. Shpanko, A. V. Shloma, A. G. Rudskaya, and A. O. Galatova Broadband Optical Limiting of Single-Walled Carbon Nanotubes with Tetracarboxy-Substituted Phthalocyanine Ligand Composite . . . . . . . . . . . . 295 M. S. Savelyev, P. N. Vasilevsky, L. P. Ichkitidze, A. Yu. Tolbin, and A. Yu. Gerasimenko Mechanics of Advanced Materials Identification of the Effective Properties of PZT-Ni and PZT-Air Composites Considering a Non-uniform Partly Polarized Field . . . . . . . . . . . . . . . 305 Mohamed Elsayed Nassar and Andrey Nasedkin Finite Element Investigation of Mechanical Properties of Highly Porous Nanoscale Materials Composed of Regular Lattices from Gibson-Ashby Cells of Variable Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Alexandr Kornievsky and Andrey Nasedkin Methodology of Determination of Porosity Parameters in the Theory of Microdilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 M. I. Chebakov and E. M. Kolosova Molecular Dynamic Study of Dependency on Mechanical Characteristic of Nanocrystalline Copper over Various Temperature and Strain Rate . . . . . . . . . 345 Mahesh Kumar Gupta, Santosh Kumar Rai, Vinay Panwar, I. A. Parinov, and Rakesh Kumar Haldkar Applied Theory of Vibrations of a Composite Electromagnetoelastic Bimorph with Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 A. N. Soloviev, V. A. Chebanenko, B. T. Do, A. V. Yudin, and I. A. Parinov
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Features of Backscattering of Short Longitudinal Waves on Spatial Defects of Canonical Form Located in Elastic Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Nikolay V. Boyev Delamination Detection in a Multilayer Carbon Fiber Reinforced Plate Based on Acoustic Methods: Numerical and Experimental Study . . . . . . . . . . . . . 380 S. N. Shevtsov, V. A. Chebanenko, I. E. Andzhikovich, and N. G. Snezhina Theoretical Approaches for the Damage Identification in the Timoshenko Beam Based on Solving a Coefficient Inverse Problem . . . . . . . . . . . . . . . . . . . . . . 390 V. E. Yakovlev, A. V. Cherpakov, and S.-H. Chang Calculation of the Stress State of a Three-Layer Spherical Shell Based on Exact, Asymptotic Solutions and Solutions According to Some Applied Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Nikolay V. Boyev Determination of Stress-Strain State in Layered Structures Made of Isotropic and Transversally Isotropic Materials for the Case of Source of the Elastic Waves Inside a Layer Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 Igor P. Miroshnichenko and Ivan A. Parinov Multi-parameter Assessment of Wear Resistance of Antifriction Ion-Plasma Coatings Deposited on a Cemented Steel Substrate . . . . . . . . . . . . . . . 424 A. I. Voropaev, O. V. Kudryakov, V. N. Varavka, V. I. Kolesnikov, I. V. Kolesnikov, and E. S. Novikov Mathematical Model of the Strength of Modified Cement Stone . . . . . . . . . . . . . . 434 Irina Serebryanaya, Alexandra Nalimova, and Andrey Matrosov Influence of Man-Made Raw Materials on the Physical and Mechanical Properties of Organomineral Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Nina Buravchuk and Olga Gurianova The Effect of Variations in Coconut Fiber Ash Waste as Added Material in Mortar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Nurul Rochmah, Retno Trimurtiningrum, Bantot Sutriono, Masca Indra Triana, and Musthofa Saifa Ardana Applications of Advanced Materials Embedding Epitaxial VO2 Film to Switchable Two-Band Filter on the Surface Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 G. Ya. Karapetyan, V. E. Kaydashev, and E. M. Kaidashev
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Structural Scheme of an Electromagnetoelastic Actuator for Nanotechnology Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 S. M. Afonin Design and Analysis of Piezoaeroelastic Energy Harvester for Mid-Range Wind Velocity Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 Prateek Upadhyay and Sujoy Mukherjee A Novel Design of Motorbike Suspension System for Energy Harvesting . . . . . . 512 Tejkaran Narolia, Vijay K. Gupta, and Ivan A. Parinov Energy Harvesting Devices Based on Solid Solutions of Barium Titanate-Zirconate and Silver Niobate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 D. V. Volkov, A. A. Pavelko, A. S. Korolkova, I. A. Verbenko, A. A. Martynenko, and L. A. Reznichenko Aspect Ratio Optimization of Piezoceramic Piezoelements for Maximizing Energy Conversion in Energy Harvesting Applications . . . . . . . . . . . . . . . . . . . . . . 532 P. A. Abramov, M. G. Konstantinova, N. A. Shvetsova, I. A. Shvetsov, A. N. Reznichenko, and A. N. Rybyanets Renewable Energy: Charcoal Briquettes from Coconut Shells . . . . . . . . . . . . . . . . 541 Erni Puspanantasari Putri Behavior of Magnetic Nanoparticles in the Phantom of the Biological Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 L. P. Ichkitidze, O. V. Filippova, M. V. Belodedov, G. Yu. Galechyan, M. S. Savelyev, A. Yu. Gerasimenko, D. V. Telyshev, and S. V. Selishchev Simulation of the Interaction of the Keratoprosthesis with the Cornea . . . . . . . . . 563 Arkadiy Soloviev, Nadezda Glushko, Alexander Epikhin, and Maria Germanchuk Image Reconstruction Algorithm of Optoacoustic Signal Transformation Based on Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 I. B. Starchenko, D. A. Kravchuk, N. N. Chernov, and D. V. Orda-Zhigulina Effective Method of Pipeline Transport Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . 577 I. E. Andzhikovich, O. V. Bocharova, I. B. Mikhailova, and A. S. Turchin Modeling of the Temperature Profile When Heating Fuel Oil with Water Vapor in a Single-Pass Tubular Heat Exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Dmitry Fugarov, Olga Purchina, Inna Popova, and Anastasia Purchina
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Optimization of Parameters of a Hydraulic Mobile Pumping Unit for Hydraulic Fracturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 A. V. Bill, S. O. Kireev, M. V. Korchagina, and A. R. Lebedev Designing a Mobile High-Pressure Well Service Pump Unit on a Vehicle Chassis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 D. V. Karabanov, S. O. Kireev, M. V. Korchagina, A. R. Lebedev, and A. V. Efimov Temporal Mapping of Coastal Areas Using Landsat Satellite Imagery . . . . . . . . . 616 D. A. Safitri, F. Saves, L. E. Fatmawati, and H. Widhiarto Teyeng Batik: A New Innovation for Making Motifs on Batik Using Iron Rust Stains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 Erni Puspanantasari Putri and Bonifacius Raditya Sri Pramana Putra Analysis of Balanced Furniture Product Requirements with Fuzzy Goal Programming Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 Jaka Purnama, Sajiyo, Erni Puspanantasari, Dian Rahma Aulia, and Novi Ariyan Pratama Correction to: Molecular Dynamic Study of Dependency on Mechanical Characteristic of Nanocrystalline Copper over Various Temperature and Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mahesh Kumar Gupta, Santosh Kumar Rai, Vinay Panwar, I. A. Parinov, and Rakesh Kumar Haldkar
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Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
Contributors
P. A. Abramov Southern Federal University, Rostov-on-Don, Russia S. M. Afonin National Research University of Electronic Technology, MIET, Moscow, Russia O. A. Ageev Southern Federal University, Taganrog, Russia D. Alekseenko Faculty of Chemistry, Southern Federal University, Rostov-on-Don, Russia Anastasia Alekseenko Faculty of Chemistry, Southern Federal University, Rostov-onDon, Russia Sabah M. M. Ameen Department of Physics, College of Science, University of Basrah, Basrah, Iraq K. P. Andryushin Research Institute of Physics, Southern Federal University, Rostovon-Don, Russia I. E. Andzhikovich Southern Federal University, Rostov-on-Don, Russia Pavel E. Antipov Don State Technical University, Rostov-on-Don, Russia Musthofa Saifa Ardana Department of Civil Engineering, Universitas 17 Agustus 1945, Surabaya, Indonesia P. A. Astafev Research Institute of Physics, Southern Federal University, Rostov-onDon, Russia Yana Astravukh Faculty of Chemistry, Southern Federal University, Rostov-on-Don, Russia Dian Rahma Aulia Universitas 17 Agustus 1945 Surabaya, Surabaya, East Java, Indonesia S. P. Avdeev Southern Federal University, Taganrog, Russia E. M. Bayan Chemistry Department, Southern Federal University, Rostov-on-Don, Russia Yu. Bayan Faculty of Chemistry, Southern Federal University, Rostov-on-Don, Russia S. V. Belenov Chemistry Department, Southern Federal University, Rostov-on-Don, Russia M. V. Belodedov Bauman Moscow State Technical University, Moscow, Russia Y. V. Berestneva Federal Scientific Centre of Agroecology, Complex Melioration and Protective Afforestation of the Russian Academy of Sciences, Volgograd, Russia
xviii
Contributors
E. R. Beskopylny Chemistry Department, Southern Federal University, Rostov-onDon, Russia A. V. Bill Don State Technical University, Rostov-on-Don, Russia O. V. Bocharova Southern Scientific Center of the Russian Academy of Sciences, Rostov-on-Don, Russia A. R. Borzykh Faculty of Physics, Southern Federal University, Rostov-on-Don, Russia Nikolay V. Boyev I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal University, Rostov-on-Don, Russia Igor Zh. Bunin N.V. Melnikov’s Institute of Comprehensive Exploitation of Mineral Resources, Russian Academy of Sciences (ICEMR RAS), Moscow, Russia M. A. Bunin Institute of Physics, Southern Federal University, Rostov-on-Don, Russia Nina Buravchuk I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal University, Rostov-on-Don, Russia V. V. Burkhovetskij A. A. Galkin Donetsk Institute for Physics and Engineering, Donetsk, Russia S.-H. Chang Department of Microelectronics Engineering, National Kaohsiung University of Science and Technology, Kaohsiung, Taiwan (R.O.C.) M. I. Chebakov I.I. Vorovich Institute of Mathematics, Mechanics, and Computer Science, Southern Federal University, Rostov-on-Don, Russia V. A. Chebanenko I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal University, Rostov-on-Don, Russia A. S. Chekhova Physics Faculty, Southern Federal University, Rostov-on-Don, Russia D. Chen School of Optoelectronic Engineering, Qilu University of Technology (Shandong Academy of Sciences), Jinan, China N. N. Chernov Southern Federal University, Taganrog, Russia A. V. Cherpakov I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal University, Rostov-on-Don, Russia M. Danilenko Faculty of Chemistry, Southern Federal University, Rostov-on-Don, Russia A. A. Davydova L. M. Litvinenko Institute of Physical-Organic and Coal Chemistry, Donetsk, Russia B. T. Do Le Quy Don Technical University, Hanoi, Vietnam I. V. Donets Faculty of Physics, Southern Federal University, Rostov-on-Don, Russia
Contributors
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Olga V. Doroshenko Institute for Mathematics, Mechanics and Informatics, Kuban State University, Krasnodar, Russia A. S. Doroshkevich Joint Institute for Nuclear Research, Dubna, Russia A. V. Efimov Don State Technical University, Rostov-on-Don, Russia Alexander Epikhin Rostov State Medical University, Rostov-on-Don, Russia A. B. Eresko Joint Institute for Nuclear Research, Dubna, Russia L. E. Fatmawati Department of Civil Engineering, Universitas 17 Agustus 1945 Surabaya, Surabaya, Indonesia O. V. Filippova Institute for Bionic Technology and Engineering, Sechenov First Moscow State Medical University, Moscow, Russia Sergey I. Fomenko Institute for Mathematics, Mechanics and Informatics, Kuban State University, Krasnodar, Russia Dmitry Fugarov Don State Technical University, Rostov-on-Don, Russia A. O. Galatova Institute of Physics, Southern Federal University, Rostov-on-Don, Russia G. Yu. Galechyan Institute for Bionic Technology and Engineering, Sechenov First Moscow State Medical University, Moscow, Russia A. Yu. Gerasimenko Sechenov First Moscow State Medical University, Moscow, Russia Maria Germanchuk V.I. Vernadsky Crimean Federal University, Simferopol, Russia E. V. Glazunova Research Institute of Physics, Southern Federal University, Rostovon-Don, Russia V. A. Glazunova A. A. Galkin Donetsk Institute for Physics and Engineering, Donetsk, Russia Nadezda Glushko Don State Technical University, Rostov-on-Don, Russian Federation V. V. Gnatovskaya L. M. Litvinenko Institute of Physical-Organic and Coal Chemistry, Donetsk, Russia Mikhail V. Golub Institute for Mathematics, Mechanics and Informatics, Kuban State University, Krasnodar, Russia Ie. Ie. Gorbenko Federal State Budgetary Educational Institution of Higher Education «Lugansk State Pedagogical University», Lugansk, Russia Mahesh Kumar Gupta Department of Mechanical Engineering, SRM Institute of Science and Technology, Delhi-NCR Campus, Ghaziabad, UP, India
xx
Contributors
Vijay K. Gupta Department of Mechanical Engineering, Rabindranath Tagore University, Raisen, India Olga Gurianova I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal University, Rostov-on-Don, Russia E. Yu. Gusev Southern Federal University, Taganrog, Russia Rakesh Kumar Haldkar I. I. Vorovich Mathematics, Mechanics and Computer Sciences Institute, Southern Federal University, Rostov-on-Don, Russia L. Han Laser Institute, Qilu University of Technology (Shandong Academy of Sciences), Jinan, China L. P. Ichkitidze Sechenov First Moscow State Medical University, Moscow, Russia V. A. Isaev Department of Theoretical Physics and Computer Technologies, Kuban State University, Krasnodar, Russia E. M. Kaidashev Laboratory of Nanomaterials, Southern Federal University, Rostovon-Don, Russia Kirill K. Kanishchev Institute for Mathematics, Mechanics and Informatics, Kuban State University, Krasnodar, Russia D. V. Karabanov Don State Technical University, Rostov-on-Don, Russia G. Ya. Karapetyan Laboratory of Nanomaterials, Southern Federal University, Rostov-on-Don, Russia V. E. Kaydashev Laboratory of Nanomaterials, Southern Federal University, Rostovon-Don, Russia Irina A. Khabarova N.V. Melnikov’s Institute of Comprehensive Exploitation of Mineral Resources, Russian Academy of Sciences (ICEMR RAS), Moscow, Russia Artur D. Khanazaryan Institute for Mathematics, Mechanics and Informatics, Kuban State University, Krasnodar, Russia Ivan O. Kharchevnikov Don State Technical University, Rostov-on-Don, Russia S. O. Kireev Don State Technical University, Rostov-on-Don, Russia V. S. Klimin Southern Federal University, Taganrog, Russia I. V. Kolesnikov Rostov State Transport University, Rostov-on-Don, Russia V. I. Kolesnikov Rostov State Transport University, Rostov-on-Don, Russia E. M. Kolosova I.I. Vorovich Institute of Mathematics, Mechanics, and Computer Science, Southern Federal University, Rostov-on-Don, Russia M. G. Konstantinova Southern Federal University, Rostov-on-Don, Russia M. V. Korchagina Don State Technical University, Rostov-on-Don, Russia
Contributors
xxi
Alexandr Kornievsky I.I. Vorovich Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Rostov-on-Don, Russia A. S. Korolkova Research Institute of Physics, Southern Federal University, Rostovon-Don, Russia S. A. Kovrigina Southern Federal University, Rostov-on-Don, Russia Ekaterina Kozhokar Faculty of Chemistry, Southern Federal University, Rostov-onDon, Russia D. A. Kravchuk Southern Federal University, Taganrog, Russia O. V. Kudryakov Don State Technical University, Rostov-on-Don, Russia Yuliya A. Kuprina Research Institute of Physics, Southern Federal University, Rostovon-Don, Russia Vasilina A. Lapitskaya A.V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus, Minsk, Belarus A. R. Lebedev Don State Technical University, Rostov-on-Don, Russia Alla Lebedinskaya Academy of Architecture and Arts, Southern Federal University, Rostov-on-Don, Russia A. M. Lerer Faculty of Physics, Southern Federal University, Rostov-on-Don, Russia M. A. Lugovaya Southern Federal University, Rostov-on-Don, Russia T. G. Lupeiko Department of Chemistry, Southern Federal University, Rostov-on-Don, Russia D. I. Makarev Southern Federal University, Rostov-on-Don, Russia N. V. Makinyan Federal Research Centre, The Southern Scientific Centre of the Russian Academy of Sciences, Rostov-on-Don, Russia S. V. Malokhatko Southern Federal University, Taganrog, Russia A. A. Martynenko Research Institute of Physics, Southern Federal University, Rostovon-Don, Russia Andrey Matrosov Don State Technical University, Rostov-on-Don, Russia Ya. Yu. Matyash Federal Research Centre, The Southern Scientific Centre of the Russian Academy of Sciences, Rostov-on-Don, Russia D. K. Mauer Chemistry Department, Southern Federal University, Rostov-on-Don, Russia E. S. Medvedeva Department of Chemistry, Southern Federal University, Rostov-onDon, Russia V. S. Menshchikov Chemistry Department, Southern Federal University, Rostov-onDon, Russia
xxii
Contributors
I. B. Mikhailova Southern Scientific Center of the Russian Academy of Sciences, Rostov-on-Don, Russia Igor P. Miroshnichenko Don State Technical University, Rostov-on-Don, Russia Ilya A. Moroz Membrane Institute, Kuban State University, Krasnodar, Russia M. O. Moysa Research Institute of Physics, Southern Federal University, Rostov-onDon, Russia Sujoy Mukherjee Discipline of Mechanical Engineering, Design and Manufacturing, PDPM Indian Institute of Information Technology, Jabalpur, India A. V. Nagaenko Institute of High Technology and Piezo Technic, Southern Federal University, Rostov-on-Don, Russia Alexandra Nalimova Don State Technical University, Rostov-on-Don, Russia Tejkaran Narolia Mechanical Engineering Discipline, Design and Manufacturing, PDPM Indian Institute of Information Technology, Jabalpur, India Andrey Nasedkin Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Rostov-on-Don, Russia Mohamed Elsayed Nassar Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt A. V. Nazarenko Federal Research Center the Southern Scientific Center of the Russian Academy of Sciences, Rostov-on-Don, Russia Alexander V. Nazarenko Research Institute of Physics, Southern Federal University, Rostov-on-Don, Russia Andrey L. Nikolaev Don State Technical University, Rostov-on-Don, Russia Alexey Nikulin Faculty of Chemistry, Southern Federal University, Rostov-on-Don, Russia Daria Nizhnik Don State Technical University, Rostov-on-Don, Russia E. S. Novikov Rostov State Transport University, Rostov-on-Don, Russia A. E. Oparina Department of Chemistry, Southern Federal University, Rostov-onDon, Russia D. V. Orda-Zhigulina Federal Research Center Southern Scientific Center of the Russian Academy of Sciences, Rostov-on-Don, Russia O. N. Oskolkova L. M. Litvinenko Institute of Physical-Organic and Coal Chemistry, Donetsk, Russia Ilya Pankov Research Institute of Physical Organic Chemistry, Southern Federal University, Rostov-on-Don, Russia
Contributors
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Yu. Pankova Faculty of Chemistry, Southern Federal University, Rostov-on-Don, Russia Vinay Panwar Mechanical Engineering Department, Netaji Subhas University of Technology, Dwarka, New Delhi, India K. Paperzh Faculty of Chemistry, Southern Federal University, Rostov-on-Don, Russia Ivan A. Parinov I. I. Vorovich Mathematics, Mechanics, and Computer Science Institute, Southern Federal University, Rostov-on-Don, Russia A. A. Pavelko Research Institute of Physics, Southern Federal University, Rostov-onDon, Russia A. S. Pavlenko Research Institute of Physics, Southern Federal University, Rostov-onDon, Russia A. V. Pavlenko Federal Research Centre, The Southern Scientific Centre of the Russian Academy of Sciences, Rostov-on-Don, Russia Angelina Pavlets Faculty of Chemistry, Southern Federal University, Rostov-on-Don, Russia E. I. Petrova Southern Federal University, Rostov-on-Don, Russia E. A. Pilipenko Federal State Budgetary Educational Institution of Higher Education «Lugansk State Pedagogical University», Lugansk, Russia V. V. Polyakov Southern Federal University, Taganrog, Russia Inna Popova Don State Technical University, Rostov-on-Don, Russia Novi Ariyan Pratama Universitas 17 Agustus 1945 Surabaya, Surabaya, East Java, Indonesia Anastasia Purchina Don State Technical University, Rostov-on-Don, Russia Olga Purchina Don State Technical University, Rostov-on-Don, Russia Jaka Purnama Universitas 17 Agustus 1945 Surabaya, Surabaya, East Java, Indonesia Erni Puspanantasari Universitas 17 Agustus 1945 Surabaya, Surabaya, East Java, Indonesia Olga Pustovalova Southern Federal University, Rostov-on-Don, Russia Bonifacius Raditya Sri Pramana Putra Department of Informatics Engineering, Universitas 17 Agustus 1945 Surabaya, Surabaya, Indonesia Erni Puspanantasari Putri Department of Industrial Engineering, Universitas 17 Agustus 1945 Surabaya, Surabaya, Indonesia Santosh Kumar Rai Inochi Care PVT LTD., Okhla phase 1, New Delhi, India E. V. Raksha Joint Institute for Nuclear Research, Dubna, Russia
xxiv
Contributors
Y. A. Reizenkind Research Institute of Physics, Southern Federal University, Rostovon-Don, Russia X. Ren School of Optoelectronic Engineering, Qilu University of Technology (Shandong Academy of Sciences), Jinan, China A. N. Reznichenko Southern Federal University, Rostov-on-Don, Russia L. A. Reznichenko Research Institute of Physics, Southern Federal University, Rostovon-Don, Russia Nurul Rochmah Department of Civil Engineering, Universitas 17 Agustus 1945, Surabaya, Indonesia A. G. Rudskaya Southern Federal University, Rostov-on-Don, Russia Angela Rudskaya Department of Physics, Southern Federal University, Rostov-onDon, Russia A. N. Rybyanets Southern Federal University, Rostov-on-Don, Russia Evgeniy V. Sadyrin Don State Technical University, Rostov-on-Don, Russia D. A. Safitri Department of Civil Engineering, Universitas 17 Agustus 1945 Surabaya, Surabaya, Indonesia Sajiyo Universitas 17 Agustus 1945 Surabaya, Surabaya, East Java, Indonesia M. S. Savelyev Sechenov First Moscow State Medical University, Moscow, Russia F. Saves Department of Civil Engineering, Universitas 17 Agustus 1945 Surabaya, Surabaya, Indonesia M. V. Savoskin L. M. Litvinenko Institute of Physical-Organic and Coal Chemistry, Donetsk, Russia S. V. Selishchev National Research University of Electronic Technology, Zelenograd, Moscow, Russia Irina Serebryanaya Don State Technical University, Rostov-on-Don, Russia S. N. Shevtsov Southern Scientific Center of RAS, Rostov-on-Don, Russia L. A. Shilkina Research Institute of Physics, Southern Federal University, Rostov-onDon, Russia A. V. Shloma Southern Federal University, Rostov-on-Don, Russia Watheq F. Shneen Department of Physics, College of Science, University of Basrah, Basrah, Iraq S. P. Shpanko Southern Federal University, Rostov-on-Don, Russia I. A. Shvetsov Southern Federal University, Rostov-on-Don, Russia N. A. Shvetsova Southern Federal University, Rostov-on-Don, Russia
Contributors
xxv
E. N. Sidorenko Southern Federal University, Rostov-on-Don, Russia N. G. Snezhina Don State Technical University, Rostov-on-Don, Russia A. N. Soloviev Crimean Engineering and Pedagogical University the name of Fevzi Yakubov, Simferopol, Russia Arkadiy Soloviev Crimean Engineering and Pedagogical University the Name of Fevzi Yakubov, Simferopol, Russia Arkady Soloviev Fevzi Yakubov Crimean Engineering and Pedagogical University, Simferopol, Russia I. B. Starchenko Don State Technical University, Rostov-on-Don, Russia D. V. Stryukov Federal Research Centre, The Southern Scientific Centre of the Russian Academy of Sciences, Rostov-on-Don, Russia P. V. Sukhov L. M. Litvinenko Institute of Physical-Organic and Coal Chemistry, Donetsk, Russia Bantot Sutriono Department of Civil Engineering, Universitas 17 Agustus 1945, Surabaya, Indonesia D. V. Telyshev National Research University of Electronic Technology, Zelenograd, Moscow, Russia A. Yu. Tolbin Institute of Physiologically Active Compounds at Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry RAS, Chernogolovka, Russia V. Yu. Topolov Department of Physics, Southern Federal University, Rostov-on-Don, Russia Masca Indra Triana Department of Civil Engineering, Universitas 17 Agustus 1945, Surabaya, Indonesia Retno Trimurtiningrum Department of Civil Engineering, Universitas 17 Agustus 1945, Surabaya, Indonesia A. S. Turchin Southern Federal University, Rostov-on-Don, Russia Prateek Upadhyay Discipline of Mechanical Engineering, Design and Manufacturing, PDPM Indian Institute of Information Technology, Jabalpur, India V. N. Varavka Don State Technical University, Rostov-on-Don, Russia P. N. Vasilevsky National Research University of Electronic Technology, Moscow, Russia I. A. Verbenko Research Institute of Physics, Southern Federal University, Rostov-onDon, Russia D. V. Volkov Research Institute of Physics, Southern Federal University, Rostov-onDon, Russia
xxvi
Contributors
Sergey S. Volkov Don State Technical University, Rostov-on-Don, Russia G. K. Volkova A. A. Galkin Donetsk Institute for Physics and Engineering, Donetsk, Russia M. G. Volkova Chemistry Department, Southern Federal University, Rostov-on-Don, Russia A. I. Voropaev Rostov State Transport University, Rostov-on-Don, Russia S. Wang Laser Institute, Qilu University of Technology (Shandong Academy of Sciences), Jinan, China Z. Wang Laser Institute, Qilu University of Technology (Shandong Academy of Sciences), Jinan, China H. Widhiarto Department of Civil Engineering, Universitas 17 Agustus 1945 Surabaya, Surabaya, Indonesia V. E. Yakovlev I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal University, Rostov-on-Don, Russia A. V. Yudin I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal University, Rostov-on-Don, Russia W. Zhang School of Optoelectronic Engineering, Qilu University of Technology (Shandong Academy of Sciences), Jinan, China K. M. Zhidel Research Institute of Physics, Southern Federal University, Rostov-onDon, Russia Sergei V. Zubkov Research Institute of Physics, Southern Federal University, Rostovon-Don, Russia
Processing Techniques of Advanced Materials
Control Over Morphological Characteristics of the Pt/C Catalysts Obtained by the Liquid-Phase Synthesis Yu. Bayan1 , K. Paperzh1(B) , M. Danilenko1 , D. Alekseenko1 , Yu. Pankova1 , I. Pankov2 , and A. Alekseenko1 1 Faculty of Chemistry, Southern Federal University, 7 Zorge St., Rostov-on-Don 344090,
Russia [email protected] 2 Southern Federal University, Research Institute of Physical Organic Chemistry, 194/2 Stachki St., Rostov-on-Don 344090, Russia
Abstract. One of the key components of low-temperature fuel cells with a protonexchange membrane is an electrocatalyst contained in the porous electrodes. The most widespread synthesis methods are the liquid-phase ones that allow controlling the microstructure of the catalyst and, thus, its functional parameters. We have investigated the influence of various conditions of the liquid-phase synthesis on the morphological and electrochemical characteristics of the resulting platinum– carbon catalysts. An increase in the synthesis temperature has been established to allow for the narrowing of the size dispersion and the decrease in the average size of platinum nanoparticles. It has been found that the presence of a carbon support during the synthesis makes it possible to enhance the uniformity of the Pt NPs’ distribution over the surface of the support and decreasing the average particle size. As a result, the values of the active surface area grow almost 1.2–2 times compared to the homogeneous synthesis, during which the resulting colloid of platinum particles is deposited on the carbon support after the reduction. When using the molar ratio of hydroxyl groups/platinum in the range from 5 to 20 during the synthesis, the resulting Pt/C catalysts are characterized by an active surface area of more than 85 m2 ·g−1 Pt . The possibility of scaling the synthesis method to obtain at least 1 g of the catalyst, which is not inferior in functional parameters to its commercial analog both before stress testing and after its completion, is also shown. Keywords: Pt-based Electrocatalysts · Catalyst Morphology · Particle Size · Platinum Nanoparticles · Electrochemically Active Surface Area
1 Introduction The development and implementation of low-temperature fuel cells with a protonexchange membrane (PEMFCs), which are deemed as highly efficient devices that convert chemical energy into electrical one, are among the current trends in the field © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 3–15, 2024. https://doi.org/10.1007/978-3-031-52239-0_1
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of alternative energy [1–3]. A wide scope for use of the PEMFC-based devices creates the demand for high-quality components that ensure high performance, fast start-up, and long service life. The key component of PEMFCs is porous catalytic layers consisting of a catalyst, with certain cell reactions proceeding therein that produce electricity and water from fuels – hydrogen and oxygen [4]. The higher the functional characteristics of the electrocatalyst, the greater the power and the service life of PEMFCs [3–5]. The functional characteristics should be referred to the electrochemically active surface area (ESA), activity in the oxygen reduction reaction (ORR) and resistance to the degradation during the operation of the catalyst. To date, the materials based on platinum nanoparticles (NPs), distributed over the surface of the carbon support, are a widespread type of electrocatalysts for the cathode and anode of PEMFCs [6]. To achieve higher functional characteristics of the electrode materials, it is necessary to control the microstructure of electrocatalysts, that is the average size of platinum NPs and the features of their size and spatial distribution over the surface of the support [7]. In general, the enhancement of the catalyst’s microstructure leading to an increase in the ESA and the ORR activity of the catalyst consists in decreasing the size of NPs and narrowing their size dispersion [8]. The most common synthesis methods are the liquid-phase ones that allow controlling the microstructure of the catalyst and, therefore, its functional parameters [9–14]. During the synthesis of the Pt/C catalysts, the methods based on the chemical reduction of Pt (IV) in liquid media are most often applied using various reducing agents, including sodium borohydride [11], formaldehyde [15, 16], formic acid [17], ethylene glycol (EG) [17] and oleic acid [18]. The choice of a reducing agent is usually coupled with the appropriate temperature conditions, pH value, solvent composition, use of stabilizing agents and duration of the synthesis stages. An increase in the temperature of the reaction medium during the liquid-phase synthesis of platinum particles is known to be conducive to the acceleration of the formation of a new phase and its growth [14]. When the growth rate prevails over the rate of nucleation, the average particle size increases. When the nucleation rate is higher than the growth rate, the average size of NPs decreases. These rates are different for various reaction media during the synthesis of platinum NPs. This feature makes it possible to obtain platinum particles of a given size. Therefore, it is almost impossible to foresee a decrease or increase in the particle size when the temperature mode changes. Thus, for different systems, it is necessary to study the influence of temperature conditions on the particle size. Moreover, the size of NPs is affected by surfactants, which are most often used to decrease the particles’ size and prevent their coalescence [19]. Being adsorbed on the surface of the growing NP, these surfactants block the surface for platinum ions, significantly inhibiting their further growth. Therefore, the difference between the rates of growth and nucleation is eliminated, and the growth rate becomes less than the rate of nucleation. The previous studies [9, 19–21] show that in the reaction medium, in the presence of ethylene glycol and formaldehyde, these components decompose, and CO is released. The CO molecules, being firmly adsorbed on the surface of growing platinum NPs, may inhibit the growth of particles, which leads to a decrease in their average size.
Control Over Morphological Characteristics of the Pt/C
5
The compliance with an appropriate pH mode or OH:Pt molar ratio during the synthesis is most often deemed as a necessity for the reaction to proceed using reducing agents [22]. It is also noted in a series of publications that the hydroxyl groups contained in the reaction mixture are capable of acting as surfactants, preventing the further growth of NPs [22]. With an increase in temperature of the reaction medium, the rate of nucleation should also increase, and due to the action of such molecules as CO and OH groups, the growth rate should decrease. In conjunction, the influence of these factors may lead to a decrease in the average size of platinum NPs. Another factor controlling the morphology of the Pt/C catalyst during its synthesis may be the presence/absence of a carbon support during the process of NPs’ nucleation/growth. It is known that in the presence of a heterogeneous phase in the reaction system, the nucleation process may proceed both in the volume of a solution [23], at the support–solvent interphase, and in the active centers of carbon black [23, 24]. At the same time, the rapid sorption of the formed platinum particles by carbon should lead to a decrease in the particle aggregation and a corresponding increase in the ESA values. Despite many publications, concerned with the use of a particular method of the chemical reduction to obtain Pt/C materials, an unambiguous choice of the optimal technique has not yet been made. Conducting a systematic study of the influence of various parameters of the Pt/C liquid-phase synthesis on the morphological characteristics of the materials obtained is an essential fundamental direction in the field of production of nanostructured materials. The choice of the liquid-phase synthesis method using various reducing agents as the principal method to obtain Pt/C materials is associated with the possibility of a wide variation of synthesis conditions to study the effect on the catalysts’ characteristics. The applied significance of studying the liquid-phase synthesis methods is associated with the prospects of their scaling for use both in the laboratory and on a production scale. Therefore, the objective of this study is to search for optimal conditions for the synthesis of the Pt/C electrocatalysts with increased functional characteristics. An important aspect of this study is to reveal the dependence of the size of platinum particles and their aggregation on the temperature mode, the OH:Pt molar ratio and the presence/absence of a carbon support during the liquid-phase synthesis.
2 Experimental Section 2.1 Preparation of the Pt/C Catalysts 2.1.1 Preparation of the Platinum–Carbon Catalysts in Homogeneous and Heterogeneous Media at Different Temperatures The NaOH solution with the NaOH:Pt molar ratio = 10 was added to ethylene glycol (99.8%, Vekton, Russia). Next, HCOH (37%, Khimikon, Russia) was added to the reaction mixture, after which stirring and heating were started. By reaching the set synthesis temperature, the H2 PtCl6 ·6H2 O (TU 2612-034-00205, 067-2003, mass fraction of Pt 37.6%, Aurat, Russia) solution was introduced into the system, and with constant stirring, the system was kept at a fixed temperature for 2 h. At the reaction completion, the carbon suspension (Vulcan XC-72, Cabot Corporation) in ethylene glycol was added for
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the formed Pt NPs to be deposited on carbon as a result of the synthesis. The separation of the Pt/C materials was carried out by filtering the mixture with repeated rinsing with isopropyl alcohol and water. The materials were dried in a desiccator over P2 O5 for 24 h at room temperature. The resulting Pt/C materials are hereinafter referred to as P60, P70, P80 and P90, where the specified number corresponds to the synthesis temperature in Celsius degrees. Additionally, the Pt/C synthesis was carried out using formic acid as a reducing agent. Ethylene glycol and HCOOH solution (85%, Vekton, Russia) were placed in a roundbottomed flask, after which heating and stirring were started. By reaching 80 °C, the H2 PtCl6 ·6H2 O solution was introduced into the system. After the complete introduction of the components, the reactor was kept at a fixed temperature for 1 h. Similarly, to the process described above, the carbon suspension was prepared, which was added to the colloidal solution of Pt NPs after the set period. The separation of the product was carried out by filtration with repeated rinsing with isopropyl alcohol and water. The materials were also dried in a desiccator over P2 O5 . The resulting Pt/C sample is hereinafter referred to as PA80. To study the effect of the presence/absence of a carbon support during the synthesis on the average particle size, the Pt/C synthesis was carried out under similar conditions described above at a temperature of 80 °C, but with the introduction of a carbon support before heating. The materials obtained are hereinafter referred to as P80C and PA80C. 2.1.2 Preparation of the Platinum–Carbon Catalysts by the Formaldehyde Method in the Presence of Various Amounts of Alkali The synthesis of Pt/C was carried out similarly to that described above with the introduction of a carbon support before heating. The alkali solution were added in such a way that the OH:Pt molar ratio should amount to 0, 2, 5, 10, 15 and 20. The obtained Pt/C samples are hereinafter referred to as Pt/C-0, Pt/C-2, Pt/C-5, Pt/C-10, Pt/C-15 and Pt/C-20, where the specified number indicates the OH:Pt molar ratio used during the synthesis. 2.2 Study of the Composition and the Structural and Morphological Characteristics of the Pt/C Catalysts The mass fraction of Pt (ω(Pt)) in Pt/C was determined by the gravimetry method related to the Pt mass residue left after burning Pt/C. The error margin was ±3%. The X-ray powder diffraction (XRD) method was used to study the structural characteristics of the Pt/C materials obtained as described in [14]. The X-ray diffraction patterns for the samples studied were recorded in the range of angles of 20° ≤ 2θ ≤ 55° by step-by-step scanning with a detector movement step of 0.02°. The calculation of the average crystallite size was carried out by the Scherrer formula, as described in [25, 26]. The error margin in calculating DXRD by the Scherrer equation was ±8%. The size of Pt NPs as well as the features of their size and spatial distributions were studied by transmission electron microscopy (TEM). The TEM micrographs were obtained as described in [14]. The histograms of the platinum NPs’ size distribution in the catalysts were plotted, based on the results of the size determination for at least 400
Control Over Morphological Characteristics of the Pt/C
7
particles, randomly selected in the TEM micrographs at different sections of the sample. The error margin was ±0.2 nm. 2.3 Study of the Electrochemical Behavior of the Catalysts The preparation of the catalytic layer at the end face of the rotating disk electrode (RDE) was carried out by applying the suspension of the Pt/C catalysts (catalytic “inks”) in the amount of 8 μL of the inks per electrode [14]. The functional characteristics of the catalysts were studied in a three-electrode cell by voltammetry at a temperature of 23 °C using the VersaSTAT3 potentiostat. A platinum wire was used as the auxiliary electrode, the reference electrode being a saturated silver chloride electrode. The 0.1 M perchloric acid solution (60%, Sigma-Aldrich) was used as the electrolyte. All the potentials used in this study were considered with regard to the reversible hydrogen electrode (RHE). Before measuring the ESA values and the ORR activity, the working electrode with a deposited catalytic layer of the studied Pt/C catalyst was subjected to the electrochemical activation as described in [27]. To assess the values of the ESA, two cyclic voltammograms (CVs) were recorded in the potential range of 0.04–1.2 V or 0.04–1.0 V with a scanning rate of 20 mV·s−1 . The ESA was calculated by the charge amount consumed for the hydrogen adsorption/desorption [28]. The error margin was 10%. The aggregation degree (DA, %) was calculated as the ratio of the geometrically calculated ESA [29] by the average size of Pt NPs to the measured ESA. To determine the ORR activity, a series of voltammograms were measured [28]. The ORR catalysts’ behavior was assessed by the values of mass (Imass ) activities. The error margin was 10%. To assess the degradation degree (DD, %) of the electrocatalysts, the accelerated stress testing (AST) method was used, that is the voltametric cycling in the potential range of 0.6–1.0 V (RHE) during 20000 cycles [24, 30]. Every 4000 cycles, two CVs were recorded at a potential sweep rate of 20 mV·s−1 in the potential range from 0.04 to 1.2 V (RHE) to assess the residual ESA. The ESA calculation for the second CV was performed as described above. The degradation degree of the catalysts was calculated by a change in the ESA values after stress testing (ESA after AST).
3 Results and Discussion 3.1 Effect of the Temperature Mode on the Average Size of Pt NPs and the ESA We have synthesized platinum NPs by the chemical reduction in the liquid phase at different temperatures from 60 to 90 °C. According to the results of the gravimetric analysis, the ω(Pt) in the synthesized samples is 18 ± 0.5%. The X-ray diffraction analysis was used to obtain the X-ray diffraction patterns of the Pt/C (Fig. 1a). The broadening of the peaks, corresponding to the reflections of the platinum facets 111 and 200, is associated with a smaller crystallite size of no more than 1.5 nm [31, 32]. The average size of platinum crystallites was calculated, which slightly decreases in the order: P60 > P70 = P80 > P90 (1.5, 1.4, 1.4 and 1.3 nm). The average size of NPs and their size distribution have been estimated by the TEM micrographs of the synthesized Pt/C materials (Fig. 1b, c, d, e). With an increase in
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Fig. 1. (a) X-ray diffraction patterns of the Pt/C materials; (b, c, d, e) TEM micrographs of the obtained Pt/C samples and histograms of the NPs’ size distribution in the corresponding materials P60 (a, b) and P80 (c, d).
temperature of the reaction medium by 20 °C, the average size of NPs slightly decreases, with their size dispersion narrowing (Fig. 1, histograms of the NPs’ size distribution). Another descriptor characterizing the average size of Pt NPs and the NPs’ agglomeration degree is the ESA. It is known that the smaller the size of NPs, the higher the value of the ESA [22, 31]. At the same time, the presence of agglomerates, even with a small average size of individual NPs, may reduce the value of the ESA [2, 31]. Figure 2a shows cyclic voltammograms of the obtained platinum–carbon samples. The values of the currents in the hydrogen region (0.04–0.35 V (RHE)) and the ESA grow in the order: P60 < P70 < P80 < P90, which correlates well with the change in the average size of platinum crystallites (Fig. 2).
Fig. 2. (a) Cyclic voltammograms of the synthesized Pt/C samples. The potential sweep rate is 20 mV·s−1 . The electrolyte is the 0.1 M HClO4 solution saturated with Ar. (b) Histograms of the ESA values calculated from cyclic voltammograms.
As a result of the study carried out, it can be concluded about the influence of temperature on the morphological and electrochemical characteristics of the catalysts, obtained by the liquid-phase formaldehyde method with the homogeneous nucleation of NPs: with an increase in temperature of the reaction system from 60 to 90 °C during the synthesis, the average size of crystallites and the average size of platinum NPs slightly decrease by about 0.1–0.2 nm, and the ESA values grow by 10–12 m2 ·g−1 Pt . The detected
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effect of temperature during the liquid-phase synthesis with the homogeneous nucleation makes it possible to choose the optimal temperature, which ranges from 80 to 90 °C, to obtain the electrocatalysts with the increased ESA values. 3.2 Effect of the Presence of a Carbon Support During the Pt/C Synthesis on the Average Size of Pt NPs and the ESA A comparative study of the structure of the P80, PA80 and P80C, PA80C Pt/C samples, obtained by introducing the carbon suspension into the system after and before the synthesis, respectively, has been carried out. The mass fraction in the obtained samples ranges from 17 to 20% (Table 1). Table 1. Structural-morphological and electrochemical characteristics of the Pt/C. Sample
DXRD (nm)
ωPt (% wt.)
ESA (m2 ·g−1 Pt )
Sgeom (m2 ·g−1 Pt )
DA (%)
P80
1.4
17.0
89
122
30
P80C
1.4
19.6
106
117
10
PA80
3.1
20.0
27
76
64
PA80C
2.7
18.5
59
82
30
For the materials, obtained under conditions of the formic acid synthesis (PA80 and PA80C), the most intense reflection peak of the 111-platinum facet is slightly narrower (Fig. 3a) than for the P80 and P80C materials, obtained by the formaldehyde method, which indicates a larger particle size.
Fig. 3. (a) X-ray diffraction patterns of the Pt/C materials. TEM micrographs of the obtained samples and histograms of the NPs’ size distribution in the corresponding materials P80C (a), PA80 (b), PA80C (c) and JM20 (d).
During the synthesis in formic acid (PA samples series), the size of platinum crystallites decreases, and during the formaldehyde synthesis (P series), it practically does not change with the introduction of the carbon support before the reduction (Table 1). The larger particle size in the PA80 and PA80C samples obtained under conditions of the formic acid synthesis, compared to P80 and P80C, is also confirmed by the TEM
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method. The size dispersion is almost 1.5 times wider, and the average particle size is 1–1.4 nm larger than those of the formaldehyde-synthesized materials (Fig. 3). The materials obtained by the formaldehyde synthesis method are only characterized by a smaller average size of platinum NPs, compared to the commercial analog JM20 (Figs. 1 and 3). It is worth noting that the aggregation degree of particles (Table 1), calculated from the ratio of the values of the determined ESA and the geometric surface area of Pt [29, 31], decreases with the synthesis being carried out in the carbon suspension (Table 1). The absence of any changes in the size of the formed NPs indicates that there is no significant contribution of the heterogeneous nucleation in the systems, containing formaldehyde. The presence of the carbon support in the initial reaction medium contributes to the rapid sorption of the formed platinum NPs, which prevents the aggregation of the particles formed. In the PA80C sample, obtained under conditions of the formic acid synthesis in the carbon suspension, the size of Pt NPs is smaller than that of PA80 (Table 1). This may be a reflection of the contribution of the heterogeneous nucleation to the process of the phase formation of Pt NPs. Under conditions of the heterogeneous nucleation, the activation energy for the nucleation of a new phase is known to decrease [33]. This effect is due to a decrease in the energy, required for the formation of new interphase surfaces during the nucleation at the already existing interphase boundary. Additionally, we have conducted a study of the uniformity of the distribution of platinum NPs over the support surface [25, 29] using the TEM micrographs. For the samples obtained with the introduction of the carbon support into the reaction mixture before the reduction, a higher proportion of individual NPs is observed compared to the samples obtained by the subsequent deposition of NPs on the colloid substrate (Fig. 4).
Fig. 4. Histograms of the NPs’ distribution by the number of intersections with their neighbors. The proportion of NPs for the P80, P80C, PA80, PA80C and JM20 samples.
The P80C material is characterized by the uniformity of the NPs’ distribution over the support surface close to the commercial analog JM20. At the same time, in both the cases, the synthesis in the carbon suspension is conducive to obtaining the samples, characterized by the higher ESA values, compared to the analogs, synthesized with the introduction of the carbon support into the reaction
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medium after the reaction completion (Table 1, Fig. 5). The reduction of Pt(IV) directly in the carbon suspension, prepared before heating of the reaction mixture contributes to the narrowing of the size dispersion and an increase in the uniformity of the spatial distribution of Pt NPs (Figs. 1 and 3).
Fig. 5. Cyclic voltammograms of the synthesized Pt/C samples PA80 and PA80C (a), P80 and P80C (b).
Thus, carrying out the synthesis in the carbon suspension leads to an increase in the ESA of the Pt/C materials. In the event of the formaldehyde synthesis, this is due to the rapid sorption of Pt NPs, which prevents their aggregation in the solution, and in the case of the formic acid synthesis, this may also be due to the contribution of the heterogeneous nucleation to the Pt phase formation process. It is noteworthy that the formaldehyde and formic acid methods for synthesizing the electrocatalysts have a lot of differences, for example the acidity of the medium at which the reduction process proceeds. 3.3 Effect of the Alkali Content in the Reaction Medium (OH:Pt Ratio) on the Average Size of Pt NPs and the ESA To assess a possible effect of the amount of alkali during the formaldehyde synthesis on the size of the Pt NPs formed, a systematic study has been conducted. The OH:Pt ratio has been maintained from 0 to 20 in the reaction mixture before the heating and the beginning of the reduction process. Based on the results of the evaluation of the Pt mass fraction in the obtained catalysts, it can be concluded that the Pt content decreases with an increase in the amount of alkali (from 0 to 20) in the reaction mixture from 20.4 to 17.1%. The broadening of the reflection peaks of the 111 and 200 platinum facets due to the smaller size of crystallites is most pronounced, and DXRD , respectively, is minimal for the materials obtained with the OH:Pt ratio = 5, 10, 15 and 20 (about 1.5 nm) (Fig. 6a). The absence (OH:Pt = 0) and presence of a small amount (OH:Pt = 2) of OH groups in the reaction medium lead to the formation of the catalysts with a larger average size of platinum crystallites (more than 3 nm). When all the Pt NPs in Pt/C have the same shape and are in direct contact with the electrolyte, the ESA of platinum only depends on the size of NPs and the degree of their aggregation. The analysis of the measured CVs (Fig. 6b) and the calculation
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Fig. 6. (a) X-ray diffraction patterns of the Pt/C materials; (b) cyclic voltammograms of the synthesized Pt/C samples with different OH:Pt molar ratios; (c) histograms of the ESA values, calculated from cyclic voltammograms.
of the charge amount, consumed for the electrochemical adsorption and desorption of hydrogen (Fig. 6b), have shown that the ESA values in general correlate well with the DXRD . The catalysts with a platinum crystallite size of 1.5–1.3 nm exhibit the highest values of the ESA (Fig. 6c). The increase in the ESA values from 39 to 84–92 m2 g−1 Pt is due to a decrease in the DXRD by almost two times from 4.3 to 1.5–1.3 nm. It should be noted that the increase in the OH:Pt ratio up to 20 leads to a slight decrease in the ESA value and at the same time to a significant decrease in the mass fraction of Pt – by more than 4% of the theoretically calculated one (21%). The presence of hydroxyl groups in the reaction medium during the formaldehyde synthesis for the OH:Pt ratio = 5 and more has a crucial effect on the ESA value. According to the results of studying the effect of the number of hydroxyl groups during the formaldehyde synthesis on the characteristics of the resulting electrocatalysts, it can be concluded that the optimal OH:Pt ratio should be at least 5 and no more than 15. 3.4 Comparison of the Obtained Catalyst with Its Commercial Analog Taking into account the above requirements for the formaldehyde synthesis, we have obtained the Pt/C home-made catalyst in the amount of 1 g at a temperature of 80 °C in the reaction system, with the introduction of the carbon support before the phase formation and the OH:Pt molar ratio = 10. The platinum mass fraction in the resulting Pt/C home-made was 20%. The commercial Pt/C JM20 with a platinum loading of 20% was used as comparative catalyst. The ESA values of the commercial catalyst JM20 were by 23 m2 ·g−1 Pt less than those of the Pt/C home-made material (Fig. 7a, Table 2). According to the potentiodynamic curves of the ORR at different rotation speeds, the Koutetsky–Levich dependence has been plotted (Fig. 7b, c), using which the electrokinetic parameters of the reaction have been estimated. The mass activity in the ORR of the Pt/C home-made material has proved to be by 13 A·g−1 Pt higher than that of the commercial analog. According to the results of the AST, we may point out the ongoing degradation of both the home-made material and its commercial analog. The decrease in the ESA values occurs almost linearly for both the samples (Fig. 7d). The degradation degree of the studied catalysts has turned out to be close, however, the Pt/C home-made sample exhibits the higher initial and residual values of the ESA at the end of stress testing, compared to the JM20 (Table 2).
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Table 2. Morphological and electrochemical characteristics of the catalysts. Sample
ESA, m2 ·g−1 Pt
Imass , A·g−1 Pt
ESAafter AST , m2 ·g−1 Pt
DD, %
Home-made Pt/C
105
270
64
39
JM20
82
257
49
40
Fig. 7. (a) Cyclic voltammograms of the Pt/C samples; (b) potentiodynamic polarization curves of the ORR, RDE rotation speed is 1600 rpm. O2 atmosphere; (c) dependence of the current strength on the rotation speed of the RDE in the Koutetsky–Levich coordinates; (d) change in the ESA during the AST in the mode.
4 Conclusions (I) An increase in temperature during the formaldehyde synthesis in an alkaline medium makes it possible to reduce the size of NPs during the homogeneous nucleation. (II) The presence of the carbon support during the synthesis allows increasing the contribution of the heterogeneous nucleation of platinum NPs in the event of the formic acid method and accelerating the sorption of particles in the case of the formaldehyde method, thus enhancing the uniformity of the NPs’ distribution over the surface of the support, decreasing the average particle size and reducing the degree of aggregation. (III) Varying the OH:Pt molar ratios makes it possible to control the size of the platinum NPs obtained and, thus, the ESA values. (IV) The formaldehyde method of synthesizing the NPs under heterogeneous nucleation conditions (when the support is introduced into the reaction system before the temperature rises and the reduction begins) at 80 °C and the OH:Pt molar ratio = 10 allows obtaining at least 1 g of the catalyst per one production cycle. The functional parameters (both initial and residual after stress testing) of the electrocatalyst, obtained by the scaled Pt/C synthesis method, are comparable and even higher than those of the commercial analog. To obtain the materials by the formaldehyde synthesis method with a platinum crystallite size of less than 1.5 nm, an average NPs’ size of less than 2.5 nm, their narrow size dispersion (from 1 to 4 nm) and low aggregation, as well as the ESA values of more than 85 m2 ·g−1 Pt , it is necessary to use a temperature mode of at least 80 °C, the OH:Pt molar ratio of not less than 5 and not more than 20 and the carbon support introduced into the reaction medium before the phase formation and the heating.
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Acknowledgement. The authors are grateful to Mr. A.V. Maltsev for the support in translation and editing processes. The authors are also grateful to the Shared Use Center “High-Resolution Transmission Electron Microscopy” (SFedU) for conducting the TEM studies.
Funding. This research was financially supported by the Ministry of Science and Higher Education of the Russian Federation (State Assignment in the Field of Scientific Activity No. FENW-2023-0016).
References 1. O’Hayre, R.P., et al.: Fuel Cell Fundamentals, 2nd edn. Wiley, New York (2009) 2. Yaroslavtsev, A.B., et al.: Nanostructured materials for low-temperature fuel cells. Russ. Chem. Rev. 81(3), 191–220 (2012) 3. Sui, S., et al.: A comprehensive review of Pt electrocatalysts for the oxygen reduction reaction: nanostructure, activity, mechanism and carbon support in PEM fuel cells. J. Mater. Chem. A 5(5), 1808–1825 (2017) 4. Thompsett, D.: Catalysts for the proton exchange membrane fuel cell. In: Hoogers, G. (ed.) Fuel Cell Technology Handbook. Boca Raton (2003) 5. Zhang, J., et al.: Preparation and characterization of Pt/C catalysts for PEMFC cathode: effect of different reduction methods. React. Kinet. Catal. Lett. 83(2), 229–236 (2004) 6. Holton, O.T., et al.: The role of platinum in proton exchange membrane fuel cells. Platin. Met. Rev. 57(4), 259–271 (2013) 7. Leontyev, I.N., et al.: Catalytic activity of carbon-supported Pt nanoelectrocatalysts. Why reducing the size of Pt nanoparticles is not always beneficial. J. Phys. Chem. 115(13), 5429– 5434 (2011) 8. Mayrhofer, K.J.J., et al.: The impact of geometric and surface electronic properties of Ptcatalysts on the particle size effect in electrocatalysis. J. Phys. Chem. B 109(30), 14433–14440 (2005) 9. Schröder, J., et al.: Anion dependent particle size control of platinum nanoparticles synthesized in ethylene glycol. Nanomaterials 11(8), 2092 (2021) 10. Kim, H., et al.: Preparation of platinum-based electrode catalysts for low temperature fuel cell. Catal. Today 87(1–4), 237–245 (2003) 11. Quinson, J., et al.: From platinum atoms in molecules to colloidal nanoparticles: a review on reduction, nucleation and growth mechanisms. Adv. Coll. Interface. Sci. 268, 102300 (2020) 12. Danilenko, M.V., et al.: The effect of a gas atmosphere on the formation of colloidal platinum nanoparticles in liquid phase synthesis. Colloid Polym. Sci. 301(5), 433–443 (2023) 13. Danilenko, M.V., et al.: CO effect on the dynamics of platinum nucleation/growth under the liquid-phase synthesis of Pt/C electrocatalysts. J. Electrochem. Soc. 169(9), 092501 (2022) 14. Paperzh, K.O., et al.: UV radiation effect on the microstructure and performance of electrocatalysts based on small Pt nanoparticles synthesized in the liquid phase. Colloids Interface Sci. Commun. 45, 100517 (2021) 15. Guterman, V.E., et al.: Synthesis of nanostructured Pt/C electrocatalysts and effects of ambient atmosphere composition and an intermediate support on their microstructure. Inorg. Mater. 52(1), 23–28 (2015) 16. Alekseenko, A.A., et al.: Application of CO atmosphere in the liquid phase synthesis as a universal way to control the microstructure and electrochemical performance of Pt/C electrocatalysts. Appl. Catal. B 226, 608–615 (2018) 17. Rodrigues, T.S., et al.: Synthesis of colloidal metal nanocrystals: a comprehensive review on the reductants. Chem.–Eur. J. 24(64), 16944–16963 (2018)
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18. Arán-Ais, R.M., et al.: Electrochemical characterization of clean shape-controlled Pt nanoparticles prepared in presence of oleylamine/oleic acid. Electroanalysis 27(4), 945–956 (2015) 19. Mathiesen, J.K., et al.: Insights from in situ studies on the early stages of platinum nanoparticle formation. J. Phys. Chem. Lett. 12, 3224–3231 (2021) 20. Danilenko, M.V., et al.: Nucleation/growth of the platinum nanoparticles under the liquid phase synthesis. Colloids Surf. A 630, 127525 (2021) 21. Quinson, J., et al.: Solvent-dependent growth and stabilization mechanisms of surfactant-free colloidal Pt nanoparticles. Chem.– Eur. J. 26(41), 9012–9023 (2020) 22. Quinson, J., et al.: Investigating particle size effects in catalysis by applying a size controlled and surfactant-free synthesis of colloidal nanoparticles in alkaline ethylene glycol – the case study of the oxygen reduction reaction on Pt. ACS Catal. 8(7), 6627–6635 (2018) 23. Thanh, N.T.K., et al.: Mechanisms of nucleation and growth of nanoparticles in solution. Chem. Rev. 114(15), 7610–7630 (2014) 24. Paperzh, K.O., et al.: Advanced methods of controlling the morphology, activity, and durability of Pt/C electrocatalysts. ACS Appl. Energy Mater. 5(8), 9530–9541 (2022) 25. Langford, J.I., et al.: Scherrer after sixty years: a survey and some new results in the determination of crystallite size. J. Appl. Crystall. 11(2), 102–113 (1977) 26. Vega, L., et al.: Nanostructuring determines poisoning: tailoring CO adsorption on PtCu bimetallic nanoparticles. Mater. Adv. 3, 4159–4169 (2022) 27. Paperzh, K.O., et al.: The integrated study of the morphology and the electrochemical behavior of Pt-based ORR electrocatalysts during the stress testing. Int. J. Hydrogen Energy 48(49) (2023) 28. Paperzh, K.O., et al.: Stability and activity of platinum nanoparticles in the oxygen electroreduction reaction: is size or uniformity of primary importance? Beilstein J. Nanotechnol. 12, 593–606 (2021) 29. Guterman, V.E., et al.: Microstructure and size effects in Pt/C and Pt3 Ni/C electrocatalysts synthesized in solutions based on binary organic solvents. Appl. Catal. A 453, 113–120 (2013) 30. Rotating Disk-Electrode Aqueous Electrolyte Accelerated Stress Tests for PGM. https:// www.energy.gov/sites/prod/files/2015/08/f25/fcto_dwg_pgm_electrocatalyst_support_aque ous_ast.pdf 31. Alekseenko, A.A., et al.: Effect of wet synthesis conditions on the microstructure and active surface area of Pt/C catalysts. Inorg. Mater. 51(12), 1258–1263 (2015) 32. Ritubarna, B., et al.: Detection of ambient oxidation of ultrasmall supported platinum nanoparticles with benchtop powder X-ray diffraction. Catal. Lett.. Lett. 147, 1754–1764 (2017) 33. Laszlo, G., et al.: Heterogeneous nucleation of/on nanoparticles: a density functional study using the phase-field crystal model. Chem. Soc. Rev. 43, 2159 (2014)
Study of Various Approaches to the Synthesis of PtCo/C Electrocatalysts for Fuel Cells E. R. Beskopylny, S. V. Belenov(B) , D. K. Mauer, and V. S. Menshchikov Chemistry Department, Southern Federal University, Rostov-on-Don, Russia [email protected]
Abstract. A comparative analysis of different methods for the bimetallic PtCo/C fuel cells electrocatalysts synthesis was carried out. The structure and catalytic activity of materials obtained by borohydride synthesis method, synthesis on a cobalt-containing carbon substrate, polyol synthesis method and modified polyol synthesis method were studied using a complex of modern methods. Some of the obtained PtCo/C materials turned out to be comparable in activity to commercial analogues. Based on the results of a synthesis methods, comparative analysis used, the synthesis method using a composite cobalt-containing carbon support can be recommended as the most promising for further optimization. Keywords: Fuel Cells · PtCo/C · Electrocatalyst
1 Introduction One of the most serious problems of the 21st century is the problem of environmental pollution. Transport, powered by fossil fuels, contributes a significant share to environmental pollution [1]. One of the promising types of power sources is hydrogen-air fuel cells with a proton exchange membrane (PEMFC). An important task today is to increase the catalyst activity and stability, as well as to reduce the platinum content in it while maintaining high functional characteristics to reduce the PEMFC cost. Increasing the activity of catalysts, as well as reducing their cost, can be achieved in several ways, one of which is alloying platinum with transition d-metals, such as Cu, Co, Ni, Fe, etc. [2, 3]. There are several reasons for the increased activity of platinum alloys: structural effect [4], electronic effect [5], leaching of the alloying component and reducing the platinum contained in nanoparticles. According to [8], among the alloys with the compositions Pt3 Ti, Pt3 V, Pt3 Fe, Pt3 Co, Pt3 Ni, as well as pure Pt, the most catalytically active is the Pt3 Co alloy, the activity of which exceeds the activity of pure platinum by approximately 2–3 times. This is explained by the optimal adsorption energy of oxygen-containing particles for this alloy composition. The high catalytic activity of PtCo/C materials has been experimentally confirmed in several studies. Thus, the activity of PtCo/C exceeded that of Pt/C by 4.6 times in [9], and by 8.7 times in [10]. In addition, it is important to note that commercially produced PEMFC vehicles use PtCo/C catalysts [11]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 16–24, 2024. https://doi.org/10.1007/978-3-031-52239-0_2
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To obtain bimetallic catalysts on a carbon support, there are many different synthesis methods [2]. One of the simplest methods is the borohydride synthesis method, where a solution of sodium borohydride serves as a reducing agent for metal precursors. Due to its high reducing ability, this method is suitable for the synthesis of a wide range of mono- and polymetallic nanoparticles (including Pt, Co, Ni, Cu, Fe) [12–14]. The disadvantages of this method include the too high reaction rate and the difficulty of controlling the shape of nanoparticles [15]. Another common synthesis method is the impregnation-annealing method. To carry it out, the carrier is dispersed in a metal precursors solution, then the mixture is dried and heat-treated in a tubular furnace in a reducing atmosphere, resulting in the reduction of metal precursors with the bimetallic nanoparticles formation [9, 10]. Also, the “impregnation-annealing” method is suitable for preparing bimetallic catalysts from Pt/C catalysts [16, 17]. Under high-temperature annealing conditions, nanoparticles tend to agglomerate, which leads to the larger particle’s formation with a wide size distribution and causes the disadvantage of this method, namely the difficulty of obtaining catalysts with high performance characteristics [18, 19]. Another widely known method for the metal nanoparticles synthesis is the polyol synthesis method. Reduction is carried out in polyhydric alcohols (polyols), and alcohols are both reducing agents and stabilizers for the growth of nanoparticles due to the chelating effect. The advantages of the method include the polyhydric alcohols high boiling point, which allows synthesis to be carried out at temperatures from 200 to 320 °C without increasing pressure. It is noteworthy that the polyol’s ability to dissolve substances is like that of water [20]. This simplifies the reagents selection for synthesis. Bimetallic PtM nanoparticles with high functional characteristics were synthesized by the polyol method in several works [21, 22]. The purpose of this work is a comparative study of various methods for the platinumcobalt electrocatalysts synthesis from the viewpoint of the composition, structure and electrochemical activity of the resulting materials.
2 Experimental Part 2.1 Preparation of the Catalysts For comparative study, four methods for the synthesis of PtCo/C materials were chosen: (i) Borohydride synthesis method. A water and ethylene glycol (EG) mixture in a 1:1 ratio in an amount of 120 ml was added to the Vulcan-XC72 carbon carrier sample. The resulting mixture was dispersed using an ultrasonic homogenizer. Then, with constant stirring on a magnetic stirrer, the calculated amount of cobalt (CoSO4 ·7H2 O) and platinum (H2 PtCl6 ·6H2 O) precursors solutions was added. After 5 min of stirring, an aqueous NaBH4 solution with concentrations of 0.5 M, 0.1 M and 0.05 M was added to the mixture in an amount equal to ten times the metal content in the mixture. The synthesis was carried out for 40 min at room temperature.
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(ii) Synthesis on a cobalt-containing support was carried out similarly to the borohydride method, replacing the carbon support with a composite carbon material (Cox Oy /C), the preparation procedure for which is described in more detail in [23]. No cobalt precursor was added during synthesis. (iii) Polyol synthesis method. 150 ml of EG was added to the Vulcan-XC72 carbon carrier. The resulting mixture was dispersed using an ultrasonic homogenizer. Next, calculated amounts of cobalt (CoSO4 ) and platinum (H2 PtCl6 ) precursors solutions were added to the suspension with constant stirring on a magnetic stirrer and the mixture was quickly heated to 160 °C. The synthesis was carried out for 3 h, after which the mixture was cooled to room temperature with constant stirring. (iv) The modified polyol synthesis method was carried out similarly to the polyol synthesis method, adding a 0.1 M NaBH4 aqueous solution to the mixture in the first 40 min of synthesis. In all cases, after the synthesis, the reaction mixture was filtered on a Buchner funnel. The filter cake was washed repeatedly with water and isopropanol, and then dried for 2 days in a desiccator with P2 O5 . 2.2 Studying of the Catalysts Structure Diffractograms of the obtained materials were recorded in the angle range 2θ = 15°−75° with a detector displacement step of 0.04° on an ARL X’TRA diffractometer (Thermo Scientific, Switzerland) using filtered CuKα radiation (λ = 0.154056 nm) at room temperature. The elemental Co:Pt ratio in the samples was determined by X-ray fluorescence analysis (TXRF) on a spectrometer with total external reflection of X-ray radiation RFS-001 (Research Institute of Physics, Southern Federal University). The electrochemical characteristics of the samples were determined in a threeelectrode cell with a platinum counter electrode and a silver chloride reference electrode in 0.1 M HClO4 . To activate the material surface, 100 potential sweep cycles were performed in the range from 0.04 to 1.2 V at a speed of 200 mV/s. The determination of the platinum electrochemically active surface area (ECSA) was carried out by recording 2 cycles with a potential sweep from 0.04 to 1.2 V at a rate of 20 mV/s based on the amount of electricity spent on the adsorption and desorption of hydrogen. The determination of activity in the oxygen reduction reaction (ORR) was carried out by applying the Koutecký–Levich equation to linear voltammograms obtained on a rotating disk electrode (RDE) in the potential range from 0.1 to 1.2 V at different rotation speeds (400, 900, 1600, 2500 rpm) in a 0.1 M HClO4 solution saturated with O2 .
3 Results and Discussions 3.1 Structural Characteristics of Pt3 Co/C Materials, Synthesized by the Borohydride Method Using aqueous solutions of NaBH4 at concentrations of 0.5 M, 0.1 M and 0.05 M, several materials with a theoretical mass fraction of platinum of 20% were obtained. Based on the gravimetric analysis results, it was found that the platinum mass fractions
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19
in the samples differed from those theoretically predicted and amounted from 11.4% to 20.2%. The difference between the theoretical and gravimetric mass fraction of platinum is apparently due to the incomplete reduction of platinum and/or cobalt when using more concentrated solutions of sodium borohydride in a water-ethylene glycol mixture. In the powder X-ray diffraction of all Pt3 Co/C materials, reflection of the carbon phase and the metal phase with a face-centered cubic lattice is observed, while the reflections in diffractogram are between the reflection positions, corresponding to the Pt and Co phases, which indicates the solid solution formation of Pt and Co. The average crystallite size calculated by the Scherrer equation was from 2.2 to 2.5 nm, which indicates an insignificant effect of the reducing agent concentration on the average crystallite size. Thus, an increase in the concentration of the NaBH4 solution leads to an increase in the proportion of cobalt in the composition of the resulting materials, but at the same time the loss of metals during synthesis increases. The borohydride synthesis method did not make it possible to obtain materials with both low metal losses and a high proportion of cobalt in the composition of nanoparticles, so it was not advisable to study the electrochemical characteristics of materials obtained by this method. 3.2 Structural Characteristics of PtCo/C Materials, Synthesized on a Cobalt-Containing Substrate By synthesis on a cobalt-containing carbon substrate, a lot of materials with theoretical compositions PtCo, Pt2 Co and Pt3 Co with platinum mass fractions of 20.0%, 31.3% and 42.8%, respectively, were obtained (see Table 1, materials CS1, CS2, CS3, respectively). Based on the results of gravimetric analysis, it was found that the mass fractions of platinum were 21.2%, 33.4% and 41.8%, respectively, which is close to the theoretical mass fractions. Table 1. Structural characteristics of PtCo/C samples of various compositions synthesized on Cox Oy /C substrate Sample
ω(Pt), %
Composition of the metal component
Metal crystallite size, nm
Theory
TGA
Theory
TXRF
XRD
CS1
20.0
21.2
PtCo
Pt62 Co38
Pt80 Co20
2.3
CS2
33.3
33.4
Pt2 Co
Pt74 Co26
Pt84 Co16
2.5
CS3
42.8
41.8
Pt3 Co
Pt77 Co23
Pt88 Co12
2.6
In the powder X-ray diffraction of all materials (Fig. 1), carbon phase and the metal phase reflections are observed, while the metal phase reflections are located between the reflection positions, corresponding to the Pt and Co phases, which indicates the solid solution of Pt and Co formation. With an increase in the cobalt content in the material, the metal phase maximum reflections shift to the higher-angular 2 theta region (Fig. 1), which indicates the inclusion of a cobalt larger amount in the solid solution (Table 1). For each of the samples,
20
E. R. Beskopylny et al.
the XRF results showed a larger cobalt content than the calculation of the bimetallic nanoparticles solid solution composition according to Vegard’s law (Table 1) based on XRD data. This probably indicates the incomplete inclusion of cobalt in the solid solution and the possible presence of some cobalt in the resulting samples in the X-ray amorphous oxides form [23, 24]. The average crystallite size is calculated using the Scherrer equation, for the obtained PtCo/C catalysts of various compositions, ranged from 2.3 to 2.6 nm (Table 1), which indicates a slight influence of the catalyst’s composition on the average crystallite size. Thus, the method for preparing PtCo/C materials using a cobaltcontaining carbon substrate makes it possible to synthesize platinum-cobalt catalysts of various compositions with low metal losses.
Fig. 1. Powder X-ray diffraction of PtCo/C samples obtained on a Cox Oy /C substrate
3.3 Electrochemical Characteristics of PtCo/C Materials Synthesized on a Cobalt-Containing Substrate Cyclic voltammograms (Fig. 2) of PtCo/C catalysts, obtained on the cobalt-containing substrate basis have the appearance characteristic of platinum-containing catalysts on a carbon support, and have not maxima, corresponding to the metal’s dissolution. Measured using CV in the hydrogen adsorption/desorption region (Fig. 2), the materials ECSA values with theoretical compositions PtCo, Pt2 Co and Pt3 Co are respectively 30, 21 and 22 m2 /g (Pt), which is slightly lower compared to commercial Pt/C catalysts (Table 2). The various compositions PtCo/C catalysts mass activities at 0.9 V potential, measured using linear voltammetry (Fig. 3), were 369, 149 and 223 A/g (Pt), respectively, which in some cases is slightly higher than the commercial samples activity (Table 2). Linear voltammograms of the CS1 material at different speeds of disk electrode rotation and the Koutecký–Levich dependence, since which the activity was calculated for these materials, are presented in Fig. 3.
Study of Various Approaches to the Synthesis of PtCo/C
21
Fig. 2. Cyclic voltammograms of various compositions PtCo/C samples, synthesized on a Cox Oy /C substrate in a 0.1 M HClO4 solution sat. Ar.
Fig. 3. (a) Linear sweep voltammograms of the CS1 material, obtained at different rotation speeds in a 0.1 M HClO4 solution sat. O2 ; (b) Koutetsky-Levich dependences of samples synthesized on Cox Oy /C substrate and commercial samples
Thus, PtCo/C catalysts of various compositions, obtained on the cobalt-containing substrate basis, despite a smaller surface area, compared to commercial Pt/C, demonstrate high specific cathode currents and, consequently, high activity in the ORR. It has been established that the material with the highest cobalt content exhibits the greatest activity.
22
E. R. Beskopylny et al.
Table 2. Electrochemical characteristics of various compositions PtCo/C samples synthesized on Cox Oy /C substrate and commercial Pt/C catalysts Sample
ECSA Hads/des , m2 /g (Pt)
Mass-activity at 0.9 V, A/g (Pt)
Specific activity at 0.9 V, A/m2 (Pt)
n(e) at 0.9 V
CS1
30
369
12.3
3.8
CS2
21
149
7.1
3.3
CS3
22
223
10.2
4.0
JM20
69
254
3.7
3.7
JM40
60
187
3.1
4.0
3.4 Structural Characteristics of Pt3 Co/C Materials, Synthesized by the Polyol Method A series of Pt3 Co/C materials with a theoretical platinum mass fraction of 20% were obtained by the polyol synthesis method in neutral and alkaline media. Thus, both in neutral and alkaline environments, significant metals losses are observed during synthesis. Synthesis in an alkaline medium, as opposed to synthesis in a neutral medium, made it possible to reduce both platinum and cobalt, as well as to obtain a material with a smaller crystallite size. However, it is possible that in this case the smaller crystallite size is provided not by the synthesis advantages in an alkaline medium, but by the less complete metals reduction in an alkaline medium, which naturally led to a smaller crystallite size. 3.5 Structural Characteristics of Pt3 Co/C Materials, Synthesized by the Modified Polyol Method Using the modified polyol synthesis method, a series of Pt3 Co/C materials with a platinum theoretical mass fraction of 20% were obtained in neutral and alkaline media. Thus, the introduction of NaBH4 into the reaction mixture during polyol synthesis made it possible to almost eliminate the loss of metals completely. The influence of the pH of the medium on the nanoparticles composition and the crystallites size remained the same, as in the polyol synthesis method, that is, the cobalt content of Pt3 Co/C in the solid solution is higher in the material, obtained in an alkaline medium, and the average crystallite size is smaller.
4 Conclusions During the work, a comparative analysis of four methods for the bimetallic PtCo/C fuel cells electrocatalysts synthesis was carried out, namely borohydride synthesis method, synthesis on a cobalt-containing carbon substrate, polyol synthesis method and modified polyol synthesis method. Comparison of the obtained materials properties made it possible to formulate several features and patterns of these synthesis methods:
Study of Various Approaches to the Synthesis of PtCo/C
23
(i) the nanoparticles solid solution composition does not coincide with the calculated one, based on the precursors loading, due to the incomplete inclusion of the doping component in the solid solution composition for all synthesis methods used; (ii) the borohydride synthesis method does not allow obtaining materials with both a high cobalt contained in nanoparticles and a low losses level; (iii) synthesis, using a composite carrier, makes it possible to obtain materials of various compositions with a low loss level; (iv) during polyol synthesis, the reduction of cobalt occurs much better in an alkaline medium, and the addition of sodium borohydride reduces the loss of metals. Some of the obtained PtCo/C materials turned out to be comparable in activity to commercial analogues. Based on the results of a synthesis methods comparative analysis used, the synthesis method using a composite cobalt-containing carbon support can be recommended as the most promising for further optimization. Acknowledgments. This research was financially supported by the Ministry of Science and Higher Education of the Russian Federation (State assignment in the field of scientific activity №FENW-2023-0016).
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13. Chen, Y.: ChemInform abstract: chemical preparation and characterization of metal-metalloid ultrafine amorphous alloy particles. ChemInform 30(17) (1999) 14. Dragieva, I.D., Stoynov, Z.B., Klabunde, K.J.: Synthesis of nanoparticles by borohydride reduction and their applications. Scr. Mater. 44, 2187–2191 (2001) 15. Gu, J., Zhang, Y.W., Tao, F.: Shape control of bimetallic nanocatalysts through well-designed colloidal chemistry approaches. Chem. Soc. Rev. 41, 8050–8065 (2012) 16. Rao, C.S., Singh, D.M., Sekhar, R., Rangarajan, J.: Pt–Co electrocatalyst with varying atomic percentage of transition metal. Int. J. Hydrogen Energy 36, 14805–14814 (2011) 17. Mani, P., Srivastava, R., Strasser, P.: Dealloyed binary PtM3 (M = Cu Co, Ni) and ternary PtNi3M (M = Cu Co, Fe, Cr) electrocatalysts for the oxygen reduction reaction: performance in polymer electrolyte membrane fuel cells. J. Power. Sources 196, 666–673 (2011) 18. Song, T.W., et al.: Small molecule-assisted synthesis of carbon supported platinum intermetallic fuel cell catalysts. Nat. Commun. 13, 1–11 (2022) 19. Yang, C.L., et al.: Sulfur-anchoring synthesis of platinum intermetallic nanoparticle catalysts for fuel cells. Science 374, 459–464 (2021) 20. Dong, H., Chen, Y.C., Feldmann, C.: Polyol synthesis of nanoparticles: status and options regarding metals, oxides, chalcogenides, and non-metal elements. Green Chem. 17, 4107– 4132 (2015) 21. Li, B., et al.: Highly active carbon-supported Pt nanoparticles modified and dealloyed with Co for the oxygen reduction reaction. J. Power. Sources 270, 201–207 (2014) 22. Zagoraiou, E., Shroti, N., Daletou, M.K.: Development of Pt-Co catalysts supported on carbon nanotubes using the polyol method—tuning the conditions for optimum properties. Mater Today Chem. 16, 100263 (2020) 23. Mauer, D., et al.: Gram-scale synthesis of CoO/C as base for PtCo/C high-performance catalysts for the oxygen reduction reaction. Catalysts 11, 1539 (2021) 24. Pryadchenko, V.V., et al.: Bimetallic PtCu core-shell nanoparticles in PtCu/C electrocatalysts: Structural and electrochemical characterization. Appl. Catal. A Gen. 525, 226–236 (2016)
Advantages of Bimetallic Electrocatalysts for Cathodes in a Proton Exchange Membrane Fuel Cell Angelina Pavlets1(B) , Ekaterina Kozhokar1 , Yana Astravukh1 Alexey Nikulin1 , and Anastasia Alekseenko1
, Ilya Pankov2
,
1 Faculty of Chemistry, Southern Federal University, 7, Zorge Str, Rostov-on-Don 344090,
Russia [email protected] 2 Research Institute of Physical Organic Chemistry, Southern Federal University, 194/2, Stachki Str, Rostov-on-Don 344090, Russia
Abstract. Production of highly efficient platinum-containing catalysts for PEMFC cathode is an urgent task for the development of hydrogen energy. Fifteen bimetallic catalysts for the ORR were synthesized by various modified borohydride synthesis methods using de-alloying treatment. Of these, 33.3% PtNi/C and 33.3% PtCu/C materials were subjected to acid treatment, 13.3% PtNi/C and 20% PtCu/C were subjected to electrochemical leaching. A comprehensive assessment of the composition and structure of the catalysts was carried out using TEM, STEM, XRD, TXRF, and EDX methods. The electrochemical behavior of the materials was estimated by cyclic and linear voltammetry. The resulting bimetallic catalysts exceed the commercial Pt/C analogue in the ORR mass activity by 1.2–4.3 times. Most of the studied PtM/C catalysts (66.6%) exceed the DOE target 2025 (440 A/gPt ). According to the results of the study, PtCu/C catalysts, subjected to electrochemical de-alloying, are considered promising for use in PEMFC MEA. Keywords: Bimetallic Catalysts · ORR · Acid Treatment · Electrochemical Treatment
1 Introduction The reaction of oxygen electroreduction (ORR), carried out at the proton-exchange membrane fuel cell (PEMFC) cathode, is one of the most studied in the field of electrocatalysis [1, 2]. However, its sluggish kinetics still pose a serious limitation to the efficiency of fuel cells [3]. The transition from platinum-carbon to more complex bimetallic systems is driven by the need to increase the activity of catalysts in the ORR, increase stability and reduce costs due to PMG savings [4–6]. Bimetallic nanoparticles can have different architectures: an alloy (“solid solution”) or an ordered solid solution (intermetallic compound), core-shell, onion structure (layerby-layer), gradient (gradual change in components), nanoframe, dendrites [7]. The composition of the metal component of electrocatalysts can vary widely, often from Pt3 M to PtM3 . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 25–37, 2024. https://doi.org/10.1007/978-3-031-52239-0_3
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The widespread use of electrocatalysts, based on PtM (M = Co, Ni, Cu) NPs, is hampered by the selective dissolution of atoms of the alloying component, due to the low thermodynamic stability PtM alloy [8, 9]. This dissolution occurs during the operation of the membrane electrode assembly (MEA) in an acidic environment at the cathode of the fuel cell and leads to poisoning of the polymer membrane due to the deposition of d-metal on the end groups of the membrane [10]. To prevent the dissolution of the alloying component during the PEMFC operation, various approaches are used, the main ones of which are acid and electrochemical treatment of bimetallic catalysts to remove base metal from the surface of the catalyst [7]. During acid treatment, the composition of the metal component of electrocatalysts can change to Ptx M, and the bimetallic NPs are characterized by a de-alloyed structure (with a defective surface). Selecting samples for testing in the MEA is an urgent task, since this type of testing is resource intensive. A preliminary assessment of the electrochemical behavior of dealloyed bimetallic catalysts in a three-electrode cell using RDE is a rapid method for determining the main functional characteristic of materials, namely the ORR activity [11]. To predict the characteristics and select from a set of samples the most promising ones for further testing in the MEA, we studied a set of bimetallic de-alloyed PtNi/C and PtCu/C catalysts of similar composition with different morphological characteristics.
2 Materials and Methods 2.1 Materials Dihydronium hexachloroplatinate(2– ) (H2 PtCl6 ·6H2 O, hydrogen hexachloroplatinate (IV); extra pure grade, Aurat, Russia, platinum mass fraction 37.6%), nickel (II) chloride (NiCl2 ·6H2 O, nickel chloride, 6-aqueous; extra pure grade, JSC Vekton), copper (II) sulfate (CuSO4 ·5H2 O, copper sulfate pentahydrate; extra pure grade, JSC Reakhim, Russia), carbon support Vulcan XC-72 (Cabot Corporation), ethylene glycol (EG; extra pure grade, JSC Vekton, Russia), ethanol (C2 H5 OH; extra pure grade, Sigma-Aldrich), bidistilled water, sulfuric acid (H2 SO4 ; pure for analysis, JSC Himikon, Russia), nitric acid (HNO3 ; pure for analysis, JSC Himikon, Russia), sodium tetrahydridoborate(1– ) (NaBH4 , sodium borohydride; extra pure grade, JSC Vekton), sodium hydroxide (NaOH; pure for analysis, JSC Himikon, Russia), N, N-Diethylethanamine ((C2 H5 )3 N, triethylamine (TEA); extra pure grade, JSC Vekton), perfluorinated resin (Nafion DE1020, 10% aqueous dispersions, DuPont), chloric (VII) acid (HClO4 , perchloric acid; extra pure grade, Sigma-Aldrich), propan-2-ol ((CH3 )2 CHOH, isopropanol; pure for analysis, EKOS-1). 2.2 Synthesis Method All synthesized materials were obtained with a theoretical Pt loading of 20%. The PtNi-AT-1 was synthesized by a liquid-phase step-by-step borohydride method [12] in an inert atmosphere in a water-ethanol medium. In the first step, the carbon support was dispersed in 30 ml of ethanol using ultrasonication. After that, a nickel precursor
Advantages of Bimetallic Electrocatalysts for Cathodes
27
(NiCl2 ·6H2 O) was added to the suspension and brought to 100 ml with ethanol. The mixture was brought to 70 °C and purged with argon throughout the synthesis. After reaching the required temperature, the reducing agent NaBH4 was added dropwise to the suspension. After 30 min, the Pt precursor (H2 PtCl6 ·6H2 O) and sodium borohydride were added to the solution. After 30 min, another portion of the reducing agent was added to the synthesis, after which the suspension was cooled and filtered. The filtered catalyst was dried in a desiccator. Acid treatment was carried out in 0.1 M HNO3 for 3 h with stirring and room temperature. The theoretical ratio of metals in the catalyst before treatment is PtNi3 . The PtNi-AT-2 catalyst was prepared similarly to PtNi-AT-1, but after adding a nickel precursor, the required amount of triethylamine was added to the suspension and the mixture was brought to 100 ml with ethanol. Acid treatment was carried out in a similar way. The PtNi-AT-3 was synthesized similarly to PtNi-AT-2, but with using double the amount of triethylamine. The PtNi-AT-4 sample was prepared as described in [13] with the exception that H2 PtCl6 ·6H2 O was used as a platinum precursor. Briefly, platinum and nickel precursors (NiCl2 ·6H2 O) were dissolved in a water-ethanol mixture. Then the carbon support was added, and the mixture was dispersed by ultrasound. Next, the reducing agent NaBH4 was added dropwise at a rate of 5 ml/min. The system was kept under stirring for 1 h, then the catalyst was filtered and dried in a vacuum oven at T = 80 °C). Acid treatment was carried out by keeping the resulting sample in 1 M HNO3 for 3 h with stirring and room temperature. The theoretical ratio of metals in the catalyst before treatment is PtNi3 . The PtNi-AT-5 was obtained by co-reduction of platinum and nickel precursors in an EG-water medium. The reducing agent NaBH4 dissolved in water was added to the system immediately. Acid treatment was carried out similarly to the PtNi-AT-4 sample. The theoretical ratio of metals in the catalyst before treatment is PtNi3 . Similarly, a sample with the composition PtNi1 was prepared and subjected to electrochemical treatment to obtain the PtNi-ET-1 catalyst. The PtCu-AT-1 sample was obtained similarly to the PtNi-AT-5 sample, as described in [14]. The salt CuSO4 ·5H2 O was used as a copper precursor. The theoretical ratio of metals in the catalyst before treatment is PtCu3 . The PtCu-ET-1 was obtained using the electrochemical treatment of the initial PtCu/C sample. Sample PtCu-AT-2 was obtained similarly to sample PtCu-AT-1. The theoretical ratio of metals in the catalyst before treatment is PtCu1 . The original sample was also subjected to electrochemical treatment and labeled as PtCu-ET-2. Samples PtCu-AT-3 and PtCu-AT-4 were obtained by treating the PtCu/C sample in 1 M H2 SO4 and 1 M HNO3 , respectively. The starting PtCu/C was prepared as described in [15]. Briefly, the first stage consisted of obtaining the copper NPs on the carbon support through the reduction of copper. The second stage included the codeposition of platinum and copper on the formed copper cores by reducing the metals from the solution of the corresponding precursors. Next, platinum and copper were again deposited on the previously obtained PtCu/C materials, decreasing the copper concentration, and increasing the concentration of platinum in the stock solution. Thus,
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a gradient approach to synthesizing shell-core type NPs has been implemented. Initial composition of the precursor material PtCu1 . The sample PtCu-AT-5 was obtained similarly to the predecessor of samples PtCuAT-3 and PtCu-AT-4 as described in [16] the exception that at the first stage of synthesis, the NPs’ cores were formed not from copper, but from platinum. The resulting sample with the theoretical composition PtCu1 was treated in 1 M HNO3 . The analogue, treated electrochemically, is labeled as PtCu-ET-3. The sample PtNi-ET-2 was obtained by electrochemical treatment of the initial PtNi/C catalyst with the theoretical composition PtNi1 . Initial PtNi/C was synthesized as described in [17]. In short, we tried to form a core from nickel and then deposit platinum and nickel on the Ni cores in several stages, similarly to samples PtCu-AT-3 and PtCu-AT-4. 2.3 Methods to Study the Composition and the Structure of the Catalysts The composition and structure of the catalysts were studied by thermogravimetry (TGA), X-ray fluorescence (XRF), X-ray diffraction (XRD), energy dispersive X-ray spectroscopy (EDX), transmission electron microscopy (TEM), and scanning transmission electron microscopy (STEM) as described in detail in [18]. 2.4 Electrochemical Methods Electrochemical measurements were carried out in a standard three-electrode cell on a rotating disk electrode using a potentiostat VersaSTAT3 (Ametek, USA). A saturated silver chloride was used as a reference electrode, and a platinum wire acted as a counter electrode. Solution of 0.1 M HClO4 was used as the electrolyte. The activation stage of electrocatalysts consists of potentiodynamic cycling of the catalytic layer for 100 cycles, in the potential range of 0.04–1.0 V with a scan rate of 200 mV/s. Cycling was carried out in an argon atmosphere, presaturating the electrolyte for 30 min. After the activation, the electrolyte was replaced with a fresh one and again purged with argon for 30 min. Next, two cyclic voltammograms were recorded in the potential range of 0.04–1.0 V with a scan rate of 20 mV/s. The amount of electricity that is spent on the adsorption/desorption of a hydrogen monolayer was determined and the ESA was calculated as described in [18]. To determine the catalytic activity of the materials, voltammograms with linear potential sweep (LSV) were recorded in an argon atmosphere with a rotation speed of RDE of 1600 rpm in the potential range of 0.1–1.1 V with a scan rate of 20 mV/s. Then the cell was saturated with oxygen for 40 min at RDE rotation of 700 rpm. Current-voltage curves were recorded at several RDE rotation speeds: 400, 900, 1600, 2500 rpm. Then the current was determined at E = 0.9 V. The dependences were plotted in KoutetskyLevich coordinates, as described in [18] the kinetic current was calculated. By dividing the obtained value by the platinum loading on the electrode and by the ESA values, the mass and specific current were obtained, respectively. A commercial Pt/C catalyst (HiSPEC3000, Johnson Matthey) with a mass fraction of platinum of 20% was used as a reference sample for comparison.
Advantages of Bimetallic Electrocatalysts for Cathodes
29
3 Results and Discussion Using various modified borohydride synthesis methods with subsequently acid (AT marking) or electrochemical treatment (ET marking) fifteen bimetallic catalysts were obtained. Of these, five PtNi/C and five PtCu/C materials were subjected to acid treatment, two PtNi/C and three PtCu/C materials were subjected to electrochemical treatment (Table 1). Table 1. Composition, structural and electrochemical characteristics of PtM/C and Pt/C catalysts Sample
PtNi AT-1 PtNi AT-2 PtNi AT-3 PtNi AT-4 PtNi AT-5 PtCu AT-1 PtCu AT-2 PtCu AT-3 PtCu AT-4 PtCu AT-5 PtNi ET-1 PtNi ET-2 PtCu ET-1 PtCu ET-2 PtCu ET-3 JM20
Type of treatment 0.1М HNO3 0.1М HNO3 0.1М HNO3 1М HNO3 1М HNO3 1М HNO3 1М HNO3 1М H2SO4 1М HNO3 1М HNO3 0.1М HCLO4 0.1М HCLO4 0.1М HCLO4 0.1М HCLO4 0.1М HCLO4 -
Initial composition PtNi3
Massfraction of Pt, % 20.0
Proposed NPs’ structure
Composition (TXRF)
Dcr, nm (XRD)
Dav, nm (TEM)
ESA m2/ gPt
Imass, А/gPt
PtNi3
20.0
PtNi3
20.0
PtNi3
20.9
Coreshell Coreshell Coreshell Alloy
PtNi0.8
4.6
10.6
30
320
PtNi0.6
5.9
15.9
30
440
PtNi0.8
4.8
-
26
301
PtNi0.4
6.0
-
34
403
PtNi3
23.2
Alloy
PtNi0.5
2.1
4.0
43
421
PtCu3
23.3
Alloy
PtCu0.5
2.6
3.4
39
645
PtCu1
19.7
Alloy
PtCu0.3
3.0
2.9
41
554
PtCu1
19.2
Gradient
PtCu0.5
2.9
4.1
37
569
PtCu1
19.4
Gradient
PtCu0.5
2.5
3.2
45
672
PtCu1
17.5
Gradient*
PtCu0.3
2.7
3.8
54
550
PtNi1
21.0
Alloy
PtNi0.4
-
-
31
506
PtNi1
19.6
Gradient
PtNi0.1
-
-
35
328
PtCu3
21.7
Alloy
PtCu0.3
3.7
4.1
47
1100
PtCu1
17.9
Alloy
PtCu0.3
-
5.1
43
800
PtCu1
20.4
Gradient*
PtCu0.3
-
4.7
46
600
Pt
20
-
-
2.7
3.5
80
254
*Gradient on Pt-core
The work considered various structures of nanoparticles: alloy, core-shell, gradient (Table 1). To prepare de-alloyed materials, PtM 1 and PtM 3 initial compositions of bimetallic catalysts were used. (Table 1). The mass fraction of platinum, determined from the results of thermogravimetry, is close between the obtained catalysts and ranges from 17.5 to 23.3 wt.% (Table 1). After the de-alloying, the composition of the bimetallic catalysts changes differently from PtM 0.1 to PtM 0.8 . Apparently, this depends on how much d-metal was initially included in the nanoparticles structure. It should be noted
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A. Pavlets et al.
that the amount of the selectively dissolved component is quite large, which indicates the need for additional treatment of bimetallic materials. X-ray diffraction analysis was carried out for all acid-treated samples and one electrochemically treated sample PtCu-ET-1 (Fig. 1). To do this, the sample was applied to a graphite plate, activated by cycling, cleaned off and submitted for analysis. The procedure was repeated 3 times to collect the required amount of material.
Fig. 1. XRD patterns of PtM/C and Pt/C materials.
In the X-ray diffraction patterns of the obtained PtM/C samples and commercial Pt/C, reflections of the carbon support (002) about 25°, and metal phases related to the (111) and (200) faces are observed (Fig. 1). All bimetallic samples are characterized by a shift of the maximum of the most intense peak (111) to the region of large angles 2θ compared to pure platinum, which indicates the formation of a Pt-M alloy [14]. The average crystallite size calculated using the Scherrer equation for all studied samples varies in the range from 2.1 to 6 nm (Table 1). It should be noted that PtNi/C catalysts after acid treatment are characterized by larger crystallite sizes compared to similar PtCu/C samples. Also interesting is the fact that samples PtCu-AT-2 and PtCu-ET-2, obtained from the same precursor material, are characterized by different Dcr : electrochemical treatment leads to enlargement of crystallites by 0.7 nm, compared to chemical treatment (Table 1). This is probably due to repeated dissolution and reprecipitation of platinum during electrochemical activation [19].
Advantages of Bimetallic Electrocatalysts for Cathodes
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TEM photographs for PtM/C and Pt/C samples show nano-sized metal particles, distributed on the surface of the carbon support (Figs. 2, 3 and 4). For all presented materials, besides PtNi-AT-1 and PtNi-AT-2, the nanoparticles have a spherical shape. For samples PtNi-AT-1 and PtNi-AT-2, an agglomerated moss-type structure of NPs is observed (Fig. 3). For PtCu-AT-4, obtained by the method of step-by-step reduction of precursors, a clearly defined structure of core-shell is observed for most NPs, which may represent porous NPs (Fig. 4). For all other bimetallic samples, because of de-alloying treatment the porous structure of NPs is not observed, which is due to the impossibility of the Kirkendall effect processing for such small-sized NPs [20].
Fig. 2. TEM images and histograms of NPs’ size distribution of the PtNi-AT-1, PtNi-AT-2 PtNiAT-5 and commercial Pt/C samples.
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For PtCu/C samples after electrochemical treatment, the formation of dumbbellshaped NPs is observed, caused by adhesions between neighboring spherical NPs (Fig. 4). The average NPs’ size of the studied materials varies from 2.9 to 15.9 nm (Table 1). It should be noted that to calculate the average NPs’ size, non-aggregated NPs were processed. In this regard, the processing did not consider the presence of agglomerates, the size of which varies from 10 to 50 nm. Elemental mapping of PtM/C samples and line scanning of individual NPs confirms the bimetallic composition of the NPs and the localization of Pt and d-metal at the same sites on the carbon support (Fig. 5).
Fig. 3. TEM images and histograms of NPs’ size distribution of the PtCu-AT-1 - PtCu-AT-5 samples.
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Fig. 4. TEM images and histograms of NPs’ size distribution of the PtCu-ET-1 - PtCu-ET-3 samples.
From the CV curves shown in Fig. 6, the electrochemically active surface area of the catalysts was calculated. The ESA of bimetallic materials varies from 26 to 54 m2 /gPt (Table 1). The commercial JM20 sample is characterized by a high ESA, which is ensured by its morphology, namely a low degree of NPs’ agglomeration. Comparing the ECA of platinum-nickel and platinum-copper samples, it is worth noting that the values for the latter are on average 10 m2 /gPt higher, which is also ensured by the less agglomerated structure of the PtCu/C catalysts. Note that for bimetallic samples there is no correlation between the average size of crystallites or nanoparticles and the ESA values. Next, the catalytic activity of the obtained PtM/C and commercial Pt/C materials was studied on a rotating disk electrode (Fig. 7). For the convenience of presenting the results of electrochemical tests in Fig. 7, the ORR LSV curves are presented and the Koutetsky-Levich dependence plots. The calculated mass activity values in the ORR are given in Table 1. To process the large number of data obtained, plots of the dependence of mass activity on various parameters were constructed (Fig. 8). Figure 8 shows markers for PtNi/C, PtCu/C, and commercial Pt/C catalysts. Most of the resulting de-alloyed bimetallic catalysts (10 out of 15 samples) exceeded the DOE target 2025 (440 A/gPt ) (Fig. 8a). The increased catalyst activity in the ORR is associated with the ligand effect, regardless of the structure type and method of the NPs’ de-alloying. There are several reasons for the positive effect of the alloying component on the functional characteristics of electrocatalysts, including: a decrease in the Pt-Pt interatomic distance in a nanoparticle (NP) facilitates the process of oxygen molecules adsorption during the ORR; a change in the energy of free d-orbitals, facilitating the process of O2 adsorption on the catalyst surface [3].
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Fig. 5. STEM images and EDX mapping of the corresponding areas (a, d, g, j, m, p) for some PtM/C samples. The line scanning of individual NPs’ (c, f, i, l, o, r) and STEM images of the corresponding areas (b, e, h, k, n, q).
Fig. 6. 2 CVs of some PtM/C catalysts. 20 mV/s, 0.1M HClO4, atmosphere of Ar.
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Fig. 7. LSV curves of PtNi/C (a) and PtCu/C (b) after acid treatment; PtNi/C and PtCu/C after electrochemical treatment (c); dependence in Koutecki-Levich coordinates for the corresponding materials (d–f).
It is difficult to estimate the influence of the average size of crystallites and nanoparticles on the catalytic activity (Fig. 8b, c). However, we note that most samples, characterized by activity above 400 A/gPt , have an average crystallite size of 2.5 to 3 nm, and an average NPs’ size of 2.5 to 4.5 nm.
Fig. 8. Dependence of mass activity on d-metal content (a), average crystallite size (b), average nanoparticle size (c) and ESA (d).
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Among the studied samples, the ones after electrochemical treatment exhibit the highest activity for both PtNi/C and PtCu/C catalysts (Table 1). The PtNi/C catalysts have a lower mass activity in the ORR, compared to the PtCu/C, but still exceed that of the commercial Pt/C analogue. The highest mass activity among acid-treated platinumnickel samples is characterized by the PtNi-AT-2 sample, which was obtained in the presence of TEA in a one-fold excess. Samples, obtained in the same way, but without TEA (PtNi-AT-1) or with a two-fold excess of it (PtNI-AT-3), are characterized by 1.5 times less activity in the ORR than the similar PtNi-AT-2. In the group of platinum-copper acid-treated samples, all samples are characterized by similar activity of 550–650 A/gPt . When considering the dependence of mass activity on the ESA, a trend is observed: the higher the ESA value, the higher the activity of the catalyst in the ORR can be expected (Fig. 8d).
4 Conclusions Electrochemical treatment of PtNi/C and PtCu/C makes it possible to obtain more active materials, compared to acid treatment. PtCu/C catalysts after electrochemical de-alloying are of greatest interest for testing in MEA. For a more in-depth study of the influence of morphological characteristics on activity in the ORR, it is necessary to conduct detailed TEM and SEM studies on many different areas of the catalyst surface the size and spatial distribution of NPs to assess more accurately the proportion of agglomerates and their sizes, the structure of NPs. Acknowledgements. The research was supported by the Russian Science Foundation (grant No. 23-79-00058), implemented in Southern Federal University. Special thanks to the «HighResolution Electron Microscopy» Center of collective use at Southern Federal University.
References 1. Wang, Y.J., et al.: Unlocking the door to highly active ORR catalysts for PEMFC applications: polyhedron-engineered Pt-based nanocrystals. Energy Environ. Sci. 11, 258–275 (2018) 2. Si, F., et al.: Electrochemical oxygen reduction reaction. In: Rotating Electrode Methods and Oxygen Reduction Electrocatalysts, pp. 133–170 (2014) 3. Nørskov, J.K., et al.: Origin of the overpotential for oxygen reduction at a fuel-cell cathode. J. Phys. Chem. B 108(46), 17886–17892 (2004) 4. Sui, S., Wang, X., Zhou, X., Su, Y., Riffat, S., Liu, C.: A comprehensive review of Pt electrocatalysts for the oxygen reduction reaction: nanostructure, activity, mechanism and carbon support in PEM fuel cells. J. Mater. Chem. A Mater. 5, 1808–1825 (2017) 5. Sharma, G., et al.: Novel development of nanoparticles to bimetallic nanoparticles and their composites: a review. J. King Saud Univ. Sci. 31, 257–269 (2019) ˇ c, V.: Electrocatalysts for the oxygen reduction reaction: from 6. Martínez-Hincapié, R., Coli´ bimetallic platinum alloys to complex solid solutions. ChemEngineering 6, 19 (2022) 7. Belenov, S., Alekseenko, A., Pavlets, A., Nevelskaya, A., Danilenko, M.: Architecture evolution of different nanoparticles types: relationship between the structure and functional properties of catalysts for PEMFC. Catalysts 12, 638 (2022)
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8. Moriau, L.J., et al.: Resolving the nanoparticles’ structure-property relationships at the atomic level: a study of Pt-based electrocatalysts. iScience. 24, 102102 (2021) 9. Gatalo, M., et al.: Comparison of Pt–Cu/C with benchmark Pt–Co/C: metal dissolution and their surface interactions. ACS Appl. Energy Mater. 2, 3131–3141 (2019) 10. Zhao, Y., Mao, Y., Zhang, W., Tang, Y., Wang, P.: Reviews on the effects of contaminations and research methodologies for PEMFC. Int. J. Hydrogen Energy 45, 23174–23200 (2020) 11. Tovini, M.F., Hartig-Weiß, A., Gasteiger, H.A., El-Sayed, H.A.: The discrepancy in oxygen evolution reaction catalyst lifetime explained: RDE vs MEA - Dynamicity within the catalyst layer matters. J. Electrochem. Soc. 168, 014512 (2021) 12. Yang, D., et al.: Monodispersed Pt3 Ni nanoparticles as a highly efficient electrocatalyst for PEMFCs. Catalysts 9, 588 (2019) 13. Chattot, R., Asset, T., Bordet, P., Drnec, J., Dubau, L., Maillard, F.: Beyond strain and ligand effects: microstrain-induced enhancement of the oxygen reduction reaction kinetics on various PtNi/C nanostructures. ACS Catal. 7, 398–408 (2017) 14. Pavlets, A.S., et al.: Effect of the PtCu/C electrocatalysts initial composition on their activity in the de-alloyed state in the oxygen reduction reaction. Int. J. Hydrogen Energy 47, 30460– 30471 (2022) 15. Alekseenko, A.A., Pavlets, A.S., Mikheykin, A.S., Belenov, S.V., Guterman, E.V.: The integrated approach to studying the microstructure of de-alloyed PtCu/C electrocatalysts for PEMFCs. Appl. Surf. Sci. 631, 157539 (2023) 16. Pavlets, A.S., et al.: A novel strategy for the synthesis of Pt–Cu uneven nanoparticles as an efficient electrocatalyst toward oxygen reduction. Int. J. Hydrogen Energy 46, 5355–5368 (2021) 17. Pavlets, A., Alekseenko, A., Kozhokar, E., Pankov, I., Alekseenko, D., Guterman, V.: Efficient Pt-based nanostructured electrocatalysts for fuel cells: one-pot preparation, gradient structure, effect of alloying, electrochemical performance. Int. J. Hydrogen Energy (2023) 18. Paperzh, K.O., Pavlets, A.S., Alekseenko, A.A., Pankov, I.V., Guterman, V.E.: The integrated study of the morphology and the electrochemical behavior of Pt-based ORR electrocatalysts during the stress testing. Int. J. Hydrogen Energy 48, 22401–22414 (2023) 19. Pavlišiˇc, A., Jovanoviˇc, P., Šelih, V.S., Šala, M., Hodnik, N., Gaberšˇcek, M.: Platinum dissolution and redeposition from Pt/C fuel cell electrocatalyst at potential cycling. J. Electrochem. Soc. 165, F3161–F3165 (2018) 20. Wang, J.X., et al.: Kirkendall effect and lattice contraction in nanocatalysts: a new strategy to enhance sustainable activity. J. Am. Chem. Soc. 133, 13551–13557 (2011)
Carbon Nanoparticles from Thermally Expanded Cointercalates of Graphite Nitrate with Organic Substances E. V. Raksha1(B) , A. A. Davydova2 , V. A. Glazunova3 , Y. V. Berestneva4 , A. B. Eresko1 , O. N. Oskolkova2 , P. V. Sukhov2 , V. V. Gnatovskaya2 , G. K. Volkova3 , V. V. Burkhovetskij3 , A. S. Doroshkevich1 , and M. V. Savoskin2 1 Joint Institute for Nuclear Research, Dubna, Russia
[email protected]
2 L. M. Litvinenko Institute of Physical-Organic and Coal Chemistry, Donetsk, Russia 3 A. A. Galkin Donetsk Institute for Physics and Engineering, Donetsk, Russia 4 Federal Scientific Centre of Agroecology, Complex Melioration and Protective Afforestation
of the Russian Academy of Sciences, Volgograd, Russia
Abstract. Thermally expanded graphite (TEG) is a vermicular-structured carbon material that can be prepared by heating expandable graphite compounds. Liquid phase exfoliation of TEG obtained from graphite nitrate cointercalation compounds (GNCCs) with organic substances allows obtaining dispersions of carbon nanoparticles, namely few-layer graphenes as well as small graphene structural fragments. This paper presents the results of complex investigations of structural features of triple GNCCs with acetic acid, formic acid, ethyl acetate and acetonitrile as well as TEGs on their base. X-Ray powder diffraction and scanning electron microscopy were used for studied GNCCs and TEGs characterization. GNCCs were used as a source for thermally expanded graphite which can be considered as perspective precursor for graphene and related structures. Morphology of the carbon nanoparticles formed from corresponded TEGs by liquid phase exfoliation in tert-butanol assisted with sonication is discussed. The most promising results were observed for the TEG sample obtained from a triple GNCC with acetic acid and acetonitrile. Keywords: Graphite Nitrate · Cointercalation · Liquid Phase Exfoliation · Carbon Nanoparticles · Few-Layer Graphenes
1 Introduction Thermally expanded graphite (TEG) is a vermicular-structured carbon material that can be prepared by heating expandable graphite compounds up to 1000 °C. TEG is actively used for the production of graphite paper [1], graphite foil [2], protection coatings [3], fillers for polymer composites [4–6], adsorbent [7, 8], cathodes for aluminum-ion batteries [9] or anodes for lithium-ion batteries [10, 11], components of catalysts [12, 13], for energy storage, phase change and sensor materials [14–16] etc. The effect of various © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 38–47, 2024. https://doi.org/10.1007/978-3-031-52239-0_4
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factors and conditions of TEG obtaining on its morphology and properties is actively studied [2, 17, 18]. The main method for producing thermally expanded graphite in laboratory practice is still the thermal expansion of graphite intercalated compounds (GICs) [19, 20]. GICs are compounds with a regular layered structure in which chemical particles (intercalants) are intercalated between the carbon layers. Graphite foam or TEG is formed due to the effect of thermal shock, microwave radiation or electric current on the GIC. Usually, TEG production is carried out in several stages: the synthesis of thermally expanding GIC, its stabilization, and heat treatment. The honeycomb microstructure of TEG is formed as a result of heat treatment of GIC. The volume of the cellular structure significantly exceeds the volume of the original GIC. The cell walls are a multilayer structure of graphite layers, the width of the walls is comparable to the size of graphite crystallites along the c-axis [21]. The structural features of thermally expanded graphite make it possible to suggest it as a source for carbon nanostructures. Methods for obtaining nanoparticles with different morphologies based on TEG have been proposed [8, 22, 23]. This paper presents the results on the production of few-layer graphenes dispersions by exfoliation of a pre-thermally expanded graphite nitrate cointercalation compounds (GNCCs) with organic substances in tert-butanol assisted with sonication. Organic cointercalates, used for GNCCs obtaining, were acetic and formic acids, ethyl acetate, acetonitrile as well as their combinations. Liquid phase exfoliation makes it possible to obtain suspensions of few-layer graphene particles containing also single-layer graphene. The formation of few-layer graphene particles was confirmed by transmission electron microscopy. At this study, we consider the following frameworks of the problem: • Synthesis and characterization of triple graphite nitrate cointercalation compounds (GNCCs) with organic compounds as precursors for thermally expanded graphite; • Characterization of thermally expanded graphite samples as promising origin for carbon nanoparticles obtaining; • Liquid phase exfoliation of thermally expanded graphites, obtained from different precursors.
2 Experimental 2.1 Synthesis of Graphite Nitrate Cointercalation Compounds and Thermally Expanded Graphite Preparation Graphite nitrate (GN) as well as graphite nitrate cointercalation compounds were prepared as a starting material for the generation of thermally expanded graphite. GN and GNCCs were synthesized in a thermostatic reactor using nitric acid (98%) and natural flake graphite GT-1 (Zavalie Graphite Works, Kirovograd region, Ukraine). A mixture of graphite (1 g) and HNO3 (0.6 ml) was stirred for 10 min at 20 °C to obtain a GN. Triple GNCCs were obtained by treatment of as-prepared GN with corresponding organic cointercalants (1: 1 by volume) and reaction system was stirred for 10 min. Resulted GNCCs were separated by filtration and dried at 20 °C until the sample mass became constant. Thermally expanded graphite samples were obtained from GNCCs in the thermal shock mode of heating. About 0.2 g of the GNCC was inserted into the pre-heated cuvette
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and has been kept in the furnace for 120 s at 900 °C. Then the cuvette with expanded graphite was removed from the furnace, the contents were gently transferred to a beaker for further use. The thermal expansion coefficients (K V , cm3 ·g−1 ) for obtained GNCCs were determined by heating of the samples within the thermal shock mode at 900 °C as described in [23]. 2.2 GNCCs and TEG Sample Characterization Methods X-ray diffraction (XRD) measurements were performed on DRON 3 diffractometer (CuKα radiation, λ = 1.54181 Å). Surface morphology of the studied TEGs was estimated by scanning electron microscopy (SEM) using a complex analytical scanning electron microscope JSM 6490 LV (JEOL). 2.3 Dispersions of Carbon Nanoparticles Dispersions of carbon nanoparticles were prepared by liquid phase exfoliation of the TEGs. Sonication of the TEGs samples in tert-butanol (22 kHz and 470 W) was used for enhancement of the exfoliation process. The duration of the sonication was 1 h. Morphology of the prepared nanoparticles was estimated by transmission electron microscopy (TEM). Only freshly prepared samples of carbon nanoparticles dispersions were used for TEM studies. The TEM images of the dispersions samples were obtained using JEOL JEM-200 transmission electronic microscope.
3 Results and Discussions 3.1 Graphite Nitrate Cointercalation Compounds as Precursors of Thermally Expanded Graphite The general strategy for obtaining carbon nanoparticles used in this work included following main stages: (i) natural graphite intercalation, (ii) cointercalation of graphite nitrate, (iii) thermal expansion of GNCC and (iv) liquid phase exfoliation of TEG (for general scheme see Fig. 1). At the first stage, graphite nitrate with a regular structure was obtained. GN was further stabilized by treatment with organic compounds, namely acetic acid or its combination with formic acid, ethyl acetate and acetonitrile. As a result, stable binary and ternary GNCCs were obtained. For comparison purposes, compounds in which acetic acid was one of the cointercalants were studied. For ternary compounds, additional co-intercalants were formic acid, ethyl acetate, or acetonitrile. Structure of the synthesized binary and triple GNCCs was characterized by X-ray powder diffraction method. In obtained samples compounds of the IVth (α-phase) and of the IInd (β-phase) stages of intercalation were presented. From the results of X-ray diffraction analysis values of interplanar space filled with cointercalants species (d i ) were estimated for studied GNCCs and listed in Table 1. The determined d i values for GNCCs are increased up to 6.532–8.072 Å as compared with 3.359 Å for initial graphite. It should be noted that the d i values for ternary compounds are greater than corresponding one for a binary GNCC with acetic acid.
Carbon Nanoparticles from Thermally Expanded Cointercalates I - Intercalation
Graphite
HNO3
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III - Thermal Expantion
Graphite nitrate
CH3COOH
900 °C GNCC
t-BuOH
TEG
CH3CN
II - Co-Intercalation
Carbon nanoparticles 1h sonication IV - Exfoliation
Fig. 1. Step-by-step scheme of the carbon nanoparticles obtaining. Table 1. Structural characteristics of the GNCCs along with coefficients of thermal expansion of the studied compounds. GNCCs
Cointercalants
IV-α compound
II-β compound
2θ (°)
d i (Å)
2θ (°)
d i (Å)
K V (cm3 ·g−1 )
1
CH3 C(O)OH
24.88
7.813
27.04
6.532
350
2
CH3 C(O)OH + HC(O)OH
24.76
7.899
26.88
6.590
380
3
CH3 C(O)OH + CH3 C(O)OC2 H5
24.52
8.072
26.88
6.590
300
4
CH3 C(O)OH + CH3 CN
24.80
7.870
26.84
6.605
350
Formation of GNCCs with organic compounds leads to additional reorganization of graphite matrix as well as to enhancement of ability to thermal expansion as compared with graphite nitrate: values of coefficient of thermal expansion (K V ), determined for graphite nitrate and studied GNCCs, are 249 cm3 ·g−1 [24] and within 300–380 cm3 ·g−1 , respectively (Table 1). Thus, the investigated GNCCs can act as effective precursors of thermally expanded graphite. 3.2 Pre-organized TEGs for Carbon Nanoparticles Thermally expanded graphite samples based on GNCCs were obtained using thermal shock (at 900 °C) method of heating. Coefficient of thermal expansion (KV ), estimated for studied GNCCs were within 300–380 cm3 ·g−1 (see Table 1) and ternary GNCC with acetic and formic acids was found to demonstrate the highest KV value. The structure of the obtained TEGs was studied by powder X-ray diffraction. Reflexes typical for the graphite phase are observed on the X-ray diffraction patterns of studied samples. The absence of uncharacteristic reflexes for crystalline graphite on TEGs XRD patterns (Fig. 2) was established, residual intercalation compounds were also not detected, that is there are no reflexes typical for the initial GNCCs. Broadening, splitting and shift of the reflexes towards smaller angles on the X-ray diffraction pattern occur upon GNCCs thermal expansion. TEG is less ordered as compared with initial graphite (Fig. 3).
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Fig. 2. X-ray diffraction patterns of investigated TEGs samples in I (0.5) – 2θ coordinates (numbers on figure correspond to precursor GNCCs listed in Table 1).
Fig. 3. X-ray diffraction pattern in I (0.5) – 2θ coordinates and representative SEM images of natural graphite (GT-1), used as a starting material for GNCCs and TEGs obtaining.
Figure 4 shows typical SEM images of studied TEGs at the same magnifications. The width of visible pores in such structures varies from 1.0 to 20 μm. It should be noted that individual few-layered fragments are already present in the samples. The change in the microstructure of graphite during the formation of GNCCs and subsequent thermal expansion is determined by the nature of graphite, intercalants, the stage of intercalation, and thermal expansion conditions [2, 17, 21]. TEGs based on studied GNCCs are characterized by a clear cellular structure. There are no significant differences in the structure and morphology of the studied samples. All TEGs obtained from binary
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and triple GNCCs with organic compounds demonstrate a structure pre-organized for the generation of few-layer graphenes. (a)
(b)
(c)
(d)
Fig. 4. Representative SEM images of studied TEG samples at the same magnifications. Correspondence to the original GNCCs listed in Table 1 is the following: (a) – 1; (b) – 2; (c) – 3; and (d) – 4.
3.3 Morphology of Carbon Nanoparticles from Expanded Graphite Nitrate Cointercalants Dispersions of carbon nanoparticles were prepared by liquid-phase exfoliation of the TEGs samples in tert-butanol, assisted with sonication (22 kHz, 470 W, 1 h). The dispersions obtained were investigated by TEM method. Overall scheme of carbon nanoparticles obtaining is summarized in Fig. 1. An analysis of the dispersions obtained in tertbutanol by the TEM method revealed that carbon nanoparticles of various morphologies can be obtained by this technique. Most commonly, a mixture of particles is formed, which includes graphene-like particles of various sizes (both single-layer and few-layer) and small amorphous carbon nanoparticles. Morphology of the observed nanoparticles was revealed to depend on the intercalant’s nature in GNCCs, used for the TEGs obtaining. Figure 5 show representative TEM images of carbon nanoparticles, obtained by TEGs exfoliation in tert-butanol induced
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by ultrasound. During exfoliation under the same conditions (frequency and power of sonication, duration of exposure, medium) for TEGs obtained from studied GNCCs, the formation of few-layer graphenes with different planar sizes was observed. The TEM analysis of dispersion, based on thermally expanded binary GNCC-1 (with acetic acid), showed the presence of few-layer graphenes with a thickness from 2 to 10 layers and with planar dimensions up to 5 μm. Exfoliation of thermally expanded triple GNCC-2 under these conditions leads to the formation of few-layer graphenes with sizes of 1– 5 μm. Processing of TEG samples based on GNCC-3 and GNCC-4 makes it possible to obtain dispersions of few-layer graphenes with planar dimensions up to 20–50 μm (fragments are presented in Fig. 5c, d). In the case of thermally expanded GNCC-4, individual particles up to 75 μm were observed. (a)
1
(b)
1
(c)
1
(d)
1
Fig. 5. Representative TEM images of carbon nanoparticles, obtained in tert-butanol by sonication of the TEGs samples and corresponding SAED patterns. Correspondence to the original GNCCs listed in Table 1 is the following: (a) – 1; (b) – 2; (c) – 3; and (d) – 4.
In the case of TEG, based on GNCC-2 with acetic and formic acids, the resulting dispersion, in addition to few-layer graphenes, contained also a significant amount of small amorphous graphene-like particles, as well as particles with obvious surface defects (Fig. 6).
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1
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1
Fig. 6. TEM images of carbon nanoparticles obtained in tert-butanol by sonication of the TEG on the base of GNCC-2 with acetic and formic acids.
Since the exfoliation of graphite via sonication involves formation of surface defects (kinks, bends) [25], the presence of small amorphous particles in the dispersion can be caused by the destruction of nanoparticles with a large number of surface defects. The appearance of oxygen-containing groups on the surface due to partial oxidation also contributes to the destruction of nanoparticles with the formation of small polyaromatic fragments [26]. It was shown previously [27] that exfoliation under similar conditions of a ternary GNCC with acetic and formic acids leads to the generation of few-layer graphenes, nanoscrolls as well as small graphene-like particles in the system. The presence of a large number of visible defects on the surface of the resulting nanoparticles was also noted. These data are in good agreement with the results of exfoliation of TEG, based on GNCC with acetic and formic acids, observed in present work. Therefore, the formation of particles in the system with a large lot of visible surface defects may be due to the morphological features of the initial GNCC, which was used to obtain TEG.
4 Conclusions The structure of TEGs based on graphite nitrate cointercalation compounds with organic substances is pre-organized for the formation of few-layer graphene particles. The presence of few-layer graphenes in the exfoliation products of TEG samples in tert-butanol medium has been established by the transmission electron microscopy method. It was shown that varying the cointercalates nature in the precursor GNCCs makes it possible to regulate planar sizes of few-layer graphenes as well as degree of formed amorphous nanoparticles. Exfoliation of TEGs, based on triple GNCCs with acetic acid and ethyl acetate as well as with acetic acid and acetonitrile, lead to generation of the few-layer graphenes with planar sizes up to 20–50 μm. Formed amorphous nanoparticles in this case were of small account. The presented results revealed TEGs, based on graphite nitrate cointercalation compound with acetic acid and its derivatives to be a promising source of carbon nanoparticles. Exfoliation of TEGs in liquid media by sonication can be simple and effective route for carbon nanoparticles obtaining under mild conditions.
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References 1. Pandey, I., Tiwari, J.D., Saha, T.K., Khosla, A., Furukawa, H., Sekhar, P.K.: Black carbon paper based polyanthraquinone coated exfoliated graphite for flexible paper battery. Microsyst. Technol. 28(1), 59–67 (2022) 2. Ivanov, A.V., et al.: Gas permeability of graphite foil prepared from exfoliated graphite with different microstructures. J. Mater. Sci. 56(2), 4197–4211 (2021) 3. Pajarito, B.B., Cayabyab, C.A.L., Costales, P.A.C., Francisco, J.R.: Exfoliated graphite/acrylic composite film as hydrophobic coating of 3D-printed polylactic acid surfaces. J. Coat. Technol. Res. 16, 1133–1140 (2019) 4. Berdyugina, I.S., Steksova, Y.P., Shibaev, A.A., Maksimovskii, E.A., Bannov, A.G.: Thermal degradation of epoxy composites based on thermally expanded graphite and multiwalled carbon nanotubes. Russ. J. Appl. Chem. 89, 1447–1453 (2016) 5. Yang, S., Wang, Q., Wen, B.: Highly thermally conductive and superior electrical insulation polymer composites via in situ thermal expansion of expanded graphite and in situ oxidation of aluminum nanoflakes. ACS Appl. Mater. Interfaces. 13(1), 1511–1523 (2021) 6. Sobolˇciak, P., et al.: Thermally conductive polyethylene/expanded graphite composites as heat transfer surface: mechanical, thermo-physical and surface behavior. Polymers 12(12), 2863 (2020) 7. Kartel, M.T., Tolmachova, V.S., Cherniuk, O.A., Hrebelna, Yu.V., Sementsov, Yu.I.: Expanded graphite as the best sorbent for hydrocarbons. IOP Conf. Ser.: Earth Environ. Sci. 1126, 012025 (2023) 8. Voitash, A.A., et al.: Thermally expanded graphite: sorption properties and carbon nanoparticles obtaining. In: Parinov, I.A., Chang, S.H., Kim, Y.H., Noda, N.A. (eds.) PHENMA 2021. Springer Proceedings in Materials, vol. 10, pp. 47–52. Springer, Cham (2021). https://doi. org/10.1007/978-3-030-76481-4_5 9. Dong, X., et al.: A graphitized expanded graphite cathode for aluminumion battery with excellent rate capability. J. Energy Chem. 66, 38–44 (2022) 10. Chen, X., et al.: A novel approach for synthesis of expanded graphite and its enhanced lithium storage properties. J. Energy Chem. 59, 292–298 (2021) 11. Zhang, D., Zhang, W., Zhang, S., Ji, X., Li, L.: Synthesis of expanded graphite-based materials for application in lithium-based batteries. J. Energy Storage 60, 106678 (2023) 12. Chen, X., et al.: Three-dimensional catalyst systems from expanded graphite and metal nanoparticles for electrocatalytic oxidation of liquid fuels. Nanoscale 11, 7952–7958 (2019) 13. Lan, R., Su, W., Li, J.: Preparation and catalytic performance of expanded graphite for oxidation of organic pollutant. Catalysts 9(3), 280 (2019) 14. Murugan, P., Nagarajan, R.D., Shetty, B.H., Govindasamy, M., Sundramoorthy, A.K.: Recent trends in the applications of thermally expanded graphite for energy storage and sensors – a review. Nanoscale Adv. 3, 6294 (2021) 15. Wang, Z., Huang, G., Jia, Z., Gao, Q., Li, Y., Gu, Z.: Eutectic fatty acids phase change materials improved with expanded graphite. Mater. (Basel Switz.) 15(19), 6856 (2022) 16. Yao, Y., Cui, Y., Deng, Z.: Phase change composites of octadecane and gallium with expanded graphite as a carrier. RSC Adv. 12(27), 17217–17227 (2022) 17. Ivanov, A.V., Maksimova, N.V., Kamaev, A.O., Malakho, A.P., Avdeev, V.V.: Influence of intercalation and exfoliation conditions on macrostructure and microstructure of exfoliated graphite. Mater. Lett. 228, 403 (2018) 18. Bannov, A.G., et al.: Highly porous expanded graphite: thermal shock vs. programmable heating. Mater. (Basel Switz.). 14(24), 7687 (2021) 19. Saidaminov, M.I., et al.: Thermal decomposition of graphite nitrate. Carbon 59, 337–343 (2013)
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20. Berestneva, Yu.V., Raksha, E.V., Voitash, A.A., Arzumanyan, G.M., Savoskin, M.V.: Thermally expanded graphite from graphite nitrate cointercalated with ethyl formate and acetic acid: morphology and physicochemical properties. J. Phys.: Conf. Ser. 1658, 012004 (2020) 21. Chen, P.-H., Chung, D.D.L.: Viscoelastic behavior of the cell wall of exfoliated graphite. Carbon 61, 305–312 (2013) 22. Jin, S., et al.: Low-temperature expanded graphite for preparation of graphene sheets by liquid-phase method. J. Phys: Conf. Ser. 188, 012040 (2009) 23. Raksha, E.V., et al.: Carbon nanoparticles from thermally expanded graphite: effect of the expansion conditions on the derived nanoparticles morphology. In: Parinov, I.A., Chang, S.H., Soloviev, A.N. (eds.) Physics and Mechanics of New Materials and Their Applications. Springer Proceedings in Materials, vol. 20, pp. 14–23. Springer, Cham (2023). https://doi. org/10.1007/978-3-031-21572-8_2 24. Davydova, A.A., et al.: Synthesis and properties of graphite nitrate cointercalation compounds with carboxylic acid esters. Russ. J. Inorg. Chem. 66, 324–331 (2021) 25. Li, Z., et al.: Mechanisms of liquid-phase exfoliation for the production of graphene. ACS Nano 14(9), 10976 (2020) 26. Tapia, J.I., Larios, E., Bittencourt, C., Yacamán, M.J.: Carbon nano-allotropes produced by ultrasonication of few-layer graphene and fullerene. Carbon 99, 541–546 (2016) 27. Raksha, Elena, et al.: Carbon nanoparticles from graphite nitrate cointercalation compounds with carboxylic acids. In: Parinov, I.A., Chang, S.-H., Kim, Y.-H., Noda, N.-A. (eds.) PHENMA 2021. SPM, vol. 10, pp. 37–45. Springer, Cham (2021). https://doi.org/10.1007/ 978-3-030-76481-4_4
Comparative Study of Photocatalytic Activities of Sn- or F-doped and Sn-F Co-doped TiO2 Nanomaterials M. G. Volkova(B)
and E. M. Bayan
Chemistry Department, Southern Federal University, 7, Zorge Str., Rostov-on-Don 344090, Russia [email protected]
Abstract. Sn-doped TiO2, F-doped TiO2 and Sn-F co-doped TiO2 materials (5.0 at.% Sn, 5.0 at.% F) were synthesized by sol-gel method and calcined at different temperatures. All of obtained materials are nanoscale, particles size is 12–40 nm, which depends on composition and calcination temperature. High photocatalytic activity is characteristic for all materials both under UV and visible light irradiation. Sn-F co-doped TiO2 nanomaterials are characterized by higher photocatalytic activity, than Sn- or F-doped TiO2 . Keywords: Sol-gel Method · Nanomaterials · Titanium Dioxide · Photocatalyst
1 Introduction Nanoscale titanium dioxide is currently one of the most common materials for various applications due to its relative cheapness and safety. Powder materials, based on TiO2 in various modifications, are widely used for water purification [1, 2]. Since the band gap (3.2 eV) does not allows one to use TiO2 -based materials as photocatalysts when irradiated with visible light, various attempts have been made to narrow the band gap to use the solar energy. Among the most common methods are particle size reduction, obtaining materials with different structures [3], doping with metals [4] or nonmetals [5], co-doping with cationic and anionic additives [6]. When choosing an additive, ideas about the proximity of ionic radii are used. For example, the ionic radius of Ti4+ is 0.605 Å, so Cr3+ (0.615 Å), Co2+ (0.65 Å), Sn4+ or Ni2+ (0.69 Å), etc. can serve as a reasonable choice as a modifying additive. In the case of doping with Sn4+ ions, doping with an isovalent cation occurs, in addition, SnO2 , like TiO2 , crystallizes in the rutile structure [7], which contributes to the inclusion of Sn4+ in the titanium dioxide crystal lattice and can lead to the formation of the rutile phase at lower temperatures compared to other materials. Different authors have shown the successful use of Sn4+ for titanium dioxide modification [8–10]. The introduction of nonmetals into TiO2 based materials leads to a change in the electronic structure and a decrease in the energy of the band gap. In accordance with the proximity of the electronic structure and radii, doping with fluorine [5], nitrogen [11] and carbon [12] is the most preferable. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 48–55, 2024. https://doi.org/10.1007/978-3-031-52239-0_5
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As anionic additives, one of the most promising materials is F− due to the proximity of the ionic radii of fluorine and oxygen. This circumstance allows us to expect the inclusion of fluorine in the oxygen position in the structure of titanium dioxide. For example, the introduction of 5 wt.% F− allowed to decompose methylene blue (MB) by 84% when exposed to laser irradiation (280 nm) for 60 min [13]. Some authors have noted that doping with nonmetals has a preference over metals doping due to the higher stability of the obtained materials, simpler doping process and greater photocatalytic activity [14]. Material containing 5 mol.% of fluoride anions [15] decomposed 65% of MB in 4 h of visible light irradiation. Increasing the concentration of the additive to 10 mol.% shows the best photocatalytic activity because 91% of MB is destroyed under the same conditions. The authors have shown that the improvement of photocatalytic activity is achieved with an increase in the introduced fluoride anions, which is explained by an increase in the adsorption of MB and the absorption of visible light. However, with an increase in the concentration to 15%, photocatalytic activity deteriorates somewhat, which may be due to a decrease in the recombination of photoelectron – hole pairs in TiO2 during F− doping due to the creation of oxygen vacancies and Ti3+ surface states. Another promising way to increase photocatalytic activity is co-doping. Among the options for co-doping, cation-anion co-doping seems to be the most promising due to the creation a defective structure that improves the recombination rate of electronhole pairs. For example, previously, some authors have performed a study of the Sn-FTiO2 materials properties, but the authors [16] have studied materials, containing high concentrations of additives. It was shown that by reducing the particle sizes to 55.82 nm for undoped TiO2 and 42.17 nm for 4Sn-20F-TiO2 and increasing the surface area, doped materials exhibit higher photocatalytic activity: MB was destroyed by 60 and 86%, respectively, whereas methyl orange by 65 and 77% under visible light irradiation for 150 min. According to the authors, higher values for MB are achieved due to its nature, and the simultaneous introduction of tin and fluorine ions leads to an increase in the photocatalytic activity by creating intermediate energy levels between the valence band and the conduction band, which provide capture centers for photogenerated electrons. Therefore, the aim of this study was to compare the morphology, particle size and photocatalytic activity of materials, doped with a small amount of tin (4+), or fluorine (1−) ions with the co-doped material Sn-F-TiO2 .
2 Research Method 2.1 Materials Titanium tetrachloride TiCl4 , tin chloride pentahydrate SnCl4 ·5H2 O, sodium fluoride NaF and 25% NH3 ·H2 O were all chemically pure grade. The concentration of the introduced tin (4+) and fluorine (1−) ions was 5 at.%. 2.2 Preparation of Materials The synthesis was carried out by the sol-gel method according to the previously described method [17] by hydrolysis of TiCl4 . Precipitation of titanium and tin hydroxides was
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carried out together by adding an ammonia solution to a neutral medium. After washing from chloride anions, the gel was dried together with sodium fluoride at 100 °C. The composition of the materials is shown in Table 1. Calcination was carried out in a muffle furnace with a heating rate of 10°/min at temperatures of 500–900 °C with an interval of 100 °C. Thus, materials containing 5 at. % Sn4+ and/or F− , were calcined at 500, 600, 700, 800, 900 °C and designated as Sn-TiO2 -600, F-TiO2 -600 and Sn-F-TiO2 -500, Sn-F-TiO2 -600, Sn-F-TiO2 -700, Sn-F-TiO2 –800, Sn-F-TiO2 –900. 2.3 Characterization The phase composition was determined by X-ray diffraction (XRD) phase analysis using diffractometer ARL X’TRA on Cu Kα1 radiation (a wavelength of 0.1540562 nm). The photocatalytic activity was assessed by the residual concentration of a model pollutant, namely an organic azo dye of the cationic type of methylene blue (MB). To assess the photocatalytic activity, the concentration of MB was 20 mg/l, both for studying in the UV and in the visible light range. Medium pressure mercury lamp (125 W) was used as a UV radiation source. A commercial photocatalyst Degussa P25, containing 20% rutile and 80% anatase, was selected as a comparison material. Photocatalytic activity in the visible light range was studied under irradiation with a fluorescent light lamp (40 W, the wavelength is about 450 nm). Residual concentration of MB was studied using the spectrophotometer UNICO 1201, l = 10 mm, wavelength was 670 nm. The relationship between concentration and absorption is described by the equation y = 0.0891x + 0.0489 (R2 = 0.9881), where y is the absorption of the solution, x is the residual concentration of the model pollutant (mg/L). The photodegradation of MB was defined as C/C 0 , where C is the residual concentration and C 0 is the initial concentration.
3 Results and Discussion All the obtained nanomaterials were studied by the XRD method. Typical XRD-patterns are shown in Fig. 1, all the data of the XRD study are summarized in Table 1. XRD patterns of synthesized nanomaterials with various additive contents, calcined at 600 °C, are shown in Fig. 1a. For synthesized materials Sn-TiO2 , F-TiO2 and Sn-FTiO2 , 7 peaks were found in the range of 2 theta from 20 to 60°, corresponding to the planes (011), (013), (004), (112), (020), (015), (121), what indicates the formation of an anatase phase (ICSD 202242). The crystallinity of the anatase phase monotonically decrease with an increase in the concentration of the introduced additives. XRD patterns of the Sn-F-TiO2 material, calcined at various temperatures are shown in Fig. 1b. The anatase structure is found up to 600 °C; heat treatment at 700 °C leads to the formation of a mixture of the anatase and rutile phases. The anatase phase crystallinity increases with an increase in the calcination temperature, which is typical for these materials [17, 18]. With an increase in the calcination temperature, regardless of the additives introduced, an increase in the particle sizes, calculated by the Scherrer equation, is observed (Table 1), which may be due to an increase in the phase crystallinity, as well as an increase in the proportion of agglomerated particles.
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Fig. 1. XRD patterns of synthesized nanomaterials: (a) 1 – F-TiO2 -600, 2 – Sn-TiO2 -600, 3 – SnF-TiO2 -600, anatase and rutile, and XRD patterns of synthesized material Sn-F-TiO2 (b), calcined at different temperatures. Anatase hkl peaks are black, rutile – red.
Table 1. Dependence of the phase composition and particle sizes of Sn-F-TiO2 nanomaterials on the calcination temperature. Calcined temperature, °C
Sample after calcination
Structure
Crystallite size, nm
500
Sn-F-TiO2 -500
A
14
600
Sn-F-TiO2 -600
A
15
700
Sn-F-TiO2 -700
R
30
800
Sn-F-TiO2 -800
R
40
900
Sn-F-TiO2 -900
R
45
The introduction of Sn4+ and F– leads to a decrease in the temperature of the anataserutile phase transition, but to a lesser extent than for doped materials. For example, for Sn-TiO2 and F-TiO2 materials, a mixture of anatase and rutile is shown only at a calcination temperature of 900 °C. For F-doped materials, the pure rutile phase is observed only when the calcination temperature increases to 1000 °C [5]. A decrease in the temperature of the anatase-rutile phase transition during co-doping may be due to the tetragonal crystal structure of the rutile type is thermodynamically stable for SnO2 and an increase in the tin concentration in the TiO2 structure accelerates the transition of TiO2 to the rutile modification [8]. Thus, all materials contain a phase, attributed to titanium dioxide or a mixture of phases, representing crystalline modifications of TiO2 (Table 1). The formation of tin fluoride or other compounds in this system is not observed. Then the photocatalytic activity was studied for nanomaterials, calcined at 600 °C, since the samples are not sufficiently crystallized at 500 °C, and the particle size and the proportion of agglomerated particles increase at 700 °C and above, which leads to a decrease in photocatalytic activity. It should be noted that the degradation process of the model pollutant under UV light irradiation takes up to 30 min (Fig. 2) and a
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sharp decrease in the concentration of the pollutant (more than 50%) after 10 min of contact with the catalyst. All materials are characterized by a higher photocatalytic activity than the commercial photocatalyst Degussa P25, and the best photocatalytic activity is the nanomaterial, co-doped with tin and fluorine ions. The photocatalytic activities of Sn-doped and F-doped materials are close, and the F-TiO2 material is more photocatalytically active. These results allow us to conclude the advantage of F-anion doping and Sn-F-co-doping over cationic doping with Sn4+ ions.
Fig. 2. Photocatalytic activity of materials calcined at 600 °C under UV light irradiation.
When irradiated under visible light, the tendency is almost completely saved: the best photocatalytic activity is shown for the co-doped materials (see Fig. 3). F-doped materials have slightly worse photocatalytic activity, the lowest is shown for Sn-TiO2 . However, regardless of the type of additive, it is possible to achieve the destruction of the model dye by more than 70% in 3 h of visible light irradiation. According to the photocatalytic activity data, rate constants were calculated for materials from the ratio ln(C/C 0 ) = kt, where C 0 is the initial concentration of the pollutant, C is the concentration at time t, k is a rate constant The data are shown in Table 2. It is shown that the highest rate constant k is 0.126 for the Sn-F-TiO2 material, when irradiated with UV light, which is more than 2 times greater than that for the commercial photocatalyst Degussa P25. Compared with materials doped only with tin (4+) or fluorine (−) ions, this value is also 2 and 1.5 times higher, respectively, which confirms the advantage of co-doping. In the visible light range, the tendency shown persists, namly the largest constant (0.021) is shown for the material Sn-F-TiO2 . However, in this case, for Sn-doped and F-doped materials, the values of the constants are close, the difference is about 20%.
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Fig. 3. Photocatalytic activity of materials calcined at 600 °C under visible light irradiation.
Table 2. Rate constant values for MB degradation by TiO2 -nanomaterials under UV and visible light activation. Material
Rate constant, k (min−1 ) UV
Visible
Degussa P25
0.057
–
5Sn-5F-TiO2
0.126
0.021
5Sn-TiO2
0.067
0.012
5F-TiO2
0.112
0.014
4 Conclusion Thus, nanoscale powder materials with a particle size of 12–45 nm were synthesized. For Sn-TiO2 and F-TiO2 materials, the anatase structure is preserved during heat treatment up to 800 °C, whereas for Sn-F co-doped materials, the anatase-rutile transition temperature occurs already at 700 °C. All synthesized nanomaterials under UV light irradiation have a higher photocatalytic activity than a commercial photocatalyst. When comparing the photocatalytic activity of doped and co-doped materials, it was found that the best properties are shown for Sn-F-TiO2 material. In the visible light range, the co-doped material also exhibits the best photocatalytic properties, which allows us to conclude the advantage of cation-anion doping.
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Acknowledgement. The authors are grateful to Southern Federal University for the opportunity to perform their initiative research.
References 1. Lee, S.Y., Park, S.J.: TiO2 photocatalyst for water treatment applications. J. Ind. Eng. Chem. 19(6), 1761–1769 (2013) 2. Al-Mamun, M.R., Kader, S., Islam, M.S., Khan, M.Z.H.: Photocatalytic activity improvement and application of UV-TiO2 photocatalysis in textile wastewater treatment: a review. J. Environ. Chem. Eng. 7(5), 103248 (2019) 3. Bayan, E.M., Lupeiko, T.G., Volkova, M.G., Kostenikova, A.S., Pustovaya, L.E., Fedorenko, A.G.: Hydrothermal two-step synthesis of titanate nanotubes. In: Parinov, I.A., Chang, S.-H., Gupta, V.K. (eds.) PHENMA 2017. Springer Proceedings in Physics, vol. 207, pp. 79–86. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78919-4_6 4. Ahmetovi´c, S., et al.: Examination of the doping effects of samarium (Sm3+ ) and zirconium (Zr4+ ) on the photocatalytic activity of TiO2 nanofibers. J. Alloy. Compd. 930, 167423 (2023) 5. Bayan, E.M., Lupeiko, T.G., Kolupaeva, E.V., Pustovaya, L.E., Fedorenko, A.G.: Fluorinedoped titanium dioxide: synthesis, structure, morphology, size and photocatalytic activity. In: Parinov, I.A., Chang, S.-H., Jani, M.A. (eds.) Advanced Materials. Springer Proceedings in Physics, vol. 193, pp. 17–24. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-560 62-5_2 6. Kanakaraju, D., anak Kutiang, F.D., Lim, Y.C., Goh, P.S.: Recent progress of Ag/TiO2 photocatalyst for wastewater treatment: doping, co-doping, and green materials functionalization. Appl. Mater. Today 27, 101500 (2022) 7. Wategaonkar, S.B., et al.: Influence of Tin doped TiO2 nanorods on dye sensitized solar cells. Materials 14(21), 6282 (2021) 8. Alves, A.K., Berutti, F.A., Bergmann, C.P.: Visible and UV photocatalytic characterization of Sn–TiO2 electrospun fibers. Catal. Today 208, 7–10 (2013) 9. Xiufeng, Z., Juan, L., Lianghai, L., Zuoshan, W.: Preparation of crystalline Sn-doped TiO2 and its application in visible-light photocatalysis. J. Nanomater. 2011, 1–5 (2011) 10. Sun, J., Wang, X.L., Sun, J.Y., Sun, R.X., Sun, S.P., Qiao, L.P.: Photocatalytic degradation and kinetics of Orange G using nano-sized Sn(IV)/TiO2 /AC photocatalyst. J. Mol. Catal. A 260(1–2), 241–246 (2006) 11. Bayan, E.M., Pustovaya, L.E., Bayan, Y.A., Lupeiko, T.G.: Optimization of the synthesis conditions of N-doped TiO2 nanoparticles. In: Parinov, I.A., Chang, S.H., Soloviev, A.N. (eds.) Physics and Mechanics of New Materials and Their Applications. Springer Proceedings in Materials, vol. 20, pp. 24–33. Springer, Cham (2023). https://doi.org/10.1007/978-3-03121572-8_3 12. Pi˛atkowska, A., Janus, M., Szyma´nski, K., Mozia, S.: C-, N-and S-doped TiO2 photocatalysts: a review. Catalysts 11(1), 144 (2021) 13. Yu, C., Jimmy, C.Y., Chan, M.: Sonochemical fabrication of fluorinated mesoporous titanium dioxide microspheres. J. Solid State Chem. 182(5), 1061–1069 (2009) 14. Ghaffari, M., Huang, H., Tan, P.Y., Tan, O.K.: Synthesis and visible light photocatalytic properties of SrTi(1–x) Fex O(3−σ ) powder for indoor decontamination. Powder Technol. 225, 221–226 (2012) 15. Yu, W., et al.: Enhanced visible light photocatalytic degradation of methylene blue by F-doped TiO2 . Appl. Surf. Sci. 319, 107–112 (2014)
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16. Ancy, K., et al.: Visible light assisted photocatalytic degradation of commercial dyes and waste water by Sn–F co-doped titanium dioxide nanoparticles with potential antimicrobial application. Chemosphere 277, 130247 (2021) 17. Bayan, E.M., Lupeiko, T.G., Pustovaya, L.E., Volkova, M.G.: Synthesis and photocatalytic properties of Sn–TiO2 nanomaterials. J. Adv. Dielectrics 10(01n02), 2060018 (2020) 18. Bayan, E.M., Lupeiko, T.G., Pustovaya, L.E.: Optimization of synthesis of nanosized titanium dioxide powder materials from Peroxo titanium complex. Russ. J. Phys. Chem. B 13, 383–388 (2019)
Silicon-on-Silicon Oxide Metalens: Design and Fabrication Aspects E. Yu. Gusev1(B) , S. P. Avdeev1 , S. V. Malokhatko1 , V. S. Klimin1 , V. V. Polyakov1 , S. Wang3 , X. Ren2 , D. Chen2 , L. Han3 , Z. Wang3 , W. Zhang2,3 , and O. A. Ageev1 1 Southern Federal University, Taganrog, Russia
[email protected]
2 School of Optoelectronic Engineering, Qilu University of Technology (Shandong Academy of
Sciences), Jinan, China 3 Laser Institute, Qilu University of Technology (Shandong Academy of Sciences), Jinan, China
Abstract. A metalens operating at 4 μm is numerically designed. By adjusting the diameters of the pillar array from 500 to 1000 nm, the metalens achieves 2π-phase modulation with over 75% transmission efficiency. Further analysis indicates that the designed metalens achieves expected axial dispersion, but the transmission efficiency shows strong dependency on the incident angle. Then fabrication aspects are shown. Firstly, conditions of silicon-on-silicon oxide structure fabrication are studied in terms of plasma deposition and annealing temperature and their effect on the physical and mechanical (roughness, grain size, stress) and optical properties (refractive index) of the layers. Both should be adjusted to provide enhanced features of metalens. In particular, refractive index could be tuned in range from 1.45 to 1.9 and from 3.71 to 5.15 for silicon oxide and silicon layer, respectively. Finally, experimental dependences of microstructure formation on the silicon surface by plasma chemical etching technique in a combined discharge of fluoride plasma are presented. Thus, structures with a height of 246 nm and a relatively low roughness of 0.63 nm were obtained. Keywords: Metasurface · Metalens · Transmission Efficiency · Phase Modulation · Roughness · Stress · Refractive Index · Plasma Deposition · Plasma Etching
1 Introduction The large size, weight and high cost of optical systems limits the possibilities of implementing optical measurements in future intelligent and autonomous applications in many industries [1]. Metasurfaces are considered as candidates for replacing conventional refractive lenses with ultrathin miniature optical elements [2]. So modern micro- and nanofabrication techniques should be used for their fabrication. The work aims to combine design modeling and simulation approaches and plasma fabrication techniques for creation of metasurfaces and metalenses based on them.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 56–67, 2024. https://doi.org/10.1007/978-3-031-52239-0_6
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2 Design Principles Before designing metasurface devices, it is necessary to first determine the materials, required to prepare the metasurface, including the substrate material and the materials of the nano elements. All metasurface devices, designed in this work, operate in transmission mode, so the principles of high refractive index and low loss should be followed when selecting nanomaterials. The loss of all dielectric metasurface can be almost negligible, and it can be compatible with existing CMOS, directly using semiconductor manufacturing methods for production and processing [3, 4]. At present, the commonly used high refractive index materials for all dielectric metasurface include silicon [4], germanium [5], tellurium [6], etc. Various problems in current optical systems can be perfectly solved by using all dielectric metasurface devices [7, 8]. Metasurface devices cannot only reduce the volume of the system, make the structure of the system simple, but also have high transmission efficiency. To obtain efficient of all dielectric metasurface devices, the complex refractive index (n = n + ik, where n is the refractive index, k is the extinction coefficient) of the selected material must meet two conditions: high refractive index and low loss, namely n > 2 and k = 0. When the absorption loss can be ignored, the extinction coefficient approaches 0, and the transmission efficiency is higher. High refractive index ensures that the incident wave can achieve a phase coverage of 0 – 2π when passing through the metasurface. In the near-infrared band, the refractive index of silicon is relatively high, about 3.5. The light in the near-infrared band can almost completely pass through the silicon medium, with a loss close to 0 [9], meeting the material selection rules for nano micro elements. When the distance of light propagation in a uniform medium with a refractive index of n is r, the phase change generated can be expressed as: φ=
2π nr. λ
(1)
Nanopillars are selected as the unit structure of metasurface devices as it can simplify the design and manufacturing. In the near-infrared band, nanopillars can support both electric and magnetic dipole Mie-type resonances [10]. By changing the geometric parameters of silicon nanopillars, these two dipole resonances can overlap at the same frequency, and the electric and magnetic fields cancel each other out in backscattering, which suppresses backscattering to some extent. Therefore, these two dipole resonance mechanisms provide the basis for achieving phase shift of 0 – 2π under high transmission with almost zero reflection. In addition, the cross-section of cylindrical nanopillars exhibits central symmetry, and metasurface devices composed of nanopillar arrays exhibit polarization insensitivity [10]. The period (P), height (H), and radius (R) of silicon nanopillar arrays are three basic design parameters that control the amplitude and phase of all dielectric metasurface transmission. In order to maintain high transmission efficiency, these parameters need to be designed. The structural unit is shown in Fig. 1. The absorption loss of silicon in this structural unit can be ignored. The difference in refractive index between the silicon nanopillar and the silicon dioxide substrate is relatively high, allowing the beam to generate the required phase modulation after passing through the metasurface device.
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Fig. 1. Unit cell for the designed metalens.
On a metasurface based on transmission phase, each structural unit acts as an element. At a certain height, the effective refractive index neff can be adjusted by adjusting the radius R of the nanopillar. When there are no nanopillars on the substrate, the effective refractive index is the smallest, which is the same as the refractive index of the surrounding medium, usually air (n = 1). When the nanopillar fills the entire unit (2R = P), the effective refractive index is the highest, equal to the refractive index of the nanopillar material. Ignoring the Fabry Perot effect, the cumulative phase shift, generated by the light from the structural unit, is: φ=
2π neff H . λ
(2)
When the working wavelength and the height of the nanopillar are determined, the phase shift can be controlled by adjusting the filling factor (FF), that is adjusting the radius of the nanopillar. To ensure that the filling factor FF of the structural unit is between [0, 1] and achieve a phase difference of φ = 2π, the height of the nanopillar must meet the following requirements during design: H>
λ neff
(3)
Adjusting the effective refractive index of the structural unit can reduce the thickness of the device to some extent. When the effective refractive index is maximum, the height H of the nanopillar is minimum: neff = n − 1
(4)
In the formula, λ represents the working wavelength, and n is the refractive index of the nanopillar. The spacing between adjacent nanopillars is represented by P, also known as the lattice constant. In the design, the lattice constant P needs to meet the Nyquist sampling criterion, namely: P
> C FE (the case of enrichment of the semiconductor surface with majority charge carriers [3, 12]), it is almost equal to the capacitance of the ferroelectric layer. Considering the case that the thickness of the SBN-60 film significantly exceeds the thickness of STO, and their dielectric constants are of the same order of magnitude, it can be argued that this capacitance corresponds to the SBN-60 layer only. Taking into account the amplitude and phase of the local piezoresponse (see Fig. 4), the obtained C(U) dependences of
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Fig. 4. (a) Phase and (b) amplitude dependences of the local piezoresponse for SBN-60/Si heterostructure.
Fig. 5. C(U) dependence of the Ag/SBN-60/Si(001) heterostructure at room temperature and at different bias voltages.
the Ag/SBN-60/Si(001) heterostructure (Fig. 5) make it possible to determine the state of the MSEP structure due to the manifestation of the field effect [12] and calculate the relative dielectric constant of the ferroelectric layer from the C FE values. All C(U) dependences of the Ag/SBN-60/Si(001) heterostructure in the analyzed temperature range and fields had the form of a high-frequency capacitance-voltage characteristic of the MFES structure. Examples of the C(U) dependences of the Ag/SBN60/Si(001) heterostructure at different temperatures are shown in Figs. 6 a-e. In the positive field region, the MFES structure goes into a low capacitance state (C min ~ 2.1 pF), due to the contribution of the semiconductor (the surface of the semiconductor goes into the depletion and inversion state). In the negative region, the capacitance of the structure increases and reaches a plateau, which is equal to the capacitance of the
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FE film (the surface of the semiconductor is in an enriched state). The C max value was subsequently used to calculate the relative dielectric constant ε of the SBN-60 layer at f = 10 kHz. The resulting ε(T ) dependence, together with similar dependences for the SBN-61 single crystal [13] and the SBN-60/Pt/MgO(001) film [14], is presented in Fig. 6 f.
Fig. 6. Dependences of the C(U) of Ag/SBN-60/Si(001) heterostructure at (a) (−190) °C, (b) (−140) °C, (c) (−70) °C, (d) 0 °C, (e) 50 °C; and (f) dependences of ε(T ) for SBN61 single crystal, SBN60/Pt/MgO(001) and Ag/SBN-60/Si(001) heterostructures at 10 kHz frequency.
For the SBN-61 single crystal, in the region of the ferroelectric → paraelectric (PE) phase transition a maximum in the ε(T ) dependence was recorded at T m –76 °C. For SBN60 films on a Pt/MgO(001) substrate and on a Si(001) substrate, the ε(T ) dependence is
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blurred, while the Tm value increases up to 117 °C for epitaxial SBN60/Pt/MgO(001) film, and Tm value decreases down to 53 °C for polycrystalline textured SBN60/Si(001). It is seen that the best temperature stability of ε in the (–190… + 200) °C temperature range is observed in the Ag/SBN-60/Si(001) heterostructure. The ε values of the SBN60 films varied from 500 to 832. From the viewpoint of practical application, it should be noted that the ε values at temperatures below 20 °C exceed those in a single crystal, which is a consequence of the size effects in ferroelectric films (primarily, increased temperature blurring of the ferroelectric-paraelectric phase transition region).
Fig. 7. (a) Topography and (b) surface potential obtained immediately after polarization of the SBN-60/Si(001) heterostructure.
The ferroelectric nature of the SBN-60 film was also demonstrated by KM technique. In the contact mode of current lithography, a “herringbone tree” image was polarized on a 10 × 10 μm2 fragment of the film surface by + 10 V. Then the surface potential signal of the sample was recorded by the KM technique (see Fig. 7). The KM data shows that it is able to form a clear polarized region according to a given pattern, which confirms the ferroelectric properties of the SBN-60/Si(001) film. The obtained polarized image was visualized for 1.5 h, but after 2 h no traces of polarization were recorded on the surface potential scans by KM technique. There was a gradual reverse polarization switching to the initial state over time. Moreover, the unpolarized film surface, as well as after depolarization, has negative potential values, which may be associated with spontaneous polarization directed from the film surface to the substrate.
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4 Conclusion Based on the performed studies, we can make the following conclusions: 1. According to X-ray diffraction and scanning probe microscopy, it has been established that SBN60 films, grown on a Si(001) substrate by RF cathode sputtering in an oxygen atmosphere, are c-oriented, single-phase and pure. There was slight unit cell strain in the SBN60 film along the polar axis in comparison with the bulk material. 2. The microstructure analysis shows that the SBN60 film surface is characterized by good homogeneity with a 17.6 nm roughness. There were no pores, cavities, impurity phases, and other surface defects. The vertical and lateral piezoresponse data of the SBN60 film showed that the polarization was mainly directed vertically, and there were practically no signals of lateral polarization. 3. The ferroelectric nature of SBN60 film on Si substrate at room temperature was confirmed by piezoresponse force microscopy and polarization switching spectroscopy mode, as well as by the possibility of creating stable polarized states in the SBN60 film surface. 4. According to the capacitance-voltage features of the MFES structure based on the SBN60 film, the dependence of ε(T), at f = 10 kHz in the temperature range 5. of (–190… + 200) °C was calculated and it was shown that the resulting films had good temperature stability for relative dielectric constant ε, and at low temperatures the values of ε exceed those in the SBN61 single crystal. 6. It is advisable to use the obtained results for the synthesis, research, and development of functional elements, based on pure nanosized SBN60 films, grown on semiconductor substrates. Acknowledgements. This work was funded as part of a state order to the Federal Research Centre The Southern Scientific Centre of the Russian Academy of Sciences, project No 122020100294– 9. The work was carried out using the equipment of the Centre of Collective Use of SSC RAS (https://ckp-rf.ru/catalog/ckp/501994/).
References 1. Gritsenko, V.A., Islamov, D.R.: Physics of dielectric films: mechanisms of charge transport and physical basis of memory devices. Parallel, Novosibirsk, Russia (2017). (In Russian) 2. Mikolajick, T., et al.: Next generation ferroelectric materials for semiconductor process integration and their applications. J. Appl. Phys. 129, 100901 (2021) 3. Zhang, J.J., Sun, J., Zheng, X.J.: A model for the C-V characteristics of the metal–ferroelectric–insulator–semiconductor structure. Solid-State Electron. 53, 170–175 (2009) 4. Shvartsman, V.V., Lupascu, D.C.: Lead-free relaxor ferroelectrics. J. Am. Ceram. Soc. 95, 1–26 (2012) 5. Pavlenko, A.V., Zinchenko, S.P., Stryukov, D.V., Kovtun, A.P.: Nano-sized Films of BariumStrontium Niobate: Features of Obtaining in a High-frequency Discharge Plasma. Structure and Physical Properties. SSC RAS Press, Rostov-on-Don, Russia (2022). (In Russian) 6. Ivanov, S., Kostsov, E.G.: Uncooled thermally uninsulated array element based on thin strontium barium niobate pyroelectric films. IEEE Sens. J. 20, 9011–9017 (2020)
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7. Mukhortov, V.M., et al.: The field effect in a metal–ferroelectric–semiconductor system of multilayer ferroelectric films with various structure types. Phys. Solid State 60, 1786–1792 (2018) 8. Olsen, G.H., Aschauer, U., Spaldin, N.A., Selbach, S.M., Grande, T.: Origin of ferroelectric polarization in tetragonal tungsten-bronze-type oxides. Phys. Rev. B 93, 180101 (2016) 9. Stryukov, D.V., Matyash, Y.Y., Strilec, N.V., Pavlenko, A.V., Malomyzheva, N.V.: Fabrication of the BiFeO3/Sr0.6Ba0.4Nb2O6/SrRuO3/MgO(001) heterostructure, as well as features of its crystal structure, surface morphology, and physical properties. Phys. Solid State 65, 1783– 1791 (2023). (In Russian) 10. Damodaran, A.R., et al.: New modalities of strain-control of ferroelectric thin films. J. Phys.: Condens. Matter 28, 263001 (2016) 11. Pavlenko, A.V., Ilyina, T.S., Kiselev, D.A., Stryukov, D.V.: Phase composition, crystal structure, dielectric and ferroelectric properties of Ba2 NdFeNb4 O15 thin films grown on a Si(001) substrate in an oxygen atmosphere. Phys. Solid State 65, 587–593 (2023). (In Russian) 12. Gurtov, V.A.: Solid-State Electronics: Tutorial. PetrGU Press, Petrozavodsk, Russia (2004). (In Russian) 13. Pavlenko, A.V., Stryukov, D.V., Kovtun, A.P., Anokhin, A.S., Ivleva, L.I., Lykov, P.A.: Synthesis, structure, and dielectric characteristics of Sr0.61Ba0.39Nb2O6 single crystals and thin films. Phys. Solid State 61, 244–248 (2019) 14. Makinyan N.V., Pavlenko A.V.: Dielectric characteristics of heteroepitaxial Sr0.60Ba0.40Nb2O6 thin films grown on a Pt(001)/MgO(001) substrate. Phys. Solid State 65, 1957–1963 (2023). (In Russian)
Influence of the Elemental Composition and Thermodynamic Background on the Processes of Phase Formation in Solid Solutions of the Triple System NaNbO3 − KNbO3 − BiFeO3 M. O. Moysa1(B) , L. A. Shilkina1 , K. P. Andryushin1,2 , A. S. Pavlenko1 , and L. A. Reznichenko1 1 Research Institute of Physics, Southern Federal University, 194, Stachki Avenue,
Rostov-on-Don 344090, Russia [email protected] 2 Kh. Ibragimov Complex Institute of the Russian Academy of Sciences (CI RAS), Grozny, Russia
Abstract. Using conventional ceramic technology, solid solutions (SS) of the system (1 – x)(Na0.5 K0.5 )NbO3 – xBiFeO3 with x = 0.05, 0.20, 0.55, 0.95 were obtained by solid phase synthesis in two stages. It has been established that the structure of the studied solid solutions critically depends on technological regulations: solid solutions close in composition to (Na0.5 K0.5 )NbO3 are characterized by the highest sintering temperatures, low relative densities and their discrepancy in trial and experimental samples at one and the same temperature; and those close in composition to BiFeO3 are characterized by low sintering temperatures, high relative densities and practically identical relative densities of trial and experimental samples. Two phase transitions have been identified: (i) in fact the decomposition of SS, takes place in the interval 0.00 < x < 0.05, (ii) transition from the C-phase to the Rh-phase occurs in the interval 0.55 < x < 0.95. Keywords: KNN · BFO · Solid Solutions · Ceramics · Crystal Structure · Solid Phase Synthesis
1 Introduction In recent years, there has been a rapid increase in the speed of development of industrial production, invariably accompanied by an increase in man-made impacts on humans and the environment. In this regard, several countries have adopted legislative acts [1, 2], restricting the use of toxic elements that negatively affect living organisms in the real sectors of world economies, including the element bases of electronics and electrical engineering. In ferroelectric materials science, it is lead, which belongs to the number of extremely harmful elements, leading the list of priority, especially toxic substances of the first hazard class. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 78–84, 2024. https://doi.org/10.1007/978-3-031-52239-0_8
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The base of the vast majority of modern ferroelectric materials, serially mastered by world practice and used in solid-state electronics, are, leadmost often,‘ multicomponent Bα“ )O3 ]n , , the most containing oxides [3–8] of the form Pb(Zr,Ti)O3 − 4n=1 [Pb(B1−α common technology of which includes solid phase synthesis and sintering at high temperatures. All operations of the technological process are inevitably associated with environmental pollution and harmful effects on living organisms: work with dusty dispersed crystalline substances in the preparation of charges. The latter also include procedures for heat treatment of charges during synthesis and sintering, accompanied by evaporation of volatile (lead) components; mechanical processing of ceramics; discharge of wastewater from the preparation of charges and grinding of synthesized products. Anthropogenic lead emissions have exceeded natural emissions these days, and technophilicity was 2 × 109 . Lead compounds belong to poisons with cumulative effects, the accumulation of which in the body has a toxic effect on a person, leading to damage of almost all vital organs: the central and peripheral nervous system, blood, skin, etc. [9]. In this regard, it becomes relevant to reduce environmental damage, including the environmental burden on human health during the production of ferroelectric materials by excluding aggressive elements from the composition of chemical compositions. To this end, we conducted experiments on the selected promising lead-free object (system (1 – x)(Na0.5 K0.5 )NbO3 (KNN) – xBiFeO3 (BFO)) to establish the influence of the elemental composition and thermodynamic background (preparation conditions) on the processes of phase formation in solid solutions (SS) of this system, one of the extreme components of which (KNN) is a ferroelectric, the other (BFO) is a multiferroic with high Curie (1123 K) and Neel (643 K) temperatures.
2 Objects, Methods of Obtaining and Studying Samples The objects of study were SS of the (1 – x)(Na0.5 K0.5 )NbO3 – xBiFeO3 system with x = 0.05, 0.20, 0.55, 0.95. The samples were obtained by solid phase synthesis in two stages and sintered using conventional ceramic technology: T synth.1 = (1120 – 1170) K, τ = 4 h, T synth.2 = (1120 – 1190) K, τ = 4 h; T sint. = (1140 – 1430) K, τ sint = 2 h depending on the composition. The selection of optimal conditions for the preparation of SS was carried out on a series of test samples (Ø 12 mm × 2 mm) obtained at different firing temperatures, with X-ray control of the phase composition and density of sintered samples. Samples of the experimental batch in the form of columns Ø 12 mm and height 36 mm were sintered at the selected optimal T sint , providing a higher density of blanks, their purity or minimizing the number of parasitic phases. The experimental density of ceramic samples, ρ exp , was determined by hydrostatic weighing in octane. At the same time, ρ exp was calculated using the formula: ρ exp = (ρ oct m1 )/(m2 – m3 + m4 ), where ρ oct is the density of octane, m1 is the mass of the dry billet, m2 is the mass of the billet, saturated with octane, m3 is the mass of the saturated billet, suspended in octane, m4 is the mass of the suspension for the billet. The relative density was calculated by the formula: ρ rel = (ρ exp /ρ x-ray ) × 100%. X-ray studies were carried out by powder diffraction using DRON-3 and ADP diffractometers (CoKα radiation; Bragg–Brentano focusing scheme). Structural parameters
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were calculated according to standard methods [10]. Crushed ceramic objects were studied, which made it possible to exclude the influence of surface effects, stresses and textures that occur during the production of ceramics. The accuracy of determining the parameters of the perovskite cell: linear δa = δc = δb = ± 0.05%; angular δα = δβ = ±5% and volume δV = ± 0.07%. The X-ray density, ρ X-ray , was found by the formula: ρ X-ray = 1.66M/V, where M is the weight of the formula unit in grams, V is the volume of the perovskite cell in Å3 .
3 Results and Discussions Figure 1, 2, 3 and 4 shows radiographs of SS in the range of angles, 2θ = (20–85)° and individual peaks 200, 220, 111 on an enlarged scale, depending on the T sint. (trial samples (TS) and samples of experimental batches (EB)). Figure 1, 2, 3 and 4 shows that the radiographs vary not only depending on the content of bismuth ferrite, but also on the T sint. of ceramics within the same composition. The conditions of isomorphism are not fulfilled in this system (the conditions of crystal chemical isomorphism in the A-sublattice by electronegativity are not fulfilled [11]), therefore, the structure of the studied SS critically depends on technological regulations. Our long-term experience with ferroceramics with a perovskite-type structure has shown that in single-phase SS, there is consequence of an increase in T sint . It is an increase in their uniformity, which on the radiograph is expressed in a decrease in the width of the X-ray peaks.
Fig. 1. X-ray patterns of test samples of the SS of the system under study with x = 0.05, sintered at different temperatures, and the experimental batch at the selected optimal sintering temperature (lower X-ray patterns).
SS of (Na0.5 K0.5 )NbO3 with 0.05 BiFeO3 is a mixture of two phases–monoclinic (M) and cubic (C). In the result of the superposition of diffraction reflections, corresponding to each phase, the overall peak profile varies with an increase in T sint. , depending on the quantitative ratio of phases. The coexistence of the M- and K-phases is identified at
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T sint. = 1373 K in the trial sample and in the experimental batch at T sint. = 1423 K. With other T sint. The blurring of diffraction peaks does not give a clear idea of the symmetry of the SS and it is designated as pseudocubic (PSC).
Fig. 2. X-ray patterns of test samples of the SS of the system under study with x = 0.20, sintered at different temperatures, and the experimental batch at the selected optimal sintering temperature (lower X-ray patterns).
Fig. 3. X-ray patterns of test samples of the SS of the system under study with x = 0.55, sintered at different temperatures, and the experimental batch at the selected optimal sintering temperature (lower X-ray patterns).
On radiographs of SS with x = 0.20 (Fig. 2) at T sint. = 1373 K, the bimodality of the sizes of coherent scattering regions (CSR) is visible. So, in SS with x = 0.20, the CSR size in the direction [100], calculated from the half-width of the X-ray peak 200 using
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Fig. 4. X-ray patterns of test samples of the SS of the system under study with x = 0.95, sintered at different temperatures, and the experimental batch at the selected optimal sintering temperature (lower X-ray patterns).
[12], is 521 Å, and calculated from the integral width of the peak is − 291 Å. In SS with a high content of BiFeO3 , this effect is clearly not observed. Bimodality, as is known from the theory of catastrophes, indicates the loss of stability of the system. Indeed, the test sample is SS with x = 0.20 at T sint. = 1413 K decays into two SS with cell parameters a1 = 3.998 Å and a2 = 3.9773 Å, and the transition to the experimental batch (larger volume) at the same temperature converts the SS into a single-phase modulated in the direction of [110] state. The two-phase nature of this SS is preserved at higher T sint. And when moving to a larger volume of the experimental batch. The breakdown of SS into two fractions is observed at concentrations of BiFeO3 equal to 0.20 and 0.55. The most homogeneous SS is formed at x = 0.95. Indeed, the increase of T sint. Reduces the width of diffraction reflections by 50 K and leads to the absence of their splitting. Thus, the decay of SS (the coexistence of several isosymmetric phases) with a change in T sint. Within the same composition indicates that a continuous series of SS in the quasi-binary section of the system under study (1 – x)(Na0.5 K0.5 )NbO3-x BiFeO3 is not formed. Two phase transitions have been established: the first transition, which is the decay of SS, takes place in the interval 0.00 < x < 0.05, the second transition from the C-phase to the rhombohedral Rh-phase occurs in the interval 0.55 < x < 0.95. It can be noted that the increase in T sint. Promotes the decay of SS, which should not be the case when a continuous series of SS is formed with complete solubility of the components. The phase composition, cell parameters and ceramic density of the studied SS are present in Table 1.
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Table 1. Phase composition, cell parameters and ceramic density of the studied SS x
T sint , K
I pr /Iper *
Symmetry
a, Å
β/α (o )
V, Å3
ρ exp. g/cm3
ρ x-ray. g/cm3
ρ rel %
1
2
3
4
5
7
8
9
10
11
0.05
1373
–
C+M
3.9806
–
63.08
3.27
4.71
69.42
1403
–
PSC
3.9795
–
63.02
3.34
4.71
70.84
1413
3
PSC
3.9825
–
63.16
3.23
4.70
68.66
1423
–
PSC
3.9808
–
63.08
3.51
4.71
74.52
1423
1
C+M
3.981
-
63.12
3.18
4.71
67.550
1373
–
C
3.9847
–
63.27
3.74
5.25
71.23
1403
traces
PSC
3.9820
–
63.14
4.15
5.26
78.87
1413
–
≈ 40 C1 ≈ 60 C2
3.9976 3.9772
–
63.87 62.91
4.20
ρ= 5.25
80
1413
–
C
3.9833
–
63.20
3.95
5.25
75.15
1343
4 Pch
C
3.9816
–
63.11
5.84
6.56
89.01
1358
12 Pch
C
3.9780
–
62.95
5.28
6.58
80.27
1373
5 Pch
C
3.9824
–
63.16
5.31
6.55
81.0
1343
17 Pch
C
3.9827
–
63.17
5.24
6.55
79.95
1143
4
Rh
3.9652
89.45
62.34
6.11
8.14
75.04
1173
4
Rh
3.9676
89.46
62.46
6.33
8.13
77.89
1223
3
Rh
3.9659
89.48
62.38
7.00
8.14
86.02
1243
Melted
–
–
–
–
–
–
–
1223
3). The microgeometrical properties for the sample 3 were obtained on the 2 × 2 µm scanning field. During the test the tungsten probe was used, scanning velocity was 1.19 µm/s, resolution was 270 × 270 points. Therefore, results for the sample 3 were also added to the Tables 1 and 2 with the mark *.
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Note, that the smallest values of Ra , Rz and Rt were observed for the sample 2 (obtained at 3 sccm of N2 ). Analysis of the results revealed a decrease in the average maximum roughness height from sample 1 to sample 2. A sharp increase in microgeometric parameters from sample 2 to sample 3 is explained by the composition of the coating, namely a transition from a mixture of titanium nitride and titanium to a columnar structure of titanium nitride, clearly visible in the SEM image (Fig. 2c). The decrease in Ra from sample 3 to sample 4 (Fig. 3) can be explained by a decrease in the size of micropillars, located closely to each other, which is clearly visible in the SEM image (Fig. 2d).
Fig. 2. AFM surface topography of the samples: (a) 1 (1 sccm of N2 ); (b) 2 (3 sccm of N2 ); (c) 3 (5 sccm of N2 ); (d) 4 (7 sccm of N2 ).
The Aztec 3.3 SP1 (Oxford Instruments, Abingdon, UK) software, which was used to analyze the EDX spectra, made it possible to recalculate the composition of chemical elements and remove elements that could be erroneously determined from the calculations [34, 35]. The composition of the coatings included oxygen in a typical amount (see Table 3). Oxygen, most likely, was adsorbed on the surface of the coatings during the transfer of samples from the magnetron chamber to the SEM. The surface topography of the samples, obtained by using AFM, confirmed this assumption. A more developed coating structure corresponds to a larger amount of adsorbed oxygen. Table 4 shows
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Fig. 3. Dependence of Ra , Rz and Rt values on the stoichiometric ratio of nitrogen to aluminum in TiN coatings. Table 3. Element composition for the samples 1–4. Sample number
Ti (atomic %)
O (atomic %)
1
84.75
11.66
N (atomic %) 3.59
2
65.84
7.95
26.21
3*
46.37
13.22
40.41
4
46.41
15.71
37.88
the recalculation of the elements composition of coatings without considering adsorbed oxygen. Table 4. Element composition of the samples 1–4 without oxygen. Sample number
Al (atomic %)
N (atomic %)
1
91.44
2
70.4
29.6
3
53.96
46.04
4
55.43
44.57
8.56
Analysis of the EDX results revealed an almost linear dependence of the elemental composition of the coatings during the deposition process at a nitrogen flow up to 5 sccm. When the argon flow is more than 5 sccm, a process of saturation and stabilization
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Fig. 4. Dependence of the elemental composition of the deposited coatings on the nitrogen flow during the deposition process.
of the nitrogen concentration in the coating is observed simultaneously with a decrease in the deposition rate. Thus, a nitrogen flow equal to 5 cubic meters is ideal for obtaining titanium nitride coatings with close to ideal stoichiometry and maximum deposition rate under these conditions. (Fig. 4). Note, that this material behavior is typical for reactive magnetron sputtering.
4 Conclusion In this work, we investigated a set of TiN coatings, deposited at different nitrogen contents (nitrogen flow). The deposition method was reactive magnetron sputtering. The influence of this factor on the elemental composition of the coatings and the microstructure of the coatings has been studied. Stable modes of obtaining TiN coatings with nitrogen content from 8.56 to 46.04 atomic percent with an average deposition rate from 6.8 to 8.86 angstroms per second were demonstrated. The results show linear dependence of the elemental composition of the coatings during the deposition process at a nitrogen flow up to 5 sccm. When the argon flow is more than 5 sccm, a process of saturation and stabilization of the nitrogen concentration in the coating is observed in parallel with a decrease in the deposition rate. Thus, a nitrogen flow equal to 5 cubic meters is ideal for obtaining titanium nitride coatings with close to ideal stoichiometry and maximum deposition rate under these conditions. This fact can be used to predict the required composition of the gas mixture under the given process parameters to obtain TiN coatings of the required thickness and with the required elemental composition. The next step of this study will be to establish relationships between the elemental composition, mechanical and tribological properties of TiN coatings. As a result, this will make it possible to obtain functionally graded coatings with a smooth change in properties with depth.
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Acknowledgements. Lapitskaya V. A. acknowledges the support of the Belarus Republican Foundation for Fundamental Research (grant No. T23RNF-132), A. L. Nikolaev acknowledges the support of the Russian Science Foundation (grant No. 23-49-10062, https://rscf.ru/project/ 23-49-10062/). The experiments were conducted in the Nanocenter of Research and Education Center ”Materials”, Don State Technical University (http://nano.donstu.ru).
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Influence of Synthesis Conditions on the Characteristics of Antimony Sulfoiodide A. E. Oparina(B)
, E. S. Medvedeva , and T. G. Lupeiko
Department of Chemistry, Southern Federal University, Rostov-on-Don 344090, Russia [email protected]
Abstract. In this paper, a semiconductor ferroelectric antimony sulfoiodide with photocatalytic activity was obtained using an optimized hydrothermal method. The influence of the precursors, time and temperature of hydrothermal treatment was studied. As a result, phases differing in purity, length and shape of particles were obtained. The sample, aged for 2 h, contained a significant number of impurities in the form of antimony sulfide. Samples, aged for 4 and 5 h, had noticeably fewer impurities. The phases, obtained after 6, 10 and 24-h exposure, had a sufficiently large yield and well-formed long rods. The results of X-ray phase analysis showed the presence of the target phase in all cases, while the longer the synthesis time, the better the crystallization of the SbSI rods. The results of scanning electron microscopy showed that the particles have the shape of a rod, which consists of smaller needles, and with increasing synthesis time their length increases. The needles thickness varies from one to several tens of microns. The study of photocatalytic properties has shown that the rods exhibit mild photocatalytic activity, compared to powder materials, which is consistent with theoretical concepts. Keywords: Hydrothermal Method · Antimony Sulfoiodide · Optimization · Photocatalytic Properties
1 Introduction The hydrothermal synthesis method is one of the practically significant methods for obtaining different materials. Even though it often requires maintaining certain temperatures and pressures the method remains used at the present time. The variability of external conditions, such as temperature, pressure, and duration of synthesis, as well as internal concentrations and forms of reagents, the proportion of mixture components, allows one to obtain different morphologies of the same substance. This is undoubtedly one of the main advantages of this method. Such antimony sulfoiodide morphologies as nanorods, nanowires and nanofilms, ranging in size from 20 to 300 nm, are grown by hydrothermal method, and used in different fields: from piezoelectric nanogenerators to creating of special textiles capable to harvest energy from human movements [1–5]. SbSI has a rather narrow band gap [6], therefore it has photocatalytic properties within the visible radiation spectrum. Photocatalytic activity in semiconductor materials
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is associated with the electron transitions into the conduction band. When a semiconductor absorbs a quantum of light, whose energy is more than energy of band gap, an electron (e− ) passes from the valence band to the conduction band and a “hole” is formed in its place, called an exciton (h+ ). Then the electron-exciton charge carrier pairs either recombine or migrate to the surface of the semiconductor and participate in oxidation-reduction reactions with adsorbed water and oxygen molecules [7]. As a result of these processes, various reactive oxygen species are formed: superoxide radicals (•O− 2 ), hydroxyl radicals (•OH), hydrogen peroxide (H2 O2 ) and singlet oxygen (O2 ) [8]. The type of dominant radical determines the photoactivity and the path of the substrate decomposition. The success of photoactivity is directly related to the particle size, which can be controlled by various external factors, such as temperature and synthesis time. It is assumed that doping of antimony sulfoiodide contributes to distortion of its crystal structure, it leads to changes of the specific surface area and, consequently, its photoactivity. Thus, the photocatalytic properties are caused by the nanostructured state of materials and depend on all factors, affecting the size, characteristics, and structure of nanoparticles. In addition, SbSI has ferroelectric properties, a low Curie temperature and a large polarization of the structure [9], which expands the possibilities of its application. Therefore, the optimization of the hydrothermal synthesis method is still relevant. In this paper, the influence of various precursors, time, and temperature of hydrothermal treatment on the phase composition, shape, crystallites size and photocatalytic activity of antimony sulfoiodide was studied to establish optimal parameters for the synthesis of high photocatalytic activity materials.
2 Materials, Synthesis and Research Methods 2.1 Materials In the current syntheses, all reagents and solvents are of analytical purity, and they were obtained from commercial sources with analytical grade and used without further purification. During the typical synthesis method 1M hydrochloric acid solution HCl, antimony chloride SbCl3 , thiourea (NH2 )2 CS and potassium iodide KI were used. 2.2 Synthesis Hydrothermal synthesis was carried out as follows: hydrochloric acid solution, crystalline antimony chloride, thiourea and potassium iodide were placed in the autoclave in the ratio of 1:1:5, respectively. Then heated in air to 160–165 ºC and kept for 2, 4, 5, 6, 10 and 24 h. At the end of the synthesis time, the autoclave was cooled to room temperature and the product was first washed with dilute hydrochloric acid, then distilled water to a neutral medium and dried at room temperature in air. As a result, small needles of dark red color with a characteristic metallic luster were obtained. At the same time, the longer the exposure time, the less the target product contained impurity orange antimony sulfide Sb2 S3 .
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2.3 Research Methods Scanning electron microscopy (SEM) and X-ray powder diffraction (XRD) were used to analyze the obtained materials. Scanning electron microscopy (SEM) was performed using a scanning electron microscope JSM-6390LA with the JEOLJED-2300 X-ray microanalysis system. In this method the studied sample is scanned with an electron beam and the intensity of the emitted quanta is measured. This intensity is converted into an electrical signal: the electrons emitted by the electron gun are accelerated to an energy from 2 to 40 keV. A set of magnetic lenses and deflecting scanning coils forms a small-diameter electron beam that is unfolded into a raster on the sample surface. When this surface is irradiated with electrons, three types of radiation are excited: reflected or back-scattered electrons, secondary electrons, X-rays, and then they are recorded and converted into an electrical signal. Thus, a scanned relief surface of the sample is obtained in the form of blackand-white image. The closer the particle is to the beam, the brighter it will appear in the resulting image. X-ray powder diffraction was performed on an ARLX’TRA X-ray powder diffractometer, Thermo ARL (Switzerland) and applied for qualitative phase analysis. Data on the interplane distances d and the relative intensities I of the diffraction pattern observed reflections are used. This is so-called set d/I of an individual substance, which is characteristic like human fingerprints. The experimental XRD patterns are compared with reference patterns from databases, and the resulting compound is identified. The most significant are the first 3 or 5 peaks with maximum intensities (bright). They are called “strong lines”, as well as several low-angle peaks with a minimum angle of 2 and a maximum d, called “long lines”. These reflections are compared in the experimental and reference XRD patterns. For a high-quality XRD pattern, there must be a 2 angle agreement of at least 0.05◦ . The agreement in intensities is usually only qualitative, the brightness of the peaks and their height are compared. The antimony sulfoiodide crystals have orthorhombic symmetry and the parameters of the crystal lattice at 25 °C are presented in Table 1. The value of Ps , Curie temperature, permittivity and band gap are also presented. The unit cell contains four molecules of antimony sulfoiodide, located along the polar c-axis. Table 1. Crystal lattice parameters of antimony sulfoiodide SbSI at 25 °C a, Å
b, Å
c, Å
Ps , μC/cm2
TC, K
ε
Band gap, eV
8.517
10.111
4.095
20
295
50000
1.8–2.0
As is known, SbSI is a ferroelectric. This is illustrated by that when cooling from a symmetric nonpolar modification, antimony and sulfur atoms shift along the c-axis, their own dipole moments cease to be compensated and spontaneous polarization occurs (Fig. 1). Colorimetry is based on indirect measurement of the substance’s concentration in colored solutions, which absorb the light, transmitted through them, the stronger the
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Fig. 1. The comparison of atomic positions of antimony coordination polyhedral (yellow – S2− , grey – Sb3+ , pink – I− ).
greater the concentration of the coloring substance. Measurements are carried out in a beam of non-monochromatic, but polychromatic narrow-spectral light, formed by a light filter. The use of various light filters with narrow spectral ranges of the transmitted light makes it possible to determine separately the concentrations of different components of the same solution. During measuring, the values of the optical density of solution are taken directly. Then, to switch to concentrations, a calibration curve is used in the coordinates “optical density – solution concentration, mg/l”, and the residual concentration of the coloring agent in the samples is determined. An organic dye of the cationic type, methylene blue (C16 H18 CIN3 S), was used as a photometric reagent. The characteristic coloring of aqueous solutions of dyes is due to the presence of a conjugated bonds system with auxochromic groups and, consequently, delocalization of bonds and charges of atoms. The longer the conjugation chain in a substance molecule, the more intense and deeper its color. Thus, the principle of photocatalysis in this case consists in the destruction of chromophore groupings of dye molecules by releasing active oxygen species: 2 C16 H18 CIN3 S + 51 O2 = 2 HCI + 2 H2 SO4 + 6 HNO3 + 32 CO2 + 12 H2 O As a catalyst, SbSI rod crystals were used, obtained by hydrothermal method at a 6 h exposure time of 160 ºC. The residual concentration of photometric reagent was determined according to the calibration curve.
3 Results and Discussion In the first experiment, several hydrothermal syntheses were carried out at a temperature of 160 ºC with a change in the synthesis time from 2 to 24 h. The obtained materials differed in phase purity, particle length and shape. The sample obtained after 2 h of synthesis, contained a significant amount of reaction intermediates, orange antimony
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sulfide and plate crystals antimony iodide. Fine particles of dark red color with a slight metallic luster were obtained after washing (Fig. 2 a, b). Before washing, the materials, synthesized within 4 h, consisted of small rods, interspersed with agglomerates resembling sea urchins, which disintegrated after washing (Fig. 2 c, d). The phase, obtained after 5 h of exposure, also had agglomerates of smaller needles, although the bulk was formed by rods (Fig. 2 e, f).
Fig. 2. Real photo (a, c, e) and SEM images with different scale (b, d, f) of SbSI, obtained after 2 (a, b), 4 (c, d) and 5(e, f) hours of exposure.
Materials, obtained after 6, 10 and 24 h of exposure, had a sufficiently large yield and well-formed long rods. At the same time, the longer the exposure time, the longer the rods were obtained (Fig. 3). The results of scanning electron microscopy showed that the particles really have a rod shape, while the rods consist of much smaller co-directional needles, and with increasing synthesis time, their length increases. Agglomerates of rods present before washing clusters of smaller ones, in many places emerging from the common center of crystallization, resembling spruce branches. The rods thickness varies from 1 to several tens of microns. The results of X-ray phase analysis showed the target phase presence in all cases, while the longer the synthesis time, the better the crystallization of SbSI crystals (Fig. 4). This indicated an increase in the severity, height, and sharpness of intensity peaks. It is assumed that the XRD patterns of materials, obtained after 10 and 24 h of exposure,
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Fig. 3. SEM images of different SbSI, obtained after 6 (g, h, i, j), 10 (k, l) and 24 (m, n) hours of exposure.
have a less clear appearance due to the complexity of samples preparation, which are long and strong needles. Then, the materials, obtained via the hydrothermal method at 160 ºC during different synthesis time, were examined for photoactivity. To do this, the 0.03 g weights from each phase were placed in cups, containing 10 ml of methylene blue dye solution. The samples were stirred in the dark for 20 min and irradiated with simulated visible light for an hour. The result is presented in the graph of the dependence of the residual concentration of methylene blue on the time of synthesis in the autoclave (Fig. 5). According to the results, the optimal synthesis time is 6 h. The resulting phase has sufficient purity from intermediate reaction products and forms long rods, and the photocatalysis efficiency is approximately 35%. This efficiency is commensurate with the efficiency of the phases obtained after 10 and 24 h of synthesis.
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Fig. 4. XRD diffractograms of samples, obtained during hydrothermal synthesis at 160 ºC and different synthesis times.
Fig. 5. Photocatalytic degradation of methylene blue.
4 Conclusion Thus, it was found that the introduction of potassium iodide instead of ammonium iodide as donors of iodine atoms results in the production of the target phase. In accordance with the results of the SbSI hydrothermal synthesis optimization, it is recommended to
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use hydrochloric acid solution, crystalline thiourea and potassium iodide to vary the size of crystallites during synthesis. Changing the exposure time leads to the production of rod crystals of different lengths. At the same time, to investigate the behavior of rods in the processes of heterogeneous catalysis, for example, photocatalysis, the optimal synthesis time in the autoclave at a temperature of 160 ºC is 6 h. This time makes it possible to obtain sufficiently long formed crystallites that exhibit photoactivity at the level of phases, obtained with a longer exposure.
References 1. Rau, H., Rabenau, A.: Crystal syntheses and growth in strong acid solutions under hydrothermal conditions. Solid State Commun. 5(5), 331–332 (1967) 2. Wang, C., et al.: Synthesis of novel SbSI nanorods by a hydrothermal method. Inorg. Chem. Commun. 7(4), 339–341 (2001) 3. Chen, G., Li, W., Yu, Y., Yang, Q.: Fast and low-temperature synthesis of one-dimensional (1D) single-crystalline SbSI microrod for high performance photodetector. RSC Adv. R. Soc. Chem. 28(5), 21859–21864 (2015) 4. Yang, Q., et al.: The synthesis of SbSI rodlike crystals with studded pyramids. J. Cryst. Growth 4(233), 774–778 (2001) 5. Shen, J., et al.: SbSI microrod based flexible photodetectors. J. Phys. D Appl. Phys. 34(53), 345106 (2020) 6. Nowak, M., Kauch, B., Szperlich, P.: Determination of energy band gap of nanocrystalline SbSI using diffuse reflectance spectroscopy. Rev. Sci. Instrum. 4(80), 4–7 (2009) 7. Gundeboina, R., Perala, V., Muga, V.: Perovskite material-based photocatalysts. In: Arul, N., Nithya, V. (eds.) Revolution of Perovskite. Materials Horizons: From Nature to Nanomaterials, pp. 251–287. Springer, Singapore (2020). https://doi.org/10.1007/978-981-15-1267-4_9 8. Wang, C., et al.: SbSI nanocrystals: an excellent visible light photocatalyst with efficient generation of singlet oxygen. ACS Sustain. Chem. Eng. 9(6), 12166–12175 (2018) 9. Nakamura, M., Hatada, H., Kaneko, Y., Ogawa, N., Tokura, Y., Kawasaki, M.: Impact of electrodes on the extraction of shift current from a ferroelectric semiconductor SbSI. Appl. Phys. Lett. 23(113), 232901 (2018)
Study of Nanosecond Repetitively Pulsed Discharges Effects on the Technological Properties of Sulfide Minerals from Low-Grade Copper-Nickel Ores Igor Zh. Bunin(B) and Irina A. Khabarova N.V. Melnikov’s Institute of Comprehensive Exploitation of Mineral Resources, Russian Academy of Sciences (ICEMR RAS), 4, Kryukovsky Tupik, Moscow 111020, Russia [email protected]
Abstract. The pulsed energy impacts are promising methods for the pretreatment of refractory mineral raw materials (refractory ores and concentration products) to increase the disintegration, softening, and liberation performance of finely disseminated mineral complexes, as well as to increase the contrast of surface properties for minerals with similar physicochemical and technological properties. In this work, we used analytical electron microscopy, electrode potential testing, sorption and flotation measurements to study changes in the surface morphology, electrochemical, physicochemical, and flotation properties of the natural pyrrhotite, pentlandite and chalcopyrite exposed to non-thermal action of the repetitive highpower nanosecond electromagnetic pulses (HPEMP) and low-temperature plasma of dielectric barrier discharge (DBD) in air at atmospheric pressure. For monomineral flotation of pyrrhotite and pentlandite, we established the optimal mode of preliminary electromagnetic impulse treatment of minerals (t treat. = 10 s), at which the contrast of their flotation properties increases in the mean on ~20%. Short-term (10 s) treatment of pyrrhotite by DBD caused the shift of its electrode potential to the region of negative values occurred, which causes the effect of a decrease in the pyrrhotite sorption and flotation activity. In the result of short-term (10 s) HPEMP pretreatment of chalcopyrite, the increased floatability of sulfide mineral from 75% up to 92% due to a greater amount of a collector, accumulated at sulfide surface, and its higher electrode potential established. Thus, the advantages of using the short-term (t treat. = 10−30 s) impulse energy impacts for improvement of the structural, physicochemical and technological properties of iron, copper, and nickel sulfides are shown. Keywords: Sulfide Minerals · Nanosecond Pulsed Discharges · Surface Properties
1 Introduction The structure and quality of reserves of Norilsk copper-nickel sulfide ores, containing platinum group metals (PGMs), indicate that as rich ores are mined, disseminated ores will become the main ore raw material, determining the development prospects of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 103–113, 2024. https://doi.org/10.1007/978-3-031-52239-0_11
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entire Norilsk Industrial District (Region, NID). The absence of world analogues in terms of industrial significance indicates the uniqueness of this geological and industrial object [1]. About 65% of the reserves of platinum metals and up to 50% of the reserves of copper, nickel and cobalt are concentrated in the disseminated copper-nickel ores of the NID deposits. Disseminated ores are highly valuable raw materials, the reserves of which are many times greater than the reserves of rich and cuprous ores. In terms of sulfide mass, disseminated ores are 3–3.5 times more enriched in PGM than high-grade ores. Moreover, per 1 ton of nickel in disseminated ores, the sulfur contains 3–4 times less than in rich ores [2]. In terms of PGM concentration, the disseminated ores of the Norilsk copper-nickel deposits exceed all copper-nickel and almost all platinum-metal deposits in the world and are comparable to the deposits of the Bushveld complex (South Africa) [3]. It is important to note that the demand and growth in the cost of precious metals is constantly growing. Pyrrhotite, chalcopyrite, cubanite, pentlandite (main); pyrite, mackinawite, violarite, sphalerite, galena, nickelite, argentopentlandite, vallerite, marcasite (minor); magnetite, titanomagnetite, ilmenite, chrome spinels, hematite (rarely) (oxide minerals) are oreforming minerals in the ore part of disseminated ores [1, 3]. Mineral varieties of the pyrrhotite type are characterized by the presence of non-stoichiometric iron sulfides of the pyrrhotite group and extremely close intergrowth (fine association) of sulfide minerals [1, 4]. Platinoids are concentrated predominantly in pyrrhotite, pentlandite and chalcopyrite; quantitatively, palladium minerals predominate over platinum minerals [3]. Platinum up to (70–80%, rel.) is distributed in mineral forms (sperrylite, isoferroplatinum, rustenburgite-autokite, etc.), palladium up to 65% (rel.) is found in isomorphic impurities of nickel-bearing minerals [1, 2]. Disseminated ores are characterized by a low content of non-ferrous metals. Therefore, the technical and economic indicators of finely disseminated ores processing are determined by the extraction of noble metals from ore to obtain enrichment products of improved quality. Pyrrhotite is the predominant mineral in sulfide copper-nickel ores of deposits in Russia and the world. This polymorphic sulfide mineral of iron (Fen Sn+1 ) degrades the quality of copper and nickel concentrates. Also, the need to solve the problem of utilizing the increased content of sulfur dioxide in the metallurgical processing arises. The flotation activity of pyrrhotite is lower than that of chalcopyrite and pentlandite, and the separation of pyrrhotite into a separate product improves the quality of the nickel concentrate [5, 6]. The use of electromagnetic pulse effects (electric discharge technologies) [7–18] as preparatory operations preceding the flotation process makes it possible to increase the efficiency of the flotation separation of sulfide minerals with similar physicochemical properties due to the directed (contrast) change in the phase composition and physicochemical properties of the sulfides surface [11, 14–16, 19, 20]. Therefore, in relation to disseminated sulfide and cuprous ores of NID, we believe that the use of Pulsed Power Technologies capabilities as operations for preparing mineral raw materials for subsequent comminution and flotation processes is quite promising. In this paper, we carried out a comparative study of the mechanisms of nonthermal action of high-power nanosecond electromagnetic pulses (HPEMPs) [8–11] and
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low-temperature plasma of dielectric barrier discharge (DBD) [12, 16, 21] in air at atmospheric pressure effect on the structural, physicochemical, and flotation properties of pyrrhotite, pentlandite and chalcopyrite extracted (isolated) from low-grade coppernickel ore. For this purpose, we used analytical electron microscopy (SEM–EDX), potentiometric titration (electrode potential measurements), sorption and flotation experiments and other methods.
2 Experimental 2.1 Minerals We used in our experiments the pyrrhotite and pentlandite ((Fe, Ni)9 S8 ; pentlanditepyrrhotite sample), extracted from copper-nickel pyrrhotite-containing ore (NID, Russia) and chalcopyrite (CuFeS2 ), taken from Dalnegorskoe mine field (the Primorsky Region, Russia). Chemical composition of minerals, wt.%: Fen Sn+1 − Fe (59.75), S (39.15), Cu (0.04), Ni (0.03); (Fe, Ni)9 S8 − Fe (32.92), S (33.67), Ni (18.3), Cu (4.7); CuFeS2 − Cu (28.55), Fe (27.53), S (29.17), Pb (3.19), Zn (1.91), Ca (0.70), Sb (0.08), Mn (0.05), and Ag (1700 ppm). The mineral samples were represented by individual grains of − 100 + 63 µm in size and plane-parallel polished sections with 10 × 10 × 4.5 mm3 in size. 2.2 Research Technique Electromagnetic pulse treatment of mineral samples was carried out on original laboratory generator equipments (ICEMR RAS; FON, Ryazan). The conditions for processing of mineral samples by high-voltage nanosecond pulses are given in [22]; the parameters of the generator of high-voltage sub-nanosecond pulses, initiating a dielectric barrier discharge in air at atmospheric pressure, are given in [21]. The nanosecond electromagnetic pulse generator (HPEMP) operates at a frequency of 100 Hz (pulse repetition rate), the output pulse amplitude is ~25 kV, the duration of the leading edge of the pulse varies from pulse to pulse within 2–5 ns, and the pulse duration varies within 4–10 ns. Video pulses of a bipolar shape are generated, pulse energy ~0.1 J, electric field strength in the inter-electrode gap is (0.5 −1) × 107 V/m, time range of the pulsed treatment of the mineral samples is t treat. = 10−150 s. We consider different modes of the existence and development of a dielectric barrier discharge (DBD) upon a change in applied stress and the frequency of pulse repetition. We establish the operating parameters of pulses that initiate a discharge at which the greatest changes in the structural-sensitive properties of minerals are observed: the length of the leading edge of a pulse is 250−300 ns, the length of pulse is 8 µs, the electrode voltage in the barrier discharge cell is 20 kV, and the frequency of pulse repetition is 15 kHz. The range of change in the duration of the DBD-treatment of the samples is t treat. = 10−150 s. The flow of a discharge current in a discharge cell was limited by one dielectric layer, and the sizes of the electrodes exceeded the length of the inter-electrode space (~5 mm). In the process of the low-temperature plasma processing of minerals, the gas temperature in the working zone of the discharge cell of the barrier discharge did
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not exceed the temperature of the dielectric barrier and remained on the order of room temperature during 10−60 s. The surface mineral structure and qualitative chemical surface composition of minerals were studied via analytical scanning electron microscopy on a LEO 1420VP (energydispersive microanalysis with spectrometer EDX Oxford INCA Energy 350), and a scanning electron microscope Hitachi Tabletop Microscope TM 4000 Plus. We measured the electrode potential (E, mV) of sulfides by potentiometric titration with simultaneous monitoring of the potential of the mineral and the pH of the medium (pH 5−12). The microhardness of the chalcopyrite samples was determined according to Vickers (HV, MPa) using a PMT-3M microhardness tester. The load on the indenter was 50–100 g, and the period of loading by indenter was 10–15 s. Electrode potential (E, mV) was measured by the potentiometric titration with simultaneous control of the mineral potential and medium pH. The medium pH within 5–12 was adjusted by additions of lime solution. The working electrodes 10 × 10 × 4.5 mm3 in size was made of pure mineral specimens with no inclusions and defects. The saturated chlorine-silver electrode was taken as a reference electrode. The virgin mineral specimen without HPEMP or DBD treatment (t treat. = 0 s) was studied to establish a relationship between E and medium pH, hereafter it was subjected to the electromagnetic pulse treatment and respective measurements were repeated. The adsorption of potassium butyl xanthate (PBX) at the mineral particle surface was evaluated from the residual concentration of the agent by UV-spectrophotometry at Shimadzu UV-1700 spectrophotometer. The effect of HPEMP-irradiation on the flotation activity of sulfide minerals was assessed by the yield of minerals into the foam product in the presence of the following flotation reagents: for pyrrhotite (pentlandite) − butyl xanthate 50 mg/L, sodium dimethyldithiocarbamate 150 mg/L, and methyl isobutyl carbinol at pH 10.5; for chalcopyrite − butyl xanthate 50 mg/L, and methyl isobutyl carbinol at pH 9.5 (CaO). Flotation experiments were carried out in a laboratory flotation machine with a 20 ml chamber on weighed portions of 1 g of minerals with 100 + 63 µm in size. The agitation time with the reagents was 1 min, and the flotation time was 2.5 min.
3 Results and Discussion 3.1 Effect of Nanosecond Electromagnetic Pulses and Dielectric Barrier Discharge on Morphology, Chemical Composition and Microhardness of Minerals Surface In the result of the generation of a pulse-periodic sequence of nanosecond spark discharges (HPEMP) in the air gap between the active electrode and the surface of the processed mineral samples, the formation of ozone occurs with a concentration of at least 0.2 mg/L. This physicochemical process intensifies the processes of sulfides surface oxidation [11, 14, 15, 19, 20]. After exposure to HPEMP (t treat. = 10−30 s), new formations of complex morphology on the pyrrhotite surface were found (Fig. 1a), which can be attributed to new formations of hydrophobic elemental (S0 ) and polysulfide (S2− n ) sulfur, iron oxides and, presumably, insoluble polysulfide’s. With an increase in the processing time (t treat. = 50−100 s), we observed the destruction of the surface films, the
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formation and unification of microtraces (autographs) of the current channels of the spark discharge. The results of SEM-EDX studies of the surface structure of pentlandite particles (Fig. 1b) also indicate the appearance of defects (breakdown channels and microcracks) and new phases on the mineral surface in the result of electric pulse treatment (HPEMP). In the X-ray spectrum (EDX) of the pyrrhotite and pentlandite surface, a peak, corresponding to oxygen, can also be traced from the localization areas of such new formations, which indicates the appearance of new hydroxides phases. These are hydroxides of iron in case of pyrrhotite and pentlandite, and hydroxides of nickel (pentlandite) (Fig. 2). On the chalcopyrite surface, low-dimensional films of iron-(Me)-deficient sulfides and oxides (hydroxides) were formed. With an increasing in the duration of processing (t treat. = 50–100 s), the films destroyed, microimprints (autographs) of the current channels of the spark discharge formed (Fig. 1c). As a result of processing mineral samples with a low-temperature plasma of a dielectric barrier discharge, characterized by high electron temperatures and a low temperature of the working gas, as well as the action of an strong electric field, micro-discharges in the DBD cell and ozone formed in electric discharges, the following changes in the surface morphology of sulfide minerals occurred. (a)
(b)
(c)
(d)
(e)
(f)
Fig. 1. (a), (b), (c) Surface morphology, new products and microdefects on (a) pyrrhotite, (b) pentlandite and (c) chalcopyrite, formed in the result of exposure to HPEMP within t treat. = 10−30 s; (d) pyrrhotite and (e), (f) chalcopyrite surface after exposure to a DBD in air (t treat = 30–50 s); scale bars: (a) 3 µm; (b), (e), (f) 10 µm, and (c), (d) 30 µm.
For pyrrhotite (pentlandite), we observed the formation of microcracks and channels of electrical breakdown, as well as the removal of microcrystalline fragments of mineral matter (microchips) from the sample surface (Fig. 1d) due to mass transfer under the action of an electric field, and, possibly, because of the effect of ponderomotive forces. In contrast to pyrrhotite, on the chalcopyrite surface, electrical breakdown microchannels
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(a)
(b)
(c)
(d)
Fig. 2. SEM-images of new products (iron oxides or hydroxides) on a surface of (a) pyrrhotite and (b) pentlandite in the result of HPEMP treatment; X-ray spectra from (c) pyrrhotite and (d) pentlandite surface; scale bars: (a), (b) 10 µm.
and paths with a complex network structure are formed (Fig. 1e); the new spherical particles 1 µm in size are observed (Fig. 1f). According to SEM-EDX data, we identified the micro and nanophases of metal oxides and/or hydroxides. Microstructural changes on the sulfide surface, caused by nanosecond electromagnetic pulses impact (HPEMP), lead to softening of the samples surface. For example, the microhardness of chalcopyrite decreased from 490 to 280 MPa in the initial and modified states at t treat. = 50 s, respectively; the relative change (decrease) in microhardness HV was ~43%. In the result of the impact of the electric field, microdischarges in the DBD cell, and the ozone formed in electric discharges, regular and irregular defects formed on the chalcopyrite surface, which reduced the microhardness of chalcopyrite by ~30% (from 490 to 340 MPa at t treat. = 50 s). 3.2 Effect of HPEMP and DBD on Physicochemical and Flotation Properties of Sulfide Minerals To improve the technology of flotation separation of minerals with similar physicochemical properties, experimental data on changes in the electrochemical properties (electrode potential) and hydrophobicity of the surface of minerals (flotation activity) in the result of energy effects are of great interest. The electrode potential is one of the most important parameters used to assess the electrochemical properties of the surface of
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minerals, which have a great influence on the process of interaction of mineral particles with flotation reagents [23]. As a result of short-term (t treat. = 10 s) treatment of pyrrhotite samples by highvoltage nanosecond pulses, a shift of the electrode potential of the mineral in the direction of negative values occurred. The maximum difference in the values of the electrode potential before and after the electric pulse treatment equal to 75 mV and was achieved in an alkaline medium at pH 10. This result is consistent with the experimental data on the effect of HPEMP on the sorption ability of the mineral, namely, a sharp shift in the electrode potential of pyrrhotite to the region of negative values, caused a decrease in the sorption of the anionic collector on the mineral. The minimum sorption of the flotation reagent butyl xanthate (decrease by 17%) on the surface of pyrrhotite was also found under a short-term pulse treatment. Preliminary electric pulse treatment of pyrrhotite for (t treat. = 10 s) caused a decrease in the hydrophobicity of the surface and floatability of the mineral in the presence of a flotation reagent (dimethyldithiocarbamate), which corresponds to the data on the highest content of oxidized ferric iron on the mineral surface. The greatest depression of pyrrhotite was achieved for mineral treated during 10 s; we found a decrease in the flotation yield of the mineral from 21% (t treat. = 0 s) to 14% (t treat. = 10 s). The electrode potential of pentlandite (t treat. = 10 s) shifted towards an increase in positive values at pH 5.5–9.0; and in the range of pH 9.0–11, the potential of pentlandite slightly decreased (by ~5−10 mV). A shift in the electrode potential of pentlandite to the region of more positive values indicates an increase in the hydrophobicity of the nickel mineral surface. We found the maximum sorption of the butyl xanthate (PBX) on pentlandite under the treatment time t treat. = 10 s. A shift in the electrode potential of pentlandite to more positive values and an increase in the amount of elemental sulfur on the mineral surface contributed to the fixation of xanthate on the sulfide. As a result of HPEMP pre-treatment of pentlandite during 10 s, its flotation activity increased from 53% (t treat. = 0 s) to 68%. We performed comparative studies of pyrrhotite and pentlandite flotability treated by nanosecond pulses series. Pre-treatment of pyrrhotite and pentlandite with nanosecond electromagnetic pulses (t treat. = 10 s) and the use in the flotation process of sodium dimethyldithiocarbamate (SDMDC) in combination with potassium butyl xanthate (PBX) led to an increase in the selectivity of flotation separation of these sulfide minerals. The difference in the recovery of minerals into the froth flotation product was 32% without HPEMP pre-treatment and 54% with pulsed effect, respectively. It should be especially noted that, the pre-treatment short-term (t treat. = 10 s) mode of sulfides, at which the maximum nickel mineral recovery into the flotation froth product was established, practically coincides with the range, where we observed the minimum pyrrhotite floatability. Dielectric barrier discharge processing caused an increase in the positive values of the electrode potential of pyrrhotite by 10−65 mV in the range of pH 5−9.5. However, for the mode of short-term (t treat. = 10 s) treatment of the mineral and, at pH 9.5–12, the largest changes in the electrode potential values of pyrrhotite were established. The shift of the electrode potential to the region of negative values (–60 mV) occurred, which caused the effect of a decrease in the sorption and flotation activity of pyrrhotite.
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The HPEMP treatment provoked the growth of the electrode potential of chalcopyrite by 25 mV on the average in pH range from 6 to 10. In the alkaline pH range pH 10−11, HPEMP treatment (t treat. = 10 s) of the chalcopyrite specimens contributed to reduction in the electrode potential on the average by 20 mV. The maximum absolute difference of the electrode potentials (ϕ) before and after electric pulses treatment is ~70 mV at pH 11. The increase in the time of the HPEMP treatment up to t treat. = 100 s resulted in the maximum of ϕ ~45 mV at pH 11. UV-spectroscopy data show the improvement of PBX sorption at chalcopyrite surfaces, treated by HPEMP. The maximum PBX sorption, higher by 22% at chalcopyrite surface, is detected at HPEMP treatment mode of t treat. = 100 s. The above results comply with the data on the influence of HPEMP treatment on the electrochemical properties of sulfide mineral: the shift of electrode potential of chalcopyrite to the more positive value domain leads to the growth of sorption of an anionic collector at mineral. In the range of the short-term (t treat. = 10 s) HPEMP treatment, we established the increased floatability of chalcopyrite from 75% up to 92% due to a greater amount of a collector accumulated at sulfide surface and its higher electrode potential. The increased time of impulse treatment provided no less than 90% chalcopyrite recovery into the froth flotation products. The improvement and directed change of sulfide minerals flotoactivity with the parameters of the pulsed and plasma effect proves the reasonability of the energy pulsed electric discharges before flotation of finely disseminated refractory ores and concentrated products. In this regard, we conducted additional studies on a low-sulfide product (charge) of finely disseminated and cuprous copper-nickel ores from the NID region. The sulfide content in the copper-nickel ore charge is about 5–6% (chemical compositions, wt.%: Cu (0.9), Ni (0.5), Fe (11.5), S (3.1)). Sulfides are mainly represented by chalcopyrite, pyrrhotite and pentlandite. The content of rock minerals (serpentine, aluminosilicates, oxides, hornfels and skarns) in the charge is 94−95%. In the result of the preliminary electromagnetic pulse treatment of a low-sulfide product of disseminated and cuprous ores (NID), the increase in the recovery of non-ferrous metals (copper and nickel) into the froth flotation product amounted to 1.7−4.1%, while simultaneously improving the quality parameters of flotation by 1.5−2%. Thus, the advantages of using the short-term (t treat. = 10−30 s) energy impacts for structural and chemical modification of the surface and physicochemical properties of iron, nickel and copper sulfides are shown. The irradiation with nanosecond high-power electromagnetic pulses and low-temperature plasma of DBD causes directed changes of the chemical and phase composition of sulfides particles surface with active participation of the ozone that is dissolved in the thin water film on the surface of mineral particles and initiates redox processes, leading to the occurrence of surface neoformations, both insoluble and soluble in water. It should be noted that relatively low energy consumption (less than 3 −4 kWh · ton−1 of the processed material), technological features of the proposed methods of energy impacts (standard conditions for temperature and pressure, the use of air as a plasma-forming gas), and high efficiency (selectivity) of influence on the functional properties of geomaterials indicate on the prospects of application of nanosecond HPEMP and low-temperature plasma of DBD to the deep processing of refractory sulfide polymetallic ores.
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4 Conclusions As a result of exposure to high-voltage nanosecond pulses (HPEMP), a sharp shift in the electrode potential of pyrrhotite to the region of negative values caused a decrease in the sorption of the anionic collector on the mineral. An increase in the content of oxidized ferric iron on the pyrrhotite surface resulted in a decrease in the surface hydrophobicity and flotation properties of the mineral. Dielectric barrier discharge treatment caused the shift of the electrode potential to the region of negative values, namely to (–60 mV) in the range of pH 9.7−12, which causes the effect of a decrease in the sorption and flotation activity of pyrrhotite. A shift in the electrode potential of pentlandite to more positive values and an increase in the amount of elemental sulfur on the mineral surface contributed to the fixation of xanthate on the sulfide. In the result of HPEMP pre-treatment of pentlandite during 10 s, its flotation activity increased from 53% (t treat. = 0 s) to 68%. Short-term (t treat. = 10 s) mode of sulfides pre-treatment by high-power nanosecond pulses, at which the maximum pentlandite recovery into the flotation froth product was established, practically coincides with the range, where we observed the minimum pyrrhotite floatability. The difference in the recovery of minerals into the froth flotation product was 32% without HPEMP pre-treatment and 54% with pulsed effect, respectively. The alteration of electrochemical properties of chalcopyrite resulted from the preliminary HPEMP treatment and the growth of the positive magnitude of the electrode potential of this mineral with the resultant improved adsorption of an anion collector (xanthate) and the floatability of chalcopyrite. The optimized mode of the preliminary HPEMP treatment of chalcopyrite was established within 10–30 s, at which the floatability of mineral improves by 10–15% on the average in the process for the monomineral flotation. The obtained experimental result indicates the advantages of the using of shortterm (t treat. = 10−30 s) electromagnetic pulse treatments (HPEMP and DBD in air under standard conditions) to increase the efficiency of the flotation separation of sulfide minerals with similar physicochemical properties in order to improve efficiency of refractory polymetallic sulfide ores processing. Acknowledgements. This work was supported in part by the President of the Russian Federation under contract number NSh−7608.2016.5 (academician V.A. Chanturiya’s Scientific School). We are grateful to the academician, Professor V.A. Chanturiya for his useful scientific advice, and to the candidate of geological and mineralogical sciences E.V. Koporulina and to the candidate of technical sciences N.E. Anashkina for their help in the experimental studies.
References 1. Genkin, A., Distler, V., Gladyshev, G., Filimonova, A., et al.: Sulfide Copper-Nickel Ores of the Norilsk Deposits. Nauka, Moscow (1981). (in Russian) 2. Alekseeva, L., Kaytmazov, N., Salaikin, Yu., et al.: Disseminated ores of Norilsk – a new approach to enrichment technology. Non-Ferrous Metals (7), 26–31 (2007). (in Russian) 3. Ryabikin, V., Torgashin, A., Shklyarik, G., Osipov, R.: Disseminated ores of the Norilsk copper-nickel deposits – a promising source of platinum metal raw materials. Non-Ferrous Metals (7), 16–21 (2007). (in Russian)
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4. Khramtsova, I., Gogotina, V., Baskaev, P., et al.: Development of technology for enrichment of rich and cuprous ores to obtain high-quality copper and nickel concentrates. Non-Ferrous Metals (7), 32–37 (2007). (in Russian) 5. Mantsevich, M., Malinsky, R., Khersonsky, M., Lapshina, G.: Search for methods to improve the grade of copper-nickel concentrates. Min. Inf. Anal. Bull. (7), 359–363 (2008). (in Russian) 6. Lesnikova, L., Kotenev, D., Datsiev, M., Bragin, V.: Advantages of Talnakh concentrator flexible processing for nickel concentrate quality control. Non-Ferrous Metals (6), 21–25 (2015). (in Russian) 7. Mesyats, G.: Pulse Power Engineering and Electronics. Nauka, Moscow (2004). (in Russian) 8. Chanturiya, V., et al.: Breakdown of refractory gold-bearing ores by the action of powerful electromagnetic pulses. Dokl. Akad. Nauk. Doklady Earth Sci. 366(5), 680−683 (1999). (in Russian) 9. Cherepenin, V.: Relativistic multiwave oscillators and their possible applications. Phys.Uspekhi 176(10), 1124−1130 (2006). (in Russian) 10. Chanturia, V.A., Bunin, I.Z.: Non-traditional high-energy processes for disintegration and exposure of finely disseminated mineral complexes. J. Min. Sci. 43(3), 311–330 (2007). https://doi.org/10.1007/s10913-007-0032-4 11. Chanturiya, V., Bunin, I., Ryazantseva, M., Filippov, L.: Theory and applications of highpower nanosecond pulses to processing of mineral complexes. Miner. Process. Extr. Metall. Rev. 32(2), 105–136 (2011) 12. Avtaeva, S.: Barrier Discharge. Research and Application. Lambert Academic Publishing, Saarbrücken (2011) 13. Kurets, V., Solovyov, M., Zhuchkov, A., Barskaya, A.: Electrodischarge Technologies for Processing and Destruction of Materials. Tomsk Polytechnic University Press, Tomsk (2012). (in Russian) 14. Chanturiya, V.: Scientific substantiation and development of innovative processes for complex processing of mineral raw materials. Gorn. Zhurnal (11), 7−13 (2017). (in Russian) 15. Bunin, I., Ryazantseva, M., Samusev, A., Khabarova, I.: Composite physicochemical and energy action on geomaterials and aqueous slurries: theory and practice. Gorn. Zhurnal (11), 77−83 (2017). (in Russian) 16. May, F., Hamann, S., Quade, A., Bruser, V.: Froth flotation improvement by plasma pretreatment of sulfide minerals. Miner. Eng. 113(11), 95–101 (2017) 17. Rostovtsev, V.I., Bryazgin, A.A., Korobeinikov, M.V.: Improvement of milling selectivity and utilization completeness through radiation modification of mineral properties. J. Min. Sci. 56(6), 1000–1009 (2020). https://doi.org/10.1134/S1062739120060125 18. Huang, W., Chen, Y.: The application of high voltage pulses in the mineral processing industry − a review. Powder Technol. 393(11), 116–130 (2021) 19. Ivanova, T.A., Bunin, I.Z., Khabarova, I.A.: On the characteristic properties of oxidation of sulfide minerals exposed to nanosecond electromagnetic pulses. Bull. Russ. Acad. Sci. Phys. 72(10), 1326–1329 (2008). https://doi.org/10.3103/S1062873808100067 20. Chanturiya, V., Bunin, I., Ryazantseva, M.: XPS study of sulfide minerals surface oxidation under high-voltage nanosecond pulses. Miner. Eng. 143(11), 105939 (2019) 21. Bunin, I.Z., Chanturiya, V.A., Ryazantseva, M.V., Koporulina, E.V., Anashkina, N.E.: Changes in the surface morphology, microhardness, and physicochemical properties of natural minerals under the influence of a dielectric barrier discharge. Bull. Russ. Acad. Sci. Phys. 84(9), 1161–1164 (2020). https://doi.org/10.3103/S1062873820090099
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Modelling of Phase Transitions in the Process of Cryopreservation of Biological Material Andrey Matrosov1(B) , Arkady Soloviev2 , Irina Serebryanaya1 Olga Pustovalova3 , and Daria Nizhnik1
,
1 Don State Technical University, 1, Gagarin Square, Rostov-on-Don 344010, Russia
[email protected]
2 Fevzi Yakubov Crimean Engineering and Pedagogical University, Simferopol, Russia 3 Southern Federal University, 8a, Milchakova St, Rostov-on-Don 344090, Russia
Abstract. This study focuses on the advancement of new techniques for cryopreservation of reproductive cells at low temperatures, including sturgeon fish. The approach is based on smart freezing process control. To ensure cryoprotectant and reproductive cells are not damaged prior to vitrification, a piezoactuator is used to stir the mixture. This technique facilitates the intensive mixing of cryoprotectant, promoting the diffusion of the substance through the cell membrane. During direct vitrification, researchers observe a phase transition phenomenon in the presence of crystal formation. This study constructs a mathematical model to investigate this process while comparing numerical experiments to in situ experiments. The results obtained have implications for selecting optimal parameters required for the low-temperature preservation of fish reproductive cells. Keywords: Cryopreservation · Thermal Conductivity · Crystallisation · Phase Transition
1 Introduction Currently, it is imperative to enhance and innovate low-temperature preservation technologies for fish reproductive cells [1–4], with particular emphasis on sturgeon [5]. These innovative technologies will not only facilitate artificial reproduction but will also enable cryobanking [6, 7] for conserving the genetic diversity of precious fish species [8]. The primary challenge encountered during cryopreservation (reducing the temperature from +4 °C to the temperature of deep freezing in liquid nitrogen, −196 °C) is safeguarding cells from cryodamage during their freezing [9–11]. Cryoprotectants may be employed for this goal, providing safeguards for cells while they are undergoing cooling. To enhance the penetration rate of the cryoprotectant inside the cell during the equilibration stage, a mechanical impact is applied to the suspension, containing the reproductive cells and the cryoprotectant. This technique has been previously reported in literature [12, 13]. This acoustic stimulus is generated by attaching a piezoactuator to the bottom of a laboratory beaker, creating a semi-passive bimorph. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 114–121, 2024. https://doi.org/10.1007/978-3-031-52239-0_12
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Field experiments demonstrate that the application of acoustic influence during equilibration increases the survival rate of sturgeon spermatozoa by 20%, compared to sperm frozen, using the traditional method [14]. Schematically, the process of vitrification (rapid freezing, more than 10 K/min) [15, 16] can be represented as follows. A vessel with suspension (reproductive cells placed in a cryoprotectant) is placed in liquid nitrogen. Such almost instantaneous freezing leads to a sharp temperature jump and crystallisation of intracellular water and formation of intracellular ice. It is the formation of ice that is the main cryo-damaging factor. This is due to the appearing ice causes an increase in the internal volume of membrane structures (nucleus, Golgi apparatus, mitochondria, endoplasmic network, lysosomes, cytoplasmic membrane, etc.) and their subsequent destruction. However, crystallisation of water present in the cell can be prevented, if a cryoprotectant is placed inside the cell. Cryoprotectant in sufficiently high concentrations modifies the crystallisation of water, turning it into an amorphous vitreous mass. This makes it possible to prevent ice formation and avoid damaging effects [17, 18] almost completely. An important step on the way to creating an intelligent control of the process of freezing reproductive cells of fish is a detailed study of the phase transition occurring in the process of vitrification. The mathematical model describing this process is called Stephan’s problem [19]. The aim of the present work is: (1) Construction of a continuum axisymmetric mathematical model of the crystallisation process considering the phase transition, but without explicit separation of the phase boundary. It is assumed that thermophysical properties on the surface, corresponding to the crystallisation temperature change abruptly during the transition from the region with the temperature above the crystallisation temperature to the region in which the temperature is lower. On this phase change surface, a temperature source, corresponding to the heat release during crystallisation, is defined; (2) Subsequent discretisation of the constructed model on the base of the finite element method and carrying out the corresponding numerical experiments. The heat source is assumed to be volumetric with a narrow Gaussian distribution. In this work, we use the FlexPDE finite element method of solving differential equations scenario models (SFedU license).
2 Research Method The mathematical model of crystallisation is founded upon the unsteady heat conduction equation for an even medium featuring two conditions, namely liquid and solid phases: ∇ · (−L · ∇T ) + cρ where
cρ = L=
∂T −S =0 ∂t
(1)
C1 , T > Tc C2 , T ≤ Tc
(2)
L1 , T > Tc L2 , T ≤ Tc
(3)
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S = Qwδ(T − Tc )
(4)
In formulae (1)–(4) the following notations are used: T is the temperature; C1 ,C2 is the heat capacities of liquid and solid phases; L1 , L2 are the thermal conductivities of liquid and solid phases; Tc is the temperature of crystallisation; S – internal heat source, associated with the crystallisation process in the vicinity of the surface with temperature Tc , ∇ is the nabla operator, δ is the Dirac delta function. The derived equations must be supplemented with appropriate boundary and initial conditions.
3 Finite Element Modelling Let us assume an axisymmetric cylindrical container with thin walls is filled with suspension at room temperature. The upper cap of the container is made of heat-insulating material, and it is placed in liquid nitrogen. Since the thickness of the glass walls of the laboratory beaker with suspension is insignificant, their thermal resistance can be neglected.
Fig. 1. Setting of boundary conditions
Because of this, the boundary conditions take the following form: on the lower end and the lateral cylindrical boundary the constant temperature of (–196 °C) is set, the upper end, closed by a thick cover, will be considered as thermally insulated (Fig. 1). The finite element method (FEM) is chosen as the method for solving the initial boundary value problem. A solution script was developed in the FlexPDE computer software. The obtained finite element mesh contains 2780 cubic triangular finite elements with 1471 nodes.
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4 Results and Discussion With the passage of time, the problem enters the stationary regime, that is the entire volume of suspension passes into the solid phase, since constant temperature and thermal insulation are set at the boundary. The interfacial boundary starts to move away from the boundaries, where constant temperature is set. Thus, Fig. 2 shows the heat map of temperature distribution at the beginning of the process (a), in the presence of two phases (b), when the entire volume of suspension has passed into the solid phase (c). Figure 3 demonstrate temperature distribution along the radius for two stages of the process.
Fig. 2. Temperature distribution in the crystallisation process: (a) the beginning of the process; (b) the presence of two phases; (c) the entire volume in the solid phase
Figures 4 and 5 show the temperature distributions as functions of time from the cylinder walls to the points lying on the vessel axis. The analysis shows that in the region of the phase boundary there are some features that arise due to the temperature jump. Thus, Fig. 4 shows the time dependences of temperature in the middle of the region vertically at points along the radius. These graphs show horizontal sections, corresponding to the temperature jump and phase transition. In points on the vessel axis sufficiently distant from the bottom end, this temperature jump is pronounced, as for example, in Fig. 5 (curves in orange and red colors) in the vicinity of the crystallisation temperature. The data of such temperature behavior can be used in assessing the quality of freezing of biological cells, as the temperature in these areas, having dropped to the crystallisation temperature, rises sharply by almost ten degrees of Celsius.
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Fig. 3. Temperature distribution along the radius: (a) at the beginning of the process; (b) in the presence of two phases
Fig. 4. Temperature distribution at selected points along the radius as a function of time
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Fig. 5. Temperature distribution in points on the symmetry axis as a function of time
5 Conclusion In the present work, a mathematical model of slurry freezing is constructed considering: (i) temperature jump at the solid-liquid phase boundary; (ii) phase transition accompanied by crystal formation. The study examines the axisymmetric scenario, where the temperature of liquid nitrogen is applied to the circular cylinder’s lateral surface and base, while its upper end is thermally insulated. The location and presence of the boundary are defined by the temperature of crystallisation and the varying thermophysical properties of the solid and liquid phases. The initial-boundary heat conduction problem is examined, wherein the heat capacity and thermal conductivity undergo a sudden change at the phase boundary. The internal sources of heat that are released during crystallisation are taken into consideration at the phase boundary. The proposed model was included in the software package of scenario models designed for solving differential equations by the finite element method FlexPDE. When solving the nonstationary problem, spatial temperature distributions at different time intervals were obtained. The time dependence of these temperature distributions at individual points of the internal volume of the cylinder was also plotted.
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The numerical analysis findings reveal the existence of certain anomalies in the spatial and temporal temperature profiles, surrounding the crystallisation boundary. Specifically, during the progression of this boundary, instances occur wherein the temperature remains constant for a period, followed by sudden temperature shifts. The results of the calculations show qualitative coincidence with the field experiment [20], which indicates the adequacy of the proposed model. Acknowledgements. This research was supported by the Russian Science Foundation (RSF), grant No. 21-16-00118.
References 1. Froehlich, H.E., Koehn, J.Z., Holsman, K.K., Halpern, B.S.: Emerging trends in science and news of climate change threats to and adaptation of aquaculture. Aquaculture 549, 737812 (2022). https://doi.org/10.1016/j.aquaculture.2021.737812 2. Jaiswal, A.N., Vagga, A.: Cryopreservation: a review article. Cureus 14(11), e31564 (2022). https://doi.org/10.7759/cureus.31564 3. Martinez-Páramo, S., et al.: Cryobanking of aquatic species. Aquaculture 472, 156–177 (2017). https://doi.org/10.1016/j.aquaculture.2016.05.042 4. Lee, Y.H., Park, J.Y., Lee, I.Y., Zidni, I., Lim, H.K.: Effects of cryoprotective agents and treatment methods on sperm cryopreservation of stone flounder, Kareius bicoloratus. Aquaculture 531, 735969 (2021). https://doi.org/10.1016/j.aquaculture.2020.735969 5. Assylbekova, A.S., Barinova, G.K., Aubakirova, G.A., et al.: Cryopreservation of reproductive cells of male Russian sturgeon. Exp. Biol. 3(92), 132–138 (2022). https://doi.org/10.26577/ eb.2022.v92.i3.011 6. Diwan, A.D., Harke, S.N., Gopalkrishna, Panche, A.N.: Cryobanking of fish and shellfish egg, embryos, and larvae: an overview. Front. Mar. Sci. 7, 251 (2020). https://doi.org/10. 3389/fmars.2020.00251 7. Di Iorio, M., et al.: The role of semen cryobanks for protecting endangered native salmonids. Front. Mar. Sci. Sec. Mar. Biotechnol. Bioprod. 9, 1075498 (2022). https://doi.org/10.3389/ fmars.2022.1075498 8. Rajan, R., Matsumura, K.: Development and application of cryoprotectants. In: Iwaya-Inoue, M., Sakurai, M., Uemura, M. (eds.) Survival Strategies in Extreme Cold and Desiccation. AEMB, vol. 1081, pp. 339–354. Springer, Singapore (2018). https://doi.org/10.1007/978981-13-1244-1_18 9. Igna, V., Telea, A., Florea, T., Popp, R., Grozea, A.: Evaluation of Russian sturgeon (Acipenser gueldenstaedtii) semen quality and semen cryopreservation. Animals 12(16), 2153 (2022). https://doi.org/10.3390/ani12162153 10. Huang, X., et al.: Effects of cryopreservation on acrosin activity and DNA damage of Russian sturgeon (Acipenser gueldenstaedtii) semen. CryoLetters 42(3), 129–136 (2021) 11. Niu, J., et al.: Effects of cryopreservation on sperm with cryodiluent in viviparous black rockfish (sebastes schlegelii). Int. J. Mol. Sci. 23(6), 3392 (2022). https://doi.org/10.3390/ijm s23063392 12. Andreev, A.A., Sadikova, D.G., Ponomareva, E.N., Krasilnikova, A.A., Belaya, M.: Method of reduction of low temperature jump of cryoprotectant solutions. Patent RU 2540598 C2 (2015). (in Russian) 13. Ponomareva, E., Firsova, A., Krasilnikova, A., Kovalenko, M., Rudoy, D.: Acousticmechanical effect on the sperm of sturgeon fish using piezoactuators. In: E3S Web of Conferences, vol. 363, p. 03020 (2022). https://doi.org/10.1051/e3sconf/202236303020
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14. Ponomareva, E.N., Krasilnikova, A.A., Belaya, M.M., Kovalenko, M.V.: Preservation of biological diversity by cryopreservation methods: experience of the Southern Scientific Center of the RAS. Mar. Biol. J. 7(3), 80–87 (2022). https://doi.org/10.21072/mbj.2022.07.3.07 15. Ponomareva, E.N., Krasilnikova, A.A., Tikhomirov, A.M., Firsova, A.V.: New biotechnological methods for cryopreservation of reproductive cells of sturgeon. South Russ. Ecol. Dev. 11(1), 59–68 (2016). (in Russian). https://doi.org/10.18470/1992-1098-2016-1-59-68 16. Ponomareva, E.N., et al.: Mathematical simulation of the acoustic effect on a cryoprotectant with fish sperm at equilibration. Biophysics 67(4), 549–558 (2022). https://doi.org/10.1134/ S0006350922040170 17. Matrosov, A.A., Nizhnik, D.A., Ponomareva, E.N., Soloviev, A.N.: Modeling the impact of piezoactuator on fish reproductive cells in the cryoprotective environment. In: State and Prospects of Agroindustrial Complex “Interagro-2022”, Rostov-on-Don, pp. 266–269 (2022). (in Russian). https://doi.org/10.23947/interagro.2022.266-269 18. Matrosov, A., Nizhnik, D., Pakhomov, V., Soloviev, A.: Diffusion of cryoprotectant through the membrane of reproductive cells during equilibration. In: Parinov, I.A., Chang, S.-H., Soloviev, A.N. (eds.) Physics and Mechanics of New Materials and Their Applications (PHENMA), vol. 20, pp. 508–514. Springer, Cham (2023). https://doi.org/10.1007/978-3031-21572-8_44 19. Blagin, A.V., Blagina, L.V., Knyazev, S.Yu., Shcherbakova, E.E., Nefyodova, N.A.: Crystallization processes in gradient fields. In: Mathematical Models and Experiment, Rostov-onDon (2021). (in Russian) 20. Ponomareva, E., Firsova, A., Kuzov, A., Korchunov, A., Kovalenko, M., Rudoy, D. Investigation of ice particle formation points during fish sperm cryopreservation. In: E3S Web of Conferences, vol. 431, p. 01042 (2023). https://doi.org/10.1051/e3sconf/202343101042
Physics of Advanced Materials
Using Quantum Dot Structure and Suitable Material for Increasing Propagation Length of Surface Plasmon Polariton Watheq F. Shneen1,2(B)
and Sabah M. M. Ameen1
1 Department of Physics, College of Science, University of Basrah, Basrah, Iraq
[email protected], [email protected] 2 College of Science, University of Kufa, An-Najaf, Iraq
Abstract. Hybrid surface plasmon polariton waveguides (HSPPWGs) are used to transmit and manipulate nanoscale optical signals as plasmonic waveguides. This increases optical communication network bandwidth and data transmission rates. Due to their unique properties, HSPPWGs are versatile photonics and integrated optics platforms. In order to achieve better results, the study used several proposed structures and simulated them using the COMSOL Multiphysics software with MATLAB. The gain in the quantum dots significantly improved the results with a choice of certain semiconductor active material. The aim is to reach more than ten times the original length (without material gain) and have good quantum confinement according to the FoM test and -confinement factor, which causes a jump in the calculated propagation length of the hybrid waveguide. Three cases result for propagation lengths in the first case High QDisk 2 nm indicates a maximum L p value of 37 μm at λ = 1550 nm and 30 μm at λ = 1300 nm. The second case involves increased propagation length in the 1300 – 1600 nm wavelength range, with L p = 55 μm at λ = 1550 nm and L p = 48 μm at λ = 1300 nm. In the third case, using AlGaAs 3.0238 in QD 3 nm height, confinement was 0.345, 0.370 for -confinement and 85.110 for FoM, while L p was 68 μm and 70 μm for wavelengths of 1300 nm and 1550 nm. Keywords: Hybrid Surface Plasmon Polariton Waveguides (HSPPWGs) · Quantum Confinement · Material Gain · FoM
1 Introduction The diffraction limit is one of the fundamental limitations of light itself [1–3]. The diffraction of light waves can limit the size and length of surface plasmons, which are electromagnetic waves that propagate along the surface of a metal and interact with light waves. Effective coupling between light and surface plasmons often requires careful design and optimization to overcome the diffraction limit and maximize the interaction between the incident light and the metal-dielectric interface. One way to overcome this limitation is an inclusion of quantum dots, which can give a material gain to overcome and reduce the effect of absorption in the metal, thereby increasing the propagation length [4, 5]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 125–139, 2024. https://doi.org/10.1007/978-3-031-52239-0_13
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Ridge-plasmonic waveguides have a metallic ridge or strip on top of a dielectric substrate. The ridge is what keeps the plasmons in place, and the dielectric substrate is what allows the waveguides to be put together. Plasmons are confined within the narrow gap between the metallic ridge and the dielectric substrate. This means that the ridge geometry provides lateral confinement while the dielectric substrate assists in vertical confinement. Dielectric-loaded plasmonic waveguides typically support hybrid modes, combining both plasmonic and dielectric waveguide characteristics. They allow for efficient coupling between the guided mode and external dielectric waveguides or optical devices. While ridge waveguides predominantly support surface plasmon polariton (SPP) modes, which are highly confined to the metal-dielectric interface. These modes offer strong field confinement and are suitable for subwavelength scales. SPPs propagate along the interface until the surface mode intensity decreases by 1/e. This direction’s imaginary wavenumber component gives the propagation length: Lp =
1 , 2Im[β]
(1)
where β is the propagation constant defined as β = k 0 neff and k 0 is the wave number in a vacuum. This definition is well established and can be found consistently in the relevant studies. The effective refractive index of the SPP mode determines mode confinement in plasmonic waveguides, which depends on waveguide material and dimensions. The decrease in the overall refractive index contrasts between the metal and the dielectric resulting in weaker confinement of the SPP mode. With weaker confinement, the SPP mode tends to spread out and its spatial extent increases, leading to a larger mode size. In other words, the SPPs become less tightly bound to the metal surface and are more likely to propagate away from the interface. When reducing the dielectric layer surrounding the metal core, the mode confinement and the effective refractive index will be increases. Therefore, waveguide geometries that can improve SPP mode metal-dielectric interface interaction like ridge, tapered, and corrugated waveguides can increase mode confinement and effective refractive index. The plasmonic waveguide geometry and material properties must be carefully engineered to maximize the effective refractive index of the SPP mode, while minimizing scattering and absorption losses to achieve strong mode confinement. To measure it accurately, we must define mode confinement using the full plasmonic field distribution. 1 W (r)dA, (2) Aeff = max{W (r)} A∞ where W (r) represents the energy density in this instance [6]: d [ω ∈ (r)] 1 1 2 2 W (r) = Re E(r)| + 2 μ0 H(r)| . 2 dω
(3)
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We must use an appropriate definition of collateral to determine the form of FoM L A because the propagation length (L p ) is well specified in [7, 8] as FoM = λ0p / Aeff0 . The ratio of the L p to the diameter of the mode field is also known as FoM: Lp FoM = A 2 πeff
(4)
For the normalized mode field area (A = Aeff /A0 ) [7]. The normalized modal area is defined as (Aeff /A0 ) [9], where A0 represents the diffraction-limited area in free space, A0 = λ2 / 4. Here λ is the vacuum wavelength [10, 11]. The relation of confinement Γcon appears as activelayer Enorm Γcon = . (5) total Enorm where Enorm is the electric field norm. Here norm means the amplitude of the electric field: Enorm = Ex2 + Ey2 + Ez2 . The Γcon value ranges between 0 and 1, then the confinement is better when the value approaches 1, and as it approaches zero the quantum confinement weakens. So, by making a comparison between the initial value and the values after applying the additions or modifications to the hybrid plasmonic structure and if the value decreased or increased and approached 1, it means that the quantum confinement has weakened or increased, respectively. Using SPPWGs in optical chip can revolutionize data communication and processing in microelectronics and computing enabling highspeed and low-power data transmission over nanoscale distances. When plasmons are used in electric circuits or in a circuit that looks like an electric circuit, they combine the small size of electronics with the amount of data that can be stored in photonic integrated circuits. Different structures for plasmonic waveguides have been suggested such as layered structures, metallic nanowires, arrays of metallic nanoparticles, hybrid wedge plasmonic waveguides, and others. By using the COMSOL Multiphysics RF Module’s Electromagnetic Waves, Frequency Domain (emw) interface to study different setups like metal-dielectric-metal (MDM) or dielectric-metal-dielectric (DMD) in different designs of the structures, DMD and MDM remain the basis for analysis and study. Orfanidis’ analytical solutions are available in [1, 12, 13]. When light is propagated through a plasmonic waveguide with embedded QDs as the QDs absorb and re-emit light, enhance the photonic material interaction and increase the effective refractive index of the waveguide. This effect can lead to an increase in the propagation length and a reduction in the propagation losses, as the light is effectively trapped in the waveguide by the QDs [5, 8, 14, 15]. The QDs in plasmonic waveguides have been used to significantly improve the propagation length and reduce losses. For example, a recent study demonstrated a hybrid plasmonic waveguide with embedded QDs that achieved a propagation length of up to 10 μm [16–18] an order of magnitude higher than conventional plasmonic waveguides. Quantum dots (QDs) can enhance the gain and extend the propagation length SPPs in plasmonic waveguides. The high metal absorption and scattering losses typically limit the propagation length of SPPs so QDs can be embedded in a dielectric layer beneath the
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metal film to enhance the gain and extend the propagation length of SPPs [19]. When a laser excites the QDs, they can efficiently transfer energy to the SPPs and amplifying the SPPs along the metal-dielectric interface [20–22]. To calculate the gain, we first need to solve the material gain equation for g(ω) [23], which contains a number of parameters in turn: 2
∞ 2 g(ω) = n cεπ em2 ω dE |Menv |2 eˆ · pcv D E Lg E , ω fc E , Fc − −∞ b 0 0 (6) i
fv E , Fv Given that the relatively long carrier dephasing time in QDs will result in significant memory effects, we also assume a Gaussian function as the line shape function for the QD gain change. The Gaussian line shape function Lg better fits the experimental gain, refractive index, and linewidth enhancement factor spectra than the Lorentzian line shape function. fc and fv denote the quasi-Fermi distribution functions for the conduction and valence bands, respectively.
⎤ 1
fc E , Fc = ∗ ei Eg +Eei +(m∗ r /me )(E −Ehi )−Fc /kB T ⎦
1+e (7) 1
fv E , Fv = ∗ ∗ ei Ehi −(mr /m )(E −E )−Fv /kB T hh hi 1+e where quasi-Fermi levels Fc and Fv represent the energy levels of the conduction and valence bands, respectively; Eg denotes the bandgap energy of a self-assembled QD; the confined energy of the QD states in the conduction and valence bands, denoted by Eei ei = E + E − E ; the refractive index change of and Ehi , respectively, is discrete: Ehi g ei hi a self-assembled QD is expressed as 2
∞ 2 2 ˆ · pcv D E Lr E , ω fc E , Fc ne (ω) = n cεπ em2 ω −∞ dE |Menv | e b 0 0 , (8) i
−fv E , Fv where all the parameters used in ne (ω) are the same as defined in g(ω). Except there is the lineshape function for the refractive index change Lr . The complex refractive index n is related to the material gain [24, 25] as n = nr − ig/2k0 .
(9)
A material’s complex refractive index, n, can be expressed as n = nr + ini , , where the free-space wave number is related to the mode wavelength by (k0 = 2π λ ), nr is the real part of the refractive index, k0 , being the wave vector of the incident light, and i is the imaginary unit. This equation is known as the Kramers – Kronig relation [26– 28], and it relates the real and imaginary parts of the complex refractive index. The imaginary part of the refractive index, which is related to the material gain, is given by: Im(n) = ni = − g2 k0 . This equation shows that the imaginary part of the refractive index is directly proportional to the material gain, with a proportionality constant of 1 − 2k0 . The relationship between the complex refractive index and material gain of QD-containing materials can be used to design and optimize optoelectronic devices, such as QD lasers, QD solar cells, and QD sensors. By adjusting the material properties
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of the QD-containing materials, such as the size and composition of the QDs, researchers can tune the optical properties of the material to achieve specific functionalities. The Hamiltonian of the quantum disk, characterized by its radius a and height h, can be expressed in the cylindrical coordinate system (ρ, φ, z) as follows: 1 ∂2 ∂2 2 1 ∂ 1 ∂ (10) + 2 2 + 2 +V H =− ∗ 2m ρ ∂ρ ρ ∂ρ ρ ∂φ ∂z The effective mass within the disk region is denoted as m∗ = m∗d , while within the barrier region it is represented as m∗ = m∗b . In a similar manner, the electric potential within the disk is denoted as Vd , while the electric potential within the barrier is denoted as Vb . By assuming the separation of variables in the solution of the Hamiltonian, it is possible to obtain an approximate wave function for the quantum disk. This can be achieved by solving the well-established problems of the two-dimensional circular potential well in the ρ − φ direction and the one-dimensional square potential well in the z-direction. In the direction of ρ − φ, a solution can be expressed in the following form eimφ C1 Jm (pρ), ρ ≤ a (11) (ρ, φ) = √ 2π C2 Km (qρ), ρ > a
where p = 2m∗d Eρ − Vd / and q = 2m∗b Vb − Eρ /. After performing mathematical manipulation on the wave equation, associated with the dot and the barrier, it is found that there exists a relationship between the constants p and q. (pa)2 +
m∗d 2m∗d (Vb − Vd )a2 2 (qa) = . m∗b 2
(12)
The barrier-dot interface is continuous, then we obtain: C1 Jm (pa)
= ∗C 2 Km (qa) ∗ C1 p/md Jm (pa) = C2 q/mb Km (qa).
(13)
where Jm (pq) and Km (qp) are the Bessel function of the first kind and the modified Bessel function of the second kind, respectively. By eliminating C1 and C2 , the eigen-equation is obtained: qaKm (qa) pa · Jm (pa) = ∗ . ∗ md Jm (pa) mb Km (qa)
(14)
The quantized constants p and q can be determined by obtaining pa and qa through a graphical solution utilizing Eqs. (12) and (14). To facilitate calculations, the potential in the disk is assumed to be Vd = 0. Under this assumption, the transverse eigen-energy Ep can be derived as Ep =
2 (pa)2 . 2mttI∗ a2
(15)
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The solution for the wave function for the z-dependence is the same as that of a quantum well. The quantized constants of kz and α can be obtained by finding kz (h/2) and α(h/2) from a graphical solution, where kz = 2m∗α (Ez − Vd )/ and α = 2m2b (Vb − Ez )/π . The constants kz and α are related as 2 2m∗d (Vb − Vd ) h 2 h h 2 m∗d kz + ∗ α = 2 mb 2 2 2 and
m∗ α 2h = m∗b kz 2h tan kz 2h for even solution d ∗ m α 2h = − m∗b kz 2h cot kz 2h for odd solution.
(16)
(17)
d
If the potential in the disk is taken as Vd = 0, the longitudinal eigen-energy Ez is obtained as Ez =
π 2 (kz (h/2))2 . 2m∗d (h/2)2
(18)
The collective eigen-energy of the quantum disk, Ed [23] can be approximated as the sum of the eigen-energies, associated with transverse and longitudinal modes: Ed = Eρ + Ez .
(19)
2 Simulation and Results The idea of the research is based on the establishment of quantum dots within the active layer close to the metal in order to benefit from the gain in increasing the SPP at the boundary between the metal and the next active layer. The active layer is a semiconductor, consisting of InP in its initial state, after which a semiconductor of trinary materials compounds locates such as InGaAs, AlGaAs, and quaternary composition InGaAsP, which can be added within the composition, where each material has its own refractive index in each case of doping. Figure 1 represents the general structure that constitutes the studied structure and illustrates the layers of quantum dots within the hybrid structure of HSPPWG, as well as a preliminary diagram of the energy levels in a quantum dot, in which the rough initial shape is also used before suggesting change and development as it appears in the context of research. A material’s conductivity and carrier transport properties are influenced by the quantity of electronic and hole states available in the conduction and valence bands. An increased density of electron states in the conduction band results in greater energy levels that electrons can occupy, resulting in improved electrical conductivity and increased electron mobility. The quantity of hole states present in the valence band denotes the potential energy levels that electrons can occupy. The quantity and arrangement of hole states influence the optical absorption, emission, and recombination processes. Efficient light absorption and emission can be achieved when there exists an adequate number of hole states.
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Fig. 1. (a) Diagram of HSPPWG and the method of building quantum dot layers; (b) shape of a quantum dot using the ComSol MultiPhysics simulation; (c) scheme of energy levels in a quantum dot.
The energy band diagram of the active region of the p-doped quantum dot is illustrated in Fig. 1. According to Harrison’s model, the relative alignment of the band edge is computed between the InAs quantum dot, the InGaAs quantum well-wetting layer, and the GaAs barrier layer [23]. The study involves the computation of the displacement of the conduction and valence band edges resulting from applying biaxial compressive strain. The biaxial strain model utilized in this study is limited in its ability to accurately represent the nonuniform strain distribution of a self-assembled quantum dot in three dimensions, as it assumes a uniform strain distribution within the quantum dot (see [29, 30]). The modal gain spectra depicted in Fig. 1 are fitted using a quantum disk model with a radius of 14 nm and a height of 2 nm. When calculating the respective quasiFermi level for the conduction band and the valence band, it is necessary to consider three electron states in the conduction band and twelve heavy holes (hh) states in the valence band: (e1-hh1), (e2-hh2) and (e3-hh3) of the QD optical transitions, either WL has one transition ewl-hwl [23]. Figure 2 demonstrates the structure from the COMSOL MultiPhysics software and Table 1 presents the structure parameters. The material gain depends on n2D. The number of quantum dots (QDs), size, shape, composition, carrier density [33], and confinement within the QDs, all these parameters affect the material gain of QD. Additionally, surface densities of the electrons (n2D) in the QD layer affect the material gain. The n2D is a crucial factor that establishes the density of carriers that can fit inside QDs and increases the SPP’s gain. In particular, the product of n2D and the optical gain coefficient (g) determines the relationship between the material gain and the SPP of QD. The material gain in the SPP for QD is directly
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Fig. 2. Hybrid structure is taken from the COMSOL MultiPhysics software.
Table 1. Parameters of hybrid structure Parameters
Values
Width of ridge
0.5 μm
Main wire width
3000 nm
Static Refractive index In1−x Gax Asy P1−y (y = 1) [31]
3.7336
Refractive index Gax In1-x As (x = 0.4) [32]
3.5279
Static Refractive index In1−x Gax Asy P1−y (y = 0)
3.5355
Refractive index Gax In1-x As (x = 0)
3.3767
Refractive index Gax In1-x As (x = 0.5)
3.1779
Refractive index InP
3.1669
Refractive index InGaAsP
3.3636
Gold Refractive index
0.187 + 10.3457i
Wavelength
1.55 μm
Metal thickness
110 nm
InP thickness
160 nm
Core thickness
0.5 μm
related to the optical gain coefficient (g) and the n2D product. During the carrier injection and extraction processes, it is possible to control the number of electrons and holes on the surface of QDs by fine-tuning these factors, such as the amount of doping as in this study and also the bias voltage. It is performed by modifying the value of the refractive index of the InP layer according to the previous parameters of Table 1 and according to appropriate material refractive indices for the materials [31, 34–38]. If replacing the material of the third layer in the plasmonic structure with a change in the material refractive index, the value of L p is equal to 23 μm at a ridge width of 500 nm for the traditional case (refractive index
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Fig. 3. Propagation length as function of ridge width for the two InGaAsP refractive indices 3.3636 and 3.2515 for TM00 and TM01 .
3.3636) of TM00 mode. The comparison (see Fig. 3) shows a decrease in the length value in the second case (refractive index 3.2515), but with stronger quantum confinement.
Fig. 4. (a) Refractive index, (b) propagation length, (c) figure of merit and (d) confinement factor as a function of wavelengths without QD.
In traditional case, before the introduction of the quantum dots, in Fig. 4b the propagation length of the 500 nm edge width was approximately 23 μm, and the amount of FoM as evident in Fig. 4c is equal to 58 for wavelength 1550 nm and the -confinement is 0.36 in Fig. 4d.
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Fig. 5. Material gain in different n2D (3.2, 8.3, 9.1, and 17.8) × 1010 cm−2 TM00 and TE00 modes for QDisk height equal to 3 nm as function of wavelength.
Fig. 6. Refractive index change for different n2D (3.2, 8.3, 9.1, and 17.8) × 1010 cm−2 TM00 and TE00 modes for QDisk height equal to 3 nm as function of wavelength.
The calculations, made by using QDisk, takes the gain value as shown in Fig. 5 and the change in refractive index in Fig. 6, then adding the values to the effective refractive index equation: n = nm + n + i(g/2k0 ), k0 = 2π/λ. Due to the electromagnetic field’s absorption and scattering by the metal and surrounding materials, SPPs suffer a loss as they move along plasmonic waveguides. However, an active layer with high material gain QD layer is positioned close to the plasmonic waveguide. In that case, the QDs’ released photons may be coupled into the SPPs, with corresponding amplification and increasing their propagation length of the Hybrid Surface Plasmon Polariton Waveguide (HSPPWG). For the wavelength range from 1200 nm to 1600 nm, in Fig. 7, it looks like, the increase happens gradually as the amount of n2D increases in height of QDisk 2nm, reaching its highest point at n2D = 17.8 × 1010 cm−2 . This value was found to be the most effective. As shown in Fig. 7, its maximum value of L p = 37 μm for a wavelength
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of 1550 nm, while L p = 30 μm for a wavelength of 1300 nm. Gain-assisted propagation of SPPs is proven in this study, which is also called SPP amplification, by adding a gain medium (QD layer) that can give the SPPs a net gain. The stimulated photon emission of the gain medium produces this net gain, which makes up for the energy loss, resulting from the SPPs’ absorption and scattering. The material gain of quantum dots determines the surface densities of electrons and holes n2D and p2D for each QD layer and enhances the propagation length of the SPP. When propagating through plasmonic waveguides, the material gain of QD can also impact surface plasmon polaritons (SPPs). Smaller QDs can accommodate more carriers and have a higher energy density, resulting in a higher material gain.
Fig. 7. Propagation length as a function of wavelength for different n2D values (3.2, 8.3, 9.1, and 17.8) × 1010 cm−2 of the fundamental in HSPPWG TM mode with active layer consists of QDisk with 3 nm height.
Similarly, a higher confinement potential can raise n2D and improve SPP gain. In turn, carrier concentrations and the wave functions of the electronic and hole states affect the surface densities of electrons n2D and holes p2D in QDs. A higher foaming ratio indicates that the confinement of the SPP increases with increasing propagation length due to the gain of the QD. A higher FoM indicates a more favorable balance between the desired SPP enhancement and the detrimental effects of losses. A higher FoM is desirable as it allows for efficient SPP propagation and minimizes the degradation of the SPP signal over longer distances. From Fig. 8a, the gotten -confinement for n2D 17.8 × 1010 cm−2 in QD 3 nm height, refers to good confined in wavelength 1300 nm and 1550 nm. The FoM is present in Fig. 8b, for n2D 17.8 × 1010 cm−2 in QD with 3 nm height. The value of FoM in the case of wavelength 1550 nm is approximately 105, which is higher than its value in the initial state without the quantum dots, whose value was less than 60. The values of FoM in most cases of wavelengths are greater than the initial state.
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Fig. 8. (a) -confinement and (b) FoM as functions of wavelength for n2D values of 17.8 × 1010 cm−2 of the fundamental in HSPPWG TM mode with active layer consists of QDisk with 3 nm height.
Fig. 9. L p for n2D (9.1, 17.8) × 1010 cm−2 in height of QD = 3 nm for the width of active layer equal to 1000 nm.
Figure 9 shows an increasing in propagation length of the HSPPWG structure, shown in Fig. 2, in the wavelength range (1300–1600) nm: L p = 55 μm at λ = 1550 nm and L p = 48 μm at λ = 1300 nm for n2D 17.8 × 1010 cm−2 . Figures 10 a, c by using refractive index of AlGaAs equal to 3.0238 in QD with 3 nm height found, for wavelengths of 1300 nm and 1550 nm, the confine was 0.345 and 0.370 for -confinement and 85 and 110 for FoM, respectively; at the same time, L p = 68 μm and 70 μm for a wavelength of 1300 nm and 1550 nm, respectively.
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Fig. 10. Characteristic of HSPP structure using different values of refractive index of AlGaAs (3.0238, 3.0953, and 3.01669) for QD with 3 nm height: (a) propagation length, (b) figure of merit, (c) -confinement.
3 Conclusion By changing the size and shape of the QDs, the interaction with any wavelength of emitted light can be controlled and specified. The refractive index of the active layer plays a significant role in determining the material gain and propagation length for the Surface Plasmon Polariton (SPP) within the Hybrid Waveguide (HWG). Reducing the refractive index of the active layer from its initial value alters the confinement mode area of the plasmonic mode, leading to an increase in L p . More layers of QDs increase the value of material gain. To achieve a redshift in wavelength, the size of the QDs needs to be increased in specific proportions. Superior outcomes can be achieved by combining different shapes of quantum dots in the layers. Additionally, the amount of quantum confinement can be enhanced by manufacturing dots of various sizes as needed within the layers of quantum dots.
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Specific Heat Capacity of Light Rare-Gas Crystals in the Model of Deformable Atoms Ie. Ie. Gorbenko1(B) , E. A. Pilipenko1,2 , and I. A. Verbenko3 1 Federal State Budgetary Educational Institution of Higher Education «Lugansk State
Pedagogical University», 2, Oboronnaya Street, Lugansk 291011, Russia [email protected] 2 Federal State Budgetary Scientific Institution «Galkin Donetsk Institute for Physics and Engineering», 72, R. Luxemburg Street, Donetsk 283048, Russia 3 Research Institute of Physics, Southern Federal University, 194, Stachki Avenue, Rostov-on-Don 344090, Russia
Abstract. The model of deformable and polarizable atoms presents ab initio calculations of the specific heat capacity of Ne and Ar crystals under and without pressure. The short-range potential takes into account both the three-body interaction and the deformation of the electron shells of dipole-type atoms in the pair and three- body approximations. The specific heat capacity was studied in a large temperature and pressure range using a dynamic matrix, based on an ab initio short-range repulsive potential and integration over ten main value points of the Chadi – Cohen method. The values of the specific heat capacity of crystalline Ne and Ar increase both under pressure and without it, if we consider the contributions of three-body forces, associated with both the overlap of the electron shells of atoms and the deformation of the electron shells. The comparison of our calculations with those of other authors and experimental data is satisfactory. Keywords: Rare-gas Crystals (RGCs) · Three-body Interaction · Many-body Interaction · Electron Shells · Heat Capacity · High Pressure
1 Introduction Rare-gas crystals (RGC) (without He) form an fcc crystal with a single atom in the unit cell. They are the simplest molecular crystals held together by Van der Waals forces. Despite the simplicity and seemingly reliable concepts of the interatomic potential, RGCs invariably attract the close attention of theorists. Firstly, this is due to the emergence of new technologies that make it possible to achieve high pressures in laboratory conditions [1, 2], which has led to intensive research into the various properties of compressed RGSs under extreme conditions. Secondly, with the difficulty of predicting their structure, fcc and hcp designs are practically isoenergetic [3], since “subtle” effects such as many-body interactions and deformation of electron shells affect their stability. Calculations of the equation of state, phonon spectra, thermodynamic characteristics, pressure-induced phase transitions in RGCs and other crystals are carried out in three © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 140–150, 2024. https://doi.org/10.1007/978-3-031-52239-0_14
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directions: (i) first-principles calculations (see [3–7] and references there); (ii) symmetry analysis, based on group theory (see, for example, [8, 9]); (iii) empirical calculations, for which the parameters are taken from experiment (see, for example, [10–12]). This division is largely arbitrary. The symbiosis of modern methods of density functional theory (DFT) with research in the second direction gives numerical results. In this case, the calculation methods are based on symmetry analysis. DFT calculations involve the use of a huge number of fitting parameters and approximations (especially for the exchange-correlation potential). Therefore, at present they are almost no different from semi-empirical ones [6]. In our works [13, 14], using non-empirical version of the quantum mechanical model of deformable and polarizable atoms (Tolpygo model, see [15] and references there), the dynamic matrix was calculated considering both one and another type of three-particle interaction, that is, due to the overlap of electronic shells and due to the deformation of electronic shells. Thanks to this, for the QIG we calculated the phonon frequencies at the necessary points of the Brillouin zone and, using the Chadi-Cohen method, the energy of zero-point vibrations in a wide pressure range. Obtaining numerical values of the specific heat capacity of compressed fcc-Ne and fcc-Ar, as well as studying the influence of three-particle interaction in the short-range potential of repulsion and deformation of the electron shells of atoms in the pair and three-particle approximations on these characteristics in a wide temperature and pressure range is the goal of this work. In order to quantify the effective contributions, four models will be used in the work: (i) model M3, based on pair interaction in a short-range repulsive potential [16]; (ii) model M3a, in which the deformation of the electron shells of the atom is added to the pair interaction in the pair approximation [16]; (iii) model MT0 , in which three-body forces are added to the pair interaction due to the overlap of electron shells; and (iv) model MT2 , in which three-body forces are added to the pair interaction due to the overlap of electron shells and deformation of the electron shells of dipole-type atoms in the pair and three-body approximations.
2 Specific Heat Capacity with Inclusion of the Three-Body Interaction and Deformation of Electron Shells The cohesive energy E coh and lattice heat capacity C V in the harmonic approximation are expressed by the following equations: 3 Ecoh = NA 3 d kωλ (k) 21 + nλ (k) , (2π ) −1 λ (1) nλ (k) = exp ωkBλT(k) − 1 , where N A is the Avogadro number, = 2a3 is the RGC unit cell volume in the fcc phase, a is the lattice parameter equal to the cube half-edge length, ωλ is the λ-branch frequency, k is the dimensionless wave vector. ωλ (k) 2 R
3 CV = nλ (k)(nλ (k) + 1) , (2) d k (2π )3 kB T λ
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where R is the universal gas constant and k B is the Boltzmann constant. To find phonon frequencies at any point in the Brillouin zone, you need to use the dynamic matrix [13]. In the Tolpygo model, used in this work, in addition to the core displacements, described by dipole momentum components pα = euα , internal degrees of freedom P, characterizing the electron shell states, are introduced. Therefore, the equations, that determine natural frequencies, can be divided into two groups:
2 M ωλk Aαβ pβ + Bαβ Pβ , (3) pα = β
2 mωλk Pα =
β
∗ Bβα pβ + Cαβ Pβ ,
(4)
where M is atomic mass, m is a some “fictive” mass of the order of the electron shell mass, since the diagonalization of the 6 × 6 matrix is simpler technically than the procedure of excluding all vector P components from the second group of the equations at m = 0, as the adiabatic approximation requires. Therefore, we will use the matrix:
(1) (2) D D , (5) D= D(3) D(4) each element of which is the 3 × 3 matrix: ⎛ ⎞ A11 (k) A12 (k) A13 (k) D1 = ⎝ A21 (k) A22 (k) A23 (k) ⎠, A31 (k) A32 (k) A33 (k) ⎛ ⎞ B11 (k) B12 (k) B13 (k) D2 = ⎝ B21 (k) B22 (k) B23 (k) ⎠, B31 (k) B32 (k) B33 (k) ⎛ ⎞ C11 (k) C12 (k) C13 (k) D3 = ⎝ C21 (k) C22 (k) C23 (k) ⎠. C31 (k) C32 (k) C33 (k)
(6)
(7)
(8)
To obtain the frequency of the required dimensionality, each of the matrix elements 2 should be multiplied by a dimension factor 2λ = ae 3 (e is the electron charge). Then e2 δαβ ((H0 + δH )μ(k) + (G0 + δG)να (k) + Fξ (k) + Eζα (k) + Vt ϑα (k)) ; Aαβ (k) = 3 a + 1 − δαβ (G0 + δG)ταβ (k) + Bχαβ (k)
(9) Bαβ (k) =
e2 a3
δαβ (hμ(k) + gνα (k)) + 1 − δαβ gταβ (k) ;
Cαβ (k) =
e2 −1 , δ A − ϕ (k) αβ αβ a3
(10) (11)
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where δαβ is the Kronecker symbol; μ(k) = 3 −
cos kγ cos kδ ; να (k) = 2 − cos kα cos kγ ; ταβ (k) = sin kα sin kβ ; γ =α ξ (k) = 3 − cos 2kγ ; ζα (k) = 1 − cos 2kα ; 1 2
γ =δ
γ
ϑα (k) = 1 − cos kα+1 cos kα+2 ; k = aK = π q. √ √ Here, H0 a 2 and G0 a 2 are the first and the second derivatives of the short-range pair repulsion potential for the equilibrium distances of the first neighbors; by analogy, for the second neighbors F = H0 (2a) and E = G0 (2a); B determines the Van der Waals interaction, h and g are the parameters of the deformation of the electron shells of dipole-type atoms in the pair approximation, χαβ (k) are the functions of k originated from the Van der Waals forces, ϕαβ (k) are the coefficients of the electric field, induced by the dipole Pl system, A is the dimensionless polarizability of atom. Parameters δG, δH, and V t describe the three-body short-range forces, caused by the overlap of electron shells (undeformed) of atoms (see [15] and Refs. there). The three-body forces, considered in [17], arise due to the deformation of the electron shell of the atom. Let us take them into account by redefining the parameters g and h. Then, using the method of diagonalization of the dynamic matrix D, we obtain phonon frequencies, taking into account the three-body interaction and deformation of the electron shell at any point in the Brillouin zone. The integrals over the Brillouin zone (1)–(2) can be calculated using the Chadi – Kohen method [18]. This method is in the replacement of the integral over the Brillouin zone by the sum of the values of the integrand in specific points (mean-value points, found by the group-theoretical methods [19]). In [18], the authors proposed a method for generating these points, based on two mean-value points k1 and k2 , to determine the desired function f (k) in the crystal: (12) f (k) = 41 3f (k1 ) + f (k2 ) ; k1 = 43 , 14 , 41 , k2 = 41 , 41 , 41 . To obtain the necessary accuracy in calculations, we need to know the values of the function f (k) at a large number of points k. Points k1 and k2 are used in [18] to generate ten stable mean-value points from which the zone average f (k) is determined with a high degree of accuracy.
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Coordinates and corresponding weights of these points for the fcc structure: 7 3 1 3 7 1 1 3 k1 = ; ; , α1 = ; k2 = ; ; ; α2 = ; 8 8 8 16 8 8 8 32 5 5 1 3 5 3 3 3 k3 = ; ; , α3 = ; k4 = ; ; ; α4 = ; 8 8 8 32 8 8 8 32 5 3 1 5 1 1 3 3 ; ; , α5 = ; k6 = ; ; ; α6 = ; k5 = 8 8 8 16 8 8 8 32 3 3 3 1 3 3 1 3 ; ; , α7 = ; k8 = ; ; ; α8 = ; k7 = 8 8 8 32 8 8 8 32 3 1 1 3 1 1 1 1 ; ; , α9 = ; k10 = ; ; ; α10 = . k9 = 8 8 8 32 8 8 8 32
(13)
The heat capacity and thermal vibrations of the RGC lattice at p = 0 in the classical Tolpygo model (the theory parameters are determined from the minimum of the meansquare deviation for ωλk ) were calculated in [20] using 408 mean-value points. Figure 1 shows the temperature dependence of the heat capacity C V (T ) of Ne and Ar, obtained from experiment at p = 0 [21, 22]. There are also our calculations using Eq. (2) in models MT2 , M3a and the classical version of the Tolpygo theory (model M1) [20]. In addition, in Fig. 1 we also indicated calculations by other authors [23].
Fig. 1. Temperature dependence of the heat capacity C V (T)/R of Ne and Ar crystals at zero pressure. ––, –– are our calculations in the MT2 and M3 models respectively; —is the calculation in the classical Tolpygo model (M1) [20]; – – – and ···· are the calculations based on the Lennard – Jones potential and extended Lennard – Jones potential respectively [23]; ✩ are the experimental data, obtained according to C p in Ne [21] and Ar [25]; – the same for Ne [22]; ★ is the direct measurement for Ar. Arrows mark the melting points T m0 = 24.6 K for Ne and T m0 = 83.8 K for Ar.
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As can be seen from Fig. 1, up to a temperature of 10 K for Ne all theoretical results are close to each other and good agreement with experimental data is observed. At temperatures from 10 K to the melting temperature T m0 , the best agreement with experiment is demonstrated by the M1 model (using parameters from [20]), as well as our calculations in the MT2 model. Moreover, calculations [23] are also close to experiment (based on the ab initio pair extended Lenard – Jones (ELJ) potential also give the results close to the experiment) [24]. From Fig. 1, it is seen that all the theoretical calculations are close to each other and agree well with the experiment at T ≈ 10 K. At temperatures from 10 K to melting temperature T m0 , the calculations in model M1 (with the parameters from [20]) and our calculations in model MT2 give the best agreement with the experiment. The calculations in [23] based on the ab initio pair extended Lenard–Jones (ELJ) potential also give the results close to the experiment [24]. As can be seen from our results (Fig. 1), quantity C V (T )/R, calculated with the allowance for three-body forces, appeared due to the overlap of electron shells. A deformation of electron shells of dipole-type atoms in the pair and three-body approximations (MT2 ) is almost superposed on the calculation in model M1 (the plots are intersected in several points), approximating it to the experimental values as compared to the calculations in model M3a. Let us consider in more detail the agreement of our C V (T )/R calculations with the experimental data, calculated from C p and C V [22, 25] and presented in Table 1, for compressed Ne and Ar. Comparing the errors (relative to experiment) of the γMT2 and γM3 models, it is clear that the contributions of three-body forces due to the overlap of the electron shells of atoms and the deformation of the electron shells in the pair and three-body approximations (MT2 ) reduce the error by an average of ~ 9% in Ne and by ~ 2% in Ar. It should be noted that in crystalline Ar, calculations in the MT2 and MT0 models are in better agreement with experimental data in the temperature range from 10 K to 45 K. Thus, consideration of three-body forces in Ar is important even at zero pressure and low temperature. Table 1. Temperature dependence of C V /R and relative error γi crystalline Ne and Ar at p = 0 T, K
γi , %
C V /R Exp
MT2
MT0
M3a
M3
MT2
MT0
M3a
M3
3
0.016
0.011
0.011
0.009
0.009
34.229
34.229
44.653
44.653
4
0.041
0.031
0.031
0.026
0.026
26.447
26.495
38.111
38.111
5
0.091
0.067
0.067
0.056
0.056
26.334
26.411
38.481
38.514
6
0.162
0.126
0.126
0.105
0.105
22.595
22.694
35.307
35.375
7
0.253
0.208
0.207
0.175
0.175
17.785
17.908
30.784
30.875
8
0.366
0.310
0.310
0.264
0.264
15.118
15.260
27.798
27.913
Ne
(continued)
146
Ie. Ie. Gorbenko et al. Table 1. (continued) γi , %
T, K
C V /R Exp
MT2
MT0
M3a
M3
MT2
MT0
M3a
M3
9
0.490
0.429
0.428
0.369
0.368
12.379
12.538
24.620
24.757
10
0.606
0.558
0.557
0.485
0.484
7.971
8.146
19.930
20.087
11
0.778
0.692
0.691
0.609
0.607
11.062
11.234
21.781
21.940
12
0.930
0.827
0.826
0.735
0.734
11.004
11.176
20.936
21.102
13
1.078
0.961
0.959
0.861
0.860
10.861
11.030
20.069
20.236
14
1.208
1.089
1.087
0.985
0.983
9.784
9.951
18.409
18.578
15
1.327
1.212
1.210
1.105
1.103
8.619
8.782
16.709
16.878
16
1.432
1.328
1.326
1.219
1.217
7.267
7.425
14.875
15.043
17
1.532
1.437
1.435
1.328
1.325
6.204
6.357
13.344
13.508
18
1.628
1.539
1.536
1.430
1.427
5.512
5.658
12.193
12.353
19
1.719
1.633
1.631
1.526
1.523
4.984
5.123
11.233
11.387
20
1.798
1.720
1.718
1.615
1.612
4.320
4.453
10.181
10.330
21
1.894
1.801
1.799
1.698
1.696
4.915
5.040
10.349
10.491
22
1.984
1.876
1.873
1.776
1.773
5.477
5.595
10.523
10.658
23
2.053
1.945
1.942
1.848
1.845
5.279
5.391
10.010
10.139
24
2.083
2.008
2.006
1.914
1.912
3.595
3.703
8.107
8.232
11.897
12.027
20.836
20.962
Ar 10
0.396
0.404
0.402
0.348
0.348
2.162
1.712
12.036
12.165
15
0.963
0.966
0.961
0.875
0.873
0.266
0.233
9.208
9.368
20
1.464
1.468
1.461
1.371
1.369
0.262
0.168
6.350
6.496
25
1.841
1.843
1.837
1.755
1.753
0.100
0.243
4.677
4.798
30
2.106
2.112
2.106
2.036
2.034
0.276
0.003
3.308
3.406
35
2.307
2.304
2.299
2.240
2.238
0.139
0.356
2.893
2.973
40
2.442
2.443
2.438
2.389
2.388
0.050
0.126
2.135
2.199
45
2.522
2.546
2.542
2.501
2.499
0.936
0.791
0.847
0.901
50
2.588
2.624
2.620
2.585
2.584
1.362
1.241
0.114
0.160
55
2.633
2.683
2.681
2.651
2.650
1.926
1.822
0.683
0.644
60
2.652
2.730
2.728
2.702
2.701
2.959
2.870
1.893
1.859
65
2.658
2.768
2.766
2.743
2.743
4.136
4.058
3.210
3.180
70
2.678
2.798
2.796
2.777
2.776
4.471
4.404
3.665
3.640 (continued)
Specific Heat Capacity of Light Rare-Gas Crystals in the Model
147
Table 1. (continued) γi , %
T, K
C V /R Exp
MT2
MT0
M3a
M3
75
2.704
2.823
2.821
2.804
2.803
4.414
4.354
3.708
3.685
80
2.729
2.844
2.842
2.827
2.826
4.201
4.148
3.579
3.559
83
2.794
2.854
2.853
2.838
2.838
2.161
2.113
1.593
1.575
1.864
1.790
3.744
3.788
MT2
MT0
M3a
M3
Figure 2 shows the specific heat capacity of Ne and Ar at p = 0 as a function of temperature, calculated in models MT2 and M3 [26]. Figure 2 demonstrates that as pressure increases, heat capacity decreases in the corresponding temperature ranges. The calculated heat capacity does not reach the 3R limit, and, in addition, the form changes curve when calculating in both models. With increasing pressure, three-body interaction and deformation of the electron shells of atoms slightly increases the heat capacity of Ne and Ar even at p = 0 (Fig. 2).
Fig. 2. Temperature dependence of heat capacity C V (T )/R of Ne and Ar crystals at varied compression u: –◯–, –– are our calculations in M3 and MT2 models.
Quantitative analysis of the average specific heat capacity C V (T )/R (at temperatures from 25 K to 400 K) in the compression range from 0 to 0.7 (p = 169.6 GPa) for Ne and from 0 to 0.6 (p = 97.9 GPa) for Ar showed that the contribution of the deformation of dipole-type electron shells in the pair and three-body approximations increases from 0.03% to 6.6% in Ne and from 0.08% to 7.7% in Ar, the contribution of three-body forces, associated with the overlap of electron shells, increases from 1% to 3% in Ne and from 1% to 7.4% in Ar, and the contribution of both types of three-body forces due to both overlap and deformation of electron shells increases from 1% to 5.7% in Ne and from 1.1% to 12.3% in Ar.
148
Ie. Ie. Gorbenko et al.
3 Conclusions Now we do not have sufficient theoretical and experimental information on the crystal lattice dynamics of the RGC at high pressures. Nonempirical calculations of C V (T ) at p = 0.5 GPa, 1 GPa and 4 GPa were carried out within the framework of DFT in the local density approximation (see [27, 28] and Refs. There) only for Xe. Ref. [23] presents empirical calculations of the phonon spectrum at zero pressure for Ne, Ar and Kr, using the ELJ potential, obtained in [24]. In [29], analytical expressions were obtained for the energy of zero-point vibrations, the contribution of anharmonicity to the binding energy and the Grüneisen parameter in the Einstein model for the RGCs series. The authors note that at present there are many unresolved questions, to answer which it is necessary to find an approximate analytical formula for the dynamic matrix of the ELJ potential, including phonon dispersion and three-body potential, and applicable at high pressure. Here, based on our previously obtained phonon frequencies for neon and argon, which are in a crystalline state in compressed and uncompressed states, their heat capacity was calculated depending on temperature. In the short-range potential, the influence of three-body interaction and deformation of the electron shells of atoms in the pair and three-body approximations was revealed. A comparison of our calculations in the MT2 model and the M3a model (Fig. 1) showed that the contribution of three-body forces, due to both the overlap and deformation of the electron shells, makes the agreement with experiment better. The investigated contribution of three-body interaction and deformation of dipole-type electron shells in the pair and three-body approximations in light RGCs occurs even at p = 0, and in that temperature range, in which the heat capacity has not yet become equal to 2.99R, and from pressure p = 5.1 GPa (or compression u = 0.4) for neon occurs throughout the entire temperature range. Table 1 shows that the relative error of our results decreases with increasing temperature. Based on calculation of the average error values for different models and their comparison, we can say that the contribution of deformation in the paired approximation reduces the error by 0.13% for neon and by 0.04% for argon; (γM3a , γM3 ), and considering three-body interaction, it reduces by 0.13% for neon and by 0.07% for argon (γMT2 , γMT0 ), and the contribution of three-body forces, both due to overlap and associated with deformation (γMT2 , γM3a ) reduces by 9% and 2% for neon and argon, respectively. It is fundamental for us to compare calculations at all stages both in the classical version of the Tolpygo model (model M1) [20] and for non-empirical (models M3, M3a, MT0 and MT2) versions. The closeness of the results of calculations of the energy of zero-point vibrations in the MT0 and MT2 models [16, 17] with the results of calculations of the specific heat capacity C V (T ) gives grounds to assert that the contribution of threebody forces, due to both the overlap and deformation of the electron shells of the atom, is extremely important for light RGCs. So, the results of our calculations of the specific heat capacity of fcc-Ne and fcc-Ar taking into account three-particle forces and deformation of electron shells of dipole-type atoms in a pair and three-body approximation are in good agreement with experimental data and are extremely close to the calculation results in the classical version of the Tolpygo model.
Specific Heat Capacity of Light Rare-Gas Crystals in the Model
149
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22. Fenichel, H., Serin, B.: Low-temperature specific heats of solid neon and solid xenon. Phys. Rev. 142, 490 (1966) 23. Moyano, G.E., Schwerdtfeger, P., Rosciszewski, K.: Lattice dynamics for fcc rare gas solids Ne, Ar, and Kr from ab initio potentials. Phys. Rev. B 75, 024101-1-6 (2007) 24. Schwerdtfeger, P., Gaston, N., Krawczyk, R.P., Tonner, R., Moyano, G.E.: Extension of the Lennard-Jones potential: Theoretical investigations into rare-gas clusters and crystal lattices of He, Ne, Ar, and Kr using many-body interaction expansions. Phys. Rev. B 73, 064112-1-19 (2006) 25. Crawford, R.K., Lewis, W.F., Daniels, W.B.: The vacancies-in-solid model applied to solid argon. J. Phys. C 9, 1381 (1976) 26. Troitskaya, E.P., Chabanenko, V.V., Gorbenko, Ie.Ie., Kuzovoi, N.V.: Ab initio calculations of phonon frequencies and associated properties of crystalline Ne under pressure. Low Temper. Phys. 35, 815 (2009) 27. Tsuchiya, T., Kawamura, K.: First-principles study of systematics of high-pressure elasticity in rare gas solids, Ne, Ar, Kr, and Xe. J. Chem. Phys. 117, 5859–5865 (2002) 28. Dewhurst, J.K., Ahuja, R., Li, S., Johansson, B.: Lattice dynamics of solid xenon under pressure. Phys. Rev. Lett. 88, 075504 (2002) 29. Schwerdtfeger, P., Burrows, A., Smits, O.R.: The Lennard Jones potential revisited – analytical expressions for vibrational effects in cubic and hexagonal close-packed lattices. arXiv:2012. 05413v1 [cond-mat.mtrl-sci]
Advanced Piezo-Active 2–1–2 Composites with Large Parameters for Hydroacoustic and Energy-Harvesting Applications V. Yu. Topolov(B) Department of Physics, Southern Federal University, 5 Zorge Street, 344090 Rostov-on-Don, Russia [email protected]
Abstract. Novel piezo-active 2–1–2 composites with two single-crystal components are studied to show their large hydrostatic parameters and figures of merit concerned with the hydrostatic, longitudinal, and transverse piezoelectric effects. The 2–1–2 composite contains layers of domain-engineered [0 1 1]-poled relaxorferroelectric (1 – x)Pb(Zn1/3 Nb2/3 )O3 – xPbTiO3 single crystal (x = 0.0475– 0.09) and layers that contain aligned piezoelectric Li2 B4 O7 single crystal rods in polyethylene, and these layers are distributed regularly and in the form of an elliptic cylinder. Due to the microgeometry of the composite and the piezoelectric properties of the single-crystal components, large values of the hydrostatic figure of merit ∗ g ∗ ∼ 10−10 Pa−1 dh∗ gh∗ > 2 · 10−10 Pa−1 and longitudinal figure of merit d33 33 are achieved in specific volume-fraction and aspect-ratio range, at relatively small volume fractions of the single-crystal rods. An effect of the aspect ratio of the rod base on the piezoelectric performance and figures of merit if the composite is discussed. Large figures of merit of the studied composites can promote hydroacoustic and piezoelectric energy-harvesting applications of the studied 2–1–2 composites. Keywords: Composite · Piezoelectric Properties · PZN–xPT · Hydrostatic Figure of Merit · Energy-harvesting · Aspect Ratio
1 Introduction A development in the field of piezo-active composites is associated with novel composite structures and components [1–4] which promote large hydrostatic, energy-harvesting, and other parameters for various applications that are based on an energy conversion. As is known, laminar composites with 2–2 connectivity (the notation in terms of work [5]) are characterised by a relatively simple composite structure and possibilities to modify it [4, 6–9] owing to an incorporation of a new component, pores, and so on. The modification of the laminar composite structure can lead to improved effective electromechanical (that is elastic, piezoelectric and dielectric) properties and related parameters [4, 6–10] of the 2–2-type composites and therefore can influence their effective electromechanical coupling factors, figures of merit, anisotropy factors, hydrostatic parameters etc. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 151–166, 2024. https://doi.org/10.1007/978-3-031-52239-0_15
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Numerous results of studies show that the piezo-active 2–2-type composites based on relaxor-ferroelectric single crystals (SCs) [6, 10–12] with specific domainengineered structures are of value and research interest due to their SC components that exhibit the outstanding electromechanical properties [13–17] including large piezoelectric coefficients d 3j and electromechanical coupling factors k 3j . Widespread domainengineered relaxor-ferroelectric (1 – x)Pb(Zn1/3 Nb2/3 )O3 – xPbTiO3 (PZN–xPT) and (1 – y)Pb(Mg1/3 Nb2/3 )O3 – yPbTiO3 (PMN–yPT) SCs with the perovskite-type structure and compositions near the morphotropic phase boundary are suitable as components of the piezo-active 2–2-type composites [4, 6, 10–12, 18]. These SC components are often poled [13–17] along one of the following perovskite unit-cell directions: [0 0 1], [0 1 1] or [1 1 1], and for such SC components, full set of electromechanical constants were found during measurements (see, for instance, Refs. [13–17]). A novel three-component SC/SC/polymer composite with 2–1–2 connectivity was put forward very recently [8], and the main component of this composite was domainengineered lead-free SC poled along [0 0 1] of the perovskite unit cell. To the best knowledge, a systematic study on the 2–1–2 composites based on [0 1 1]-poled SCs was not yet carried out in detail, and only a few examples of the effective properties and related parameters of the 2–1–2 composites based on [0 1 1]-poled relaxor-ferroelectric SCs were discussed recently [9]. The aim of the present chapter is to analyse the piezoelectric properties and related figures of merit of the 2–1–2 composites wherein the main component is [0 1 1]-poled PZN–xPT SC, and the molar concentration x is related to compositions near the morphotropic phase boundary [13, 17–19].
2 Structure of the Composite, Its Effective Properties, Figures of Merit, and Components The 2–1–2 composite shown in Fig. 1 is regarded as a system of the parallel-connected layers of two types [9], and the layers are arranged periodically along the OX 1 axis of the rectangular coordinate system (X 1 X 2 X 3 ). The composite interfaces x 1 = const are continuous along the OX 2 and OX 3 axes. The layer of the first type (LFT) in the composite represents relaxor-ferroelectric domain-engineered SC with the spontaneous (1) s ||OX3 and main crystallographic axes X, Y and Z (inset 1 in polarisation vector P Fig. 1). This SC component is poled along [0 1 1] of the cubic (perovskite) unit cell. We use the notation ‘SC-1’ for this SC component that forms each LFT. The layer of the second type (LST) represents a system of the aligned SC rods (SC-2 in our notation to be used in the text of the present chapter), and these rods are continuous along the OX 3 axis and periodically arranged in a polymer matrix (inset 2 in Fig. 1). The LST is characterised as a SC/polymer composite with 1–3 connectivity in terms of work [5]. The rod in the LST is in the form of an elliptic cylinder that is described by the equation (x1 /a1 )2 + (x2 /a2 )2 = 1 in the coordinate system (X 1 X 2 X 3 ) (inset 3 in Fig. 1), where a1 and a2 are semiaxes of the ellipse. The aspect ratio of each rod base is given by ηc = a1 /a2 . The poling axis of the composite sample (see Fig. 1) is OX 3 , and electrodes on this sample are perpendicular to the OX 3 axis. The effective electromechanical properties of the 2–1–2 composite shown in Fig. 1 are found in two stages in accordance with the algorithm and formulae [8, 9]. The first
Advanced Piezo-Active 2–1–2 Composites with Large Parameters
153
Fig. 1. Schematic of a 2–1–2 composite. m and 1 – m are volume fractions of the LFTs and LSTs, respectively (reprinted from Topolov [9], with permission from Taylor and Francis). In inset 1, orientations of domains and main crystallographic axes X, Y and Z in SC-1 (LFT) are shown. In inset 2, the SC-2/polymer LST is shown. mc and 1 – mc are volume fractions of SC-2 and polymer, respectively, in the LST. In inset 3, the elliptic cross section of the SC-2 rod and its unit-cell vectors a, b and c are shown. a1 and a2 are semiaxes of the ellipse (or the SC-2 rod base).
stage is associated with the application of the effective field method [6, 20] to evaluate the effective properties of the LST, that is the 1–3 SC/polymer composite with the regular distribution of the rods (see inset 2 in Fig. 1). In the second stage, the matrix method [6, 21] is used to evaluate the effective electromechanical properties of the 2–1–2 composite shown in Fig. 1. Both the effective field method and matrix method taken an electromechanical interaction into account. This interaction takes places between the structural elements of the composite (including the LST as an independent composite) due to the presence of the piezoelectric components. On the output, the matrix of the effective properties of the 2–1–2 composite has the form: ∗E ∗ t s d ∗ C = , d ∗ ε∗σ where superscript ‘t’ denotes the matrix transposition. The full set of the effective elec∗E at electric tromechanical properties in the C ∗ matrix contains elastic compliances sab ∗ ∗σ field E = const, piezoelectric coefficients dij and dielectric permittivities εfh at mechanical stress σ = const. These effective properties, written in the general form as Π * (m,
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V. Yu. Topolov
mc , ηc ), are found in the longwave approximation [4, 6, 10, 18]. Such an approximation holds on condition that a wavelength of an external acoustic field is much more than a width of each layer of the composite sample shown in Fig. 1. Taking the effective electromechanical properties Π * (m, mc , ηc ) of the 2–1–2 composite into account, we analyse its hydrostatic piezoelectric coefficients: ∗ ∗ ∗ ∗ ∗ ∗ dh∗ = d31 + d32 + d33 and gh∗ = g31 + g32 + g33 ,
(1)
hydrostatic figure of merit: ∗ 2 Qh = dh∗ gh∗ ,
(2)
and energy-harvesting figures of merit: 2 ∗ ∗ Q3j = d3j∗ g3j
(3)
that are associated with either the longitudinal piezoelectric effect (at j = 3) or transverse piezoelectric effect (at j = 1 and 2). The piezoelectric coefficients dh∗ and gh∗ from Eqs. (1) are used to characterise the hydrostatic activity and sensitivity [4, 6], respectively. 2 The hydrostatic figure of merit Qh∗ from Eq. (2) is of importance at estimations of a signal/noise ratio of a hydrophone [4, 6, 10, 22] at hydrostatic loading. The energy 2 ∗ from Eq. (3) are used [4, 6, 7, 10, 18] to estimate harvesting figures of merit Q3j the signal/noise ratio of a transducer by taking the piezoelectric effect (longitudinal or 2 2 ∗ from Eqs. (2) and transverse) into consideration. The figures of merit Qh∗ and Q3j (3) are also used to describe a mechanical-to-electric energy conversion in piezoelectric materials and to show their effectiveness in the context of the energy conversion and energy harvesting [4, 6, 7, 10]. The main piezoelectric component of the 2–1–2 composites studied by us is domainengineered PZN–xPT SC poled along [0 1 1] of the perovskite unit cell. Domain orientations in such a poled state are shown in inset 1 of Fig. 1. Full sets of electromechanical constants of PZN–xPT SCs at 0.0475 ≤ x ≤ 0.09 [16, 17, 26] are shown in Table 1, and these PZN–xPT compositions are located close to the morphotropic phase boundary [19]. It should be added that domain-engineered PZN–xPT SCs poled along [0 1 1] were considered [18] as main components of the advanced 2–2-type composites wherein an appreciable orientation effect was described. The second piezoelectric component of each 2–1–2 composite from our present study is Li2 B4 O7 (LBO) SC [23] that exhibits a unique anisotropy of the piezoelectric and elastic properties (see the full set of electromechanical constants in Table 2). The SC components from the LFT and LST differ by symmetry, see data in Tables 1 and 2. The LST of the composite contains polyethylene (PE) that is regarded as a piezo-passive isotropic component [4, 24] with constants listed in Table 2.
Advanced Piezo-Active 2–1–2 Composites with Large Parameters
155
E , piezoelectric coefficients d and dielectric Table 1. Room-temperature elastic compliances sab ij σ of domain-engineered [0 1 1]-poled PZN–xPT SCs with macroscopic mm2 permittivities εpp symmetry
Constants
x = 0.0475, Ref. [17]
x = 0.055, Ref. [17]
x = 0.065, Ref. [17]
x = 0.07, Ref. [16]
x = 0.09, Ref. [26]
E , s11
34.26
39.04
46.99
67.52
73.07
E , s12
−47.45
−56.67
−74.01
−60.16
−63.98
E , s13
26.05
32.65
39.51
3.355
4.256
E , s22
125.35
137.40
170.69
102.0
125.6
E , s23
−74.22
−85.47
−96.76
−54.47
−68.04
E , s33
51.32
60.68
61.47
62.02
67.49
E , s44
14.68
15.46
15.04
15.45
15.12
E , s55
277.78
294.12
333.33
291.5
299.3
E , s66
125.98
155.29
169.08
14.08
16.54
d 15 , pC/N
4037
4187
4871
1823
2012
d 24 , pC/N
134
207
121
50
118.7
d 31 , pC/N
750
858
1191
478
476.0
d 32 , pC/N
−1852
−1985
−2618
−1460
−1705
d 33 , pC/N
1185
1319
1571
1150
1237
σ /ε ε11 0
8000
8500
9500
8240
8740
2600
3050
1500
1865
2075
3900
4000
5600
3180
3202
10−12 Pa−1 10−12 Pa−1
10−12 Pa−1 10−12 Pa−1
10−12 Pa−1 10−12 Pa−1 10−12 Pa−1
10−12 Pa−1 10−12 Pa−1
σ /ε ε22 0 σ /ε ε33 0
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E , piezoelectric coefficients e and dielectric Table 2. Room-temperature elastic moduli cab ij ξ
permittivity εpp of LBO SC (4 mm symmetry) and PE (isotropic medium) Constants
LBO SC, Ref. [23]
PE, Refs. [4, 24]
E , 1010 Pa c11 E , 1010 Pa c12 E , 1010 Pa c13 E , 1010 Pa c33 E , 1010 Pa c44 E , 1010 Pa c66 e15, C/m2 e31, C/m2 e33, C/m2 ξ ε11 /ε0 ξ ε33 /ε0
13.5
0.0778
0.357
0.0195
3.35
0.0195
5.68
0.0778
5.85
0.0292
4.67
0.0292
0.472
0
0.290
0
0.928
0
8.90
2.3
8.07
2.3
Taking the microgeometry of the composite (Fig. 1) and symmetry of the aforementioned components into account, we represent the matrix of the piezoelectric coefficients dij∗ as ⎡
∗ 0 0 0 0 d15 ∗ ∗ ⎣ ||d | = 0 0 0 d24 0 ∗ d∗ d∗ 0 0 d31 32 33
⎤ 0 0 ⎦. 0
The matrix of the piezoelectric voltage coefficients g ∗ of the composite has the similar form, and elements of the g ∗ matrix can be evaluated by using the formula [25] dfk∗ = εfj∗σ gjk∗ .
(4)
In Eq. (4), the dielectric permittivity εfj∗σ and piezoelectric coefficient dfk∗ are taken as elements of the C ∗ matrix. Considering Eqs. (2) and (4) as well as the symmetry of the components of the composite, one can write its hydrostatic figure of merit as ∗ 2 2 ∗σ Qh = dh∗ /ε33 . (5) Comparing the piezoelectric performance of [0 1 1]-poled PZN–xPT SCs from Table 1, we note the large piezoelectric coefficients d 15 and |d 32 |. Our analysis of the effective piezoelectric properties of the 2–1–2 composite suggests that the piezoelectric (1) coefficient d32 of PZN–xPT SC in the LFT influences the effective piezoelectric coef∗ ficient d32 of the composite to a large extent. This is accounted for by the continuous
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distribution of SC-1 as a piezoelectric component along the OX 2 and OX 3 axes, see ∗ can lead to a large negative contribuFig. 1. Therefore, the piezoelectric coefficient d32 tion in the hydrostatic piezoelectric coefficient dh∗ from Eqs. (1), and this will promote decreasing dh∗ of the composite. To avoid this decreasing dh∗ and its influence on the 2 hydrostatic figure of merit Qh∗ from Eq. (2), we consider the rotation of the main crystallographic axes X and Y of PZN–xPT SC around the Z axis (see inset 1 in Fig. 1) by 90° clockwise [9]. Such an orientation of the X and Y axes in each LFT leads to (1) an influence of the negative d32 of PZN–xPT SC on the lateral piezoelectric effect (1) in the composite along the OX 1 axis, and then the positive d31 of PZN–xPT SC will influence the lateral piezoelectric effect in the composite along the OX 2 axis (Fig. 1). Along the OX 2 axis shown in Fig. 1, the continuous distribution
∗ of
the LFT with the high
is achieved due to the piezoelectric activity is observed. Moreover, a decrease of d31 interfaces x 1 = const and due to the heavily oblate shape of each rod base in the LST, i.e., on condition ηc 1 (see inset 3 in Fig. 1). Such a shape of the rod base promotes (2),E (2),E a small elastic compliance s11 of the LST in comparison to its s22 , and the large (2),E ∗ of the composite and s22 is important to achieve the large piezoelectric coefficient d32 the related large positive contribution in the hydrostatic piezoelectric coefficient dh∗ . Considering the pointed orientation of the main crystallographic X and Y axes of SC-1 in the LFT (see inset 1 in Fig. 1), we represent the electromechanical properties Π (1) of each LFT in the (X 1 X 2 X 3 ) system as follows: (1),E
(1)
d32
(1),E
(1),E
(1),E
(1),E
E ,s E ,s E ,s E ,s E , s11 = s22 = s12 = s23 = s11 = s12 12 13 22 23 (1),E (1),E (1),E (1),E (1) E E E E s33 = s33 , s44 = s55 , s55 = s44 , s66 = s66 , d31 = d32 , (1) (1) (1) (1),σ σ , ε (1),σ = ε σ , = d31 , d33 = d33 , d15 = d24 , d24 = d15 , ε11 = ε22 22 11 (1)σ σ . and ε33 = ε33
(6)
In Sect. 3, we show and discuss new results on the effective parameters (1)–(3) of the 2–1–2 composites based on [0 1 1]-poled PZN–xPT SCs.
3 Results and Discussion 2 3.1 Hydrostatic Figures of Merit Q∗h of Composites Based on PZN–xPT 2 Figure 2 shows that the largest hydrostatic figure of merit Qh∗ from Eq. (2) is achieved in
the 2–1–2 composite at x = 0.065. This is due to the larger piezoelectric coefficients
(1) (1)
d32 and d33 of PZN–0.065PT SC (see Table 1) as a main component of the composite. ∗ > 0 and d ∗ > 0 which This component promotes the large piezoelectric coefficients d33 32 give a decisive contribution [9] in the hydrostatic piezoelectric coefficient dh∗ from Eqs. 2 (1) and in the hydrostatic figure of merit Qh∗ from Eqs. (2) and (5). The diagram in Fig. 2 was built at the relatively small volume fractions of PZN–xPT SC, namely, m = 0.1 and 0.2. At these and smaller m values, the dielectric permittivity ∗σ of the composite is small in comparison to ε (1),σ of PZN–xPT SC, and therefore, ε33 33 2 ∗σ cannot lead to a significant decrease of the hydrostatic figure of merit Qh∗ ∼ 1/ε33 in accordance with Eq. (5).
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2 Fig. 2. Molar-concentration (x) dependences of the hydrostatic figure of merit Qh∗ (in 10–12 Pa−1 ) of 2–1–2 PZN–xPT SC/LBO SC/PE composites at m = const (reprinted from Topolov [9], with permission from Taylor and Francis).
3.2 Volume-Fraction (m) Dependence of Hydrostatic Parameters Graphs in Fig. 3 show the non-monotonic volume-fraction (m) behavior of the hydrostatic ∗σ of the composite is an increasing parameters (1) and (2). The dielectric permittivity ε33 function as is known from studies on the parallel-connected 2–2-type composites [4, 6, ∗σ ε (1),σ . The small ε ∗σ 9], and at small volume fractions of SC-1 (that is at m 1), ε33 33 33 ∗ values promote large values of gh from Eqs. (1) and sharp max gh∗ , see curve 2 in Fig. 3. In comparison to this feature, max dh∗ is less pronounced and observed at relatively large volume fractions m of PZN–xPT SC, see curve 1 in Fig. 3. 2 Maxima of Qh∗ from Eq. (2) are located in an intermediate region of volume fractions m, see curve 3 in Fig. 3. This is associated with the influence of the dielectric ∗σ on Q ∗ 2 in accordance with Eq. (5) and the inequality ε ∗σ ε (1),σ permittivity ε33 33 33 h 2 that holds at m 1. The less sharp character of max [ Qh∗ ] in comparison to max gh∗ 2 2 (see curves 2 and 3 in Fig. 3) can be accounted for by the relationship Qh∗ ∼ dh∗ that follows from Eq. (5).
Advanced Piezo-Active 2–1–2 Composites with Large Parameters
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Fig. 3. Volume-fraction (m) dependences of the hydrostatic piezoelectric coefficient dh∗ (in pC/N), 2 piezoelectric coefficient gh∗ (in mV. m/N) and figure of merit Qh∗ (in 10–12 Pa−1 ) of the 2–1–2 PZN–0.065PT SC/LBO SC/PE composite ar mc = 0.03 (graph a) or mc = 0.05 (graph b) in the LST.
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3.3 Links Between Piezoelectric Properties and Figures of Merit Relationships between the piezoelectric properties and energy-harvesting figures of merit 2 ∗ ∗ (see curves 5 and 6 in Fig. 4, a) and Q3j are illustrated by Fig. 4. Maxima of g3j 2 ∗ (see curves 2 and 3 in Fig. 4, b) obey the following similar inequalities: Q3j ∗ ∗ ∗ 2 ∗ 2 > max Q32 . maxg33 > maxg32 and max Q33
(7)
∗ 2 increases monotonously (see curve 1 in Fig. 4, b) However, the figure of merit Q31 ∗ (see curve 4 in Fig. 4, b) and cannot be involved at the non-monotonic dependence of g31 in formulae (7) or similar relations. The reason of such an
inconsistent volume-fraction ∗ value and monotonic increase behavior is concerned with the relatively small min g31 ∗ (see curves 1 and 4 in Fig. 4, a). The presence of the interfaces x = const in the of d31 1 ∗ composite (see Fig. 1) strongly influences the piezoelectric effect associated with both d31 ∗ . Therefore, the volume-fraction dependence of Q ∗ 2 evaluated in accordance and g31 ∗ 2 31 ∗ 2 dependences, see Fig. 4, with Eq. (3) differs from the volume-fraction Q33 and Q32 b. It should be added that the inequality: 2 ∗ 2 < max Qh∗ (8) max Q33 holds, see curve 3 in Fig. 3, a and curve 3 in Fig. 4, b. Condition (8) is in contradiction ∗ 2 2 with the inequality max [ Q33 ] > max [ Qh∗ ] that is valid [4, 6, 27, 28] in many 2–2-type composites based on perovskite ferroelectric ceramics or domain engineered SCs poled along [0 0 1]. The validity of condition (8) is accounted for specifics of the piezoelectric effect in the [0 1 1]-poled SC component, for example PZN–0.065PT in the present case. As seen in Table 1, the piezoelectric coefficients d 3j of [0 1 1]-poled PZN–0.065PT SC obey the inequality d 31 < d 33 < |d 32 | that differs from conditions d 31 = d 32 and |d 32 | < d 33 which hold in [0 0 1]-poled PZN–xPT, PMN–yPT [6, 10, 14] and other SCs. 3.4 Maxima of Figures of Merit at Variations of the Aspect Ratio ηc Our results, shown in Figs. 2, 3 and 4, are obtained for the 2–1–2 composite with the SC rods at the aspect ratio of their bases ηc = 100. Changes of the aspect ratio ηc led to changes of the elastic properties of the LST, and this influences the piezoelectric response of the composite along the OX 1 and OX 2 axes. We remind the reader that each SC rod is oriented as shown in inset 3 of Fig. 1, and such an orientation is inseparably linked with orientations of the OX 1 and OX 2 axes. 2 2 ∗ Table 3 contains data on local maxima of figures of merit Q3j and Qh∗ which are evaluated for the composite at various aspect ratios ηc from 50 (at a1 a2 , as shown in inset 3 of Fig. 1) to 0.01 (at a1 a2 ,). The volume fraction of the SC rods in the LST is mc = 0.03, that is the polymer component occupies a large area on the ∗ 2 (X 1 OX 2 ) plane (see Fig. 1). The figure of merit Q33 concerned with the longitudinal
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∗ (a, in pC/N) and g ∗ (a, Fig. 4. Volume-fraction (m) dependences of piezoelectric coefficients d3j 3j 2 ∗ . –12 −1 Pa ) of the 2–1–2 PZN–0.065PT SC/LBO in mV m/N) and figures of merit Q3j (b, in 10 SC/PE composite.
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∗ 2 piezoelectric effect does not undergo changes of local max[ Q33 ] because the rod base changes the semiaxes a1 and/or a2 in the perpendicular direction (see insets 2 and 3 in ∗ 2 Fig. 1). The figure of merit Q31 is characterised by the monotonic dependence (see, for example curve 1 in Fig. 4, b). A decrease of the aspect ratio ηc from 50 to 1 does ∗ 2 not lead to appreciable changes of values of local max[ Q32 ], see the 2nd column of ∗ 2 Table 3. Within the same ηc range, local max[ Qh ] undergoes changes, however its value decreases by ca. 10%, as shown in the 3rd column of Table 3. It is important to underline that the aforementioned local maxima at ηc ≥ 1 are observed at the almost equal volume fractions m ≈ 0.05–0.06.A further decrease of ηc from 1 to 0.01 causes ∗ 2 2 noticeable changes of local maxima of both Q32 and Qh∗ (see Table 3) as well as displacements of locations of these maxima. The aspect ratio ηc 1 corresponds to an elliptic cylinder with the base semiaxes that obey the inequality a1 a2 . 2 2 ∗ Table 3. Local maxima of figures of merit Q3j and Qh∗ (in 10–12 Pa−1 ) of the 2–1–2 0.065PT SC/LBO SC/PE composite at mc = 0.03 and variations of the aspect ratio ηc ηc
∗ 2 max[ Q32 ]
2 max[ Qh∗ ]
50
105 (at m = 0.064)
306 (at m = 0.060)
20
105 (at m = 0.063)
292 (at m = 0.058)
10
106 (at m = 0.063)
283 (at m = 0.052)
5
106 (at m = 0.063)
278 (at m = 0.052)
2
105 (at m = 0.063)
274 (at m = 0.052)
1
105 (at m = 0.064)
272 (at m = 0.052)
0.5
104 (at m = 0.065)
269 (at m = 0.053)
0.1
96.9 (at m = 0.073)
255 (at m = 0.055)
0.05
89.9 (at m = 0.083)
241 (at m = 0.059)
64.2 (at m = 0.140) 182 (at m = 0.073) ∗ 2 Note. Local max[ Q33 ] = 174 × 10–12 Pa−1 is achieved at the volume fraction m = 0.070 irrespective of the aspect ratio ηc in the LST. 0.01
(2),E
Variations of the aspect ratio ηc lead to changes of the elastic compliances sab in (2),E the LST. In Fig. 5, the aspect-ratio dependence of sab is graphically represented for (2),E (2),E (2),E ab = 11, 12, …, and 33. The shear elastic compliances s44 , s55 and s66 do not influence the piezoelectric response associated with the piezoelectric coefficients d3j∗ or (2),E
∗ of the composite, where j = 1, 2 and 3. The behavior of s g3j ab (2),E
(2),E
shows that only s11
and s12 (see curves 1 and 2 in Fig. 5) undergo noticeable changes on decreasing ηc , ∗ ( j = 1 and 2) and and stimulates changes of the piezoelectric coefficients d3j∗ and g3j ∗ 2 ∗ 2 ∗ 2 figures of merit Q31 , Q32 and Qh of the composite. At ηc < 1, a decrease of
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∗ 2 2 Q32 and Qh∗ is caused by a lower piezoelectric activity of the composite along the ∗ to d ∗ , Q ∗ 2 OX 2 axis. In this case, a contribution from the piezoelectric coefficient d32 h h ∗ 2 and Q32 [see Eqs. (1)–(3)] becomes smaller due to increasing the elastic compliances
(2),E (2),E s11 and s12 (see Fig. 5) in the LST.
(2),E
Fig. 5. Elastic compliances sab (in 10–12 Pa−1 ) of the LBO SC/PE LST at mc = 0.03 (reprinted from Topolov [9], with permission from Taylor and Francis).
3.5 Comparison to Data on Some 2–2-Type Composites The studied 2–1–2 composites based on [0 1 1]-poled PZN–xPT SCs are characterised 2 2 by large hydrostatic figures of merit Qh∗ (Fig. 2), and the Qh∗ values at the relatively small volume fraction m = 0.1 or 0.2 are by approximately an order-of-magnitude larger 2 than Qh∗ of conventional 2–2 ferroelectric ceramic/polymer composites [4, 6, 29]. The ∗ 2 2 Qh values at m = 0.1 or 0.2 (see Fig. 2) are larger than Qh∗ of 2–2 composites based on [0 1 1]-poled PMN–yPT SCs [11]. The 2–1–2 composite based on [0 1 1]-poled PZN– 0.065PT SC with the set of electromechanical constants from Eqs. (6) is of value due to 2 the large hydrostatic parameters dh∗ , gh∗ and Qh∗ , and energy-harvesting figure of merit ∗ 2 Q33 (see Figs. 3 and 4, and Table 3). These parameters indicate advantages of the 2–1–2 composite over 2–0–2–0 composites [27, 28] that contain the porous ferroelectric 2 PZT-type ceramic and porous polymer layers. The Qh∗ values of the 2–1–2 composite 2 based on [0 1 1]-poled PZN–0.065PT SC are larger than Qh∗ of a 2–1–2 lead-free composite [8] wherein the main SC component is replaced by [0 0 1]-poled domain engineered SC from solid solutions of alkali niobates-tantalates [30]. Despite a high
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piezoelectric sensitivity of lead-free [0 0 1]-poled domain-engineered SC [30], this SC 2 component from work promotes the smaller dh∗ and Qh∗ values of the 2–1–2 composite from work [8].
4 Conclusion In the present chapter, results on the novel high-performance 2–1–2 composites based on [0 1 1]-poled PZN–xPT SCs have been shown, and specifics of the hydrostatic piezoelectric response and figures of merit of these composites have been discussed. Molar concentrations x have been chosen near the morphotropic ohase boundary to provide a high piezoelectric activity of PZN–xPT SCs (Table 1). The due orientation of the main crystallographic axes of this SC component in the LFT (inset 1 in Fig. 1) and the related set of its electromechanical constants from Eqs. (6) lead to the large hydrostatic piezo 2 electric coefficients dh∗ , gh∗ and figure of merit Qh∗ from Eqs. (1) and (2). In the LST, changes of the aspect ratio ηc of the SC rod base (insets 2 and 3 in Fig. 1) influence dh∗ , 2 ∗ 2 ∗ 2 gh∗ , Qh∗ , and energy-harvesting figures of merit Q31 and Q32 of the composite. ∗ 2 −10 Large values of the hydrostatic figure of merit Qh > 2 · 10 Pa−1 and energy ∗ 2 ∗ 2 –10 −1 −10 harvesting figures of merit Q33 ~ 10 Pa and Q32 ∼ 10 Pa−1 are achieved in the 2–1–2 composite based on [0 1 1]-poled PZN–0.065PT SC (Figs. 2, 3 and 4) at relatively small volume fractions mc of the LBO SC rods in the LST. The effect of the aspect ratio ηc of the rod base and the influence of the elastic properties of the LST on the piezoelectric performance and figures of merit if the composite has been analysed at small volume fractions mc of SC-2 in the LST. The best sets of the effective parameters (1)–(3) of the composite are achieved at aspect ratios ηc 1 and volume fractions 2 ∗ 2 of LBO SC mc 1 in the LST. Due to the large figures of merit Qh∗ , Q33 and ∗ 2 Q32 , the studied 2–1–2 composites can be of value as active elements in modern hydroacoustic and piezoelectric energy-harvesting devices or systems. Acknowledgements. The author would like to thank Prof. Dr. A. E. Panich and Prof. Dr. I. A. Parinov (Southern Federal University, Russia), Prof. Dr C. R. Bowen (University of Bath, UK), and Prof. Dr. P. Bisegna (University of Rome Tor Vergata, Italy) for their interest in the field of modern piezo-active composites. This research was supported by the Southern Federal University (research topic “Development and Materials-Science Substantiation of the Creation of Materials and Products Based on Piezoelectric Ceramics Using Additive Technologies”, contract No. 176/22-D, July 11th, 2022).
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Novel Lead-Free 2–2-Type Composites with High Piezoelectric Sensitivity and Strong Hydrostatic Response: Examples of 2–1–2 Connectivity V. Yu. Topolov(B)
and S. A. Kovrigina
Southern Federal University, 5, Zorge Street, Rostov-on-Don 344090, Russia [email protected]
Abstract. A modification of the well-known laminar composite structure with 2–2 connectivity leads to changes of the effective physical properties and their dependences on the composite content and microgeometric characteristics. The present chapter reports new results on novel three-component 2–1–2 composites with a high piezoelectric sensitivity and large hydrostatic parameters. In the 2– 1–2 composite, there are parallel-connected layers of two types. The first type represents a ferroelectric domain-engineered [0 0 1]-poled lead-free single crystal based on solid solutions of alkali niobates-tantalates with the perovskite-type structure. In the second type of the layers, ferroelectric ceramic rods in the form of an elliptic cylinder are regularly aligned in a large polymer matrix, and the ceramic component is chosen among the perovskite-type compositions being either leaffree or lead-containing. The hydrostatic piezoelectric coefficients d h * and gh * , related figure of merit d h * gh * , and their non-monotonic behaviour are studied by taking an orientation effect into account. This orientation effect is caused by rotations of the ceramic rod bases in the polymer medium. The aforementioned hydrostatic parameters are compared to those of the conventional two-component ceramic-polymer composites, and some advantages of the studied lead-free 2–1–2 composites are discussed. These novel composites are suitable for hydroacoustic applications. Keywords: 2–2-type Composite · 2–1–2 Composite · Hydrostatic Piezoelectric Coefficient · Figure of Merit (FOM) · Orientation Effect
1 Introduction The piezo-active 2–2-type composites are widespread [1–4] and attractive due to their important electromechanical properties, large energy-harvesting figures of merit (FOM), hydrostatic parameters [5–7] etc. As is known, the 2–2 connectivity pattern [8] means a system of layers of two types (or two components of the composite), and each of these layers is distributed continuously along two axes of a rectangular coordinate system (X 1 X 2 X 3 ). Modifications of the traditional laminar structure of the 2–2 composite [5, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 167–179, 2024. https://doi.org/10.1007/978-3-031-52239-0_16
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8] are concerned with adding a third component or with forming a porous structure in one of the layer types [4, 7]. The conventional piezo-active 2–2 composite contains the ferroelectric (FE) ceramic and polymer layers that are usually parallel-connected [1, 2, 5] to facilitate a poling process. In the 2000–2010s, numerous examples of a performance of the 2–2-type composites based on FE single crystals (SCs) were discussed (see, for instance, Refs. 3, 6, 7). Very recently, a novel three-component lead-free composite with 2–1–2 connectivity was put forward, and its piezoelectric performance and hydrostatic parameters were analysed in work [9]. The 2–1–2 composite is characterised by the system of the parallel-connected layers of two types or by three components, namely, FE SC, piezoelectric SC and polymer. As shown in work [9], a new orientation effect in the 2–1–2 composite is associated with rotations of the piezoelectric SC rod bases in the polymer medium, and this influences the elastic properties and piezoelectric anisotropy in the SC/polymer layers and, therefore, the piezoelectric performance and hydrostatic response of the 2–1–2 composite as a whole. However, to date no attempts were made to analyse examples of the 2–1–2 composite containing materials with two FE components, for instance, SC and ceramic in the adjacent layers. In fact, the chapter reports our results (i) on the orientation effect in the context of the hydrostatic piezoelectric coefficients and FOM of the 2–1–2 FE SC/FE ceramic/ polymer composite and (ii) on the comparison of the hydrostatic parameters of the similar 2–1–2 FE SC/FE ceramic/polymer composites that differ by their FE ceramic components.
2 Model of the Piezo-Active 2–1–2 Composite, Its Effective Properties, Hydrostatic Parameters and Components 2.1 Model Concepts, Effective Electromechanical Properties and Related Hydrostatic Parameters The 2–1–2 FE SC/FE ceramic/polymer composite studied by us represents the system of the parallel-connected layers of two types (Fig. 1). These layers are arranged periodically along the OX 1 axis of the rectangular coordinate system (X 1 X 2 X 3 ), and planar interfaces between the layers obey the condition x 1 = const. In the layer of the first type (LFT), there is ferroelectric domain-engineered SC with the spontaneous polarisation vector P s (1) || OX 3 , see inset 1 in Fig. 1. The crystallographic directions [h k l] shown in inset 1 of Fig. 1 are related to the perovskite unit cell. The main crystallographic axes X, Y and Z of domain-engineered SC in each LFT are oriented as follows: X || [1 0 0], Y || [0 1 0] and Z || [0 0 1]. The layer of the second type (LST) is regarded as a system of the aligned FE ceramic rods that are continuous along the OX 3 axis and periodically arranged in a large polymer matrix, see inset 2 in Fig. 1. The remanent polarisation vector P r (2) of each ceramic rod is parallel to the OX 3 axis. The LST can be characterised as a composite with 1–3 connectivity in accordance with concepts [8]. Each rod in the LST is in the form of an elliptic cylinder with semiaxes a1 and a2 of the rod base. The aspect ratio of each rod base is ηc = a1 /a2 . In the LST, a rotation of each ceramic rod around OX 3 in the polymer medium is described in terms of the angle γ (see inset 3 in Fig. 1). The poling axis of the composite sample as a whole (Fig. 1) is OX 3 .
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Fig. 1. Schematic of a 2–1–2 composite. m and 1 – m are volume fractions of the LFTs and LSTs, respectively. In inset 1, the domain-engineered SC component is schematically shown. In inset 2, the FE ceramic/polymer LST is shown. Where mc and 1 – mc are volume fractions of FE ceramic and polymer, respectively. In inset 3, the orientation of the elliptic cross section of the FE ceramic rod in the LST is shown, where a1 and a2 are semiaxes of the ellipse (or the ceramic rod base), and γ is the rotation angle.
This coordinate axis is parallel to both P s (1) and P r (2) , and perpendicular to electrodes applied to this sample. We note that the present orientation of the P s (1) and P r (2) vectors in the adjacent layers of the composite sample facilitates the poling process in an external electric field E || OX 3 . Effective electromechanical (i.e. elastic, piezoelectric and dielectric) properties of the 2–1–2 composite shown in Fig. 1 are found in two stages as follows. In the first stage, the effective field method [5, 10] is applied for evaluating the effective electromechanical properties of the LST (or the 1–3 ceramic/polymer composite). An electromechanical interaction between the FE ceramic rods (that are piezoelectric) in a large polymer matrix (that is either isotropic or transversely isotropic) is taken into account. The effective
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electromechanical properties of the LST shown in inset 2 of Fig. 1 are given by the matrix −1 (2) (p) K = K + mc K (c) −K (p) [I + (1−mc ) S · K (p) (K (c) −K (p) )]−1 .
(1)
In Eq. (1), || K (c) || and || K (p) || are 9 × 9 matrices with electromechanical constants of FE ceramic and polymer, respectively, mc is the volume fraction of FE ceramic in each LST, || I || is the identity matrix, and || S || is the matrix that contains the electroelastic Eshelby tensor components [11]. The || S || matrix depends on the electromechanical properties of the polymer component (i.e., elements of || K (p) ||) and the aspect ratio ηc of the FE ceramic rod base. A full set of electromechanical properties of the LST is given by the || K (2) || matrix that contains elastic moduli cab (2),E at electric field E = const, piezoelectric coefficients eij (2) and dielectric permittivities εpp (2),ξ at mechanical strain ξ = const. The electromechanical properties of the LST depend on the volume fraction mc of FE ceramic in the LST and on the aspect ratio ηc of the FE ceramic rod base. After this procedure of evaluations according to Eq. (1), the electromechanical properties of the LST are represented in the coordinate system (X 1 X 2 X 3 ) by taking the rotation mode into account, see inset 3 in Fig. 1. In the second stage of our revaluations, the matrix method [5, 7, 12/] is applied to the laminar structure of the 2–1–2 composite (Fig. 1). An electromechanical interaction between the piezoelectric LFTs and piezoelectric LSTs is taken into account. The effective electromechanical properties of the composite are described by the 9 × 9 matrix ∗ C = (2) C (1) · M m + C (2) (1 −m) · [M m + I (1 −m)]−1 . In Eq. (2), || C (1) || and || C (2) || characterise the electromechanical properties of the LFT and LST, respectively, m is the volume fraction of the LFT, || M || is concerned with the electric and mechanical boundary conditions [5, 7, 12] at interfaces x 1 = const (Fig. 1), and || I || is the identity matrix. A transition from the set of electromechanical constants involved in || K (2) || from Eq. (1) to the set of electromechanical constants from || C (2) || in Eq. (2) is carried out by using formulae [13] for a piezoelectric medium. The || C * || matrix from Eq. (2) is written in the general form as ∗E ||s || ||d ∗ ||t ||C ∗ || = (3) ||d ∗ || ||ε∗σ || ∗E at electric field E = const, piezoand contains the full set of the elastic compliances sab ∗ ∗σ at mechanical stress σ = const. electric coefficients dij and dielectric permittivities εfh Superscript ‘t’ in Eq. (3) denotes the matrix transposition. The effective electromechanical properties [or elements of the || C * || matrix from Eqs. (2) and (3)] can be also written as Π * (m, mc , γ , ηc ). The properties of the 2–1–2 composite (Fig. 1) are found in the longwave approximation [5, 7], i.e. on condition that a wavelength of an external acoustic field is much more than a width of each layer of the composite sample. Based on the effective electromechanical properties Π * (m, mc , γ , ηc ) of the 2–1–2 composite [or matrix elements of || C * || from Eq. (2)], we find and analyse the hydrostatic piezoelectric coefficients ∗ ∗ ∗ ∗ ∗ ∗ dh∗ = d31 + d32 + d33 and gh∗ = g31 + g32 + g33 ,
(4)
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and related hydrostatic FOM
∗ 2 Qh = dh∗ gh∗
(5)
of the composite. The piezoelectric voltage coefficients g3j * from Eqs. (4) are evaluated by using the formula dfk∗ = efj∗s gjk∗
(6)
that illustrates the relationship between the piezoelectric and dielectric properties [17]. In Eq. (6), d fk * and εfj *σ are elements of the || C * || matrix from Eq. (2). The piezoelectric coefficients d h * and gh * from Eqs. (4) characterise the hydrostatic activity and sensitivity, respectively. At hydrostatic loading, the related FOM (Qh * )2 from Eq. (5) is important to estimate a signal/noise ratio of a hydrophone [19] and to describe an effectiveness of a mechanical-to-electric energy conversion due to the piezoelectric effect. 2.2 Components of 2–1–2 Composites The main component of the studied 2–1–2 composites is lead-free FE SC that forms the LFT. Among potential components to be of interest in the sense of the ecofriendly and piezoelectric characteristics, we choose the [0 0 1]-poled domain-engineered [Lix (K1-y Nay )1-x ](Nb1-z Taz )O3 :Mn (KNNTL-Mn) SC [15] with the perovskite-type structure, see the full set of electromechanical constants in Table 1. Herewith we note that the longitudinal piezoelectric coefficient d 33 of this lead-free [0 0 1]-poled KNNTL-Mn SC [15] is larger than d 33 of many conventional poled FE ceramics with the perovskitetype structure, e.g. ZTS-19, modified PbTiO3 , PZT-4, PZT-5, and PZT-5A ceramics [5, 6]. In Table 2, the full sets of electromechanical constants of poled FE ceramics with the perovskite-type structure [16–18] are given. These constants are involved in the | K (c) || matrix from Eq. (1). For comparison, we consider the sets of constants of two lead-free compositions (see KNNLT and KNN–KCN in Table 2) and a lead-containing modified PbTiO3 . These ceramic components exhibit a smaller piezoelectric effect in comparison to KNNTL-Mn from the LFT. It should be noted that the transverse piezoelectric coefficient e31 of the aforementioned ceramic components can be either negative or positive. In general, the FE ceramic components from Table 2 represent different examples of the anisotropy of the piezoelectric and elastic properties that can influence the electromechanical properties of the LST to a certain degree. We add that the piezoelectric activity of these FE ceramic components from the LST is lower than the piezoelectric activity of KNNTL-Mn SC from the LFT. For instance, poled KNNTL ceramic is characterised by the longitudinal piezoelectric coefficient d 33 = 174 pC/N, and this is the largest d 33 value related to the FE ceramic components listed in Table 2. In Table 3, we show the elastic and dielectric properties of polyethylene (PE) being the piezo-passive polymer component [7, 19] of the studied 2–1–2 composites. The set of constants of PE from Table 3 is included in the || K (P) || matrix from Eq. (1) to find the effective electromechanical properties of the LST.
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E (in 10–12 Pa−1 ), piezoelectric Table 1. Experimental room-temperature elastic compliances sab
σ [15] of lead-free [0 0 1]-poled domaincoefficients d ij (in pC/N), and dielectric permittivity εpp
engineered KNNTL-Mna SCb E s11
E s12
E s13
E s33
E s44
E s66
d31
d33
d15
σ /ε ε11 0
σ /ε ε33 0
33.4
–7.36
–25.8
57.7
12.8
13.5
–260
545
66
400
650
a KNNTL-Mn SC is [Li (K x 1−y Nay )1−x ](Nb1-z Taz )O3 :Mn, where x = 0.06, y = 0.1–0.3, z =
0.07–0.17, and the level of Mn doping is 0.25 mol. % [15] b Electromechanical constants are related to the main crystallographic axes of the SC sample (see inset 1 of Fig. 1) with macroscopic symmetry 4mm [15]
E (in 1010 Pa),a piezoelectric coeffiTable 2. Experimental room-temperature elastic moduli cpq ξ
cients efp (in C/m2 ) and dielectric permittivities εff of FE ceramics Constants
KNNLT [16]b
KNN–KCN [17]c
Modified PbTiO3 [18]d
E c11
10.5
14.6
14.33
E c12 E c13 E c33 E c44
4.1
5.6
3.220
3.9
4.7
2.413
13.7
12.9
13.16
3.1
4.6
5.587
e31
–0.6
–2.2
0.4584
e33
19.9
8.4
6.499
e15
6.3
5.7
5.923
424
310
210
361
190
140
ξ ε11 /ε0 ξ ε33 /ε0
a FE ceramics poled along the OX axis are characterised by ∞mm symmetry, so that the elastic 3 E – cE )/2 [13]. modulus c66 obeys the condition c66 = (c11 12 b Chemical composition is Li 0.03 (K0.48 Na0.52 )0.97 (Nb0.8 Ta0.2 )O3 [16]. c Chemical composition is (K 0.45 Na0.55 )NbO3 doped by K4 CuNb8 O23 [17]. d Chemical composition is (Pb 0.9625 La0.025 )(Ti0.99 Mn0.01 )O3 [18].
3 Piezoelectric Properties and Hydrostatic Parameters of Composites In Sect. 2, we present and discuss our new results on the effective piezoelectric properties and hydrostatic parameters (4) and (5) of the 2–1–2 KNNTL-Mn SC/FE ceramic/PE composites, where the FE ceramic component in the LST is one of those listed in Table 2. This means that the composites containing either KNNLT or KNN–KCN ceramic in the
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Table 3. Experimental room-temperature elastic moduli cab (in 1010 Pa) and dielectric permittivity εpp of PE [7, 19] Polymer
c11
c12
εpp /ε0
Polyethylene (high-density)
0.0778
0.0195
2.3
LST are lead-free in contrast to the composite containing modified PbTiO3 ceramic in the same LST. In the LST of the composite, we consider the cylindrical FE ceramic rods with the large aspect ratio of the rod base ηc = 100. This enables one to achieve a large elastic anisotropy even at small volume fractions mc of the rods [9] in the LST. A variation of the volume fraction m of the FE SC component in the composite and changes in the rotation angle γ in the LST are important to achieve maxima of the hydrostatic parameters from Eqs. (4) and (5) at ηc = const and mc = const. Below we show graphs concerned with the orientation effect due the rotation of the rod bases in the LST (see inset 3 in Fig. 1) at fixed volume fractions m of the FC SC component. We consider the hydrostatic parameters of the 2–1–2 composite at m = 0.05–0.30, i.e., at volume fractions of the LSTs from 0.70 to 0.95. Graphs in Fig. 2 show changes in the hydrostatic parameters (4) and (5) of the leadfree KNNLT-Mn SC/KNNTL ceramic/PE composite at small volume fractions mc of KNNLT in the LST. An increase of the volume fraction mc of the ceramic component in the LST leads to a decrease of the hydrostatic piezoelectric coefficients d h * , gh * and FOM (Qh * )2 of the composite (cf. Figure 2, a and 2, c, Fig. 2, c and 2, e, Fig. 2, b and 2, d, and so on). This decrease is observed even at the relatively small volume fractions mc = 0.03–0.05. A simple comparison of the elastic properties of the ceramic and polymer components (see Tables 2 and 3) suggests that the volume-fraction level mc = 0.03–0.05 is of interest because the elastic moduli cab (2),E of the LST become by about an orderof-magnitude larger in comparison to cab of PE that is regarded as a matrix component in the LST. Such specifics of the elastic properties of the LST and the rotation of the ceramic rod bases therein influence together and lead to a non-monotonic behaviour of the hydrostatic piezoelectric coefficients and d h * and gh * of the composite at variations of the rotation angle γ . Based on the composite structure (Fig. 1) and symmetry of the components, one can state that the hydrostatic parameters Π h * obey the condition: Πh∗ (m, mc , γ , ηc ) = Πh∗ (m, mc , 180o −γ , ηc ). This enables us to analyse the orientation effect in the studied 2–1–2 composites at rotation angles γ from 0° to 90°. The change of the sequence of curves 1–6 related to gh * in comparison to d h * (cf. Figure 2, a and 2, b) is concerned with an effect of the combination of the piezoelectric and dielectric properties [5–7], and this effect is noticeable at the small volume fraction m of SC. Taking the composite structure, components and symmetry into account, we represent a relationship between the hydrostatic piezoelectric coefficients from Eqs. (4) in accordance with Eq. (6) as follows: ∗σ . gh∗ = dh∗ /ε33
(7)
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Fig. 2. Orientation (γ ) dependences of hydrostatic piezoelectric coefficients d h * (in pC/N, graphs a, c and e) and gh * (in mV. m/N, graphs b, d and f) of the 2–1–2 KNNTL-Mn SC/KNNLT ceramic/PE composite at the volume fraction of FE ceramic mc = 0.03 (graphs a and b), mc = 0.04 (graphs c and d). or mc = 0.05 (graphs e and f) in the LST.
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Graphs in Fig. 2 enable us to conclude that the presence of two related lead-free components (namely, KNNTL-Mn SC and KNNLT ceramic in the adjacent layers, see chemical compositions in footnotes in Tables 1 and 2) does not lead to a serious worsening of the piezoelectric performance of the composite. The aforementioned lead-free components play different roles in the 2–1–2 composite structure (Fig. 1) and influence the piezoelectric hydrostatic response in different ways. Table 4 illustrates the electromechanical properties of the LST where one of the three FE ceramic components from Table 2 is present, and the volume fraction mc of the ceramic component is small. It should be noted that even at mc = 0.03 or 0.05, the LST is characterised by an appreciable elastic anisotropy, and only the elastic compliance s22 (2),E undergoes minor changes when changing the ceramic component at mc = const (see Table 4). Such a behaviour of s22 (2),E is inextricably linked to the orientation of the ceramic rod base (see inset 3 in Fig. 1) at the rotation angle γ = 0°. In our opinion, the rotation of the rod base leads to changes in the elastic anisotropy of the LST, however this anisotropy remains noticeable and influences the piezoelectric coefficients d 31 * and d 32 * from Eqs. (4) and, therefore, the hydrostatic parameters of the composite. In Fig. 3, examples of the orientation (γ) dependence of the hydrostatic FOM (Qh * )2 are graphically represented for the three studied 2–1–2 composites. In these composites, the FE ceramic component in the LST is either lead-free (KNNLT or KNN-KCN) or lead-containing, based on PbTiO3 (see chemical compositions in footnotes to Table 2). A simple comparison of Fig. 3, a–c to Fig. 2 shows that the sequence of curves 1–6 related to gh * remains similar in the case of (Qh * )2 . By analogy with the hydrostatic piezoelectric coefficient gh * from Eq. (7), we write the expression for the hydrostatic FOM (Qh * )2 as follows:
∗ 2
2 ∗σ Qh = dh∗ /ε33 .
(8)
The appreciable effect of the combination of the piezoelectric and dielectric properties in the 2–1–2 composite influences not only its hydrostatic piezoelectric coefficient gh * , but also the hydrostatic FOM (Qh * )2 from Eqs. (5) and (8) largely at volume fractions m 450 °C.
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Fig. 4. (a) Dependences of ε/ε0 and tan δ of Pb(Zr0.96 Ti0.04 )O3 solid solution on temperature and time. The insets show the evolution of the induced piezoresonant spectra; (b) the induced piezoresonant spectra, obtained at T = 97 °C.
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4 Conclusion In the result of the studies, it was established that thermostatting at T > 450 °C irreversibly shifts T 1mc in accordance with Fig. 2, curve 4. Subsequent hardening only leads to a further increase in T 1mc . Apparently, such a thermal effect leads to a decrease in the number of clusters of the PbTiO3 type [11] due to the generation of groups of ions in the form of interlayers of elements of binary oxides ZrO2 and TiO2 , coherently embedded in the crystalline matrix phase [13]. The possibility of reducing the temperature hysteresis of AFE ↔ FE phase transition by increasing T 1mc with a targeted choice of its value on the temperature scale by changing the concentration x has also been shown; it has great prospects for creating highly sensitive pyroelectric and piezoelectric sensors, as well as converters of thermal and mechanical energy into electrical one. Acknowledgements. The study was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation (State task in the field of scientific activity in 2023). Project No. FENW-2023–0010/(GZ0110/23–11-IF).
References 1. Shirane, G., Suzuki, K., Takeda, A.: Phase transitions in solid solutions PbZrO3 and PbTiO3 (II) X-ray study. J. Phys. Soc. Japan 7, 12 (1952) 2. Shilkina, L.A., et al.: Formation of a cluster structure in the PbZr1-x Tix O3 system. Phys. Solid State 58, 551–556 (2016) 3. Vakhrushev, S.B., et al.: Electric field control of antiferroelectric domain pattern. Phys. Rev. B 103, 214108 (2021) 4. Cordero, F., et al.: Merging of the polar and tilt instability lines near the respective morphotropic phase boundaries of PbZr1-x Tix O3 . Phys. Rev. B 87, 094108 (2013) 5. Andryushina, I.N., et al.: The PZT system (PbZr1-x Tix O3 , 0.0 ≤ x ≤ 1.0): specific features of recrystallization sintering and microstructures of solid solutions (Part 1). Ceramics Int. 39, 753 (2013) 6. Andryushina, I.N., et al.: The PZT system (PbTix Zr1–x O3 , 0 ≤ x ≤ 1.0): the real phase diagram of solid solutions (room temperature) (Part 2). Ceramics Int. 39, 1285 (2013) 7. Andryushina, I.N., et al.: The PZT system (PbTix Zr1− x O3 , 0 ≤ x ≤ 1.0): high temperature X-ray diffraction studies. Complete x – T phase diagram of real solid solutions (Part 3). Ceramics Int. 39, 2889 (2013) 8. Pavelko, A.A., et al.: Refined phase portrait of the rhombohedral region of the x – T diagram of the Pb(Zr1-x Tix )O3 system and singularities of dielectric spectra of its solid solutions. Phys. Solid State 57, 2431–2440 (2015) 9. Zakharov, Yu.N., et al.: Control of the temperature hysteresis and diffuse dielectric anomaly in the temperature range of the ferroelectric-antiferroelectric phase transition in PbZr1-x Tix O3 (0.03 ≤ x ≤ 0.05) ceramics. Phys. Solid State 48, 1077–1078 (2006) 10. Talanov, M.V., Pavelko, A.A., Reznichenko, L.A.: Low- and high-field electromechanical responses of relaxor-based multicomponent ceramics for application in multiregime actuators. J. Adv. Dielectr. 10, 2060004 (2020) 11. Morozov, E.M., et al.: On the ferroelectric–antiferroelectric transition in lead zirconate with small additions of titanium and germanium. Crystallogr. Rep. 23, 119–123 (1978)
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12. Klimnchenko, E.N., et al.: The influence of the internal field induced by defects on the interval of existence of the ferroelectric phase of lead-containing antiferroelectrics of the OSP type. In: Proceedings of the 7th International Symposium, pp. 131–132 (2004) 13. Samoˇılenko, Z.A., et al.: Characteristics of the clustered structure of Pb(Li, La)(Zr1-y Tiy )O3 in the antiferroelectric-ferroelectric transition region. Tech. Phys. 43, 171–175 (1998)
Dielectric, Piezoelectric and Pyroelectric Properties of Ceramic Solid Solutions Based on PZT System A. A. Pavelko1(B) , A. A. Martynenko1 , K. P. Andryushin1 E. V. Glazunova1 , A. V. Nagaenko2 , L. A. Shilkina1 , and L. A. Reznichenko1
,
1 Research Institute of Physics, Southern Federal University, Rostov-on-Don 344090, Russia
[email protected] 2 Institute of High Technology and Piezo Technic, Southern Federal University,
Rostov-on-Don 344090, Russia
Abstract. Using the method of two-stage solid-phase synthesis, involving mechanical activation of the synthesized powders and subsequent sintering using conventional ceramic technology, ferroelectric multicomponent solid solutions, based on the PZT system, were produced, characterized by fairly high Curie temperatures and pyroelectric coefficients, high coefficients of pyroelectric sensitivity and resistance to vibration interference, which makes it possible to classify them as pyroelectric materials, are promising for use as working elements of sensors for pyroelectric receivers of radiant (thermal) energy. Keywords: Ferroelectrics · Lead Zirconate Titanate · Ceramics · Solid Solutions · Phase Transition · Pyroelectric Coefficients · Dielectric Properties · Piezoelectric Properties
1 Introduction The search for new pyroelectric materials that have a set of parameters (minimum values of specific heat capacity, relative dielectric constant and dielectric losses in combination with a high pyroelectric coefficient) is an important task of modern physical materials science, the solution of which will improve the characteristics of existing pyroelectric detectors. Based on the totality of parameters, the best pyroelectric materials, in most cases, are ferroelectrics (FEs), however, difficulties may arise on the way to their creation due to high technological requirements during their synthesis and sintering. A promising basis for pyroelectric materials is multicomponent solid solutions, based on the PbZrO3 – PbTiO3 system [1–3], including relaxor components [4–7], such as PbNb2/3 Zn1/3 O3 – PbTiO3 and PbNb2/3 Mg1/3 O3 – PbTiO3 . The present work is devoted to the development and study of the electrophysical and, first, pyroelectric properties of such multicomponent solid solutions.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 197–208, 2024. https://doi.org/10.1007/978-3-031-52239-0_19
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2 Research Method 2.1 Materials and Research Methods The objects of study were ceramic samples of solid solutions PbTiO3 – PbZrO3 – PbNb2/3 Zn1/3 O3 – PbNb2/3 Mg1/3 O3 + MnO2 , based on the PbZrO3 – PbTiO3 (PZT) system. The production of solid solutions was carried out by two-stage solid-phase synthesis followed by sintering using conventional ceramic technology. The selection of optimal technological modes was carried out on a series of experimental samples with X-ray control of the phase composition and relative density (ρ rel. ) of the samples. The optimal synthesis modes were T sint1 = 820 °C, T sint2 = 850 °C, τ sint1 = τ sint2 = 5 h. Mechanical activation (MA) of the synthesized solid solution powders was carried out at the stage of manufacturing press powders, prepared for sintering. MA was carried out in a high-energy grinding planetary ball mill AGO-2 (Russia). Grinding was carried out in an alcohol medium for 15 min., the drum rotation speed was 1820 rpm. Optimal sintering mode: T sp = 1150 °C, sintering time τ sp = 3 h. Search measurement samples were made in the form of disks with a diameter of 12 mm and a thickness of 1 mm. The electrodes were applied by burning silver-containing paste twice. X-ray structural studies were carried out by powder diffraction on a DRON-3 diffractometer using CoKα radiation. The cell parameters were calculated using the standard method [8], the error in measuring the parameters was: a = b = c = ± 0003 Å, V = ±0.05 Å3 , where a, b, c are the cell parameters, V is the cell volume. Experimental density of samples ρ exp was measured by hydrostatic weighing in octane. X-ray density was calculated using the formula: ρ X-ray = 1.66 × M/V, where M is the molecular weight per cell, V is the cell volume. The relative density was determined using the formula: ρ rel = (ρ exp /ρ X-ray ) × 100%. To study the microstructure of the samples, a JSM-6390L scanning electron microscope (Japan) with a microanalyzer system from Oxford Instruments (UK) was used. The resolution of the microscope was 1.2 nm at an accelerating voltage of 30 kV (image in secondary electrons), the accelerating voltage range was from 0.5 to 30 kV, the magnification was from × 10 to × 1000000, and the beam current was up to 200 nA. The polarization of the samples was carried out in a chamber with polyethylene siloxane liquid at ~140 °C. Heating to the specified temperature was carried out according to a linear law for 0.5 h, accompanied by an increase in the constant electric field from 0 to 3 kV/mm. The samples were kept under these conditions for 25 min, after which they were cooled under the field to room temperature. The dielectric, piezoelectric and pyroelectric properties of the obtained samples were studied in a wide temperature range. The study of relative dielectric constant (ε/ε0 ) and dielectric losses (tanδ) was carried out in the temperature ranges T = 20–350 °C and measuring signal frequencies f = 20–106 Hz on a specially designed setup, including a high-temperature furnace, an SRS PTC10 thermal controller with a sensor temperature PT-100, as well as an LCR meter Agilent E4980A. Piezoelectric characteristics were measured automatically on the same bench using a Keysight E4990A impedance meter; the resonance-antiresonance method was used to calculate the piezoelectric parameters [9]. The relative dielectric constant of polarized samples (ε33 T /ε0 ), piezoelectric modulus (|d 31 |), piezoelectric coefficient (piezosensitivity) (|g31 |), electromechanical coupling
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coefficient of planar vibration modes (K p ), mechanical quality factor (QM ), Young’s modulus (Y 11 E ) and speed of sound (V 1 E ) were calculated. The errors in measuring electrophysical parameters have the following values: ε33 T /ε0 ≤ ±1.5%, K p ≤ ± 2.0%, |d 31 | ≤ ±4.0%, QM = ≤ ± 12%; Y 11 E ≤ ± 07%. The piezoelectric modulus d 33 was determined in the quasi-static mode on samples in the form of a disk using d 33 meter YE2730A. To study the pyroresponse, various methods were used: the quasi-static method, the temperature wave method, and the radiation method [10]. To study the pyroelectric effect using the quasi-static method, we used the setup described above, in which, using an Agilent 4339B high-resistance meter, we measured the current iQS flowing through a pyroelectric sample with a continuous change in its temperature at a rate of b = 1 °C/min, from which the pyroelectric coefficient was determined as pQS = iQS /(Ab), where A is the surface area of the sample. The disadvantage of this method is the inability to separate currents of a pyroelectric nature from thermally stimulated currents that arise when the temperature of crystals with defects increases. To solve this problem, dynamic methods are used, one of which is the temperature wave method [11, 12]. The method is based on the fundamental difference in the temperature behavior of pyroelectric (ip ) and thermally stimulated (iTS ) currents: ip is reversible, and its value is proportional to the rate of temperature change: ip = pAb, while iTS within small temperature intervals is proportional to the temperature itself: iTS = i0 + RT, where R is a constant coefficient. In our case, in a specially designed temperature chamber using an SRS PTC10 thermal controller, a sinusoidal temperature wave was set according to the law: T = T0 + T1 sin(ωt),
(1)
in this case, the wave amplitude T 1 was set in the range from 0.25 to 0.5 °C, and the average temperature T 0 was selected from the range 30–200 °C. In parallel, also using an Agilent 4339B high-resistance meter, the iHW current generated by the sample was measured, which can be written as follows: iHW = i0 + RT0 + RT1 sin(ωt) + pHW AT1 ωcos(ωt). The total current contains a non-pyroelectric contribution, in-phase with the temperature wave, and a pyroelectric component, leading the temperature wave by 90°. The sum of these oscillations gives the resulting current wave, directly measured in experiment, which can be represented in the following form: iHW = idc + i1 sin(ωt + φ),
(2)
1/2 where idc = i0 + RT0 , i1 = ip2 + in2 , ip = pHW AT1 ω = i1 sin(φ), in = RT1 = i1 cos(φ). If the recorded current is of a purely pyroelectric nature, R = 0 and φ = π/2. If the sample generates both pyroelectric and non-pyroelectric currents, R = 0 and φ < π/2. By dividing the measured current into two components in this way, it is possible to determine the value of the desired pyroelectric coefficient pHW .
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We also used the radiation dynamic method, first described by Chynoweth [13]. In this method, the surface temperature of the sample was continuously varied by a small amount around the average temperature by illuminating the sample with a sinusoidally varying flux of infrared radiation, while the average temperature varied continuously in the range from 20 to 240 °C. As a result of this temperature change, the sample generated a pyroelectric current, the rms value of which was measured using an SR830 lock-in amplifier (Stanford Research Systems). This method has many shortcomings, among which the main one is the difficulty of determining the absolute value of the pyroelectric coefficient pIR , caused by the need for independent measurements of the heat capacity and thermal conductivity of the sample, the density of the incident radiation flux, and the absorption capacity of the irradiated electrode, due to which one cannot be sure of the actual temperature change in the sample. However, the great advantage of the method is that the described temperature change mode is close to the operating mode of pyroelectric detectors, which is of great importance from the viewpoint of the potential practical application of the materials under study. Due to these features, in this work pIR is given in relative units.
3 Results and Discussion Figure 1 shows X-ray diffraction patterns of the material, obtained under different conditions, in the angle range 2θ = 20°–80°, as well as individual reflections 111 and 200 on an enlarged scale. X-ray phase analysis of the samples, the results of which are given in Table 1, showed that all solid solutions had a perovskite structure and there were no impurity phases. The symmetry of the crystal lattice is rhombohedral (Rh) since it is based on the solid solutions of the PZT system from the Rh region of its phase diagram. The X-ray lines in the X-ray diffraction pattern of the material are double, which indicates that the solid solution is inhomogeneous and consists of two solid solutions with similar cell parameters. (110)
(a)
(b)
(100) (111)
No MA
(211)
(200)
(c)
(210)
(200)
111 111
(220)
MA
20
30
40
50
2 , (°)
60
70
80
44
45
2 , (°)
46 51
52
2 , (°)
53
Fig. 1. X-ray diffraction patterns of the material in the angle range 2θ = 20°–80° (a), as well as individual reflections 111 (b) and 200 (c) on an enlarged scale, obtained using different conditions.
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Table 1. Structural parameters and densities of the obtained material. Acti-vation Sintering Phase mode MA
No MA
Symmetry a, Å
T sint = 100 Rh 1150 °C, Perovskite 3h 2.5 ZrO2 T sint = 100 Rh 1150 °C, Perovskite 1.5 h traces of ZrO2
α, (o ) V, Å3 ρ exp , ρ X-ray , ρrel, g/cm2 g/cm2 %
4.075 89.78 67.66 6.82
8.08
84.40
4.079 89.75 67.89 6.01
8.05
74.63
Fig. 2. Photographs of the microstructure of the ceramic sample.
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Typical images of the microstructure of the ceramics, obtained at different magnifications are presented in Fig. 2. The grain landscape is determined by the following indicators: phasing of the crystal structure, porosity, presence, and nature of the defects. Features of the phase structure are reflected in such elements of the grain structure as: (i) (ii) (iii) (iv) (v)
average grain size; type of grain size distribution; nature and degree of uniform grain size (heterogeneity); the shape of the grains and the degree of its perfection; type of packaging (dense/loose, mosaic, chaotic, combined, block, radial-ring, with aggregation elements, etc.), presence and nature of texture.
When describing porosity, the types of pores (open, closed, intergranular, intragranular), their average size, dispersion (range of size changes), dispersion (fineness), shape, nature of distribution, origin (technological, secondary, etc.) are analyzed. Defectiveness is determined by the localization of cracks: transgranular, that is via the grain body, intercrystalline, inclusions of various natures, etc. Following the above scheme for describing the grain structure of ferroelectric ceramics, let us consider the given fragments of the microstructures of the material under study. Belonging to the group of ferroelectric rigid materials, this material has a fine-grain (with an average grain size, Dav. , ~ 4…6 μm), densely packed, mosaic-chaotic, uniformly-grain structure with grains of irregular (most often elongated) geometric shape. A feature of the grain landscape of the ceramics under study is the formation of areas with fused grains (Fig. 2 d, solid curved line), between which pores are visible, most likely of artificial origin, namely “breakouts”, formed during the mechanical processing of samples, which indicates a reduced strength of the boundaries of such formations. In addition, attention is drawn to the presence of areas in the grain field (Fig. 2b, d, highlighted with solid circles) with tiny unformed grains, indicating, most likely, local fluctuations in composition, caused by the same local violations of “average” technological regulations. Minor inclusions of impurity phases are visible (Fig. 2b, highlighted by dashed circles), localized in the pores. Figure 3 shows the dependences of ε/ε0 and tanδ of the sample on temperature and frequency of the measuring signal. The phase transition (PT) from the ferroelectric to paraelectric state occurs at temperature ~294 °C. Also, there is a strong dispersion of both dielectric constant and dielectric losses. To quantify the depth of dispersion, the parameter E = 100% × (ε/ε0.25Hz – ε/ε0.1MHz )/(ε/ε0.1MHz ) was calculated, the dependences of which on temperature are shown in the inset of Fig. 3a. The dispersion ε/ε0 begins to increase at temperatures of 100–150 °C and reaches maximum values in the PT region. The dispersion tanδ appears at lower temperatures, and at temperatures above the PT it grows exponentially. Figure 4, as an illustration of the application of the temperature wave method, shows the experimental dependences of temperature T and pyroelectric current i on time for the sample, measured at T0 = 30 °C (a, b) and 200 °C (c, d), as well as the results of approximations by expressions (1) and (2), which made it possible to calculate the pyroelectric coefficients pHW , given in Table 2. The table also shows the values of εT 33 /ε0 , d33 , |d31 |, as well as the technical coefficients pHW × ε0 /εT 33 and pHW /|d31 |, characterizing the
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Fig. 3. Dependences of ε/ε0 (a) and tanδ (b) of the material on temperature. The arrows show the direction of temperature change. In the inset of (b) markers depict the dependence ln ω(T m ), where T m are the temperatures of the maxima ε/ε0 obtained at the frequency ω = 2πf , and the solid lines illustrate the fulfillment of the Vogel-Fulcher law (3).
Fig. 4. Time dependences of temperature T and pyroelectric current i for the sample, measured at T 0 = 30 °C (a, b) and 200 °C (c, d). The dotted line in (c) is the result of applying a Fourier filter to the i(t) relationship. The fragments of the dependences T (t) and i(t), shown on the right (b, d), illustrate the correspondence of the temperature wave and the pyroelectric current wave (hollow markers) to expressions (1) and (2) (solid curves).
detectability and resistance of the pyroelectrics to vibration interference (a decrease in the piezoelectric modulus helps to reduce the piezoelectric polarization that occurs on the electrodes of the pyroelectric under various mechanical influences), respectively.
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Material
εT 33 /ε0
d 33 × 1012 , C/N
|d 31 | × 1012 , C/N
phw × 104 , C/(m2 · K)
(phw × ε0 /εT 33 ) × 106 , C/(m2 · K)
(phw /|d 31 |) × 10−6 , N/(m2 · K)
This work
348
118
34
2.6
0.8
7.6
PZT-5A
1700
374
171
4.7
0.3
2.8
For comparison, the parameters of the well-known industrial material PZT-5A are given. Figure 5 shows the dependences of εT 33 /ε0 , |d 31 |, pHW , pIR and pQS of polarized sample on temperature. As the phase transition approaches the paraelectric state, the pyroelectric coefficient pQS increases by an order of magnitude, and therefore in the insets it is presented on a scale that makes it possible to compare its values with the results, obtained by the temperature wave method.
Fig. 5. Temperature dependences of the electrophysical parameters of the polarized sample: εT 33 /ε0 (1), |d 31 | (2), pHW (3), pIR (4), pQS (5).
The material has high T C and is also characterized by a strong diffusion of the ε/ε0 (T ) maximum, accompanied by a significant dispersion of the dielectric constant, reaching 60% in the FE PT region. Since X-ray phase analysis did not reveal the presence of impurities with which the observed features could be associated, we carried out a dielectric relaxation analysis according to the method, described in [14–16]. This approach made it possible to record shifts in the maxima of the ε/ε0 (T ) dependences, illustrated by the inset in Fig. 3b. There is the dependence of the logarithm of frequency ω = 2πf on the temperature (T m ) of the maximum of ε/ε0 (T, ω) dependence. Two
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regions can be distinguished in the ln ω(T m ) dependence, differing in different relaxation patterns. The first region (I), in which high temperature (high frequency) is satisfactorily described by the Vogel-Fulcher law: ω = ω0 exp [Ea /k(T −TVF )],
(3)
where E a is the average height of the potential barrier, T is the absolute temperature, k is Boltzmann’s constant, ω0 is the pre-exponential factor and T VF is the “freezing” temperature of the relaxation process. The relaxation width in this region is about 1°; as noted in [17], such weak relaxation can be observed in ferroelectric relaxors, in which the characteristic temperature, associated with the destruction of the classical domain structure, is quite high and close to the T m region. In the materials under consideration, the relaxor components PbNb2/3 Zn1/3 O3 and PbNb2/3 Mg1/3 O3 may be responsible for such a behavior. The second (II), low-temperature region, at first glance, has a regular activation character, however, the Arrhenius law fit gives inadequate activation energy value (E a ≈ 20 eV), while the use of the Vogel-Fulcher law allows one to obtain more reliable relaxation parameters, the values of which for both regions are shown in Table 3. It should be noted that all coefficients were calculated with a large error, which is associated both with the small number of experimental points per region and with the weakness of the described relaxation processes. Table 3. Results of approximation of relaxation processes of the dependences ε/ε 0 (T, ω) of the material by the Vogel-Fulcher law. Region
E a , eV
ω0 , rad/s
T VF , K
Region I
5.54 × 10–3 ± 2.44 × 10–3 3.56 × 10–2 ± 1.36 × 10–2
6.68 × 1013 ± 1.58 × 1013 1.54 × 1011 ± 2.27 × 1010
290.96 ± 0.02
Region II
267.97 ± 3.45
Thus, low-temperature relaxation cannot be associated with Maxwell-Wagner polarization [18], which arises in the result of charge accumulation at grain boundaries, extended defects, or pores, which could be expected, based on an analysis of the grain structure of the studied ceramics (Fig. 2). Most likely, the observed processes are associated with the relaxation of domain walls, the dynamics of which can be “frozen” during their interaction with defects in the crystal structure [19], due to which such relaxation is usually described by the Vogel-Fulcher law with similar parameters [20]. The contribution of domain walls sandwiched by defects can also explain the observed dispersion of the dielectric constant in the FE PT region, as well as some features of the pyroelectric response of the object. Firstly, experiments carried out using the temperature wave method revealed some features in the formation of current waves with increasing average temperature. Thus, Fig. 4a, b shows the experimental result, close to the model one: the experimentally measured temperature and current waves are well described by expressions (1) and (2) with parameters T 0 = 30.00 °C, T 1 = 0.32 °C, ω = 0.0151 rad/s, idc = − 192 × 10−13
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A, i1 = 1.40 × 10−10 A, φ = −1.32 rad (−75.64°). However, such correspondence is not always satisfied and in the general case idc can also depend on time according to some law, which can be observed in Fig. 4 c, which shows the dependence i(t) measured at T 0 = 200 °C. In such cases, most often, idc can be described by an exponential decay function, which indicates the relaxation nature of the observed phenomenon, which can be associated both with temperature activation of free charge carriers and with mechanisms of a different nature, for example, relaxation of domain walls. Since the measurements are carried out in a limited time range, it is not possible to establish specific parameters of such a dependence, and therefore we processed the experimental data in accordance with the following procedure: the measured current wave was smoothed using a Fourier filter, the smoothed curve represented the dependence of the average value of the wave (idc (t), dotted line in Fig. 4c), corresponding to the relaxation component of the current, which was subsequently subtracted from the original experimental data. In the result of the subtraction, a purely pyroelectric component remained (Fig. 4 d), necessary for calculating the pyroelectric coefficient. Such a contribution appeared in the measured thermally stimulated current only at T 0 = 200 °C, which somehow correlates with the results of the relaxation analysis presented above. The pHW and pQS coefficients presented in Fig. 5 in the temperature range of 100– 200 °C are in good agreement, while in the range of 30–50 °C there is a significant discrepancy in these parameters. This may be due to differences in thermodynamic conditions under which the measurement is carried out. The quasi-static method assumes, albeit slow, but continuous heating of the sample, while in the temperature wave method, the temperature stabilizes in the vicinity of T 0 for a long time. As a result, the pyroelectric response can be significantly influenced by the slow relaxation processes of domain walls, described above, which leads to a significant discrepancy between the pyroelectric coefficients pHW and pQS . These features of the formation of the dielectric response were reflected in the practically important technical coefficients pHW × ε0 /εT 33 and pHW /|d 31 | (Table 2), which at room temperature exceed similar coefficients of known industrial materials, compensating for the initially insufficiently high pyroelectric coefficients of the studied material.
4 Conclusion Samples of pyroelectric material PbTiO3 – PbZrO3 – PbNb2/3 Zn1/3 O3 – PbNb2/3 Mg1/3 O3 + MnO2 based on the PbZrO3 – PbTiO3 system were obtained in the form of ceramics using a two-stage solid-phase synthesis followed by mechanical activation of the synthesized powders and sintering using conventional ceramic technology. Analysis of the dielectric spectra of the obtained material showed that the phase transition from the ferroelectric to the paraelectric state is accompanied by a sequence of relaxation processes, associated with the presence of relaxor components in the composition of the objects and with the critical dynamics of domain boundaries, interacting with structural defects. Obtained ceramics have exceptional pyroelectric properties, pyroelectric parameters pHW , pHW × ε0 /εT 33 (pyroelectric detectivity) and pHW /|d 31 | (pyroelectric resistance
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to vibration interference) at room temperature are 2.6 × 10–4 C/(m2 ·K), 0.8 × 10–6 C/(m2 ·K) and 7.6 × 106 N/(m2 ·K), respectively. Such excellent pyroelectric properties are maintained in the temperature range of 30–200 °C, which makes this material promising from the viewpoint of practical applications. The results of this work can be used in the future in the development of pyroelectric detectors used for remote measurement of the temperature of heated bodies, including moving ones (rolling mills, high-frequency furnaces for hardening or tempering steel; plastic production; food industry, etc.); for measuring power systems; control of overheating of rubbing parts, etc.); in protection systems (from fire, from intruders), which determines the relevance and significance of the results obtained in the work. Acknowledgements. The study was supported by the grant of the Russian Science Foundation No. 23-12-00351, https://rscf.ru/project/23-12-00351/
References 1. Andryushina, I.N., et al.: The PZT system (PbZr1-x Tix O3 , 0.0 ≤ x ≤ 1.0): specific features of recrystallization sintering and microstructures of solid solutions (Part 1). Ceramics Int. 39, 753 (2013) 2. Andryushina, I.N., et al.: The PZT system (PbTix Zr1–x O3 , 0 ≤ x ≤ 1.0): the real phase diagram of solid solutions (room temperature) (Part 2). Ceramics Int. 39, 1285 (2013) 3. Andryushina, I.N., et al.: The PZT system (PbTix Zr1− x O3 , 0 ≤ x ≤ 1.0): high temperature Xray diffraction studies. Complete x–T phase diagram of real solid solutions (Part 3). Ceramics Int. 39, 2889 (2013) 4. Andryushina, I.N., et al.: Crystal structure, microstructure and electrophysical properties of highly sensitive ferroactive materials based on the Pb(Zr1-x Tix )O3 system. Mater. Sci. Eng. B 283, 115804 (2022) 5. Pavelko, A., Shilkina, L., Reznichenko, L.: Phase states and electrophysical properties of multicomponent perovskite solid solutions on the base of PMN-PT and PZT systems. J. Adv. Dielectr. 10, 2060011 (2020) 6. Pavelko, A.A.: Piezodielectric properties of PMN–PZT–PT solid solutions under the action of high temperatures. Bull. Russ. Acad. Sci. Phys. 78, 802–803 (2014) 7. Zakharov, Y.N., et al.: Field-induced enhancement of pyroelectric response of PbMg1/3 Nb2/3 O3 -PbTiO3 and PbFe1/2 Nb1/2 O3 -PbTiO3 solid solution ceramics. Ferroelectrics 399, 20–26 (2010) 8. Fesenko, E.G., Danziger, A.Ya., Razumovskaya, O.N.: New Piezoceramic Materials. Russian State University Press, Rostov-on-Don (1983) (In Russian) 9. IEEE Standard on Piezoelectricity ANSI/IEEE Std 176–1987, New-York (1988).https://doi. org/10.1109/IEEESTD.1988.79638 10. Whatmore, R.: Characterisation of pyroelectric materials. In: Cain, M. (ed.) Characterisation of Ferroelectric Bulk Materials and Thin Films, vol. 2, pp. 65–86. Springer, Dordrecht (2014). https://doi.org/10.1007/978-1-4020-9311-1_4 11. Garn, L.E., Sharp, E.J.: Use of low-frequency sinusoidal temperature waves to separate pyroelectric currents from nonpyroelectric currents. Part I. Theory. J. Appl. Phys. 53, 8974–8979 (1982) 12. Garn, L.E., Sharp, E.J.: Use of low-frequency sinusoidal temperature waves to separate pyroelectric currents from nonpyroelectric currents. Part II. Experiment. J. Appl. Phys. 53, 8980–8987 (1982)
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13. Chynoweth, A.G.: Dynamic method for measuring the pyroelectric effect with special reference to barium titanate. J. Appl. Phys. 27, 78 (1956) 14. Pavelko, A.A.: Effect of Li2CO3 modification on the formation of the ferroelectric properties of PbFe0.5Nb0.5O3 ceramic targets and thin films prepared by RF cathode sputtering. J. Alloys Compd. 836, 155371 (2020) 15. Pavelko, A.A., Pavlenko, A.V., Reznichenko, L.A.: Effect of lithium carbonate modification on the ferroelectric phase transition diffusion in lead ferroniobate ceramics. J. Adv. Dielectr. 12, 2160021 (2022) 16. Talanov, M.V., Pavelko, A.A., Kamzina, L.S.: Domain-wall freezing in Cd2 Nb2 O7 pyrochlore single crystal. Mater. Res. Bull. 145, 111548 (2022) 17. Li, F., et al.: Local structural heterogeneity and electromechanical responses of ferroelectrics: learning from relaxor ferroelectrics. Adv. Func. Mater. 28, 1801504 (2018) 18. Turik, A.V., Radchenko, G.S.: Maxwell-Wagner relaxation in piezoactive media. J. Phys. D Appl. Phys. 35, 1188 (2002) 19. Tagantsev, A.K., Cross, L.E., Fousek, J.: Domains in Ferroic Crystals and Thin Films. Springer, New York (2010) 20. Huang, Y.N., et al.: Domain freezing in potassium dihydrogen phosphate, triglycine sulfate, and CuAlZnNi. Phys. Rev. B: Condens. Matter Mater. Phys. 55, 16159–16167 (1997)
Structural Order/Disorder Phenomena Investigation in Pb-Containing Complex Perovskites with Relaxor Ferroelectric Properties Alla Lebedinskaya1(B) and Angela Rudskaya2 1 Academy of Architecture and Arts, Southern Federal University, 39, Budennovskiy Avenue,
Rostov-on-Don 344002, Russia [email protected] 2 Department of Physics, Southern Federal University, 5, Zorge Street, Rostov-on-Don 344090, Russia
Abstract. This article presents the results of studying the samples, obtained on the base of lead magnoniobate (PMN) and described by the general formula (1 – x)PbMg1/3 Nb2/3 O3 – xPbMg1/2 Nb1/2 O2.75 , where the value of parameter x varies from 0 to 1 with a step of 0.1. Experimentally, it was shown that all compounds from this series are ferroelectric-relaxors. With an increase in the value of the parameter x, the proportion of nonstoichiometric PMN with an artificially set oxygen deficiency increased. Measurements of the dielectric constant of the synthesized samples and the tangent of the dielectric loss angle showed violation of the Curie-Weiss law, which indicates the diffusion of the phase transition. Keywords: Relaxor · Ferroelectric · Perovskite · Lead Magnoniobate · PMN
1 Introduction Lead-based perovskite materials with the general formula Pb(B’x B”1-x )O3 have become the constant subject of extensive research in recent years due to their unique relaxor properties [1–5]. Unlike classical ferroelectrics, in which, during a phase transition from a polar to a non-polar state, a sharp change in dielectric constant occurs according to the Curie-Weiss law, relaxors exhibit a significantly diffuse dielectric peak in a wide temperature range, and also have a wide range of dependence on frequency dispersion. This behavior of relaxors, according to research [6, 7], is associated with a partially ordered distribution of various ions at equivalent crystallographic positions in the B-sublattice, which leads to disorder in composition and polar heterogeneity (the so-called polar nanoregions, PNR) at the atomic level. However, a clearer understanding of the nanoscale structural description of relaxors is still lacking, and a direct connection between local order/disorder and relaxor behavior has not yet been established. Many Pbbased relaxors are complex perovskites in which two or more element types occupy the B-site, resulting in a partial cation-ordered B-site structure [7]. Several cationic ordering models have been proposed for these materials [2, 3, 8]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 209–215, 2024. https://doi.org/10.1007/978-3-031-52239-0_20
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Among oxide perovskites of complex composition Pb3 B’B”2 O9 , lead magnoniobate PbMg1/3 Nb2/3 O3 (PMN), which is still a conventionally model object for studying relaxor properties, is of great interest to researchers. The space charge model was first proposed for PMN. In this model, it was assumed that atoms in the B-sublattice in the ratio Nb:Mg = 1:1 are distributed alternately on (111) planes, which leads to the appearance of an unbalanced electric charge. However, most of the results clarifying the distribution of atoms in the B-sublattice are based only on chemically doped materials, and studies of pure perovskites such as Pb(B 1/3 B 2/3 )O3 are very limited [2, 8]. Complex compositions, developed on the base of lead magnoniobate, to study the cause of the appearance of relaxor properties, understand the mechanism of their occurrence and influence on their intensity [8–11]. Interest in the properties and methods of obtaining ceramic samples is due to that those single crystals, which, as a rule, have high characteristics, are very expensive and unique in preparation. However, due to defects that occur during their production and further processing, the use of crystals is often limited. The advantage of ceramic materials, compared to single-crystalline materials, is their availability, practicality and the identity of the functional properties observed in them upon transition to the nanocrystalline state, compared to single crystals. When synthesizing solid solutions, based on PMN, the main problem that needs to be solved is associated with the appearance in the samples of a piezoelectrically inactive pyrochlore phase together with the perovskite phase, which does not exhibit practically important ferroelectric properties.
2 Materials and Methods 2.1 Fabrication of Samples and X-ray Structural Studies Ferro-ceramic samples were synthesized from a mixture of PbO, MgO, and Nb2 O5 oxides, selected in the appropriate proportions [11–13], corresponding to the composition: (1 – x)PbMg1/3 Nb2/3 O3 – xPbMg1/2 Nb1/2 O2.75 , where x varied from 0 to 1 with a step of 0.1. The synthesis of the samples was carried out according to the usual ceramic technology for 3 h at a temperature of 890 °C. It should be emphasized that both synthesis and sintering of the samples were carried out under almost completely identical conditions, which excludes the possibility of the influence of changes in the conditions of the technological process on the properties of the ceramic [9–13]. After each stage of the synthesis, X-ray structural studies were carried out on an URS-50 IM X-ray diffractometer using Cukα - radiation with Cukβ1 filtration by a nickel filter. Lines with h2 + k 2 + l 2 = 14 (321) and 24 (422) were recorded, the measurement accuracy was 0.003 Å in the first case and 0.001 Å in the second case. X-ray diffraction profiles of all samples were processed using the PowerCell computer program. 2.2 Investigation of Dielectric Properties of Samples Measurements of the dielectric constant of the obtained samples and the tangent of the dielectric loss angle were carried out at the frequency of the measuring field f = 1 kHz
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in the range of research temperatures from 25 to 150 °C with the heating rate of the samples v = 1 °C / min, filmed forward and backward. The temperature dependences of the electrical capacity (C) and electrical conductivity (G) of the samples were measured. The calculation of the required values was carried out according to the formulae: ε=
4π h C S
(1)
G ωC
(2)
tanδ =
where E is the dielectric constant, tan δ is the tangent of the dielectric loss, h and S are the thickness and surface area of the sample, C and G are the measured electrical capacity and electrical conductivity of the sample, ω = 2π f , where f = 1 kHz. 2.3 Determination of Phase Composition of the Samples with Using the Retrieve Crystallographic Database Determination of the phase composition of samples of the system of complex oxides (1 – x)PbMg1/3 Nb2/3 O3 – xPbMg1/2 Nb1/2 O2.5 (the presence of foreign phases in the form of a mixture of simple oxides, perovskite and pyrochlore phases) was carried out using the Retrieve crystallographic database. At the initial stage of determining the phase composition, we used models of magnesium niobate (Mg3 Nb6 O11 , MgNb2 O6 , Mg0.652 Nb0.598 O2.25 ), models of magnesium oxide (MgO), models of niobium oxide (NbO, NbO2 , Nb2 O5 , Nb12 O29 , Nb4 O5 ), models of lead niobate (PbNb2 O6 , Pb1.5 Nb2 O6.5 , Pb17 Nb17 O59.5 , Pb2.31 Nb2 O7.31 , Pb2.44 Nb2 O7.44 , Pb3 Nb2 O8 ), models of lead oxide (Pb2 O3 , Pb2 O, PbO, Pb3 O4 ), models of lead magnoniobate (PbMg1/3 Nb2/3 O3 ) and pyrochlore model (Pb1.86 Mg0.24 Nb1.76 O6.5 ). At the final stage of clarifying the phase composition of samples of a system of complex oxides, seven models were used.
3 Experimental Results and Discussion Table 1 shows the structural parameters of perovskite PMN cells. It should be noted that as x increases, there is a slight decrease in the volume of perovskite cells from V = 66.35 Å3 at x = 0 to V = 66.21 Å3 at x = 1, which indicates the formation solid solution of (1– x)PbMg1/3 Nb2/3 O3 – xPbMg1/2 Nb1/2 O2.75. . For compositions with x from 0 to 0.5, in addition to the perovskite phase, reflections of impurity phases (pyrochlore and two modifications of Nb2 O5 ) were recorded on the diffraction patterns. From the relative intensities of the lines, it was found that the impurities are concentrated mainly on the surfaces of the samples (Fig. 1). The half-widths of X-ray diffraction reflections can provide information about the crystallinity of the resulting samples. The narrower the line, the larger the crystallite size. At x = 0.7, the narrowest X-ray diffraction reflections can be observed, which means the most well-formed crystallites. Widest line for sample exists at x = 0.5. Careful examination of the obtained diffraction patterns of the samples did not reveal any superstructural
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A. Lebedinskaya and A. Rudskaya Table 1. Structure parameters in nonstoichiometric PMN compositions
(1– x)PbMg1/3 Nb2/3 O3 – xPbMg1/2 Nb1/2 O2.75
a
x = 0.0
4.048(1)
66.35(2)
x = 0.1
4.047(1)
66.26(2)
x = 0.2
4.048(1)
66.31(2)
x = 0.3
4.048(1)
66.26(2)
x = 0.4
4.046(1)
66.25(2)
x = 0.5
4.045(1)
66.20(2)
x = 0.6
4.046(1)
66.22(2)
x = 0.7
4.044(1)
66.13(2)
x = 0.8
4.047(1)
66.30(2)
x = 0.9
4.046(1)
66.25(2)
x = 1.0
4.046(1)
66.21(2)
per
,Å
V per , Å3
Fig. 1. Dependence of the concentration and perovskite phase (blue line) of the unit cell parameter (orange line) on x
reflections for all the studied sample compositions, which indicates the absence of the effect of ordering atoms of different types in the B-sublattice, leading to a doubling of the unit cell parameters. Figures 2 and 3 show the dependences of the dielectric constant (E) and tangent of the dielectric loss (tan δ) at temperatures T = 25 °C and 50 °C. These characteristics are nonlinear as x increases, and the greatest heterogeneity is observed at the 0.5 < x < 0.7. As is known, lead magnoniobate belongs to the class of ferroelectric relaxors [14–16]. For all samples, it was found that the Curie-Weiss law is not fulfilled, which indicates a
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Fig. 2. Values of the dielectric constant (E) at temperatures T = 25 °C and 50 °C
Fig. 3. Values of tangent of the dielectric loss (tan δ) at temperatures T = 25 °C and 50 °C
smearing of the phase transition. It is generally accepted that in the case of ferroelectrics with a diffuse phase transition, instead of the usual Curie-Weiss law 1/E = (T – T 0 )/C w , one should use the formula 1/E = A + B(T – T 0 )m , where A and B are some constants, T 0 is the Curie temperature, m is taken for different compositions from 1.5 to 2.0. For lead magnoniobate, it is customary to consider m = 2. From the temperature dependences E(T ) at temperatures from 25 to 150 °C, the values of the Curie temperature, the constants A and B were obtained by calculation. The values of these parameters are given in Table 2. The calculated Curie temperature turned out to be significantly different for different sample compositions, which once again confirms the dependence of the ceramic properties on stoichiometry [17].
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A. Lebedinskaya and A. Rudskaya Table 2. A and B constants, and Curie temperature for different compositions
(1 – x)PbMg1/3 Nb2/3 O3 – xPbMg1/2 Nb1/2 O2.75
A, × 10–4
B, × 10–8
T 0, oC
x = 0.0
0.376355
1.523466
−58.26
x = 0.1
1.964185
3.096179
−3.68
x = 0.2
0.612425
3.028753
−48.53
x = 0.3
1.870444
3.515499
0.98
x = 0.4
2.544109
7.887078
−15.78
x = 0.5
0.935968
2.134559
22.07
x = 0.6
1.384200
2.972646
31.70
x = 0.7
1.236226
2.657120
8.40
x = 0.8
1.069574
2.442676
2.32
x = 0.9
1.433186
2.951184
12.31
x = 1.0
2.102110
3.865964
6.50
4 Conclusion An experimental study of the synthesis of non-stoichiometric compositions (1 – x)PbMg1/3 Nb2/3 O3 – xPbMg1/2 Nb1/2 O2.75 (0 ≤ x ≤ 1, x = 0.1) showed that the formation of the perovskite phase most likely corresponds to the effects characteristic for nanocrystalline materials. The study established that the properties of the resulting ferroelectric ceramics based on lead magnoniobate depend significantly on the stoichiometry. In ceramics, a structure with a given ratio of magnesium and niobium is not realized. Some of these substances go into the formation of the perovskite structure, and some are released in the form of impurity phases. Analysis of currently available experimental data on the effect of deviations in oxygen content from stoichiometric on the physical properties of samples with a perovskite structure does not provide an accurate idea of the observed changes in structure and properties. A diffuse phase transition was observed in all synthesized samples. Thus, studies of the nature and conditions for the formation of the structural and electromagnetic properties of ceramic samples with substitutions of different valent magnesium and niobium ions in the B-sublattice and deviations from stoichiometry in oxygen are promising both from the viewpoint of determining the dependence and mutual influence of structural parameters on physical properties, as well as in connection with the urgent need to create promising functional materials and control their characteristics.
References 1. Pandya, S., et al.: Pyroelectric energy conversion with large energy and power density in relaxor ferroelectric thin films. Nat. Mater. 17(5), 432 (2018) 2. Bokov, A.A., Ye, Z.-G.: Dielectric relaxation in relaxor ferroelectrics. J. Adv. Dielectr. 2(2), 1241010 (2012)
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3. Samara, G.A.: The relaxational properties of compositionally disordered ABO3 perovskites. J. Phys. Condens. Matter 15, R367 (2003) 4. Delre, E., Spinozzi, E., Agranat, A.J., Conti, C.: Scale free optics and diffractionless waves in nanodisordered ferroelectrics. Nat. Photonics 5, 39 (2011) 5. Smolensky, G.A., Isupov, V.A., Agranovskaya, A.I., Popov, J.V.: New ferroelectrics of complex composition. In: Physics of Dielectrics, Moscow Academy of Sciences of the USSR, vol. 336 (1960). (In Russian) 6. Verbaere, A., Piffard, Y., Ye, Z.-G., Husson, E.: Lead magnoniobate crystal structure determination. Mat. Res. Bull. 27, 1227–1234 (1992) 7. Viehland, D., Jang, S.J., Cross, L.E., Wuttig, M.: Freezing of the polarization fluctuations in lead magnesium niobate relaxors. J. Appl. Phys. 68(6), 2916 (1990) 8. Wang, C., et al.: Determination of chemical ordering in the complex perovskite Pb(Cd1/3Nb2/3)O3. IUCrJ 5, 808–815 (2018) 9. Kolodiazhnyi, T., Padchasri, J., Yimnirun, R.: Effect of temperature and stoichiometry on the long-range 1:2 cation order in BaZn1/3 Ta2/3 O3 . J. Eur. Ceram. Soc. 38(4), 1524–1528 (2018) 10. Davies, P.K., Akbas, M.A.: Chemical order in PMN-related relaxors: structure, stability, modification, and impact on properties. J. Phys. Chem. Solids 61(2), 159–166 (2000) 11. Kupriyanov, M.F., Fesenko, E.G.: News of the USSR academy of sciences. Phys. Ser. 31(7) 1086–1090 (1967) (In Russian) 12. Surovyak, Z., et al.: Crystal chemical aspect of reconstructive pyrochlore-perovskite phase transitions in complex oxides Izvestiya RAN. Phys. Ser. 66(6), 867–870 (2002) 13. Hwang, J., Rao, R.R., Giordano, L., Katayama, Y., Yu, Y., Shao-Horn, Y.: Perovskites in catalysis and electrocatalysis. Science 358(6364), 751–756 (2017) 14. Vakhrushev, S.B., Kvjatkovskiy, B.E., Naberezhnov, A.A., Okuneva, N.M., Tolerverg, B.P.: Glassy phenomena in perovskite-like crystals. Ferroelectrics 90, 173–176 (1989) 15. Cowley, R.A., Gvasaliya, S.N., Lushnikov, S.G., Roessli, B., Rotaru, G.M.: Relaxing with relaxors: a review of relaxor ferroelectrics. Adv. Phys. 60(2), 229–327 (2011) 16. Lebedinskaya, A.R., Kasparova, N.G.: Features of the structural model of the low-temperature phase of lead magnesium niobate—relaxor ferroelectric. In: Parinov, I.A., Chang, S.-H., Kim, Y.-H. (eds.) Advanced Materials. SPP, vol. 224, pp. 267–276. Springer, Cham (2019). https:// doi.org/10.1007/978-3-030-19894-7_20 17. Gusev, A.I.: Nonstoichiometry, Disorder, Short-range and Long-range Order in a Solid. Fizmatlit, Moscow (2007). (In Russian)
Switching Processes and Ferroelectric Hysteresis in Porous PZT Type Piezoceramics I. A. Shvetsov(B)
, N. A. Shvetsova , E. I. Petrova , D. I. Makarev , and A. N. Rybyanets
Southern Federal University, Rostov-on-Don, Russia [email protected]
Abstract. In present work, particular aspects of the large-signal switching properties and electromechanical hysteresis of the PZT-type porous piezoceramics were investigated by comparison with the dense piezoceramics of the same composition. Large-signal ferroelectric polarization and strain hysteresis loops were recorded at the bipolar electric fields in the range of 0–5 kV/ mm and in the frequency range of 0.01–5 Hz. Measurements and analysis were performed by means of the Electromechanical Measurement System (STEPHV) and Electromechanical Response Characterization Program (STEP), combining large signal modelling of the mechanical and electrical behavior of ferroelectric materials. Keywords: Porous Piezoceramics · Ferroelectric Hysteresis · Electromechanical Hysteresis · Large-Signal Switching Process · Domain Walls
1 Introduction Ferroelectrics are a technologically important class of materials that are used in sensors, actuators, and ultrasonic transducers [1, 2]. Except in some special applications, electromechanical hysteresis is undesired in high-precision sensor, actuator, and capacitor applications [3, 4]. The control, description and understanding of the electromechanical and ferroelectric hysteresis are an important and difficult matter both from a practical and theoretical point of view. Thus, the study of hysteresis can provide valuable information about various physical processes occurring in ferroelectric materials, for example, space charge relaxation, domain orientation process, pinning of domain walls, and defect ordering. The interest towards porous ceramics has grown rapidly in recent years with the increasing demand for specific properties and features that generally cannot be achieved by their dense counterparts [5, 6]. Porous ferroelectric ceramics possess usually remarkably less polarizability than dense ceramics; instead, they display high tenability of various physical properties [7, 8]. However, the effect of porosity on the polarization switching behavior of ferroelectrics, which is the fundamental physical process determining their functional properties, remains poorly understood. In part, this is due to the complex effects of porous structure on the local electric field distributions within these materials. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 216–223, 2024. https://doi.org/10.1007/978-3-031-52239-0_21
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In our previous papers [9, 10], we have proposed a new method for studying ferroelectric hysteresis and relaxation process in PZT type ferroelectric ceramics, induced by a weak dc electric field, based on the measurement and analysis of piezoresonance spectra (PRAP) [11, 12]. At high levels of excitation, porous ferroelectric materials are expected to be highly non-linear, primarily due to switching of domains (extrinsic nonlinearity). This can be evidenced either as mechanical displacement (strain) output (S-E loops) or as electric displacement (polarization) output (P-E loops). In present work, particular aspects of the large-signal switching properties and electromechanical hysteresis of the PZT-type porous piezoceramics were investigated by comparison with the dense piezoceramics of the same composition.
2 Experimental PZT type dense and porous piezoelectric ceramics of the composition Pb0.95 Sr0.05 Ti0.47 Zr0.53 O3 + 1% Nb2 O5 with relative porosity of 24% and average pore size of 10–20 µm were chosen as the object of the study. Porous PZT samples were fabricated, using the pore former burning-out method [13]. A dense PZT samples with the same chemical composition were fabricated by conventional sintering method. Polarized and nonpolarized discs of porous and dense piezoceramics (∅ 12 mm and thickness of 0.75 mm) were used for the experiments. Piezoceramic elements were polarized in air by applying to silver electrodes dc electric field (~1 kV/cm) at heating above Curie temperature (~290 °C) and cooling to a room temperature. Studied PZT-type ferroelectric ceramics belongs to the tetragonal boundary of MPT region and characterized by the presence of 90° and 180° tetragonal domains. When the piezoceramic is polarized, all 180° domain reorientations are realized completely, and 90° only partially. The initial polarization of the piezoceramics leads to a shift in the hysteresis (P-E) loop and distortion of the deformation (S-E) loop but allows one to analyze and determine the main parameters of the switching processes [14]. Ferroelectric polarization and strain loops were recorded at the bipolar electric fields in the range of 0–5 kV/ mm and in the frequency range of 0.01–5 Hz using a sinusoidal waveform. Measurements and analysis were performed by means of the Electromechanical Measurement System (STEPHV) and Electromechanical Response Characterization Program (STEP) from TASI Technical software Inc. [15], combining large signal modelling of the mechanical and electrical behavior of ferroelectric materials. STEPHV includes the instrumentation components necessary to perform large signal electromechanical response measurements using STEP. Key components of STEPHV include: enclosure with interlocks; Sawyer-Tower circuit integrated with high voltage protection electronics; strain measurement component; voltage A/D and D/A component for controlling and monitoring the measurement; voltage amplification component. This system includes a contact DVRT strain measurement with 10 nm resolution. The ferroelectric analysis module provides STEP with common models, applied to ferroelectric materials [16, 17]. The model, used in STEP to fit electric displacement D versus electric field E, was proposed by Chen et al. [16]. They present a function for the polarization as a function of electric field. In most useful materials, the dielectric constant
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of the material is much larger than the permittivity of free space and the assumption D = P is a very good approximation. The data, acquired by STEP, is typically electric displacement data therefore the electric displacement D is used instead of the polarization (that is D = f (E)). The modified model of Piquette and Forsythe [17] is used to model ferroelectric behavior by adding a coercive field. To fit strain (S-E loop) STEP use the equation for strain S AVG = QD2 and S HYST = a1 D + a2 D3 + a3 D5 and the model of Chen et al. [16] to find a general equation for strain as a function of electric field that model the “butterfly” loops.
3 Results and Discussions Figure 1 shows the hysteresis loops of dense and porous piezoceramics samples, measured at frequency 0.01 Hz.
Fig. 1. Hysteresis loops of the dense (a) and porous (b) piezoceramics samples, measured at frequency 0.01 Hz.
The results of STEP analysis shows that coercive field, remnant, and saturation displacement values are much smaller, and the hysteresis loop is more saturated for porous materials. In fact, coercive field value E C are 1.31 MV/m and 1.0 MV/m for dense and porous specimens, respectively. Remnant Dr and saturation DS electric displacement values are 0.25 C/m2 and 0.35 C/m2 for dense samples and 0.18 C/m2 and 0.23 C/m2 for porous specimens. The difference in Dr and DS of dense and porous ceramics is due to and in proportion with the porosity of the samples (p = 24%), which reduces the polarization values (P = D, because of D = ε0 E + P, and ε0 is very small). A significant decrease in the coercive field for porous piezoceramics is associated with the features of the micro- and domain structure of the porous piezoceramic [18]. The removal of internal mechanical stresses in the coral-like branched ceramic skeleton and the change in the grain size of porous piezoceramics facilitate 90° domain rotations and lead to a decrease in the coercive field. To evaluate the effect of frequency on the switching processes, hysteresis loops of dense and porous piezoceramics at different frequencies were measured. Figures 2 and
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3 show the hysteresis loops of dense and porous piezoceramics samples, measured at frequency of 0.1 and 1 Hz.
Fig. 2. Hysteresis loops of the dense (a) and porous (b) piezoceramics samples, measured at a frequency of 0.1 Hz.
Fig. 3. Hysteresis loops of dense (a) and porous (b) piezoceramics samples measured at frequency 1 Hz.
As with the measurement on the frequency of 0.01 Hz, coercive field, remnant, and saturation polarization values in these cases are much smaller for porous materials. In these cases, coercive field values E C for dense and porous piezoceramic specimens are 1.45 MV/m, 1.075 MV/m at 0.1 Hz, and 1.55 MV/m, 1.15 MV/m at 1 Hz, respectively. Remnant Dr and saturation DS electric displacement values are 0.26 C/m2 , 0.27 C/m2 for dense samples, and 0.18 C/m2 , 0.19 C/m2 for porous piezoceramic specimens at 0.1 Hz, and 0.28 C/m2 , 0.37 C/m2 for dense samples and 0.19 C/m2 , 0.24 C/m2 for porous piezoceramic specimens at 1 Hz, respectively. Comparison of the obtained values shows that the coercive field E C as well as remnant Dr and saturation DS electric displacement values for dense and porous ceramics increase significantly with frequency. The increase in coercive field E C with frequency was experimentally observed in single crystals, bulk ceramics, and thin films, and can
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be attributed to 180° and non-180° domain nucleation-switching processes [19]. The increase in remnant Dr and saturation DS electric displacement with increasing frequency for both dense and porous piezoceramics is due to a more complete polarization switching of domains because of the removal of non-180° domain walls pinching by defects under bipolar electric field influence. Figures 3 and 4 show corresponding to Figs. 3 and 4 displacement (strain) loops of the dense and porous piezoceramics as a function of the applied electric field measured at frequencies of 0.01 and 1 Hz.
Fig. 4. High electric field induced strains (S-E loops) for the dense (a) and porous (b) piezoceramics measured at 0.01 Hz frequency.
Complete polarization switching is obtained in all cases at high electric field as evidenced by butterfly-like shape of the curves. It can be observed (Figs. 4, 5 and 6) that porous piezoceramic samples, despite significantly lower remnant and saturated polarization values, demonstrate high induced strains S. In addition, “butterfly” loops (S-E loops) for porous piezoceramics have a significantly asymmetric shape. The asymmetry of the S-E loops in Figs. 4(b), 5(b), and 6(b) was probably due to the porous piezoceramics samples possess a residual polarization due to preliminary sequential cycling at smaller fields and lower coercive field of the porous samples. Nevertheless, S-E dependencies can be reliably used to obtain non-linear properties of porous ferroelectric materials. The maximum magnitude of the field-induced strains S, accompanying the polarization switching for the dense and porous piezoceramics at 0.01 Hz are ~ 4.5 × 10–3 and ~ 6.5 × 10–3 , respectively. The corresponding S values, measured at 0.1 Hz and 1 Hz for the dense and porous piezoceramics are ~ 4.2 × 10–3 , ~ 6 × 10–3 , and ~ 4 × 10–3 , ~ 5·10–3 respectively. Thus, with frequency grows the field-induced strains decrease in both cases. Considering significant asymmetry of S-E loops for porous piezoceramics, we can state that despite lower remnant and saturated polarization values the field-induced strains S for dense and porous piezoceramics are approximately the same. This is related to the equality of the piezomodules d 33 of porous and dense piezoceramics due to microand mesostructural features of porous piezoceramics, namely the presence of a rigid
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Fig. 5. High electric field induced strains (S-E loops) for the dense (a) and porous (b) piezoceramics, measured at 0.1 Hz frequency.
Fig. 6. High electric field induced strains (S-E loops) for the dense (a) and porous (b) piezoceramics, measured at 1 Hz frequency.
continuous piezoceramic skeleton and a quasi-rod structure in the direction of residual polarization of piezoelectric element. The results obtained in this work are in good agreement with the results of studying the effect of a microstructure features on complex electromechanical characteristics of porous piezoceramics [5, 18].
4 Conclusion Analysis of large-signal ferroelectric and strain hysteresis loops by means of Electromechanical Response Characterization Program (STEP) made it possible to obtain full sets of parameters characterizing the switching process and electromechanical hysteresis behavior of the porous and dense PZT-type piezoceramics and understand the effect of porosity on the polarization-field and strain-field responses of ferroelectric materials. As a result of the study, it was found that porous piezoceramic samples, despite significantly lower remnant and saturated polarization values, demonstrate high field
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induced strains comparable to dense ceramics. It was shown that the differences in switching behavior of dense and porous piezoceramics are due to the specific features of the domain- and micro-structure of porous piezoceramics. The resulting information provides new insights in the interpretation of the physical properties of porous ferroelectric materials to inform future effort in the design of ferroelectric materials for piezoelectric sensor, actuator, energy harvesting, and ultrasonic transducer applications. Acknowledgements. The study was financially supported by the Russian Science Foundation Grant No. 22–11-00302, https://rscf.ru/project/22-11-00302/ at the Southern Federal University.
References 1. Scott, J.F.: Applications of modern ferroelectrics. Science 315(5814), 954 (2007). https://doi. org/10.1126/science.1129564 2. Kim, T.Y., Kim, S.K., Kim, S.-W.: Application of ferroelectric materials for improving output power of energy harvesters. Nano Convergence 5(1), 1–16 (2018). https://doi.org/10.1186/ s40580-018-0163-0 3. Damjanovic, D.: Hysteresis in piezoelectric and ferroelectric materials. In: Mayergoyz, I., Bertotti, G. (eds.) The Science of Hysteresis, pp. 337–465. Elsevier Inc., UK (2005) 4. Damjanovic, D.: Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics. Rep. Prog. Phys. 61, 1267 (1998). https://doi.org/10.1088/0034-4885/61/ 9/002 5. Schultheiß, J., et al.: Orienting anisometric pores in ferroelectrics: piezoelectric property engineering through local electric field distributions. Phys. Rev. Mater. 3, 084408–084411 (2019). https://doi.org/10.1103/PhysRevMaterials.3.084408 6. Rybyanets, A.N.: Porous Ceramics and Piezocomposites: Modeling, Technology, and Characterization. Nova Science Publishers Inc., NY (2017) 7. Shvetsova, N.A., et al.: Microstructure characterization and properties of porous piezoceramics. J. Adv. Dielectr. 12(2), 2160006–2160011 (2022). https://doi.org/10.1142/S2010135X 21600067 8. Rybyanets, A.N., et al.: Electric power generations from PZT composite and porous ceramics for energy harvesting devices. Ferroelectrics 484(1), 95 (2015). https://doi.org/10.1080/001 50193.2015.1060065 9. Shvetsov, I.A., et al.: Application of the impedance spectroscopy method for the study of relaxation processes in ferroelectric ceramics. Ferroelectrics 561(1), 69 (2020). https://doi. org/10.1080/00150193.2020.1736917 10. Shvetsov, I.A., et al.: Piezoelectric hysteresis in ferroelectric ceramics in a weak electric field. Ferroelectrics 605(1), 93 (2023). https://doi.org/10.1080/00150193.2023.2169015 11. PRAP (Piezoelectric Resonance Analysis Program). TASI Technical Software Inc. (www.tas itechnical.com) 12. Rybianets, A. et al.: Accurate evaluation of complex material constants of porous piezoelectric ceramics. In: Proceedings - IEEE Ultrasonics Symposium, vol. 1. no.4152245, pp. 1533–1536 (2006). https://doi.org/10.1109/ULTSYM.2006.389 13. Rybyanets, A.N.: Porous piezoceramics: theory, technology, and properties. IEEE Trans. UFFC 58(7), 1492 (2011). https://doi.org/10.1109/TUFFC.2011.1968 14. Fabbri, G., et al.: Characterization techniques for porous piezoelectric materials. Ferroelectrics 293, 291 (2003). https://doi.org/10.1080/00150190390238649
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15. STEP (Electromechanical Response Characterization Program). TASI Technical Software Inc. (www.tasitechnical.com) 16. Chen, D.Y., et al.: A simple unified analytical model for ferroelectric thin film capacitor and its application for non- volatile memory operation. In: Proceedings of 1994 IEEE International Symposium on Applications of Ferroelectrics, pp. 25–30 (1994). https://doi.org/10.1109/ ISAF.1994.522289 17. Piquette, J.C., Forsythe, S.E.: A nonlinear material model of lead magnesium niobate (PMN). J. Acoust. Soc. Am. 101, 289 (1997). https://doi.org/10.1121/1.418009 18. Shvetsova, N.A., et al.: Microstructural features and complex electromechanical parameters of Pb0.95 Sr0.05 Ti0.47 Zr0.53 O3 + 1% Nb2 O5 porous piezoceramics. Ferroelectrics 591(1), 143–149 (2022). https://doi.org/10.1080/00150193.2022.2041932
Microstructure, Complex Electromechanical Parameters and Dispersion in Porous Piezoceramics M. G. Konstantinova(B) , P. A. Abramov, N. A. Shvetsova , I. A. Shvetsov , M. A. Lugovaya , and A. N. Rybyanets Southern Federal University, Rostov-on-Don, Russia [email protected]
Abstract. In this work, the microstructure features and complex electromechanical parameters of porous piezoelectric ceramics based on the PZT system with different relative porosities and pore sizes are studied. Complex elastic, dielectric, and electromechanical parameters of porous piezoceramics were measured on the thickness mode of vibration of standard samples and analyzed using the piezoresonance analysis program (PRAP). It was shown that the microstructural features of porous piezoceramics determine the character of the porosity dependences of real and imaginary parts of dielectric, piezoelectric and electromechanical properties of porous piezoelectric ceramics. Keywords: Porous Piezoceramics · Microstructure · Complex Electromechanical Parameters · Piezoresonance Analysis Program
1 Introduction Porous piezoceramics based on various piezoceramic compositions are currently widely used in ultrasonic transducers and sensors for various purposes, for example, in medical ultrasound, non-destructive testing, underwater acoustics and other practical applications [1]. Porous piezoceramics is a heterogeneous medium with a unique microstructure that provides original effective macro-properties [2–4]. The use of porous ceramics is based on its structural and functional characteristics, determined by the peculiarities of the microstructure, determined by the raw materials, compositions, and methods of manufacturing ceramics [5–7]. Intensive R&D and technological works, carried out in recent years, as well as the development of new manufacturing methods, have made it possible to organize the industrial production of porous piezoceramics of various compositions with controlled porosity and reproducible electrical properties [8, 9]. However, despite numerous studies and long-term observations of the influence of pores on the properties of materials, many aspects of the relationships between the microstructure and electromechanical parameters of porous piezoceramics still remain insufficiently studied. In addition, the reasons for the dependence of complex parameters on porosity, as well as the microstructural and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 224–231, 2024. https://doi.org/10.1007/978-3-031-52239-0_22
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physical mechanisms of dielectric, elastic and electromechanical losses and dispersion in porous piezoceramics, remain unclear. In [10, 11], the features of the microstructure and electrical properties of porous piezoceramics of the PZT system were studied, measured by standard methods without considering losses in the material. In this work, the features of the microstructure and the porosity dependence of complex elastic, dielectric, and electromechanical parameters of PZT type porous piezoceramics with different relative porosity are studied.
2 Objects of the Study and Methods of Measurements The objects of study were piezoelectric ceramics of the PZT system of the composition Pb0.95 Sr0.05 Ti0.47 Zr0.53 O3 + 1% Nb2 O5 with different relative porosities in the range of 0–40% and an average pore size of 10–20 µm. To produce dense piezoceramics, the traditional methods of solid-phase synthesis and sintering were used. Experimental samples of porous piezoceramics were obtained using a modified method of burning out of pore-forming agents [9]. Porosity and pore size distribution were determined using simple but effective methods of stereology and hydrostatic weighing, as well as from the results of weighing and measuring the geometric dimensions of a sample of porous piezoceramics, followed by calculation of the relative porosity, using the formula p = (ptheor − pexp )/ptheor , where ptheor is the X-ray density of dense piezoceramics, and pexp is the measured density of porous piezoceramics. For the experiments, thin disk-shaped piezoelements (with 20 mm diameter, and 1 mm thickness) made of dense and porous piezoceramics were used. Piezoceramic elements were polarized in air by applying a dc electric field (~1 kV/mm) to fired silver electrodes, heating above the Curie temperature (~290 °C) and cooling under dc field to room temperature. Microstructural studies were carried out on the chipped surfaces of porous piezoceramic samples using scanning electron microscopes (JEOL JSM-6390LA and TM-100, Hitachi). Complex elastic, dielectric, and electromechanical parameters of piezoceramic elements were measured on standard samples using precision impedance analyzer Agilent 4294A and Piezoelectric Resonance Analysis Program (PRAP) [12]. Analysis of the experimental piezoresonance spectra for the thickness mode of vibrations of disk samples made it possible to obtain dependences on the porosity of the complex dielectric, piezoelectric and electromechanical parameters of porous ceramics in the range of relative porosity 0–40%. The PRAP software package [12–14], used in this work, analyses piezoresonance spectra to determine the complex parameters of the piezoelectric material. The PRAP software uses a generalized form of the Smith method, as well as a generalized ratio method for the radial vibration mode [13], valid for materials with any mechanical quality factor, which makes it possible to determine the parameters of a piezoelectric material for an arbitrary resonance mode.
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3 Results and Discussions 3.1 Microstructure Study SEM micrographs illustrating the main features of the microstructure of dense and porous (relative porosity 37%) piezoceramics are presented in Fig. 1.
(a)
(b)
Fig. 1. Microstructure of dense (a) and porous (b) piezoceramics with a relative porosity of 37% of the same composition Pb0.95 Sr0.05 Ti0.47 Zr0.53 O3 + 1% Nb2 O5 .
Dense piezoceramics, produced by traditional sintering, are characterized by a chaotic packing of irregularly shaped grains with an average size of 7 microns and a residual porosity of about 6%. Porous piezoceramics are characterized by a chaotic distribution of irregularly shaped pores with dimensions of 10–20 microns. From the micrographs, it is clear that the microstructure of porous piezoceramics is characterized by a dense packing of grains of a regular polyhedral shape with an average size of 3.5 µm, which is significantly smaller than the grain size, observed for dense piezoceramics of the same composition. These micrographs also demonstrate the presence of a rigid coral-like framework with a continuous quasi-rod structure in the porous piezoceramics. Namely this branched quasi-rod mesostructure of porous piezoceramics in the direction of residual polarization determines the main features of its electromechanical properties. 3.2 Study of Electromechanical Properties Figure 1, 2, 3, 4 and 5 show the dependences of the complex elastic, dielectric, and electromechanical parameters of the studied porous piezoceramics of the composition Pb0.95 Sr0.05 Ti0.47 Zr0.53 O3 + 1% Nb2 O5 on the relative porosity. D/ D// The dependences of the real C33 and imaginary C33 parts of the elastic modulus on the relative porosity P for the studied porous piezoceramics are shown in Fig. 2. D/ The real part of the elastic modulus C33 decreases quite quickly with increasing porosity due to a decrease in the rigidity of the porous ceramic framework. The correD// sponding decrease in the imaginary part of the elastic modulus C33 is due to an increase
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D on the relative Fig. 2. Dependences of the real and imaginary part of the elastic modulus C33 porosity P for the studied porous piezoceramics.
in the damping of resonant oscillations due to Rayleigh scattering of elastic waves on the pores and mesostructural elements of the porous piezoceramic framework (λ > > D, where λ is the wavelength, D is the size of the pores or structural elements). The depenD/ D// dences of C33 and C33 on porosity P are nonlinear, and the decrease in the real part of D// C33 slows down with increasing P in the result of the restructuring of the micro- and mesostructure of the porous piezoceramics. The mechanical quality factor of the thickD/ D// ness vibration mode Qt = C33 /C33 decreases with increasing porosity, much faster than the mechanical quality factor of the radial vibration mode [9], which is associated with the microstructural features of the piezoceramic framework noted above. S/ S// The dependences of the real ε33 and imaginary ε33 parts of the dielectric constant of a mechanically clamped piezoelectric element on the relative porosity P of porous piezoceramics are shown in Fig. 3. S/ The real part of the complex dielectric constant ε33 of porous piezoceramics decreases almost linearly with increasing porosity (Fig. 3), which is caused by a significant difference in the dielectric constants of piezoceramics and air. The corresponding S// decrease in the imaginary part of the dielectric constant ε33 is due to a decrease in mobility and fixation of 90° domain walls at grain boundaries (Fig. 1), located on a more developed pore surface, as well as a decrease in grain size. T // T / The dielectric loss tangent tan δ = ε33 /ε33 decreases slightly with increasing porosity (Fig. 3) in the result of the mentioned fixation of domain walls due to the more developed internal surface of the porous piezoceramics (Fig. 1b). It should be noted that the complex dielectric constant of highly porous piezoceramics depends on air humidity and measurements must be carried out while controlling relative humidity.
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S on the relative Fig. 3. Dependences of the real and imaginary part of the dielectric constant ε33 porosity P for the studied porous piezoceramics. /
//
The dependences of the real kt and imaginary kt parts of the complex coefficient of electromechanical coupling of the thickness mode of vibrations on porosity P for the studied piezoceramics are shown in Fig. 4. The real part of the complex electromechan/ ical coupling coefficient kt increases with porosity over the entire range of porosity P to values, approaching the value of the electromechanical coupling coefficient of the / longitudinal vibration mode of the piezoceramic rod k33 (Fig. 4). The increase in kt values is due to the weakening of the mechanical clamping of the piezoceramic framework of porous ceramics in the transverse direction, typical for dense piezoceramics, and the formation of a continuous quasi-rod structure in the direction of the thickness of the disk piezoceramic element. The relationship between the thickness, radial and longitudinal electromechanical is determined by the known approximate ratio kt2 ≈ coefficients coupling
2 + k 2 / 1 − k 2 . A sharp decrease in the electromechanical coupling coefficient k33 p p kp , caused by a violation of the electromechanical coupling of the piezoceramic frame of porous ceramics in the transverse direction [9, 10], with a slight decrease in k33 , leads to / // an observed increase in kt . The increase in the imaginary part kt of the complex coefficient of electromechanical coupling of the thickness mode of vibrations from porosity P is due to an increase in electromechanical losses, caused by the interaction of resonant vibrations of the piezoelectric element with micro- and mesostructural components of porous ceramics. / // The dependences of the real e33 and imaginary e33 parts of the piezoelectric constant on the relative porosity p for porous piezoceramics are shown in Fig. 5. The real and imaginary parts of the piezoelectric constant e33 decrease almost linearly with increasing porosity. As was shown earlier [9, 10], the piezomodulus d 33 for porous piezoceramics
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Fig. 4. Dependences of the real and imaginary parts of the electromechanical coupling coefficient kt on the relative porosity P for the studied porous piezoceramics.
in a wide porosity range has an almost constant value, which is explained by the presence of a mesostructural quasi-rod piezoceramic frame in the direction of the residual polarization (thickness) of the sample. The decrease in the relative surface area of the porous piezoceramic sample in this case is compensated by an increase in the pressure, applied to the quasi-rod ceramic frame. At the same time, the piezoelectric constant e33 can be expressed as follows [2]: D S 1/2 D is the elastic modulus, ε S is the dielectric constant ε33 , where C33 e33 = kt C33 33 of the clamped sample, and kt is the electromechanical coupling factor of the thickness mode of vibration of the piezoelectric element. The resulting behavior of the piezoelectric constant e33 with increasing porosity is determined by the competing influence of the indicated dielectric, elastic, and electromechanical parameters. As it was shown above, D and dielectric constant ε S the real and imaginary parts of the elastic modulus C33 33 quickly decrease, while the real and imaginary parts of the electromechanical coupling coefficient kt increase with increasing porosity (Figs. 2 and 3). / // Thus, the observed decrease in e33 and e33 with increasing porosity P (Fig. 5) is D and dielectric associated with the mentioned rapid decrease in the elastic modulus C33 S , due to the noted microstructural and electromechanical features of porous constant ε33 piezoceramics.
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Fig. 5. Dependences of the real and imaginary parts of the piezoelectric constant e33 on the relative porosity P for the studied porous piezoceramics.
4 Conclusion As a result of SEM analysis of the microstructure, it was established that the real structure of porous piezoceramics, based on PZT solid solutions of the composition Pb0.95 Sr0.05 Ti0.47 Zr0.53 O3 + 1% Nb2 O5 in the porosity range from 6% to 40%, changes from dense medium with inclusion of isolated pores to the structure of a matrix medium with a rigid coral-like framework and a continuous quasi-rod structure in the direction of the residual polarization of the piezoelectric element. It is shown that the dependences of the complex elastic, dielectric and electromechanical characteristics of porous piezoceramics are determined by the following microand mesostructural features: (i) a significant difference in the dielectric constants of piezoceramics and air; (ii) a decrease in grain size, as well as a decrease in mobility and fixation of 90° domain walls at grain boundaries, located on a more developed pore surface; (iii) reducing the rigidity of the porous ceramic frame; (iv) weakening of the mechanical clamping of the piezoceramic framework of porous ceramics in the transverse direction, typical for the dense piezoceramics, and the formation of a continuous quasi-rod structure in the direction of the thickness of the disk piezoelement; (v) an increase in electromechanical losses, caused by the interaction of resonant vibrations of the piezoelectric element with micro- and mesostructural components of porous piezoceramics.
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Acknowledgements. The work was financially supported by the Ministry of Science and Higher Education of the Russian Federation [State assignment in the field of scientific activity, Southern Federal University, 2023, Project No. FENW-2023–0015/(GZ0110/23–08-IF)].
References 1. Rybyanets, A.N.: Porous piezoelectric ceramics - a historical overview. Ferroelectrics 419(1), 90–96 (2011). https://doi.org/10.1080/00150193.2011.594751 2. Wight, J.: Cellular Ceramics – Structure, Manufacturing. Properties and Applications. WileyVCH, Weinheim (2005) 3. Zeng, T., et al.: Effects of pore shape and porosity on the properties of porous PZT 95/5 ceramics. J. Eur. Ceram. Soc. 27, 2025–2029 (2007). https://doi.org/10.1016/j.jeurceramsoc. 2006.05.102 4. Pabst, W., et al.: Processing, Microstructure, Properties, Applications and Curvature-based Classification Schemes of Porous Ceramics. Nova Science Publishers Inc., NY (2017) 5. Rybyanets, A.N., et al.: Electric power generations from PZT composite and porous ceramics for energy harvesting devices. Ferroelectrics 484(1), 95–100 (2015). https://doi.org/10.1080/ 00150193.2015.1060065 6. Studart, A.R., et al.: Processing routes to macroporous ceramics – a review. J. Am. Ceram. Soc. 89, 1771–1789 (2006). https://doi.org/10.1111/j.1551-2916.2006.01044.x 7. Lee, S.H., et al.: Fabrication of porous PZT-PZN piezoelectric ceramics with high hydrostatic figure of merits using camphene-based freeze casting. J. Am. Ceram. Soc. 90, 2807–2813 (2007). https://doi.org/10.1111/j.1551-2916.2007.01834.x 8. Rybyanets, A.N.: Porous Ceramics and Piezocomposites: Modeling, Technology, and Characterization. Nova Science Publishers Inc., NY (2017) 9. Rybyanets, A.N.: Porous piezoceramics: theory, technology, and properties. IEEE Trans. UFFC. 58(7), 1492–1507 (2011). https://doi.org/10.1109/TUFFC.2011.1968 10. Shvetsova, N.A. et al.: Microstructural features and complex electromechanical parameters of Pb0.95 Sr0.05 Ti0.47 Zr0.53 O3 + 1% Nb2 O5 porous piezoceramics. Ferroelectrics 591(1), 143–149 (2022). https://doi.org/10.1080/00150193.2022.2041932 11. Shvetsova, N.A. et al.: Microstructure characterization and properties of porous piezoceramics. J. Adv. Dielectr. 12(2), 2160006–1–2160006–4 (2022). https://doi.org/10.1142/S20101 35X21600067 12. PRAP (Piezoelectric Resonance Analysis Program). TASI Technical Software Inc. (www.tas itechnical.com) 13. Smits, J.G.: Iterative method for accurate determination of the real and imaginary parts of the materials coefficients of piezoelectric ceramics. IEEE Trans. Sonics Ultrason. SU-23(6), 393–402 (1976). https://doi.org/10.1109/T-SU.1976.30898 14. Rybianets, A. et al.: Accurate evaluation of complex material constants of porous piezoelectric ceramics. In: Proceedings - IEEE Ultrasonics Symposium, vol. 1. no. 4152245, pp. 1533–1536 (2006). https://doi.org/10.1109/ULTSYM.2006.389
The Behavior of Dielectric Properties in the Solid Solutions Based on Na0.5 Bi0.5 TiO3 and Na0.5 K0.5 NbO3 E. V. Glazunova1(B) , A. S. Chekhova2 , L. A. Shilkina1 , I. A. Verbenko1 A. V. Nazarenko3 , L. A. Reznichenko1 , and V. A. Isaev4
,
1 Research Institute of Physics, Southern Federal University, Rostov-on-Don 344090, Russia
[email protected]
2 Physics Faculty, Southern Federal University, Rostov-on-Don 344090, Russia 3 Southern Scientific Center of the Russian Academy of Sciences, Rostov-on-Don 344006,
Russia 4 Department of Theoretical Physics and Computer Technologies, Kuban State University,
Krasnodar 350040, Russia
Abstract. In this work, the ceramics of (1 – x)Na0.5 K0.5 NbO3 – xNa0.5 Bi0.5 TiO3 were prepared by conventional solid state method. At room temperature in studied system, two morphotropic phase transitions occur near the extreme components of the system: (i) from tetragonal to pseudocubic at 0.1 < x < 0.2 and (ii) from pseudocubic to rhombohedral at x > 0.9. The best storage properties are observed near these transitions: W eff = 0.6 J/cm3 and η = 63%, at x = 0.2; W eff = 0.1 J/cm3 and η = 75% at x = 0.9 for E = 25 kV/cm and E = 27 kV/cm, respectively. Therefore, the ceramic with 0.1 < x < 0.2 and 1.0 < x ≤ 0.9 is promising candidate lead-free materials for energy storage devices. Keywords: Ceramics · BNT · NKN · Energy Storage Density
1 Introduction The rapid development of electronics has led to the fact that materials with high energy storage density, low losses and good temperature stability are necessary for modern technologies [1–5]. Ceramic dielectric materials are good candidates in this sense [6– 9]. They have excellent thermal stability and are well suited for application at high temperatures [10–12]. The density of stored recoverable energy W and the efficiency η of a dielectric capacitor can be determined by the loops of dielectric hysteresis according to the formulae: P P EdP(1), and Weff = Pmax EdP(2), where Pmax is a polarization value Wmax = Pmax r 0 at the maximum of E, Pr is a residual polarization. The η value was calculated by the forWeff ×100% (3). Thus, dielectric materials having a high electric breakdown mula: η = W max field, a huge Pmax value and a low Pr must have increased stored energy [13, 14]. Potential candidates for the role of dielectric capacitors for high energy storage and conversion © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 232–241, 2024. https://doi.org/10.1007/978-3-031-52239-0_23
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are antiferroelectric and relaxor materials, compared to linear nonpolar dielectrics [13– 16]. In a weak electric field, antiferroelectric ceramics often have insignificant Pr and small hysteresis losses due to the absence of ferroelectric domains, and to achieve high Pmax , the application of significantly higher electric fields is necessary [12, 17, 18]. For example, V.P. Cao et al. [19] found that there is a ferroelectric/relaxor phase transition in Bi0.5 Na0.5 TiO3 –SrTiO3 ceramics and obtained a relatively high W rec of 0.65 J/cm3 at 65 kV/cm. Yang et al. [20] also modified BNT-ST ceramics and obtained high values of W rec (3.4 J/cm3 ) and η (90%) at 310 kV/cm. Similarly, an energy density of 1.2 J/cm3 at 100 kV/cm was obtained in complex ceramics of (1 − x)(Bi0.5 Na0.5 TiO3 – BaTiO3 ) –xNaTaO3 [21]. In the relaxation ceramics (K0.5 Na0.5 )NbO3 – Sr(Sc0.5 Nb0.5 )O3 , an energy equal to 2.02 J/cm3 at 295 kV/cm was obtained [22]. H. Wang et al. also achieved a high accumulation density of recoverable energy of 4.08 J/cm3 at a voltage of 300 kV/cm in samples of (1 − x)(K0.5 Na0.5 )NbO3 – xBi(Mg2/3 Nb1/3 )O3 [23] with η = 62%. From the examples given, it can be seen that high values of stored energy can be achieved in multicomponent systems by modifying them and only with the application of high fields, which is a critical disadvantage, since for industrial applications it is necessary to develop a material with sufficient energy for the operation of devices and its conversion efficiency at relatively low applied fields. Currently, Bi0.5 Na0.5 TiO3 (BNT) is one of the most studied piezoelectric systems with a perovskite-type structure, which has a large spontaneous polarization exceeding 40 μC/cm2 as well as Na0.5 K0.5 NbO3 (NKN), which has a Curie temperature T C comparable to the temperature of commercial lead zirconate titanate (PZT) and relatively good piezoelectric and electromechanical properties, but only in complex multicomponent compositions based on NKN, or obtained by complex and costly methods [25–27]. Both materials are actively studied in the literature but have a few open questions and problems. For example, in BNT there is still no clear idea of the structure at temperatures below 200 °C. According to the literature data, the structure in this temperature range is rhombohedral R3c [28], monoclinic Cc [29, 30] or a mixture of both [31], depending on thermal, electrical, and mechanical processing. There is also no clear opinion about the structure in the range of 200 – 320 °C, in some works [32, 33] the formation of an intermediate orthorhombic phase of Pnma with an AFE character was noted, but many researchers consider it as a nonpolar or weakly polar phase [28, 34, 35]. The creation of systems, based on NKN, also has a number of problems, related to both the difficulties of achieving high density of materials and stoichiometry control, and the low piezoelectric properties of pure NKN ceramics [25, 36, 37]. The great advantage of these materials is their non-toxicity in comparison with the PZT-materials used in industry, which has become particularly relevant in recent decades. There are a few works in the literature devoted to systems based on BNT and NKN, which contain either separate compositions or a very narrow concentration range close to one of the components [1, 2, 4, 25]. In addition, it was reported in the article [14] that sodium niobate stabilizes the AFE phase BNT, which makes the NKN-BNT system a promising basis from the viewpoint of developing materials for energy storage.
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2 Research Method 2.1 Sample Preparation The objects of the study are solid solutions of the system (1 – x)Na0.5 K0.5 NbO3 – xNa0.5 Bi0.5 TiO3 [(1 – x)NKN – xBNT] (0.9 ≤ x ≤ 1.0, x = 0.10). NaHCO3 (chemically pure) and KHCO3 (pure for analysis), Nb2 O5 (qualification “pure”), Bi2 O3 (chemically pure), TiO2 (pure) were used as following raw materials. Samples were obtained by double solid-phase synthesis, T sint1 = 850 – 950 °C, T sint2 = 900 – 970 °C (depending on the composition); τ sint1,2 = 4 h, followed by sintering using conventional ceramic technology (CCT), T sintering = 1115 – 1155 °C (depending on the composition), τ sintering = 2 h. The search for optimal synthesis and sintering modes was carried out on a series of laboratory samples with a variation of firing temperatures at each stage of the production process. Before sintering, the synthesized press powder was subjected to mechanical activation (MA) in a spherical planetary mill AGO-2 (manufactured by Novosibirsk), Grinding was carried out in special drums (inner diameter 63 mm, grinding balls made of ZrO2 with a diameter of 8 mm, total weight 200 g) in an alcoholic medium for 20 min, the rotation frequency of the drum was 1800 rpm. Mechanical activation was carried out to reduce the grain size and improve the sinterability of the samples. To measure the electrophysical parameters, the samples were made in the form of disks (Ø 12 × 1.2) mm2 . Metallization was performed by double burning of a silvercontaining paste at a temperature of T = 750 °C for 0.5 h. 2.2 Experimental Methods X-ray phase analysis was performed on a DRONE 3.0 diffractometer using CoKα radiation (Bragg – Brentano focusing). The calculations of the parameter a and volume V of the perovskite unit cell were carried out according by standard methods [38]. Accuracy of cell parameters determination: linear a = c = b = ± (0.002 –0.004) Å; angles α = β = ± 0.05°; volume V = ± 0.05 Å3 . The study of ceramic chips was carried out using a scanning electron microscope Carl Zeiss EVO 40 (Germany) in the CCP “Joint Centre of Scientific and Technological Equipment of UNC RAS (research, development, approbation)” (No. 501994). Dielectric hysteresis loops were obtained using a measuring setup, assembled according to the Sawyer-Tower scheme at room temperature, a frequency of 50 Hz and an electric field value of from 700 to 2900 V for few samples. The energy density was obtained from the curves of the dependence of polarization on the electric field (P-E) by integrating the area between the axis of polarization and the curve P-E. With the help of integration, the values of effective energy and efficiency were calculated. The stored energy density, W max , and energy efficiency, W eff , energy storage efficiency, η, were calculated by the formulas (1) – (3).
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3 Results and Discussion Figure 1a shows diffractograms of the solid solutions of system (1 – x)NKN – xBNT, studied at room temperature. X-ray phase analysis of the solid solutions study system showed the absence of impurities in all the samples studied. It is shown that solid solution with x = 0.1 has a tetragonal distortion of the cubic cell. With an increase in the content of BNT up to x = 0.9, the symmetry changes to pseudocubic (due to very small distortions of the unit cell). The solid solution with x = 0.5 also has cubic symmetry, but all diffraction peaks are double and are accompanied by modulation, which indicates spinodal decay, that is, the occurrence of two cubic phases with different cell parameters.
Fig. 1. Dependences of the diffraction patterns (a); parameter a, theoretical (X-ray) – 1, experimental – 2, and relative, ρ rel , densities of ceramics (b), and fragments of the microstructure (c) on the concentration of the x-component of the (1 – x)NKN – xBNT system.
Analysis of the splitting of diffraction peaks of pure BNT showed rhombohedral distortion of the unit cell. Figure 1b shows the change in the unit cell parameters. It can be seen that in the (1 – x)NKN – xBNT system, a decrease in the unit cell parameters is observed with an increase in the concentration of BNT, which is due to the replacement of a larger K+ cation (RK+ = 1.33) with a smaller Bi3+ (RBi3+ = 1.20). In the studied
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substitution system in the B-position, the isomorphism conditions R(Nb5+ − Ti4+ ) = 3%, EO(Nb5+ − Ti4+ ) = 0.1 are fully satisfied.
Fig. 2. P-E loops (left), the variation of Pmax and Pr , W max , W eff , W loss and η (right) of the solid solutions (1 – x)NKN – xBNT system with x = 0.2 and x = 0.3 at different electric fields at room temperature and 50 Hz.
In the A-position in sodium-potassium niobate was R(K+ – Na+ ) = 35.7%; EO(K+ – Na+ ) = 0.1; with the introduction of BNT became R(K+ – Bi3+ ) = 10.8%; EO(K+ – Bi3+ ) = 1.2; R(Na+ – Bi3+ ) = 22.4%; EO(Na+ – Bi3+ ) = 1.1. Thus, replacing K+ with Bi3+ leads to a gain in bond length, but to an increase in the difference between the electronegativity of Bi3+ and (Na+ , K+ ) ~ 1.2. In addition, due to the unshared electron pair, the Bi3+ cation has lower symmetry than the K+ cation and higher polarizability, which also expands the possibilities of isomorphism. The density of solid solutions relative to the theoretical (X-ray) increases as the content of BNT increases, and in solid solutions with x > 0.8 acquires values close to the theoretical. Fragments of the microstructure of some ceramics of the system under study obtained on a ceramic chip are shown in Fig. 1c. In solid solutions with x = 0.1, a dense, heterogeneous, multi-grained microstructure is formed. The grain size varies from 1 to 3
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Fig. 3. P-E loops (left), the variation of Pmax and Pr , W max , W eff , W loss and η (right) of the solid solutions (1 – x)NKN – xBNT system with x = 0.7 – 1.0 at different electric fields at room temperature and 50 Hz.
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microns. The shape of the grains is mainly cubic. At x = 0.5, the microstructure becomes more heterogeneous, the shape of the grains is preserved. The microstructure is predominantly fine-grained with a grain size of 0.3 – 0.5 microns with some inclusions of large crystallites from 1.5 to 3 microns. At x = 1.0, it is noticeable that the microstructure becomes almost a continuous medium, which indicates the comparable strength of the grain and grain boundaries. On the microstructure of BNT, closed pores are visible, in which grains close to spherical shape are traced. Thus, the behavior of the microstructure corresponds to the behavior of the density of solid solutions. Figures 2 and 3 present the P-E loops and energy storage properties (W max , W eff , W loss and η) of the solid solutions (1 – x)NKN – xBNT system at different electric fields at room temperature. Analysis of Figs. 2 and 3 shows that all loops do not reach saturation under applied fields. At x = 0.1 not possible to obtain the P-E loops due to the high conductivity of the ceramics. Analysis of loops in the solid solutions at x = 0.2 shows that at E = 25 kV/cm, maximum energy storage properties are observed: W eff = 0.6 J/cm3 and η = 63% (Fig. 2) at W loos = 0.3 J/cm3 . Wherein Pr achieves 20 μC/cm2 . At x = 0.3, the P-E loops are widening due to growing of the conductivity of this solid solution. Thus, the energy storage properties at x = 0.3 decrease W eff = 0.3 J/cm3 and η = 30%. At following increase of the concentration x-component near midpoint of the phase diagram at the range 0.4 ≤ x ≤ 0.6, the conductivity of the solid solutions grows and does not allow us to obtain the P-E loops. The energy storage properties, obtained for solid solutions close to BNT (0.7 ≤ x ≤ 1.0) are shown in Fig. 3. At x = 0.7, the wide hysteresis loops form with high loos energy. Wherein Pr achieves 40 μC/cm2 , W eff = 0.2 J/cm3 and η = 25% at W loos = 0.5 J/cm3 . At increasing x to 0.8 and 0.9, the P-E loops become slimmer and slimmer with increasing x. Thus, the (1 – x)NKN – xBNT ceramics may possess good energy storage properties. The values of Pr at x = 0.8 significantly decrease to 4 μC/cm2 , the energy storage properties such as W eff and W loss become close each other and equal 0.12 J/cm3 and 0.13 J/cm3 , respectively, η = 50%. At x = 0.9, the values of Pr decrease to 0.9 μC/cm2 . The density storage energy W eff = 0.1 J/cm3 and the W loss attains the lowest value of 0.03 J/cm3 , and the η reaches up to 75% at 27 kV/cm. Further in BNT occurs widening of the P-E loops, the value of Pr rises to 1.2 μC/cm2 . The storage density decreases W eff = 0.03 J/cm3 and becomes closer to W loss , and η = 45%. Thus, the solid solutions close to NKN possess maximum storage density but have a high energy dissipation. The better energy storage efficiency, obtained in the solid solution close to BNT (with x = 0.9).
4 Conclusion In summary, the (1 – x)NKN – xBNT ceramics were produced by solid state reaction method. A nonmonotonic change in the properties of energy storage is observed as x increases with maxima near morphotropic phase transitions at 0.1 < x < 0.2 and at x > 0.9. The obtained values of W eff = 0.6 J/cm3 , W loss = 0.3 J/cm3 and η = 63% at x = 0.2; W eff = 0.1 J/cm3 , W loss = 0.03 J/cm3 and η = 75% at x = 0.9 are higher than in BNT, where W eff ~ W loss = 0.03 J/cm3 and η = 45%, which indicates of the promising
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of their further research by modifying and further optimization producing technology to improve their energy storage properties. Acknowledgements. The study was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation (State task in the field of scientific activity in 2023). Project No. FENW-2023–0010/(GZ0110/23–11-IF), using the equipment of the Center for Collective Use “Electromagnetic, Electromechanical and Thermal”.
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17. Hao, X., et al.: A comprehensive review on the progress of lead zirconate-based antiferroelectric materials. Prog. Mater. Sci. 63, 1–57 (2014) 18. Xu, C., et al.: High charge-discharge performance of Pb0.98 La0.02 (Zr0.35 Sn0.55 Ti0.10 )0.995 O3 antiferroelectric ceramics. J. Appl. Phys. 120, 074107 (2016) 19. Cao, W.P., et al.: Large electrocaloric response and high energy-storage properties over a broad temperature range in lead-free NBT-ST ceramics. J. Eur. Ceram. Soc. 36, 593–600 (2016) 20. Yang, L., et al.: Ultra-high energy storage performance with mitigated polarization saturation in lead-free relaxors. J. Mater. Chem. A 7, 8573–8580 (2019) 21. Xu, Q., et al.: Structure and electrical properties of lead-free Bi0.5 Na0.5 TiO3 -based ceramics for energy-storage applications. RSC Adv. 6(64), 59280–59291 (2016) 22. Qu, B.Y., Du, H.L., Yang, Z.T.: Lead-free relaxor ferroelectric ceramics with high optical transparency and energy storage ability. J. Mater Chem C 4, 1795–1803 (2016) 23. Shao, T., et al.: Potassium–sodium niobate based lead-free ceramics: novel electrical energy storage materials. J. Mater Chem A 5, 554–563 (2017) 24. Schutz, D., et al.: Lone-pair-induced covalency as the cause of temperature and field-induced instabilities in bismuth sodium titanate. Adv. Funct. Mater. 22, 2285–2294 (2012) 25. Zuo, R., Fang, X., Ye, C.: Phase structures and electrical properties of new lead-free Na0.5 K0.5 NbO3 –Bi0.5 Na0.5 TiO3 ceramics, Appl. Phys. Lett. 90, 092904 (2007) 26. Li, J.F., Wang, K., Zhang, B.P., et al.: Ferroelectric and piezoelectric properties of fine-grained Na0.5 K0.5 NbO3 lead-free piezoelectric ceramics prepared by spark plasma sintering. J. Am. Ceram. Soc. 89, 706–709 (2006) 27. Zuo, R.Z., Fu, J.: Rhombohedral-tetragonal phase coexistence and piezoelectric properties of (NaK)(NbSb)O3 –LiTaO3 –BaZrO3 lead-free ceramics. J. Am. Ceram. Soc. 94, 1467–1470 (2011) 28. Rao, B.N., Datta, R., Chandrashekaran, S.S., et al.: Local structural disorder and its influence on the average global structure and polar properties in Na0.5 Bi0.5 TiO3 . Phys. Rev. B, 88, 224103 (2013) 29. Aksel, E., Forrester, J.S., Jones. J.L., et al.: Monoclinic crystal structure of polycrystalline Na0.5 Bi0.5 TiO3 . Appl. Phys. Lett. 98, 152901 (2011) 30. Aksel, E., Forrester, J.S., Nino, J.C., et al.: Local atomic structure deviation from average structure of Na0.5 Bi0.5 TiO3 : Combined x-ray and neutron total scattering study. Phys. Rev. B, 87, 104113 (2013) 31. Rao, B.N., Fitch, A.N., Ranjan, R.: Ferroelectric-ferroelectric phase coexistence in Na1/2 Bi1/2 TiO3 . Phys. Rev. B 87, 060102 (2013) 32. Dorcet, V., Trolliard, G., Boullay, P.: The structural origin of the antiferroelectric properties and relaxor behavior of Na0.5 Bi0.5 TiO3 . J. Magn. Magn. Mater. 321,1758–1761 (2009) 33. Trolliard, G., Dorcet, V.: Reinvestigation of phase transitions in Na0.5 Bi0.5 TiO3 by TEM. Part II: second order orthorhombic to tetragonal phase transition, Chem. Mater. 20, 5074–5082 (2008) 34. Zvirgzds, J.A., Kapostin, P.P., Zvirgzde, J.V., Kruzina, T.V.: X-ray study of phase-transitions in ferroelectric Na0.5 Bi0.5 TiO3 . Ferroelectrics, 40, 75–77, (1982) 35. Suchanicz, J.: Investigations of the phase transitions in Na0.5 Bi0.5 TiO3 . Ferroelectrics, 172, 455–458 (1995) 36. Wattanawikkam, C., Chootin, S., Bongkarn, T.: Crystal structure, microstructure, dielectric and piezoelectric properties of lead-free KNN ceramics fabricated via the combustion method. Ferroelectrics 473(1), 24–33 (2014)
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Ferroelectric and Dielectric Properties of Solid Solution Ceramics Based on Bismuth Ferrite and Lead Ferroniobate Multiferroics with Germanium Dioxide Additive K. M. Zhidel1(B)
, A. V. Pavlenko1,2 , E. V. Glazunova1 and L. A. Reznichenko1
,
1 Research Institute of Physics, Southern Federal University, Rostov-on-Don 344090, Russia
[email protected] 2 Federal Research Centre, The Southern Scientific Centre of the Russian Academy of Sciences,
Rostov-on-Don 344006, Russia
Abstract. Ferroelectric and dielectric characteristics of multiferroics solid solutions ceramics of the (1 – x)BiFeO3 − xPbFe0.5 Nb0.5 O3 composition with x = 0.3, modified superstoichiometrically with GeO2 in amounts (0.5–1.0) wt.%, were studied. It is noted that the temperature dependencies of the real part of the complex dielectric permittivity and the dielectric loss tangent exhibit anomalies in the vicinity of the ferroelectric phase transition, while the nature of their changes in the vicinity of the latter allows us to classify the studied objects as ferroelectrics with a diffuse phase transition. The ferroelectric properties of the modified objects at room temperature indicate that it was not possible to obtain saturated dielectric hysteresis loops for all samples in the analyzed field range. This is apparently due to high coercive fields indicating improved electrical strength of ceramics. In this regard, it is advisable to use materials, based on 0.7BiFeO3 − 0.3PbFe0.5 Nb0.5 O3 solid solutions, developed during the research, to create and study new multifunctional media with special electrophysical parameters, promising for applications in innovative areas of electronics. Keywords: Multiferroics · Solid Solutions · Dielectric Properties · Ferroelectric Properties · Modification
1 Introduction Promising multifunctional media include the (1 – x)BiFeO3 − xPbFe0.5 Nb0.5 O3 system of solid solutions [1–6], in which the extreme components are high-temperature multiferroics, that is materials combining several types of ordering [7]. Multiferroics with coexisting electric and magnetic ordering are currently among the most intensively studied objects in materials science [8] due to the wide range of their possible applications, including the production of alternating and constant magnetic field sensors, memory elements, and spintronics devices [9–11]. Lead ferroniobate (PbFe0.5 Nb0.5 O3 , PFN) is a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 242–251, 2024. https://doi.org/10.1007/978-3-031-52239-0_24
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well-known multiferroic with a perovskite-type structure (general chemical formula— A(B0.5 B 0.5 )O3 ) with a diffuse phase transition from the paraelectric (PE) phase to the ferroelectric (FE) phase at the Curie temperature, T C , ~ 370 K. FE and antiferromagnetic (AFM) ordering coexist in it below the Néel temperature, T N , ~ 120–150 K [12]. Bismuth ferrite (BFO) is also a promising multiferroic (T C ~ 1103 K, T N ~ 643 K) possessing G-type AFM ordering [13]. Both materials are currently being considered as the basis for new magnetoelectric structures. However, their widespread use is limited by a set of factors. For bismuth ferrite, this is the difficulty of obtaining it in a single-phase state and the extremely high coercive field (E C ), required to reorient the ferroelectric domains. Moreover, both PFN and BFO are characterized by high electrical conductivity, due to the presence in their structure of variable valency ions (Fe3+ /Fe2+ , Nb5+ /Nb4+ ) and oxygen vacancies. In addition, the high volatility of PbO and Bi2 O3 oxides is a serious technological problem in the synthesis of solid solutions based on them. Nevertheless, modification [14–17] or construction of solid solutions [18, 19] based on BFO or PFN make it possible to stabilize the structure and improve the characteristics of the resulting ceramics [14–19], which was implemented in the present work. Germanium dioxide, GeO2 , glass former was chosen in this purpose instead due to it has a complete polymerized 3-D tetrahedral structure with Ge in four-fold coordination (Ge4+ ). It has been reported [20] that Ge4+ cations have the peculiarity to segregate along the grain boundary, resulting in the reduction of the jump of mobile ions. As reported by Secco and Yoshida [20], a lower ionic hopping frequency is achieved, which in turn leads to the decrease in the electrical conductivity of the ceramics due to the mechanisms noted above. Moreover, the GeO2 can promotes important changes in microstructural and structural parameters [21–25], namely, grain boundary thickness increase of ceramics, as exemplified by CaCu3 Ti4 O12 , leading to increase in dielectric constant and reduction in dielectric loss [26]. The similar idea can also be applied to the BFO – PFN system. Thereby, it seems reasonable to establish the regularities of the formation of ferroelectric and dielectric characteristics for samples of the 0.7BiFeO3 − 0.3PbFe0.5 Nb0.5 O3 binary system of solid solutions, modified superstoichiometrically with GeO2 in amounts of (0.5–1.0) wt.%, which became the purpose of this work.
2 Obtaining and Research Methods The objects of the study were ceramic samples of (1 – x)BiFeO3 − xPbFe0.5 Nb0.5 O3 (BFO − PFN, x = 0.3) binary system of solid solutions, superstoichiometrically modified with GeO2 in quantities of (0.5–1.0) wt.%, which was introduced at the stage of mixing the initial reagents. The samples were obtained by single or double solid-phase synthesis followed by sintering using conventional ceramic technology under the following procedures: 0.7BFO − 0.3PFN (SS1): T 1 = 1073 K, τ 1 = 10 h; T sint = 1223 K, τ = 2 h; 0.7BFO − 0.3PFN + 0.5 wt.% GeO2 (SS1 + 0.5 wt.% GeO2 ): T 1 = 1073 K, T 2 = 1123 K, τ 1,2 = 10 h; T sint = 1173 K, τ = 2 h; 0.7BFO − 0.3PFN + 1.0 wt.% GeO2 (SS1 + 1.0 wt.% GeO2 ): T 1 = 1073 K, T 2 = 1123 K, τ 1,2 = 10 h; T sint = 1173 K, τ = 2 h. The initial reagents were PbO, Nb2 O5 , Bi2 O3 , Fe2 O3 , GeO2 with a basic substance content of at least 99%. The charge material was sintered in the form of columns with
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a diameter of 12 mm and a height of 15 mm. The samples were made by cutting the sintered columns into disks with a diameter of 10 mm and a height of 1 mm. To carry out electrophysical measurements, after mechanical processing, silver electrodes were applied to the flat surfaces of ceramic disks by stepwise burning of silver-containing paste. Dielectric hysteresis loops (P(E) dependencies) at a frequency of 50 Hz at room temperature were obtained on a Sawyer – Tower oscillographic facility. This made it possible to estimate the residual polarization, PR , and coercive field, E C , of the test samples. Measurements of the relative dielectric constant (ε’/ε0 ) and the dielectric loss tangent (tan δ) in the temperature range T = (300–900) K and the frequency of the alternating measuring electric field f = (2 × 102 –2 × 106 ) Hz with an amplitude of 1 V were conducted on an automated measuring complex based on the Agilent 4980A LCR meter and the Varta TP703 thermocontroller (temperature stabilization accuracy was ~ 1 K), connected to a specialized shielded thermal chamber for heating the sample with a built-in thermocouple. The complex was managed by the Kalipso 2.0.0.27 software.
3 Experimental Results and Discussion The introduction of germanium dioxide additives, both 0.5 wt.% and 1.0 wt.%, superstoichiometrically at the stage of preparation 0.7BFO − 0.3PFN ceramics, contributed to a decrease in the sintering temperature by 50 K (see Table 1). Moreover, ceramics containing 1.0 wt.% GeO2 had a density of 90.40% of the theoretical one. However, the addition of exactly 0.5 wt.% GeO2 to the solid solution ceramics decreased its relative density, ρ rel , in comparison with the value for the initial composition. Table 1. Synthesis and sintering temperatures, relative ceramic densities of test solid solutions. Composition
T 1, K
T 2, K
T sint , K
ρ rel , %
SS1
1073
−
1223
90.25
SS1 + 0.5 wt.% GeO2
1073
1123
1173
84.90
SS1 + 1.0 wt.% GeO2
1073
1123
1173
90.40
It follows that solid solution ceramic samples with the composition SS1 + 1.0 wt.% GeO2 have a high relative density, which indicates the high quality of the prepared compositions, the reliability and validity of the results achieved. Figure 1 shows the results of studying the dielectric hysteresis loops of manufactured ceramics at room temperature. In the analyzed field range, we were unable to obtain saturated dielectric hysteresis loops for all samples, which indicate high coercive fields in this material, as it also occurred in the initial solid solution (Fig. 1a). The introduction of 1.0 wt.% GeO2 into the SS1 allowed higher electric field values (up to 20 kV/cm) to be applied to the sample. It was possible to observe the dependencies confirming the ferroelectric properties only
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in composition SS1 + 1.0 wt.% GeO2 (Fig. 1c), in which the dielectric hysteresis loops close to saturation with the residual polarization value of the order of 3.2 μC/cm2 and a coercive field of the order of 9 kV/cm were fixed in the field E = 20 kV/cm. In the initial solid solution, as well as in SS1 + 0.5 wt.% GeO2 , in fields less than 13 kV/cm, dependencies close to an ellipse were fixed (Fig. 1 a,b), and with a further increase in the field, dielectric breakdown of the sample and its destruction occurred.
Fig. 1. Dielectric hysteresis loop family of ceramics: (a) SS1; (b) SS1 + 0.5 wt.% GeO2 ; (c) SS1 + 1.0 wt.% GeO2 at voltage amplitudes: U = 500, 700, 900, 1000, 1100, 1200, 1300, 1500, 1700, 2000 V.
These results indicate that the use of GeO2 as a modifying additive leads to a significant increase in the electrical strength of the material while maintaining high values of polarization and coercive field. Figure 2 shows the dependencies of ε’/ε0 (T, f ) and tan δ(T, f ) on the temperature of solid solution ceramics based on the composition 0.7BFO − 0.3PFN. For all samples, the ε’/ε0 (T ) dependencies at frequencies f = (2 × 102 − 2 × 6 10 ) Hz show a wide maximum in the vicinity of (700–750) K, associated with a diffuse phase transition FE → PE [27, 28], and in the region T = (500–600) K, they demonstrate “humps”, associated with the FE → FE phase transition in materials [29]. When f decreases in the interval (102 –104 ) Hz, a sharp increase of ε’/ε0 (T ) is observed on the dependences ε’/ε0 (T ) after the maximum due to the increase of the electrical conductivity of ceramics. The latter is also clearly seen from the dependencies tan δ (T, f ), in which a sharp increase in tan δ is observed already at T = (400–450) K. According to the Vogel – Fulcher law the dependence of the maxima ε’/ε0 (T ) on frequency, T m1 ( f ), indicates that the tested multiferroics belong to ferroelectric relaxors from the viewpoint of electrical order. To determine the values of the Burns temperature, T b , (the temperature at which polar clusters appear in the material), the dependencies (ε’/ε0 )−1 (T ) were studied at the maximum measuring frequency (Fig. 3). It can be seen, that in all cases, at T > T m1 , the dependencies (ε’/ε0 )−1 (T ) have linear sections illustrating the fulfillment of the Curie – Weiss law in the PE phase. The calculated values of T b , as well as the values of T m1 and dielectric constants at the corresponding temperatures at the maximum measuring frequency are given in Table 2.
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Fig. 2. Dependencies of ε’/ε0 (T, f ) and tan δ (T, f ) for solid solution ceramics: (a) SS1; (b) SS1 + 0.5 wt.% GeO2 ; (c) SS1 + 1.0 wt.% GeO2 (the f growth direction is indicated by arrows).
Fig. 3. Dependencies for solid solutions ceramics of pure and modified compositions: (a) ε’/ε 0 (T, f ) ( f = 106 Hz) in the temperature range T = 300–900 K; (b) (ε’/ε 0 )−1 (T ) at f = 106 Hz. The solid line is an illustration of the fulfillment of the Curie – Weiss law.
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Table 2. Dielectric characteristics of the tested solid solution ceramics. Composition
T m1 , K
ε’/ε0 (T = T m1 ) ε"/ε0 (T = T m1 )
ε’/ε0 (RT)
ε"/ε0 (RT)
T b, K
SS1
737
9049
9578
326
16
874
SS1 + 0.5 wt.% GeO2
694
5761
5274
334
12
868
SS1 + 1.0 wt.% GeO2
762
6638
6719
340
18
880
In all cases, the values of the Burns temperature, T b , were more than 100 K higher than T m1 , which indicates a strong diffusion of the FE → PE phase transition region, however, the highest values of both temperatures were observed in the SS1 + 1.0 wt.% GeO2 sample. Figure 4 shows the dependencies of the real and imaginary parts of the complex dielectric constant for solid solution ceramics of pure and modified compositions at room temperature (RT) on frequency, and Cole – Cole diagrams (ε” vs. ε’) are shown in Fig. 5 in the vicinity of anomalies on the ε’/ε0 (T, f ) dependences.
Fig. 4. Dependencies for solid solutions ceramics of pure and modified compositions: (a) ε’/ε0 ( f ) and (b) ε"/ε0 ( f ) at room temperature and different frequencies.
It can be seen that with increasing frequency, ε’/ε0 decreases monotonically in the initial and modified 1.0 wt.% GeO2 compositions, and for the SS1 + 0.5 wt.% GeO2 composition, the value of ε’/ε0 weakly depends on f . In the dependence ε”/ε0 ( f ), as the frequency increases, there is a monotonic decrease, and then an increase and the formation of a maximum in the vicinity of f = 106 Hz. Taking into account the results of Ref. [30], this indicates a relaxation dispersion of the dielectric constant in all ceramic samples already at room temperature, while the shape of the ε”/ε0 ( f ) curve in the low frequency region indicates a significant contribution to the dielectric response from the electrical conductivity of the material. Since in almost all cases the dependencies ε”(ε’) show signs of the two circles arcs formation, the resulting dielectric response, as
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in Ref. [31], is most likely contributed by two relaxation processes: a high-frequency process, caused by relaxation of the ferroelectric polarization, and a low-frequency one, the contribution of which increases significantly as the temperature increases, which is associated with the effects of Maxwell – Wagner interlayer polarization.
Fig. 5. Cole – Cole diagrams (ε” vs. ε’) of modified ceramic solid solutions for different temperatures in the frequency range f = (2 × 102 –2 × 106 ) Hz.
Thus, the results obtained indicate that the introduction of superstoichiometric germanium oxide at the stage of 0.7BFO − 0.3PFN ceramics production, namely 1.0 wt.%, leads, on the one hand, to a decrease in the optimal sintering temperature of ceramics while maintaining high temperatures of phase transformations in the ferroelectric subsystem, and, on the other hand, to a decrease in the conductivity of ceramics and greater manufacturability of the material. The latter manifested itself in the fact that the modified solid solutions proved to be the best in the fabrication of large-sized blocks, which can be used as blanks for the fabrication of piezoelements operating in a wide range of temperatures, and ceramic disks used as targets in the production of nanoscale films.
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Further, it was the modified solid solutions that we used to prepare the corresponding heterostructures.
4 Conclusions Results of the ferroelectric and dielectric properties study of the samples 0.7BiFeO3 − 0.3PbFe0.5 Nb0.5 O3 system of solid solutions, superstoichiometrically modified with GeO2 in amounts of (0.5–1.0) wt.% and obtained using conventional ceramic technology, indicate that when studying P(E) dependencies in solid solutions, it was not possible to fix saturated hysteresis loops, which indicates high coercive fields in this material. However, this indicates that the use of GeO2 , as an additive, leads to an increase in the electrical strength of the material while maintaining high PR and E C values. Objects are characterized by a significant change in the temperature of the maximum dielectric constant, T m1 , and the values of ε’/ε0 (T, f ). In the temperature range (300–900) K, strong frequency dispersion ε’/ε0 (T, f ) is observed in solid solution objects, and at temperatures above the expected phase transition, the contribution of through conductivity increases, which causes the manifestation of the effects of Maxwell – Wagner polarization and its corresponding dielectric relaxation. The materials developed during the work, based on solid solutions containing bismuth ferrite and lead ferroniobate, are advisable to use in the creation and study of new multifunctional media with special electrophysical parameters, promising for applications in micro- and nanoelectronics, microwave technology, and spintronics . Acknowledgments. The study was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation (State task in the field of scientific activity in 2023). Project No. FENW-2023–0010/(GZ0110/23–11-IF). The equipment of the Center for Collective Use of the Research Institute of Physics of the Southern Federal University “Electromagnetic, electromechanical and thermal properties of solids” was used.
References 1. Dadami, S.T., et al.: Structural, dielectric and conductivity studies of PbFe0.5 Nb0.5 O3 – BiFeO3 multiferroic solid solution. J. Alloys Compd. 724, 787–798 (2017). https://doi.org/ 10.1016/j.jallcom.2017.07.126 2. Li, H., et al.: Coexistence of relaxor behavior and ferromagnetic order in multiferroic Pb(Fe0.5 Nb0.5 )O3 – BiFeO3 solid solution. J. Mater. Chem. C. 8(38), 13306–13318 (2020). https://doi.org/10.1039/D0TC03505J 3. Dadami, S.T., Shivaraja, I., Deshpande, S.K., Rayaprol, S., Angadi, B.: BiFeO3 induced enhancement in multiferroic properties of PbFe0.5 Nb0.5 O3 . Ceram. Int. 44(16), 20449–20456 (2018). https://doi.org/10.1016/j.ceramint.2018.08.039 4. Stoch, A., Stoch, P.: Magnetoelectric properties of 0.5BiFeO3 – 0.5Pb(Fe0.5 Nb0.5 )O3 solid solution. Ceram. Int. 44(12), 14136–14144 (2018). https://doi.org/10.1016/j.ceramint.2018. 05.013 5. Zhidel, K.M., Pavlenko, A.V.: Structure and optical properties of STO/Si and BFOPFN/STO/Si heterostructures obtained by RF-cathode sputtering. Ferroelectrics 590(1), 180–187 (2022). https://doi.org/10.1080/00150193.2022.2037949
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25. Reznitchenko, L.A., et al.: Phase equilibrium and properties of solid solutions of PbTiO3 – PbZrO3 –PbNb2/3 Mg1/3 O3 –PbGeO3 system. Inorg. Mater. 45(2), 173–181 (2009). https:// doi.org/10.1134/S0020168509020125 26. Amaral, F., Rubinger, C.P.L., Valente, M.A., Costa, L.C., Moreira, R.L.: Enhanced dielectric response of GeO2 -doped CaCu3 Ti4 O12 ceramics. J. Appl. Phys. 105(3), 034109 (2009). https://doi.org/10.1063/1.3075909 27. Smolenski, G.A., Chupis, I.E.: Ferroelectromagnets. Soviet. Physics Uspekhi. 25, 475–493 (1982). https://doi.org/10.1070/PU1982v025n07ABEH004570 28. Patel, J.P., Singh, A., Pandey, D.: Nature of ferroelectric to paraelectric phase transition in multiferroic 0.8BiFeO3 – 0.2Pb(Fe1/2 Nb1/2 )O3 ceramics. J. Appl. Phys. 107(10), 104115 (2010). https://doi.org/10.1063/1.3428410 29. Krainik, N.N., Khuchua, N.P., Berezhnoy, A.A., Tutov, A.G.: On the nature of phase transitions in BiFeO3 –PbFe0.5 Nb0.5 O3 solid solutions. Soviet Physics, Solid State. 7(1), 132–142 (1965) 30. Bogatin, A.S., Turik, A.V.: Relaxation polarization processes in dielectrics with high through conductivity. Feniks, Rostov-on-Don (2013) 31. Pavlenko, A.V., Zhidel, K.M., Kubrin, S.P., Kolesnikova, T.A.: High-temperature 0.5BiFeO3 – 0.5PbFe0.5 Nb0.5 O3 multiferroic: microstructure, ferroelectric properties, and Mössbauer effect. Ceram. Int. 47(15), 21167–21174 (2021). https://doi.org/10.1016/j.ceramint.2021. 04.120
Resonant Microwave Response in Strontium Titanate Single Crystals P. A. Astafev1(B) , A. A. Pavelko1 , K. P. Andryushin1 , A. R. Borzykh2 , A. M. Lerer2 , Y. A. Reizenkind1 , I. V. Donets2 , A. V. Pavlenko1,3 , and L. A. Reznichenko1 1 Research Institute of Physics, Southern Federal University, Rostov-on-Don 344090, Russia
[email protected]
2 Faculty of Physics, Southern Federal University, Rostov-on-Don 344090, Russia 3 Federal Research Centre, The Southern Scientific Centre of the Russian Academy of Sciences,
Rostov-on-Don 344006, Russia
Abstract. Materials, based on strontium titanate, have many applications in microwave technology. The main methods for achieving target characteristics when developing such materials are modification of the composition and creation of low-dimensional structures. Due to the high cost of single crystals of suitable shape for measuring electrical parameters using standard methods, ceramics and composites are being considered as an alternative. The work proposes to use a set of simpler measurement techniques that allow for a primary analysis of the microwave response of single-crystal materials and simplify further modeling and calculations. The results of calculating the resonance response of a model single-crystal sample of strontium titanate are presented. The influence of the features of measurement techniques on the final results is analyzed. Conclusions are drawn about the feasibility of conducting comprehensive studies using various techniques. The results obtained will help in further calculations and modeling of the electrical parameters of single-crystal samples of functional materials. Keywords: Microwave · Strontium Titanate · Ferroelectric · Microstrip Lines · Waveguide · Single Crystal
1 Introduction Strontium titanate in various solid states is one of the most attractive electrically active materials for use in microwave technology due to its high dielectric constant and low losses in the microwave range. Thin films based on this material have attracted the most attention in recent decades. It is noted [1] that thin films of Srn+1 Tin O3n+1 can serve as a replacement for Bax Sr1-x TiO3 , thin films of which have high losses due to defects. The dielectric constant of Bax Sr1-x TiO3 thin films strongly depends on the surface effects that occur at the boundary of the BaTiO3 and SrTiO3 layers [2], so the task of achieving the highest dielectric constants with the lowest losses is non-trivial. It is proposed to use thin films © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 252–265, 2024. https://doi.org/10.1007/978-3-031-52239-0_25
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of (SrTiO3 )nSrO on a DyScO3 substrate as a basis for voltage-tunable microstrip resonators [3]. It is noted that such resonators can have a very high frequency tuning speed (tuning time less than 1 ns), however, the highest sensitivity to tuning is achieved at low temperatures (about 200 K). Due to many ferroelectric materials currently being developed have certain disadvantages and do not always satisfy all the requirements of microwave technology, researchers continue to search for new compositions [4, 5], devoid of the previously mentioned disadvantages, while having all the advantages of existing commercial ferroelectrics. Due to the high cost of producing single crystals and single crystal thin films, based on strontium titanate, ceramics are being considered as an alternative. However, its dielectric constant at room temperature in the microwave range is on average lower than that of single crystals and is about 200 [6]. To effectively search for new ferroelectric materials with compromise properties, it is necessary to develop new, simpler methods for studying them, requiring as few expensive technological operations as possible when creating materials and preparing samples for research. One of the widely used methods for studying the electrical characteristics of materials is the resonance method. For example, the work [4] describes resonant methods for measuring the dielectric constant and loss tangent in single crystals of quantum paraelectrics KTaO3 and SrTiO3 at low temperatures, and also presents the measurement results. The value of the dielectric constant of SrTiO3 is presented at a frequency of 3 GHz at temperatures from 5 to 300 K and at room temperature is approximately 300. The phenomenon of a pronounced dependence of the dielectric constant on temperature in the microwave range for SrTiO3 single crystals at low temperatures can also be used when measuring temperature with very high resolution [5]. In [6], the dielectric constant of dense (relative density 98.8%) SrTiO3 ceramics at a frequency of 36.2 GHz in a wide temperature range (90–300 K) was calculated, based on measurements of the impedance of a section of a short-circuited rectangular waveguide with a material sample located at the end. The use of powder or composites with a polymer matrix does not allow one to accurately determine the dielectric constant of dense ceramics or single crystals, although it expands the range of possible measurement techniques. For example, composite materials based on SrTiO3 in an epoxy resin matrix are considered [7] as promising microwave absorbers. The dielectric constant in the frequency range 130 MHz–10 GHz was calculated based on measurements of the S-parameters of a cylindrical sample with a hole filling a section of a coaxial line. The work [8] proposes to use a composite material filled with crushed SrTiO3 ceramics in a polymer matrix to create flexible dielectric waveguides. Dielectric constant spectra of this material were obtained in the frequency range 500 MHz–5 GHz, based on measurements of the reflectance of the material surface using an open-ended coaxial probe. For a composite with a volume fraction of SrTiO3 of 60%, the dielectric constant does not exceed 20 up to 3 GHz, in the vicinity of 4.5 GHz it increases to 40, and then a sharp decrease is observed to almost zero. A relatively small number of works with experimental results of studying microwave spectra of dielectric constant indicate fundamental difficulties in their measurements. Meanwhile, the use of a set of methods for experimental study of the microwave response
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of functional materials will make it possible to obtain more reliable data, subsequently improving the quality of modeling and calculations. Therefore, it was decided to analyze and compare the resonant microwave response of functional materials using several different measurement techniques. A single crystal of strontium titanate was chosen as a model sample, since similar materials are well described in the literature. In connection with the above, the purpose of this work was to detect resonant responses in a single-crystal sample of strontium titanate in microwave transmission lines of various configurations.
2 Obtaining and Research Methods The object of the study was a rectangular sample of SrTiO3 single crystal, 5 × 5 × 0.5 mm3 in size. Radiophysical measurements were performed using a P9375A “Keysight” vector network analyzer (VNA), which has an operating frequency range of 300 kHz–26.5 GHz. 2.1 Microstrip Measuring Cells To measure the microwave response of the samples, four measuring cells were used, 3 of which were microstrip and 1 was waveguide. The two microstrip measurement cells were straight sections of microstrip lines (MSLs) of various lengths (4 and 5 cm) on glass fiber reinforced epoxy (FR4) substrates with SMA 3.5 coaxial connectors at both ends. The third microstrip measuring cell consisted of 2 parallel MSLs located at a distance of 14 mm from each other. The dielectric constants of the substrates were different and evaluated as to 6 for the first measuring cell and 5.2 for the second and third ones. The maximum operating frequency range of the connectors is 18 GHz. To achieve the best impedance matching with the measuring setup (a characteristic impedance of 50 ) and the lowest level of losses, the width of the MSL was selected considering the thickness of the MSL, the dielectric constant of the substrate and its height using the Hammerstad-Jensen method [9]. The use of the first two measuring cells was due to the need to assess the influence of the effective dielectric constant of the MSL on the results of calculating the dissipation coefficient of the samples, as well as on the intensity of the resonant maxima and their position in the frequency domain. To study the resonance properties, the sample was located next to the MSL on the substrate in the region of a weak electric field, which made it possible to isolate the resonance absorption maxima against the background of a general low absorption level. Two parallel MSLs in the third measuring cell are located at a large distance from each other and are weakly coupled. Placing a resonator between them enhances the coupling at resonant frequencies, which leads to signal transmission from one MSL to another. This makes it possible to unambiguously identify the resonant nature of loss maxima. The quality of contact of MSL coaxial connectors, VNA microwave cables and measuring cells was assessed by analyzing the dependences of S-parameters on the frequency of an empty measuring cell (Fig. 1).
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Fig. 1. Amplitudes of S-parameters of empty measuring cells.
Periodic minima and maxima of the reflection coefficients (S 11 , S 22 ) are caused by the interference of reflected waves from the coaxial connectors of the measuring cells. Assuming the interference nature of the minima and maxima of the reflection coefficients, a numerical estimate of their position in the frequency domain was carried out considering the calculated effective dielectric constant [10], which was 4.7 for the first MSL and 4 for the second and third MSLs (calculated minima and maxima are indicated by spherical markers in Figs. 1 a and b). The frequency bandwidth with a satisfactory degree of matching is about 7 GHz in the first case and 5 GHz in the second (the level of reflection coefficients does not exceed –20 dB). 2.2 Method for Calculating the Dissipation Coefficient of a Sample on an MSL The coefficient of dissipation of electromagnetic wave energy by a sample located next to the MSL was calculated using the formula: Di = di − d0
(1)
where Di is the scattering coefficient of the sample, d i is the loss coefficient in the MSL with the sample, d 0 is the loss coefficient in the empty MSL. Both loss coefficients, in turn, were calculated using the following formula [11–14]: di,0 = 1 − |S11(i,0) |2 − |S21(i,0) |2
(2)
where |S11(i,0) |2 and |S21(i,0) |2 are the squared moduli of the elements of the scattering matrix (amplitude of S-parameters) of the MSL with a sample (index “i”) and empty MSL (index “0”), expressed in relative units, which have the physical meaning of power reflection coefficient and power transmittance coefficient, respectively. These parameters were determined during the experiment. The sample dissipation coefficient in our case indicates how much of the energy was dissipated when the sample was introduced into the measuring cell. It consists of the following main contributions: – energy absorbed by the sample;
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– energy absorbed by the MSL substrate; – energy scattered into free space at the sample boundary. The second contribution arises from the fact that the electromagnetic wave, reflected by the sample, experiences losses in the MSL substrate. The maximum dissipation coefficient of samples can be caused by two mechanisms: – the transmitted and reflected waves are attenuated in the sample due to active losses in the sample material; – part of the energy of the incident wave is absorbed in the sample at resonance. Due to the location of the samples in the region of a weak electromagnetic field, the second mechanism will have a greater contribution to the level of the dissipation coefficient maxima. 2.3 Waveguide Measuring Cell The waveguide measuring cell was a cross-section of a rectangular waveguide of the “WR-90” type (a cross-section of 23 × 10 mm2 ) and a length of l = 58 mm with a transparency range of 8.2–12.4 GHz. The waveguide was connected to the VNA using coaxial-waveguide junctions (CWJs) from the X11644A calibration kit. Figure 2 shows the dependence of the amplitude of the S-parameters on the frequency of the empty waveguide.
Fig. 2. Comparison of the dependence of the S-parameter amplitude on the frequency of the empty waveguide and the position in the frequency domain of the calculated reflection maxima and minima. For clarity, the intensity of the maxima was taken to be (–30 dB), and the minima equal to (–50 dB) (intensities were not calculated).
The reflection coefficient dips at 10 and 12 GHz are likely due to interference of reflected waves at the coupling points between the CWJs and the waveguide, which arise due to small differences in the cross-sections of the CWJs and the waveguide (the cross-sectional dimensions of the CWJs are 22.86 × 10.16 mm2 ). Similar to the case with the MSL, a numerical assessment of the position of the interference maxima and
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minima in the frequency domain was carried out. All calculated interference maxima and minima are shown in Fig. 2 in comparison with experimental data, and their positions are in good agreement with experiment. The two reflection coefficient maxima, observed in the experimental data, are 3rd and 4th order interference maxima, and the three minima are 2nd, 3rd and 4th order interference minima. To measure the sample in the waveguide, 3 foam inserts were made. The first insert is designed to position a rectangular sample perpendicular to the wide wall in the center of the waveguide, namely in the E-plane (Fig. 3a); the second insert is designed to position a rectangular sample parallel to the wide wall in the center of the waveguide, namely in the H-plane (Fig. 3b); the third insert is designed to position a rectangular sample in the cross-sectional plane of the waveguide, namely in the transverse plane (Fig. 3c).
Fig. 3. Layout of the insert with the sample in the waveguide: a – E-plane, b – H-plane, c – transverse plane.
The S-parameters of the foam inserts were measured in the waveguide without a sample. Their presence increases the level of reflection coefficient maxima by approximately 5 dB, while the reflection coefficient minima are slightly shifted towards low frequencies (Fig. 4).
Fig. 4. Dependence of the amplitude of the S-parameters of a waveguide with foam inserts of various configurations on frequency.
All three foam inserts almost completely fill the waveguide. Therefore, it is possible to estimate the value of the dielectric constant of the foam plastic by the position of the maxima and minima of reflection in the frequency domain, using the same method that was used previously when estimating their position in the MSL, taking into account the
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assumption of the absence of dispersion of the dielectric constant in the foam. In this approximation, the dielectric constant of the foam was 1.02. 2.4 Method for Calculating the Dissipation Coefficient of a Sample in a Waveguide The coefficient of energy dissipation of an electromagnetic wave by a sample in a waveguide was calculated as the difference between the loss coefficient of a waveguide with a sample in a foam insert and the loss coefficient of an empty waveguide (using Formula 1). Unlike the MSL, the dissipation coefficient in the case of a waveguide consists of two contributions: – energy absorbed by the sample; – energy absorbed by the foam inserts. The calculation showed that the value of the dissipation coefficient in foam inserts does not exceed 1.5% relative to the power level of the incident wave in the entire frequency range under study for all three inserts (Fig. 5). Negative values of the dissipation coefficient indicate an improvement in the matching of the measuring cell with the device at some frequencies when a foam insert is introduced inside the waveguide.
Fig. 5. Dependence of the dissipation coefficient of foam inserts with various configurations on frequency.
3 Experimental Results and Discussion Measurements of a rectangular sample of single-crystalline SrTiO3 in microstrip measuring cells with a single MSL and different values of effective dielectric constant (4.7 and 4.0) showed the presence of pronounced resonance maxima (Fig. 6). The relative positions of the resonant maxima in the first and second measuring cells are slightly different, which is likely due to the presence of dispersion in the effective value of the dielectric constant. The small half-width of the maxima and their high intensity are maintained over a wide frequency range, up to 18–20 GHz.
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Fig. 6. Dependence of the dissipation coefficient (D) of a single-crystal rectangular SrTiO3 sample in the first (a) and second (b) microstrip measuring cells on frequency.
When the sample is placed between two parallel MSLs, only one obvious resonance maximum is observed in the third microstrip measuring cell (Fig. 7). Due to the absence of other clearly defined maxima, it is difficult to correlate its position with the position of maxima previously observed on single MSLs. The absence of other maxima in this case is probably due to the weak coupling of the sample with the MSL (sample width is 5 mm, the distance between the MSL is 14 mm). Conventionally, we will assume that this maximum is the first of those observed in previous measurements (its frequency is very close to the frequency of 6.509 GHz for the first maximum in Fig. 6b). To study the influence of the accuracy of the sample position in the measuring cell on the measurement results, a series of measurements of the sample dissipation coefficient were performed in all three microstrip measuring cells with sequential displacement of the sample along the MSL (Fig. 8). When the sample is displaced along the MSL, in all three cases a periodic increase and decrease in the intensity of the dissipation coefficient are observed throughout the entire frequency range under study (Fig. 9). This is probably due to the phenomenon of the appearance of standing waves in the tested systems: when the sample is located at the nodes of the standing wave, intensity minima are observed, and maxima are observed at the antinodes. A sample, located in the H-plane of the waveguide (see Fig. 3b), does not introduce any significant perturbations, compared to measurements of the waveguide with an insert (Fig. 10, compared to Fig. 4b). This is evidenced by both measurements of the amplitude of the S-parameters (Fig. 10) and the phase difference (Fig. 11). The dependences of the phase difference between the incident and transmitted signal on the frequency (ϕ21 , ϕ12 ) of both the waveguide with a foam insert without a sample and with a sample practically coincide. The dependences of the phase difference between the incident and
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Fig. 7. Dependence of the dissipation coefficient (D) of a single-crystal rectangular SrTiO3 sample located between two parallel MSLs (third microstrip measuring cell) on frequency.
Fig. 8. Layout of sample location in a microstrip measuring cell with a single MSL (a) and two parallel MSLs (b).
Fig. 9. Dependences of the dissipation coefficient (D) of a single-crystal rectangular SrTiO3 sample on frequency ( f ) and coordinates (x).
reflected signals on frequency (ϕ11 , ϕ22 ) were measured with very low accuracy. This is due to the reflected signal power being too low, which prevents the VNA phase detector from correctly detecting the phase difference. Therefore, it is difficult to judge
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the differences between the phase differences of the incident and reflected signal (ϕ11 , ϕ22 ). The small effect of the sample on the S-parameters is probably due to its small height, located in a region with a high electric field density.
Fig. 10. Dependence of the amplitude of the S-parameters of a waveguide with a sample in the H-plane on frequency.
Fig. 11. Dependences of the phase difference of S-parameters on the frequency of a waveguide with a foam insert without a sample (a) and with a foam insert with a sample (b).
At the same time, a sample, located in the E-plane of the waveguide (see Fig. 3a) has several obvious resonant responses (transmission coefficient minima in Fig. 12a). Resonance minima with frequencies close to 9.7, 10.4 and 11.3 GHz were also found in measurements of a rectangular SrTiO3 sample in the transverse plane of the waveguide (Fig. 12b). The position of the resonant responses is more clearly distinguishable in the dependences of the dissipation coefficient on frequency. In the frequency range under study, for a sample, located in the E-plane of the waveguide, 5 resonant dissipation maxima were detected at frequencies of 9.565, 9.71, 10.368, 11.286 and 11.479 GHz (Fig. 13a). Calculation of the dependence of the sample dissipation coefficient on frequency, when measuring the sample in the transverse plane of the waveguide (Fig. 13b), showed the
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Fig. 12. Dependence of the amplitude of the S-parameters of a waveguide with a single-crystal rectangular sample of SrTiO3 in the E-plane (a) and in the transverse plane (b) on frequency; the circles highlight the resonant minima.
Fig. 13. Dependence of the dissipation coefficient of a single-crystal rectangular SrTiO3 sample in the E-plane (a) and in the transverse plane (b) on frequency.
absence of “side” dissipation maxima at frequencies of 9.565 and 11.479 GHz, in contrast to the measurements in the first insert (Fig. 13a). The remaining three resonant maxima are located at frequencies of 9.756, 10.444 and 11.367 GHz. Three measurements were taken in the first insert, and three measurements in the third insert. Repeating the same measurements multiple times was necessary to collect statistics. Each measurement was performed with complete disassembly and reassembly of the fixture to eliminate the influence of inaccuracies in the location of the foam insert in the waveguide and the sample inside the foam insert. Comparing all the measurements obtained, it is clear that in all cases the resonant dissipation maxima have approximately the same frequency (Fig. 14). The maximum at a frequency of 9.633 GHz is absent both in measurements in the transverse plane of the waveguide and in all measurements on the MSL; therefore, this maximum is probably associated with another vibration mode
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of the sample, which is excited only when located in the E-plane of the rectangular waveguide.
Fig. 14. Comparison of the dependences of the dissipation coefficient of a single-crystal rectangular SrTiO3 sample on frequency in the E-plane (D1(*) ) and in the transverse plane (D3(*) ) of the waveguide. The index in brackets indicates the measurement number.
Let us present the results of all measurements in the form of Table 1, averaging the frequencies of the resonant maxima over several measurements for each of the five detected resonances. A dash indicates the absence of resonance, and n/d indicates the absence of data due to the limited measurement range. Table 1. Averaged resonance frequencies of a single-crystal rectangular SrTiO3 sample. Measuring cell
Average values of resonance frequencies, GHz 1
2
Single MSL, εeff = 4.7
6.87
8.08
9.09
9.64
Single MSL, εeff =4
6.51
7.86
8.96
Parallel MSL, εeff =4
6.53
-
-
E-plane waveguide
9.769
3
4
10.44
H-plane waveguide
n/d
-
-
Waveguide transverse plane
n/d
-
9.763
5
6
7
9.94
10.41
11.21
9.54
9.78
10.36
11.13
-
-
11.386 10.44
-
11.55 11.364
n/d
-
n/d
-
n/d
It should be noted that 2–6 resonant maxima in the case of measurements in a waveguide are significantly shifted to higher frequencies, which is likely due to wavelength dispersion in the waveguide.
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4 Conclusions The difference in the effective dielectric constant of the MSL, as well as the presence of wavelength dispersion in the waveguide, leads to a shift in the position of the maximum dissipation coefficient of the samples in the frequency domain, which should be considered when calculating and modeling the electrical parameters of materials with high dielectric constants, in particular, those based on strontium titanate. Complex measurements using various equipment make it possible to clarify the nature of the maxima of the coefficient of dissipation of electromagnetic energy by a sample of the substance under study, and in some cases to determine, which mode of oscillation of the resonator certain maxima belong to. The above helps in choosing a model and in numerical modeling of the system to further determine the electrical characteristics of the material. The highest intensity of the resonant maxima of the sample is achieved when it is placed at the antinodes of a standing wave that arises in the system under study. This makes it possible in some cases to detect low-intensity maxima. A sample of single-crystal SrTiO3 has relatively low losses in the microwave range and has a good resonant response, which indicates the feasibility of using the material under study in bulk dielectric resonators. Acknowledgments. The study was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation (State task in the field of scientific activity in 2023). Project No. FENW-2023–0010/(GZ0110/23–11-IF). The equipment of the Center for Collective Use of the Research Institute of Physics of the Southern Federal University "Electromagnetic, electromechanical and thermal properties of solids" was used.
References 1. Lee, C.-H., et al.: Exploiting dimensionality and defect mitigation to create tunable microwave dielectrics. Nature 502, 532–536 (2013) 2. Liu, M., et al.: Interface engineered BaTiO3 /SrTiO3 heterostructures with optimized highfrequency dielectric properties. ACS Appl. Mater. Interfaces 4, 5761–5765 (2012) 3. Hagerstrom, A.M., et al.: Sub-nanosecond tuning of microwave resonators fabricated on Ruddlesden-popper dielectric thin films. Adv. Mater. Technol. 3, 1800090 (2018) 4. Geyer, R.G., Riddle, B., Krupka, J., Boatner, L.A.: Microwave dielectric properties of singlecrystal quantum paraelectrics KTaO3 and SrTiO3 at cryogenic temperatures. J. Appl. Phys. 97, 104111 (2005) 5. Gallop, J.C., Hao, L.: Applications of coupled dielectric resonators using SrTiO/sub 3/pucks: tuneable resonators and novel thermometry. IEEE Trans. Instrum. Meas. 50, 526–530 (2001) 6. Petzelt, J., et al.: Dielectric, infrared, and Raman response of undoped SrTiO3 ceramics: evidence of polar grain boundaries. Phys. Rev. B 64, 184111 (2001) 7. Choi, H.D., Shim, H.W., Cho, K.Y., Lee, H.J., Park, C.S., Yoon, H.G.: Electromagnetic and electromagnetic wave-absorbing properties of the SrTiO3 –epoxy composite. J. Appl. Polym. Sci. 72, 75–83 (1999) 8. Xiang, F., Wang, H., Yao, X.: Dielectric properties of SrTiO3 /POE flexible composites for microwave applications. J. Eur. Ceram. Soc. 27, 3093–3097 (2007)
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9. Hammerstad, E., Jensen, O.: Accurate models for microstrip computer-aided design. B. In: 1980 IEEE MTT-S International Microwave Symposium Digest, pp. 407–409. IEEE (1980) 10. Gupta, K.C., Garg, R., Chadha, R.: Computer aided design of microwave circuits. NASA STI/Recon Technical Report A 82, 39449 (1981) 11. Capwell, J., Weller, T., Markell, D., Dunleavy, L.: Automation and real-time verification of passive component s-parameter measurements using loss factor calculations. Microwave J. 47, 82–88 (2004) 12. Cojocaru, V., Markell, D., Capwell, J., Weller, T., Dunleavy, L.: Accurate simulation of rf designs requires consistent modeling techniques. High Frequency Electronics, Summit Technical Media, LLC (2002) 13. Micheli, D., Apollo, C., Pastore, R., Marchetti, M.: X-Band microwave characterization of carbon-based nanocomposite material, absorption capability comparison and RAS design simulation. Compos. Sci. Technol. 70, 400–409 (2010) 14. Wang, Z., Zhao, G.-L.: Electromagnetic wave absorption of multi-walled carbon nanotube– epoxy composites in the R band. J. Mater. Chemistry C. 2, 9406–9411 (2014)
Crystal Structure, Microstructure, Piezoelectric and Dielectric Properties of Piezoceramic Bi7 Sr2 TiNb5 O27 Sergei V. Zubkov1(B) , Ivan A. Parinov2 , Alexander V. Nazarenko1,3 , and Yuliya A. Kuprina1 1 Research Institute of Physics, Southern Federal University, Stachki, Avenue, 194,
Rostov-on-Don 344090, Russia [email protected] 2 I. I. Vorovich Mathematics, Mechanics, and Computer Science Institute, Southern Federal University, Rostov-on-Don 344090, Russia 3 Southern Scientific Center of the Russian Academy of Sciences, Rostov-on-Don 344090, Russia
Abstract. A new layered perovskite-like oxide Bi7 Sr2 TiNb5 O27 was synthesized by the method of high-temperature solid-state reaction, in which partial substitution of bismuth (Sr2+ ) atoms in the dodecahedra of the perovskite layer (A-positions) by Bi3+ ion took place. X-ray structural studies have shown that compound was single-phase and had the structure of Aurivillius-Smolensky phases (ASPs) with close parameters of orthorhombic unit cell, corresponding to space group A21 am. The dependence of the relative permittivity ε/ε 0 , the tangent of loss tan σ at different frequencies on temperature and piezoelectric constant d 33 were measured. Keywords: Aurivillius-Smolensky Phases · Bi7 Sr2 TiNb5 O27 · Activation Energy · Curie Temperature · Piezoelectric Modulus · Depolarization Temperature
1 Introduction In 1949, while studying the Bi2 O3 – TiO2 system, V. Aurivillius established the formation of Bi4 Ti3 O12 oxide with a perovskite-type structure [1]. Ten years later, G. Smolensky’s group [2] discovered the ferroelectric properties of Bi2 PbNbO9 , which belongs to this family of compounds, after which an intensive stage of studying these compounds began, which can rightfully be called Aurivillius – Smolensky phases (ASPs). Subsequently, about ten new Aurivillius – Smolensky phases were obtained and almost all of them turned out to be ferroelectrics [3, 4]. Aurivillius-Smolensky phases form a large family of bismuth-containing layered compounds of the perovskite type, the chemical composition of which is described by the general formula Bi2 Am–1 Bm O3m+3 . The crystal structure of ASPs includes alternating layers of [Bi2 O2 ]2+ , separated by m perovskitelike layers [Am−1 Bm O3m+1 ]2– , where A-ions have large radii (Bi3+ [5], Ca2+ , Gd3+ [6], © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 266–274, 2024. https://doi.org/10.1007/978-3-031-52239-0_26
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Sr2+ , Ba2+ , Pb2+ , Na+ , K+ , Y3+ [7], Ln3+ , Nd3+ [5], Lu3+ [8] (lanthanides) and demonstrate dodecahedral coordination; the B-positions in oxygen octahedra are occupied by highly charged (≥3+) cations of small radius (Ti4+ , Nb5+ , Ta5+ [9], W6+ , Mo6+ , Fe3+ , Mn4+ , Cr3+ , Ga3+ , etc.). The value of m is determined by the number of perovskite layers [Am−1 Bm O3m+1 ]2− , located between fluorite-like layers [Bi2 O2 ]2+ along the pseudotetragonal c-axis, and can be an integer or half-integer number in the range m = 1 – 5. Substitutions of atoms in positions A and B significantly affect the electrical properties of ASPs. There are large changes in the dielectric constant and electrical conductivity; in addition, the Curie temperature T C can also vary over a wide range. Thus, the study of cation-substituted compounds of ASPs is of great importance in the development of materials for various technical purposes.
2 Experiment A polycrystalline samples of ASPs Bi7 Sr2 TiNb5 O27 (m = 2) were synthesized by a solid-phase reaction of the corresponding oxides Bi2 O3 , SrCO3 , Nb2 O5 . All starting compounds were of analytical grade. After weighing according to the stoichiometric composition and thorough grinding of the original oxides with the addition of ethyl alcohol, the pressed samples were calcined at a temperature of 800 °C for 4 h. The samples were fired in a laboratory muffle furnace in air and then crushed, ground repeatedly and pressed into tablets with a diameter of 10 mm and a thickness of 1.0 –1.5 mm, followed by final synthesis at a temperature of 1050 °C (3 h). The X-ray pattern was recorded on a Rigaku Ultima IV diffractometer with a Cu X-ray tube. The CuKα1,α2 radiation was isolated from the general spectrum using a Ni filter. The radiograph was measured over a 2θ-angle range from 10° to 60° with a scan step of 0.02° and an exposure (intensity recording time) of 4 s per point. The lattice parameters were refined using the full-profile Rietveld analysis method in the Full Prof program. The refinement was carried out in the Space Group A21 am No. 36 (4). It has been established that the samples were pure and doping with Ti and increasing the concentration of Bi3+ ions led to a decrease in lattice parameters (Table 1) and, accordingly, to a decrease in its volume. To measure the dielectric constant and electrical conductivity, electrodes were applied to the flat surfaces of ASPs samples in the form of disks with a diameter of 10 mm and a thickness of about 1.5 mm using Ag paste; the samples were annealed at a temperature of 600 °C (10 min). The temperature and frequency dependences of the dielectric characteristics were measured using an E7-20 immittance meter in the frequency range from 100 kHz to 1 MHz and in the temperature range from room temperature to 750 °C. The sample was polarized in an oil bath at 150 °C at a voltage of 35 kV/cm for 30 min. The images were taken using a Carl Zeiss EVO 40 scanning electron microscope (Germany) at the Shared Use Center of the Southern Scientific Center of the Russian Academy of Sciences. The study was carried out on transverse chips of manufactured ceramics. In the absence of an additional conductive layer, grain blurring and multiple charge accumulation effects were observed. Therefore, to analyze the chip surface, the conductive layer was deposited using an SC7620 Mini Sputter Coater magnetron sputtering unit. Before deposition, the samples were not previously subjected to
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Table 1. Rietveld analysis results for Bi7 Sr2 TiNb5 O27 (Space Group A21 am; a = 5.482 Å, b = 5.492 Å, c = 25.1143 Å; V = 757.81 Å3 ; d x = 7.357 g/cm3 ) Name
No
Position
x
y
z
Sr
38
4a
1
1
0
Bi
83
8b
0.1995
0.781
−0.4593
Nb
41
8b
0.4152
0.7513
−0.481
O
8
8b
0.3416
0.6782
−0.5059
O
8
8b
0.2529
0.9968
−0.7274
O
8
8b
0.588
0.9677
−0.7858
O
8
8b
0.9304
0.9832
−0.7345
O
8
4a
1
0.3009
0
mechanical treatment. The study was carried out in the high accelerating voltage mode (EHT = 20 kV). To increase the resolution, the probe current was I probe = 55 pA, and the working distance was WD = 8–9 mm.
3 Results and Discussion The powder X-ray diffraction pattern of the studied Bi7 Sr2 TiNb5 O27 solid corresponds to single-phase ASPs with m = 2 and does not contain additional reflections. It was found that the synthesized ASPs crystallize in the orthorhombic system with the space group of the unit cell A21 am (No. 36). Figure 1 shows an experimental powder X-ray diffraction pattern of the compound under study.
Fig. 1. X-ray diffraction patterns of Bi7 Sr2 TiNb5 O27 ceramic
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Based on X-ray diffraction data, the parameters and volume of the unit cell of Bi7 Sr2 TiNb5 O27 were determined and shown in Table 2. Table 2. Parameters and volume of the unit cell of Bi7Sr2TiNb5O27 a0 , Å
b0 , Å
c0 , Å
V, Å3
c, Å
at , %
δc, %
δb0 , %
5.482
5.492
25.1143
757.81
3.767
3.878
−2.8
1.8
a0 , b0 , c0 , V are the unit cell parameters; at is the parameter of the tetragonal period, c is the height of the octahedron along c-axis, δc is the deviation from the cubic shape, δb0 is the rhombic distortion. Table 2 shows the parameters of orthorhombic δb0 and tetragonal δc deformation; = 3c /(8 average tetragonal period at , and average thickness of one perovskite layer c ; c√ 0 + 6m) is the thickness of a single perovskite-like layer, at = (a0 + b0 )/(2 2) is the average value of the tetragonal period; a0 , b0 , c0 are the lattice parameters; δc = (c − at )/at is the deviation of the cell from the cubic shape, that is lengthening or shortening from the cubic shape; δb0 = (b0 – a0 )/a0 is the orthorhombic deformation [10]. The negative value of δc (see Table 2) indicates that the oxygen octahedra in the perovskite-like layer are in a compressed state, which is consistent with the previously obtained result [11]. The obtained unit cell parameters of the studied ASP sample Bi7 Sr2 TiNb5 O27 are close to those previously determined for SrBi2 Nb2 O9 : a = 5.55 Å, b = 5.48 Å, c = 25.261 Å [12]. The tolerance factor t was introduced by V.M. Goldschmidt [13] as a geometric criterion that determines the degree of stability and distortion of the crystal structure: √ t = (RA + RO )/[ 2(RB + RO )], (1) where RA and RB are the radii of cations in positions A and B, respectively; RO is the ionic radius of oxygen. In this work, the tolerance factor t was calculated by using the ionic Shannon radii [14] for the corresponding coordination numbers (CN) (O2– (CN = 12): RO = 1.40 Å; Sr2+ (CN = 12): RSr2+ = 1.44 Å; Nb5+ (CN = 6): RNb5+ = 0.6 Å; Ti4+ (CN = 6): RTi4+ = 0.605). Shannon did not give the ionic radius of Bi3+ for coordination with CN = 12. Therefore, its value was determined from the ionic radius with CN = 8 (RBi3+ = 1.17 Å), multiplied by an approximation factor of 1.179; then for Bi3+ (CN = 12), we obtained RBi3+ = 1.38 Å. To assess the degree of distortion of the ideal perovskite structure, we determined the tolerance factor t from (1), which is presented in Table 3. The analysis showed that this ceramic has a high density, practically without pores. The cleavage took place mainly through the ceramic grains, with boundaries of grains are quite distinguishable. The crystallites have the shape of a flat prism, due to the anisotropic growth rate along axes a, b due to the existence of hard layers (Bi2 O2 )2+ , that is characteristic for Aurivillius-Smolensky phases (Fig. 2). Figure 2 shows images of Bi7 Sr2 TiNb5 O27 ceramics, obtained using a laser microscope.
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Table 3. Dielectric characteristics of Sr2 Bi7 Nb2 O9 : Curie temperature T C , piezoelectric modulus d 33 , tolerance factor t, relative dielectric constant ε/ε 0 , activation energy E n Compound
TC
d 33 , pC/N
t-factor
ε/ε0 (T ) (at 10 kHz)
E 1 /E 2 , eV
Bi7 Sr2 TiNb5 O27
555
14
0.865
750
1.3/0.004
Fig. 2. Microstructure of the cleavage surface of Bi7 Sr2 TiNb5 O27 at different magnifications; marker: left – 10 μm, right – 5 μm.
Fig. 3. Schematic representation of possible chips of idealized grains in the plane (ab)
By analyzing grain sizes, length and width were measured separately (see Fig. 2, arrows). The spread of lengths is quite large (from 3 μm to 14 μm), which can be explained by the chaotic arrangement of grains in the bulk of the ceramic. In this case, the grain itself may be chipped in a random place (see Fig. 3), and, considering the
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natural scatter in grain sizes, the chipped projections may have a continuous spectrum of lengths. The thickness of the crystallites varied in the range of 1–3 μm. This also corresponds to ASPs ceramics. The nature of the arrangement of the grains is such that they stick together in the direction normal to the ab-plane, forming “stacks” of thin (~400 nm) plates (see Fig. 2, highlighted areas). Figure 4 shows the temperature dependences of the relative dielectric constant ε(T ) and the dielectric loss tangent for Bi7 Sr2 TiNb5 O27 at a frequency from 100 kHz to 1 MHz.
Fig. 4. Temperature dependences of the relative dielectric constant ε/ε 0 and loss tangent tan σ for Bi7 Sr2 TiNb5 O27 ASPs at a frequency from 100 kHz to 1 MHz.
The maximum dielectric constant, corresponding to the ferroelectric-to-paraelectric phase transition (T C ), is clearly observed in the dependence of the relative dielectric constant ε/ε0 on the temperature of the synthesized compound Bi7 Sr2 TiNb5 O27 (at frequencies from 100 kHz to 1 MHz). The magnitude of the peak relative dielectric constant decreases, compared to the previously studied compound Bi2 SrNb2 O9 . With increasing temperature, dielectric losses increase, have a clearly defined maximum at all measured frequencies, and then sharply decrease. The maximum dielectric loss is usually 5 °C ahead of the peak dielectric constant, although this is not in all cases. With a further increase in temperature above the Curie temperature, the dielectric losses increase sharply. It should also be noted that the value of the dielectric loss tangent is approximately the same as the dielectric loss tangent with the Bi2 SrNb2 O9 compound. Dielectric losses are very small in the temperature range up to 400 °C for Bi7 Sr2 TiNb5 O27 . The activation energy E a was determined from the Arrhenius equation: σ = (A/T ) exp[−Ea /(kT )],
(2)
where σ is the electrical conductivity, k is a Boltzmann constant, A is a constant, E a is the activation energy. A typical dependence of ln σ on 1/T (at a frequency of 100 kHz), which was used to determine the activation energy E a , is shown in Fig. 5 for ASPs Bi7 Sr2 TiNb5 O27 . The compounds Bi7 Sr2 TiNb5 O27 have two temperature ranges in which the activation energy E a differs significantly in value. At low temperatures, electrical conductivity is determined primarily by impurity defects with very low activation
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energy of the order of several hundredths of an electron volt. For Bi7 Sr2 TiNb5 O27 , we observe a region with pronounced impurity conductivity in the temperature range from 20 to 400 °C. Figure 5 shows the dependence of the logarithm of conductivity ln σ on temperature. When doping Bi7 Sr2 TiNb5 O27 , the activation energy E a remains virtually unchanged in the low- and high-temperature regions, as follows from Table 2. The constancy of the activation energy indicates the constancy of the nature of conductivity in the synthesized compound.
Fig. 5. Dependence of ln σ on 10000/T for the sample Bi7 Sr2 TiNb5 O27 at a frequency of 100 kHz.
Fig. 6. Dependence of d 33 vs thermal depolarization temperature.
The values of piezoelectric modulus (d 33 ) of Bi7 Sr2 TiNb5 O27 ceramic in dependence on thermal depolarization temperature are shown in Fig. 6. The values of d 33 increase first and then decrease with a thermal depolarization. The d 33 of Bi7 Sr2 TiNb5 O27 ceramics has an optimal value of 14 pC/N at a depolarization temperature of 350 °C, which is
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not much more, than for pure Bi2 SrNb2 O9 . It has been reported that the piezoelectric properties can be attributed to the crystal distortion, caused by atomic substitution. The crystal distortions of the (Ti, Nb)O6 octahedron and (Bi2 O2 )2+ layer become larger with increasing Bi3+ ion values. This result implies that replacement of strontium ions Sr2+ leads to larger crystal distortion, where an enhanced piezoelectric modulus is obtained. Moreover, it has been reported that larger grain size makes domain motion easier [15]. The Bi7 Sr2 TiNb5 O27 ceramic has a larger piezoelectric modulus because of the larger grain size shown in Fig. 2. The d 33 values of Bi7 Sr2 TiNb5 O27 compound decrease rapidly above the Curie temperature T C because of the ferroelectric-paraelectric phase transition. At low-annealing temperature regions, the d 33 values have a slightly increasing trend. The Bi7 Sr2 TiNb5 O27 ceramic still retains a high piezoelectric modulus (d 33 = 11 pC/N) after annealing at 550 °C, showing the potential for application in high-temperature environments.
4 Conclusions Layered bismuth ceramic Bi7 Sr2 TiNb5 O27 was synthesized using a conventional solidstate reaction. The X-ray diffraction pattern shows that the sample has a single layered structure of the Aurivillius-Smolensky phase. The temperature of the paraelectric-ferroelectric phase transition for Bi7 Sr2 TiNb5 O27 has been determined. For Bi7 Sr2 TiNb5 O27 , the Curie temperature is T C = 555 °C. The optimal sintering temperature is determined as equal to T = 1050 °C. From SEM, it is clear grain growth has a characteristic preferred direction in ab-plane. A decrease in grain oxygen vacancies leads to a decrease in conductivity and, therefore, to a decrease in tan σ losses and an increasing in d 33 . The ceramic composition Bi7 Sr2 TiNb5 O27 demonstrates optimal electrical properties with a high d 33 value of 14 pC/N, a high T C value of 555 °C and a low tan σ value. Moreover, d 33 of Bi7 Sr2 TiNb5 O27 ceramics remains at 66% of the original value at annealing temperatures up to 550 °C. All these results indicate that Bi7 Sr2 TiNb5 O27 ceramics is a new lead-free piezoelectric material with high Curie temperature T C . Acknowledgements. The equipment of SFedU was used. The study was financially supported by the Russian Science Foundation (grant No. 21-19-00423) in the Southern Federal University.
References 1. Aurivillius, B.: Mixed bismuth oxides with layer lattices: I. Structure type of CaBi2 B2 O9 . Arkiv. Kemi. 1(54), 463–480 (1949) 2. Smolensky, G.A., Isupov, V.A., Agranovskaya, A.I.: Ferroelectrics of the oxygen-octahedral type with layered structure. Soviet Phys. Solid State 3, 651–655 (1961) 3. Subbarao, E.C.: Crystal chemistry of mixed bismuth oxides with layer-type structure. J. Am. Ceram. Soc. 45, 166–169 (1962) 4. Subbarao, E.C.: Ferroelectricity in mixed bismuth oxides with layered-type structure. J. Chem. Phys. 34, 695–696 (1961)
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5. Zubkov, S.V., Parinov, I.A., Nazarenko, A.V., Kuprina, Yu.A.: Crystal structure, microstructure, piezoelectric and dielectric properties of high-temperature piezoceramics Bi3-x Ndx Ti1.5 W0.5 O9 (x = 0, 0.1, 0.2). Phys. Solid State 64, 1475–1482 (2022) 6. Zubkov, S.V.: Crystal structure and dielectric properties of layered perovskite-like solid solution Bi3–x GdTiTaO9 (x = 0, 0.1, 0.2, 0.3) with high Curie temperature. In: Physics and Mechanics of New Materials and Their Applications. Abstracts & Schedule. Kitakyushu, Japan, pp. 317–318 (2021) 7. Zubkov, S.V., Vlasenko, V.G.: Crystal structure and dielectric properties of layered perovskitelike solid solutions Bi3−x Yx TiNbO9 (x = 0, 0.1, 0.2, 0.3) with high Curie temperature. Phys. Solid State 59, 2325–2330 (2017) 8. Zubkov, S.V., Shevtsova, S.I.: Crystal structure and dielectric properties of layered perovskitelike solid solutions Bi3−x Lux TiNbO9 (x = 0, 0.05, 0.1) with high Curie temperature. In: Parinov, I.A., Chang, S.-H., Long, B.T. (eds.) Advanced Materials. SPM, vol. 6, pp. 173–182. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45120-2_15 9. Zubkov, S.V.: Crystal structure and dielectric properties of layered perovskite-like solid solutions Bi3−x Gdx TiNbO9 (x = 0, 0.1, 0.2, 0.3) with high Curie temperature. J. Adv. Dielectrics 10, 2060002 (2020) 10. Isupov, V.A.: Crystal chemical aspects of the bismuth-containing layered compounds of the Am−1 Bi2 Bm O3m+3 type. Ferroelectrics 189, 211–227 (1996) 11. Vlasenko, V.G., Zubkov, S.V., Shuvaeva, V.A., Abdulvakhidov, K.G., Shevtsova, S.I.: Crystal structure and dielectric properties of Aurivillius phases A0.5 Bi4.5 B0.5 Ti3.5 O15 (A = Na, Ca, Sr, Pb; B = Cr, Co, Ni, Fe, Mn, Ga). Phy. Solid State 56, 1554–1560 (2014) 12. Zubkov, S.V., Parinov, I.A., Nazarenko, A.V., Kuprina, Y.A.: Microstructure, crystal structure, piezoelectric and dielectric properties of piezoceramic SrBi2 Nb2 O9 . In: Parinov, I.A., Chang, S.H., Soloviev, A.N. (eds.) Physics and Mechanics of New Materials and Their Applications. SPM, vol. 20, pp. 155–162. Springer, Cham (2023). https://doi.org/10.1007/978-3031-21572-8_13 13. Goldschmidt, V.M.: Geochemische Verteilungs gesetze der Elemente. J. Dybwad, Oslo, 1923 (1927) 14. Shannon, R.D.: Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Cryst. A32, 751–767 (1976) 15. Chen, H., Shen, B., Xu, J., Kong, L., Zhai, J.: Correlation between grain sizes and electrical properties of CaBi2 Nb2 O9 piezoelectric ceramics. J. Am. Ceram. Soc. 95(11), 3514–3518 (2012)
The Relationship of Electrical Conductivity, Morphology and Protective Properties of Organic Films with the Type of Inhibitor and Its Concentration E. N. Sidorenko1(B) , S. P. Shpanko1 , A. V. Nazarenko2 , M. A. Bunin3 , and A. V. Shloma1 1 Southern Federal University, 5, Zorge Street, Rostov-on-Don, Russia
[email protected]
2 Federal Research Center the Southern Scientific Center of the Russian Academy of Sciences,
41, Chekhov Avenue, Rostov-on-Don, Russia 3 Institute of Physics, Southern Federal University, 194, Stachki Avenue, Rostov-on-Don, Russia
Abstract. The protective effect of an organic compound of imidazole class on the corrosion of mild steel in sulfuric acid was studied. Modification of the organic base with bromine anions increases the protection effect. The blocking-activation mechanism of the protective effect of benzimidazole hydrobromide was determined by gravimetry, impedancesometry and polarization measurements. The interaction of adsorbed particles in the adsorption layer was found to be increased with increasing organic salt concentration. The features of the cellular microstructure of the films have been studied by scanning electron microscopy. AFM imaging study showed that the Fe coatings have a globular character and depend on the type of inhibitor and its concentration. It has been established that the most reliably protect Fe from corrosion are the coatings representing aggregates of large, accreted globules with maximum electrical conductivity. Keywords: Corrosion · Inhibitor · Adsorption · Isotherm · Morphology · Structure · Film · Electrical Conductivity · Microstructure
1 Introduction One of the most important tasks of modern industry is the development of research in the field of protection of metal materials from corrosion destruction. This is primarily due to the intensification of the most metal-intensive industries, as well as tougher operating conditions of metal products [1]. Inhibition of the corrosive environment with suitable compounds is one of the most versatile and economical methods of protection. The most important directions in the development of the inhibition theory are the study of the mechanism of protective action, as well as the physicochemical properties of protective films, depending on the inhibitor composition and experimental conditions [2, 3]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 275–284, 2024. https://doi.org/10.1007/978-3-031-52239-0_27
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Numerous studies have established that the most effective inhibitors are organic compounds, containing functional groups with nitrogen, sulfur, and oxygen atoms, as well as substances with unsaturated bonds [4]. Such compounds include nitrogenous heterocycles, the study of which has recently received quite a lot of attention [5]. Especially interesting are the studies aimed at investigating the morphology and electrical conductivity of organic films, obtained on metals in inhibited media [6, 7]. This is interesting, due to the organic materials have a set of features such as: structural disorder, molecular structure, the presence of conjugated chemical bonds in the molecules, a large set of traps, hopping character of conductivity, the value of intrinsic conductivity, close to the conductivity of semiconductors. It has already been found that the characteristics of surface films are determined largely by the nature and composition of the inhibitor, the individual features of the corroding metal and the reaction medium. Work in this direction continues. The aim of this study is to investigate the relationship between the inhibition mechanism, microstructure features and electrical conductivity of organic films on steel surface in sulfuric acid medium.
2 Experimental Results and Discussion The protective effect of organic films on the surface of steel samples in acidic environment has been studied. The surface morphology of films, obtained under different conditions, was studied by raster electron and scanning probe microscopy. Their static and dynamic electrical conductivities were measured. A new organic compound of imidazole class was investigated as an inhibitor. 2.1 Inhibition of the Steel Corrosion in Sulfuric Acid The benzimidazole derivative (surfactant, SAS), see Fig. 1 and its salt with bromine anion SAS•HBr were investigated for corrosion of St-3 steel in one molar solution of sulfuric acid.
Fig. 1. SAS scheme
The SAS concentration was 10–3 mol/l. SAS•HBr concentration was varied in the range of 1 × 10–3 –4 × 10–3 mol/l. Film formation time was 1–5 days.
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Since the area and time of corrosion tests of all samples are constant, the corrosion inhibition coefficient K is determined only by the mass loss of samples m: K = m0 /m
(1)
where m0 and m are the mass changes of samples in pure and in inhibited sulfuric acid, respectively. The degree of protection Z was determined as Z = [(K−1)/K] × 100%
(2)
The results of the experiment are summarized in Table 1. The surface activity of adsorption-type inhibitors is estimated by the decrease in the double electric layer capacitance (DEL) and the degree of shielding of the metal surface by the inhibitor components θ. The capacitance of the DEL was measured at 1 kHz by an AC bridge in a molecular hydrogen saturated electrolyte. The degree of shielding was calculated, based on the decrease in the capacitance of DEL in the presence of inhibitor C, compared to background C 0 (Table 1) as θ = [(C0 − C)/C0 ] × 100%
(3)
Table 1. Dependence of inhibition and adsorption parameters on the nature of the additive and SAS•HBr concentration Composition of the medium
K
Z, %
C/S, F/m2
θ, %
Background 1M H2SO4
–
–
0.442
–
SAS
4.5
77.8
0.426
3.5
1 × 10–3
8.1
87.7
0.374
15.5
2 × 10–3
11.9
91.6
0.368
16.7
3 × 10–3
13.4
92.5
0.353
20.1
4 × 10–3
16.9
94.1
0.326
26.2
SAS•HBr C SAS·HBr , mol/l
Polarization tests were carried out on degreased electrodes by galvanostatic method. A platinum plate served as an antielectrode, and a silver chloride electrode was used as a reference electrode (Table 2). It was found that the quantitative protection indices K and Z depend on the nature of the inhibitor and increase in the transition from the organic base to its salt the higher the salt concentration (Table 1). Capacitive and electrochemical dependencies under similar conditions were measured to interpret the obtained results. According to literature data [4, 8], the mechanism of inhibition of metal corrosion in acidic media is mainly due to the blocking component (θ-factor) and changes in the structure of DEL (energetic or ψ-factor). According
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to impedance measurements, the transition from the organic base of the surfactant to its salt SAS•HBr increases the blocking effect by more than 4 times. The increase in the concentration of benzimidazole bromide is also accompanied by an increase in the degree of filling of the steel surface with organic surfactant cations. Earlier studies have established that inorganic anions adsorbed on the positive surface of iron play the role of anionic bridges between the metal surface and organic salt cations [5]. An increase in the concentration of C SAS•HBr leads to a stronger shielding effect, which is accompanied by an increase in the degree of protection. Different empirical adsorption isotherms are traditionally used to describe adsorption phenomena [4, 9, 10]. The results of graphical processing of capacitive measurements in the coordinates of the corresponding adsorption isotherms are given in Table 2. Table 2. Correlation coefficients r in coordinates of the performance of the corresponding adsorption isotherms. The inhibitor is SAS•HBr Isotherms
Langmuir
Temkin
Isotherm coordinates
θ /(1 − θ ) = ƒ (C SAS•HBr )
θ = ƒ(lg C SAS•HBr )
r
0.9463
0.8804
Isotherms
Freundlich
Frumkin
Isotherm coordinates
lgθ = ƒ (lg C SAS•HBr )
lg(1 − θ)C SAS•HB /θ] = ƒ(θ)
r
0.9064
0.9705
It was found that the adsorption of benzimidazole hydrobromide on the steel surface obeys the Frumkin adsorption isotherm, which describes adsorption on a uniform surface taking into account the interaction of adsorbed particles in the adsorption layer (Fig. 2, Table 2). According to the obtained experiment, the attractive interaction between organic surfactant cations and bromine anions is mainly realized in the adsorption layer, increasing with increasing salt concentration. Additional information on the inhibition mechanism was obtained from electrochemical measurements (Table 2, Fig. 3). Figure 3 shows the cathodic and anodic polarization curves of steel in the background solution and in the presence of the investigated inhibitors. It can be seen from the figure that as the inhibitor concentration increases, the cathode polarization curves naturally shift to the negative potential region (E), increasing the polarization of the cathode reaction (E k ). The polarization of the anodic reaction does not change. Thus, the investigated organic compound is a cathodic-type inhibitor. This is also evidenced by the shift of the corrosion potential E cor to the region of negative potential values compared to the background (Table 3). The tangents of the inclination angles of the Tafel sections of the cathode polarization curves bk and the polarization of the electrode in the cathode potential region E k (i = 8 × 10−3 A/cm2 ) were calculated for the studied systems (Table 3). In the presence of inhibitors, both these parameters are larger than those with background and they increase when the SAS is replaced by SAS•HBr and further with increasing salt concentration.
The Relationship of Electrical Conductivity, Morphology
Fig. 2. Froomkin adsorption isotherm. The inhibitor is SAS•HBr
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Fig. 3. Cathodic (1 – 5) and anodic (1* – 5*) polarization curves of Fe electrode in pure (1, 1*) and inhibited H2 SO4 in the presence of SAS (2, 2*) and at different SAS•HBr concentrations, mol/l: 3, 3* − 10–3 ; 4, 4* − 2 × 10–3 ; 5, 5* − 4 × 10–3
Table 3. Dependence of electrochemical parameters on the nature and concentration of the inhibitor Composition of the medium
E cor , V
bk
E k , V i = 8 × 10−3 A/cm2
1M H2SO4
−0.374
0.088
–
SAS
−0.383
0.11
0.057
−0.385
0.133
0.086
SAS•HBr C SAS•HBr , mol/l 1 × 10–3 2 × 10–3
−0.385
0.153
0.132
4 × 10–3
−0.384
0.157
0.159
Thus, it has been established that the protective properties of the studied inhibiting systems are related both to the mechanical shielding of the steel surface from the corrosive medium and to the change in the structure of DEL, which manifests itself in a change in the electrochemical parameters of the corrosion process. 2.2 Microstructure of the Investigated Films To elucidate the reasons for changes in the protective characteristics of the films, the microstructure of their surface was investigated under similar conditions. The scanning electron microscopy (SEM) method on a Carl Zeiss EVO 40 instrument was used for this purpose. Raster images of the film surfaces were obtained using an Everhart-Thornley SE secondary electron detector at an accelerating voltage of EHT. Figure 4 shows SEM images of the surface of the formed films obtained under different conditions. Typical
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of the morphology of all surfaces is a cellular (or grain) structure, expressed to varying degrees, which has been observed previously in similar films [7, 11]. The cells are usually oblong in shape and separated by narrow channels with a width of 0.4–0.7 µm. Figure 4.1 shows a close-up of one such cell. This film (C SAS•HBr = 1 × 10–3 mol/l) is characterized by a crushed structure with a cell size of 2−8 µm with the presence of larger cells up to 15−20 µm long.
Fig. 4. SEM surface images of samples obtained in sulfuric acid electrolyte in the presence of SAS•HBr salt of different concentrations of C SAS•HBr , mol/l: (1) 1 × 10–3 , (2) 2 × 10–3 , (3) 3 × 10–3 , (4) 4 × 10–3 and SAS base C SAS = 10–3 mol/l: 10 µm marker is at 5a, 2 µm marker is at 5b; t = 1 day.
With increasing concentration of C SAS·HBr , films with larger cells are formed, apparently, with a more perfect structure. The cells become larger and their length reaches 30–50 µm (Fig. 4.2, 4.3). The microstructure of SAS films is also of the cellular type (Fig. 4.5a). However, the channels separating the grains are not constant in width, are wider (1.7–3 µm) and have breaks. The cell surfaces look heterogeneous, with a large set of pores and high roughness. This is clearly seen in Fig. 4.5b, when the surface of the same film is viewed with a high magnification microscope, where numerous volumetric micro formations are observed. The surface morphology of the inhibited samples with different adsorption film formation time t was further investigated by scanning probe microscopy. Two-dimensional (2D) and three-dimensional (3D) images of the surface were obtained using a Veeco Multimode VS scanning probe microscope in contact mode using standard techniques, probe NP-10, k = 0.32 N/m. The obtained scans 40 × 40 µm of films with adsorption time t = 1 and 3 days (C SAS = 10–3 mol/l) are present in Fig. 5. The surface of the films looks different, but individual regions-cells with blurred boundaries are also observed. At higher magnification, individual fragments (Figs. 6a, 6b) show that the entire surface of the substrate is covered with globules of various sizes of columnar shape and their embryos.
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(a)
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(b)
Fig. 5. 2D and 3D AFM images of the surface of samples (40 × 40 µm2 ) as a function of inhibition time t: (a) 1 day, (b) 3 day; C SAS = 10–3 mol/l
(a)
(b)
(c)
Fig. 6. Single fragments (10 × 15 µm2 ) of 3D AFM images of the surface of SAS samples, C SAS = 10–3 mol/l, at inhibition time t: (a) 1 day, (b) 3 day, (c) 5 day (SAS + 0.03 KCl)
The size of the globules increases with increasing t. On the films, obtained with application of composite inhibitor for 5 days, a coating of large aggregates of fused globules of crystalline nature is observed (Fig. 6c). It is such coatings that more reliably protect the metal from corrosion.
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2.3 Electrical Conductivity of Protective Films Since the films obtained under different conditions have individual local structure, their electrical conductivity is different. To determine the values of static resistance (R) and conductivity (G) of the obtained films and to clarify the conduction mechanisms, their voltammetric characteristics (VACs) were measured. The voltage from a regulated stabilized source was applied normal to the film surface. The surface area of the samples was equal to 1 cm2 . Typical VACs of the films taken at low mechanical stress (P = 104 Pa) are shown in Fig. 7. As for other similar organic films with disordered structure [3, 12], a jump-like character of I(U) dependences is observed in the obtained VACs of all films. This feature of the VAC indicates a series of “soft” microbreakdowns, occurring between regions of increased electrical conductivity in the films [13, 14] and a hopping type of electrical conductivity of the films, independent of the type and concentration of the inhibitor.
Fig. 7. Voltammetric characteristics of films, obtained: (1) by adsorption of SAS and at different concentrations of SAS•HBr C SAS•HBr , mol/l: (2) 1×10-3 and (3) 4×10-3 ; t = 1 day, P = 104 Pa.
Fig. 8. Dependences of conductivity of films, obtained by adsorption of SAS: (1) and at different concentration of SAS•HBr C SAS•HBr , mol/l : (2) 1×10-3 and (3) 4×10-3 on the value of uniaxial pressure; t = 1 day.
The average values of static conductivity G were determined from the VAC: G = I/U. SAS films have underestimated values of G = 0.3 S, whereas conductivity of SAS•HBr films are larger and equal to 2 and ~ 3 S. Moreover, the conductivity of the films increases with the growth of CSAS•HBr . That is, by changing the conditions of film growth, it is possible to obtain films with different specific conductivity. Thus, the organic coatings on Fe with the lowest structural disorder, with the highest degree of protection and corrosion inhibition coefficient have the highest conductivity. The static conductivities of the films coincide with their low-frequency dynamic conductivities and depend significantly on the applied mechanical stress. Figure 8 shows the dependences of the dynamic conductivity of the investigated films at a frequency of 1 kHz on the magnitude of the applied uniaxial compression. With increasing pressure P, the conductivity of the films grows, apparently, due to the increase in the charge concentration at their deformations. The reverse course of the G(P) dependence does not coincide with the direct one. After depressurization, the conductivity decreases only
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by 20–30% relative to the maximum values due to residual deformations of the films. The conductivity of the films is restored after 2–3 h.
3 Conclusion Thus, by the methods of gravimetry, impedansometry and electrochemical measurements, it was established that the protective properties of the studied inhibiting systems are related both to the mechanical shielding of the steel surface obeying the Frumkin adsorption isotherm and to the change in the structure of DEL. The cellular and globular microstructure of organic films was determined by raster electron and scanning probe microscopy. Its parameters depend on the type of inhibitor, its concentration and time of film formation. Films with high electrical conductivity and consisting of large aggregates of fused globules have the greatest protective properties.
References 1. Semenova, I.V., Florianovich, G.M., Khoroshilov, A.V.: Corrosion and Corrosion Protection, 3rd edn. Physmatlit, Moscow (2010) 2. Kuznetsov, V.: Organic Ingibitors of Corrosion of Metals. Plenum Publishing Corporation, New York (1996) 3. Shpanko, S.P., Sidorenko, E.N., Kuznetsova, L.E., Sosin, E.A.: In: Parinov, I.A., Chang, S.-H., Kim, Y.-H. (eds.) Proceedings of the 2018 International Conference on «Physics, Mechanics of New Materials and Their Applications», vol. 224, pp. 123–130. Springer, Cham (2019) 4. Reshetnikov, S.M.: Inhibitors of acid corrosion of metals. Chemistry, Leningrad (1988) 5. Shpanko, S.P., Sidorenko, E.N., Abdulvakhidov, K.G., Grineva, D.A.: In: Parinov, I.A., Chang, S.-H., Long, B.T. (eds.) Proceedings of the 2019 International Conference on «Physics, Mechanics of New Materials and Their Applications», pp. 137–146. Nova Science Publishers, New York (2020) 6. Bogatin, A.S., Sidorenko, E.N., Shpanko, S.P., Kovrigina, S.A., Abdulvakhidov, K.G., Nosatschev, I.O.: In: Parinov, I.A., Chang, S.-H., Kim, Y.-H., Noda, N.-A. (eds.) Proceedings of the 2020 International Conference on «Physics, Mechanics of New Materials and Their Applications», vol. 10, pp. 223–232. Springer, Cham (2021) 7. Sidorenko, E.N., Bunin, M.A., Shpanko, S.P.: In: Parinov, I.A., Chang, S.-H., Soloviev, A.N. (eds.) Physics and Mechanics of New Materials and Their Applications - Proceedings of the International Conference PHENMA 2021–2022, vol. 20, pp. 232–240. Springer, Cham (2023) 8. Antropov, L.I.: Theoretical Electrochemistry, 4th edn. Higher School, Moscow (1984) 9. Shpanko, S.P., Sidorenko, E.N.: Composite inhibitors for acid corrosion of mild steel. In: Proceedings of II International Conference on Actual Issues of Electrochemistry, Ecology and Corrosion Protection, pp. 210–214. Chesnokova Publishing House, Tambov (2021) 10. Damaskin, B.B., Petri, O.A., Batrakov, V.V.: Adsorption of Organic Compounds on Electrodes. Nauka, Moscow (1968) 11. Sidorenko, E.N., Shpanko, S.P., Bunin, M.A., Debelova, T.I.: In: Parinov, I.A., Chang, S.-H., Long, B.T. (eds.) Proceedings of the 2019 International Conference on «Physics, Mechanics of New Materials and Their Applications», pp. 129–136. Nova Science Publishers, New York (2020)
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12. Shpanko, S.P., Semenchev, A.F., Lyanguzov, N.V., Anisimova, V.A.: Anticorrosive properties and physicochemical characteristics of organic films based on heterocyclic compound of imidazole class. Surface Physicochemistry Prot. Mater. 53(2), 210–217 (2017) 13. Vlasov, D.V., Apresyan, L.A.: Cascade model of conduction instability and giant fluctuations in polymer materials and nanocomposites. Am. J. Mater. Sci. Adv. Mater. Sci. Appl. 2(2), 60–65 (2013) 14. Sidorenko, E.N., Shpanko, S.P., Prikhodko, G.I.: Electrical conductivity of anticorrosive adsorption films. Izvestiya RAN. Ser. Phys. 80(11), 1547–1549 (2016)
Negative Electrical Capacitance of the Organic Anticorrosion Films E. N. Sidorenko1(B) , S. P. Shpanko1 , A. V. Shloma1 , A. G. Rudskaya1 , and A. O. Galatova2 1 Southern Federal University, Zorge 5, Rostov-on-Don, Russia
[email protected] 2 Institute of Physics, Southern Federal University, Stachki 194, Rostov-on-Don, Russia
Abstract. Experimental data on the dielectric spectra of the organic films were studied and analyzed. An equivalent circuit consisting of two series-parallel RCcircuits is used for computer modeling and interpretation of experimental data of films with negative electrical capacitance in the high-frequency range. The critical frequency of changing the sign of the electrical capacitance has been determined. The negative electrical capacitance of the films under study is due to two factors: high-frequency processes of strong relaxation polarization with inverse electric field strength and low-frequency processes of hopping-type conductivity. Keywords: Film · Negative Electrical Capacitance · Conductivity · Dispersion · Inhibitor
1 Introduction Organic films have recently attracted much attention due to the possibilities of their practical application for protecting metal structures from corrosion and as a building material for the fabrication of circuit elements and functional units of molecular electronics [1, 2]. The high protective properties and negative electrical capacitance of the films arouse increased scientific interest to them [3]. Negative electrical capacitance has been discovered in various materials such as ferroelectrics, semiconductors, organic compounds, composites and others. Several models have been proposed to explain the nature of negative electrical capacitance [4–6]. Various variants of their practical application are considered. On their basis, for example, transistors with ultra-low power consumption can be obtained, due to which the power consumption of all modern digital circuits will be drastically reduced [7, 8]. However, the issue has not been fully studied and work in this direction continues due to the promising use of the materials with negative electrical capacitance.
2 Problem Formulation The aim of this study is to prove the negativity of the electrical capacitance of organic anticorrosion films in a wide frequency range. Moreover, we perform computer modeling of the films with negative electrical capacity, using an equivalent circuit, consisting of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 285–294, 2024. https://doi.org/10.1007/978-3-031-52239-0_28
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two series-parallel RC circuits, interpret the obtained experimental data and compare them with the calculated ones.
3 Experimental Results and Discussion 3.1 Film Samples Organic films were obtained on the surface of rolled low-alloy steel by adsorption method from inhibited sulfuric or hydrochloric acid solution. The inhibitor was an organic compound derived from benzimidazole (surface-active substance – SAS). Two types of inhibitors were used for the study (Fig. 1). Their concentration was maintained constant and equal to 10−3 mol/l. Combined inhibitors with different concentrations of inorganic additives were also used for the study. The adsorption time of the inhibitors was also a variable factor. OH N NH N
Cl Cl 2HCl
N
SAS(1)
SAS(2)
Fig. 1. Schemes of the SAS inhibitors
The samples with silver electrodes had dimensions of 1 × 1 cm2 . The thickness of the films was several tens of microns. The protective anticorrosion properties improved with increasing adsorption time when inorganic compounds were added to SAS. The surfaces of the films have a cellular structure. The cells are oblong in shape, up to 20– 50 µm long, separated by channels 2–5 µm wide. The more perfect the cellular structure, the greater the protective effect of the film. Studies of the film morphologies using the AFM method showed that the films have the crystalline forms of the fused cone-shaped globules. The static and dynamic conductivities ( f = 1 kHz) of the films, measured in the transverse direction, are equal to 0.1–5 S and depend on the external pressure P. The electrical capacitance of the films is also sensitive to the influence of P and is negative in the frequency range under study. 3.2 Negative Electrical Capacitance of the Organic Films To prove the existence of the materials with negative capacitance, work [9] shows schemes of series connection of a resistor or industrial capacitor with a sample having negative capacitance. Voltage was applied to the circuit input and the voltages on the
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circuit elements were examined. These voltages were distributed specifically, in accordance with the presence of a circuit element with negative electrical capacitance in the circuit. In this work, with a similar goal, to show the negative sign of the capacitance of the studied anticorrosion films, another simple experiment is proposed. Capacitance measurements of a set of industrial capacitors were previously made at three different frequencies, and then the capacitances of circuits, consisting of a film sample and each capacitor connected in parallel, were measured at the same frequencies. The electrical capacitance of the film, which has a negative sign, is determined graphically at each frequency. All capacitance measurements were performed using an E7-30 Immittance Meter at room temperature. The organic film on the surface of steel St-3, obtained in a HCl solution with a combined inhibitor SAS (2) + KBr, was studied. The concentrations of SAS (2) and KBr in the combined inhibitor are equal, respectively: C SAS (2) = 10−3 mol/l and C KBr = 0.3 mol/l. The time for obtaining an organic film is one day. A typical frequency dependence of the film capacitance was previously obtained (Fig. 2). Therefore, at frequencies of 25, 50 and 105 Hz, the film capacitance is (−163), (−27) and (–2) µF, respectively. At the same frequencies, the capacitances of a set of industrial capacitors C C with different values of their capacitances were measured. The graphs of the dependences of the capacitance C of the circuits with various capacitors on the capacitance of the capacitors C C at three frequencies are presented in Fig. 3. C is the algebraic sum of the film capacitance and the capacitance of the parallel connected capacitor.
Fig. 2. Frequency dependence of the capacitance of the SAS (2) + 0.3 KBr film sample, t = 1 day.
Fig. 3. Dependences of the total capacitance of the parallel circuits of the capacitor and the SAS (2) + 0.3 KBr (C ) film on the capacitance of the capacitors (C C ). Measuring field frequency: (1) 25 Hz, (2) 100 Hz and (3) 105 Hz, t = 1 day.
In the upper half-plane of Fig. 3, C > 0 and the capacitances of the capacitors C C are larger than the modulus of the film capacitance C f . In the lower half-plane C < 0 and the modulus of the film capacitance C f is greater than the capacitance of the capacitors. At C = 0, C f = −C C . Consequently, from the points of intersection of the graphs with the abscissa axis, it is possible to determine the values of the film capacitance modules at different frequencies. The obtained results of film capacitances practically coincide with the results of direct measurements of C f .
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Thus, the fact of intersection of C (C C ) dependences with the abscissa axis do not give grounds to doubt the negativity of the capacity of the investigated anticorrosion films. 3.3 Analysis of the Equations of Frequency Dependences C (ω) and G (ω) Using the Impedance Representation of the Relaxation Polarization of the Films with Negative Capacitance To describe the relaxation polarization of the films with negative capacitance in an inverse field relative to the external field, an equivalent electrical circuit of the film was used (Fig. 4), which was also used earlier in [10].
Fig. 4. Equivalent circuit of the film to describe the relaxation polarization process at an internal electric field strength of inverse direction.
The presence of the resistor R1 and the capacitor C 1 , connected in parallel in the circuit is due to the existence in the film of through conductivity and high-frequency polarization (for example, ion displacement polarization), respectively. Resistor R2 and capacitor C 2 of negative capacitance, connected in series, are necessary to describe the relaxation polarization. In this case, the frequency dependence of the real part C of the complex electrical capacitance has the form [10]: C =
(ωC2 R2 )2 C1 + C1 −C 2 1 + (ωC2 R2 )2
(1)
The analysis of formula (1) shows that C → (C 1 − C 2 ) for ω → 0, and C → −C 2 for C 1 105 rad/s), where C decreases by more than two orders of magnitude with increasing frequency.
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However, in the low frequency range (ω < 105 rad/s), the value of C is equal to a constant value (Fig. 8, curve 1). Such characteristics are typical for the films, obtained using a SAS (1) inhibitor [11] or a SAS (1) inhibitor, mixed with KCl or thiourea [12, 13].
Fig. 8. Frequency dependences of the capacitance C (a) and the imaginary part of the complex conductivity G (b): points correspond to experiment; lines correspond to theoretical calculation for the film experiencing different pressures P: (1) 3 × 104 Pa; (2) 7 × 104 Pa; (3) 1.7 × 105 Pa; Inhibitor: SAS (1) + KCl.
For the film, obtained in the presence of the combination inhibitor SAS (1) + KCl, and an adsorption time of 3 days, at different uniaxial pressures, the experimental results are presented as points in Fig. 8. Here, the solid lines show the theoretically calculated dependences, which practically coincide with the experimental ones. The calculation was performed with C 2 and R2 equal to: 1.2 × 10−5 F and 0.27 – curve 1; 2.5 × 10−5 F and 0.185 – curve 2; 3.7 × 10–5 F and 0.16 – curve 3. Here C 1 = 10−9 F for all C 2 and R2 pairs. The plots of the dependences G (ω) in the frequency range ω = (3 – 5) × 107 rad/s cross the abscissa axis (Figs. 7 and 8b). This frequency is called critical ωk [5], at which the signs of C and G change. Starting from this frequency, in the region ω < ωk , due to the contribution of the relaxation polarization processes with inverse electric field strength: C < 0 and G < 0. All frequency dependences of the C (ω) films, obtained in the presence of SAS (2) are different from those, described above for SAS (1). As in the previous case, for the films, obtained with the combination inhibitor SAS (2)•HBr + 0.02KBr, the C (ω) dependences (Fig. 9a) also show a multiple increase in capacitance with a decrease in frequency in the high-frequency region. However, in the low frequency region (ω < (2 – 3) × 103 rad/s) the value of capacitance C is not constant and increases tenfold with decreasing ω (Fig. 9a). These changes in C can be considered as a manifestation of infra-low-frequency (ILF) dispersion. Computer modeling was performed at the following values of C 2 and R2 : 7 × 10−6 F and 1.0 – curve 1; 5 × 10−5 F and 0.4 – curve 2; 1.5 × 10−4 F and 0.25 – curve 3. Here C 1 = 10−9 F for all C 2 and R2 pairs. In the frequency range ω > (2 – 6) × 103 rad/s, the experimental results almost completely coincide with the theoretically calculated ones.
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Fig. 9. Frequency dependences of the capacitance C (a) and the imaginary part of the complex conductivity G (b): points correspond to experiment; lines correspond to theoretical calculation for the film experiencing different pressures P; P is equal to: (1) 3 × 104 Pa; (2) 7 × 104 Pa; (3) 1.5 × 105 Pa. Combination inhibitor is SAS (2)•HBr + 0.02KBr.
Previously [6], ILF dispersion was observed in the frequency region of (0.1 – 1) kHz in the study of C70 fullerite films having negative dielectric permittivity (ε0 < − 1000) in the temperature range of 170–270 K. The authors explain the negative sign of the capacitance and the dispersion of the dielectric permittivity by the influence of the Lorentz correction to the local field, acting on the conduction electrons. The authors of [5] believe that the transition in the CaCu3 Ti4 O12 ceramics samples from gigantic positive values of dielectric permittivity (or capacitance) to negative values in the region of infralow frequencies ( f < 10−2 Hz) is caused by the hopping nature of the conductivity of ceramics with a disordered structure. To simulate and interpret experimental data in this frequency range, the authors used a parallel RC-circuit. The organic films under study are also disordered heterogeneous systems. They are also characterized by hopping-type conductivity [14]. Their current-voltage characteristics contain many kinks and S-shaped sections [15]. The fragments of polymer chains in organic materials represent regions of the electronic states localization. Electronic transport of such materials is based on the theory of hopping conduction with a variable hop length between localization centers (traps) with random energies [16, 17]. Therefore, the fact that the negative capacitance values of the investigated films increase in the region of ILF (Fig. 9a) can be considered as a contribution of hopping-type conductivity due to filling and emptying of traps. In the absence of highly developed relaxation processes in the high-frequency range in these films, the critical frequency of changing the signs of the electrical capacitance would be observed in the region of low – infra-low frequencies, as well as for CaCu3 Ti4 O12 ceramics [6] or fullerite films [14]. In the films, obtained using the SAS (1) compound, the contribution of hopping conductivity is not observed in the frequency range investigated, but appears to be similarly present in the lower frequency region. In the frequency range f > 20 Hz, it is observed on the same films only, if they are exposed to uniaxial mechanical pressure P and increases with increasing P due to a shift in the center of the ILF dispersion [18].
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4 Conclusion Thus, the obtained results indicate that the high values of negative electrical capacitance of the investigated films are due to high-frequency processes of strong relaxation polarization in internal fields with inverse strength and low-frequency hopping-type conduction processes. Acknowledgements. The work was supported by the Ministry of Science and Higher Education of the Russian Federation; the state task in the field of scientific activity No. FENW-2022-0001.
References 1. Kuznetsov, V.: Organic Ingibitors of Corrosion of Metals. Plenum Press, New York and London (1996) 2. Plotnikov, G.S., Zaitsev, V.B.: Physical Foundations of Molecular Electronics. Moscow State University, Moscow (2000) 3. Shulman, J., Xue, Y.Y., Tsui, S., Chen, F., Chu, C.W.: General mechanism for negative capacitance phenomena. Phys. Rev. B 80, 134202 (2009) 4. Yadav, A.K., Nguyen, K.X., Hong, Z., et al.: Spatially resolved steady-state negative capacitance. Nature 565, 468–471 (2019) 5. Turik, A.V., Bogatin, A.S.: Non-Debye relaxation and resonance phenomena in dielectric spectra of CaCu3 Ti4 O12 family functional ceramic materials. Funct. Mater. Lett. (Singapore) 8(4), 155003(1–4) (2015) 6. Makarov, V.V., Sherman, A.B.: Low-frequency dispersion of the negative dielectric permittivity in C70 films. Phys. Solid State 44(11), 2101–2105 (2002) 7. Íñiguez, J., Zubko, P., Luk’yanchuk, I., Cano, A.: Antiferroelectric negative capacitance from a structural phase transition in zirconia. Nat. Rev. Mater. 4, 243–256 (2019) 8. Malvika, C.B., Mummaneni, K.: A review on a negative capacitance field-effect transistor for low-power applications. J. Electron. Mater. 51, 923–937 (2022) 9. Alam, M.A., Si, M., Ye, P.D.: A critical review of recent progress on negative capacitance field-effect transistors. Appl. Phys. Lett. 114, 090401(1–5) (2019) 10. Bogatin, A.S., Sidorenko, E.N., Shpanko, S.P., Kovrigina, S.A., Abdulvakhidov, K.G., Nosatschev, I.O.: Physics and Mechanics of New Materials and Their Applications. In: Parinov, I.A., Chang, S.-H., Kim, Y.-H., Noda, N-A. (eds.) Proceedings of the International Conference, PHENMA 2020, vol. 225, pp. 203–213. Springer, Cham (2021) 11. Sidorenko, E.N., Shpanko, S.P., Kabirov, Yu.V., Gavrilyatchenko, V.G., Volkov, A.V.: Physics and Mechanics of New Materials and Their Applications. In: Parinov, I.A., Chang, S.-H., Jani, M.A. (eds.) Proceedings of the International Conference, PHENMA 2016, pp. 353–360. Nova Science Publishers, New York (2017) 12. Shpanko, S.P., et al.: Physics and Mechanics of New Materials and Their Applications. In: Parinov, I.A., Chang, S.-H., Jani, M.A. (eds.) Proceedings of the International Conference, PHENMA 2016, pp. 345–352. Nova Science Publishers, New York (2017) 13. Shpanko, S.P., Sidorenko, E.N., Kuznetsova, K.G., Abdulvakhidov, D.C., Obuhov, D.S.: Physics and Mechanics of New Materials and Their Applications. In: Parinov, I.A., Chang, S.H., Kim, Y.-H. (eds.) Proceedings of the International Conference, PHENMA 2018, pp. 13–19. Nova Science Publishers, New York (2019) 14. Sudar, N.T.: Physical Foundations of Molecular Electronics. St. Petersburg State Polytechnic University, St. Petersburg (2011)
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15. Shpanko, S.P., Sidorenko, E.N., Kuznetsova, L.E., Sosin, E.A.: Physics and Mechanics of New Materials and Their Applications. In: Parinov, I.A., Chang, S.-H., Kim, Y.-H. (eds.) Proceedings of the International Conference, PHENMA 2018, vol. 224, pp. 123–130. Springer, Cham (2019) 16. Kapralova, V.M.: Physics of Macromolecules. St. Petersburg State. Polytechnic University, St. Petersburg (2012) 17. Venediktov, V.A., Zvyagin, I.P.: Bulletin of Moscow University, Moscow. Series 3. Physics. Astronomy, vol. 1, pp. 89–92 (2011) 18. Shpanko, S.P., Sidorenko, E.N., Abdulvakhidov, K.G., Grineva, D.A.: Physics and Mechanics of New Materials and Their Applications. In: Parinov, I.A., Long, B.T., Chang, S.H., Wang, H.Y. (eds.) Proceedings of the International Conference, PHENMA 2019, pp. 137–146. Nova Science Publishers, New York (2020)
Broadband Optical Limiting of Single-Walled Carbon Nanotubes with Tetracarboxy-Substituted Phthalocyanine Ligand Composite M. S. Savelyev1,2(B)
, P. N. Vasilevsky2 , L. P. Ichkitidze1,2 , A. Yu. Tolbin3 and A. Yu. Gerasimenko1,2
,
1 Sechenov First Moscow State Medical University, 8-2, Trubetskaya St., Moscow 119991,
Russia [email protected] 2 National Research University of Electronic Technology, 1, Shokin Sq. 1, Zelenograd, Moscow 124498, Russia 3 Institute of Physiologically Active Compounds at Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry RAS, 1, Severnyi Proezd, Chernogolovka 142432, Russia
Abstract. The broadband optical limiting of single-walled carbon nanotubes (SWCNTs) with tetracarboxy-substituted phthalocyanine ligand composite was assessed. For this purpose, optical studies of the spectra and nonlinear response were carried out. The value of the normalized transmittance in experiments on optical limiting, provided with the method of a fixed absorber arrangement, has been evaluated. The description of the nonlinear change in the transmitted energy fluence was carried out using the radiative transfer equation (RTE) for the case of a threshold dependence of the absorption coefficient on intensity. The determination of the nonlinear absorption coefficient and the threshold total pulse energy was performed on the optical limiting measurement data for a pulse. Knowing these parameters, the attenuation of the pulse was simulated by two methods: Z-scan with an open aperture and when the absorber was immovable at the focus of the lens. Keywords: Ultraviolet (UV) · Optical Limiting · Phthalocyanine · Composite · Nonlinear Absorption · Carbon Nanotubes (CNTs)
1 Introduction When developing laser protection means, one should ensure the attenuation of such radiation in the entire visible wavelength range while ensuring the safety of vision. Laser radiation limiters based on nonlinear optical effects are irreplaceable in case of danger of exposure to pulses with a duration of less than 1 μs [1] when it is necessary to maintain the functioning of the device at the same wavelength. At certain values of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 295–302, 2024. https://doi.org/10.1007/978-3-031-52239-0_29
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total energy, such pulses have the ability to influence the structures of eye tissues up to irreversible changes in biological tissues [2] or damage detection systems based on sensitive light detectors [3]. To evaluate the possibility of using materials as part of broadband laser radiation limiters, based on single-walled carbon nanotubes (SWCNTs) with tetracarboxy-substituted phthalocyanine ligand (phthalocyanine@SWCNTs) composite, the present investigation was carried out at three wavelengths: 355, 532, and 1064 nm. When studying nonlinear optical properties, the special model was applied that takes into account the presence of a limiting threshold total pulse energy. To determine the optical parameters of the material, a rectangular pulse shape was used [4]. Unlike the case of a Gaussian pulse, the nonlinear absorption coefficient can be expressed explicitly.
2 Experimental Part 2.1 Materials 2,9(10),16(17),23(24)-tetrahydroxyphthalocyanine ligand [5] (25 mg) was added to a suspension of carboxylated SWCNTs (50 mg) in dimethylformamide (50 mL), and the mixture was subjected to ultrasonic treatment for 30 min. After that, xylylene dibromide (10 mg) and 1,8-Diazabicyclo(5.4.0)undec-7-ene (0.1 ml) were added. The mixture was boiled for 30 min, after which the residue was centrifuged and washed several times with tetrahydrofuran to remove unreacted oligomers. The structure of the phthalocyanine@SWCNTs composite is shown in Fig. 1; an ultrasonic homogenizer was used to prepare dispersions. Thus, the attachment of phthalocyanine molecules to the carboxylated nanotubes was performed during nucleophilic reaction of tetrahydroxyphthalocyanine with xylylene dibromide and functionalized carbon nanotubes. For the measurements, the concentration of all samples did not exceed 1/50 mg/ml. O
O O
O
O
N
N NH N
N HN N
O O
O
N
O
O
O O
SWCNTs Fig. 1. Chemical structure of composite.
The optical spectra of the samples were determined using UV-Vis-Nir spectroscopy (Thermo Fisher Scientific™) to monitor the optical density over a wavelength range that
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included all three wavelengths at which nonlinear optical limiting studies are conducted with the material. The control of the structure of carbon nanotubes before the manufacture of composite was controlled by Raman spectroscopy (LabRAM Horiba spectrometry complex). Measurements of nonlinear optical properties were carried out using a laser complex including an LS2147 Nd:YAG source with a third harmonic generator capable of generating radiation at three wavelengths: 355, 532, and 1064 nm. 2.2 Spectral Measurement of the Materials Prepared The optical density of dispersions of initial SWCNTs and ones after purification in water was determined using a spectrophotometer (Fig. 2). In accordance with these data, such materials slightly absorb light at wavelengths of 355, 532 and 1064 nm, at which the possibility of using the material to attenuate laser radiation when nonlinear optical effects occur was assessed.
Fig. 2. UV-Vis-NIR spectra of dispersions of initial and purified carbon nanotubes.
Raman spectroscopy is helpful for studying the structure of single-walled carbon nanotubes. Figure 3a shows the region of the RBM mode from 100 to 250 cm−1 ; in accordance with these data, the diameter of single-walled nanotubes is about 1.4 nm [6, 7]. The G-band, which is associated with optical vibrations of two adjacent carbon atoms in the nanotube lattice, has a clear doublet structure, which affects the presence of semiconducting SWCNTs (Fig. 3b). This band is split into longitudinal optical (LO, G+) tangential and transverse optical (TO, G−) tangential vibrational modes. In the spectra, the D band is small, which indicates a low level of defects in the nanotube under study. The I D /I G band intensity ratio decreased by 6.5 times after purification, indicating the effectiveness of purification. Cleaning from oxygen-containing groups can significantly increase the nonlinear optical response, so such groups must also be removed from the composite [4].
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Fig. 3. Raman spectra of SWCNTs: a – RBM mode region, b – TO and LO mode region and D – band.
2.3 Spectral Measurement of the Materials Prepared The nonlinear absorption coefficient β and the threshold total pulse energy U thr were determined using the radiation transport equation (RTE) [4, 8]. The setup for the experiment is shown in Fig. 4. In the case of a beam with a flat top, the nonlinear absorption coefficient is determined by the formula: √ π πτw02 z02 . (1) β= 2 Uthr z02 − U0 z02 − U0 z 2 d where w0 is the beam radius at the waist, U 0 is the incident total pulse energy, z0 is the Rayleigh length, z is the sample displacement relative to the lens focus, and d is the optical path length in the sample.
Fig. 4. Scheme of setup for studying optical limiting parameters.
The setup for measuring nonlinear optical effects includes a solid-state Nd:YAG crystal with generators for the second and third harmonics. To carry out measurements at three wavelengths, it was necessary to realize the possibility of following rays with different wavelengths along the same optical path. A motorized polarizer (MP) is used to control the incident total energy of a single pulse. Before entering the main optical path, the laser radiation of the fundamental and third harmonics hits an MP designed to operate at the appropriate wavelength, after which it enters the measurement circuit through an optical system including mirrors and flip-up mirrors. The second harmonic
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is directed using one mirror to the corresponding MP and, without the use of additional optical elements, is already started for research. To carry out measurements using Zscanning methods with an open aperture and determine the optical limiting, the setup includes two collecting lenses, between them there is a motorized slide on which the test sample is fixed.
3 Results and Discussion Figure 5 shows the results of studies of phthalocyanine@SWCNTs composite at the fundamental, second and third harmonics of the laser. As the laser wavelength increases from 355 to 1064 nm, the waist radius increases too, which required the use of increased total pulse energies compared to UV. According to the optical spectra and measurements carried out far from the waist, when nonlinear effects do not yet appear, the recorded values of linear absorption are given in Table 1. At a wavelength of 355 nm, the highest nonlinear absorption is also observed; for this reason, without destroying the walls of the quartz cell, it is possible to use large values of the incident total pulse energy when measuring the optical limiting of phthalocyanine@SWCNTs composite (Fig. 6).
Fig. 5. Results of measurements using the Z-scan method with an open aperture.
The most dangerous UV radiation, which corresponds to the highest photon energy, is greatly attenuated when it hits the material in a waist of ~1.5 J/cm2 . Similar normalized transmittance is achieved at ~2.5 J/cm2 and 17.5 J/cm2 for wavelengths of 532 and 1064 nm, respectively. To compare the limiting effectiveness, the previously created correlation models were used to calculate the corresponding descriptors. Thus, with regard to the aggregation of dispersions [9] the highest efficiency was calculated for the UV region: (2) σ1 = lg α−2 β−1 .
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Fig. 6. Alteration of transmittance on increasing the input fluence on studying optical limiting with composite of phthalocyanine@SWCNTs.
Table 1. Optical limiting parameters for composite of phthalocyanine@SWCNTs. λ, nm
U, μJ
d, mm
α, cm−1
w0 , μm
β, cm·GW−1
F th mJ/cm2
355 ± 1
100 ± 5
3.0 ± 0.1
1.8 ± 0.1
16 ± 1
150 ± 3
50 ± 10
532 ± 1
330 ± 30
3.0 ± 0.1
1.5 ± 0.1
19 ± 1
73 ± 3
100 ± 10
1064 ± 1
1690 ± 90
5.0 ± 0.1
0.9 ± 0.1
45 ± 1
42 ± 2
100 ± 10
The stronger σ1 is shifted to the negative numbers, the lesser does the aggregation affect the optical limiting. Applying the second model [10], which is based on correlation with quantum chemical nonlinearities, we obtain the same result: σ2 =
kA β , DR · Fth
(3)
in which k A is the attenuation coefficient, and DR is the dynamic range. The higher σ2 , the stronger the total limitation efficiency. In accordance with the results obtained, the attenuation coefficient for all wavelengths studied is over 6. The minimum dynamic range is 45 at a pulse duration of 20 ns (Table 2). It also follows that the efficiency of optical limiting with composites is generally higher when compared with simple dye solutions, based on data presented in [10].
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Table 2. Performance evaluation for composite of phthalocyanine@SWCNTs λ, nm
kA
DR
σ1 , a.u
σ2 , a.u
355 ± 1
6.4 ± 0.1
45 ± 4
2.7 ± 0.1
435 ± 25
532 ± 1
9.3 ± 0.1
71 ± 5
2.2 ± 0.1
96 ± 5
1064 ± 1
11.9 ± 0.1
581 ± 21
1.5 ± 0.1
9±1
4 Conclusion The present study demonstrated that phthalocyanine@SWCNTs composite is capable of exhibiting attenuation of laser radiation in a wide range of wavelengths, which is confirmed by the results of measurements at the fundamental, second and third harmonics of a neodymium laser. The threshold model for the case of a rectangular pulse shape was used to determine the nonlinear absorption coefficient, and in this case an explicit solution to the RTE for this coefficient can be obtained. Our composite can be used as nonlinear absorbers in design of the optical limiters to improve the protection of eyes and photodetectors compared to other analogs and achieve an attenuation coefficient of 6 or higher. Unlike most nonlinear materials, which are capable of attenuating laser radiation only for certain wavelengths, the developed composite operates in the range from 355 to 1064 nm. Moreover, the efficiency of optical limiting in the UV region is found to be significantly higher. Acknowledgements. The research concerning investigation of nonlinear optical and structural characteristics of composite of SWCNTs and phthalocyanine was financially supported by the Ministry of Science and Higher Education of the Russian Federation under grant agreement No. 075-15-2021-596 (Sechenov University). Synthesis of phthalocyanine@SWCNTs composite was carried out with the financial support of the Russian Science Foundation (Grant No. 21-73-20016).
References 1. Geis, M.W., Bos, P.J., Liberman, V., Rothschild, M.: Broadband optical switch based on liquid crystal dynamic scattering. Opt. Express 24, 13812 (2016) 2. Seifert, E., et al.: Investigations on retinal pigment epithelial damage at laser irradiation in the lower microsecond time regime. Investig. Opthalmol. Vis. Sci. 62, 32 (2021) 3. Westgate, C., James, D.: Visible-band nanosecond pulsed laser damage thresholds of silicon 2D imaging arrays. Sensors 22, 2526 (2022) 4. Savelyev, M.S., Vasilevsky, P.N., Shaman, Y., Tolbin, A., Gerasimenko, A., Selishchev, S.V.: Limitation of laser radiation power by carbon materials with a nonlinear optical threshold effect at a flat-top pulse shape. Tech. Phys. 68, 476 (2023) 5. Vasilevsky, P.N., et al.: Nonlinear optical response of dispersed medium based on conjugates single-walled carbon nanotubes with phthalocyanines. Photonics 10, 537 (2023) 6. Araujo, P.T., Pesce, P.B.C., Dresselhaus, M.S., Sato, K., Saito, R., Jorio, A.: Resonance Raman spectroscopy of the radial breathing modes in carbon nanotubes. Phys. E Low-Dimensional Syst. Nanostructures 42, 1251–1261 (2010)
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7. Levshov, D.I., Slabodyan, Y.S., Tonkikh, A.A., Michel, T., Roshal’, S.B., Yuzyuk, Y.I.: Specific features of tangential modes in Raman scattering spectra of semiconducting single-walled carbon nanotubes with a large diameter. Phys. Solid State 59, 594–600 (2017) 8. Savelyev, M.S., Gerasimenko, A.Y., Podgaetskii, V.M., Tereshchenko, S.A., Selishchev, S.V., Tolbin, A.Y.: Conjugates of thermally stable phthalocyanine J-type dimers with single-walled carbon nanotubes for enhanced optical limiting applications. Opt. Laser Technol. 117, 272– 279 (2019) 9. Tolbin, A.Y., Shestov, V.I., Savelyev, M.S., Gerasimenko, A.Y.: Deviation from Beer’s law and relationship of aggregation with the effective TPA coefficient: UV-vis studies on a series of low symmetry monophthalocyanines bearing a cyclotriphosphazene substituent. New J. Chem. 47, 1165–1173 (2023) 10. Tolbin, A.Y.: An efficient method of searching for correlations between unlimited datasets to provide forecasting models. Mendeleev Commun. Commun. 33, 419–421 (2023)
Mechanics of Advanced Materials
Identification of the Effective Properties of PZT-Ni and PZT-Air Composites Considering a Non-uniform Partly Polarized Field Mohamed Elsayed Nassar1,3(B)
and Andrey Nasedkin2
1 Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering,
Menoufia University, Menouf 32952, Egypt [email protected] 2 Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, 8a, Milchakov Str., 344090 Rostov-on-Don, Russia [email protected] 3 Faculty of Computers and Artificial Intelligence, AlRyada University for Science and Technology (RST), El-Sadat City 32897, Egypt
Abstract. This study compares the equivalent properties of a PZT-Ni composite to those of a standard PZT-Air porous piezocomposite. The effective properties of both composites were estimated principally numerically using the finite element method and the Hill-Mendel principle, with fully homogeneous and highly inhomogeneous polarized piezoceramic matrix models. The numerical findings concerning the fully polarized model were verified analytically using Mori-Tanaka homogenization technique. The elastic moduli of PZT-Ni and PZT-Air composites and the dielectric permittivity of PZT-Air composite exhibited no significant dependence on polarization inhomogeneity; however, the piezoelectric coefficients showed a considerable dependence on the chosen polarizations model. The incorporation of metal inclusions into the piezoceramic matrix increases the polarization field at the interface, which improves the composite’s homogenized dielectric permittivity moduli. The PZT-Ni composite’s improved dielectric permittivity boosts its efficiency in lighting electronics, voltage controllers, and multilayer small-volume high-performance capacitors. Keywords: Porous Piezoceramics · Metal Inclusion · Piezoelectric Ceramic-Metal Composite · Homogenization Problem · Effective Properties · Finite Element Analysis · Mori-Tanaka Homogenization
1 Introduction Piezoelectric composites have a wide range of potential applications as energy harvesters, sensors, and actuators due to their ability to transform mechanical energy into electrical energy and vice-versa [1–3]. Occasionally, the piezocomposites’ mechanical or physical properties are significantly better than their single-phase counterparts. Most piezocomposites are made of porous piezoelectric ceramics or piezoelectric ceramics and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 305–320, 2024. https://doi.org/10.1007/978-3-031-52239-0_30
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polymers [4–6]. Less research has been conducted on improving the characteristics of piezoelectric materials utilizing piezoelectric-metal composites. This could be because researchers anticipated that the presence of significant amounts of metal in the piezoelectric matrix would degrade the piezoelectric characteristics due to the accumulation of impurities at the grain boundaries [7]. However, incorporating metal, such as platinum, into the piezoceramic matrix can increase mechanical properties, particularly fracture resistance and dielectric permittivity, compared to the homogeneous PZT ceramics [7, 8]. This raises opportunities for using these composites in lighting electronics, voltage controls, and multilayer small-volume high-performance capacitors [7, 9]. To overcome the problem of spontaneous aggregation of nanoparticles, Rybyanets et al. employed polymeric nano- or microgranules, packed with metal-containing nanoparticles (here, nickel), to transport metallic particles into ceramic powders [10, 11]. Transporting nanoparticles into the ceramic matrix is done using polymeric nano and microgranules, which are commercially available [12]. As a result, they constructed a piezoceramic/metal composite. This piezoceramic-metal composite is denoted here PZT-Ni; whereas, the ordinary porous piezocomposite system is denoted as the OPS or the PZT-Air. Microgranules are simpler to manipulate than nanoparticles and can be used to create homogeneously dispersed systems. Therefore, a simple periodic representative volume element (RVE) was employed in this work. Although it is anticipated that the polarization vectors within the composite will be greatly affected by the presence of metal inclusions within the dielectric medium. Up to the authors’ knowledge, there is no research on how the equivalent properties of these composites depend on the heterogeneity of the polarization field. The effective characteristics of composites can be estimated using homogenization approaches if the properties and proportions of the various constituents as well as the structure of the composite are known. The homogenization of two-phase piezoelectric composites with simple structures can be accomplished using analytical techniques such as the Mori-Tanaka method [13, 14], the self-consistent method [15, 16], the HashinSthrikman [16, 17], and Halpin–Tsai bounds estimates [18, 19]. Numerical homogenization techniques, e.g., unit-cell finite element model, are preferred over the analytical techniques due to their ability to account for complex composite structures like partially polarized fields and have virtually no restrictions on geometry, material properties, or the number of phases. Here, we used the analytical Mori-Tanaka homogenization technique to authenticate the numerical results under uniform polarization of the piezoceramic matrix. Here, we investigate the equivalent properties of the PZT-Ni and the OPS as functions of the inclusion’s volume proportion while taking polarization field homogeneity or non-uniformity into account. A simple periodic RVE with only one spherical inclusion was created, and the boundary value problem was solved using the finite element method (FEM) while taking into account linear essential boundary conditions, which are identical to periodic boundary conditions [20–23]. Finally, the effective moduli were calculated from the FEM findings using the Hill-Mendel principle. The equivalent boundary-value problem for the analogous dielectric medium was solved to determine the heterogeneously scattered polarization vectors in the piezoelectric phase. These polarization vectors were then used to construct the element coordinate system for each
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finite element in the piezoelectric phase. To consider the partially polarized model, we additionally adopted a simplified method of linear approximation of the moduli values from the corresponding values for unpolarized ceramics to the values for fully polarized ceramics depending on the moduli of the polarization vectors with “superpolarization” capabilities within limited extent. The results demonstrated that the equivalent PZT-Ni properties are more influenced by the polarization field heterogeneity than those of the OPS (PZT-Air).
2 Mathematical Model The mathematical model for the boundary value homogenization problem consists of the RVE, the linear piezoelectric constitutive equations, interphase and external boundary conditions, and formulae for calculating effective moduli from the averaged components of solutions to these problems. 2.1 Representative Volume Element (RVE) Model A simple periodic RVE consisting of a cube of the utilized piezoceramic matrix PZT5H m containing a spherical inclusion/pore i or p (i ≡ p ) at its center was constructed as seen in Fig. 1(a). The inclusion’s radius can be obtained using the relationship: ri = L 3 3vp /(4π ), where ri , vi or vp (vi = |i |/||), and L are respectively the inclusion’s radius, inclusion’s/porosity volume fraction, and the RVE’s external edge length. The ten-nodal tetrahedral SOLID227 elements with electro-elastic/piezoelectric analysis options and 40 DOFs were applied to create the finite element model of both composites. The free mesh option, which can govern the maximum permitted element edge length, was used to generate the finite element mesh. The back half of the RVE is illustrated in Fig. 1(b). When the maximum length of the element edge was restricted to L/8, the computational results were reasonable. The pore in the OPS is supposed to be a piezoelectric material with minimal piezoelectric and elastic characteristics and a vacuum dielectric permittivity. Nickel was chosen because of its mechanical rigidity and low cost [24, 25]. Nickel’s longitudinal elastic stiffness is approximately 225% of E ). PZT-5H’s highest elastic modulus (c11 E = 12.6 · 1010 , cE = The material properties of the piezoceramics PZT-5H are: c11 12 E E E 10 10 10 10 7.95 · 10 , c13 = 8.41 · 10 , c33 = 11.7 · 10 , c44 = 2.3 · 10 (N/m2 ), e31 = −6.5, S = 1700ε , ε S = 1470ε , where ε = 8.85 · 10−12 e33 = 23.3, e15 = 17 (C/m2 ), ε11 0 33 0 0 (F/m) is the vacuum dielectric permittivity. The metal inclusion in the PZT-Ni is assumed m = κ (C/m2 ), very high to be a piezoelectric material with negligible piezomoduli eiα S m −10 dielectric permittivity moduli εii = χ ε0 , where κ = 10 , χ = 1012 , and elastic properties of Nickel, that is Young’s modulus Em = 20.5 · 1010 (N/m2 ) and Poisson’s ratio μm = 0.31. The vacuum pore was modeled as a piezoelectric material with the E v = κ cE , e v = κ following elastic, piezoelectric, and dielectric permittivities: cαβ αβ iα S v 2 (C/m ), and εii = ε0 , respectively.
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Fig. 1. (a) Cubic RVE and corresponding faces and (b) back half of the finite element mesh for OPS and PZT-Ni.
2.2 Boundary Value Homogenization Problems The constitutive equations of the linear piezoelectric theory are written in standard tensor notation as follows [26, 27]: E σij = cijkl skl − ekij Ek ,
Di = eikl skl + εijS Ej .
(1)
The traditional equilibrium equations, the quasi-electrostatic equation, the straindisplacement equations, and the electric field-electric potential equation are considered in the homogenization problem as follows:
skl =
σij,j = 0, Di,i = 0,
(2)
1 (uk,l + ul,k ), Ej = −ϕ,j , 2
(3)
where σij , skl , Di , and Ej are the components of the stress tensor, the strain tensor, the E electric flux density vector, and the electric field vector, respectively; the quantities cijkl are the components of the elastic stiffness tensor (semi-symmetric fourth-order, that is E = cE = cE ); e are the components of piezoelectric moduli tensor, symmetric cijkl kij jikl klij w.r.t. the last two indices (ekij = ekji ); εijS constitute the components of symmetric secondorder tensor of dielectric permittivities; ui are the components of the displacement vector, and ϕ is the electric potential. The comma in subscript indicates partial differentiation, and over repeating indices, summation from 1 to 3 is assumed. Using the conventional approach of the efficient moduli theory [15, 21–23], we applied the linear essential displacement and electric potential boundary conditions as follows: uk = xk s0kl ,
ϕ = −xk E0k ,
xk ∈ ,
(4)
where s0kl = s0lk and E0k are the array of constant strains and the constant electric field components, which are applied at the boundary = ∂ . The full contact continuity
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conditions are satisfied at the interphase boundaries between the piezoceramic matrix m and the inclusion i or p : ni σijm = ni σiji , ujm = uji , x ∈ i ,
(5)
ni Dim = ni Dii , ϕ m = ϕ i , x ∈ i ,
(6)
where superscripts m and i indicate field values in the matrix and in the inclusion, respectively, i = ∂ i are the boundary of the inclusion’s volume i , ni is the components of the unit normal to the boundary i , and x = {x, y, z} = {x1 , x2 , x3 } is the spatial coordinates vector. Further, instead of tensor notations for elastic stiffness moduli and piezomoduli, we E and e [26, 27], where Greek indices can use conventional Voigt’s matrix notations cαβ jα vary from 1 to 6. 2.3 Modeling the Non-uniform Polarization Field Although the polarization process is nonlinear, we used a simplified linear model because the nonlinear techniques are complex and require separate precision experiments [28– 30]. These simplified methods approximate the polarization process linearly assuming that the polarization vectors can rotate with respect to the primary polarization axis [31–33]. The model utilized here takes the magnitude of the polarization vectors into account besides its direction. A better specification of the element coordinate systems was created here, but this model was first presented in [34]. In the PZT-Ni, due to the presence of high and low permittivity regions, the equivalent properties are predicted to highly depend on the polarization field heterogeneity. The polarization process of the OPS and the PZT-Ni is simulated in a simplified linear formulation by solving the electrostatic problem for a dielectric composite to ascertain the inhomogeneous field of residual polarization. During the polarization process, the piezoelectric, vacuum, and metal phases are assumed to be isotropic dielectric materials. The conductive/metal phase is filled with an isotropic dielectric material with very high dielectric permittivity. The same RVE presented in Fig. 1 was used to model the electrostatic problem with the exception that the finite elements SOLID123 or SOLID227 were specified for Electrostatic-Structural analysis. To solve the electrostatics problem, the upper and lower (Z + and Z − ) are coated with electrodes of negligible thickness. Then, an E-field with a value Ec sufficient to polarize a homogeneous piezoceramic material is applied to the electrodes. The electrostatic problems are listed as follows:
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∇ · D = 0,
D = ε E,
E = −∇ϕ,
x ∈ ,
(7)
under the interface conditions (6) and the following boundary conditions: ϕ = −LEc , x ∈ Z + , ϕ = 0, x ∈ Z − ,
(8)
n · D = 0, x ∈ q ,
(9)
where Z + and Z − are the electro-plated surfaces at x3 = −L 2 and x3 = L 2, q = X − ∪ X + ∪ Y − ∪ Y + , q is the outer surfaces of the RVE without electrodes, ε = ε(x) is the piecewise constant dielectric permittivity function (at x ∈ m , ε = εi , T + ε T )/3; in the metal inclusion of the PZT-Ni x ∈ , ε is extremely εi = (2ε11 i 33 high, for example, ε = 1010 ε0 ; at the pore x ∈ p , ε = ε0 , where ε0 is the vacuum E , and formulae for compliance moduli sE dielectric permittivity; εjjT = εjjS + ejα ejβ sαβ αβ for piezoceramics are given in Appendix. From this electrostatic problem (7)–(9) and (6), the polarization vector was calculated at the central point of finite element ek ⊆ m as follows: Pek = Dek − ε0 Eek ,
(10)
where Dek and Eek are, respectively, the electric induction and the electric field strength vectors at the center of each finite element ek . The simplest technique of considering a non-uniformly polarized model assumes that the direction of the residual polarization vector coincides with the direction of the induced polarization Pek . Then an element coordinate system (ECS) Ox1ek x2ek x3ek can be created for each finite element ek such that the axis Ox3ek is directed along the vector Pek . The unit vectors of these element coordinate systems can be determined using the formulae: ek ek ek P ek P ek P ek ek ek P ek P ek P P P P P 2 1 3 1 i1ek = , − 1ek , 0 , i2ek = , 2 3 , − 12 , 2 , 3 , , i3ek = ek ek P ek P ek P ek P ek P ek P ek P ek P12 P12 P12 12 (11) ek ek = (P1ek )2 + (P2ek )2 such that P12 where P12 2 (P1ek )2 + (P2ek )2 + (P ek 3 ) .
=
0, P ek
=
ek P
=
ek = 0, then instead of formulae (11), the unit vectors iek , iek , iek of the ECS are If P12 1 2 3 assumed to coincide with the unit vectors of the original Cartesian coordinate system i1 , i2 , i3 , respectively. It should be noted that homogenization problems involving non-uniform polarization are handled on a finite element mesh, where each polarized piezoceramic element (SOLID227) typically possesses its own moduli cEek , eek , and εSek . When accounting for the nonuniform polarization field considering only the direction of the polarization vectors Pek , these material properties cEek , eek , and εSek can be determined from the
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original properties of the piezoceramic matrix using the following transformation tensor from the ECS Ox1ek x2ek x3ek to the global Cartesian coordinate system Ox1 x2 x3 : cEek = crEek , eek = erek , εSek = εSek r , crEek = Hek · cE · Hek T , erek = Qek · e · Hek T , εSek = Qek · εS · Qek T , r
(12) (13)
where crEek , erek , εSek r are the moduli of main piezoceramic phase (matrix) of the composite (cE = cE(m) , e = e(m) , εS = εS(m) ) in a rotated ECS, recalculated for the global coordinate system Ox1 x2 x3 , Q ek is the matrix of direction cosines ([Qek ]ij = qijek , qijek = ii · ijek = cos(ii , ijek )), and the components of the 6 × 6 matrix Hek are expressed through the components qijek as follows in [35]. This is the first simplified method for analyzing a partially polarized model: e ek ek ek ek Hijek = (qijek )2 , i, j = 1, 2, 3, Hk44 = q22 q33 + q23 q32 , ek ek ek ek ek ek ek ek ek ek H55 = q11 q33 + q13 q31 , H66 = q11 q22 + q12 q21 , ek ek ek ek ek ek Hi4ek = 2qi2 qi3 , Hi5ek = 2qi1 qi3 , Hi6ek = 2qi1 qi2 , ek ek ek ek ek ek H4iek = 2q2i q3i , H5iek = 2q1i q3i , H6iek = 2q1i q2i , ek ek ek ek ek ek ek ek ek ek H45 = q21 q33 + q23 q31 , H54 = q12 q33 + q32 q13 , ek ek ek ek ek ek ek ek ek ek H46 = q31 q22 + q32 q21 , H64 = q13 q22 + q23 q12 , ek ek ek ek ek ek ek ek ek ek H56 = q11 q32 + q12 q31 , H65 = q11 q23 + q21 q13 .
The second stage of the proposed technique considers that in a piezoceramic composite matrix, generally speaking, at each point x ∈ m the value of the residual polarization vector may differ from the saturation polarization value, and assumes that the elastic compliance moduli sEek , piezomoduli dek , and dielectric permittivities εTek at constant stress (superscript “T ”) of partially polarized piezoceramics are linear functions of the modulus of the residual polarization vector. Thus, we can assume that the material properties of partially polarized ceramics in each finite element ek ⊆ m change linearly from the values in the depolarized state (si , di = 0, εi ) to the values of these moduli in the saturation state (srEek , drek , εTek r ), as indicted in the following relationships:
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sEe k = (1 − χp ) si + χp srEek , dek = χp drek , εTe k = (1 − χp ) εi + χp εTek r , e P /psat , Pe ≤ κp psat , k ke χp = P ≥ κp psat , , κp ,
(14) (15)
k
srEek = (crEek )−1 , drek = erek · srE ek , εTr ek = εSr ek + drek · erek T , si = (ci )−1 , (16) where psat is the polarization value in the saturation state, which, within the framework of the adopted model, is determined by the formula: psat = (εi − ε0 )Ec ; κp = 1.2 is the coefficient that determines the possibility of “superpolarization” of piezoceramics, which can arise due to large values of the polarization field in the vicinity of electric i are the stiffness moduli of non-polarized ceramics, defined field concentrators; cαβ as the Hill-averaged constants of a polycrystalline transversely isotropic material with E and compliance moduli sE [36]. Note that this model, due to stiffness moduli cαβ αβ the coefficient, expands similar models for calculating the moduli of inhomogeneously polarized piezoceramics presented in [35, 36]. 2.4 Computing the Effective Moduli Solutions of the problems (7)–(9) and (6) are not required when fully/homogeneous polarized models are investigated. To explore the equivalent characteristics of both composites (PZT-Air or PZT-Ni) considering partial/inhomogeneous polarization, at the first step, the electrostatic problem (7)–(9) and (6) is solved using the finite element method, then the moduli of inhomogeneously polarized piezoceramics are calculated using formulae (8)–(15) for each finite element ek of the matrix m , and then the piezoelectric boundary value problems (1)–(6) are solved with piezoelectric finite elements with modified material properties. Then, to determine the effective properties of piezoceramic composites for partially or fully polarized models, it is enough to solve five boundary value problems (1)–(6). These problems I–V differ in boundary conditions (4), and from the solutions of these individual problems, all effective material moduli can be found. Problem I. The linear displacement of the external boundaries of the volume in the direction of the x1 -axis is specified, for example, according to Fig. 1(a) the normal displacements of X − , Y + , Y − , Z + , and Z − surfaces were fixed and the X + surface was exposed to the normal displacement:
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E eff eff ui = x1 δ1i S0 , ϕ = 0, x ∈ ⇒ c1j = /S0 , j = 1, 2, 3, e31 = /S0 ; (17)
Problem II. The linear displacement of the external boundaries of the volume in the direction of the x3 -axis is specified as E eff eff ui = x3 δ3i S0 , ϕ = 0, x ∈ ⇒ cj3 = /S0 , j = 1, 2, 3, e33 = /S0 ; (18)
Problem III. A linear shear displacement of the external boundaries of the volume is specified along the x2 - and x3 -axes as E eff eff ui = (x3 δ2i + x2 δ3i )S0 /2, ϕ = 0, x ∈ ⇒ c44 = /S0 , e15 = /S0 ; (19)
Problem IV. At the outer boundaries of the volume , an electric potential linear in x1 -axis is specified as eff S eff ui = 0, ϕ = −x1 E0 , x ∈ ⇒ e15 = −/E0 , ε11 = /E0 ;
(20)
Problem V. At the outer boundaries of the volume , an electric potential linear in x3 -axis is specified as eff S eff ui = 0, ϕ = −x3 E0 , x ∈ ⇒ e3j = −/E0 , j = 1, 3, ε33 = /E0 . (21) Here = (1/||) (...) d , δij is the Kronecker delta, S0 , E0 are arbitrary nonzero constants, characterizing macroscopic values of deformations and E-field strength, respectively. As can be seen from (17)–(21), in total, from the solutions of problems I–V, a complete set of 10 effective moduli of the piezoceramic composite is determined, and the same piezoelectric moduli are found twice and, up to the error of numerical methods, should have the same values [23].
3 Results and Discussion The section presents the effective moduli of the PZT-Ni and OPS (PZT-Air) as functions of inclusion volume. All effective properties are displayed, compared to the employed S ) = ε Seff /ε S . The equivalent proppiezoceramic matrix properties., for example, r(ε33 33 33 erties of the PZT-Ni and the OPS are shown respectively in red and blue colors. Both systems were investigated numerically considering homogeneous and heterogeneous polarization field. For both systems, the findings of the homogeneous and inhomogeneous polarization field are represented in continuous and small dashing lines, respectively. The analytical Mori-Tanaka homogenization technique [13, 14] was used to confirm the results when a homogeneous polarization field was considered. The results obtained using Mori-Tanaka method are outlined in dotted lines with circular markers
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and the same color of the anticipated composite. When discussing the results, we mainly concentrated our attention on the PZT-Ni because the OPS was well studied in [15, 37–39]. Because the used method of considering a partially polarized field by Eqs. (11)– (16) allows us to take into account the inhomogeneity of the polarization field both in direction and magnitude, it gives more practical results than the method of taking into account the inhomogeneity of polarization only in direction [33] according to (11)–(13). We were satisfied with exhibiting the findings of the more complete new inhomogeneous polarization approach to obtain clear figures.
E ), r(cE ), and r(cE ) as functions of the inclusion’s Fig. 2. Relative elastic stiffness moduli r(c33 13 44 volume proportion. Continuous lines: fully polarized field model; dashed lines: partially polarized field model (inhomogeneous polarization using (14)–(16)); and dotted lines with circular markers: equivalent properties using Mori-Tanaka method.
E ), r(cE ) and r(cE ) of both composFigure 2 depicts the elastic stiffness moduli r(c33 13 44 ites as functions of inclusion’s volume fraction. Contrary to the OPS, the elastic moduli of the PZT-Ni increase when vp rise due to the higher elastic properties of Nickel incluEeff , transverse cEeff , and shear cEeff elastic moduli sions. When vp = 0.4, longitudinal c33 13 44 of the PZT-Ni are improved by 307%, 303%, and 221% from the analogous values of the OPS, and 42%, 15%, and 69% from the analogous values of the pure piezoceramic
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PZT5H. Considering the inhomogeneous polarization field has little effect on the elastic characteristics of both systems. Also, very good agreement between the analytical and numerical results is noted.
S ), r(ε S ), and r(ε T ) as functions of the Fig. 3. Relative dielectric permittivity moduli r(ε11 33 33 inclusion’s volume proportion. Continuous lines: fully polarized field model; dashed lines: partially polarized field model (inhomogeneous polarization using (14)–(16)); and dotted lines with circular markers: equivalent properties using Mori-Tanaka method.
The effective dielectric permittivity moduli of both composites are illustrated in Fig. 3. The presence of metal inclusions with extraordinary dielectric permittivity results in extremely high polarization field, electric flux density, and electric field intensity in the vicinity of the metal inclusions [7, 9, 40]. Therefore, the equivalent dielectric permittivity moduli of the PZT-Ni increase with the growth of the metal’s volume fraction. Considering polarization field inhomogeneity has an insignificant effect on all dielecSeff of the PZT-Ni; but tric permittivity properties of the OPS and on the modulus ε11 Seff T eff significantly influences effective moduli ε33 and ε33 of the PZT-Ni. For vp = 0.4, Seff and ε T eff of the PZT-Ni decrease about 26.2% and 25.6%, the effective moduli ε33 33 respectively, when partially polarized field was considered.
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Fig. 4. Relative piezoelectric moduli r(e31 ), r(e33 ), and r(e15 ) as functions of the inclusion’s volume proportion. Continuous lines: fully polarized field model; dashed lines: partially polarized field model (inhomogeneous polarization using (14)–(16)); and dotted lines with circular markers: equivalent properties using Mori-Tanaka method.
Figure 4 depicts the piezoelectric stress coefficients vs vp for both composites. Because of the vacuum inclusion’s insignificant piezomoduli, the effective piezomoduli of the OPS diminish in absolute values as porosity increases. For uniformly polarized eff and shear eeff piezomoduli of the PZT-Ni piezoceramics, the equivalent longitudinal e33 15 eff | of the PZTmonotonically increase, while the equivalent transverse piezomodulus |e31 Ni slightly decreases with the υp rise. The analytical results are in good agreement with eff and eeff . When considering the numerical results particularly for the piezomoduli e33 15 eff eff of a heterogeneously polarized field, the corresponding e33 is unaffected, and the e15 eff | the PZT-Ni decreases. Nevertheless, the homogenized transverse piezomodulus |e31 decreases considerably in comparison to the analogous values of the fully polarized eff |, eeff , and eeff of the PZT-Ni, for instance, model. The homogenized piezomoduli |e31 33 15 are lowered by about 98.9%, 2.5%, and 18.4% from the corresponding values of homogeneous polarization when the inhomogeneous polarization field is taken into account at vp = 0.4. The piezoelectric strain coefficients diα are very important in practice in assessing the effectiveness of piezoelectric transducers in actuator applications. They can be identified
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Fig. 5. Relative piezoelectric moduli r(d31 ), r(d33 ), and r(d15 ) as functions of the inclusion’s volume proportion. Continuous lines: fully polarized field model; dashed lines: partially polarized field model (inhomogeneous polarization using (14)–(16)); and dotted lines with circular markers: equivalent properties using Mori-Tanaka method. eff , via the following relationship: deff = eeff · (cEeff )−1 . The relationships concerning d31 eff eff eff d33 , and d15 are detailed in Appendix. According to these formulae, although e33 eff eff eff increases, |d31 | and d33 decrease, when vp increases because of the decrease of |e31 | and eff increases, d eff reduces with the increase in v due all compliance moduli. Although e15 p 15 E eff . The piezomoduli d eff to the reduction in s44 , which has a more significant effect on d15 31 eff significantly reduces in absolute values, when polarization field heterogeneity and d33 eff . When considering the polarization field inhomogeneity, due to their dependence on e31 eff |, d eff , and d eff are reduced about 25.6%, 14.8%, and at vp = 0.4, the piezomoduli |d31 33 15 eff , d eff , and d eff of the PZT-Ni 16%, as seen in Fig. 5. The decrease of the piezomoduli d31 33 15 in absolute values indicates that the PZT-Ni cannot be applied efficiently for piezoelectric actuator applications. Also, according to the relationships of the transduction coefficient 2 /ε T , which is an important energy harvesting figure of merit [41], and the TCiα = diα ii eff , d eff , d eff and the dielectric permittivity modulus change trends of the piezomoduli d31 33 15 T ε33 , the PZT-Ni is not recommended also for sensing/energy harvesting applications.
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4 Conclusions This work explores the effective properties of a piezoceramic/nickel (metal) composite (PZT-Ni) in comparison with an ordinary porous piezocomposite (OPS or PZT-Air), considering fully and partially polarized models. The analytical Mori-Tanaka homogenization technique was applied to verify the numerical findings, when fully/homogeneously polarized model was studied, and the results demonstrated good agreement. The main findings of the paper are listed as follows. The greater elastic properties of the metal improve the homogenized elastic stiffness properties of the PZT-Ni composite. The insertion of metal inclusion in the PZT-Ni produces high E-field intensity, E-flux density, and E-polarization at the interface between the metal inclusion and the piezoceramic matrix, which in turn enhances the equivalent dielectric permittivity of the PZT-Ni. The elastic moduli of both composites insignificantly depend on the considered type of polarizaS eff and ε T eff tion (full or partial polarization). The dielectric permittivity coefficients ε33 33 of the PZT-Ni decrease significantly when a partially polarized model is considered. eff , the effective piezomoduli of the PZT-Ni were strongly influenced by Except for e33 the implied polarizations model. The great improvement of dielectric permittivity of the PZT-Ni improves its efficiency in lighting electronics, voltage controls, and multilayer small-volume high-performing capacitors. Acknowledgement. This research was supported by the Russian Science Foundation, grant number No. 22-11-00302.
Appendix
E )2 E )2 cE cE − (c13 cE cE − (c13 E E E E 2 E E
, s12
, c11 − 2(c13 = c33 + c12 ) , s11 = 11 33E = − 12 33E E E c11 − c12 c11 − c12
E E
E c11 + c12 c13 1 E E E E E E , s44 + e33 s13 s13 = − , s33 = = E , d31 = e31 s11 + s12 c44 E E E d33 = 2e31 s13 + e33 s33 , d15 = e15 s44
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Finite Element Investigation of Mechanical Properties of Highly Porous Nanoscale Materials Composed of Regular Lattices from Gibson-Ashby Cells of Variable Geometry Alexandr Kornievsky
and Andrey Nasedkin(B)
I.I. Vorovich Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Rostov-on-Don 344090, Russia [email protected]
Abstract. In this research, a finite element model of regular highly porous foam structure based on the macro or nano sized Gibson-Ashby cell is proposed. To account for the dimensional effect, the Gurtin-Murdoch model was used, which considers the surface stresses arising on the boundaries of the cell. The computer model construction, finite element meshing, and the numerical solution of the homogenization problem were carried out in the ANSYS software package. Here, the influence of the cell geometry at a fixed porosity was initially investigated. The obtained results confirm that the effective moduli of highly porous structures composed of Gibson-Ashby cells depend not only on porosity, but also on the geometric configuration. For example, at the same porosity, cells with thicker edges have greater rigidity. Numerical experiments were also carried out at the nanoscale, that is, model considered surface stresses. The relative stiffness moduli of nanoscale structures are significantly higher than similar values of regular size structures. Moreover, the dimensional effect has a greater influence on the effective properties of a highly porous material than the cell configuration. For example, the relative effective Young’s modulus of a regular size structure with thick edges is smaller than that of a nanoscale structure with thin edges. In this paper, cells with a complex geometric structure are considered, so a representative volume loses the isotropic properties of the material. Therefore, a study of the anisotropic effective properties based on the Zener coefficient was also carried out. Keywords: Nanomechanics · Surface Stress · Gibson-Ashby Cell · Regular Lattice · Homogenization · Effective Modulus · Finite Element Method
1 Introduction Recently, technologies to produce new porous materials have made great strides forward. This is due to the use of 3D printing and high-precision measurements to create materials with high performance characteristics. Highly porous materials were studied in this paper. Now their use is gaining popularity in many areas, such as construction, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 321–334, 2024. https://doi.org/10.1007/978-3-031-52239-0_31
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medicine, energy, etc. Since foam materials have good noise insulation, thermal insulation, sufficient strength and at the same time have a low weight [1–3], the interest in them has been growing. To date, many papers devoted to the study and description of the properties of such structures have been published. Both analytical and computer models have been proposed, but they tend to find effective properties that depend primarily on the physical constants of the starting material and the porosity of the representative volume element (RVE). However, it is obvious that the elastic properties of porous materials depend not only on the volume fraction of voids, but also on their location and the geometric structure of the RVE. For highly porous materials, the most popular is the Gibson-Ashby analytical model [4, 5], based on solutions of problems about tension-compression and problems about bending of beams in a cell of a special shape (Fig. 1). Firstly, such popularity is associated with the high accuracy of the Gibson-Ashby formulae, which have been repeatedly confirmed both experimentally and by other numerical models based on this cell. Secondly, this popularity is due to the ease of calculation. Thus, according to the classical model for an open Gibson-Ashby cell, which is subjected primarily to flexural deformations, the value of the effective Young’s modulus is directly proportional to the square of the mass fraction of the material in RVE. But this model, like some others, based on it, considers the dependence of the effective Young’s modulus only on the porosity of the RVE. Some studies considered cases of cells with changed dimensions along the coordinate axes, which naturally affected the anisotropy of the RVE [6–8]. Irregular structures from Gibson-Ashby cells were also studied in [9, 10] and others. More detailed investigations of the mechanical properties of Gibson-Ashby cells depending on their geometric structure were carried out in [11, 12] and others. In this paper, like [11], we consider a finite element model, based on lattices, composed of Gibson-Ashby cells. At the same time, the geometric dimensions of Gibson-Ashby cells can vary within wide limits, maintaining their sufficiently high porosity. Thus, the effective properties of the RVE here should depend not only on porosity, but also on the geometric features of the cells [11]. In addition to geometry features, the effective properties are also affected by the cell size. When a representative volume has dimensions, corresponding to the nanoscale, then the usual approaches of mechanics require modifications, since nanoscale materials have some additional properties that distinguish them from materials of ordinary sizes. For example, experimental and theoretical studies confirm that nanomaterials, unlike conventional ones, can have higher stiffness [13, 14]. One of the models that describes this effect is the Gurtin-Murdoch model, according to which surface stresses arise at the interface between the material and pores [13–16]. This model for nanostructured foam materials has been used in a fairly large number of works [17–28]. At the same time, in [19, 22, 23, 27, 28] the Gibson-Ashby cell was considered based on the theories of Bernoulli-Euler and Timoshenko beams with surface stresses and various analytical formulae were obtained. In [25], a nanoscale Gibson-Ashby cell with surface stresses was immersed in a liquid. In [21], a lattice of regular Gibson-Ashby cells of classical shape with surface stresses was studied, based on finite element homogenization methods.
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This paper considers a combination of the Gibson-Ashby model for regular lattices with variable geometry [11] and the Gurtin-Murdoch model, which considers the surface effect for nanodimensional cells of a composite structure [21]. Thus, here we present a new model, based on highly porous cells of different geometric structures, additionally considering stresses at the internal boundaries of the RVE. The homogenization problem, as in [11, 21], is solved using the finite element method in the ANSYS APDL package.
2 Homogenization Approaches Following [29–31], we describe the setting of the elastic problem with interface stresses for anisotropic inhomogeneous bodies. A representative volume element (RVE) of composite material is considered as a union of two regions V = Vf ∪ Vp , where Vf is the material of the first (main) phase or matrix of the composite, Vp is the pores, which we will consider filled with the second material with negligible elastic moduli. In this RVE, we consider the static problem of elasticity theory for the displacement vector u = u(x) = {u1 (x), u2 (x), u3 (x)} with special boundary conditions on the outer boundary = ∂V . For the subsequent application of the finite element method, we write the classical equations of the linear static problem of elasticity theory in the following form: L∗ (∇) · T = 0, T = c · S, ⎤ ∂ 1 0 0 0 ∂ 3 ∂2 L∗ (∇) = ⎣ 0 ∂2 0 ∂3 0 ∂1 ⎦, 0 0 ∂ 3 ∂2 ∂1 0
(1)
⎡
S = L(∇) · u,
(2)
where T = {σ11 , σ22 , σ33 , σ23 , σ13 , σ12 } is an array consisting of stress tensor components σij , S = {ε11 , ε22 , ε33, 2ε 23 , 2ε 13 , 2ε 12 } is an array consisting of strain tensor components εkl = uk,l + ul,k /2, c is the 6 × 6 matrix of elastic stiffnesses cαβ , (. . . )∗ is the transposition operation. The homogenization problem is set as following: knowing the material moduli and geometric characteristics of the composite phases, determine the effective properties eff cαβ of an “equivalent” homogeneous medium, and then, if necessary, obtain from them eff
other effective elastic moduli, such as, for example, the effective Young’s modulus Ei , eff eff effective shear moduli Gij and effective Poisson’s ratios νij . Here, in the case of an isotropic homogeneous medium, it is sufficient to determine any two elastic moduli, for example, the effective Young’s modulus E eff and Poisson’s ratio ν eff . In accordance with the Gurtin-Murdoch model, it is assumed that the surface stresses exist on nano-sized interphase boundaries s = Vf ∩ Vp and the following condition is satisfied: L∗ (n) · [T] = L∗ (∇s ) · Ts , x ∈ s ,
(3)
where [T] = T(1) − T(2) is the stress jump on the phaseboundary; ∇s is a surface nabla ∂ , where r is the coordinate, operator, which is defined by the equality ∇s = ∇ − n ∂r measured along the normal n to s ; Ts = {σs11 , σs22 , σs33 , σs23 , σs13 , σs12 } is an array
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consisting of components σsij of the surface stress tensor. For stresses Ts , assuming the absence of residual surface tensions, Hooke’s “surface” law is used: Ts = cs · Ss , Ss = L(∇s ) · us , us = A · u, A = I − nn∗ ,
(4)
where cs is the 6 × 6 matrix of surface elastic stiffness moduli csαβ ; Ss = {εs11 , εs22 , εs33 , 2ε s23 , 2ε s13 , 2ε s12 } is the array consisting of components εskl of the surface strain tensor; us is the vector of surface displacements, I is the 3 × 3 identity matrix. To find all the components of the effective stiffness matrix ceff of the anisotropic composite under consideration, boundary conditions in displacements were specified on the external boundary of the RVE: u = L∗ (x) · S0 , x ∈ ,
(5)
where S0 = {S01 , S02 , S03 , S04 , S05 , S06 } is the array of some constant values. In the general case of anisotropy, it is necessary to solve six problems (1)–(5), setting only one of the components of the array S0 to be nonzero: u = L∗ (x) · S0 , x ∈ ,
(6)
where δβζ is the Kronecker symbol, the index ζ takes some specific integer from 1 to 6; S0 is some constant value. Solutions of problems (1)–(6) with fixed ζ = 1, 2, . . . 6 in the aggregate allow us to find effective moduli for all six columns of the matrix ceff : eff
cαζ = /ε0 ,
(7)
where ε0 = S0 for ζ = 1, 2, 3; ε0 = 2S0 for ζ = 4, 5, 6; the correspondence formulae between Greek and Latin indices α ⇐⇒ (ij) are used: 1 ⇐⇒ (11), 2 ⇐⇒ (22), 3 ⇐⇒ (33), 4 ⇐⇒ (23)∼(32), 5 ⇐⇒ (13)∼(31) and 6 ⇐⇒ (12)∼(21); and the symbol denotes averaging over the RVE: = V + < (. . . )> , V =
1 |V |
V
(. . . )dV , =
1 |V |
(8)
s
(. . . )d s .
(9)
As can be seen, boundary value problems (1)–(6) differ from conventional elastic problems in the presence of boundary conditions (3), (4) for nanoscale structures on the interface s , which are typical for the Gurtin-Murdoch surface stress model. In addition, to determine effective moduli from (7)–(9), it is necessary to use not only integration over the volume with averaging V , as in conventional macro-scale homogenization problems, but also integration over phase interfaces with averaging . Note that, as was substantiated in [21], these homogenization problems can only be solved in a region Vf , filled with the main material.
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3 Construction of Representative Volume Element (RVE) According to the Gibson-Ashby approach, foam-like materials can be represented as an array of cubic cells that consist of thin beams. A finite element model, based on the Gibson-Ashby cell, is proposed in the current work (Fig. 1a). To simplify further meshing of the RVE, the initial model is composed of twenty-four internal edges Vint , twelve external edges Vext and twenty cubic volumes Vc , which connect the internal and external edges. Thus, the volume of the cell can be calculated as a sum: Vf = 24Vint + 12Vext + 20Vc .
(10)
Let us consider the geometric characteristics of the presented model (Fig. 1b). Let the cell size along each of the coordinate axes is equal to L. Each cell consists of an inner frame and external edges. Let us define the size of the inner frame a as the distance between the midpoint of the external edges. Let the thickness of the edges is equal to h. In the original paper [5] the frame size is constant and assumed to be equal a = L/2, hence the porosity p of the RVE depends only on the thickness of the edges, that is, p = p(h).
Fig. 1. Original Gibson-Ashby cell of 95% porosity: (a) space view, (b) xy-plane view.
Now we can extend that finite element model by considering not only the changes of the edge thickness, but also the size of the inner frame. Then the porosity of the RVE will be a function of two parameters p = p(a, h). It was noted before that the cell volume is calculated by formula (10) and consists of volumes of three types Vint , Vext and Vc , the values of which can be calculated as 2 (11) Vint = a2 − h h2 , Vext = (L−a−h)h , Vc = h3 . 2 On the other hand, the cell volume can be calculated using porosity of the cubic RVE with side L: Vf = (1 − p)V = (1 − p)L3 .
(12)
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Substituting formulae (11), (12) into Eq. (10), and taking L = 1, we obtain the dependence: p(a, h) = 1 + 10h3 − 6h2 (a + 1).
(13)
Obviously, the same values for porosity p can be obtained as a combination of different frame size and edge thickness. Because of the cell geometric features, the sizes of the parameters a and h can vary within certain limits. Let us define limitations on the edge sizes of the cell. To set limitations we introduce parameter δ > 0, which corresponds to the smallest possible length of the internal or external edge (Fig. 2).
Fig. 2. Projections of a Gibson-Ashby cell of 95% porosity on the xy-plane: (a) the largest frame size, (b) the smallest frame size.
Figure 2 shows two cells with the same porosity of 95% and different geometric configurations. For example, Fig. 2a shows a cell with the largest size of the inner frame and the smallest thickness of the edge, respectively. Based on this figure and taking L = 1, we can determine that the largest frame size is equal to amax = 1 − h − 2δ. Substituting amax to formula (13), we obtain equation 16h3 − 12h2 + 12h2 δ + 1 = p. The solution of this equation under the condition h ∈ (0; 1/2) is hmin . That is why, the limit values of the size of the cell shown in Fig. 2a can be obtained as follows: 1 + 16h3min − 12h2min + 12h2min δ = p, hmin ∈ 0; 21 , (14) amax = 1 − hmin − 2δ. The same way, based on Fig. 2b, we can determine the geometric sizes of the cell with the smallest frame: 1 − 2h3max − 12h2max δ − 6h2max = p, hmax ∈ 0; 21 , (15) amin = 2hmax + 2δ.
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Now let us define the value of the parameter δ. The most interesting are cells with extremely large and small frame sizes, so it is necessary that δ > 0 tends to zero. However, if we choose too small δ, then the computer model of the cell will contain flat volumes. These elements impose additional difficulties for numerical calculations related either to accuracy or to the calculation time. Empirically, it is taken that δ = 0.01L. The computer model described above is valid for finding the effective properties of high-porous materials in macro scale. Let us expand it to a nanoscale. The GurtinMurdoch model [16] is chosen as a model that considers the dimensional effect. This approach additionally considers surface stresses arising at the boundaries of the material and pores. These surface stresses can be considered as membrane elements located on the surfaces of the RVE edges. There are some difficulties in the finite element model with membrane elements simulating surface stresses [21]. To calculate the effective moduli of the RVE, it is necessary to know the physical characteristics of both bulk elements and membrane ones. However, there is no accurate and reliable data on surface moduli. In addition, in this finite element model, the membranes have a certain thickness h˜ m and the bulk Young’s modulus Em . Denote the surface Young’s modulus by Es , then the equation Es = h˜ m Em must be satisfied for the membrane elements. Then we introduce the proportionality coefficient kss = Em /E, where E is the Young’s modulus of the material which the cell is composed of. Thus, Es = h˜ m kss E. For further calculation convenience, we will use the dimensionless thickness of the membrane hm = 1, hence: Es = kss E.
(16)
Then, by changing the values of the kss coefficient, it is possible to obtain different values for the surface module Es . A similar problem was considered for the original Gibson-Ashby cell (a = L/2) at the nanoscale in [26–28]. In these works, an analytical model of the dependence of the effective Young’s modulus of a nanoporous foam-like material on the cell edge thickness was obtained. That formula is based on the problem of bending a beam with surface stresses. Comparing the values, obtained by the analytical formula, and numerically by finite element model, we obtain an empirical formula which connects the proportionality coefficient kss with the real value of the edge thickness h (nm): (4/9)
h = a/kss
+ b,
(17)
where a = 0.2305177888, b = −0.5306719104. After the construction of the cell is completed, we generate a lattice, which consists of copies of the original cell. Figure 3 shows lattices made up of cells with different configurations, but the same porosity of 85%. Thus, a finite element model for finding effective moduli of foam materials of various geometric configurations, considering surface stresses arising at the nanoscale, according to the Gurtin-Murdoch model is proposed.
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Fig. 3. Lattices with 85% porosity and different configurations of cell geometry: (a) the largest frame size, (b) the smallest frame size.
4 Computational Experiments and Results Computational experiments were carried out for the RVE with the specified material properties of Au by analogy with the selected material in papers [26–28]. This material is isotropic and has the following mechanical characteristics: Young’s modulus E = 70 (GPa), Poisson’s ratio ν = 0.42. Numerical experiments were carried out for regular 3 × 3 × 3 lattices with porosity varying from 65% to 95% in 5% increment. The effective properties of the RVE depend not only on the porosity, but also on the geometric configuration of the cell. Therefore, numerical experiments were carried out on three types of cells: original cells (a = L/2) and cells with edges of extremely large and extremely small thickness, in accordance with formulae (14) and (15). Based on previous numerical experiments, conducted on the original nanoscale cell [21], we set the kss coefficient. Let the proportionality coefficient kss , which determines the values of surface stresses, took the values kss = 0.0001 and kss = 0.1. According to formula (17) these coefficient values correspond to the real cell thicknesses: hreal ≈ 10 (nm) and hreal ≈ 0.1 (nm). In the first case, the influence of surface stresses can be neglected, since the cell size is large enough. In the second case, the edge thickness is two orders smaller. At such a scale, surface stresses have a significant impact on the effective moduli of the composite. In this paper, the combined influence of both geometry and surface effects was evaluated. Figures 4, 5, 6, 7 and 8 show graphs of the dependencies of relative moduli on porosity for 3 × 3 × 3 lattices, composed of Gibson-Ashby cells. Relative moduli are defined as the ratio of the effective module to the module of a solid material, that is, eff eff r(E) = EE is the relative Young’s modulus, r(ν) = ν ν is the relative Poisson’s ratio, eff eff cαβ c r(G) = r(c44 ) = c44 = is the relative shear modulus, and r c αβ cαβ is the relative 44 components of the stiffness matrix. There are two types of lines: solid and dashed. Both
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types are also divided by color. Solid lines correspond to calculations with a coefficient kss = 0.0001, that is, cells have negligible values of surface stresses. These lines describe the effective properties of the RVE on a macro scale. Dashed lines are based on the results of calculations at kss = 0.1, so, they correspond to the nanoscale model. The blue lines represent relative moduli of cells with the thinnest edges, the red lines correspond to the original cells (a = L/2) and the yellow lines correspond to the thickest edge cells.
Fig. 4. Dependencies of the relative Young’s modulus on porosity for different cell configurations and kss values.
Figures 4, 5, 6, 7 and 8 show the expected trends. Namely, effective moduli increase both with the proportionality coefficient increasing and with an increase in the thickness of the edges. Here we determine which parameters have a stronger influence on the effective moduli of the composite. Let us consider the red solid line in Fig. 4. It corresponds to the lattice, composed of the original Gibson-Ashby cells at the macro scale. Regarding the selected case, consider the yellow solid line and the red dashed line. The first one is obtained in the result of increasing the thickness of the cell edge. The second one is obtained because of adding surface stresses. The red dashed line is located above the yellow solid line for all porosity values. Thus, it can be concluded that the influence of the dimensional effect on the elastic moduli of porous composites is higher than the influence of the geometric structure.
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Fig. 5. Dependencies of the relative stiffness modulus r(c11 ) on porosity for different cell configurations and kss values.
Fig. 6. Dependencies of the relative stiffness modulus r(c12 ) on porosity for different cell configurations and kss values.
Similar trends are observed for effective stiffness moduli r(c11 ) and shear moduli r(G). The effective values of the Poisson’s ratio also depend more on the surface stresses than on the geometric structure of the composite. However, this influence works in the opposite direction. For example, in Fig. 8, the red dashed line is below the solid yellow one. In Fig. 6, the graphs describe the behavior of the relative effective stiffness moduli r(c12 ). However, in this case, the influence of surface stresses is negligible. Here, with
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Fig. 7. Dependencies of the relative shear modulus on porosity for different cell configurations and kss values.
Fig. 8. Dependencies of the relative Poisson’s ratio on porosity for different cell configurations and kss values.
a porosity of more than 80%, the geometric configuration of the cell has more influence than the surface stresses. Since the RVE has a complex geometric shape, we use the Zener ratio Aeff = eff eff eff 2c44 /(c11 − c12 ) to quantify the anisotropy [11, 21]. Figure 9 shows that with increasing porosity, this structure uniformly loses isotropic properties. Moreover, from the arrangement of solid and dashed lines, it can be concluded that the surface effect does not affect the anisotropy of the material. However, the geometric configuration of the cell and the porosity of the structure have more influence on this property.
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Fig. 9. Dependencies of the Zener ratio on porosity for different cell configurations and kss values.
5 Conclusions In this paper a finite element model for finding effective moduli of highly porous foam materials with a regular structure is proposed. This model is based on the Gibson-Ashby cell, but it has important distinctive features. Firstly, not only the cells of the original shape are considered, but also cells with different geometric configurations. Secondly, based on the Gurtin-Murdoch approach, this model has been expanded to nano scales. The analysis of numerical results showed that the dimensional effect has a greater influence on the effective moduli of the RVE than the geometric configuration of the cell. Moreover, the size effect increases the differences between the stiffness moduli for cells with different configurations. However, the anisotropic properties of the material depend more on the geometry of the cell than on surface stresses. Acknowledgement. This research was supported by the Russian Science Foundation, grant number No. 22-11-00302.
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Methodology of Determination of Porosity Parameters in the Theory of Microdilation M. I. Chebakov(B)
and E. M. Kolosova
I.I. Vorovich Institute of Mathematics, Mechanics, and Computer Science, Southern Federal University, Rostov-on-Don 344006, Russia [email protected]
Abstract. Cowin and Nunziato developed the theory of microdilatation, which allows one to effectively investigate various problems for poroelastic bodies, containing unfilled pores, based on analytical approaches. The theory of microdilatation is based on a system of differential equations, which contains several parameters, the values of which are not available in reference books, which limits the use of this theory in practice. Some publications propose schemes for determining these parameters. This paper proposes a method for determining such important parameters of the theory of microdilatation as stress parameter with porosity change and void stiffness coefficient. To determine the values of these parameters, a scheme has been developed based on a comparison of solutions plane contact problems for an elastic body in the form of a strip, the deformation of which is described by the Cowin-Nunziato theory of microdilatation, and for the corresponding elastic body in the form of a rectangle containing real, unfilled, uniformly distributed voids in the shape of circles. The first problem was studied, based on a numerical-analytical method for solving the resulting integral equation, the second problem was studied by using the finite element method. Control of the correctness of the obtained parameter values was carried out by comparing the results of studying similar contact problems with other values of the geometric parameters of the strip and rectangle. Keywords: Contact problems · Cowin-Nunziato Theory · Microdilatation · Porosity Parameters
1 Introduction In present time, porous materials are actively used in many industries, construction, and medicine. There are two ways to model materials with pores: (i) use the geometry of such bodies with obvious voids of the needed shapes and sizes with the required density and in this case solve problems within the framework of the elasticity theory, or (ii) take geometry without visible pores, but when solving such problems use equations for poroelastic bodies. In the first case, the finite element method can be used; in the second case, for example, the theory of microdilatation can be used, developed by CowinNunziato [1, 2] for poroelastic bodies with empty pores. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 335–344, 2024. https://doi.org/10.1007/978-3-031-52239-0_32
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In the case of static problems for an isotropic material, the microdilatation theory operates on five material parameters: Young’s modulus, Poisson’s ratio, the voids diffusion coefficient, stress parameter with porosity change, and void stiffness coefficient. Now, there is no unified approach to determine the last three parameters. An experimental technique for finding these constants was proposed in [3]. In [4], the methods were developed for identifying the porosity parameters of the microdilatation theory, based on an analysis of the structure parameters and rigidity of lattice structures. An example of an auxetic cellular metamaterial is given that corresponds to the theory of microdilatation in a wide range of parameters. The properties of Cowin-Nunziato composite media with voids, as well as the properties of conventional poroelastic materials with different structures porosity, can be determined, for example, by the effective modulus method in combination with finite element modeling of representative volumes. Such technology for porous composites with a different type of connectivity of physical and mechanical fields was described in [5, 6]. The purpose of this investigation is to create a methodology for determining such parameters of the Cowin-Nunziato theory as stress parameter with porosity change, and void stiffness coefficient. The first parameter is determined in the computational experiment using the method proposed in [3]. The second is found by selection it on the base of the comparison of calculation results for two contact problems. As the first contact problem, we consider the interaction of the stamp with poroelastic strip, the deformation of which is described by the Cowin-Nunziato equations. As the second contact problem, we consider the interaction of a similar stamp with an elastic body in the rectangle shape containing uniformly distributed empty pores in the circle shape with a given radius and their number. In both problems the stamp base has a flat shape, the same dimensions of the contact area, the thickness of the strip and the rectangle, while the rectangle width is selected from the condition that its side edges do not influence on the contact stresses. The first problem is investigated, based on a numerical-analytical approach by solving the resulting integral equation for contact stresses, using the collocation method. The solution of the second problem is based on the finite element method. To verify the correctness of the found parameter values, the calculation results were compared for similar contact problems with the found microdilatation parameters, when the strip and rectangle have other consistent geometric dimensions.
2 Statement and Solving Methods of the Contact Problems The governing equations of the Cowin-Nunziato microdilatation theory for the elastic isotropic medium with voids relative to the displacement vector u in general form in the static case have the form [2]: μu + (λ + μ)∇ div u + β∇φ = 0, αφ − ξφ − β divu = 0,
(1)
Methodology of Determination of Porosity Parameters
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where λ and μ are Lamé coefficients, and α is the voids diffusion coefficient; β is a parameter of the connection between microdilatation and macrodilatation properties (stress parameter with porosity change), and ξ is the void stiffness coefficient; the function φ describes the change in the volume fraction of pores. In the case β = 0, we have elastic deformation without pores. To determine the porosity parameters of the microdilatation theory, we will use the scheme, proposed in [3]. For this, we consider two contact problems for solids with similar geometries, the properties of which are described by different relations in the mechanics of deformable solids. As the first problem (Problem 1, see Fig. 1), we consider in rectangular coordinates (x, y) symmetric along the x, the problem about interaction between a stamp and strip 0 ≤ x ≤ h, the properties of which are described by the Cowin-Nunziata relations. We assume that the strip lies on a rigid base without friction, the stamp is also indented in the strip without friction to a depth δ; contact area is |x| ≤ a.
Fig. 1. To the formulation of problem 1.
Fig. 2. To the formulation of problem 2.
As the second problem (Problem 2, see Fig. 2), we consider in rectangular coordinates (x, y) the plane contact problem of elasticity theory about the interaction between a stamp and rectangle (|x| ≤ b, 0 ≤ y ≤ h) with uniformly distributed voids of the circle shape with radius R; contact area is |x| ≤ a. Here, as in the first problem, it is assumed that the stamp moves translationally in the negative direction of the y axis to a depth δ. Under conditions of the plane deformation in rectangular coordinates (x, y), Problem 1 is reduced to solving the system of differential equations [7]: 2 2 ∂φ ∂ 2u 2∂ u 2 ∂ w +H = 0, + c + 1 − c 2 2 ∂x ∂y ∂x∂y ∂x 2 2 ∂ 2w ∂φ 2∂ w 2 ∂ u +H = 0, + c + 1 − c 2 2 ∂y ∂x ∂x∂y ∂y 2 l12 ∂ 2φ ∂u ∂w 2 ∂ φ l1 + = 0. + 2 − 2φ− ∂x2 ∂y ∂x ∂y l2
(2)
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The components of the stress tensor are expressed through the displacement functions u, w and the function φ as follows: ∂w σx ∂u = + 1 − 2c2 + H φ, λ + 2μ ∂x ∂y (3) ∂u ∂w τxy σy ∂u ∂w = 1 − 2c2 + + H φ, = + , λ + 2μ ∂x ∂y μ ∂x ∂y β μ 1−2ν = 2(1−ν) ; H = λ+2μ ; l12 = αβ ; l22 = αξ ; ν is a Poisson’s ratio; α, β and where c2 = λ+2μ ξ are material parameters, characterizing the porosity properties of the strip. The boundary conditions of the assigned problem at y = h and y = 0 will be written in the form:
w = −δ + η(x) (y = h, |x| ≤ a), σy = 0 (y = h, |x| > a), τxy = 0, τxy = 0, w = 0,
∂φ = 0, (at y = h), ∂y
∂φ = 0, (at y = 0), ∂y
(4) (5) (6)
where δ is the displacement of the stamp; η(x) is the shape of the stamp base; |x| ≤ a is the contact area. The translational displacement of the stamp in the negative direction of the y-axis is given (Fig. 1). It is required to determine the value of the contact stresses: q(x) = σy (x, h) under the a stamp base and the value of the force: P = −a q(x)dx, applied to the stamp depending on the depth of movement δ of the stamp. Let us use the scheme for solving this problem, which was proposed in [7]. Preliminarily assuming that the contact stresses σy (x, h) = q(x) under the stamp are known, and using the representation of unknown functions in system (2) in the form of the Fourier transform to determine the displacements w(x) of the boundary y = h of the poroelastic strip after satisfying the boundary conditions (5), (6), we obtain similar to [7] the following relation: 2 w(x) = θπ ∞ k(x, τ ) =
a q(τ) k(x, τ) d τ, 0
L(u) cosux cosuτdu u
0
μ θ= . 1−ν
(7)
The function L(u), constructed using analytical transformations, has the following form: L(u) =
L1 (u) , L2 (u)
L1 (u) = −uS 2 W 2 N12 sh(Sh)[1 − ch(2uh)]c2 ,
(8)
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L2 (u) = Su[B1 ch(Sh) + B2 sh(Sh)], B1 = 2Nc2 u3 WN1 [1 − ch(2uh)], B2 = −SWN1 N S 2 − u2 c1 + Wc2 sh(2uh) − 2ShuW 2 N1 c2 + N1 .
The following designations are introduced here: S = l2−1 l22 u2 − N + 1, N =
l22 l1−2 H = β2 ξ−1 (λ + 2μ)−1 , W = S 2 − u2 , c1 = 1 + 2c2 , c2 = c2 − 1, N1 = N − 1; N is a dimensionless quantity, characterizing the porosity of the strip. In [8], it is shown that 0 ≤ N < 1 − c2 = 0.5(1 − ν)−1 . Note, that the larger N , the greater the porosity of the strip material. The function L(u) at infinity and zero demonstrates the following behavior: lim L(u) = 1 (u → ∞), lim L(u)u−1 = A (u → 0), A=−
h(ν − 1)(N − 1) . 2(ν − 1)[2N (ν − 1) + 1]
(9)
By using the boundary condition (4), we obtain to determine contact stresses an integral equation (IE) with a logarithmic kernel: a
π q(τ ) k(x, τ ) d τ = − θδ (0 ≤ x ≤ a). 2
(10)
0
To solve the IE, we will use the collocation method [9]. With this aim, we will divide = a , a the segment [0, a] into n segments l j j−1 j , aj = εj (j = 1, .., n), ε = a/n and
take the set of points τj = aj + aj−1 /2 = ε(j − 1/2) (j = 1, 2, ..., n). We will assume, that the contact stresses on the segment lj are constant and equal qj . Then the integral equation can be reduced to solving the system of equations: ε
n
aij qj =
j=1
∞ aij =
πθ δ (i = 1, 2, ...n − 1), 2
L(u) cos uτi sin uaj − sin uaj−1 du, u
(11)
(12)
0
a P=2
q(x)dx = 2ε 0
where qj = q(τj ).
n j=1
qj ,
(13)
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It can be shown [10], that: aij = a1, j−i+1 (2 ≤ i ≤ n − 1, i ≤ j ≤ n), aij = a1, i−j+2 (2 ≤ i ≤ n − 1, 1 ≤ j ≤ i − 1).
(14)
Relations (14) make it possible to reduce the number of calculations by n times. It is enough to calculate the elements of only the first row of the systems matrices (11) and (12), and the other elements are expressed through them. The boundary conditions of Problem 2 at y = h, y = 0 and |x| = b will be written in the form: w = −δ + η(x) (y = h, |x| ≤ a), σy = 0 (y = h, a < |x| < b), τxy = 0, uy = 0 (aty = 0), τxy = 0, σy = 0 (|x| = b).
(15)
We also assume that there are no stresses at the pore boundaries. In calculations, we select the half-width value of the rectangle b from the condition that it does not influence on the value of contact stresses within a given accuracy. We will denote the distance between the centers of any two adjacent pores horizontally and vertically by d, and the distance between the edges of the rectangle and the centers of the nearest pores will be considered equal to d/2. In this case, the number of pores n in the rectangle, depending on its parameters, will be equal to n = 2bh/d 2 . Let us enter the volume fraction of pores (reference porosity) in the rectangle and define it as the ratio of the total pores area to the area of the entire rectangle f = /(2bh). Then the radius of the voids will be expressed through f and n as follows: nπ R2 R = 2f bh/(nπ ). To find the contact stresses and the value of the applied force P in problem 2, we use the finite element method using the ANSYS software. The finite element model was built using 8-node finite elements (FE) 2-nd order PLANE82 with a plane strain option, response, and contact FE TARGE169 and CONTA175 were used to model the contact, which ensured the necessary calculation accuracy.
3 The Scheme for Finding Microdilatation Parameters In accordance with the assigned problem, to determine the microdilatation parameters α, β, and ξ we will use the scheme posed in [3]. As in [3], we will assume the parameter α = 105 N . The parameter β is determined using the computational experiment for an individual pore within the framework of the elasticity theory when the value of the microdilatation function φ is specified as the difference in porosity f before and after application of the load. When carrying out the calculations, the ANSYS finite element software was used, which made it possible to find the parameter β for materials with specified properties and pore distribution parameters. As the calculations result, the parameter values β = β∗ 1011 were obtained depending on f . As an example, values of β∗ are given in Table 1 for steel, copper, and aluminum. Note, that Young’s modulus and Poisson’s ratio were considered respectively equal to E = 2 · 1011 Pa, ν = 0.3 for steel; E = 1 · 1011 Pa, ν = 0.32 for copper, and E = 0.7 · 1011 Pa, ν = 0.3 for aluminum.
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Table 1. Values of β∗ for steel, copper, and aluminum depending on f. f
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Steel
2.62
2.77
2.93
3.12
3.33
3.57
3.84
4.16
4.54
4.99
Copper
1.46
1.54
1.63
1.73
1.85
1.98
2.13
2.31
2.52
2.78
Aluminum
9.18
9.70
1.03
1.09
1.17
1.25
1.34
1.46
1.59
1.75
Based on the results given in Table 1, the interpolation polynomial was constructed as
β = a5 f 5 + a4 f 4 + a3 f 3 + a2 f 2 + a1 f + a0 · 1011 (0.05 ≤ f ≤ 0.5),
(16)
the coefficients of which are given in Table 2. Table 2. Coefficients of the interpolation polynomial (16) for steel, copper, and aluminum. ai
a5
a4
a3 10.707
Steel
22.063
−16.670
Copper
13.730
−11.062
Aluminum
8.6552
−6.9754
a2
a1
a0
0.66696
2.7192
2.4824
6.7233
0.24369
1.5154
1.3802
4.2385
0.15298
0.95473
0.86955
To find the parameter ξ depending on the volume fraction of pores f , taking into account the relation ξ = β2 N −1 (λ + 2μ)−1 and the values of β, we need to find N. To do this, we will use the solution of the two problems stated above. First of all, let us solve Problem 2 and find the values of the contact stresses q(x) and the force P acting on the stamp, for the given value of its displacement δ = 0.01, depending on the volume fraction of pores f and their number n, for given geometric dimensions of the rectangle, the value of the contact area a, its elastic parameters Young’s modulus E, Poisson’s ratio ν. Further we proceed to solving Problem 1, assuming that the parameters β and α are known, and the strip thickness h, the contact area size a, the stamp displacement δ and the elastic constants E and ν are taken the same as in Problem 2. In the process of solving Problem 1, we select the value of N such, that the value of the force P, applied to the stamp coincides with the value of the force P, obtained in Problem 2. Knowing N, the parameter ξ can be found using the formula ξ = β2 N −1 (λ + 2μ)−1 . Such procedure for fixed parameter f was carried out for different quantity of voids n in Problem 2 with a = 1, h = 1, b = 5. For the cases of steel, copper and aluminum, the results are shown in Table 3. Note, that in the calculations, the number of pores n was chosen from the condition that with further its increase, the result does not change. Maximum of n = 4096.
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f
0.05
0.1
0.2
0.3
0.4
0.5
N (Steel)
0.261
0.376
0.499
0.564
0.608
0.640
N (Copper)
0.280
0.400
0.526
0.590
0.633
0.664
N (Aluminum)
0.261
0.376
0.499
0.564
0.608
0.640
As can be seen from Table 3, the values of the parameter N for steel and aluminum turned out to be the same and differ from the parameter values for copper. Taking this into account, calculations were carried out to find the values of parameter N for materials with different values of Young’s modulus, Poisson’s ratio, and parameter f . Table 4 shows some results of the corresponding calculations. To control the accuracy of the found parameters, we consider Problems 3 and 4, like Problems 1 and 2, respectively, when h and b take other values. We change the volume fraction of pores f from 0.05 to 0.5 with a step of 0.05 and set the corresponding elastic constants and porosity parameters of the microdilatation theory, found solving Problems 1 and 2. The calculations results of the force value
when P ∗ = 2P 1 − ν2 /E are shown in Table 5 for problems 3 and 4 at h = 0.5, which show that the value of the force P ∗ in Problems 3 and 4 is practically the same. Calculations Table 4. Values of parameter N for materials with different values of Young’s modulus, Poisson’s ratio and parameter f. E · 10−11
ν
β · 10−11
N
f = 0.1 2
0.3
2.7718
2
0.45
1
0.3
1.3858
0.376
1
0.45
5.5434
0.702
11.087
0.376 0.702
f = 0.3 2
0.3
2
0.45
3.5677
1
0.3
1.7838
0.564
1
0.45
7.1351
0.838
2
0.3
4.9985
0.640
2
0.45
1
0.3
2.4993
0.640
1
0.45
9.9963
0.877
14.270
0.564 0.838
f = 0.5 19.993
0.877
Methodology of Determination of Porosity Parameters
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are given for steel with parameters E = 2 · 1011 Pa, ν = 0.3. A similar situation is observed for copper and aluminum. Note, that in Table 5, the values of P* for problem 4 are given in column 5 with the number of pores n = 1024, and in column 6 – with n = 4096. Table 5. Value of P* for problem 3 and 4. f
β·1011
N
P* (Problem 3)
P* (Problem 4)
P* (Problem 4)
1
2
3
4
5
6
0.05
2.621
0.261
7.450
7.434
7.435
0.1
2.770
0.376
6.585
6.567
6.456
0.3
3.566
0.564
4.187
4.180
4.2160
0.5
4.995
0.640
2.506
2.550
2.560
4 Conclusions 1. The methodology has been developed for determining the parameters of the CowinNunziato microdilatation theory based on the comparison of solutions of two contact problems modeling poroelastic materials with unfilled pores on the base of the different approaches: the analytical method using the relations of the microdilatation theory and the finite element method. 2. The calculations showed that the parameter value N and, respectively, the parameter α does not depend on the elasticity modulus E, but depends only on the Poisson’s ratio ν, in this case, the values of parameters β and ξ are influenced by both the elastic modulus and Poisson’s ratio. Acknowledgements. The study was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation (State task in the field of scientific activity, scientific project No. FENW-2023-0012).
References 1. Nunziato, G.W., Cowin, S.C.: A nonlinear theory of elastic materials with voids. Arch. Ration. Mech. Anal. 72, 175–201 (1979) 2. Cowin, S.C., Nunziato, G.W.: Linear theory of elastic materials with voids. J. Elast. 13, 125–147 (1983) 3. Bishay, P.L., Repka, M., Sladek, V., Sladek, J.: On the characterization of porosity-related parameters in micro-dilatation theory. Acta Mech. 228, 1631–1644 (2017) 4. Lurie, S.A., Kalamkarov, A.L., Solyaev, Y.O., Ustenko, A.D., Volkov, A.V.: Continuum microdilatation modeling of auxetic metamaterials. Int. J. Solids Struct. 132–133, 188–200 (2018)
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5. Kudimova, A.B., Nadolin, D.K., Nasedkin, A.V., Nasedkina, A.A., Oganesyan, P.A., Soloviev, A.N.: Models of porous piezocomposites with 3–3 connectivity type in ACELAN finite element package. Mater. Phys. Mech. 37(1), 16–24 (2018) 6. Nasedkin, A.V., Nasedkina, A.A., Rybyanets, A.N.: Numerical analysis of effective properties of heterogeneously polarized porous piezoceramic materials with local alloying pore surfaces. Mater. Phys. Mech. 40(1), 12–21 (2018) 7. Scalia, A.: Contact problem for porous elastic strip. Int. J. Eng. Sci. 40, 401–410 (2002) 8. Scalia, A., Sumbatyan, M.A.: Contact problem for porous elastic half-plane. J. Elasticity. 60, 91–102 (2000) 9. Belotserkovsky, S.M., Lifanov, I.K.: Numerical Methods in Singular Integral Equations and Their Applications to Aerodynamics, Theory of Elasticity and Electrodynamics. Nauka, Moscow (1985). (in Russian) 10. Chebakov, M.I., Kolosova, E.M.: Contact interaction between parabolic punch and elastic strip bonded to poroelastic half-plane. In: Parinov, I.A., Chang, S.-H., Kim, Y.-H., Noda, N.-A. (eds.) PHENMA 2021. SPM, vol. 10, pp. 321–330. Springer, Cham (2021). https://doi. org/10.1007/978-3-030-76481-4_27
Molecular Dynamic Study of Dependency on Mechanical Characteristic of Nanocrystalline Copper over Various Temperature and Strain Rate Mahesh Kumar Gupta1 , Santosh Kumar Rai4 , Vinay Panwar2 , I. A. Parinov3 and Rakesh Kumar Haldkar3(B)
,
1 Department of Mechanical Engineering, SRM Institute of Science and Technology,
Delhi-NCR Campus, Modinagar, Ghaziabad 201204, UP, India [email protected] 2 Mechanical Engineering Department, Netaji Subhas University of Technology, Dwarka, New Delhi 110078, India [email protected] 3 I. I. Vorovich Mathematics, Mechanics and Computer Sciences Institute, Southern Federal University, Rostov-on-Don 344090, Russia [email protected], [email protected] 4 Inochi Care PVT LTD., Okhla phase 1, New Delhi 110017, India [email protected] Abstract. Nanocrystalline copper is one of the best suited materials for integrated chips industries due to its high mechanical stability and less resistivity. In the present work, the molecular dynamic simulation approach is employed to analyze the mechanical properties and potential energy of single crystal nano copper. The response of temperature as well as strain rate on selective properties has been explored. Nanocrystalline copper deformation under virtual uniaxial tensile test has demonstrated that the increase in strain rate from 1.0 × 108 /s to 5.0 × 1010 /s provides a significant change in mechanical properties of nano copper. In addition, the outcomes reveal that the mechanical properties of crystalline nano copper degraded with temperature response under uniaxial tensile loading. With increasing temperature from 50 to 500 K, potential energy response shows an increase in instability of structure at higher temperatures. The results may help to accelerate functional applications of nanocrystalline copper at high temperatures subjected to a different levels of strain rates. Keywords: Nano copper · Energy · Molecular Dynamic Simulation · Temperature · Single Crystal
1 Introduction Nanocrystalline materials are popular due to their excellent properties compared to corresponding bulk material. In recent times, electronic technology targeting smaller and simpler parts, the integration circuit (IC) chips are in the same continuation as density is The original version of this chapter has been revised. A correction to this chapter can be found at https://doi.org/10.1007/978-3-031-52239-0_61 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024, corrected publication 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 345–354, 2024. https://doi.org/10.1007/978-3-031-52239-0_33
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increasing it approaches to more compact in size accordingly. Applications of electroplated copper are not just limited to integrated chips but rather used in applications such as printed wiring boards, and semiconductors through silicon due to low resistivity and high thermal conductivity [1]. In the last, some decades nanocrystalline materials was one of the very emerging fields for scientists and researchers. In recent times nanocrystalline materials are of critical interest due to their very important characteristics as they have a high surface area to volume ratio. They are having increased strength and other properties due to more surface area compared to their volume, which is not in the case of the same counterpart of bulk form. Nanocrystalline materials are superior with their bulk form in ductility, toughness, density, electrical resistivity, thermal conductivity, excellent wear resistance, and high fatigue strength. Nanocrystalline materials have outstanding properties due to their smaller grain size [2–4]. The unique properties of nanomaterials are due to their unique size, shape, and structure. Nanomaterials are widely used in all disciplines of materials applications due to their excellent properties. They are suitable candidates in almost all commercial as well as domestic applications including medical components, energy-based applications at higher temperatures. Temperature and strain rate changes arouse due to several factors during these applications and play a significant role in determining the response of nanocrystalline material. Nanocrystalline material generally refers to the grain size from 1–100 nm. Nanocrystalline material is generally known for its excellent properties as high hardness and yield strength. The mechanical behaviour of copper in the bulk form are much lower than in the nanocrystalline form [5, 6]. These properties are the result of a greater area percentage of grain boundaries to grains, that are essentially responsible to act as a hurdle to the atomic movement or dislocation motion. The other factors which are also accountable for the excellent properties of materials like lattice parameter, grain orientation, types of grain boundary, grain size and impurities present in the nanoscale [7, 8], therefore excluding the effect of the listed parameters for a more precise understanding of thermal and strain rate effect on nanocrystalline copper. Using molecular dynamic simulation, the reaction of nanocrystalline platinum to the temperature effect was investigated. It was hypothesized that when the temperature raises, mechanical parameters, including ultimate strength, yield strength, and flow stress decrease [9]. Chen P. et al. investigated the impact of temperature and grain size on polycrystalline copper. The findings show that elastic modulus decreases with temperature increase and that grain boundary sliding, and rotation of grain is the process causing deformation at the nanoscale. In contrast, the consequences of dislocation nucleation and migration are longer than those seen at the nanoscale size [10]. A molecular dynamic study on copper nanowire was done by Liang W. and Zhou M. [11] to study the tensile deformation mechanism and strain rate effect on mechanical properties. The study suggested that Young’s modulus is independent of strain rate, the size of the specimen and yields strength decreases with increasing strain rate. The ductility of copper increases with the increasing size of the specimen and it can be due to more opportunities available for dislocation motion. According to Lu L. et al. [12], who reported an anomalous strain rate effect, nanocrystalline copper becomes more ductile as the strain rate rises. Horstemeyer, M. F. and Baskes, M. I. [13] investigated the impact of size and time scale on the plastic deformation of bulk copper, and they claimed that samples with less than
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1000 atoms have a weak dependence on strain rate due to fewer possibilities for slip deformation. Large samples are yielding strength independent at lower strain rates, but they follow a linear relationship at larger strain rates. The molecular dynamic simulation has been extensively studied to examine the deformation behaviour of nanocrystalline metals and alloys. The molecular dynamic modelling has yielded useful knowledge regarding the mechanism of deformation and sensitivity to strain rate [14–16], temperature [17–21], grain size [17, 18], size of the cell of polycrystalline structure, and size of nanowire of metals, bulk materials.
2 Simulation Model and Methods To evaluate the underlying change in mechanical properties with temperature and strain rate variation in nanocrystalline structure, an atomistic simulation method is a crucial tool [22]. Atomistic simulation has been used to investigate how the behaviour of single crystal nanocrystalline copper changes. A single crystal of nanocrystalline copper was used for the molecular dynamic simulation. With lattice parameters of 3.61, a copper single crystal is a face-centered cubic (FCC). It has been decided to use a single nanocrystalline copper crystal with the same sizes in all three directions of 16 × 16 × 16 nm3 . Figure 1a shows a model that has been created with identical specifications. The crystal x-direction has a preset strain rate applied to it, while the y- and z-directions are unaffected. While varying the strain rate in the x-direction, the crystal response was observed. The crystal is oriented as follows in each of the three directions: {100}, {010}, and {001}. A high strain rate is necessary for computational approaches like molecular dynamic (MD) simulation to decrease the time steps. The requirement for high-frequency thermal oscillation of molecules limits MD simulation. To obtain high rates of deformation, strain levels should be high for the use of practically computer systems available. The potential function, developed by Zhou et. al, was used for the present simulation that is embedded atom method (EAM) potential [23]. In this MD simulation, equilibrium boundary conditions have been used. Boundary conditions applied for simulation were periodic in all three directions to maintain the integrity of the system during deformation at different strain rates. The time step for the simulation was one femtosecond (fs). Prior to the loading condition model needs to be relaxed with an ensemble NVT, that is number of atoms – volume – temperature, for 50 pico-seconds. The conjugate gradient (CG) technique has been employed for energy minimization. Number of atoms – pressure – temperature (NPT) ensemble was used for 50 picoseconds to maintain 0 pressure before deformation. Crystal has been gradually deformed in the x-direction by keeping NPT in the y- and z-direction to ensure a one-directional loading condition. Another important observation has been made when an in-depth real-time analysis of the uniaxial tensile test has been performed and found a change in the FCC structure of copper atoms at subsequent time steps. Prior to deformation, the percentage of FCC copper atoms was 100, which means there are no other defects present in crystal in the form of BCC or HCP structures. However, as soon as the crystal is put under uniaxial tensile simulation, the composition of FCC copper drops to 2.1% at a strain of 5%. It is followed by the next increase in this composition to 8.7% at a strain of 10%. The remaining atoms, therefore, get converted to BCC or HCP atoms, which are present in the form of strained atoms (see Fig. 1).
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In this paper, investigation of the mechanical behaviour of a single crystal, nanocopper has been done with the molecular dynamic simulation. The mechanical behaviour of a single crystal nanocrystalline copper with temperature ranges, which is below room temperature at 50 K to more than room temperature at 500 K at a fixed strain rate of 5.0 × 109 s−1 was studied. Along with temperature response on the engineering stressstrain curve of nano copper single crystal, a strain rate variation from a low strain rate of 1.0 × 108 s−1 to a high strain rate up to 5 × 1010 s−1 has also been studied at 300 K. Additionally potential energy response of the crystal with temperature and strain rate has also been analysed in the present work. This study helps to explore applications of single crystal copper in the different temperature ranges and with the highest strain rates up to 5 × 1010 s−1 .
Fig. 1. Computational model of copper: (a) before relaxation, (b) after minimization, (c) 5% strain, (d) 10% strain
3 Results and Discussion Engineering stress-strain response of a single crystal nano copper has been plotted in Fig. 2a with a fixed strain rate of 5.0 × 109 s−1 and variation in temperature from below room temperature, that is 50 K to above room temperature, that is 500 K. The change of stress-strain curve can be noticed from Fig. 2a, and it can be depicted that as the temperature varies from the lower side to upper side the area under the stress-strain curve decreases. The change in the engineering curve, shifted in the lower direction and covered a lesser area with an increment in temperature, can be observed in Fig. 2a. The variation of ultimate tensile strength and yield strength of single crystal nano copper has been plotted in Fig. 2b and corresponding variation in elastic modulus (Young’s modulus) with temperature has been plotted in Fig. 2c.
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3.1 Effect of Temperature on Mechanical Properties and Elastic Constant at a Fixed Strain Rate of 5.0 × 109 s−1 Nanocrystalline single crystal copper under uniaxial tensile loading at various temperatures and fixed strain rate is plotted in Fig. 2 as mentioned earlier. The data, obtained from the stress-strain curve of nanocrystalline copper in Fig. 2a, are analysed to refine associated properties such as yield strength, ultimate strength. From Fig. 2b temperature effect on maximum stress capacity and yield strength can be depicted. There is an inverse relationship, observed between mechanical properties and temperature. At 50 K, yield strength of a single crystal nano copper is 7.82 GPa. As the temperature increased from 50 K to room temperature, yield strength approaches 6.07 and at 500 K its value reaches 3.86 GPa. The ultimate strength is initially 14.11 GPa at a temperature of 50 K. With the variation in temperature from 50 K to 100 K, correspondingly ultimate tensile strength decreases to 12.0 GPa. At room temperature (300 K) the ultimate tensile strength is 7.18 GPa and as temperature increases by 200 K above room temperature, the ultimate tensile strength decreases to 4.64 GPa at 500 K. The inverse relation of mechanical properties of a single crystal nano copper with temperature can be explained, with the rise in temperature bonding strength of the atoms decreases and consequently the stress, required
Fig. 2. Single crystal nano copper at discrete temperatures range under fixed strain rate: (a) stressstrain behaviour, (b) variation in yield strength and ultimate tensile strength, (c) variation in elastic constant
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to break the material, also decreases. Low stress is required to break the bond resulting in less strength of the material at high temperatures. Figure 2c depicts the change of Young’s modulus with the discrete temperatures of 50 K, 100 K, 200 K, 300 K, 400 K and 500 K at a fixed strain rate of 5.0 × 109 s−1 . Young’s modulus is 66.88 GPa at 50 K and decreases to 52.12 GPa at room temperature. From 50 K to 500 K decrement in Young’s modulus is found to be almost 40%. This could be seen clearly in Fig. 2c. The engineering stress-strain curve of the corresponding model at a range of temperatures correctly addresses the superior mechanical properties of nanocrystalline copper at low (50 K) temperature and thereby shows a drop in mechanical properties by increasing temperature up to 500 K in subsequent runs of uniaxial tensile loading. By increasing the temperature internal energy of the crystal system becomes high resulting in vigorous vibrations of atoms in the crystal system. Due to vigorous vibration movement at atomic level becomes easier which subsequently decreases the amount of stress necessary for separation of the dislocations from their equilibrium positions [24]. This could be an important parameter in deciding the selection of nanocrystalline copper, where the operating temperature of the actual application scenario is pre-known.
Fig. 3. Single crystal nano copper at 300 K under different strain rates: (a) stress-strain behaviour, (b) variation in yield and ultimate strength, (c) variation in elastic constant
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3.2 Effect of Strain Rates on Mechanical Properties and Elastic Constant at a Fixed Temperature of 300 K Figure 3a depicts the engineering stress-strain relation of single crystal copper at 300 K. The strain rates used in the uniaxial tensile simulation are in the range from 1 × 108 s−1 to 5 × 1010 s−1 . Figure 3a represents the shape of engineering stress-strain curve in the elastic region. The curve follows a similar pattern at different strain rates but attains higher values when the strain rate is high. At a higher strain rate, curve shifts upward, and it is interesting to note that the curve reaches the peak value before dropping in a steep manner and finally this drop becomes steady. Figure 3b represents yield strength and ultimate tensile strength in relation to strain rate. Figure 3c represents variation in Young’s modulus with strain rate at room temperature. In Fig. 3, there is a rise in the mechanical properties of nanocrystalline copper with varying strain rates from 1 × 108 s−1 to 5 × 1010 s−1 . Yield and ultimate tensile strength increased from 5.24 to 6.95 GPa and from 6.48 to 9.45 GPa, respectively. The response of elastic constant (Young’s modulus) with the variation of strain rate can be depicted in Fig. 3c, where Young’s modulus at 1 × 108 s−1 equal to 52.7 GPa approaches 59.2 GPa at a higher strain rate of 5 × 1010 s−1 . It can be inferred from the above discussion that strain rates have not as much impact on Young’s modulus as in other mechanical properties. The effect of increasing strain rate in nano copper shows the upward shift of curve. This is because a shorter time is given at a higher value of strain rate for the dislocation movement to consume less amount of energy or work done by the surroundings and it also avert the single crystal nano copper to reach its equilibrium positions. It is caused by lesser time available for dislocation movement at higher strain, which have led to increase in the required stress value [25]. Thus, it is convenient to say that higher strain rates are responsible for the higher mechanical properties in nanocrystalline copper. 3.3 Potential Energy Dependence of Single Crystal Nano Copper on Strain Rate and Temperature Potential energy variation with the temperature rise at a particular strain rate of 5.0 × 109 s−1 is depicted in Fig. 4. The graphs reveal that potential energy has a negative value of −57.9 × 103 eV at the temperature of 50 K. It means that this great energy (57.9 × 103 eV) is necessary to produce a single vacancy of the crystal system to make this energy to 0 eV. However, there is a drop in this required energy when the temperature is increased. At 500 K, the potential energy becomes −56.8 × 103 eV and 56.8 × 103 eV of energy is necessary to produce a single vacancy. Moreover, Fig. 4 shows basic structure of the potential energy curve; its shape does not disturb with changing temperature rather than shifting in upward direction due to variation in potential energy of the system. The response of potential energy with variation in strain rate is another aspect of this study and this dependence is shown in Fig. 5. It is important to analyse the behaviour of the potential energy curve with the strain rate parameter. In low strain rates such as 1.0 × 108 s−1 , the potential energy curve shows a minute constant negative slope in subsequent time steps. While varying in strain rate from low to high value of 5 × 108 s−1 , the potential energy shows approximately constant curve with a little rise at later time steps. By further increase in strain rate, potential energy shows a sudden increase
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to the maximum value and a drop with a waviness at later time steps. The shape of the curve is completely different from Fig. 5, where the relative change in the potential energy with temperature is shown. Figure 5 depicts that potential energy decreases by raising temperature from 50 to 500 K. Temperature rise is responsible for the change in atomic vibrations of crystals. At lower temperatures, the atomic vibrations are less, correspondingly the kinetic energy of the atom is also less and by increasing temperature, the kinetic energy increases, resulting in decrement of the potential energy of the crystal.
Fig. 4. Potential energy response of single crystal nano copper at 5 × 109 s−1
Fig. 5. Potential energy response of single crystal nano copper at 300 K
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4 Conclusion The single-crystal copper has been investigated to reveal the relation of mechanical properties and potential energy with temperature and strain rate, using the molecular dynamics approach. The response of mechanical behaviour of nano copper on temperature (from 50 to 500 K) and strain rate (from 1 × 108 s−1 to 5 × 1010 s−1 ) was investigated on a cubic sample of 16 nm by using EAM potential. Results of uniaxial tensile loading showed a great influence of temperature variation on the mechanical behaviour of nanocrystalline copper. Rise in temperature of nano copper reveals that mechanical properties change inversely. The change in temperature from 50 to 500 K results in a decrease of Young’s modulus by 38%, from 66.88 to 41.27 GPa, yield strength by 50%, and ultimate tensile strength by 67%, respectively. Apart from this, the strain rate was also checked for its influence on mechanical properties. It was found that with the increase in strain rate, there is a marginal change or increase in Young’s modulus by 12%, an increase in yield strength by 68%, and an increase in ultimate tensile strength by 48%, respectively. This increment is due to a better response of the material at a higher strain rate. The potential energy is found to decrease from 57.9 × 103 to about 56.8 × 103 eV by increasing the temperature, whereas in the case of strain rate, this dependence is different at low and high strain rates. For lower strain rates the potential energy does not fluctuate much but at higher strain rates the potential energy initially rises before dropping in a wavy pattern. Hence with this analysis, an in-depth understanding of mechanical responses of nanocrystalline copper have been studied at various parameters which will provide a base for carrying out further studies in this field. Acknowledgements. The study was financially supported by the Ministry of Science and Higher Education of the Russian Federation, State task in the field of scientific activity, scientific project No. FENW-2023-0012.
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9. Li, J., et al.: Molecular dynamics simulation of mechanical properties of nanocrystalline platinum: grain-size and temperature effects. Phys. Lett. A 383(16), 1922–1928 (2018) 10. Chen, P., Zhang, Z., Liu, C., An, T., Yu, H., Qin, F.: Temperature and grain size dependences of mechanical properties of nanocrystalline copper by molecular dynamics simulation. Modelling Simul. Mater. Sci. Eng. 27, 065012 (2019) 11. Liang, W., Zhou M.: Response of copper nanowire in dynamic tensile deformation. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 218(6), 599–606 (2004) 12. Lu, L., Li, S.X., Lu, K.: An abnormal strain rate effect on tensile behavior in nanocrystalline copper. Scr. Mater. 45, 1163–1169 (2001) 13. Horstemeyer, M.F., Baskes, M.I.: Atomistic finite deformation simulations: a discussion on length scale effects in relation to mechanical stresses. Trans. ASME J. Eng. Mater. Technol. 121(2), 114–119 (1999) 14. Zhou, K., Liu, B., Shao, S., Yao, Y.: Molecular dynamics simulations of tension–compression asymmetry in nanocrystalline copper. Phys. Lett. A 381, 1163–1168 (2017) 15. Zhang, T., Zhou, K., Chen, Z.Q.: Strain rate effect on plastic deformation of nanocrystalline copper investigated by molecular dynamics. Mater. Sci. Eng. A 648, 23–30 (2015) 16. Xiang, M., Cui, J., Tian, X., Chen, J.: Molecular dynamics study of grain size and strain rate dependent tensile properties of nanocrystalline copper. J. Comput. Theor. Nanos. 10, 1215–1221 (2013) 17. Alam, M.F., Shahadat, M.R.B.: Temperature and strain rate dependent mechanical properties of ultrathin metallic nanowires: a molecular dynamics study. AIP Conf. Proc. 19(80), 030015 (2018) 18. Wang, W., Yi, C., Fan, K.: Molecular dynamics study on temperature and strain rate dependences of mechanical tensile properties of ultrathin nickel nanowire. Trans. Nonferr. Met. Soc. China 23, 3353–3361 (2013) 19. Ma, B., Rao, Q., He, Y.: Molecular dynamics simulation of temperature effect on tensile mechanical properties of single crystal tungsten nanowire. Comp. Mater. Sci. 117, 40–44 (2016) 20. Gan, Y., Chen, J.K.: Molecular dynamics study of size, temperature and strain rate effects on mechanical properties of gold nanofilms. Appl. Phys. A 95, 357–362 (2009) 21. Li, W., Li, K., Fan, K., Zhang, D., Wang, W.: Temperature and pressure dependences of the elastic properties of tantalum single crystals under tensile loading: a molecular dynamics study. Nanoscale Res. Lett. 13, 118 (2018) 22. Pan, Z., Li, Y., Wei, Q.: Tensile properties of nanocrystalline tantalum from molecular dynamics simulations. Acta Mater. 56, 3470–3480 (2008) 23. Zhou, X.W., Johnson, R.A., Wadley, H.N.G.: Misfit-energy-increasing dislocations in vapordeposited CoFe/NiFe multilayers. Phys. Rev. B 69, 144113 (2004) 24. Li, Z., Gao, Y., Zhan, S., Fang, H., Zhang, Z.: Molecular dynamics study on temperature and strain rate dependences of mechanical properties of single crystal Al under uniaxial loading. AIP Adv. 10, 075321 (2020) 25. Zeng, F., Gao, Y., Li, L., Li, D.M., Pan, F.: Elastic modulus and hardness of Cu–Ta amorphous films. J. Alloys Compd. 389(1), 75–79 (2005)
Applied Theory of Vibrations of a Composite Electromagnetoelastic Bimorph with Damping A. N. Soloviev1,2
, V. A. Chebanenko2,3(B) , B. T. Do4 , A. V. Yudin2,3,5 , and I. A. Parinov2
1 Crimean Engineering and Pedagogical University the name of Fevzi Yakubov,
Simferopol 295015, Russia 2 I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal
University, Rostov-on-Don 344006, Russia [email protected] 3 Federal Research Center “Southern Scientific Center of the Russian Academy of Sciences”, Rostov-on-Don 344006, Russia 4 Le Quy Don Technical University, Hanoi 100000, Vietnam 5 M. I. Platov South-Russian State Polytechnic University, Novocherkassk 346428, Russia
Abstract. In this paper, we considered the plane problem of steady-state bending vibrations of a hinged-supported bimorph in an alternating magnetic field, considering damping. The bimorph under study consisted of a multilayer electromagnetoelastic composite modeled using the effective modulus approach. Within the framework of Kirchhoff’s hypotheses, an applied theory was constructed, which considered the quadratic distributions of electric and magnetic potentials along the thickness of the bimorph and considered heterogeneity in the longitudinal direction. In addition, the theory introduced linear viscoelasticity. Using the resulting theory, distributions of electric, magnetic, and mechanical fields were constructed. Comparison of the obtained results, based on the theory, with the results of the finite element model in the COMSOL Multiphysics package showed a good convergence of the results. Next, based on the obtained theory, the behavior of plate de-flection in the vicinity of the first resonance was studied at various values of the damping coefficient, which showed that the obtained theory can be used to analyze the bandwidth of converters, based on constructing the amplitude-frequency characteristics of bimorphs made of an electromagnetoelastic composite. Keywords: Smart materials · Electromagnetoelastic · Viscoelasticity · Forced Vibrations · Damping
1 Introduction Piezoelectric materials are widely used in a variety of applications, including mechanical engineering and aerospace, where they are used as actuators, sensors, and generators. Devices based on them play an important role in structural monitoring, energy harvesting, spurious vibration suppression, and noise cancellation applications. A more detailed overview of this topic can be found in [1, 2]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 355–364, 2024. https://doi.org/10.1007/978-3-031-52239-0_34
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The multilayer configuration is the basis for devices, based on piezoactive materials, operating on bending modes. This configuration includes a set of layers with different electrical and mechanical properties, which allows the creation of a wide range of actuators [3] and sensors [4]. There are also antiferromagnetic and ferrimagnetic crystals in which piezomagnetism is manifested. One of the widely studied piezomagnetic materials is CoFe2 O4 [5]. The combined manifestation of piezoelectric and piezomagnetic effects leads to the occurrence of a magnetoelectric coupling [6]. This opens opportunities for the creation of composite materials, based on CoFe2 O4 and BaTiO3 , which have both piezoelectric and piezomagnetic properties [7], united by magnetoelectric coupling. Previously, we developed a theory of oscillations of a bimorph, consisting of electroelastic and magnetoelastic layers, which showed good agreement with the data of finite element modelling (FEM) [8]. In the next paper [9], we presented an applied theory that considers the nonlinear distribution of electric and magnetic potentials in the longitudinal and transverse directions for a cantilever bimorph in an alternating magnetic field. This theory has been successfully compared with the results of finite element modeling. In this study, a plane problem of stationary bending vibrations of a hinged-feathered bimorph in an alternating magnetic field is considered with account of damping. The studied bimorph consists of a multilayer electromagnetoelastic composite modeled using the efficient modules approach. To solve this problem, an applied theory is constructed on the base of [9], which takes into linearly viscoelastic behavior. With the help of the obtained theory, the distributions of electric, magnetic, and mechanical fields, as well as the behavior of deflection in the vicinity of the first resonance, were constructed. The results were compared with the results of the finite element (FE) model in COMSOL Multiphysics.
2 Problem Statement This paper analyzes the problem of steady-state plane oscillations of an infinitely wide bimorphic plate, considering the linear-viscous elastic behavior of the material. This plate consists of two identical layers with electromagnetoelastic properties (see Fig. 1). Both layers have large surfaces that are covered by electrodes and polarized along their thickness. It is assumed that the bimorph layers are perfectly bound both mechanically and electrically and magnetically. The bimorph is hinged at both ends, with all other surfaces free from mechanical forces. The upper and lower boundaries of the plate are ˜ while the magnetic potential at the boundary between affected by the magnetic flux B, the layers is assumed to be zero. The electric potential is assumed to be zero at all electrodes. The end surfaces of the plate are insulated from magnetic and electric fields.
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Fig. 1. Diagram of the studied bimorph
To describe the vibrations of this plate, we will use general Eqs. (1) and the corresponding constitutive relations (2) and (3) for the electromagnetoelastic material, as shown in [10]: ¨ ∇ · σ + ρf = ρ u, ∇ · D = σ , ∇ · B = 0. ε = 21 ∇u + (∇u)T , E = −∇ϕ, B = −∇φ. σ = c : ε − e T · E − hT · H D=e :ε+κ·E+α·H B = h : ε + αT · E + μ · H.
(1)
(2)
(3)
In these expressions, the following notations are used: tensors of mechanical stresses σ and deformations ε, vectors of electric induction D and electric field strength E. In addition, the equations take into account the vectors of magnetic induction B and magnetic field strength H, material density ρ, elastic moduli tensor c, piezoelectric module tensor e, piezomagnetic module tensor h, the dielectric constant tensor m, the magnetoelectric module tensor α, the magnetic permeability tensor μ, the mass force density vector f , the bulk density of electric charges σ and the displacement vector u, and the electric ϕ and magnetic potentials ξ . It is important to note that we consider electric and magnetic fields to be stationary because we assume that elastic waves propagate much more slowly than electromagnetic waves. For the completeness of the formulation of the problem, it is also necessary to consider the relevant boundary and initial conditions. A complete description of the general formulation of the problem for an electromagnetoelastic material is presented in [10].
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3 Solution To solve this problem, let us derive an applied theory. To do this, we will use the variational Eq. (4) for the case of steady-state oscillations [10]. This equation generalizes Hamilton principle within the framework of the theory of electroelasticity, considering magnetic components as well. In the case of planar deformation without the presence ˜ the of surface and volumetric loads or charges, and in the presence of magnetic flux B, variational equation takes the form: ¨ ¨ ˜ dl = 0, δ H˜ dS − ρω2 ui δui dS + Bδξ (4) S
∂S
S
where δ H˜ = σij δεij − Di δEi − Bi δHi , S is the region of the bimorph. To simplify further calculations, let us use Kirchhoff’s hypotheses. According to these hypotheses, the condition σ33 = 0 must be met in the entire region of the bimorph, and the displacement vector is described as follows: u1 (x1 , x3 ) = −x3 w,1 , u3 (x1 , x3 ) = w(x1 ),
(5)
where w(x1 ) is the function, defining the deflection of the plate. Using the condition σ33 = 0, we can exclude ε33 from consideration. In addition, for the mechanical field, we will accept the hypothesis of a single normal. To account for the linear viscoelastic behavior of the material, let us add to the constitutive relations of the mechanical stress tensor (3) the term ωγ c˜ 11 ε11 , where ω is the circular frequency, and γ is the damping coefficient. After all operations, the constitutive relations (3) for the plane problem within the framework of Kirchhoff hypotheses will take the following form: T E −h ˜ T H3 σ11 = c˜ 11 ε11 (1 + iωγ ) − e˜ 31 3 31 T D3 = e˜ 31 ε11 + ˜33 E3 + α˜ 33 H3
˜ E + μ B3 = h˜ 31 ε11 + α˜ 33 ˜ 33 H3 . 3
(6)
The following notation is introduced here, which arose after eliminating ε33 : c˜ 11 = c11 −
2 c13 c33 ,
e˜ 31 = − c13c33e33 + e31 , h˜ 31 = − c13c33h33 + h31 , e2
h2
33 α˜ 33 = − e33c33h33 − α33 , ˜33 = − c33 − 33 , μ˜ 33 = − c33 − μ33 33
(7)
Within the framework of this model, two scalar potentials are considered: electric and magnetic. We will consider their quadratic shape, which depends on the longitudinal and transverse coordinates. This will make it possible to consider the boundary conditions on large surfaces for electric and magnetic fields within the framework of the accepted Kirchhoff hypotheses. Since the problem is symmetrical about the neutral axis, we will consider only the upper layer. Let us accept the general form of the scalar potential for the upper layer of the bimorph in the form: 4˜x32 x˜ 3 2˜x3 x˜ 3 2˜x3 − 1 + g1 (x1 ) 1 − 2 + g2 (x1 ) +1 (8) π (x1 , x˜ 3 ) = g0 (x1 ) H H H H H
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Here the relative coordinate x˜ 3 = x3 − H /2 for the top layer is introduced. The functions g0 (x1 ), g1 (x1 ) and g2 (x1 ) are responsible for the potential value at the lower boundary, in the middle of the layer and at the upper boundary, respectively. Since we are considering a situation where the electric potential at the electrodes can reach zero, its thickness distribution for the first oscillatory models is described as a nonlinear function. According to the results of the finite element analysis, this thickness distribution can be characterized as a quadratic relationship, as shown in [11]. To account for these conditions in the expression for the electric potential ϕ, we assume the following assumptions in (8): g0 (x1 ) = V0 = const, g1 (x1 ) = (x1 ), g2 (x1 ) = V2 = const.
(9)
Here, the function (x1 ) is an unknown variable. The magnetic potential at the inner boundary of the piezomagnetic layer is assumed to be zero, while its distribution at the outer boundary is unknown. For this reason, for the magnetic potential ξ , we assume the following assumptions in (8): g0 (x1 ) = M0 = const, g1 (x1 ) = 2 (x1 ), g2 (x1 ) = 3 (x1 ).
(10)
In this case, the functions 2 (x1 ) and 3 (x1 ) are the unknown variables. After substituting into Eq. (4), it undergoes thickness integration, and as a result, the coefficients for the independent variations δw, δ, δ2 and δ3 are set to zero. This results in a system that includes four differential Eqs. (11)–(14) and five internal force, electrical and magnetic factors (15)–(19) to obtain the boundary conditions. In what follows, for convenience of notation, x1 will be omitted. 32α˜ 33 16˜ 33 32 ˜33 16α˜ 33 16˜ 11 H 16˜ 33 V0 + V2 − − 2 + 3 − 3H 3H 3H 3H 3H 15 16α11 H 2α11 H 4˜e31 H 16α˜ 33 2 − 3 − w + M0 = 0 − 15 15 3 3H 32μ˜ 33 16 16α˜ 33 32α˜ 33 16μ˜ 33 16α11 H V0 + V2 − − 2 + 3 − 3H 3H 3H 3H 3H 15 16μ11 H 2μ11 H 4h˜ 31 H 16μ˜ 33 2 − 3 − w + M0 = 0 − 15 15 3 3H −
16μ˜ 33 2α˜ 33 14α˜ 33 16α˜ 33 14μ˜ 33 2α11 H V0 − V2 + + 2 − 3 − 3H 3H 3H 3H 3H 15 2μ11 H 4μ11 H 5h˜ 31 H 2μ˜ 33 2 − 3 + w − 2B0 − M0 = 0 − 15 15 3 3H 4˜e31 H 4h˜ 31 H 5h˜ 31 H 3 + 3 2 − 3 3 + 2p3 H 3 2ρ ω2 H 3 2 − 3 w + 2ω ρwH + 2H 3c˜ 11 (1 + iωγ )wIV
=0
(11)
(12)
(13)
(14)
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ˆ 1 = 16 11 H + 16α11 H 2 + 2α11 H 3 D 15 15 15
(15)
16α11 H 16μ11 H 2μ11 H + 2 + 3 Bˆ 1 = 15 15 15
(16)
2α11 H 2μ11 H 4μ11 H + 2 + 3 Bˆ 2 = 15 15 15
(17)
˜ ˆ 1 = e˜ 31 H V0 − 5˜e31 H V2 + 4˜e31 H + 4h31 H 2 M 3 3 3 3 ˜ 31 H 5h˜ 31 H 2H 3 c˜ 11 h 3 + (1 + iωγ )w + M0 − 3 3 3
(18)
3 2 3 ˜ ˜ ˆ 1 = − 4˜e31 − 4h31 H 2 + 5h31 H 3 − 2H c˜ 11 (1 + iωγ )w + 2ρ ω H w Q 3 3 3 3 3 (19)
Since our problem involves viscoelastic behavior, it is necessary to introduce complex modules for each variable to further solve this system: w = wr + iwi = r + ii 2 = r2 + ii2 3 = r3 + ii3
(20)
After that, the system will break down into 8 equations and 10 boundary conditions to find the imaginary and real part of each variable. The resulting system will be solved by the shooting method.
4 Numerical Experiment For the numerical experiment, a bimorph (see Fig. 1) with a length of L = 10 mm and a thickness of one layer of H = 0.6 mm was chosen. This bimorph consists of multilayer composites that include alternating piezoelectric and piezomagnetic layers. The mechanical and physical characteristics of such a composite are determined by the values of the effective material constants, described in [7]. The main materials, used in the composite, are CoFe2 O4 and BaTiO3 in a ratio of 20% and 80%, respectively. The material constants for this composite are presented in Table 1.
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Table 1. Material Properties Material constant
Value
Material constant
Value
Material constant
Value
−0..2
α33 , nN · s/(V · C)
235
0.5
h31 , N/(A · m)
15
e15 , C/m2
8.8
h33 , N/(A · m)
−6
72
11 , nF/m
9.4
μ11 , mN · s2 /C2
12.5
E , GPa c33
200
33 , nF/m
30
μ33 , mN · s2 /C2
200
E , GPa c44
39
α11 , nN · s/(V · C)
−0.23
ρ, kg/m3
, C/m2
5847
e31
E , GPa c11
185
e33 , C/m2
E , GPa c12
90
E , GPa c13
To verify the resulting theory, the results will be compared with the solution of a similar problem in the COMSOL Multiphysics FE package (CKP SSC-RAS No. 501994). Previously, in [9], we presented a technique for solving electromagnetoelastic problems in COMSOL. To account for viscoelastic behavior in COMSOL, the constitutive relationship for the stress tensor should be modified as follows: σ = c : ε · (1 + iωγ ) − eT · E − hT · H
(21)
Let us consider the forced oscillations of the bimorph under the influence of magnetic B˜ = 5 µWb with a frequency of f = 1 kHz at a damping coefficient γ = 10−6 . Next, let us compare the results obtained from the applied theory with the results of the finite element analysis of a similar problem in COMSOL. 10-14
6
Imaginary
10-11
0
5
Theory FEM
-0.2
4
Real
-0.4
3 -0.6
2 1 0
-0.8
Theory FEM
0
0.002 0.004 0.006 0.008
0.01
-1
0
0.002 0.004 0.006 0.008
0.01
Fig. 2. Comparison of the bimorph deflection distribution, based on applied theory and FEM: imaginary and real parts.
Analysis of Fig. 2 allows us to conclude that the theory describes the distribution of mechanical fields quite well. It can also be noted that the real part will make the greatest contribution to the absolute value.
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2.6
Imaginary
2.4
Real
10-5
-1.5
Theory FEM
-2
2.2 2
-2.5
1.8
-3
1.6 1.4 1.2
-3.5
Theory FEM
0
0.002 0.004 0.006 0.008
0.01
-4
0
0.002 0.004 0.006 0.008
0.01
Fig. 3. Comparison of the electric potential distribution along the length of the bimorph in the center of the upper layer on the base of applied theory and FEM: imaginary and real parts
Considering Fig. 3 there is an error in the distribution of the electric potential along the length in the fixing regions. This may be due to the Kirchhoff hypotheses do not consider the shear stresses that occur in the fixing region. To take them into account, it may be necessary to introduce new hypotheses or build a boundary layer solution. Figure 4 shows that based of the applied theory, in the imaginary part of the distribution there are significant errors in the distribution of the magnetic potential along the length of the bimorph in the center of the upper layer. However, the real part describes the distribution quite well. Due to the real part gives a much larger contribution to the absolute value, this discrepancy can be considered insignificant. Once it has been shown that the theory describes the behavior of the electromagnetoestic bimorph with sufficient accuracy when compared with FE analysis, the effect of the damping coefficient on the behavior of deflection in the vicinity of the first resonance can be investigated. To do this, consider the values of deflection in the center of the bimorph under the influence of an alternating magnetic field in the vicinity of the first resonance. 10-10
2.6
Imaginary Theory FEM
2.4
10-4
7.8
Real Theory FEM
7.7
2.2 7.6 2 7.5
1.8 1.6
0
0.002 0.004 0.006 0.008
0.01
7.4
0
0.002 0.004 0.006 0.008
0.01
Fig. 4. Comparison of the distribution of magnetic potential along the length of the bimorph in the center of the upper layer on the base of applied theory and FEM: imaginary and real parts.
Applied Theory of Vibrations of a Composite Electromagnetoelastic Bimorph Imaginary
5
= 5e-09 = 1e-08 = 5e-08 = 1e-07
0.8 0.6 0.4 0.2 0 2.75
2.8
2.85
Frequency, Hz
2.9 10
4
10-9
Real
1
Displacement, m
10-8
Displacement, m
Displacement, m
1
0 = 5e-09 = 1e-08 = 5e-08 = 1e-07
-5 2.75
2.8
2.85
Frequency, Hz
2.9 4
10
10
-8
363
Absolute = 5e-09 = 1e-08 = 5e-08 = 1e-07
0.8 0.6 0.4 0.2 0 2.75
2.8
2.85
Frequency, Hz
2.9
104
Fig. 5. Frequency response of the displacement of the midpoint of the bimorph in the vicinity of the first resonance, depending on different values of the damping coefficient: imaginary and real parts, and absolute value
Considering the data of Fig. 5, as the damping coefficient increases, the amplitude of oscillations in the vicinity of the first resonance decreases. The imaginary part has a peak at the resonance frequency, and the real part has a gap. In addition, the imaginary part makes the greatest contribution to the absolute value. From the analysis of the numerical results, it can be concluded that the developed applied theory can be used to solve stationary problems of bending vibrations of a bimorph made of an electromagnetoelastic composite, considering viscoelastic behavior. Based on this theory, it is possible to build the amplitude-frequency characteristics of such devices.
5 Conclusions In this chapter, we considered the plane problem of steady-state bending vibrations of a hinged-supported bimorph in an alternating magnetic field, considering damping. The bimorph under study consisted of a multilayer electromagnetoelastic composite, modeled using the effective modulus approach. Within the framework of Kirchhoff hypotheses, an applied theory was constructed, which considered the quadratic distributions of electric and magnetic potentials along the thickness of the bimorph and considered heterogeneity in the longitudinal direction. In addition, the theory introduced linear viscoelasticity. Using the resulting theory, distributions of electric, magnetic, and mechanical fields were constructed. Comparison of the obtained results based on the theory with the results of the finite element model in the COMSOL Multiphysics package showed a good convergence of the results. However, for the longitudinal distribution of the electric potential, some discrepancies were observed, possibly associated with tangential stresses in the fixing area, which are not considered by the theory within the framework of the accepted hypotheses. Next, based on the obtained theory, the behavior of plate deflection in the vicinity of the first resonance was studied at various values of the damping coefficient, which showed that the obtained theory can be used to analyze the bandwidth of converters based on constructing the amplitude-frequency characteristics of bimorphs made of an electromagnetoelastic composite.
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Acknowledgement. This work was carried out with the financial support of the Russian Science Foundation grant No. 21-19-00423 at the Southern Federal University.
References 1. Gaudenzi, P.: Smart Structures: Physical Behaviour, Mathematical Modelling and Applications. Wiley, New York-Chichester-Brisbane-Toronto (2009) 2. Qader, I.N., Kok, M., Dagdelen, F., Aydogdu, Y.: A review of smart materials: researches and applications. El-Cezerî J. Sci. Eng. 6(3), 755–788 (2019) 3. Plotnikova, S.V., Kulikov, G.M.: Shape control of composite plates with distributed piezoelectric actuators in a three-dimensional formulation. Mech. Compos. Mater. 56(5), 557–572 (2020) 4. Janeliukstis, R., Mironovs, D.: Smart composite structures with embedded sensors for load and damage monitoring – a review. Mech. Compos. Mater. 57(2), 131–152 (2021) 5. Amrillah, T., Hermawan, A., Wulandari, C.P., Muthi’Ah, A.D., Simanjuntak, F.M.: Crafting the multiferroic BiFeO3-CoFe2O4 nanocomposite for next-generation devices: a review. Mater. Manuf. Process. 36(14), 1579–1596 (2021) 6. Fiebig, M.: Revival of the magnetoelectric effect. J. Phys. D Appl. Phys. 38(8), R123–R152 (2005) 7. Challagulla, K.S., Georgiades, A.V.: Micromechanical analysis of magneto-electro-thermoelastic composite materials with applications to multilayered structures. Int. J. Eng. Sci. 49(1), 85–104 (2011) 8. Parton, V.Z., Kudryavtsev, B.A.: Electromagnetoelasticity of Piezoelectric and Electroconductive Bodies. Nauka, Moscow (1988). (in Russian) 9. Binh, D.T., Chebanenko, V.A., Duong, L.V., Kirillova, E., Thang, P.M., Soloviev, A.N.: Applied theory of bending vibration of the piezoelectric and piezomagnetic bimorph. J. Adv. Dielectr. 10(3), 2050007 (2020) 10. Soloviev, A.N., Do, B.T., Chebanenko, V.A., Parinov, I.A.: Flexural vibrations of a composite piezoactive bimorph in an alternating magnetic field: applied theory and finite element simulation. Mech. Compos. Mater. 58(4), 471–482 (2022) 11. Soloviev, A.N., Chebanenko, V.A., Parinov, I.A., Oganesyan, P.A.: Applied theory of bending vibrations of a piezoelectric bimorph with a quadratic electric potential distribution. Mater. Phys. Mech. 42(1), 65–73 (2019)
Features of Backscattering of Short Longitudinal Waves on Spatial Defects of Canonical Form Located in Elastic Bodies Nikolay V. Boyev1,2(B) 1 I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal
University, Rostov-on-Don, Russia [email protected] 2 Admiral Ushakov Maritime State University, Novorossiysk, Russia
Abstract. Within the framework of the geometric theory of diffraction, based on an explicit analytical expression, the amplitude of displacements in the backreflected high-frequency longitudinal wave from the surfaces of cavity defects and non-planar cracks in elastic bodies was studied. A detailed numerical analysis of the amplitude of displacements in the case of back-reflection from a cylinder, sphere, triaxial ellipsoid, one-cavity and two-cavity hyperboloids, elliptical and hyperbolic paraboloids for various values of geometric parameters of boundary surfaces was carried out. Graphs of the dependence of the amplitude of displacements on the distance of the source and receiver of the wave from the defect surfaces are constructed. The focal points of the amplitude of displacements in the reflected wave from each type of defects have been established. Keywords: Longitudinal Wave · Transverse Wave · Diffraction · Defect · Wave Back-Reflection
1 Introduction The development of methods for solving dynamic problems of the mechanics of deformable solids and structures has a wide application in modern technology [1–5]. In this paper, an important applied problem of high-frequency dynamics of elastic media containing defects in the form of cavities and non-planar cracks is investigated. The main methods for calculating scattered wave fields on obstacles of complex shape in an elastic medium are based on replacing the boundary surfaces of non-planar reflectors with a set of faces of an inscribed or circumscribed polyhedron near the reflecting surface of the obstacles. Such an approach, of course, greatly simplifies the numerical calculations of the wave field characteristics within the framework of the geometric theory of diffraction (GTD). When calculating the diffraction of short waves on the boundary surfaces of cavity defects of complex non-convex shape, it is necessary to take into account the reflected waves. If the reflective boundary surface of an obstacle is replaced by a polyhedron, then the displacements in the re-reflected finite number of times of the high-frequency longitudinal elastic wave are determined by the inverse of the sum of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 365–379, 2024. https://doi.org/10.1007/978-3-031-52239-0_35
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the distances from the wave source to the first point of the mirror reflection, between the successive points of the mirror reflection, from the last point of the mirror reflection to the point of reception. At the same time, such substitution, firstly, distorts the trajectory of the reflected rays, and secondly, does not consider the curvature of the surfaces due to the local parameters of curvature of the surfaces at the mirror reflection points, located at the defect boundary. Such an approach requires justification of the correctness of replacing the boundary surface with a polyhedron with a sufficient number of faces. To overcome the described features, it is convenient to apply Kirchhoff’s physical theory of diffraction to solve this problem [6–16]. The use of differential geometry methods [17] in combination with asymptotic estimation of Kirchhoff diffraction integrals makes it possible to obtain explicit asymptotic expansions of the amplitude of displacements in scattered wave fields. The study is based on the explicit expression of the amplitude of displacements in a single-reflected high-frequency wave. In this paper, an explicit expression of the principal term of the asymptotic displacements in a longitudinal wave, reflected from the boundary surface, was obtained within the framework of GTD, based on the asymptotic assessment of the two-fold Kirchhoff diffraction integral by the method of two-dimensional steady-state phase [18]. The explicit expression for displacements at the point of wave reception contains all the geometric parameters of the problem: the distances from the point of mirror reflection to the source and receiver of the wave, the direction of the incident wave, the principal curvatures of the surface at the point of mirroring. The explicit form of displacements in the reflected wave makes it possible to establish the phenomenon of focusing the wave amplitude when it is reflected from smooth concave boundary surfaces of defects.
2 Problem Statement A case is considered when a spherical monochromatic high-frequency wave falls from the point x0 of an infinite elastic medium onto the boundary surface of a cavity or a non-planar crack, located in it (Fig. 1). The wave is generated by the force Qeiωt , concentrated at the point x0 , where ω is the frequency of oscillations. At the same time, the displacements at the point y of elastic space are determined by the Kupradze matrix [19]: k, j = 1, 2, 3, U˜ j(k) (y, x0 ) = U˜ jp(k) (y, x0 ) + U˜ js(k) (y, x0 ) , ∂2 1 eikp R0 (k) U˜ jp (y, x0 ) = − , R0 = |x0 − y|, 4πρω2 ∂ yk ∂ yj R0 eiks R0 eiks R0 ∂2 1 (k) 2 ˜ k δkj + Ujs (y, x0 ) = 4πρω2 s R0 ∂ yk ∂ yj R0
(1)
where ρ is the density of the elastic material, λ, μ are Lame coefficients, kp = ω /cp , ks = ω/cs , cp , cS are the wave numbers and velocities of the longitudinal and transverse waves, and δkj are Kronecker symbols. The purpose of the work at the first stage of solving the problem is to study the amplitude characteristics of a scattered field on the surface of a cavity or a non-planar crack free from stresses.
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3 Solution Method The dependence of the characteristics of the problem on time is monochromatic, for example, for displacements in an infinite elastic medium, it has the form: u˜ (x1 , x2 , x3 , t) = Re u˜ (x1 , x2 , x3 ) exp (−i ω t) . The Kupradze matrix determines non-zero displacements in longitudinal (p-wave) and transverse (s-wave) waves at the point y of elastic space in the radial direction q˜ = (x0 y)|x0 y|−1 . In the high-frequency mode of oscillations at kp → ∞ and ks → ∞ in directions q˜ and q˜ 1 , we write out asymptotic representations of displacements in an incident spherical wave:
1 eikp R 0 1+O = Qq q˜ 4π ρ ω2 R0 kp R0
1 ks2 eiks R0 (s) u˜ q1 (y) = Qq1 q˜ 1 1+O 4π ρ ω2 R0 ks R0 kp2
(p) u˜ q (y)
(2) (3)
˜ and the magnitudes Qq and Qq1 where, the tangential direction q˜ 1 is perpendicular to q, are the projections of the force Q on the directions q˜ and q˜ 1 . The components of the displacement vector in the wave reflected from the free boundary surface of the spatial defect at the point x of the elastic medium are determined by the Somigliana integral [20]: ¨ ˜ (k) (y, x) ] · u˜ (y) dSy Ty [ U (4) u˜ k (x) = S
˜ (k) ˜ (k) + μ n × rot U ˜ (k) (y, x) = 2μ ∂ U + λ n div U ˜ (k) Ty U ∂n
(5)
˜ (k) (y, x) matrix is obtained from the matrix U ˜ (k) (y, x0 ) in relation where the Kupradze U (1) by substituting x0 for x and R0 by R = |y − x|, Ty is the force vector at the point ˜ y, u(y) is the vector of the total field of displacements on the boundary surface of the obstacle, n is the normal to the surface of the defect S. In the vectors of total displacement on the boundary surface of the defect and in the force vector Ty at the point y, there are two terms defined by the longitudinal (p) and transverse (s) waves. ¨ ˜ p(k) (y, x)] + Ty [U ˜ s(k) (y, x)] · u˜ (y; p) + u˜ (y; s) dSy (6) T˜ y [U u˜ k (x) = S
where u˜ (y; p) is the vector of total displacement at the boundary S, formed when the longitudinal wave falls on the boundary of the defect S; u˜ (y; s) is the vector of total displacement at the boundary S, formed when the transverse wave falls on the boundary of the obstacle S. In the expanded notation of the dot product (6), four terms can be distinguished, which describe p − p both s − s reflections p − s and s − p transformations.
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It should be noted that in the classical geometric theory of diffraction, developed in the problems of scalar acoustics [21] and in the problems of the dynamical theory of elasticity [22], a distinction is made between high-frequency asymptotics in the local and global sense. The asymptotic solution, which will be obtained below, is of a local nature and gives the principal asymptotic term of the amplitude of the diffracted field in the small neighborhood of any ray exiting the point x0 , reflected from the boundary surface of the defect at the point y∗ and arriving at the point of elastic space x. Taking into account the initial problem of studying the reverse reflection of a wave, let us consider in detail the propagation in an elastic medium of the longitudinal (p) ˜ in which p− wave component (2) of an incident spherical wave in a fixed direction q, interacts with the free surface of a cavity or non-planar crack and is reflected from it. In this case, the longitudinal wave will be reflected in the p− wave (p − p reflection) and transform into a transverse s− wave (p − s transformation). Reception points x at p − p reflection and x˜ at p − s transformations will be different and will be located on the rays along which the reflected longitudinal and transverse waves propagate. In this case, the y∗ intersection of the smooth surface of an obstacle with the direction of q˜ = (x0 y∗ )|x0 y∗ |−1 will be the point of mirror reflection falling p− waves (2). Let us obtain the formula for the amplitude of the reflected p− wave at the receiving point x.
4 The Case of p-p Reflection The direction of the wave incidence q˜ = {− cos α, − cos β, − cos γ } is attributed to the right Cartesian coordinate OX1 X2 X3 system at the point y∗ , the axis OX3 of the application of which coincides with the external normal n (directed towards the elastic medium) to the boundary surface of the defect, and the OX1 and OX2 axes coincide with the tangents to the lines of curvature of the obstacle surface at the point y∗ (Fig. 1). In this coordinate system, the vector q˜ 1 has coordinates {−ctg γ cos α, −ctg γ cos β, − sin γ }, and the normal to the boundary surface has coordinates n = {0, 0, 1}.
Fig. 1. Reflection of a high-frequency longitudinal wave into a longitudinal wave (p−p reflection) and transformation of a longitudinal wave into a transverse wave (p − s transformation) on the free surface of a cavity or a non-planar crack, located in an elastic medium.
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The coordinates of the displacement vector in a scattered longitudinal wave are determined by the first term of formula (6): ¨ (p) ˜ p(k) (y, x) ] · u˜ (y; p) dSy u˜ k (x) = Ty [U (7) S
which we will write in the expanded form: (k)
(k) 2 ˜ ∂ U˜ 3p ∂ U˜ m p (p) μ u˜ m (y; p) u˜ k (x) = + ∂ y3 ∂ ym m=1 S
(k) ∂ U˜ ˜ p(k) u˜ 3 (y; p) dSy + 2 μ ∂ y3p3 + λ div U
(8)
Below we obtain a relation that represents the principal term of the asymptotic representation of the longitudinal displacement amplitude in the wave, reflected from the boundary surface at high vibration frequencies. For the asymptotic estimation of the diffraction integral (7) at kp → ∞ , we use the following asymptotic representations at kp → ∞ : ˜ p(k) (y, x) = i div U (k) ∂ U˜ j
∂ ym
=i
−1 eikp R ∂ R 1 + O k R p 4 π ρ ω2 R ∂ yk kp3
(9)
−1 eikp R ∂ R ∂ R ∂ R 1 + O k k, j, m = 1, 2, 3 (10) R p 4 π ρ ω2 R ∂ ym ∂ yk ∂ yj kp3
y (y1 , y2 , y3 ); x (x1 , x2 , x3 ); y ∈ S, ∂R ∂R ∂R y1 − x1 y2 − x2 y3 − x3 = − cos α, = − cos β, = cos γ . = = = ∂ y1 R ∂ y2 R ∂ y3 R Here {− cos α, − cos β, cos γ } are the guiding cosines of the vector yx. After substituting (9) and (10) into (8), we get: (p)
u˜ k (x) = i
kp3 4 π ρ ω2
¨ S
˜ (y)
∂ R eikp R dSy , ∂ yk R
∂R ∂R ∂R 2 ∂R ˜ (y) = 2μ u˜ 1 (y; p) + u˜ 2 (y; p) + 2μ + λ u˜ 3 (y; p) ∂ y1 ∂ y2 ∂ y3 ∂ y2
Let us move on to the local spherical coordinate system r, θ, ψ in the neighborhood of the y∗ point. In this coordinate system, the components of the displacement vector are reduced to the form:
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kp3
(p)
u˜ r (x) = i
4 π ρ ω2
(p) u˜ θ (x)
= 0,
¨
˜ (y)
eikp R dSy , R
(11)
S (p) u˜ ψ (x)
= 0,
˜ (y) = −2 μ cos α u˜ 1 (y; p) + cos β u˜ 2 (y; p) cos γ + 2 μ cos2 γ + λ u˜ 3 (y; p) In the asymptotic estimation of the Kirchhoff diffraction integral in formula (8), the components of the total displacement field u˜ k (y; p), k = 1, 2, 3 in the integral expression should be chosen as the solution of the local diffraction problem of the reflection of a plane incident p− wave from the plane boundary of an elastic half-space [23, 24]: ⎛ ⎞ 2 k k p s (p) 1 − 2 sin2 γ V˜ ps (y)⎠ u˜ mq (y) , m = 1, 2 u˜ m (y; p) = ⎝1 + V˜ pp (y) − kp sin γ ks (12)
(p) u˜ 3 (y; p) = 1 − V˜ pp (y) − tg γ V˜ ps (y) u˜ 3q (y),
where V˜ pp and V˜ ps are the coefficients of p − p reflection and p − s transformation [23, 24]: 2 1 4 ctg γ ctg γ1 − 1 − ctg2 γ1 V˜ pp = z 4 (13) V˜ ps = ctg γ 1 − ctg2 γ1 z 2 z = 4ctg γ ctgγ1 + 1 − ctg2 γ1 Substituting (12) and (2) into (11), we obtain an integral representation of the radial displacement: (p) u˜ r =
⎧ ⎡ ! " ⎪ ¨ k2 ⎨ " kp2 kp Qq p ⎢ ˜ pp + ks #1 − − sin γ 1 + V − sin 2 γ i 2 · ⎣ 2 ⎪ 4 π μ ks 2π kp ks ⎩
k2 + s2 − 2 sin2 γ kp
S∗
⎤ kp2 ks2
⎥ sin2 γ V˜ ps ⎦
'
eikp (R0 +R) − cos γ 1 − V˜ pp + sin γ V˜ ps dSy R0 R
By substituting the relations (13) into the integral expression of the diffraction integral, it is analytically proved that:
Features of Backscattering of Short Longitudinal Waves
( kp2 2ks2
+
ks2 kp2
) kp2 k 2 s − sin 2 γ − sin γ 1 + V˜ pp + kp 1 − k 2 sin γ V˜ ps s
* − 2 sin2 γ − cos γ 1 − V˜ pp + sin γ V˜ ps = cos γ V˜ pp .
371
(14)
The proof of the written relation (14) is carried out by the following transformations of the expression in parentheses of the left part, taking into account Snell’s law: kp sin γ = ks sin γ1 , where {− cos α1 , − cos β1 , cos γ1 } is the direction of the reflected s− wave. ks 2 sin2 γ cos γ 1 + V˜ pp − 2 sin γ cos γ cos γ1 V˜ ps kp ' 2 k ks2 s 2 2 + 2 sin γ − 2 cos γ 1 − V˜ pp + − 2 sin γ sin γ V˜ ps kp kp2 ( 2 2 kp2 ks2 k sin γ kp s 2 = 2 cos γ V˜ pp + 2 4 sin γ − 2 + kp ks kp cos γ ks2 * sin γ sin2 γ 2 ˜ ps × −2 cos γ cos γ1 + − 2 sin γ V sin γ1 sin2 γ1 sin2 γ1 k2 = s2 cos γ V˜ pp + 4 sin2 γ1 − 1 + tg γ kp sin2 γ * sin2 γ sin γ 2 ˜ ps cos γ cos γ1 + − 2 sin γ × −2 V sin γ1 sin2 γ1
k2 = s2 cos γ V˜ pp + 4 sin2 γ1 − 1 + −2 sin γ1 cos γ1 + 1 − 2 sin2 γ1 tg γ V˜ ps kp 2
ks2 sin2 γ1 2 2 ˜ = 2 cos γ Vpp + 3 − ctg γ1 4 ctg γ ctg γ1 + 1 − ctg γ1 kp Z − 2 ctg γ1 + 1 − ctg 2 γ1 tg γ × 4 ctg γ 1 − ctg 2 γ1 =
k2 ks2 cos γ V˜ pp + V˜ pp = 2 s2 cos γ V˜ pp . 2 kp kp
The proven relation allows us to obtain the following basic representation of the (p) radial displacement u˜ r (x) (after taking the slowly changing functions as the sign of the integral in the high-frequency approximation):
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(p)
u˜ r (x) = φpp
Qq kp2 kp cos γ i 2· V˜ pp y∗ 4 π μ ks 2 π L0 L
¨
eikp φpp dS
(15)
S
+ + + + = |x0 − y| + |y − x|, L0 = +x0 − y∗ +, L = +y∗ − x+
The asymptotic estimation of the integral in the representation of radial displacement (15) by the method of two-dimensional stationary phase is given in [9, 10] when solving the problem of the reflection of an acoustic wave from the surface of a solid obstacle. Using this estimate, we write down the principal term of the radial displacement amplitude in the p− wave, reflected from the defect boundary. (pp) π 2 exp i k δ + L) + + 2 (L p 0 k 2 Q 4 p ˜ q (p) u˜ r (x) = (16) Vpp y∗ cos γ )+ + 4 π μ ks2 + (pp) + L0 L +det D2 + (pp)
where D2 is a Hessian matrix of size 2 × 2 of a symmetric structure (pp) (pp) di j = dj i ; i, j = 1, 2 , and δ2 = sign D2 is the difference between the number of (pp) positive and negative eigenvalues of the matrix D2 . Calculating the determinant in the denominator of the expression (16) and considering the case: d21 = d12 , we write out the formula (16) in an expanded form: (pp) B × V˜ pp (y∗ ) exp i kp (L0 + L) + π4 δ2 + 2 (p) u˜ r (x) = ,+ +, +(L0 + L)2 + 2L0 L(L0 + L) k2 sin2 α + k1 sin2 β cos−1 γ + 4L2 L2 K + 0 B=
Qq kp2 4 π μ ks2
(17)
Here {− cos α, − cos β, − cos γ } is a vector that determines the direction of incidence of the ray x0 − y∗ in the selected local Cartesian coordinate system. For further study, it is convenient to transform the obtained formula (17) by highlighting the local geometric parameters of the defect surface at the mirror reflection point: (pp) B × V˜ pp (y∗ ) exp i kp (L0 + L) + π4 δ2 + 2 (p) , u˜ r (x) = ) (L0 + L)2 + 2 L0 L(L0 + L) 2 H cos 2 γ + k˜ sin2 γ cos−1 γ + 4 L20 L2 K B=
Qq 4π μ
kp2 ks2
(18)
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In formula (18), K = k1 k2 is the Gaussian curvature, and H = (k1 + k2 ) 2 is the average curvature of the surface of the defect at the point of mirror reflection y∗ . Magnitude k˜ is the curvature of the normal cross-section of the surface of the defect by the plane of the ray x0 − y∗ − x. Curvature k˜ is defined by Euler’s formula k˜ = cos β α ˜ k1 cos 2 φ˜ + k2 sin 2 φ˜ (cos φ˜ = cos sin γ , sin φ = sin γ ), expressing the curvature of an arbitrary normal cross-section of the surface in terms of principal curvatures k1 , k2 , and ˜ which is tangent to the surface in this normal cross-section with the first the angle φ, principal direction at the specular point. Formula (18) gives the principal term of the asymptotic amplitude of radial displacement at kp L0 >> 1, kp L >> 1, kp R1 >> 1, kp R2 >> 1. Based on the formula obtained, we will draw conclusions. The amplitude of the radial displacement in the reflected wave during p − p reflection is inversely proportional to the radical, located in the denominator, which depends on the local geometric characteristics of the surface at the point-of mirror reflection y∗ : Gaussian curvature K = k1 k2 , average curvature H = (k1 + k2 ) 2, curvature of the normal section of the surface at the point of mirror reflection y∗ by the plane ray x0 − y∗ − x; on the distances of the points, involved in the formation of the beam, and on the angle γ between the direction of incidence and reflection of the longitudinal wave and the normal to the surface at point y∗ . The structure of the function in the denominator of formula (18) for a reflected longitudinal wave is the same as in the case of a one-time reflected acoustic wave [9, 10]. This means that at kp L0 >> 1, kp L >> 1, kp R1 >> 1, kp R2 >> 1 the characteristic qualitative and quantitative features of the dependence of the amplitude of radial displacements (18) in the longitudinal wave on the geometric parameters of the problem are the same as in the acoustic case. The presence of a coefficient V˜ pp (y∗ ) in the numerator of formula (18) indicates that the qualitative features of the reflection of a longitudinal wave into a longitudinal wave in the case of a surface remain the same as in the case of the reflection of a wave from a tangent plane to a surface at the point of mirror reflection. For formula (18), let us distinguish two limiting cases. If k1 = k2 = 0, then from (18), it follows the known result [23] for radial displacement in a wave, reflected from a free boundary plane of half-space: (p)
u˜ r (x) =
Qq kp2 V˜ pp y∗ (L0 + L)−1 exp i kp (L0 + L) 4 π μ ks2
(19)
In the case of backscattering of a high-frequency longitudinal wave (V˜ pp = −1) in the far field, the expression (18) for radial motion modulo is the same as the expression for pressure in the scalar case [6]: . π (pp) (p) δ (20) u˜ r (x) = −0.5 Qq i L−2 R R exp i 2 k L + 1 2 p 0 0 4 2
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The relation (18) is obtained for the case when a high-frequency longitudinal wave falls on a boundary surface convex from the side of the elastic medium. If the longitudinal wave falls on a concave surface, then the principal curvatures k1 and k2 are taken as negative. The expression for radial displacement (18) differs from that of [9, 10, 25] for the pressure p(x) in the reflected high-frequency acoustic wave in the scalar case only in the presence of the reflection coefficient V˜ pp in the case of an elastic medium. The formula for the pressure of a one-time reflected high-frequency wave from an acoustically solid obstacle was obtained in the monograph [25], based on the Keller’s geometric theory of diffraction (GTD), and in [9, 10], based on the estimation of the diffraction integral by the method of two-dimensional steady-state phase. This means that the principal term of the asymptotic of the diffraction integral in the formula for radial displacement (18) coincides with the calculations of displacement in the reflected wave according to the GTD. At the same time, it is important to make a significant remark. If we investigate in detail the phenomenon of wave focusing, we can do so only on the basis of the diffraction integral (15) of Kirchhoff’s physical theory of diffraction. In this case, when asymptotically estimating the integral, it is necessary to use the asymptotic expansion of the phase function, considering the infinitesimal increments of arcs of higher orders. Based on the general expression (18), we study the backward reflection of a longitudinal wave from cavity defects and non-planar cracks in elastic media. Within the framework of a three-dimensional problem, we consider the back reflection of a spherical elastic longitudinal wave, caused by a point source of pressure (2), placed at point A, from the characteristic points B of the mirror reflection of the following canonical surfaces: triaxial ellipsoid (21a), single-sheet hyperboloid (21b), two-sheet hyperboloid (21c), elliptic paraboloid (21d), hyperbolic paraboloid (21e) (Fig. 2): x2 a2
+
y2 b2
+
z2 c2
= 1 (a) x2 a
+
x2 a2 y2 b
+
y2 b2
= 2z
2 z2 = 1 (b) ax 2 c2 2 2 (d) xa − yb = 2z
−
+
y2 b2
(e)
−
z2 c2
= −1 (c)
(21)
The source and receiver of the wave are located at point A. Point B is the point of mirror reflection of the longitudinal wave from the surface of a cavity obstacle or a non-planar crack. In this case, the normal to the surface of the defect at point B lies on the line BA. The purpose of the study in the second part of the problem is to give a qualitative and quantitative analysis of the amplitude of displacements in the back-reflected wave depending on the distance of the source and receiver of the wave from the surface of the reflectors under consideration. The analytical expression of the amplitude of displacements in the reflected longitudinal wave was obtained on the base of the study of the problem in the general formulation (Fig. 1).
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Fig. 2. Back-reflection of a high-frequency longitudinal wave: (a) from a point (0, 0, − c) of a triaxial ellipsoid, (b) from a point (0, − b, 0) of a single-cavity hyperboloid, (c) from a point (0, 0, c) of a two-cavity hyperboloid, (d) from a point (0, 0, 0) of an elliptical paraboloid, (e) from the point (0, 0, 0) of the hyperbolic paraboloid.
Let us write out the principal term of the asymptotic of the diffraction integral in the case of back-reflection (γ = 0), (V˜ pp = −1) the longitudinal wave at the selected points of the canonical boundary surfaces of defects under consideration: (pp) −B × exp i kp (L0 + L) + π4 δ2 + 2 (p) u˜ r (x) = , (L0 + L)2 + 4 L0 L(L0 + L) H + 4 L20 L2 K B=
Qq kp2 4 π μ ks2
in which K = k1 k2 is the Gaussian curvature and H = (k1 + k2 )/2 is the mean curvature of the surface at the specular reflection point y∗ .
5 Numerical Results For each type of defect, the principal curvatures of the boundary surfaces at the points of specular reflection B were calculated. Numerical analysis for each type of defect was carried out for several variants of geometric parameters. In Figs. 3, 4, 5 for singlecavity (Fig. 3), double-cavity (Fig. 4) hyperboloids and a hyperbolic paraboloid (Fig. 5), graphs of the magnitude of the displacement amplitude at the point of receiving the back reflected wave are plotted on the distance between the source and receiver of the wave
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for the surfaces of the canonical-shaped defects under consideration. The constructed graphs are compared with graphs of the amplitude of displacements in reflected waves from flat reflectors, located in tangent planes to the boundary surfaces of defects at the points of specular reflection.
Fig. 3. Reverse reflection of a high-frequency longitudinal wave from single-cavity hyperboloids (1, 2) and flat reflector (3).
Within the framework of the problem of replacing non-planar reflectors with a set of flat reflectors, this allows for a quantitative analysis of the error in calculating the pressure during such a replacement, depending on the distance between the source and the wave receiver. The number of positive values of the main curvature of the boundary surface of the defect, that is, the number of concave main normal sections of the reflector surface, determines the number of vertical asymptotes in the graph of the displacement amplitude. This is due to the determinant of the Hessian matrix, located in the denominator of the pressure expression, becomes zero, which corresponds to a surge in the displacement amplitude, that is, focusing the amplitude of the reflected longitudinal wave. As the positive principal curvature of the normal section decreases, the corresponding focusing point moves away from the point of specular reflection (Fig. 3). Figures 3, 4, 5 indicate the coordinates of the intersection points of the graphs. The abscissas of these points divide the horizontal axis into several intervals. The closer the curve is to the x-axis, the greater the scattering of the wave on the surface of the reflector in the interval under consideration.
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Fig. 4. Reverse reflection of a high-frequency longitudinal wave from two-cavity hyperboloids (1, 2) and flat reflector (3).
Fig. 5. Reverse reflection of a high-frequency longitudinal wave from hyperbolic paraboloids (1,2) and a flat reflector (3).
6 Conclusion An explicit expression is obtained for the amplitude of displacements when a longitudinal wave is reflected into a longitudinal wave from the boundary surface of a cavity defect and a non-planar crack, located in an elastic medium. This model of the problem can be used in ultrasonic non-destructive testing during practical irradiation from the surface of an elastic body with a longitudinal wave of a cavity defect, located in an elastic body of finite dimensions. A detailed numerical analysis of the amplitude of longitudinal displacement in the case of backward reflection was carried out for defects of the canonical shape: cylinder, sphere, triaxial ellipsoid, single-sheet and double-sheet hyperboloids, elliptic
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and hyperbolic paraboloids for various values of the parameters of the boundary surfaces. Graphs of the dependence of the amplitude of longitudinal displacement on the distance of the source and receiver of the wave from the surfaces of obstacles were constructed. For elliptic, hyperbolic and parabolic points of boundary surfaces, depending on the combinations of signs and values of the principal curvature of the surfaces, acoustic wave focusing points are established at the points of mirror reflection. For the considered canonical surfaces, a comparative analysis of the amplitudes of reflected waves from nonplane surfaces and flat reflectors, located in tangent planes to the boundary surfaces of defects at the points of specular reflection of short elastic longitudinal waves, was carried out.
References 1. Popov, A.N., Zelenkov, G.A., Papulov, D.S.: J. Phys. Conf. Ser. 2061(1), 012114 (2021) 2. Bashkatov, V.A., Khudyakov, S.A., Ignatenko, A.V.: J. Phys. Conf. Ser. 2061(1), 012054 (2021) 3. Gerasidi, V.V., Lisachenko, A.V.: J. Phys. Conf. Ser. 2061(1), 012056 (2021) 4. Kozenkova, G.L., Talamanov, V.N., Kozenkov, V.A., Kondratyev, S.I., Khekert, E.V., Modina, M.A.: J. Phys. Conf. Ser. 2061(1), 012068 (2021) 5. Senchenko, V., Lopatina, V., Butsanets, A.: E3S Web of Conferences, vol. 258, p. 02005 (2021) 6. Shenderov, E.L.: Wave Problems in Hydroacoustics. Sudostroenie, Moscow (1972). (in Russian) 7. Henl, H., Maue, L., Vestpfal, K.: Diffraction Theory. Mir, Moscow (1964). (in Russian) 8. Sumbatyan, M.A., Boyev, N.V.: J. Acoust. Soc. Am. 95(5), 2346 (1994) 9. Boyev, N.V., Sumbatyan, M.A.: Doklady Akademii Nauk SSSR 392(5), 614–617 (2003). (in Russian) 10. Boyev, N.V.: Acoust. Phys. 50(6), 651–656 (2004) 11. Boyev, N.V.: In: Parinov, I.A., Chang, S.-H., Topolov, V.Yu. (eds.) Proceedings of the 2015 International Conference on “Physics, Mechanics of New Materials and Their Applications”, pp. 353–358. Nova Science Publishers, New York (2016) 12. Boyev, N.V.: In: Advanced Structured Materials, vol. 59, p. 173. Springer, Heidelberg (2017) 13. Boyev, N.V.: Mater. Phys. Mech. 37(1), 60 (2018) 14. Boyev, N.V., Sumbatyan, M.A., Zampoli, V.: In: Advanced Structured Materials, vol. 109, p. 199. Springer, Heidelberg (2019) 15. Boyev, N.V.: In: Parinov, I.A., Chang, S.-H., Kim, Yu.-H., Noda, N.-A. (eds.) Springer Proceedings in Materials, vol. 10, pp. 273–288. Springer, Cham (2021) 16. Boyev, N.V., Sumbatyan, M.A., Brigante, M.: J. Sound Vib. 523, 116723 (2022) 17. Rashevsky, P.K.: Course of Differential Geometry. Fizmatlit, Moscow (1956). (in Russian) 18. Fedoryuk, M.V.: The Pass Method. Nauka, Moscow (1977). (in Russian) 19. Kupradze, V.D.: Potential Methods in Elasticity Theory. Fizmatlit, Moscow (1963). (in Russian) 20. Novatsky, V.: Elasticity Theory. Mir, Moscow (1975). (in Russian) 21. Borovikov, V.A., Kinber, B.E.: Geometrical Theory of Diffraction. Svyaz, Moscow (1978). (in Russian) 22. Achenbach, J.D., Gautesen, A.K., McMaken, H.: Ray Methods for Waves in Elastic Solids with Applications to Scattering by Cracks. Pittman, New York (1982) 23. Brekhovskikh, L.M.: Waves in Layered Media. Nauka, Moscow (1973). (in Russian)
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24. Grinchenko, V.T., Meleshko, V.V.: Harmonic Oscillations and Waves in Elastic Bodies. Naukova Dumka, Kiev (1981). (in Russian) 25. McNamara, D.A., Pistorius, C.W.I., Malherbe, I.A.G.: Introduction to the Uniform Geometrical Theory of Diffraction. Artech House, Norwood (1990)
Delamination Detection in a Multilayer Carbon Fiber Reinforced Plate Based on Acoustic Methods: Numerical and Experimental Study S. N. Shevtsov1(B)
, V. A. Chebanenko1 and N. G. Snezhina3
, I. E. Andzhikovich2 ,
1 Southern Scientific Center of RAS, Rostov-on-Don 344006, Russia
[email protected]
2 Southern Federal University, Rostov-on-Don 344006, Russia 3 Don State Technical University, Rostov-on-Don 344000, Russia
Abstract. In this paper, we present the results related to the development of a technique for low-frequency acoustic non-destructive testing of layered composite structures. The main goal was to detect and pinpoint potential bundles in such structures. To do this, we used as an object of research a manufactured carbon fiber panel consisting of eight layers, with the addition of inclusions between the fourth and fifth layers. By performing a finite element analysis of the effects of an omnidirectional piezoelectric transducer mounted in the center of the panel, we found that the amplitudes of the out-of-plane velocities increase significantly near the lamination regions. Our technique includes the use of a laser vibrometer and a simple method of velocity analysis that allows us to detect and accurately locate lenticular delaminations within the geometry under study. It was found that the results of the simulation were sufficiently like to the experiment. In addition, we also considered a model case, where the defect is located between the first and second layers of the panel. It is noted that the behavior of the velocity field in this case differs from the previous variant, but the average amplitude of the out-of-plane velocity undergoes a slight decrease relative to the case of the defect occurring in the center of the thickness. Keywords: Composite Material · Composite Technology · Acoustic NDE · Delamination · Computer Modeling
1 Introduction With the increasing use of composites fiber reinforced polymer (CFRP) in industries such as aircraft, shipbuilding, and automotive, there is a need to ensure a high level of quality and reliability of structures. This is especially important under conditions of high cyclic loads to which composite materials are subjected. However, defects may occur during the production of these structures, which cannot always be detected by visual inspection [1, 2]. These defects include various anomalies, such as delamination within composite materials, which can occur due to the presence of air or vapors during the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 380–389, 2024. https://doi.org/10.1007/978-3-031-52239-0_36
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resin curing process [3–9]. In addition, peel-up and push-down delaminations of layers at the edges of parts when drilling holes [1, 5, 10, 11] and, also, accidental damage that occurs during transportation are dangerous types of defects [5, 10, 12–14]. To solve this problem, reliable non-destructive evaluation (NDE) methods are needed to detect and localize such defects. One such method is acoustic analysis, which allows one to detect changes within a material, based on the propagation of acoustic waves. However, there are some difficulties in the development of acoustic methods. Early studies included modal analysis of beams [3, 6, 15] and thin plates [4]. More recently, studies have focused on the analysis of certain modes of Lamb waves, such as modes A0 and S0 in beams [7] and plates [12], as well as modes SH0 in plates [13]. These methods achieve higher sensitivity and resolution by analyzing the characteristics of reflected waves and the time the wave spends, passing through delamination [12, 13]. However, the application of these methods to complex composite structures with curved surfaces is limited. In this regard, new designs of non-contact sensors and actuators are being developed. Such developments include integrated piezoelectric plate active sensors (PWAS) [11, 16], embedded sensor networks [17], macrofiber composite (MFC) sensors [18], multichannel MEMS sensor arrays [19], conventional broadband laser Doppler vibrometers (LDV) [11], and scanning sensors (SLDV) [8, 17]. In parallel with the development of experimental methods, mathematical models, and methods for analyzing the propagation of acoustic waves in layered composites with defects are being improved [2, 15, 17, 20, 21]. These methods, often based on a finite element (FE) approach, provide a more detailed understanding of the processes within the material. Developments in this area provide a basis for the analysis and improvement of acoustic methods for non-destructive testing of laminated composites with various structural characteristics: quasi-isotropic [13, 20], transverse-isotropic [9, 17], orthotropic materials [10, 14], cross-layer laminated materials [8, 12, 21], as well as materials with wave attenuation characteristics [16, 22]. In this work, we focused on the study of the possibility of diagnosing lenticular delamination, which often occurs in the production of polymer composites with multilayer reinforcement. An important factor was the development of a method that does not require complex experiments or large-scale computer simulations with composite structures without structural defects, and which could be applied in practice. We have applied a frequency range of acoustic waves limited to 30 kHz, as this is the most accessible range for the use of simple laser vibrometers in production environments. Below there are results of our experiments and numerical studies, related to the propagation of wave packets in a flat plate, made of carbon fiber, and having artificially created delamination. Within the framework of numerical experiments, two cases of the defect occurring in thickness were considered: under the top layer and in the middle. These studies contribute to the development of non-destructive testing methods and improve the quality of composite structures.
2 Experiment We investigated a square CFRP panel with a configuration of [−45°, 45°, 90°, 0°]S , consisting of eight layers. The dimensions of this panel are 62 × 62 cm2 with a thickness of 2 mm. In its center there are lenticular delaminations with a diameter of 15 and
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30 mm, located along the diagonals of the square at 22 cm from its center. The thickness of the delaminations is 0.1 and 0.3 mm, respectively. A porous material is placed around the perimeter of the plate to absorb the reflected waves (see Fig. 1b). The elastic characteristics of the CFRP material were obtained experimentally in accordance with the requirements of ASTM standards and refined numerically [22].
Fig. 1. Sample under study: (a) diagram and (b) full-scale sample
Fig. 2. Map of speed measurement points in the panel
To generate wave packets on the panel under study, we used a STEMINC piezoelectric transducer SMD50T21F45R with a diameter of 50 mm and a thickness of 2.1 mm, installed in the center of the panel. The control signal was generated using a RIGOL DG1022 signal generator and amplified by Apex Microtechnology PA94 piezo drivers. Measurement of the out-of-plane velocity on the surface of the panel during the propagation of Lamb waves was carried out using a Polytec PDV 100 laser vibrometer. Signals from the vibrometer were recorded with the LeCroy 422 WaveSurfer digital oscilloscope, which made it possible to average the data and find the maximum absolute values of the out-of-plane velocity, when the wave packets passed through the observed areas of the surface. The obtained data was processed using scripts in MATLAB.
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We measured the out-of-plane velocity on the surface of a square panel along lines, drawn through its center and directed at different angles to the diagonals (0°, ±2°, ±5°, ±10°), close to the artificially created delaminations (see Fig. 2).
3 Finite Element Model of the System Under Study To prevent damage to the layers of the composite material adjacent to the delamination, the geometry of this material was modeled in two stages. First, the upper surface of the lenticular inclusion was created from two parts, using the equations of the ellipse and the square of the hyperbolic tangent. This created top surface was then combined with its specular reflection from the horizontal plane to create a lens-shaped solid (see Fig. 3). This solid was inserted between adjacent layers of composite material and cut from them by a Boolean operation. To achieve the desired thickness of delamination, the internal pressure inside thin empty inclusion was gradually increased and maintained constant.
Fig. 3. Three-dimensional delamination model
COMSOL Multiphysics version 6.1 finite element analysis software was used to simulate the propagation of acoustic waves, caused by the PZT disc in the center of the carbon fiber panel. The model included the related equations of anisotropic linear dynamics of an elastic body and piezoelectricity. The geometry of the object fully corresponded to the real prototype (see Fig. 4a). The panel consisted of 8 layers (see Fig. 4b) with the properties, given in Table 1. Tensor of elastic moduli of each layer had an angle of rotation in accordance with the layout [−45°, 45°, 90°, 0°]S . To construct the finite element mesh, geometry features such as a thin panel, local delamination, and the contact area of the panel with a disc piezoelectric transducer were considered. The main type of finite element was triangular. A high-density triangular grid partition was created on the midplane both under the PZT disk and at the lower boundary of the empty inclusion (see Fig. 4c). The created 2D mesh was then swept onto the top and bottom layers of the panel, creating two layers of prismatic elements on each layer (see Fig. 4d). Because of limitations in the range of Lamb wave propagation due to both geometric and material factors, we resorted to using the Rayleigh model to describe the attenuation. The values of the Rayleigh damping parameters, such as mass damping (α M ) and stiffness damping (α K ), were taken from the source [22]. To reduce the effects of reflected waves from the edges of the panel, the coefficients α M and α K have been increased exponentially
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In-plane Young’s modulus, E x
120 GPa
In-plane Young’s modulus, E y
9.5 GPa
Transversal Young’s modulus, E z
9.5 GPa
In-plane shear modulus, Gxy
4.5 GPa
Interlaminar shear modulus, Gxz
4.5 GPa
Interlaminar shear modulus, Gyz
3.2 GPa
Poisson’s ratios, ν xy , ν xz
0.3
Poisson’s ratio, ν yz,
0.48
Mass density, ρ
2000 kg/m3
Fig. 4. FE model of the experiment: (a) panel geometry and actuator, (b) layer-by-layer plate partitioning, (c) triangular finite element mesh, (d) FE thickness partitioning.
with distance and are more noticeable at distances greater than 30 cm from the center of the panel. When modeling the non-stationary process of generating elastic waves, using a piezoelectric actuator and their further propagation in a layered panel, we used the linear elastic body and piezoelectricity interfaces built into COMSOL.
4 Numerical and Experimental Results About 4 ms after the simulation was began, when the delamination had reached the required thickness and stabilized, a signal was applied to the PZT actuator, the shape of which is shown in Fig. 5.
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Fig. 5. Time dependence of the electric potential, applied to the PZT actuator.
Then the process of wave propagation begins. The results of the problem of wave propagation in a plate with a defect in the middle (between the 4th and 5th layers) are presented in Fig. 6.
Fig. 6. Out-of-plane velocity field for the case of a defect located between layers 4 and 5: (a) before the wave has reached the defect zone, (b) after themain wave peak has passed through the defect zone, (c) enlarged defect area.
This figure shows the non-planar velocity fields of points on the surface of the panel at two points in time: the first before the wave has reached the defect zone, and the second after the wave peak has passed through this zone but has not yet been reflected from the plate boundary. It is worth noting that due to the asymmetry of the velocity front and the square shape of the plate, the initial signs of the wave reflection process at the right and left edges of the plate are visible in Fig. 6b. In Fig. 6c, where the region with the defect is presented in more detail, it is clearly seen that the field of out-of-plane velocities of wave A0 undergoes distortion, when passing over the defect: zones of local amplitude extremes appear. This confirms the possibility of conducting experimental diagnostics of delamination using a laser vibrometer, which measures the out-of-plane velocity on an oscillating surface. As the COMSOL system provides a wide range of data collection and analysis capabilities, we increased the number of speed measurement points when comparing
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finite element analysis data with experimental data (see Fig. 7). This would be done to consider possible errors in the positioning of the laser beam on the surface of the panel.
Fig. 7. Map of velocity measurement points on the panel during FE simulation
Figure 8 shows maps of the level lines of the maximum vibration velocity moduli, both experimentally measured and calculated by the finite element method. The position of the defect is indicated by a red circle.
(a)
(b)
Fig. 8. Maps of maximum vibration velocity modulus level lines: (a) experiment and (b) FE simulation.
In Fig. 8, the differences between the images can be explained by imperfect measurement accuracy and possible deviations within the panel. It is important that both images show general patterns in the propagation of waves in the defective zone, including an increase in the amplitude of the vibration velocity when the Lamb A0 wave passes through the defective zone, by a factor of about 1.5. Beyond the defect, distortions in wave propagation are also observed. After the finite element model data have shown good convergence with the experiment, consider the case, where the defect lies near the surface (between 1st layer and
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2nd layer). As in the previous case, we will construct the distribution of the field of out-of-plane velocities for two moments of time, when the wavefront has not reached the defect and when the maximum of the wave packet has passed through the defect.
Fig. 9. Out-of-plane velocity field for the case of a defect located between layers 1 and 2: (a) before the wave has reached the defect zone, (b) after themain wave peak has passed through the defect zone, (c) enlarged defect area.
Analyzing Fig. 9, we can conclude that, as in the case of a defect, located in the middle of the thickness, the wave field in the defect area undergoes distortion. However, in this case we have four local extrema in the defect zone. This may be due to the unlike the first case, the delamination lies much closer to the surface, which means the flexural stiffness will be lower in this area. To quantify the effect of the delamination position on the A0 wave, the velocities were averaged over the period when the wave passes the defective area along the diagonal of the panel (direction 0°). Also, for comparison, a similar non-stationary problem was solved for a defect-free panel.
Out-of-plane velocity, m/s
0.03 0.025
defect under 1st layer defect under 4th layer no defect
0.02 0.015 0.01 0.005 0 4.2
4.25
4.3
4.35
4.4
Time, s
4.45
4.5
4.55
4.6 10-3
Fig. 10. Average values of out-of-plane vibration velocity for the period of passage of the wave packet through the defect zone
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The data in Fig. 10 show that the maximum amplitude of the out-of-plane velocity is reached, when the defect lies in the middle of the panel thickness. If the defect is located under the top layer, there is a speed drop of 8%. This difference in velocity is not sufficient to determine the position of the defect in depth with sufficient accuracy, but it is still sufficient to detect the defect in the panel.
5 Conclusions This chapter presents the results of a study of low-frequency acoustic NDE method of layered composite structures for the detection and accurate localization of delamination. As an object of research, we used a manufactured CFRP panel, consisting of 8 layers, with inclusions between 4th and 5th layers. After conducting a finite element analysis of the Lamb waves, generated by omnidirectional piezoelectric transducer, mounted in the center of the panel, we found an increase in the amplitudes of out-of-plane velocities near the delaminations. We also introduced a simple method for controlling velocities and used a laser vibrometer to detect and accurately locate delaminations within the geometry under study. The experimental results confirmed our theoretical conclusions. Also, based on the model, the case of the defect between 1st and 2nd layers was considered. It turned out that the behavior of the velocity field differs from the previous case, but the averaged amplitude of the out-of-plane velocity has decreased slightly. Acknowledgements. The authors wish to acknowledge the Russian Academy of Science which provides financial support for project No. AAAA–A16–116012610052-3, and the Joint Center for Scientific and Technological Equipment of SSC RAS No. 501994 for providing a necessary equipment.
References 1. Babu, J.: Assessment of delamination in composite materials. J. Eng. Manuf. 230(11), 1990– 2003 (2016) 2. Chaupal, P., Rajendran, P.: A review on recent developments in vibration-based damage identification methods for laminated composite structures. Compos. Struct. 311, 1–13 (2023) 3. Pardoen, G.C.: Effect of delamination on the natural frequencies of composite laminates. J. Compos. Mater. 23(12), 1200–1215 (1989) 4. Chattopadhyay, A., Nam, C., Kim, Y.: Damage detection and vibration control of a delaminated smart composite plate. Adv. Compos. Lett. 9(1), 096369350000900101 (2000) 5. Tan, P., Tong, L.: Delamination detection of composite beams using piezoelectric sensors with evenly distributed electrode strips. J. Compos. Mater. 38(4), 321–352 (2004) 6. Baba, B.O., Gibson, R.F.: The vibration response of composite sandwich beam with delamination. Adv. Compos. Lett. 16(2), 096369350701600204 (2007) 7. Hu, N., Liu, Y., Li, Y., Peng, X., Yan, B.: Optimal excitation frequency of Lamb waves for delamination detection in CFRP laminates. J. Compos. Mater. 44(13), 1643–1663 (2010) 8. Migot, A., Giurgiutiu, V.: Numerical and experimental investigation of delamination severity estimation using local vibration techniques. J. Intell. Mater. Syst. Struct. 34(9), 1057–1072 (2023)
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9. Nabiswa, E.J., Chierichetti, M.: Structural health monitoring of composite structures using guided Lamb waves. In: AIAA SciTech Forum, vol. 1709 (2023). Springer, Heidelberg 10. Lorriot, T., Wargnier, H., Wahl, J.C., Proust, A., Lagunegrand, L.: An experimental criterion to detect onset of delamination in real time. J. Compos. Mater. 48(18), 2175–2189 (2014) 11. Shpak, A.N., et al.: Influence of a delamination on Lamb wave excitation by a nearby piezoelectric transducer. J. Intell. Mater. Syst. Struct. 32(3), 267–283 (2021) 12. Ramadas, C., Hood, A., Padiyar, J., Balasubramaniam, K., Joshi, M.: Sizing of delamination using time-of-flight of the fundamental symmetric Lamb modes. J. Reinf. Plast. Compos. 30(10), 856–863 (2011) 13. Mei, H., James, R., Giurgiutiu, V.: Damage detection in laminated composites using pure SH guided wave excited by angle beam transducer. In: Proceedings of SPIE Health Monitoring of Structural and Biological Systems XIV, pp. 65–76. Springer, Heidelberg (2020) 14. Zhang, H., Sun, J., Rui, X., Liu, S.: Delamination damage imaging method of CFRP composite laminate plates based on the sensitive guided wave mode. Compos. Struct. 306, 116571 (2023) 15. Ihesiulor, O.K., Shankar, K., Zhang, Z., Ray, T.: Validation of algorithms for delamination detection in composite structures using experimental data. J. Compos. Mater. 48(8), 969–983 (2014) 16. Gresil, M., Giurgiutiu, V.: Prediction of attenuated guided waves propagation in carbon fiber composites using Rayleigh damping model. J. Intell. Mater. Syst. Struct. 26(16), 2151–2169 (2015) 17. Allen, J.C.P., Ng, C.T.: Damage detection in composite laminates using nonlinear guided wave mixing. Compos. Struct. 311, 116805 (2023) 18. Daraji, A.H., Ye, J., Hale, J.M.: Optimization of active SHM system based on optimal number and placement of piezoelectric transducers. J. Intell. Mater. Syst. Struct. 34(4), 425–439 (2023) 19. Volker, A.W.F., et al.: Non-contact MEMS-sensor array inspection of composites and metallic parts using Lamb waves. J. Nondestruct. Eval. Diagnost. Prognost. Eng. Syst. 6(4), 041002 (2023) 20. Zeng, X., et al.: Lamb wave-based damage localization and quantification algorithms for CFRP composite structures. Compos. Struct. 295, 115849 (2022) 21. Yu, Y., Liu, X., Wang, Y., Wang, Y., Qing, X.: Lamb wave-based damage imaging of CFRP composite structures using autoencoder and delay-and-sum. Compos. Struct. 303, 116263 (2023) 22. Shevtsov, S., et al.: On the directivity of Lamb waves generated by wedge PZT actuator in thin CFRP panel. Materials 13(4), 907 (2020)
Theoretical Approaches for the Damage Identification in the Timoshenko Beam Based on Solving a Coefficient Inverse Problem V. E. Yakovlev1(B) , A. V. Cherpakov1,2 , and S.-H. Chang3 1 I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal
University, 8a, Milchakov Street, Rostov-on-Don, Russia [email protected] 2 Don State Technical University, 1, Gagarin Square, Rostov-on-Don, Russia 3 Department of Microelectronics Engineering, National Kaohsiung University of Science and Technology, Kaohsiung, Taiwan (R.O.C.)
Abstract. The modal analysis problem for a beam performing bending vibrations is considered. The defect in the beam is modeled as a change in the cross-section area and the moment of inertia. The damage identification is based on the recovery of these coefficients by using additional information about resonant frequencies and eigenmodes. The solution of such a coefficient problem is conducted to minimize a special misfit functional. The paper presents the construction of this functional, considering the specificity of the modal analysis problem. The trust region method was used to solve the optimization problem. The gradient and the Hessian of the misfit functional were obtained on the sensitivity analysis of the forward problem. Keywords: Damage Identification · Timoshenko Beam · Coefficient Inverse Problem · Modal Analysis
1 Introduction The damage identification in the beam can be easily performed based on the analysis of the curvature of the eigenmodes [1]. With this approach, localization of all defects is only possible if enough measurements are taken along the entire length of the beam. Complete information about the eigenmodes is also required in methods as modal assurance criterion [2], strain energy-based methods [3] and other [4–7]. In practice, there are various limitations that prevent measurements from being taken in the entire body. They can be caused by the total number of sensors placed, their dimensions, the influence of sensors on the vibrations, the impossibility to install sensors due to the isolation of the surface. There are approaches where only resonant frequencies are used for damage identification [8, 9]. Thus, a sufficiently small single defect was considered in [9]. Due to this, using the expression for the correction of resonant frequencies, the problem of finding © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 390–398, 2024. https://doi.org/10.1007/978-3-031-52239-0_37
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the beam’s parameters of defect was reduced to solving a system of transcendental equations. However, it can lead to non-uniqueness of the solution by using only resonance frequencies and eigenmodes should also be considered.
2 Problem Statement and Methods 2.1 Research Aim The aim of this work is to construct approach to identify both the location and the magnitude of defects from additional information on the eigenmodes, obtained in only a small number of measurement points. 2.2 Statement of the Modal Analysis Problem The eigenvalue problem for the Timoshenko beam with free boundary condition has the following form: −λρAw − ∂x [κAG(wx − θ)] = 0, −λρI θ − ∂x (EI θx ) − κAG(wx − θ) = 0, (EI θx )(0, t) = 0, (κAG(wx − θ))(0, t) = 0, (EI θx (l, t)) = 0, (κAG(wx − θ))(l, t) = 0,
(1)
where w is the bending displacement, θ is the rotation angle of the normal to the midface of beam, ρ is the density, E is Young’s modulus, I is the moment of inertia, A is the cross-section area, κ is a shear coefficient, G = 0.5E/(1 + v) is the shear modulus, ν is a Poisson’s ratio, l is the beam length. Discretization of problem (1) by finite element method leads to the generalized eigenvalue problem: (K − λi M)vi = 0,
(2)
where M and K are the real, symmetric mass matrix and stiffness matrix, respectively, λi = (2π fi )2 is an i-th eigenvalue, f i is the i-th resonant frequency, vi is the i-th eigenvector, which includes the values of w and θ. 2.3 Optimization Problem Formulation To solve the coefficient inverse problem, a functional satisfying the following properties should be used: (i) values of the largest and smallest resonant frequencies should equally affect the misfit functional (see [8]); (ii) since eigenmodes is unique up to a constant multiple they should be normalized; (iii) the misfit functional should be differentiable;
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This property is possessed by the functional: f (k) = L(k) + αR k − kprior , diT di + λ2i L(k) = i
i
1 1 di (k) = v˜ i (k) − ∗ v˜ iobc , λi (k) = ai ai
λi (k) −1 , λobc i
(3)
where k is unknown coefficient, solution of the inverse problem, k prior is a priori information about the solution, v˜ i is the vector of i-th eigenmode of oscillations measured at ˜ iobc are the i-th eigenvalue and the vector of i-th eigenmode of the given points, λobc i ,v oscillations, obtained from the experimental results, ai , ai∗ are normalizing multipliers, R(k) is the stabilizing norm, α is the regularization parameter. The parameter α is chosen to obtain certain properties of the solution. If the problem requires finding two unknown coefficients k 1 and k 2 , then the functional (3) has the form: f (k1 , k2 ) = L(k1 ) + L(k2 ) + α1 R k1 − k1prior + α2 R k2 − k2prior where there are already two regularization parameters α1 and α2 . Their values should usually be related to the values of the coefficients k 1 and k 2 . An example of the use of the misfit functional with two regularization parameters can be found in [10]. 2.4 Minimization of the Misfit Functional For efficient minimization of the functional (3), information about the gradient and the Hessian is required. This can be obtained by considering an expansion of the misfit function (3) in the vicinity of the vector k0 [11]: 1 f (k 0 + δk) = f (k 0 ) + ∇f T δk + δkT Hδk + O δk3 , 2 T T ∇f = 2 di Jvi + λi Jλi + α∇R k − kprior , (4) i
H ≈ 2 JT J + α C ,
where δk is the perturbation in the coefficient, ∇f is gradient, H is the Hessian, J is the Fréchet derivative matrix, consisting of the matrices Jvi and Jλi , C is the Hesse matrix of the stabilizing norm R(k). The matrix Jvi and matrix row Jλi express the dependencies: 1 δ˜vi = Jvi δk, δλi = Jλi δk. ai The matrix 2JT J in (4) is the so-called approximate Hessian of the functional L(k). Obtaining the gradient and approximate Hessian, based on the matrix J, is discussed in [11]. Using the trust regions method [12], the approximate Hessian is sufficient to perform an efficient minimization of the functional (3), which will be demonstrated in the following.
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2.5 Obtaining the Fréchet Derivative To obtain the sensitivity of the modal analysis problem, the standard procedure from [3, 8] is used. It is based on discretization of the forward problem and linearization of the dependence of its solution on the coefficients. As a result, the variation problem is obtained, from which the required relations are expressed. Thus, if the coefficient is included in the mass and stiffness matrix, then at the perturbed coefficient, Eq. (2) takes the form: (K + δK − (λi + δλi )(M + δM))(vi + δvi ) = 0.
(5)
It is further assumed that for each finite element, the value of the coefficient is constant. Then the perturbation in the mass matrix and stiffness matrix can be represented as δM =
Ne e=1
δke Me , δK =
Ne
δke Ke ,
e=1
where e is a number of FE grid, Ne is a number of grid elements, δke is a perturbation of the coefficient in element e, Ke and Me are the assembled local matrices N × N, N is a number of grid nodes. By opening the brackets in (5) and discarding the elements of the second order of smallness, the perturbed problem is obtained as δKvi − λi δMvi − δλi Mvi + (K − λi M)δvi = 0.
(6)
To obtain the required expressions from (6), it is necessary to consider that the following properties are valid due to the symmetry of the matrices K and M: viT (K − λi M) = 0,
(7)
vjT Mvi = 0, i = j.
(8)
Due to (7), by multiplying the expression (6) on the left side by the transposed vector viT , the dependence between the perturbation of eigenvalue and the perturbation of the coefficients is expressed as δλi =
1 T T v . δKv − λ v δMv i i i i i viT Mvi
(9)
To obtain the Fréchet derivative for the eigenmodes from Eq. (6), their perturbation is represented as
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N
δvi =
δcj vj
(10)
j=1, j=i
where N is a total number of eigenvectors. Substituting (10) into (6), multiplying this expression from the left by vjT , and considering (7) and (8), it can be obtained: 1 T vjT δKvi − λi vjT δMvi δcj = λi − λj vj Mvj
(11)
Multiplying the row vector from (11) and the column vector from (10) and adding row (9) gives the Fréchet derivative matrix, which provides opportunity to compute the gradient and the Hessian of the functional (3).
3 Algorithm for Solving the Coefficient Inverse Problem for the Moment of Inertia First, to check the efficiency of the described algorithm, the problem of recovery of one unknown coefficient I(x) will be solved. The parameters in problem (1) for the model without defect were chosen in accordance with the real hollow beam of rectangular cross-section: E = 1.9 · 1011 , ρ = 7700, l = 1.13, v = 0.3, A0 = bh − (b − 2t2 )(h − 2t1 ), I0 =
1 3 bh − (b − 2t2 )(h − 2t1 )3 12
b = 28.5 · 10−3 , h = 26 · 10−3 , t1 = t2 = 2.5 · 10−3 . The shear coefficient was calculated for a given cross-section using formulae from [13]. The recoverable moment of inertia I(x) is a piecewise constant function and deviates from the value I 0 in a certain region. Thus, the a priori information and the initial approximation for the functional (3) was chosen as constant I 0 . To reduce problem (1) to the form (2), linear finite element approximation was used. The averaged scheme [14] was chosen because it allows more accurate calculation of the first resonant frequencies. The stabilizing norm R(k) in the functional (3) was chosen in accordance with the so-called Tikhonov regularization in the Sobolev space W12 [15]: l k 2 + (∂x k)2 dx
RW1 (k) = 2
0
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The reconstruction of the piecewise constant coefficient I(x) from the first nine resonant frequencies is presented in Fig. 1. The results of numerical experiment have shown that in the absence of noise, a smooth solution is obtained without using regularization (α = 0). However, such a result is observed only when the first resonant frequencies are used. When their number increases, the identification becomes difficult. The solution of the coefficient inverse problem is non-unique. In the result of using symmetric boundary conditions, symmetry in the solution of the identification problem arises. This problem can be solved by also considering oscillations under other boundary conditions. However, this is unjustified because the authors have found that only the mathematical model of a beam with a free fixation agrees well with the real experiment. Adding information about the resonant frequencies to the information about the eigenmodes provides opportunity to solve the problem of uncertainty in the solution. Thus, Fig. 2 shows the identification of the coefficient I(x), using four resonant frequencies and four measurements of the displacement eigenmodes w, obtained at these frequencies.
Fig. 1. Identification of the coefficient by using first nine resonant frequencies
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Fig. 2. Identification of the coefficient by using four resonant frequencies and four measurements of the eigenmodes of the displacement
In addition to a single zone, this approach also allows one to recover a coefficient with several degradation zones. Figure 3 shows the identification of two areas using eigenmodes information at six points: x = 0.1, 0.3, 0.5, 0.7, 0.9, 1.1.
Fig. 3. Identification of two regions using eigenmodes information at six points
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4 Defect Identification Based on Recovery of Two Coefficients The presence of a defect in a beam structure leads to a change in two coefficients: moment of inertia and cross-sectional area. Figure 4 shows the recovery of the coefficients I(x) and A(x). As can be seen, the recovery of several coefficients at once is a more difficult task. In the figure, two zones of destruction can be easily recognized, but around these zones there is a noticeable increase in the coefficient relative to its a priori value.
Fig. 4. Identification by recovering the coefficients I(x) and A(x)
5 Conclusion In this paper, the use of a special functional that considers the specificity of the inverse modal analysis problems is proposed. As it was shown, minimization of such a functional gave a stable solution using information on the first resonant frequencies even in the absence of regularization. However, an identification only by using the resonant frequencies of one type led to non-uniqueness of the solution. Therefore, the information about the eigenmodes was also used. It allowed one to easily identify different damages. It is shown that the considered approach, based on the solution of the coefficient inverse problem, has several advantages in comparison with approaches where the defect is modeled by a small quantity of parameters. When using it, do not need to have information about the number of defects. The parameters of several defects at once can be determined quite accurately by analyzing the recovered coefficients. Moreover, this approach provides opportunity to solve the inverse problem considering the non-uniqueness of the solution. This is extremely useful when performing identification in practice, where the uniqueness of the solution cannot be ensured due to limitations in the amount of available information.
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Acknowledgements. The work was supported by the grant No. 22-29-01259 of the Russian Science Foundation in the Don State Technical University, https://rscf.ru/project/22-29-01259/.
References 1. Dahak, M., Touat, N., Kharoubi, M.: Damage detection in beam through change in measured frequency and undamaged curvature mode shape. Inverse Prob. Sci. Eng. 27(1), 89–114 (2019). https://doi.org/10.1080/17415977.2018.1442834 2. Pastor, M., Binda, M., Harˇcarik, T.: Modal assurance criterion. Procedia Eng. 48, 543–548 (2012). https://doi.org/10.1016/j.proeng.2012.09.551 3. Fan, W., Qiao, P.: Vibration-based damage identification methods: a review and comparative study. Struct. Health Monit. 10(1), 83–111 (2011). https://doi.org/10.1016/j.ymssp.2011. 11.010 4. Soloviev, A.N., Parinov, I.A., Cherpakov, A.V., Chaika, Y.A., Rozhkov, E.V.: Analysis of oscillation forms at defect identification in node of truss based on finite element modeling. Mater. Phys. Mech. 34(2), 192–197 (2018). https://doi.org/10.18720/MPM.3722018_12 5. Gillich, N., et al.: Beam damage assessment using natural frequency shift and machine learning. Sensors 22, 1118 (2022). https://doi.org/10.3390/s22031118 6. Lyapin, A., Shatilov, Y.: Vibration-based damage detection of the reinforced concrete column. Procedia Eng. 150, 1867–1871 (2016). https://doi.org/10.1016/j.proeng.2016.07.184 7. Soloviev, A. N., Parinov, I. A., Cherpakov, A. V., Esipov, Yu. V.: Experimental vibration diagnostics of the floor plate set of building construction. In: Long, B.T., Kim, Y.-H., Ishizaki, K., Toan, N.D., Parinov, I.A., Vu, NPi. (eds.) MMMS 2020. LNME, pp. 255–260. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-69610-8_35 8. Lebedev, I.M., Shifrin, E.I.: Solution of the inverse spectral problem for a rod weakened by transverse cracks by the Levenberg—Marquardt optimization algorithm. Mech. Solids 54, 857–872 (2019). https://doi.org/10.3103/S0025654419060025 9. Vatul’yan, A.O., Osipov, A.V.: One approach to the determination of the parameters of a defect in a rod. Russ. J. Nondestruct. Test. 50, 649–658 (2014). https://doi.org/10.1134/S10 61830914110084 10. Burstedde, C., Ghattas, O.: Algorithmic strategies for full waveform inversion: 1D experiments. Geophysics 74(6), WCC37–WCC46 (2009). https://doi.org/10.1190/1.3237116 11. Pratt, R.G., Shin, C., Hick, G.J.: Gauss-Newton and full Newton methods in frequency– space seismic waveform inversion. Geophys. J. Int. 133(2), 341–362 (1998). https://doi.org/ 10.1046/j.1365-246X.1998.00498.x 12. More, J.J., Sorensen, D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 3, 553–572 (1983). https://doi.org/10.1137/0904038 13. Cowper, G.: The shear coefficient in Timoshenko’s beam theory. J. Appl. Mech. 33(2), 335– 340 (1996). https://doi.org/10.1115/1.3625046 14. Kim, K.O.: A review of mass matrices for eigenproblems. Comput. Struct. 46(6), 1041–1048 (1993). https://doi.org/10.1016/0045-7949(93)90090-Z 15. Tikhonov, A.N., Arsenin, V.A.: Solution of Ill-Posed Problems. Halsted Press, New York (1997)
Calculation of the Stress State of a Three-Layer Spherical Shell Based on Exact, Asymptotic Solutions and Solutions According to Some Applied Theories Nikolay V. Boyev1,2(B) 1 I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal
University, Rostov-on-Don, Russia [email protected] 2 Admiral Ushakov Maritime State University, Novorossiysk, Russia
Abstract. The stress state of a three-layer spherical shell was calculated on the basis of exact, asymptotic solutions and solutions according to some applied theories. As applied theories, we took theories, based on the theory of a single normal for the entire package on the theory of a broken element, and on the theory that considers transverse shear. A comparative analysis of the stresses, obtained from all five solutions, is given. The asymptotic solution, constructed in the work, based on three-dimensional equations, indicates methods for constructing refined applied theories for inhomogeneous thicknesses, including layered plates and shells. Keywords: Layered Plates · Layered Shells · Three-layer Spherical Shell · Applied Theories · Exact Solution · Asymptotic Solution
1 Problem Statement Static and dynamic problems of mechanics are used to study a wide range of technical problems [1–5]. This paper examines layered thin-walled structures, which are widely used in modern technology. The complexity of their calculation is associated with the structure of the materials used, their shape, physical and mechanical properties. The main method for calculating layered plates and shells is the selection or construction of a well-founded applied theory, which must adequately consider the structure, physical and mechanical properties of the layers and thin-walled structure, as well as the conditions of influence on its operation. Some classic monographs on the general theory of plates and shells [6–14], as well as layered thin-walled structures [15–19], should definitely be mentioned. Here is a link to a current review on the theory of layered plates and shells [20]. Works in which the accuracy of applied theories is assessed on the base of a three-dimensional solution, including an exact analytical solution of the problem, remain relevant. In this paper, we will assess the applicability of applied theories of layered shells using the example of a closed three-layer spherical shell = [r1 , r2 ] × [0, π ] × [0, 2π ] with a light filler. Let us consider the connection of layers to be rigid (see Fig. 1). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 399–411, 2024. https://doi.org/10.1007/978-3-031-52239-0_38
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On the spherical surfaces of the shell r = rj (j = 1, 2) normal forces are specified as σr (r1 ) = 0 = σrθ (r1 ), σr (r2 ) = APn (cos θ ), σrθ (r2 ) = 0 Here A is a constant, Pn (cos θ ) is a polynomial of Legendre of degree n.
Fig. 1. Shell under study
2 Solution Method Three theories most common in technical applications are taken as applied ones. The first theory, based on the Kirchhoff-Love hypotheses for the entire package (single normal theory), was developed in [6]. The second is based on the adoption of the Kirchhoff-Love hypotheses for each load-bearing layer, and for the filler – constancy in shear thickness. We will call this theory by the theory of the broken element. The equilibrium equations of this theory for shallow shells were obtained in [21]. The third theory generalizes the first by taking into account transverse shear deformations and its comparison is constructed in [22]. The results of calculations, based on these theories, are compared with the exact solution of the problem and the asymptotic one, obtained by directly integrating the equations of the linear theory of elasticity, considering in the characteristics of the stress-strain state the first two terms of their expansions in terms of the thin-walled shell parameter.
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2.1 Exact Solution Let us consider the construction of an exact solution for a closed three-layer spherical shell (see Fig. 1). In the case of radial inhomogeneity, the equations of the linear theory of elasticity allow separation of variables, if ur and urθ sought in the form: ur = y1 (ξ )Pn (cos θ), uθ = y2 (ξ )
dPn (cos θ ) dθ
(1)
Representation (1) allows us to write the stresses σr and σrθ in the form: σ (ξ ) = ε−1 e−εξ y3 (ξ ), y3 (ξ ) = 2μ(1 − ν)−1 (1 − ν)y1 + εν 2y1 − n(n + 1)y2 τ (ξ ) = ε−1 e−εξ y4 (ξ ), y4 (ξ ) = μ y2 + ε(y1 − y2 ) Let us introduce a vector-column, which for convenience we will write as y = {y1 , y2 , y3 , y4 }. Then, for each layer, the equations of the theory of elasticity can be represented as dys = Bs ys , s = 1, 2, 3. dξ
(2)
where Bs are the matrices that depend on the νs , n, ε. νs 1−2νs − 2νs ε 0 1−νs 1−νs n(n + 1)ε 2(1−νs ) −ε ε 0 1 Bs = − 2(1+νs ) ε2 − 1+νs n(n + 1)ε2 − 1−3νs ε n(n + 1)ε 1−νs 1−νs 1−νs n(n+1) 1+νs 2 νs 2 −2ε − 1−νs ε 2 1−νs − 1 ε − 1−νs ε The solution of the vector differential Eq. (2) is easy to write as ys = eBs ξ cs , ξs−1 ≤ ξ ≤ ξs
(3)
Here cs is a column vector of arbitrary constants, which is defined in terms of boundary conditions, and a function of the matrix is defined by the relation [23]: f (Bs )a =
4 k=1
a, ψ sk
f (λsk )ask ϕ sk , ask =
ϕ sk , ψ sk
(4)
where λsk are the characteristic numbers of the matrix Bs , ψ sk are the eigenvectors of the matrices, corresponding to Bs and its conjugation Bs∗ . The characteristic numbers of the matrices Bs and Bs∗ = BsT coincide λsk = λ∗sk = λk , k = 1, 2, 3, 4:
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λk = βk ε, β1 = n, β2 = −(n + 1), β3 = n − 2, β4 = −(n + 3) ϕsk1 = (1 − 2νs )[2(1 − 2νs )n(n + 1) − (1 − 2νs )βk (1 + βk )] ϕsk2 = (1 − 2νs )[4(1 − νs ) + βk ] ϕsk3 = 2ε(1 − 2νs )−1 {[2νs + (1 − νs )βk ]ϕsk1 − νs n(n + 1)ϕsk2 } ϕsk4 = ε[ϕsk1 − (1 − βk )ϕsk2 ] ψsk1 = 2ε(1 − 2νs )−1 {[(1 − 3νs ) + (1 − νs )βk ]ψsk3 − νs ψsk4 } ψsk2 = ε[−n(n + 1)ψsk3 + (2 + βk )ψsk4 ] ψsk3 = (1 − 2νs )[2(1 − 2νs )n(n + 1) − (1 − 2νs )βk (1 + βk )] ψsk4 = −(1 − 2νs )n(n + 1)(4νs − 3 + βk )
Vectors ϕ sk , ψ sm have the property of biorthogonality ϕ sk , ψ sm = 0 in case, if k = m. In formula (4), ask are the coefficients of the vector’s a decomposition with respect to the eigenvectors of the ϕ sk of matrix Bs . Along with the known boundary values: y3 (−1) = 0 = y4 (−1) and y3 (1) = A, y4 (1) = 0, we will consider y1 (−1) = u0 , y2 (−1) = v0 , y1 (1) = uh , y2 (1) = vh . For the first layer, the solution (3) is quite determined by setting y 0 = y(−1) = {u0 , v0 , 0, 0}, and the solution for each subsequent layer is determined by the value of the solution in the previous layer on the contacting surface. Satisfaction of the boundary conditions on the outer sphere yields a linear algebraic inhomogeneous system of the fourth order, with respect to the unknown components u0 , v0 , uh , vh of the vectors y0 and yh = {uh , vh , A, 0}, which can be written in the following form: 4
yh , ψ 3t − ctk y0 , ψ 1k = 0
exp[(1 − ξ2 )λt + (ξ1 + 1)λk ]
ctk = ϕ 1k , ψ 1k
(5)
k=1 4
ϕ 1k , ψ 2m ϕ 2m , ψ 3t
exp[(ξ2 − ξ1 )λm ] ϕ 2m , ψ 2m k=1
Here t = 1, 2, 3, 4; ξ1 , ξ2 are the radial coordinates of the contacting surfaces (see Fig. 1). From the system (5) the components u0 , vo of the vector yo are determined, after which all the characteristics of the stress-strain state are known in each layer. The proposed method for constructing a three-dimensional solution for a three-layer shell can be generalized to a shell with an arbitrary number of layers. 2.2 Asymptotic Solution This solution is constructed by the method of asymptotic integration of the equations of the theory of elasticity. The fundamentals of the asymptotic method used here were developed for transversely inhomogeneous plates by Yu. A. Ustinov and presented in his monograph [20]. Since the problem is axisymmetric, following [19], we seek a solution to the Lame equations in variables in the form: ur (r, θ ) = r u(r, θ ), uθ (r, θ ) = r
∂(r, θ ) ∂θ
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With respect to a pair of functions {u, }, we obtain a system of differential equations, which we write out in the variables ξ, θ :
κu + ε(2λ + κ)u + ελ∗ + εc∗ + εu + ε2μ 2u + ε∗ = 0
(6) + εu + εu + ε3μ + εu + ελ 3εu + u + ε∗ + +ε2 2μ(u + + ∗ ) = 0 and the boundary conditions:
κu + ε(2λ + κ)u + ελ∗ ξ =±1 = f ± (θ ), + εu ξ =±1 = 0.
(7)
Here ∂2 ∂ κ = λ + 2μ, f + (θ ) = f (θ ) = APn (cos θ ), f − (θ ) = 0, ∗ = ∂θ 2 + ctgθ ∂θ . The solution of the boundary value problem (6), (7) is found in the form of series by ε: u(ξ, θ ) = ε−1 (u0 + εu1 + ...), (ξ, θ ) = ε−1 (0 + ε1 + ...) To determine {ui , i }, we get a sequence of equations: ∗ 0 + 20 = −
1 f , 2 μ(0)
(8)
(0)
u0 = 0 +
1 g1 f 2 μ(0) g (0) 2
u1 =
1 f 2 g (0) 2
ξ
1 ˆ1 g2 d ξ + ∗ 0 ξ + uˆ 1 , 1 = −u0 ξ + 2
0
1 (9) (Q1 + 2Q2 ), 2μ(0) 1 (0) (0) g , Qi = Aqi Pn (cos θ), i = 1, 2 Q + g Q u1 = 1 + 1 2 1 2 2μ(0) g2(0) ˜ (0) (0) 1 ξ ξ˜ g1 μ(1) n1 g1 2 4 q1 = 2 (0) 1 + μ(0) − μ(0) + (0) μ g2 g2 d ξ + κ μg2 d ξ d ξ˜ g2 g2 −1 −1 ˜ 0 (1) ξ 1 (0) (1) ξ˜ (1) (1) g2 g g μ 3 2μ 1 λ q2 = 2 − (0) + μ(0) + 3 (0) − (0) 2μg2 d ξ − μg2 g2 d ξ d ξ˜ − 1(0) 1(0) n1 κ ˆ 1 + 2 ˆ1 =− ∗
g2
n1 = n(n + 1),
g2
(i) gj
g2 −1
−1
1 =
0
(i)
2μgj ξ d ξ , μ i
−1
2μ
1 =
μξ i d ξ , j = 1, 2, −1
2(1 − ν) 1 1+ν 2ν μ, κ = μ, g1 = , g2 = . λ= 1 − 2ν 1 − 2ν 1−ν 1−ν
g2
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N. V. Boyev
As functions ui , i , (i = 0, 1), it is sufficient to take the partial solutions of Eqs. (8) and (9): θ) 0 = − AP2μn (cos , (0)(0)n1 g 1 1 1 u0 = 2μ(0) (0) − n1 Pn (cos θ ), g2
A ˆ 1 = − (0) (q + 2q2 )Pn (cos θ ), 2μ n1 1 (0) (0) 1 g1 q1 + g2 q2 − uˆ 1 = 2μA(0) (0) g2
1 n1 (q1
+ 2q2 ) Pn (cos θ).
Thus, functions ui , i , (i = 0, 1) have been found that make it possible to find the main stresses considering the first two terms of the asymptotic expansions in terms of the shell thinness parameter ε. 2.3 Equations of Shell Theory, Obtained Under the Assumption that the Kirchhoff-Love Hypotheses are Satisfied for the Entire Package At this point in the paper, we will use the equations of the linear theory of multilayer anisotropic shells, based on the Kirchhoff-Love hypotheses, and obtained by S.A. Ambartsumyan in [6]. In the case of a three-layer spherical shell with symmetrical (both geometrically and mechanically) location of layers relative to the median surface of the shell, the system of equations of the shell theory in forces and moments takes the form: ⎧ ⎨ ctgθ (T1 − T2 ) + ∂T1 + 1 ctgθ (M2 − M1 ) + ∂M1 = 0 ∂θ R0 ∂θ (10) ⎩ T1 + T2 + 1 ∂ (M2 − M1 ) cos θ + sin θ ∂M1 + ZR0 = 0 R0 sin θ ∂θ ∂θ T1 = T2 =
8hmid 1−νmid 8hmid 1−νmid
M1 = − 89
M2 = − 89 ε1 = κ1 =
h3mid 1−νmid
8hext 1−νext (ε1 + νext ε2 ), 8hext 1−νext (ε2 + νext ε1 ), h3 −h3mid p(κ1 + νmid κ2 ) + 1−νext (κ1 + νext κ2 ) ,
h3mid 1−νmid
p(κ2 + νmid κ1 ) +
p(ε1 + νmid ε2 ) + p(ε2 + νmid ε1 ) +
1 R0 1 R20
h3 −h3mid 1−νext
(11)
(κ2 + νext κ1 ) ,
∂u 1 ∂θ − w , ε2 = R0 (uctgθ − w), ∂ ∂w 1 ∂w ∂θ u + ∂θ , κ2 = R2 ctgθ u + ∂θ ,
(12)
0
Here u is the tangential displacement, w is the displacement for the points of the median surface of the shell; R0 = 0.5(r1 + r2 ) is the normal component of the external load, carried to the median surface; hext , hmid , νext , νmid are the thicknesses and Poisson’s ratios of the outer and middle layers, respectively; 2h is the shell thickness. Substituting the relations (12) into (11) and the latter into the original Eqs. (10) and find u and w in the form: u = be
dPn (cos θ ) , w = ae Pn (cos θ ), dθ
Calculation of the Stress State of a Three-Layer Spherical Shell
405
with respect to constants ae , be , we obtain a linear algebraic system of the second order: a11 ae + a12 be = 0, a11 = s2 + s3 , a12 = s1 + s3 , a21 ae + a22 be = ε0 (1 + 0.5ε0 )2 A, a21 = 2l1 + l2 , a22 = l1 + n(n + 1)l2 , −1 n n s1 = αmid mid )Kext , pKmid + (1 − νmid )(1 − νext ) (2 − α−1 − ν + ν − α , s2 = − αmid p(1 + νmid ) + (1 − νmid )(1 ) (1 )(2 ) ext ext mid −1 n 2 3 n 3 −1 s3 = 48 ε0 αmid pKmid + 8 − αmid (1 − νmid )(1 − νext ) Kext , l1 = − αmid (1 + νmid )(1 − νmid )−1 p + (2 − αmid )(1 + νext )(1 − νext )−1 , −1 −1 n 3 (1 − ν n 3 l2 = ε02 n(n + 1)48−1 αmid mid ) pKmid + 8 − αmid (1 − νext ) Kext , −1 −1 −1 λmid = hmid h , λext = hext h , ε0 = 2hR0 , n =1−ν n Kmid mid − n(n + 1), Kext = 1 − νext − n(n + 1). 2.4 Equations of Broken Element Theory Let us use the equations of shell theory, obtained by E.I. Grigolyuk in [21]. The derivation of these equations is based on the following assumptions: (i) for the load-bearing layers, the Kirchhoff-Love hypotheses are accepted; (ii) for the filler, a constant shift across the thickness is assumed; (iii) the deflection of all layers is assumed to be the same. Let us write out the equilibrium equations of this theory as ⎧ 1 1 ⎪ ctgθ (T1 − T2 ) + ∂T =0 + R10 ctgθ (M2 − M1 ) + ∂M ⎪ ∂θ ∂θ ⎪ ⎪ ⎪ ⎨ ctgθ (t1 − t2 ) + ∂t1 + 1 ctgθ (m2 − m1 ) + ∂m1 − (h1 +h3 +2h2 )2 μ2 R0 α − ∂w = 0 ∂θ R0 ∂θ 4h2 ∂θ ∂M1 ∂ 1 ⎪ ⎪ T + T + − M cos θ + sin θ (M ) 1 2 2 1 ⎪ R0 sin θ ∂θ ∂θ ⎪ ⎪
⎩ (h1 +h3 +2h2 )2 1 ∂ ∂ R0 ∂θ (α sin θ) − ∂θ sin θ ∂w + ZR0 = 0 − R0 sin θ 4h2 ∂θ (13) Ei hi Ei hi (ε + νi ε2i ), T2i = 1−ν 2 (ε2i + νi ε1i ), 1−νi2 1i i 3 Ei hi Ei h3i M1i = − 12 1−ν M2i = − 12 1−ν 2 (κ1i + νi κ2i ), 2 (κ2i + νi κ1i ), i i 3 −u1 ) , T = Tj1 + Tj3 , Mj = Mj1 + Mj3 , j = 1, 2 α = h12(u
+h3 + 2h2 j
1 tj = 4 Tj1 − Tj3 (h1 + h3 + 2h2 ), mj = 41 Mj1 − Mj3 (h1 + h3
T1i =
(14) + 2h2 )
Here i = 1, 3 is the number of the load-bearing layer, hk , (k = 1, 2, 3) is the thickness of the layers, and Tj1 , Tj3 , Mj1 , Mj3 (j = 1, 2) are forces and moments in the loadbearing layers in the first and third layers, respectively. The layers are numbered as shown in Fig. 1. i − w , ε2i = R10 (ctgθ ui − w) ε1i = R10 ∂u ∂θ
(15) ∂ 1 ∂w κ1i = R12 ∂θ ui + ∂w ∂θ , κ2i = R2 ctgθ ui + ∂θ , i = 1, 3 0
0
If, in each rigid layer, the forces and moments (14) are expressed by (15) in terms of the displacements of their median surfaces, and the solution of the transformed system (13) will be searched in the form: w = aL Pn (cos θ ), u1 = b1
dPn (cos θ ) dPn (cos θ ) , u3 = b3 , dθ dθ
406
N. V. Boyev
then, with respect to the constants aL , b1 , b3 , we obtain an inhomogeneous algebraic system of the third order: ⎧ ⎪ aL −mi (1 + νi ) + ˜li Kin + bi mi + ˜li Kin = 0 ⎪ ⎪ ⎪ i=1,3 i=1,3 ⎪ ⎪ ⎪ ⎪ ⎪ (α1 +α3 +2α2 ) n ⎪ ˜ aL σi −mi (1 + νi ) + li Ki − p(1 − ν1 ) 2ε0 α1 α2 ⎪ ⎪ ⎪ ⎪ i=1,3 ⎪ ⎨ 1 + bi σi mi + ˜li Kin + p ε1−ν (b1 − b3 ) = 0 2 0 α1 α2 ⎪ i=1,3 ⎪ ⎪ ⎪ ⎪ 2 ⎪ +α +2α (α ) n 1 3 2 ⎪ aL −2mi (1 + νi ) + l˜i n1 Ki − p n1 ⎪ 8α1 α2 ⎪ ⎪ i=1,3 ⎪ ⎪ ⎪ ⎪ +α3 +2α2 ) 1−ν1 ⎪ bi −mi (1 + νi ) + ˜li Kin − p(b1 − b3 ) (α14ε (1 − ν1 )n1 = A 2ε ⎩ +n1 0 α1 α2 0 α1 i=1,3
n1 = n(n + 1), σ1 = 1, σ3 = −1, Kin = 1 − νi − n1 α3 m1 = 1, m3 = α3 1−ν1 , ˜l1 = 1 ε2 α 2 , ˜l3 = 1 ε2 3 1−ν1 α1 1−ν3
12 0 α1 1−ν3
12 0 1
2.5 Calculation Based on Theory that Consider Transverse Shear Deformations At this point, we will proceed from the equations of the theory of shells, which consider the transverse shear deformation of the element, and were obtained by A.G. Teregulov in [22]. For analysis, we will use the linear version of these equations: ⎧ ∂T 1 + ctgθ (T1 − T2 ) = 0 ⎪ ⎪ !" ⎨ ∂θ ∂ 1 sin θ ctgθ (M2 − M1 ) − ∂M =0 T1 + T2 + R10 − sin1 θ ∂θ (16) ∂θ ⎪ ⎪ ∂H 1 1 ⎩ −N1 + 2 + ctgθ (H1 − H2 ) = 0 R0
∂θ
T1 = (λ + 2μ)0 e1 + λ0 e2 − (λ + 2μ)1 κ1 − λ1 κ2 + (λ + 2μ)0ϕ ω1 + λ0ϕ ω2 , T2 = λ0 e1 + (λ + 2μ)0 e2 − λ1 κ1 − (λ + 2μ)1 κ2 + λ0ϕ ω1 + (λ + 2μ)0ϕ ω2 , M1 = (λ + 2μ)1 e1 + λ1 e2 − (λ + 2μ)2 κ1 − λ2 κ2 + (λ + 2μ)1ϕ ω1 + λ1ϕ ω2 , M2 = λ1 e1 + (λ + 2μ)1 e2 − λ2 κ1 − (λ + 2μ)2 κ2 + λ1ϕ ω1 + (λ + 2μ)1ϕ ω2 , H1 = (λ + 2μ)0ϕ e1 + λ0ϕ e2 − (λ + 2μ)1ϕ κ2 − λ1ϕ κ2 + (λ + 2μ)2ϕ ω1 + λ2ϕ ω2 , H2 = λ0ϕ e1 + (λ + 2μ)0ϕ e2 − λ1ϕ κ1 − (λ + 2μ)1ϕ κ2 + λ2ϕ ω1 + (λ + 2μ)2ϕ ω2 , 1
uθ N1 = A1313 ∗ (17) In this theory, it is assumed that the law of the distribution of transverse shear stresses over the thickness of the shell is given: 0 σrθ (θ, z) = σrθ (θ )f (z)
(18)
As a result, the tangential displacements are approximated as ∂w 1 + ϕ(z) u, ϕ(z) = uθ = u −z θ θ ∂θ 0
z f (z)dz 0
(19)
Calculation of the Stress State of a Three-Layer Spherical Shell
Using (19), we get the relationships: 0 0 1 ∂ uθ 1 e1 = Ro ∂θ − w , e2 = Ro ctgθ uθ −w , ω1 = 0 0 1 ∂ ∂w 1 ∂w κ1 = R2 ∂θ ∂θ + uθ , κ2 = R2 ctgθ ∂θ + uθ 0
1
1 ∂ uθ Ro ∂θ
, ω2 =
407
1 1 Ro ctgθ uθ
0
(20) Here z is the thickness coordinate, measured from the median surface. With the help of relations (17) and (19) we obtain from (16) a system of differen0
1
θ
θ
tial equations with respect to the deflection w and the terms u and u of the tangential 0
displacement uθ , the solution of which we look for in the form of w = ac Pn , u = θ
0
1
1
0
1
dP dP b d θn , u = b d θn , constants Pn , b, b satisfy a linear algebraic system of the third θ order: ⎧ 0 1 ⎪ ⎪ a11 = −2(λ + μ)0 + ε0 2μ1 − n1 (λ + 2μ)1 ⎪ ⎨ a11 ac + a12 b +a13 b = 0, 0 1 a21 ae + a22 b +a23 b = −A, a12 = 2(μ0 + ε0 μ1 ) − n1 (λ + 2μ)0 + ε0 (λ + 2μ)1 ⎪ ⎪ ⎪ 0 1 ⎩ a13 = 2μ0ϕ − n1 (λ + 2μ)0ϕ a31 ae + a32 b +a33 b = 0, 2 a21 = −4(λ + μ)0 + ε0 n1 n1 (λ + 2μ)2 − 22μ2 a22 = n1 −2 (λ + μ)0 + ε0 (λ + 2μ)1 + ε0 μ2 + n1ε0 (λ + 2μ)1 + ε0 (λ + 2μ)2 a23 = n1 −2(λ + μ)0ϕ + ε0 n1 (λ + 2μ)1ϕ − 2ε0 μ1ϕ a31 = −2(λ + μ)0ϕ − ε0 n1 (λ + 2μ)1ϕ + 2ε0 μ1ϕ
a32 = 2 μ0ϕ + ε0 μ1ϕ − n1 (λ + 2μ)0ϕ + ε0 (λ + 2μ)1ϕ a33 = 2μ2ϕ − n1 (λ + 2μ)2ϕ − A1313 , n1 = n(n + 1) ∗
μn =
1 −1
μ(ζ )ζ n d ζ, μnϕ =
1 −1
μ(ζ )n (ζ )d ζ, ϕ(ζ ) =
ζ
f (ζ )d ζ,
0
0 (ζ ) = ϕ(ζ ), 1 (ζ ) = ζ ϕ(ζ ), 2 (ζ ) = ϕ 2 (ζ )
The formulae for (λ + μ)n , (λ + 2μ)n , (λ + μ)nϕ , (λ + 2μ)nϕ are similar. = A1313 ∗
1 2
1
μ−1 (ζ )f 2 (ζ )d ζ, ζ =
−1
z h
Here the function selected is f (ζ ) = ζ 2 − 1. 2.6 Comparative Analysis of Numerical Results Numerical analysis was carried out for a closed three-layer spherical shell of symmetrical structure relative to the median surface (see Fig. 1). Let us write down the indicated stresses σθ obtained from the considered theories.
408
N. V. Boyev
Stresses, derived from an exact solution, have the form: ! 2μ n σθtd = ε−1 e−εξ 1−2ν νy1 + εy1 − (1 − ν)n1 y2 Pn − ε(1 − 2ν)y2 ctgθ dP dθ
4 y ,ψ gk (ξ )ϕ1ki , gk (ξ ) = e(ξ +1)λk ϕ0 ,ψ1k , 1 ≤ ξ ≤ ξ1 yi (ξ ) = yi (ξ ) = yi (ξ ) =
k=1 4
0
4
1k
ϕ ,ψ
gmk (ξ )ϕ2mi , gmk (ξ ) = gk (ξ1 )e(ξ −ξ1 )λm ϕ 1k ,ψ2m , ξ1 ≤ ξ ≤ ξ2 2m
m=1 k=1 4 4 4
gtmk (ξ )ϕ3ti , gtmk (ξ ) = gmk (ξ2 )e(ξ −ξ2 )λt
t=1 m=1 k=1
(21)
2m
ϕ ,ψ
2m 3t , ϕ 3t ,ψ 3t
ξ2 ≤ ξ ≤ 1, i = 1, 2
Stresses, derived from the asymptotic solution, have the form: # $ g2 dPn 1 1 σθas = μA ε−1 2μ(0)n(2−n + + ctgθ P n (0) dθ μ(0) (2−n1 ) 1) g2 % & ξ ξ (0) g2 n1 1 2ν + (0) Pn 2μg d ξ − g g d ξ + g ξ + q 2 2 2 1 (0) 2μ(0) 1 κ(1−2ν) g2 g2 0 ' (0) −1 $ g1 q1 +2q2 dPn 1 1 n1 Pn + 2ctgθ d θ + O(ε) + 2μ(0) (0) − 2−n1 ξ + 2−n1
(22)
g2
Stresses, obtained on the base of a solution, constructed on the assumption of the fulfillment of the Kirchhoff-Love hypotheses for the entire package, have the form: 2μ n n σθsn = 1−ν − (ae + n1 be )Pn + be ctgθ dP + 21 ε0 ζ (ae + be ) ctgθ dP + n1 Pn d θ d θ ! dPn 1 n + ε ζ + b +ν −ae Pn + be ctgθ dP (a )ctgθ 0 e e dθ 2 dθ (23) where i = 1, 2, 3 is the number of the layer, z is the coordinate, measured from the median surface. Stresses, derived from a solution, based on the theory of broken element„ have the form: 2μi dPn dPn − + ε + n b + b ctgθ ζ + b + n P ctgθ σθbr = 1−ν (a )P (a ) L 1 i n i 0 i L i 1 n d θ d θ i ! dPn n + ε ζ + b +ν −aL Pn + bi ctgθ dP (a )ctgθ 0 i L i dθ dθ b2 = 21 b1 + b3 − 21 ε0 (α1 − α3 )aL , ζi = zhi , zi ∈ − 21 hi , 21 hi , ζi ∈ − 21 αi , 21 αi (24) Stresses, derived from a solution, based on a theory, that considers transverse shear deformations, have the form: # 0 0 1 shift 2μ n σθ = 1−ν b +ζ ε0 ac + b + ϕ(ζ ) b −ctgθ dP − n P 1 n dθ $' (25) 0 0 1 dPn − ac Pn + ν b +ζ ε0 ac + b + ϕ(ζ ) b ctgθ d θ − ac Pn To clarify the influence, on the one hand of geometric and mechanical factors (the first series of calculations), on the other hand the variability of the external load (the
Calculation of the Stress State of a Three-Layer Spherical Shell
409
second series) on the accuracy of the description of the mechanical stress field by applied theories, two series of calculations were performed. In the first series, the load parameters were assumed as A = 1.0, n = 3, and ν1 = ν2 = ν3 = 0.3 were taken for Poisson’s ratios. With this load, for different values of ε0 , parameter ranges were constructed (at θ = 450 ) in the coordinate system (α2 , p˜ ), p˜ = − lg p, at which the considered theories in each layer and for the entire structure as a whole give an error of no more than 3%. For each of the four taken values of ε0 = 0.001, 0.005, 0.01, 0.05, five values of p = 10−K (K = 1, 2, .., 5) were recorded. Next, for all accepted p, various theories (22)–(25) were compared with the three-dimensional solution (21), when calculating with respect to the parameter α2 (0 ≤ α2 ≤ 2) with a step of 0.2. For each ε0 , tables of maximum relative errors were compiled in each of the three layers at fixed p. At the same time, the results of calculations according to various theories were compared in each layer at six points equally spaced in thickness. Tho values of α2 have also been established, for which the maximum error was achieved in the interval 0.02 ≤ α2 ≤ 1.98. The results of the first series of calculations allow us to draw the following conclusions: 1. As ε0 increases, the range of applicability of applied theories narrows. 2. In general, the most accurate calculation in practically important cases is provided by the asymptotic theory (with the retention of the first two terms of the expansion), as well as the theory of a single normal [6]. 3. At the same time, there are ranges of values of the parameters p, ε0 , α2 in which, for more rigid layers, calculations using the broken element theory [22] give less error than calculations using other theories. 4. Shells with a rather soft filler and certain geometric characteristics were discovered, for which the true stresses in the middle layer turn out to be from two to three orders of magnitude less than the stresses, calculated according to technical theories. Let us now dwell on the second series of calculations, in which we studied the influence on the work of applied theories of the value of the n-degree of the Legendre polynomial Pn (cos θ ), which in this case characterizes the load variability parameter. The comparative study was carried out at ε0 = 0.005, with n consistently set to 3, 6, 10; ε0 = 0.005 (n = 3, n = 6, n = 10). As n increases, a narrowing of the zones of applicability for all applied theories is visible. For n = 10, the calculation of the entire structure as a whole with an error of no more than 3% essentially cannot be carried out on the basis of technical theories, since only for p˜ = 1 in a narrow zone near α2 = 1.8, only theory 2.2 can be used. If we compare the areas of satisfactory operation of theories in the cases n = 3 and n = 6, then in the latter case theory 2.4 turns out to be completely unacceptable. As p˜ increases, the regions of required accuracy of theories 2.1 and 2.2 narrow noticeably. As for theory 2.3, in this case, it is subject to fewer changes, compared to other theories. When n = 10, none of the layers, fits within the permissible error of the theory 2.1 and 2.4. In the outer rigid layer, only theories 2.2 and 2.3 work for p˜ = 1, 2, 3. In the internal rigid layer at p˜ = 3 and p˜ = 4 there are small regions α2 , where theory 2.3 gives satisfactory results. Theory 2.1 does not go beyond 3% error only in a narrow region near α2 = 1.8 at p˜ = 4 in the third layer.
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If we compare results at n = 3, 6, 10 (ε0 = 0.005) in the tables of maximum relative errors for stress calculations, we should pay attention to the fact that at n = 10 the maximum errors in the first rigid layer are of the same order as the filler, and in some cases and much more. Analysis of the results of the second series of calculations indicates that with an increase in the load variability index, the range of applicability of simplified theories narrows and for ε0 > 0.005 at n = 10 for the shell as a whole, none of the considered applied theories for maximum stresses provides 3% accuracy.
3 Conclusions The stress state of a three-layer spherical shell was calculated on the base of exact, asymptotic solutions and solutions according to some applied theories. A comparative analysis of the stresses obtained from all five solutions is given. The asymptotic solution constructed in the work, based on three-dimensional equations, indicates methods for constructing refined applied theories for inhomogeneous thicknesses, including layered plates and shells.
References 1. Popov, A.N.: Reconstruction of various navigational scenarios of the “Ever Given” ship, including grounding in the Suez Canal using the bridge simulator with up-to-date electronic navigation charts. J. Phys: Conf. Ser. 2061(1), 012114 (2021) 2. Bashkatov, V.A., Khudyakov, S.A., Ignatenko, A.V.: Practical confirmation of mechanical balancers’ effectiveness to reduce vibration of marine main diesel engines. J. Phys: Conf. Ser. 2061(1), 012054 (2021) 3. Gerasidi, V.V., Lisachenko, A.V.: Analyzing vibration parameters of a modern high-speed engine during operation. J. Phys: Conf. Ser. 2061(1), 012056 (2021) 4. Kozenkova, G.L., Talamanov, V.N., Kozenkov, V.A., Kondratyev, S.I., Khekert, E.V., Modina, M.A.: Study of pneumatic sources of elastic waves for marine seismic exploration. J. Phys: Conf. Ser. 2061(1), 012068 (2021) 5. Senchenko, V., Lopatina, V., Butsanets, A.: Calculating the longitudinal and vertical displacements of a moving object by digital image processing methods. In: E3S Web of Conferences, vol. 258, p. 02005 (2021) 6. Ambartsumyan, S.A.: General Theory of Anisotropic Shells. Nauka, Moscow (1974). (In Russian) 7. Vorovich, I.I.: Mathematical Problems of Nonlinear Theory of Thin Shells. Nauka, Moscow (1989). (In Russian) 8. Goldenveizer, A.L.: Theory of Elastic Shells. Nauka, Moscow (1976). (In Russian) 9. Lehnitsky, S.G.: Anisotropic Plates. Gostekhizdat, Moscow (1957). (In Russian) 10. Mushtari, Kh.M., Galimov, K.Z.: Nonlinear Theory of Elastic Shells. Tatknigizdat, Tatarstan (1957). (In Russian) 11. Novozhilov, V.V.: Theory of Thin Shells. Sudpromgiz, Leningrad (1962). (In Russian) 12. Ogibalov, P.M., Koltunov, M.A.: Shells and Plates. Moscow State University Publishing, Moscow (1969). (In Russian) 13. Pikul, V.V.: Current state of shell theory and prospects for its development. Izv. AN. Solid State Mechanics 2, 145–168 (2000). (In Russian)
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14. Chernykh, K.F.: Linear Theory of Shells. Leningrad State University, Leningrad (1964). (In Russian) 15. Bolotin, V.V., Novichkov, Y.: Mechanics of Multilayer Structures. Mashinostroenie, Moscow (1980). (In Russian) 16. Grigolyuk, E.I., Chulkov, P.P.: Stability and Vibrations of Three-Layer Shells. Mashinostroenie, Moscow (1973). (In Russian) 17. Kobelev, V.N., Kovarsky, L.M., Timofeev, S.I.: Calculation of Three-Layer Structures. Handbook. Mashinostroenie, Moscow (1984). (In Russian) 18. Myachenkov, V.I., Grigoryev, I.V.: Calculation of Composite Shell Structures on Computers: Handbook. Mashinostroenie, Moscow (1981). [in Russian] 19. Ustinov, Yu.A.: Mathematical Theory of Transversely Inhomogeneous Plates. CVVR, Rostovon-Don (2006) (In Russian) 20. Annin, B.D., Volchkov, Yu.M.: Non-classical models of plate and shell theory. Appl. Mech. Tech. Phys. 57(5), 5–14 (2016) (In Russian) 21. Grigolyuk, E.I.: Equations of three-layer shells with a lightweight filler. Izv. AN USSR. OTN. 1, 77–84 (1957). (In Russian) 22. Teregulov, A.G.: On the theory of multilayer anisotropic shells. In: Research on the Theory of Plates and Shells, vol. 6, no. 7, pp. 762–767. Kazan University Publishing, Kazan (1970). (In Russian) 23. Bellman, R.: Introduction to Matrix Theory, 2nd edn. Nauka, Moscow (1976). (In Russian)
Determination of Stress-Strain State in Layered Structures Made of Isotropic and Transversally Isotropic Materials for the Case of Source of the Elastic Waves Inside a Layer Package Igor P. Miroshnichenko1(B) and Ivan A. Parinov2 1 Don State Technical University, Rostov-on-Don, Russia
[email protected] 2 Southern Federal University, Rostov-on-Don, Russia
Abstract. The article describes matrix representations of the displacements, stresses, and deformations in layered isotropic and transversely isotropic structures for the case of elastic wave sources, located inside a layer package. The calculated relationships are based on the generalized method of scalarization of dynamic elastic fields, developed by V.P. Sizov [1] and implemented in original software. The noted results were used to simulate the process of diagnosing the state of plane-layered structures made of isotropic and transversally isotropic materials using the acoustic active method of non-destructive testing. Keywords: Stress-Strain State · Layered Structures · Isotropic Materials · Transversally Isotropic Materials · Generalized Scalarization Method · State Diagnostics
1 Introduction The generalized method of scalarization of dynamic elastic fields in transversely isotropic media [1], proposed by V.P. Sizov, is currently quite well-known and is successfully used in solving various actual practical problems for determining and theoretically calculating the stress-strain state in layered structures made of isotropic and transversally isotropic composite materials under pulsed influences with a given spatial and temporal distribution. The works [2, 3] describe the constitutive equations, solution methods and examples of solving various problems for plane-layered structures made of transversally isotropic composite materials under pulsed influences. The works [2, 4, 5] provide the constitutive equations, solution methods and examples of solving various problems for layered cylindrical structures made of transversally isotropic composite materials under pulsed influences. The works [2, 6, 7] present the constitutive equations, solution methods and features of solving various problems for layered elliptical structures made of transversally isotropic composite materials under pulsed influences. In [8], the application of the method [1] was proposed for solving problems of the propagation of surface elastic waves in a rotating transversally isotropic half-space. In [9], the application of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 412–423, 2024. https://doi.org/10.1007/978-3-031-52239-0_39
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the method [1] was expanded to solve practical problems, considering various options for the relative position of the axis of material symmetry of a transversally isotropic material and the interface boundaries of layers. In the above-mentioned works, a pulse action with a given spatial and temporal distribution is applied to the external and/or internal surfaces of the structures under consideration, while research for the cases of the location of sources of elastic waves inside a layer package is currently interesting and relevant for practical problems. This paper is devoted to the application of the generalized method of scalarization of dynamic elastic fields in transversely isotropic media, developed by V.P. Sizov [1], to determine the stress-strain state in structures made of isotropic and transversely isotropic materials for the case of the location of elastic wave sources inside a layer package.
2 Problem Statement In the general case, sources of elastic waves can be located not only on the outer and/or inner surfaces of a layered structure, but also inside a stack of layers, for example, in the case of acoustic emission.
(a)
(b) Fig. 1. Calculation schemes
Let us consider n-th layer, where the sources are located, with boundaries S 1 and S 2 _ (see Fig. 1). The coordinate line x 1 (which corresponds to the unit vector e perpendicular 1
to the interface between the layers) will be denoted by z. Let us consider a vector ψn (z) (we will omit the index “n” below) that determines the field of the v-th harmonic in the layer n, considering the reaction of multilayer systems, connected, respectively, to the outer boundary zs2 and to the inner boundary zs2 . The vector ψ(z) corresponds to the vector ψn , but is not identical to it. This is a three-block column vector, the elements of
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which represent the complex amplitude of the v-th harmonic ψv+ + ψv− in the observed sections z = z of the layer n [2, 4, 5]. We consider the amplitudes of source potentials in a boundless homogeneous medium and their distribution in space to be known. Let us assume that they are located on a surface z0 parallel to the boundary between the layers. Let us represent them by expansion into an integral or Fourier series in functions F(x2 , x3 ) and denote the column ∼
ψ0 = (ϕ0 w0 v0 ) by the three-block vector. We will look for the vector ψ at observation points, located on the surface inside the layer. Let us use the method of multiple reflection from arbitrary boundaries of a solid layer. To do this, let us imagine the propagation of waves from points z0 to points z as ∼
a sequential transformation of sources ψ by the layer’s environment and its boundaries. 0
In this case, two cases are possible: (i) when z < z0 and (ii) when z > z0 .
3 Matrix Representation of Fields in a Layer with Sources ∼
Let us consider first case (see, Fig. 1a). Waves, excited by sources ψ , initially propagate 0
from surface z0 to surface zs1 and to surface zs2 . Then they fall to zs2 , repeatedly reflecting from surfaces zs1 and zs2 . The contribution to the total amplitude ϕ(z ) of these waves will be denoted by ϕ ∗ (z ) and ϕ ∗∗ (z ), respectively. Then we represent the vector ψ as a sum: ψ(z ) = ϕ ∗ (z ) + ϕ ∗∗ (z )
(1)
Let us determine the value of ψ ∗ (z ). The field at points z consists of an infinite ∞ sequence of partial waves ψi∗ , that is: ψ ∗ (z ) = ψi∗ . i=1
The first partial wave is the direct propagation of the wave from sources z0 to observation points. The contribution of this wave can be written by introducing the operator ∼
F − (z z0 ), acting by which on the amplitude ψ of the sources, we obtain: 0
∼
ψ1∗ (z ) = F − (z z0 ) ψ
(2)
0
Here the superscript “−” indicates that the wave travels from points z0 to points z against the z-axis. The operator F − (z z0 ) characterizes the transformation of the field by the medium enclosed between the surfaces z0 and z , and transfers the field amplitudes ∼
ψ from points z0 to points z as it would happen in an infinite homogeneous medium. 0
The second partial wave is characterized by a single reflection from the surface zs1 (Fig. 1a). We write the transformation that occurs with the amplitude ψ1∗ (z ) in the form (1)
of a sequential action of three operators: F + (z zs1 ) F − (zs1 z ). Here the operator F − (zs1 z ) transfers the amplitude from points z to points zs1 (1)
against the z-axis; the operator characterizes the action of the boundary zs1 ; the
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operator F + (z zs1 ) describes the propagation of waves from points zs1 to points z . Thus, the contribution of the second partial wave can be represented as follows: (1)
∼
(1)
ψ2∗ (z ) = F + (z zs1 ) F − (zs1 z )ψ1∗ (z ) = F + (z zs1 ) F − (zs1 z )F − (z z0 ) ψ 0
∼
(1)
= F + (z zs1 ) F − (zs1 z ) ψ
(3)
0
In the last equality, the sequential action of the operators F − (z z0 ) and F − (zs1 z ) is ∼
replaced by the action of a single operator F − (zs1 z0 ), transforming the amplitudes ψ 0
from points z0 to points zs1 . The third partial wave is composed of a wave ψ2∗ that runs towards the surface zs2 , reflected from it, and again passing through the layer zs2 z , but in the opposite direction, hitting the surface z . By introducing the appropriate operators and arranging them in a certain order, we write: (2)
ψ3∗ (z ) = F − (z zs2 ) F + (zs2 z )ψ2∗ (z ) (2)
∼
(1)
= F − (z zs2 ) F + (zs2 z )F + (z zs1 ) F − (z z0 ) ψ 0
(2)
∼
(1)
= F − (z zs2 ) F + (zs2 zs1 ) F − (zs1 z0 ) ψ
(4)
0
(2)
Here the operator characterizes the reflective properties of the boundary zs2 . The following partial waves are obtained by repeated reflection from the boundaries zs1 and zs2 , three times, etc. In this case, all transformations can be written by sequential action of the entered operators. The totality of these partial waves makes up the following series: ψ ∗ (z ) =
∞
∼
ψi∗ (z ) =F − (z z0 ) ψ +F + (z zs1 ) 0
i=o (2)
+ F − (z zs2 ) F + (zs2 zs1 )
∞
∼
(1)
n Q21 F − (zs1 z0 ) ψ 0
n=0 ∞
(1)
∼
(Q21 )n F − (zs1 z0 ) ψ
(5)
0
n=0
where (1)
(2)
Q21 = F − (zs1 zs2 ) F + (zs2 zs1 ).
(6)
The second set of partial waves ψ ∗∗ (z ), which contributes to the overall complex amplitude ψ(z ), can be written by analogy as a series: (1)
ψ ∗∗ (z ) = F + (z zs1 ) F − (zs1 z0 )
∞ n=0
(2)
∼
n Q21 F + (zs2 z0 ) ψ 0
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+ F − (z zs2 )
∞
∼
(2)
n Q12 F + (zs2 z0 ) ψ
(7)
0
n=0
where (2)
(1)
Q12 = F + (zs2 zs1 ) F − (zs1 zs2 ).
(8)
Let us denote: P12 =
∞
n Q12 ;
P21 =
n=0
∞
n Q21 .
(9)
n=0
Then, combining expressions (5) and (7), we obtain the following relation for the vector of complex amplitudes: ψ(z ) = ψ − (z ) + ψ + (z ) = {F − (z z0 ) (2)
(1)
+ F − (z zs2 )[ F + (zs2 zs1 )P21 F − (zs1 z0 ) ∼
(2)
(1)
+ P12 F + (zs2 z0 )]} ψ +F + (z zs1 ){P12 F − (zs1 z0 ) 0
(1)
∼
(2)
+ F − (zs1 zs2 )P12 F + (zs2 z0 )} ψ , z < z0
(10)
0
Here the first term combines the contribution of all waves, traveling against the z-axis, and the second term combines the contribution of waves, traveling along the z-axis. Carrying out similar reasoning, we obtain a formula for complex amplitudes in the case when the observation points are located along the z-axis further from the origin of the source points: ψ(z ) = ψ + (z ) + ψ − (z ) = {F + (z z0 )+ (1)
(2)
+ F + (z zs1 )[ F − (zs1 zs2 )P12 F + (zs2 z0 ) (1)
∼
(2)
+ P21 F − (zs1 z0 )]} ψ +F − (z zs2 ){P12 F + (zs2 z0 ) 0
(2)
(1)
∼
+ F + (zs2 zs1 )P21 F − (zs1 z0 )} ψ , z > z0
(11)
0
Let us give a general description of the operators introduced above. They operate on column vectors ψ, whose coordinates are the complex amplitudes of various types of waves. The set of all ψ constitutes a complex vector space. It is convenient to represent operators in this space in the form of square matrices. The operators F(z1 z2 ) are associated with the layer of the material environment, enclosed between the surfaces z = z1 and z = z2 . They characterize the transformation of amplitudes by this layer. Since the medium is considered linear and homogeneous, this transformation is reduced to a change in the modulus and argument of each harmonic
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of each wave type and does not lead to the appearance of new harmonics and wave types. Due to, the matrix F(z1 z2 ) must be diagonal. Also note the following properties of operators F(z1 z2 ): (1). The first, obvious property, which has already been used in deriving formulae (7)−(11), is based on that the transformative action of the layer is equivalent to the sequential action of smaller layers into which the original layer z1 z2 can be divided; that is the following relation is valid: F(z1 z2 )F(z2 z3 )F(z3 z4 ) = F(z1 z4 ).
(12)
(2). The second property relates to the operator F(z1 z0 ), transforming sources; it can be written in the following form: F + (zs2 z0 ) = 0,
z0 = zs2 ;
F − (z zs2 ) = 0,
z = zs2 ;
F − (zs1 z0 ) = 0,
z0 = zs1 ;
F + (z zs1 ) = 0,
z = zs1 .
(13)
This property comes from that the operators F act on a set ψ and map this set into themselves, and vectors are defined only within the layer zs2 zs1 . If the sources are located at the boundary, then the waves enter the layers from one side, namely from the boundary into the layer. (3). The next property looks like this: F + (z1 z2 ) = 0,
z1 < z2 ;
F − (z zs2 ) = 0,
F − (z1 z2 ) = 0, F(z1 z2 ) = E,
z1 > z2 ; z1 = z2
F + (z zs1 ) = 0,
(14)
(4). Regarding the reflection operators , we can say that they characterize the reflective properties of the boundary for a given harmonic, when one harmonic can excite harmonics of different types of waves only of its number. It is convenient to represent the operator in the form of a square matrix, the elements of which are reflection coefficients ij . We define each coefficient as the ratio of the amplitude of a reflected wave of j-type to the amplitude of an incident wave of i-type, that is: (1) def
ψj+
(2) def
ψj−
ij
ψi
ij
ψi+
=
=
−;
.
(15)
Waves of v-type are independent from waves of ϕ-type and w-type for the considered boundary geometries and do not transform into each other. In this case, the matrix G for each harmonic has the form: ⎞ ⎛ ϕϕ wϕ 0 ⎟ ⎜ (16) = ⎝ ϕw ww 0 ⎠. 0
0
vv
(5). The matrices Q12 and Q21 are connected by a similarity transformation, which directly follows from expressions (6) and (8): (2)
(2)
Q12 = [ F + (zs2 zs1 )]Q21 [ F + (zs2 zs1 )]−1 ; (1)
−
(1)
−
Q21 = [ F (zs1 zs2 )]Q12 [ F (zs1 zs2 )]
−1
(17)
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Matrices P12 and P21 are also similar: (2)
(2)
(1)
(1)
P12 = [ F + (zs2 zs1 )]P21 [ F + (zs2 zs1 )]−1 ;
(18)
P21 = [ F − (zs1 zs2 )]P12 [ F − (zs1 zs2 )]−1 . To prove the first expression, we multiply the second relation of (9) from the left (2)
hand by F + (zs2 zs1 ) and get: (2)
(2)
(1)
(2)
F + (zs2 zs1 )P21 = F + (zs2 zs1 )[E + F − (zs1 zs2 )] F + (zs2 zs1 ) + ...] (2)
(1)
(2)
(2)
= [E + F + (zs2 zs1 ) F − (zs1 zs2 ) + ...] F + (zs2 zs1 ) + ...] = P12 F + (zs2 zs1 ).
(2)
Multiplying this relation on the right hand by the matrix [ F + (zs2 zs1 )]−1 , we obtain the first expression of (18). Similarly, multiplying the first relation of (9) on the left hand (1)
(1)
by F − (zs1 zs2 ) and on the right hand by [ F − (zs1 zs2 )]−1 , the validity of the second formula of (18) is proved. (6). The operators are sums (9). These sums are Neumann series and converge if ||Q|| < 1. This condition is always satisfied if the medium has absorption. Then, summing series (9), we obtain: ∞ Qn = (E − Q)−1 . (19) P= n=0
where E is the identity matrix; Q is the matrix formed in accordance with (6) and (8). (1)
(2)
Let us find an expression for calculating the matrices and . The elements of matrices are equal to the ratio of reflected waves to incident ones (15). Therefore, if we represent for the vector ψn in formula (2.120) [2, 4, 5], the field, falling on a layered structure with a boundary zs1 , in the form of spatial harmonics with unit amplitude, then the reflected field will be equal to . Let the incident field is longitudinal, that is ϕ − = I , then the reflected field will be equal to the first column of the matrix . The vector of complex amplitudes ψn in this case will have the form: ⎛ ⎞
+ (1) ψ ⎟ ⎜ ϕi (20) ψn = = ⎝ ⎠, − ψ E1i (1)
where is the first column of the matrix , and E1i is the first column of the identity ϕi
matrix E, that is:
⎛ (1)⎞ ⎟ ⎜ ϕϕ ⎛ ⎞ ⎜ ⎟ 1 ⎟ ⎜ (1) ⎜ (1)⎟ ⎜ ⎟ = ⎜ ⎟; E1i = ⎝ 0⎠. ⎜ ϕw ⎟ ϕi ⎜ ⎟ 0 ⎝ (1)⎠ ϕv
(21)
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Let us substitute expression (21) into formula (2.120) [2, 4, 5], where we set n = 1. Then we obtain the following relation for the vector of states at points on the internal surface of the object: ⎛ ⎞ ↓ B1
(1)⎜
= R⎝
(1)
⎟ ⎠, E1i ϕi
where
⎛ (1)
↓
↑−1
R = C1 C1
L1n Cn 23
(22)
(1)
R ⎜ (1) ⎜ =⎜ ⎝ (1) R (3)
(1)
⎞
R⎟ (2)⎟ ⎟. (1)⎠ R
(23)
(4)
↓
Components for matrices C and 23 are given in [2, 4, 5]; C1 is the matrix, calculated ↑−1 for the internal boundary (down arrow) of layer 1;C1 is the matrix inverse to matrix, ↓ ↑−1 ↓ ↑−1 ↓ calculated for the outer boundary (up arrow) of layer 1; L1n = C2 C2 C3 C4 ...Cn ; C n is the matrix, calculated for the area inside the n-layer. If the inner surface is free, then, due to the boundary conditions, the stress on it is zero. Expression (22) will take the form: ⎞ ⎛ (1) (1) ⎞ ⎛ ⎞ ⎛ (1) (1) (1) R(1) R(2) ⎠ ⎝ ϕi ⎠ ↓ · . (24) B1 = R ⎝ ϕi ⎠ = ⎝ (1) (1) E1i E1i R(3) R(4) Then we have: (1)
(1)−1 (1)
ϕi = R (3) R (4) E1i
(25)
Consistently assigning unity to all possible types of waves that can propagate against the z-axis to the surface S 1 (this must be done for each harmonic), we obtain formulae (1)
like (25) for various elements of the matrix . The set of these formulae can be written (1)
as an expression for the matrix as follows: (1)
(1)−1 (1)
= R (3) R (4) . ↓
(26) (1)
In the case of a rigid internal surface, we have U1 = 0 and expression for has the following form: (1)
(1)−1 (1)
r = R (1) R (2)
(27)
Similarly, using expressions (2.122) [2, 4, 5], we obtain formulas for the operator (2)
: (2)
(2)−1 (2)
= R (3) R (4) ,
(28)
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(2)−1 (2)
r = R (3) R (4) , where
(29)
⎛ (2)
R =
⎞ (2) (2) R R ⎝ (1) (2) ⎠ (2) (2) R(3) R(4)
↑
↑ = CN C ↓−1 L−1 nN Cn 23 .
(30) (1)
(2)
Thus, formulae (26)−(29) allow us to find all elements of the matrices and . As an example, consider the case, when the sources are located on the surface zs2 of the outer layer. In this case, z0 = zs2 , z < z0 , therefore, formula (30) should be used. Then, considering property (13) of the operators F, we obtain: (2)
∼
(1)
ψ(z ) = {F − (z zs2 ) + F − (z zs2 ) F + (zs2 zs1 )P21 F − (zs1 zs2 )} ψ + ∼
(1)
+ F + (z zs1 )P21 F − (zs1 zs2 ) ψ (1)
(31)
(1)
Replacing P21 F − (zs1 zs2 ) with P21 F − (zs1 zs2 ) (the validity of such a replacement follows from the second formula of (18), we write expression (31) in the form: ∼
(2)
ψ(z ) = F − (z zs2 ) ψ +[F − (z zs2 ) F + (zs2 zs1 ) + F + (zs2 zs1 )] ∼
(1)
(2)
× F − (zs1 zs2 )P12 ψ = {F − (z zs2 )[E + F + (zs2 zs1 ) (1)
∼
(1)
× F − (zs1 zs2 )P12 ] + F + (z zs1 ) F − (zs1 zs2 )P12 ] ψ (1)
∼
+ F + (z zs1 )P21 F − (zs1 zs2 ) ψ Using equality E + QP = P, let us present the expression for ψ(z ) in a simpler form: (1)
∼
ψ(z ) = [F − (z zs2 ) + F + (z zs2 ) F − (zs2 zs1 )]P12 ψ .
(32)
From here, we write the expression for a six-block column vector ψN for points on the surface z = zs1 and z = zs2 , respectively: ⎛ ⎞
+ (1) − ψ F (zs1 zs2 )P12 ψ ⎠ ψN (zs1 ) = =⎝ ; (33) − ψ− z F (z z )P ψ s1 s2 12 s1 ⎛ ⎞ ∼
+ P12 ψ ψ ⎠. =⎝ (34) ψN (zs2 ) = ∼ (1) ψ− z + − F (zs1 zs2 ) F (zs1 zs2 )P12 ψ s2 Substituting expression (34) into (2.120) [2, 4, 5] for n = N , we can obtain a relation for calculating the state vector at the considered point of the n-th layer through the values
Determination of Stress-Strain State in Layered Structures ∼
421
(2)
of the sources ψ. Note that the expression for calculating the operator in accordance with (35) in this case takes the form: (2)
−1 = −C(4) C(3) ,
(35)
where C(4) and C(3) are the matrix blocks:
C(1) C(2) ↑ CN = . C(3) C(4)
(36)
In the case when the sources are located on the surface zs1 (z = zs1 , z > z), we obtain an expression like (32): ∼
(2)
ψ(z ) = [F + (z zs1 ) + F − (z zs2 ) F + (zs2 zs1 )]P12 ψ .
(37)
From expressions (32) and (37), it follows that if there is no reflection from the (1)
(2)
system of layers adjacent to the layer with sources ( Γ = 0 in the first case and Γ = 0 in the second), then the field, as it should be, is determined only by waves traveling from ∼
the sources ψ and there cannot be an increase in the field in this case. If Γ = 0, then ∼
the field increases with increasing factor Pψ in the absence of active losses P → ∞ in the material, if the frequency of the exciting field, on which the matrix elements depend, takes certain values. These cases correspond to resonances of thickness vibrations. The equation for determining the resonant frequencies follows from expression (19), when we equate the determinant of the matrix (E − Q) to zero, that is: det(E − Q) = 0.
(38)
In this equation, we can use either the matrix Q12 or Q21 . This follows from the equality of invariants of similar matrices through which could be present det(E − Q). For matrices 2 × 2, we have: det(E − Q) = 1 − SpQ + det Q = 0.
(39)
The above matrix relations provide a fundamental solution to the problems of studying displacements, stresses, and deformations in layered structures. After determination of the potential functions using formulae (1.70) − (1.72) [2, 4, 5] for isotropic layers or formulae (1.133), (1.142), (1.143) [2, 4, 5] for transversely isotropic layers, it is possible to determine the components of the fields of displacements, stresses, and deformations, and, further, using the Fourier transform, one can find the spatial and temporal expressions of the desired fields. The results described above were implemented in original software [10–15] to determine the stress-strain state in plane-layered structures, made of isotropic and transversally isotropic materials, and were used to simulate the process of diagnosing their condition using the acoustic active method of non-destructive testing.
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4 Conclusions The paper describes a matrix representation of displacements, stresses and deformations in layered structures made of isotropic and transversely isotropic materials for the case of elastic wave sources, located inside a layer package. The calculated relationships are based on the generalized method of scalarization of dynamic elastic fields, developed by V.P. Sizov [1] and implemented in original software [10–15]. The noted results were used to simulate the process of diagnosing the state of plane-layered structures made of isotropic and transversally isotropic materials using the acoustic active method of non-destructive testing. Acknowledgement. I. A. P. thanks the Ministry of Science and Higher Education of the Russian Federation for scientific project No. FENW-2023–0012.
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10. Miroshnichenko, I.P., Sizov, V.P.: Modeling of displacements and stresses in construction layer under pulsed action on its surface. Russian Certificate of State Registration of Computer Program 2018664987. Accessed 27 Nov 2018 (In Russian) 11. Miroshnichenko, I. =P.: Modeling of displacements and stresses in construction conjugate layers under pulsed action on surface. Russian Certificate of State Registration of Computer Program 2019663431. Accessed 16 Oct 2019. (In Russian) 12. Miroshnichenko, I.P.: Modeling of displacements and stresses in package of conjugate layers of construction under pulsed action on surface. Russian Certificate of State Registration of Computer Program 2019663432. Accessed 16 Oct 2019. (In Russian) 13. Miroshnichenko, I.P.: Modeling the process of diagnosing the state of structural material using the acoustic active method of non-destructive testing. Russian Certificate of State Registration of Computer Program 2023668798. Accessed 04 Sept 2023. (In Russian) 14. Miroshnichenko, I.P.: Modeling the process of diagnosing the state of materials of two-layer construction using the acoustic active method of non-destructive testing. Russian Certificate of State Registration of Computer Program 2023669088. Accessed 07Sept 2023. (In Russian) 15. Miroshnichenko, I.P.: Modeling the process of diagnosing the state of package of construction materials by using the acoustic active method of non-destructive testing. Russian Certificate of State Registration of Computer Program 2023669356. Accessed 13 Sept 2023. (In Russian)
Multi-parameter Assessment of Wear Resistance of Antifriction Ion-Plasma Coatings Deposited on a Cemented Steel Substrate A. I. Voropaev1
, O. V. Kudryakov2(B) , V. N. Varavka2 , V. I. Kolesnikov1 I. V. Kolesnikov1 , and E. S. Novikov1
,
1 Rostov State Transport University, 2, Rostovskogo Strelkovogo Polka Narodnogo
Opolcheniya Square, Rostov-on-Don 344038, Russia 2 Don State Technical University, 1, Gagarin Square, Rostov-on-Don 344000, Russia
[email protected]
Abstract. The work solves the problem of multi-parameter optimization of vacuum ion-plasma coatings based on the use of a database, which was formed by the authors over several years of experimental and applied work in the field of increasing the antifriction properties of engineering products. Three types of monolayer vacuum ion-plasma coatings (thin films) with a thickness of 0.5–3.5 μm were considered as the object of study: coatings of two nitride systems TiAlN and CrAlSiN, as well as a diamond-like carbon coating (DLC) with a gradient distribution of carbon electronic configurations sp2 and sp3 in depth. The coatings were applied to a substrate made of structural steel 12Cr2Ni4, subjected to carburization with a followed hardening and low tempering. Wear-resistant cemented surface layers of steel products are widely used in mechanical engineering as a contact surface in loaded friction units. Therefore, when conducting a comparative analysis, the mechanical and tribological properties of the cemented surface of the samples (without coatings) were used as a standard and were considered in the optimization process along with coatings. The work used a multi-parameter optimization technique in the form of an integral assessment of each material by constructing radial (ray) diagrams, which were based on a complex of eight characteristics, combining 4 physical-mechanical and 3 tribological properties, as well as the coating thickness parameter. The results of the analysis showed a consistent pattern: in terms of a complex of eight parameters, all coatings are significantly superior to the reference cemented layer. The CrAlSiN coating has the highest integral property indicator. Keywords: Wear-resistant Thin Films · Tribological Properties of Coatings · Multi-parameter Optimization
1 Introduction In the process of previous research, the team of article authors compiled a database of vacuum ion-plasma coatings, including various characteristics of the elemental and phase composition of coatings, their structure, quality and surface topography, physical, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 424–433, 2024. https://doi.org/10.1007/978-3-031-52239-0_40
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mechanical, tribological, adhesive and other properties [1–8]. One of the tasks of analyzing the created database is currently the selection of coatings that are optimal for a particular area of application. Most existing methods for selecting materials for specific products or operating conditions are appropriate for use when the choice is clearly determined by one or two basic properties of the material, which determine the reliability and durability of the product [9]. If there are several such leading properties or they are completely unknown, then deterministic methods most likely will not give the expected unambiguous result. Such an uncertain situation often arises in scientific work, when a new material is being created and the scope of its application is not yet completely clear, or when it is necessary to characterize a known material for a new field of use, without resorting to expensive bench or full-scale tests. In the latter case, a qualitative or quantitative correlation is established experimentally between test data and laboratory measurement data of several (that is, a complex) of known properties of the material, to further characterize and compare materials according to this set of properties measured in the laboratory. To solve scientific and applied problems for the integral assessment of the properties of materials, that is, to compare materials according to a set of properties, a method for constructing radial (ray) diagrams has been developed, examples of practical implementation of which can be found in [10, 11]. The initial stage of constructing ray diagrams comes down to determining the group (complex) of material properties, on the base of which the diagram is subsequently constructed. The selection of material properties can be carried out from any sources or databases, but the properties must satisfy several requirements: (i) they must be relevant for solving the task, that is, be directly related to the area of operation of the product for which the material is selected; (ii) all properties of the material must correspond to the same structural state (or processing state) of the material; (iii) the complex must include at least 6–8 properties (the maximum number of properties is not limited); (iv) for a more objective final assessment, it is recommended, if possible, to select heterogeneous properties for the complex, including mechanical, physical, technological, economic (cost) and other properties. The database of ion-plasma coatings generated by the authors makes it possible to satisfy all of the listed requirements for using the technique of constructing ray diagrams for the purpose of optimal selection of coatings for tribological purposes. In this regard, the main objective of this work is to construct and analyze radial 8-ray ( j = 1…8) diagrams for four optimization objects (i = 1…4), namely three different ion-plasma coatings and the surface layer of the substrate, used as a comparative standard.
2 Research Method 2.1 Materials and Research Methods The studied nitride coatings of the TiAlN and CrAlSiN systems were obtained by cathodic arc evaporation using a vacuum platform PLATIT π80 . Carbon DLC-coating was obtained by laser evaporation of graphite using a BRV600 platform. Tile samples
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with dimensions of 50 × 30 × 5 mm, made of case-hardened 12Cr2Ni4 steel (casehardening depth 0.9…1.5 mm; hardness of the case-hardened surface HRC ≥ 59) were used as a substrate. Previous studies have shown a weak dependence of the mechanical and tribological properties of TiAlN system coatings on their structure. For example, of the four mechanical characteristics studied (see below, H, E, H/E and H 3 /E 2 ), only the H 3 /E 2 ratio demonstrated a structural dependence [7]. Taking these data into account, it is advisable not to include such structural characteristics of coatings as, for example, layering or layer thickness in the optimization parameters. Therefore, to maintain uniformity during optimization, only coatings with a monolayer structure were considered. To study the physical and mechanical properties of coatings on the nano- and microscale, the Nanotest 600 measuring platform was used. Using the continuous indentation method [12–14], the elastic modulus E and the hardness of coatings H were determined. For measurements in the microrange (load less than 2N, indentation depth more than 0.2 μm), a tetrahedral Vickers indenter was used; for work in the nanorange (indentation depth no more than 0.2 μm), triangular Berkovich indenter was used. Test conditions and processing of the obtained data were carried out in accordance with ISO 14577-1:2002. In addition to the structure-dependent hardness H and the structureindependent elastic modulus E, determined by the indentation method, the H/E and H 3 /E 2 ratios are often used to characterize coatings. The first of them determines the resistance of coatings to elastic deformation, and the second determines the resistance of coatings to plastic deformation [15, p. 608; 16]. In particular, the experimental data for determining the specified mechanical characteristics of the substrate material (a cemented layer of 12Cr2Ni4 steel) were: H = 6.42 GPa; E = 200 GPa; H/E = 0.0321; H 3 /E 2 = 0.0066 GPa. Tribological tests of coatings were carried out on a TRB friction machine (Anton Paar Tritec) in accordance with DIN 50324 and ASTM G99 methods. A “pin-plate” test setup was used with the plate (coated sample) reciprocating at a frequency of 10 Hz and an amplitude of 800 μm. The normal force on the pin was 10 N. The counter sample was a 6.35 mm diameter ball made of WC–Co cermet fixed in a pin. Due to the fact that the ball is stationary in the pin, the tests are classified as sliding friction. The test duration was 50,000 cycles. The main tribological parameters determined are the friction coefficient μ, the intensity of volumetric wear of the sample J (mm3 /N/m) and the counterbody J K (mm3 /N/m). Eight characteristics were selected as optimization parameters when constructing radial 8-ray diagrams: four physical-mechanical (H, E, H/E, H 3 /E 2 ) and three tribological properties (friction coefficient μ, volumetric wear rate of sample J and counterbody J K ) described above, as well as the coating thickness h. Values of h were measured on cross-sections of coated samples using a ZEISS Crosbeam 340 scanning electron microscope, in parallel with the identification of local chemical composition determined using an X-Max EDAX energy dispersive detector (Oxford Instruments) integrated into the microscope. The value of h for coatings is more than an order of magnitude less than the depth of the cemented layer of the substrate, since a vacuum ion-plasma coating cannot have a large thickness due to the application technology, and also because it should not violate the geometry of tribocontact surfaces.
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2.2 Multiparametric Assessment Methodology After the optimization objects (i = 1…4) and their optimization parameters, that is, properties ( j = 1…8) are determined, a radial diagram is constructed according to known rules. The number of rays of the diagram corresponds to the number of properties, and all angles between the rays must be equal. Each axis of the diagram shows one of the properties of each material in the units of measurement of this property. For example, if four materials are selected, then there should be four points on each property axis. When four points are plotted on each axis, all points corresponding to the same material are sequentially connected to each other. Thus, the resulting diagram should have four polygons corresponding to each of the selected materials. A ray diagram constructed in this way is a “diagram in absolute units of properties”. The essence of the integral assessment of the properties of materials using ray diagrams is to calculate and compare the areas of the figures obtained on the diagram. However, it is impossible to make such an assessment from a “diagram in absolute units of properties”, since the scale of each axis (that is, each side of the polygon) is different. Therefore, it is necessary to bring the resulting diagram to a single (relative) scale. To do this, you need to perform two operations: (i) go to the scale of relative units of measurement on each axis of the diagram; (ii) rank all properties (axes), depending on their significance for the operating conditions of a given product. Converting a diagram from absolute to relative units is done differently for “progressive” and “regressive” properties. For the first of them: the higher the property value, the better the final result; for the latter, there is the other way around: the overall result is better if the property has a minimal value. For the “progressive” property, the absolute values of all points on the corresponding axis (X i ) are divided by their maximum value (X max ): X i / X max . Then the relative maximum on this axis of the diagram will have a value of 1.0, and the remaining points will have values less than one. For the “regressive” property, the conversion of absolute values into relative ones is carried out as: X min / X i , where X min is the minimum absolute value of all points X i on the corresponding axis. Then the maximum relative value on this axis will also be 1.0, but it will correspond to the absolute value of X min . After performing the first of the above operations, applied to all points of the diagram (on each axis separately), we obtain a single dimensionless (relative) scale of measurement along all axes of the diagram. Such a diagram is called a “diagram in relative units of properties”. In order to complete its construction, calculations are carried out on it, and the optimal material is selected; it is necessary to perform a second operation, namely ranking of properties. It is carried out by introducing weighting coefficients (significance coefficients) for each property qj , where j is the set of axes (properties) in the diagram. All relative values located on the axis of a given property must be multiplied by the corresponding weighting coefficient qj of this property. The process of assigning significance coefficients always has some degree of subjectivity. Often their values are chosen a priori, based on expediency and practical experience, by justifying their values in terms of the functionality of the material. Mathematical methods do not solve the problem of the subjective factor, the influence of which in the
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procedure for applied selection of materials is apparently not possible to exclude. However, in the process of optimizing the experimental results of scientific research, when the influence of a particular property on the final result of a comprehensive assessment is uncertain and in the absence of quantitative dependencies, the weighting coefficients can be taken equal to or close to 1.0. In accordance with the methodology for selecting materials using radial (ray) diagrams, the area of the polygon on the constructed “diagram in relative units of properties” can be used as an integral statistical estimate for comparing the complex of properties of each material: the larger the area of the polygon of the material (coating), the higher the complex of its properties. The area of a polygon S is calculated as the sum of the areas of its n triangles, using the following expression: n−1 2π 1 , (1) S = · a1 · an + (ai · ai+1 ) · sin 2 n i=1
where n is the number of rays (properties) of the diagram; ai is the relative value of the property in the diagram; ai and ai+1 are the sides of the triangle lying on adjacent rays of the diagram.
3 Results and Discussion Based on the analysis of the available experimental data base of vacuum ion-plasma coatings of TiAlN, CrAlSiN and DLC systems, deposited on a substrate of cemented steel 12Cr2Ni4, the average initial property values for constructing radial diagrams were determined. They are presented in Table 1. It should be noted that the thickness of the cemented layer in Table 1 is taken equal to 30 μm. In reality, with gas cementation it can reach several millimeters. The use of real values of the thickness of chemical-thermal treatment (CTT) layers practically brings this optimization parameter to a zero value (in the diagrams in relative units) and makes its use meaningless. Therefore, as an optimization parameter h for CTT layers, it is advisable to use values that are comparable with the h values of the coatings under study and, on the other hand, the artificiality of which does not affect the final optimization result. Radial (ray) diagram in absolute values of properties, constructed according to the data in Table 1, is shown in Fig. 1. For the reasons discussed above, the “absolute” diagram in Fig. 1 is not analytical or informative but is given for illustrative purposes. Therefore, without stopping to analyze it, let us move on to the diagram in relative units of measurement of the properties of coatings, deposited on a substrate of 12Cr2Ni4 steel, which was constructed in accordance with the methodology of Sect. 2.2 according to the data in Table 2 and shown in Fig. 2. The results indicate that in terms of a set of 8 properties, all coatings are superior to the reference cemented steel layer. In this case, the undisputed leader is the nitride coating of the CrAlSiN system. If we compare this result with individual properties, then only the mechanical characteristics H, H/E and H 3 /E 2 , as well as the parameter h, give a
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Table 1. Generalized experimental data on the properties of the studied coatings on cemented steel 12Cr2Ni4. Coating type Cemented layer
H, GPa
6.42
E, GPa
H/E
H 3 /E 2 , GPa
μ
J × 10−7 , mm3 /N/m
JK × 10−7 , mm3 /N/m
h, μm
200
0.0321
0.0066
0.9
12.3
4.46
30
TiAlN
25.5
316.1
0.081
0.166
0.756
12.91
4.79
3.6
CrAlSiN
24.1
251.3
0.0959
0.2885
0.82
0.79
1.66
1
DLC
12.4
170.6
0.0727
0.0655
0.5
10.46
2.75
0.4
Fig. 1. Radial diagram in absolute units of measurement of the properties of coatings, applied to case-hardened steel 12Cr2Ni4; contour execution of polygons.
similar prioritization. However, the main tribological parameter, namely wear of samples J, does not correspond to the result of complex optimization, since the wear resistance of
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Table 2. Relative values of the properties of the studied coatings on 12Cr2Ni4 steel, used for the optimization procedure. Coating type
H
E
H/E
H 3 /E 2 μ
J
JK
h
Area S on the diagram
Cemented layer 0.252 0.633 0.335 0.023
0.556 0.064 0.372 0.013 0.162
TiAlN
1
0.661 0.061 0.347 0.111 1.033
CrAlSiN
0.945 0.795 1
DLC
0.486 0.54
1
0.845 0.575 1
0.758 0.227
0.61
1
1
0.076 0.604 1
1
0.4
1.96 0.807
Fig. 2. Radial diagram in relative units for optimization of coatings, applied to case-hardened steel 12Cr2Ni4; contour execution of polygons.
the cemented layer is almost the same level as that of TiAlN and DLC coatings. In terms of another important tribological parameter, namely the friction coefficient μ, the DLC coating is unrivaled. Even if the priority of the specified properties J and μ is increased by introducing weighting coefficients, the final result of the optimization performed (the order of the coatings according to the optimization criterion S) will not change.
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Distribution of values of the complex optimization parameter S (shown in the histogram in Fig. 3) illustrates the feasibility of using coatings to improve the complex properties of the contact friction surface of case-hardened steel. It should be noted that the introduction of weighting coefficients of properties, for example, reducing the importance of strength properties (all except H 3 /E 2 ), as well as J K and h by 20–30%, reduces the parameter S in general, but does not change the relative location of coatings in Fig. 3. l(zO d )rm T y p cfe o ita n sg
1,96
2 1,8 1,6 1,4
Optimiz ation c riterion (ra dia l dia gra m a re a )
1,2
1,033
1
0,807
0,8 0,6 0,4 0,162 0,2 0 Cemented layer
TiAlN
CrAlSiN
DLC
T ype s of c oatings
Fig. 3. Comparative data for the optimization parameter of coatings: the areas of figures in the radial diagram in Fig. 2.
In general, the most optimal solution is to use an ion-vacuum nitride coating of the CrAlSiN system. Its use increases both the strength characteristics of the steel surface (H, E, N/E, N 3 /E 2 ) and tribological characteristics (μ, J, J K ), and most importantly, it ensures a stable significant increase in the wear resistance of the tribological interface. At the same time, the thickness of the coating remains in the range of “thin films”, which does not violate the geometry of the tribological connections used and does not require their additional design modification or new strength calculations.
4 Conclusion In accordance with the results of performed multi-parameter optimization, to improve the performance of contact friction surfaces made of case-hardened steel 12Cr2Ni4, it is recommended to use a vacuum ion-plasma coating of the CrAlSiN system in the form of a thin film thickness 1.0…1.5 μm. To increase tribological properties (in particular,
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to reduce the coefficient of friction), it is recommended to use a combined coating in the form of the specified CrAlSiN coating with additional application of a DLC coating film as an outer layer. Determining the optimal set of material properties using the radial (ray) diagram method is aimed at predicting the reliability of experimental results obtained in the laboratory in relation to the results of operational, full-scale or bench tests. Therefore, the use of this method requires feedback in the form of a comparative assessment of the adequacy of the forecast. Only if there is a satisfactory inverse correlation, it is appropriate to recommend the use of laboratory measurements instead of complex or expensive empirical tests. Therefore, based on the results of the optimization performed, technological recommendations were developed for bench testing of loaded spline couplings, including at critical loading parameters. Acknowledgements. This work was supported by the Russian Science Foundation (grant No. 21-79-30007).
References 1. Kudryakov, O.V., Varavka, V.N., Kolesnikov, I.V., Novikov, E.S., Zabiyaka, I.: DLC coatings for tribotechnical purposes: features of the structure and wear resistance. IOP Conf. Ser. Mater. Sci. Eng. 1029(1), 012061 (2021) 2. Kudryakov, O.V., Varavka, V.N., Zabiyaka, I.Y., Sidashov, A.V., Novikov, E.S.: Synthesis, electronic structure, microstructure, and properties of vacuum ion-plasma coatings based on carbon. In: Parinov, I.A., Chang, S.-H., Kim, Y.-H., Noda, N.-A. (eds.) PHENMA 2021. SPM, vol. 10, pp. 197–206. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-76481-4_17 3. Kudryakov, O.V., Varavka, V.N., Manturov, D.S.: Effective wear resistance parameters of ion-plasma tribological coatings. Defect Diffusion Forum 410, 444–449 (2021) 4. Kolesnikov, V.I., Kudryakov, O.V., Zabiyaka, I., Novikov, E.S., Voropaev, A.I.: Influence of the electronic structure of carbon (diamond-like) thin films on tribological characteristics. J. Phys: Conf. Ser. 1954, 012018 (2021) 5. Kolesnikov, V.I., Vereskun, V.D., Kudryakov, O.V., Manturov, D.S., Popov, O.N., Novikov, E.S.: Technologies for improving the wear resistance of heavy-loaded tribosystems and their monitoring. J. Frict. Wear 41(2), 169–173 (2020) 6. Kolesnikov, V.I., Novikov, E.S., Kudryakov, O.V., Varavka, V.N.: The degradation mechanisms in ion-plasma nanostructured coatings under the conditions of contact cyclic loads. J. Phys: Conf. Ser. 1281, 012036 (2019) 7. Kolesnikov, V.I., Kudryakov, O.V., Zabiyaka, I., Novikov, E.S., Manturov, D.S.: Structural aspects of wear resistance of coatings deposited by physical vapor deposition. Phys. Mesomech. 23, 570–583 (2020) 8. Kolesnikov, V.I., Kudryakov, O.V., Varavka, V.N., Manturov, D.S., Novikov, E.S.: Relationship between the adhesive properties of vacuum ion-plasma TiAlN coatings and wear resistance in friction. J. Frict. Wear 42(5), 317–326 (2021) 9. Sedunov, A.N., Mal’tsev, I.M.: Software for choosing grades of machine-building steel. Metal Sci. Heat Treat. 52(7), 1–2 (2010) 10. Kudryakov, O.V., Varavka, V.N.: Integrated indentation tests of metal-ceramic nanocomposite coatings. Inorg. Mater. 51(15), 1508–1515 (2015)
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11. Varavka, V.N., Kudryakov, O.V., Ryzhenkov, A.V.: Multilayered nano-composite coatings for anti-erosive protection. In: Parinov, I.A. (ed.) Piezoelectrics and Nanomaterials: Fundamentals, Developments and Applications, pp. 105–132. Nova Science Publishers, New York (2015) 12. Oliver, W.C., Pharr, G.M.: An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7(6), 1564–1583 (1992) 13. Fischer-Cripps, A.C.: Nanoindentation. Springer, New York (2004). https://doi.org/10.1007/ 978-0-387-22462-6 14. ISO/FDIS 14577–1:2002; Metallic materials—Instrumented indentation test for hardness and materials parameters—Part 1: Test method 15. Cavaleiro, A., de Hosson, J.T.: Nanostructured Coatings. Springer, New York (2007) 16. Roy, M.: Surface Engineering for Enhanced Performance against Wear. Springer-Verlag, Wien (2013). https://doi.org/10.1007/978-3-7091-0101-8
Mathematical Model of the Strength of Modified Cement Stone Irina Serebryanaya , Alexandra Nalimova , and Andrey Matrosov(B) Don State Technical University, 1 Gagarin Square, 344010 Rostov-on-Don, Russia [email protected]
Abstract. The present work investigates the strength properties of modified cement stone and polymer-cement compositions made from it. The control of the strength properties is achieved by recipe regulation of the mineral component of the multi-component binder. This makes it possible to significantly increase not only the strength of the materials, based on them, but also their reliability and durability. Dependencies, describing the influence of complex polymer additive consisting of superplasticiser, stabiliser and redispersible polymer powder on the strength of cement stone were obtained; coefficients reflecting the influence of polymers on the strength of cement stone were determined; functions of coefficient values depending on the dosage of additives were obtained. Based on the data obtained, a mathematical model of the strength properties of cement stone under the influence of complex additives was created. The developed multiplicative model for evaluating the influence of modifiers on the properties of polymer-cement composites allows to determine and predict the strength properties of polymer-cement composites to design composites with predetermined constructional properties. Keywords: Polymer-Cement Composition · Strength Properties · Modifier Additives · Multiplicative Model
1 Introduction Modern requirements for the design and construction of buildings and structures, reduction of labor and material intensity have led to the need for the development of modern high quality construction materials, including mortars and concretes, based on cement binders. At the same time, special attention is paid to the strength properties of the material, as these properties mainly influence the technological and operational properties of concrete and reinforced concrete structures. Modified concrete mixes can form the base of almost any structure, including loadbearing structures. Of particular interest are self-compacting mixes, modified with various additives. For cement systems, modifiers are substances that improve the technological properties of mortar and concrete mixtures and, consequently, the structural and technical properties of mortar and concrete [1–6]. Such modifiers are polymer-cement compositions. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 434–442, 2024. https://doi.org/10.1007/978-3-031-52239-0_41
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Due to the relevance of the study of the influence of various modifiers on the properties of mortar and concrete mixtures, a significant number of publications are devoted to this topic [7–12]. The results of these and other studies show that: (i) with the increase of polymer content in the mortar composition, there is a slight decrease in compressive strength, whilst the absolute value of strength reduction is limited; (ii) flexural strength increases with increasing polymer content; (iii) tensile strength and strain increase with increasing polymer content in self-levelling mortars; (iv) the higher the percentage of polymer content in the self-levelling compounds, the higher the adhesion and abrasion resistance characteristics. As it is known, depending on the country of production, the properties of Portland cement can differ significantly in chemical, mineralogical composition, and other physical and mechanical properties from those of domestic (locally produced) Portland cement. Therefore, it is necessary to conduct research on the influence of exported polymer modifiers on the properties of domestic binders, used in the production of mortar and concrete mixes. The aim of this work is to create a mathematical model of strength properties of cement stone under the influence of complex additive. Such a model will make it possible to determine and predict the strength characteristics of polymer-cement compositions to design future compositions with predetermined construction and technical characteristics on their base.
2 Research Method The use of super plasticizing additives alone in polymer cement compositions for mortar and concrete mixtures are not sufficient. Cast mortar mixtures are subject to delamination. Moreover, the mortar mixture is applied to porous surfaces, most often concrete screeds, with a thickness of up to 10 mm. This makes it necessary to solve the problem of water retention of the mixture. This issue can be resolved through the application of stabilizing additives, derived from cellulose ethers. High adhesion of the mortar to the substrate is achieved by using redispersal polymer powders. It is the combination of three polyfunctional additives that provide the main technological and construction-technical properties of mortar mixtures and mortars. There is a standard method of determining the effectiveness of additives for concrete and mortars by comparing the quality indicators of mixtures and concrete of control and basic compositions [13]. This methodology assumes the evaluation of the effectiveness of plasticizing additives by increasing the mobility (workability), without considering their effect on strength and deformation properties. Taking this into account, the effectiveness of various superplasticizers (SPs) should be evaluated using methods that allow for considering the influence of SPs on the basic properties of cement stone. This necessitated the development of an improved methodology for evaluating the effectiveness of superplasticizers. The methodology that allows for quantitative assessment of superplasticizer effectiveness is based on the method of reducing the primary standardized indicator to an equal
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value [14]. Based on this methodology, the efficiency of superplasticizers is evaluated by comparing the calculated values of efficiency criteria to the baseline [15]. The formulae (1) for strength and shrinkage are used to convert the strength and shrinkage deformations to equal values of W/C (Water/Cement), respectively: Ac = R/0.48(W /C)−1.3885 εc = ε(1.98 W /C + 0.18)
(1)
where AC is the reduced strength of cement stone; R is the actual strength of cement stone at the age of 28 days; εc is the shrinkage of cement stone to W/C value of cement stone with superplasticizer; ε is the actual shrinkage of cement stone at the age of 120 days. In contrast to all previously proposed methods of efficiency assessment, which take into account either the water-reducing effect or the complex effect of the additive on the strength of cement stone, this method allows us to divide the effect of superplasticizer application into two components: (i) the water-reducing effect; (2) the effect of the additive on the reactivity of cements, that is on the process of hydration and formation of cement stone strength. In this study, the strength characteristics of cement stone, namely compressive strength (Rcs ) and flexural strength (Rfs ), were measured for 40 × 40 × 160 mm3 beam specimens according to the standard method [16]. Mathematical planning and processing techniques were utilized, employing appropriate software, in the analysis of experimental data. A confidence probability of 0.95 was achieved using a sufficient set of control twin samples, with an error margin of no more than 10%. Portland cement of CEM I 42.5 class was used in this work. The list of additives of different actions used in the experiment and their technical characteristics are given in Table 1. Table 1. Specifications of additives Additive brand
Purpose of use
Chemical basis
Form of release
Color
Recommended dosage (% by weight of binder)
Manufacturer
Melment F 10
Super plasticizer
Melamine formaldehyde
Powder
White
0.2–1.5
Hoechst Chemie, Germany
Vinnapas RE 523 Z
Regulator of water-holding properties of the solution
Vinyl acetate copolymer with ethylene
Powder
Yellowish
0.8–2.5
Wacker Chemie AG, Germany
Bermokoll E230
Stabilizer
Ethylhydroxyethylcellulose
Powder
White
0.4–0.8
Sweden
In order to study the effect of a complex polymer additive comprising of a superplasticizer (Melment F10), stabilizer (Bermokoll E230) and RDP (Vinnapas) on the strength
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characteristics of cement stone, a three-factor experiment according to the Box-Benkin plan (B3) was implemented. The levels of variation of the main factors are presented in Table 2. Table 2. Main factor variation levels Main factors of variation
Code
Levels of variation
Variation interval
lower
basic
upper
Bermakoll content (MC), %
x1
0
0.2
0.4
0.2
Vinnapas content (VA), %
x2
0
2.5
5.0
2.5
Content of Melment (SP), %
x3
0
0.6
1.2
0.6
The additives utilized and their respective quantities were selected, based on the guidelines, provided by the associated manufacturers.
3 Results and Discussion The research results of the additive effect on the strength characteristics of modified cement stone are shown in Table 3. The following regression equations were obtained during statistical processing of the results: 2 2 R28 CS = 61.57 + 1.32b1 + 1.02b3 + 2.156b1 − 1.744b2 + 2.251b1 b2 + 1.711b2 b3 2 2 2 R28 fs = 5.037 − 0.143b1 − 0.258b2 + 0.804b1 − 0.571b2 + 1.079b3 + 0.153b1 b3 + 0.265b2 b3
Figure 1 shows the relative compressive and flexural strength values of cement stone at 28 days of age in combination with different polymer additives, compared to cement stone without additives. The formulations 1–15, shown in the figure, are summarized in Table 3. Table 3. Compositions of mixtures and dynamics of cement stone deformations No. of composition
Dosage, %
Strength of cement stone
MC
VA
SP
W/C
Rcs 28 , MPa
Rfs 28 , MPa
1
0
0
0
0.28
62.4
7.5
2
0.4
5
1.2
0.22
60.2
6.12
3
0
5
1.2
0.22
68
6.75
4
0
0
1.2
0.22
62.8
6.45 (continued)
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I. Serebryanaya et al. Table 3. (continued)
No. of composition
Dosage, %
Strength of cement stone
MC
VA
SP
W/C
Rcs 28 , MPa
Rfs 28 , MPa
5
0.4
0
0
0.28
59.6
5.95
6
0.4
5
0
0.28
64
5.5
7
0
5
0
0.28
55
5.75
8
0.4
0
1.2
0.22
60
6.5
9
0.4
2.5
0.6
0.24
70.4
6.45
10
0
2.5
0.6
0.24
58
5.5
11
0.2
5
0.6
0.24
63.6
4.45
12
0.2
0
0.6
0.24
57
4.75
13
0.2
2.5
1.2
0.22
67.4
5.5
14
0.2
2.5
0
0.28
56
7.0
15
0.2
2.5
0.6
0.24
59.6
4.5
Fig. 1. Relative strength of cement stone aged 28 days with complex polymer additives, compared to cement stone without additives.
The following conclusions can be drawn from the obtained results of the study of the influence of modifier additives on the strength characteristics of cement stone:
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(i) a significant increase (by 30%) in the strength of cement stone with super plasticizers (compositions Nos. 3, 4, 9, 10) due to the reduction of water-cement ratio from 0.28 to 0.22 (by 20%) was revealed: (ii) cement stone without super plasticizers (compositions Nos. 5 and 14) but containing MC and VA at equal value of W/C with the reference composition (composition No. 1) have lower strength. Thus, polymeric additives–modifiers in the adopted combinations led to a decrease in the strength of cement stone. The strength reduction of modified mortars and concretes can be explained by the fact that the additives, being in the volume of the material, are distributed on the surface of the cement and aggregate grains and thus worsen the adhesion between them [17]. However, it is possible that the involvement of air could also result in a reduction in strength, given the surface-active nature of SP and MC. Other causes of strength reduction may also occur, which require further detailed study. The authors have proposed a model describing the influence of polymer additives on the strength properties of cement stone. This correlation is established by incorporating coefficients that consider the impact of polymer supplements, depending on the quantity employed: R = k1 k2 k3 R0 where k1 is the coefficient that considers the influence of SP; k2 is the coefficient, considering the influence of MC; k3 is the coefficient, considering the influence of VA; R0 is the compressive strength of cement stone without additives at the age of 28 days. The coefficients k1 , k2 , and k3 are correction coefficients that consider the change in strength of cement compositions, compared to the strength of cement due to the influence of polymer additives. Because of the research conducted, we have obtained the dependencies that describe the impact of functional polymeric additives on the strength properties of cement stone. Based on the obtained dependencies, functions of the values of coefficients k1 , k2 , and k3 , depending on the dosages of additives, were found. The obtained dependencies are presented in Table 4. Table 4. Experimental dependencies of compressive strength Influence factor W/C
Function R
Coefficient
Reference sample
1
SP content is 0–1.2%
R = 8.4528 x −1.3885 R = 7.656 x −1.3885
MC content is 0–0.4%
R = 6.0768 x −1.3885
kMC = −0.75(MC) + 1
VA content is 0–5%
R = 9.3744 x −1.3885
kVA = 0.02(VA) + 1
kSP = −0.0833(SP) + 1
Figure 2 shows the dependence of the reduced strength on the B/C of the cement stone. The following designations are used:
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Ref is the cement stone without additives; MC is the cement stone with MC = 0.4%; VA is the cement stone with additive VA = 5%; SP is the cement stone with SP = 1.2%. As can be seen from the presented dependencies, the use of super plasticizer and MC leads to a decrease in the strength of cement stone (curves SP and MC), which is probably due to the negative effect of these polymers on the processes of cement hydration. At the same time, it should be noted the positive influence of redispersal polymer powder (curve BA).
Fig. 2. Dependence of reduced strength of cement stone with different modifiers on W/C.
The combined effect of polymer additives on the strength properties of cement stone using multiplicative models is confirmed by a few experiments. Comparison of theoretical values of coefficients (1) with the obtained experimental data is presented in Table 5.
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Table 5. Theoretical and experimental values of coefficients Influence factor W/C
Value coefficients k
SP content is 1.2%
0..9
MC content is 0.4%
0.7
VA content is 5%
1.1
Value of the complex indicator k1 , k2 , k3 according to the formula (1)
experimental data
0.693
0.690
Deviation, %
0.435
4 Conclusion Because of the research conducted, the potential to manage the strength properties of polymer-cement mixtures has been identified. Such control is achieved by means of recipe regulation of the mineral component of a multicomponent binder. In addition, dependencies, describing the effects of a complex polymer additive, consisting of super plasticizer, stabilizer, and redispersal polymer powder on the strength characteristics of cement stone have also been obtained. The coefficients, reflecting the influence of polymers on the strength of cement stone, have been proposed and the functions of the coefficients’ values, depending on the dosage of additives have been obtained. The values of the proposed coefficients are compared with the values, obtained from a series of specially conducted experiments. A multiplicative mathematical model for assessing the influence of modifiers on the strength properties of polymer-cement compositions has been developed, with the help of which it is possible to regulate the strength of modified cement compositions and, on this base, to design future compositions with predetermined characteristics.
References 1. Al Makhadmeh, W., Soliman, A.: On the mechanisms of shrinkage reducing admixture in alkali activated slag binders. J. Build. Eng. 56, 104812 (2022). https://doi.org/10.1016/j.jobe. 2022.104812. 2. Yuan, Q., Xie, Z., Yao, H., Huang, T., Fan, M.: Hydration, mechanical properties, and microstructural characteristics of cement pastes with different ionic polyacrylamides: a comparative study. J. Build. Eng. 56, 104763 (2022). https://doi.org/10.1016/j.jobe.2022. 104763. 3. Ghosal, M., Chakraborty, K.: Superplasticizer compatibility with cement properties – a study. Mater. Today 56(1), 568–573 (2022). https://doi.org/10.1016/j.matpr.2022.02.386 4. Qin, D., Dong, C., Zong, Z., Guo, Z., Xiong, Y., Jiang, T.: Shrinkage and creep of sustainable self-compacting concrete with recycled concrete aggregates. J. Mater. Civil Eng. 34(9), 04022236 (2022). https://doi.org/10.1061/(ASCE)MT.1943-5533.0004393
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5. Zhang, H., Wang, Y., Geng, Y., Wang, Q.: Drying shrinkage model for recycled fine and coarse aggregate concrete considering compressive strength of matrix concrete. J. Build. Struct. 41(12), 156–164 (2020). https://doi.org/10.14006/j.jzjgxb.2020.0111 6. Serebryanaya, I.A., Shlyakhova, E.A., Egorochkina, I.O., Serebryanaya, D.S.: Repair of compounds for restoration of reinforced concrete supports of power lines. In: 2nd ScientificPractical Conference of Russian and Croatian Scientists in Dubrovnik, NUST «MISIS», pp. Moscow, pp. 79-80 (2020). 7. Liu, S., Sun, J., Zhang, J., Xie, Z., Yu, Z.: Effect of graphene oxide on the mechanical property and microstructure of clay-cement slurry. Materials 16(12), 4294 (2023). https://doi.org/10. 3390/ma16124294 8. Stepanov, S., Krasilnikova, N., Makarov, D.: Cement stone, modified by galvanic sludge. In: IOP Conference Series: Materials Science and Engineering 890(1), 012086 (2020). https:// doi.org/10.1088/1757-899X/890/1/012086. 9. Zheng, Y., Sun, D., Feng, Q., Peng, Z.: Nano-SiO2 modified basalt fiber for enhancing mechanical properties of oil well cement. Colloids Surfaces A: Physicochemical Eng. Aspects 648, 128900 (2022). https://doi.org/10.1016/j.colsurfa.2022.128900 10. Kozlova, I., Samchenko, S., Zemskova, O.: Physico-chemical substantiation of obtaining an effective cement composite with ultrafine GGBS admixture. Buildings 13(4), 925 (2023). https://doi.org/10.3390/buildings13040925 11. Iranmanesh, R., et al.: Effect of modified nano-graphene oxide and silicon carbide nanoparticles on the mechanical properties and durability of artificial stone composites from waste. Polycyclic Aromat. Compd. 1-13 (2023). https://doi.org/10.1080/10406638.2023.2214287 12. Serebryanaya, I.A., Shlyakhova, E.A., Egorochkina, I.O., Serdyukova, A.V., Sarkisyan, R.G.: Investigation of the technological factors influence on the formation of the repair compositions properties. In: IOP Conference Series: Materials Science and Engineering, vol. 1083, p. 012036 (2021). https://doi.org/10.1088/1757-899X/1083/1/012036 13. Russian GOST 30459-2008: Admixtures for concretes and mortars. Determination and estimate of the efficiency, Standardinform, Moscow (2010). (In Russian) 14. Nesvetaev, G., Nalimova, A., Chmel, G.: Evaluation of the effectiveness of super plasticizers for high-strength and high-quality concrete. Bull. High. Educ. Institutions 9(537), 38–41 (2003). (In Russian) 15. Nalimova, A.V., Serebryanaya, I.A., Matrosov, A.A.: Mathematical model of deformations of cement stone under the influence of complex additive. In: Parinov, I.A., Chang, SH., Soloviev, A.N. (eds.) Physics and Mechanics of New Materials and Their Applications. Springer Proceedings in Materials, vol. 20, pp. 391–399. Springer, Cham (2023). https://doi.org/10.1007/ 978-3-031-21572-8_32 16. Russian GOST 24544-2020: Concretes. Methods of shrinkage and creep flow determination, Standardinform, Moscow (2021). (In Russian) 17. Chernykh, T.N., Trofimov, B.Y., Kramar, L.Y.: Effect of cellulose ethers on the properties of mortar mixtures and mortars. Constr. Mater. 4, 42-43 (2004). (In Russian)
Influence of Man-Made Raw Materials on the Physical and Mechanical Properties of Organomineral Compositions Nina Buravchuk(B) and Olga Gurianova I. I. Vorovich Mathematics, Mechanics and Computer Science Institute, Southern Federal University, Rostov-on-Don, Russia [email protected]
Abstract. The properties of burnt mine rocks are described. Compositions of asphalt concrete mixtures and asphalt concrete, based on materials from burnt mine rocks, have been developed, their physical and mechanical properties have been determined. Improvement of properties of experimental compositions of asphalt concrete is shown. The basics of the formation of the structure of asphalt concrete on materials from burnt rocks are outlined. Tests of experimental asphalt concrete mixes for the device of paving on experimental sites are carried out. Samples-cores from experimental sections of road surfaces were tested. The expediency of using mineral materials from burnt mine rocks in asphalt concrete mixtures and asphalt concrete has been confirmed. Keywords: Burnt Mine Rocks · Crushed Stone · Screening · Crushing · Finely Dispersed Fraction · Asphalt Mixes · Asphalt Concrete · Physical and Mechanical Properties · Formation of Structure
1 Introduction One of the main environmental problems of the industry is the problem of waste accumulation and disposal, which is especially relevant for the coal-mining regions of Donbass. As a result of the activities of industrial enterprises of the coal industry, a large amount of waste is generated in the form of mine rocks, coal enrichment waste, slag, etc. The rock, raised to the surface, is dumped into rock dumps and waste heaps. Dumps occupy vast areas, sharply worsen the ecological state in coal-mining areas: dust and gas contamination of the air basin, pollution of water resources, land, subsoil, violate the natural landscape of the area. Significant funds are spent on transport and storage of waste, and agricultural land is torn away. Waste fuel and energy complex pose a serious threat to the environment of our planet. This problem needs to be addressed. Scientists around the world are conducting research in this direction. Modern scientific and technological progress makes it possible to utilize numerous production wastes on a large scale by replacing traditional primary types of raw materials and materials with waste and use them in various industries with a significant economic effect. Much has been said about © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 443–456, 2024. https://doi.org/10.1007/978-3-031-52239-0_42
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this waste for many years, but little progress has been made in solving the problem. The volume of utilization of mining waste is 5–6%, although the potential opportunities here are much wider. The volumes of utilization of this technogenic raw material should be significantly increased. The available domestic and foreign developments [1–9], producing building materials from the fuel and energy waste, indicate the technical and economic interest that this type of raw material causes. This is due to (i) the huge reserves of mine rocks in all coal-mining regions, (ii) the aggravated environmental situation near mine dumps, and (iii) the search for new mineral raw materials that are equivalent in quality and properties to traditional but more accessible and cheaper. However, there has not yet been a wide involvement in the production of stocks of raw materials from dumps. The main reason for this is the heterogeneity of the composition and properties of rocks, the lack of information about their environmental safety, the need for additional costs for their enrichment during processing. The use of technogenic raw materials is one of the strategic ways to reduce the extraction of natural raw materials, solve the environmental problem of improving the state of the environment in industrialized areas. Numerous dumps, waste heaps, ash dumps, in which huge amounts of mine rocks and ash and slag waste are stored, are a characteristic feature of the Rostov region, on the territory of which there are significant reserves of coal and thermal power facilities. The fuel and energy waste are universal materials for use in various industries. Both dumps of empty mine rocks and ash and slag wastes stored on the earth’s surface have a certain value. According to the chemical and phase-mineralogical composition, this technogenic raw material is identical to the natural mineral raw material. Ash and slag waste and mine rocks have good prospects for their wide application and conservation of natural resources. The disposal of these wastes is a promising direction for the development of small and medium businesses. The use of ash and slag waste is more common in comparison with burnt-mine rocks. Based on technogenic raw materials, it is possible to obtain new materials with desired properties. Technogenic raw materials can be disposed of only if not only their technological properties are considered, but also the environmental and hygienic properties of the waste and, above all, radiation safety. This will ensure the receipt of environmentally friendly products based on them. Based on the results of numerous studies, it has been established that ash and slag waste and coal mining waste available in the Rostov region belong to the 5th hazard class. All dump rocks and ash and slag mixtures can be used without limitation in the production of building materials and in the road industry since the value of their specific effective activity does not exceed 370 Bq/kg. Road construction is the most material-intensive area for the use of technogenic raw materials. In this industry, the issue of both improving the quality of the structural layers of pavement and replacing natural materials, especially those imported from other regions, with local, cheaper, and more accessible raw materials is acute. Increasing the durability of roads and reducing operating costs for their maintenance is always relevant. 1.1 Purpose of the Study The purpose of this work was to establish the possibility of using materials from burnt mine rocks (crushed stone, crushing screenings and finely ground burnt rock) in the composition of asphalt concrete mixtures and the influence of their specific features
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on the formation of the structure and properties of organomineral composites (asphalt concrete). 1.2 Research Area We study: (i) composition and properties of burnt-mine rocks; (ii) methods of preparation of materials from burnt mine rocks for use in the composition of asphalt concrete mixtures; (iii) features of the formation of the structure and properties of asphalt concrete on materials from burnt mine rocks.
2 Research Methods The object of research was the burnt rocks of the Mayskaya and No. 26 mines, stored in dumps in the Rostov region. Technological samples were taken from the dump for testing. The preparation of burnt mine rocks, consisted of crushing, fractionation and grinding. Analytical and laboratory studies were carried out to obtain information on the chemical and material composition of raw materials, physical and mechanical properties of materials from burnt-mine rock. Compositions of asphalt-concrete mixtures with materials from burnt-mine rocks have been developed. Production tests of the use of materials from burnt-mine rocks in asphalt concrete mixtures were carried out. The implementation of a complex of physicochemical, technological studies and production tests makes it possible to determine the effectiveness and expediency of using materials from burnt-mine rocks in the composition of concrete mixtures for asphalt concrete.
3 Results and Discussion When coal is mined, accompanying rocks with an admixture of coal are brought to the surface and stored in dumps. By origin, mine rocks belong to sedimentary rocks (III genetic type). In terms of basic physical and chemical properties, they are close to clays fired at 800–1000 °C. According to the state of aggregation, the studied rocks belong to solid formations - loose (lump-like lumpy, dispersed and finely dispersed). They belong to the class of coarse clastic rocks, which are unconsolidated deposits of fragments, between which there are no structural bonds (pieces of rock, crushed stone, pebbles, grass, gravel, sand), thermally altered in the dump under conditions of long-term firing at a temperature of 600–1000 °C. The rocks in the dumps are subjected to prolonged thermal action in an oxidizing environment. Self-ignition of rocks in dumps occurs relatively evenly and lasts for several years. In the process of long-term self-ignition of mine rocks in dumps, the mineral part undergoes transformations, associated with dehydration, dissociation, oxidation, polymorphic changes, sintering, amorphization, and crystallization of newly formed compounds. Most of the burnt mine rocks available in the Rostov region are clayey according to their lithological composition [10]. The content of sandstones is less than 10%. Dump rocks of mine No. 26 were stockpiled during the mining of coal seams, located in coal-bearing rocks, mainly of clay composition. The dump of the Mayskaya
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- 9,83 - 10,62
- 7,25
- 4,45
- 4,25
- 3,58
- 2,56 - 2,29
- 2,23
- 1,93
- 1,82 - 1,67
- 1,54
- 3,35
mine belongs to the clayey-sandy type with a sandstone content of 10 to 30%. According to the lithological composition, the dumps are dominated by clayey-argillite rocks with an admixture of siltstones and sandstones. The main rock-forming mineral of the clay substance is hydromica. The content of siliceous rocks is approximately 10–13% by volume. Most of the minerals that make up the rocks are modified in the process of self-firing. The products of roasting clay minerals in burnt mine rocks are represented by amorphous silica and alumina, metakaolinite [11, 12]. Hydromicas, siderite, quartz, and organic matter are present. The following crystalline minerals were found in the mineral composition of burnt rocks (Fig. 1): β-SiO2 with d = 3.35; 2.29; 2.12; 1.82; 1.67Å; β-cristobalite with d = 2.81; 2.53; 2.49; 1.59; 1.54Å; γ-tridymite with d = 4.29; 3.21; 2.81; 2.49Å; γ-Al2 O3 with d = 2.70; 2.45; 2.29; 1.97Å; hematite with d = 2.69; 2.2Å; mullite with d = 2.20; 1.69; 1.59Å, kaolinite, feldspars, calcite, carbonates are present. Iron is found in the form of pyrite, marcasite, and hematite.
2
Fig. 1. X-ray diffraction patterns of burnt rock: 1 – dump of Mayskaya mine; 2 - dump of mine No. 26
According to the data of thermal analysis, endothermic effects are recorded due to the release of hygroscopic and loosely bound water at temperatures of 100 and 200 °C. The exoeffect at a temperature of 340 °C is associated with the transition of amorphous iron to crystalline iron. At 480 °C (endoeffect) the removal of constitutional water and the destruction of the crystal lattice occur. The endoeffect at 573 °C is associated with the polymorphic transformation of β-quartz. The absence of effects characteristic of clay minerals on the thermogram clearly indicates that the rocks were fired at temperatures, exceeding the decomposition temperature of kaolinite. Peaks on the thermograms appear in the result of processes, occurring in that part of the rock that underwent insufficient roasting in the dump or burned at a lower temperature. Well-fired rocks are thermally
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inert and almost no thermal effects are detected on the curves of differential thermal analysis. In burnt argillites and siltstones, mainly clay matter is exposed to thermal effects: it is partially amorphized and covered with films of iron hydroxides. In sandstones, rock fragments are mainly affected. The textural and structural features of sandstones and siltstones remained unchanged. The greatest changes in the mineral composition of the mass under thermal action occur during its transportation up to the formation of completely new types of rocks. In burnt mine rocks, two groups of minerals can be distinguished [10]: minerals of the original rocks and newly formed ones. The first group includes quartz, feldspars, mica, rock fragments, hydromica, kaolinite, carbonates, iron compounds in the form of pyrite, marcasite, hematite. The second group includes: mullite, forsterite, cordierite, spinel, tridymite, cristobalite, sillimanite, andalusite, glass. Unburned fuel particles, present in small quantities as impurities, are metamorphosed to varying degrees, differ from the initial state and are in the form of coke and semicoke and graphitized carbonaceous matter. They are resistant to oxidation and durable when exposed to moisture and temperature changes. According to the content of rocks classified as harmful impurities, the studied rocks do not exceed the requirements of regulatory documents and can be used to product crushed stone, sand, fine fractions without restrictions. The cementing rock mass is composed of brown pelitic material, fired and partially vitrified. In the chemical composition of the average samples of the rock mass of the studied mine dumps, no significant fluctuations in the chemical composition of the rocks were found (Table 1). According to the chemical composition of the rocks, they belong to semi-acidic raw materials with a high content of coloring oxides. According to the content of acid oxides, the studied rocks can be characterized as siliceous-aluminous, sulfur-medium sulfur (up to 3%), organic carbon (0.6 –2.7%) - low-carbon. The presence of losses on ignition indicates incomplete burnout of organic impurities. Table 1. Chemical composition of burnt rocks of mine dumps Mine
SiO2
Al2 O3
Fe2 O3
CaO
MgO
TiO2
K2 O+Na2 O
SO3 general
LI
Mayskaya
59.73
19.22
5.87
1.29
2.69
1.00
3.00
2.78
4.40
No. 26
56.65
21.79
6.60
1.63
2.07
0.96
2.98
2.71
4.94
Note: LI − loss on ignition
By activity in accordance with the classification of Knizhina G.I. [13], these rocks belong to group I: mudstones are active and highly active. In terms of chemical, granulometric and phase-mineralogical composition, these wastes are largely identical to natural mineral raw materials, used in building materials technology. The instability of the properties of technogenic raw materials requires mandatory preliminary preparation to bring its properties and quality to the indicators that apply to natural raw materials. To obtain conditioned raw materials from burnt mine rocks,
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they should be processed, the basis of which is strength enrichment. The original burntmine rocks were subjected to crushing and grinding to obtain fractionated crushed stone, crushing screenings and a fine fraction. From the experience of operation of crushing and screening complexes at the dumps of mines “Maiskaya” and No. 26, it follows that after the second stage of crushing burnt mine rocks in a hammer crusher, crushed stone is obtained mainly with a cube-shaped grain shape, and there are also lamellar and needle particles. Aggregates from burnt mine rocks (fractionated crushed stone and crushed screenings) are not inferior in quality to similar products from traditionally used raw materials, and even have some advantages [1, 14]. The surface of the particles is not rounded, torn, rough. The surface of the particles does not contain clay and other clogging impurities. This feature of such fillers increases the adhesion of crushed stone particles with binders of inorganic and organic origin. As a result of thermal action, the rock particles acquired a porous structure. The presence of porosity of the particles is associated with a reduced bulk density, in comparison with natural crushed stone, an increased sorption capacity with respect to the binder, and improved thermal insulation properties of structural layers with such filler. Crushed stone withstands tests for the stability of the structure against all types of decay (ferrous and silicate). The strength of crushed stone from burnt mine rocks corresponds to the crushability grade 600, 800, 1000, 1200, the frost resistance grade F50, F100, F150, F200, and the abrasion grade I1. Materials from burnt mine rocks – crushed stone, crushing screenings and a finely dispersed rock fraction were tested in the compositions of asphalt concrete mixtures used for the construction of structural layers of pavement. Screenings crushing burnt rocks in terms of grain composition correspond to the sand-gravel mixture. The sandy fraction of screenings from crushing mine rocks in terms of physical and mechanical properties can correspond to sands of I or II class. According to the fineness modulus, sands are classified as coarse and medium. Screenings crushing burnt-mine rocks are not subject to frost heaving, practically do not get wet and do not swell in water. According to the filtration capacity, screenings are water permeable. Under laboratory conditions, an experimental determination was made of the adhesion of a binder film with crushed stone particles after their treatment with bitumen according to the method [15]. The adhesion was evaluated visually by the surface area of the crushed stone, pre-treated with bitumen, on which the bitumen film was preserved after keeping the samples in boiling distilled water. The results showed good adhesion of bitumen with crushed stone from burnt mine rocks: at least 95% of the crushed stone area is covered with a binder film. In asphalt concrete mixtures intended for the upper layers of road surfaces of IV, V categories of roads, aggregates from burnt mine rocks of a continuous grain composition were used, prepared from fractions of St. 5 to 15 mm and crushing screenings from 0 to 10 mm. Screenings crushing by grain composition (from 0 to 10 mm) were a mixture of sand and crushed stone fractions. The content of the latter ranged from 25 to 30%. Fine screening fraction (0–0.071 mm) and finely dispersed burnt mine rock perform the function of mineral powder in the composition of asphalt- concrete mixtures. Mixture preparation and testing of samples was carried out in accordance with the requirements of regulatory documents for asphalt concrete. The physical and mechanical properties of
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asphalt concrete were tested on cylinder samples with a diameter and height of 50 mm, molded on a press with a two-sided application of a pressing pressure of 40 MPa. For testing, the compositions of fine-grained asphalt concrete of dense type B grade II and sandy dense type G of grade II were selected. Table 2 shows the grain composition of the mineral part of the studied asphalt mixtures for dense asphalt concrete of types B and G. The bitumen content ranged from 7.0 to 9.0%. When selecting the composition of asphalt concrete, the optimal amount of bitumen was calculated, based on the residual porosity established for this type of asphalt-concrete, and was refined during test batches according to the results of strength. Table 2. Grain composition of mineral part of mixtures for upper layers of coatings, % by weight Type and type of mixtures and Grain size, mm, no more asphalt concrete 20 15 10 5 2.5 1.25 0.63 0.315 0.16 0.071 Dense B
Dense G
100 85 70
55 38
29
20
14
9
6
100 88 81
60 46
33
24
17
13
7
100 96 86
65 52
38
27
22
16
10
–
–
100 95 68
45
28
18
11
8
–
–
100 97 77
56
39
26
19
12
–
–
100 98 85
67
50
35
24
16
Table 3. Physical and mechanical properties of asphalt concrete of types B and G Properties of asphalt-concrete
Type B brand II
Type D brand II
Test results Average density, g/cm3
2.21–2.30
2.22–2.37
Porosity of the mineral part, %
15.8–17.3
15.3–16.0
at test temperature +50 °C
1.8–2.7
1.7–7.8
+20 °C
5.5–8.0
8.3–8.7
0 °C
11.0–11.2
11.4–11.6
Water saturation, in % by volume
1.7–1.9
1.6–1.7
Water resistance
0.99–1.07
1.03–1.15
Water resistance with long-term water saturation
1.10–1.18
1.13–1.23
0.81–0.83
0.80–0.82
(ii) shear adhesion at temperature of +50 °C, MPa 0.35–0.37
0.36–0.38
Compressive strength, MPa:
Shear resistance according to: (i) coefficient internal friction
(continued)
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Properties of asphalt-concrete
Type B brand II
Type D brand II
Test results Average density, g/cm3
2.21–2.30
2.22–2.37
Crack resistance in terms of tensile strength at a temperature of 0 °C and strain rate 50 mm/min
4.2–4.6
4.0–4.2
The results of the physical and mechanical properties of asphalt concrete for materials from burnt mine rocks are given in Table 3. The technology of preparing asphalt concrete mixtures with burnt mine rocks did not fundamentally differ from the traditional one. Only an increase in the viscosity and stiffness of the mixture was noted. This is due to the surface roughness of the crushed particles. The surface roughness and the presence of acute-angled particles impart rigidity to the mixture. The increased viscosity makes it difficult to mix the mixture and additional efforts are required to evenly distribute the components of the asphalt concrete mixture throughout its preparation. In addition, it is necessary to increase the consumption of binder for fine-grained asphalt concrete by 2–3% compared to the consumption of bitumen for asphalt concrete mixtures based on natural stone materials. However, this is not an obstacle to the use of materials from burnt mine rocks in asphalt concrete mixtures. It is only necessary to correct the technological methods for the preparation of asphalt concrete mixtures, for example, to select plasticizing additives, the optimal temperature of the mixture and adjust the mixing mode of the components, to improve the granulometric composition of aggregates. For uniform distribution and enveloping of the surface of the mineral part of the mixture, the mixing time was increased by 10–15%. It is possible to reduce the stiffness of the asphalt concrete mixture and reduce the consumption of bitumen by introducing finely dispersed additives, for example, TPP ash, ground burnt mine rocks. A good effect is achieved with the introduction of up to 30% sand into the mixture. Possessing hydraulic activity and high dispersion, these additives in asphalt concrete will perform not only the role of a plasticizer, but also the function of a fine micro-filler. Ashes and ground burnt mine rocks can replace the traditionally used mineral powder. Asphalt concrete containing active mineral fillers from ash or ground burnt mine rock is highly frost-resistant. Fly ash and ground mine rock can be recommended as a mineral powder in asphalt concrete for road surfaces, not only in regions with a temperate climate, but especially in harsh climatic conditions and in areas of high humidity. Testing of experimental compositions of asphalt concrete mixtures was carried out on sections of roads and sidewalks. The experimental mixtures were produced at the asphalt plant of the Gukovskoe Mine Construction Department No. 1 (Gukovo, Rostov Region). To assess the quality of asphalt mixtures, containing materials from burnt mine rocks, cylinders were made from mixes, selected at an asphalt plant. The results of comparative tests of experimental asphalt concrete, made in laboratory conditions and in production, are given in Table 4.
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From the experimental compositions of asphalt concrete mixtures, sidewalks, and road sections of industrial facilities in the city of Gukovo, Rostov Region, were paved. The condition of the experimental road sections and the territory of the industrial site was monitored for 3 years from the date of their manufacture. During field observations, the evenness of the coating and the state of the surface (presence of shells, sagging, cracks, quality of seams, etc.) were controlled. Every year, cores in the form of 50 × 50 cm2 cuttings were drilled from the experimental sites to determine changes in the physical and mechanical properties of asphalt concrete over time during operation. The results of such determinations are shown in Table 5. When sampling, the thickness of the layers was measured and their adhesion to each other and to the base was visually assessed. Comparison of the physical and mechanical properties of cores and reshaped samples indicates that the properties of asphalt concrete improve during compaction during operation. For asphalt concrete on materials from burnt mine rocks, high values for strength, water resistance, frost resistance (Figs. 2 and 3) and low values for water saturation were revealed. First, this can be explained by the good adhesion of the active aggregate grains to the organic binder. The bitumen film becomes denser and is less subject to aging from external influences than the bitumen film on the surface of traditional aggregate. Secondly, this is due to the absence of clay and clogging impurities on the surface of aggregates, leading to the optimal grain composition. Thirdly, due to the presence of active components that have free energy and can exhibit pozzolanic activity. The aggregates from burnt mine rocks independently form a monolith and influence on additional strengthening of the material structure, which manifests itself over time, after reaching the design strength. Comparative experimental studies [16] of the fatigue strength of asphalt concretes on materials from burnt mine rocks and traditional ones on crushed stone, sand and limestone micro-filler led to the conclusion that asphalt concretes on materials from burnt mine rocks in terms of endurance to cyclically repeating mechanical loads are not only inferior to the reference ones, but also surpass them. A survey of the constructed experimental sites showed an increased resistance of asphalt concrete pavements on materials from burnt mine rocks to the effects of traffic loads. The pavements retain evenness and solidity, individual rare transverse cracks do not affect the operating conditions of the road. The results of studying the state of experimental sites and the properties of asphalt concrete cores taken in different periods of road operation confirm the suitability of materials from burnt mine rocks for use as part of asphalt concrete mixtures and asphalt concrete. The performed studies on the use of materials from burnt mine rocks in the composition of asphalt concrete show its good adhesion to bitumen, which contradicts the ideas about the interaction of bitumen with acidic rocks. The adsorption activity of a mineral material depends not only on the amount of silicon dioxide. An important role is played by the nature of the surface of mineral particles, the presence of active surface centers, characterized by an increased energy potential [17, 18]. The appearance of such centers on the surface of burnt mine rocks is due to the specifics of their origin. In the process of self-baking, dehydration of clay minerals and the appearance of metakaolinite and a certain number of active modifications of aluminum oxides, soluble or active forms of
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Table 4. Physical and mechanical properties of asphalt concrete on materials from burnt mine rocks The name of indicators
Value of indicators of asphalt concrete laboratory factory
Average density of asphalt concrete, g/cm3 Average density of mineral part, g/cm3
2.39–2.41 2.23
True density of mineral part, g/cm3
2.63–2.68 2.73
2.00–2.10 2.07
True density of asphalt concrete, g/cm3
2.37–2.48 2.35
Porosity of mineral part, %
21.8–23.3 17.0
Residual porosity of asphalt concrete, %
2.7–3.1
Water saturation of asphalt concrete, %
1.38–2.16 3.3
Asphalt concrete swelling, %
0.44–0.51 0.48
2.9
Ultimate compressive strength of dry samples, MPa: at temperature of 50 °C
1.8–2.7
1.4
at temperature of 20 °C
5.5–8.4
6.9
at temperature of 0 °C
12.4–13.0 10.1
Ultimate compressive strength of water-saturated samples, MPa: at a temperature of 20 °C
5.45–8.8
6.2
at a temperature of 20 °C
3.85–9.4
6.0
Water resistance coefficient
0.99–1.05 0.9
Water resistance coefficient for long-term water saturation
0.7–1.19
0.87
Adhesion of bitumen to the surface of the mineral part of asphalt concrete
excellent
excellent
Ultimate compressive strength at long-term water saturation of samples, MPa:
Table 5. Properties of cores-cuttings from asphalt concrete pavement and over molded samples Cores-cuttings
Over molded samples
Average density asphalt concrete, g/cm3
Water saturation, % by volume
Compressive strength, MPa (at t = 20 °C)
Water resistance
Water resistance with long-term water saturation
Average density asphalt concrete, g/cm3
Water saturation, % by volume
Compressive strength, MPa (at t = 20 °C)
Water resistance
Water resistance with long-term water saturation
2.39
2.15
8.8
1.11
0.96
2.23
1.07
8.2
0.98
0.99
2.41
1.93
9.1
1.18
1.03
2.32
1.05
8.5
0.96
1.10
2.42
1.87
9.7
1.27
1.09
2.30
1.07
8.8
1.01
1.13
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1.4
Water resistance coefficient
1.2 1.0 0.8
2
0.6 0.4
1
0.2 0
25
50 75 100 125 Duration of water saturation, days
150
175
Fig. 2. Change in the water resistance coefficient of asphalt concrete during long-term water saturation: 1 - type B; 2 - type G
Compressive strength, MPa
8.0 7.5 7.0
2
6.5 6.0
1 5.5 0
10 20 30 40 Number of freeze and thaw cycles
50
Fig. 3. Change in the asphalt concrete strength from the duration of tests: 1 - type B; 2 - type G
silicic acid, and iron oxides occurred. The formation of these components occurs during self-firing due to the disruption of the molecular bonds of aqueous aluminosilicates and other minerals in the process of dehydration and activation of quartz during surface corrosion and dispersion of particles during pyroprocesses. During the roasting of rocks, destructive processes occur, leading to a violation of the crystal lattice of clay minerals. During weathering, microcracks appear in rock grains. These factors lead to the emergence of a certain energy potential in the products of self-firing and weathering. The active components of burnt mine rocks give them pozzolanic activity. Such materials do not harden on their own and, when mixed with water, they do not seize and do not turn
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into a stone-like state. However, they can exhibit pozzolanic activity and interact with inorganic and organic binders. Due to the presence of active components that have free energy and can exhibit binding properties, materials from burnt mine rocks are actively involved in the formation of the structure of asphalt concrete during the hardening of asphalt concrete mixtures. It is associated with the presence of a set of structural defects that arise during physical and chemical processes, occurring during high-temperature firing of coal-containing mine rocks. The presence of these active centers on the particle surface has a positive effect on the interaction of mineral components with bitumen, and, consequently, on the basic physical and mechanical properties of the organo-mineral composition (asphalt concrete), namely strength, frost resistance, heat and water resistance, including long-term stability, increase. The endurance of asphalt concrete largely depends on the adhesion of the binder to the surface of the mineral aggregates. The adhesion of the binder to the surface of the mineral particles of the burnt mine rock is good, and the strength in the contact zone is a serious obstacle to the formation of microcracks. It is possible to obtain high-quality asphalt concrete composites only by creating an optimal structure between the organic binder and mineral components, which is determined by physical, physico-chemical, and mechanical interactions at the phase boundary. When forming the structure of asphalt concrete, in addition to the active components of the burnt mine rock, it is necessary to consider the properties of the surface of mineral particles. It is known that the surface of all solid materials contains acidic and basic centers of the Lewis and Brönsted types [19], which obviously determine its activity with respect to binders (inorganic and organic). However, their influence on the processes of interaction with bitumen is practically not considered. According to studies [17–19], contrary to the traditional opinion, the surface of acidic silica-containing mineral materials is not inert with respect to bitumen components. It is necessary to consider the presence on the surface of mineral materials of not only the thinnest colloidal films of oxides and hydroxides and amorphous silica, but also the presence of active surface centers that can adsorb almost all organic compounds found in bitumen and thereby ensure strong adhesive contacts between the binder and the surface mineral materials. The greatest contribution to this interaction will be made by acid Lewis, acid and basic Brønsted centers. The main role in the formation of the structure of asphalt concrete is played by the strength of contacts between particles. The organic binder and active components of the finely dispersed burnt mine rock act as a binder at the stage of structure formation during the hardening of the composition. The formation of the structural framework takes place in several stages. The initial interaction between dispersed particles is due to long-range van der Waals forces and the formation of coagulation structures. The further process of structure formation is characterized by maximum contact and transformation of coagulation structures into condensationcrystallization structures, which provide the mechanical strength of the composition. The use of finely dispersed burnt mine rock as a micro-filler and aggregates from technogenic raw materials makes it possible to obtain a high-quality fine-grained dense structure of the composition. The emerging structure of the asphalt concrete composition on materials from burnt mine rocks is characterized by a mixed, coagulation-condensation type of bond [20]. The adhesion forces of such bonds are more durable than coagulation ones,
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and the composite system with such bonds has improved physical, mechanical and deformation properties.
4 Conclusion In the decision to improve the quality of asphalt concrete, one of the important factors is the strength of the structural bonds between the components. The strength of contacts is determined by the physical and chemical interaction between the organic binder and the mineral component, namely aggregates. Active components in the composition of burnt mine rocks and energy centers on the surface of dispersed systems determine their reactivity and participation in the formation of contacts with binders. As a result, a dense and perfect fine-grained structure of asphalt concrete is formed. In such a structure, the accumulation of local destruction zones and the formation of cracks in asphalt concrete under loads and climatic influences are less likely. These features make it possible to replace natural mineral raw materials with materials from burnt mine rocks with a positive effect. This is confirmed by laboratory studies and a positive result of production tests of the developed asphalt concrete mixtures in road surfaces, sidewalks, and industrial sites. The use of burnt mine rocks in road construction makes it possible to improve the quality of asphalt concrete, extend its service life by improving its physical and mechanical properties, reduce the consumption of natural resources, and use cheaper road building material. The implementation of research results in practice will solve some environmental problems: (i) reduce the number of quarrying for the extraction of natural raw materials, (ii) preserve the natural landscape of the earth’s surface, (iii) free land from dumps, and (iv) reduce pollution of the water and air basins. Acknowledgements. The study was financially supported by the Ministry of Science and Higher Education of the Russian Federation, State task in the field of scientific activity, scientific project No. FENW-2023–0012.
References 1. Lyapin, A.A., Parinov, I.A., Buravchuk, N.I., Cherpakov, A.V., Shilyaeva, O.V., Guryanova, O.V.: Improving Road Pavement Characteristics. Applications of Industrial Waste and Finite Element Modelling. Series: Innovation and Discovery in Russian Science and Engineering. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-59230-1 2. Abdrakhimova, G.S.: Use of waste from the fuel and energy complex - burnt rocks and chromite ore enrichment waste in the production of porous aggregates based on a liquid-glass composition. Coal. 7, 67–69 (2019) 3. Vlasova, V.V., Artemova, O.S., Fomin, E.Y.: Determination of directions for the effective use of waste from thermal power plants. Ecol. Ind. Russia 21(11), 36–41 (2017). (In Russian) 4. Fedorova, N.V., Shaforost, D.A.: Prospects for the use of fly ash from thermal power plants in the Rostov region. Thermal Power Eng. 1, 53–58 (2015). (In Russian) 5. Chumachenko, N.G., Tyurnikov, V.V., Seikin, A.I., Bannova, S.E.: Possibilities of using burnt rocks in construction. Ecol. Ind. Russia 19(11), 41–46 (2015). (In Russian)
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6. Rimkevich, V.S., Pushkin, A.A., Churushova, O.V.: Complex processing of ash from thermal power plants. Mining Inf. Anal. Bull. 6, 250–253 (2015). (In Russian) 7. Shpirt, M.Y., Artemiev, V.B., Silyutin, S.A.: Use of solid waste from coal mining and processing. Moscow, Mining LLC “Cimmerian Center” (2013). (In Russian) 8. Naganathan, S., Mohamed, A.Y.-O., Mustapha, K.N.: Performance of bricks made using fly ash and bottom ash. Constr. Build. Mater. 96, 576–580 (2015). (In Russian) 9. Zhang, L.: Production of bricks from waste materials - a review. Constr. Build. Mater. 47, 643–655 (2013) 10. Gipich, L.V.: Peculiarities of the material composition of dump rocks of the mine dumps of the Eastern Donbass and new directions for their use. PhD Thesis (Technical Sciences). Rostov-on-Don. Rostov State University (1998). (In Russian) 11. Nefedov, A.P., Krivoborodov, Y.R., Kosov, D.Y.: Use of metakaolin in cement production. In: Proceedings of the IV International Conference on Concrete and Reinforced Concrete. Moscow, MGSU IV, pp. 122–128 (2014). (In Russian) 12. Mansour, M., Abadla, M., Jauberthie, R., Messaoudene, J.: Metakaolin as a pozzolan for high performance mortar. Cement Warno Beton 17(2), 102–108 (2012) 13. Knigina, G.I., Zavadsky, V.F.: Microcalorimetry of mineral raw materials in the production of building materials. Moscow, Stroyizdat (1987). (In Russian) 14. Buravchuk, N.I., Gurianova, O. V.: Research and application of materials from burnt rocks of mine dumps to product mineral aggregates. Ecol. Ind. Russia 24(10), 26–32 (2020). (In Russian) 15. Lysikhin, A.I.: Road surfaces and bases with the use of bitumen and tar. Moscow, Stroyizdat (1962). (In Russian) 16. Mironov, A.A., Bazuev, V.P.: Fatigue properties of asphalt concrete from industrial waste. In: Design and Construction of Highways and Bridges in Siberia”. Tomsk, Tomsk University Press, pp. 80–86 (1992). (In Russian) 17. Yadykin, V.V.: Influence of active surface centers of silica-containing mineral components on interaction with bitumen. Univ. News. Constr. 9, 75–79 (2003). (In Russian) 18. Yadykin, V.V.: Increasing the efficiency of asphalt and cement concrete based on technogenic raw materials. Sci. Technol. Road Ind. 1, 45–47 (2004). (In Russian) 19. Yadykin, V.V.: Control of the process of formation and quality of building composites, considering the surface of dispersed raw materials. Moscow, Association of Building Universities (2009). (In Russian) 20. Rebinder, P.A.: Surface phenomena in disperse systems. Physical and chemical mechanics. Selected Works. Moscow, Nauka (1979). (In Russian)
The Effect of Variations in Coconut Fiber Ash Waste as Added Material in Mortar Nurul Rochmah(B) , Retno Trimurtiningrum , Bantot Sutriono , Masca Indra Triana , and Musthofa Saifa Ardana Department of Civil Engineering, Universitas 17 Agustus 1945, Surabaya, Indonesia [email protected]
Abstract. Indonesia is the largest coconut fiber producing country in the world and has coconut plantation land with an area of close to 3.74 ha. The plantation products will be used to meet human needs, while the rest of the utilization will become waste. The process of destroying waste naturally usually takes place slowly, causing a pile of waste. Coconut fiber ash contains minerals consisting of silica, alumina, and iron oxides. This shows that coconut fiber ash has the potential to be used as a construction material. The previous research shows that the use of coconut fiber ash in large quantities as a cement substitute has the potential to reduce compression strength, so it cannot be utilized in large quantities. In this research, coconut fiber ash would be used as an added material and not as a cement substitute to utilize the waste as much as possible. This study would examine the influence of mortar characteristics on the addition of coconut fiber ash. The variation in the percentage of coconut fiber ash used is 0%, 2.5%, 5%, 7.5% and 10% to the weight of cement. The composition of mortar’s mixture would be based on SNI 03-6825-2002. The studied characteristics of the mortar are the following: density (unit weight), absorption and compression strength. From the result, it is concluded that the addition of coconut fiber in mortar mixture increases the density and the water absorption percentage. The optimum value of compression strength test result is 32.61 MPa, obtained from the mixture with the addition 2.5% of coconut fiber ash. Keywords: Absorption · Coconut Fiber Ash · Compression Strength · Density · Mortar
1 Introduction Indonesia is the largest coconut fiber producing country in the world and has coconut plantations with an area of nearly 3.74 ha [1–3]. The plantation products will be used to meet human needs, while the remaining results will become waste. Agricultural and plantation waste can take the form of unused waste materials and waste materials resulting from processing. The natural process of destroying waste is usually slow, causing piles of waste that can disturb the surrounding environment and even have an impact on human health [4]. Indonesian coconut production averages 15.5 billion pieces/year or the equivalent of 1.8 million tons of coir fiber and 3.3 million tons of coir dust [5]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 457–467, 2024. https://doi.org/10.1007/978-3-031-52239-0_43
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Coconut fiber contains organic elements such as fiber, cellulose, and lignin. Coconut fiber ash contains minerals consisting of silica, alumina, and iron oxides [4]. Based on testing of coconut fiber ash, carried out by the Center for Environmental Health Engineering (BBTKL), the composition of silica compounds (SiO2 ) was found equal to 67.55% [3]. Based on research, conducted by Alexander and Mukhlis, coconut fiber ash contains chemical compounds (in weight percent units) consisting of the element SiO2 (42.98%), Al2 O3 (2.26%) and Fe2 O3 (1.66%) [6]. This shows that coconut fiber ash has the potential for using as a construction material. The silica content in coconut fiber ash has the potential to react with lime Ca(OH)2 , produced from the cement hydration process to form a binder [4]. In previous research, conducted by Hendra Taufik, et al. [13], coconut fiber ash had the potential for using as a partial replacement material for cement. They used variations of coconut fiber ash equal to 0%, 2.5%, 5%, 7.5%, 10% of the cement weight. The research results showed that there was an increase of 5.5% in the compression strength of mortar aged for 28 days with a 2.5% coconut fiber ash. The use of coconut fiber ash in large quantities shows a decrease in the compression strength of the test object [4]. In this research, coconut fiber ash (CFA) will be used as an additional material and not as a cement substitute and will examine the effect of construction material characteristics on the addition of coconut fiber ash. This is because the use of coconut fiber ash in large quantities as a cement substitute has the potential to reduce compression strength, so that coconut fiber ash cannot be used in large quantities. This research used mortar as the specimens, where mortar is a mixture of materials that consist of fine aggregate (sand) as a filler, cement with/without other cementitious material and water as a binder. Mortar characteristics, which will be studied, include volume weight, compression strength and mortar absorption.
2 Research Method 2.1 Materials Materials, used in this research, are Portland composite cement (PCC) from PT. Semen Gresik, coconut fiber from East Java, Lumajang sand as fine aggregate and water. Before use as raw materials in the mortar’s mixture, these materials should be tested, and the result should be compared with the applicable standard. The purpose of material testing is to ensure the quality of the materials, used in mixture. Quality mortar is mortar, whose constituent ingredients are of good quality and meet the specified requirements. The following tests of the material are carried out [7–12]: • • • • • • • •
XRF test for Coconut fiber ash (CFA) Sand sieve analysis experiment (ASTM – C136-06) Sand specific gravity experiment (ASTM – C128-01) Sand moisture experiment (ASTM – C556-71) Sand infiltration water experiment (ASTM C128) Sand volume weight experiment (ASTM – C29-78) Sand volume development test (bulking) (ASTM C29/C29M-97) Sand cleanliness test against mud using wet method (ASTM-C117-95)
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Table 1. Mortar specimens Mixtures
Coconut fiber ash (CFA) (%)
Mortar Compression strength
Absorption
CFA-0
0
9
2
CFA-2.5
2.5
9
2
CFA-5
5
9
2
CFA-7.5
7.5
9
2
CFA-10
10
9
2
45 Specimens
10 Specimens
Total number of specimens
Fig. 1. Wet curing of mortar specimens
• Sand cleanliness test against mud using dry method (ASTM- C117-95) The variation is based on the percentage of coconut fiber ash, used in the mixture. the coconut fiber ash in this research is used as the additional material in mortar. Variations in the percentage of coconut fiber ash used are 0%, 2.5%, 5%, 7.5% and 10% of the cement weight. The use of the percentage of mixed variations is due to the previous research, which showed that mixture with 2.5% of coconut fiber ash, experiencing an increase in compression strength, while mixture with percentage of coconut fiber ash above 10% will experience a decrease in compression strength. The specimens for compression strength and absoption tests are cubes with size 5 cm × 5 cm × 5 cm. the cubes were cured by submerged into the water at room temperature. The test specimens are showed
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Fig. 2. Mortar samples for compression strength tests
in Table 1. Figures 1 and 2 show the curing process of mortar specimens and the mortar sample for compression strength test, respectively. 2.2 Mixture Composition of Mortar Specimens The mix design calculation is based on SNI 03–6825-2002 with the title “Compression Strength Testing Method of Portland Cement Mortar for Civil Works.” SNI 03–68252002 arrange the manufacture of cube-shaped test specimens with each side size of 5 cm with the following composition: (i) 500 g of Portland cement; (ii) 1.375 g of sand (fine aggregate); (iii) 242 ml of water. Table 2 shows the proportion of each material in the mixture for cube-specimens. Table 2. Mixture composition for cube-specimens Mixtures Coconut fiber ash (%) Cement (g) Coconut fiber ash (g) Sand (g) Water (ml) CFA-1
0
500
0
1.375
242
CFA-2
2.5
500
12,5
1.375
242
CFA-3
5
500
25
1.375
242
CFA-4
7.5
500
37,5
1.375
242
CFA-5
10
500
50
1.375
242
2.3 Preparing Process of Coconut Fiber Ash Material The coconut fiber waste was obtained from Lumajang, East Java. The coconut fiber ash was obtained in the result of burning process of coconut fiber waste. Figures 3, 4, 5 and 6 present the burning process of coconut fiber waste. The process in making the coconut fiber ash include the following steps:
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Fig. 3. Dry coconut fiber waste
Fig. 4. Burning process of coconut fiber waste
(i) gather dried coconut fiber, whose weight at least 100 kg in total; (ii) prepare two barrels for burning coconut fiber; (iii) fill the barrel with coconut fiber until it is full; if the barrel is already full, press down on it as much as you can so it compressed, and then fill it up again with coconut fiber until it is completely full (the burning barrel should not have any air spaces); (iv) to start the burning process, use a wooden or iron stick to elevate a small portion of the coconut fiber so that the lower portion is exposed to the fire; (v) close the combustion barrel after it starts to smoke. (vi) continue steps (iii) and (iv) until all 100 kg of coconut fiber have been completely burned. This should take about 30–45 min;
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(vii) after burning the coconut fiber ash is collected and left in ambient temperature until the temperature of coconut fiber ash is not very high. (viii) the coconut fiber ash is sieved by using sieve No.100 to get a finer material.
Fig. 5. Coconut fiber ash before sieving
Fig. 6. Coconut fiber ash after sieving
2.4 Making Specimen of Mortar This study used a cube-shaped test specimen with a side size of 5 × 5 × 5 cm3 . The stages of making test objects are: 1. Combine the sand and cement in the mixture in a container in a ratio 1:3, until they are evenly distributed.
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2. Up to three times should be done after the initial stage. 3. Coconut fiber ash adds as an additional material in the following percentages: 2.5%, 5%, 7.5%, and 10%. 4. Mix each of the added ingredients thoroughly to ensure even distribution. 5. Add 242 cc of water and whisk for around 10 min to ensure that mix is dispersed evenly. 6. Put the equally dispersed mortar mixture into a pipe with a diameter from 9 to 10 cm and a height of 15 cm to evaluate the mortar’s workability. 7. Make the dough mix inside the pipe after inserting it, rock it with a sand SSD rocking tool. 8. After the mixture has blended and completely filled the pipe, lift the pipe so that the dough may be removed in order to assess the mortar’s workability. 9. Put mortar mixture into mortar mold. 10. When the mortar has dried, take the specimen from mortar mold (Fig. 7).
Fig. 7. Mortar making process
2.5 Testing of Specimens 2.5.1 Density (Unit-Weight) Density is defined as the weight per unit volume in [14]. To determine the true value of the concrete’s specific gravity before usage, a test for specific gravity is conducted. SNI 1973–2008 states that the following formula is used to determine unit weight: D=
Mc − Mm , Vm
(1)
where D is the unit-weight of concrete (kg/m3 ), Mc is the concrete-filled weighing container weight (kg), Mm is the container’s weight (kg), Vm is the container’s volume (m3 ).
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2.5.2 Water Absorption A test, called water absorption, is used to figure out how much water exists in concrete. In other words, the goal of this test is to determine the test object’s level of water absorption. In (SNI 03-6882-2002), the formula used to calculate the water content of concrete is: Absorption =
Mb − Mk × 100% Mk
(2)
where Mb is the mass of saturated specimen (g), Mk is the mass of dry specimen (g). 2.5.3 Compression Strength Test The ability of a test object to endure a specific compressive force, applied by the mortar compression test machine is known as mortar compression strength. The object is positioned and loaded using the compression testing machine until it collapses or cracks. Based on that, the peak load operates. The following formula can be used to evaluate the compression strength of mortar, based on SNI 03-6825-2002: fc = p/A
(3)
where fc is the mortar compression strength (MPa), A is the cross-section area of the test sample (mm2 ), P is the compression load (N) (Fig. 8).
Fig. 8. Compressive strength test
3 Results and Discussions 3.1 Density (Unit-Weight) Density of mortar is calculated my ratio of mass to volume of specimens. Figure 9 shows the influence of coconut fiber ash as an additional material on density of mortar.
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Fig. 9. Influence of coconut fiber ash on density of mortar
The addition of coconut fiber ash in the mixture increased the density (volumeweight) of mortar. It is because of the effect of coconut fiber ash in mixture as an additional material. So that, the addition of coconut fiber ash in the mixture increases the unit mass of mixture. Based on the test results, the mixture CFA-10 has the maximum density (mass per unit volume) of 2.384 kg/m3 . The anomaly result obtained from mixture CFA-8 is due to the error in casting and handling process of mortar mixture.
Fig. 10. Influence of coconut fiber ash on absorption value
3.2 Water Absorption Water absoption testing was performed for the 28-days mortar specimens. Figure 10 shows the influence of coconut fiber ash on absorption value of mortar. The addition
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of coconut fiber ash in the mortar mixture tends to increase the water absorption in the sample. The result is appropriate with previous research, conducted by Arinmawa et al. (2020) that the water, absorbed by the mortar cube, increases as the amount of coconut fiber ash content increases [15]. Mixture, which has highest value of water absorption (1.0546%) is obtained by the mixture with 10% of coconut fiber ash. The mortar absorption with a 2.5% coconut fiber ash content showed the lowest value of 0.702%, compared to the percentage of other coconut fiber ash content. 3.3 Compression Strength Compression strength test result used the average values of compression strength for 28day mortar. Figure 10 shows the influence of coconut fiber ash on compression strength of mortar specimens.
Fig. 11. Influence of coconut fiber ash on compression strength
Figure 11 shows that the addition of coconut fiber ash to the mortar influences the compressive strength test result. The mixture which has highest compressive strength value is mixture with 2.5% of coconut fiber ash (32.61 MPa). The addition of coconut fiber ash more than 2.5% of cement weight decreases the compression strength. Hendra Taufik et al. (2013) have found that the ash content of coconut fiber with a percentage of 0%, 2.5%, 5%, 7.5%, and 10% will increase the optimum compression strength at the percentage of 2.5%, while above 10% there will be a decrease in compression strength. The increased value of compression strength test is caused by the silica content on the coconut fiber ash, so that it reacts with calcium hydroxide, emerged from the hydration process of cement and water to form silica gel of C-S-H.
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4 Conclusion For density test, the addition of coconut fiber ash in the mixture increased the density (volume-weight) of mortar. CFA-10 has the maximum density (mass per unit volume) of 2.384 kg/m3 . For water absorption test, the addition of coconut fiber ash in the mortar mixture tends to increase the water absorption in the sample. Mixture which has highest value of water absorption (1.0546%) is obtained by the mixture with 10% of coconut fiber ash. For compression strength test, the optimum value is 32.61 MPa, obtained for the mixture with the 2.5% addition of coconut fiber ash. The increasing value of compressive strength test is caused by the silica content due to the coconut fiber ash that react with calcium hydroxide and form silica gel of C-S-H. Acknowledgement. This research was carried out in Civil Engineering Department Universitas 17 Agustus 1945 Surabaya.
References 1. Affandy, N.A., Bukhori, A.I.: Pengaruh Penambahan Abu Serabut Kelapa terhadap Kuat Tekan Beton. UkaRst 3(2), 150–158 (2019). (In Indonesian) 2. Kementan: Prospek dan Arah Pengembangan Agribisnis Kelapa (2005). (In Indonesian) 3. Bayuaji, R., Kurniawan, R.W., Yasin, A.K., Fatoni, H.A.T., Lutfi, F.M.A.: Material Inovatif Ramah Lingkungan: Pemanfaatan Komposit Abu Serabut Kelapa dan Fly Ash pada Pasta Semen. J. Aplikasi 13(1), 15–20 (2015). (In Indonesian) 4. Taufik, H., Djauhari, Z., Muhandis, M.: Pengaruh Pemakaian Abu Serabut Kelapa (ASK) sebagai Subtitusi Semen pada Mortar. Universitas Riau, Riau (2016). (In Indonesian). 5. Ariatma, A. A., et al.: Pemanfaatan Limbah Serabut Kelapa di Desa Korleko Kecamatan Labuhan Haji Kabupaten Lombok Timur. J. Warta Desa 1, 36–43 (2019). (In Indonesian) 6. Fenriana, A.M., Sartika Nisumani, U.S.M.: Pengaruh Penambahan Abu Serabut Kelapa dan SIKACIM Concrete Additive terhadap Kuat Tekan Beton. Jurnal Gradasi Teknik Sipil 6(2), 74–81 (2022). (In Indonesian) 7. ASTM C29/C29M-07: Standard Test Method For Bulk Density (Unit Weight) and Voids in Aggregate. United States, ASTM International (2009) 8. ASTM C 128 – 01: Standard Test Method for Density, Relative Density (Specific Gravity), and Absorption of Fine Aggregate. United States, ASTM International (2001) 9. ASTM C 29/C 29 M– 97: Standard Test Method for Density, Relative Density (Specific Gravity), and Absorption of Fine Aggregate. United States, ASTM International (2001) 10. ASTM C 566 – 97: Standard Test Method for total Evaporable Moisture Content of Aggregate. United States, ASTM International (1997) 11. ASTM C 136-06: Standard Test Method for Sieve Analysis of Fine and Coarse Aggregates. United States, ASTM International, vol. 04 (2006) 12. ASTM C 117 – 95: Standard Test Method for Materials Finer than 75-µm (No. 200) Sieve in Mineral Aggregates by Washing. United States, ASTM International (1995) 13. Taufik, H., Widya, B., Muhandis, M.: Pengaruh Pemakaian Abu Serabut Kelapa (Ask) Sebagai Substitusi Semen Pada Mortar. J. Saintis 13(2), 1–12 (2013). (In Indonesian) 14. Sni 03-6825-2002: Standar Nas. Indones. Metod. Penguji. kekuatan tekan mortar semen Portl. untuk Pekerj. sipil (2002). (In Indonesian) 15. Arimanwa, M.C., Anyadiegwu, P.C., Ogbonna, N.P.: The potential use of coconut fibre ash (CFA) in concrete. Int. J. Eng. Sci. (IJES) 9(1), 68–75 (2020)
Applications of Advanced Materials
Embedding Epitaxial VO2 Film to Switchable Two-Band Filter on the Surface Acoustic Waves G. Ya. Karapetyan(B) , V. E. Kaydashev, and E. M. Kaidashev Laboratory of Nanomaterials, Southern Federal University, 200/1 Stachki Ave, Rostov-on-Don 344090, Russia [email protected]
Abstract. The paper proposes a tunable bandpass filter, the operating frequency of which in the range of 600–800 MHz is regulated by voltage. It is performed on a piezoelectric substrate made of lithium niobate and contains four parallel acoustic channels formed by two input interdigital transducers (IDTs), two multistrip directional couplers and two output IDTs with the ability to switch each pair of acoustic channels with its independent vanadium dioxide film switch provides attenuation between channels of at least 70 dB. By using a special switch design based on a vanadium dioxide film and a coupler, we were able to reduce the control voltage to 0.1 V. The width of the vanadium dioxide film is no more than two periods of input IDT, which eliminates an increase in the SAW attenuation of s with an increase in the operating frequencies of the filter. Keywords: Surface Acoustic Wave (SAW) · Vanadium Dioxide · VO2 Film · Interdigital Transducer (IDT)
1 Introduction Currently, cellular communication systems use frequency filters of surface acoustic waves (SAW) with a fixed central frequency without the possibility of adjustment. However, miniature narrowband filters with a central frequency adjustable by voltage are highly desired. To meet the demand for ever-growing data traffic in wireless communications, multiband wireless interface modules are required, which can cover many frequency ranges. Currently, these interface modules are built using switched filter blocks based on many discrete filters and multiplexing switches at the input and output, which leads to an increase in module size, cost, and energy consumption. There is a need for an adaptive filtering design in which frequency channels can be added or removed while maintaining the smallest number of required elements. For modern mobile communication systems, a necessary requirement is to support operation in several frequency bands, which is achieved through the implementation of tunable input cascades. To create such cascades, it is necessary to develop highfrequency tunable and switchable SAW filters, capable of changing the bandwidth and center frequency, having high selectivity and low insertion losses. In addition, such a filter should have small overall dimensions and low power consumption. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. A. Parinov et al. (Eds.): PHENMA 2023, SPM 41, pp. 471–485, 2024. https://doi.org/10.1007/978-3-031-52239-0_44
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Considering the switchable SAW filters known to us, for example, patent [1] presents a tunable SAW filter, the characteristics of which can be programmed. It contains a piezoelectric substrate, on the surface of which there are input and output unidirectional three-phase interdigital transducers (IDT), and groups of electrodes of each phase are connected to a generator (or receiver) through controlled phase shifters and amplifiers. This allows one to change the phase and amplitude characteristics of the filter, changing the phase and gain of the amplifiers and program the necessary characteristics in advance. A tunable filter, based on frequency-selective matrices, is also known [2]. These matrices represent an n-channel SAW filter in which different channels are switched using switches. In this case, the input IDTs of each channel have a small aperture, and the output IDTs are common to all channels and have an aperture at least n times larger. This results in a significant increase in insertion loss as the SAWs emitted by the transmitting IDT only fall on a small portion of the receiving IDT. That result in reradiation of SAWs by that IDT in locations, where there is no SAW from the transmitting IDT, which is a disadvantage of such a tunable filter. In addition, tunable resonators, based on surface acoustic waves and SAW filters with digital-to-analog transducers, are known [3], in which external elements for tuning the filter are excluded. A constant bias voltage is applied to the IDT electrodes, located on the semiconductor layer, which leads to a shift in the center frequency of the IDT, and, consequently, to a shift in the frequency of the SAW resonator. In this case, the tunable filter does not contain amplifiers, phase shifters, or switches. However, the manufacture of such filters requires multilayer structures to form areas under each IDT electrode, where the SAW speed will change under the influence of voltage, applied to the electrodes. This complicate the manufacturing technology of such filters, since it requires precise alignment of the semiconductor layers with the IDT electrodes, the dimensions of which may be less than 1 micron. A multichannel filter, based on surface acoustic waves with a tunable frequency controlled by voltage, is also known [4], in which the input and output IDTs are several partial IDTs, connected in parallel with different periods. This IDTs are calculated in such a way that the frequency response of such a filter is comb shaped. The partial IDTs are located on the semiconductor layer. There is a bottom electrode on the bottom surface of the substrate. By applying a voltage between the bottom electrode and one of the IDT combs, it is possible to change the SAW speed in the layer, thereby changing the center frequency of the partial IDTs. Moreover, the voltages, supplied to the input and output IDTs, can be different. This leads to the fact that the central frequencies of partial IDTs, located in the same acoustic channel may differ, which leads to a change in the attenuation in the amplitude-frequency response in this acoustic channel. In this filter, the attenuation can change between different channels up to 50–55 dB. The layer, in which the SAW speed changes under the influence of voltage, is located under each IDT electrode, and the IDT is applied directly to this layer. This complicates the manufacturing technology of such filters, since it requires precise alignment of the semiconductor layers with the IDT electrodes, the dimensions of which can be less than 1 micron. Another approach to obtain tunable filters is the use of resonator SAW filters, which use additional capacitances, by changing which, we can change the amplitude-frequency
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characteristics of the filter [5]. The basic structure of a tunable filter is the same as that of a conventional ladder filter. The varicaps are connected to SAW resonators in series and parallel filter branches. The varicaps, connected in parallel with SAW resonators, shift their anti-resonant frequencies, while varicaps, connected in series, shift their resonant frequencies. Thus, the position of the upper and lower boundaries of the filter passband is adjusted independently. To achieve a large tuning range in this type of filter, it is necessary to create a broadband SAW resonator. Thus, it is possible to obtain differences in attenuation in tunable channels of only 7–10 dB. To further miniaturize such filters, to rebuild the filter on SAW resonators, it was used in [6] not variable capacitances controlled by voltage (varicaps), but vanadium oxide films, the resistance of which changes sharply at temperatures above the phase transition temperature. This resistance varies from 235 to 0.8 . Switching occurs by heating the film with a special miniature heater through which current is passed. The main advantage of using VO2 switches over MEMS or solid-state switches is that VO2 switches lend themselves to monolithic integration with SAW filters on a single chip. The two-channel filter has four states: (i) both channels are closed (the resistance of the vanadium dioxide film is 235 ), the attenuation is 12 dB in both channels; (ii) both channels are open (the resistance of the vanadium dioxide film is 8 ), the attenuation is 3.9 dB in both channel; (iii) the high-frequency channel is open (943 MHz), and the low-frequency channel is closed (the resistance of the vanadium dioxide film is 235 in the first channel and 8 in the second channel), the attenuation is 3.9 dB in the first channel and 10 dB in the second channel; (iv) the low-frequency channel is open (765 MHz), and the high-frequency channel is closed (the resistance of the vanadium dioxide film is 8 in the first channel and 235 in the second channel). The attenuation is 10 dB in the first channel and 3.9 dB in the second channel; The advantages of the above-described filters on SAW resonators are their low losses, but the disadvantage is that it is possible to obtain a difference in attenuation in tunable channels of no more than 7–10 dB. Another approach to obtain tunable amplitude-frequency characteristics in SAW filters is to use tunable light-induced periodic structures along the SAW propagation path. In this case, if the grating period coincides with the SAW length, intense reflection occurs. By changing the grating period, we can change the central frequency of reflection within a wide range. Thus, the patent [7] proposes a tunable SAW filter containing a piezoelectric sound guide with a strong piezoelectric effect and at the same time an acousto-photorefractive effect. There are broadband input and output IDTs on the sound guide surface. In this case, under the influence of laser radiation, a photoinduced grating is created on the surface of the sound guide between the IDTs, which is formed by the interference of two beams of the same laser. In places where there is light, the SAW speed changes, which creates a reflective grating on the surface, the period of which varies depending on the interference period of the laser beams. By changing this period, we can change the period of the reflective grating and thereby the SAW reflection frequency. In this
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way, the center frequency of the filter can be varied within the bandwidth of the input and output IDT. The disadvantage of such a tunable filter is the presence of a laser and a system for creating interference fringes on the surface of the sound guide. This drawback can be eliminated by creating a periodic structure between the IDTs in the form of a directional coupler, the period of which differs from the length of the SAW, so that the reflection band of the SAW from such a periodic structure would be outside the filter passband. For example, a two-channel tunable filter, based on surface acoustic waves [8], as the closest in technical essence to the design we propose, contains a piezoelectric sound guide with one input and two output interdigital transducers applied to it, forming two parallel acoustic channels, and multistrip directional coupler, located between the input and output IDTs. To simplify the design and increase reliability, control electrodes connected to a voltage source, are located on both sides of the multistrip directional coupler, and a semiconductor film is applied to the surface of the multistrip directional coupler and the control electrodes. The output IDTs have different center frequencies, and the total bandwidth of the output IDTs does not exceed the bandwidth of the input IDT. The SAW is excited by the input transducer and, propagating along the piezoelectric sound guide, enters the multistrip directional coupler. The step and number of electrodes of the multistrip directional coupler are selected in such a way that in the absence of a control voltage, all the energy, emitted by the input IDT, is re-radiated to the output IDT. The voltage applied to the control electrodes creates a drift of charge carriers in the direction of propagation of the acoustic wave, and the effective electromechanical coupling coefficient is a function of the electric field strength in the semiconductor film. The use of lithium niobate substrate with indium antimonide semiconductor film with a charge carrier mobility of 1250 cm2 /V·s and a thickness of 0.5 µm as a piezoelectric sound guide makes it possible to obtain complete signal switching between channels, when the control voltage changes from 0 to 60 V. The disadvantage of this design is the high voltage (up to 60 V) that must be supplied to the control electrodes. In addition, the difference in attenuation in different acoustic channels does not exceed 20 dB, even if the effective electromechanical coupling coefficient changes by a factor of 10. In this case, complete switching occurs only in one acoustic channel, and in the other acoustic channel, switching occurs in a frequency band less than the filter bandwidth. Both bands may contain signals with the same amplitudes, which are not acceptable. In addition, since the semiconductor film is 0.5 µm thick, it can introduce significant attenuation to the SAW propagation at frequencies above 1000 MHz, which means that the operating frequencies of such a tunable filter are limited.
2 Theoretical Study of Switchable/Tunable Filter on SAW The purpose of this research is to develop a new design and calculate a switchable filter on surface acoustic waves that eliminates the above shortcomings. Theoretical studies of a SAW switch containing a vanadium dioxide film were carried out, based on our experimental studies, synthesis processes and properties of VO2 thin films on YX/128° cut LiNbO3 piezoelectric substrate (piezoelectric guide).
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Fig. 1. Switchable two-band filter on surface acoustic waves, top view.
The result of this study will be an increase in the difference in attenuation values between acoustic channels to 70 dB and a decrease in the control voltage to 0.1 V. A switchable two-band filter, based on surface acoustic waves (Fig. 1), contains a piezoelectric sound guide 1 of a rectangular shape, on the working surface of which, on one side, a series-connected first 2 and second 3 input IDTs are formed, having the same period. Two multistrip directional couplers (MSCs) 4 and 5 in the form of interdigitated structures (IDS) are located on the same longitudinal axis. MSC 4 has in the middle part an electrical contact with an acoustic channel switch, made in the form of a transverse strip of VO2 film 6 with a TiO2 sublayer, formed on the working surface of the piezoelectric sound guide 1. The multi-strip directional coupler 5 has in the middle part an electrical contact with another switch of acoustic channels, made in the form of a transverse strip of VO2 film 7 with a TiO2 sublayer, formed on the working surface of the piezoelectric sound guide 1. On the other hand, couplers 4 and 5 on the piezoelectric sound guide 1 form two output interdigital transducers 8 and 9 with different central frequencies. The location of multi-strip directional couplers 4 and 5 is clearly shown in Fig. 1. The interdigital structure buses the multistrip directional coupler 4 are connected through a
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Fig. 2. Switchable two-band surface acoustic wave filter, top view, where horizontal lines indicate acoustic channels 1, 2, 3, 4.
limiting resistor 10 to the control voltage source 11. The IDS MSC buses 5 are connected through a limiting resistor 12 to the control voltage source 13. The first MSC 4, interacting with the acoustic channel film switches 6, forms the first and second parallel acoustic channels (Figs. 1 and 2). The second MSC 5, interacting with the film switch 7 of the acoustic channels, forms the third and fourth parallel acoustic channels, as depicted by the lines in Fig. 2. The bus of the first input IDT 2 is connected to the filter input through a matching inductance 14, all IDTs are made unidirectional with internal reflectors [9]. The first input IDT 2 and the first output IDT 8 have the same period of the interdigitated structure. The period of the interdigital structure of the first MSC 4 is from 0.5 to 1.5 periods of one of the IDTs of the first acoustic channel. The period of the second MSC 5 is from 0.5 to 1.5 periods of the input IDT 3 of the third acoustic channel. The width of the vanadium dioxide film 4 and 7 is 1–2 periods of the input IDT 3, and the thickness of the vanadium dioxide film is 100–200 nm. The period of the first input IDT 2 is the same as the period of the first output IDT 8 and differs from the same periods of the second input IDT 3 and the second output IDT 9. These periods are determined by the center frequencies of the lower and upper passbands of the switched filter. A switchable two-band filter on surface acoustic waves works as
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Fig. 3. Amplitude-frequency characteristics (frequency responses) of a switchable two-band filter on SAW, measured in the absence of control voltage on MSC 4 and 5; curves 1, 2, 3, 4 – frequency responses of the filter in the low-frequency band; in the high-frequency band; MSC 4 in the first and second acoustic channels; MSC 5 in the third and fourth acoustic channels.
Fig. 4. Frequency responses of a switchable two-band filter on SAW, measured when the coupler in the low-frequency band is open (the control voltage is not applied to the high-frequency band coupler, but the control voltage is supplied to the low-frequency coupler); curves 1, 2, 3, 4 – frequency response of the filter in the lower frequency band; in the high-frequency band; in the 1st acoustic channel; in the 2nd acoustic channel (when voltage IDS 4 is supplied with voltage, but IDS 5 has no voltage), respectively.
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follows. The signal applied to the filter input is fed to input IDTs 2 and 3, connected in series through inductance 14 to the filter input to compensate for the influence of the impedance of IDT 2 or IDT 3, if the input signal is in one of the filter bands. This leads to a decrease in insertion attenuation, since the voltage drops across the capacitive component of the IDT, whose operating frequencies are outside the frequency band of the input signal, the impedance is capacitive in nature. Next, SAW excited by input IDTs 2 and 3 enter acoustic channels 1 and 3 and go to IDS 4 and 5 (Figs. 1, 2), which transfer SAW to acoustic channels 2 and 4. If the IDS buses are not supplied with the voltage necessary to prevent the interdigitated structures 4 and 5 from driving the SAW from one acoustic channel to the other, they drive the SAW to the other acoustic channel. The SAW ends up on the output IDTs 8 and 9 and signals appear at their outputs (see Fig. 3).
Fig. 5. Frequency responses of a switchable two-band filter on SAW, measured when the coupler in the high-frequency band is open and the low-frequency band is closed (the control voltage is not applied to the low-frequency band coupler, but the control voltage is supplied to the highfrequency coupler): curves 1, 2, 3, 4 – frequency response of the filter in lower frequency band; in the high-frequency band; in the 1st acoustic channel; in the 2nd acoustic channel (when voltage is applied to IDS 5, but not to IDS 4), respectively.
If the control voltage from voltage source 11 is applied to IDS 4, then the vanadium dioxide film 6 heats up, a phase transition occurs, its resistance drops sharply and IDS 4 almost stops transferring SAW from acoustic channel 1 to acoustic channel 2, and the signal level is at the output of IDT 8 drops by 70 dB. In this case, the film on IDS 7 does not heat up since voltage source 13 does not supply the required voltage to the IDS 5 buses and IDS 5 transfers SAW from acoustic channel 3 to acoustic channel 4 and a signal appears at the IDT 9 output (Fig. 4). If the voltage, required for the IDS to transfer the SAW from one acoustic channel to other, comes from voltage source 13, the IDS 5 will not transfer the SAW from acoustic channel 3 to acoustic channel 4. The IDS 4 buses are not supplied with the necessary
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Fig. 6. Frequency responses of a switchable two-band filter on SAW, measured when both couplers are closed (control voltage is applied to the couplers): curves 1, 2, 3, 4 – frequency responses of the filter in the lower frequency band; in the high-frequency band; the coupler, located in acoustic channels 1 and 2; the coupler, located in acoustic channels 3 and 4.
voltage from source 11, the signal appears at the IDT 8 output, and at the IDT 9 output the signal is suppressed by 70 dB. This happens in the same way as in the previous case, since, due to heating, the resistance of film 7 drops sharply and IDS 5 does not transfer SAW from acoustic channel 3 to acoustic channel 4 (Fig. 5). If control voltages from sources 11 and 13 are supplied to the IDS buses, then IDS 4 and 5 do not transfer SAW from one acoustic channel to another and the signals at the output of IDT 8 and 9 drop by 70 dB (Fig. 6). Frequency response, IDS and MSC in the third and fourth acoustic channels are also similar in the second and first acoustic channels, if the voltage U = 0.1 V is applied to IDS MSC 4, and there is no voltage on the IDS MSC 5 buses. In contrast to the design that is closest in essence to the proposed one, the coupler is not controlled on the base of a change in the electromechanical coupling coefficient due to the applied control voltage, but due to the vanadium dioxide film changes its resistance in the region of the phase transition, which leads to the short circuit of the coupler electrodes. This, at a certain film resistance, stops transferring SAW from one acoustic channel to other and transfers almost completely SAW to other acoustic channel if the film resistance is greater than a certain value. The voltage of sources 11 and 13 is controlled by the operator or a special control device according to a specific algorithm, depending on the problem being solved, in which frequency channels can be added. A surface acoustic wave incident on an multistrip coupler can be represented as symmetrical and asymmetrical modes [10, 11]. In this case, the asymmetric mode does not interact with it, since it induces equal opposite currents in them, which are mutually compensated.
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The symmetric mode perceives these IDS as one partial IDT with an aperture equal to the sum of the IDT apertures in different acoustic channels. In this case, at the output of this IDT, the SAW experiences an additional phase shift, which can be determined from the transmission coefficient of the SAW under the IDT. Let us assume that the IDS has only 1 period. The IDT SAW then receives an additional phase shift at its output, provided Ga < < ωC T : ϕ = Ga/ωCT ,
(1)
where Ga is the active component of IDT conductivity, Ga = 8f0 k 2 CT N
sin X 2 X
0 , X = π N f −f f0 ,
(2)
where f 0 is the central frequency of IDT (acoustic synchronous frequency), C T is the capacitance of partial IDT, N is a set of pairs of electrodes of partial IDT (N = 1), k 2 is a square of the electromechanical coupling coefficient. If there are M Ssuch IDS, then the total phase shift of the symmetric mode will be: sinX 2 4 . (3) ϕtotal = k 2 NM π X Then the frequency response of such a coupler at the output of a channel, where input IDT is absent, has the form: 2 2 2 sin X NMk H (f ) = sin . (4) π X At the output of the channel, where the input IDT is located, the SAWs are shifted by 90° in respect to the other channel and the frequency response of the coupler has the form: 2 2 2 sinX NMk H 1(f ) = cos . (5) π X It is obvious that at ϕ total = π the symmetric and asymmetric modes will be in antiphase and the SAWs will completely transfer from one acoustic channel to other. At the center frequency of IDT, X = 0, and the function (sin X)/X = 1. Then for 100% SAW transition from one acoustic channel to other, NM = π 2 /(4k 2 ) = N 0 M 0 . In this case, at the center frequency in the channel, where the input IDT is located, SAW will not hit the receiving IDT, and in another acoustic channel, SAW emitted by the input IDT will hit the receiving IDT. If a load in the form of conductivity Y is connected to one IDS, then the phase shift of the symmetrical mode on it will have the form: ϕ=
Ga · ωCT (ωCT )2 + Y 2
.
(6)
From this expression, it follows that the phase shift will depend on the magnitude of the load and will be less, the greater the load of capacitive conductivity IDS (ωCT ).
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Then, with a fixed number of IDS electrodes, when there is a 100% transition of SAW from one acoustic channel to other at zero load (Y = 0), it may turn out that SAW will almost not transit to other acoustic channel, if Y > > ωCT . Provided that Y < > ωC T and surface acoustic waves pass through the coupler without passing into a parallel acoustic channel, since in this case H( f ) ≈ 0, H 1 ( f ) ≈ 1; and H 1 ( f ) ≈ 0, H( f ) ≈ 1 at Y > ωC T to Y