Foam Extrusion - Principles and Practice 1-56676-879-9

Citation preview

FOAM EXTRUSION

Principles and Practice Edited by

S.-T. Lee, Ph.D.

Sealed Air Corporation

CRC PR E S S Boca Raton London New York Washington, D.C. © 2000 by CRC Press LLC

Library of Congress Cataloging-in-Publication Data

Main entry under title: Foam Extrusion: Principles and Practice

Full Catalog record is available from the Library of Congress

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher.

The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2000 by CRC Press LLC Originally Published by Technomic Publishing No claim to original U.S. Government works International Standard Book Number 1-56676-879-9 Library of Congress Card Number 00-104702 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

To the Lord who gives us life, abundant life, and even eternal life.

© 2000 by CRC Press LLC

Contents

Foreword Preface Acknowledgments List of Contributors 1.

INTRODUCTION SHAU-TARNG LEE

1.1

Thermoplastic Foam

1.2

Foam Extrusion

1.3

Recent Developments

1.4

Outline of the Book

1.5

References

2. STATISTICAL THERMODYNAMICS OF GAS SOLUBILITY IN POLYMERS ROBERT SIMHA and PIERRE MOULINIÉ

2.1

Introduction

2.2

Thermodynamics

2.3

Statistical Thermodynamics

2.4

Methodology

2.5

Discussion

2.6

Outlook

© 2000 by CRC Press LLC

2.7

Nomenclature

2.8

References

3. RHEOLOGY OF THERMOPLASTIC FOAM EXTRUSION PROCESS RICHARD GENDRON and LOUIS E. DAIGNEAULT

3.1

Introduction: The Importance of Rheology in Foaming Processes

3.2

Shear Rheology of Blowing Agent-Charged Polymeric Systems

3.3

Extensional Rheology for Extrusion Foaming of Polymers

3.4

Conclusion

3.5

References

4. FOAM NUCLEATION IN GAS-DISPERSED POLYMERIC SYSTEMS SHAU-TARNG LEE

5.

4.1

Introduction

4.2

Equilibrium Considerations

4.3

Conventional Nucleation Theories

4.4

Cavitation

4.5

Foam Extrusion Nucleation

4.6

Summary

4.7

Nomenclature

4.8

References

FOAM GROWTH IN POLYMERS N. S. RAMESH

5.1

Introduction

5.2

Importance of this Study

5.3

Literature Review

5.4

Foam Growth Experiment

5.5

Foam Growth Modeling

5.6

Foam Growth Equations

5.7

Boundary Conditions

5.8

Comparison of Theory with Experiment

5.9

Conclusions

5.10

Nomenclature

5.11 References

© 2000 by CRC Press LLC

6.

POLYMERIC FOAMING SIMULATION: BATCH AND CONTINUOUS MASAHIRO OHSHIMA

6.1

Introduction

6.2

Batch Foaming

6.3

Continuous Foaming

6.4

Conclusions

6.5

Nomenclature

6.6

References

7. PROCESS DESIGN FOR THERMOPLASTIC FOAM EXTRUSION LEONARD F. SANSONE

8.

7.1

Introduction

7.2

High-Density Structural Foam Process

7.3

Low-Density Foam Process

7.4

Die Design Procedures for Foam Extrusion

7.5

References

FOAM EXTRUSION MACHINERY FEATURES WILLIAM C. THIELE

8.1

Preface Regarding Extruders for Foaming

8.2

Basic Properties of Extruders

8.3

Basic Unit Operations in Foam Processes

8.4

Extruder Types, Support Devices, and Where Subprocesses are Placed

8.5

General Extruder Observations

8.6

References

9. MIXING DESIGN FOR FOAM EXTRUSION: ANALYSIS AND PRACTICES CHI-TAI YANG and DAVID I. BIGIO

9.1

Introduction

9.2

Thermoplastic Foam Extrusion Processes

9.3

Mixing—Theories and Experiments

9.4

Mixing Practices in Single- and Twin-Screw Extruders

9.5

Process Challenges

9.6

Summary

© 2000 by CRC Press LLC

9.7

Nomenclature

9.8

References

10. FOAMING AGENTS FOR FOAM EXTRUSION THOMAS PONTIFF

10.1

Introduction

10.2

Physical Foaming Agents

10.3

Chemical Foaming Agents

10.4

References

11. CONTINUOUS PRODUCTION OF HIGH-DENSITY AND LOW-DENSITY MICROCELLULAR PLASTICS IN EXTRUSTION CHUL B. PARK

11.1 Introduction 11.2 Previous Studies on Batch and Semicontinuous Microcellular Processing 11.3 Background on Microcellular Plastics Processing 11.4 Formation of a Single-Phase Polymer/Gas Solution 11.5 Microcellular Nucleation Control 11.6 Suppression of Cell Coalescence 11.7 Promotion of Large Volume Expansion 11.8 Experimental Set-Up 11.9 Experiments and Discussion 11.10 Summary and Conclusions 11.11 Nomenclature 11.12 References 12. FOAM EXTRUSION OF POLYETHYLENE TEREPHTHALATE (PET) MARINO XANTHOS and SUBIR K. DEY

12.1

Introduction

12.2

Review of Pet Chemistry and Processing Characteristics

12.3

Foaming With Physical Blowing Agents

12.4

Foaming With Chemical Blowing Agents

12.5

Concluding Remarks

12.6

Acknowledgements

12.7

References

© 2000 by CRC Press LLC

Foreword

I

AM pleased to introduce this volume “Foam Extrusion: Principles and Practice” to the growing list of excellent technical monographs by the Technomic Publishing Company. Dr S.T. Lee, the editor of and one of the main contributors to this volume is an acknowledged original research contributor in the area of Extrusion Foaming Mechanisms, as well as a leading industrial practitioner of Foam Extrusion. We at the Polymer Processing Institute have, through our long and active involvement in this field, witnessed the impressive growth of foam extrusion both as an engineering discipline and industrial practice. In our view this present book represents the maturing of Foam Extrusion as a significant processing technique. The editorial approach and contents of this work place appropriate emphasis on the fundamental phenomena of gas dissolution in polymers and its effects on melt rheology and on the complex and still not fully elucidated dynamic processes of foam nucleation and growth. With this background it attends to important aspects relating to the effects of the screw and the design as well as process variables on foam extrusion. The contributions under the heading of “Practices” are technologically both informative and significant, as they treat comprehensively specific materials, process and products. This volume will undoubtedly be used widely by and serve Foam Extrusion research, development and production practitioners. At the same time it will find its way in the list of important reference texts for graduate courses in polymer processing and structuring of polymeric materials and products.

Costas G. Gogos President, Polymer Processing Institute Professor Emeritus, Stevens Institute of Technology Research Professor, New Jersey Institute of Technology © 2000 by CRC Press LLC

Preface

T

texture of matter has been modified technologically since ancient times. Human survival depended on softening rice and other foods using yeast, water, and heat, to make them compatible with digesting tissues. The effect of these techniques was to create expanded structure, which is widely found in many natural systems, ranging from tree pulp to marine organisms. With the emergence of plastics engineering, it was recognized that expanded structure could be readily achieved in polymers, especially in thermomorphology reversible plastics. Since early in the twentieth century, polymer synthesis was greatly upgraded to enhance polymer structure, and methods have been developed to make broad ranges of polymeric products. Industrial foams were developed near the middle of this century. This book brings dissimilarly natured foaming and extrusion under one cover, in other words, it puts scientific principles and engineering practice together. Starting with fundamentals, gas molecules in polymers, then moving to separation, gaseous voids in polymers, scientific foundations are laid in such a way that the microscopic transition from nuclei to a void (nucleation) and the macroscopic movement from a void to an object (formation) are plausibly addressed. However, the detailed mechanism of converting from a dispersed to a growth state is not fully explained, but the proposed path in this book brings forth insights into this interesting area. Indeed, one of the underlying theories of gaseous inflation even engages the attention of cosmologists seeking to describe the very early stage of the Big Bang. Together with the science of foaming, this book continues into another popular technology, extrusion. The art of processing to match the foaming requirements is addressed from the operation perspective: processing and shaping. The last section of this book presents interesting foam extrusion HE

© 2000 by CRC Press LLC

developments, showing how scientific findings can be applied to the engineering field, from principles to batch experimentation to continuous foaming. An understanding of foaming principles and how to apply them to engineering practice will benefit industrial and academic readers in building a coherent and solid confidence in foam extrusion so challenges can be approached proactively in the new millennium. This book can also be used as a supplementary textbook for a graduate polymer, engineering, or science majors course.

© 2000 by CRC Press LLC

Acknowledgments

P

rofessor J. A Biesenberger of Stevens Institute of Technology, who passed away in January, 1998, showed encouragement and offered valuable advice even during his later stage of illness, and is particularly remembered. Professor C. B. Park of the University of Toronto, a chapter contributor, is acknowledged for his invaluable collaboration in critical decision making in 1996. I would like to thank all the contributors and reviewers for their efforts in preparing the manuscripts, and providing useful comments to make this book the best possible. My special thanks go to Dr. David Todd of the Polymer processing Institute, Dr. Paul Handa of the Canadian Research Council, Dr. N. S. Ramesh of Sealed Air Corporation, and Dr. Marino Xanthous of the New Jersey Institute of Technology, for their help during the editing of this book. I appreciate my employer, Sealed Air Corporation, and their support of this undertaking, especially Mr. Donald Tate. My secretary, Sandy Porporino, demonstrated superb in making faithful contacts throughout the editing of the book. To my wife, Mjau-Lin, her unconditional support is beyond what words can express. I thank God for this precious opportunity to not only work with a group of technical experts who provided direct and indirect support to make this book possible, but also to learn to humble myself in acknowledging how much yet needs to be explored.

© 2000 by CRC Press LLC

List of Contributors

David I. Bigio Department of Mechanical Engineering University of Maryland College Park, MD 20742

Shau-Tarng Lee Sealed Air Corporation 301 Mayhill St. Saddle Brook, NJ 07663

Louis E. Daigneault IPEX, Inc. Port of Montreal Building Wing 3, 1st Floor, Cité du Havre Montreal, Quebec H3C 3R5 Canada

Pierre Moulinié Bayer Corporation Corporate Polymer Research 100 Bayer Road Pittsburgh, PA 15205-9741

Subir K. Dey Polymer Processing Institute GITC Building, Suite 3901 New Jersey Institute of Technology Newark, NJ 07102-1982 Richard Gendron National Research Council for Canada Process Development 75 de Mortagne Boucherville, Quebec J4B 6Y4 Canada

© 2000 by CRC Press LLC

Masahiro Ohshima Department of Chemical Engineering Kyoto University Kyoto 606-8501 Japan Chul B. Park University of Toronto Department of Mechanical Engineering 5 King’s College Rd. Toronto, Ontario M5S 1A4 Canada

Thomas Pontiff Techmer PM 299 Huntersridge Road Winchester, VA 22602

William Thiele American Leistritz Extruder Corp. 169 Meister Avenue Sommerville, NJ 08876

N. S. Ramesh Sealed Air Corporation 10 Old Sherman Tpk. Danbury, CT 06810

Marino Xanthos Polymer Processing Institute GITC Building, Suite 3901 New Jersey Institute of Technology Newark, NJ 07102-1982

Leonard F. Sansone Sussex Plastics Engineering P.O. Box 152 Andover, NJ 07821 Robert Simha Dept. of Macromolecular Science Case Western Reserve University Cleveland, OH 44106

© 2000 by CRC Press LLC

Chi-Tai Yang Sealed Air Corporation 301 Mayhill St. Saddle Brook, NJ 07663

CHAPTER 1

Introduction SHAU-TARNG LEE

1.1 THERMOPLASTIC FOAM

F

OAM can be defined as a gaseous void surrounded by a much denser continuum matrix, which is usually in a liquid or solid phase. It exists widely in nature, in cellulositic wood, marine organisms, and other phenomena, and it can be made using synthetic processes (i.e., foamed plastics). The presence of gas voids can be outside, encapsulation, or inside, irreversible volume expansion. In most cases, a gas phase possesses dramatically different properties and structures (or states) than the surrounding solid phase, as opposed to different property and similar structure (or state) blends, to make a lighter heterogeneous composite structure [1]. A material property and density chart is shown in Figure 1.1 [2, 3]. It seems to follow a linear band in the log–log scale. Foamed material evidently extends the solid property lower limit. When very tiny voids are evenly dispersed in the solid matrix without seriously disrupting its continuity, the parent property hardly varies, even with less weight. As cell size, quantity, and its distribution varies, a much different composite property spectrum can thus be established. In other words, foamed material’s performance/weight ratio can markedly vary from foam-free material. Polymer synthesis and processing have shown dramatic improvements since the mid twentieth century. Foaming methods started to be transferred from lab scale to industrial scale. Foamed plastics have thus been used in many applications. Table 1.1 shows some established markets and foamed plastic attributes. Although the foamed plastic industry is highly fragmented, its demands continue to grow as indicated in Table 1.2, quoted from the 1997 Freedonia market report [4, 5]. In 1996, over six billion pounds (three metric tons) of synthetic foamed plastics

© 2000 by CRC Press LLC

FIGURE 1.1 Log-log of thermal conductivity and Young’s modulus vs. material density. (Data collected from Cellular Solids, Gibson & Ashby, Pergamon Press, 1988, and Properties of Polymers, Van Krevelen, Elsevier, 1990.)

were consumed in the United States alone. Apart from urethane foam, it was slightly over three billion pounds. A constant growth between 3 to 4% in quantity and over 6% in sales has been projected for the U. S. and the global market into the twenty-first century. Nowadays, it is almost unlikely to live without encountering foamed plastics directly or indirectly on a daily basis. Although it is a small portion of the market compared with the nonfoamed plastics industry, as illustrated in Table 1.3 [6], it may have a great future if its potential is fully tapped. Foamed plastics can be classified in different ways, for instance, by nature as flexible and rigid, by dimension as sheet and board, by weight as low density and high density, by structure as open cell and closed cell, and by cell size as foam and microcellular. In essence, standard nomenclature for foam including cell structure, density, and materials, such as from IUPAC, is extremely desirable to minimize communication confusions. In any event, its bulky nature limits it from being used for extensive transportation to make local produce more economical. It certainly develops specific niches geographically to make effective communication more critical for this already diversified industry. In this respect, technicality appears to be a common ground and, thus, a good starting point for creating a strong foundation from which the foam industry can grow toward a more prosperous future. Moving to technical domain, the type of polymer, the type of blowing agent, the expansion technique, and the post-foaming curing dictate foam formation and its morphology, and, thereof, the properties. It is not surprising to find that

© 2000 by CRC Press LLC

TABLE 1.1

Functions

Markets for Foamed Plastics.

Markets

Cushioning

Furniture Transportation Construction Construction Automotive

Insulations

Protection

Packaging

Strength/weight

Athletic Construction Marine, Medical Decoration Household Automotive Athletic Thermoforming Packaging Electrical

Impact Absorption Thermal/Chemical Electrical

TABLE 1.2

Attributes

Polymers

Energy absorption

Flexible PU PE, ABS

Low thermal conductivity Sound absorption Soft and flat surface Cushioning Strength and softness

Rigid PU, PS, PE Rigid vinyl

Sharp energy absorption Thermal strength Chem. and electrical inertness

RIM PU, PS bead, PE and PP sheet RIM PU, x-linked PE, PS, PVC, flexible PU Phenolics, Acrylics bead PP, x-linked PE PS, X-linked PE Flexible vinyl Epoxy, Silicones Rubber

U.S. Foamed Plastics Demand: Past and Future (in Millions). % Annual Growth

Item

1987

1996

2001

96/87

01/96

Total Foamed Plastics Demand (lbs) Urethanes Polystyrene Other Polymers Foamed Plastics Demand ($)

4,558

6,325

7,420

3.7

3.2

2,363 1,316 879 6,850

3,325 1,676 1,324 12,100

3,910 1,900 1,610 16,200

3.9 2.7 4.7 6.5

3.3 2.5 3.6 6.0

Source: The Freedonia Group, Inc. (1997) [3]. Note: Global consumption of foamed PS was 3.8 billion lbs (1.9 million tons) in 1996 [4].

TABLE 1.3

PE PS PP PVC PF Others Total

1993 Foamed Resin Consumption Ratio in Japan (Tons). Unformed

Formed

Formed/Total

954,780 600,245 866,782 1,313,514 28,302 843,636 4,607,259

50,006 270,037 14,429 8,713 64 41,782 385,031

5.0% 31.0% 1.6% 0.7% 0.2% 4.7% 7.7%

Source: Plastics Age, 40, Dec. (1994) [5].

© 2000 by CRC Press LLC

foaming method and foaming formula are closely related. Even with a fixed polymer/gas system and the same foaming method, the amount of voids (blowing agent percentage), their dispersion and distribution, and void interconnection have profound impact upon its properties and thereby applications. While focusing on foaming, the independent variables are gas type, gas content, processing conditions, and foaming dynamics, and the dependent parameters are foam structure, morphology, and properties. Although deep foaming understanding is still not readily available, some earlier publications laid down a good foundation. For instance, a solid technical perspective was attempted by C. J. Benning in Plastic Foams, published in 1969 [1]. An extensive collection of thermoplastic foaming technology can be found in Polymeric Foams [7] and Thermoplastic Foams [8]. Both are handbooks in nature. The former covers the existing foaming technologies classified by materials, the latter is based on general foaming processes and provides good coverage of mechanical and chemical details of conventional foam processing. The major weight of foamed plastics is polymer. Its long chain structure and, in certain cases, functional groups, display unique viscoelastic property versus temperature characteristics as illustrated in Figure 1.2 for polystyrene

FIGURE 1.2 Er(10), stress measured at 10 seconds after constant strain, vs. temperature for crystalline polystyrene, for amorphous atactic polystyrene samples A (Mn ⫽ 140,000) and C (Mn ⫽ 217,000), and for lightly cross-linked atactic polystyrene. Shaded area is ideal for foam processing. (Adapted from A. V. Tobolsky, Properties and Structure of Polymers, John Wiley and Sons, New York, 1960)

© 2000 by CRC Press LLC

Thermoplastic and Thermoset General Comparison.

TABLE 1.4

X-links

Morphology

Melt Viscosity

Structure

Thermoplastic

No

Thermally reversible

Thermoset

Yes

Thermally irreversible

Arrhenius dependency Independent of temp

Amorphous to Crystalline Amorphous

TABLE 1.5

Common Foaming Technologies and Relevant Polymers. Technologies

Applicable Thermoplastics

Extrusion Molded Beads Injection Molding Reactive Injection Mechanical Blending

PS, PVC, PE, PP, PVOH PS, PP, PE ABS, PC, PPO PU, UF PU, UF, Elastomer

[9], in which application and processing ranges are determined. Regarding polymer’s thermal response, it is categorized as thermoplastic and thermoset as listed in Table 1.4. Due to its thermal reversible structure, thermoplastic materials became quite favorable in making common goods through various plasticating processes. By considering polymer’s chemical nature, including reactive functional groups and solvent compatibility, a reactor or a processor can be selected and set to accommodate the necessary reaction or processing reTABLE 1.6

History of Foam Extrusion [9–17].

Time

Authors or Companies

1931 1941

Munters, G. and Tandberg, J. G. Johnson, F. L.

1944

Dow Chemical

1948

Colombo, R. (L. M. P.)

1962

Rubens, L. C., et al

1966

Boutillier, P. E.

1967

L. M. P.

1972

Parrish, R. G. (DuPont)

1990

Shell/Petlite

© 2000 by CRC Press LLC

Contents Foamed Polystyrene Foamed Polyethylene Extruded Styrene Foam Twin-Screw Processing Extruded Ethylene Foam PVC Foam Extrusion (Celuka Process) Twin-Screw Foaming Extruded Propylene Foam PET Foam Extrusion

Reference US patent 2,023, 204 US patent 2,256, 483 [8] [9] US patent 3,067, 147 Fr. patent 75594 BP 1,184,688 It. patent 795,793 BP 1,152,306 US patent 3,637, 458 [17]

quirements so that adequate reactive processing can be accomplished. In the past decades, material and machinery evolution showed amazing mutual benefits for fostering polymer processing industries. Table 1.5 summarizes available foaming methods. In styrene, olefin, and vinyl foam produce, extrusion appears to be the primary technology, due not only to its continuous nature, but also to its capability of handling thermoplastic material’s thermal reversible characteristics. It thus became a common practice to implement gas phase into polymeric melt for foaming. A brief foam extrusion history is presented in Table 1.6 [10–18]. It appears that the major foundation was laid in the 1960s in the U.S. and Europe.

1.2 FOAM EXTRUSION Foaming plastics has been developed as an extension of the extrusion application while extruder evolution is primarily based on its function optimization. Figure 1.3 shows the foamed ethylene setup prepared in 1941 [12]. Since the 1950s, the extruder has been recognized not only as inherently effective in converting thermal energy and mechanical power into processing heat for polymer phase change, but also as being efficient in generating adequate positive pumping force for fast material transport. Considering thermoplastic’s melting, molding and forming nature, the plastic extruder turned out to be an excellent processing unit for converting thermoplastics into simple geometry products. Mixing and cooling were well implemented in the extruder, and it started to meet the critical processing conditions for foaming and has been widely adopted for production since the 1970s. History shows a solid synergy from thermoplastic foaming and extrusion. Extrusion can be designed in such a way that various functions become possible in a single processor, including reactive extrusion, devolatilization, x-linking, etc. Furthermore, screw design can be tailored to match polymer processing characteristics (shear and thermal sensitivities) to make the extruder a very useful processing unit. From a physical chemistry viewpoint, foam extrusion is simply a change of states and mass transfer as illustrated in Table 1.7. However, considering kinetics, degree of state change and residence time for adequate mixing make extruder design and processing intriguing areas for scientists and engineers of various disciplines. As for foaming, gas phase injection, mixing, and dispersion undoubtedly add complexity to thermoplastic extrusion. Figure 1.4 shows the relevant mechanisms for thermoplastic foam extrusion, in which both thermal activation (chemical blowing agent) foaming and gas dissolution (physical blowing agent) foaming are included. Since polymeric melt is heavily dependent on processing history, downstream foaming cannot be viewed without paying close attention to upstream kinetics. Also pointed out in Figure 1.4 are the polymer processing and foam stabilization issues [19]. When low-density

© 2000 by CRC Press LLC

FIGURE 1.3

© 2000 by CRC Press LLC

Foamed polyethylene process (U. S. Patent 2,256,483).

TABLE 1.7

States and Conditions in Foam Extrusion Process.

Location

Prior Extrusion

In Extrusion

At Die Tip

Materials

Resin, Blowing Agent Feeding

Gas/Melt

Gas/Polymer

PostExtrusion Air/Polymer

Melting, Mixing, Cooling Molten High Pressure High Temperature

Foaming

Aging

Gas/Melt Low Pressure High Temperature

Gas/Solid Low Pressure Low Temperature

Mechanisms State Conditions

Solid, Liquefied Low Pressure Low Temperature

foam and/or high-speed operation are imminent, the operation window narrows. The balance of mechanical energy input and thermal energy transfer and the balance of heating and cooling are critical concerns for an effective and efficient low-density foam extrusion. In other words, developing a melt/gas solution in the right foaming range as illustrated in Figure 1.5 is a key

FIGURE 1.4 Thermoplastic foam extrusion characteristics for amorphous, semicrystalline, and crystalline structures.

© 2000 by CRC Press LLC

FIGURE 1.5

© 2000 by CRC Press LLC

Foam extrusion units and their mechanisms.

TABLE 1.8

Type Long Single Extruder L/D*: 38–42

Tandem Extruders L/D*: 24–32 and 28–30

Common Extrusion Processes. Advantages

• Less leaking point • Less investment • Independent melting/ cooling control

• Can process high-melt

Disadvantages

• Narrow melting/cooling • • • • •

control Precise screw design Long screw length More leaking points High investment More power consumption

polymers Twin-Screw Extruder L/D: around 25

• Easy to control • Good mixing • Good heat transfer

• Cooling limited • Narrow melting/cooling range

* L/D: Screw length/Barrel diameter from [19]

subject in foam extrusion [20]. A desirable morphology can thus be developed for applications. Plastic properties, system parameters, processing setup, foaming, and posthandling consist of the necessary elements of a professional foam extrusion technology. They should be viewed in a balanced way to avoid erroneous analysis. Nonetheless, as plastic synthesis and processing techniques continue to grow, so does the foaming technology and, thereof, the number of applications of thermoplastic foams. Table 1.8 shows common extrusion processes for foaming. Each extrusion process, of course, has its strengths that benefit certain operations and its weaknesses that heighten business and technical concerns. Basically, the optimal extruder/polymer combination remains a challenge in some existing industries and is definitely a challenge in developing new foaming technology. A proper understanding of polymers, including concepts such as melting, flow, foaming, and forming, facilitates machinery design optimization and product property evaluation. On the other hand, an adequate knowledge of extruders can minimize unnecessary errors in selecting the right type and size of machinery during lab-to-plant or batch-to-continuous scale-up.

1.3 RECENT DEVELOPMENTS A chemical blowing agent or a physical blowing agent or both are needed to generate gas phase for expansion at a lower pressure in foam extrusion. A physical blowing agent is generally preferred for foam under 0.2 g/cm3, termed low-density foam. Before the 1980s, CFC was preferred, mainly because of its soluble, volatile, and nontoxic nature. However, its stability and reactivity with ozone raised substantial concern about ozone depletion. The

© 2000 by CRC Press LLC

Montreal Protocol was signed in 1987 and became effective in 1989. It essentially called for global cooperation to phase out halogenated hydrocarbons manufacturing to minimize the continued ozone layer damage. According to AFEAS (Alternative Fluorocarbons Environmental Acceptability Study), the major producers reduced CFC production from 980 metric tons in 1986 to 95 metric tons in 1996 [21]. There was no doubt that a variety of gas operation technologies were affected. Foaming involved product exposure to the atmosphere and was classified as an open system, as opposed to refrigeration and air-conditioning that are closed systems, and it is definitely one of most concern. Besides, since the early 1990s, global warming known as the greenhouse effect, became another major issue. The Intergovernmental Panel on Climate Change (IPCC) reported scientific assessment on warming potentials of halogenated hydrocarbons relative to carbon dioxide. Moreover, regulations were proposed to reduce the emission of not only fluorine-based, but also carbon dioxide and methane volatile chemicals at the Kyoto conference in 1997. Since the late 1980s, foam producers in the developed countries began concentrated research and development efforts to switch to friendly alternatives in support of protecting the ozone layer. Most of them found substitutes for halogenated blowing agents in the early 1990s. As a result, the foam industry not only survived the challenge but continues to constantly grow, and enhance the foaming knowledge basis. However, considering warming potentials, volatile carbon dioxide faces not only technical hurdles but also legislation concerns. For general foam industries, other than ozone and warming issues, legislation and revisions from environmental agencies and safety and health agencies remain viable issues in almost every country. Disposal, waste stream control, and usage of recycled plastics still require a deep understanding of foaming technology in order to continue to enjoy the status of being user and public friendly. Since the mid 1980s, the Massachusetts Institute of Technology (MIT) has presented a series of papers on the microcellular foaming concept, stemming from the industrial challenge on improved performance/weight to enhance polymer value. Although early works focused on batch process, it demonstrated nice foam structure under 5 ␮m cell with supercritical carbon dioxide blowing agent [22], which created a deep interest from industry and academia. Foaming method, cell structure, and morphology have been intensively investigated by several institutes, and lately, Trexel reported success in commercial scale tests for polystyrene foam sheet. For the first time in foam history, it attracted a wide and deep dedicated effort from academia and industry. Up to 1998, over twenty Ph.D. dissertations directly related to microcellular foaming have been published worldwide. The number continues to increase. As a result, some interesting insights into the mysterious foaming, fine cell and flat sheet with volatile blowing agent, have been shed. Continued commitment is still

© 2000 by CRC Press LLC

necessary to assure more success in commercializing developed foam technology and in developing new foam technology. As polyolefin resin producers successfully made liner low-density polyethylene via metallocene catalyst to claim better control on monomer distribution and molecular weight distribution [23], another opportunity to enhance foaming technology has arisen by tailoring resin structure to fit extrusion processes for optimal foaming. Improved foam properties can thus be envisioned by eliminating the “slack” portion of semicrystallines. Moreover, ethylene-styrene interpenetrating (ESI) polymers were made with the same catalyst technology to pose the possibility of making foam with “combined” benefits from each monomer to widen its application window by using one polymer [24]. Penetrating into the existing market becomes a real test for this developmental project. Concerted efforts are necessary to succeed in this fast-paced society. High melt strength polypropylene is another example for foaming. It was first presented by Himont [25], in which, by chain extension, semicrystalline polypropylene demonstrates a wider processing window and a much better foaming structure over the nonextended conventional PP. As a result, more polypropylene research and development has been initiated. It is anticipated to have more grades of better structured PP in the market for foaming. Recently, a polyethylene terephethalate (PET) resin supplier reported solid state polymerization technology enhancement to allow improved polymer strength PET for foaming [26]. In brief, resin structure development opens interesting possibilities to further enhance foamed plastic strength/weight performance. Extruder manufacturers have attempted to make bigger and better extruders. Bigger tandem lines and twins with improved screw design to minimize heat generation without sacrificing mixing and pumping capabilities are expected to enhance the foaming window by enlarging exit dimension to make more pressure drop with less shear heat generation possible. As a result, bulkier foamed product can be made for bulky and advanced applications. In essence, material and machine need close association to make useful synergies for foaming.

1.4 OUTLINE OF THE BOOK This book is divided in four parts, starting with the fundamentals: Chapters 2 and 3 on thermodynamics and rheology of the gas/polymer system. The next section, Chapters 4 to 6, is focused on foaming including nucleation, growth, and their correlation with experimental and operational observations. Machinery issues are covered in Chapters 7 through 9, in which process and die design, mixing design and practice, and twin-screw foam extrusion are adequately addressed. Finally, applications are stressed in Chapters 10 through 12. A blowing agent overview is given and microcellular foam extrusion and poly-

© 2000 by CRC Press LLC

ester foam extrusion are well discussed. Hopefully, this comprehensive coverage will generate further insights on the complex foam extrusion subject to allow engineers to move into extensive applications with confidence and to allow scientists to view a wider horizon. When we get to fundamentals, we have to confess that a sizable gap exists in understanding the detailed mechanisms at the molecular level and applying them properly in foam extrusion for a friendly production with business sense. Thermoplastics’ viscoelastic properties vary depending upon the amount of blowing agent dissolved and the surrounding temperature and pressure. Phase separation occurs when surrounding conditions change to a point where gas phase starts to appear and continues to grow until it is balanced by polymeric tension and adjacent cells. The latent heat, bubble expansion work, and melt elasticity [27] are involved in a nonisothermal condition. Moreover, extrusion makes the already complex and dynamic foaming, nucleation, growth, and coalescence, even more complex. It is my sincere hope that this book will serve as a valuable step in advancing the understanding of true foaming problems for experienced foam experts and of how to apply the fundamentals in foaming for newcomers.

1.5 REFERENCES 1. C. J. Benning, Introduction in Polymeric Foams, Wiley-Interscience of John Wiley and Sons, New York, 1969. 2. L. J. Gibson and M. F. Ashby, of Cellular Solids: Structure & Properties, Pergamon Press, Elmsford, New York, 1988. 3. D. W. Van Krevelen, p. 533 of Properties of Polymers, Elsevier, New York, 1990. 4. “Foamed Plastics,” pub. Freedonia Group, Inc., Cleveland, Ohio, 1997. 5. W. D. Back, “Foamable Polystyrol-EPS,” Kunststoffe, 86, 10, 1996. 6. Plastics Age, 40, Dec. 1994, presented by Y. Kitamori, “Foamed Polyolefin Process Development,” in Thermoplastic Foam Conference sponsored by Ind. Tech. Res. Ins., Taipei, Taiwan, 1995 7. D. Klempner and K. C. Frisch, editors of Polymeric Foams, Hanser, New York, 1991. 8. J. L. Throne, Thermoplastic Foams, Sherwood, Hinckley, Ohio, 1996. 9. A. V. Tobolsky, Properties and Structure of Polymers, John Wiley and Sons, New York, 1960. 10. G. Munters and J. G. Tandberg, “Heat Insulation,” U.S. patent 2,023,204, 1935. 11. F. L. Johnson, “Synthetic Spongy Material,” U.S. patent 2,256,483, 1941. 12. R. N. Kennedy, “Extruded Expanded Polystyrene,” Section XII of Handbook of Foamed Plastics, ed. by R. J. Bender, Lake, Libertyville, IL, 1965. 13. M. Martelli, “Twin Screw Extruders—A Separate Breed,” SPE Journal, 27, 25–30, 1971. 14. L. C. Rubens, J. D. Griffin and D. Urchick, “Process of Foaming and Extruding Polyethylene Using 1,2-dichlorotetrafluoroethane as the Foaming Agent,” U.S. patent 3,067,147. 15. P. E. Boutillier, “Extrusion of Plastics Material,” French patent 75594, appl. in 1966, British patent 1 184 688, 1970.

© 2000 by CRC Press LLC

16. L. M. P. “Process and Apparatus for Manufacturing Foamed Articles of Thermoplastic Materials,” Italian patent 795,793, appl. in 1967 and British patent 1,152,306, 1969. 17. R. G. Parrish, “Microcellular Foam Sheet,” U.S. patent 3,637,458, 1972. 18. Y. Kitamori, “PET Foams and Applications,” Proc. of Thermoplastic Foams Technical Conference, spon. by Industrial Technology Research Institute, Taipei, Taiwan, 1995. 19. S. T. Lee, “A Fundamental Study of Thermoplastic Foam Extrusion with Physical Blowing Agents,” Ch. 13 of Polymeric Foams: Science and Technology, ed. by K. C. Khemani, Amer. Chem. Soc. Symposium Series 669, 1997. 20. K. D. Kolossow, Chap. 13: Extrusion of Foamed Intermediate Products with Single-Screw Extruders in Plastics Extrusion Technology, ed. by F. Hensen, Hanser, New York, 1988. 21. AFEAS, “Production, Sales and Atmospheric Release of Fluorocarbons through 1996,” AFEAS Program Office, Washington, D. C., 1998. 22. J. E. Martini-Vvedensky, N. P. Suh and F. A. Waldman, “Microcellular Closed Cell Foams and Their Method of Manufacture,” U.S. patent 4,473,665. 23. S. Lai and G. W. Knight, “Dow Constrained Geometry Catalyst Technology (CGCT): New Rules for Ethylene (a-olefins Interpolymers-Controlled Rheology Polyolefins,” 51st Ann. Tech. Conf. sponsored by Soc. Plas. Eng. Preprint 1188–1192, 1993. 24. S. V. Karande and B. I. Chaudhary, “INSITE Technology Based Ethylene Styrene Interpolymers for Foams Applications,” 1–5, FoamPlas ’98, sponsored by Schotland Business Research, Inc., New Jersey. 25. M. B. Bradley and E. M. Phillips, “Novel Foamable Polypropylene Polymers,” 48th Ann. Tech. Conf. sponsored by Soc. Plas. Eng. Preprint 717–720, 1990. 26. H. Al Ghatta, “Process for the Production of High Molecular Weight Polyester Resins,” U. S. patent 5,376,734, Dec. 1994. 27. C. Sagui, L. Piche, A. Sahnoune and M. Grant, “Elastic Effects in the Foaming of Thermoplastics,” Physics Review E, 58, 4, 4654–4657, 1998.

© 2000 by CRC Press LLC

CHAPTER 2

Statistical Thermodynamics of Gas Solubility in Polymers ROBERT SIMHA PIERRE MOULINIÉ

2.1 INTRODUCTION

L

EGISLATION against the use of blowing agents with ozone depletion poten-

tial has forced many foam producers to change their processes. Because gas solubility influences the nucleation and growth of cells in foams, it is one of the primary concerns when changing physical blowing agents. To that end, recent experimental efforts have focused on the measurement of solubility of gases in molten polymers at high temperatures and pressures often encountered during foam extrusion. As a complement to experimental data being reported for solubility at these conditions, efforts have also been made to make use of equations of state (EOS) to model gas solubility. This work reports on the use of the Simha-Somcynsky theory to model gas solubility. Much like Flory-Huggins or Sanchez-Lacombe theories, the Simha-Somcynsky theory stems from treating molecules as segments on a lattice. In the case of a mixture, the lattice contains both species, which are divided into nearly equally sized segments, as illustrated in Figure 2.1 [1]. Unlike the other theories, however, the Simha-Somcynsky theory allows for a pressure- and temperature-dependent fraction of vacancies or holes that are to express freevolume within the lattice, which account for molecular disorder in the lattice model. Moreover, the equations derived from the Simha-Somcynsky theory include temperature- and pressure-independent parameters that account for intra- and intermolecular interactions within the mixture’s components. The Simha-Somcynsky theory has been successfully applied to model various fluids and fluid mixtures [2]. Recently, the theory was applied to treat the solubility of gases in molten polymers [3, 4]. In this work, the application of the

© 2000 by CRC Press LLC

FIGURE 2.1 Mixture of two molecules placed on a lattice. Both have been decomposed into segments (black circles: four segments, white circles: three segments). Unoccupied sites are also placed on the lattice in Simha-Somcynsky theory.

theory to gas solubility in molten polymers is illustrated. Various trends observed theoretically and experimentally are also discussed.

2.2 THERMODYNAMICS The equilibrium between two phases 1 and 2 at a specified temperature and pressure is determined by the equality of the chemical potentials ␮i1 and ␮i2 for all constituents i in the two phases, in the presence of other constituents j, defined as follows [5]: ␮ik ⫽ ( ⭸Gk>⭸nik )P,T, nik; j ⫽ i, k ⫽ 1, 2

© 2000 by CRC Press LLC

(1)

where Gk is the Gibbs free energy of phase k, and nik is the number of moles of constituent i in phase k. In our case, the vapor phase 1 contains the single component 1, and ␮11 is simply the Gibbs free energy of the vapor. This quantity is to be balanced by ␮12, the chemical potential of the condensed vapor dissolved in the polymer. The functional dependence of both potentials on T and P and of ␮12 additionally on composition then determines the equilibrium composition, i.e., the solubility of the vapor. For ␮12 we have the following relation [5]: ␮12 ⫽ Gm ⫹ (1 ⫺ x1)⭸Gm/⭸x1

(2)

with x1 the mole fraction of component 1 and Gm the free energy of the mixture. Conventionally, other composition units are employed in the present context. For example, employing the volume of gas over the volume of substrate, referring to STP conditions, we have the following: S ⫽ x1/(1 ⫺ x1)(22,400/p)(M2VS2)⫺1

(3)

where M2 and Vs2 are the molar mass and the specific volume, respectively, of the polymer. Our task, therefore, is to develop explicit expressions for the Gibbs free energies of the vapor and the mixture. This is what statistical thermodynamics is expected to accomplish.

2.3 STATISTICAL THERMODYNAMICS 2.3.1 THE VAPOR The following equation is convenient for obtaining the chemical potential using equations of state: ⬁

⌬␮ ⫽ ␮ ⫺ ␮o ⫽

冮 (p ⫺ RT/V)dV ⫹ (pV ⫺ RT)

(4)

V

Where ␮o is the chemical potential of gas when treated ideally. Many equations of state for gases allow the evaluation of the integral in Equation (4) to obtain ⌬␮ for a gas using an equation of state. Once the chemical potential difference has been obtained, the actual chemical potential of the gas can be calculated by adding the ideal state contribution, ␮o. For a molecule with three volume-dependent degrees of freedom and mass m, the chemical potential ␮11 can be determined from: ␮11 ⫽ RTᐉn e

© 2000 by CRC Press LLC

p(Nah)3 f ⫹ ⌬␮ kT(2␲mRT)3/2

(5)

The logarithm term, containing Planck’s constant h and Avogadro’s number Na, clearly represents the chemical potential of an ideal gas [6]. We note the underlying assumption that rotational and vibrational degrees of freedom in a polyatomic vapor molecule are not perturbed by intermolecular interactions. We pursue this point in the next section.

2.3.2 THE CONDENSED PHASE: SINGLE CONSTITUENT To model the structure of a melt, consider a lattice. Flexible polymer chains are assumed to meander through this lattice such that each of their segments executes thermal motions in a cell formed by its neighbors and located in average positions that are defined by the lattice sites. These motions are subject to the restraints of intersegmental attractions and repulsions. This is the cell model originally devised by Lennard-Jones and Devonshire for rare gas-type liquids and extended by Prigogine [7] to fluids of chain molecular constituents. This model, one will argue, possesses too much order for disordered fluids. On the other hand, a lattice description offers mathematical advantages in connection with the intrinsically difficult problem of the dense, disordered state. In an attempt to retain the lattice picture as far as possible, Simha and Somcynsky [1] formulated the lattice-hole model. That is, a volume (or pressure) and temperature-dependent fraction h ⫽ 1 ⫺ y of lattice sites is unoccupied in order to simulate disorder. Thus, additional entropy arises from the mixing of occupied sites and holes. The characteristic elements of the model include the intersegmental or intermolecular interactions and chain flexibility, where appropriate. This latter feature is to acknowledge the fact that a flexible chain is capable of “soft” (low frequency) internal motions in addition to “stiff” bond and bond angle deformations. Such soft motions are subject to intermolecular, and thus, volume-dependent perturbations. These elements are accounted for by an assumed 6–12 LennardJones pair potential involving a maximum attraction energy ␧* and a repulsion volume ␷* defined by the location of the maximum. Chain flexibility is to be characterized by a constant factor 3c, which represents the number of volume-dependent degrees of freedom [7]. For a linear C-C backbone “s-mer” with unrestricted internal bond rotation, 3c ⫽ s ⫺ 3 ⫹ 6, which includes the six translational and rotational motions of the chain as a whole. For a real chain with more or less complex substituents, c becomes a disposable parameter, expected to be of the order of s [see Equation (6)]. In terms of these quantities, we can define a characteristic temperature T*, volume V*, and pressure P* as follows: T* ⫽ qze*/(ck); V* ⫽ Na␷*/m; P* ⫽ qze*/s␷*

© 2000 by CRC Press LLC

(6)

with qz ⫽ s (z ⫺ 2) ⫹ 2, V* a specific volume for a molar segmental mass m, k the Boltzmann constant, and z ⫽ 12 the lattice coordination number. In terms of these quantities, reduced variables of state can be defined. The partition function formalism of statistical mechanics evaluates the model to yield a free energy and an EOS. We refer to Reference [1] for details and state the result, expressed in reduced variables, viz: 苲 苲 /T 苲 ⫽ (1 ⫺ ␩)⫺1 ⫹ 2yQ2(1.011Q2 ⫺ 1.2945)/T 苲 PV

(7)

苲 with Q ⫽ (yV)⫺1; ␩ ⫽ 2⫺1/6yQ1/3. 苲 苲 苲 Equation (7) is incomplete as a P⫺V⫺T relation, since it contains the quan苲 苲 tity y. This quantity is expressed as a function of V and T by the physical condition that in thermodynamic equilibrium, y must minimize the Helmholtz free energy for a specified volume and temperature. This results in the following equation: s/3c[(s ⫺ 1)/s ⫹ y⫺1ln(1 ⫺ y)] ⫽ (1 ⫺ ␩)⫺1(␩ ⫺ 1/3) 苲) ⫹ yQ2(2.409 ⫺ 3.033Q2)/(6T

(8)

苲 苲 The solution of Equation (8) yields y as a function of V and T. We observe that for large s and provided s/3c is constant, the coupled Equations (7) and (8) 苲苲苲 contain only scaled variables of state. That is, a universal PVT surface ensures; a principle of corresponding states is satisfied. Equations (4) and (5) illustrate the significance of the EOS for the chemical potential and, ultimately, the solubility of the vapor. The same applies to the condensed phase. Clearly, here the scaling parameters defined in Equation (6) must be known for the particular polymer and condensed vapor pair in order to proceed. These are to be obtained by the superposition of the theoretical scaled 苲 苲苲 PV T onto the experimental PVT surface. The numerical accuracy of the theory and the values of T*, P*, and V* have been discussed in detail for some 50 polymers by Rodgers [8]. Over a pressure range of maximally 2,000 bar, the overall average deviation ⌬V is found to be ⫾7 ⫻ 10⫺4 cm3/g and only ⫾4 ⫻ 10⫺4 for a range of 500 bar. However, before these evaluations could be performed, something had to be done about the factor 3c/s appearing in Equation (8). We observe from Equation (6) the connecting relation as follows: (P*V*/T*)m ⫽ (c/s)R, with R ⫽ Nak

(6⬘)

There are three “intrinsic” quantities, i.e., the number s of segments in the chain, the molar segmental mass m, and the flexibility factor c, where s ⭈ m is the molar mass of the molecule. This leaves a freedom of choice. If the

© 2000 by CRC Press LLC

segment is identified with the chemical repeat unit, then s is given for a specified molar mass, and c becomes an adjustable parameter, to be determined together with the scaling constants to yield the best fit, consistent with Equation (6⬘). As an alternative, we may adopt a priori a value for c/s, e.g., 1/3 for a long chain. Both procedures have been applied in the past and, it should be noted, will not affect the quantitative success of the theory. In the present context, the second and more convenient method will be employed. The molar mass m in Equation (6⬘) then indicates the size of the segment yielding one volumedependent degree of freedom. For a complex structure, such as a polycarbonate, this will be but a fraction of the monomer unit. As for the small species, the balance of chemical potentials requires the input of the c-value used in the vapor phase. For a rare gas, we have s ⫽ c ⫽ 1. In a polyatomic species, there are additional rotational and vibrational degrees of freedom. These would require a consideration of intermolecular, volume-dependent perturbations and thus a detailed consideration of the molecular structure. The inherent difficulties make this impracticable and beyond the scope of the present theory. We assume instead that the molecule is a unit with three external degrees of freedom. It is characterized then for our purposes by the intermolecular parameters ␧* and ␷*, see Equation (6). The kinetic term then becomes identical with the first temperature factor in Equation (5). 2.3.3 THE CONDENSED PHASE: MULTICONSTITUENT The generalization of the foregoing results has been obtained making the assumption of random mixing [2]. The scaled EOS, Equation (7), and the minimum condition, Equation (8), in scaled coordinates retain their validity, and the parameters become averages over the composition. The new elements are the cross interactions between unlike species to be added to the self interactions. Specifically, for a binary system we obtain [2] the following: 具s典 ⫽ x1s1 ⫹ x2s2; 具c典 ⫽ x1c1 ⫹ x2c2 The repulsion volume 〈␷*〉 and the maximum attraction 〈␧*〉 of the mixture are given in terms of the self, 11 and 22, and cross interactions, 12, are given by the equations 具␷*典2 ⫽ B4/B2; 具 e*典 ⫽ B22/B4 and * *2,4 2 * *2,4 B2,4 ⫽ X21e*11␷*2,4 12 ⫹ 2X1(1 ⫺ X1)e12␷12 ⫹ (1 ⫺ X1) e22␷22

© 2000 by CRC Press LLC

(9)

with X 1 ⫽ x1q1z/8qz9 . The quantity 〈qz〉 is the average of qz, defined by Equation (7) and the average 〈s〉. Equation (9), combined with the definitions, Equation (6), applied to constituents and to the mixture averages, determines the EOS of the mixture. Equation (9) can be read in two ways. Provided the EOS of the mixture and constituents has been measured, all interactions can be extracted. On the other hand, if information about the cross-interaction parameters is available or assumed, the EOS of the mixture can be computed. It is the latter route we must take here. The theory of intermolecular forces suggests the geometric mean proposition e*12 ⫽ ␦e(e*11e*22)1/2

(10)

with ␦e close to unity. Moreover, we set *1/3 3 ␷*12 ⫽ ␦v(␷*1/3 11 ⫹ ␷22 ) /8

(11)

This represents an arithmetic average of lengths, when ␦v is unity. Thus, when the EOS data for the polymer and the small component in the vapor and condensed phases have been evaluated, ␦e and ␦v are the only disposable parameters. We recall the extensive amount of polymer data at hand [8]. Also, for a number of gases, the virial coefficients over ranges of temperature have been measured.

2.3.4 THE FREE ENERGY OF THE MIXTURE For the free energy Gm, we have the following expression [3, 9]: Gm/RT ⫽ x1ln x1 ⫹ (1 ⫺ x1)ln(1 ⫺ x1) ⫹ ln(y/s) ⫹ s(1 ⫺ y)ln(1 ⫺ y)/y ⫹ (s ⫺ 1)ln [e/(z ⫺ 1)] ⫺ c[ln(v*/Na) 苲 ⫹ (1 ⫺ ␩)3/Q] ⫹ cyQ2(1.011Q2 ⫺ 2.409)/2T (12) 3 3 2 2 ⫺ c1x1ln[2␲m1RT/(Nah) ]⫺ c2(1 ⫺ x1)ln[2␲m2RT/(Nah) ] 2 2 苲 /s ⭈ m ⫹ c[(1 ⫺ ␩)⫺1 ⫹ 2yQ2(1.011Q2 ⫺ 1.2045)/T For simplicity, we have omitted the average symbol in s and ms and note that 具s典 具m典→s ⭈ m ⫽ x1s1m1 ⫹ x2s2m2

© 2000 by CRC Press LLC

We recognize in Equation (12) in the first two lines the contributions from the entropy of mixing holes and sites occupied by the two constituents, followed by equation of state terms, and finally, kinetic energy contributions. The thermodynamic prescription ␮11 ⫽ ␮12, with Equations (5), (2), and (12), then yields the equilibrium mole fraction x1 and finally the solubility S, Equation (3). The lengthy expression for ␮12 has been derived [10]. However, it is more convenient to proceed by numerical differentiation of Equation (12).

2.4 METHODOLOGY It is initially necessary to obtain the scaling parameters (i.e., T*, P*, V*) for the gases and the polymers, since SS equations of state are required for each component of a given mixture. The scaling parameters were determined by fitting Equations (8) and (9) to experimental liquid densities, with T* and V* as dependent variables. Nonlinear least squares fitting can be done with commercially available software packages. Saturated vapor-liquid density data were used for determining the scaling parameters for CO2[11]. Scaling parameters for the polymers were adapted from literature values reported by Rodgers [8] or determined with experimental p-V-T data. As required by the theory, the polymer segment sizes were adjusted such that the molar repulsion volumes of the segments (v*) matched those of the gas molecules: m1v*1 ⫽ m2v*2

(13)

This often leads to segment sizes that are smaller than the polymer repeat unit. The chemical potentials of the gaseous phase can be determined with equations of state. An example of such a calculation is described for HFC 134a in research by McLinden et al. [12] The chemical potential of the gas and the scaling parameters were then entered into computer programs developed to solve Equation (12) for a given pressure and temperature. Thus, solubility curves can be generated by performing such calculations for several temperatures and pressures.

2.5 DISCUSSION 2.5.1 COMPARISON BETWEEN EXPERIMENT AND THEORY Once the equations of state for the mixture were developed, the disposable parameters ␦e and ␦v were adjusted to reconcile the theoretical equations with experimental data. Experimental data reported by Sato et al. were used to fit the mixture equations for CO2 [13]. Values of 1.065 were both obtained for ␦e

© 2000 by CRC Press LLC

FIGURE 2.2 Solubilities of carbon dioxide in polystyrene. Lines: theoretical calculations; Symbols: data reported by Sato et al. [13].

and ␦v. Figure 2.2 illustrates the computed CO2 solubilities along with experimental data. Except for a difference observed at higher pressures for solubilities at 373°K, the agreement between the experimental solubilities and theoretical predictions is very good. Furthemore, a decrease in CO2 solubilities with increasing temperatures is correctly predicted by the SS equations. The difference observed at higher pressures is due to the concavity of the computed solubility curve. At 453°K, however, the solubility curve is linear over the same pressure range. Figure 2.3 shows calculations for the solubility of HFC 134a in PS, previously reported by Simha and Xie along with experimental data reported by Daigneault et al. [4,14]. Interestingly, the solubility curves show similar behaviour to those for CO2, where the solubility curves become more linear as temperature increases. In the case of 134a, it was believed that this was due to the pressure approaching the saturated vapor condition. [4] Solubility predictions at 373°K for CO2, however, show that this is not the case, since it is well above its critical temperature (Tc for CO2 ⫽ 304.2°K). As will be shown later, it is believed that this is the result of competing temperature and pressure effects on computed solubility, which will be discussed later.

© 2000 by CRC Press LLC

FIGURE 2.3 Solubilities determined for HFC 134a in polystyrene at three different temperatures. Lines: Theoretical calculations; Symbols: Data reported by Daigneault et al. [14]. Figure adapted from Reference [4].

Figure 2.4 illustrates solubilities predicted for N2, CH4, CO2, and Ar in polybutadiene (PBD) at different temperatures, previously reported by Xie and Simha [3]. With respect to the critical temperatures of the gases, solubility decreases with decreasing critical temperature. Except for N2, a decrease in solubility is predicted with increasing temperature. This positive temperature dependence is also observed experimentally for N2 in natural rubber [15] and polystyrene [13]. These temperature dependences reflect competing efforts of reduction in polymer-solvent attractions and enhancement of free-volume available with increasing temperature. At this stage, a discussion as to the behavior of the mixture with changes in temperature and pressure is useful. Solubility does not only depend on the free-volume available, but also on the interaction between the gas and the polymer and interactions within the gas molecules. As the vapor pressures of a liquid increase with temperature, the tendency to reside in the condensed phase (polymer) decreases. Also, as greater pressures are exerted on the condensed phase, the free-volume tends to decrease. Ultimately, the dominant factor accounts for the temperature-pressure behavior. Hence, for N2, inter-

© 2000 by CRC Press LLC

FIGURE 2.4 Dependence of theoretical solubility (symbols) on temperature at 1 bar for carbon dioxide (CO2), methane (CH4), argon (Ar), and nitrogen (N2) in polybutadiene. Data obtained from Reference [3].

molecular interactions are low enough that solubility is sensitive to changes in the polymer. The temperature dependence of solubility has been noted to show positive and negative behavior for noble gases, with solubility decreasing at higher temperatures for Ar and Xe. Thus, molecular size also influences the pressure-temperature behavior of solubility [16]. Figures 2.2 and 2.3 show that at lower temperatures, the theory predicts an eventual decrease in free-volume due to the pressures exerted on the polymer liquid. In light of the solubility measurements for CO2 and HFC 134a, experimental data have yet to demonstrate the existence of a maximum in the solubility-pressure curves. 2.5.2 INFLUENCE OF EQUATION OF STATE PARAMETERS FOR CO2 The scaling parameters P*, T* and V* previously reported by Xie and Simha [3] were extracted from low pressure data between 5 and 7.5 bar, and an 苲 equation that allows for simplifications for low pressures (i.e., P ⬇ 0) [4]: 苲 ⫽ A(s, c) ⫹ B(s, c)T 苲3/2 ln V

© 2000 by CRC Press LLC

(14)

TABLE 2.1.

CO21 CO22

Scaling Parameters Determined for CO2.

m (g/mol)

V* (cc/g)

T* (K)

P* (bar)

44.01 44.01

0.586 0.623

2960 3043

9542 9227

1. Derived using all p-V-T data. 2. Derived from limited low T data.

Thus, linear regression of low-pressure V-T data yielded the scaling parameters for CO2. In this work, however, the scaling parameters obtained for CO2 made use of liquid p-V-T data between 217–304°K. The scaling parameters obtained by each procedure are compared in Table 2.1. Figure 2.5 compares the fit of predicted volumes to actual volumes at different temperatures and pressures. In both cases, the standard error of prediction for V is within 0.5%. The Xie-Simha parameters give a better fit in the low temperature region, but result in divergences above 290°K. Thus, the scaling parameters are sensitive to the data used to determine them. The solubilities

FIGURE 2.5 Specific volume of carbon dioxide as a function of temperature along the saturated vapor curve. Line: actual data [11]; Open symbols: using parameters reported by Xie and Simha [3]; Closed symbols: derived using complete saturated vapor curve.

© 2000 by CRC Press LLC

FIGURE 2.6 Solubilities for carbon dioxide in polystyrene determined with scaling parameters shown in Table 2.1; Open symbols: determined with parameters reported by Xie and Simha [3]; Closed symbols: this work.

computed at 373°K and 453°K from the two sets of scaling parameters are shown in Figure 2.6. Although differences are observed with respect to the calculated solubilities, the shapes of the diagrams are unaffected. The segment size for PS was set such that the molar volumes, v*, of the gas and effective polymer repeat units were identical. In this case, CO2 is roughly half the size of the chemical PS repeat unit. Because 3c/s has been kept fixed in the computations, segment sizes are decreased at the expense of flexibiity of the segments. Although the number of segment(s) depends on the total molecular weight of the polymer, Xie and Simha demonstrated that this effect becomes negligible at high molecular weights. In this work, the molecular weight was set to 124,800. The disposable parameters ␦e and ␦v are supposed to reflect specific interactions that may occur between the gas and polymer. In the case of ␦e, added attractions between the gas and the polymer are incorporated into the equations when ␦e is greater than one. Figure 2.7 illustrates solubility calculations with systematic changes in ␦e. A significant increase in solubility is observed as ␦e is increased. As expected, solubility increases with ␦e, indicating that attractive forces can increase the solubility of a gas agent in a polymer. Nevertheless, a

© 2000 by CRC Press LLC

FIGURE 2.7 at 373°K.

Effect of ␦e parameter on computed solubilities of carbon dioxide in polystyrene

15% increase in ␦e increases solubility at least sixfold, showing that the EOS are very sensitive to changes in these scaling parameters. We note that two measurements suffice to determine ␦e and ␦v. Allowance for T or p dependences of these parameters merely introduces empiricism into the theory. A more thorough understanding of the siginificance of these parameters will be possible as more experimental solubility data becomes available. 2.5.3 SOLUBILITY COMPUTATIONS WITH OTHER STATISTICAL-THERMODYNAMIC MODELS There are other statistical thermodynamic models available for modeling polymer liquids. A review by Rodgers [8] compares the predictions of several equations of state applicable to polymer melts. Several theories have been extensively used for mixtures, such as Sanchez-Lacombe. As with SimhaSomcynsky theory for gas solubility in polymers, the polymer-gas system is treated as a liquid mixture, whose EOS is derived by applying mixing rules to the scaling parameters of the pure constituents. A detailed discussion on the Sanchez-Lacombe EOS and the mixing rules for each scaling parameter when

© 2000 by CRC Press LLC

modeling gas solubility can be found elsewhere [17–19]. Among the mixing rules, a binary interaction parameter ␦ij is also included to reconcile computations with experimental data. Finally, solubility predictions are computed by equating the composition-dependent chemical potentials of the mixture and of the gaseous phase [17–19]. Good quantitative predictions of gas solubility have been demonstrated by Sanchez-Lacombe and Panayiotou-Vera theories for CO2 in PDMS or 1,1-difluoroethane in PS [17]. Good correlations were also obtained with Sanchez-Lacombe theory for N2 and CO2 in PS [13,20]. An important consideration when making use of the Sanchez-Lacombe model, however, is that the binary interaction parameter ␦ij is temperature dependent. Figure 2.8 shows solubility predictions using the Sanchez-Lacombe theory for CO2 in PS at different temperatures. These predictions are based on correlations found by Sato et al. with their experimental solubility data [13]. The correlations show that the binary parameter ␦ij must be determined for each solubility isotherm. Computations using Simha-Somcynsky theory shown in Figures 2.2 and 2.3, however, showed reasonable agreement with ex-

FIGURE 2.8 Solubilities computed with Sanchez-Lacombe theory for carbon dioxide in polystyrene at 373, 413, and 453°K, using parameters reported by Sato et al. [13].

© 2000 by CRC Press LLC

TABLE 2.2 Scaling Parameters Reported by Sato et al. (I) [13] and Kwag et al. (II) [20] for Solubility Computations with S-L Theory.

Substance

P* (bar)

␳* (g/cc)

T* (K)

CO2(I) CO2(II) PS(I) PS(II)

72.03 45.80 38.70 35.70

1.580 1.430 1.108 1.105

269.5 330.0 739.9 735.0

perimental data using single values for ␦␧ and ␦v. This is in the spirit of a selfconsistent theory. As shown earlier, vapor-liquid equilibria are often used for obtaining scaling parameters for the gas. Calculations using Simha-Somcynsky theory demonstrated that the data range used for determining scaling parameters has

FIGURE 2.9 Solubilities for carbon dioxide in polystyrene at 150°C computed with SanchezLacombe theory. Computations with scaling parameters listed in Table 2.2: parameter set I (solid line) and parameter set II (dashed line).

© 2000 by CRC Press LLC

an important influence on solubility computations. Table 2.2 compares two sets of scaling parameters reported for pure CO2 and PS, determined independently [13,17]. In the case of CO2, the difference in the scaling parameters is significant. The values for CO2(I) were determined with only a portion of the saturated vapor curve, below 269°K. As mentioned previously, it is desirable to maximize the data range of pure component data since it minimizes variations when extrapolating the pure component EOS to higher T and p conditions used for solubility computations. Figure 2.9 illustrates Sanchez-Lacombe computations for CO2 in PS at 150°C, using both sets of parameters listed in Table 2.2. The predictions with parameter set (I) are based on a ␦ij value of ⫺0.118, calculated with an empirical temperature dependence from the ␦ij values reported in Figure 2.8; predictions with parameter set (II) were done with a ␦ij value of 0.0855, reported in the literature [20]. These computations show that very different solubilities were obtained by each, demonstrating that there is still a need for developing accurate and consistent techniques for gas solubility measurements.

2.6 OUTLOOK The quantitative success of Simha-Somcynsky theory toward gas solubility in polymer melts suggests further experimental and theoretical directions. The most obvious issue facing the theory is the existence of the maxima predicted in the solubility isotherms. Several extensions of the theory are currently being explored. Among those of interest to foam process scientists are the treatment of blowing agent mixtures as well as foam aging properties. The issue of foam aging raises the issue of solubility in the glass-transition region (Tg) and in the glassy state, as well as the shift of the glass-transition temperature by the dissolved gas. Glassy-state solubilities are important when considering the permeation of gases into and out of a finished foam. Furthermore, the glassy state also exhibits the well-known features of formation history-dependent properties and time-dependent properties of the glassy state. Although conceptually more involved for glassy polymers, the Lattice-Hole theory has been applied to a polymer equation of state [21]. The nonequilibrium character is acknowledged in this treatment by the elimination of the equilibrium condition, Equation (8).

2.7 NOMENCLATURE ␮ik ␮o Na c

Chemical potential of component i dissolved in matrix k Chemical potential of an ideal gas Avogadro’s number Volume-dependent degrees of freedom

© 2000 by CRC Press LLC

s z h k ␧* G n p P* 苲 P y h V V* v* ␷* 苲 V R T T* 苲 T xi S ␦e ␦v

Number of segments dividing a molecule Coordination number for a given atom (z ⫽ 12 in this work) Planck’s constant Boltzmann’s constant Maximum attraction energy Gibb’s free energy Molar quantity Pressure Scaling pressure Reduced pressure Occupied volume fraction Free-volume fraction Volume Scaling volume (cc/g) Scaling volume (cc/mol) Measure of the segmental repulsion volume Reduced volume Gas constant Temperature Scaling temperature Reduced temperature Mole fraction of component i Solubility (expressed in grams of gas per gram of polymer) Correction constant for geometric mean averaging [see Equation (10)] Correction constant for arithmetic mean averaging assumption for length scales [see Equation (11)]

2.8 REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Simha, R. and T. Somcynsky. Macromolecules, 2 (1969) 342. Jain, R. K. and R. Simha. Macromolecules, 17 (1984) 2663. Xie, H. and R. Simha. Polymer Int., 44 (1997) 348. Simha, R. and H. Xie. Polymer Bull., 40 (1998) 329. Guggenheim, E. A. Thermodynamics. Amsterdam: 1959, North Holland Publishing Co., pp. 215–216. Kestin, J. and J. D. Dorfman. A course in Statistical Thermodynamics. New York: 1971, Academic Press, pp. 313. Prigogine, I. The Molecular Theory of Solutions. Amsterdam: 1957, North Holland Publishing Co. Rodgers, P. A. J. Appl. Polym. Sci., 48 (1993) 1061. Jain, R. K. and R. Simha. Macromolecules, 13 (1980) 1501. Nies, E., A. Stroeks, R. Simha, and R. K. Jain. Colloid Polym. Sci., 268 (1990) 731.

© 2000 by CRC Press LLC

11. Angus, S., B. Armstrong, and K. M. de Reuck, eds. International Thermodynamic Tables of the Fluid State:Carbon Dioxide. New York: 1976, Pergamon. 12. McLinden, M. O., J. S. Gallagher, L. A. Weber, G. Morrison, D. Ward, A. R. H. Goodwin, M. R. Moldover, J. W. Schmidt, H. B. Chae, T. J. Bruno, J. F. Ely, and M. L. Huber. ASHRAE J., 3282 (RP-588), p. 263 13. Sato, Y., M. Yurugi, K. Fujiwara, S. Takishima, and H. Masuoka, Fluid Phase Equilibria, 125 (1996) 129. 14. Daigneault, L. E., Y. P. Handa, B. Wong, and L.-M. Caron. Proc. SPE ANTEC ‘97 (1997) 1983. 15. Van Amerogen, G. J., Rubber Chem. & Tech., 37 (1964) 1065. 16. Curro, J. G., K. G. Honnell, and J. D. McCoy. Macromolecules, 30 (1997) 145. 17. Garg, A., E. Gulari, and C. W. Manke. Macromolecules, 27 (1994) 5643. 18. Sanchez, I. C. and P. A. Rodgers. Pure and Appl. Chem., 62 (1990) 2107. 19. Sanchez, I. C. and R. H. Lacombe. Macromolecules, 11 (1978) 1145. 20. Kwag, C., L. J. Gerhardt, V. Khan, E. Gulari, and C. W. Manke. ACS PMSE Preprints, 74 (1996) 183. 21. McKinney, J. E. and R. Simha. Macromolecules, 9 (1976) 430.

© 2000 by CRC Press LLC

CHAPTER 3

Rheology of Thermoplastic Foam Extrusion Process RICHARD GENDRON LOUIS E. DAIGNEAULT

3.1 INTRODUCTION: THE IMPORTANCE OF RHEOLOGY IN FOAMING PROCESSES

B

ECAUSE the production of extruded thermoplastic foams, such as foam sheets or boards, is largely linked to the rheology of the mixture of a polymeric matrix with a physical blowing agent that is kept dissolved until the melt is allowed to foam at the die exit, then Section 3.2 will deal mostly with the rheological behavior, in a shear field, of this one-phase system. The knowledge of the rheology of such polymer/dissolved blowing agent mixtures is critical because of the following:

(1) It improves our fundamental understanding of the process, and it is a valuable information source for the optimization of the processing conditions. Not all blowing agent/polymer pairs behave the same way with regard to their plasticization effect. The extent of the viscosity reduction can be reflected through the temperature at which the mixture can be processed, the lowering of the torque exerted on the machine, or the increase of the maximum throughput at which the extrusion line can be run. (2) It is a must for computer-assisted die design, since it is the key element for the numerical simulation of the flow. Viscosity results first should be translated into a rheological model compatible with the flow simulation software package. This model should take into account the effect of the many variables on the viscosity levels, such as type and concentration of the blowing agent, temperature, pressure, and shear-thinning behavior. (3) Since the foam plastic industry will be facing new HCFC regulations that will force it to look for replacements for the present ozone-depleting

© 2000 by CRC Press LLC

chemicals, the knowledge of the rheology of polymer/blowing agent systems is a key parameter in the development of new foam compounds. Not only should the blowing agent replacements yield adequate foam properties, but the new systems should typically exhibit the same processibility as their predecessors to limit the extent of equipment modifications. For example, it is not surprising to find an increased interest for carbon dioxide, CO2, as a potential replacement blowing agent because of safety and cost considerations. For this tailoring of foam product properties by new combinations of polymers and blowing agents, the extensional rheology of the polymer in the twophase system is critical as discussed in Section 3.3. A better understanding of the role that extensional rheology plays during the foaming process is a must because of the following: (1) Not all polymer resins are suitable for foaming, and a clear understanding of the prerequisites for foaming in terms of the rheological behavior should help in the design of alternative resins and optimization of existing ones. (2) Foam properties are largely linked to cellular morphology. Besides the nucleation stage, the following steps, cell growth and stabilization, which are governed largely by the extensional rheology, should dictate the final foam structure.

3.2 SHEAR RHEOLOGY OF BLOWING AGENT-CHARGED POLYMERIC SYSTEMS 3.2.1 RHEOMETRICAL CONSIDERATIONS Unfortunately, the rheology of mixtures of a polymeric matrix and a physical blowing agent cannot be directly measured on standard laboratory rheological equipment such as cone and plate or capillary rheometers. A closed pressurized rheometer is necessary because the system must be kept under pressure at all times in order to prevent degassing. Even so, capillary and slit die viscometers are the most frequently used apparatuses for the study of such systems of polymers and blowing agents. During the extrusion of the foam, the measurements can be conducted in-line (directly in the process stream) or on-line (a sampling stream is diverted from the process flow line and transferred to the measuring apparatus) using the rheometer attached at the end of the extruder. In some other cases where the measurements are conducted off-line, standard capillary viscometers can be modified to accommodate the pressure requirements.

© 2000 by CRC Press LLC

TABLE 3.1

Viscosity Calculations for Capillary and Slit Die Rheometers.

Geometry

Capillary (diameter Dc and length Lc; ⌬Pc is the pressure drop across the capillary)

Shear stress, j12

␴12 ⫽

Apparent shear rate, ␥᝽ app (at volumetric flow rate Q)

h P1 ⫺ P2 dc d 2(1 ⫹ (h>w)) L

(a)

6Q

(b)

32Q

␥᝽ app ⫽

␲D3c

w h2

n ⫽ ␦log ␴12>␦log ␥᝽ app

Power-law index, n Actual shear rate at the wall, ␥᝽

␴12 ⫽ c

⌬PcDc 4Lc

␥᝽ app ⫽

Slit (h is the height of the slit, w its width, L the distance between P1 and P2)

␥᝽ ⫽ a

Viscosity, ␩

3n ⫹ 1 ᝽ b ␥app 4n

␥᝽ ⫽ a ␩⫽

␴12 ␥᝽

2n ⫹ 1 ᝽ b ␥app 3n

(c) (d) (e)

For either slit or capillary rheometers, herein also referred to as dies, the viscosity measurements are derived from the pressure drop observed through the channel of a known geometry, at a given volumetric flow rate (see Table 3.1). It has been observed that, under given conditions of pressure and temperature, bubble nucleation may start inside the die as a result of insufficient blowing agent solubility at the concentration of interest [1, 2]. The formation of gas bubbles strongly affects the shear stress and thus the pressure profile along the die axis, and deviations from linearity are encountered (see Figure 3.1). Taking this problem into account, the following alternative protocols have been proposed: measurements based only on the linear portion of the pressure profile using several pressure transducers [3, 4], back pressure assembly such as a valve [5] or a gear pump [6, 7, 8] attached at the exit of the die and set to maintain the pressure inside the die above the critical pressure required for bubble nucleation, and the use of very long length-to-diameter (L/D) capillaries to minimize the relative contribution of the nonlinear portion of the pressure drop [9]. Off-line measurements require particular care to the sample preparation and to the loading of the sample into the apparatus, because the concentration of the blowing agent must be determined precisely and kept constant up to the measuring cell or die, while the pressure is maintained above the critical degassing value to prevent phase separation [10, 11]. A much simpler experimental protocol can be followed when conducting the measurements on an extrusion line: the extruder is first used to form a single-phase system from the two components, polymer and blowing agent, and second to pump the mixture

© 2000 by CRC Press LLC

FIGURE 3.1 Axial pressure profile along a die of constant cross section: beyond entrance effects, a linear pressure drop is followed by a deviation from linearity as bubbles are nucleating.

through the slit or capillary die, while maintaining the pressures at relatively high levels [5, 12]. Both single- and twin-screw extruders can be used to that purpose, provided that a single-phase solution is formed upstream from the measuring die; this will be a function of the mixing efficiency of the extruder as well as the residence or equilibrium time in the extruder. Some on-line measurements have also been obtained using a commercial slit die rheometer [6, 7, 8, 13]. The material flows from the process up a conduit into a gear pump as shown in Figure 3.2. The material is pumped via the inlet gear pump through the rheometer head and is returned to the process via the exit gear pump and the return conduit. The output from the process extruder is not affected. The instrument head is fitted with a die that is rectangular in cross section (a slit die). Three pressure transducers are mounted to be flush with the face of the die as shown in Figure 3.2. This instrument can be made to function as a constant stress or a constant rate machine. The pressure drop between the transducers P1 and P2 is measured, and the volume throughput to give a given stress or shear rate is calculated from the known rotational

© 2000 by CRC Press LLC

FIGURE 3.2 Schema of a commercial on-line rheometer (Rheometric Scientific). P1, P2 and P3 are the pressure transducers, and T1 and T2 are the thermocouples.

speed of the inlet gear pump and its capacity. A third pressure transducer is located near the exit of the slit die to control the absolute level of pressure within the slit channel. This happens through the manipulation of two independent gear pumps that control the flow in the measurement stream. With this kind of on-line rheometer, one should be aware that sampling delays must be expected, so a complete purge of the sampling conduits must be performed before any characterization test is conducted. 3.2.2 EFFECT OF THE BLOWING AGENT ON THE SHEAR VISCOSITY Literature is scarce on the rheological evaluation of mixtures of polymer resins with physical blowing agents. This may be attributed to the complexity of the experimental setup, as described in the previous section. Table 3.2 lists the conditions under which the rheological evaluation of systems in the literature has been made. This table includes only foamable systems based on a

© 2000 by CRC Press LLC

TABLE 3.2

Physical Foaming Agent Rheology Studies.

Blowing Agent Polymer

Type

LDPE LDPE LDPE LDPE LDPE LDPE LDPE LLDPE PP PP PP PP

R-12 R-114 R-114 R-12 ⫹ R-114 R-22 R-22 n-butane R-114 R-12 R-114 CO2 Phenol alcohol R-12 R-114 CO2 R-11 R-12 R-12 R-11 ⫹ R-12 R-134a R-134a R-142b R-152a CO2 CO2 CO2 Phenol alcohol

PB PB PDMS PS PS PS PS PS PS PS PS PS PS PS PS

Concentration (wt%)

T (°C)

Method

References

5–20 10–20 0–20 10–20 0–20 21 15 10–15 10 5–20 0–4.9 1–5

110–160 110–160 120 110–140 110 100 100 140–160 160–180 150–170 150–220 180–240

CD/E CD/E NM CD/E CD/E CD/PV CD/PV CD/E CD/E CD/E OLR MFI

3, 12 3, 12 14 3 15 16 16 12 12 12 7 17

10–20 10–20 0–20.7 10–15 5–15 0–12 5–15 0–4.3 0–15 0–15 0–10 0–5 0–4 0–3.7 1–5

140–160 150–170 50–80 150–170 140–170 135–246 140–170 150 130–156 114–175 150 150 220 150–200 180–240

CD/E CD/E PCR CD/E CD/E NM CD/E PCR OLR OLR PCR PCR CD/E OLR MFI

12 12 10, 18 4, 12 4, 12 14 4 11 6 6 11 11, 18 5 7 17

*Methods: CD/E: capillary die mounted on the extruder; CD/PV: capillary die mounted on a pressure vessel; PCR: pressurized capillary rheometer; OLR: commercial on-line rheometer; MFI: melt flow indexer; NM: not mentioned.

physical blowing agent. Chemical blowing agents are not included because most decompose to water or carbon dioxide or nitrogen or combinations of these [1, 9, 19, 20, 21, 22, 23, 24]. They often yield other compounds contributing to the polymer plasticization. The extent of their influence on rheology also depends on the fraction of decomposition. The key element of the viscosity response is that the dissolved gas or liquid induces plasticization of the polymer, which can be translated in terms of the apparent decrease of the viscosity. Typical viscosity results are presented in Figure 3.3(a) and Figure 3.3(b): these examples relate to mixtures of PS with

© 2000 by CRC Press LLC

FIGURE 3.3 Viscosity of PS with HCFC 142b: (a) at two nominal temperatures, 125°C and 150°C; (b) for different combinations of composition and temperature yielding similar viscosities.

HCFC 142b, at different temperatures and compositions, as indicated on the graphs. These results are representative of the plasticization behavior encountered in all investigated systems. That range of shear rates is also typical of such viscosity measurements using slit or capillary dies, spanning generally between 1 and 1000 s ⫺1. In this range of rates, most polymers’ viscosity behavior falls in, or near, the power-law region. Figure 3.3(a) illustrates the effect of composition on the viscosity level, at roughly constant temperatures (125°C and 150°C). The range of compositions investigated is such that the viscosity readings fall within the operating window of the rheological apparatus. For this reason, it is rare to find, for such experiments, results over a wide composition range at a constant temperature. Figure 3.3(b) illustrates this limitation by showing the plots of similar viscosity curves for different blowing agent concentrations (from 0 to 15 wt%) where the temperature was selected in order to maintain nearly identical viscosity levels. For these examples, the temperature varies from 187°C for the pure PS to 114°C when adding 15 wt% of HCFC 142b. Simple observations can be made from these two figures:

• •

At high levels of shear, the slopes on the graphs are identical. All curves thus exhibit the same power-law index. Globally, the shape of the curves remains approximately the same for any temperature or composition. From Figure 3.3(a), one can roughly estimate the degree of plasticization in terms of the viscosity reduction measured either at constant stress or at

© 2000 by CRC Press LLC

•

constant rate. Adding 2.5 wt% of HCFC 142b to PS lowers the viscosity by a factor of 4.0 at constant stress or by a factor of 1.6 at constant rate. The curves of Figure 3.3(b) suggest the equivalence principle that the incorporation of 2.5% of HCFC 142b to PS is equivalent to an increase of the temperature by roughly 10°C. This result may be linked to two interesting concepts: first, running at a concentration of 10 wt% of HCFC 142b, one would expect to be able to run the process at a temperature 40°C lower than that achievable with pure PS only; second, if we relate the viscosity reduction to the lowering of the mixture’s glass transition temperature, Tg, as we will do in an upcoming section, then the addition of HCFC 142b blowing agent can be translated into the change of the glass transition temperature of PS by approximately ⫺4°C per weight percent of HCFC 142b. An exact number is provided through a fine numerical analysis, a procedure detailed in the following pages.

These examples, and most of the results shown in the references listed in Table 3.2, raise the question of how the viscosity reduction resulting from the addition of a physical blowing agent should be presented and analyzed. Empirical evaluation of the viscosity reduction, as a function of the blowing agent concentration, could be reported in terms of apparent plasticization effect. Nevertheless, such reporting would not help our fundamental understanding of the plasticization behavior. In addition, it would not be possible to generalize these results with a formalism helpful in predicting the viscosity for new mixtures of polymers and blowing agents. Details regarding the way of reporting the results, their limits, their uses, and the extent of applicability, are given in the following sections. 3.2.2.1 Apparent Plasticizing Effect Evaluated at Constant Rate Rate-controlled viscosity measurements have been conducted by Han and coworkers on numerous systems of polymers and blowing agents (see Table 3.2). They chose to translate the degree of plasticization observed into a viscosity reduction factor (VRF) defined as follows: VRF ⫽

Viscosity of the mixture of molten polymer and blowing agent Viscosity of the molten polymer

(1)

This factor was calculated under set temperature and shear rate. Han et al. have reported the following [3, 4, 12]: (1) The value of VRF, for a given system of polymer and blowing agent, with set blowing agent concentration, was practically independent of the shear rate and temperature, within the range of measurements.

© 2000 by CRC Press LLC

(2) For a given system of polymer and blowing agent, the value of VRF decreases as the blowing agent concentration increases. (3) The VRF values differ for different polymers or blowing agents used. Figure 3.4 illustrates these empirical findings. One should be cautious about conclusions drawn from such constant rate experiments. Even though these results are very attractive due to their relative simplicity, and even if such experiments are seemingly conducted with ease while reflecting a production concern of constant throughput, attention should be paid to the conditions under which the data were obtained before making

FIGURE 3.4 Viscosity reduction factors, at constant rate, as a function of the concentration of blowing agent for various polymers. These results were obtained at temperatures of 150°C and 160°C [12].

© 2000 by CRC Press LLC

generalizations. Empirical findings based on larger experimental ranges may be more insightful in developing a global understanding of the rheological behavior of mixtures of polymers and blowing agents. First, the measurements should be made under the same conditions under which the foam will be produced: for instance, in the case of polystyrene, the addition of the physical blowing agent may yield viscosity reductions that can accommodate extrusion temperatures typically as low as 120°C. In this case, useful results should cover a broader temperature range to suit the conditions encountered throughout the process. This is particularly true for amorphous polymers. Then, at very low temperatures, the viscosity of the amorphous polymer phase alone, used as the reference, may be experimentally difficult to obtain, which precludes the general use of a VRF parameter. The lower limit for processing semicrystalline polymers is, however, still dictated by the temperature of crystallization, which is only slightly influenced by the presence of the blowing agent [25]. Second, most of the results obtained by Han et al. are limited to the powerlaw region, where the viscosity functions, straight lines on a log-log plot, are parallel. For this particular range of shear rates, where all the results belong to this power-law dependency, one can expect to find a constant ratio between the viscosity of the mixture and that of the pure polymer, irrespective to the shear rate. However, it can be anticipated that extrapolation to low shear rates, prior to the power-law regime but still within a pertinent rate range for foam applications, would yield erroneous estimates. Viscosity curves covering a broader range in shear rates should exhibit low-shear-rate Newtonian plateau regions, or at least some curvature in the viscosity curves that would tend to level off at low rate values. Classical viscosity scaling principles apply a double shift to both ␩ and ␥᝽ axes and do not reduce the scaling on a constant rate basis. Scaling at constant stress is thus the better procedure. 3.2.2.2 Apparent Plasticizing Effect Evaluated at Constant Stress The results shown in Figure 3.5(a) for the PDMS/CO2 system at 50°C [10] are a good illustration of the concentration-dependent viscosity curves encountered for physically foamable polymeric systems. The viscosity curves for different gas concentrations exhibit similar slopes in the power-law regions, and Newtonian plateau regions can be observed at low shear rates for mixtures containing a high level of the blowing agent. Because of the pronounced similarity between these curves, classical viscoelastic scaling principles are envisaged, with curve shifting performed at constant stress. Analogous to the timetemperature superposition shift factor aT, the concentration-dependent viscosity curves can be shifted according to a shift factor aC that will depend on the concentration of the blowing agent, while the temperature is maintained constant. This results in the master curve illustrated in Figure 3.5(b). Again,

© 2000 by CRC Press LLC

FIGURE 3.5 Viscosity curves for PDMS/CO2 at 50°C for different compositions: (a) viscosity versus shear rate; (b) same curves reduced to a master curve, using a composition-dependent shift factor aC [10].

this procedure necessitates sets of data obtained at a constant temperature over the range of interest of the blowing agent concentration. Mixtures of PS with various blowing agents at 150°C are compared in Figure 3.6(a). Results shown in Section 3.2.2.1, previously expressed on the basis of constant rate, have been translated here using the power-law index to conform to the constant stress criterion. Several observations can be made from this figure:

• •

•

There is an excellent agreement between the results obtained from different laboratories [6, 7, 11] using different techniques for the few systems reported by more than one author: notably, HFC 134a and CO2. The viscosity ratios reported on a constant stress basis span over a much larger scale than when reported on a constant rate basis. We might say that this represents the true nature and amplitude of the plasticization effect. The plasticization effect, reported as a function of the percent by weight of blowing agent, is more drastic for BA of low molecular weights: the largest variation in the viscosity is encountered for the carbon dioxide (MW ⫽ 44.0 g/mol), while the least pronounced effects are observed for fluorocarbons R-11 and R-12 (MW respectively of 137.4 and 120.9 g/mol). HCFC 142b and HFC 134a behave similarly, and they have close molecular weights (100.5 and 102.0 g/mol, respectively).

© 2000 by CRC Press LLC

FIGURE 3.6 Plasticization of PS expressed on a constant stress basis in terms of reduced viscosity (␩mixture/␩PS ⫽ a C): (a) for various blowing agents, at 150°C [6, 7 ,11, 12]; (b) with CO2 as the blowing agent, at different temperatures [5, 7, 11].

However, it can be anticipated that the plasticization at one particular temperature may not be applicable at a different temperature, especially for amorphous polymers like PS where the viscosity dependency to the temperature follows an exponential fit. This is illustrated in Figure 3.6(b), where the viscosity reduction of the mixture of PS with CO2 is given for different temperatures [5, 7, 11]. As the temperature is increased, the plasticization effect is less pronounced, thus indicating a significant effect of the temperature on viscosity reduction. A proper way of reporting the plasticization effect should be to isolate the contributions of the blowing agent and of the temperature in order to generalize at any condition. 3.2.3 FREE VOLUME AND GLASS TRANSITION TEMPERATURE 3.2.3.1 Theory for Concentrated Polymer Solutions A strong correspondence exists between the rheology of a concentrated polymer solution and that of a pure polymer melt. The influences of molecular mass, temperature, and shear rate on the viscosity are quite analogous. The nature and the concentration of the diluent are the key characteristics that shift the behavior of the concentrated polymer solutions. In thermoplastic foam processing, the polymer is often chosen to exhibit high viscous properties, which are beneficial during the bubble nucleation and

© 2000 by CRC Press LLC

growth phases: the molecular weights of such polymers are quite high. On the other hand, the concentration of the blowing agent, which acts as the diluent, is kept to relatively low levels of usually less than 20 wt%. For these reasons, the concentration of polymer beyond which the slope of log ␩ against log MW changes abruptly is normally not a preoccupation because it is exceeded. This particular case for mixtures of polymers and blowing agents will simplify the rheological picture. It is generally accepted that the addition of a solvent decreases the viscosity of a polymer due to the following two causes [26]: (1) A decrease of the viscosity of the pure polymer due to an increase of its free volume, which translates into a decrease of the glass transition temperature. (2) A real dilution effect, with the resulting solution viscosity falling between that of the pure polymer and that of the pure solvent. Each contribution will be examined separately. 3.2.3.1.1 Increase of the Free Volume and Decrease of the Glass Transition Temperature The viscosity of molten polymers at different temperatures can be related to the free volume fraction f, which is set equal to (v ⫺ v0)/v, with v being the total volume and v0 the occupied volume: ln ␩ ⫽ ln A ⫹ B/f

(2)

In the liquid state, that is, above the glass transition temperature, the dependence of the free volume can be defined as follows: f ⫽ fg ⫹ ␣f (T ⫺ Tg)

(3)

where fg is the free volume fraction at Tg, and ␣f is the thermal expansion coefficient of the free volume. The combination of the above two equations yields the well-known Williams-Landel-Ferry (WLF) equation [27]: log ␩/␩g ⫽ ⫺

c1(T ⫺ Tg) c2 ⫹ T ⫺ Tg

(4)

c1 and c2 are constants reflecting the temperature-dependent free-volume fraction. This equation relates the variation of the viscosity to the temperature in reference to the glass transition temperature of the polymer system. Equation (4) is valid for a mixture of a polymer and a solvent. c1 and c2 are reported to be independent of the solvent fraction [26]. Tg is the sole parame-

© 2000 by CRC Press LLC

ter that will reflect the composition of the mixture. The glass transition temperature of a polymer is lowered when a liquid or a gas of low molecular weight is dissolved into the polymer. The measurements of the depression of the glass transition temperature based on traditional laboratory methods such as for Differential Scanning Calorimetry (DSC) are fraught with uncertainty because polymer samples containing the dissolved blowing agent are difficult to prepare and prone to foam at usual low test pressures. A number of methods are proposed in the literature to calculate the mixture Tg of two components. Unfortunately, most are based on the respective component Tg [26, 28], whereas, more often than not, the glass transition temperature of the blowing agent is not known. As a quick approximation, the blowing agent Tg can be set to 2/3 of the melting temperature on an absolute scale [29]. An estimate of Tg can also be obtained by the following theoretical relation developed by Chow [30]. It should be noted that this relation is particularly useful since it does not require knowledge of the Tg of the diluent. ln

Tg ⫽ ⌿[(l ⫺ ␪) ln(1 ⫺ ␪) ⫹ ␪ ln␪] Tg0

(5)

where

␪⫽

and

␻/Md z(l ⫺ ␻)/Mp

⌿⫽

zR Mp ⌬Cp

(6)

(7)

Tg0 is the glass transition temperature for the pure polymer and Tg for the solution whose weight fraction of diluent is ␻. Md and Mp are, respectively, the molecular weight of the diluent and that of the polymer repeat unit. ⌬Cp is the change in specific heat of the polymer at its glass transition. z is a coordination number, and R is the gas constant. A value of z ⫽ 2 was found to be appropriate for mixtures of polystyrene with diluents of molecular weights in the order of 100 g/mol [30]. For the mixture of PS with CO2, a value of z ⫽ 1 is preferred [31]. Other properties for polystyrene are Mp ⫽ 104.15 g/mol and ⌬Cp ⫽ 0.3209 J/(g ⭈ K) [30, 31]. 3.2.3.1.2 Dilution Effects The description of the dilution effects is often reported as ␩s, the viscosity of the solution, being proportional to the ␣th power of ␾p, the volume fraction of

© 2000 by CRC Press LLC

the polymer, with ␣ varying between values of 3.5 and 5 [26]. Setting ␩p as the viscosity of the polymer, the dilution effect can be written as follows: log ␩s ⫽ log ␩p ⫹ ␣ log ␾p

(8)

High fractions of the diluent are also expected to have some effects on the shear-thinning behavior usually observed for the polymers of interest. Following are definitions of the three different regions of viscosity dependency with the shear rate: (1) At low shear rates, the viscosity is independent of the rate of deformation, and the polymer behaves as a Newtonian fluid (Newtonian plateau region) with a viscosity expressed as the zero-shear viscosity, ␩0. (2) At high shear rates, ␩ is a decreasing function of the rate, and this dependency can be expressed as a power-law dependency (power-law region): ␩ ⫽ k ␥᝽ n⫺1

(9)

with n defined as the power-law exponent. (3) For intermediate shear rates, a transition zone is observed that links the Newtonian plateau region to the power-law region. The width of the transition zone is a function of the polydispersity of the polymer. Shear-thinning is then observed in the last two regions, and the onset of viscosity dependence on shear rate could be specified by a characteristic shear rate ␥᝽ 0 , or its counterpart, a characteristic shear stress ␴0. It can also be expressed as a characteristic time ␶␩, that is equal to the reciprocal of ␥᝽ 0 : ␴0 ⫽ ␩0␥᝽ 0 ⫽ ␩0/␶␩

(10)

Typically, one can define such a characteristic value at the point where ␩ has been reduced to some arbitrary fraction of ␩0. For polymer solutions, the shear-thinning behavior can be modified in two ways [32]: (1) If the polymer is sufficiently diluted, the power-law exponent will cease to be the one observed for the pure polymer melt and will become a function of the diluent concentration due to the severe modifications in the polymer coil entanglements. This was observed for moderate polymer concentrations, but concentrations usually found in foam applications do not fall into this category. For this reason, the power-law exponent for mixtures of

© 2000 by CRC Press LLC

polymer with a physical blowing agent should remain that of the pure polymer melt. (2) The onset of the shear-thinning behavior, as defined previously using either ␴0, ␶␩, or ␥᝽ 0 , will vary with the modification in the level of entanglement due to the presence of the diluent. The shear-thinning behavior for polymer solutions will start at higher shear rate values than that of pure polymer melt, and ␴0 should be found proportional to ␾p. In summary, with a practical concentration limit of 20 wt% physical blowing agent for foam applications, expandable mixtures could be interpreted as concentrated polymer solutions given that the blowing agent acts as a low molecular weight solvent and that the blowing agent is dispersed on a molecular scale, so that the limit of solubility of the blowing agent, under the given pressure and temperature conditions, has not been reached. Furthermore, in addition to the global lowering of the viscosity with the addition of the blowing agent, and since foams rely on polymers of high molecular weights, we should also consider the effect of the diluent on the entanglement network that may be translated into some modifications of the shear-thinning zone, notably in the transition zone located between the Newtonian plateau region and the power-law region. Essentially, the presence of the diluent molecules close to the polymeric chains will act as a lubricant, easing the flow displacement of polymer macromolecules. 3.2.3.2 Case for Amorphous Polymers A simple model based on the previous theory is herewith presented. The results are from various studies conducted with polystyrene, in turn mixed with several blowing agents, using a commercial on-line slit die rheometer [6, 7, 8]. The link is made between the rheological behavior and the key processing variables such as the type of blowing agent, its concentration, the processing conditions of pressure and temperature, and the rate prevailing in the flow. 3.2.3.2.1 Computation of the Glass Transition Temperature The rate dependency of the viscosity can be modeled using a modified Carreau equation [33]: ␩⫽

␩0 (1 ⫹ ((␩0/␴0)᝽␥)m1)m2

(11)

where ␩0 is the zero-shear viscosity, ␴0 is the characteristic shear stress, and m1 and m2 are parameters related to the curvature of the viscosity curve and the slope in the power-law region.

© 2000 by CRC Press LLC

The procedure to numerically determine the temperature and composition dependency of the viscosity was the following. Since the curves can be superimposed by shifting, parameters m1 and m2 are constant for a given polymer. They were evaluated quite accurately off-line for the polymer without blowing agent. By fitting the data to the above modified Carreau equation, the ␩0 and ␴0 values were obtained for each specific set of experiments obtained under given conditions of temperature and composition. For the several examples previously presented in Figure 3.3(a) and Figure 3.3(b), a master curve could be constructed using reduced variables. Shifting vertically and horizontally the

FIGURE 3.7 Viscosity curves of Figure 3.3 for mixtures of PS with HCFC 142b, expressed as a master curve using reduced axes.

© 2000 by CRC Press LLC

viscosity curves shown in Figure 3.3(a) and Figure 3.3(b) translates into master curve reduced axes, ␩/␩0 and (␩0/␴0) ⭈␥᝽ (see Figure 3.7). The zero-shear viscosity values, ␩0, were first corrected for the dilution effect that is reported to be proportional to the 3.5 power of the polymer concentration. For amorphous polymers such as PS, at temperatures less than 100°C above their glass transition temperature, the WLF equation [Equation (4)] is used. For PS, the parameters of the WLF equation are c1 ⫽ 13.7 and c2 ⫽ 50.0 [26]. The resulting ␩0 values were then fitted through this WLF equation. As stated before, this equation is governed by two variables: T and Tg. T is the experimental temperature, and Tg was used as a fitting parameter

FIGURE 3.8 Zero-shear viscosity values for mixtures of PS and HCFC 142b at different compositions and temperatures and fitted to the WLF equation [Equation (4)].

© 2000 by CRC Press LLC

function of the concentration of the blowing agent. WLF fits are graphically illustrated in Figure 3.8 for the different compositions of HCFC 142b that were investigated. The above procedure has been applied to mixtures of PS with various blowing agents. The resulting values of Tg are plotted in Figure 3.9 as a function of the blowing agent concentration. The lowering of Tg with the presence of the blowing agent was anticipated because it acts as a plasticizer. The Tg estimates obtained from the predictive model developed by Chow [30] are also plotted in Figure 3.9. Agreement is quite good between estimates and experimental results for most of the cases, especially at low concentrations. These

FIGURE 3.9 Glass transition temperatures of PS as a function of the blowing agent type and concentration. Symbols are for experimental results and lines are for estimates using the Chow equation [Equation (5)].

© 2000 by CRC Press LLC

FIGURE 3.10 mixtures.

Characteristic stress, ␴0, as a function of the composition, for PS/HCFC 142b

results can also be reported in terms of the lowering of the Tg per blowing agent unit (see Table 3.3). The resulting ␴0 obtained for the example using a mixture of PS with HCFC 142b are plotted in Figure 3.10 as a function of the blowing agent concentration. Despite the significant scatter in data, results shown in that figure exhibit a trend in accordance with the theory stating that ␴0 decreases as the level of entanglement is slightly reduced by the diluent. 3.2.3.2.2 Effect of the Pressure on the Viscosity It is a well-known fact that the viscosity of polymers is pressure sensitive. This was also verified for the case of mixture of polymer with blowing agents.

© 2000 by CRC Press LLC

TABLE 3.3

Depression of the Glass Transition Temperature of Polystyrene. Blowing Agent

⌬Tg /Unit BA (°C/wt%)

HCFC 142b HFC 134a CO2 Pentane [34]

⫺4.5 ⫺4.1 ⫺6.8 ⫺8.0

The effect of pressure on the viscosity can be translated into a shift in the Tg value in the WLF equation: T⬘g ⫽ Tg ⫹ D3P

(12)

where P is the pressure of measurement, Tg is the glass transition temperature of the mixture at P ⫽ 0, T⬘g is the glass transition temperature at P, and D3 is a pressure coefficient. Viscosity measurements are shown in Figure 3.11(a) for a mixture of 7.5 wt% of HFC 134a with PS at 140°C, expressed as shear stress versus apparent shear rate. Each curve corresponds to a different level of pressure set in the rheometer slit through the variation of the exit pressure P3. The levels of pressure chosen are high enough to satisfy the condition of bubble-free flow.

FIGURE 3.11 Shear response of a mixture of 7.5 wt% of HFC 134a with PS at 140°C, at different levels of pressures: (a) expressed in terms of shear stress versus shear rate; (b) same results grouped on the basis of constant-stress measurements, as a function of the average pressure.

© 2000 by CRC Press LLC

The stress-rate curve is shifted to the left as the pressure is increased inside the measuring cell, which corresponds to the expected increase of viscosity with pressure. This variation is rather small but is still significant. Analyzed on the basis of constant-stress measurements, as in Figure 3.11(b), results show a steady but constant decrease of the shear rate as the pressure is increased, for all stress levels. This behavior can be modeled through the variation of the glass transition temperature, Tg, with pressure [Equation (12)]. The coefficient for the pressure dependency D3 was found to be a decreasing function of the blowing agent concentration, as displayed in Figure 3.12, with values ranging between 0.8 K ⭈ MPa⫺1 for PS with no blowing agent to 0.3 K ⭈ MPa⫺1 for PS containing 15 wt% of HCFC 142b.

FIGURE 3.12 Coefficient for the pressure dependency, D3, as a function of the blowing agent concentration, for PS/HCFC 142b mixture.

© 2000 by CRC Press LLC

3.2.3.2.3 Vapor-Liquid Equilibrium The solubility of a blowing agent into a given polymer is a function of the pressure and the temperature. Experiments conducted at a given temperature should be made above a critical pressure at which bubble nucleation occurs in order to make measurements on a single-phase system. The pressure conditions in the measuring cell, externally controlled through the use of a valve or a gear pump, affect the rheological response by modifying the amount of gas dissolved. This is illustrated in Figure 3.13 for a mixture of 10% 134a with PS at 130.5°C. This figure exhibits an unusual response com-

FIGURE 3.13 Shear rate response with pressure at constant shear stress, for a mixture of PS with 10 wt% of 134a at 130.5°C, undergoing phase separation.

© 2000 by CRC Press LLC

pared to results shown in Figure 3.11(b). Viscosity is no longer increased by stepping up the pressure. On the contrary, it is reduced as the pressure is increased, and this translates into an increase of the apparent rate at constant stress level. This behavior can be explained by the extra amount of blowing agent dissolved with a further increase of the pressure that accordingly plasticizes the melt to a lower viscosity value. The plasticization behavior is preponderant over that of the pressure dependency of the viscosity. The plasticization behavior is illustrated in Figure 3.14 for mixtures of PS with HFC 134a, at 180°C with a mean pressure inside the die of P ⫽ 10.8 MPa, where the apparent decrease of the viscosity measured at a constant stress level

FIGURE 3.14 Plasticization behavior of mixtures of PS with HFC 134a at 180°C and mean pressure of 10.8 MPa.

© 2000 by CRC Press LLC

of ␴12 ⫽ 39.8 kPa is displayed. The decrease of the viscosity can be fitted through an exponential relationship with the blowing agent content. On this semi-log plot, the linear trend can be extended to a certain critical composition, at which point the viscosity tends to level off. Under such temperature and pressure conditions, the maximum concentration of HFC 134a that can be dissolved in PS was found to be approximately 7.6 wt%. Conducting experiments over a wide composition range can thus provide information on the vapor-liquid equilibrium for a given combination of polymer and blowing agent under controlled processing conditions of pressure and temperature. 3.2.3.3 Case for Semicrystalline Polymers The above proposed equations are not applicable to semicrystalline polymers, such as PE and PP, processed at temperatures well above Tg ⫹ 100°C. For example, the glass transition temperature of PP is ⫺20°C. A generalized modified Arrhenius-WLF equation has been successfully used to fit the viscosity data obtained at different pressures and temperatures for various amorphous and semicrystalline polymers [35]: ␩0 ⫽ A exp e

B ␬P ⫹ f T ⫺ Tr T ⫺ Tr

(13)

in which A [Pa ⭈ s], B [K], ␬ [K ⭈ Pa⫺1] and Tr [K] are constants. In the case of amorphous polymers, the above equation may be rearranged under the classical form of the WLF equation [Equation (4)]. For semicrystalline polymer, Tr should be set to zero, and Equation (13) reduces to a classical Arrhenius form: ␩0 ⫽ D⬘exp e

E␩ 1 1 a ⫺ bf R T (Tg ⫹ D3P)

(14)

The viscosity is again related to the glass transition temperature, with pressure effects also taken into account through the pressure coefficient D3. The knowl-

TABLE 3.4

Properties to Calculate Viscosity Dependence on Temperature for Semicrystalline Polymers [28, 36].

Polymer

E␩ (kJ/mol)

Tg (K)

Polyethylene (LDPE) Polypropylene

25 40

195 253

© 2000 by CRC Press LLC

edge of the energy of activation, E␩, is required, and selected values are listed in Table 3.4 for the two most frequently used thermoplastic semicrystalline polymers in foam applications. Using Equation (14) and applying a similar procedure to that described in Section 3.2.3.2, Tg values were calculated from the experimental data for systems PP/HCFC 142b and PP/CO2. The results are reported in Figure 3.15. For similar blowing agents, the plasticization behavior expressed in term of the decrease of Tg is less drastic for PP when compared to the results obtained for PS, previously shown in Figure 3.9.

FIGURE 3.15 Glass transition temperatures for PP as a function of the blowing agent composition for HCFC 142b and CO2.

© 2000 by CRC Press LLC

FIGURE 3.16 Lowering of the temperature of crystallization of LDPE due to the presence of various blowing agents [25].

The temperature of crystallization is a very important constraint for the processing of semicrystalline polymers. The presence of a blowing agent may lower the melting point. Figure 3.16 illustrates this fact through experimental results on LDPE with various physical blowing agents [25]. The magnitude of the depression of the crystallization temperature was reported to be a function of the molecular weight of the blowing agent and its solubility. However, it cannot be anticipated from these results that the processing window of such polymers is greatly widened compared to the enormous changes encountered with amorphous polymers.

© 2000 by CRC Press LLC

3.2.4 TWO-PHASE SYSTEMS: BUBBLE NUCLEATION AND FOAMING 3.2.4.1 Effect of Shear on the Cell Nucleation The bubble point in vapor-liquid equilibria studies is a first-order equilibrium transition. In theory, nucleation occurs at the point at which the pressure drops to the thermodynamic vapor pressure. In practice, however, it has been found that supersaturation is not a sufficient condition for observing bubble nucleation. Bubble formation takes place more readily when a mixture of polymer and blowing agent is exposed to a shear field. Experimental evidence of the influence of shear stress on the conditions of bubble nucleation was given by Sahnoune et al. [37]. A nonintrusive ultrasonic technique detected the onset of bubble formation for mixtures of PS and HCFC 142b. Bubble growth induces a strong increase in ultrasonic signal attenuation caused by the scattering of the ultrasonic wave. In-line experiments were conducted on a flowing polymer submitted to shear. Off-line experiments were conducted on a confined sample for which pressure and temperature were controlled. A comparison of flow conditions in Figure 3.17 at the same temperature shows that a shear-free field yields much lower degassing pressures than those obtained under given shear stress. Using a light-scattering technique, Han and Han observed the same phenomenon for mixtures of PS with R-11 [38]. Bubble nucleation occurred in the center of a slit flow channel at a total normal stress level greater than the thermodynamic equilibrium pressure. They have also observed that the onset of bubble nucleation changed with the position in the direction perpendicular to the flow direction and this can be explained by the velocity profile and stress distributions in the slit channel. They suggested that the bubble nucleation may be induced by flow, which would be the primary mechanism near the center of the flow, and by shear stress, for positions near the die wall. Nucleation behavior was also investigated by Lee for mixtures of LDPE with CFC 12 and CFC 114, using talc as nucleating agent [39, 40]. It was observed during experiments conducted at constant talc concentration that a greater number of cells were formed at higher levels of shear rates, as shown in Figure 3.18. It has been proposed that the presence of shear stress can help to pull the gas phase out of the solid cavities provided by the nucleating agent. The shear force should act as a “catalyst” to lower the energy barrier between the stable gas cavity and the unstable bubble phase. The capillary number, Ca, which is the ratio of shear force to surface tension force, is pivotal in characterizing the foam nucleation behavior. It is defined as follows: Ca ⫽ r 2 ␩␥᝽ / 4 R ␴

© 2000 by CRC Press LLC

(15)

FIGURE 3.17 Comparison of the bubble nucleation process for flowing versus static mixtures of PS with 12.6 wt% of HCFC 142b at 135°C, measured using an ultrasonic technique [37]. The flow condition shear stress was ␴12 ⫽ 25 kPa. Take note that the x-axis is reversed.

with r the curvature of the interface between the gas and the polymer melt, ␩ the viscosity of the polymer, ␥᝽ the shear rate, R the cavity mouth radius and ␴ the surface tension. It can be assumed that r ⫽ R is required to initiate nucleation. Under given circumstances, for example, Ca ⬎ 1, the stress term given by the product ␩␥᝽ would overcome the surface tension. 3.2.4.2 Shear Viscosity of the Foam Phase Most of the literature that deals with the rheology of cellular materials is based on aqueous foams made from surfactant solutions and air [41]. Studies

© 2000 by CRC Press LLC

FIGURE 3.18 Cell count as a function of shear rate for mixture of LDPE/CFC-12 using a talc content of 0.275 wt% [39].

performed on cellular plastics, though scarcely mentioned in the literature, are more within the scope of this work. It is, however, worthwhile to mention that the complexity of such studies on the viscosity of a cellular structure depends on the many factors that influence the flow behavior of the foams, including ratio of the bubble size to that of the channel, size distribution of the bubbles, bubble aspect ratio due to flow-induced anisotropy, pressure, and interaction between the wall and the fluid. Fortunately, in order to achieve good quality foam of low density, processes based on thermoplastics must be controlled so that bubble nucleation occurs

© 2000 by CRC Press LLC

close to the exit of the die, if not outside the die. This infers that the extrusion process, from the point where the physical blowing agent is dissolved within the molten polymer up to the die lips, sees only a single-phase polymer system whose rheology was covered in the previous sections. The shear rheology of the two-phase cellular materials is thus of minor importance in most polymer applications. There are rheological properties more significant in the last process step of bubble growth and stabilization.

3.3 EXTENSIONAL RHEOLOGY FOR EXTRUSION FOAMING OF POLYMERS 3.3.1 DEFINITIONS During the foam extrusion process, bubble growth takes place outside the forming die and involves extensional, or elongational, flow. Nucleation generates a two-phase structure, where gas bubbles are surrounded by polymer walls. As the bubbles grow during foaming, these walls are stretched. This stretching is similar to that which occurs during film blowing or blow molding. A common feature of these processes is that they are controlled by the extensional rheology. This two-phase system implies that the polymer being stretched has lost some of the blowing agent diluent, the gas now within the cells. Although the gas depletion process is time dependent, we may assume that the gas concentration within the polymer has rapidly dropped to zero. Thus, the extensional deformation is applied to the neat polymer. The extensional rheology that should concern us at this point during the bubble growth and cell stabilization is, therefore, that of the pure polymer. Even if extensional rheology has enjoyed increasing popularity over the years, the amount of scientific literature still lies well behind that devoted to shear rheology. It is now recognized that this rheological behavior is the key to understanding and controlling processes such as film blowing and blow molding. Most publications on this topic are, in fact, motivated by the requirements of these major processes. The importance of the extensional rheology to foam processing is, unfortunately, still not fully appreciated. Three types of extensional flow are known: uniaxial, planar, and biaxial extension. Even though biaxial stretching is the main mode of deformation in processes like film blowing and polymer foaming, the relatively few rheological studies presented in the literature have been primarily devoted to the uniaxial extensional viscosity, ␩E. The reason for that lies in the availability of reliable techniques and the experimental difficulties associated with these methods.

© 2000 by CRC Press LLC

TABLE 3.5

Definitions for Uniaxial Extensional Rheometry.

Velocity of extensional deformation

U

Hencky rate of strain

e᝽

Elongational stress Hencky strain

␴E ␧

Stress growth function Strain hardening Steady state, Trouton, viscosity

␩E⫹ SH ␩T

U ⫽ e᝽ L 1 dL U e᝽ ⫽ ⫽ L dt L ␴E ⫽ F / A e ⫽ ln(L/L0) ⫽ e᝽ t ␩E⫹ (e᝽ 0,t) ⫽ ␴E⫹ (t)/e᝽ 0 ⫹ ⫹ ᝽ SH(e,t) ⫽ ␩E,obs. /␩E, linear ᝽ ⫽ 3␩ lim ␩T (e) 0 ᝽

(a) (b) (c) (d) (e) (f) (g)

e→0

Extensional rheology is analogous to shear rheology: a uniform rate of deformation (strain rate e᝽ for extensional deformation) is applied to the polymer, and a resulting stress (elongational stress ␴E) is measured. Definitions of the various parameters are summarized in Table 3.5. Experiments have shown that when material is subjected to deformation at a constant strain rate, the stress evolves with time [and thus, with strain; see Equation (d) in Table 3.5] until either the specimen breaks or the stress reaches a steady-state value. This behavior, where the stress is a function of time, ␴ ⫽ ␴(t), is known as “transient,” and it provides valuable information of the polymer behavior. It is usually presented in the form of a “stress growth function,” as illustrated in Figure 3.19 for LDPE. This information is of industrial relevance, because in many processes, the polymer melt is exposed to extensional flow fields over a limited period of time. For these applications, the transient elongational flow properties are more relevant than the steady-state behavior. Figure 3.19 shows that for LDPE at high extensional strains, the extensional viscosity tends to increase well above the linear viscoelastic curve. This behavior is known as “strain hardening” (SH), and it is associated with a rapid increase of the elongational viscosity at large strain. It can be quantitatively expressed as the ratio of the measured value of the stress growth function to that calculated from viscoelastic principles [see Equation (f) in Table 3.5). SH is generally related to the inability of the macromolecules to disentangle quickly enough to follow the exponential deformation. Long chain branching, as in LDPE or modified PP, is the molecular structure parameter that would explain the occurrence of SH. Other sources of SH have been identified, such as polydispersity (or molecular weight distribution, MWD) and bimodality of the molecular weight distribution, with the presence of a high molecular weight polymer (HMW) component. In this latter case, the presence of long times in the relaxation spectrum may be part of the explanation for the observed increase in the transient extensional viscosity with time [42].

© 2000 by CRC Press LLC

FIGURE 3.19 Typical stress growth functions at 200°C for a branched polyethylene, LDPE. The strain rates are indicated. These measurements were performed on a Rheometrics Extensional Rheometer, Model RER-9000.

The SH deviation from the linear-viscoelastic growth curve generally occurs at a given value of strain e᝽ that is nearly independent of the extensional strain rate e᝽ [43]. This implies that if SH should be part of the rheological behavior to make a polymer foamable, then the stretching process associated with bubble growth must exceed that particular strain level. Also, the strain hardening effect is usually amplified as the strain rate is increased. Thus, for high enough extensional strain rates, the polymer melt may behave like a perfectly elastic solid [43]. At low strain rates, SH tends to diminish and, accord-

© 2000 by CRC Press LLC

ing to the Trouton rule [Equation (g) in Table 3.5], it should vanish at low enough deformation rates. The total amount of deformation experienced by the polymer melt during foaming may be approximated as a Hencky strain of 3 to 4. Also, the biaxial extensional flow that takes place during the bubble growth occurs over a limited period of time, typically only a few seconds, which yields strain rates in the range of 1 to 5 s⫺1. The magnitude of strain hardening has also been reported in terms of the type of deformation. Uniaxial deformation yields the highest SH, while SH is barely visible for biaxial stretching [44, 45, 46]. This observation, combined with the fact that polymer foaming is based on biaxial deformation, raises the following question: should the magnitude of SH be excessively large in uniaxial deformation to be significant in biaxial deformation in order to meet the prerequisites for polymer foaming, or is the magnitude of the SH in uniaxial deformation a false indicator for polymer performance? In the latter case, it has been suggested that the indicator be changed for some other rheological value, such as yield strain, and measured under biaxial deformation [46]. At low rates of elongation and long deformation times, the extensional stress should reach a steady-state value. Under these conditions, the extensional, “Trouton” viscosity ␩T is obtained that can be related to the Newtonian shear viscosity through a factor of 3 [see Equation (g) in Table 3.5]. Such relation between shear and extensional viscosity can also be stated in the two other types of deformation, namely for planar (␩P) and biaxial (␩B) viscosities: lim ␩p / ␩s ⫽ 4; ᝽ ␥→0 ᝽ e,

lim ␩B / ␩s ⫽ 6

(16, 17)

᝽ ␥→0 ᝽ e,

“Melt strength” is a fashionable term in the foam business vocabulary. The melt strength refers to the maximum tensile strength (stress at break) measured during the continuous drawing of an extrudate from a die. This index yields only qualitative estimates of the extensional rheology, and it can be compared on many aspects to its shear rheology counterpart, melt index (MI). Because melt strength is relatively easy to measure, it has often been used for quality control of resins used for film blowing or blow molding. Even though measurement of the melt strength may appear attractive because it is easy to perform and it implies large deformations and high strain rates, it also has several drawbacks. The melt strength does not yield consistent quantitative results in terms of the extensional viscosity as stated previously, and it suffers from lack of uniformity for the temperature and the stress involved. Moreover, the viscous and elastic components are both compounded in the rate of strain and no information on the transitional behavior can be obtained.

© 2000 by CRC Press LLC

3.3.2 METHODS AND RHEOMETERS Contrary to the shear rheology that is well documented for a very broad range of polymers, the extensional rheology has suffered from the complexity required in the experimental setup to generate valuable information for this type of deformation under well-controlled conditions. Since the extensional viscosity is defined as the ratio of stress ␴E and rate of strain e᝽ , the experiments should be conducted with one of these variables held constant. In either case, the measured elongational viscosity depends on time, temperature, and the deformation rate. The constant stress method implies shorter experimental times and is, therefore, more convenient to use for steady-state measurements. The constant rate of strain method, however, provides more information about the transitional behavior. Very few commercially available instruments provide extensional viscosity measurements. In the past, the Rheometrics Extensional Rheometer (RER, commercialized by Rheometric Scientific [47], and originally developed by Münstedt [48]) was based on experiments conducted through variation of the specimen length by separating the two ends of the specimen. Typically, a constant extensional strain rate was obtained by exponentially varying the velocity of the ends with time. The polymer sample was held in an oil bath and maintained at constant temperature. The Hencky strain was limited to 3.1. This instrument has been replaced recently by the Elongation Rheometer for Melts RME (developed by Meissner [49], and commercialized by Rheometric Scientific). It is based on the following approach: the constant gauge length rheometers draw the specimen between rotary clamps at fixed points in space, maintaining the specimen length but changing its volume. The rotary clamps rotate with constant angular velocity. The samples of rectangular cross section, prepared from compression-molded plaques, float on a cushion of inert gas. Hencky elongation of up to seven can be achieved. The measurements of the extensional viscosity associated with convergent flow, often reported as the Cogswell method, can be easily obtained from rheological capillary experiments [50, 51]. The Cogswell method extracts information related to the extensional viscosity from the Bagley correction, ⌬P0, usually computed to correct shear viscosity data obtained by capillary rheometry. This parameter can also be obtained from the pressure measurements using the so-called orifice flow (L/D ⫽ 0). Thus, the elongational stress ␴E is related to the pressure drop in the orifice die ⌬P0, or to the Bagley correction, through: 3 ␴E ⫽ (n ⫹ 1) ⌬P0 8

© 2000 by CRC Press LLC

(18)

The strain rate e᝽ is given by: e᝽ ⫽

4␴12 ␥᝽ 3(n ⫹ 1)⌬P0

or ␩␥᝽ ⫽ 2␩E e᝽

(19)

where n is the power-law slope of shear stress vs. shear rate in capillary flow, ␥᝽ is the shear rate in a capillary of equivalent diameter, ␴12 is the shear stress and ␩ and ␩E, the shear and extensional viscosity, respectively. At higher rates of deformation (e.g., above 1 s⫺1), this method was found to be quite accurate. At lower strain rates, however, converging flow measurements tend to underestimate the steady-state response of the extensional viscosity because of the transient nature of stress growth and the uncontrollable time scale of the convergent flow measurements. Improvements of this method have been achieved through the lubrication of the flow using a low viscosity fluid near the die wall and through the use of a hyperbolic-shaped die that imposes a constant elongational strain rate [52]. Several other methods have been developed to measure extensional viscosity [53, 54], such as the bubble inflation rheometer in which the polymer is biaxially deformed as a bubble through the displacement of hot silicone oil [55], and the lubricated squeeze flow method that is applicable to planar and biaxial deformations. Such an instrument, developed by Macosko et al. [56], was commercialized as the Multifunction Axial Rheometer System, Mars III, by Polymics [57]. Many instruments that provide melt strength measurements are available on the market. For example, most modern capillary rheometer manufacturers offer melt strength measurement accessories. This test is often referred to as a Rheotens experiment, named after the tensile testing apparatus developed by Meissner [43, 58, 59] and commercialized by Goettfert [60]. A typical test consists of a continuous increase of the haul-off speed of the extrudate while monitoring the tension. At the breaking of the melt, the maximum values of the rotation speed of the clamp and its force are linked to the extensibility and tensile strength (melt strength) of the polymer. 3.3.3 ELONGATIONAL RHEOLOGY OF POLYMERS Even though reliable experimental methods have been developed, literature is still scarce on rheological responses under extensional deformation. Several attempts have been made to correlate molecular structure to elongational viscosity. On the other hand, information that explicitly links the elongational viscosity behavior of polymers to their foamability is still in its infancy. Some key elements, notably the strain hardening phenomenon observed mainly for branched polymers and identified as a critical property for foamability, will be reviewed in the following pages.

© 2000 by CRC Press LLC

3.3.3.1 Polyolefins The initial growth of the bubbles initially requires a low viscosity. Subsequently, a high extensional viscosity is advantageous such that the cell walls— the melt membranes between the bubbles—may withstand the stretching force experienced during the later stages of bubble growth. Linear polyolefins do not possess such characteristics, and attempts to produce foams with these polymers generally yield cell collapsing. The resulting high open-cell content prevents their use in many applications. Polyolefins that have long chain branchings (LCB), such as LDPE which is preferred for polyolefin foams, behave differently. Their stress growth functions, illustrated in Figure 3.19, exhibit strain hardening that helps with the stabilization of the cellular structure. Comparison of the elongational responses between polymers that yield opposite foam qualities, for example LDPE versus LLDPE (Figures 3.19 and 3.20, respectively), highlights the prerequisites for good foam production. A linear polyethylene such as LLDPE generally does not show any SH behavior, which makes it a very poor candidate for foam extrusion. This underlines the importance of the elongational viscosity in foam processing. Synthesis technology applied to the production of LDPE makes it possible to produce macromolecules with different chain structures. Tubular technology yields a “comb” type branching, while vessel or autoclave technology leads to a more complex, “tree” type of structure for the side branching (see Figure 3.21). More recently, the introduction of the metallocene technology yields new pathways for the synthesis of macromolecules with precisely controlled composition, molecular weight distribution, and structure. Another example is that linear resins such as LLDPE and HDPE, designed to show bimodality in their MWD, exhibit elongational rheological behaviors that may be suitable for foam applications [61]. The impact of the degree of branching on melt strength for LDPE has received some attention [62, 63, 64, 65], and the results yield unambiguous conclusions: an increase in the LCB content translates directly into an increase of the melt strength, with the molecular weight or the melt flow rate maintained constant. Typically, for resins having the same MI, the melt strength will follow that order: linear HDPE or LLDPE ⬍tubular LDPE (“comb” type) ⬍vessel (“tree” type), as shown in Figure 3.22(a). Figure 3.22(b) illustrates the unique relation between melt strength and maximum stretching ratio, irrespective of the structure of the polyethylene resin. Linear polypropylene (PP) was known as a resin that was difficult to foam due to its narrow temperature processing window, the lower end controlled by the melting point of the polymer, and the upper bound linked to the low viscosity that causes cell collapse [66]. This poor performance, which makes cell growth unstable during the foaming process, is linked to the

© 2000 by CRC Press LLC

FIGURE 3.20 Typical stress growth functions at 200°C for a linear polyethylene, LLDPE. The strain rates are indicated. These measurements were performed on a Rheometrics Extensional Rheometer, Model RER-9000.

absence of strain hardening, as shown in Figure 3.23(a) for a conventional linear PP. Recent developments of resins applicable for foam processing put the emphasis on the molecular structure prone to improve their melt strength with the same shear viscosity or melt index. For example, at Montell, the development of a technology that induces long chain branching in polypropylene [66] was a breakthrough in the mid 1980s. As shown in Figure 3.23(b), the “High Melt Strength” PP (HMS-PP) has SH behavior, comparable to LDPE, shown in

© 2000 by CRC Press LLC

FIGURE 3.21 Schematic drawings of branching in macromolecules of LDPE produced by different technologies: (top) tubular technology yields “comb” type of branching; (bottom) vessel technology yields “tree” type of branching.

Figure 3.19. Another example can be given for a polypropylene-polystyrene graft copolymer that has been developed by melt reaction, to overcome the weakness of the strain hardening of the linear polypropylene [67]. The graft copolymer showed SH and high foamability, with density reduction factor as high as 30. Increasing the melt strength then became the name of the game, and innovative resins dedicated for foam applications are more likely to make their emergence on the market. Such resin developments have been presented for HDPE [68] and PET [69, 70, 71]. Blending of polymers is another method of incorporating SH effects into the extensional rheological behavior. Studies on LLDPE/LDPE blends have shown that these immiscible blends can yield systems having an intermediate melt index (which can be related to the shear flow behavior), and synergistically higher melt strength (which should be derived from higher strain hardening responses) than predicted from the additivity rule. In contrast with branched/linear PE blends, the blends of either linear/linear or branched/ branched PE type did not show the same effects. For melt strength synergism,

© 2000 by CRC Press LLC

FIGURE 3.22 (a) Melt strength as a function of melt index, for polyethylenes having different degrees of long chain branching: linear PE (HDPE, LLDPE), LDPE by tubular technology (“comb” type), and LDPE by vessel technology (“tree” type) [65]. (b) Melt strength as a function of the maximum stretch ratio; all polyethylenes (HDPE, LDPE, LLDPE) are confounded into the same relationship [63].

FIGURE 3.23 Typical stress growth functions at 180°C for different polypropylenes: (a) conventional linear PP and (b) Montell high-melt-strength (branched) PP, HMS-PF-814. The strain rates are indicated. These measurements were performed on a Rheometrics Extensional Rheometer, Model RER-9000.

© 2000 by CRC Press LLC

a mixture of linear and branched species of similar molecular weights may therefore be required [72]. A similar study on LLDPE/LDPE blends, whose objective was to propose new and easier-to-process blend formulations for the film blowing process, has shown that addition of as little as 2 wt% of LDPE to LLDPE already generated some SH and improvement of the bubble stability during film blowing [73, 74]. 3.3.3.2 Polystyrene Strain hardening is commonly observed for PS, although it disappears at low strain rates [44]. By contrast, branched polyolefins show SH regardless of the strain rate. This suggests that different mechanisms of entanglement are responsible for SH in these two polymers. In LDPE, it originates from physical entanglement of macromolecules (enhanced by their side branching), while in PS, it comes from the ␲⫺␲ interactions between the aromatic rings. The relation between the Trouton viscosity ␩T and the zero-shear viscosity ␩0 holds in both uniaxial and biaxial deformation for polystyrene melts [44]. Since it is well known that ␩0 can be correlated to the molecular weight, the same type of relationship was consequently found for ␩T [75, 76]. The timetemperature superposition was found applicable despite the nonlinear elongational flow properties of polystyrene; it was not clear, however, to what extent the temperature dependence was still valid over the entire range of strain [77]. Presence of a high molecular weight component in a bimodal MWD has been reported to increase the strain hardening behavior, with HMW fraction as low as 0.8 wt% [42]. For bimodal MWD resin, the rheological response mimics that of branched polyolefins in terms of the SH behavior. It was concluded that the existence of relaxation times above 1,000 s, provided by the HMW fraction, should govern the SH in uniaxial elongational flow [42]. Bimodal MWD or broadening the MWD should then be beneficial for increasing the SH behavior. Contrary to the case of polyolefins, where SH was found essential for their foamability, the role of strain hardening has not yet been clearly defined for PS. Polystyrene resins are easy-to-foam materials. Their processability may be explained in terms of the high degree of plasticization due to the presence of the blowing agent, and their high energy of activation, which should contribute to the rapid solidification of the polymer matrix and, therefore, to the cell stabilization. 3.3.4 ELONGATIONAL RHEOLOGY OF BLOWING AGENT-CHARGED POLYMERIC SYSTEMS Before closing this section on extensional rheology, it should be pointed out that this type of flow prevails within the forming die, where the polymer/

© 2000 by CRC Press LLC

blowing agent mixture still exists under a single-phase system. The pressure drop experienced by the polymeric mixture near the die exit has two contributions, shear and extensional. The shear rheology of a blowing agent-charged polymeric system was extensively covered in the first part of this chapter. Several experimental setup concerns were also underlined. It has also been reported that measurement of the extensional viscosity is rather complex and is associated with several experimental difficulties. For these reasons, practically no study of polymer/physical blowing agent mixtures under extensional deformation has been reported. One exception is the experiment conducted using the extrudate haul-off measurement method on a mixture of LDPE and physical blowing agent (11 wt% isopentane) [78, 79]. The results indicated a reduction of over 75% in the extensional viscosity, accompanied with a twofold increase in the stretch ratio at break. In general, it might be expected that the plasticizing effect observed for shear rheology should also be encountered in extensional rheology, on the basis of their relationship in the linear viscoelatic domain (Trouton rule). However, it is not clear if this translation from shear to extensional still prevails for deviations from linear behavior, such as the strain hardening effect. More studies would be required to elucidate this aspect.

3.4 CONCLUSION The main rheological characteristic of single-phase mixtures of polymers and blowing agents is the plasticization phenomenon. It is quantified mostly by the lowering of the Tg that is well documented for amorphous polymers. Predictions of semicrystalline polymer/blowing agent mixture viscosities are also calculable above Tm. The degree of plasticization is, however, less significant than that of amorphous polymers. The need for understanding the rheological behavior of broader ranges and types of polymers and blowing agents can only be filled by a wider proliferation of viscosity measuring apparatuses such as the pressurized rheometers in present use. We need not close the door on new model developments such as those based on free volume theory. These may provide a further link between measurable macroscopic properties such as viscosity or solubility and inherent microscopic phenomena such as molecular packing. Insofar as rheology is only one of many variables influencing the foaming process beyond the single-phase portion, two-phase systems discussed in this chapter dealt mostly with bubble nucleation and cell stabilization. Experimental evidence shows that a shear field enhances nucleation. Work is still under way to quantify this observation. In the bubble growth and stabilization regime, low viscosity for bubble nucleation but increasing elongational viscosities during cell formation are desir-

© 2000 by CRC Press LLC

able traits of foaming resins. The elongational viscosity, especially its nonlinear response at high strain, has been identified as one of the critical properties that semicrystalline polymers should possess to be foamable. The extensional viscosity behavior, through this strain hardening effect, appeared to be more sensitive to high molecular weight components than the shear viscosity. The goal that follows is to increase our understanding of the phenomenon and its incidence on the morphology of the foam, that is, the cell size and density. This will lead to better modeling, allowing the tailoring of polymer elongational viscosities to optimize the foam density and morphology. However, it must be pointed out that extensional flow is only part of the complex process. The final strategy must encompass the other elements, viz., solubility of the foaming agent, shear flow behavior inside the extruder, nucleation, shaping and cooling, etc. Unfortunately, there is still information missing on several aspects of the foaming process that prevent direct incorporation of the extensional flow measurements into a process model.

3.5 REFERENCES 1. Han, C. D. and C. A. Villazimar. 1978. “Studies on Structural Foam Processing. I. The Rheology of Foam Extrusion,” Polym. Eng. Sci., 18:687–698. 2. Kraynik, A. M. 1981. “Rheological Aspects of Thermoplastic Foam Extrusion,” Polym. Eng. Sci., 21:80–85. 3. Han, C. D. and C. -Y. Ma. 1983. “Rheological Properties of Mixtures of Molten Polymer and Fluorocarbon Blowing Agent. I. Mixtures of Low-Density Polyethylene and Fluorocarbon Blowing Agent,” J. Appl. Polym. Sci., 28:831–850. 4. Han, C. D. and C. -Y. Ma. 1983. “Rheological Properties of Mixtures of Molten Polymer and Fluorocarbon Blowing Agent. II. Mixtures of Polystyrene and Fluorocarbon Blowing Agent,” J. Appl. Polym. Sci., 28:851–860. 5. Lee, M., C. B. Park and C. Tzoganakis. 1999. “Measurements and Modeling of PS/Supercritical CO2 Solution Viscosities,” Polym. Eng. Sci., 39:99–109. 6. Gendron, R., L. E. Daigneault and L. M. Caron. 1999. “Rheological Behavior of Mixtures of Polystyrene with HCFC 142b and HCF 134a,” J. Cell. Plast., 35:221–246. 7. Gendron, R. and L. E. Daigneault. 1997. “Rheological Behavior of Mixtures of Various Polymer Melts with CO2,” SPE Antec Tech. Papers, 43:1096–1100. 8. Gendron, R. and A. Correa. 1998. “The Use of On-Line Rheometry to Characterize Polymer Melts Containing Physical Blowing Agents,” Cell. Polym., 17:93–113. 9. Blyler, L. L. Jr. and T. K. Kwei. 1971. “Flow Behavior of Polyethylene Melts Containing Dissolved Gases,” J. Polym. Sci.: Part C, 35:165–176. 10. Gerhardt, L. J., C. W. Manke and E. Gulari. 1997. “Rheology of Polydimethylsiloxane Swollen with Supercritical Carbon Dioxide,” J. Polym. Sci., Polym. Phys. Ed., 35:523–534. 11. Kwag, C., C. W. Manke and E. Gulari. 1999. “Rheology of Molten Polystyrene with Dissolved Supercritical and Near-Critical Gases,” J. Polym. Sci.: Part B: Polym. Phys., 37:2771–2781. 12. Ma, C. -Y. and C. D. Han. 1982. “Measurement of the Viscosities of Mixtures of Thermoplastic Resin and Fluorocarbon Blowing Agent,” J. Cell. Plast., 18:361–370.

© 2000 by CRC Press LLC

13. Rheometric Scientific, Inc. 1996. Process Control Rheometer PCR-620, One Possumtown Rd, Piscataway, NJ 08854, USA. 14. Han, C. D. 1981. Multiphase Flow in Polymer Processing. New York, Academic Press, Ch. 6. Dispersed Flow of Gas-Charged Polymeric Systems:257–340. 15. Lee, S. -T. and N. S. Ramesh. 1996. “Gas Loss During Sheet Formation,” Adv. Polym. Technol., 15:297–305. 16. Lee, S. -T. 1997. “A Fundamental Study of Thermoplastic Foam Extrusion with Physical Blowing Agents,” ACS Symp. Ser., 669:195–205. 17. Marusenko, V. V. 1992. “Flowability of Foaming Compositions Based on Thermoplastics and Phenol Alcohol,” Int. Polym. Sci. Technol., 19/3:86–87. 18. Kwag, C., L. J. Gerhardt, V. Khan, E. Gulari and C. W. Manke. 1996. “Plasticization of Polymer Melts with Dense or Supercritical CO2,” Proc. ACS Div. Polym. Mater. Sci. Eng., 74:183–185. 19. Nikolaeva, N. E., O. Y. Sabsai, A. Y. Malkin and M. L. Fridman. 1985. “Rheological Characteristics of the Extrusion of Articles from Foamed Thermoplastics,” Int. Polym. Sci. Technol., 12/12:51–53. 20. Fridman, M. L., O. Y. Sabsai, N. E. Nikolaeva and G. R. Barshtein. 1989. “Rheological Properties of Gas-Containing Thermoplastic Materials During Extrusion,” J. Cell. Plast., 25:574–595. 21. Kim, K. U., B. C. Kim, S. M. Hong and S. K. Park. 1989. “Foam Processing with Rigid Polyvinylchloride,” Int. Polym. Process., 4:225–231. 22. Kim, B. C., K. U. Kim and S. I. Hong. 1986. “Foam Extrusion of Rigid PVC. III. The Rheological Properties of Unexpanded and Expandable Formulations,” Polymer (Korea), 10:324–331. 23. Mitrofanov, A. D., Y. T. Panov and N. I. Kashcheeva. 1990. “Effect of Chemical Blowing Agents on the Rheological Properties of Polystyrene,” Int. Polym. Sci. Technol., 17/7:26–28. 24. Sakino, K. and K. Ito. 1982. “Study on Extrusion Melt Flow of Foamed Polyvinylchloride,” Reports on Progress in Polym. Phys. in Japan, 25:195–198. 25. Dey, S. K., C. Jacob and J. A. Biesenberger. 1994. “Effect of Physical Blowing Agents on Crystallization Temperature of Polymer Melts,” SPE Antec Tech. Papers, 40:2197–2198. 26. Ferry, J. D. 1970. Viscoelastic Properties of Polymers, 2nd ed., New York, Wiley. 27. Williams, M. L., R. F. Landel and J. D. Ferry. 1955. “The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids,” J. Am. Chem. Soc., 77:3701–3707. 28. van Krevelen, D. W. 1990. Properties of Polymers, Amsterdam, Elsevier, 875pp. 29. Lee, W. A. and G. J. Knight. 1970. “Ratio of the Glass Transition Temperature to the Melting Point in Polymers,” Br. Polym. J., 2:73–80. 30. Chow, T. S. 1980. “Molecular Interpretation of the Glass Transition Temperature of PolymerDiluent Systems,” Macromolecules, 13:362–364. 31. Chiou, J. S., J. W. Barlow and D. R. Paul. 1985. “Plasticization of Glassy Polymers by CO2,” J. Appl. Polym. Sci., 30:2633–2642. 32. Graessley, W. W. 1974. “The Entanglement Concept in Polymer Rheology,” Adv. Polym. Sci., 16:1–179. 33. Utracki, L. A. and B. Schlund. 1987. “Linear Low Density Polyethylenes and Their Blends: Part 2. Shear Flow of LLDPE’s,” Polym. Eng. Sci., 27:367–379. 34. Throne, J. L. 1996. “Is Your Thermoplastic Polymer Foamable?” Proceedings of Foam Conference 96, Somerset, NJ, 1–14.

© 2000 by CRC Press LLC

35. Kadijk, S. E. and B. H. A. A. van den Brule. 1994. “On the Pressure Dependency of the Viscosity of Molten Polymer,” Polym. Eng. Sci., 34:1535–1546. 36. Yoo, H. J., S. W. Shang, J. L. Chin and J. Shekhtmeyster. 1993. “Development of On-line Melt Flow Monitor (MFM) for Polypropylene Melt Flow Rate (MFR) Measurement,” SPE Antec Tech. Papers, 39:1444–1447. 37. Sahnoune, A., L. Piché, A. Hamel, R. Gendron and L. E. Daigneault. 1997. “Ultrasonic Monitoring of Foaming in Polymers,” SPE Antec Tech. Papers, 43:2259–2263. 38. Han, J. H. and C. D. Han. 1988. “A Study of Bubble Nucleation in a Mixture of Molten Polymer and Volatile Liquid in a Shear Flow Field,” Polym. Eng. Sci., 28:1616–1627. 39. Lee, S. -T. 1993. “Shear Effects on Thermoplastic Foam Nucleation,” Polym. Eng. Sci., 33:418–422. 40. Lee, S. -T. 1994. “More Experiments on Thermoplastic Foam Nucleation,” SPE Antec Tech. Papers, 40:1992–1997. 41. Heller, J. P. and M. S. Kuntamukkula. 1987. “Critical Review of the Foam Rheology Literature,” Ind. Eng. Chem. Res., 26:318–325. 42. Minegisshi, A., A. Nishioka and T. Takahashi. 1998. “Uniaxial Elongational Flow Behavior of PS/UHMW-PS Melts,” Proceedings of PPS-14, Yokohama, Japan: 289–290. 43. Meissner, J. 1972. “Development of a Uniaxial Extensional Rheometer for the Uniaxial Extension of Polymer Melts,” Trans. Soc. Rheol., 16:405–420. 44. Takahashi, M., T. Isaki, T. Takigawa and T. Masuda. 1993. “Measurement of Biaxial and Uniaxial Extensional Flow Behavior of Polymer Melts at Constant Strain Rates,” J. Rheol., 37:827–846. 45. Koyama, K. and A. Nishioka. 1998. “Comparison of the Characteristics of Polymer Melts Under Uniaxial, Biaxial and Planar Elongational Flows,” Proceedings of PPS-14, Yokohama, Japan: 234–235. 46. Tajiri, T., K. Obata and R. Kamoshita. 1998. “The Effect of Elongational Properties on Thickness Uniformity in Blow Molding,” Proceedings of PPS-14, Yokohama, Japan: 681–682. 47. Rheometric Scientific, Inc., One Possumtown Road, Piscataway, NJ 08854, U.S.A. 48. Münstedt, H. 1979. “New Universal Extensional Rheometer for Polymer Melts. Measurements on a Polystyrene Sample,” J. Rheol., 23:421–436. 49. Meissner, J. and J. Hostettler. 1994. “A New Elongational Rheometer for Polymer Melts and Other Highly Viscoelastic Liquids,” Rheol. Acta, 33:1–21. 50. Cogswell, F. N. 1972. “Converging Flow of Polymer Melts in Extrusion Dies,” Polym. Eng. Sci., 12:64–73. 51. Cogswell, F. N. 1978. “Converging Flow and Stretching Flow: a Compilation,” J. Non-New. Fl. Mech., 4:23–38. 52. Pendse, A. V. and J. R. Collier. 1996. “Elongational Viscosity of Polymer Melts: A Lubricated Skin-Core Flow Approach.” J. Appl. Polym. Sci., 59:1305–1314. 53. Dealy, J. M. 1978. “Extensional Rheometers for Molten Polymers; A Review,” J. Non-New Fl. Mech., 4:9–21. 54. Meissner, J. 1987. “Polymer Melt Elongation—Methods, Results, and Recent Developments,” Polym. Eng. Sci., 27:537–546. 55. Michaeli, W. and K. Hartwig, 1996. “Biaxial Elongational Rheometry,” Kunststoffe, 86:85–88. 56. Chatraei, S. H., C. W. Macosko and H. H. Winter. 1981. “Lubricated Squeezing Flow: A New Biaxial Extensional Rheometer,” J. Rheol., 25:433–443. 57. Polymics, 2820 East College Avenue, Suite G, State College, PA 16801, U.S.A.

© 2000 by CRC Press LLC

58. Meissner, J. 1971. “Dehnungsverhalten von Polyälthylen-Schmelzen,” Rheol. Acta, 10:230–242. 59. Wagner, M. H., B. Collignon and J. Verbeke. 1996. “Rheotens-Mastercurves and Elongational Viscosity of Polymer Melts,” Rheol. Acta, 35:117–126. 60. Goettfert, U.S.A., 488 Lakeshore Parkway, PO Box 10844, Rock Hill, SC 29730, U.S.A. 61. Münstedt, H. and S. Kurzbeck. 1998. “Elongational Properties of Polyolefin Melts,” Proceedings of International Symposium on Elongational Flow of Polymeric Systems, Yamagata U., Yonezawa, 13–15 June 1998, 75–84. 62. Münstedt, H. and H. M. Laun. 1981. “Elongational Properties and Molecular Structure of Polyethylene Melts,” Rheol. Acta, 20:211–221. 63. Romanini, D. 1982. “Synthesis Technology, Molecular Structure, and Rheological Behavior of Polyethylene,” Polym.-Plast. Technol. Eng., 19:201–226. 64. Attalla, G. and D. Romanini. 1983. “Influence of Molecular Structure on the Extensional Behavior of Polyethylene Melts,” Rheol. Acta, 22:471–475. 65. Ghijsels, A., J. J. S. M. Ente and J. Raadsen. 1990. “Melt Strength Behavior of PE and Its Relation to Bubble Stability in Film Blowing,” Intern. Polym. Proc., 5:284–286. 66. Bradley, M. B. and E. M. Phillips. 1990. “Novel Foamable Polypropylene Polymers,” SPE Antec Tech. Papers, 36:717–720. 67. Murata, T., T. Noma, J. Takimoto and K. Koyama. 1998. “Extrusion Foaming Processability and Elongational Flow Behavior of Polypropylene-Polystyrene Graft Copolymers,” Proceedings of PPS-14, Yokohama, Japan, 447–448. 68. Firdaus, V., P. P. Tong and K. K. Cooper. 1996. “A Developmental HDPE Foam Resin,” SPE Antec Tech. Papers, 42:1931–1936. 69. Boone, G. 1996. “Expanded Polyesters for Food Packaging,” Proceedings of Foam Conference 96, Somerset, NJ, 145–157. 70. Al-Ghatta, H. and T. Severini. 1996. “Production of Foam Grade PET,” SPE Antec Tech. Papers, 42:1846–1849. 71. Johnston, W. F. 1997. “Cellular PET for Bakery and Other Packaging Applications,” Proceedings of Foamplas ‘97, Mainz, Germany, 335–345. 72. Ghijsels, A., J. J. S. M. Ente and J. Raadsen. 1992. “Melt Strength Behavior of Polyethylene Blends,” Int. Polym. Proc., 7:44–50. 73. Schlund, B. and L. A. Utracki. 1987. “Linear Low Density Polyethylenes and Their Blends: Part 3. Extensional Flow of LLDPE’s,” Polym. Eng. Sci., 27:380–386. 74. Schlund, B. and L. A. Utracki. 1987. “Linear Low Density Polyethylenes and Their Blends: Part 5. Extensional Flow of LLDPE Blends,” Polym. Eng. Sci., 27:1523–1529. 75. Münstedt, H. 1975. “Viscoelasticity of Polystyrene Melts in Tensile Creep Experiments”, Rheol. Acta, 14:1077–1088. 76. Münstedt, H. 1980. “Dependence of the Elongational Behavior of Polystyrene Melts on Molecular Weight and Molecular Weight Distribution,” J. Rheol., 24:847–867. 77. Li, L., T. Masuda, M. Takahashi and H. Ohno. 1988. “Elongational Viscosity Measurements on Polymer Melts by a Meissner-type Rheometer,” J. Soc. Rheol. Jpn, 16:117–124. 78. Lee, S. T. and T. Kimble. 1999. “Rod Strength Analysis for Polyethylene/Gas Systems,” SPE Antec Tech. Papers, 45:2078–2083. 79. Ramesh, N. S. and S. T. Lee. 1999. “Blowing Agent Effect on Extensional Viscosity Calculated from Fiber Spinning Method for Foam Processing,” Proceedings of Foams ‘99, Parsippany, NJ, 85–96.

© 2000 by CRC Press LLC

CHAPTER 4

Foam Nucleation in Gas-Dispersed Polymeric Systems SHAU-TARNG LEE

4.1 INTRODUCTION

N

phase formation is a common physical and/or chemical phenomena and includes actions such as boiling, foaming, solidification, decomposition, etc. [1–3]. It is a key element in quite a few phase separation processes, for instance, evaporation, crystallization, devolatilization [4], and foam extrusion [5,6]. Focusing on physical phenomena, new phase formation, also known as nucleation, can originate from self-structural adjustment or from foreign “seeds” as a way to release an outside change-induced load. Its unstable nature easily lends itself to unstable phase separation. Foaming basically involves bubble nucleation and bubble growth (phase separation) to make a foamed product that can be defined as visible gas cells dispersed in a denser continuum matrix. Foaming can occur with denser medium in a dynamic state or a static state, ranging from boiling to wave foaming to plastic foam. Bubble formation can be caused by a variety of sources including heat, vacuum, motion, reaction, and cavitation. Table 4.1 lists the common phenomena and possible mechanisms. Boiling is a common phenomenon in which micro vapor bubbles in liquid are formed through a homogeneous and/or heterogeneous mechanism. In principle, gas molecules disperse in the liquid phase and become energetic enough to overcome surrounding confinement to expand into a visible size. Superheat is a typical example, in which the vapor pressure exceeds the ambient pressure to a point where bubble formation becomes the effective way to minimize the chemical potential difference. Crystallization is another wellknown subject; under supersaturation, a tiny nucleus forms and expands in size to join with a nearby developing nucleus to cause phase transformation. EW

© 2000 by CRC Press LLC

TABLE 4.1

Phase Separation Phenomena and Its Mechanisms.

Phenomena

Mechanisms

Boiling Plastic Foaming Cavitation Devolatilization Wave Foaming

Heat Heat and/or Reaction or Pressure Reduction Pressure Variation Vacuum, Inert Gas Seeding Hydrodynamic Pressure

Both phenomena are induced by energy variation, yet kinetics are greatly affected by heterogeneity. Similarly, a pressure-induced superheat, often in a liquid/gas or melt/gas system, requires an expanded view on energy sources. To be precise, a clean and smooth surface simply provides surface energy to reduce the foaming superheat threshold. As for a porous surface, it not only offers surface energy but also provides residence for gas molecules in the cavities. The former is heterogeneous nucleation, and the latter is cavitation. Both will be further discussed from the thermoplastic foam extrusion perspective in

FIGURE 4.1 cavitation.

Schematics of various gas bubble formations; homogeneous, heterogeneous, and

© 2000 by CRC Press LLC

the following sections. Figure 4.1 presents a schematic drawing to differentiate homogeneous, heterogeneous, and cavitation nucleation. It should be pointed out that there is a difference between nucleation and nucleation rate. The former is a thermodynamic phenomenon focusing on the first bubble formation, and the latter is a kinetic mechanism indicating the speed of the bubble formation. If a sharp thermodynamic instability can be implemented to make impulse nucleation, a swarm of bubbles will come into existence within a very brief period. Nucleation, #/cm3, and nucleation rate, #/cm3/s, are, therefore, very similar in quantity. A sharp pressure drop is a very common practice in foam extrusion in making multibirths out of one nucleating site unlikely. Although quality foam products can be routinely and consistently produced, how the invisible gas clusters get together to overcome the surrounding tension to evolve into visible bubbles (i.e., ⬎ 0.1 mm) remains unsettled in many aspects [7]. It is conceivable to start this chapter with equilibrium considerations, then proceed to conventional nucleation theories, and conclude with modifications made when applied to actual foaming processes. We will then cover flow- and thermal-induced cavitation from thermo- and hydrodynamic viewpoints. This chapter ends with applications to thermoplastic foam extrusion.

4.2 EQUILIBRIUM CONSIDERATIONS The gas solubility in liquid has been addressed by splitting the solution energy and entropy into two parts: the formation of cavities and the diffusion of the gas molecules into them [8]. The activity of the gas cavity depends on the nature of the gas/liquid system and the surrounding temperature and pressure. An equilibrium state is established when the gas diffusion rates, in and out of the cavity, are equal. When gas dissolves in a polymeric melt, a vapor pressure is established. According to the Flory-Huggins equation: ln(a) ⫽ ln(Vg) ⫹ Vp ⫹ ␹ V2p

(1)

where a is activity, defined as the partial pressure over the vapor pressure, p/Po, and ␹ is the interaction parameter. Assuming a dominant polymer phase, Equation (1) becomes: p/Po ⫽ Vs exp(1 ⫹ ␹)

(2)

and Henry’s law constant, Kw, can be expressed as follows: Kw ⫽ p/Wg ⫽ ␳p/␳g Po exp(1 ⫹ ␹)

© 2000 by CRC Press LLC

(3)

TABLE 4.2

Selected Henry’s Law Constants at 177°C.

System

Kw, atm

Reference

Polystyrene/Styrene

30 50 280 595 600 118 190 504 85 115

Werner, 1981 [9] Biesenberger and Todd, 1983 [4] Chaudhary and Johns, 1998 @ 160°C [10] Durill and Griskey, 1966 [11] Lee, 1996 [7] Gorski et al., 1983 [12] Lee, 1996 [7] Lee, 1996 @ 187.8°C [7] Werner, 1981 [9] Biesenberger and Todd, 1983 [4]

LDPE/I-butane LDPE/CHCIF2 LDPE/CC12F2 LDPE/C2H3CIF2 LDPE/C2H4F2 PMMA/MMA HDPE/Hexane

where W and ␳ represent the weight fraction and density, respectively. Table 4.2 [4, 7, 9–12] lists several common gas/polymer Henry’s law constants. The temperature dependence of Henry’s law constant can be presented as follows [13]: Kw ⫽ Kwo exp(⌬Hs/R(1/To ⫺ 1/T))

(4)

where ⌬Hs represents the heat of solution, R represents the gas constant, and subscript o is the reference state. By knowing the equilibrium constants at two different temperatures, the constants of Equation (4) can be determined in order to calculate the Henry’s law constant at other temperatures. In foam extrusion, the loading of the blowing agent can be calculated via decomposition or preset injection. In general, foam extrusion in nature has a sharp pressure release at the die end, that, from the processing viewpoint, is continuous foaming; however, from the nucleation perspective, it is a typical one-point (impulse) nucleation that is confirmed by uniform cell structure in extruded foam products. In other words, as mentioned earlier, nucleation rate (#/cm3/sec.) and total nucleation (#/cm3) are interchangeable for foam extrusion with a sharp pressure drop at the die tip, assuming nucleation is completed in fraction of second. At a given die flow geometry, with known Kw and Wg, one can easily figure out the vapor pressure to determine the pressure ratio (PR) that is defined as follows: PR ⫽ Vapor Pressure / Die Flow Pressure

(5)

Referring to Figure 4.2, as PR is greater than one, the bubble nucleus tends to expand into visible foams. When PR is equal to one, it is a good reference point for nucleation. A general guideline in die design is to keep the nucleation point as close to the die end as possible to not only prevent premature foaming,

© 2000 by CRC Press LLC

FIGURE 4.2 Pressure profiles in parallel and tapered plates’ gas/melt flow with temperatures as an ordinate. The dotted line represents vapor pressure, and PR ⫽ Flow Pressure/Vapor Pressure.

but also to allow optimal free expansion. However, when tiny gas bubbles form inside the die, due to its high pressure nature, the die design becomes extremely critical in allowing controlled expansion without disrupting flow integrity prior to exit. It is also noted that the overall die pressure can help calculate pressure-induced slippery flow in the extruder to determine its net pumping capacity. Although extrusion is a dynamic state and surrounding conditions are not constant, equilibrium considerations, at least, allow us to establish a general idea of nucleation that can easily extend into real design. It is essential to view the dynamic nucleation process from the equilibrium viewpoint. Considering a static state, mechanical equilibrium can be written as follows: Pme ⫽ P ⫹ 2␴/Rb

(6)

P represents external pressure, and ␴ and Rb represent surface tension and bubble radius, respectively. The critical radius can be calculated while reducing P to generate superheat. Knowing the number of cells in a unit volume formed, Nb/V, a mass balance equation, or chemical equilibrium, can easily be established: Pce ⫽ Po ⫺ 4/3␲ R3bcr (Nb/V) Pb K⬘w

© 2000 by CRC Press LLC

(7)

where, K⬘w ⫽ Kw/RT, and R and T denote ideal gas law constant and temperature, respectively. Subscript o represents initial state. In equilibrium Pb ⫽ Pme ⫽ Pce, we obtain a radius quadratic equation: R4bcrXP ⫹ R3bcrX2␴ ⫺ Rbcr (Po ⫺ P) ⫹ 2␴ ⫽ 0

(8)

in which, X ⫽ (Nb/V)4/3␲K⬘w, and P is the surrounding pressure. It was pointed out [14, 15] that a perturbation is needed to make unstable growth pos-

FIGURE 4.3 Computational results of critical bubble radius for 4,880 ppm styrene in polystyrene under 5 mmHg. ␧ represents a small perturbation to move equilibrium from unstable A to stable B.

© 2000 by CRC Press LLC

sible, as illustrated in Figure 4.3. Shafi and Flumerfelt [16] proposed a Taylor expansion based on the Peclect number to figure out perturbation quantity. Since foam extrusion involves dynamic variables to make automatic perturbations, bubble growth appears to be a natural consequence. However, viewing the stable equilibrium radius, the corresponding volume expansion becomes unrealistic. That means spherical shape and equal bubble volume, common assumptions for bubble growth, cannot coexist. In fact, with respect to packing theory, 75% gas void, approximately three times expansion, is the limit for the aforementioned assumptions, over which contact with neighboring bubble and subsequent distortion become inevitable. In any event, this dynamic process involves other real parameters, such as bubble interaction and diffusion loss, to make it impossible for the stable bubble radius to exist. Also noted in the equilibrium equation is the critical role played by bubble number density, Nb/V, which is as important in foam nucleation and growth.

4.3 CONVENTIONAL NUCLEATION THEORIES When variation of internal or external conditions occurs, the system itself automatically starts to adjust to the “disturbances.” Sometimes it simply reestablishes a stable state. Other times, a sharp change is necessary, such as bubbling. Since gas molecules tend to adjust themselves faster than liquid molecules, foaming becomes a way to alleviate the thermodynamic “load.” This thermodynamic phenomenon is demonstrated in Figure 4.4, the pressurevolume diagram. It shows that enough superheat is needed to make phase change possible, as indicated, from a stable state to an unstable state. At modest superheat, metastable state appears. Cahn [17] postulates a wave function to describe “new phase” molecule distribution, when the wavelength increases to a point where a new phase automatically appears. Nucleation itself appears to be a “process” rather than a “point.” In quenching and crystallization [18], lengthy phase separation appears to correlate well with the spinodal decomposition concept. Moreover, Keller [19] noted in the polyethylene morphology experiments that two crystallization mechanisms, stable chain folding growth and metastable spontaneous thickening growth, compete with each other depending upon the temperature drop rate. In a relatively brief bubbling process, impulse nucleation, thermodynamics, and kinetics in a metastable state are not separable. Here, we would like to proceed with the conventional nucleation theories, followed by modifications and cavitation, and ending with extrusion nucleation. 4.3.1 HOMOGENEOUS NUCLEATION In the classical theory of nucleation, the nucleation rate is governed by the rate at which invisible gas clusters are energized by effective diffusion as a

© 2000 by CRC Press LLC

FIGURE 4.4 region.

Phase diagram: bimodal and spinodal. The shaded areas represent a metastable

result of supersaturation to exceed the critical radius [1, 20–22]. According to Gibbs [2], a gas cluster containing n molecules can be expressed as follows: C (n) ⫽ N exp(⫺W(n)/kT)

(9)

where W(n) is the minimum work to sustain a bubble, N is the number of molecules per unit volume of the metastable state, and k and T are the Boltzmann constant and absolute temperature, respectively. By multiplying a frequency factor, B, the rate of nucleation can be expressed as follows: J ⫽ B N exp (⫺W(n)/kT)

© 2000 by CRC Press LLC

(10)

In the metastable region, the total work includes surface area generation, size expansion, and evaporation. W ⫽ ␴A ⫺ (Pg ⫺ Pl) Vb ⫹ n (␮g ⫺ ␮l)

(11)

where ␴, A, V, P, and ␮, denote the surface tension, bubble surface area, volume, pressure, and gas molecules’ chemical potential, respectively. The subscripts l and g represent liquid and gas, respectively. Assuming it is in a spherical shape, Figure 4.5 illustrates that a minimum work must be obtained to make size expansion irreversible. It corresponds to the critical bubble size. However, under critical size, it is hardly possible to get sufficient work to resist the surrounding surface tension force that in theory, can compress the gas cluster out of existence. If the cluster can be sustained in a nonspherical shape (i.e., string), where the surface tension force is not at a maximum, the cluster could survive. At equilibrium, the chemical potentials, ␮g and ␮l, are equal, and W becomes: W ⫽ 4␲r2␴ ⫹ (4/3) ␲r3(Pg ⫺ Pb)

(12)

⭸W/⭸r ⫽ 0 for the critical condition, or considering the Laplace equation for mechanical equilibrium, one obtains: Rbcr ⫽ 2␴/(Pg ⫺ Pl)

(13)

Minimum work becomes the following: Wmin ⫽ 16␲␴3/(3(Pg ⫺ Pl)2)

(14)

Blander and Katz [3] obtained the following equation for the rate of nucleation: J ⫽ N (2␴/(␲m))1/2 exp (⫺16␲␴3/(3kT(Pb ⫺ Pl)2)

(15)

where m represents the mass of a gas molecule. In polymer processing, it is reasonable to assume Pl equal to P, representing the surrounding pressure while the gas/melt resides in the barrel. Then, the difference between Pb and P becomes superheat. SH ⫽ Pb ⫺ P

(16)

At a slight superheat (i.e., Pb barely over P), diffusion is able to reestablish equilibrium before reaching the corresponding Rbcr; in other words, the nucle-

© 2000 by CRC Press LLC

FIGURE 4.5

Works for bubble expansion from Vo to pass Vc for continual expansion.

ation rate becomes negligible. Equation (16) suggests at least two ways to induce enough thermodynamic instability for nucleation, pressure release, and temperature increase. Since thermoplastic polymer is a poor thermal conductor and will decompose at a high temperature, the former appears favorable. Knowing Kw and Wg, we are also able to calculate SH and then the nucleation rate. As indicated in Table 4.3, a modest superheat is not sufficient to bring forth gas bubble formation. It is noted in Figure 4.6 by Tadmor et al. [23] that nucleation shows too strong a sensitivity toward temperature increase to

© 2000 by CRC Press LLC

TABLE 4.3 1

Homogeneous Nucleation Rate Calculation; LDPE/CFC-12.

Temp. (°c)

Temp. (°k)

Kw (atm)

⌬P (atm)

␳ (g/ml)

␴2 (erg/cm2)

A ⫻ 1032 (#/ml ⫺ s)

B

J3 (#/ml ⫺ s)

60 70 80 100 100 100 120

333 343 353 373 373 373 393

64.6 68 78 102 126 136 109

12.9 13.6 14.6 19.4 12.6 1.1 20.8

0.91 0.91 0.91 0.91 0.91 0.91 0.905

5.12 4.0 2.9 0.7 0.7 0.7 —

7.07 6.25 5.32 2.61 2.61 2.61 —

286 119 38.2 0.3 0.69 90 —

0 0 1.4 ⫻ 1016 1.95 ⫻ 1032 1.31 ⫻ 1032 0 —

(Reprinted with permission from Lee, Polym. Eng. Sci., 33, 1993.) 1 Henry’s law constant [7]. 2 Reference [3]. 3 J ⫽ A exp(2B).

© 2000 by CRC Press LLC

FIGURE 4.6 Nucleation rate calculations vs. foaming temperature for 5,000 ppm styrene in polystyrene under 10 mmGg and 760 mmHg, respectively. (Plot based on results given by Tadmor et al. in their presentation at the first PPS at Akron, 1985 [22].)

realistically correlate with actual foaming. SH can be established by dissolving gas in the polymer and then applying pressure drop or temperature rise. Normally, SH can be precisely controlled in a batch process. Since polymer always contains residual catalyst or unreacted monomers or contaminants, from the thermodynamic perspective, it is hardly possible to justify homogeneous nucleation in polymeric melt foaming. However, when the pressure gradient and the surface tension dominate, it is not surprising to find good agreement with homogeneous predictions. Table 4.4 presents PS/N2 nucleation results that suggest that SH is a viable homogeneous nucleation parameter. The homogeneous equation was tested by batch microcellular experiments during which an inert-gas-saturated amorphous polymer was exposed to a

© 2000 by CRC Press LLC

TABLE 4.4

Superheat Effects on Nucleation for PS/N2.

Saturation Pressure Mpa

Foaming Pressure Mpa

Superheat Mpa

Cell Density #/cm3

4.0 7.0 10.5 14.0

0.1 0.1 0.1 0.1

3.9 6.9 10.4 13.9

107 2⫻107 8⫻107 8⫻108

(Data collected with permission from Kumar and Suh, Polym. Eng. Sci. 30, 1990.)

lower pressure and a higher temperature (above Tg) to induce fine cell formation. Figure 4.7 presents the comparison. It is not surprising to find orders of magnitude difference from predictions. First, an amorphous polymer can hardly be structurally characterized by the homogeneous approach. Second, pressure reduction and temperature increase generally happened simultaneously in actual processing and affected gas activity and polymer chain mobility effects. The thermodynamic-based model especially cannot cover the latter. Modification becomes necessary to improve agreement with batch foaming experimental results [24].

FIGURE 4.7 Homogeneous nucleation comparison between theory and experiment; cell density vs. external pressure for PS/N2. (Replot from V. Kumar’s Ph.D. Dissertation, MIT, 1998 [23].)

© 2000 by CRC Press LLC

4.3.2 MODIFIED NUCLEATION THEORIES In the conventional nucleation theories, the rate of nucleation is governed by the rate of diffusion (or vaporization) of gas molecules from the surrounding liquid through the interphase. For general liquids, Kagan [25], in 1960, included the hydrodynamic and heat transfer effects for vaporization to obtain the corrected formula: J ⫽ N␴/␩(␴/(kT))1/2(1 ⫺ Pl/Pb) exp (⫺16␲␴3/(3kT(Pb ⫺ Pl)2)

(17)

Blander and Katz [3] and Blander [26] developed a similar correction to account for diffusion and viscosity-controlled nucleation rate: JD ⫽ ND(Cb ⫺ Cl)(kT/␴)1/2 exp(⫺16␲␴3/(3kT (Pb ⫺ Pl)2)

(18)

J␩ ⫽ N␴/␩(␴/kT)1/2(Pb/(Pb ⫺ Pl)) exp(⫺16␲␴3/(3kT(Pb ⫺ Pl)2) (19) Ruengphrathuengsuka [27] developed a general nucleation expression for a non-Newtonian fluid, including the hydrodynamic, heat, diffusion, and viscous effects. J ⫽ N/(1 ⫹ ␦D ⫹ ␦␩)(2␴/␲/m(1 ⫺ YRbcr/␴))1/2 exp(⫺16␲␴3/(3kT((Pb ⫺ Pl)2 ⫺ Y)) (20) in which Y ⫽ ␳g␮g⫺l/Mg. The parameters, ␳g, ␮g, and Mg represent the gas density, gas chemical potential in liquid phase, and gas molecular weight, respectively. The surface tension and the cluster pressure (or bubble pressure), Pb, appear to show strong influences on the nucleation rate calculation. With respect to a higher saturation pressure, the cluster pressure becomes higher; in other words, a higher solubility is obtained. Moreover, as suggested in Figure 4.8, a higher pressure tends to lower the gas/melt surface tension that was obtained through Wihelmy technique (net force divided by contact area) by Ruengphrathuengsuka [27]. Figure 4.9 shows a higher cell density at increased saturation pressure or foaming temperature, and the temperature has a similar impact in the batch design LDPE/nitrogen experiments. Similar results were reported by microcellular HIPS/CO2 extrusion experiments by Park et al. [28]. At increased temperature, the solubility becomes lower, and this helps nucleation by leaving less residual gas in the polymer. Combined with higher volatility, a higher diffusion coefficient, and lower surface tension, it is not surprising to find the temperature increase a primary parameter for foaming. Nucleation density could affect thin foam sheet density. As pointed out by Lee and Ramesh [29], the foaming efficiency, defined as the actual foam density

© 2000 by CRC Press LLC

FIGURE 4.8 Surface tension variation vs. saturation pressure for LDPE/N2. (Reprint from Ruengphrathuengsuka’s Ph.D. Dissertation, Texas A&M, 1992 [26].)

over the theoretical density, becomes higher at a higher saturation pressure or a higher cell density because gas loss from the sheet is reduced. Unlike semicrystalline LDPE, Goel and Beckman [30] experimented amorphous polymethylmethacrylate (PMMA) with carbon dioxide, in which a less sensitive thermal dependency of cell density was observed. At increased temperatures, the reduced surface tension and solubility are obvious factors for more nucleation. However, increased diffusion can cause significant surface evaporation when the surface-to-volume ratio is high, for example, thin sheet. In PET microcellular experiments, Baldwin et al. [31] noted that the skin has less cell density than the center, and when foaming temperature was above 100°C, thermal dependency became insensitive. Kumar et al.’s PVC foam experiments employed 2 mm thick sheet rather than 0.4 mm [32]; the tempera-

© 2000 by CRC Press LLC

FIGURE 4.9 Pressure and temperature effects on LDPE/N2 nucleation. (Replot from Ruengphrathuengsuka’s Ph.D. Dissertation, Texas A&M, 1992 [26].)

ture showed a greater impact as illustrated in Figure 4.10. Temperature appears to draw different, and sometimes, competing mechanisms into the nucleation phenomena. Considering semicrystalline polymer, the temperature effects on crystallinity formation make it a critical nucleation parameter. A brief thermal cause and effect summary is presented in Table 4.5. In the polystyrene/toluene nucleation analysis, Han and Han [33] proposed the free energy change: ⌬Fp ⫽ ⌬F ⫺ ⌬Fs ⫺ ⌬Ft

(21)

where ⌬F represents the free energy change required for single-component phase transformation, and subscripts s and t represent supersaturation and polymer-solvent interaction. Considering an empirical equation for the fre-

© 2000 by CRC Press LLC

FIGURE 4.10 Cell density vs. foaming temperature for PVC/CO2 nucleation. (Plot based on data from V. Kumar et al., ANTEC, 1992 [31].)

quency factor, the semiempirical nucleation rate for polystyrene and toluene is proposed as follows: J ⫽ 4.71 ⫻ 1034MD/(4␲Rbcr)exp(⫺(42,344/T ⫹ ⌬Fp/(nkT))

(22)

where D represents the diffusion coefficient. With viscosity and surface tension dependent on the temperature and the diffusion coefficient dependent

TABLE 4.5

Foaming Temperature Effects on Nucleation.

Surface Tension Shear Viscosity Elastic Viscosity Critical Superheat, SHc Diffusion Solubility Volatility *Enhance bubble growth.

© 2000 by CRC Press LLC

High Temperature

Nucleation

Decrease Decrease Decrease Decrease Increase Decrease Higher

Increase *Decrease No Effect Increase No Effect Increase More

TABLE 4.6

Nucleation Comparison for PS/Toluene; Experiments and Predictions.

Temperature (°C)

Concentration of polystyrene (wt%)

N ⫻ 10⫺15 (number/m3) (Experimental)

N ⫻ 10⫺19 (number/m3) (Theoretical)

150 150 150 170 180

60 50 40 50 50

17.33 4.374 1.481 7.480 12.29

40.00 16.76 2.531 16.76 14.05

(Reprinted with permission from Han and Han, J. Polym. Sci. B: Polym. Phys., 1990.)

on the free volume and temperature, J was calculated, and its comparison with experimental results is shown in Table 4.6 [33]. Although a reasonably good agreement was observed, Han and Han claimed a more satisfactory model to describe pre-nucleation pockets and a model for coalescence is still needed. Patel et al. [34] used small-angle X-ray analysis to detect the nitrogen solubility in LDPE melt by correlating cluster distribution to report a good agreement with Henry’s law constant. In hydrocarbon homogeneous nucleation, Kwak et al. [35, 36] postulated cluster formation in the supersaturated state and investigated its stability by assuming face center packing and constant dispersion force. Either one needs justification for concentrated polymer. Further insight on cluster formation and transformation into a critical pocket will be extremely important in comprehending the complex nucleation dynamics. Instead of using the Sugden expression for the surface tension [33] or the Eotvos expression [23], Ruengphrathuengsuka [27] performed experiments to investigate its variation with temperature and pressure for the LDPE/nitrogen system as presented in Figure 4.11, in which the surface tension decreased over 50% after loading with a gas blowing agent. Shafi et al. [37] presented an “influence” volume approach to account for growth-induced gas diffusion in the rate nucleation process, in which nucleation continued until all melt was influenced to a point below the prescribed threshold. The modified homogeneous nucleation equation is as follows: J ⫽ N(2␴/␲mB)1/2exp(⫺16␲␴3/(3kT(Pb ⫺ Pl ⫺ I ⫹ Pbln⍀/Z)2) (23) Where, I, ⍀, and Z denote elasticity number, activity coefficient of dissolved gas, and compressibility factor of gas in the melt, respectively. Further experiments are necessary to verify the postulation. Because gas molecule is in the range of Angstrom, thousands of gas molecules without compression are required to make up a micro-sized hole. Fast diffusion into a gas pocket or a combination of gas pockets at a pressure drop are plausible routes for homo-

© 2000 by CRC Press LLC

FIGURE 4.11 Surface tension variation vs. temperature for LDPE and LDPE/N2. (Reprint from Ruengphrathuengsuka’s Ph.D. Dissertation, Texas A&M, 1992 [26].)

geneous nucleation and more and precise experiments are necessary for the gas/melt system. In polymer/gas batch foaming, the discrepancies between prediction and experimental results still exist. In fact, material impurities (residual catalyst, unreacted monomers, dusts, etc.) and machinery contact in foam extrusion make heterogeneous nucleation a more realistic mechanism to consider. 4.3.3 HETEROGENEOUS NUCLEATION Heterogeneous nucleation accounts for the surface energy when nucleation occurs at the interface of a liquid and a clean surface. Its interfacial phenomena is depicted in Figure 4.12, and Blander and Katz [3] proposed its work as follows: W ⫽ ␴lgAlg ⫹ (␴sg ⫺ ␴sl)Asg ⫺ ⌬P Vb ⫹ n(␮g ⫺ ␮l)

© 2000 by CRC Press LLC

(24)

FIGURE 4.12

Heterogeneous nucleation schematics.

Subscripts lg, sg, and sl represent liquid-gas, solid-gas, and solid-liquid interphases, respectively. Because of the presence of solid surface, a chemical equilibrium between gas and melt is not useful for bubble formation. Considering the interfacial contact, the nucleation rate J is obtained as follows: J ⫽ N2/3(1 ⫹ cos␪)/2(2␴/(␲mF))1/2 exp(⫺16␲␴3/(3kT(Pb ⫺ Pl)2) (25) where F, the geometry factor, is defined as (2 ⫹ 3cos␪ ⫺ (cos␪)3)/4. Computational results of PS/S are tabulated in Table 4.7, in which differences from reality are hardly justifiable. Attaim [38] noted the presence of a critical upper limit determined by thermodynamic parameters and greatly affected by kinetic factors. Modification became necessary to improve agreement with experimental observations. In addition to the chemical potential, the surface energy and deformation energy [39], Colton and Suh [40] proposed to correct chemical potential by subtracting L-J energy to describe Polystyrene/Zinc stearate and carbon dioxide nucleation. As indicated in Figure 4.13, orders of magnitudes difference from experimental results are difficult to justify. Colton [41] applied the linear mixing rule to calculate the surface tension of semicrystalline polypropylene for a mixed-mode nucleation. Nucleated polypropylene and copolymer had better agreement with experimental data than non-nucleated and talc-filled polypropylene. Ramesh et al. [42] reported a better cell density correlation by adding rubber particles in the amorphous polymer and in polystyrene, and they postulated that preexisting cavities are the main sources for nucleation. It was found that the size of the rubber particles in the polystyrene phase follow the log-normal distribution. By applying proper empirical constants, good nucleation agree-

© 2000 by CRC Press LLC

TABLE 4.7

Heterogeneous Calculations for PS/S.

Temp. (°C)

Temp. (K)

(atm)

SH (atm)

␳b (g/cm3)

␴c (erg/cm2

A ⫻ 1032 (#/cm3 ⭈ sec)

B

Jd (#/cm3 ⭈ sec)

150 171 190 210 230 250 280 300 325 350

423 444 463 483 503 523 553 573 598 623

5.0 7.9 10.5 15.0 21.5 34.8 52.6 109.9 181.3 403.4

0.012 0.026 0.039 0.062 0.095 0.161 0.25 0.54 0.89 2.00

0.79 0.76 0.74 0.72 0.70 0.67 0.64 0.62 0.59 0.55

18.1 15.6 13.9 12.1 10.4 8.6 6.2 4.7 2.9 1.2

9.66 8.62 7.91 7.18 6.47 5.64 4.58 3.86 2.88 1.70

1.2 ⫻ 1010 1.5 ⫻ 109 4.5 ⫻ 108 1.1 ⫻ 108 2.9 ⫻ 107 5.5 ⫻ 106 8.2 ⫻ 105 7.4 ⫻ 104 6.0 ⫻ 103 7.4 ⫻ 101

0 0 0 0 0 0 0 0 0 2.2

a

a Kw

Kw(T) are from Werner, 1981 [9]. ␳(T) are found using Watson’s expansion method. ␴(l) are calculated using Eötvös’ equation. d J ⫽ A exp(⫺B), A ⫽ 3.1 ⫻ 1035(␳2␴/M)1/2, B ⫽ 1.2 ⫻ 105␴3/(T ⭈ SH2) M: molecular weight of styrene. b c

© 2000 by CRC Press LLC

FIGURE 4.13 Microcellular nucleation comparison between theory and experiment for PS/CO2. (Replot with permission from Colton and Suh, Poly. Eng. Sci., 27, 7, 1987 [39].)

ment was observed as illustrated in Figure 4.14. Also noted is the submicron particle size for effective nucleation. Lee [43] reported that even for low-level gas dissolution in a polymeric melt system, neither homogeneous nor heterogeneous nucleation theories were able to describe foam nucleation. Table 4.7 contains polystyrene and styrene heterogeneous nucleation calculations. When gas dissolution increases, intra- and intermolecular forces become important. Nucleation modeling will be more complicated. Nonetheless, Jemison et al. [44], based on the heterogeneous results on methanol and water, concluded that the favorable conditions for satisfying nucleation kinetics were not realistic. Therefore, these theories were not able to offer a solid ground for plastic foam formation in foam extrusion or in foam devolatilization.

4.4 CAVITATION Cavitation is a physical phenomenon that most often occurs in the areas of discontinuity in the fluid due to external disturbance-induced pressure variation. Bubble formation occurs when the pressure reduction is greater than the critical value or its difference overcomes surrounding confinement. As pointed

© 2000 by CRC Press LLC

FIGURE 4.14 Cell density vs. particle size effects for HIPS/CO2 Nucleation: computation and experiment. The solid line represents computational results. (Replot with permission from Poly. Eng. Sci., 1944 [42].)

out by Rayleigh in 1917 [45], this phenomenon involves physical and flow properties and sometimes the chemical properties of the fluid. It is a very complex research topic, and a clear understanding has not yet been reached. For simple liquids, Knapp et al. [46] ascribe bubble formation to vibrations, vortices, or local pressure variations that accompany flow acceleration at high flow velocities, for instance, upon pouring or shaking a just-opened soda can or wine bottle, pressure variations induce more superheats to already supersaturated solution to cause vigorous foaming. In fact, long-chain polymer is known for its viscoelastic and creeping nature, and it tends to maintain flow continuity [47]. In other words, flow-induced significant acceleration is not anticipated for viscous polymeric melt. It is not, however, unusual to observe quench-induced cavitation in melt solidification, as reported on LDPE by Ainslie [48].

© 2000 by CRC Press LLC

Besides, a hydrophobic solid surface surrounded by liquid can provide residence for undissolved gas as illustrated in Figure 4.15, which becomes a nucleus for bubbling when favorable thermodynamic conditions occur. Chin [49] postulated that existing cavities appeared to be a plausible mechanism for nucleation in foam devolatilization. It should be pointed out that cavitation is different from classical heterogeneous nucleation that has no preexisting “gas seeds” in the interface. It is also different from gas entrainment during motion when hydrodynamic force exceeds the surface tension force to cause forced nucleation [50]. Microvoids and “holes” are postulated to exist in statistical equilibrium throughout the liquid as a result of random thermal fluctuations [51, 52]. The radius of these holes is around 10 ⫺8 cm, and the probability of spontaneous formation of holes capable of expanding is negligible. Free volume is a wellestablished concept in polymers. It successfully characterizes the polymer melt viscosity by applying “holes” theories. Lee [43], by applying the Turnbull and Cohen probability model [53], pointed out the improbability of forming holes with a radius close to the critical bubble radius. One can argue that, in a dynamic state, it is not impossible to observe free volume distribution shifting to extremes to generate nucleable holes. However, foam extrusion is too far above the glass transition temperature to cause significant variation of free volume distribution. As illustrated in Figure 4.16 from devolatilization experiments [54], very low deformation is sufficient to enhance nucleation. In the visualization experiments, as vacuum was established in a staged manner, new

FIGURE 4.15 Stabilization of gas-solid pocket in hydrophobic crevice: (a) liquid saturated with gas, interface with an equilibrium contact angle ␪e ⬎ ␲/2 ⫹ ␣; (b) liquid undersaturated with gas, liquid advances when ␪A (advance angle) ⬎ ␪e, gas solution proceeds to establish ␪e; (c) liquid supersaturated with gas, liquid recedes when ␪R (recede angle) ⬍ ␪e, gas phase expands till ␪R ⫽ ␪e.

© 2000 by CRC Press LLC

FIGURE 4.16 Nucleation visualization experiment (Methyl Chloride in PDMS—photographs taken through glass tube: (a) vacuum applied at 21 sec, (b) low RPM commenced at 50 sec, (c) pool volume increase, (d) bubble disappearance. (Reprinted with permission from Biesenberger and Lee, Polym. Eng. Sci., 27, 1, 1987 [53].)

bubbles came into existence when a new lower vacuum was applied. That suggests that a given superheat can induce new bubbles into formation. In the rubber industry, Gent and Tompkins [55] presented theoretical results of expanding a small hole in a highly elastic solid. Considering surface and elastic energy, the inflation pressure is as follows: P ⫽ 2␴/R ⫹ G(2.5 ⫺ 0.5 (R/R0)⫺4 ⫺ 2(R/R0)⫺1)

(26)

where R and G represent cavity radius and elastic constant of the polymer, respectively, and subscript 0 denotes initial state. Inflation pressure can be over 50 atm for a submicron hole (i.e., 100 A radius), which appears to be reasonable for foam extrusion. In making high-impact polystyrene (HIPS), polybutadiene is mixed in the PS. The former has a higher thermal expansion coefficient. It can be conceived that after cooling to room temperature, a thermal-induced stress may exceed interfacial cohesion to cause microvoids. This was confirmed by Kekkula et al. [56] by using a Transmission Electron Microscope (TEM) to investigate rubber particle morphology as shown in Figure 4.17. Noteworthy is the void dimension in the submicron domain. Ramesh et al. [57] used the microvoids in the HIPS to control cell nucleation

© 2000 by CRC Press LLC

when saturated with carbon dioxide gas and then exposed to an elevated temperature. They reported that particle size of minimum submicron (0.2 micron) is necessary to function as an effective nucleator as indicated in Figure 4.14. Lately, in the simulation of cavity growth in polymer-rubber blend, Steenbrink and Giessen [58] reported that there is a threshold value for rubber modulus at which the cavity would grow. Harvey et al. [59], in their foam formation in organisms study, proposed that the undissolved gas nuclei could exist in submicroscopic, hydrophobic cracks and interstices in microscopic solid surface cavities. Various possible states of cavity are presented in Figure 4.15. This mechanism offers two great advantages: first, it explains the pre-nuclei state, and second, it provides a physically conceivable way of distributing nuclei throughout liquids. Also noted was the submicron size of crevice, 10⫺5 to 10⫺4 cm. Atmospheric and industrial dust may provide such particles. In devolatilization experiments, Biesenberger and Lee [60] found that in the absence of a nucleating agent, polydimethylsiloxane melt showed minimum foaming after vacuum was applied, however, a slight deformation drove out swarms of bubbles immediately. They combined the metastable state and cavity concept to ascribe shear as a detaching mechanism for gas bubble from solid cavity. However, one can argue that shear-induced

FIGURE 4.17 Transmission electron micrograph (TEM) of high-impact polystyrene (HIPS); rubber particles in polystyrene (Reprint from Keskkula et al., Polymer, 27, 1986 [55].)

© 2000 by CRC Press LLC

FIGURE 4.18 Cavity model—five successive stages for gas bubble formation: (a) stable gas cavity, (b) at pressure reduction, (c) metastable cavity, (d) under shear and (e) unstable gas bubble. (Reprinted with permission from Lee and Biesenberger, Polym Eng. Sci, 29, 1989.)

© 2000 by CRC Press LLC

viscosity reduction is favorable for gas cluster expansion. It was pointed out by Lee [7] that a slight deformation appears to be sufficient to drive out vigorous foaming. Deagneault et al. [61] reported solubility drop under high shear for polystyrene. Solubility reduction evidently contributes to bubble formation. Again, for low shear and combined with an almost instant foaming response, the solubility reduction does not appear to be a plausible mechanism. The proposed five successive steps for bubble formation are illustrated in Figure 4.18, in which stable gas cavity evolves into metastable cavity when the surrounding condition changes. The pressure drop rate is evidently a key factor in developing metastable cavities. In theory, a successive gradual pressure drop can release the whole pressure for a long time without generating any bubbles. On the contrary, at a sharp pressure drop, it is easier to have the meniscus fall outside the static contact region to form metastable cavities. At step (d), a shear force simply distorts the gas cavity into a lower interfacial tension state that favors cavity expansion. When it exceeds the critical state, it grows into a spherical bubble. This model offers a reasonable and qualitative description for correlating shear effects with foaming in a melt flow. Another hypothesis is worth mentioning before we move to foam extrusion. That is the “string” theory. It assumes that gas molecules form strings dispersed in the melt, and sufficient diffusion time is lapsed for homogeneity. The shape of the string is determined by total pressure and the surrounding polymeric molecular structure. When the pressure drop rate gets higher, it has less time for the string to group together, and it is easier to form nucleating sites at various locations of strings as long as the critical condition is met. Although this model needs further work to correlate with observations, it, at least, explains the pressure drop rate effects on cell density.

4.5 FOAM EXTRUSION NUCLEATION Han and Han [62] studied nucleation of polystyrene and trichlorofluoromethane (CFC-11) extrusion, in which a slit die with side glass windows was established for laser detection of bubble images. They reported two kinds of nucleation: flow induced at the center and shear induced at the die wall. At higher CFC-11 loading, as shown in Figure 4.19, the nucleation point is close to the theoretical prediction, PR (pressure ratio) equal to one. Center cells were shown to be larger in size than those of the side. Earlier nucleation appeared to be a reasonable explanation. Lee [7] investigated foam nucleation on an extruder with a specially designed die to allow die gap adjustment externally. In the LDPE dichlorodifluoromethane (CFC-12) extrusion experiments, a physical nucleator was added to control cell size and its distribution. It can form a conglomerate as a potential nucleating site [63]. Shear-enhanced foam formation was qualitatively well correlated with preexisting cavity theories. Force balance on metastable

© 2000 by CRC Press LLC

FIGURE 4.19 Nucleation experiments: (a) pressure profile and (b) nucleation sites in the slit flow channel for PS/CFC-11 (4 wt%) at 180°C. (Reprinted with permission from Han and Han, Polym. Eng. Sci., 24, 28, 1988 [61].)

cavity as presented in Figure 4.20, a replot of Figure 14.8(d), demonstrates dominant shear force and surface tension force, which gives the following capillary number: Ca ⫽ Rbcr␩␥/(4␴)

(27)

where ␩ and ␥ represent the melt viscosity and the average shear rate, respectively. Figure 4.21 suggests a critical radius, around 0.3 micron, by extrapolating laser experimental results. Also noted in Table 4.8, it is relatively insensitive to the temperature and the weight of solvent. When plotted, the cell density vs. Ca in a semi-logarithmic scale, a straight-line mode in Figure 4.22, instead of concave curve in the linear-linear cell density vs. ␥ chart, underscores the important role played by shear force in the metastable state. Its loglinear ordinates are similar to the general form of the rate equation: J ⫽ K1 exp(⫺K2)

(28)

where K1 and K2 are, generally, material parameter with frequency factor and system parameter, respectively. Evidently, the straight-line mode suggests that

© 2000 by CRC Press LLC

FIGURE 4.20 Metastable cavity under shear: capillary number (Ca) defined as shear force over tension force, Ca ⫽ (a) R1␩␥/(4␴), (b) after distortion, R2␩␥/(4␴).

FIGURE 4.21 Bubble radius vs. time for PS/Toluene nucleation at an equilibrium pressure of 2,859 Kpa and at three temperatures (0°): (O) 150, (⌬) 170 and (□) 180. (Reprinted with permission from John Wiley & Sons, Inc., Figure 16 of Han and Han, J. Poly. Sci. B: Poly Phys., 28, 1990a [68].)

© 2000 by CRC Press LLC

TABLE 4.8

Critical Radius Results.

Polymer Conc. (wt %)

Temperature (°C)

Critical Pressure (kPa)

Critical Time (s)

Critical Radius (␮m)

60 60 50 50 50 40

150 170 150 170 180 150

271.0 303.0 315.1 381.4 408.5 486.3

0.302 0.273 0.277 0.248 0.238 0.214

0.24 0.28 0.29 0.31 0.32 0.33

(Reprinted with permission from Han and Han, J. Polym. Sci., B: Polym. Phys., 1990.)

the shear force is an energy source for nucleation. The use of a chemical blowing agent is a common practice for nucleation without adding a physical nucleator. This becomes an interesting comparison; as illustrated in Figure 4.23, shear enhancement increases dramatically by adding a physical nucleator [63]. How to incorporate the shear energy into the free energy change term becomes another interesting subject for foam nucleation. Dey and Todd [64] conducted visual experiments by implementing a sapphire window to videotape gas/melt flow. They reported that a two-phase flow is detrimental for consistent foaming, and when the extrusion melt temperature is reduced to close to the melting point, a cloudiness appeared prior to swarms of nucleation. Sites of crystallization become potential nucleation sites for foaming, however, the actual processing temperature has to be kept much higher than the melting point of the polymer to facilitate viscous flow in the extruder. It is a common practice to add a nucleating agent to assist cell size formation and its distribution. The extrusion process generally provides adequate thermal and mechanical energies to polymers to form a homogeneous melt to evenly disperse in a narrow flow region to build up enough pressure for optimal free expansion. In foam extrusion, a higher temperature seems to be a favorable factor in generating bubbles. However, the melt strength needed to sustain bubbles becomes a limiting parameter. Thermal effects on morphology are presented in Figure 4.24 [65]. The cell structure, or cell integrity, appears to be better when the processing temperature is decreased. This also suggests that the nozzle temperature and the melt temperature can independently control the surface cell structure and core cell structure, especially when a large die flow region is used. Park et al. [28] designed various die nozzles for HIPS/CO2 extrusion experiments in which the nucleation phenomena at similar total pressure drop and different pressure drop rates were studied. As illustrated in Figure 4.25, finer cells were associated with a sharp pressure gradient rate that was attributed to the gas diffusion and its characteristic length. However, considering the preexisting cavities, a near straight line could be obtained in the log-linear plot of

© 2000 by CRC Press LLC

FIGURE 4.22 Nucleation Results for LDPE/CFC-12; log (cell density) vs. Capillary number at different talc nucleating agent levels. (Reprinted with permission from Lee, Polym, Eng. Sci., 33, 1993.)

cell density vs. capillary number. In the conventional nucleation theory, the degree of superheat defined in Equation (16), the difference between system pressure and surrounding pressure, is a more critical parameter than the pressure drop rate. However, as illustrated in Figure 4.26 [66, 67], after gas dissolution is completed, surrounding pressure increase can make fugacity increase to suggest that superheat is no longer solely dependent on the amount of gas

© 2000 by CRC Press LLC

FIGURE 4.23 Shear effects for various nucleators with and without fluoropolymer (400 ppm FP) at different nucleating agent levels—FPN is an endothermic chemical blowing agent from Reedy International. (Reprinted from Lee, ASME, Cell. Micro. Mat., 53, 1994 [62].)

dissolution. In foam extrusion, after a given amount of gas is injected to dissolve in the molten polymer, extruder pressure can be easily built up much higher than the system vapor pressure. The additional hydrostatic pressure will not improve solubility. However, it establishes additional “mechanical” superheat, which can be expressed as follows: SH ⫽ SHce ⫹ SHme

(29)

A higher operation pressure can certainly make a higher total superheat at the cost of more power consumption. The resulting higher pressure drop rate can increase the number of metastable sites by promoting gas activity under the cavity membrane. In general, a high pressure drop inherently causes high shear in the processing. The shear energy can virtually develop more heat for the flow system to, in turn, increase gas volatility. Shear force can, thus, enhance the transfer from the metastable to the unstable state. In addition, a sharp pressure drop rate and high shear tend to make a viscoelastic polymer “swell,” in

© 2000 by CRC Press LLC

FIGURE 4.24 Thermal effects on cell morphology for HIPS/CO2 foams; microstructures at various melt (Tc) and nozzle temperatures (Tn). (Reprinted with permission from Behravesh et al., ANTEC, 1998.)

© 2000 by CRC Press LLC

FIGURE 4.25 Cell density vs. pressure drop rate for HIPS/CO2. (Replot with permission from Park et al., ASME, Plas. And Plas. Comp., Vol. 46, 1993.)

FIGURE 4.26 Hydrostatic pressure effect on polymer/gas fugacity. (Reprinted from Handa, NRC lecture notes, 1998.) © 2000 by CRC Press LLC

FIGURE 4.27 Rod nucleation results for LDPE/HCFC-142b; skin vs. core. (Reprinted with permission from Lee and Kim, ANTEC preprint, 1998 [67].)

other words, to bring forth its expansion potential, which is another favorable condition for nucleation. In the laser detection and visual experiments with polystyrene/CFC-11, Han and Han [62] observed early nucleation in the center of a slit die. Center cells appeared larger than side cells at the die exit. To prevent early center formation, Lee and Kim [68] designed a sharp tapered end attached to a pipe die for LDPE/HCFC-142b experiments. It provided a sharp pressure drop at die end to minimize premature foaming for over 25 times expansion. Center nucleation was examined against skin nucleation for pressure-controlled and shearcontrolled comparison. In Figure 4.27, the skin shows more nucleation than the center. The latter is pressure drop rate controlled, and the former experiences high shear. Again, the cavity model appeared fitting in describing the differences.

4.6 SUMMARY Although foam extrusion involves efficient energy transfer and effective material transport, it adds more complexity to the already complex dynamic nucleation. This chapter adopts a realistic approach, starting with fundamental

© 2000 by CRC Press LLC

principles to evolve to correlate with extrusion parameters. Nucleation is the beginning of a process to stabilize a supersaturated state en route to a stable state. Applying equilibrium theories, one can calculate the chemical superheat, SHce, that is determined by the amount of gas dissolution and the subsequent pressure drop and/or temperature rise. The nucleation rate equations developed previously account for chemical superheat and surface tension confinement. They fail to explain parameter variation and surrounding phase-induced transport issues. It is virtually not surprising to find that orders of magnitude differ between the experimental results and predictions in batch foaming. Modifications were made to include the transport, rheological, thermal, and physical properties, and to cover the gas dissolution and its interaction with polymeric melt. However, differences from observations still exist. The cavity model and mechanical contribution to superheat were proposed to underscore the importance of metastable state in foam nucleation. The stable-metastableunstable process appears to be plausible for foam nucleation. Although qualitative agreement was observed, quantitative agreement should be explored on the existing theoretical and experimental bases. In polymer/gas analysis, it is worthwhile to establish a useful phase diagram covering a gas-enriched phase to a polymer-concentrated phase. The former is rarely available for a gas/melt foaming system. It will offer a logical “path” from the formation of gas clusters and its transformation from a stable to a metastable and, ultimately, to an unstable state. Energy terms involved in this separation kinetics can be identified and, hopefully, quantified for modification and implementation in the nucleation equations to make them more realistic. Based upon superheat (SH) and phase separation, Figure 4.28 illustrates a simplified sketch in which SHc denotes critical superheat for nucleation, whereas SHT denotes the total superheat determined by gas dissolution and pressure reduction. Phase separation and spinodal decomposition concepts, as indicated in Figure 4.4, strongly suggest that a modest superheat, or SHc, be necessary for nucleation. In actual foam extrusion, SH is a function of position, or time, in the pressure-controlled die flow region. Knowing the critical bubble size, submicron, as reported by Han and Han [69], critical superheat can be calculated as follows: SHc ⫽ 2 ␴ / Rcr

(30)

Assuming a negligible hydrostatic pressure contribution, the nucleation point can be determined inside the die. From that point on, positive SH continues to increase until SHT at the die end, during which growth starts and continues to progress outside the die into free expansion. However, it is desirable to figure out the effects of hydrostatic pressure not only in its contribution to superheat, but also in its drop rate to convert into energy terms for an analytical expression. When surface tension variation with temperature, gas concentration, and

© 2000 by CRC Press LLC

FIGURE 4.28 Nucleation in nonparallel plate pressure flow; flow pressure and vapor pressure (or fugacity) variation in flow direction.

pressure is available [70], Equation (29) can become a more realistic foundation for rate equation development. A further issue is the secondary nucleation out of surface stretch as observed by Albalak et al. [71]. It is more mechanical than chemical-induced nucleation. An overall view is necessary to cover chemical and mechanical contribution to superheat. A narrow die flow is necessary to build up high enough pressure to overcome critical superheat to the die end for free expansion, in other words, to keep PR (pressure ratio) above one until the die end is reached. On the other hand, the corresponding shear heat generally makes the heat-sensitive nucle-

© 2000 by CRC Press LLC

ation even more sensitive. That means that temperature variation in flow direction and its distribution in cross-flow direction can make nucleation deviate from impulse (or one-point) nucleation. For maximum expansion, melt/gas needs to be cooled to enhance melt strength before entering the die. The interfacial phenomenon, melt/die surface, becomes another interesting processing issue. As the nucleation experimental technique continues to be upgraded, currently, ultrasonic techniques have been used to enhance foaming [72] and have been applied to monitor foaming [73]. Combined with a high-speed camera on the side window for dynamic nucleation experiments, insight into the precritical state can be established. Total pressure, pressure drop rate, and shear energy appear to qualitatively correlate with nucleation via cavity model. Nucleation potential (superheat) increases as total pressure, dissolved gas partial pressure, or processing pressure increases. The nucleation barrier seems to be reduced by pressure drop rate and shear energy to make more bubbles come into existence from a metastable state. A detailed development including cluster formation and transformation and interfacial contact variation under nonisothermal heat transfer is necessary for comprehending the dynamic foam nucleation. The “string” concept may play a role in cluster formation. Nonetheless, the fundamental foaming principles, engineered parameters, and visualization techniques presented in this chapter are useful tools in correlation with observations and actual machine design for better nucleation control.

4.7 NOMENCLATURE A a B C(n) Ca D F ⌬F G J Kw Kw⬘ K1 K2 k M m

Surface area, cm2 Activity Frequency factor for nucleation equation Gas cluster containing n molecules Capillary number, R␩␥/(4␴) Diffusion coefficient, cm2/sec. Geometry factor Free energy change Elastic constant Nucleation rate, #/cm3/sec. Henry’s law constant, atm Dimensionless Henry’s law constant Material constant for simplified nucleation equation System constant for simplified nucleation equation Boltzmann constant Molecular weight, g Mass of a gas molecule

© 2000 by CRC Press LLC

N Nb/V n P PR P0 ⌬P p R SH T V Wg W(n) X Y

Number of molecules per unit volume Cell density, #/cm3 Number of molecules Pressure or external pressure, Pa or psi Pressure ratio; flow pressure/vapor pressure Vapor pressure, Pa or psi Pressure difference, Pa or psi Partial pressure, Pa Radius, cavity or bubble, cm Superheat, Pa Temperature, °K Volume, cm3 Weight fraction of gas phase Work to sustain a pocket with n molecules Constant, (Nb/V)3/4␲Kw⬘ Constant

Greek letters: ␹ Interaction parameter for Flory-Huggins equation ␧ A small perturbation ␥ Shear rate, 1/sec. ␩ Viscosity, poise or Pa-sec. ␮ Chemical potential ␪ Contact angle, degree ␳ Density, g/cm3 ␴ Surface tension, dyne/cm Subscripts b Bubble phase c Critical state bcr Critical bubble radius ce Chemical equilibrium D Diffusion controlled g Gas phase l Liquid phase lg Liquid-gas interface me Mechanical equilibrium min Minimum p Polymer phase sg Solid-gas phase sl solid-liquid phase t Polymer-solvent interaction T Total ␩ Viscosity controlled

© 2000 by CRC Press LLC

4.8 REFERENCES 1. Zeldovich, J. B., “On the Theory of New Phase Formation; Cavitation,” Acta Physicochem (USSR), 18, 1, 1943. 2. Gibbs, W., The Scientific Papers, Vol. 1, Dover, New York, 1961. 3. Blander, M. and Katz, J. L., “Bubble Nucleation in Liquids,” AIChE J., 21, 5, 833–848, 1975. 4. Biesenberger, J. A. and Todd, D. Section I: Fundamentals in Devolitilization of Polymers, ed. by Biesenberger, J. A., Hanser, New York, 1983 5. Kennedy, R. N. Sec. XII: Extruded Expanded Polystyrene in Handbook of Foamed Plastics ed. by Bender, R. J., Lake Publishing Corp., 1965. 6. Rubens, L. C., Griffin, J. D. and Urchick, D., “Process of Foaming and Extruding Polyethylene Using 1,2-dichlorotetrafluoroethane as the Blowing Agent,” U. S. Patent 3,067,147, 1962. 7. Lee, S. T., Chap. Six: “A Fundamental Study of Foam Devolatilization,” Polymer Devolatilization, ed. by Albalak, R. J., Marcel Dekker Inc., New York, 1996. 8. Hildebrand, J. H. and Scott, R. L., Chap. XV. Solubility of Gases in Liquids, The Solubility of Nonelectrolytes, Reinhold Publishing Corp., New York, 1964. 9. Werner, H. W., “Devolatilization of Polymers in Multi-Screw Devolatilizers,” Kunststoffe, 71, 18, 1981. 10. Chaudhary, B. I. and Johns, A. I., “Solubilities of Nitrogen, Isobutane and Carbon Dioxide in Polyethylene,” J. Cell. Plas., 34, 312–328, 1998. 11. Durrill, P. L. and Griskey, R. G., “Diffusion and Solution of Gases into Thermally Softened or Molten Polymers: I. Development of Technique and Determination of Data,” AIChE J., 12, 1147–1151, 1966. 12. Gorski, R. A., Ramsey, R. B. and Dishart, K. T., “Physical Properties of Blowing Agent Polymer Systems: I. Solubility of Fluorocarbon Blowing Agents in Thermoplastic Resins,” Proc. SPI 29th Ann. Tech. Mark. Conf., p. 286, 1983. 13. Stiel, L. J. and Harnish, D. F., “Solubility of Gases and Liquids in Molten Polystyrene,” AIChE J., 22, 1, 117–122, 1976. 14. Lee, S. T., “Computational Analysis of Bubble Behavior in the Devolatilization of Polymer Melt,” Master’s Thesis, Chem. Engr., Dept., Stevens Inst. of Technology, 1982. 15. Amon, M. and Denson, C. D., “A Study of the Dynamics of Foam Growth: Analysis of the Growth of Closely Spaced Spherical Bubbles,” Poly. Eng. Sci., 24, 13, 1026–34, 1984. 16. Shafi, M. A. and Flumerfelt, R. W., “Initial Bubble Growth in Polymer Foam Process,” Chem. Eng. Sci., 52, 4, 627–633, 1997. 17. Cahn, J. W., “Phase Separation by Spinodal Decomposition in Isotropic Systems,” J. Chem. Phys., 42, 1, 93–99, 1965. 18. Bates, F. S. and Wiltzius, P., “Spinodal Decomposition of a Symmetric Critical Mixture of Deuterated and Protonated Polymer,” J. Chem. Phys., 91, 5, 3258–3273, 1989. 19. Keller, A., “An Approach to Phase Behavior in Polymers,” Macromol. Symp., 98, 1–42, 1995. 20. Volmer, M. and Weber, A., Z. Phys. Chem., 119, 227, 1926. 21. Farkas, L., “The Velocity of Nucleus Formation in Supersaturated Vapors,” Z. Phys. Chem. (Leipzig), 125, 236, 1927. 22. Becker, R. and Doring, W., Ann. Physik, 24, 719, 1935. 23. Tadmor, Z., Albalak, R. J. and Canedo, E., “Polymer Melt Devolatilization Mechanisms,” 1st Polym. Proce. Lecture, Akron, Ohio, 1985. 24. Kumar, V., Ph.D. Dissertation, MIT, 1988.

© 2000 by CRC Press LLC

25. Kagan Y., “The Kinetics of Boiling of a Pure Liquid,” Russ. J. Phys. Chem., 34, 42–46, 1960. 26. Blander, M., “Bubble Nucleation in Liquids,” Adv. Colloid. Interf. Sci., 10, 1, 1979. 27. Ruengphrathuengsuka, W., “Bubble Nucleation and Growth Dynamics in Polymer Melts,” Ph. D. Dissertation, Chem. Eng. Dept., Texas A&M Univ., 1992. 28. Park, C. B., Baldwin, D. F. and Suh, N. P., “Effect of the Pressure Drop Rate on Cell Nucleation in Continuous Processing of Microcellular Polymers,” Polym. Eng. Sci., 35, 432–440, 1995. 29. Lee, S. T. and Ramesh, N. S., “Study of Foam Sheet Formation III: Effects of Foam Thickness and Cell Density,” Annual Conference, ASME, 1996. 30. Goel, S. K. and Beckman, E. J., “Generation of Microcellular Polymeric Foams Using Carbon Dioxide. I: Effect of Pressure and Temperature on Nucleation,” Polym. Eng. Sci., 14, 1137, 1994. 31. Baldwin, D. F., Park, C. B. and Suh, N. P., “A Microcellular Processing Study of Poly(Ethylene Terephthalate) in the Amorphous and Semi-Crystalline States. Part I: Microcell Nucleation,” Poly. Eng. Sci., 36, 11, 1437–1445, 1996. 32. Kumar, V., Weller, J. E. and Montecillo, R., “Microcellular PVC,” Ann. Tech. Conf. (ANTEC) preprint, 1452–1456, 1992. 33. Han, J. H. and Han, C. D., “Bubble Nucleation in Polymeric Liquids. II.Theoretical Considerations,” J. Poly. Sci. B: Poly. Phys., 28, 743, 1990b. 34. Patel, A., Stivala, S. S. and Biesenberger, J. A., “Small Angle X-ray Scattering Studies on Solubility of Nitrogen in LDPE Melt,” ACS Annul Meeting, 1992. 35. Kwak, H. Y. and Panton, R. L., “Gas Bubble Formation in Nonequilibrium Water-Gas Solutions,” J. Chem. Phys., 78, 9, 5795–5799, 1983. 36. Kwak, H. Y. and Lee, S., “Homogeneous Bubble Nucleation Predicted by a Molecular Interaction Model,” J. Heat Transfer, 113, 714–721, 1991. 37. Shafi, M. A., Lee, J. G. and Flumerfelt, R. W., “Prediction of Cellular Structure in Free Expansion Polymer Foaming Processing,” Poly. Eng. Sci., 36, 14, 1950–1959, 1996. 38. Attaim, A., “Bubble Nucleation in Viscous Material Due to Gas Formation by a Chemical Reaction: Application to Coal Pyrolysis,” AIChE J., 24, 1, 106–115, 1978. 39. Martini, J. E., Master’s Thesis, Mech. Eng. Dept., Mass. Inst. Technology, 1981. 40. Colton, J. S. and Suh, N. P., “The Nucleation of Microcellular Thermoplastic Foam with Additives, Part I: Theoretical Considerations,” Poly. Eng. Sci., 27, 485–503, 1987. 41. Colton, J. S., “Making Microcellular Foams from Crystalline Polymers,” Plastics Engineering, 8, 53, 1988. 42. Ramesh, N. S., Rasmussen, D. H. and Campbell, G. A., “The Heterogeneous Nucleation of Microcellular Foams Assisted by the Survival of Microvoids in Polymer Containing Low Glass Transition Particles. Part II: Experimental Results and Discussion,” Polym. Eng. Sci., 34, 22, 1698–1706, 1994. 43. Lee, S. T., “Study of Foam-Enhanced Devolatilization; Experiments and Its Theories,” Ph.D. Dissertation, Chem. Eng. Dept., Stevens Inst. of Technology, 1986. 44. Jemison, T. R., Rivers, R. J. and Cole, R., “Incipient Vapor Nucleation of Methanol from an Artificial Site-Uniform Superheat,” AIChE Annual Meeting, Chicago, 1980. 45. Rayleigh, L., “On the Pressure Developed in a Liquid during the Collapse of a Spherical Cavity,” Phil. Mag., 34, 94–98, 1917. 46. Knapp, R. T., Paity, J. W. and Hammitt, F. G., Chap. Three: “Cavitation Inception,” Cavitation, McGraw-Hill, New York, 1970. 47. Wojs, K. and Sitka, A., “Cavitation Phenomenon in Newtonian and Non-Newtonian Fluids,” Inzynieria Chemiczna I Procesowa, 18, 2, 321–336, 1997.

© 2000 by CRC Press LLC

48. Ainslie, C. P., “Cavitation Phenomena in Polymer Melts,” Master’s thesis, Loughborough University of Technology, 1973. 49. Chin, J., “Simulation of Devolatilization in Polymer,” Master’s Thesis, Chem. Engr. Dept., Stevens Inst. of Technology, 1982. 50. Bolton, B. and Middleman, S., “Air Entrainment in a Roll Coating System,” Chem. Eng. Sci., 35, 597–601, 1980. 51. Frenkel, J., Kinetic Theory of Liquids, Clarendon Press, Oxford, 1946. 52. Fisher, J. C., “The Fracture of Liquids,” J. Appl. Phys., 19, 1062, 1948. 53. Turnbull, D. and Cohen, M. H., “On the Free Volume Model of the Liquid-Gas Transition,” J. Chem. Phys., 52, 6, 3038, 1970. 54. Biesenberger, J. A. and Lee, S. T., “Fundamental Study of Polymer Melt Devolatilization. III: More Experiments on Foam-Enhanced DV,” Poly. Eng. Sci., 27, 7, 510–517, 1987. 55. Gent, A. N. and Tompkins, D. A., “Surface Energy Effects for Small Holes or particles in Elastomers,” J. Poly. Sci., 2, 7, 1483–1487, 1969. 56. Kekkula, H., Schwara, M., and Paul, D. R., Polymer, 27, 210, 1986. 57. Ramesh, N. S., “Investigation of the Foaming Characteristics of Nucleation and Growth of Microcellular Foams in Polystyrene Containing Low Glass Transition Particles,” Ph.D. Thesis, Chem. Eng. Dept., Clarkson University, 1992. 58. Steenbrink, A. C. and Van der Giessen, E., “A Numerical Study of Cavitation and Yield in Amorphous Polymer-Rubber Blends,” J. Eng. Mater. and Tech., 119, 256–261, 1996. 59. Harvey, E. N., Barnes, D. K., McElroy, W. D., Whiteley, A. H., Pease, D. C. and Kooper, K. W., “Bubble Formation in Animals: I. Physical Factors,” J. Cell. and Compar. Physiology, 24, 1, 1–22, 1944. 60. Biesenberger, J. A. and Lee, S. T. “Visulization of Foamed Devolatilization Experiments” Video Tape in Polymer Processing Institute, Hoboken, New Jersey, 1987. 61. Daigneault, L., Handa, Y. P., Wong, B. and Caron, L. M., “Solubility of Blowing Agents HCFC-142b, HFC-134a, HFC-125 and Isopropanol in Polystyrene,” Ann. Tech. Conf. (ANTEC) preprint 1983–1987, 1997. 62. Han, J. H. and Han, C. D., “A Study of Bubble Nucleation in a Mixture of Molten Polymer and Volatile Liquid in a Shear Field,” Poly. Eng. Sci., 28, 24, 1616–1627, 1988. 63. Lee, S. T., “Nucleation in Thermoplastic Foam Nucleation,” in Cellular and Microcellular Materials, ed. by Kumar, V. and Seeler, K. A., ASME, 1994. 64. Todd, D. and Dey, S., Private Communication. 65. Park, C. B., Behravesh, A. H. and Venter, R. D., “Extrusion of Low Density Microcellular HIPS Foams Using CO2,” Polym. Eng. Sci., 38, 1812–1823, 1998. 66. Enns, T., Scholander, P. F. and Bradstreet, E. D., “Effect of Hydrostatic Pressure on Gases Dissolved in Water,” J. Phys. Chem., 69, 2, 389–391, 1965. 67. Handa, P. and Zhang, Z., “New Pathways to Microcellular and Ultramicrocellular Polymeric Foams,” Porous, Cellular and Microcellular Materials, ed. by Kumar, V., ASME, 1998. 68. Lee, S. T. and Kim, Y., “Shear and Pressure Effects on Extruded Foam Nucleation,” Soc. Plas. Eng. Conf., ANTEC preprint 3527–3532, 1998. 69. Han, J. H. and Han, C. D., “Bubble Nucleation in Polymeric Liquids. I. Bubble Nucleation in Concentrated Polymer Solutions,” J. Poly. Sci. B: Poly. Phys., 28, 711, 1990a. 70. Kwok, D. Y., Cheung, L. K., Park, C. B. and Neumann, A. W., “Study on the Surface Tensions of Polymer Melts Using Axisymmetric Drop Shape Analysis,” Polym. Eng. Sci., 38, 5, 757764, 1998.

© 2000 by CRC Press LLC

71. Albalak, R. J., Tadmor, Z. and Talmon, Y., “Polymer Melt Devolatilization Mechanisms,” AIChE J., 36, 9, 1313–1320, 1990. 72. Tukachinsky, A., Tadmor, Z. and Talmon, Y., “Ultrasound-enhanced Devolatilization of Polymer Melt,” AIChE J., 39, 359, 1993. 73. Sahnoune, A., Piche, L., Hamel, A., Gendron, R., Daigneault, L. E. and Caron, L. M., “Ultrasonic Monitoring of Foaming in Polymers,” Soc. Plas. Eng. Ann. Conf. (ANTEC), preprint 2259–2263, 1997.

© 2000 by CRC Press LLC

CHAPTER 5

Foam Growth in Polymers N. S. RAMESH

5.1 INTRODUCTION

T

foams have diverse applications in everyday life. Insulation, surface protection, sports, recreation, and cushioning are some of the major applications of olefinic and styrenic foam products. Extrusion has been conventionally used for producing low-density foam sheet and rods with physical blowing agents in the last decades. Foam nucleation, foam growth, and cell coalescence are the three major events in the foaming process. The dynamic nature of thermoplastic foam formation introduces extra variables, greatly increasing the analysis complexity of this extrusion process. This chapter covers a condensed review of papers dealing with foam growth models and experiments. The latest approach to model foam extrusion is also presented. A fundamental study focusing on the influence of blowing agent on bubble growth during thermoplastic foam extrusion is presented. The extruded molten mixture expands and cools simultaneously when exposed to ambient conditions. The bubble growth is influenced by the concentration-dependent blowing agent diffusion coefficient, the transient cooling of the expanding foam, the influence of blowing agent on polymer viscosity, and the escape of blowing agent from the surface of the foam. Previous models in the literature do not consider these significant influences. A modified model is presented accounting for those more subtle effects. In addition, a new experimental technique is described to collect experimental bubble growth data. HERMOPLASTIC

© 2000 by CRC Press LLC

5.2 IMPORTANCE OF THIS STUDY In general, the foaming process involves three important steps: nucleation, bubble growth, and bubble coalescence. Another chapter in this book dealt with the phenomenon of nucleation. This chapter deals with foam growth in polymers. Bubble collapse and coalescence in polymers are not desirable, and the foaming process is tuned to avoid that phenomenon. It is well known that the cell size, cell size distribution, cell geometry, type of cells, and foam density play major roles in controlling the mechanical properties of a foam. The fundamentals of a bubble growth study are important because the properties of cellular materials directly depend on the shape and structure of the cells. Hence, it is necessary to be able to predict and control the cell size during the foam growth of bubbles in the mold or in a free foam sheet expansion to achieve the desired mechanical properties.

5.3 LITERATURE REVIEW Several investigations addressing the theoretical and experimental analyses of bubble growth and collapse in fluids and polymers have been available in the literature since 1917. The historical development of important models is listed in Table 5.1. Most of the models can be classified into two groups: single bubble growth model and cell model (swarm of bubbles growing without interaction). The references [1–29, 36] are given at the end of this chapter. 5.3.1 SINGLE BUBBLE GROWTH MODELS (1917–1984) Between 1917 and 1984 [1–17], the published models focused on the growth or collapse of a single bubble surrounded by an infinite sea of fluid with an infinite amount of gas available for growth. Although these models gave several insights into bubble growth phenomena, their practical application in industry was severely limited, because in real life, the foaming process involves the growth of numerous bubbles expanding in close proximity to one another with a limited supply of gas. This led to the development of a new model called the “Cell Model” that will be discussed in the next section. 5.3.2 CELL MODEL (1984–1998) The cell model has been widely used to describe devolatilization and batch microcellular and foam extrusion processes. Gross motion, no bubble motion, and motion without shear have different implications on the cell model.

© 2000 by CRC Press LLC

The concept of a cell model was first introduced by Amon and Denson [18] in 1984. The study involved the growth of a group of gas bubbles separated by a thin film of polymer and dissolved gas during the injection molding process. The foam was divided into spherical microscopic unit cells of equal and constant mass, each consisting of a liquid envelope surrounding a single bubble, and the gas available for growth is thus limited. Because of this more realistic assumption, the cell model yielded a final radius, while other single bubble growth models showed growth of bubble radius with time, infinitely. This fundamental improvement caused more interest in this area of research, and several studies have emerged since then. However, the Newtonian viscosity equation was used to describe the rheology of the polymer due to the complexity of the problem. Later, it was modified to account for the non-Newtonian viscoelastic effects to make it more suitable for polymeric systems [19]. Cell models [18–29, 34] can be broadly classified into two groups: cell model for closed system with no blowing agent and gas loss effects and modified cell model for foam extrusion with blowing agent and gas loss effects. They are described in the next section. Recently, Shafi, Joshi, and Flumerfelt [32] did good work and proposed a model by combining nucleation and foam growth processes to study the effects of operating conditions on bubble growth dynamics in polymeric foams. The final bubble size distribution depended upon nucleation rate and bubble growth dynamics. However, that model assumed Newtonian behavior for the polymer and neglected important features of the “Cell Model” that limits its application to real-life processes. For example, the previous studies [21–23] show that viscoelastic phenomena are important in changing bubble growth characteristics. The gas loss and blowing agents are important in matching typical foam extrusion processes for manufacturing thin sheets. 5.3.2.1 Viscoelastic Cell Model for the Injection Molding Process 5.3.2.1.1 Arefmanesh and Advani Model The validity of the cell model was initially tested by experimental work by Amon and Denson [18], Arefmanesh et al. [20–21], Ramesh and Malwitz [25], and Lee and Ramesh [24]. The first two research groups concentrated on experiments to verify prediction of the cell model during the injection molding process, where the bubble growth occurs in a closed system. Amon and Denson [18] used low-density polyethylene with a chemical blowing agent, whereas Arefmanesh et al. used [20] polycarbonate with a chemical blowing agent. The assumption of no gas loss from the mold was adequate, since the foam growth occurred in a closed mold. These studies showed a qualitative

© 2000 by CRC Press LLC

TABLE 5.1

Historical Development of Bubble Growth Models. Transport Phenomena in Bubble Growth Process

Author(s)

Year

Momentum Transfer

Heat Transfer

Rayleigh, Lord Epstein and Plesset Scriven

1917 1950 1959

inertial — yes

isothermal isothermal nonisothermal

Barslow and Langlois

1962

viscous

isothermal

Darby Yang and Yeh

1964 1966

— viscous

yes conduction

bubble surface diffusion — —

Street Gent and Tomkins

1968 1969

viscoelastic elastic

isothermal isothermal

yes

Stewart

1970

elastic

isothermal

Street, Fricke, and Reiss Rosner and Epstein Zana and Leal

1971

viscous

conduction

bubble surface diffusion yes

1972 1975

— viscoelastic

conduction

yes yes

Villamizar and Han

1978

—

Patel Han and Yoo

1980 1981

viscous yes

Papanastasiou, Scriven, and Macosko

1984

viscoelastic

—

Upadhyay

1985

viscoelastic

nonisothermal

yes

Lenov, viscoelastic

Amon and Denson

1984, 1986

viscous

conduction

yes

Newtonian

Arefmanesh and Advani

1991

viscoelastic

conduction

diffusion at bubble interface

viscoelastic Maxwell

Ramesh

1991, 1992 1993

viscoelastic

conduction

yes

viscoelastic

conduction

yes

1995

viscoelastic

nonisothermal

⫹gas loss

Maxwell, power law convected Maxwell Maxwell

1995, 1996

viscoelastic

nonisothermal

Ramesh and Malwitz

1997

viscoelastic

nonisothermal

Ramesh and Malwitz

1998

viscoelastic

nonisothermal

⫹gas loss from sheet surface ⫹gas loss from sheet surface ⫹surface evaporation

Lee, Ramesh, and Campbell Ramesh and Malwitz Lee and Ramesh

© 2000 by CRC Press LLC

Mass Transfer — yes yes

—

— isothermal isothermal

— yes yes —

Rheological Model — — Newtonian

Newtonian

Newtonian Ostwald Model Oldroyd NeoHookean NeoHookean Power Law — Newtonian, Oldroyd B — Newtonian Viscoelastic Dewitt BKZ type viscoelastic

Maxwell

Maxwell

Maxwell

Influence of Blowing Agent on Theoretical (T) Experimental (E) T T T

T

Bubble Growth Medium

water waterethylene glycol vinylidene copolymer

E T

— —

T T

— —

Blowing Agent (B/A)

Bubble Count Study

Polymer Viscosity

Diffusion Coefficient

Reference No.

— air vapor

— — —

— — —

single single single

1 2 3

nitrogen

—

—

single

4

— —

— —

— —

single single

5 6

— air

— —

— —

single single

7 8

air

—

—

single

9

—

single

10

T

rubber

T

rubber

T T

iron melts —

nitrogen —

— —

— —

single single

11 12

E

polystyrene HDPE, PC — polystyrene

chemical B/A — chemical B/A —

—

Constant

single

13

— —

Constant Constant

single single

14 15

—

constant

16

—

constant

single, bubble growth collapse single

—

constant

swarm

18, 19

—

constant

swarm

20, 21

—

constant

swarm

22, 23

—

constant

swarm

24

yes

constant

swarm

25

yes

constant

swarm

26, 27, 34

concentrationdependent diffusion concentrationdependent diffusion

Swarm

28

swarm

29

T T and E

—

yes

T

hydroxyl propyl cellate in water

T and E

polystyrene, polycarbonate

T and E

LDPE

T and E

polycarbonate

T and E

polystyrene

T and E

LDPE

T and E

PVOH

T and E

LDPE

T

LDPE

n-butane

yes

T and E

LDPE

butane

yes

© 2000 by CRC Press LLC

physical blowing agent chemical blowing agent chemical blowing agent FLC-95 physical N2, CO2 HCFC water, methanol HCFC-22 HCFC-142b

17

agreement between experimental data and theoretical results and have led to a better understanding of the processing of polymeric foam materials. Ramesh et al. [22–23] have tested the validity of the cell model by conducting a cell growth experiment during microcellular foaming process using a scanning electron microscope as shown in Figure 5.1(a). Polystyrene was used with physical blowing agents such as nitrogen and carbon dioxide. In all of the above cases, the blowing agent loss was considered to be negligible, and, therefore, the no gas loss boundary condition was adequate to predict the foam

(a) FIGURE 5.1 (a) Comparison of theory with the experiment for the foam growth in polystyrene at 378 K when carbon dioxide was used as a blowing agent; (b) comparison between experimental foam density data and simulation results at various final cell density, #/cm3; and (c) comparison between experimental foam density data and simulation results for HCFC-22 and HCFC 142b at two different levels.

© 2000 by CRC Press LLC

(b)

(c)

© 2000 by CRC Press LLC

growth kinetics reasonably well. However, in the polymeric foam extrusion process of interest here, the foam sheet or rod is allowed to expand freely under atmospheric conditions. The assumption of no gas loss to surroundings no longer applies for the actual processing. Therefore, such gas escape must be accounted for, especially with thin foam shapes. Furthermore, the previous models do not include further complicating effects such as the influence of blowing agent on polymer viscosity and the concentration dependence of the TABLE 5.2

Comparison of Bubble Growth in Batch and Continuous Processes. Bubble Growth Study In

Process Parameters/Properties 1. Process 2. Key steps

3. Polymer studied 4. Polymer type 5. Blowing agent 6. Foaming temperature 7. Blowing agent surface evaporation 8. Diffusivity of blowing agent 9. System 10. Polymer Viscosity 11. Bubble size, ␮m 12. Cell count/cm3 of foam 13. Foam density range, Kg/m3 14. Foam expansion time, second 15. Experimental study on foam growth

16. Model used

17. Comparison—Theory vs. Experiment 18. Figure/(Reference)

© 2000 by CRC Press LLC

Microcellular Batch Foaming Process

Conventional Foam Extrusion

Batch Gas saturation and heating the polymer in a constant temperature bath Polystyrene Amorphous Nitrogen, Carbon dioxide Isothermal No

Continuous Direct gas injection into the extruder

LDPE Semicrystalline n-butane Nonisothermal Yes

Concentration independent Pellet expansion Plasticized due to blowing agent ⬇20 108 200–300

Concentration dependent Rod expansion Plasticized due to blowing agent ⬇1,000 200 30

60–120 seconds

1–2 seconds

Microscopic Study

Cell model with isothermal, no gas loss effect and blowing agent effect on diffusion (Figure 5.1) Reasonable

Measurement using the video technique, cell growth study via video techniques Modified cell model with blowing agent and gas loss effects (Figure 5.7). Modified boundary condition. Good

Figure 5.1/[22]

Figure 5.7

blowing agent on diffusion during the bubble growth process. Table 5.2 shows the comparison of bubble growth studies done by Ramesh [22] for a batch microcellular process and a continuous extrusion process. This chapter shows improvement on the former models to more closely reflect thermoplastic foaming dynamics including the author’s ongoing research [22–25] for the past eleven years on modeling foam growth in the extrusion process. 5.3.2.2 Modified Viscoelastic Cell Model for the Foam Extrusion Process 5.3.2.2.1 Ramesh, Lee, Malwitz Model Recently, Lee and Ramesh [34] studied the effects of foam sheet thickness and nucleation cell density on thermoplastic foam sheet extrusion. Lowdensity polyethylene was used with HCFC-22 and HCFC-142b to produce foam sheets of various thicknesses and nucleation characteristics. They used a 70 mm counterrotating twin-screw extruder to produce foams. The results are shown in Figures 5.1(b) and 5.1(c). It was concluded that solubility, rheology, and gas loss transport mechanisms play an important role in determining the foaming efficiency during foam sheet extrusion. While predicting the foam density as a function of final cell density (or number of cells/unit volume) and foam sheet thickness, they used mathematical equations from packing theory to calculate the ratio of bubbles present closer to the surface to the bubbles present in the core portion of the foam sheet where there is no gas loss. The value of this ratio, of course, strongly depends on the nucleation density (which is defined as the number of bubbles nucleated per unit volume of the polymer) and the thickness of the foam sheet. With this theoretical background, an attempt was made to check the validity of the modified model by comparing it with the experimental data. Although the improved cell model predicts the experimental data very well, between 919 kg/m3 to 60 kg/m3, it was observed that the agreement decreases for increased expansion ratios and volatility of blowing agent, where the nonspherical nature of the bubble and bubble-bubble interaction phenomena become pronounced. Higher nucleation rates and extrusion of thicker foam sheets seem to enhance foam efficiency. Correlation with cell geometry and nonspherical bubble dynamics and consideration of interaction between bubbles during the growth process would greatly benefit in understanding the low-density foam process and would assist in product design. In this chapter, a new approach has been taken to model bubble growth during the foam extrusion process. The objective is to present a new model describing the thermoplastic foam formation process, where the foam is expanded unrestricted nonisothermally and the blowing agent plays an important role in changing the rheological and other physical properties.

© 2000 by CRC Press LLC

The following major changes have been made to historical models to account for blowing agent effects on foam dynamics: (1) Gas loss to surroundings from a foam sheet or rod from the cells that are adjacent to the surface. Mathematically, the gas loss boundary condition is different. (2) Blowing agent plasticizes the polymer. Therefore, the reduced viscosity variations as a function of concentration during foam expansion are used for simulation based on experimental rheological data. (3) The concentration-dependent diffusion coefficient is included to model more accurately the actual nature of the blowing agent-polymer binary system. (4) Transient cooling of the foam is accounted for as a nonisothermal foam expansion problem under ambient process conditions. Temperature effects on viscosity, diffusivity, and blowing agent pressure are considered, for example. The non-Newtonian nature of the polymer/blowing agent mixture was predicted using the upper convected Maxwell Model. The convected Maxwell is simple and, therefore, it is widely used to study the influence of viscoelasticity on flow calculations. More details are listed in the literature [35]. Experimental data were gathered to describe the rheology of the polymer/blowing agent mixture and its influence on foam development.

5.4 FOAM GROWTH EXPERIMENT The experimental objective was to establish the bubble growth data during foam processing. Since bubbles tend to grow quickly, typically within 2 to 3 seconds, to the final radius for low-density foams below about 30 kg/m3, it is very difficult to measure their growth with real time. On the other hand, it is relatively easy to measure the diameter of the expanding rod of foam when it exits the die and grows until it reaches the final equilibrium diameter. Hence, it was decided to conduct a foaming experiment to produce a cylindrical rod of foam so that its expansion can be easily measured. To accomplish this task, the experimental procedure described below was followed. Figure 5.2 shows the experimental setup for collecting the bubble growth data. A low-density 2MI polyethylene resin having a specific gravity of 0.919 was blended with 0.25% talc and metered into a Haake twin-screw corotating extruder. The use of talc yields fine and uniform cellular structure. The blowing agent was added at the midsection of the barrel to allow complete mixing with the molten polymer. Then, the plasticized melt was cooled before being extruded through a rod capillary nozzle. When the gas-charged polymer

© 2000 by CRC Press LLC

FIGURE 5.2

Experimental setup for measuring rod expansion data.

exited the die, nucleation of bubbles occurred due to thermodynamic instability, and the strand expanded due to the growth of bubbles assisted by diffusion of gas into the cells. The cylindrical rod continued to expand until it reached the final steady diameter. The bubble growth (rod expansion) stopped due to depletion of gas in the polymer and stiffening of the cell walls. A digital caliper was used to measure the rod diameter as it expanded with traveling distance from die exit. The local velocity of the strand was determined using a video camera system to interpret the rod expansion data in terms of time. An ink dot was introduced on the strand at the die exit as a video marker. The time it took for the dot to move along the travel distance was monitored to determine local velocity. The obtained experimental data are plotted in Figure 5.7.

5.5 FOAM GROWTH MODELING As foam rod exits from a die, one-dimensional heat transfer is assumed. Heat conduction becomes the dominant heat transfer mechanism. The heat loss effects on foam growth are found to be less important when the diameter of the foam rod exceeds 10 mm. The average temperature of the thin foam as it is exposed to ambient conditions can be found from the standard heat transfer equation in the literature [30]. The foam rod formation is pictured in Figure 5.2. The schematic diagram of the cell model is shown in Figure 5.3. The appropriate bubble growth modeling equations are listed here.

© 2000 by CRC Press LLC

FIGURE 5.3

Gas loss boundary condition in the cell model.

5.6 FOAM GROWTH EQUATIONS To define gas pressure inside the bubble, the integrated momentum equation reduces to the following: Pg ⫺ P⬁ ⫺

2␴ dr f ⫽0 ⫹ 兰R R (␶␥␥ ⫺ ␶␪␪) r R

(1)

Rheological Equations: ␶⫹

␩*0 ␶(1) ⫽ ⫺␩*0␥ G

(2)

Ev 1 1 ␩*0 ⫽ ␩0 exp c a ⫺ b d *f(c) Rg T T0

(3)

Polymer viscosity with blowing agent f(c) ⫽ viscosity reduction factor ⫽ Viscosity of polymer at same temperature

(4)

where:

Growth of Radius: d ⭸c (␳gR3) ⫽ 3␳DR2 c d dt ⭸r r⫽R

© 2000 by CRC Press LLC

(5)

Experimental Parameters for Bubble Growth Calculations (from our Own Experiment and the Resin Data from the Supplier).

TABLE 5.3

Process Variables 1. Surface tension of low-density polyethylene (LDPE) 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Molecular weight of butane (blowing agent) Initial blowing agent concentration Foaming temperature, To Ambient temperature, Ts Henry’s law constant, Kw Density of LDPE No. of bubbles per 1 cm3 of foam (from experiment) No. of bubbles across strand diameter Diffusion constant parameter, A Diffusion constant parameter, B Diffusion constant parameter, C

Values mN m 58 kg/kg mole 6.0% by weight 383 K 293 K 2.235 ⫻ 108Pa 920 kg/m3 136 5 0.5334 21.93 7,090 K 30

Concentration-Dependent Diffusion Equation: ⭸c ⭸c 1 ⭸ ⭸c ⫹ Vr ⫽ 2 aDr2 b ⭸t ⭸r ⭸r r ⭸r

(6)

where diffusivity is a function of blowing agent concentration, and it changes within the polymer envelope during the gas-diffusion process. The diffusivity also depends strongly on the temperature. Mathematically, D(c,T) ⫽ [1 ⫹ Ac]10⫺7e(B⫺c/T)

(7)

Where A, B, and C are constants fitted to the experimental data that depend on the nature of the blowing agent. The values of A, B, and C need to be obtained experimentally. For this work, the values used based on our experimental data and the data provided by the resin supplier are listed in Table 5.3.

5.7 BOUNDARY CONDITIONS The initial and boundary conditions are as follows:

© 2000 by CRC Press LLC

c(r,o) ⫽ co ⫽ KwPgo

(8)

c(R,t) ⫽ KwPg

(9)

For bubbles situated in the core of thick sheet expansion, ⭸c ⫽0 ⭸r

(10)

For bubbles on the surface of foam sheet that undergoes gas loss: D

⭸c ⫽ km(cs ⫺ c⬁) ⭸r

(11)

where km ⫽ 2

D B ␲t

(12)

km is defined as the mass transfer coefficient derived from the penetration theory [30] and “t” is the foam growth time. The surface evaporation or gas loss condition can be mathematically expressed in Equation (11). R(t ⫽ 0) ⫽ R0

(13)

The boundary condition, Equation (10), represents a closed-cell system where there is no gas loss or escape to its surroundings. This is true in the injection molding process because the foam expansion occurs inside the mold. But, in the practice of producing foams with unrestricted rise, gas loss is encountered from the bubbles that are adjacent to the top and bottom surfaces of the foam. The influence of gas loss on foaming efficiency is especially profound when the extruded shape is thin. Some of the gas close to the surface, instead of diffusing into the cell, diffuses to the surface and transpires into the atmosphere. As a result, the final surface cell size is smaller than that of the core cell. As the final foam density is lower, or the shape is thinner, more gas escape is anticipated. In other words, high concentration of blowing agents and/or high surface-to-volume foam sheets accelerate gas loss from a sheet surface, thereby lowering the blowing agent effectiveness. This phenomenon is compounded at lower atmospheric temperatures that decrease the foam growth rate. A similar situation is encountered when a small diameter foam rod is extruded. The above system of equations was solved numerically by an iterative numerical scheme. In order to facilitate the numerical simulation, a definite value of initial radius, Ro, is required. Bubble growth originates from thermodynamic and mechanical instability in the system with perturbed initial radius [18, 32, 33, 36]. Hence, the selection of initial radius is important to achieve

© 2000 by CRC Press LLC

great accuracy in the predicted results. Great care is needed to model successfully. This principle is not clearly emphasized in the literature so far. The bubble growth seems to be a strong function of the value of the perturbed initial radius during the induction period that is less than 8% of the total expansion time. Beyond this initial growth time, i.e., greater than 8% of the total growth time, the foam growth rate is found to be independent of the growth radius according to Denson [18], Upadhyay [17], Ramesh [22], and Arefmanesh [20]. For all practical reasons, it is better to calculate the initial radius from fundamentals of mechanical equilibrium principles that are involved with the calculation of a critical bubble radius. The value of the critical bubble radius, Rc, can be easily calculated from the force balance at the bubble 2␴ wall that can be written as Rc ⫽ , where ␴ is the surface tension and ⌬P is ⌬P the pressure difference between the liquid polymer phase and the dissolved gas phase. For example, in the case of Figure 5.1, the critical radius for the polystyrene-CO2 system ranged from 0.1␮m to 1␮m depending on the various foaming conditions. Hence, for the entire simulation, the value 1␮m was chosen for the initial radius, Ro. Solving the model yields the bubble radius as a function of time. However, to achieve accurate results, a careful evaluation of rheological and diffusion properties is needed. Rheological properties for LDPE with butane were obtained using Haake capillary rheometer. An experimental setup is shown in Figure 5.4. Dissolution of blowing agent tends to plasticize the molten polymer and decrease the polymer viscosity. Rheological results for LDPE/butane system are shown in Figure 5.5, from which a viscosity reduction factor, f(c), for appropriate blowing agent concentration was determined for the bubble growth simulation. The concentration and temperature effects on diffusion of butane in LDPE are shown in Figure 5.6. These results are obtained by fitting data presented by Coonahan [31]. The diffusion coeffi-

FIGURE 5.4

© 2000 by CRC Press LLC

Schematic diagram of the capillary rheometer.

FIGURE 5.5 butane data).

Influence of blowing agent on viscosity (the dotted line corresponds to 15%

cient of butane in LDPE as a function of temperature and concentration can be expressed in the following form: D (c, T) ⫽ ([1 ⫹ Ac] 10⫺7) e[B⫺C/T]

(14)

where A, B, and C are constants fitted to the experimental diffusion data shown in Figure 5.6. Although the modeling equations are the same whether a physical or a chemical blowing agent system is used, the appropriate blowing agent gas properties must be used to run the simulation.

FIGURE 5.6

© 2000 by CRC Press LLC

Concentration effect on gas diffusivity [31].

FIGURE 5.7

Comparison of experiment versus theory.

The other simulation parameters used for all cases in this study are listed in Table 5.3. The model yields the result in terms of bubble radius as a function of time. The growing bubble reaches an equilibrium bubble radius once the gas is depleted. Finally, the bubble diameter was multiplied by the number of cells that are simultaneously growing across the diameter of the expanding rod to calculate the rod diameter. The simulated results are plotted in Figure 5.7.

5.8 COMPARISON OF THEORY WITH EXPERIMENT Figure 5.7 shows the comparison of Ramesh et al.’s model with Arefmanesh et al.’s model [20–21] and the experimental data. The solid line denotes the current model, dashed line denotes a previous model. The prediction of the foam strand diameter by the new model seems to agree well with the experimental data when the concentration and surface gas loss effects are included. The previous model predicts longer growth time due to neglecting concentra-

© 2000 by CRC Press LLC

tion effects. Furthermore, the predicted induction time is longer and the shape of the curve does not seem to match the characteristics exhibited by the experimental data. The previous model also predicts a higher strand diameter than experimentally observed because it assumes no gas loss from the bubbles that are close to the surface. A significant difference in strand diameter leads to serious discrepancies in predicting the final foam density, as it depends on the square of the rod diameter. Therefore, analysis including blowing agent concentration and surface escape effects in the thermoplastic foam expansion model is essential to accurately predict foam growth rate and density.

5.9 CONCLUSIONS An overview of bubble growth models and experiments is presented in this chapter; Ramesh et al.’s modified model presents a new approach for modeling bubble growth during extrusion processing of foams that for the first time includes blowing agent concentration and temperature effects on physical properties during foam formation and gas loss. Gas loss, blowing agent, and transient cooling effects are found to be important. Predictions by this new model agree well with experimental data. A simple but useful experimental technique is presented to collect the foam growth data. The predictions of previous foam growth models lacking the influence of blowing agent concentration effects in these binary systems differ significantly from experimental observations in extruded thermoplastic foam formation in an open system. This is another step toward arriving at a more thorough understanding of the concentration and gas loss interactions of the free-form foam process. Although the bubble growth discussion dates back to 1917 in the literature, still more modeling and experimental skills are required to describe polymeric foam systems.

5.10 NOMENCLATURE c cs co c∞ D Ev f(c) G Kw Pg Pgo

Gas concentration Surface gas concentration Initial gas concentration Surrounding gas concentration Diffusion coefficient, cm2/sec Activation energy for viscosity equation Viscosity reduction factor Elastic modulus Henry’s law constant Gas pressure at time t Initial gas pressure inside the bubble

© 2000 by CRC Press LLC

P∞ Ro Rg R r Rf Ro T To Ts t Vr ␩*0 ␶rr, ␶∞ ␶(1) ␥ ␳ ␳g ␴

Surrounding pressure Initial bubble radius Gas constant Gas-polymer interfacial radius Radial coordinate Cell outer radius Initial bubble radius Foaming temperature, °K Initial sheet temperature, °K Surrounding room temperature, °K Foam growth time Radial component of velocity Modified zero shear viscosity according to equation Stress in radial and circumferential directions Convected time derive of stress tensor Rate of strain tensor Density of polymer Density of blowing agent Surface tension of the polymer, N/m

5.11 REFERENCES Lord Rayleigh, Phil. Mag., 6th Series, 34:94 (1917). P. S. Epstein and M. S. Plesset, J. Chem. Phys., Vol. 18, 1505 (1950). L. E. Scriven, Chem. Eng. Sci., Vol. 10, 1 (1959). E. J. Barslow and W. E. Langlois, IBM J., 329 (1962). R. Darby, Chem. Eng. Sci., Vol. 19, 39 (1964). Wen-Jei Yang and H. C. Yeh, AICHE J., Vol. 12, 927 (1966). J. R. Street, Tran. Soc. Rheol., Vol. 12, 110 (1968). A. N. Gent and D. A. Tomkins, J. Appl. Phys., 2520 (1969). C. W. Stewart, J. Polym. Sci., A-28, 937 (1970). J. R. Street, A. L. Fricke and L. Phillip Reiss, Ind. Eng. Chem. Fundam., Vol. 10, 54 (1971). D. E. Rosner and M. Epstein, Chem. Eng. Sci., Vol. 27, 69 (1972). E. Zana and L. G. Leal, Ind. Eng. Chem. Fundam., Vol. 14, 175 (1975). C. A. Villamizar and C. D. Han, Polym. Eng. Sci., Vol. 18, 699 (1978). R. D. Patel, Chem. Eng. Sci., Vol. 35, 2352 (1980). C. D. Han and J. J Yoo, Polym. Eng. Sci., Vol. 21, 518 (1981). C. Papanastasiou, L. E. Scriven, and C. W. Macosko, J. Non-Newt. Flu. Mech., Vol. 16, 53 (1984). 17. R. K. Upadhyay, Adv. In Polym. Tech., 5,1, 55 (1985). 18. M. Amon and C. D. Denson, Polym. Eng. Sci., Vol. 24, 1026 (1984). 19. M. Amon and C. D. Denson, Polym. Eng. Sci., Vol. 26, 255 (1986).

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

© 2000 by CRC Press LLC

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

A. Arefmanesh, Ph. D. Thesis, Dept. of Mech. Eng., Univ of Delaware (1991). A. Arefmanesh and S. Advani, Rheo. Acta., Vol. 30, 274 (1991). N. S. Ramesh, G. A. Campbell and D. H. Rasmussen, Polym. Eng. Sci., Vol. 31, 1657 (1991). N. S. Ramesh, Ph.D. Thesis, Dept. of Chemical Engineering, Clarkson University, Potsdam, NY (1992). S. T. Lee, N. S. Ramesh and G. A. Campbell, SPE ANTEC J., 3033 (1993). N. S. Ramesh and Nelson Malwitz, SPE ANTEC J., 2171 (1995). S. T. Lee and N. S. Ramesh, Adv. Polym. Tech., Vol. 15, 297 (1996). S. T. Lee, N. S. Ramesh and G. A. Campbell, Polym. Eng. Sci., Vol. 36, 2477 (1996). N. S. Ramesh and N. Malwitz, Polymeric Foams, ACS Ser. 669, Chapter 14, 206 (1997). N. S. Ramesh and N. Malwitz, SPE ANTEC J., 1907 (1998). J. R. Welty, C. E. Wicks and R. E. Wilson, Fundamentals of Heat and Mass Transfer, John Wiley and Sons, NY, 556–557 (1969). V. C. Coonahan, Ph.D. Thesis, Dept of Chem. Eng., Univ. of Maryland (1971). M. A. Shafi, K. Joshi and R. W. Flumerfert, Chem. Eng. Sci., Vol. 52, 635 (1997). S. T. Lee, PhD. Thesis, Stevens Institute of Techology, ChE Dept (1986). S. T. Lee, N. S. Ramesh, “Cellular and Microcellular Materials,” ASME ‘96, 71–80 (1996). R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Second Edition, Vol. 1: Fluide Mechanics, pp. 345–346, John Wiley & Sons (1987). M. A. Shafi and Flumerfelt, Chem. Eng. Sci., 628–643 (1996).

© 2000 by CRC Press LLC

CHAPTER 6

Polymeric Foaming Simulation: Batch and Continuous MASAHIRO OHSHIMA

6.1 INTRODUCTION

T

use of polymeric foams is rapidly expanding due to their excellent properties including light weight, high strength/weight ratio, and superior insulating abilities. The foaming methods can be divided into two categories: physical and chemical. In the former, dissolving a gas (such as nitrogen or carbon dioxide) into polymer at a specified temperature and pressure creates foams. Once the dissolution is completed, either the pressure is reduced (as in extrusion foaming) or the temperature is increased in order to supersaturate the gas (as in compression foaming). In chemical foaming, foams are created by decomposing a chemical blowing agent that has been incorporated into a polymer. At a certain temperature and pressure, a chemical reaction (usually a molecular decomposition) releases the foaming gas. In either method, the nucleation and subsequent bubble growth create the cellular structure in the polymer. A basic understanding of the cause-and-effect relationships between the processing conditions and bubble size and foam density are important to control the cellular structure and to produce foamed product of high quality. Many researchers have been studying nucleation and bubble growth in polymers and developing several models. Two chapters of this book, 4 and 5, have already been dedicated to the modeling of foaming. One deals with the modeling of the nucleation phenomena, and another describes the modeling of bubble growth dynamics. In order to simulate bubble size distribution and density, simultaneous consideration of nucleation and bubble growth dynamics is indispensable. This chapter deals with foaming simulation that combines nucleation and bubble growth models to estimate the effects of different paHE

© 2000 by CRC Press LLC

rameters such as viscosity, dissolved gas concentration, and pressure release rate. The chapter is roughly divided into two sections: one is for the simulation of batch foaming, and the other is for continuous foaming, i.e., foaming in the flow. In each section, basic models of nucleation and bubble growth dynamics are described together with a total mass balance equation for all bubbles that plays a role of jointing nucleation models together with bubble growth models, followed by some results of parametric computations to illustrate the relative importance of the parameters.

6.2 BATCH FOAMING The classical work on bubble nucleation and growth goes back to Zeldovich [1]. The pioneering modeling of the growth of a single gas bubble in a polymer matrix was carried out by Street et al. [2]. The latter authors introduced the concept of a finite influence volume around each bubble. They also considered the effects of heat, mass, and mass transfer on bubble growth. During the intervening years, numerous models have been published [3–5, 12–19]. Recently, extending the influence volume approach, Shafi et al. developed a model for free expansion polymer foaming that includes simultaneous nucleation and bubble growth [6–9]. In this section, employing their models as the basis, modeling as well as simulation of simultaneous nucleation and bubble growth for batch physical foaming is described. The physical foaming processes can be divided into four basic steps: (1) Dissolution of gas into polymer at elevated pressure (2) Nucleation of bubbles in a supersaturated solution of gas in a molten polymer matrix, by releasing the pressure or increasing the temperature (3) Growth of bubbles in molten polymer to an equilibrium size (4) Stabilization of the foam structure by lowering the temperature below the melting point or the glass transition temperature 6.2.1 NUCLEATION MODEL Shafi et al. proposed models for steps (2) and (3), above [6]. It was assumed that, initially, the equilibrium dissolved gas concentration is co at pressure PD0. To describe bubble nucleation, the models of heterogeneous and homogeneous nucleation rates were given by the following [6, 8, 9]: Jhet (t) ⫽ fhet exp a

© 2000 by CRC Press LLC

⫺16␲␥3F b NS (1) 3kBT{PD0 ⫺ PC ⫺ I{Gk,␣} ⫹ PD0 ln⍀/Z2L}2

Jhom(t)

⫽ fhom exp a

⫺16␲␥3 b LC(t) (2) 3kBT{PD0 ⫺ PC ⫺ I{Gk,␣} ⫹ PD0 (ln⍀)/Z2L}2

where Jhet and Jhom represent the rates of heterogeneous and homogeneous nucleation, respectively, NS denotes the number of heterogeneous nucleation sites, fhet and fhom are frequency factors for the heterogeneous and homogeneous nucleation, respectively, kB is Boltzmann’s constant, T denotes temperature, co is concentration of the dissolved gas, t is the time since the beginning of foaming, C(t) is the average concentration of gas dissolved in the polymer and Pc is the ambient pressure, I{Gk,␣}, ⍀, Z2L, and L denote the elasticity number, activity coefficient of dissolved gas, the compressibility factor and Avogadro’s number, respectively, and ␥ and F ⫽ F(⌰) denote surface tension and the wetting factor for the heterogeneous nucleation that depends only on the wetting angle ⌰. The total nucleation rate at time t was given by the following: Jtotal(t) ⫽ Jhom(t) ⫹ Jhet(t)

(3)

The derivation is based on the classical approach that predicts the critical radius for bubble nucleation: r* ⫽ 2␥bp/⌬P, where ␥bp is the interfacial tension coefficient between the bubble and polymer matrix and ⌬P is the pressure differential between the inside bubble pressure and the ambient pressure: ⌬P ⫽ PD ⫺ Pc. Since the free energy of nucleation is proportional to the bubble volume, the theory predicts high sensitivity: ⌬G*het ⬃ ␥3bp/⌬P2. However, in the newer theory, the classical pressure drop, ⌬P, is reduced by incorporating the effects of viscoelasticity and nonideal solution effects [6]. Additional details and the history of nucleation models can be found in the nucleation modeling chapter of this book (Chap. 4). It is important to recognize that at the moment of phase separation, as defined by the spinodal conditions, ␥bp might be equal to 0. As the concentration of gas molecules in the polymeric matrix decreases, the interfacial tension coefficient is expected to increase toward its equilibrium value. However, it is difficult to assess the dynamic concentration at the nucleating bubble wall and, consequently, the value of the parameter ␥bp [10] is also difficult to assess. 6.2.2 BUBBLE GROWTH MODELS The main difference between the old foaming simulation models and the recent one is the careful analysis of the bubble growth kinetics that start immediately after the bubble is nucleated. The motivation for inclusion of the

© 2000 by CRC Press LLC

growth kinetics is the presence of the well-known exclusion volume—the volume of the material around each bubble wherein the concentration of the dissolved gas is below the equilibrium concentration, co. The bubble growth mechanism was described using the equations of motion, mass balance over the bubble, and gas diffusion in the surrounding polymer phase [11]: # R 2␥ 4␩ ⫽ PD ⫺ PC ⫺ R R

(4)

d 4␲ PDR3 ⭸c a b ⫽ 4␲R 2D ` dt 3 RgT ⭸r r⫽R

(5)

# ⭸c RR2 ⭸c D ⭸ 2 ⭸c ⫹ 2 ⫽ 2 ar b ⭸t ⭸r r ⭸r r ⭸r

(6)

The initial conditions are as follows: c(r,0) ⫽ c0 ⫽ kHPD0

(7)

The boundary conditions are as follows: c(⬁,␶) ⫽ co

(8)

c(R,␶) ⬅ cR (␶) ⫽ kHPD(␶)

(9)

where r is the coordinate and ␶ is the time since the bubble was born, D and ␩, respectively, denote the diffusion and viscosity coefficients, PD(␶) is pressure # inside the bubble at time ␶, and R and R represent the radius of the bubble and its growth rate, respectively. By Equation (6), the polymer surrounding the bubble is dealt with as a Newtonian fluid. This might severely limit practical use, especially for situations in which the Deborah number is not negligible [2]. One could take nonNewtonian behavior of polymer into account as is done by the cell model [5, 12, 13, 14]. However, for the sake of computational simplicity, Equation (6) was employed for all simulation calculations illustrated in this chapter. In the simulation, it was also assumed that the dissolved gas concentration c(r, ␶) around a bubble depends on the radial position, r, and time, ␶, and that the gas concentration, c(R, ␶), at the surface is determined by the gas pressure inside the bubble according to Henry’s law: PD(t) ⫽ k⫺1 H c(r, ␶). Basically, simultaneous nucleation and bubble growth simulation is carried out in the following way (Figure 6.1): every moment, the number of bubbles

© 2000 by CRC Press LLC

(a) FIGURE 6.1

(b)

Simultaneous nucleation and bubble growth calculation.

nucleated is calculated using Equations (1) and/or (2) as a function of average gas concentration in the polymer matrix, C(t), ambient pressure, Pc(t), and temperature. Then, for each bubble born at time t, the bubble size growth is calculated by Equations (4)–(6) with the boundary conditions, Equations (7)–(9) [Figure 6.1(a)]. By taking the total mass balance between residual gas in polymer and gas consumed in the growth of all bubbles, the average gas concentration in the polymer is recalculated and used for nucleation calculation at the next moment, t ⫹ ⌬t [Figure 6.1(b): ⌬t is the simulation time step]. At the next moment, retaining the bubble growth calculation for all bubbles already born, the nucleation and bubble growth calculation for newly nucleated bubbles is carried out in a similar way to that carried out in the previous time step. In Shafi’s study as well as in the simulations performed in this chapter, it is assumed that whenever the bubble is born, i.e., whatever value average gas concentration takes, the pressure in the newly born bubble, PD0, is always determined by the initial gas concentration, i.e., PD0 ⫽ k⫺1 H co. However, the assumption can be modified so that PD0 becomes k⫺1 C(t) with modifications of H gas concentration profile calculation. The average dissolved gas concentration, C(t), in polymer was calculated following the influence volume approach. 6.2.3 INFLUENCE VOLUME Assuming that a bubble is born at time t⬘, and the operating trajectory of pressure release is given, one can calculate the radius, the pressure inside the bubble, and the gas concentration profile in the polymer surrounding the bubble at time t by solving Equations (4) to (6) with ␶ ⫽ t ⫺ t⬘. Consequently,

© 2000 by CRC Press LLC

one can assign an “influence volume” to the bubble, inside which no nucleation can take place. This is schematically presented in Figure 6.2. In Figure 6.2, the volume defined as Vs(␶) ⫽ 4␲[S3(␶) ⫺ R3(␶)]/3, with ␶ ⫽ t ⫺ t⬘, is called the influence volume at time t for the bubble born at time t⬘. S(␶) is the radial position at which the dissolved gas concentration is equal to the nucleation threshold, cs. As the bubble grows, the concentration gradient propagates radially in the surrounding polymer, thus, the influence volume also grows. Instead of solving the partial differential equation with a moving boundary, the gas concentration profile in the surrounding polymer can be obtained using a moment (integral) method, i.e., the weighted residual method proposed in the literature [15]. The integral method assumes that the gas concentration profile can be approximated by a polynomial function: c(r) ⫺ cR r3 ⫺ R3 ⫽ 1 ⫺ (1 ⫺ x)Nd⫹1 where x ⫽ 3 co ⫺ cR rcb ⫺ R3 When Nd ⫽ 3, the relation between the volume Vs (r ⫽ rs) and Vcb (r ⫽ rcb) of the bubble born at the time t⬘ can be given by: Vs(t ⫺ t⬘) ⫽ e 1 ⫺ a

FIGURE 6.2

© 2000 by CRC Press LLC

1/4 co ⫺ cs b f Vcb(t ⫺ t⬘) co ⫺ cR (t ⫺ t⬘)

Influence volume of a bubble.

(10)

Vcb is the volume enclosed by the radial position outside of which the dissolved gas concentration is equal to co. Its value is calculated by using the following equation: Vcb(t ⫺ t⬘) ⫽

20␲{PD(t ⫺ t⬘)R(t ⫺ t⬘)3 ⫺ PDoR(0)3} 3RgT(co ⫺ cR(t ⫺ t⬘))

(11)

where R(0) ⫽ 2␥/(PDo ⫺ Pc). Equation (11) is derived directly from the mass balance of gas between the bubble and surrounding polymer: 4 PD(t ⫺ t⬘)R(t ⫺ t⬘)3 4 PD(0)R(0)3 ␲ ⫺ ␲ ⫽ 3 RgT 3 RgT

rcb

冮 {c

o

⫺ c(r,t)}4␲r2dr (12)

R

In the simulation, the nucleation threshold of the dissolved gas concentration is assumed to be equal to cs ⫽ 0.95 co, where co is the initial dissolved gas concentration. Every bubble has an influence region; therefore, subtraction of the total influence volume of every bubble from the initial polymer volume gives the total volume of the noninfluence region, VL(t), where the new bubble can generate: t

冮 V (t⬘)J

VL(t) ⫽ VL(0) ⫺

L

total(t⬘)Vs(t

⫺ t⬘)dt⬘

(13)

0

where the initial value of VL, VL(0), is equal to the polymer volume. The average dissolved gas concentration in the noninfluence region, C(t), is then expressed by the following: t

C(t)VL(t) ⫽ coVL(0) ⫺

冮

VL(t⬘)Jtotal(t⬘)

4␲ PD(t ⫺ t⬘)R(t ⫺ t⬘)3 dt⬘ 3 RgT

0 t

⫺

冮 V (t⬘)J L

0

total(t⬘)

•

S(t⫺t⬘)

冮 R(t⫺t⬘)

4␲r2c(r,t ⫺ t⬘)dr ¶ dt⬘ (14)

The first term on the right-hand side denotes the initial amount of dissolved gas, and the second expresses the total amount of gas existing in all bubbles, while the third represents that dissolved in the influence volume. To solve

© 2000 by CRC Press LLC

Equation (14), S(␶), which is the radius of influence volume, is defined as follows: 4 3 1/3 S(t ⫺ t⬘) ⫽ e a ␲R(t ⫺ t⬘)3 ⫹ Vs(t ⫺ t⬘)b f 3 4␲

(15)

6.2.4 BUBBLE SIZE AND DENSITY AT THE FINAL EQUILIBRIUM One of the assumptions made by Shafi et al. is that bubbles are spherical, not mutually interacting, and not coalescing. Thus, at time tf, when the total influence volume becomes equal to the polymer volume [i.e., VL(t) becomes zero], the nucleation stops and the number of bubble becomes constant. However, the bubbles keep growing by consuming the gas dissolved in Vs(t) until an equilibrium is reached. After the nucleation stops, for t ⬎ tf , the bubbles and the influence volume share the total amount of gas initially dissolved in Vs(tf ⫺ ti), which is the influence volume at time t for the bubbles born at time ti. From the mass balance of the gas, the final size of the bubbles born at time ti can be calculated as follows: coVs(tf ⫺ ti) ⫽ C(⬁)Vs(tf ⫺ ti) ⫹

4 PD(⬁ ⫺ ti)R(⬁ ⫺ ti)3 ␲ 3 RgT

(16)

Substituting the force balance at equilibrium and C(⬁) ⫽ kHPD(⬁ ⫺ ti), PD(⬁ ⫺ ti) ⫽ Pc (⬁) ⫹

2␥ R(⬁ ⫺ ti)

(17)

into the above equation, the resulting equation is given by coVs(tf ⫺ ti)

⫽ aPc(⬁) ⫹

2␥ 4 R(⬁ ⫺ ti)3 b akHVs(tf ⫺ ti) ⫹ ␲ b R(⬁ ⫺ ti) 3 RgT

(18)

Solving the nonlinear algebraic equation provides the final size of bubbles born at time ti, R(⬁ ⫺ ti). 6.2.5 SIMULATION RESULTS AND DISCUSSIONS Using the relations presented in the previous sections, several numerical simulations can be performed using Equations (1)–(18). The parameter values used for the simulation are listed in Table 6.1. These were mostly taken from

© 2000 by CRC Press LLC

TABLE 6.1

T(°C) ␩ (Pa.s) D(m2.s⫺1) kH(mol.N⫺1.m⫺1) fhom fhet F ␥(N m⫺1) I(Gka) (Pa) Z2L ⍀x

List of Parameter Values.

188 6,542 2.0 ⫻ 10⫺9 3.61 ⫻ 10⫺5 2.0 ⫻ 1020 3.97 ⫻ 1011 0.1 1.12 ⫻ 10⫺2 3.96 ⫻ 105 4.46 0.497

[measured] [estimated by experiments] Reference [6] Reference [6] References [6], [25] Reference [6] Reference [6] Reference [6]

Reference [6]. The parameter values of viscosity and diffusion coefficients were obtained from the author’s experiments. In the author’s experience, the simulation results were very sensitive to the parameter values in the nucleation rate equations, such as the surface tension, ␥, and the wetting factor F. 6.2.5.1 Effect of Pressure Releasing Rate on the Bubble Size and Density Figure 6.3 shows a simulation result for a LDPE and nitrogen system. It is assumed that at a pressure of 12.7 MPa, nitrogen dissolves into a molten LDPE

FIGURE 6.3

© 2000 by CRC Press LLC

Effect of pressure release rate on bubble size.

FIGURE 6.4

Effect of dissolved gas concentration.

and the system reaches equilibrium. Physical foaming is conducted isothermally by pressure release. For the simulation, it was also assumed that the pressure is released as a function of time, such that it is linearly decreased at a constant rate, kp: Pc(t) ⫽ e

Pc(0) ⫺ kpt: Pc(⬁):

for for

Pc(0) ⫺ kpt ⬎ Pc(⬁) Pc(0) ⫺ kpt ⱕ Pc(⬁)

By changing the value of kp, the effect of pressure release rate on bubble size and density was simulated. The simulation result illustrates that the average bubble size R␯ becomes smaller and the number of bubbles becomes larger as the rate of pressure release increases. These simulation results could qualitatively explain the experimental results that are shown in Figure 6.3 by open squares. 6.2.5.2 Effect of Dissolved Gas Concentration on the Bubble Size and Density Assuming the dissolved N2 gas concentration profile established in the molten LDPE, the simulation is performed for the case in which the pressure is released from 10.1 to 0.1 MPa at the rate of kp ⫽ 10.1. Figure 6.4 shows the simulation result. In this simulation, concentration dependence of the surface

© 2000 by CRC Press LLC

tension was not taken into account. The simulation shows that the average bubble size becomes smaller and the number of bubbles becomes larger as the initial concentration is increased. This is what we call a “pizza-sharing situation”: when one has to share a pizza with friends, your piece becomes smaller as the number of the friends you have to share with increases. To perform a more precise simulation, one has to consider concentration, pressure, and temperature dependencies of physical parameters such as shear viscosity [20], surface tension [10], solubility constant, and diffusion coefficient [5, 18]. 6.2.5.3 Interaction between Homogeneously Nucleated and Heterogeneously Nucleated Bubbles [24] During industrial foaming processes, a nucleating agent such as talc is often used to increase bubble density. The presence of the nucleating agent provides heterogeneous nucleation sites for foaming but does not preclude homogeneous nucleation. Homogeneous nucleation may still occur in the regions where local density of the heterogeneous nuclei is low. However, homogeneous nucleation affects heterogeneous nucleation by reducing gas concentration in the polymeric matrix. Due to the competition between these two nucleation mechanisms, the bubble density might be reduced as the concentration of heterogeneous nucleation sites increases. Colton and Suh [3] reported this phenomenon for autoclave-prepared polystyrene microcellular foams. They described the phenomenon by employing the following heterogeneous and homogeneous nucleation rate equations: Jhet ⫽ Nsfhet exp(⫺⌬G*het /kBT)

(19)

Jhom ⫽ (co ⫺ Jhet␪nb)fhom exp(⫺⌬G*hom /kBT)

(20)

where nb is the number of gas molecules in a bubble, and ␪ is the time since the first heterogeneous nucleation occurred. The term of (co ⫺ Jhet ␪ nb) in the homogeneous nucleation rate equation, Equation (20), expresses the concentration of gas molecules left after the consumption of dissolved gas by the heterogeneous nucleation. However, their theory, i.e., Equations (19) and (20), could not describe the bubble size and foam density under the conditions of coexistence of the two nucleation mechanisms. Using the nucleation and bubble growth models mentioned above, the interaction between the two types of nucleation mechanisms could be simulated. Figure 6.5 illustrates the computational results. Change of the bubble density, Nb, is shown in the figure as a function of the number of the heterogeneous nucleating sites, Ns, for four rates of the pressure release. The models could

© 2000 by CRC Press LLC

FIGURE 6.5

Mixed modes of nucleation.

predict that the increasing number of heterogeneous nucleation sites reduces bubble density locally. As the rate of pressure release increases, this reduction of bubble density diminishes and eventually disappears for high rates of pressure release, kp ⬎ 150 MPa/s. The temporary drop of Nb versus Ns can be better comprehended by considering the pressure release diagram of Figures 6.6 (a, b, c), for a system with a relatively low concentration of nucleating agent. When pressure is released gradually, the heterogeneous nucleation precedes the homogeneous one because of the free energy difference. Until the homogeneous nucleation starts, the heterogeneously nucleated bubbles alone consume the dissolved gas in the polymer. As the pressure release rate, kp, decreases, the time difference between onsets of the two nucleation mechanisms becomes larger and, consequently, gas consumption before the onset of the homogeneous nucleation increases. In consequence, the density drop becomes more evident as the pressure release rate is reduced. Figure 6.7 (a, b, c) shows the volume occupied by each size of bubble (4␲R3Nb/3) for three levels of the heterogeneous nucleation concentration, viz., Ns ⫽ 103, 107, and 1012. At lower concentrations of the heterogeneously nucleated sites [see Figure 6.7(a)], the size of the heterogeneously nucleated bubble is larger than that of the homogeneously nucleated ones, and this indicates the possibility that when two nucleation mechanisms occur simultaneously, the bubble size distribution may have two peaks. On the other hand, when Ns is large [see Figure 6.7(c)], the population of the heterogeneously nu-

© 2000 by CRC Press LLC

FIGURE 6.6 Dynamics of bubble density in mixed modes of nucleation: (a) kp ⫽ 10.1, (b) kp ⫽ 20.2, and (c) kp ⫽ 40.4 Mpa.s ⫺1.

FIGURE 6.7

© 2000 by CRC Press LLC

Volume of bubbles in mixed modes of nucleation.

cleated bubbles dominates the system. These phenomena could be observed as long as the heterogeneous nucleation rate had lower free energy than the homogeneous one and increased monotonically by the number of heterogeneous nucleation sites.

6.3 CONTINUOUS FOAMING 6.3.1 VISUAL OBSERVATION Compared with the modeling of batch foaming, relatively few publications feature models and simulation of continuous foaming processes such as extrusion foaming and injection foaming. One of the difficulties in simulating continuous foaming results from the fact that flow induces many changes in the nucleation mechanism and physical properties. As described in the nucleation model chapter of this book, Han and Villamizar [17] observed the bubble formation in a shear field by using a light scattering technique and reported several kinds of nucleation mechanisms: flow-induced and shear-induced nucleation, nucleation by thermal fluctuations, and nucleation by cavitation. Recently, Tujimura et al. conducted a visualization observation of bubble nucleation in foam extrusion die for a PP-isobutane with talc system [21]. Using a long-distance microscope and high-speed video, they observed the dynamic behavior of bubble nucleation in a shear field of a slit-die equipped with a quart glass window and obtained the pictures illustrated in Figures 6.8 and 6.9. Figure 6.8 shows the pictures of foaming at three different inlet pressures. The

FIGURE 6.8 Visual observation of foaming behavior in the die at various extrusion pressures (inlet flow pressure): (a) 3, (b) 5, and (c) 7 Mpa. Courtesy of Kanegafuchi Chemical.

© 2000 by CRC Press LLC

FIGURE 6.9 The magnified fields of bubble nucleating behavior at various talc concentrations: (a) 0%, (b) 0.1%, and (c) 1%.

white part indicates the foam, and it starts at a distance of 5–15 mm from the die exit. By increasing the inlet pressure of polymer flow, the onset position of foaming is shifted toward the die exit as can be seen in Figure 6.8. Figure 6.9 shows the magnified pictures of the onset position of foaming. Those were taken under the three different talc concentrations. They reported two characteristics of bubble formation: formation of a wedge-shaped bubble film, where nucleation of one bubble seems to induce other nucleation and forms a wedgeshaped gas film; and formation of a single, bullet-shaped bubble that seems to be induced by the talc and runs in the gas film. 6.3.2 FOAM MODELS IN THE SHEAR FIELD No perfect model is available to quantitatively simulate the foaming phenomena observed in Figures 6.8 and 6.9. However, several attempts have been made to simulate the bubble growth in a shear field [12, 13, 17, 19]. As a result, the current models could qualitatively predict the onset location of bubble formation, average bubble sizes, and density. Murayama et al. proposed a scheme for estimating the onset position of foam formation in the extrusion die of the PP-isobutane system, which was the identical system from which Figures 6.8 and 6.9 were taken [22]. They used models composed of a classical nucleation model, Patel’s bubble growth model, which is a single bubble growth model in the finite polymer sea, and Baldwin’s fluid model [19]. The basic idea behind their modeling is the integration of macroscopic flow models with the models of microscopic behavior of the bubble as described by Equations (1)–(18). Baldwin et al. modified the non-Newtonian fluid model of the flow running through a constant cross-sectional slit of height

© 2000 by CRC Press LLC

2B and width W [23] so that the model could consider the change in bulk viscosity and flow against the volume fraction of bubble (Figure 6.10). ⫺

n dPc 2 ⫹ 1/n 1 ⫽ c Q d mf 2 dz B 2WB

(21)

where ⫺dPc/dz is the pressure drop along the flow of the z-direction. mf and n, respectively, are the power law coefficient and index of the bulk viscosity of the foamed polymer fluid: # ␩ ⫽ mf␥n⫺1 ⫽ m c 1 ⫺

Vg # d ␥n⫺1 Vp ⫹ Vg

(22)

m is the constant. Vg and Vp are the total volume rate of the gas phase (bubbles) and that of polymer, respectively. Q denotes the volumetric flow rate of foamed polymer fluid and is given by the following: Q ⫽ (Vg ⫹ Vp)

(23)

Using Equations (21)–(23) with Equations (1)–(18), foaming in the flow can be simulated in the following manner: Using Figure 6.10 as a reference, the slit flow can be divided into N elements of length ⌬z. The polymer pressure and polymer volume flow rate at the first element are given as a boundary condition. When the pressure Pc(i ⌬z) and volumetric flow rate Q(i⌬z) at the up-

FIGURE 6.10

© 2000 by CRC Press LLC

Flow through a slit.

stream of the i-th element are given, the time period ⌬ti needed for flow to run through the i-element can be calculated by dividing 2 B W ⌬z by the volumetric flow rate Q(i⌬z). Then, the number and growth of bubbles nucleated during the period is calculated using Equations (1)–(18). Substituting these calculation results into Equations (21)–(23) gives the local gas phase volume, Vg, volumetric flow rate, Q, and pressure drop, ⌬P(i ⌬z). These values are used as Q((i ⫹ 1) ⌬z) and Pc((i ⫹ 1) ⌬z) ⫽ Pc(i ⌬z) ⫹ ⌬P(i ⌬z) for the next calculation at the neighbor element. 6.3.3 SIMULATION AND EXPERIMENTAL RESULTS Figure 6.11 is a result given by Murayama et al. [22]. The dots represent the experimental data. The average bubble sizes are measured from the picture and are plotted along the location from the die exit. In the simulation, the location where the nucleation rate exceeds a threshold value, Js, is regarded as the onset point of nucleation. The dotted line represents the simulation results by the models of Equations (1)–(12), which are nucleation rate plus single bubble growth models in an infinite polymer matrix (an infinite amount of gas). The

FIGURE 6.11 Chemical.

A simulation result of foaming in a shear field. Courtesy of Kanegafuchi

© 2000 by CRC Press LLC

solid line represents the results by their models, Equations (1)–(12), with a mass balance equation [i.e., simplified Equation (14)]: t

C(t)VP ⫽ co VP ⫺ VP

冮

4␲ J(␶)PD(t ⫺ ␶)R(t ⫺ ␶)3 d␶ 3 RgT

(24)

0

Apparently, the simulation results conducted under the assumption of an infinite amount of gas show larger bubble size and greater bubble density than the experimental data. By introducing the mass balance equation, Equation (24), the models could simulate a situation in which the gas available for bubble nucleation and growth runs out in the middle of the die, and, therefore, the bubble size reaches a plateau value. Because the total volume of the region where nucleation could take place is always equivalent to polymer volume, Vp, their simulation overestimates the number of bubbles that could nucleate by the time gas runs out. As a result, the calculated position where the gas runs out and reaches the plateau value is different from that observed in the experiments. It might be interesting to simulate these results using the influence region models, i.e., Equations (1)–(18) and Equations (21)–(23) mentioned in the previous section. Even though a discrepancy existed between the experimental data and simulated values, the onset position of bubble foaming as illustrated in Figure 6.8 could be successfully predicted by introducing the threshold value, Js, for the nucleation rate.

6.4 CONCLUSIONS Simulation schemes for batch and continuous foaming are described, based on Shafi and Flumerfelt’s recent work. The models used in batch foaming the simulation can take into account nucleation and bubble growth simultaneously. These models were combined with the macroscopic fluid model when continuous foaming was simulated. The foaming models described in this section can deal with only two steps of foaming processes (nucleation and bubble growth). To perform at consistent foaming simulation, the models for the remaining two steps (i.e., gas dissolution and foam structure stabilization processes) are also indispensable.

© 2000 by CRC Press LLC

6.5 NOMENCLATURE B C c0 cR cs D fhom fhet ⌬G* I(Gk,a) Jhom Jhet Jtotal kB kH kp Nb Ns Pc PD PD0 Q r R Rg R# ␯ R tf t⬘ T Vcb Vs VL Vp Vg W z Z2L ␩ ␥

Half of the slit height (m) Average dissolved gas concentration (mol m⫺3) Initial dissolved gas concentration (mol m⫺3) Dissolved gas concentration at a bubble surface (mol m⫺3) Threshold of dissolved gas concentration (mol m⫺3) Diffusivity of gas in polymer melt (m2 s⫺1) Frequency factor of homogeneous nucleation rate (s⫺1) Frequency factor of heterogeneous nucleation rate (s⫺1) Critical free energy of nucleation (J) Elasticity number of nucleation (Pa) Homogeneous nucleation rate (m⫺3 s⫺1) Heterogeneous nucleation rate (m⫺3 s⫺1) Total nucleation rate (m⫺3 s⫺1) Boltzmann constant (J K⫺1) Solubility coefficient (mol m⫺3 Pa⫺1) Rate of pressure release (Pa s⫺1) Bubble density (m⫺3) Number of heterogeneous nucleation sites (m⫺3) Pressure in continuous phase or ambient pressure (Pa) Pressure in the bubble (Pa) Initial bubble pressure (Pa) Volumetric flow rate of fluid (m3 s⫺1) Radial coordinate (m) Radius of a bubble (m) Gas constant Average bubble radius Bubble radius growth rate (m s⫺1) Time nucleation stops (s) Time at which a bubble is nucleated (s) Temperature (K) Volume of concentration boundary (m3) Influence volume (m3) Noninfluence volume (m3) Volumetric flow rate of polymer (m3 s⫺1) Volumetric flow rate of gas (m3 s⫺1) Slit width (m) Distance (m) Compressibility factor of dissolved gas solute in polymer melt (⫺) Viscosity (Pa s) Surface tension (J m⫺2)

© 2000 by CRC Press LLC

t ⍀

Time (s) Activity coefficient of dissolved gas solute in polymer melt (⫺)

6.6 REFERENCES 1. Zeldovich, Ya. B., Zh. Eksp. Theor. Fiz., 12, 525 (1942); Acta Physico-chim. USSR, 18, 1 (1943). 2. Street, J. R., Arthur, L. F. and Reiss, L. P., Ind. Eng. Chem. Fundam., 10, 54–64 (1971). 3. Colton, J. S. and Suh, N. P., Polym. Eng. Sci., 27, 7, pp. 500–503 (1987). 4. Ramesh, N., Rasmunssen, S. and Campbell, G. A., Polym. Eng. Sci., 34, pp. 1685–1698 (1994). 5. Goel, S. K. and Beckman, E. J., AIChE J., 41, pp. 357–367 (1995). 6. Shafi, M. A., Lee, J. G. and Flumerfelt, R. W., Polym. Eng. Sci., 36, 14, pp. 1950–1959 (1996). 7. Lee, J. G. and Flumerfelt, R. W., J. Appl. Polym. Sci., 58, pp. 2213–2219 (1995); J. Colloid Interface Sci., 184, 335 (1996). 8. Shafi, M. A., Joshi, K. and Flumerfelt, R. W., Chem. Eng. Sci., 52, 4, pp. 635–644 (1997). 9. Joshi, K., Lee, J. G., Shafi, M. A. and Flumerfelt, R. W., J. Appl. Polym. Sci., 67, 1353 (1998). 10. Utracki, L. A., private communication (1998). 11. Barlow, E. J. and Langlois, W. E., IBM J. Res. Develop., 6, pp. 329–337 (1962). 12. Amon, M. and Denson, C. D., Polym. Eng. Sci., Sept, 24, 13, pp. 1026–1034 (1984). 13. Afefmanesh, A., Advani, S. G. and Michalelides, E. E., Polym. Eng. Sci., Oct. 30, 20, 1330–1656 (1990). 14. Ramesh, N. S., Rasmussen, D. H. and Campbell, G. A., Polym. Eng. Sci., Mid-Dec, 31, 23, pp. 1657–1664 (1991). 15. Rosner, D. E. and Epstein, M., Chem. Eng. Sci., 27, pp. 69–88 (1972). 16. Patel, R. D., Chem. Eng. Sci., 35, pp. 2352–2356 (1980). 17. Han, C. D. and Villamizar, C. A., Polym. Eng. Sci., July, 18, 9, pp. 687–698, pp. 699–710 (1978). 18. Ramesh, N. S. and Malwitz, N., ANTEC ‘95, pp. 2171–2174 (1995). 19. Baldwin, D. F., Park, C. B. and Suh, N. P., MD-Vol. 53, Cellular and Microcellular Materials, ASME, pp. 85–107 (1994). 20. Park, C. B., Lee, M. and Tzoganakis, C., PPS-14, Yokoyama, pp. 277–282 (1998). 21. Tujimura, I., Zenki, T. and Ishida, M., PPS-14, Yokoyama, pp. 113–114 (1998). 22. Murayama, T., Ikeda, J. and Ishida, M., JSPP ‘98, Osaka, pp. 169–170 (1998). 23. Bird, R. B., Armstrong, R. C. and Hassager, O., Dynamics of Polymeric Liquid, John Wiley & Sons, New York (1987). 24. Ohshima, M., Inamori, K., Takada, M. and Tanigaki, M., PPS-NA, Toronto, pp. 36–37 (1998). 25. Ruengphrathuengsuka, W., Texas A & M, Ph.D. Thesis (1992).

© 2000 by CRC Press LLC

CHAPTER 7

Process Design for Thermoplastic Foam Extrusion LEONARD F. SANSONE

7.1 INTRODUCTION

D

EPENDING on the polymer used and the end-use application, thermoplastic

foams are produced with densities ranging from 3% to 50% of the polymer density. Higher density foams utilize chemical blowing agents (CBA) that decompose to release gas that is dissolved in the melt. As the pressure is reduced in passing through the die, the gas is released to form the foam. Lower density foams utilize liquid or gaseous blowing agents injected into a plasticating extruder. These blowing agents dissolve in the melt and have a strong plasticizing action causing a large reduction in melt viscosity. An attempt to form a foam under these conditions results in excessive expansion, cell rupture, and collapse of the structure. This effect necessitates the reduction of melt temperature to avoid overblowing and cell rupture. Although heat can be extracted by cooling the downstream zones of the extruder barrel, the conflicting requirements for design of a plasticating extruder versus the design for heat extraction limit the production rate. Higher production rates can be obtained by using a second extruder operating in tandem with the plasticating extruder. The primary function of the second extruder is to cool the melt to a temperature range where good quality foam may be formed. Good instrumentation is necessary to provide adequate control needed for extrusion of foam products. Modern self-tuning temperature controllers provide stable operating temperature. A retractable melt thermocouple mounted in the die adaptor allows measurement of the temperature distribution in the melt stream. The device allows positioning of the thermocouple tip during operation to obtain the melt temperature profile. Use of a high-speed recorder or computer data acquisition can reveal time-dependent temperature

© 2000 by CRC Press LLC

fluctuations caused by poor mixing or surging. Fixed melt thermocouples can provide misleading information because they cannot indicate the temperature extremes in the melt stream. A pressure transducer located near the screw tip provides vital information to confirm proper design of the die and operation of the extruder. An extruder screw is a pressure-sensitive pump. At low pressures, work input will be inadequate to provide proper mixing, while at high pressures, excessive melt temperature may require reduction of output rate.

7.2 HIGH-DENSITY STRUCTURAL FOAM PROCESS High-density structural foams find applications competing with products traditionally made from wood. PVC is the leading plastic used in these applications due to its favorable physical properties and cost. The CBA can be dispersed in the compound during the pelletizing process, but care must be taken to avoid excessive melt temperature that will cause the CBA to decompose prematurely. The utilization of PVC dryblend powder circumvents this problem and, in addition, eliminates the cost of pelletizing. Both single- and twin-screw extruders are successfully used in forming PVC foams from dryblend powder. High-density foams can also be formed from PE, ABS, PS, PP, and other polymers. These materials are generally not available in powder form and are best handled by using blowing agent concentrates or pellets that have been compounded with the blowing agent incorporated during the pelletizing process. Selection of the CBA depends on the processing temperature of the polymer. If the decomposition temperature of the blowing agent is too low, it will decompose prematurely in the extruder, and the gas will be vented through the feed throat. Conversely, if the decomposition temperature is too high, the decomposition temperature will not be reached, and gas will not be evolved. The decomposition temperature can be adjusted by using an activator that induces decomposition at a lower temperature. It should be noted that many common additives used in polymer formulation act as activators so that formulation changes may affect the decomposition temperature. Suppliers of CBA compounds have developed materials suited to various polymers and can provide recommendations for required concentration and operating conditions. When using polymers in powder form, air entrapment can cause large blisters in the extrudate. This can be avoided by using a metered feed to allow air to escape in the feed throat or by using a two-stage screw with vacuum venting. Another approach is to use a small-diameter tube inserted in the feed throat to act as an air vent. Residual moisture will also cause large blisters and should be removed by vacuum venting or hopper drying.

© 2000 by CRC Press LLC

Experience in extruding solid, unfoamed products with the particular polymer chosen for foam extrusion is helpful in setting up operating conditions for foam extrusion. However, additional constraints needed to form good quality foam may require alteration of operating conditions typically used for extrusion of solid products. 7.2.1 SCREW DESIGN FOR SINGLE-SCREW EXTRUDERS A screw that provides good mixing and control of melt temperature for a given polymer will provide good results for structural foam extrusion. Some adjustment of barrel zone temperatures may be required to initiate blowing agent decomposition at the right time and to provide the optimum melt temperature as the polymer enters the die. As indicated by Rauwendaal [1], barrier screws fitted with a mixing section give excellent results. Extruder manufacturers have developed screw designs specific to various polymers and can provide recommendations for foam extrusion. 7.2.1.1 Setting Operating Conditions on Single-Screw Extruders 7.2.1.1.1 Feed Throat The feed throat is typically cored to provide cooling of this section. Cooling is beneficial for low-melt-point polymers that may stick to the inside surface if the feed throat is allowed to reach a high temperature by conduction from zone 1 of the extruder barrel. It should be recognized that typically, two to three screw flights are within the feed throat section, and they comprise part of the screw feed section. Screw and barrel temperatures, bulk density, and pellet temperature affect feeding of solids. Feeding is optimized with low friction on the screw and high friction on the barrel. High-melting-point polymers may not require feed throat cooling and may actually show improved feeding by allowing the feed throat section to come to a higher temperature. Cold pellets, caused by storage in an outside silo during winter, may cause erratic feeding due to the reduction of the friction coefficient on the barrel surface. This may be alleviated by preheating the pellets using a hopper dryer or by heating the feed throat by circulating a hot fluid under controlled temperature conditions. 7.2.1.1.2 Barrel Zone 1 Care must be taken in setting zone 1 barrel temperature to avoid premature decomposition of the blowing agent that will result in loss of density reduction. However, excessively low zone 1 temperature may result in poor mixing and poor extrudate quality. In addition, some polymers show a very sensitive relationship between zone 1 temperature and feed efficiency. Studies relating

© 2000 by CRC Press LLC

friction coefficients to metal surface temperature and material temperature have been described by Gregory [2]. Other studies have included the effects of pressure and surface speed. Darnell and Mol [3] showed that minimizing friction on the screw and maximizing friction on the barrel surface optimizes solids feeding. Polymers typically show an increase in friction coefficient as the surface temperature is increased, reaching a peak and then diminishing as the temperature is increased further. Some polymers such as polystyrene (PS) have a very sharp peak, making control of zone temperature critical. Others, such as high-density polyethylene (HDPE) have a very broad peak, allowing a broad temperature range to be utilized while maintaining good feeding efficiency. Materials such as polypropylene (PP) have a very flat response to temperature and exhibit low feed efficiency. It should be noted that compound additives such as slip agents and lubricants often alter the frictional characteristics drastically. Although most polymers show adequate feed efficiency, those with poor feeding characteristics can be improved by cooling the feed section of the screw. Improper zone 1 temperature can result in surging due to erratic feeding. The best approach is to start with a low zone 1 temperature, observe extrudate quality, pressure stability, and foam density, and then adjust zone 1 temperature as required. If limits on zone 1 temperature are found to be narrow, zone 2 temperature can be raised to promote melting and mixing. It is important to reach the decomposition temperature prior to the metering section of the screw in order to allow the released gas to be solvated and dispersed uniformly. 7.2.1.1.3 Downstream Barrel Zones A reverse temperature profile on the barrel promotes early melting that improves mixing and reduces the temperature gradient in the melt stream as it enters the die. As indicated above, excessive temperature in zone 1 may cause the blowing agent to decompose prematurely with loss of gas through the feed throat. When high temperatures are employed in either zone 1 or 2, it is usually necessary to set the downstream zone temperatures below the desired melt temperature to avoid excessive melt temperature as the polymer enters the die. Extraction of heat in the downstream zones raises the melt viscosity, imparting more shear work to the polymer resulting in improved mixing and melt homogeneity. A retractable melt thermocouple is useful in measuring the temperature gradient in the melt stream as it passes through the adaptor section. It is not uncommon to find temperature gradients as large as 40°F in the melt stream due to improper operating conditions or a worn screw and barrel. Large temperature gradients will result in underblown or overblown portions of the extrudate. Excessive melt temperature will result in overblowing, subsequent cell rupture, and collapse of the structure. Low melt temperature causes the foam

© 2000 by CRC Press LLC

density to be too high. Temperature gradients can be reduced by good screw design, proper barrel zone temperatures, or the use of a static mixer. The static mixer provides a uniform temperature distribution as well as a more uniform distribution of the dissolved gas. Care must be taken in selecting a static mixer when processing materials with poor thermal stability as some designs utilize geometries resulting in excessive residence time. 7.2.2 SCREW DESIGN FOR TWIN-SCREW EXTRUDERS Although twin-screw extruders can be employed in the processing of a wide variety of polymers, their main application in foam extrusion is in the production of PVC products utilizing dryblend powders. The elimination of the cost of pelletizing and the ability to maintain controlled melt temperature at a high production rate provides manufacturing cost reductions that offset the higher equipment capital cost compared to a single-screw extruder. Intermeshing, counterrotating twin-screw machines for powder extrusion of PVC compounds are available with conical or parallel screw geometries, and they typically employ metered feeding. To accommodate the need to compress the powder material and extract trapped air and moisture, the screws are designed with feed, compression, first metering, venting, compression, and a second metering section. As with single-screw extruders, barrel heat is arranged in zones in order to optimize the temperature for each section of the extruder. In addition, the temperature of the screws is controlled over the entire length by circulating a heat transfer fluid though the bores of the screws. Compression is obtained by reducing the available volume in the screw channels by changing the pitch and number of thread starts. In the case of conical screws, the reduction of diameter also contributes to the compression as the material is conveyed forward. After the first metering section, the volume is increased so that this section runs partially empty to allow venting of air and moisture. The volume is then reduced and conveyed into the second metering section that builds pressure to pump the material through the die. The screw flights are designed to have clearances between the crown of the flight and the root of the opposing screw and adjacent flanks, typically in the range of 0.040 to 0.120 inches. 7.2.2.1 Setting Operating Conditions for Twin-Screw Extruders Since there are substantial gaps between the screw flights, counterrotating twin-screw extruders are not positive displacement pumps, but they achieve 60% to 85% volumetric pumping efficiency in the feed section under flood feed conditions. Metering the feed so that the screw flights are just visible will reduce the output slightly but will allow entrapped air in the powder to escape through the feed throat.

© 2000 by CRC Press LLC

The CBA can be incorporated in the dryblend powder or added at the feed throat using a screw feeder. A CBA concentrate will provide good dispersion when added with a feeder but will also increase material costs. Employing metered feeding allows start-up of the extruder in a starved condition to initially operate at low power input. After observing melt temperature and product quality, feed rate can be increased as required. As the filled length is increased, power input, shear work, and mixing are increased. In addition, increasing screw temperature and zone 1 barrel temperature will move the fusion point back in the extruder, thus increasing mixing length and melt temperature. The first metering section acts as a dynamic throttle, causing a pressure buildup that in turn drives more material through the flank and crown/root gaps in the compression and metering sections where high shear rates are imposed. Depending on the design of the screws and feed rate, the flow rate of the first metering section may be 25% above the theoretical drag flow rate. Shear rates in the gaps are typically in the range of 200 to 600 reciprocal seconds, while the rates in the screw channel may be only 5 to 10 reciprocal seconds. The intermittent high shear rate, followed by conveying at low shear rate provides good dispersion of compound additives without causing excessive melt temperature. In addition, the large surface area of the screws allows substantial extraction of heat through screw cooling. Heat transfer fluid temperatures are typically in the range of 275°F to 325°F for PVC powder compounds. Depending on the lubricity of the compound, higher temperatures may cause it to stick to the screw surface and may also cause subsequent thermal degradation. As with single-screw extrusion, care must be taken to avoid premature rise in the melt temperature, resulting in decomposition of the CBA and loss of gas through the feed throat or the vent section. The preferred condition at the vent is a semi-fused crumb. If the material is still in powder form at the vent, air leaking from the feed section will cause blow-over when vacuum is applied. If a well-fused ribbon is formed, blow-over is avoided, but the melt temperature may be high enough to cause premature CBA decomposition. Depending on screw design and compound characteristics, this may be alleviated by operating with an open vent or at reduced vacuum levels in the range of 2 to 5 inHg. Starve feeding to the extent that the crowns of the screw flights are just visible allows release of much of the air before the material is conveyed to the vent section. The melt temperature should reach the decomposition temperature of the CBA in the compression section prior to the second metering section. This provides time for the released gas to be dissolved and dispersed in the melt under high pressure. Failure to reach the decomposition temperature just prior to the second metering section will result in inadequate density reduction. Attempts to correct this by increasing the die temperature will result in density reduction

© 2000 by CRC Press LLC

on the surface of the extrudate and a higher density core due to the poor thermal conductivity of the polymer. This approach can also cause overblowing of the surface material, causing cell rupture and a very rough appearance. As with single-screw extrusion, a retractable melt thermocouple installed in the die adaptor and a pressure transducer mounted near the screw tips are vital instruments needed to control operating conditions. The best operating conditions are obtained by operating at a sufficient high pressure to raise the melt temperature in the second metering section by shear work. Normally, the barrel zone temperature for the metering section will be set at 10°F to 30°F below the desired melt temperature for extrusion of PVC. The variation in melt temperature through the melt stream as measured by incrementally moving the thermocouple position provides information useful in setting barrel zone and screw temperatures. High temperature at the centerline of the adaptor flow channel can be corrected by reducing screw temperature and/or raising the metering barrel zone temperature. Excessively low operating pressure will result in poor mixing, large temperature gradients in the melt stream, and large density gradients in the product. Conversely, excessively high pressure will cause high melt temperature resulting in overblowing and cell collapse. Due to the substantial gaps in the intermeshing region of the screws, the metering section of the extruder performs as a pressure-sensitive pump. As pressure is increased, the fill point of the screws moves back toward the vent section. Additional increase in the pressure will eventually cause vent flooding.

7.3 LOW-DENSITY FOAM PROCESS Foams with densities as low as 1.5 lb/cu ft can be produced by utilizing liquid or gaseous blowing agents commonly termed physical blowing agents. These blowing agents must be soluble in the polymer at high pressure and temperature and must be at least partially insoluble when pressure is reduced. When the applied pressure falls below the level of the partial pressure of the blowing agent, bubble nucleation is initiated. Commonly used physical blowing agents include HCFC 141b and 142b, hydrocarbons propane, butane, and pentane, and argon or carbon dioxide gas. Cell nucleators such as fine-particle calcium carbonate or talc are used at 1 to 2% concentration in order to provide a fine cell structure. Physical blowing agents are metered into the extruder under high pressure and are dissolved in the melt. Studies by Han [4, 5] indicate that at the concentrations used to obtain low-density products, physical blowing agents have a strong plasticizing action on the melt, greatly reducing the viscosity. The high melt temperature and intensive shear required to properly disperse the blowing agent prevents direct extrusion into a die as the high partial pressure of the blowing agent and the low viscosity will cause

© 2000 by CRC Press LLC

overblowing, cell rupture, and collapse of the extrudate. Although a screw can be designed to minimize work input in the downstream region, conflicting requirements for cooling versus intensive mixing and limited surface area for heat extraction severely limit the production rate capability. For this reason, commercial systems often employ a second extruder in tandem with the plasticating extruder. The second extruder is typically one size larger than the plasticating extruder and functions as a heat exchanger designed to optimize extraction of heat in order to reduce the melt temperature into a range where a satisfactory foam can be formed. 7.3.1 SINGLE-SCREW DESIGN FOR LOW-DENSITY FOAM PLASTICATING EXTRUDER A barrier type screw with an intensive mixing section downstream of the blowing agent injection point is recommended. An L/D ratio of at least 32:1 is recommended in order to provide sufficient time and shear to obtain good dispersion of the blowing agent and cell nucleator. A screw design found to provide good performance with the polymer used for solid extrusions will be satisfactory for use in the plasticating extruder in a tandem foam extrusion system. 7.3.2 SETTING OPERATING CONDITIONS FOR THE PLASTICATING EXTRUDER The recommendations for operation of the feed throat as described for highdensity structural foams apply here as well. The setting of zone 1 barrel temperature differs in that we are not concerned with premature decomposition of a CBA and thus have considerably more latitude in setting zone 1 barrel temperature. The first concern is to utilize a temperature that provides steady solids feeding. Second, a reverse temperature profile with zone 1 set higher than the required melt temperature provides early melting that enhances mixing. It is important to assure that melting is completed before reaching the injection port. If the solid bed extends to the injection port, the blowing agent will flow through the bed and escape through the feed throat. A gas detector in the feed hopper is useful to assure that there is no backflow of the blowing agent, especially if combustible blowing agents are used. If zone 1 temperature is limited by feeding performance, melting can be accelerated by using a high temperature in zone 2 to assure that a melt seal has been formed. The blowing agent is injected downstream of the barrier section and prior to the mixing section. Blowing agents in the liquid state can be metered by using a variable speed multiplex piston pump. The metering pump capacity should be selected to provide multiple small volumes in order to approach continuous flow. Calculating the number of screw revolutions per injection pulse will in-

© 2000 by CRC Press LLC

dicate if the injection rate has approached continuous flow. The injection pump should only be started after assuring that the screw is filled and pressure at the screw tip has reached the normal operating level in order to avoid forward or backward flow of the blowing agent. The injection nozzle should be fitted with a check valve to avoid clogging before the injection pump is started. Metering of gaseous blowing agents requires instrumentation to measure pressure, temperature, and volume flow rate in order to control the mass flow rate. In order to minimize the amount of heat that must be extracted in the cooling extruder, the downstream zones should be held at a temperature below the melt temperature. In addition, the melt pipe connecting the plasticating extruder to the cooling extruder should be designed to minimize the pressure drop. Higher pressure requires higher screw speed to maintain the same output rate. This, in turn, causes a larger temperature rise in the melt stream. In addition, there is a melt temperature rise in the feed pipe which is as follows: ⌬T ⫽ ⌬P/␳ Cp (under adiabatic conditions)

(1)

where: ⌬T ⫽ average temperature rise, °F; ⌬P ⫽ pressure drop, psi; ␳ ⫽ melt density , lb/in3; and CP ⫽ specific heat capacity, BTU/lb °F. 7.3.4 COOLING EXTRUDER SCREW DESIGN FOR LOWDENSITY FOAM Since the polymer has been melted and mixed prior to entering the cooling extruder, the cooling extruder screw design is significantly different from that of a plasticating extruder screw. There is no need to have a feed section such as that required for a plasticating screw. Since the pumping characteristics of a melt are determined by its viscous properties, we are not concerned with the effects of bulk density and frictional characteristics as with the case for solids feeding. In order to effectively reduce the melt temperature, we must minimize the energy input to the melt due to mechanical work. By examining the equations for work input and output rate for a single-flighted, single-screw extruder pumping a polymer melt, we can develop a design and operating mode that provides the highest throughput with the minimum energy expended. Considering a small element along the screw axis, the net output rate is as follows: Qnet ⫽

Fd ␲2 D2 N h (1 ⫺ e/ t) sin␾ cos␾ ⫺Fp ␲ D h3 (1 ⫺ e/ t) sin2␾ dP (2) 2 12 ␮ dL

where Qnet ⫽ volumetric flow rate, in3/sec; Fd ⫽ drag flow correction factor; D ⫽ screw diameter, inch; N ⫽ screw speed, revolutions/sec; h ⫽ channel

© 2000 by CRC Press LLC

depth, inch; e ⫽ screw flight width parallel to screw axis, inch; t ⫽ screw pitch, inch; ␾ ⫽ screw helix angle, degrees; Fp ⫽ pressure flow correction dP factor; ␮ ⫽ melt viscosity, lbf sec/in2; and ⫽ Pressure gradient along dL screw, psi/inch. If the pressure rise along the screw is zero, the output rate, termed the drag flow rate, is as follows: Qd ⫽

Fd ␲2 D2 N h (1 ⫺ e/ t) sin␾ cos␾ 2

(3)

This condition is obtained by adjusting the screw speed of the plasticating extruder to maintain the pressure at the inlet to the cooling extruder equal to the pressure at the exit. The power input over a small segment where conditions can be considered isothermal is as follows: ⌬E ⫽ [ ␲3 D3 N2 ␮ ( 1 ⫺ e / t )

(cos2 ␾ ⫹ 4 sin2 ␾) h ␲2 D2 N2 ␮L e] ⫹ dL ⫹ Q⌬P (4) ␦ tan ␾

where ⌬E ⫽ power input, in lbf/sec; ␮L ⫽ melt viscosity over flight land, lbf sec/in2; and ␦ ⫽ radial screw/barrel clearance, inch. Since there is no pressure gradient along the screw, the term Q⌬P is zero under drag flow conditions. The power input is: term 1 ( cos2 ␾ ⫹ 4 sin2 ␾ ) ⌬E ⫽ c ␲3 D3 N2 ␮ ( 1 ⫺ e/t ) h ⫹

term 2

␲2 D2 N2 ␮L e d ⌬L (5) ␦ tan ␾

Term 1 is the power expended by shear work in the downstream and crossstream directions. Term 2 is the power expended by shear work in the leakage flow over the flight land. With a new screw and barrel, this term is small in comparison to term 1. However, as the radial screw/barrel clearance, ␦, becomes large, the power expended at the flight land increases, causing large temperature gradients in the melt. In addition, the metering performance of the screw deteriorates causing poor tolerance control. At a given flow rate, we can minimize term 1 by optimizing the helix angle, ⌽. The power input per unit volume is calculated by dividing power input by Qd, the drag flow rate. By increasing the pumping efficiency, the screw speed

© 2000 by CRC Press LLC

can be reduced while maintaining the same output rate. It is necessary to take into account the effect of screw speed on the viscosity of the polymer. This can be accomplished by employing the power law relationship, ␮ ⫽ ␮0 (N/N0)(n⫺1) in Equations (2) and (4). When the energy input per unit volume is compared for a screw using the customary “square pitch” (helix angle of 17.66 degrees) with a screw using a helix angle of 30 degrees, a 20% reduction in energy input per unit volume can be attained while running the screw at 64% of the speed needed to provide the same output using a square pitch screw. In order to extract sufficient heat from the melt in order to drop the temperature to a range where quality foam can be formed, the barrel temperature must be well below the melt temperature. From the standpoint of extracting heat, a shallow channel depth would be beneficial because polymers have poor thermal conductivity. However, it can be seen from Equation (4) that reducing the channel depth, h, causes additional power input. We have two conflicting variables and find that there is an optimum channel depth depending on the screw design, operating conditions, and polymer properties. In addition, there is an optimum barrel temperature. If the barrel temperature is set too low, the polymer viscosity is increased excessively, causing more current draw on the extruder motor and resulting in increased melt temperature. Due to mechanical work input, heat is generated in the small element studied at the rate of: qm ⫽ c ⌬E

(6)

where c is a factor converting mechanical power to heat flow rate. In order to reduce the melt temperature, this heat must be extracted. The total heat flow rate required is: q ⫽ qm ⫹ ␳ Qd Cp ( T1 ⫺ T2 )

(7)

where: T1 ⫽ melt temperature entering the cooling extruder and T2 ⫽ melt temperature leaving the cooling extruder. To effectively remove the heat at a high rate, it is important to optimize the design of the barrel cooling system. Older extruders have used copper tubes wrapped into grooves in the barrel surface. This limits heat transfer due to poor contact with the barrel caused by differential thermal expansion, low contact area, and eventual corrosion. The best design utilizes split aluminum blocks containing resistance heater elements and stainless steel tubing for circulation of cooling water. They must be closely fitted and tightly clamped to the barrel surface. As the melt temperature is reduced, the viscosity is increased and mechanical heat generation is increased making it more difficult to extract additional

© 2000 by CRC Press LLC

heat as the exit end of the extruder is approached. Good instrumentation and data recording allow the operator to evaluate various operating strategies to arrive at optimum operating conditions. Recommended instruments include pressure transducers at the screw tips of both extruders. In addition, a pressure transducer should be located at the entry port of the cooling extruder to allow the operator to equalize the entry pressure with the exit pressure. Retractable melt thermocouples should be mounted in the adaptors of both extruders to allow adjustments to minimize temperature gradients in the melt stream and record any fluctuations with time. In order to optimize cooling performance, both extruders should be equipped with wattmeters. The cooling system zones should be equipped with flow meters and thermocouples to measure the heat extraction rate at each zone. A computer control system can acquire this information, process it, and provide a basis for adjusting operating conditions. Operating conditions can be stored for various products, and startup procedures can be set to coordinate the operation of the two extruders and the blowing agent injection pump.

7.4 DIE DESIGN PROCEDURES FOR FOAM EXTRUSION The primary function of an extrusion die is to form a product from molten polymer on a continuous basis while meeting property and dimensional requirements. A secondary function is to provide the optimum pressure at the screw tip to assure that the extruder properly mixes the polymer and delivers it to the die at a uniform rate. Since both single- and twin-screw extruders are pressure-sensitive pumps, excessive pressure will result in high melt temperature, frequently requiring a reduction of output rate to bring the process into control. If the pressure is too low, mixing may be inadequate, resulting in poor appearance or inadequate physical properties. A die system includes the items downstream of the screw tip. Sections include an adaptor, a transition or distribution section where the flow channel is altered from the circular bore of the adaptor, and an orifice where the shape of the product is formed. If filtering is required for the product, the effects of the screen pack and breaker plate must be considered. Not only must the total pressure drop across the die system be controlled, the distribution of the pressure drops within the die must also be considered in order to provide the best control of the shape and dimensions of the product. This involves calculating the pressure drops in each region of the die, taking into account the geometry of each section and applying the appropriate flow analysis. The design is optimized by maximizing the pressure drop in the orifice region and minimizing the pressure drops in the other regions. The region having the highest pressure drop has the greatest influence on the flow distribution. The pressure drop in the orifice region is controlled by setting the

© 2000 by CRC Press LLC

land length in order to provide a sufficient pressure drop. The viscous properties of the polymer must be considered as well as the production rate and the melt temperature required. Foam extrusion adds another requirement in that the pressure gradient and flow rate through the die must be great enough to prevent premature foam formation within the die. Foam formation is a nucleation and growth process. When the pressure within the die drops below the partial pressure of the blowing agent, the nucleation process is started. Nucleation is initiated at sites formed by additives such as talc, calcium carbonate, or other additives. Initiation of nucleation takes time as evidenced by extrudates issuing from the die at high speed with no visual evidence of foam formation for 1 or 2 inches. After nucleation, bubbles grow in size, increasing the product volume and reducing the density. The concentration of blowing agent and melt temperature determine the amount of density reduction. The expansion is limited by cooling of the gas as it expands and subsequent extraction of heat from the polymer. The reduction in gas pressure and increase in the viscosity of the polymer terminates the expansion process. Thin-gauge products can be reheated in order to achieve post-expansion to provide additional density reduction. If the extrusion rate is too low or the melt temperature is too high, foam will form within the die, cells will be ruptured by the shearing action, and the structure will collapse. To avoid this, it is necessary to maintain good control of the melt temperature and the temperature gradient within the melt stream. Extremes of melt temperature will result in voids due to insufficient blowing, high-density regions, or collapsed regions due to overblowing, The capillary rheometer is the instrument most commonly used to obtain rheology data for polymers. It utilizes a cylindrical chamber and piston to drive the polymer at a controlled rate though a small bore capillary. By measuring the force on the piston and computing the volumetric flow rate, the shear stress and apparent shear rate can be determined. Due to the geometry of the instrument, the Bagley [6] correction procedure must be used to deduct the entry pressure loss from the shear stress calculation. The Rabinowitsch [7] correction factor is applied to account for the pseudoplastic nature of polymer melts. Since the polymer is melted by conduction in this instrument, typically requiring 6 to 8 minutes, polymers containing blowing agent will start to release gas prior to the start of the test. The dissolved gas has a strong plasticizing action resulting in the reduction of viscosity. Variable amounts of gas loss provide erratic and misleading viscosity data. In addition, data obtained at low shear rates will exhibit extrudates issuing from the capillary that have expanded within the capillary and collapsed, again giving misleading data. The problems associated with the capillary rheometer can be avoided by using a slot rheometer as shown in Figure 7.1. The device employs a slot die fitted with a melt thermocouple and three pressure transducers to measure the

© 2000 by CRC Press LLC

FIGURE 7.1

Slot rheometer for foam compound rheology.

pressure gradient. The first transducer is typically placed 1/3 of the land length of the slot downstream of the entry to avoid measuring pressure in the entry region. The use of three transducers allows comparison of the local pressure gradients to assure that the first transducer is downstream of the entry region. If a structural foam compound utilizing CBA is evaluated, a conventional 3⁄4⬙ or 1⬙ laboratory extruder may be used to melt and mix the material and drive it through the slot die at various conditions of melt temperature and flow rate. The melt temperature, mass flow rate, and appearance of the extrudate are recorded. Sample densities are measured for correlation with operating conditions to provide information useful in die design and developing full-scale operating conditions. Because it is difficult to maintain the same melt temperature over a wide range of throughput rates, it is necessary to use multivariate statistical analysis to determine the rheological parameters. Shear stress and shear rate are calculated from the slot rheometer data using the following equations: ␶⫽

Rh⌬P ⌬L

2Q(a ⫹ bn) # ␥⫽ RhAn

© 2000 by CRC Press LLC

(8)

(9)

FIGURE 7.2

Flow channel geometries and nomenclature.

where: ␶ ⫽ shear stress, psi; Rh ⫽ hydraulic radius, inch ⫽ flow area/ perimeter; ⌬P ⫽ pressure drop between pressure taps, psi; ⌬L ⫽ distance # beween pressure taps, inch; ␥ ⫽ shear rate, 1/sec; Q ⫽ volumetric flow rate, in3/sec; a, b ⫽ shape factors using aspect ratio obtained from Figure 7.2, Equations (10) and (11) using coefficients in Table 7.1 and Table 7.2; n ⫽ power law exponent; and A ⫽ cross-sectional flow area, in2.

TABLE 7.1

A0 A1 A2 A3 A4 A5 A6

TABLE 7.2

B0 B1 B2 B3 B4 B5 B6

© 2000 by CRC Press LLC

Coefficients for Equation (10). SLOT

ANNULUS

0.50 ⫺1.321099 4.339961 ⫺11.399043 17.873964 ⫺14.120467 4.338235

0.250 3.525716 ⫺20.943061 61.164339 ⫺92.305040 69.031394 ⫺20.224675

Coefficients for Equation (11). SLOT

ANNULUS

1.00 ⫺0.840096 1.73808E-03 3.810358 ⫺8.594332 8.127828 ⫺2.826797

0.750 3.590286 ⫺21.350268 62.083357 ⫺93.253366 69.480584 ⫺20.302289

Equations for calculating shape factors a and b: a ⫽ A0 ⫹ A1e ⫹ A2e2 ⫹ A3e3 ⫹ A4e4 ⫹ A5e5 ⫹ A6e6

(10)

b ⫽ B0 ⫹ B1e ⫹ B2e2 ⫹ B3e3 ⫹ B4e4 ⫹ B5e5 ⫹ B6e6

(11)

Using the power law relationship, the following equation correlates shear stress with shear rate and temperature: # ␶ ⫽ m ␥n exp(⌬E/R T)

(12)

# Where ␶ ⫽ shear stress, psi; m ⫽ consistency, lbf secn/in2; ␥ ⫽ shear rate, 1/sec; n ⫽ power law exponent; ⌬E ⫽ flow activation energy, BTU/lb mol; R ⫽ gas constant, 1.9872 BTU/°R lb mol; and T ⫽ absolute temperature, Deg. Rankine. The equation is linearized by taking logarithms in order to make a linear multivariate regression: # ln ␶ ⫽ ln m ⫹ n ln ␥ ⫹ ⌬E/R T

(13)

The values of m, n, and ⌬E are then determined from the data set. 7.4.1 DIE DESIGN FOR HIGH-DENSITY STRUCTURAL FOAM PRODUCTS High-density structural foam products having density of about 50% of the polymer density find applications as a wood replacement. These products utilize a CBA to provide release of gas upon reaching a decomposition temperature. An alternate approach is to inject a gas such as carbon dioxide into the extruder. In either case, the gas must be dissolved in the melt and thoroughly mixed before the melt enters the die. Examples of two approaches for die design are given for rigid PVC foamed to a density of 40 pcf. 7.4.1.1 Free-Foaming Method The free-foaming method employs a die orifice smaller than the product size and a vacuum calibrator to form the shape of the product and to hold dimensional tolerances. Design for a simple rectangular shape is provided as an example. Product: 0.25⬙ ⫻ 2.00⬙ rectangular profile with 1/64⬙ radius corners. Density: 40 pcf ⫹/⫺ 3 pcf Material: Rigid PVC Melt density at 400 F: 0.045 lb/in3 Consistency at 400 F: 3.86 lbf secn/in2 Power law exponent: 0.433

© 2000 by CRC Press LLC

Production rate: 80 lb/hr at 2300 psi Adaptor: The adaptor has a 2.000 inch ⫻ 60° included angle cone section and 0.50 inch bore, 4 .000 inches long Volume expansion ratio: (1728 ⫻ 0.045 lb/in3 )/40 pcf ⫽ 1.944 Width expansion ratio: (1.944)1/3 ⫽ 1.248 Die orifice width: 2.0 / 1.248 ⫽ 1.600 Thickness expansion ratio: (1.944)1/3 ⫽ 1.248 Die orifice gap: .25/1.248 ⫽ 0.200 Orifice aspect ratio: 0.200/1.600 ⫽ 0.125 The pressure drops across the adaptor cone and bore are found using the Kozicki [8] relationship:

⌬P ⫽

mL W(a ⫹ bn) n c d Rh 1,800␳RhAn

(14)

where ⌬P ⫽ pressure drop across section, psi; L ⫽ length of section, inch; m ⫽ consistency, lbf secn/in2; Rh ⫽ hydraulic radius ⫽ area/perimeter of section, inch; W ⫽ throughput rate, lb/hr; a, b ⫽ shape factors from Figure 7.2 and Equations (10) and (11); n ⫽ power law exponent; ␳ ⫽ melt density, lb/in3; and A ⫽ cross-sectional flow area, in2. ⌬P (cone) ⫽ 48 psi ⌬P (0.50 bore) ⫽ 692 psi For an aspect ratio of 0.125 from Equations (10) and (11), a ⫽ 0.384 and b ⫽ 0.901. The die orifice with a 1.000 inch land is evaluated: ⌬P(1.000 land) ⫽ 259 psi. This pressure drop is insufficient to provide good flow distribution and is inadequate to prevent premature foam nucleation. By utilizing a 1.25 inch preland with a 0.100 inch gap, the pressure distribution becomes: ⌬P (cone) ⫽ 48 psi ⌬P (0.50 bore) ⫽ 692 psi ⌬P (0.100 preland) ⫽ 1130 psi ⌬P (1.250 land) ⫽ 324 psi Total ⫽ 2,194 psi The transition between the adaptor bore and the preland has been neglected as it will make only a small contribution to the total pressure drop. Similarly, the cone portion of the pressure drop is not seen to be significant. A properly designed extruder screw will typically operate well within a range of ⫹/⫺15% of the optimum pressure so that the total pressure drop of 2,194 psi is adequate. The steep pressure gradient in the preland and land sections and the small cross-sectional flow area assures that premature nucleation will not occur. On issuing from the die orifice, we like to observe foaming starting 1⁄8 to 1⁄4 inch from the die face. The partially foamed extrudate is fed into a water-

© 2000 by CRC Press LLC

cooled vacuum calibrator where foam expansion is completed. A dense skin can be formed by blowing air on the surface of the extrudate as it enters the calibrator. However, excessive cooling will restrain expansion, resulting in failure to fill the calibrator. The calibrator dimensions are typically about 1% larger than the finished product size to allow for thermal shrinkage. Very good tolerances can be held on the part as the calibrator does the final forming of the extrudate. Excessive discrepancies in the flow distribution will not affect the dimensions but will cause density variation in the product that may be detrimental. 7.4.1.2 Constrained Foaming Method The rectangular slat described above can also be produced using a process designated Celuka® developed by Ugine Kuhlmann, S. A. Paris, France. The technique involves using a die with a core plate. The vacuum calibrator is mated to the die face plate with 0.010 to 0.015 inch standoff. For PVC foam, the internal dimensions of the calibrator are about 1% larger than the finished product to allow for thermal shrinkage. The die orifice is designed with dimensions 0.020 inch less than the calibrator to assure that the extrudate enters the calibrator freely. The following analysis demonstrates the design procedure. Product: 0.25⬙ ⫻ 2.00⬙ rectangular profile with 1/64 inch radius corners Density: 40 pcf ⫹/⫺ 3 pcf Material: Rigid PVC Melt density at 400 F: 0.045 lb/in3 Consistency at 400 F: 3.86 lbf sec/in2 Power law exponent: 0.433 Production rate: 80 lb/hr at 2300 psi Adaptor: The adaptor has a 2.000 inch ⫻ 60° included angle cone section and 0.50 inch bore, 4 .000 inches long Volume expansion ratio: (1728 ⫻ 0.045 lb/in2 )/40 pcf ⫽ 1.944 Calibrator size: 0.2525 ⫻ 2.020 Die orifice dimensions: 0.2325 ⫻ 2.000 Orifice flow area: Calibrator area/1.9442/3 ⫽ 0.2525 ⫻ 2.020/1.9442/3 ⫽ 0.3275 in2 Core plate size: 0.075 ⫻ 1.842 Orifice perimeter: 2 ⫻ (0.2325 ⫹ 2.000 ) ⫹ 2 ⫻ (0.075 ⫹ 1.842) ⫽ 8.299 inch Orifice hydraulic radius: 0.3275/8.299 ⫽ 0.03946 inch Shape factors: a ⫽ 0.5, b ⫽ 1.0 The pressure drops across the adaptor cone and bore are found using the Kozicki relationship, Equation (14). ⌬P (cone) ⫽ 48 psi

© 2000 by CRC Press LLC

⌬P (0.50 bore) ⫽ 692 psi The die orifice with a 1.000 inch land is evaluated: ⌬P(1⬙ land) ⫽ 892 psi. Total ⌬P with 1 inch land ⫽ 1,632 psi or 668 psi less than 2,300 psi required operating pressure. If the land is increased to 1.75 inch: ⌬P(1.75⬙ land) ⫽ 1,561 psi Total ⌬P ⫽ 2,301 psi As the extrudate enters the calibrator, the outer skin can be quickly cooled to form a dense, tough surface. Since foam is a poor conductor, the inside will continue to foam inward to fill the cavity formed by the core. 7.4.2 DIE DESIGN FOR LOW-DENSITY FOAM PRODUCTS Low-density foam products typically utilize the tandem extruder process where the second extruder is used to cool the melt to avoid excessive blowing and cell rupture. The die must be designed to minimize total pressure drop in order to minimize the work input to the melt in the cooling extruder. However, if the pressure is too low, the foam will be formed inside the die, resulting in cell rupture and collapse of the foam structure. The melt must be maintained at a pressure above the partial pressure of the blowing agent at the operating temperature for as long as possible. Since there is a pressure gradient in the die, at some point, the pressure inside the die will fall below the partial pressure of the blowing agent. Since foam formation is a nucleation and growth process, it is possible to avoid premature expansion if a steep pressure gradient is maintained and sufficient flow rate exists in order that the polymer exits the die orifice before cell nucleation starts. With unpigmented material, a clear melt can be observed issuing from the die. In processes where a high exit velocity can be achieved, a clear melt can be observed extending for an inch or more from the die face. The following information is needed to design the die: a. Product dimensions and foam density b. Production rate, melt temperature, and pressure at the screw tip c. Number and mesh sizes of screens and breaker plate dimensions d. Extruder adaptor dimensions e. Consistency and melt density of the polymer at the operating melt temperature f. Partial pressure of the blowing agent and the nucleation time at the operating temperature The mean nucleation time can be determined by experiment using an instrumented slot die. By adjusting the flow rate until foaming appears just at the die exit and knowing the pressure gradient in the land, the mean velocity and mean nucleation time can be calculated.

© 2000 by CRC Press LLC

The following example illustrates the design process. Product: 10⬙ ⫻ 1⬙ foam slab, 2.0 pcf density 125 lb/hr, 250F melt temperature, 900 psi maximum at screw tip. Screen pack and breaker plate not used. Adaptor bore is 0.50⬙ ⫻ 3⬙ long. Die orifice: 7.5⬙ ⫻ 0.20 Melt density: 0.02818 lb/in3 where: Consistency, m ⫽ 2.0 lbf secn / in2 at 250°F Power law exponent, n ⫽ 0.2834 Blowing agent partial pressure at 250°F ⫽ 120 psi Mean nucleation time, tm ⫽ 6.7 sec The geometry of each section of the die must be determined, and the appropriate flow relationship must be applied to compute the local pressure drop. The following relationship uses shape factors related to the geometry of the section in Equation (14) in order to compute the pressure drop. ⌬P ⫽

mL W(a ⫹ bn) n c d Rh 1,800␳RhAn

(15)

Evaluate the pressure drop across the adaptor: L ⫽ 3.0 inches; A ⫽ ␲ D2/4 ⫽ 0.1963 in2; ␳ ⫽ perimeter ⫽ ␲ D ⫽ 1.5708 inch; Rh ⫽ hydraulic radius ⫽ A / p ⫽ 0.1249 inch; a ⫽ 0.5, b ⫽ 0.75 for a circular cross section; and ⌬P ⫽ 204 psi (adaptor). In order to distribute the flow uniformly to the full width of the orifice, a coat-hanger manifold design is used. Since some pressure loss is experienced in flowing down the manifold, it is necessary to compensate for this pressure reduction by cutting each half of the manifold at an angle forming a coathanger shape. The pressure at the centerline of the die is highest, and the preland length and gap at the center must be designed to provide a pressure drop equal to that experienced in flowing to the end of the manifold. In addition, the pressure gradient down the manifold should be linear. This can be closely approximated by tapering the manifold cross section. This is necessary to achieve a linear pressure gradient because the flow rate down the manifold diminishes in a linear fashion due to material leaking out of the manifold and flowing out of the die orifice. The flow equation can be written in differential form: dP ⫽

m W(a ⫹ bn) n d dL c Rh 1,800␳RhAn

(16)

The flow rate down each side of the manifold at the centerline is W/2. The local flow rate is WX ⫽ W/2(1⫺x/Lm). The local hydraulic radius and cross-

© 2000 by CRC Press LLC

FIGURE 7.3

Geometry of tapered manifold for slot die.

sectional area must also be expressed as a function of the distance down the manifold. Figure 7.3 illustrates the geometry used for the manifold. Since the manifold is symmetrical, only one side must be evaluated. The pressure gradient down the manifold is then determined by making a numerical integration. The pressure drop and the linearity of the pressure gradient are adjusted by choosing proper values of the manifold dimensions. Performance of a numerical integration yields the valves shown in Table 7.3. Evaluate the pressure drop across one inch of land at the die orifice: L ⫽ land length, 1.0 inch; w ⫽ orifice width, 7.5 inches; h ⫽ orifice gap, 0.20 inch; A ⫽ flow area, ⫽ w ⫻ h ⫽ 1.50 in2; p ⫽ perimeter ⫽ 2(w⫹h) ⫽ 15.4 inch; Rh ⫽ hydraulic radius ⫽ A / p ⫽ 0.0974 inch; and e ⫽ aspect ratio ⫽ h/w ⫽ 0.02667. The aspect ratio approximates an infinitely wide slot so the shape factors are a ⫽ 0.5, b ⫽ 1.0 for a rectangular cross section. ⌬P ⫽ pressure drop, psi and ⌬P ⫽ 60 psi (1⬘ land). Since the pressure at the entry to die land is well below the partial pressure of the blowing agent, nucleation and cell growth may take place within the die with resultant cell rupture. By increasing the land length to 2.5 inches, the pressure at the land entry is raised to 150 psi or 30 psi above the partial pressure of the blowing agent. The distance from the die orifice where the pressure reaches 120 psi is LN ⫽ 120/150 ⫻ 2.5 ⫽ 2 inches. The mean residence time within the die after reaching 120 psi is tm ⫽ LN ⫻ 3600 ⫻ A ⫻ ␳/W; tm ⫽ 2 ⫻ 3600 ⫻ 1.5 ⫻ 0.02818/125 ⫽ 2.4 sec. The total pressure drop

© 2000 by CRC Press LLC

TABLE 7.3

Tapered Manifold Analysis Obtained by Numerical Integration. Tapered Manifold for Slot Die Temp F 250

Project lowdens1

Material LDfoam

Die Width 7.500 in

Preland Gap .080 in

m psi s 2.0000

Manifold Angle 12.7500 Degrees

Relative Distance

Pressure psi

Deviation psi

Shear Rate 1/sec

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

160 145 129 113 96 80 63 46 29 14 0

0.0 0.9 1.1 0.9 0.4 ⫺0.3 ⫺1.2 ⫺2.1 ⫺2.6 ⫺2.1 0.0

37.0 36.4 35.5 34.3 32.5 30.0 26.8 22.4 16.6 9.0 0.0

n 0.2834 Center Depth .625 in

Density lb/cu in 0.02818

Rate lb/hr 125.0

Channel Radius .200 in

To minimize deviation, adjust center depth and channel radius. To minimize manifold-preland pressure difference, adjust manifold angle and/or preland gap. Manifold Preland Delta P Delta P Difference psi psi psi 160 160 0 Angle U Angle S Preland Lc Deg. Deg. Length in in 1.7704 9./6856 0.640 0.532

across the die is the sum of the pressure drops across the adaptor, preland, and land: Total ⌬P ⫽ 204 ⫹ 160 ⫹ 150 ⫽ 514 psi. Note that it was necessary to reduce the gap in the preland to 0.080 inch to obtain a higher pressure in this region. The pressure drop of 310 psi in the region where the product shape is formed contributes to improved thickness distribution. Since the mean residence time is below 3.7 sec, and the total pressure drop is below 900 psi, the design is satisfactory. Since foaming is a three-dimensional expansion process, we might expect expansion of width and height to be in direct proportion to the orifice size. However, it is found that the edges of the die orifice tend to restrain the width expansion, causing corrugation of the extrudate. In addition, residual elastic stresses and molecular orientation can cause further distortion. Corrugation can be prevented by using rolls to level the surface of the foam extrudate as described in the Carlson [9] or Aykanian [10] patents. Other techniques include forming between parallel flat plates while allowing the edges to form freely or extruding into a rectangular jacket where all four sides are formed. Due to the difficulty in predicting the effects caused by edge restraint and elastic stresses, it may be necessary to make modifications to the die orifice to obtain the specified product dimensions.

© 2000 by CRC Press LLC

7.5 REFERENCES 1. Rauwendaal, C., Screw Design for Foam Processing, Plastics World, 38, (May 1997). 2. Gregory, R. G., Friction Coefficients of Plastics and Steel, Proc. SPE ANTEC, (1969). 3. Darnell, W. H. and Mol, E. A. J., Solids Conveying in Extruders, SPE Journal, 20, (April 1954). 4. Han, C. D. and Ma, C. Y., J. Appl. Polym. Sci., 28, 831 (1979). 5. Han, C. D. and Ma, C. Y., J. Appl. Polym. Sci., 28, 851 (1983). 6. Bagley, E. B., End Corrections in the Capillary Flow of Polyethylene, J. Appl. Phys., 28, 624 (1957). 7. Rabinowitsch, B., The Viscosity and Elasticity of Sols, Z. Physik Chem., A115, 1 (1929). 8. Kozicki, W., Chou, C. H. and Tiu, C. Non-Newtonian Flow in Ducts of Arbitrary Crosssectional Shape, Chem. Eng. Sci., 21, 665 (1966). 9. Carlson, F. A., U.S. Patent 2,857,265. 10. Aykanian, A. A., U. S. Patent 2,945,261.

© 2000 by CRC Press LLC

CHAPTER 8

Foam Extrusion Machinery Features WILLIAM C. THIELE

8.1 PREFACE REGARDING EXTRUDERS FOR FOAMING

B

single-screw and twin-screw extruders are used to make foamed products. Many processing schemes and materials are employed, which means that there is really no standard device or process method. A “thinkthrough” machinery approach follows to help define a processing system that is most likely to work for a specific foamed product. It may be somewhat generalized that high-density foams are easier to make than low-density foams. Sometimes, using blowing agents is easier than injecting gas, although the growing foam bubble is unlikely to realize from which source its gas has come. Polymers that strain harden best support foam manufacturing. Such easy generalizations do not really apply to machine systems. If, however, the process is reduced to its subprocesses (unit operations), rational choices can become apparent. These choices might include whether to use a twin-screw or single-screw extruder, whether these should be in single or cascaded format, or whether chemical blowing agent or gas injection is best. Many foaming applications can be satisfied quite well by single-screw extruder devices. An objective of this chapter is for you to sort out the fewest and simplest devices required to do your job. This chapter will first describe the capabilities and limitations of screw extruders (single-screw and twin-screw) as foam-making tools. The next section will discuss the candidate subprocesses or unit operations in foaming that relate to extruders. The final sections will comment on primary and support machinery properties and choices. OTH

© 2000 by CRC Press LLC

8.2 BASIC PROPERTIES OF EXTRUDERS Somewhere, the extruder is charged with fineness of mixing an injected gas or a blowing agent into the polymer. It may also be required to compound other ingredients into the recipe. These might include regrinds, colorants, stabilizers, nucleating agents, or modifiers (such as to make the polymer better support bubble growth). In addition to melting, pumping, and forming jobs of profile extruders, foam systems also bear mixing responsibilities of compounding extruders while delivering melt in a process window that is right for uniform bubble nucleation and growth. The properties of extruders are balanced against the basic demands of subprocesses in foaming operations. Comparing the mixing of compounding extruders versus traditional internal mixers can be a helpful part of understanding this. 8.2.1 SIMILARITIES TO CONTINUOUS MIXERS Twin-screw extruders transfer mechanical energy somewhat like internal mixers, through turning two screws or rotors. Internal mixers have intense cooling provisions to remove heat from the viscous mixing powered by the motor. Surrounding the rotors, it is possible to assign five basic activity regions. These are as follows: (1) (2) (3) (4) (5)

Screw or rotor channels with relatively mild strain rates Lobal capture region where material confronts an advancing mixer “wall” Tip acceleration region fed from lobal pools Apex regions where the twin chambers of the device meet Intermesh/proximity region where the two screws or rotors most closely meet

These exist in both internal mixers and extruders (Figure 8.1). The latter four can apply extensional and shear forces to accomplish dispersive mixing. Screws or rotors may be designed to capture or minimize these forces (Figure 8.2). Much of the mixing in foam processes is distributive, and mixers that can divide and recombine melts at high rates and at low energy per division are preferred (Figure 8.3). Again, channel shear and extensional strain rates are low, due to the deep flighted nature of twin-screw extruders and internal mixers and also due to a characteristically lower screws’ or rotors’ speeds of foam processes. While single-screw extruders cannot participate in the last two regions (for lack of a second screw), lobal capture and tip acceleration to some degree is possible. The most familiar single-screw lobal capture and tip accelerator is in a Maddoc mixing section. Pin, vane, and other types of distributive mixers can be made to work in single-screw extruders.

© 2000 by CRC Press LLC

FIGURE 8.1

FIGURE 8.2

© 2000 by CRC Press LLC

Mass transfer regions.

Wide and narrow kneader elements.

FIGURE 8.3

High division rate mixers.

8.2.2 OTHER BASIC PROPERTIES OF SINGLE-SCREW AND TWIN-SCREW EXTRUDERS AND DIFFERENCES FROM INTERNAL MIXERS Three other properties make extruders more suitable than internal mixers for foam processes. They are continuous, “small-mass,” and “longitudinal” (Figure 8.4). The need for continuous, steady output is the main reason internal mixers are not common in foam processes. Batch operations in the melt parts of the process promote inconsistencies in the product. Extruders being “small-mass” relates to the localized material volume, which is bounded by barrels and screws. It is tiny when compared to the bounded large mass in an internal mixer or even in a continuous internal mixer. Extruders have approximately five to seven times the surface-to-volume ratio as compared with internal mixers. This is very important for temperature control with foams that can have fairly narrow process windows. The small mass property also relates to tiny transport distances for mixing blowing agents (and additives). Short transport distances within screw flights or screw mixing elements promote speed and accuracy for incorporating materials into polymers. For example, nucleating agents, which may be added at below 1%, are more quickly and accurately distributed within small volumes with their short transport distances than they would within large melt domains.

© 2000 by CRC Press LLC

FIGURE 8.4

Localized mass and length of mixing devices.

Longitudinality is also a critical property. That refers to the extruder’s capability to sequentially perform subprocesses or unit operations along its length. These unit operations are mechanically powered through the main motor and gearbox driving the screw(s). The heat transfer requirements of these unit operations are satisfied by the reaction-grade heating and cooling provisions in the extruder’s barrel(s), whether the barrels are one-piece or segmented. Gearboxes can and should be supplied strong enough that the power transmission limitation lies with the screw(s), whether the machine is a singlescrew or twin-screw type. Segmented screws are highly desirable because the best screw pieces (elements) can be strung onto screw shafts to optimize unit operations to be conducted along their length. The torque limitation becomes the strength of the cross section of the shafts onto which the elements are strung to power a given process volume. As single-screw extruders power

© 2000 by CRC Press LLC

FIGURE 8.5

Construction of modular twin-screw extruder screws.

processes with only one shaft, and because their screws are deepened in the feed area, fully segmented screws are generally not practical for them. Segmented screws are practical for twin extruders (Figure 8.5). Commercial segmented twin-screw machines are generally built with screws that have an outside-to-inside diameter ratio of 1.43/1 to 1.56/1. This balances available process volume to the strength of the screw shafts. Most pipe and profile type twin-screw extruders have one-piece screws. Having no screw shafts, they can transmit high torque. Screws made to target functionality and longitudinality may not be available for the uncommon requirements of your foam process. While single-screw extruder screws are generally one piece, they can be torque limited for reasons that include transmitting power with only one screw instead of two and having a deep-cut feed section that leaves a smaller root diameter cross section available. Various unit operations in foam processes will be discussed that involve feeding, mixing, cooling, and pumping. It is important to run unit operations efficiently. That will maximize the number of jobs that can be sequenced along the screws, the quality of performing those tasks, and even the practical length of the extruder to house them. 8.2.3 SINGLE-SCREW VERSUS TWIN-SCREW EXTRUDERS IN FOAM PROCESSES Single-screw extruders can be preferred over twin-screw types usually for two reasons. (1) For a given screw diameter, a single-screw is usually half the price of a twin. (2) Single-screws are simpler to understand and maintain than twin-screws, although they are not necessarily easier to operate.

© 2000 by CRC Press LLC

Twin-screw extruders would be opted for performance reasons. Some of the special capabilities demonstrated by twin-screw extruders as compared with their single-screw counterparts include the following: (1) (2) (3) (4) (5) (6) (7)

Greater and more steady solids feeding Twin-shaft drive to power more sequenced unit operations Better dispersive mixing of nucleating agents and polymer systems Fast, even incorporation of blowing agents Better cooling and temperature control capability More thorough mass temperature homogenization Relatively strong and stable dynamic seals before gas injection points

These statements do not apply to every foaming processes. Foam-related processes in general have tighter specific requirements than do basic compounding or profile extrusion processes. It can be further generalized that the fussiness of the process is inversely proportional to the density of the foam. Specific requirements to make foamed products vary. 8.2.4 LIMITATIONS OF TWIN-SCREW FOAM EXTRUDERS Thus far, only a few limitations to using twin-screw extruders have been expressed: twin-screws are expensive and their shafts in the case of segmented screws tailored to your process restrict the torque, even with recent industry advances. There are other limitations, such as not being able to feed a solid particle that is much bigger than the flight depth. If you wish to perform an in situ reaction that requires twenty minutes, you might expect only a tenth or a twentieth of the output rate as compared with roughly a half-minute dwelltime typical of compounding processes. Substantial costs may be incurred to armor a machine against corrosive, abrasive, or adhesive wear. Fortunately, foaming processes are not generally subject to any of these additional problems. Heat transfer is an important limitation. It can also affect shaft torque demands. For example, when foaming gases are injected (whether in liquid or gas phase) and become dissolved in the polymer system, the viscosity plunges, as does usually the softening point. It is necessary to cool the material to a lower and uniform temperature to achieve an output process window (of maybe only a few degrees). The ability to do this total heat transfer, and also to avoid frictional hot spots on the screws, places constraints upon screw speed, the screws’ geometry, and the barrels’ design. Hopefully, cooling will be completed before the shafts’ torque is exhausted. Torque and heat transfer are serious boundaries. Getting enough torque to the right kinds of special screw shapes to power the subprocesses within the

© 2000 by CRC Press LLC

extruder’s volume is most often the more serious problem of the two. This is generally true for single-screw and twin-screw devices alike, with exceptions. 8.2.5 SOME DISCUSSIONS ABOUT FREE VOLUME AND SCREWS’ TORQUE Today’s single-screw and twin-screw gear technology supports very high torque possibilities. Unfortunately, the shafts onto which the screw elements are strung are torque limiting. The screw shafts at the middle of the extruder only need to carry the combined torque contributions from the screw elements of the output half of the extruder. The shafts under the feed throat screw elements carry the sum of the torque load contributions of all of the elements for the entire screw set (Figure 8.6). That position is the torque limiting point in extruder design (or should be). This is also the weak point for single-screw one-piece screws, a fact particularly illustrated for smaller machines (3.5 inch OD and under) with various “shank collections” from broken screws. It is unfortunate that the most productive foaming screw designs are prohibitively complicated and too nonstandard to be machined in one piece. Onepiece screws would avoid the torque limitations imposed by screw shafts.

FIGURE 8.6

© 2000 by CRC Press LLC

Screw shafts’ loading.

However, segmented screws are very maintenance friendly and may be reconfigured in response to new developments of the art and the need to be changed to make different products. However, at least one manufacturer, Leistritz (New Jersey, U.S.), in October, 1998, announced the ability to machine complicated one-piece shaftless custom screws, primarily for sanitary reasons, for the pharmaceutical industry. The common term, “free volume,” means the space available for processing in an extruder of a given screw diameter for a given length. It could be expressed as cc/D, where D is a length of barrels with screws equal to the diameter of a screw. It has also been expressed as liters per meter. The idea is that greater volume produces more product. That is only true, however, if the shafts (and gearbox) can transfer the needed power to the volume of the screw elements. Today, it is common for twin-screw extruders to have a flight depth based upon a compromise between mechanical energy transfer (from torque) and the available process volume. If the relationship of shaft cross section to useful volume is high, then the machine would be very strong (fat shafts and inside diameter with respect to screws’ outside diameter) but would have little producing capacity. If the volume compared to shaft area is high (skinny shafts and small inside screws’ diameter with respect to screws’ outside diameter), then there will be a large volume for doing processes but little strength to power them. The outside diameter divided by the inside diameter (“OD/ID ratio”) seems to fall between 1.43/1 and 1.56/1 as a balance between the two above extremes. High-strength splined screw shafts are required. Sometimes these shafts are produced by strain hardening/“hammering” processes. Such shafts are over 20% stronger than those produced by cutting. These issues hold true for both corotating and counterrotating extruder types. Corotating and counterrotating machines share the same OD/ID depth compromise and, therefore, the same basic extruder barrels sections for a given gearbox centerline spacing (Figure 8.7). A similar tradeoff exists in the feed section of a single-screw extruder. Making the feed depth great, and, therefore, making the metering depth after compression residually large, is a tradeoff against mechanical strength of the screw in the main feed root cross section. (Parallel twin-screw extruders have approximately the same flight depth along their entire length.) It is interesting that the OD/ID compromise between free volume and torque is reasonable for other aspects of foam processes. The viscous heating when compared with shallow devices is less. Volume output per rotation is reasonable, as are mixing and heat transfer. These are subjective statements without apology. The relationship of OD/ID describes twin-screws as inherently “deep-flighted” devices when compared with single screws. In the metering sections of plasticating single-screw foam extruders, OD/ID ratios can range

© 2000 by CRC Press LLC

FIGURE 8.7

Classical corotating and counterrotating screws.

from 1.1/1 to 1.3/1. A nonplasticating and noncompressing single-screw device used for melt cooling, however, can be as deep as OD/ID of 1.8/1 or even more. Substantial efforts have been undertaken to make shafts of a given size stronger. This has been done through stress-relieved spline designs, special metals, and new treatments of metals. Some manufacturers (Werner & Pfleiderer, Leistritz, and perhaps others) are strain-hardening shafts through a hammering or rolling process. Better torque safety and management techniques are also used. For long, segmented twin-screws, these combined efforts have raised state-of-the-art mechanical energy availability from about 2.3 watts/rpm/cc to approximately 3 watts/rpm/cc. This is the maximum motor power in watts at full torque scaled to one rpm over the standard free volume of one diameter of screws length. This represents a total torque improvement of about 30%! This is an interesting testimony to engineering and urgency. 8.2.6 OVERVIEW OF USING THE EXTRUDER First, some nonmachinery basic rules are important. They include that the polymer system, blowing agent, and nucleating agent should work right together. For example, the gas must be sufficiently soluble in the polymer system to achieve the target foam density and the polymer system must support the growth of bubbles. Second, there are machinery rules. The pieces must perform the unit operations you elect to perform to make the product. Extruders, therefore, are not introduced as the historic appliances that have been used, but as a platforms onto which functions of your process can be operated. Extruders are continuous, small-mass, and longitudinal. Unit operations or subprocesses may be sequenced along the length of their screw(s). These unit operations are mechanically powered by sections of the screw(s). The barrel(s) should have reaction-grade heat transfer, including liquid cooling and electrical heating. The screws have a weak strain-rate region, their channels. Twinscrews (like internal mixers) can employ lobal capture, barrels apexes, mixer tips, and intermesh (proximity) regions are made available for intensive

© 2000 by CRC Press LLC

mixing, while single-screws can achieve still substantial performance from screw designs of the art not involving a second screw. Screw shapes exist to do low-energy distributive mixing. Extruders are versatile, but special attention is required for torque and heat transfer boundaries in foam processes. All of these generalizations are equally valid for intermeshing corotating and counterrotating machines as well as single-screw extruders.

8.3 BASIC UNIT OPERATIONS IN FOAM PROCESSES Foam processes differ, depending upon target density, polymers, gas injection versus blowing agent, finished product shape and form (including whether part of a coextrusion), special performance properties, and the like. No example describes all of them. However, the following may help to provide a general understanding of likely subprocess issues involved with making foamed products. The example is in-line manufacturing of a sheet or a shape, inclusive of compounding a polymer system with additives and a nucleating agent in a twin-screw extruder with modular barrels and screws. The gas (or blend of gases) for foaming is injected. Variations could be envisioned for coextrusions, high-density foams, gas supply through chemical blowing agents, and the like. Downstream cooling and calibrating processes are not part of the example. Nearly all of the unit operations listed can be done by the extruder. Questions arise concerning whether they can or should all be done together in the extruder. Some of them may need to be transferred to other devices to relieve loads from the screw shafts or for other reasons. Sometimes, a subprocess might just work better in an external device. The choices will depend upon the materials’ system, the product form, the capabilities of equipment, and your own preferences. Basic common unit operations, from materials’ feeding through die forming, may include: (1) (2) (3) (4) (5) (6) (7) (8) (9)

Polymer system feeding—premix or multiple streams Melting/compounding—blend polymers, disperse nucleating agents, color Dynamic sealing—prevent gas blow-back Gas injection—steady, regulating, non-ponding Distribution—high-distribution rate rapid incorporation Cooling—bring temperature and viscosity to process window Pumping—pumping power for final die flow End homogenization—create mass and thermal homogeneity Die forming—exit homogenous morphology for right bubble growth

© 2000 by CRC Press LLC

FIGURE 8.8

Gas-injected foam screws.

A few of these could be split or resequenced in some process trains (Figure 8.8).

8.3.1 POLYMER SYSTEM FEEDING For this example, it is assumed to feed one polymer, a stabilizer, a colorant, and a nucleating agent. The gas is sufficiently soluble in the polymer to achieve the desired foam density. Conflicts about the stabilizer and colorant affecting the nucleation and bubble growth processes have been already resolved. The nucleating agent is not soluble in the polymer, which is a general qualification to perform its function. (It is possible for a little well-dispersed nonsoluble gas to act as a nucleating agent.) Twin-screw extruders are starve-fed. The screws’ design and screws’ speed determine the mass-transfer rates. The rate of feeding sets the output rate and amount of remastication. These two parameters are part of balancing the process. In general, starved screws turning fast constitute harsh processing, while fuller screws turning slowly constitute gentle processing. The energy rate to the screws is proportional to the product of screws’ speed and the torque applied to them.

© 2000 by CRC Press LLC

Foam processes tend to be moderately gentle. This is in spite of the need to disperse nucleating agents, colorants, and additives, as well as possibly blending polymers. Twin-screws are generally not flood-fed like most single-screw extruders. One or more powder or pellet feeders are part of rate and formulation control. The feeder(s) should be gravimetric to maintain your recipe with the gas, which is injected later. Metering a premix to the main port needs just one feeder. However, the mix must not segregate. Using powdered resin is one trick used to keep premixes stable. Very small amounts of powders can sometimes be made to cling onto pellets. Small amounts of pellets can sometimes be made to remain static within powders, if the powders do not frequently move. These are some techniques to stabilize premixes. If premixes are unstable, then using additional gravimetric feeders may be necessary. Separate gravimetrically fed streams may not deliver accurately at low rates. The output of foam lines can be cooling rate limited, and the component feed streams can be small. For 100 kg/hr of product, perhaps only 1⁄2 kg of talc nucleating agent might be needed. Gravimetric feeders at 1⁄2 kg/hr and similar rates need to be chosen and operated carefully, as normal units and installations may perform in an unstable manner. Additive streams may sometimes be combined to achieve a rate at which a feeder becomes stable. Some of the main polymer may also be made into powder that can then be used for establishing premixes with additives for delivery at a rate more acceptable to production gravimetric feeders. Other schemes can also be devised, but operating gravimetric feeders at least 3–5 kg/hr, and preferably, 12 kg/hr or faster, is a good policy. The main solids’ stream(s) should not be delivered from high on a service deck but rather from close to the screws to ensure reliable delivery without clinging to and releasing from a drop tube wall, “clouding away,” or fluidizing excessively. This is particularly important if the materials are powders. A compact system can also be designed to promote rapid product changeovers. It is not likely that the feed throat will appear overfilled, unless the feedstock is fluffy, fluidized, or otherwise difficult. It is important to remember that the extruder does not see “bulk density.” It sees “feed density.” The relation of bulk density to feed density follows: Df ⫽ Db ⫺ ddt ⫺ dbg ⫺ dft ⫺ ded ⫹ dec Where Df ⫽ the effective feeding density the extruder sees; Db ⫽ the measured bulk density of the material not moving; ddt ⫽ the bulk density change from turbulation dropping from the feeder to the screws; dbg ⫽ the change in bulk density from back gassing turbulation from the screws; dft ⫽ the change in bulk density by screw flight movement turbulation of the feed stock; ded ⫽ reduction in bulk density from electrostatic dispersion; and dec ⫽ the increase in bulk density provided from mechanical stuffing pressures.

© 2000 by CRC Press LLC

By means of lowering feeders, opening relief vents, installing back-gassing baffles, neutralizing electrical charges, and possibly even mechanically cramming, feeding properties may be improved in twin-screw and single-screw extruders. Twin-screws are deep flighted, and they are normally not feed limited in foam applications unless there are major fluffy components, such as foamed regrind. Normal screw types may be used in the feed throats of the various types of twin-screw extruders. Ones causing excessive friction at the feed throat should be avoided. The melting/compounding subprocess follows feeding. However, some process trains can benefit from an external melter/mixer feeding the twinscrew extruder. That device could sometimes be a single-screw extruder. 8.3.2 MELTING AND COMPOUNDING If only melting and light mixing are needed, a single-screw extruder would be adequate. However, it may be important to deagglomerate and disperse the nucleating agent. It could also be necessary to disperse a colorant and control its effects upon cell nucleation and bubble growth. Blending two polymers might be required. These and similar situations can justify the twin-screw extruder for melting and mixing. In these cases, dispersive mixing may be required in this region, as in intense compounding extruders. Some stress rate may be required to fracture agglomerates or to reduce polymer/additive phase domains. That stress rate is the product of some controlling material modulus (influenced by viscosities and the presence of particles) and a strain rate (from the shear and elongational movement of the material in the screws). ␦␴/␦␶ ⫽ Ec ⫻ ␦e/␦␶ The screw design to do this unit operation can include proven classical dispersive mixers and/or newer generation geometries. The design, however, needs to compensate for the lower 30–200 rpm speeds of many foam machines as opposed to 200–1,200 rpm and higher speeds of compounding-only machines doing similar mixing. The materials should be mixed to be ready to see gas. This relates to morphology and temperature. Ideally, the temperature should be raised some to compensate for the freeze-off effects when the gas is introduced. Unfortunately, that cannot always be done, since high viscosities may be needed to make the dynamic seals hold gas pressure and prevent blow-back. High injection pressures, even those below 100 bar, place special requirements on the mixers, which may be part of the dynamic seal system.

© 2000 by CRC Press LLC

8.3.3 DYNAMIC SEALING At low gas injection pressures, 5–20 bar, this operation can usually be built into the melting and compounding subprocess. For high pressures, dedicated seals are advised. Using an external, side arm, melter/compounder is a special case. In that configuration, a dynamic seal might be needed upstream of the melt input point to the extruder. Alternately, some pumping and mixing must be carried over to the twin-screw extruder to drive normal seals. Seals operating at high pressures upstream and before the melt input might require the sealing medium for their elements to be externally supplied from a separate source. The normal case, however, should be a designated screw element group that’s job is to allow the compounded polymer system to pass but to prevent blow-back of any gas. These elements may be cascaded discs, special flights, or other appropriate barriers as needed. They are executed similarly for counterrotating and corotating machines (Figure 8.9).

FIGURE 8.9

© 2000 by CRC Press LLC

Dynamic seals and mixing elements for gas injection.

The energy requirements to seal gas pressures over 200 bar, such as with supercritical carbon dioxide, will be very substantial. 8.3.4 GAS INJECTION The gas may be in liquid phase or in gas phase. In either case, the foaming agent pumping mechanism should provide good volumetric control. (Some reasonably good results have been achieved regulating gas flow by pressure sensing at the injector and in the extruder barrel. These have tended to be tricky or fortunate. Load cell mounting of the gas tank can be linked to control the injection accuracy in some cases. How to keep liquid pumping systems from gas locking or gas phasing and how to get volumetric control at a critical point in gas phase and maintain it seem to be regarded as proprietary practice by some processors. Pressure and temperature monitoring in the extruder barrel near the injector is important at least for process verification. That information is a good indication of whether the exit rate of melt from the dynamic seals and the gas from the injector are constant. For dual injection of two complementary gases, the injector pressures should be compared to help control the mixture. Concentration variances can occur even if the metering of the gases from the metering device is positive. This variation would be seen in the barrel’s pressure measurement near the injectors and perhaps in the lines themselves. The line from the metering device to the injector should be short, rigid, and small. A worse-case example will illustrate these factors. A bad delivery line would be long, elastic, and big. A momentary melt impingement upon the injector would require a pressure rise to blow it out. The line has considerable volume, and it will stretch, so that the time to raise the pressure will be long. Meanwhile, the melt is flowing further up the injector, which in turn requires even more pressure. When the pressure finally rises to blow out the injector, a surge of gas will occur that was preceded by a period of gas-less extrusion, of course. Delivery from small, short, rigid lines tends to be constant. Momentary melt plugging is almost instantly blown out without disturbing the process. Short, steel tubing small enough to have a measurable pressure drop along its length at the target rate is ideal. These issues are important for constant delivery of liquid-phase gases. They are much more important for gas-phase gases that are more compressible. 8.3.5 HIGH-RATE DISTRIBUTIVE MIXING The gas is soluble in the polymer system to the amount necessary to achieve the target foam density. Once the gas is dissolved, the rheology of the material

© 2000 by CRC Press LLC

is uniform. Before that has been achieved, a liquid or gas phase exists concurrently with a polymer melt phase. It is critical to prevent mass ponding of the gas or liquid. To minimize ponding, the injection should occur over very high division rate distributive mixers, with substantial pressure under the injector. As the gas exits the injectors, it is greeted by rapid passages of mixer protrusions that preclude pond forming and accelerate dissolution. The absolute worst, opposite case would be to inject at low pressure over flighted low mass transfer rate elements. Liquid ponds or large gas bag domains would be formed that would greatly retard the dissolution process and could result in dissolved gas concentration variances. The speed of dissolution may, therefore, be inversely proportional to the presence of ponded gas domains. Speed is important to get the job done accurately with minimum energy. Using minimum energy for this and other unit operations on the twin-screw extruder makes more shaft torque available for additional unit operations or for increasing productivity and/or quality. While some foam machines mix gases with simple kneaders and related elements, special elements targeted to the task can perform better. Remember that the earlier mixing zones used enough mixing stress to reduce phase and agglomerate sizes. That is not so true here. The gas, by definition, is soluble in the polymer system. High-rate, low-energy distribution can be the dominant tool to speed the dissolution of gas into the polymer. When a nonsoluble gas is used in small amounts partly as a nucleating agent, bilobal mixing elements are useful to reduce the domain sizes. For process trains utilizing twin-screw extruders, the dynamic seals, injection, and gas mixing are almost always within the twin-screw extruder, even though some unit operations may, or may not, have been shifted to outside devices. 8.3.6 COOLING When gases are dissolved into the polymer system, they function as a plasticizer. The viscosity of the melt drops, as does its softening point. These effects increase approximately as the foam density decreases. The melt must be cooled to raise its viscosity in preparation to be formed through a die. There will be a temperature, pressure, and shear relationship in the die that will hopefully cause nucleation and fine bubble growth to begin immediately upon discharge. If the melt is too warm, nucleation and bubble growth in the die can disrupt the process, and the bubbles will be too large. If too cool, nucleation and bubble growth will occur too late or bubbles may not grow or grow fully. After gas mixing, the screws should just cool and forward the melt. Unfortunately, the screws also provide something else, viscous heating. Therefore,

© 2000 by CRC Press LLC

FIGURE 8.10

Low shear cooling, mixing and pumping screws.

the heat being removed from the barrels is somewhat in competition with pumping heat being put in through the turning screws. Tight intermesh tracking screws are prone to viscous heating, both in classical corotation and in classical counterrotation. Open-meshing single-flighted elements are equally preferred for both of these extruder types. They are also polished to reduce friction (Figure 8.10). The barrels’ sections are usually longitudinally cored for liquid cooling to accomplish the total cooling in as short a length as possible. As the viscosity rises, a significant torque contribution will result for the screw shafts. Some processes will fully cool the material in the extruder. Some will partly or fully cool the material in a heat exchanger or in a slow-turning separate single-screw or twin-screw cascaded pumping extruder. In general, a process train should consist of a minimum number of devices. 8.3.7 PUMPING Powering flow through the forming die would ideally be done from the twin-screw extruder. It could alternately be done with a gear pump, a cascaded single-screw extruder, or a cascaded twin-screw extruder. At least the first three of the four options have been done in production. The pumping unit operation in an extruder might be accomplished with the same loosely meshing elements that are used for cooling. For cases in which they would be too weak, single-flighted close-meshing elements would

© 2000 by CRC Press LLC

be used in either corotation or counterrotation. If classical bilobal corotating elements are used, their recommended flight advance is approximately 1 ⫹/⫺ 20% diameters per turn. Like other unit operations on the extruder, pumping should be done as efficiently as possible to place the least torque on the screw shafts. An exception to this rule would be elements sufficiently severe to cause frictionally heated spots on the surface hot enough to cause premature foaming. 8.3.8 END HOMOGENIZATION Besides achieving a target mass temperature into the forming die, it is necessary that the temperatures of small mass domains do not significantly differ. If they do, there may be tiny regions of foaming in the die and/or regions of retarded nucleation and large bubbles in the product. Extruder pumping tends to leave hot “turtle track” domains in the melt. Near the end of the extruder, high distribution rate mixers can be used to homogenize the melt at low energy. Some processes can tolerate these at the extreme end of the screws. Some prefer some weak forwarding elements (that do not generate hot domains) to follow them. If a gear pump is used, its hot domain tracks may be blended out with a static mixer. A cascaded pumping extruder will sometimes require the same. If a heat exchanger follows the pumping, the various temperature output streams from its tubes may sometimes need to also be blended. 8.3.9 DIE FORMING Dies vary greatly with specific prefoam rheologies and product type. As foam densities become greater, die systems more resemble their nonfoam counterparts. Die forming is one of the foam unit operations that users usually consider to be proprietary art. But, a few generalities can be mentioned. The temperature control of the die must be very uniform. This helps to keep the nucleation point even across a sheet or profile cross section. The consequences of failing to do that have been discussed previously. Liquid temperature control may be needed and/or more heat zones should be used as compared with nonfoam dies. Localized internal decompressions should be avoided or minimized that could cause premature localized nucleation and bubble growth. The shear history of the material at each point exiting the die should be similar. It is known that shear is a factor in promoting nucleation. Complicated dies may require localized temperature/shear compensations. These generalities also relate to sandwich and single-sided coextrusion.

© 2000 by CRC Press LLC

8.4 EXTRUDER TYPES, SUPPORT DEVICES, AND WHERE SUBPROCESSES ARE PLACED In the example, or in your own process, the extruder will be assigned jobs to the extruder according to possibilities and preferences. Preferences are valid. Making the extruder more efficient will allow more tasks to be placed on the extruder. When this cannot be done, tasks can be diverted to external devices. 8.4.1 CLASSICAL EXTRUDER TYPES AND EFFICIENCY It is the intermeshing twin-screw, more than single-screw, extruders that seem to power the most subprocesses and have the most modularity to support efficiency improvements. There are several generic types of intermeshing twin-screws. For all of them: (1) Core patents or concepts were fully in place before 1950. (2) The screws tracked each other during rotation to wipe and/or mix. (3) Early screw shapes dominate the current machine offerings. The classical forms of these twin-screws can be nonideal for making foamed products due to the following: (1) Close-meshing elements cause unwanted viscous heating and difficulties in cooling. (2) Classical screws may demand excessive shaft torque in unit operations. (3) Dynamic seals and gas mixing can be inefficient. (4) The numbers of unit operations on a screw’s length can be small. The main classical forms of intermeshing twin-screws include: (1) Counterrotating, intermeshing • Slow speed; Profile heritage; Cincinnati-Milacron, Krauss-Maffei, etc. • High speed; Compounding heritage; Leistritz (2) Counterrotating, nonintermeshing (not significantly used in foams) • High speed; Compounding heritage; Welding Engineers (3) Corotating, intermeshing. • Low speed; Profile heritage; LMP (Columbo), Windsor • High speed; Compounding heritage; Werner & Pfleiderer, Leistritz, etc. In practice, the classical screw shapes have worked reasonably well for their purposes, particularly when supported by the current higher mechanical strengths and heat transfer. Much work has been done to characterize the classical screw shapes. Conclusions about them tend to be defended. New process

© 2000 by CRC Press LLC

shapes are often logical modifications of classical ones. In short, twin-screws, and their single-screw counterparts, tend to be burdened with historical baggage when applied to sensitive foaming processes.

8.4.2 MAKING TWIN-SCREWS MORE EFFICIENT IN FOAMING PROCESSES Limits to the number of subprocesses and/or the quality of performing them occur when a boundary condition is reached. Assuming the extruder torque and heat transfer cannot be changed (and are hopefully state-of-the-art), many things can be done to help the extruder’s capability. These include new screws’ types and staging, formulation modifications, and operating conditions. The unit operations of the example illustrate these. While feeding requires little energy, melting and mixing does. Some powdered polymer to nucleate melt, additives, or a mixture of carriers can remove some shafts’ load. Minimizing feed throat cooling or preheating can also help. Mixing elements that quickly develop an acceptable morphology for gas introduction can be useful. Dynamic sealing to prevent blow-back of injected gas is difficult at high pressures with forward flights and kneaders. Short disc stacks and special seals can replace longer, torque-consuming sealing sections. The torque to seal against high injection pressures can be very substantial. Mixing of injected “gas” must be very rapid to conserve energy. High division rate distributive mixers prevent “ponding” that would otherwise usually happen in classical mixers. By definition, the gas is soluble in the polymer for making foam. Therefore, no dispersive mixing is needed unless ponds were formed with droplets that require reduction (Figure 8.11). Classical forwarding elements are prone to viscous heating. Open-meshing forwarding elements are relatively low-energy forwarders. Therefore, the cooling rate, heat out less heat generated, with them is much greater. There is no danger of coalescence since the gas and carrier by definition in foaming are mutually soluble. Formula modification to reduce the torque load during cooling and other extruder processes can be productive. Open meshing elements are ideal for pumping as the pressure rise is distributed over a larger length of barrel for heat transfer, and hot spots are less likely to be formed with high domain temperature differentials. Powering end thermal homogenization with active low-energy distributors on the screws consumes less energy than powering static mixers from axially generated flow energy. The morphology results are similar. Foamed products have tight condition windows in their dies for mass and thermal homogenization to maintain profile shape and constancy of cell morphology.

© 2000 by CRC Press LLC

FIGURE 8.11

Twin-screw research foam extruder.

These examples are generally effective with any intermeshing twin-screw extruder. As more twins in foam processes must first blend polymers and color or stabilize the formulation, the demands to operate at higher screws speeds while still doing rapid cooling are increasing. These conflicting demands cause deviation from classical art.

8.4.3 WHICH TYPE OF TWIN-SCREW EXTRUDER? All of the improvements to screw design and unit operations’ management can be applied to all of the commercial intermeshing twin-screw extruders. Applying foaming operations’ parameters to the extruder may be more important than the type of twin-screw extruder that is chosen. Tailoring the extruder for foaming processes usually makes it more productive than taking it as an off-the-shelf appliance. This is also true for singlescrew machines. People may favor an extruder type, such as counterrotation for its accommodation of high-pressure dynamic seals. However, the issue is not usually courrotation or corotation. The factors of modularity and available compo-

© 2000 by CRC Press LLC

nents plus torque adequate to power the unit operations are generally more important. 8.4.4 WHERE TO PUT SUBPROCESSES WHEN THE EXTRUDER CANNOT DO THEM When shaft torque, heat transfer, or other boundaries do not permit all of the desired unit operations to be placed on the extruder, then some subprocesses must be designated to external devices. It is assumed that the classical twin-screw (or single-screw) device has been upgraded as fully as possi-

FIGURE 8.12 Corp.

Twin-screw foamed sheet extrusion. Courtesy of American Leistritz Extruder

© 2000 by CRC Press LLC

ble. If that has been done, then there is no choice but to relocate some of the work of the total process (Figure 8.12). Some possibilities include the following: Action External melter Powered dynamic seals External cooler External pumping External end mixing

Device Single-screw extruder Seal block and driver

Issue System control Shafts’ positioning

Single-screw extruder, etc. Gear pump, extruder Static mixer

Homogenizing, control Homogenizing, control Gear pump (alt.) to drive it

Cascade systems have favored downstream coolers. These may be slowturning single-screw extruders or heat exchangers. They could also be twinscrew devices. Somewhere, mass and thermal homogenization has to be done. Static mixers need drivers. There are no standard solutions. There is an argument to relieve the extruder of its melting work with an external melter. This can be a single-screw extruder feeding the polymer system to the twin-screw ahead of the gas injection. Advantages in dynamic sealing can also be engineered. Again, there seem to be no really standard foam processes. Differing process requirements, a small market, and tendencies toward confidentiality contribute to explaining the general lack of standard foam equipment.

8.5 GENERAL EXTRUDER OBSERVATIONS The extruder is a useful tool for foam manufacturing because it is continuous, small mass, and longitudinal. The intermeshing twin-screw classical extruder types were based upon steric tracking of their intermesh regions during rotation to produce wiping and/or mixing. That basis presents positive and adverse properties for making foams. By analyzing the needs of the step processes required to make the foamed products and not being subservient to the classical geometries and approaches of the extruders, it is possible to make foams with twin extruders with a minimum of support devices. While extruders, including single-screws, may not satisfy all requirements, they can be optimized to excel beyond what could be done in their classical formats.

© 2000 by CRC Press LLC

8.6 REFERENCES 1. Thiele, W. “Polymer Processing Advances in Counterrotating Intermeshing Twin-Screw Extruders,” Proceedings of the University of Akron, Akron, Ohio (May 24, 1994). 2. Erdmenger, R. German Patents 815,641 and 813154 filed (Sept. 1949). 3. Thiele, W. “How Twin-Screw Extruders Work,” Continuous Compounding in the 90’s RETEC, Somerset, NJ, Dec. 1 (1993). 4. White, J. Twin-Screw Extrusion. Chapter 7, Hanser, New York (1991). 5. White, J. Twin-Screw Extrusion. Chapter 10, Hanser, New York (1991). 6. Utracki, L. A. and Abdellah, A. Compatibilization of Polymer Blends, Canadian National Research Council, Boucherville, PQ May (1995). 7. Gogos, Costas, G. “New Mixing Developments,” Proceedings of the Leistritz Twin Screw Workshop, Somerville, NJ (May 23, 1995). 8. White, J. L. and Lim, S. “An Experimental Study of Flow Mechanisms, Materials Distributions and Morphology Development in a Modular Intermeshing Counterrotating Twin Screw Extruder of Leistritz Design,” Inst. of Polymer Engineering, Univ. of Akron, Akron, Ohio (Jan., 1993). 9. White, J. Twin-Screw Extrusion. Chapters 7–12, Hanser, New York (1991). 10. Booy, M. L. “Geometry of Fully Wiped Twin-Screw Equipment,” Polymer Eng. Sci., 18, 973 (1978). 11. Knights, M. and Thiele, W. “Which Twin-Screw Compounder is for You?” Plastics Technology, April (1995). 12. Thiele, W. “Introducing the Twin-Screw Extruder as a Continuous Reaction and Compatibilization Tool,” Proceedings of the National Research Council Reaction Course, NRC-CRNC, Boucherville, Quebec (December 4, 1995). 13. Thiele, William C. “Configuring Twin-Screw Extruders to Develop Target Morphologies,” Compounding ‘96 Conference, Philadelphia, Pennsylvania (August 26, 1996). 14. Thiele, William C. “Twin-Screw Extruders for Foam Processing,” Foam Conference, LCM Communications, Somerset, New Jersey (December 10, 1996). 15. Thiele, William C. “Trends and Guidelines in Devolatilization and Reactive Extrusion,” National Plastics Exposition, Society of the Plastics Industry, Chicago, Illinois (June 18, 1997). 16. Thiele, William C. “Non-Classical Approaches for Blending Polymers,” PolyBlends ‘97 RETEC, Canadian National Research Council, Boucherville, Quebec (October 10, 1997). 17. Thiele, William C. “Twin-Screw Technology for Making Pellets from Powdered Metal,” PIM 98, International Conference on Powder Injection Molding of Metals and Ceramics, Univ. of Pennsylvania, State College, PA (April 28, 1998). 18. Todd, David B., Ed. Plastics Compounding Equipment and Processing, Hanser, New York (1998).

© 2000 by CRC Press LLC

CHAPTER 9

Mixing Design for Foam Extrusion: Analysis and Practices CHI-TAI YANG DAVID I. BIGIO

9.1 INTRODUCTION

E

is known as a complex and dynamic plasticating process in converting thermoplastic raw materials into more useful products. It is capable of generating high temperature and high pressure without sacrificing pumping capacity. Foaming virtually became an interesting application of extrusion. Since other chapters focus on design and processing issues, this chapter begins with mixing theory followed by its applications to various types of extruders. Typical thermoplastic foam extrusion processes will be described briefly as they dedicate the type of processes and machinery used and how the theory of mixing is applied. Mixing is very important in any polymer processing application. Many other ingredients are added to the main polymer to meet finished product property requirements. These ingredients can be additives, modifiers, fillers, colorants, or other polymers. To break up the large agglomerates or clumps into smaller particles, dispersive mixing is achieved by generating proper levels of shear and elongational stresses in the processing equipment. On the other hand, distributive mixing relies on strain rates to spread out and homogenize all ingredients uniformly throughout the spatial polymer mixture domain. For both types of mixing, the objective is to obtain uniform dispersion and distribution of the ingredients in the main polymer matrix. Thermoplastic foam extrusion is a polymer processing application where additives, sometimes a nucleating agent and a foaming agent (chemical foaming agent or physical blowing agent) are mixed with a thermoplastic polymer in an extrusion system. A foamed product is made by proper design and operating conditions in the downstream equipment and the die. A uniform XTRUSION

© 2000 by CRC Press LLC

distribution and control of the foam cells are important to obtain the desired foamed product properties. The first step for achieving this is to have a good mixing of the foaming agent in the polymer mass. Beyond the mixing requirements of uniform dispersion and distribution, there is an added requirement for minimizing the energy input to the foaming agent. From the processing point of view, the mixing process is critical in foam extrusion in the following aspects:

• • • • •

gentle dispersive mixing of the nucleating agent and other additives with the polymer system good distributive mixing of chemical foaming agents fast incorporation and good distribution mixing of physical blowing agents good distributive mixing for homogenization of polymer/gas solution system good distributive mixing for uniform cooling of polymer/gas solution system

To obtain the required type and level of mixing in the above, the extruders for making foamed products must be capable of performing the following process functions:

• • • •

frequent flow division/splitting and reorientation efficient heat transfer streamlined flow path without hot spots forward pumping

9.2 THERMOPLASTIC FOAM EXTRUSION PROCESSES There are two distinct thermoplastic foam extrusion processes. One creates a higher density foam product and the other a lower density foam product. The thermoplastic foam extrusion process and the cellular foam structure are affected by the type of foaming/blowing agents, the evolved gas and its solubility, method of compounding, processing temperatures, and melt viscosity. Figure 9.1 shows the schematic process diagrams of these two foam extrusion processes. In cases where the polymers have weak melt strength, such as polyolefins, the foam process may be modified by inducing cross-linking during foaming. The cross-linking may increase the melt strength, that can stabilize the cell growth and obtain a better foam structure. This section briefly describes these two foam extrusion processes. It provides key information with regard to the important roles of mixing and how to design proper mixing sections to meet the various mixing capability requirements in each thermoplastic foam extrusion process.

© 2000 by CRC Press LLC

FIGURE 9.1

Schematic of thermoplastic foam extrusion processes.

9.2.1 HIGH-DENSITY FOAM PROCESS For the higher density foam product, the thermoplastic foam extrusion process utilizes a chemical foaming agent (CFA) that decomposes at a processing temperature to evolve gas (e.g., nitrogen, carbon dioxide) to form the cellular foam structure. In some cases, such as foamed PVC products used as a wood replacement, physical blowing agents may be used in preference to chemical foaming agents. The chemical foaming agents are either endothermic or exothermic when they release the gas. The heat or energy associated with the decomposition reaction affects the polymer melt temperature and the processing window. There are also chemical foaming compounds that contain endothermic and exothermic CFAs. The exothermic component provides the gas volume and pressure needed for lower densities, while the endothermic part produces a stable, fine, and uniform cell structure. Some available chemical foaming agents are listed in Modern Plastics Encyclopedia [1]. The list shows the types of chemical foaming agents, trade names and suppliers, processing temperature range, gas yield, and types of plastics recommended for use. Special caution must be taken in temperature control to avoid premature melt temperature rise before the melt seal, resulting in decomposition of the CFAs and loss of gas through the feed throat or the vent section. The chemical foaming agents are available in different forms [2]. The powder forms can be tumble-blended with resin pellets or dry-blended with resin powder. They are also available in pellet concentrates that facilitate more precise feeding and better dispersion.

© 2000 by CRC Press LLC

9.2.2 LOW-DENSITY FOAM PROCESS The lower density foam process employs physical blowing agents injected into the extruder by a high-pressure injection system. Physical blowing agents are atmospheric gases or hydrocarbon-based volatile organic liquids. Atmospheric gases used as blowing agents include nitrogen and carbon dioxide, which have low solubilities and high volatilities in polymers. Common hydrocarbonbased blowing agents are isobutane and isopentane, which have high solubilities and high diffusivities in polymers. Chlorofluorocarbons (CFCs) have been banned for use due to environmental concern. A recent development has been the “microcellular foam process,” which utilizes supercritical fluids of atmospheric gases as the blowing agents [3]. These blowing agents dissolve in the polymer melt and have a strong plasticizing effect causing a large reduction in melt viscosity. The polymer melt viscosity can be reduced by more than 50% after being mixed with the blowing agent [4]. The reduced melt viscosity could cause overblowing, cell rupture, and collapse of the extrudate. The melt viscosity has to be raised by lowering subsequent extruder barrel zone temperatures in order to have proper rheological properties to facilitate die forming and subsequent foaming. However, reduction in barrel zone temperatures for cooling may require a reduction in throughput rate to achieve the desired product temperature. A longer extruder, or more commonly, a second extruder (e.g., another single-screw extruder) in tandem, is used for the purpose of cooling the melt temperature to a range where satisfactory quality foam can be formed [5, 6]. This is not economical since either a longer screw extruder or a second extruder involves equipment investment. Therefore, rapid and uniform cooling of the polymer system is one of the process challenges in foam extrusion. Nucleating agents are normally used to provide initial nucleating sites and fine cell structure. Fine particle talc and calcium carbonate are commonly used as nucleating agents at 0.2 to 2% concentration. Special nucleating agents are available to control the size and distribution of the foam cells produced. They are made to provide higher efficiency than pure particle nucleation created by talc or calcium carbonate. In light of the above foam process descriptions, the following is a list of requirements for designing the mixing sections to obtain desired foam extruded products:

• • • •

pre-dynamic-seal mixing: dispersive mixing/melting gas/liquid injection mixing: distributive mixing with low-energy/highdivision mixing elements final homogenization mixing: distributive mixing with low-energy/highdivision mixing elements and good conveying capability cooling: distributive mixing for rapid and uniform cooling

© 2000 by CRC Press LLC

9.3 MIXING—THEORIES AND EXPERIMENTS As mentioned earlier, mixing plays several important roles in the foam extrusion process. Some of the mixing processes that may be involved in successful foam processing include the following: (1) Adding a low-viscosity liquid additive that must be included into the polymer matrix—once it is included into the matrix, it must be stretched out, broken up and distributed well enough throughout the polymer system (2) Adding a solid additive so that the agglomerates are broken up to the fundamental element or adding a high-viscosity liquid or melt so that it is broken up into drops and then distributed evenly throughout the polymer system (3) Adding a solid, so that it is not broken up but rather is distributed evenly throughout the polymer system for later action (4) Mixing and dissolving physical blowing agents Each one of these processes is achieved through some fundamental aspect of mixing. They are related but are governed by different forces and equations. The first example of a low-viscosity liquid being stretched out through the matrix, especially in the case of no to low interfacial tension, is a case where the interfacial area is being stretched by the deformation of the interface. The purpose of this section is to present the fundamentals of the mixing processes in the context of foam processing. For this discussion, “well-mixed” is defined as follows: Given a system with a global volume fraction of a minor component into the major component, in our case a foaming or nucleating agent into a polymer matrix, the system is well mixed if the same volume fraction is found in any subset of the overall system. For this definition of wellmixed, the mixing process has two functions. The first is to reduce the scale of segregation from the initial, agglomerated scale to the final or desired scale. For example, if an immiscible liquid were being injected, the liquid would be stretched and broken to a final drop size or area scale. In the case of solids addition, this would describe the process of breaking up the agglomerates down to the fundamental kernel or particle size. For each of these processes, there is extensive literature on the fundamentals of the processes and applications. The second function, after reducing the scale, is to distribute the material evenly throughout the medium. The classical methods to quantify the physics and the process of mixing are insufficient to predict the quality of distribution. The distribution is something that can be quantified as a result of a calculation, but it is not something that comes directly from the physics. This will be further discussed.

© 2000 by CRC Press LLC

9.3.1 DISTRIBUTIVE MIXING The first example for foam processing, given above, is for a liquid additive where the interfacial tension between it and the polymer is negligible. The motion of the major fluid component, the polymer, causes the dispersed phase to deform. The act of deformation causes the interface to deform and stretch. Distributive laminar mixing refers to the physical process of two fluids being blended such that the physical separation distances are reduced to a scale where diffusion or chemical reaction can occur. The mathematics and nature of distributive mixing have been expressed in terms of the kinematics of the flow [7–11] and by the continuum mechanics [12–14]. Each approach describes “mixing” as the growth of an interfacial line or interface, though each offers a unique viewpoint. The kinematic approach provides an overall view to the mixing process and an understanding as to the nature of mixing. This approach allows one to see directly whether a mixer is “linear” (i.e., the growth of interfacial area is linear with the applied shear strain) or whether it is “exponential.” In spite of the limitations of the two-dimensional assumptions required in the theoretical development, this approach enables the practitioner to have insight into the nature of the flow, even for complex three-dimensional flows. By contrast, the advantage of continuum mechanics is that it provides the mathematics necessary to examine the details of the flow. Given the current computational power readily available and given the current requirements for very tight product properties, knowledge of every aspect of the flow provides the information to relate the flow dynamics with the product properties. For example, the 1–5% of product that is not acceptable in an industrial process can make the entire batch unacceptable to the customer. Continuum mechanics can determine the aspects of the flow that could be causing that result, which can lead to appropriate changes in the flow geometry or operating conditions. 9.3.1.1 Laminar Mixing Numerous measures and indices have been proposed to characterize a mixture’s state of “mixedness.” Many of these are indirect measures, such as the mixture’s bulk electrical conductivity, the resistance of cured material to solvent or ultraviolet penetration, or some other gross property of the mixture. These measures are often directly applicable to the performance of a particular product, but while they may have direct technological application, they offer little insight into the mechanisms of mixing. Mixing in polymer melt processing is primarily the reduction of scales of segregation between immiscible fluids. The scale of a polymer mixture is typically described by either an average striation thickness or the amount of inter-

© 2000 by CRC Press LLC

facial area. Interfacial area generation was recognized by Brothman [15] as a primary mechanism for mixing. Mohr et al. [16] introduced the striation thickness (proportional to the inverse of the interfacial area), to characterize mixing due to laminar shear flow. The interfacial area, now accomplished by deterministic considerations, appeared later in the work of Spencer and Wiley [7]. Many indices have been proposed in recent years to quantify the mixed state. Most of these are simply related to interfacial area or striation thickness distributions. 9.3.1.2 Theoretical Approach 9.3.1.2.1 Interfacial Area Growth in Simple Shear The first fundamental study of mixing in laminar flow was based on interfacial area growth in a simple shear flow field. Spencer and Wiley [7] considered the deformation of an arbitrary oriented element of interface within a fluid undergoing simple shear. They found the growth of interfacial area to be a function of the magnitude of the shear and the initial orientation of the element: Af ⫽ [1 ⫺ 2s cos ␣ cos ␤ ⫹ s 2 cos ␣2]1/2 Ai

(1)

where Ai and Af are the initial and final interfacial areas, s is the magnitude of shear strain, and ␣ and ␤ are the angles defining the initial orientation of the element. This relation applies only to simple shear that has a constant velocity gradient everywhere in the flow field. This fact makes the calculation of the deformation tensor trivial or even unnecessary. Calculation of deformation tensor in complex flow fields requires a procedure for calculation of velocity gradient as well as integration of this tensor to obtain deformation tensor. Erwin [9] derived an upper bound for the increase of the length of an interface in shear flow for very large deformation. He discussed the importance of the direction of interface with respect to the principal directions of strain and derived the following relation for the growth of an element of area subject to an arbitrary three-dimensional strain of finite magnitude: Af cos ␣2 cos ␤2 cos ␥2 1/2 ⫽ c ⫹ ⫹ d 2 2 Ai ␭x ␭y ␭2z

(2)

in which the direction cosines describe a unit vector perpendicular to the element of area in its initial orientation, and ␭’s are the principal elongation ratios.

© 2000 by CRC Press LLC

9.3.1.2.2 Reorientation in Changing Flow Field In the course of an interfacial material flowing through a constantly changing flow field, the change of the flow field and the orientation of the interface relative to the flow field must be tracked. If the instantaneous growth of a material interface is given by: # ␭ ⫽ D : nn ␭

(3)

where ␭ is the interfacial line length, D is the rate of deformation tensor, and n is undeformed unit normal vector. Then, orientation of that interface relative to the flow is given by: n⫽

F ⭈N ␭

(4)

where F is force at the interface and N is deformed unit normal vector. By differentiating the above equation, the rate of change of the interface is given by: # ␭ # n ⫽ (D ⫹ W)n ⫺ n ␭

(5)

where D is the rate-of-deformation tensor and W is the rate-of-rotation tensor. The above expression can be said to quantify the reorientation of a material interface as it is deformed in the flow field. A physical understanding of the nature of reorientation of the interface can be garnered by reviewing a simplified version of this equation as applied to a Couette flow, which is given by the following: ⭸␪ ⭸u ⫽ ⫺Sin2 ␪ ⭸t ⭸y

(6)

By inspection, it can be seen that the rate-of-change of the angle is zero when the interface is either parallel or perpendicular to the flow direction. The rateof-change of the angle is maximum when the line is at a 45° and 135° angle. The rate of change becomes very small when the angle approaches 7°. In other words, in a steady shear flow, an interface, when placed in an arbitrary orientation, will quickly rotate to an orientation of approximately 7°. After that, the decrease in the angle proceeds slowly. We will discuss screw designs later in the chapter. The estimation of the time it takes for a fluid element to achieve this angle would help in determining frequency of screw disruption.

© 2000 by CRC Press LLC

9.3.1.2.3 Reorientation in Laminar Mixing Theory The previous discussion of reorientation, from continuum mechanics principles, quantified the change of orientation on an interfacial area with respect to a deforming coordinate system. The effect of the reorientation on the rate-ofstretch of an interface, although directly calculable, is never characterized. In the 1970s, using laminar mixing principles, Erwin showed that reorientating the interface, with respect to the direction of the flow, can alter the linear rate of mixing. In a simple shear flow, as the interfacial area deforms, it orients itself parallel to the direction of shear, which is least favorable for mixing. Erwin showed that subsequent reorientations produced a relationship for the final area growth, after k reorientations as follows: k Af s k ⫽ q fi a( )b Ai k i⫽1

(7)

It was shown later that the reorientation did not produce new interface itself, rather it produced an increased efficiency in the linear mixing. The total area growth in a simple shear flow, which is characterized with a linear relationship, can tend toward exponential growth rates after sufficient reorientations. The nonintermeshing twin-screw extruder has been shown to be a greaterthan-linear mixer by Bigio et al. [17]. This can be understood in terms of the previous discussion. As material moves from one screw into the apex region, it reorients as it enters into the channel of the other screw. 9.3.1.2.4 Distribution of Mixing In addition to the rate of stretching, it is important to consider the spatial distribution of stretching. A nonuniform distribution means that different portions of the system exhibit different extents of mixing at a given time (hence, different local striation thickness), resulting in poor global mixing. Until recently, it was believed that mixing in chaotic regions was always fast and efficient. This is not necessarily true, because first, the effects of islands can span regions that are much larger than the islands themselves [18], and second, for systems such as the cavity flow, segregated high stretching regions can develop within the chaotic region and persist for many periods. Since particles can remain trapped for long times within these high stretching regions, such regions can act as barriers to mixing. These regions develop as a result of singularities at the corners of the cavities [19]. Singularities are also present in the upper corners of the wavy channel flow, where the top wall slips past the vertical walls, and also in extruder flows at the locations where the screw contacts the shell.

© 2000 by CRC Press LLC

In the previous sections, we have presented the fundamental physics associated with distributive mixing. The difficulty lies in understanding how to apply the fundamentals to complex flow situations. To demonstrate this point, the reader is directed to the following exercise. Figure 9.2 shows the mixing in two different geometries. The initial condition shows a single fluid that is pigmented with titanium dioxide or carbon black and fed into a flow channel. The bottom two figures show the mixing at the end of the flow path. Both of the flow channels have wavy walls, with the mixing pattern on the right being caused by the channel with the walls with the greater amplitude. Since the volume fraction of the pigment is very small and the interfacial tension is minimal, this is a good example of distributive mixing. The question is as follows: Which one is better mixed? Which one is better mixed from the principles of laminar mixing? Based on the principles of laminar mixing theory, mixing is defined by the increase of interfacial area between the black and white fluids. The picture on

FIGURE 9.2

© 2000 by CRC Press LLC

Cross-channel images before and after mixing.

the left was caused by flow in a channel with a mildly wavy wall. It was found that the interfacial area growth was linear with the average applied fluid strain. So, the flow was essentially linear and the mixing was not very efficient. On the other hand, the area growth of the flow on the right-hand side was exponential with the applied strain. In terms of the theory, the mixing on the righthand side was greater. In fact, it can be shown that the nature of the mixing was chaotic. But, the concept of well-mixed has another connotation. How well is it distributed inside of the control volume? To that criterion, the answer is that the sample on the left is better mixed. There is a more even distribution of the area thickness, and it is better distributed inside of the volume. The sample on the right would continue to have the large unmixed area in the middle of the channel, so in fact, it would not go away, and although the area would continue to increase, there would be a large unmixed region. This region is called an “island” in the chaos literature, which refers to a region of the flow in which the fluid inside of this region does not interchange with the fluid outside of the region. One final comment about the question of which one is better mixed. The answer may depend on the process you are performing. In the case of mixing color, for example, the flow on the left is preferred; whereas, if you are interested in a reaction where the diffusion scales need to be small, the sample on the right would be preferred. 9.3.1.3 Experimental This section describes experimental investigations into the mechanism of distributive mixing in a corotating intermeshing twin-screw extruder and a counterrotating nonintermeshing twin-screw extruder. Previous work has shown the pronounced effect of percent channel fill (or percent drag flow) on distributive mixing in each machine and no effect of screw speed [17, 20]. To translate the percent channel fill variable into machine operating parameters, it implies that the specific throughput (i.e., volumetric throughput divided by screw speed, Q/N) is the more appropriate operating parameter for distributive mixing than just screw speed alone. Thus, the following experimental studies will demonstrate the importance and effect of specific throughput on distributive mixing. Experiments were run on two clear-barreled extruders. One is a fully intermeshing, 30 mm corotating twin-screw extruder (CoTSE), provided by Krupp-Werner & Pfleiderer. The other is a 20 mm, square-pitched, nonintermeshing twin-screw extruder (NITSE), provided by Welding Engineers Inc. (now called NFM/Welding Engineers Inc.). Silicone oils with viscosities of 60, 100, and 300 Pa-sec, supplied by GE Plastics, were tested in the experimental apparatus. The viscosities were suffi-

© 2000 by CRC Press LLC

cient to operate at starve-fed conditions without producing gravitational droop at the operating screw speeds. At the rate-of-strain levels employed in the experiment, the fluids performed like a Newtonian, nonviscoelastic, isothermal fluid. Screw speeds of 7 to 50 rpm were reported. A small volume of silicone oil, pigmented with carbon black, was injected on one screw and then videotaped as it was transported through the extruder. Feeding the dyed fluid onto one screw allowed for observation of the screw-to-screw transfer of the fluid and for tracking its specific path through the extruder. For partially-filled channels, the flow field is purely a function of drag flow. Drag flow is defined as the cross-sectional area of the channel times the downstream velocity, which is modeled as flat plate flow. This area of the flow field is modeled as a semi-ellipse for the CoTSE and a rectangle for the NITSE with the width being the distance between two flights perpendicular to the helical axis and the height being the distance between the screw root and the inner diameter of the barrel. Flow downstream is parallel to the helical axis, then described as the product of screw geometry, barrel diameter, rotational velocity, and cosine of the helical angle. This expression is given for each extruder: QCoTSE ⫽ ␲WHNDb cos ␪

(8)

QNITSE ⫽ ␲WHNDb fdFD cos ␪

(9)

where, W ⫽ channel width, H ⫽ channel height, Db ⫽ diameter of barrel, N ⫽ number of screw revolutions, fd ⫽ drag flow factor, FD ⫽ nip factor ⫽ 4f/(1⫹3f), f ⫽ 1 ⫺ (nip height/barrel circumference), and ␪ ⫽ helix angle. Percent channel fill is then found as the pump flow rate divided by drag flow rate. Both the flow rate, Q, and the screw speed, N, were varied to give a constant Q/N ratio over a range of screw speeds. For the CoTSE, the ratios ranged from 2.0 to 5.0 (equivalent to 30% channel fill to 70% channel fill) and for the NITSE, the ratios ranged from 1.0 to 4.0 (equivalent to 30% channel fill to 90% channel fill). Analysis of the dye distribution employed a quasi-quantitative technique of measurement for each machine. For the CoTSE, this technique was the measure of the increase in linear length of the tracer. An injected pulse stays in one channel in conveying elements for the CoTSE and thus stretches into a line, wrapping around the elements. The length of this line could be determined and plotted against strain as a measure of mixing where average total strain is defined as the integral of strain rate over time. For partially filled channels, the flow field is generated by drag alone, and the average strain rate is constant. Strain is defined as follows: ␥⫽

© 2000 by CRC Press LLC

␲3 ND2b ⭈n 4Qp

(10)

where N is the screw speed, Db is the barrel diameter, Qp is the volumetric flow rate, and n is the number of screw revolutions. For the NITSE, an injected pulse quickly enters many channels on each screw (as opposed to the CoTSE) so the measure of mixing is the number of channels containing dye (where each L/D is broken into eighths for analysis). Channels containing dye is presented as a function of screw revolutions, which is analogous to imparted strain. 9.3.1.4 Results 9.3.1.4.1 Corotating Twin-Screw Extruder (CoTSE) Figure 9.3 is a plot of averaged line length versus strain for a 30mm CoTSE. There is a pronounced effect of percent channel fill on mixing. A jump in line lengths appears between 40% and 50% fill. This is due to the way that the CoTSE channel physically filled with fluid. Since the channel of the CoTSE is near parabolic, at a given screw speed slight increases in the flow rate will tend to fill the bottom of the channel, only slightly increasing the wetted barrel

FIGURE 9.3 [20].)

Averaged line length versus strain in a 30 mm CoTSE. (Adapted from Bigio et al.

© 2000 by CRC Press LLC

FIGURE 9.4 Normalized line length versus percent channel fill in a 30 mm CoTSE. (Adapted from Bigio et al. [20].)

surface. Since the shearing force is imparted to the fluid via the wetted barrel surface, for channel fills of less than 30%, slight increases in barrel surface contact translate into only slight increases in mixing. However, at above about 30% channel fill, the bottom of the channel is filled and increases in flow rate will yield proportionate increases in the wetted barrel surface. In other words, increased flow rate yields rapid increases in mixing for percent channel fills above 30%. Figure 9.4 shows normalized line length versus percent channel fill. The normalized line length is the line length divided by strain. It can be seen that the rate of mixing is very much dependent on the degree of fill of the machine. Above about 40% fill, slight increases in the flow rate (at constant screw speed) yield rapid increases in mixing. However, below about 40% fill (especially around 30% fill), only slight increases in mixing are seen from increases in flow rate, and the rate of mixing is nearly constant. To optimize mixing in partially filled conveying sections in the CoTSE, the channels must be filled to greater than 40%. By running the machine over 40% fill, dramatic mixing results can be achieved when compared to the lower percent fills. 9.3.1.4.2 Nonintermeshing Twin-Screw Extruder (NITSE) Figure 9.5 is a plot of channels containing dye versus screw speed for the 20 mm NITSE run at various percent channel fills while keeping the screw speed

© 2000 by CRC Press LLC

FIGURE 9.5 Extruder volume containing dye versus screw speed at various percent channel fills in a 20 mm NITSE. (Adapted from Bigio et al. [17].)

constant at a given screw stagger. As can be seen, there is an increase in mixing as the degree of channel fill is increased. This is due to the nature of the screwto-screw transfer that depends on the relation between the screw stagger and the percent channel fill [17]. If the percent fill is less than the percent stagger, there is no fluid-to-fluid contact across the screws. The only mode of transfer is deposition where fluid from one screw gets deposited into the nip region between the two screws and is picked up by the opposite screw. This is a relatively inefficient mode of transfer (as is seen by the few channels containing dye for low percent channel fills). When the percent fill is greater than the screw stagger, there is direct fluid-to-fluid contact and highly efficient screwto-screw transfer. Thus, for high percent channel fills—when the percent fill is greater than the percent screw stagger—there is enhanced transfer; hence, enhanced distributive mixing. Figure 9.6 is a similar plot for a constant percent channel fill at various screw speeds. Mixing is obviously not a function of screw speed, given pure positive conveying elements. Even though the fluid sees a high shear stress at high screw speeds, the residence time decreases such that the fluid sees the high shear region for a shorter time. 9.3.2 DISPERSIVE MIXING Processing of materials for foam applications requires the successful addition of a foaming agent into the polymer matrix. The requirements for the process vary as a function of the additive and the desired application. In some cases where a physical blowing agent is used, the additive is a lower viscosity material that needs to be distributed throughout the polymer. In other cases where a chemical foaming agent is used, the material is very stress sensitive

© 2000 by CRC Press LLC

FIGURE 9.6 Extruder volume containing dye versus percent channel fill at various screw speeds in a 20 mm NITSE.

and over mixing could result in loss of efficiency. On the other extreme, the additive could be a pseudo-solid that requires strong forces to overcome the agglomeration so that it could be distributed throughout the polymer matrix. In both of these examples, force is required to achieve the desired mixing. What is similar in both of these cases is that there is a range of forces that all of the dispersed phase elements must see in order to be sufficiently distributed throughout the flow field. Similarly, there are maximum stresses and temperatures to which the chemical foaming agents must not be exposed to avoid premature decomposition. In this section, we will present some of the basic physics associated with the dispersive mixing process. In subsequent sections, the effects of screw design and operating conditions in extruders will be discussed. 9.3.2.1 Relationship of Dispersive and Distributive Mixing When people discuss mixing, they often say that they first need to break up the large particles and then distribute them throughout the flow volume. The

© 2000 by CRC Press LLC

way it is described in practice, it would seem that the two types of mixing are separate and distinct and driven by different forces. In fact, the opposite is true. In the simplest sense, the difference between distributive and dispersive mixing is that the latter requires a minimum force to overcome adhesive forces, whereas distributive mixing has no resistance to the process. The measure for mixing with dispersive mixing is morphology or solid diameter; the measure for mixing with distributive mixing is interfacial area or scale of separation. Distributive mixing is related to the shear rate; the dispersive mixing is related to the stress. But, the stress is given by the viscosity multiplied by the shear rate, or # ␶ ⫽ ␩␥

(11)

In the locations where there are large shear rates, there are also large stresses. So, the locations for good distributive mixing are also those with good dispersive mixing. The only caveat to that statement is that with dispersive mixing, there needs to be a minimum stress to overcome the adhesive forces. 9.3.2.2 Liquid-Liquid Mixing The process of mixing a liquid additive into a liquid matrix is fulfilled through the competition of forces—surface tension forces that tend to hold the drop together and viscous forces imparted by the fluid motion that tend to break them apart. The dimensionless parameters that describe the process include capillary number and viscosity ratio. Capillary number (or Weber number) is a nondimensional number that is the ratio of the viscous forces to the surface tension forces or Ca ⫽

# ␩m␥d ␴

(12)

# Where ␩m ⫽ the matrix viscosity, ␥ ⫽ the local shear rate, d ⫽ the characteristic drop length scale, and ␴ ⫽ interfacial tension. Viscosity ratio is the ratio of the disperse phase to the continuous phase or ␭⫽

␩d ␩m

(13)

In the case of mixing of Newtonian fluids, the critical capillary number, as a function of the viscosity ratio, describes the critical diameter of a drop in a flow field above which the drop can be broken into smaller, daughter, drops.

© 2000 by CRC Press LLC

FIGURE 9.7 Critical capillary number versus viscosity ratio for simple shear flow and extensional flow. (Adapted from Grace [22].)

As the viscosity ratio changes between the dispersed and continuous phases, so does the value of the critical capillary number. Taylor [21] showed that for ␭ ⬎ 3.7, the dispersed drop will not break because the matrix fluid could not impart a sufficient stress. Figure 9.7 shows the critical capillary number versus viscosity ratio curves in a simple shear flow and an extentional flow [22]. For a miscible fluid system with a hydrocarbon blowing agent, the interfacial tension is essentially zero, so the capillary number is not determinable. 9.3.2.2.1 Viscoelastic Effects Most polymers exhibit viscoelastic effects that affect the process of drop breakup. Literature reported contradictory results in the effect of viscoelasticity on drop deformation and breakup. It was generally assumed that viscoelasticity retards the process of deformation and breakup. The critical capillary numbers have been reported for viscoelastic drops in a viscous matrix [23–25]. Their results showed that for viscosity ratios on the order of one, no significant difference (e.g., a factor of 2) from Newtonian drops was found. Viscoelasticity plays a significant role only if the deformation is fast and large enough. In most drop breakup experiments, the total strain may be large, but the rate of deformation may be too small to build up a significant level

© 2000 by CRC Press LLC

of orientational stresses. In addition, the stresses that have built up could relax quickly. Then the deformed drops retract back to initial shape. Tipstreaming breakup phenomenon was observed by Milliken and Leal [23] for viscoelastic drops. Janssen [26] reasoned that this type of phenomenon was not caused by the viscoelastic behavior of the fluid. The polymer molecules added to modify and make the model fluid viscoelastic may accumulate at the interface and act as a surfactant [27, 28]. The inhomogeneities in the interfacial concentration of this surfactant and the interfacial tension cause nonuniform drop breakup observed as tipstreaming. That is, an extended drop breaks up into a major drop with a few much smaller satellite drops at both ends. Mighri et al. [29] examined the deformation and critical breakup in a simple shear flow of two viscoelastic fluids. They developed empirical relationships between the elasticity ratio, k⬘, the ratio of the drop and matrix relaxation times, and the Capillary number. Their results showed that for k⬘ ⬎ 0.37, the elastic drops deform less than Newtonian drops; whereas for k⬘ ⬍ 0.37, the drops deform more than Newtonian drops. In general, they concluded that the matrix elasticity helps to deform the drops, whereas the drop elasticity resists the drop deformation. Elasticity ratio affected drop breakup in the following manners. The critical Capillary number and breakup time increase with increasing elasticity ratio. For k⬘ ⱕ 4, the critical Capillary number increases rapidly, increasing k⬘. Above k⬘ ⬎ 4, the critical Capillary number is about 1.75 that corresponds to the maximum contribution of elasticity on drop breakup. Wu [30] studied the interfacial and rheological effects for incompatible polymer blends in a corotating intermeshing twin-screw extruder. The dispersion process was investigated using nonreactive and reactive ethylene propylene rubbers as the dispersed phase. Polyamide (nylon 6,6 resin) and poly(ethylene terephthalate) were used as the matrix phase. Based on experimental data, he established a master curve that similarly correlated the critical capillary number with viscosity ratio. The portion of the curve for viscosity ratio greater than one can be expressed as a straight line. Ca crit ⫽ 4␭0.84

for ␭ ⬎ 1

(14)

Although there lacked sufficient data available for ␭ ⬍ 1, a linear curve can be tentatively drawn for the ␭ ⬍ 1 portion. Then, the whole range of the master curve becomes V-shaped as the elasticity of the system increases. Furthermore, he found that viscoelastic drops can break up even for ␭ ⬎ 4 in the extruder. He suggested that this arose from a combination of several factors: the viscoelastic effects, complex transient shear, complex viscosity/temperature profile along the extruder, and the presence of elongational flow field in the extruder.

© 2000 by CRC Press LLC

9.3.2.3 Application to Complex Flow The fundamentals for drop breakup have been developed in simple, steadystate flows in order to distinguish the breakup mechanisms. The trouble occurs when these models are used to predict the resultant morphology in a complex flow where there is no constant value for shear rate. Even in a simple flow like the flow in a screw channel, the shear rate for a fluid element varies as it flows around the cavity. So, from a rigorous point of view, one would have to track a number of drops over various streamlines and then apply some criteria for breakup to generate a prediction for the final morphology. Bastian et al. [31] are beginning to look into developing that approach. Therefore, how much time a fluid drop is at a particular shear rate becomes a critical factor for trying to predict average drop sizes using established theory. Bigio et al. [32] applied an experimentally derived model for drop breakup in the steady shear flow of a screw channel. The result shows the conditions under which drop breakup would occur and the final average drop sizes. To further complicate the issue, in many of the reported experiments, the average particle sizes are smaller than what is predictable by the above models. What, then, is the cause of the breakup? It is possible that the source of breakup is the folding and breaking of the drop fibers as they translate from one flow field to another in a complex flow regime as might be found in the mixing elements of an extruder. Ottino [33] has shown the physics associated with this mechanism. 9.3.2.4 Solid-Liquid Breakup In the case of solid-liquid breakup for foaming agent and/or nucleating agent in the polymeric melt, the primary forces involved are particle cohesive force, flow shear, and extensional force. When the breaking force is greater than the holding force, agglomates tend to break up into smaller ones. The attractive van der Waals force between two spheres with same radius, R, is F ⫽ AR/12/z2

(15)

Where, A denotes Hamaker constant (⬵ 5 ⫻ 10⫺20 to 5 ⫻ 10⫺19 J), and z separation distance, 4 A for adhering particles. Considering void fraction, ␧, the cohesive force in the rupture cross-sectional area, S, can be expressed as follows [34]: Fc ⫽ (9/16)(1⫺e)/e(A/(12z2R)S

(16)

When the sum of breaking forces, shear force, hydrodynamic force, and extensional force, is greater than Fc, smaller particle clusters in the flow area can thus be anticipated.

© 2000 by CRC Press LLC

Since shear rate and elongation play critical roles in the breaking force, the design of mixer, flight tip clearance, and multiple passage of the high-shear region, become very important factors in solid-liquid dispersion. The criteria, Fb ⫽ Fc, can be used to calculate the solid agglomarate size. Based upon the particle density and mass rate, the solid cluster distribution becomes a straightforward calculation.

9.4 MIXING PRACTICES IN SINGLE- AND TWIN-SCREW EXTRUDERS Thiele [5] has reviewed the machinery features for thermoplastic foam extrusion. He asserted that there is really no standard device or process method to make foamed products. He used a “think-through” machinery approach to develop a processing protocol for a specific foamed product. One cannot design proper mixing sections to meet specific process demands and product requirements without understanding the common machinery employed. The above sections have reviewed the theory of mixing and the operating principles for dispersive and distributive mixing. The mixing sections in single- and twin-screw extruders utilized in thermoplastic foam extrusion are covered in the following. The screw configurations for achieving required dispersive and distributive mixing intensities are demonstrated. Table 9.1 lists common mixing elements available in various extruders. 9.4.1 MIXING SECTIONS IN SINGLE-SCREW EXTRUDERS The standard screws in single-screw extruders provide very limited mixing capability. The linear rate of mixing generated by the simple shear flow in the standard single-screw extruder is a very poor mixing mechanism [10, 35]. The extruder machine makers, screw designers, and processors have recognized this fact. Modifications have been made to the standard screws to improve their mixing capability. New screw designs to accomplish different process demands or product quality requirements have been evolving in the market. Rauwendaal [36] listed common types of screw elements for dispersive and distributive mixing in single-screw extruders. Maddock mixing section, Egan mixing section, and blister ring are some common dispersive mixing elements. Common distributive mixing elements include pin, Saxton, Dulmage, Pinapple, slotted screw, and cavity transfer mixing sections (see Figure 9.8) [36]. Rauwendaal [37] discussed the screw design for good control of melt temperature, mixing, and melting in foam extrusion systems. One of the critical steps in foam extrusion is the distributive mixing of the physical blowing agents into the polymer melt. He compared several distributive mixing elements commonly used in single-screw extruders. (see Table 9.2) [37]. The best

© 2000 by CRC Press LLC

TABLE 9.1

Extruder Type and Mixing Elements

Single-Screw Extruder

Mixing Elements in Various Extruders. Corotating Intermeshing Twin-Screw Extruder

Counterrotating Intermeshing Twin-Screw Extruder Intermeshing calendering screws, discontinuous discs, mixing rings, hexalobal mixing screw Discontinuous discs, slotted screw, pin mixer mixing ring, vane mixer, gear mixer

Dispersive Mixing

Maddock, Egan, blister rings

Discontinuous discs

Distributive Mixing

Pin mixer, Saxton, Dulmage, Pinapple, slotted screw, cavity transfer mixing element

Discontinuous discs, toothed mixing element

© 2000 by CRC Press LLC

Counterrotating Nonintermeshing Twin-Screw Extruder Cylindrical element

Staggered non-intermeshing screws, double-reverse flight element, slotted doublereverse element

FIGURE 9.8 Some common dispersive and distributive mixing elements for single-screw extruders. (Adapted from Rauwendaal [36].)

overall mixing section is the Saxton mixer that combines forward conveying capability and a streamlined geometry with frequent flow path splitting and reorientation. The cavity transfer mixer (CTM) also provides the features for frequent splitting and reorientation [38]. New special screw designs for achieving improved dispersive and distributive mixing in single-screw extruders have been available from various manufacturers. 9.4.2 MIXING SECTIONS IN COROTATING INTERMESHING TWIN-SCREW EXTRUDERS Corotating intermeshing twin-screw extruders have been widely used as plastics compounding machines. They provide flexible design of screw config-

© 2000 by CRC Press LLC

TABLE 9.2

Comparison of Distributive Mixing Elements for Single-Screw Extruders [37].

Mixers

Pressure Drop

Dead Spot

Barrel Wiped

Operator Friendly

Machining Cost

Shear Strain

Splitting, Reorienting

Pins Dulmage Saxton CTM TMR Axon D-Wave Pulsar Stratabl.

2 4 4 1 1 4 4 4 4

2 2 4 3 3 4 4 4 3

3 2 5 2 4 4 4 4 4

4 4 4 1 3 4 4 4 4

5 4 4 1 3 5 2 3 3

2 4 4 4 4 4 4 3 3

4 5 5 5 5 3 2 2 2

(Rating scale: 5 ⫽ very good, 1 ⫽ very poor)

urations and modular barrel assembly to balance the various process tasks in the machine. The common mixing elements in corotating intermeshing twinscrew extruders are discontinuous discs (so-called “kneading blocks”) and special mixing elements. The mixing zones are often arranged in combinations of mixing elements of different geometries to achieve the desired level of mixing intensity. Each discontinuous disc is characterized by its axial width, number of discs, staggering angle between the discs, and whether it is constructed for forward conveying or for reverse flow restriction. Distributive mixing is said to be enhanced by narrow discontinuous to maximize the number of flow divisions for a given machine length. Dispersive mixing utilizes wide discontinuous discs that maximize the flow of fluid that is affected by the stagnation points on every disc. The polymer stream must first be mixed distributively to allow uniform stress input. As the stagger angle increases, the amount of material that can backflow through the gaps is increased, which increases the percent of material seeing the high stress and the total residence time in the mixing section. For both types of mixing sections, reverse discontinuous discs, neutral discontinuous discs, or reverse screw elements can be used to restrict the material flow and build up pressure, thereby increasing the mean residence time, broadening the residence time distribution, and enhancing the mixing intensity. They also create a melt seal to separate the subsequent process functions. One caveat is that the mixing elements should not be arranged in such a way that excessive polymer melt temperature rise and subsequent material degradation are caused. For the same mixing sections, the mixing intensity increases as the flow is restricted by the following elements in ascending order: neutral kneading blocks, reverse discontinuous discs, and reverse screw elements. Figure 9.9 shows several screw configurations that illustrate the screw design concepts for dispersive and distributive mixing.

© 2000 by CRC Press LLC

FIGURE 9.9

Screw configurations for distributive and dispersive mixing.

One of the processing challenges in foam extrusion is the injection and rapid mixing of physical blowing agents. The distributive mixing for this function may be beyond the capabilities of narrow discontinuous discs. Special mixing elements are available for higher intensity distributive mixing. For instance, Krupp W&P’s ZSK twin-screw compounders utilize toothed distributive mixing elements such as the TME (turbine mixing element) or ZME (Zahnmischelement) design [39]. Note that other suppliers of corotating intermeshing twin-screw extruders also provide similar mixing element design. These elements are defined by the number of teeth around the circumference and the tooth angle. Both mixing elements provide the maximum amount of distributive mixing with minimal energy input by providing flow splitting and reorientation. One difference between these two mixing elements is that the TME mixing elements do not have flight advancements, so their conveying capability is poor. The polymer material may stagnate and degrade in these regions.

© 2000 by CRC Press LLC

Some conveying screw elements are normally staged between zones of TME mixing elements to provide forward conveying capability. On the other hand, ZME mixing elements have flight advancements that provide better conveying capabilities than TME elements. TME or ZME mixing elements can be arranged in combinations of alternating patterns of forward and reverse elements to balance the degree of fill, back pressure, and distributive mixing intensity. Similar to the discontinuous discs, flow restriction and pressure buildup can be utilized at the end of TME or ZME mixing zones to increase mixing intensity. ZMEs, reverse TMEs, neutral discontinuous discs, reverse discontinuous discs, and reverse screw elements can be used to provide mixing with increasing intensity. Figure 9.9 shows several screw configurations that utilize TME mixing elements for distributive mixing. As pointed out by Dreiblatt and Eise [40], the design of a screw configuration for twin-screw extruders is more of an art than a science. The above discussions merely provide general guidelines for mixing in corotating intermeshing twin-screw extruders. When designing a screw configuration for mixing, one has to consider the physical properties and compatibility of the ingredients, temperature profile and mechanical energy management, shear sensitivity of the polymer materials, and required mixing intensity. 9.4.3 MIXING SECTIONS IN COUNTERROTATING INTERMESHING TWIN-SCREW EXTRUDERS Counterrotating intermeshing twin-screw extruders are often used in processes that require short mean residence time, narrow residence time distribution, good control over material temperatures, and relatively positive pumping capacity. As a result of the above advantages, these extruders have been used in extrusion of temperature-sensitive materials, such as profile extrusion of rigid PVC, with minimum degradation. These extruders are well suited for foam extrusion processes where thermal history control is important to make quality foamed products. In counterrotating screws, the roll-off process between the screw flight and screw root and between the screw flanks creates a calender effect. The mating in the intermeshing region between the root of one screw and the flight of another screw forms a chamber shaped like a capital letter “C.” The C-shaped chambers in the classical counterrotating intermeshing twin-screw extruders provide better sealing between the two screws and efficient forward pumping capability. However, high pressure can develop in the intermeshing region due to the locked C-shaped chambers. This pressure works to push the two screw shafts apart. In addition, the wear of the screws increases as the screw speed increases. As a result, these extruders have to run at low speeds around 10 to 50 rpm, generally limited to 150 rpm [1, 42]. New design concepts have made it possible to run high speeds in the range of 300 to 500 rpm in counterrotating

© 2000 by CRC Press LLC

intermeshing twin-screw extruders [42]. The high shear and elongational flow in the classical counterrotating calender gap offer an effective dispersive mixing mechanism. Three book volumes [43–45] are good references for describing the development history, screw designs, process functions, and application examples in counterrotating intermeshing twin-screw extruders. Classical counterrotating intermeshing twin-screw extruders use slotted screws, pin mixers, and mixing rings to achieve distributive mixing, whereas the intermeshing calendering and mixing rings are utilized for dispersive mixing. Similar to corotating intermeshing twin-screw extruders, discontinuous discs are available to assemble appropriate screw configurations for dispersive and distributive mixing. Vane or gear mixers are available for distributive mixing [42]. Solid and slotted hexalobal mixing screws were invented to improve the mixing capability in counterrotating intermeshing twin-screw extruders [42, 46]. 9.4.4 MIXING SECTIONS IN COUNTERROTATING NONINTERMESHING TWIN-SCREW EXTRUDERS The Counterrotating Nonintermeshing Twin-Screw Extruder, manufactured by NFM/Welding Engineers enjoys wide usage in reactive processing, especially devolatilization and latex processing. This is due to the high free-volume capabilities and the flow path in the apex region between the two screws [47]. The extruder has been shown to have excellent distributive mixing characteristics [17, 20, 48, 49]. They showed that in the apex region, flow transfer occurs from one screw to another and in the back flow direction. The driving force was suggested to be due to a local pressure gradient that occurs when the screws are staggered that places the pushing flight of one screw (high pressure) near the trailing flight of the adjacent screw (low pressure). The transfer of 10–15% of the material in the channel results in a reorientation of the material in the channel and an enhanced mixing rate [50, 51]. As mentioned earlier, the distributive mixing in counterrotating nonintermeshing twin-screw extruders could be a greater-than-linear type. In other words, mixing can be greater than just doubling two single-screw extruders as long as the mixer is operated under conditions where the degree of fill is greater than the screw stagger. One standard element available for dispersive mixing in counterrotating nonintermeshing twin-screw extruders is a cylinder or a cylindrical compounder. The cylinders offer no forward conveying capabilities. The pressure flow through the cylinders can occur either in the annular region between the cylinder and the barrel wall or through the apex region. The pressure drop across the cylinders also provides the melt seal function. The cylinders are available with different clearance to vary the level of pressure drop and degree of shear stress. A tighter gap results in a higher material flow in the apex region. The advantage of the cylinders is that the flow that passes in the

© 2000 by CRC Press LLC

annular region experiences an equal amount of stress. The limiting factor is that the tighter the gap that is needed, the more the material will flow through the apex rather than the annular region. Hagberg et al. [52] conducted a scaleup study of dispersive mixing on cylindrical compounders using 30 mm and 51 mm counterrotating nonintermeshing twin-screw extruders. They found that dispersive mixing was scaled up with increasing extruder size. The pressure drop across the cylinder is an indication of stresses necessary for dispersive mixing. To obtain a higher dispersive mixing, one would run the extruder at the higher pressure drop conditions where tighter apex barrels with tighter cylinders are used.

9.5 PROCESS CHALLENGES 9.5.1 INJECTION AND MIXING OF PHYSICAL BLOWING AGENTS Injection and mixing of physical blowing agents is one of the challenging process tasks in foam extrusion. The process requires pumping and injection of the blowing agent in liquid form (or supercritical fluids) into the extruder at a consistent rate without much fluctuation. Once the blowing agent is injected into the extruder, the next task is to mix the blowing agent into the polymer in an efficient and uniform distributive mixing mechanism. Due to the large viscosity difference between the low-viscosity blowing agent and the high-viscosity polymer (i.e., small viscosity ratio), higher energy input is required to achieve good mixing [22, 30, 53–55]. Their data showed that droplet breakup for mixing (as characterized by a critical capillary number or Weber number) is ineffective for low-viscosity ratio fluids subject to a simple shear flow field. On the other hand, elongational flow significantly reduced the critical capillary number for improved drop breakup. Distribution mixing involves the reduction in segregation of scale, flow division, and reorientation of the minor component (blowing agent) in the major component (polymer). A good distributive mixing process is beneficial if starting with a smaller domain of the minor component. This can be achieved by using an injection nozzle with a smaller diameter port. Another alternative is to split-inject the liquid into two or more injection nozzles. Each injection nozzle handles a smaller quantity of liquid. A third option is to use a multi-injection port as described by Park et al. [56]. They utilized the multi-injection ports in the supercritical gas injection nozzle to increase the interfacial area between the gas and the polymer melt. As a result, the diffusion of gas into the polymer melt is improved and the diffusion time required to complete the polymer/gas solution formation is decreased [57]. To further improve mixing of the blowing agent, the above practices are complemented by injecting the blowing agent over distributive mixing ele-

© 2000 by CRC Press LLC

ments (e.g., narrow discontinuous discs, toothed mixing elements, gear mixers) under high pressure. The blowing agent immediately divides into smaller domains when injected over mixing elements. By contrast, the blowing agent flowing from the injection nozzle to the flighted screw elements may form large drops and coalesce into larger drops. Then, it would require more mixing elements, longer screw, and higher energy input in the downstream to achieve the same level of mixing as injection over mixing elements. In other words, the preferred distributive mixing process is one starting with as small a length or area for stretching and orientation as possible. It is important that the blowing agent is finely divided without relaxation to prevent it from forming larger drops due to coalescence. A rapid reduction in scale of domain increases the diffusion rate of the blowing agent into the polymer and shortens the diffusion time required to complete dissolution. Mass transfer properties of the materials used in foam extrusion are also important in determining the intensity of mixing required for achieving rapid dissolution. In foam extrusion, diffusivity and solubility of the blowing agent in the polymer are two key properties. Diffusivity determines the rate of diffusion of the blowing agent in the polymer. Solubility determines the equilibrium level of blowing agent dissolving in the polymer at given conditions. For example, isopentane has higher solubility than carbon dioxide in polystyrene for the same given conditions. Besides, isopentane is in a liquid phase at room temperature and, thus, can be easily metered when injected using a positive displacement pump, as opposed to the gaseous carbon dioxide. A more controllable process can be achieved. It is easier to mix isopentane with polystyrene because they are more compatible to each other. As long as the scale of segregation for mixing is reduced sufficiently, the dissolution process takes place quickly. By contrast, higher mixing intensity is required for dissolving carbon dioxide. Sometimes, the injection pressure also has to be maintained at thousands of psi’s to obtain the same level of dissolution as with isopentane. The intensity of mixing has to be maintained and also sustained and prolonged. This may occur to an undesirable degree causing excessive shearheating. As a result, the downstream cooling of the polymer system could be difficult and that would limit machine throughput. Figure 9.9 has shown some of the screw configurations for distribution mixing in a corotating intermeshing twin-screw extruder. One can assemble a distributive mixing screw by alternating the right-handed and reverse mixing elements to give the desired flow division and reorientation. For single-screw extruders and other types of extruders where modular barrel and screw assembly are not available, commercial systems utilize suitable distributive mixing elements for mixing the blowing agent. Static mixers are used after the distributive mixing elements to further improve mixing of the blowing agent. The materials flowing into a static mixer are divided by baffles, and mixing occurs by continual flow splitting and recombination. Most commonly used static mixers rely on

© 2000 by CRC Press LLC

shear flows and high shear stresses to split and redistribute the materials uniformly. They tend to cause excessive temperature rise while mixing the materials. The excessive temperature rise reduces the melt viscosity and makes cooling even more difficult. As a result, throughput and productivity are limited. Some special temperature control devices for static mixers are available that offer efficient heat transfer to reduce downstream cooling and line length [58]. A new type of static mixer that provides dispersive and distributive mixing capabilities in one device has been described in the literature [59]. As the blowing agent is injected into the extruder under high pressure, it may blow back upstream and interfere with the process. It is necessary to have a melt seal before to the blowing agent mixing section. Because liquid pumps are used to deliver and inject the blowing agents, it is important that they provide enough positive pumping capacity to overcome the pressure over the mixing elements. Otherwise, the blowing agents could back up from the extruder to the injection nozzle. The precision of the metering device is critical in supplying consistent flow rate of the blowing agents. The injection tube is ideally short, rigid, and in smaller diameter to minimize pressure loss. 9.5.2 EXTRUDER COOLING Since the viscosity of the polymer system may drop by as much as half after being mixed with the blowing agent, it is necessary to cool the polymer system and develop proper rheological properties for die forming. As mentioned earlier, the extraction of heat from the polymer for cooling conflicts with the requirements for mixing and limits the throughput capability. So far, commercial foam extrusion systems have been utilizing a longer plasticating extruder or a second oversized extruder in tandem to attain the cooling requirement before die forming. Therefore, one of the processing challenges in foam extrusion is rapid and uniform cooling of the polymer system. Several books and articles have covered the subject on heat transfer of polymer systems in extruders [36, 45, 60–63]. However, there was little engineering analysis as to how to achieve good mixing and rapid cooling for a given extruder at the same screw speed and throughput [64, 65]. It is easy to identify that there are two major heat transfer mechanisms providing the heating source for the polymer systems. They are conductive heat through barrel surfaces and the heat generated by viscous energy dissipation. For the same given extruder and high screw speed requirement for mixing, the only source of heat exchange for cooling is heat conduction through barrel surfaces. The process requirements for these two tasks conflict with each other. Han [65] developed a mathematical model and performed computer simulation of cooling extruder performance in thermoplastic foam extrusion. Based on the computation results on a 6-inch, 32 L/D single-screw extruder, he showed that the screw temperature can be 30°C higher than the barrel temper-

© 2000 by CRC Press LLC

ature while the polymer melt temperature is 20°C higher than the screw temperature. In other words, the temperature difference between the barrel and the melt is 50°C. Such a large temperature difference indicates that it is not sufficient to rely on the cool barrel temperature to reduce melt temperature effectively. In addition, the computer simulation showed that the melt temperature difference from the screw root to the highest temperature point (about 2/3 channel height from the screw root) in the cross channel can be 40 to 50°C. Each fluid element follows the velocity distribution profile and circulates inside the cross-channel melt pool while conveying downstream in the machine direction. The higher melt temperature fluid elements circulate near the center of the pool and never have the opportunity to cross the flow path to low-temperature areas near the screw and barrel surfaces. This mode of heat transfer is not efficient due to poor mixing and the relatively large distance between the maximum melt temperature region to both screw and barrel. This problem can be alleviated by using efficient mixing elements and proper process conditions. One can use mixing elements that are capable of efficient flow reorientation and surface renewal. Rapid surface renewal increases the surface area for efficient heat transfer. Flow reorientation gives an opportunity for the fluid elements inside the circulating melt pool to travel to cool surfaces near the screws and barrel for heat transfer. The screw design factors affecting surface renewal include the helix angle, number of flights, and the flight clearance [36, 37]. Efficient heat transfer is favored by multiple flights, a small flight clearance, and a large screw pitch. Todd [63] reported heat transfer in partially filled twin-screw extruders. He derived an overall heat transfer correlation equation based on a classical heat transfer approach. The equation expresses Nusselt number (hD/k) as a function of Reynolds number (D2N␳/␩) and Prandtl number (c␩/k) with no effects from degree of fill or flight clearance: (hD/k) ⫽ 0.94(D2 N␳/␩)0.28 (c␩/k)0.33 (␩/␩w)0.14

(17)

where h ⫽ film coefficient, D ⫽ barrel diameter, k ⫽ thermal conductivity, N ⫽ screw speed, ␳ ⫽ polymer melt density, ␩ ⫽ effective viscosity (␩w at wall), and c ⫽ specific heat. The above equation can be further simplified when lumping the Reynolds and Prandtl numbers together: (hD/k) ⫽ 1.02(D2 N ␳c/k)0.33 (␩/␩w)0.14

(18)

Experimental results with cooling polyethylene in a single-screw extruder showed good agreement with the above heat transfer equation [61]. Notice that this equation does not include the effects of screw flight clearance and degree of fill.

© 2000 by CRC Press LLC

TABLE 9.3

The Relative Effects of Process Parameters on Extruder Performance.

Increase in the Following Parameters Throughput Screw speed Barrel temperature Screw pitch Channel depth Number of flights Flight clearance

Mixing

Melt Temperature

Energy Input

Conveying Capacity

⫺ ⫹ ⫹ ⫺ ⫺ ⫹ ⫹

⫺ ⫹ ⫹ ⫺ ⫺ ⫺ ⫺

⫺ ⫹ ⫺ ⫺ ⫺ ⫺ ⫺

⫹ ⫹ ⫹ ⫹ ⫹ ⫺ ⫹

Table 9.3 compares the relative effects of various parameters on the process functions in foam extrusion. They show either positive or negative effects of increasing throughput, screw speed, barrel temperature, screw pitch, channel depth, number of flights, and flight clearance on mixing, melt temperature, torque (energy input), and conveying capacity (throughput). The effect on melt temperature increase is the opposite of melt cooling. It is seen that the parameters used to cool the melt temperature compete with better mixing and higher throughput.

9.6 SUMMARY This chapter presents the design of dispersive and distributive mixing for foam extrusion. Mixing plays important roles in foam extrusion by dispersivemelting the raw materials, distributive-mixing the foaming agent, homogenizing the polymer system, and cooling the polymer system for die forming. The fundamentals of dispersive and distributive mixing are covered. The mixing sections in various extruders are described: single-screw extruders, corotating intermeshing twin-screw extruders, counterrotating intermeshing twin-screw extruders, and counterrotating nonintermeshing twin-screw extruders. Two process challenges in foam extrusion are described. One is the injection and mixing of physical blowing agents. The other is rapid and uniform cooling of the polymer system. This chapter intends to give a guideline as to how to design mixing sections to achieve the various process tasks in foam extrusion rather than to give a detailed engineering analysis.

© 2000 by CRC Press LLC

9.7 NOMENCLATURE A c Ca d D fd F FD h H k⬘ n N R s S Q W z

interfacial area; Hamaker constant specific heat Capillary or Weber number characteristic drop length scale diameter; rate of deformation tensor drag flow factor force nip factor film coefficient channel height elasticity ratio number of screw revolution; undeformed unit normal vector screw speed; deformed unit normal vector radius shear strain cross-sectional area volumetric throughput channel width; rate of rotation tensor distance

Greek Letters ␣ angle of orientation ␤ angle of orientation ␧ void fraction ␥ strain ␩ viscosity ␭ viscosity ratio; interfacial line length ␭i principal elongation ratio ␪ helix angle ␳ density ␴ interfacial tension ␶ shear stress

9.8 REFERENCES 1. Modern Plastics Encyclopedia ‘99. Mid-Nov. 1998. New York, NY: McGraw Hill, pp. C78–C79. 2. Plastics Compounding Redbook. 1998. Cleveland, OH: Advanstar Communications, pp.18–20.

© 2000 by CRC Press LLC

3. Park, C. B. 2000. “Microcellular Foam Extrusion,” in Principles of Thermoplastic Foam Extrusion. S. T. Lee, ed. Lancaster, PA: Technomic Publishing Co., Inc. 4. Gendren, R. and L. Daigneault. 2000. “Rheology of Blowing Agent Charged Polymeric Systems,” in Principles of Thermoplastic Foam Extrusion. S. T. Lee, ed. Lancaster, PA: Technomic Publishing Co., Inc. 5. Thiele, W. C. 2000. “Foam Extrusion Machinery Features,” in Principles of Thermoplastic Foam Extrusion. S. T. Lee, ed. Lancaster, PA: Technomic Publishing Co., Inc. 6. Sansone, L. F. 2000. “Process Design for Thermoplastic Foam Extrusion,” in Principles of Thermoplastic Foam Extrusion. S. T. Lee, ed. Lancaster, PA: Technomic Publishing Co., Inc. 7. Spencer, R. J. and R. M. Wiley. 1951. “The Mixing of Very Viscous Liquids,” J. Colloid Sci., 6: 133. 8. Erwin, L. 1978a. “Theory of Laminar Mixing,” Polym. Eng. Sci., 18: 1044. 9. Erwin, L. 1978b. “An Upper Bound of the Performance of Plain Strain Mixer,” Polym. Eng. Sci., 18: 738. 10. Erwin, L. 1978c. “Theory of Mixing in Single Screw Extruders,” Polym. Eng. Sci., 18: 572. 11. Erwin, L. 1978d. “New Fundamental Considerations on Mixing in Laminar Flow,” SPE ANTEC Tech. Papers. 12. Aref, H. 1984. “Stirring by Chaotic Advection,” J. Fluid Mechanics, 143: 1. 13. Ottino, J. M., W. E. Ranz and C. W. Macosko. 1979. “A Lamellar Model for Analysis of Liquid-Liquid Mixing,” Chem. Eng. Sci., 34: 877. 14. Ottino, J. M., W. E. Ranz and C. W. Macosko. 1981. “A Framework for the Description of Mechanical Mixing of Fluids,” AIChE J., 27: 565. 15. Brothman, 1945. Chem. Metallurgy Eng., 52. 16. Mohr, W. D., R. L. Saxon and C. H. Jepson. 1957. “Mixing in Laminar Flow Systems,” Ind. Eng. Chem. Fundamentals, 49: 1855. 17. Bigio, D. I., W. Baim and M. Wigginton. 1991. “Mixing in Non-intemeshing Twin Screw Extruders,” Intern. Polym. Processing, VI: 172. 18. Riley, M., F. Muzzio, H. Buttner and S. Reyes. 1992. AIChE J., 41: 691. 19. Liu, M., F. Muzzio and R. Peskin. 1994. “Fractal Structure of a Dissipative Particle-Fluid System in a Time-Dependent Chaotic Flow,” Am. Phys. Soc., 50: 4245. 20. Bigio, D. I., K. Cassidy, M. Dellapa and W. Baim. 1992. “Starve-fed Flow in Co-Rotating Twin Screw Extruders,” Intern. Polym. Processing, VII: 111. 21. Taylor, G. I. 1934. “The Formation of Emulsions in Definable Fields of Flow,” Proc. Roy. Soc. Lond., A146: 501. 22. Grace, H. P. 1982. “Dispersion Phenomena in High Viscosity Immiscible Fluid Systems and Application of Static Mixers as Dispersive Devices in Such Systems,” Chem. Eng. Commun. 14: 225. 23. Milliken, W. J. and L. G. Leal. 1991. J. Non-Newt.Fluid Mech., 40: 355. 24. Milliken, W. J. and L. G. Leal. 1992. J. Non-Newt.Fluid Mech., 42: 231. 25. de Bruijn, R. A. 1989. Ph.D. Thesis, Eindhoven University of Technology, The Netherlands. 26. Janssen, J. M. H. 1993. “Dynamics of Liquid-Liquid Mixing,” Ph.D. Thesis, Eindhoven University of Technology, The Netherlands. 27. Milliken, W. J., H. A. Stone and L. G. Leal. 1993. Phys. Fluids, A5 (1): 69. 28. Stone, H. A. 1994. Ann. Rev. Fluid Mech., 26. 29. Mighri, F., P. Carreau and A. Ajji. 1999 (in press). “Influence of Elastic Properties on Drop Deformation and Breakup in Shear Flow,” J. Rheol.

© 2000 by CRC Press LLC

30. Wu, S. 1987. “Formation of Dispersed Phase in Incompatible Polymer Blends: Interfacial and Rheological Effects,” Polym. Eng. Sci., 27: 335. 31. Bastian, M., M. Stephan and H. Potente. 1999. “Morphology Development of Polymer Blends in Twin Screw Extruders,” 15th Polym. Proc. Soc. Ann. Meeting, Paper 275. 32. Bigio, D. I., C. Marks and R.V. Calabrese. 1998. “Predicting Drop Breakup in Complex Flow,” Intern. Polym. Processing, XIII: 192. 33. Ottino, J. M. 1989. The Kinematics of Mixing: Stretching, Chaos and Transport. Cambridge, MA: University Press. 34. Elmendrop, J. J. 1991. “Dispersive Mixing in Liquid Systems” in Mixing in Polymer Processing. C. Raunwendaal, ed. New York, NY: Marcel Dekker. 35. Bigio, D. I. and L. Erwin. 1987. “Criteria for the Prediction of Mixing in Laminar Mixers,” SPE ANTEC Tech. Papers, pp. 164–169. 36. Rauwendaal, C. 1994. Polymer Extrusion. New York, NY: Hanser. 37. Rauwendaal, C. 1997. “Screw Design for Foam Processing,” Plastics World. May: 38. 38. Wang, C. and I. Manas-Zloczower. 1994. Polym. Eng. Sci., 34: 1224. 39. Andersen, P. G. 1998. “The Werner & Pfleiderer Twin-Screw Co-Rotating Extruder System,” in Plastics Compounding: Equipment and Processing. D. B. Todd, ed. New York, NY: Hanser. 40. Dreiblatt, A. and K. Eise. 1991. “Intermeshing Corotating Twin-Screw Extruders,” in Mixing in Polymer Processing. C. Raunwendaal, ed. New York, NY: Marcel Dekker. 41. Raudendaals, C. 1995. “Which Twin Screw Extruder Is for You?” Plastics Formulating & Compounding. Nov./Dec.: 15. 42. Thiele, W. C. 1998. “Counterrotating Intermeshing Twin-Screw Extruders,” in Plastics Compounding: Equipment and Processing. D. B. Todd, ed. New York, NY: Hanser. 43. Janssen, L. P. B. M. 1978. Twin Screw Extrusion. New York, NY: Elsevier. 44. White, J. L. 1991. Twin Screw Extrusion: Technology and Principles. New York, NY: Hanser. 45. Todd, D. B., ed. 1998. Plastics Compounding: Equipment and Processing. New York, NY: Hanser. 46. Thiele, W. C. 1997. “Optimizing Twin-Screw Extruders for Foamed Products,” FoamTech Meeting, National Research Council Canada, Boucherville, Quebec, Canada. 47. Nichols, R. J. 1983. “Pumping Characteristics of Counter-Rotating Tangential Twin-Screw Extruders,” SPE ANTEC Tech. Papers, p. 69. 48. Howland, C. and L. Erwin. 1983. “Mixing in Counter-Rotating Twin Screw Extruders,” SPE ANTEC Tech. Papers, p. 113. 49. Bigio, D. I. and W. Baim. 1992. “Distributive Mixing of Non-Newtonian Fluids in the Partially-Filled Non-Intermeshing Twin Screw Extruders,” SPE ANTEC Tech. Papers, pp. 1809–1813. 50. Bigio, D. I. and P. Herman. 1990. “Cross Channel Flow in a Counter Rotating Non-Intermeshing Twin Screw Extruders,” SPE ANTEC Tech. Paper, pp. 143–146. 51. Bigio, D. I., M. R. Ramanthan and P. Hermann. 1993. “Pressure-Related Driving Forces in the Fully Filled Nonintermeshing Twin Screw Extruders,” Advances in Polymer Technology, 12: 353. 52. Hagberg, C. G., D. I. Bigio and M. Shah. 1995. “Scaleup of Dispersive Mixing Cylindrical Compounders in CRNI Twin Screw Extruders,” SPE ANTEC Tech. Papers, pp. 333–339. 53. Karam, H. J. and J. C. Bellinger. 1968. “Deformation and Breakup of Liquid Droplets in a Simple Shear Field,” Ind. Eng. Chem. Fundam., 7: 576. 54. Tavgac, T. 1972. Ph.D. Thesis, Department of Chemical Engineering, University of Houston, Houston, Texas.

© 2000 by CRC Press LLC

55. Lee, W. K. 1972. Ph.D. Thesis, Department of Chemical Engineering, University of Houston, Houston, Texas. 56. Park, C. B., D. F. Baldwin and N. P. Suh. 1994. “Formation and Application of Polymer/Gas Mixtures in Continuous Processing of Microcellular Polymers,” Cellular and Microcellular Materials, ASME, MD-54: 109. 57. Park, C. B. and N. P. Suh. 1996. “Filamentary Extrusion of Microcellular Polymers Using a Rapid Decompressive Element,” Polym. Eng. Sci.,36: 34. 58. Plastics World. Sept. 1980. New York, NY: PTN Publishing Co., p. 68. 59. Gramann P., B. Davis, T. Osswald and C. Rauwendaal. 1999. “A New Dispersive and Distributive Static Mixer for the Compounding of Highly Viscous Materials,” SPE ANTEC Tech. Papers, pp. 162–166. 60. Davis, W. M. 1992. “Heat Transfer in Extruder Reactors,” in Reactive Extrusion—Principles and Practices. M. Xanthos, ed. New York, NY: Hanser. 61. Guo, Y. and C. Chung. 1988. “Dependence of Melt Temperature on Screw Speed and Size in Extrusion,” SPE ANTEC Tech. Papers, pp. 132–136. 62. Larsen, H. and A. Jones. 1988. “Heat Transfer in Twin Screw Extruders,” SPE ANTEC Tech. Papers, pp. 67–70. 63. Todd, D. B. 1988. “Heat Transfer in Twin Screw Extruders,” SPE ANTEC Tech. Papers, pp. 54–58. 64. Dey, S. K., C. Jacob and D. B. Todd. 1993. “Cooling in Single Screw Extruders,” SPE ANTEC Tech. Papers, pp. 2248–2255. 65. Han, C. D. 1987. “Analysis of Performance of Cooling Extruder in Thermoplastic Foam Extrusion,” SPE ANTEC Tech. Papers, pp. 113–116.

© 2000 by CRC Press LLC

CHAPTER 10

Foaming Agents for Foam Extrusion THOMAS PONTIFF

10.1 INTRODUCTION

F

agents are a vital and necessary ingredient used in the production of extruded foamed thermoplastics. Foaming agents have several different but similar definitions. Most generally, a foaming agent is a material that, in the vapor phase, expands a thermoplastic melt upon reduction in pressure. Physical foaming agents are materials that are injected into the process as either liquids or gasses. Chemical foaming agents are materials that decompose to generate gasses during the processing. Some physical foaming agents are low boiling point liquids, such as pentane or isopropyl alcohol, that remain liquids in the thermoplastic melt while the melt is under pressure. When the pressure is reduced, the foaming agent quickly changes from liquid to vapor and comes out of solution with the polymer to expand the melt. Another type of foaming agent includes the so-called inert gasses, such as carbon dioxide or nitrogen. These materials dissolve as vapors in the plastic melt and come out of solution as vapors to expand the plastic melt. Chemical foaming agents utilize a decomposable material that produces a gas or gasses during decomposition. In this case, chemical foaming agents must first decompose, then the gas produced behaves much like a physical foaming agent but with some effects influenced by the residual material from the chemical foaming agent decomposition. OAMING

10.2 PHYSICAL FOAMING AGENTS Almost as early in the history of thermoplastics processing as the development of polystyrene thermoplastic resin is the advent of foamed polystyrene.

© 2000 by CRC Press LLC

The first public disclosure of foamed polystyrene might have been made by Munters and Tandberg, who obtained U.S. patent 2,023,204 on December 3, 1935 [1]. Early processes for production of foamed polystyrene utilized low boiling point liquids, such as methyl chloride or butylene, which was used to dissolve the polystyrene resin in a large vessel. The heated and pressurized polystyrene/solvent solution was then released to atmospheric conditions where the solvent volatilized and expanded the polystyrene. Eventually, this process was transferred to extrusion processes in which polystyrene, which was preimpregnated with a foaming agent, was extruded to produce foamed products used for food and protective packaging and flotation [2]. Later, the direct gassing process was developed in which the foaming agent was injected directly into the extruder and dissolved into the thermoplastic melt [3, 4]. The foaming agent dissolved into the melt and then was extruded to atmospheric conditions to expand and form a foam product. Several different variations of extruders have been developed for this process, including long single-screw extruders, tandem extruders, twin-screw extruders, and several specialized systems. All of the extrusion systems must be capable of melting and mixing the polymer and additives, dissolving and dispersing the physical foaming agent, cooling the melt containing the foaming agent, and maintaining sufficient pressure until extrusion from the die. These are the necessary steps for successful production of low-density foams using the direct gassing process. These methods have been used to produce a wide variety of foam products from many types of thermoplastic resins, including LDPE (low-density polyethylene), PP (polypropylene), PET (polyethylene terapthalate), and others. Over the years, many types of physical foaming agents have been used in the production of thermoplastic foams. In the early years of the technology, hydrocarbons, such as pentane, and chlorinated hydrocarbons, such as methyl chloride and CFCs were most popular. CFCs were desirable because they were nonflammable and performed well in the process. They had the disadvantages of being more expensive than hydrocarbons, and it was later determined in the 1970s and 1980s that they were contributing to the destruction of the Earth’s ozone layer. Efforts have been made to list important physical foaming agent properties. One list, proposed by James Burt of E. I. Du Pont de Nemours & Co., lists the following as important properties [5]:

• • • • • • •

boiling point: ⫺50°F to ⫹150°F inertness safety adequate solubility in molten resin low permeability through resin low solubility in solid resin economical

The boiling point range is defined by reasonable processing temperatures. If a physical foaming agent is chosen that has a higher boiling point, it is too diffi-

© 2000 by CRC Press LLC

cult to vaporize. In fact, a better upper limit for boiling point might be about 100°F (37.8°C). Even materials that are candidates to be physical foaming agents that have boiling points around 100°F (37.8°C) do not possess sufficient vapor pressures at room temperature to maintain sufficient pressure within the cells to keep them inflated. An example is n-pentane. This is a commonly used physical foaming agent in extrusion of polystyrene foam. As the foam cools to room temperature, its vapor pressure drops, creating low pressure within the foam cells. Polystyrene foam produced with n-pentane does not collapse as the polystyrene has enough rigidity to support the cellular structure. However, LDPE foam extruded with n-pentane will collapse as the temperature approaches room temperature, as the LDPE is not rigid enough to support the cellular structure with reduced internal pressure. Other factors will influence this collapsing phenomenon, and they will be discussed later. As the boiling point becomes lower, the volatility of the material increases. Higher volatility or vapor pressure requires more pressure to keep the material in the liquid phase in the polymer melt. Therefore, the use of propane requires a higher melt pressure than the use of pentane. Inertness relates to the material’s reactivity and corrosiveness to the polymer being foamed, any additives, the machinery, or surrounding environment. It is best, for obvious reasons, that the physical foaming agent be as inert as possible. Safety is important, again, for obvious reasons. The ideal physical foaming agent would be completely safe, nontoxic, inflammable, and noncorrosive. However, the use of a material that might be considered unsafe, such as an alkane hydrocarbon, is accomplished by employing appropriate safety measures. Good solubility in the polymer melt is important for a physical foaming agent. As the foaming agent dissolves into the melt, it will plasticize the melt, lowering its viscosity. This allows the melt temperature to be reduced. In an extruder, as the solubility of the foaming agent in the melt increases, the “minimum” melt pressure needed to get and to keep the foaming agent in solution decreases. If the foaming agent has poor solubility, high amounts of energy must be used to force the foaming agent into solution. This is usually accomplished by running the equipment so that the melt is kept under higher melt pressure, which facilitates the dissolving of the foaming agent into the melt. The higher pressure causes the polymer melt to be exposed to more shear heat, and therefore, it is more difficult to cool to the optimum foaming temperature. The end result is that the lowest density achievable with a poorly soluble foaming agent is higher than that for a foam produced with a highly soluble foaming agent. Once the foam is formed, enough foaming agent vapor must remain inside the cells to prevent the cells from collapsing. This is obviously more critical with flexible foams, such as LDPE, than for more rigid foams, like PS. If the foaming agent escapes from the flexible foam at a rate much greater than the

© 2000 by CRC Press LLC

rate that air diffuses into the cells, the foam can collapse due to the lower total pressure inside the cells. In the past, the preferred physical foaming agent for LDPE was dichlorotetrafluoroethane (CFC 114). It had a very slow permeation rate through the LDPE and allowed collapse-free foams to be produced. Now that CFC 114 is not available due to its impact on the environment, the most common physical foaming agents for LDPE are alkanes, such as butane and propane. These have high permeation rates through the LDPE and foams produced with them collapse. However, permeation modifiers, such as glycerol monostearate are used to slow the permeation of the foaming agent from the foam. Low solubility of the physical foaming agent in the solid resin is important to prevent the polymer to be weakened by the solvent effect. If too much foaming agent remains in solution in the cell walls, the foam structure will be made weaker and subject to creep and poor physical properties. The fact that the physical foaming agent needs to be economical is obvious. However, there are some factors that need to be considered. CFCs, like dichlorodifluoromethane (CFC 12) or CFC 114, were quite popular in spite of their cost being roughly two to five times that of alkanes, such as butane or pentane. The main reason is that the CFCs are not flammable and are therefore much easier to use and handle. To utilize alkanes, several steps must be taken to ensure safety, which can be quite costly [6, 7, 8]. Table 10.1 lists properties of the commonly used physical foaming agents for thermoplastic foam extrusion and some materials that are similar but are not used for various reasons. The properties, molecular weight, boiling point, vapor pressure, liquid density, and thermal conductivity of the vapor all affect the function of the physical foaming agent in the process. While these materials do not act as exactly ideal gasses, their behavior can generally be approximated by ideal gas behavior. Therefore, one mole of propane vapor will occupy approximately the same volume as one mole of chlorodifluoromethane (HCFC 22), assuming that their vapor pressures are about the same. Referring to Table 10.1, 44.1 grams of propane should occupy the same volume as 86.5 grams of HCFC 22. This would mean that propane is roughly two times more efficient than HCFC 22 on the basis of weight required for the same expansion. Of course, other factors (such as solubility in the polymer melt and vapor pressure at the expansion temperature) affect the efficiency and will be discussed, but molecular weight can provide an estimate of the relative efficiency of a physical foaming agent compared to other similar materials. The boiling point of the physical foaming agent is important in several ways. First of all, the boiling point (at atmospheric pressure) of the foaming agent must be below the processing temperature in order to expand the polymer. Generally, the higher the boiling point, the lower the vapor pressure of the foaming agent. The effects of vapor pressure will be discussed later. As discussed previously, once the foam is formed and cooled to room tempera-

© 2000 by CRC Press LLC

TABLE 10.1

Name Propane n-Butane i-Butane n-Pentane i-Pentane HCFC-22 HCFC-142b HFC-152a HCFC-123 HCFC-123a HCFC-124 HFC-134a HFC-143a CFC-11 CFC-12 CFC-113 CFC-114 MeCl MeCl2

Properties of Physical Foaming Agents [9]. Boiling Point (°C)

Vapor Pressure (psi)

Liquid Density (g/cc)

Vapor Thermal Conductivity (W/mK)

Flammable

Formula

CAS #

Molecular Weight

C3H8 C4H10 CH3(CH3)CHCH3 C5H12 CH3(CH3) CHCH2CH3 CHF2Cl CF2ClCH3 CHF2CH3 CHCl2CF3 CHFClCF2Cl CHFClCF3 CH2FCF3 CH3CF3 CFCl3 CF2Cl2 CFCl2CF2Cl CF2CLCF2Cl CH3Cl CH2Cl2 CO2 N2 O2

74-98-6 106-97-8 75-28-5 109-66-0 78-78-4

44.1 58.1 58.1 72.2 72.2

⫺42.1 ⫺0.5 ⫺11.7 36.1 27.0

137.89 35.26 50.53 9.90 14.23

0.49 0.57 0.55 0.621 0.615

0.0179 0.0159 0.0161 0.0141 —

Yes Yes Yes Yes Yes

75-45-6 75-68-3 75-37-6 306-83-2 354-23-4 2837-89-0 811-97-2 420-46-2 75-69-4 75-71-8 76-13-1 76-14-2 74-87-3 75-09-2 124-38-9 7727-37-9 7782-44-7

86.5 100.5 66.0 153.0 153.0 136.5 102.0 84.0 137.4 120.9 187.4 170.9 50.5 84.9 44.0 28.0 32.0

⫺40.8 ⫺9.2 ⫺24.7 27.1 28.2 ⫺12.0 ⫺26.5 ⫺46.7 23.8 ⫺29.8 47.6 3.6 ⫺24.2 40.1 ⫺78.5 ⫺195.8 ⫺183.0

151.4 49.16 86.81 13.27 12.61 55.85 96.52 182.5 15.32 94.51 6.46 30.96 82.16 8.22 N/A N/A N/A

1.194 1.11 0.899 1.46 1.467 1.356 1.207 1.089 1.476 1.311 1.565 1.456 1.098 1.322 N/A N/A N/A

0.0106 0.0108 0.0136 0.0095 0.0111 0.0106 0.0127 0.0137 0.0082 0.0100 0.0097 0.0112 0.0106 0.0084 0.0165 0.0258 0.0266

No Yes Yes No No No No Yes No No No No Yes No No No No

© 2000 by CRC Press LLC

ture, the foaming agent should have a sufficiently high boiling point to maintain sufficient pressure inside the foam cells to keep the foam from collapsing. This is very important for flexible foams, such as LDPE, but not as important for more rigid foams, such as PS. As stated before, the vapor pressure of the physical foaming agent is inversely related to the boiling point, and there must be sufficient vapor pressure at room temperature to keep the foam from collapsing. The vapor pressure is also important at the foam expansion temperature or melt temperature. The higher the vapor pressure at the expansion temperature, the higher the pressure on the melt must be to prevent it from expanding inside the die or extrusion equipment, a behavior called prefoaming. Of course, the amount of pressure required to keep the physical foaming agent in solution is also related to the solubility of the foaming agent in the melt, but the vapor pressure also has an impact. For example, in the production of LDPE foam using either propane or isobutane (foaming agents with similar solubilities in LDPE), the propane requires a higher die pressure to prevent prefoaming than does the butane. The liquid density of the physical foaming agent is important for storage considerations and for when a volumetric flow measurement device is used. The thermal conductivity of the foaming agent vapor (K-vapor) is important for thermal insulation applications. Polystyrene foam boards have been produced in the past using CFC 12 and methylene chloride. These materials in vapor phase inside the foam cells provide a structure with a much better thermal insulation capability than foam with cells containing air [10]. Even though the foaming agent vapors would escape over some years, the lower initial thermal conductivity of the PS insulation board was used as a selling point. Of course, after several years, the air replaced the foaming agent vapor, reducing the thermal insulating performance slightly. Currently, many thermal insulation applications are utilizing chlorodifluoroethane (HCFC 142b) under a special exemption from the UN Montreal protocol on halogenated hydrocarbons [11]. HCFC 142b offers some advantage over other materials, such as CO2 and alkane hydrocarbons in terms of thermal insulation performance, but this material is being phased out so processors will need to find suitable alternatives. In the 1970s, the scientific community determined that the CFCs were decomposing high in the Earth’s atmosphere and causing the ozone layer to become depleted. It was determined that this effect was basically due to the chlorine atoms in the molecule. Later, in the 1980s, programs were implemented to phase out the halogenated alkanes that contain chlorine atoms. Therefore, foam processors were required to seek alternative materials. The chemical structure of the physical foaming agent has effects on its performance. For example, isobutane is slightly less soluble in polymer melts and slightly less permeable in the polymer than n-butane because of the differences in molecular structure. Further, the presence of halogens substituted for hydrogen has a similar effect, reducing the solubility and permeability.

© 2000 by CRC Press LLC

10.3 CHEMICAL FOAMING AGENTS Chemical foaming agents (CFAs) have their history based in explosive and bakery products. Endothermic CFAs were originally used as nucleating agents for direct-gassed polystyrene foam production. Their use as effective CFAs was discovered some time later. CFAs are solid or liquid materials that decompose under certain conditions to generate vapors. It is the vapor material that then behaves like a physical foaming agent in the plastics process. Most CFAs are finely divided solid materials, and most decompose within a certain temperature range. The decomposition temperature of the CFA should be in the same range as the melt temperature of the polymer being processed. This ensures that the CFA does not prematurely decompose and that the decomposition is complete. The gas that is yielded as a result of the CFA decomposition has several effects on the process and product. Because carbon dioxide is more soluble in plastics and has a lower vapor pressure than nitrogen, it is generally easier to work with. CFAs that generate carbon dioxide generally give finer cells, lower densities, better surface appearance, and shorter cycle times than CFAs that generate nitrogen. However, for example, for plastics with high melt viscosities or for injection molding parts that are difficult to fill, nitrogen generating CFAs can be advantageous. In this case, the higher pressure generated by the nitrogen gives a more efficient and complete expansion that more effectively fills the mold. The gas yield, a measure of the volume of gas generated by a given mass, of the CFA is important in determining its relative efficiency compared with other grades or types. Most commonly used CFAs predominately generate carbon dioxide or predominately generate nitrogen. Please refer to Table 10.1 for detailed properties of nitrogen and carbon dioxide. As temperatures used for plastics processing are above the critical points of both gasses, specific volume can be used to compare the behavior of the two gasses in expansion of plastic foams. Nitrogen occupies more volume for the same mass of the material, and this fact partially illustrates why the expansion of nitrogen during the foaming of plastics is more explosive and difficult to control. Another factor that effects the efficiency or explosiveness of the expansion is the solubility of the gas in the polymer. The solubility of a gas in a molten polymer is governed by the following equation [12]. S⫽H*P Where S ⫽ solubility, H ⫽ Henry’s law constant, and P ⫽ gas pressure. The Henry’s law constant for each gas is different for each of the various polymers. For commonly used polymers, the Henry’s law constant for carbon dioxide is 1.5 to 4 times than that for nitrogen. For polyethylene, gas ⫽ H, (cc(STP)/gm atm), nitrogen ⫽ 0.111, and carbon dioxide ⫽ 0.275. Therefore, carbon diox-

© 2000 by CRC Press LLC

ide is about 2.5 times more soluble in polyethylene than nitrogen. This illustrates that it is necessary to keep more pressure on a melt that contains nitrogen than on a melt that contains carbon dioxide to keep the gas in solution. It also shows that, during expansion, nitrogen is going to come out of solution and expand the plastic melt faster, and coupled with the greater specific volume for nitrogen, the expansion of nitrogen will be more efficient in terms of volume per mass of gas but will be more difficult to control. Chemical foaming agents are generally divided into two categories, exothermic and endothermic. Exothermic CFAs generate thermal energy (heat) during their decomposition, and endothermic CFAs consume thermal energy during their decomposition. The fact that the CFA decomposition reaction generates or consumes heat generally has very little, if any, effect on the temperature of the polymer melt or the product. The main effect of the heat generation or absorption is manifested in the rate and temperature range of decomposition. Generally, once an exothermic CFA begins to decompose, it is difficult to stop it before it reaches full decomposition. This results in a faster decomposition in a narrow temperature range. Of course, when the exothermic CFA is dispersed at low levels (below about 5%) in the polymer, the reaction can be stopped by rapid cooling or insufficient temperature for processing. Endothermic CFAs, on the other hand, require additional heat to support their continuing decomposition. This results in a broader decomposition time and temperature range. The main foaming gas generated by the CFA has a great effect on its foaming and processing behavior. Considering the most commonly used CFAs, the endothermic CFAs generate carbon dioxide as the main foaming gas, and the exothermic CFAs generate nitrogen as the main foaming gas. Most CFAs generate other gasses, but they do not have as much effect on the processing and foaming behavior as the main foaming gas. Below are some of the common CFAs with some of their typical properties. 10.3.1 AZODICARBONAMIDE [13] Azodicarbonamide decomposes in a temperature range of about 205°C to 215°C. There are many materials that act as activators for the decomposition of the CFA, lowering its decomposition temperature range by up to about 40°C. Common activators include zinc oxide, zinc stearate, urea, and many tin or zinc-containing PVC stabilizers. The reaction of azodicarbonamide can leave residual materials, referred to as plateout, on die and mold tooling. Proper use of additives can minimize or alleviate this problem. Chemical formula:

© 2000 by CRC Press LLC

O O L L 2H2NCN ⫽ NCNH2

Gas composition:

65% 24% 5% 5%

Nitrogen Carbon monoxide Carbon dioxide Ammonia (NH3)

Solid decomposition products include Urazol, Biurea, Cyamelide, and Cyanuric acid. Gas yield: 220 cc/gm FDA Status: Regulation Number 175.300 (2.0% max.) 177.1210 (2.0% max.) 177.2600 (5.0% max.)

Use Resinous and polymeric coatings, (can end cements) Closure sealing gaskets in contact with food (repeated use) Rubber articles in contact with food

10.3.2 ACID/CARBONATE BLEND The most commonly used formulation for these endothermic CFAs is a blend of citric acid and sodium bicarbonate. The materials are encapsulated or coated in a material that prevents or inhibits the decomposition reaction and exposure to moisture. The materials are generally blended in stoichiometrically correct proportions. The ratio of the blend components can be modified to affect changes in the decomposition temperature range for the CFA. The equation for the decomposition can be generally written as follows: Citric acid ⫹ Sodium bicarbonate ⫽ Sodium citrate dihydrate ⫹ Carbon dioxide ⫹ Water C6H8O7 ⫹ 3NaHCO3 ⫽ (C6H5Na3O7) ⭈ 2(H2O) ⫹ 3(CO2) ⫹ H2O In practice, some of the sodium citrate is probably in the pentahydrate form, but some free water vapor is always generated. The reaction takes place in two temperature ranges, one at about 160°C and the second one at about 210°C. The gas yield is 120 cc/gm and the FDA status is that both components are generally recognized as safe (GRAS) for any food contact use. Many CFAs also generate other gas by-products aside from the main foaming gas, such as water. If the plastic being processed is sensitive to hydrolysis, decomposition by water, it is important to use either a CFA, which generates very little or no water in the decomposition, or one that can effectively bind the water chemically. Examples of hydrolytically unstable plastics include polycarbonate and polyesters.

© 2000 by CRC Press LLC

Table 10.2 lists commercially used CFAs. The proper choice of CFA not only depends upon the main foaming gas evolved but also on the decomposition temperature. Basically, in the processing equipment, the same sequence of steps must take place in order to ensure proper performance. First, the CFA is added to the process in the same fashion as other additives or colorants. The CFA, whether powder or pelletized masterbatch, can be blended with the polymer resin prior to addition to the hopper or it can be metered into the feed throat of the extruder/injection molder. The CFA additive must then be dispersed into the polymer as the polymer begins to melt. Once the polymer is melted and the CFA (and any other additives) is dispersed, sufficient melt temperature must be achieved to decompose the CFA. As the gas is evolving, the pressure on the melt must be sufficient to allow the gas to dissolve into the melt. Once the gas is formed and dissolved into the melt, the melt must be kept under sufficient pressure to keep the gas in solution. The pressure sufficient to keep the gas in solution depends on several factors, including the solubility of the gas in the polymer and the temperature of the solution. The gas should be kept in solution until the expansion of the melt is desired, generally upon exiting from the die or injection into the mold. If the melt is allowed to expand prior to this (inside the extruder barrel or injection molding machine barrel or nozzle), poor foam will result. Temperature control in several areas of the process is critical to good foam production using CFAs. If the temperature in the feed section of the process,

TABLE 10.2

Common Name Citric acid/ Sodium bicarbonate ADCA OBSH

TSH TSS DNPT 5PT SBH

Commercially Used Chemical Foaming Agents [13, 14].

Chemical Name

Azodicarbonate p,p’-Oxybis (benzene) sulfonyl hydrazide p-Toluene sulfonyl hydrazide p-Toluene solfonyl semicarbazide Dinitrosopentamethylenetetramine 5-Phenyltetrazole Sodium borohydride

Endo or Exo

Decomposition Temperature, °C

Gas Evolution, cc/gm

Main Foaming Gas Evolved

Endo

160–210

120

CO2

Exo Exo

205–212 158–160

220 125

N2 N2

Exo

110–120

115

N2

Exo

228–235

140

N2

Exo

190

190

N2

Exo Endo

240–250 *

220 2000

N2 H2

* SBH is chemically activated by exposure to water.

© 2000 by CRC Press LLC

the first or second zone, is too high, the CFA can decompose before the polymer has completely melted. In this case, the gas evolved can escape through the feed hopper and not dissolve in the melt. The temperature in the metering/mixing sections of the process should be high enough to ensure complete decomposition of the CFA. If either the decomposition of the CFA is carried out too early or not completely, the resultant product will be foamed inconsistently or not at all. Once the CFA is completely decomposed and the gas is dissolved in the melt, the melt temperature can be adjusted to suit the process so long as sufficient melt pressure is maintained to keep the gas in solution. Generally, the melt temperature is reduced, especially in extrusion processing, to increase the melt viscosity. The higher melt viscosity generally also increases the melt strength of the melt, thereby improving the foam quality and density reduction. Because of the temperature considerations described above, the CFA chosen for a particular polymer resin must be matched to the resin in terms of its decomposition temperature. The melting point and processing temperatures of a particular resin must be considered when choosing the CFA to ensure that the CFA does not predecompose or insufficiently decompose. For example, for LDPE processing that takes place at temperatures generally in the range of 150 to 180 °C, the CFA should have a decomposition temperature that is in that general range. If a CFA is used that decomposes at a higher temperature, such as 5PT, no decomposition of the CFA will be realized. CFAs are also used as high-performance nucleating agents for direct-gassed foam extrusion. The performance of endothermic CFAs for nucleation is better than so-called inactive nucleators, such as talc. Generally, endothermic CFAs are three to five times more effective than talc in nucleating direct-gassed foams, but can be up to eight times more efficient [15]. Endothermic CFAs (sodium bicarbonate and citric acid) are the most commonly utilized types for nucleation and are used over less expensive nucleators, such as talc, when finer-celled foam products are required. Exothermic CFAs (mainly azodicarbonamide) are used for nucleation of direct-gassed wire and cable insulation as the residual materials from the decomposition have the least effect on the electrical properties of the insulating material when considering all available nucleators. While the mechanism of the nucleation effect of CFAs is not entirely understood, it must be a combination effect of the residual material and the gas generated by the decomposition [16]. The future for foaming agents is quite exciting. The constant search for easy to use environmentally friendly physical foaming agents will continue. This endeavor will bring about improvements in processing equipment and polymer morphology/formulation as well as new and unique physical foaming agent materials/blends. The difficulties in using inert gasses, such as carbon dioxide or nitrogen, over the traditional physical foaming agents, like isobutane or CFCs, should be obvious from the text of this chapter.

© 2000 by CRC Press LLC

The desire to foam polymers other than polystyrene and low-density polyethylene such as polypropylene, polyethylene terapthalate, and thermoplastic elastomers continues to challenge processors to find new foaming agent alternatives. Chemical foaming agents are constantly being modified by reformulation or blending. The use of CFAs in the form of pelletized masterbatches in a polymer carrier is becoming more popular. The use of masterbatches improves the accuracy of dosing the additive to the process and also greatly improves their use in terms of handling and cleanliness. Foaming agents are a vital ingredient for the production of foamed plastics. The interaction between the foaming agent and the rest of the process can be quite complex, but when the proper choice of foaming agent is coupled with the right polymer, equipment, and processing parameters, unique and interesting products can be produced.

10.4 REFERENCES 1. Munters, C. G. and Tandberg, J. G., U. S. Patent 2,023,204. 2. Collins, F. H., “Controlled Density Polystyrene Foam Extrusion”, SPE 16th Annual Technical Conference, 1960. 3. Jacobs, W. A. and Collins, F. H., U. S. Patent 3,151,192. 4. Carlson, Jr., F. A., U. S. Patent 2,797,443. 5. Burt, J. G., “The Blowing Agent in Polystyrene Foam Sheet,” E. I. Du Pont de Nemours & Co., Inc., Technical Paper (publication and date unknown). 6. Kolosowski, P. A., U. S. Patent 5,424,016. 7. Pontiff, T. M. and Rapp, Joseph P., U. S. Patent 5,059,376. 8. Miyamoto, A., Akiyama, H. and Usuda, Y., U. S. Patent 3,808,380. 9. E. I. DuPont de Nemours & Co., Inc. (various product bulletins). 10. Levy, S., “Extruding Plastics Foam Insulation,” Plastics Machinery and Equipment, August, 1979. 11. Suh, K. W., U. S. Patent 4,916,166. 12. Throne, J. L., Thermoplastic Foams, Sherwood Publishers, Hinckley, OH, 1996. 13. Uniroyal Chemical Company, Inc., Product Bulletin: “Technology of Celogen Blowing Agents,” August, 1992. 14. Hamel, R. G. and Poulin, S. P., U. S. Patent 4,520,137. 15. Pontiff, T. M., “Nucleation in Direct Gassed PS Foam Using Chemical Foaming Agents,” Dubai Plast Pro ‘98, Maack Business Services, May 11–13, 1998. 16. Colombo, E. A., “Controlling the Properties of Extruded Polystyrene Foam Sheet,” Science and Technology of Polymer Processing, edited by N. P. Suh and N. Sung, The MIT Press, 1979.

© 2000 by CRC Press LLC

CHAPTER 11

Continuous Production of High-Density and Low-Density Microcellular Plastics in Extrusion CHUL B. PARK

11.1 INTRODUCTION

T

chapter describes a continuous extrusion process for the manufacture of high-density and low-density microcellular plastics. Microcellular plastics are foamed polymers characterized by a cell density greater than 109 cells/cm3 and a fully grown cell size on the order of 10 micrometers. The basic approach to the production of microcellular structures is to continuously form a polymer/gas solution, to nucleate a large number of bubbles using thermodynamic instability via a rapid pressure drop, to suppress cell coalescence by increasing the melt strength, and to induce a volume expansion to a desired expansion ratio. All the processing requirements and the approaches to achieve the required processing steps are described in detail. The experimental results obtained at various processing conditions are presented to elucidate the effect of each processing parameter on the cell morphology. With careful tailoring of the processing conditions, microcellular foamed plastics with a cell-population density higher than 109 cells/cm3 and a controlled volume expansion ratio in the range of 1.5 to 43 times for the high-density and low-density applications are obtained. Microcellular plastics are characterized by a cell density higher than 109 cells/cm3 and a cell size on the order of 10 micrometers. A typical fractograph of a microcellular plastic part is shown in Figure 11.1. The concept for microcellular plastics was created by Suh [1] in response to industrial need for reducing materials cost for certain polymer products without major compromise to mechanical properties. The central idea was to create a large number of bubbles, smaller than the preexisting flaws in a polymer. Typically, microcellular plastics exhibit high Charpy impact strength (i.e., up to a fivefold HIS

© 2000 by CRC Press LLC

FIGURE 11.1

Typical scanning electron micrograph of a microcellular plastic.

increase over unfoamed plastic [2–5]), high toughness (i.e., up to a fivefold increase over unfoamed plastic [5–7]), high fatigue life (i.e., up to a fivefold increase over unfoamed plastic [8]), high thermal stability [9], high light reflectability, low dielectric constant, and low thermal conductivity [10]. Because of these unique properties, a large number of innovative applications of microcellular plastics can be imagined. These include food packaging with reduced material costs, airplane and automotive parts with high strength-toweight ratio and acoustic dampening, sporting equipment with reduced weight and high energy absorption, insulative fibers/filaments for fabric, molecular grade filters for separation processes, light reflecting boards, surface modifiers for low friction, and biomedical materials. Conventional foams use a nucleating agent or a chemical blowing agent to nucleate gas bubbles. The quality of the foam produced depends on the amount and the distribution of these agents. A nonuniform distribution of the agents results in a foam that has a high concentration of gas bubbles or cells in agentrich areas and a low concentration in agent-poor areas. The density of cells is determined by the concentration of the foaming agent. The large bubbles have a lower internal pressure, and there is a steeper gas concentration gradient in the vicinity. Larger bubbles will then be energetically favored to grow at a

© 2000 by CRC Press LLC

faster rate than smaller bubbles because gas will preferentially diffuse toward the low-pressure area [11,12]. As a result, larger bubbles grow at the expense of smaller bubbles, and the bubble size distribution becomes highly nonuniform in the foam structure. Because the number and the size of the bubbles are determined by the concentration of the foaming agent, the uniformity of the cell structure and the cell density are limited by the method used to mix the agents and the polymer. In fact, it is rather hard to obtain a uniform cell structure with a high cell density in a conventional foam processing with a chemical or physical blowing agent. For conventional foams, a typical cell density is in the range of 102 to 106 cells/cm3, the cell size is larger than 100 micrometers, and the cell size distribution is very nonuniform. Over the past decade, substantial research and development have been conducted to gain knowledge about the physical phenomena governing microcellular processing of microcellular polymers. This knowledge has successfully led to the implementation of microcellular batch processes [13–25] and semicontinuous processes [26–28]. Some significant structure and property characterization studies have been carried out as well [2–9, 18, 19, 22–25, 29–35]. In order to overcome the high processing cost of microcellular batch and semicontinuous processing due to their long cycle times, continuous extrusion processes have also been developed [36–50]. This chapter briefly outlines the current progress of continuous microcellular processing and its future prospect.

11.2 PREVIOUS STUDIES ON BATCH AND SEMICONTINUOUS MICROCELLULAR PROCESSING Based on the concept of microcellular plastics [1], the first microcellular plastics technology in a form of batch processing was developed by Martini and Suh [13,14]. The basic approach of the developed batch microcellular processing was to saturate a plastic sheet with an inert gas in a high-pressure chamber and to induce thermodynamic instability by rapidly dropping the solubility of gas in the polymer. Martini performed the first experiments and analyzed theoretically the formation and growth of microcellular foams [13]. Waldman and Suh then investigated the mechanical properties and processing behavior of microcellular foams [29]. Process parameters were determined experimentally, and the strategy for process design was presented. A method of producing lightweight polyester composites was developed by Youn and Suh [15,16]. Investigations into modeling microcellular thermoplastic nucleation were made by Colton and Suh [30–33]. Their model explained the effect of various additives and processing conditions on the number of bubbles nucleated. Cha and Suh improved the batch microcellular processing by utilizing the supercritical state of CO2 to enhance its solubility and diffusivity in the

© 2000 by CRC Press LLC

plastic matrix [17,18]. As a result, much finer cell structures were obtained from various thermoplastics including PVC, PETG, PMMA, ABS, PC, TPX, LDPE, HDPE, etc. A similar approach was taken by Goel and Beckman to improve the foam structure of PMMA [19]. Colton and Suh, Baldwin et al., and Doroudiani et al. have developed special processing strategies to develop microcellular foam structures from various semicrystalline polymers [20–22]. Matuana et al. studied the microcellular foaming behavior of wood-fiber composites [23–25]. Efforts have also been made to develop semicontinuous microcellular processing with a view to realizing the microcellular processing concept to industrial production. Kumar and Suh developed a modified thermoforming microcellular process for producing microcellular foamed parts with a geometry [26, 27]. The basic approach was to decouple the shaping process from the foaming process. Kumar and Schirmer developed a solid-state microcellular batch process in which semicontinuous sheets of microcellular polymers are produced by saturating a roll of sheet with the aid of gas permeable material followed by foaming the roll of sheet as in the batch process [28].

11.3 BACKGROUND ON MICROCELLULAR PLASTICS PROCESSING Microcellular plastics have been produced using thermodynamic instability of a polymer/gas system to promote a large bubble density in the polymer matrix. Microcellular plastics processing involves the following four basic steps to utilize such thermodynamic instability: polymer/gas solution formation, microcell nucleation, suppression of cell coalescence, and cell growth. These steps are basic to microcellular processing and are applied to both batch and continuous manufacturing processes. Typically, formation of a polymer/gas solution is accomplished by dissolving an inert gas, such as carbon dioxide or nitrogen, into a polymer matrix under a high pressure. This creates a solution having a high gas concentration (typically 5 to 20 wt%) in the polymer matrix. Solution formation is governed by gas diffusion in the polymer. Diffusion processes are typically slow, resulting in long cycle times. The next phase of microcellular processing involves subjecting the polymer/gas solution to thermodynamic instability to nucleate microcells. This is achieved by lowering the solubility of gas in the solution by controlling the temperature and/or pressure. The system now seeks a state of a lower free energy that results in the clustering of gas molecules in the form of cell nuclei. Formation of cell nuclei provides a relatively small mean free distance for the

© 2000 by CRC Press LLC

gas molecules in the solution to diffuse through before reaching a cell nucleus (i.e., the gas phase). As the gas diffuses into the cells, the free energy of the system is lowered. The cell nucleation process can occur homogeneously throughout the material or heterogeneously at high energy regions such as phase boundaries [30–33]. In the ideal case, nucleation occurs instantaneously. The cell nucleation process is very important in microcellular plastics production in that it governs the cell morphology of material, and to a large extent, the properties of the material. Although a large number of nuclei have been achieved in the nucleation stage, the final cell density of produced foams might not be the same as the initial nuclei density if cell coalescence occurs. Cell coalescence is thermodynamically favored because the total surface area of cells is reduced by coalescence. When the cell density is deteriorated by cell coalescence, the mechanical and thermal properties are deteriorated as well. In order to prevent the deterioration of the properties and to fully utilize the unique properties of microcellular plastics, cell coalescence should be suppressed. When the cells are nucleated, they continue to grow as available gas diffuses into cells, provided little resistance is encountered. The cells grow and the total polymer density decreases as the gas molecules diffuse into the nucleated cells from the polymer matrix (a distance on the order of 10 microns). The rate at which the cells grow is limited by the diffusion rate and the stiffness of the viscoelastic polymer/gas solution. If the stiffness of the matrix is too high, cell growth is extremely slow. In this case, the solution temperature can be increased to lower the matrix stiffness. In general, the cell growth process is controlled primarily by the time allowed for the cells to grow, the temperature of the system, the state of supersaturation, the hydrostatic pressure or stress applied to the polymer matrix, and the viscoelastic properties of the polymer/ gas solution [42]. Microcellular plastics were produced first in a batch process [14]. Figure 11.2 shows a typical example of a batch process for producing microcellular plastics. In this process, a polymer sample is first placed in a high pressure chamber where the sample is saturated with CO2 or N2 under high pressure at ambient temperature. Then, a thermodynamic instability is induced by rapidly dropping the solubility of gas in the polymer. This is accomplished by releasing the pressure and heating the sample. This drives nucleation of a myriad of microcells, and the nucleated cells continue to grow, leading to the foam expansion. One of the disadvantages of the batch process is a very long cycle time required for gas saturation in the polymer because the diffusion rate is very low at the ambient temperature. In this context, the extrusion-based continuous microcellular process has been developed to reduce the gas saturation time, and this continuous microcellular processing is described in detail in the following sections.

© 2000 by CRC Press LLC

FIGURE 11.2 Typical batch process for microcellular foamed plastics. Reproduced with permission from Reference [2]. Copyright (c) 1998 Polymer Engineering and Science.

11.4 FORMATION OF A SINGLE-PHASE POLYMER/GAS SOLUTION One of the critical steps in the continuous production of microcellular plastics is formation of a polymer/gas solution at industrial processing rates [38, 39]. In order to design a solution formation device, the physical phenomena behind the solution formation process should be analyzed, and the critical process parameters of solution formation should be identified. Figure 11.3 shows the morphology change of polymer and gas system in the solution formation process. Initially, a soluble amount of gas is injected into a polymer melt stream, forming a two-phase polymer/gas mixture. Then, the large injected gas bubbles are broken into smaller bubbles and stretched through shear mixing. Eventually, the gas diffuses into the polymer matrix, forming a singlephase solution. 11.4.1 ESTIMATION OF GAS SOLUBILITY IN POLYMERS AT ELEVATED TEMPERATURES AND PRESSURES Only a soluble amount of gas should be injected into the polymer melt stream. Excess gas would result in formation of undesirable voids in the melt. Such voids could be detrimental to the cell structure unless hollow cores are intentionally formed in the final product [51]. The existence of voids sup-

© 2000 by CRC Press LLC

FIGURE 11.3 Schematic of the morphology change of polymer/gas systems in the solution formation process. Reproduced with permission from Reference [39]. Copyright (c) 1996 Polymer Engineering and Science.

presses homogeneous nucleation because the gas molecules preferentially diffuse to larger cells [12,13], resulting in formation of hollow cavities in the final product. In order to inject only a soluble amount of gas into the polymer matrix, the solubilities of CO2 in various polymers were estimated at 200°C and 27.6 MPa (4,000 psi), which are a typical processing temperature and pressure, respectively. Shim and Johnston’s work [52, 53] suggested that the logarithm of the CO2 solubility in polymers at elevated pressures is well correlated to the density of CO2 up to 30 MPa for a constant temperature. Durrill and Griskey’s data [54, 55] was used to derive this correlation, and the solubilities of CO2 at 27.6 MPa (4,000 psi) were extrapolated from this correlation. The estimated solubilities are summarized in Table 11.1. The solubilities of CO2 in many polymers are approximately 10 wt%.

TABLE 11.1

Estimated Solubility of CO2 in Polymers at 200°C and 27.6 MPa (4,000 psi). Polymer

CO2 weight gain (%)

PE PP PS PMMA

15 12 12 15

Source: Reproduced with Permission from Reference [39]. Copyright (c) 1996 Polymer Engineering and Science.

© 2000 by CRC Press LLC

11.4.2 ESTIMATION OF GAS DIFFUSIVITY IN POLYMERS AT ELEVATED TEMPERATURES The diffusivity of gas in a polymer was also investigated to determine the processing time required for formation of a single-phase polymer/gas solution. In general, the gas diffusivity in a polymer changes with temperature, pressure, and gas concentration [54-61] and can be approximated as follows: D ⫽ Do exp (⫺⌬ED / R T)

(1)

Since the diffusivity increases as the temperature increases, the rate of gas diffusion is enhanced by processing the mixture at a high temperature. Therefore, compared to room temperature, the gas diffusion rate for the polymer mixture is increased in the heated extrusion barrel. Only limited data is available for gas diffusion in polymers at high temperatures [54–56]. The estimated diffusivities are summarized in Table 11.2. At 200°C, a typical diffusivity of CO2 and N2 in a thermoplastic is approximately 10⫺6 cm2/sec, which is two orders of magnitude greater than a typical diffusivity of 10⫺8 cm2/s at room temperature [56]. 11.4.3 BASIC CONCEPT OF CONVECTIVE DIFFUSION: MIXING AND DIFFUSION When a metered amount of gas is delivered to the polymer melt stream, formation of a uniform and homogeneous single-phase solution from the twophase mixture can be accomplished through gas diffusion. Because gas diffusion is a very slow process, a technique for rapid solution formation has been TABLE 11.2

Polymer

PS PP PET HDPE LDPE PTFE PVC

Estimated Diffusivity of CO2 in Polymers at Elevated Temperatures. D of CO2 (cm2/s) at 188°C*

at 200°C†

— 4.2 ⫻ 10⫺5 — 5.7 ⫻ 10⫺5 — — —

1.3 ⫻ 10⫺5 2.6 ⫻ 10⫺6 2.4 ⫻ 10⫺5 1.1 ⫻ 10⫺4 7.0 ⫻ 10⫺6 3.8 ⫻ 10⫺5

Source: Reproduced with Permission from Reference [39]. Copyright (c) 1996 Polymer Engineering and Science. *Durril and Griskey [54, 55] †van Krevelen [56]

© 2000 by CRC Press LLC

developed in order to produce microcellular plastics at industrial processing rates [38, 39]. The basic strategy for rapid solution formation is to induce convective diffusion. Mass transfer by molecular diffusion is analogous to heat transfer because the heat conduction and diffusion equations have the same form [62]. The heat transfer rate is enhanced by convection. A convective flow causes fluid particles of lower (or higher) temperature to be brought into contact with the heat source, resulting in a higher temperature gradient near the source. The heat transfer rate is promoted by the higher temperature gradient. Similarly, the diffusion rate can also be enhanced by convection. Convection brings low gas concentration polymer into contact with high gas concentration bubbles. This convective flow induces a high concentration gradient that promotes diffusion. When the diffusion source is stationary and exhibits a simple shape, such as a flat plate as shown in Figure 11.4, the concentration profile is similar to the temperature profile associated with a similar heat source. Therefore, the concentration profile may be expressed from known heat transfer solutions with an appropriate change in notation. Typical examples are the concentration boundary layer on a flat plate, the concentration profile in a flow between the parallel plates, and the concentration profile in a flow through a circular pipe. The diffusion rates are promoted by a forced convective flow. When the diffusion source is also moving, the analysis of the problem becomes complicated. In investigating the local concentration profile, the first step is to trace the shape of the source boundary. Since the diffusion sources in this study are the gas bubbles that are moving along the polymer melt, the dynamic behavior of the gas bubbles and the polymer melt should be investigated first. The diffusion phenomena will then be analyzed based on the mixing behavior of the gas bubbles and the polymer melt.

FIGURE 11.4 Diffusion boundary layer on a flat plate. Reproduced with permission from Reference [39]. Copyright (c) 1996 Polymer Engineering and Science.

© 2000 by CRC Press LLC

As the gas molecules diffuse into the polymer matrix, the total volume of the gaseous phase diminishes in size until the gas completely dissolves into the matrix. Since the size of one phase component is changing, the mixing behavior of the two-phase mixture is far more complicated than simple mixing. It is very difficult to investigate the mixing behavior of the two-phase mixture and the flow fields of each phase. Since the diffusion phenomenon strongly depends on mixing behavior, the analysis of the mass transfer phenomena of diffusion is also complicated. Therefore, the diffusion of gas into the polymer matrix may not be completely analyzed. Despite the complication of modeling the dynamic behavior of the mixture of the two fluids, the diffusion rate is greatly enhanced when the diffusion source is also moving. As the degree of mixing increases, more polymer melt is brought into contact with the source of the high gas concentration that increases the effect of convective diffusion. This convective diffusion effect is enhanced through an increase in the interfacial area per unit volume, a reduction of the diffusion distance, and a redistribution of the local gas concentration profile in the polymer matrix [38, 39]. In addition, since the diffusion rate strongly depends on the mixing behavior, the diffusion time can be controlled by varying the degree of mixing. 11.4.4 ANALYSIS OF A CONVECTIVE DIFFUSION PROCESS The concept of convective diffusion can be effectively utilized to enhance the diffusion rate in an extrusion barrel [38, 39]. One technique for rapid solution formation using convective diffusion employs laminar mixing in the molten polymer shear field. Since the mixing accomplished by the simple screw motion is limited, efforts were also made to enhance the mixing effectiveness by introducing various mixing sections in the extruder [63, 64]. The idea behind the mixing section is that reorientation of the mixture during processing will enhance the effectiveness of shear mixing. Using the mixing section, the diffusion time would decrease due to the enhanced degree of mixing. Figure 11.5 shows a schematic of the convective diffusion device developed by Park [36]. The shearing action in the extruder draws small bubbles of gas into the molten polymer shear field. The mixing action of the shear field slowly disperses a source of high gas concentration (i.e., the gas bubbles) into the polymer matrix. The gas eventually diffuses into the polymer, forming a single-phase solution. The convective diffusion effect in the shear field generated by the screw motion is well described for devolatilization of polymer solution in a twin-screw extruder [65, 66]. Deformation and movement of the concentration source and diminishment of the total volume of the gaseous phase component cause it to be very difficult to analyze the mixing behavior of the two-phase mixture and the diffusion phenomena. However, we can still es-

© 2000 by CRC Press LLC

FIGURE 11.5 Polymer/gas solution formation in an extrusion barrel. Reproduced with permission from Reference [46]. Copyright (c) 1997 American CHemical Society.

timate the diffusion time for completing solution formation based on the estimated striation thickness of the gas and polymer mixture. As an order of magnitude, the diffusion distance (lD) is estimated as lD ⬇ 2D tD

(2)

where tD is the diffusion time. The time at which the diffusion distance is of the same order as the striation thickness (s) of the mixture can be estimated as the diffusion time: tD ⬇

l 2D s2 ⬇ D D

(3)

The estimated diffusion times are shown in Table 11.3 for various striation thicknesses and diffusivities. For example, if the striation thickness is less than 100 ␮m, the diffusion of gas would be completed within 2 minutes for a typical diffusivity of 10⫺6 cm2/s at 200°C. A fundamental study has been carried out to estimate the striation thickness of polymer and gas in an extruder. An order of magnitude analysis [38] predicts that when the polymer/gas system is fully mixed, the striation thickness of the mixture would be 45 ␮m. Based on this striation thickness, the required diffusion time is estimated to be tD ⬇

© 2000 by CRC Press LLC

s2 (45 ⫻ 10⫺4)2 ⬇ ⫽ 20 s D 1 ⫻ 10⫺6

(4)

TABLE 11.3

Estimated Diffusion Time at Various Striation Thicknesses and Diffusivities.

Striation Thicknesses (s)

Diffusivity (D) 10

1 ␮m 10 ␮m 50 ␮m 100 ␮m 250 ␮m 500 ␮m 750 ␮m 1000 ␮m

⫺5

2

cm /sec

1 ⫻ 10⫺3 sec 0.1 sec 2.5 sec 10 sec 63 sec 4 min 9 min 17 min

10

⫺6

2

cm /sec

0.01 sec 1 sec 25 sec 100 sec 10 min 42 min 94 min 3 hrs

10⫺7cm2/sec

10⫺8cm2/sec

0.1 sec 10 sec 4 min 17 min 2 hrs 7 hrs 16 hrs 28 hrs

1 sec 100 sec 42 min 3 hrs 17 hrs 3 days 7 days 12 days

Source: Reproduced with Permission from Reference [39]. Copyright (c) 1996 Polymer Engineering and Science.

This indicates that formation of a single-phase solution out of the two-phase polymer/gas mixture would be completed in 20 seconds. Therefore, continuous solution formation can be achieved in extrusion systems without substantially decreasing the processing rates. However, it should be noted that if the amount of gas injected is more than the solubility at the processing condition, the solution formation could not be completed. In addition, due to the limited available CO2 diffusivity data for high concentration at elevated temperatures, the concentration-dependent nature of the diffusivity could not be accounted for in this order of magnitude analysis. In order to develop a better model of continuous formation of a single-phase polymer/gas solution, fundamental research on the diffusivity and the solubility of a gas at high temperatures and pressures and the polymer/gas mixing behavior in the mixing elements should be carried out.

11.5 MICROCELLULAR NUCLEATION CONTROL The next critical step in the continuous production of microcellular plastics is promotion of high bubble nucleation rates in the polymer/gas solution. Nucleation of bubbles is transformation of small clusters of gas molecules into energetically stable pockets of molecules with distinct walls. The microcellular process requires that the nuclei density be larger than 109 cells/cm3 so that the fully grown cell size will be less than 10 ␮m. The key to producing the required cell density is inducing a very high rate of cell nucleation in the polymer/gas solution [40]. High nucleation rates have been achieved in batch processes by using thermodynamic instability of the gas and polymer system. In order to make use of

© 2000 by CRC Press LLC

FIGURE 11.6 cessing.

Solubility change of polymer and gas systems in batch and continuous pro-

thermodynamic instability in a continuous process, a rapid drop in the gas solubility as in the batch process must be induced in the polymer/gas solution. The solubility of gas in a polymer changes with pressure and temperature [52–56]. Therefore, thermodynamic instability can be induced by rapidly varying the pressure, temperature, or both. Since, in the typical range of interest, the solubility of gas in a polymer decreases as the pressure decreases, a high cell nucleation rate can be promoted by subjecting the polymer/gas solution to a rapid pressure drop. The solubility drop due to a change of pressure or temperature is illustrated in Figure 11.6. The greatest possible number of cells for a given pressure difference would be nucleated out of a given polymer/gas solution if the pressure drops instantaneously. However, in reality, the pressure drops over a finite time period as shown in Figure 11.7. It is expected that the more rapidly the pressure drops, the greater the number of cells that would be nucleated, because greater thermodynamic instability would be induced. A rapid pressure drop element consisting of a nozzle (shown in Figure 11.8) has been utilized in Park et al.’s studies to demonstrate the effect of the pressure drop rate on cell nucleation [39, 40]. When a viscous polymer/gas solution passes through the long, narrow nozzle, the pressure drops almost linearly with distance due to friction. In order to be able to maximize the pressure drop rate and, therefore, to induce the greatest thermodynamic instability, the pressure drop and the pressure drop rate are analyzed in an order of magnitude analysis. To compensate for the

© 2000 by CRC Press LLC

FIGURE 11.7 Pressure drop profile in a rapid pressure drop device. Reproduced with permission from Reference [40]. Copyright (c) 1995 Polymer Engineering and Science.

error involved in the theoretical model, an experimental calibration has been performed. 11.5.1 EFFECT OF THE PRESSURE DROP RATE ON NUCLEATION The homogeneous nucleation theory [31] predicts that the cell nucleation rate is given by Nhom ⫽ fo Co exp (⫺⌬Ghom/kT)

(5)

⌬Ghom ⫽ 16␲ ␴3bp /3 ⌬P2

(6)

where Nhom is the homogeneous nucleation rate and ⌬P is the pressure drop of the gas/polymer solution. Equations (5) and (6) predict that, for a larger pressure drop, the cell nucleation rate will increase. If an instantaneous pressure drop and instantaneous homogeneous nucleation are assumed, then the nucle-

FIGURE 11.8

Typical pressure drop element for microcellular nucleation.

© 2000 by CRC Press LLC

ation rate and the number of nucleated cells correspond. Thus, for a constant pressure drop with an instantaneous pressure drop rate, the cell density should be constant. However, in reality, the pressure drop is not instantaneous but happens over a finite time period. It is expected that the nucleation time period is affected by the time period over which thermodynamic instability is induced in the system, and that the pressure drop rate will affect the nucleation time period, and therefore, will affect the nucleation rate. To explain the pressure rate effect, one must consider microcell nucleation and growth of the cells with respect to the competition between these mechanisms [36, 40]. The basic concept is the following. During the course of the pressure drop, which instigates thermodynamic instability, some stable cells nucleate early during the residence time of the polymer in the nozzle. The gas in solution will diffuse to the nucleated cells to lower the free energy of the system. As the gas diffuses to these cells, low gas concentration regions where nucleation cannot occur are generated adjacent to the stable nuclei as shown in Figure 11.9. As the solution pressure drops further, the system will nucleate

FIGURE 11.9 Competition between nucleation of new cells and growth of existing cells. Reproduced with permission from Reference [40]. Copyright (c) 1995 Polymer Engineering and Science.

© 2000 by CRC Press LLC

additional microcells and expand the existing cells by gas diffusion or only expand the existing cells. To determine whether further microcells are nucleated, one must look closely at the depleted gas regions around the previously nucleated cells [13]. If the depleted regions of adjacent cells impinge upon each other, then no further nucleation will tend to occur. This follows since the gas preferentially diffuses to the existing cells, depleting the gas between cells. In this state, the gas concentration between the existing cells is below the critical level needed to nucleate additional cells. However, if the size of the depleted gas regions is less than the dimension between existing cells, then additional cells will tend to nucleate between existing cells. Here, the effect of the pressure drop rate on the competition of nucleation and cell growth is considered. Consider the pressure drop profiles for Nozzle i and Nozzle j as shown in Figure 11.10. At time t1, where to-t1 is an arbitrarily small time period, the pressure drop for Nozzle j, ⌬P1j, is larger than the pressure drop for Nozzle i, ⌬P1i. Since the nucleation rate is inversely sensitive to ⌬P according to Equations (5) and (6), the nucleation rate in Nozzle j is higher than the nucleation rate in Nozzle i as illustrated in Figure 11.11 (a) and (b). At the next time step, t2, the previously nucleated cells have grown, thus, reducing the available gas for nucleating additional cells. In the region of the polymer/gas solution where the gas has not been depleted, the nucleation rates

FIGURE 11.10 Comparison of pressure drop between nozzle j (narrow and short) and nozzle i (wide and long).

© 2000 by CRC Press LLC

FIGURE 11.11 Nucleation of new cells and growth of the depleted zones in Nozzle i and Nozzle j at each time step [36].

© 2000 by CRC Press LLC

for Nozzle i and Nozzle j increase exponentially. However, for any given time, the nucleation rate for Nozzle j is higher than that for Nozzle i due to the additional pressure drop experienced by Nozzle j. This is illustrated in Figure 11.11 (c) and (d). Since the nucleation rates for Nozzle j are much higher than for Nozzle i, the total nucleation time for Nozzle j will be less than for Nozzle i. This follows since the number of cells nucleated increases exponentially as the sizes of the depleted gas regions increase with time. Thus, the depleted zones impinge upon one another more rapidly for Nozzle j than Nozzle i as illustrated in Figure 11.11 (e) and (f). For Nozzle i, the depleted zones impinge upon one another at a much later time. Since nucleation for Nozzle i takes place over a long time period, the depleted zones of the previously nucleated cells must have grown significantly by the time the nucleation stops. Therefore, some of the gas that could have been used for additional nucleation was used for cell growth for Nozzle i. In other words, more gas was used for cell growth in Nozzle i. Since in Nozzle j, nucleation took place over a short time period with high nucleation rates, more gas was used for nucleation of cells. Therefore, the total number of nucleated cells should be higher for Nozzle j than Nozzle i. This competition between microcell nucleation and cell growth for gas molecules is determined by the rate of the pressure drop and results in the different total numbers of nucleated cells even from an identical polymer/gas solution [40]. Based on this concept, the pressure drop rate of a polymer melt flowing in a filamentary die is analyzed. The effect of the pressure drop rate on the cell density has been examined experimentally (see Section 11.9.1). 11.5.2 ANALYSIS OF THE PRESSURE DROP RATE IN A FILAMENTARY DIE The pressure change of flowing, non-Newtonian fluids in a nozzle is briefly analyzed in this section [40]. Assuming the viscosity of the polymer/gas solution is shear-rate dependent and described by a “power law” [67, 68], the pressure drop over the length of the nozzle for a non-Newtonian fluid can be expressed as follows [69]: 1 n q a3 ⫹ b n § 2mL £ ⫺⌬p ⫽ 3 ro ␲ ro

(7)

For a set of filamentary dies that satisfy the following: L r3n⫹1 o

© 2000 by CRC Press LLC

⫽

n ⫺⌬p ␲ c d ⫽ constant 2m q(a⫹3)

(8)

The relationship between the pressure drop and flow rate would be the same. In other words, for a given flow rate, the pressure drop across the dies that satisfy the relationship of Equation (8) would be the same. Theoretically, all of these nozzles can be replaced with each other without affecting the pre- and post-flow conditions, i.e., upstream/downstream pressures and flow rates. It may be noted that the pressure change at the sudden contraction [70] is not considered in this analysis; however, this effect is incorporated in the experimental calibration [40]. Now, the pressure drop rate in the die can be derived. The average residence time, ⌬t, of the flowing polymer/gas solution in the nozzle is expressed as follows: ⌬t ⬇

L ␲r2oL L ⫽ ⫽ vavg q q ␲r2o

(9)

where vavg is the average velocity of the polymer/gas solution in the nozzle. Using this residence time, the average pressure drop rate is estimated as follows: ⫺⌬p ⫺⌬p q ⫺dp ⬇ ⬇ dt ⌬t ␲r2oL

(10)

Since the gas solubility is approximately proportional to the pressure [52–56], the solubility drop rate can also be derived in a similar manner: dCs ⌬Csq ⬇ 2 dt ␲roL

(11)

where ⌬Cs is the change of the gas solubility in the polymer melt. Park et al. calculated the pressure and solubility drop rates for impact grade polystyrene (Novacor/Monsanto 3350 HIPS) at 221°C (430°F) flowing in three different nozzles for a pressure difference of 38.64 MPa (5,600 psi) as a typical application [40]. Table 11.4 summarizes the geometry of the three nozzles and the estimated residence time, pressure drop rate, and solubility drop rate for each nozzle at a given pressure difference. Although the length of nozzle was experimentally calibrated to compensate for the errors involved in the analysis, the calibrated values of nozzle length, residence time, pressure drop rate, and solubility drop rate were very close to the estimated ones. Since there was approximately an order of magnitude difference in the residence times between each of the nozzles, there was about an order of magnitude difference in the pressure drop rates between each nozzle, while the total pressure

© 2000 by CRC Press LLC

Estimated Geometry, Pressure Drop Rate and Solubility Drop Rate of the Nozzles for a Pressure Drop of 38.64 MPa (5,600 psi).

TABLE 11.4

Pressure Drop Rate

Solubility Drop Rate

0.666 s

0.058GPa/s

0.21 s⫺1

1.28 ⫻ 10⫺7 m3/s

0.136 s

0.284 GPa/s

1.03 s⫺1

1.28 ⫻ 10⫺7 m3/s

0.016 s

2.355 GPa/s

8.75 s⫺1

Radius

Length

Flow Rate

Nozzle 1

0.60 mm (0.024 in)

78.74 mm (3.100 in)

1.28 ⫻ 10⫺7 m3/s

0.39 mm (0.016 in)

35.75 mm (1.407 in)

0.23 mm (0.009 in)

12.75 mm (0.502 in)

§£

Nozzle

dc

Nozzle 2

DC

Nozzle 3

Residence Time

Source: Reproduced with Permission from Reference [40]. Copyright (c) 1995 Polymer Engineering and Science.

© 2000 by CRC Press LLC

drops across the length of the three nozzles were the same. Similarly, there was approximately an order of magnitude difference in the solubility drop rates between each nozzle. Since the solubility drop rates were all different, the induced thermodynamic instability for each nozzle was different. Although the solubility drops of the polymer/gas solution across the three nozzles were the same, the largest thermodynamic instability would result when the solubility drops most rapidly. Since the third nozzle with the smallest radius and the smallest length has the highest solubility drop rate, it should be the most effective microcellular nucleation element among the three nozzles. This argument was also examined through experiments. According to Equations (10) and (11), higher pressure and solubility drop rates are achieved when the nozzle radius is smaller. Theoretically, when the nozzle radius becomes zero, the pressure and solubility drop rates become infinitely large. However, the radius reduction is limited by the manufacturability of the hole and the mechanical strength of the nozzle. According to Equation (8), for the same flow rate and pressure drop, the nozzle length decreases with the radius. The nozzle length cannot be arbitrarily reduced because the nozzle must have the mechanical strength to withstand the processing pressure of extrusion.

11.6 SUPPRESSION OF CELL COALESCENCE Following Park et al.’s early stage research on microcellular nucleation in the continuous processing of microcellular plastics [36, 38–40], Baldwin et al. [41, 42] carried out a preliminary study on the cell growth control in microcellular extrusion processing. Because of the difficulty of inducing a rapid pressure drop for microcell nucleation at the die exit, it was proposed that nucleation of the microcells be controlled independently in the first stage of the die by rapidly dropping the pressure of the polymer/gas solution using a nozzle and controlling shaping and cell growth in the second stage of the die. Baldwin et al. proposed to use a high pressure to suppress premature growth in the shaping die and to prevent the bubbles from being stretched along in the shaping direction. Based on this strategy, they demonstrated the feasibility of the concept of shaping a nucleated polymer/gas solution under high pressure to prevent distortion of the bubbles and successfully produced 2 mm thick filament and 1 mm thick sheet with a microcellular foamed structure. However, the volume expansion ratios of the extruded foams were relatively low (less than two times) and coalescence of nucleated cells was observed to a large extent. It was expected that when the cells were controlled to be further grown, the cell coalescence problem would be more severe and a lower cell-population density would be obtained.

© 2000 by CRC Press LLC

In this context, the next critical step in microcellular extrusion processing is suppression of cell coalescence during cell growth. Although a large number of cell nuclei have been achieved by a high pressure drop rate, this does not guarantee that the final cell density of produced foams will be high enough to be microcellular. If the cells coalesce during cell growth, the initial cell density will be deteriorated. As nucleated cells grow, adjacent cells will touch each other. These contiguous cells tend to coalesce because the total free energy is lowered by reducing the surface area of cells via cell coalescence [11]. It may be noted that the shear field generated during the shaping process tends to stretch nucleated bubbles, and this will further accelerate cell coalescence [46]. When the cell density is deteriorated, the mechanical and thermal properties are deteriorated as well. In order to prevent deterioration of the properties and to fully utilize the unique properties of microcellular plastics, cell coalescence should be suppressed. Although Baldwin et al. attempted to prevent cell coalescence in the die by pressurizing the nucleated polymer solution during shaping, the extruded foam structure showed that many adjacent cells were coalesced and the cell density was deteriorated [41]. Maintaining a high pressure in the shaping section to prevent premature cell growth right after the cell nucleation was believed to be a good strategy because the nucleated cells could not grow under high pressure. However, considering the difficulty of maintaining a high back pressure in the shaping die for the case of a large cross section of extruded foam, it may not be realistic to prevent cell coalescence by controlling the pressure alone in the shaping die. Park et al. proposed a means for suppressing cell coalescence by increasing the melt strength of polymer via temperature control in microcellular extrusion processing [46]. The melt strength, by definition, may be considered a degree of resistance to the extensional flow of the cell wall during the drainage of polymer in the cell wall when volume expansion takes place. Therefore, the cell wall stability will increase as the melt strength increases [71]. It is known that the melt strength of polymer can be enhanced by branching, cross-linking, temperature reduction, control of molecular weight and molecular weight distribution, and blending of polymers and compatibilization of blends [72]. In the low-density microcellular extrusion process for PS and HIPS, it was proposed that the processing temperature be controlled to increase the melt strength and to prevent cell coalescence in the cell growth process. Park et al. demonstrated that the increased melt strength due to the lowered melt temperature effectively suppressed coalescence of cells, and the high cell nucleation densities obtained in the microcellular nucleation device were successfully maintained. The idea of increasing the melt strength to prevent cell coalescence at the die exit has been well utilized and practiced in conventional foam processing, particularly in the low-density foam production [71, 73]. However, temperature control for suppressing cell coalescence was not effectively used in the early stage of extrusion microcellular processing.

© 2000 by CRC Press LLC

FIGURE 11.12 Schematic of a designed heat exchanger for cooling the polymer melt. Reproduced with permission from Reference [47]. Copyright (c) 1998 Polymer Engineering and Science.

Figure 11.12 shows a schematic of the heat exchanger used to cool the flowing polymer melt. To achieve uniform cooling, static mixers that effectively promote convective heat transfer, were used. The temperature of the polymer melt flowing out of the exit of the heat exchanger was monitored using a thermocouple mounted at the die as shown in Figure 11.12. To regulate the temperature of the polymer melt, a cooling oil was circulated through the channel machined on the heat exchanger.

11.7 PROMOTION OF LARGE VOLUME EXPANSION The next critical step in the continuous production of microcellular plastics is to control volume expansion of extruded foams to a desired ratio. Because CO2 easily escapes through the exterior skin of foam during expansion, special attention needs to be paid to achieve a low-density microcellular foam of a high-volume expansion ratio in the extrusion foam processing. It may be noted that the diffusion rate of a gaseous blowing agent such as CO2 is much higher due to its smaller molecular size compared to the conventional long-chain blowing agents such as pentane or butane [74]. This section briefly describes how to prevent loss of CO2 that occurs during the microcellular foam extrusion for achieving a large volume expansion ratio [47]. 11.7.1 PREVIOUS STUDIES OF LOW-DENSITY EXTRUSION FOAM PROCESSING USING AN INERT GAS There are a number of well-known extrusion processes for production of low-density plastic foams using long-chain blowing agents such as CFCs, pentane, butane, etc. [74]. However, there are very few studies on use of an inert gas for low-density foam production. Considering the fact that conventional blowing agents are known to be environmentally hazardous, inert gases

© 2000 by CRC Press LLC

have been considered to be alternatives whether used alone or used together with the conventional agents [75–83]. Jacob and Dey [79] employed inert gases such as CO2, N2, and Ar as a blowing agent in extrusion processing of low-density PS foams. They produced low-density foams of 50 kg/m3 using CO2, which is equivalent to 20fold volume expansion. Using N2 and Ar, higher foam densities were obtained. For these low-density foams, the cell size was large on the order of 300 ␮m. Dey et al. [80] also used an inert gas for PP foam processing in extrusion. They produced a medium-density (about 300 kg/m3) foam PP sheet using CO2 as a blowing agent and SAFOAM FP (Reedy International) and talc as a nucleating agent. Shimano et al. [81] demonstrated the use of a gaseous blowing agent to develop a foamed thermoplastic resin article. They described a means to properly inject a constant quantity of gas into the extruder barrel, since one of the difficulties in injecting gas is variation of the gas flow rate due to fluctuation of the barrel pressure. Johnson et al. [82] argued that use of CO2 with pentane is advantageous because the amount of the hazardous and costly pentane blowing agent required in the process is reduced. Also, the aging process could be simplified and the emission of organic blowing agent into the atmosphere could be reduced. Lee [83] claimed that the mixture of CO2 and isobutane could be used to produce low-density thermoplastic foams in extrusion. The achieved foam densities were as low as about 20 kg/m3 in polyolefin foam products. 11.7.2 STRATEGY FOR CONTROL OF VOLUME EXPANSION As the thickness of cell walls decreases in low-density foam production, the rate at which gas diffuses out of the foam to the environment increases. Furthermore, the high diffusivity of CO2 at an elevated temperature can increase the blowing agent diffusion rate during expansion. It should be noted that the blowing agent that has diffused into the nucleated cells eventually tends to diffuse out to the atmosphere because complete separation of two phases is thermodynamically more favorable [84]. Gas escape through the thin walls will decrease the amount of gas available for growth of cells. As a result, if the cells do not freeze, they tend to collapse causing foam contraction. In the microcellular foam extrusion process, this volume contraction is due to gas loss after an initial volume expansion has been observed [47, 48]. As a consequence, the final product had a high foam density. In order to produce lowdensity foams, gas diffusion through the skin of the foam must be prevented. One way of preventing gas escape from the foam is to freeze the skin of the extrudate [47]. Since the diffusivity drops as the temperature decreases [54–61], the rate of gas escape can be substantially reduced by freezing the skin of the foam. The surface of the extrudate can be quenched by lowering the die temperature. Therefore, the basic strategy that has been taken for promot-

© 2000 by CRC Press LLC

ing large volume expansion is to freeze the foam skin by controlling the die temperature. The die temperature can be precisely controlled by circulating a cooling air or a low-temperature oil that is connected to a temperaturecontrolled oil bath. It should be noted that the temperature of the polymer melt flowing into the die also affects the amount of CO2 that escapes to the environment because the diffusion rate of CO2 in the cell walls can be retarded by lowering the temperature of polymer melt [47]. Furthermore, the increased stiffness of cell walls caused by decreasing the melt temperature will also prevent the contraction of the foam structure due to gas loss. Therefore, the heat exchanger designed to cool the polymer melt for suppression of cell coalescence will also be helpful in maintaining high-volume expansion ratios. The effects of the polymer melt temperature on cell coalescence and foam contraction are presented in detail in Section 11.9.3.

11.8 EXPERIMENTAL SET-UP Based on the proposed processing steps described above, an experimental extrusion setup has been constructed to produce high-density and low-density microcellular foams. A schematic of the equipment is shown in Figure 11.13. It consisted of a 5 hp DC motor, a speed reduction gearbox, a 3⁄4⬙ extruder (C.W. Brabender 05-25-000), and a mixing screw (Brabender 05-00-144). The L/D

FIGURE 11.13 Schematic of the overall experimental equpiment. Reproduced with permission from Reference [47]. Copyright (c) 1998 Polymer Engineering and Science.

© 2000 by CRC Press LLC

ratio and the compression ratio of the screw were 30:1 and 3:1, respectively. The other systems include a positive displacement pump for CO2 injection, a diffusion-enhancing device containing static mixers (Omega Model FMX8441S), a heat exchanger for cooling the polymer melt that contains homogenizing static mixers (Labcore Model H-04669-12), a cooling sleeve for precise control of the die temperature, and two hot-oil circulating baths (Grant Model W6-KD). Heat transfer oil (Labcore Model H-01294-40) was used in the heat exchanger to control the temperature of the flowing polymer melt and in the cooling sleeve to cool the die temperature. Three pressure transducers (Dynisco PT462E-10M-6/18) were also installed: one to measure the pressure of the extrusion barrel where the blowing agent gas is injected, one to measure the pressure of the mixing and diffusion process between the diffusion enhancing device and the heat exchanger, and one to measure the pressure of polymer melt before exiting the die.

11.9 EXPERIMENTS AND DISCUSSION A number of semicrystalline and amorphous polymers including HIPS, PS, SPS, ABS, PP, PE, and some proprietary materials have been microcellular processed in extrusion for high-density and low-density applications. The cell nucleation and growth behaviors of the extruded foams were studied. The foam samples were randomly chosen at each processing condition and characterized with a scanning electron microscope (SEM, Hitachi 510). The samples were dipped in liquid nitrogen and then fractured to expose the cellular morphology. The fractured surface was then coated with gold and the microstructure was examined using SEM. The cell density (i.e., the number of cells per unit volume of unfoamed polymer), and the volume expansion ratio (or, equivalently, the foam density) were the structural foam parameters measured. The expansion ratio of foam was determined by measuring the weight and volume of the sample. 11.9.1 NUCLEATION EXPERIMENTS A series of experiments has been conducted to investigate the nucleation behavior of extruded foams in various processing conditions [36, 38-40]. It turned out that the critical processing and materials parameters that affect cell nucleation are the type of gas, processing pressure, injected gas amount, and pressure drop rate. 11.9.1.1 Effect of Gases on Cell Nucleation First, the effect of different gases on the cell morphology was studied [38]. The gases injected in these experiments were CO2 and N2, and the processed

© 2000 by CRC Press LLC

polymers were PP (phillips 66 Marlex) and HIPS (Nova 3350). Approximately 10 wt% of CO2 and 2 wt% of N2 were injected into each polymer melt based on the estimated solubilities. The thermodynamic instability induced through the solubility drop seems sufficient to produce a microcellular structure when the maximum soluble amount of CO2 was injected. When CO2 was processed, the cell densities of PP and HIPS foams were 6 ⫻ 108 cells/cm3 and 8 ⫻ 109 cells/cm3, respectively. When N2 was processed, the cell densities were 3 ⫻ 107 cells/cm3 and 9 ⫻ 107 cells/cm3, respectively, for PP and HIPS foams. The higher cell densities of the foams obtained with CO2 injection seemed to be due to the higher solubility of CO2 in the polymers. Because of the higher solubility, it was possible to inject more gas into the polymer when CO2 was injected. It is believed that the larger amount of dissolved gas induced a greater thermodynamic instability and, thereby, a higher cell density. 11.9.1.2 Effect of the Processing Pressure on Cell Nucleation Next, the effect of the processing pressure on the cell morphology was studied [39]. In these experiments, the maximum soluble amount of CO2 was injected into the polymer melt at each processing pressure. The polymer used in these experiments was HIPS. Figure 11.14 shows the micrographs of the extruded HIPS at each processing pressure. When the processing pressures were 5.4 MPa (780 psi), 10.6 MPa (1,530 psi), 18.6 MPa (2,700 psi), and 28.3 MPa (4,100 psi), the cell densities were 7 ⫻ 105 cells/cm3, 2 ⫻ 107 cells/cm3, 2 ⫻ 108 cells/cm3, and 6 ⫻ 109 cells/cm3, respectively. The results show that the cell density increased with the processing pressure as shown in Figure 11.15. When the processing pressure was 28.3 MPa (4,100 psi), the foam structure of the extruded HIPS was microcellular, and the cell density was 6 ⫻ 109 cells/cm3. Since the solubility of gas is approximately proportional to the processing pressure [52–56], more gas could dissolve in the polymer melt when the processing pressure was higher. The increased amount of dissolved gas induced a greater thermodynamic instability and a larger cell density. Therefore, a larger cell density is expected when the processing pressure is higher. However, the processing pressure was limited by the capacities of the high-pressure gas pump and the extruder. On the other hand, a lower cell density is expected when the processing pressure is lower. In fact, when the processing pressure fell below 6.9 MPa (1,000 psi), the cell density in the samples was on the order of 106 cells/cm3. 11.9.1.3 Effect of the Injected Gas Amount on Cell Nucleation The effect of the amount of gas injected on the cell density was also studied by Park and Suh [36]. In these experiments, the processing pressure was main-

© 2000 by CRC Press LLC

FIGURE 11.14 Scanning electron micrographs of extruded HIPS at different processing pressures. Reproduced with permission from Reference [39]. Copyright (c) 1996 Polymer Engineering and Science.

tained as 27.6 MPa (4,000 psi), while the injected gas amount was varied. When 1 wt%, 5 wt%, and 10 wt% CO2 were injected, the cell densities of HIPS foams were 107 cells/cm3, 4 ⫻ 108 cells/cm3, and 6 ⫻ 109 cells/cm3, respectively. These cell densities are plotted as a function of the injected gas amount in Figure 11.16. When the injected gas amount was 10 wt%, the structure of the extruded HIPS foam was microcellular. For the processing pressure of 27.6 MPa (4,000 psi), 10 wt% is the maximum soluble amount of CO2 (see Section 11.4). Up to this amount, all the injected gas will dissolve in the polymer. This dissolved gas amount again affects the cell density of the extruded HIPS foam. The experimental results shown in Figure 11.15 predict that a microcellular structure with a cell density larger than 109 cells/cm3 can be produced when the processing pressure is higher than 22 MPa (3,200 psi). Since the gas solubility is approximately proportional to the processing pressure, the required gas amount for microcellular nucleation is estimated to be 7.5 wt%. The experimental results shown in Figure 11.16 predict that the required gas amount

© 2000 by CRC Press LLC

FIGURE 11.15 Effect of the processing pressure on the cell density of extruded HIPS. Reproduced with permission from Reference [39]. Copyright (c) 1996 Polymer Engineering and Science.

is 6.5 wt%. Similar results were obtained by Shimbo et al. for extruded microcellular PS foams [50]. 11.9.1.4 Effect of the Pressure Drop Rate on Cell Nucleation In order to investigate the effect of the pressure drop rate, the three nozzles, theoretically equivalent with respect to the pre- and post-flow conditions as described in Section 11.5, were calibrated experimentally so that they produced the same amount of flow rate for a given pressure difference [40]. This calibration was needed because the actual experimental relationship between the flow rate and the pressure drop in a nozzle was different from the theoretically predicted values. First, Nozzle 3 was chosen as a reference for the calibration of other nozzles. When Nozzle 3 was mounted, the nozzle pressure and the flow rate were measured as 38.64 MPa (5,600 psi) and 1.9 ⫻ 10⫺7 m3/s, respectively. It may be noted that the actual flow rate through Nozzle 3 was almost twice the predicted value. Next, the size of Nozzle 1 was calibrated as follows. Initially, a nozzle of a radius 0.60 mm (0.024 in) and a length 92.23 mm (3.631 in) was used. When the nozzle was mounted, the nozzle pressure and the flow rate were measured as 44.57 MPa (6,460 psi) and 1.5 ⫻ 10⫺7 m3/s, respectively. The nozzle length was then cut in stages until the flow rate and the pressure were measured to be 38.64 MPa (5,600 psi) and

© 2000 by CRC Press LLC

FIGURE 11.16

Effect of the injected CO2 amount on the cell density of extruded HIPS [36].

2.0 ⫻ 10⫺7 m3/s. This completed the calibration of Nozzle 1. The length of Nozzle 2 was also determined in a similar manner. Using the calibrated nozzles, a series of experiments was carried out for each nozzle to investigate the effects of the pressure drop rate on the cell morphology of the extruded microcellular HIPS. In all the experiments, the same amount of gas, 10 wt% CO2, was injected to form identical polymer/gas solutions. The cell densities for the nozzles from 1 to 3 were 1 ⫻ 108 cells/cm3, 1 ⫻ 109 cells/cm3, and 7 ⫻ 109 cells/cm3, respectively. The cell densities for each nozzle are plotted as a function of the pressure drop rate in Figure 11.17. The residence times of the polymer/gas system in Nozzle 1 and Nozzle 3 were 0.19 s and 0.005 s, respectively. The corresponding pressure drop rates for Nozzle 1 and Nozzle 3 were 0.18 GPa/s and 6.9 GPa/s, respectively. Since Nozzle 3 experiences a higher pressure drop rate, the nucleation rates for Nozzle 3 are higher than for Nozzle 1. Therefore, the total number of nucleated cells are larger for Nozzle 3 than for Nozzle 1, and the average cell size for Nozzle 3 (5 ␮m) is smaller than for Nozzle 1 (20 ␮m).

© 2000 by CRC Press LLC

FIGURE 11.17 Effect of the pressure drop rate on the cell density of extruded HIPS. Reproduced with permission from Reference [40]. Copyright (c) 1995 Polymer Engineering and Science.

As discussed in Section 11.5, nucleation stops when all the depleted regions of adjacent cells impinge upon each other and the gas concentration in the polymer matrix is below the critical level needed to nucleate additional cells. The pressure drop rates that the nozzles experience affect this nucleation time period. For Nozzle 1, the pressure drop rate is low, and therefore, the nucleation rates are slow. Since the nucleation rates are governed by the pressure drop rate, the nucleation time period would be shorter than the total time period over which the pressure drops. Assume that the nucleation rates for Nozzle 3 are governed by the pressure drop rate and that nucleation is completed before the pressure drops completely to the downstream pressure as shown in Figure 11.18. In this case, the nucleation time period is shorter than the residence time period in the nozzle. If the residence time period decreases, then the pressure drop rate increases. This will induce higher nucleation rates, and the nucleation time period will decrease. However, the decrease of the nucleation time period is limited because it takes time for the depleted regions to impinge upon each other. The growth of the depleted region is governed by gas diffusion in the polymer matrix. Therefore, the nucleation time period cannot be reduced infinitely small as the pressure drop rate increases infinitely. When the pressure drops

© 2000 by CRC Press LLC

almost instantaneously, the nucleation time period may be longer than the residence time period. In this case, the pressure drop rate does not affect the nucleation rate because most of nucleation takes place after the pressure drops completely. Figure 11.18 depicts the nucleation time period for Nozzle 3 to be less than the residence time period. It may be noted that this is a conservative estimate. The actual nucleation time period for Nozzle 3 may be longer than the residence time period. Additional work is required to determine the actual nucleation time period for each nozzle. The previous argument predicts that some critical pressure drop rate exists above which the pressure drop rate does not affect the competition between microcell nucleation and cell growth. This would imply that the graph in Figure 11.17 should reach a maximum constant value. Figure 11.17 shows that start of such a maximum, however, due to practical limitations in the manufacture of small bore nozzles, as described in Section 11.5, does not permit data points verifying the maximum to be obtained.

FIGURE 11.18

Comparison of residence time periods and the nucleation time periods [36].

© 2000 by CRC Press LLC

11.9.1.5 Other Materials Although HIPS foaming behavior was mainly described above, the nucleation behaviors of other materials such as PS [41, 49, 50], PP [44], SPS, and HDPE [48] were also tested. From most of these polymers, a high cell density on the order of 1 ⫻ 109 cells/cm3 was achieved by injecting CO2 on the extruder barrel. 11.9.2 CELL COALESCENCE EXPERIMENTS The feasibility of suppressing cell coalescence in the microcellular extrusion foam processing was examined [46]. Figure 11.19 shows a schematic of the filament dies used in the experiments that consisted of two stages: nucleating nozzle and shaping section. HIPS was used as the polymer material. Three sets of critical experiments were carried out. In the first experiment, only the nucleating nozzle was attached to the die without any shaping section. In this experiment, cell nucleation took place when the polymer/gas solution experienced a rapid pressure drop in the nozzle, and the nucleated cells continued to grow in the air after the extrudate exited the nucleating nozzle. In the second experiment, a shaping section was attached further to the nucleating nozzle

FIGURE 11.19 Schematic of the nucleating and shaping device. Reproduced with permission from Reference [46]. Copyright (c) 1997 American Chemical Society.

© 2000 by CRC Press LLC

TABLE 11.5

Nozzle Specifications and the Melt Temperatures Used in the Experiments.

Nucleation Stage

Experiment 1 Experiment 2 Experiment 3

Shaping Stage

Melt Temperature

Diameter mm (in.)

Length mm (in.)

Diameter mm (in.)

Length mm (in.)

(°C)

0.46 (0.018) 0.46 (0.018) 0.46 (0.018)

10.16 (0.400) 10.16 (0.400) 2.54 (0.100)

—

—

200

1.52 (0.060) 1.52 (0.060)

10.16 (0.400) 10.16 (0.400)

200 125

Source: Reproduced with Permission from Reference [46]. Copyright (c) 1997 American Chemical Society.

and the nucleated cells were induced to grow in the shaping section. In the first and second experiments, the die temperature was maintained at 200°C. In the third experiment, a shaping section was also attached to the nucleating nozzle. However, the die temperature in the experiment was significantly lowered to 125°C. Because of the increased resistance in the die at the low temperature, the length of the nucleating nozzle was reduced from 10.2 mm (0.4 inch) to 2.5 mm (0.1 inch) to maintain the same processing pressure of 28 MPa (4,000 psi) as in the first and second experiments. The dimension and the processing temperature of each nozzle are presented in Table 11.5. Figure 11.20(a) shows the microstructure of the foam obtained when no shaping section was attached to the nucleating nozzle (Experiment 1). The cell density in this case was 1 ⫻ 1010 cells/cm3, which agreed with the previous results [39, 40]. Since the processing temperature was very high (200°C), it was observed that some adjacent cells were coalesced. Figure 11.20(b) shows the microstructure of the foam obtained from the second experiment. When the shaping section was added to the die, the cell density was dropped to 2 ⫻ 108 cells/cm3. This implies that cell coalescence vigorously occurred in the shaping section at the high temperature after a large cell density was achieved at the nucleating nozzle. If we assume that 1 ⫻ 1010 cells were nucleated in the nozzle as in the other experiments [39, 40], we may conclude that around 50 bubbles were coalesced into a single bubble in the shaping section. It was believed that the shear field generated in the shaping section caused this vigorous coalescence of cells. Finally, Figure 11.20(c) shows the microstructure of the foam obtained from the third experiment in which a shaping section was also attached but the processing temperature was significantly lowered. The cell density in this case was found to be 5 ⫻ 109 cells/cm3. This high cell density indicates that cell coalescence was substantially prevented by decreasing the temperature. Because the melt strength increases as the temperature decreases [72],

© 2000 by CRC Press LLC

FIGURE 11.20 Microstructures of the foams showing the effect of processing condition on cell coalescence: (a) Experiment 1, (b) Experiment 2, and (c) Experiment 3. Reproduced with permission from Reference [46]. Copyright (c) 1997 American Chemical Society.

© 2000 by CRC Press LLC

it may be concluded that the increased melt strength by lowering the temperature prevented cell coalescence despite the shear field during shaping. This experimental result confirmed that the previously proposed scheme of independently controlling cell nucleation, followed by controlling shaping and cell growth, can be effectively utilized in the microcellular extrusion foam processing. In addition to pressurizing the nucleated polymer/gas solution [41], the melt temperature can be significantly lowered to prevent cell coalescence in the shaping section of the die. 11.9.3 EXPANSION EXPERIMENTS The feasibility of producing low-density microcellular foams by closely controlling the temperature of the polymer melt flowing into the die and the temperature of die was also examined through experiments [47]. HIPS was used in the experiments. All the experimental results are summarized in Figure 11.21. This figure shows how the melt and nozzle temperatures affect the volume expansion ratio of the extruded foams. Figure 11.22 shows the microstructures of HIPS foams at various melt temperatures and nozzle temperatures, demonstrating the effects of these temperatures on cell coalescence and foam contraction. Figure 11.21(a) shows the expansion ratio versus the nozzle temperature at the three melt temperatures. Equivalently, the foam density is plotted in Figure 11.21(b) in terms of the nozzle and melt temperatures. It was observed that the volume expansion ratio was a strong function of both the melt temperature and die temperature. When the nozzle temperature was as high as 175°C, the achieved volume expansion ratio was only about 1.5 times, regardless of the melt temperature. This means that when the die temperature was too high, most of the gas escaped through the hot skin layer of foam during expansion. On the other hand, when the melt temperature was as high as 170°C, the volume expansion ratio was also only around 1.5 times irrespective of the nozzle temperature. Even when the nozzle temperature was lowered to 110°C, the volume expansion ratio was not changed. This implies that when the temperature of polymer melt was too high, the frozen surface of the extrudate was not effectively blocking gas escape because the formed frozen skin layer must have been remelted immediately by the heat transferred from the hot polymer melt. However, when the temperature of polymer melt was lowered to 150°C, a low-density microcellular foam was successfully obtained by freezing the surface of the extrudate. In the nozzle-temperature range of 135°C to 175°C, the expansion ratio increased as the nozzle temperature decreased. In other words, the gas diffusion was blocked at the surface and more gas remained in the foam to contribute to volume expansion as the nozzle temperate was lowered. Very similar results with a larger volume expansion were obtained when the melt temperature was further decreased to 120°C.

© 2000 by CRC Press LLC

FIGURE 11.21 Effect of the nozzle temperature on (a) the Volume Expansion Ratio and (b) the foam density at various melt temperatures. Reproduced with permission from Reference [47]. Copyright (c) 1998 Polymer Engineering and Science.

When the nozzle temperature was further decreased from 135°C to 110°C, the volume expansion ratio decreased. Even though it was expected that more gas was preserved in the foam at this lower nozzle temperature than at 135°C, the increased stiffness of the frozen skin layer adversely affected volume expansion and limited the achieved expansion ratio of extruded foam. The nozzle temperature could not be lowered below 110°C because the extrudate clogged in the nozzle. The slight increase of the volume expansion ratio at a melt temperature of 150°C with the decrease in the nozzle temperature around 110°C seems to be due to the melt fracture that occurred on the extrudate. The experimental results indicate that there exists an optimum nozzle temperature to achieve a maximum expansion ratio for each temperature of polymer melt. The results shown in Figure 11.21(a) also imply that one can use two methods to achieve a desired volume expansion ratio at a fixed gas

© 2000 by CRC Press LLC

FIGURE 11.22 Microstructures of HIPS foams at various melt temperatures (Tc) and nozzle temperatures (Tn): (a) Tc ⫽ 170°C, Tn ⫽ 175°C; (b) Tc ⫽ 170°C, Tn ⫽ 135°C; (c) Tc ⫽ 170°C, Tn ⫽ 110°C; (d) Tc ⫽ 150°C, Tn ⫽ 175°C; (e) Tc ⫽ 135°C, Tn ⫽ 175°C; (f) Tc ⫽ 150°C, Tn ⫽ 110°C; (g) Tc ⫽ 120°C, Tn ⫽ 175°C; (h) Tc ⫽ 120°C; Tn ⫽ 135°C; (i) Tc ⫽ 120°C, Tn ⫽ 110°C. Reproduced with permission from Reference [47]. Copyright (c) 1998 Polymer Engineering and Science.

© 2000 by CRC Press LLC

amount. First, the amount of gas escape to obtain a desired volume expansion ratio can be adjusted by controlling the die temperature above 135°C. Second, below 135°C, the degree of stiffness of the frozen extrudate skin can be adjusted to control the expansion ratio. Similar experiments have been conducted on PS [49], HDPE [48], SPS, PP, and other proprietary materials to investigate the volume expansion behaviors of these materials. Utilizing the strategy of blocking gas loss by freezing the foam skin, a large volume expansion ratio over 30 times has been achieved with CO2 as a blowing agent. Very similar volume expansion behaviors as a function of the melt and die temperatures have been observed for all the amorphous materials. However, for some semicrystalline materials, the maximum volume expansion ratio was obtained when the melt and die temperatures were the lowest [48]. It is believed that this was due to the rheological characteristics of semicrystalline polymers above their freezing point [48].

11.10 SUMMARY AND CONCLUSIONS An extrusion process for manufacturing high-density and low-density microcellular polymers has been presented. This process is cost-effective compared to the microcellular batch processes that require a long time for gas saturation. The basic approach to the production of microcellular structures is to continuously form a polymer/gas solution, to nucleate a large number of bubbles using thermodynamic instability via a rapid pressure drop, to suppress cell coalescence by increasing the melt strength, and to induce a volume expansion to a desired expansion ratio by blocking gas loss. The main strategy for the process design was to integrate these processing steps into an extrusion process such that the overall process had independently controllable functions. The kinetics of polymer/gas solution formation, microcell nucleation by thermodynamic instability, suppression of cell coalescence, and volume expansion control were examined through experimental work. An experimental extrusion setup was built based on the proposed processing strategies. Experiments were carried out to verify the design and to identify the critical process parameters. Various semicrystalline and amorphous polymers were processed with high pressure gases. Microcellular foams of a high nuclei density were achieved from a variety of thermoplastics when CO2 was processed in the foam process. It turned out that the amount of gas dissolved in the polymer and the pressure drop rate across the die are the most critical parameters in determining the nuclei density of extruded foams. The melt and die temperatures proved to be the most critical parameters that affect the volume expansion ratio of the extruded foams. By controlling the melt and die temperatures, a very high volume expansion ratio up to 43 times has been successfully achieved even with CO2.

© 2000 by CRC Press LLC

11.11 NOMENCLATURE a Co Ci Cw C∞ D Do dCs/dt ⫺dp/dt (⫺dp/dt)i fo k L lD m Nhom n p q R ro s T Tc Tn tD to t1, t2 U∞ vavg ⌬Cs ⌬ED ⌬Ghom ⌬P ⌬p ⌬p1i

Dimensionless characteristic constant of a non-Newtonian fluid ( ⫽ 1/n) Concentration of nucleation sites, #/m3 Concentration of the source, g/g Concentration of gas molecules in solution at the cell wall interface, g/g of polymer Concentration of the free-stream or initial concentration of gas molecules in solution, g/g of polymer Diffusivity or diffusion coefficient, cm2/s Diffusion coefficient constant, cm2/s Solubility drop rate across the nozzle, 1/s Pressure drop rate across the nozzle, Pa/s Pressure drop rate across nozzle i, Pa/s Frequency factor for microcell nucleation, 1/s Boltzman’s constant, J/K Length of the nozzle, m The diffusion distance, cm Characteristic constant of a non-Newtonian fluid, N-sn/m2 Homogeneous microcell nucleation rate, cells/m3s Dimensionless characteristic constant of a non-Newtonian fluid Pressure of a flowing polymer/gas solution, Pa Volumetric flow rate, m3/s Universal gas constant ⫽ 8.314 J/mol-K Radius of the nozzle, m Striation thickness of the mixture, cm Temperature, K Melt temperature, K Nozzle temperature, K Diffusion time, s Reference time in a nozzle, s Arbitrarily small time period during nucleation, s Velocity of the free-stream, m/s Average velocity of the polymer/gas solution in the nozzle Change of the gas solubility in polymer, g/g of polymer Activation energy for diffusion of a gas in a polymer, J/mol Change in Gibbs free energy for homogeneous microcell nucleation, J Difference between initial solution pressure and nucleation solution pressure, Pa Pressure difference, Pa Pressure drop at time t1 in Nozzle i, Pa

© 2000 by CRC Press LLC

⌬p1j ⌬t ␴bp ␦C ␩

Pressure drop at time t1 in Nozzle j, Pa Average residence time of the polymer/gas solution in the nozzle, s Interfacial energy between gas bubble and polymer, N/m Concentration boundary layer, m Viscosity of a non-Newtonian fluid, Pa-s

11.12 REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

N. P. Suh, Private Communication, MIT—Industry Polymer Processing Program, 1980. S. Doroudiani, C. B. Park and M. T. Kortschot, Polym. Eng. Sci. 38, 1205, 1998. L. M. Matuana, C. B. Park and J. J. Balatinecz, Cellular Polym., 17, 1, 1998. L. M. Matuana, C. B. Park and J. J. Balatinecz, Polym. Eng. Sci., 38, 11, 1862, 1998. D. I. Collias and D. G. Baird, Polym. Eng. Sci., 35, 1178, 1995. D. I. Collias and D. G. Baird, Polym. Eng. Sci., 35, 1167, 1995. D. F. Baldwin and N. P. Suh, SPE ANTEC Tech. Papers, 38, 1503, 1992. K. A. Seeler and V. Kumar, J. Reinforced Plast. Comp., 12, 359, 1993. M. Shimbo, D. F. Baldwin and N. P. Suh, Polym. Eng. Sci., 35, 1387, 1995. L. Glicksman, Notes from MIT Summer Session Program 4.10S Foams and Cellular Materials: Thermal and Mechanical Properties, Cambridge, MA, June 29–July 1, 1992. J. H. Sanders, In: Handbook of Polymeric Foams and Foam Technology, D. Klempner and K. C. Frisch, eds., Hanser Publishers, 1991, p. 5. G. Liu, C. B. Park and J. A. Lefas, Polym. Eng. Sci., 38, 1997, 1998. J. E. Martini, S. M. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1982. J. E. Martini-Vvedensky, N. P. Suh and F. A. Waldman, U.S. Patent 4473665, 1984. J. R. Yoon, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1984. J. R. Yoon and N. P. Suh, Polymer Composites, 6, 175, 1985. S. W. Cha, N. P. Suh, D. F. Baldwin and C. B. Park, U.S. Patent 5158986, 1992. S. W. Cha, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1994. S. K. Goel and E. J. Beckman, Polym. Eng. Sci., 34, 1137 and 1148, 1994. J. S. Colton and N. P. Suh, U.S. Patent 5160674, 1992. D. F. Baldwin, C. B. Park and N. P. Suh, Polym. Eng. Sci., 36, 1437 and 1446, 1996. S. Doroudiani, C. B. Park and M. T. Kortschot, Polym. Eng. Sci., 36, 2645, 1996. L. M. Matuana, Ph.D. Thesis, University of Toronto, Toronto, Ontario, 1997. L. M. Matuana, C. B. Park and J. J. Balatinecz, J. Cellular Plast., 32, 449, 1996. L. M. Matuana, C. B. Park and J. J. Balatinecz, Polym. Eng. Sci., 37, 1137, 1997. V. Kumar, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1988. V. Kumar and N. P. Suh, Polym. Eng. Sci., 30, 1323, 1990. V. Kumar and H. G. Schirmer, SPE ANTEC Technical Papers, 41, 2189, 1995. F. A. Waldman, S. M. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1982. J. S. Colton, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1985. J. S. Colton and N. P. Suh, Polym. Eng. Sci., 27, 485, 1987.

© 2000 by CRC Press LLC

32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67.

J. S. Colton and N. P. Suh, Polym. Eng. Sci., 27, 493, 1987. J. S. Colton and N. P. Suh, Polym. Eng. Sci., 27, 500, 1987. M. R. Holl, Ph.D. Thesis, University of Washington, Seattle, 1996. J. E. Weller, Ph.D. Thesis, University of Washington, Seattle, 1996. C. B. Park, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1993. D. F. Baldwin, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1994. C. B. Park and N. P. Suh, ASME Trans. J. Manuf. Sci. Eng., 118, 639, 1996. C. B. Park and N. P. Suh, Polym. Eng. Sci., 36, 34, 1996. C. B. Park, D. F. Baldwin and N. P. Suh, Polym. Eng. Sci., 35, 432, 1995. D. F. Baldwin, C. B. Park and N. P. Suh, Polym. Eng. Sci., 36, 1446, 1996. D. F. Baldwin, C. B. Park and N. P. Suh, Polym. Eng. Sci., 38, 674, 1998. K. K. Cheung, M.A.Sc. Thesis, University of Toronto, Toronto, Ontario, 1996. C. B. Park and L. K. Cheung, Polym. Eng. Sci., 37, 1, 1997. A. H. Behravesh, Ph.D. Thesis, University of Toronto, Toronto, Ontario, 1998. C. B. Park, A. H. Behravesh and R. D. Venter, In: Polymeric Foam: Science and Technology, K. Khemani, ed., Chap. 8, ACS, Washington, 1997. C. B. Park, A. H. Behravesh and R. D. Venter, Polym. Eng. Sci., 38, 1812, 1998. C. B. Park, A. H. Behravesh and R. D. Venter, Cellular Polym., 17, 309, 1998. M. Pan, M.Eng. Thesis, University of Toronto, Toronto, Ontario, 1997. M. Shimbo, K. Nishida, S. Nishikawa, T. Sueda and M. Eriguti, In: Porous, Cellular and Microcellular Materials, V. Kumar, ed., ASME, 93, 1998. M. L. Berins, SPI Plastics Engineering Handbook, Van Nostrand Reinhold, New York, 1991. J. -J. Shim and K. P. Johnston, A.I.Ch.E. Journal, 37, 607, 1991. K. P. Johnston, Private Communication, University of Texas at Austin, 1993. P. L. Durril and R. G. Griskey, A.I.Ch.E. Journal, 12, 1147, 1966. P. L. Durril and R. G. Griskey, A.I.Ch.E. Journal, 15, 106, 1969. D. W. van Krevelen, Properties of Polymers, Elsevier, New York, 1980. W. R. Vieth, Diffusion in and through Polymers: Principles and Applications, Hanser Publishers, 1991. W. -C. V. Wang, E. J. Kramer and W. H. Sachse, J. Polym. Sci.: Polym. Phy. Ed., 20, 1371, 1982. A. R. Berens, Barrier Polymers and Structures, edited by W. J. Koros, ACS symposium series, 92, 1989. A. R. Berens and G. S. Huvard, Supercritical Fluid Science and Technology, K. P. Johnston and J. M. L. Penninger, eds., ACS symposium series, 207, 1989. W. J. Koros and D. R. Paul, Polym. Eng. Sci., 20, 14, 1980. R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, 592–625 and 642–652, 1960. L. Erwin, Polym. Eng. Sci., 18, 572, 1978. C. Raunwendaal, Mixing in Polymer Processing, Raunwendaal, C. ed., Marcel Dekker, Inc., 129–240, 1991. R. W. Foster and J. T. Lindt, Polym. Eng. Sci., 30, 424, 1990. R. W. Foster and J. T. Lindt, Polym. Eng. Sci., 30, 621, 1990. A. de Waele, Oil and Color Chem. Assoc. Journal, 6, 33, 1923.

© 2000 by CRC Press LLC

68. W. Ostwald, Kolloid-Z., 36, 99, 1925. 69. R. B. Bird, R. R. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Fluid Mechanics Volume 1, Wiley, 233, 1977. 70. M. E. Kim-E, R. A. Brown and R. C. Armstrong, J. Non-Newtonian Fluid Mech., 13, 341, 1983. 71. K. C. Frisch and J. H. Saunders, Plastic Foams, Vol. 1, Marcel Dekker, New York, 1972. 72. S. K. Goyal, Plastics Engineering, 25, February 1995. 73. S. T. Lee and N. S. Ramesh, SPE ANTEC Technical Papers, 41, 2217, 1995. 74. D. Klempner and K. C. Frisch, Handbook of Polymeric Foams and Foam Technology, Hanser Publishers, 1991. 75. F. J. Dwyer, L. M. Zwolinski and K. M. Thrun, Plastics Eng., 46, 29, 1990. 76. M. Moskowitz, Plastics World, 49, 93, 1991. 77. L. Yu-Hallanda, K. P. Mclellan, R. J. Wierabicki and C. J. Reichel, J. Cell. Plastics, 29, 589, 1993. 78. L. M. Zwolinski and F. J. Dwyer, Plastics Eng., 42, 45, 1986. 79. C. Jacob and S. K. Dey, SPE ANTEC Tech. Papers, 1964, 1994. 80. S. K. Dey, P. Natarajan and M. Xanthos, SPE ANTEC Tech. Papers, 1955, 1996. 81. T. Shimano, K. Orimo, S. Yamamoto and M. Azuma, U.S. Patent 3,981,649, 1976. 82. D. E. Johnson, C. M. Krutchen and V. Sharps Jr., U.S. Patent 4,424,287, 1984. 83. S. T. Lee, U.S. Patent 5,348,984, 1994. 84. E. P. Gyftopoulos and G. P. Beretta, Thermodynamics: Foundations and Application, MacMillan Publishing Company, 1991.

© 2000 by CRC Press LLC

CHAPTER 12

Foam Extrusion of Polyethylene Terephthalate (PET) MARINO XANTHOS SUBIR K. DEY

12.1 INTRODUCTION

E

XTRUSION foaming of most plastic resins has been carried out successfully

for some time with physical blowing agents (PBA) or chemical blowing agents (CBA) in single or tandem lines. CBAs are normally used for the production of high- and medium-density foams whereas PBAs are generally used for the production of lower-density foams—as low as 0.05 g/cc. CBAs release gaseous products at a required rate over a fairly narrow temperature range, their choice being dictated by the process temperature of the particular polymer. PBAs are atmospheric gases, volatile hydrocarbons, hydrofluorocarbons (HFC), or hydrochlorofluorocarbons (HCFC) that are metered and dissolved in the polymer melt during processing. It is believed that bubble nucleation is heterogeneous and begins inside the shaping die [1]. As the gas-laden melt emerges from the die, it experiences a sudden pressure drop; this thermodynamic instability causes a phase separation. The escaping gas leads to expansion within the fluid matrix in such a manner that individual bubbles grow and merge into cells, and through subsequent solidification, stable expanded structures are produced [2, 3]. The cell size and cell density depend on the amount of gas dissolved into the polymer [4]. More information on the physicochemical aspects of bubble nucleation and growth occurring during extrusion foaming may be found in other sections of this monograph. The favorable cost/performance characteristics of solid PET (virgin and recycled) may be extended to lower density sheet structures produced by singlelayer extrusion or coextrusion for thermoforming and lamination. Potential applications would take advantage of the combination of good mechanical properties, dimensional stability of the semicrystalline resin at temperatures up

© 2000 by CRC Press LLC

to 200°C, and recyclability. In addition to the food packaging industry where PET foams have already made advances, other applications would be intended for the building/construction, transportation, and other industries currently utilizing rigid PUR, PS, or PVC foams. By contrast to the commonly used PS and LDPE resins, extrusion foaming of the relatively high-density, semicrystalline PET is an area presenting several challenges. Difficulties are mostly related to the required high processing temperatures (260–290°C), the absence of a broad extrusion foaming window as compared to amorphous resins, the often inadequate for foaming rheological characteristics of the resin, its slow rate of crystallization and limited process stability, and the possible interference of crystal nuleation with bubble nucleation. Poor foaming characteristics are accentuated by the known sensitivity of the PET ester linkages to hydrolytic degradation at processing temperatures leading to chain cleavage and reduction in molecular weight. Hydrolytic degradation may be due to inadequate drying of the base resin or to water given off as a decomposition by-product of certain chemical blowing agents. In addition, the sensitivity of PET to thermal or thermooxidative degradation may lead to further reduction in MW and the formation of various by-products that could affect the foaming process. Significant developmental work has been conducted over the past twenty years by resin producers and converters to develop suitable resins and extrusion processes, particularly for low-density foaming. Patents have been issued to Amoco Corp., Celanese Corp., Dow Chem. Co., E.I. Dupont de Nemours & Co., Eastman Chemical Co., General Electric Co., Goodyear Tire and Rubber Co., Sekisui Kaseihin Kogyo Kabushiki Kaisha, M&G Ricerche, Rohm and Haas Co., Teijin Ltd., among others. Since the early 1990s, Shell Chemical Co. (PetliteTM) and Sekisui (CelpetTM) are among suppliers of PET foamed sheets (1–3 mm thick) with different bulk densities and low % crystallinity. Sinco Engineering is among the recent suppliers of foam-grade pellets (CobitechTM). Equipment suppliers for high-volume foam extrusion lines suitable for PET include Battenfeld Gloucester Eng. Co., Berstorff Corp., Cincinnati Milacron/Sano, Wayne Machine and Die, and Leistritz Corp. [5]. Low- to medium-density (⬍ 0.5 g/cc) polymeric sheets or boards are produced by injection of physical blowing agents (PBA) in single extruders or in an extruder of a tandem system. Flat or annular dies may be used. The rheological properties of conventional PET resins with relatively low MW and narrow MWD are not particularly suitable for low-density extrusion foaming. As a result, modified resins with higher melt viscosity, broader MWD, and high melt strength/elasticity are often required to control cell expansion and stabilize the growing bubbles. Such resins can be produced through chain extension/ branching reactions with di- or polyfunctional reagents, as, for example, pyromellitic dianhydride (PMDA). A variety of blowing agents (CFCs, HCFCs, VOC, atmospheric gases) have been reported to be effective PBAs for PET. In single extruder foaming processes, a two-stage screw is typ-

© 2000 by CRC Press LLC

ically used with the blowing agent added usually in the decompression zone of the screw [5]. ABA or AB structures, where A is unfoamed skin and B is foamed core, may be produced by coextrusion with a cap layer to reduce gas loss and, hence, achieve lower density. In a tandem line, the blowing agent is added in the first extruder, and the mixture is transferred in the second cooling extruder that conveys the melt to the die at appropriate temperature and pressure to allow expansion without coalescence. Due to the longer residence times in tandem lines, there always exists the possibility of thermal or thermooxidative degradation of certain PET resins. Medium- to high-density (⬎ 0.5 g/cc) extruded PET foams may also be produced with chemical blowing agents. The rheological characteristics of the resin are less critical than in the case of PBA foaming, and resins with lower viscosity/melt strength may be acceptable [6]. However, in contrast to other thermoplastic materials, e.g., PE, PP, PS, PVC, etc, this method is still less advanced. Following a review of the resin chemistry and current process technologies for extruding/forming solid PET sheets, the sections below will discuss the following:

• • •

available chemical modification methods to meet the rheological requirements of PET resins for foaming to low densities with PBAs available process technologies for foaming with PBAs and CBAs properties and characteristics of the extruded foams

The processes discussed in this chapter involve macrocellular foam (typically 100 ␮m or larger cell size) and do not cover the so-called microcellular foaming where resins presaturated with blowing agents, such as CO2, are heated at or above Tg and cooled rapidly to lock in the cellular morphology and prevent excessive cell growth. Such morphology is characterized by closed cells up to 25 ␮m in size with cell density of 108 cells/cm3 [7]. Continuous microcellular processes involving extrusion equipment as described in various patents, e.g. [8], are under development [9].

12.2. REVIEW OF PET CHEMISTRY AND PROCESSING CHARACTERISTICS The general information in this section on the characteristics of PET resins and extrusion processing/forming of solid sheets is extracted from the presentations of several authors in Reference [10]. This review will provide a useful background to the foaming processes that will be described in later sections. PET is prepared by transesterification from dimethyl terephthalate and ethylene glycol or direct esterification from terephthalic acid and ethylene glycol in the presence of catalysts; both reactions are followed by a polycondensation

© 2000 by CRC Press LLC

step. Typical reaction products for textile fiber applications are characterized by intrinsic viscosity, IV (typically 0.62 IV for MW ranging from 12,000 to 20,000). Higher MW can be achieved by longer polycondensation times or by solid-state polymerization. In the latter process, PET pellets are heated under a stream of hot, dry nitrogen to temperatures up to 215°C for 16 hours to reach IVs of 0.80 (MW 30,000–35,000) suitable for bottle applications. In addition to solution viscosity, polymers are characterized by percent crystallinity, glass transition temperature (about 75°C), melting temperature (approximately 260°C), color, barrier properties, and comonomer content. Overall density of PET homopolymer is high (1.35 g/cc for the amorphous phase, 1.46 g/cc for the crystalline phase). PET has a low crystallization rate compared to polymers such as PE or PP; ultimate degrees of crystallinity are also low (typically ⬍ 40%). Crystallization kinetics of PET depend primarily on temperature, MW or IV, catalyst residue, and the presence of diethylene glycol formed during synthesis. Effects of reduced IV are faster crystallization rates and reduced impact resistance. A variety of plasticizers and nucleating agents, including inorganic and organic substances, inorganic minerals, and organic polymers (particularly, polyolefins for thermoformable PET) are added at small concentrations to increase crystallinity and produce faster rates of crystallization and fine spherulitic structure [11]. Organic plasticizers, including dissolved gases (e.g., CO2), result in increased mobility of the polymer chains, reduced Tg, and overall increase of crystallization rate. Nucleating agents induce heterogeneous nucleation by increasing the density of nucleating sites. Upon cooling from the melt, crystallization half-time values for nucleated resins reach minimum values in the temperature region of 200–180°C (appropriate for thermoforming). For unnucleated resins, corresponding values in the same temperature range are significantly higher. Comonomers such as isophthalic acid, diethylene glycol, or cyclohexane dimethanol at concentrations up to 5 mole % are used to lower percent crystallinity, rate of crystallization and melting temperature without compromising the PET’s desirable properties. Degradation of PET may be thermal through chain scission reactions in essentially air-free environments. Degradation may result in increased carboxyl end group content, the formation of acetaldehyde, and eventually the formation of polyene structures. If air is present at these high processing temperatures, oxidative processes may occur creating free radicals, gel formation, etc. Predrying to very low moisture levels is required prior to processing to minimize hydrolytic degradation. Typical conditions in desiccant dryers to reduce moisture of precrystallized PET pellets to ⬍ 50 ppm moisture are 150°C for four to six hours to a dew point of ⫺40°C. The majority of PET applications include bottles, film, strapping, fibers, and sheeting. For sheet extrusion/thermoforming applications, that are particularly

© 2000 by CRC Press LLC

amenable to density reduction by foaming, suitable materials include the following: a. Amorphous PET grades (APET) for “crystal clear” packaging for use at temperatures below Tg. b. Crystalline PET grades (CPET) for opaque microwavable or dual ovenable. food containers. APET polymers are typically copolymers with IVs from 0.8 to 1.05 extruded to sheets of low crystallinity (⬍ 5%); these low crystallinity levels are retained even after thermoforming usually in cold molds (120°F), with 6 sec cycle times. Depending on the degree of draw, APET can undergo stressinduced crystallization that may also result in haze and reduced transparency. CPET homopolymers with IVs above 0.7 contain nucleating agents (typically, 3% of heat stabilized polyolefins, e.g., LLDPE, LDPE, or PP) to assist in crystallization during thermoforming. During extrusion, CPET crystallinity needs to be kept as low as possible by chilling to provide more latitude in consequent heating and forming. Cystallinity of CPET increases from ⬍ 5% in the extruded sheet to about 30–35% on the thermoformed item that is usually produced in hot molds (around 330°F), with cycle times 6–10 sec. CPET trays and containers are considered to be heat stable up to 200°C. To improve impact strength of CPET, particularly at low temperatures, core shell butyl acrylate based tougheners are employed. For tray thermoforming, an APET layer extruded on top of CPET gives improved sealability to MylarTM film and better impact resistance. Oxygen and water vapor transmission decrease through additional crystallization; reported values from materials suppliers [10] are about 30–40% lower for CPET trays versus APET sheets.

12.3 FOAMING WITH PHYSICAL BLOWING AGENTS 12.3.1 CHEMICAL MODIFICATION OF PET FOR LOW-DENSITY EXTRUSION FOAMING Effects related to melt viscoelasticity of thermoplastics can, in general, be controlled through additives or through changes in MW and MWD during reactor/post-reactor processing by chain extension, grafting, branching, controlled cross-linking or controlled degradation [12]. Modified PET resins with increased MW, viscosity, and melt strength/elasticity, can be produced through chain extension/branching reactions, primarily between the carboxyl/hydroxyl polyester end groups and di- or polyfunctional reagents containing anhydride, epoxy, oxazoline, isocyanate, carbodiimide, hydroxyl, tertiary phosphite,

© 2000 by CRC Press LLC

phthalimide, and other groups [13]. The term “melt elasticity” is widely used in polymer processing where complicated geometric effects and flow fields necessitate the use of linear viscoelastic functions and their nonlinear counterparts. Changes in melt elasticity have been related to changes in the values of parameters such as normal stress difference, storage modulus, and extrudate swell [14-16]. For certain polymers, “melt elasticity” has also been related to extensional viscosity [17] and melt strength, the latter being related qualitatively to extensional rheology [15]. Rheological modification of PET may be carried out during polycondensation, by solid stating in the presence of premixed modifier, or by post-reactor modification via reactive extrusion. Common polyester modifiers listed in technical publications and the patent literature include polyanhydrides, such as pyromellitic dianhydride (PMDA) (see for example References 18–22) and polyepoxides, such as diglycidyl esters or copolymers containing glycidyl functional groups (see for example References 22–28). Within the context of low-density foaming by PBA injection, the following is a review of modification methods that have been reported, mostly, in the patent literature. 12.3.1.1 Reactor Polycondensation in the Presence of Modifier As an example, Muschiatti [18] reported the production of highly branched, high melt strength, non-Newtonian behavior resins suitable for making lowdensity, closed-cell foams through polymerization in the presence of branching agents (polyols, polyanhydrides, polyacids). 12.3.1.2 Solid Stating in the Presence of Premixed Modifier The production of PBA extrusion foamable polyesters modified by solidstate polyaddition with a premixed modifier, e.g., PMDA, is discussed in a series of patents and publications from Sinco Engineering [19, 29–34]. These resins are distinguished from nonfoamable resins by increased melt strength, high extrudate swell, increased complex viscosity with non-Newtonian behavior in the low-frequency region, and higher storage modulus (G⬘). Branched foamable PET was also prepared by mixing with ethylene copolymers containing carboxyl, ester, and alcohol functionalities followed by solid stating to produce a modified resin with a 10-fold higher melt viscosity and higher die swell and melt strength than the unmodified resin [35]. 12.3.1.3 Post-Reactor Modification by Reactive Extrusion Extrusion modification of PET with PMDA or other branching additives in a concentrate form results in significant increase in zero shear melt viscosity, increase in melt strength and die swell, increase in molecular weight, and in-

© 2000 by CRC Press LLC

crease in polydispersity from about 4.5 to up to 11 as measured by the zaverage/number-average molecular weight ratio [20, 36]. Property changes depend on the choice of process conditions, additive concentration, and its type of carrier. Foamable PMDA modified resins had IV of 0.805 dl/g (versus 0.70 for the unmodified resin), weight average molecular weight of 85,690, and polydispersity of 10.8. Foams with 0.2 g/cc density were obtained by HCFC injection in the primary extruder of a tandem line equipped with an annular die [20]. In another patent by Hayashi et al. [22], isopentane was injected into the molten mixture of PET/PMDA to produce rod-shaped foams. Expansion of the extrudate was proportional to the amount of PMDA added (up to 0.4 %wt); density values for the PMDA-modified foams ranged from 0.35 to 0.13 g/cc versus 0.76 g/cc when no PMDA was used. Dianhydrides and metallic catalysts were used to produce hydrocarbon/inert gas extrusion foamable PET by substantially increasing its shear viscosity and melt strength [37]. PMDA concentrates in different carriers and in the presence of various additives have also been used for improved stability of the foaming process [21, 38]. A suggested reaction mechanism [36], with PMDA involves as a first-step linear extension through reaction of terminal polyester hydroxyl end groups with the anhydride functionalities and the formation of two carboxyl groups per incorporated PMDA moiety. Subsequent reactions may involve all functionalities of the PMDA molecule through esterification and transreactions to yield branched or even cross-linked structures. Combinations of PMDA/pentaerythritol/Lewis acid catalyst have also been used to produce resins with modified rheology [39]. Other modifiers such as multifunctional epoxy chain extending compounds were also used in a tandem line with CO2 to produce foams with 40 kg/m3 density [23]. Tetrafunctional epoxy reactive additives were added in a batch mixer to produce PET with increased elongational viscosity and melt strength; preliminary batch foaming experiments indicated that the modified PET foams obtained by CO2 dissolution at room temperature were closed-cell structures [40]. A suggested mechanism [27–28] for chain-extension reactions with the glycidyl functionality includes esterification of carboxyl end groups and etherification of hydroxyl end groups; secondary hydroxyls formed from these reactions may further react with carboxyl or epoxy groups leading to branching or cross-linking. A summary of possible reactions converting the linear polyesters into partially branched resins may be found in Reference [41]. Results of sequential reactive modification/foaming of low-IV recycled resins with premixed branching additives followed by CO2 injection in a single 40:1 L:D extruder are shown in Table 12.1 [6]. The recycled PET 1 and PET 5 resins having low viscosity (as suggested by low-IV and high-MFI values), and low elasticity (as suggested by low die swell values), produced unstable foams in a rod die. By contrast, their counterparts containing premixed modifiers (PET 1M and PET 5M) foamed well and produced predomi-

© 2000 by CRC Press LLC

TABLE 12.1

Materials Characteristics in CO2 Foaming [6].

Description

Nominal IV

MFI (260°C/ 2.16 kg)

Recycled post consumer pellets (Phoenix) Extruder modified PET 1—0.9 phr branching additives Recycled post-consumer pellets modified by reactive processing (pilot) Recycled post-consumer pellets modified by reactive processing (pilot) Recycled green post-consumer pellets (St. Jude) Extruder modified PET 5—2 phr branching additives

0.7

26.6

VIRGIN PET 6

Bottle grade

0.77

PET 7

Copolyester

PET 8 PET 9

Resins RECYCLED, MODIFIED/ PET 1 PET 1M

PET 2

PET 4

PET 5 PET 5M

© 2000 by CRC Press LLC

Die Swell (270 s⫺1/ 260°C) 1.29

No foam Good, uniform; high swell; density 0.2 g/cc, aver. cell size 410 ␮m Good, uniform; high swell; density 0.12 g/cc, aver. cell size 200 ␮m Uniform; high swell; density 0.12 g/cc, aver. cell size 150 ␮m No foam

N/M

4.38

2.76

0.87

6.87

2.66

0.95

4.38

2.86

0.7 N/M

38 1.85

1.23

Foam Quality/ Comments

2.97

Good, uniform; high swell; density 0.11 g/cc, aver. cell size 350 ␮m

15.9

1.29

0.8

15.3

1.43

CPET

0.95

10.4

1.39

CPET

1.0

Poor/little expansion, large bubbles Poor/little expansion, large bubbles Poor/little expansion, large bubbles Poor/little expansion, large bubbles

7.38

1.43

nantly closed-cell foams with low density and fine cell size. Similarly [42], recycled PET was sequentially modified/foamed in a twin-screw extruder with 0.25% PMDA, talc nucleating agent, and Freon 12 to produce highly expanded extrudates with density of about 0.2 g/cc. 12.3.2 RHEOLOGICAL CHARACTERISTICS OF FOAMABLE PET RESINS Extrusion foamability of PET resins is usually reported in the literature through single point, single temperature, and single shear rate measurements of parameters related to melt viscosity and melt strength/elasticity. ASTM standard methods D4440 for zero shear melt viscosity and D3835 for melt strength and die swell have been employed [35]. In an earlier publication [6], extrusion foamability of a variety of commercial and experimental resins, virgin and recycled, with different intrinsic viscosity (IV) values was evaluated in a single-screw extruder modified for carbon dioxide injection and equipped with a rod die. In addition to molecular weight (through IV), foamability was related to melt viscosity through melt flow index and melt elasticity/melt strength through die swell measurements under prescribed conditions. The characteristics of the resins and the results of the foaming experiments are summarized in Table 12.1. As mentioned earlier, the post-consumer recycled materials PET 1 and PET 5 were almost impossible to foam due to their very low viscosity and poor melt strength. The virgin materials (PET 6 to PET 9), although having higher viscosity and fairly good melt strength during extrusion, did not expand sufficiently to produce satisfactory foams. The recycled materials modified with appropriate concentrations of low MW multifunctional branching agents (PET 1M, PET 2, PET 4, and PET 5M) swelled significantly at the die exit and foamed well. They produced predominantly closedcell foams with densities from 0.1 to 0.3 g/cc and fine cell size with fairly uniform size distribution (Figure 12.1). The obtained data suggested that CO2 foamable PET resins should have die swell values, under the prescribed offline experimental conditions, approximately 100% higher than those of the poorly foamable ones; MFI values at 260°C/2.16 kg should also be lower than about 7g/10 min for improved foamability. More complete melt viscoelasticity data for two of the above resins, the unmodified PET8 (poorly foamable) and the chemically modified PET4 (well foamable), are reported below. With respect to melt viscosity, by invoking the Cox-Merz rule, combined capillary and mechanical spectrometer viscosity data are plotted in Figure 12.2 [43]. The unmodified resin has lower overall viscosity and shows typical Newtonian behavior in the low-frequency region and significant shear thinning starting at 50–100 s⫺1. In contrast, the chemically modified resin seems to behave over almost the entire shear rate region as a power law fluid starting shear-thinning at very low shear rates. Such high-

© 2000 by CRC Press LLC

FIGURE 12.1

FIGURE 12.2

Cell structure of PET foam. (Courtesy of the CRC for Polymers, Australia.)

Melt viscosity comparison of unmodified PET8 and modified PET4 at 290°C.

© 2000 by CRC Press LLC

FIGURE 12.3 290°C.

Extrudate swell versus shear rate for unmodified PET8 and modified PET4 at

shear sensitivity behavior would be typical of branched or broad MWD polymers and very high MW materials [44], also, of high melt strength PET resins modified for foaming or extrusion blow molding [29, 30, 45]. With respect to “melt elasticity” criteria, extrudate swell data of the two resins at 290°C are compared in Figure 12.3. The chemically modified resin has more than double the extrudate swell of the unmodified resin depending on the shear rate; extrudate swell is shown to increase with increasing shear rate, as expected. A region of flow instability is noted at high shear rates for the high-viscosity chemically modified resin. Storage modulus, G⬘, as another measure of melt elasticity suggests similar ranking of the two resins (Figure 12.4). The values of G⬘ of the chemically modified resin are larger than those of the unmodified by a factor of 10 at low frequencies, and they begin to converge as frequency increases. 12.3.3 PROCESS TECHNOLOGY AND APPLICATIONS 12.3.3.1 General Examples of four different extrusion foaming processes described in the literature are summarized in Table 12.2. The systems involve the use of single or tandem lines, premodified resins or resins modified during a sequential

© 2000 by CRC Press LLC

FIGURE 12.4

© 2000 by CRC Press LLC

Comparison of G⬘ and G⬙ for unmodified PET8 and modified PET4 at 290°C.

TABLE 12.2

Parameters/ Properties Resin IV: Chain extension/ branching method Foaming nucleator Extruder system PBA Product Density, g/cc Crystallinity Cell size Post-processing

Examples of PET Foam Extrusion Processes Using PBAs

Reference [46]

Reference [35]

Reference [19]

Reference [38]

⬎ 0.95 Solid stating in presence of modifier None Single-SSE (1.25⬙) CO2, N2, Ar Thin sheets (1.2 mm) from flat die Variable to 0.2 min. depending on resin ⬍ 10% or ⬎ 15% , depending on resin 0.2–0.3 mm Press lamination to thick boards easier for the low-crystallinity sheet

⬎ 0.8 Solid stating in presence of modifier Talc, TiO2, Na2CO3 Tandem-SSE (2⬙ and 2.5⬙) Isopentane Thin sheets (1.5 mm) from annular die 0.20

⬎ 0.85 Solid stating in presence of modifier Talc Single-SSE (3.5⬙) HCFC Thin sheet from annular die 0.15–0.18

⬎ 0.8 During extrusion prior to PBA injection Talc, Na2CO3 Single-TSE (1.25⬙) Freon 22 Thick 15 mm board

15.3%

⬍ 15%

N/M

0.1–0.2 mm Post-expansion at 175°C to 1.9 mm, 0.16 g/cc density and 0.2–0.3 mm cell size

0.05–0.2 mm —

0.56 mm —

© 2000 by CRC Press LLC

About 0.1

extrusion/foaming operation, different size extrusion equipment, and different blowing agents. Products are thin sheets or thicker boards, all of relatively low density (0.2 g/cc), medium to low percent crystallinity and medium-sized, closed-cell morphology. Talc is the material of choice as a foaming nucleator; it is recognized that talc could also promote crystal nucleation and subsequent crystallization to different extents during the extrusion or postprocessing steps. Control of crystallinity to low levels through cooling or material formulation (e.g., polyolefin nucleators for crystallinity) is critical when sheets are to be used in thermoforming/lamination. Details on the systems of Table 12.2 are given below. 12.3.3.1.1 Example I In a recent study [46] attempting to provide a better understanding of the parameters affecting foam extrusion with atmospheric gases, a 32 mm dia., 40 L/D Killion segmented single-screw extruder equipped with gas injection port, a 250 mm wide flat sheet die, and a three-stack chilled roll assembly (Figure 12.5) was used to produce foamed monolayer (about 1.2 mm in thickness) from resins with different rheological characteristics. Carbon dioxide, nitrogen, or argon were injected at 19D length at different pressures and mixed into

FIGURE 12.5. Schematic of single extruder foaming line equipped with gas injection. (Courtesy of the Polymer Processing Institute.)

© 2000 by CRC Press LLC

the PET melt. No additional nucleating agents were used. Process conditions depended on the type of gas and the type of resin. For the chemically modified, high-viscosity, high-melt strength resin (referred to as PET 4 in Table 12.1 and Figures 12.2–12.4) the density of the foamed extrudates was found to decrease with increasing gas injection pressure with a plateau attained at about 5,500–7,000 kPa. The results appear to be independent of the type of gas (Figure 12.6). Densities as low as 0.2–0.3 g/cc were obtained at about 5,500 kPa gas pressure regardless of the gas type. For the unmodified, lower viscosity and lower melt strength PET 8 resin, the density of the foamed extrudates was also found to decrease with increasing gas injection pressure, independently of the type of gas, but with an apparent plateau reached earlier at gas pressures about 4,000 kPa (Figure 12.6). It was not possible to produce satisfactory foams at higher gas pressures due to cell collapse. Densities at about 5,500 kPa were significantly higher (0.7–0.9 g/cc) than the corresponding values obtained with the modified resin. In the absence of gas solubility data at the processing conditions, gas injection pressures can

FIGURE 12.6 Density versus gas injection pressure for two PET resins foamed with inert gases in a flat sheet die.

© 2000 by CRC Press LLC

only be considered as indicators of concentration, and final densities are independent of the type of gas molecules. Injection in the partly filled zone of the segmented screw would initially lead to a two-phase system that further downstream could be converted in a combination of a single-phase and undissolved gas clusters, depending on screw configuration, operating conditions, temperature, etc. Such clusters could act as premature nucleators and affect the foam microstructure. The obtained trends in the density versus gas pressure curves are similar to the ones observed in the cell density versus gas concentration curves by Park and Cheung [47], who suggested that reaching the gas solubility limit was one of the reasons for the observed plateaus. It should be noted that preliminary results [48] with on-line optical monitors attached at the extruder end suggest differences in the solubilities of the three gases used in Reference [46]. An observation in this study was the appearance of corrugation parallel to the machine direction in all sheets produced from the modified PET 4 and foamed to densities lower than about 0.5 g/cc. Corrugation was not observed in the case of the 0.7–0.8 g/cc higher density PET 8 sheets. Corrugation in flat sheet dies has also been reported by other authors [49] and is attributed to the uneven directional expansion of gas for a given die/take-up configuration. Ease of expansion in two directions is accompanied by difficulty of the gas expanding in the transverse direction resulting in sheet folding. Transverse expansion becomes more difficult in the presence of the larger amounts of gas necessary to attain low density, a phenomenon that appeared to be amplified by the particular rheology of the PET 4 resin. 12.3.3.1.2 Example II As an example of extrusion in a tandem line [35], a prereacted with EVOH branched PET was fed in a system consisting of a 2 inch primary extruder operating at 87 rpm and a 2.5 inch secondary extruder operating at 16.4 rpm and equipped with a 3 inch diameter annular die. Isopentane was introduced in the first extruder at 1.6 lb/hour. Temperatures ranged from 260 to 275°C, and polymer output was 66 lb/hour. The foam was slit and collected as a 36-inchwide sheet having a density of 0.21 g/cc, a thickness of 59 mils, crystallinity 15%, and closed cells 100–200 microns in size. Further heating of the sheet at 175°C for three minutes resulted in expansion to about 75 mils, 0.16 g/cc density, and 31% crystallinity. 12.3.3.1.3 Example III Equipment and conditions for producing low-density PET foamed sheets from solid-stated, PMDA chemically modified PET resins with improved melt strength are described in Reference [19]. In a particular example from this

© 2000 by CRC Press LLC

FIGURE 12.7

PET pilot extrusion foam line. (Courtesy of Sinco Engineering.)

patent issued to M&G Richerche, a 90 mm single-screw extruder (L:D 30) equipped with an annular 40 mm diameter die was used in combination with trichlorofluoroethane and talc nucleating agent to produce sheets with properties shown in Table 12.3. Typical operating conditions were 24 rpm, 259°C melt temperature, and 9.1 MPa melt pressure. Figure 12.7 shows a pilot plant extrusion line for producing low-density PET foams intended for a variety of structural and insulation applications. Thick boards, about 35–40 mm in thickness and 230 mm wide, produced by the same technology were found to have thickness dependent percent crystallinity as a result of cooling rates. Crystallinity ranged from about 37% in the core to about 25% in the skin [50]. 12.3.3.1.4 Example IV A corotating twin-screw extruder (W&P ZSK-30) was used to produce thick insulation boards [38]. The screw profile consisted of conveying elements in the feeding/melting section, a melt seal at the sixth barrel section, and conveying elements toward the die. Freon–22 was fed in the seventh barrel section at 2 wt% with respect to a resin feed of about 19 lb/hour. Resin feed contained two concentrates (PMDA and sodium carbonate, respectively) in PET carriers.

© 2000 by CRC Press LLC

TABLE 12.3

Foamable Versus NonFoamable PET—Comparison of Resin and Extrudate Properties [19].

Resin properties IV, dl/g Melt strength, cN Complex viscosity, Poise 104 Elastic modulus,G⬘ dyne/cm2 104 Extrudate properties Density, kg/m3 Compession set, MPa Compression Modulus, MPa Tensile Strength, MPa

© 2000 by CRC Press LLC

Unmodified PET— not Foamable

Modified Foamable PET (0.71 IV ⫹ 0.15% PMDA Solid Stated ⫹ Additional PMDA in Extruder)

Modified Foamable PET (0.71 IV ⫹ 0.3% PMDA Solid Stated)

0.80 0.2 1.05 1.04

0.82 25 3.59 26.0

1.95 43 40 100

1,300–1,400 N/M N/M N/M

150–180 1.5 15.4 3.3

50–80 20 17.0 4.0

Foam was pulled at 40 inches /min to produce a board of about 0.6 inch thickness, density of 6.3 pcf, cell size 0.56 mm, compressive strength of 53 psi, and excellent moisture and thermal dimensional stability. Pressure at the die was 440–540 psi, and pressure at the gas injection point was 260–280 psi. 12.3.3.2 Extrusion Followed by Thermoforming 12.3.3.2.1 Extrusion for High-Density Thermoformable Trays The examples that follow describe extrusion foaming, thermoforming, and properties of relatively high-density (⬎ 0.7 g/cc) products. In a patent assigned to the Goodyear Rubber and Tire Co., unmodified PET (IV 0.95) with LLDPE nucleating agent was extruded in a 2.5 inch Egan extruder operating at 70 rpm with barrel temperatures 280–330°C, die temperature about 260°C, and nitrogen injected at 3,200 psi. [51]. The foamed sheet was about 0.03 inch thick having density of about 1 g/cc and low crystallinity (about 5%). Thermoforming of this CPET sheet was carried out utilizing a preheat oven time of about 15 seconds, mold time of 8–10 seconds, sheet temperature of 154°C, mold temperature of 154–136°C, top oven temperature of 300°C, and bottom oven temperature of 116°C. Thermoforming produced trays of density 0.85 g/cc due to further expansion of the nitrogen containing cells and additional crystallization to about 25–35% crystallinity. The trays were heat stable to 200°C and were considered as dual ovenable for use in the frozen food industry. As mentioned earlier, Shell Chemical Co. commercialized in the early 1990s CPET foams with relatively high densities (0.8–0.9 g/cc) in 30–40 mil thick sheet form. Petlite IITM intended to be used for thermoforming food containers met applicable FDA regulations. In a study conducted by Eastman Chem. Co. [49] on the use of expanded polyesters for food packaging applications, it was shown that CPET foams could be thermoformed on conventional PET thermoforming tools. Standard 1–1.5⬙ deep trays of foamed CPET could be formed with the same cycle times as unfoamed CPET. Sheet foam extrusion in the presence of proprietary additives and PBAs was shown to result in reduction of mechanical properties as a function of density to a 1.5–2 power. Gas barrier properties also decreased with decreasing density, for example, oxygen permeability increased from about 5 cm2.mm/m2.24 hr.atm for the solid PET to about 15 cm2.mm/m2.24 hr.atm for a 0.7 g/cc density foam. Mechanical properties at 0.7 g/cc density were as follows:

• • • •

tensile strength at break (Machine Direction, MD) about 14 MPa tensile strength at break (Transverse Direction, TD) about 9 MPa secant modulus 1%, MD 850 MPa secant modulus 1%, TD 550 MPa

© 2000 by CRC Press LLC

12.3.3.2.2 Extrusion for Low-Density Thermoformable Trays The examples that follow describe extrusion foaming, thermoforming, and properties of lower density (⬍ 0.5 g/cc) products as well as approaches used to reduce extractables to meet food regulations. In the original Sekishui patent [22], a variety of PET resins were extruded to different densities by varying the amount of butane injected downstream. Different branching additives were used in a 65 mm single-screw extruder (L:D 35) equipped with a circular die. Typical barrel temperatures were 280–290°C, die temperatures were 270–280°C, extrusion head pressure was 80–90 kg/cm2, rpm was 30–25, and extrusion rates were 24 kg/hr. The injection pressure of butane was 80 kg/cm2 for systems 1 and 2 and 40 kg/cm2 for system 3 (Table 12.4). The resulting extrudate was cut open and samples about 1.5 mm thick were tested. The results summarized in Table 12.4 indicate that the low-density foam containing PMDA ⫹ Na2CO3 had satisfactory tensile strength, high elongation at break, and high heat resistance. Sheets for thermoforming had densities 0.16–0.19 g/cc, thickness 1.5–2.6 mm, and crystallinity 10–18%. They were preheated at 175°C for 15 seconds and thermoformed in a plug assist tool for an additional 25 seconds. In subsequent publications from Sekishui Plastics [37, 52], PET plus diacid anhydride and metal compound were processed in a 65 mm, L/D 35:1 singlescrew extruder equipped with a sheet die at 270–280°C. Volatile hydrocarbons or inert gases were injected downstream. Properties of the extruded foamed sheets (Celpet™) are shown in Table 12.5. Thermoforming to trays was carried out by preheating to 150°C for 4 seconds, forming and crystallizing at 180°C for 6 seconds, and cooling to 20°C for 4 seconds. Crystallinity increased from 10 to 22–28% after thermoforming. The formed PET trays had improved heat resistance in the presence of food in microwave or electric ovens versus equivalent PP or PS foam containers. In attempts to reduce extractables of thermoformed trays produced from foamed PET modified with PMDA, several approaches involving precompounded concentrates were used. For example, in a patent assigned to Amoco Corp. [38], an Egan 4.5 single-screw extruder with seven temperature control zones was used with a flat sheet die. The extruder was modified with a gas delivery system. Barrel set temperatures ranged from 540 to 520°F. Two different concentrates containing PMDA and sodium carbonate, respectively, in PET carriers were fed through a side feed hopper. CO2 was injected into the fourth barrel segment, and foam was produced at about 600 lbs/hour. At 0.2% PMDA ⫹ 0.04% Na2CO3, the sheet density was about 30 pcf. Thermoformed trays had very small amounts of extractable unreacted PMDA in the range of 30–40 ppm, based on the weight of the tray. By contrast, foam trays believed to be prepared from a polyester foaming process in which PMDA was added directly as a powder had up to sixthfold the extractable PMDA amount.

© 2000 by CRC Press LLC

TABLE 12.4

PET Type #1, Eastman 9902 #2, Eastman 10388 #3, Teijin 8580

Effects of Modifier and PBA Amount on Properties of Extruded Foamed PET Sheets [22].

Cell Size

Tensile Strength, kg/cm2

Elongation at Break, %

Dynamic Modulus, Pa @ 150°C

6

medium

63.6

116.6

107

1.7

6

medium

39.3

64.3

4 ⫻ 106

0.9

3

medium

81.5

53.3

—

Butane, wt%

Expansion Ratio

PMDA ⫹ Na2CO3

1.7

Diglycidyl terephthalate ⫹ Na montanate PMDA ⫹ Diglycidyl terephthalate

Additives

© 2000 by CRC Press LLC

TABLE 12.5

Properties of Extruded Low-Density Foamed PET Sheets [37, 52].

Property Thickness, mm Density, g/cc Crystallinity, % Tensile strength, kgf/cm2 MD/TD Tensile strength, kgf/cm2 (⫺20°C) MD/TD Elongation at break, % MD/TD Elongation at break, %, (⫺20°C) MD/TD Tear strength, kgf/cm2 MD/TD Aver. Cell diameter, mm TD/TD/VD

Hydrocarbon Blowing Agent

Hydrocarbon Blowing Agent

Inert Gas Blowing Agent

1 0.3 10 80/65

3 0.3 — 40/25

0.9 0.4 10 105/110

85/75

25/25

—

120/110

40/20

129/72

30/30

15/20

—

54/40

25/15

70/46

0.3/0.34/0.07

—

0.2/0.23/0.05

When the levels of PMDA and sodium carbonate were increased to 0.3% and 0.06%, respectively, the resultant extruded foam sheet had a density of 9 pcf, whereas thermoformed trays from the sheet had density of about 28 pcf. In other publications by Eastman Chemical Co. describing methods to reduce extractables [20, 41], sheets with density of 0.26 g/cc at a thickness of 45 mils and low crystallinity of 4.5% were produced by extruding PET in the presence of a PP/PMDA concentrate, talc and 1,1-difluoroethane in a tandem line. Equipment was similar to that described in Reference [35]. Thermoformed trays with crystallinities of 28% were subjected to a complete extraction test protocol to evaluate for compliance with European Union and FDA guidelines for total migration, antimony migration, and PMDA migration into food simulating solvents. Results indicated that the trays complied with all limits and no migration of PMDA was detected in the extracts.

12.4 FOAMING WITH CHEMICAL BLOWING AGENTS 12.4.1 GENERAL CBAs can be divided into exothermic and endothermic blowing agents. In general, CBAs may decompose over a broad temperature range. Exothermic chemical blowing agents include azodicarbonamide, (4,4)-oxy-bis(benzene-

© 2000 by CRC Press LLC

sulfonyl hydrazide), p-toluenesulfonyl semicarbazide, and 5-phenyltetrazole. The most common CBA, azodicarbonamide, decomposes between 190°C and 230°C and generates mostly nitrogen gas; the exothermic reaction can be catalyzed so that gases are produced at lower temperatures. This decomposition rate can be modeled using an Arrhenius-type equation. Decomposition products of other exothermic blowing agents may include nitrogen, small amounts of carbon dioxide, carbon monoxide, ammonia, and miscellaneous organic compounds. Endothermic chemical blowing agents are usually blends of inorganic carbonates and polycarbonic acids. The reaction product of these ingredients is primarily carbon dioxide. The reaction temperature can be varied between 150°C and 300°C by altering the chemistry of the system. The endothermic blowing agents yield, in general, lower quantities of gases than the exothermic ones. In contrast to the decomposition products of exothermic CBAs, such gases appear to be more soluble in the polymer. This results in lower levels of die pressure required for extrusion foaming with endothermic CBAs. General advantages/characteristics of CBAs as compared to PBAs in extrusion foaming are as follows: (1) (2) (3) (4)

Some equipment modification required Suitable for high- and medium-density foams Broader operating window Finer cell sizes

Disadvantages of CBAs are as follows: (1) Possible moisture as a by-product, particularly critical with hydrolytically unstable resins (2) Changes of polymer rheology, difficulty of recycling nonconforming products or contamination due to unreacted CBA or solid residue from the reacted CBA (3) Relatively higher cost versus most PBAs 12.4.2 EXTRUSION FOAMING OF PET In contrast to other thermoplastic materials, e.g., PE, PP, PS, PVC, etc., extrusion foaming of PET using CBAs is still less advanced. PET homopolymer, being extremely moisture sensitive, requires chemical blowing agents that do not produce moisture as a reaction by-product and decompose in a controlled fashion at the high processing temperatures of PET. There exist several U.S. patents summarized in Reference [41] describing the use of CBAs for foaming PET resins or copolymers with most of the early work reported in the area of injection molding. The reference to polycarbonate (PC) resin as CBA for PET to produce relatively high-density foams is noteworthy [53]; PC may appar-

© 2000 by CRC Press LLC

ently undergo an ester interchange reaction with PET with simultaneous decomposition to generate carbon dioxide. The sections below describe recent experimental work on CBA foaming of thin PET sheeting for the production by lamination of thicker panels to be evaluated for building applications. 12.4.2.1 Parameters Affecting PET Foaming—Laboratory Scale Based on vendor’s information and following preliminary thermal analysis (DSC and TGA), three chemical blowing agents were selected (Table 12.6) in a recent work [6, 46]. The effectiveness of the blowing agents with respect to PET foaming was evaluated in two extruders: a 19 mm dia. single-screw Brabender extruder equipped with a 50-mm-wide ribbon die and a 63 mm dia. Welex coextrusion setup equipped with a 860 mm sheet die. PET resins with different rheological characteristics (intrinsic viscosity, melt flow index as a viscosity indicator, and die swell as a melt elasticity indicator) were used. The resins included a standard post-consumer grade and two recycled materials modified by reactive processing for increased melt strength (Table 12.7). The

Materials and Concentrations Used In CBA Foaming [46].

TABLE 12.6

CBA 1 Description

Decomposition Range by TGA, °C Grade/ Supplier

CBA2

CBA3

Endothermic 40% Masterbatch; 2–5 phr 180–315

Exothermic Powder; 0.5–2 phr 190–310

Endothermic 40% Masterbatch; 1–5 phr 200–270

Safoam RPC-40 (Reedy Intern.)

Expandex 5PT (Uniroyal Chem.)

HK40B (Boehringer Ingelheim)

(Reprinted from Reference [46]).

TABLE 12.7

Description

Resins Used in CBA Foaming [46].

PET 1

PET 2

PET 3

Recycled postconsumer pellets

Recycled postconsumer pellets modified by reactive processing (Pilot) 0.87 6.87

Recycled postconsumer pellets modified by reactive processing (Pilot) 1.2 N/M

2.66

N/M

Nominal IV MFI (260°C/2.16 kg) Die Swell (270 s⫺1/260°C)

0.7 26.6 1.29

(Reprinted from Reference [46]).

© 2000 by CRC Press LLC

CBA was premixed with predried PET resin and fed into the hopper of the extruder. The extrudates produced in the 50 mm die were characterized for density and the results were fitted into a statistical model. In this case, the independent variables were type of CBA, % CBA, melt temperature, and screw rpm. The dependent variables were die pressure and the density of foam. Results are shown in Figures 12.8–12.11 [6] and are summarized in Table 12.8 [46]. As expected, the density was found to be a decreasing function of IV and increasing function of melt temperature and screw rpm. The % CBA showed a minimum at approximately 1.5 wt%. CBA2 showed the most promise, followed by CBA1, followed by CBA3. In another publication [54], extruded PET foams with densities above 0.6 g/cc and uniform and fine cellular structure were produced with endothermic CBAs. Mechanical properties were found to decrease linearly with decreasing foam density. 12.4.2.2 Parameters Affecting PET Foaming—Large Scale A PCR PET resin (PET1) with nominal IV of 0.71 was chosen for the scaleup study in the Welex coextrusion line equipped with the 860 mm flat sheet die (Figures 12.12, 12.13). CBA2 and CBA1 were used in two different experiments. CBA2 was a non-free-flowing powder that was difficult to feed. Attempts were made to meter-feed this powder using single- and twin-screw

FIGURE 12.8

© 2000 by CRC Press LLC

Foam density versus resin IV.

FIGURE 12.9.

FIGURE 12.10.

© 2000 by CRC Press LLC

Foam density versus extruder rpm.

Foam density versus CBA type.

FIGURE 12.11

Foam density versus CBA concentration.

powder feeders, including a flexible oscillating wall single-screw feeder, without much success. Consistent feed rate could not be achieved over a long period of time. Attempts were also made to precompound a masterbatch into a high-melt flow modified PE in a corotating twin-screw extruder with limited success. Further experiments with the unmodified PET1 resin and the pelletized masterbatch CBA1 indicated that it was difficult to produce monolayer sheet with reasonable density reduction in the large flat shet die, since the low melt

TABLE 12.8

Summary of CBA Foaming Runs [46].

• Satisfactory foams with densities 0.5–0.9 g/cc were possible at certain material/process condition combinations.

• Average cell size was about 65 ␮m, but with a broad size distribution from about 25 to 190 ␮m. From statistical analysis: • Density decreases somewhat with increasing resin IV and die swell and appears to increase with rpm. • Lower die temperatures appear to result in overall lower foam densities. • CBA 2 yields overall lower densities. • CBA concentrations in excess of about 1.5% do not result in any further density reduction. (Reprinted from Reference [46]).

© 2000 by CRC Press LLC

FIGURE 12.12

Schematic of coextrusion line.

FIGURE 12.13 Photograph of coextrusion line equipped with 30 in wide flat die. (Courtesy of the Polymer Processing Institute.)

© 2000 by CRC Press LLC

strength of the resin caused escape and/or rapid collapse of the growing bubbles. Better results were obtained with monolayers containing chemically modified resins such as PET2 or PET3 at 10–15 phr as rheology modifiers, and ABA structures where A was unmodified PET1 and B was PET1 chemically modified with a reactive polyethylene copolymer and foamed with 1.5% CBA1. Densities in all cases were relatively high, ranging between 0.8 and 1.1 g/cc. For extruded sheets with crystallinity exceeding 10%, lamination to thicker panels about 1⬙ thickness was assisted by incorporating a dry thin adhesive film within the layers. Additional crystallization during press or oven lamination at about 150°C for few minutes resulted in samples with final crystallinities of 25–30%.

12.5 CONCLUDING REMARKS PBA extrusion foaming of PET provides a cost-effective method to produce low-density rigid cellular structures that have some of the attributes of the solid semicrystalline resin, particularly rigidity, high-temperature dimensional stability, and recyclability. Although modification of existing sheet lines for gas injection may be costly, advantages of PBAs versus the expensive, specialized CBAs may prove more cost effective. Most of the recent industrial R&D activities have focused in the extrusion forming of expanded CPET for the food packaging industry in efforts to produce lightweight materials that could be more effective versus competitive products, including rigid and expanded polyolefins, aluminum foil, and paper /pulp products. The availability of extrusion foaming technologies employing environmentally friendly atmospheric gases as PBAs, combined with the recyclability of the formed containers/trays may be considered as additional advantages for market penetration. Other markets under development are in the building/construction and transportation industries with expanded PET competing with other rigid polymeric or nonpolymeric materials in structural sandwich panels or for insulation. As with other extrusion foaming processes for semicrystalline materials, e.g., PP, product attributes depend on various parameters including the following:

• • • •

obtaining materials with proper melt strength selecting proper cell and crystal nucleators obtaining optimum cell size and distribution and spherulitic size and concentration optimizing post-processing techniques, e.g., thermoforming

Existing technologies described in this chapter have attained these goals to various extents. However, improvements are still desirable and could lead to further market penetration. A better understanding of cell nucleation/growth and its differentiation from crystal nucleation/growth in crystallizable poly-

© 2000 by CRC Press LLC

mers is essential. Such phenomena may occur concurrently at different stages of the extrusion process, competing for the amount of amorphous phase present. In addition, extrusion by gas injection produces macrocellular foams with properties not always adequate for a given application. Selection of gas cell nucleators to control type, size, shape, and distribution of gas cells coupled with the selection of the optimum PBA based on solubility and diffusivity measurements could lead to a better control of physical properties, particularly toughness. If cell size reduction can be achieved with a concurrent control of crystal size, impact strength of extrusion produced foams would greatly benefit.

12.6 ACKNOWLEDGEMENTS The authors wish to acknowledge the assistance of Dr. Victor Tan, Mr. Dale Conti and Dr. Q. Zhang of the Polymer Processing Institute and Mr. G. Quintans and Mr. Y. Li of NJIT in several aspects of the experimental work reported in this chapter. Sinco Engineering kindly supplied various pilot PET resins. Partial financial support was provided by the Multi-lifecycle Engineering Research Center (MERC) of the NJ Institute of Technology (NJIT).

12.7 REFERENCES 1. J. L. Throne, Proc. Foam Conference 96, LCM Public Relations, pp. 2–14, Somerset, NJ, December 10–12 (1996). 2. S. K. Dey, C. Jacob and M. Xanthos, J. Vinyl and Addit. Technol., 2, 1, 48 (1996). 3. S. K. Dey, P. Natarajan, M. Xanthos and M. D. Braathen, J. Vinyl and Additive Technol., 2, 4, 339 (1996). 4. D. F. Baldwin, N. P. Suh, C. B. Park and S. W. Cha, U.S 5,334,356 (1994). 5. PM&E, p. 33, May 1993—Also, equipment vendors’ literature. 6. M. Xanthos, Q. Zhang, S. K. Dey, Y. Li, U. Yilmazer and M. O’Shea, J. Cell. Plastics, 34, 6, 498 (1998). 7. Y. P. Handa and Z. Zhang, “New Pathways to Microcellular and Ultramicrocellular Polymeric Foams,” Symposium on “Porous, Cellular and Microcellular Materials,” ASME Intern. Mech. Eng. Congr. and Exp., MD-Vol. 82, 1 (1998). 8. J. Martini-Vvedensky, N. P. Suh and F. A. Waldman, US 4,473,665 (1984). 9. Trexel Inc., Technical Literature on MuCell process, Woburn, MA. 10. Proc. CPPP/PPI Workshop on “APET and CPET Sheet Coextrusion and Thermoforming for Food Tray Uses,” NJ Institute of Technology, Newark, NJ, May 27–28 (1997). Available from the Polymer Processsing Institute, NJIT, Newark, NJ: S. K. Dey, J. Schaul, “Sheet extrusion for APET and CPET;” T. J. Dolce, “PET Basics;” M. L. Druin, “Potential Applications for Barrier PET”; Eastman Chemical Co.. literature on APET and CPET resins; R. Nwana, “Thermoforming PET”; Shell Chemical Co., literature on APET and CPET resins.

© 2000 by CRC Press LLC

11. M. Xanthos, B. C. Baltzis and P. P. Hsu, J. Appl. Polym. Sci., 64, 1423 (1997). 12. B. Maxwell, Plast. Eng., Sept., p. 41 (1987). 13. S. B. Brown, pp. 161–176, in “Reactive Extrusion: Principles and Practice,” M. Xanthos, Ed., Carl Hanser Verlag, Munich, New York (1992). 14. C. D. Han, Multiphase Flow in Polymer Processing, Ch.4, Academic Press, New York (1981). 15. L. A. Utracki, Polymer Blends and Alloys, Part 3, Carl Hanser Verlag Publishers, Munich, New York (1989). 16. M. Xanthos, V. Tan and A. Ponnusamy, Polym. Eng. Sci., 37, 6, 1102 (1997). 17. M. Xanthos and D. W. Yu, Adv. Polym. Tech., 10, 3, 243 (1990). 18. L. C. Muschiatti, US 5,229,432 (1992). 19. H. Al-Ghatta, S. Severini and L. Astarita, US 5,422,381 (1995). 20. K. J. Khemani, C. H. Juarez-Garcia and G. D. Boone, US 5,696,176 (1997). 21. G. E. Rotter, W. Chiang, B. C. Tsai and J. J. Kelly, US 5,536,793 (1996). 22. M. Hayashi, N. Amano, T. Taki and T. Hirai, US 5,000,991 (1991). 23. N. Inokuchi, T. Fukumoto and Y. Mori, US 4,284,596 (1981), JP62001732 (1987), JP 63082955 (1988). 24. T. Shiwaku, K. Hijikata, K. Furui and S. Masato, US 5,523.135 (1996). 25. E. J. Deyrup, E. A. Flexman and K. L. Howe, US 4,912,167 (1990). 26. B. Guo and C. -M. Chan, Proc. 56th SPE ANTEC, 1653, (1998). 27. D. N. Bikiaris and G. P. Karayannidis, J. Polym. Sci., Part A: Polymer Chemistry, 34, 1337 (1996). 28. D. N. Bikiaris and G. P. Karayannidis, J. Polym. Sci., Part A: Polymer Chemistry, 33, 1705 (1995). 29. H. Al-Ghatta and T. Severini, Proc. 54th SPE ANTEC, 42, 1846 (1996). 30. H. Al-Ghatta, S.Severini and L. Astarita, US 5,362,763 (1994). 31. G. Ghisolfi, WO 92/17520 (1992). 32. H. Al-Ghatta, WO 92/17521 (1992). 33. G. Ghisolfi, WO 92/17522 (1992). 34. H. Al-Ghatta, S. Cobror and S. Severini, Polym. Adv.Technol., 8, 161 (1997). 35. R. L. McConnell and K. C. Khemani, US 5,482,977 (1996). 36. K. J. Khemani, Proc. 56th SPE ANTEC, 44, 1934 (1998). 37. T. Hirai and N. Amano, Proc. 51st SPE ANTEC, 39,1256 (1993). 38. G. E. Rotter, J. L. Melquist, W. Chiang, B. C. Tsai and J. J. Kelly, US 5,288,764 (1994). 39. CRC for Polymers Pty. Ltd, Intern. Patent Appl., AU Patent Appl. PO4887 (1997). 40. S. Japon, L. Boogh, Y. Leterrier and J. -A. E. Manson, in “Porous, Cellular and Microcellular Materials,” Proc. ASME Int. Mech. Eng. Congr. and Exp., MD-Vol. 82, p. 85 (1998). 41. K. C. Chemani, pp. 54–80 in Polymeric Foams—Science and Technology, K. S. Khemani, Ed., ACS Symposium Series 669, Washington D.C. (1997). 42. V. Smith and E. Trevitt, WO 90/10667 (1990). 43. J. Quintans, M. Xanthos and U. Yilmazer, “Rheological Characterization of Unmodified and Chemically Modified PET Resins,” PPS North American Meeting, August 17–19, Toronto, Ontario (1998). 44. E. A. Grulke, Polymer Process Engineering, pp. 375–377, TPS Publishing Company, Lexington, KY (1994).

© 2000 by CRC Press LLC

45. R. Edelman, F. M. Berardinelli and K. F. Wissbrun, US 4,161,579 (1979). 46. M. Xanthos, S. K. Dey and Q. Zhang, “Sheet extrusion of Foamed Polyethylene Terephthalate,” 1998 TAPPI Polymers, Laminations and Coatings Conference, pp. 645–650, TAPPI Press, Atlanta (1998). 47. C. B. Park and L. K. Cheung, Polym. Eng.Sci., 37, 1, 1 (1997). 48. Q. Zhang, M. Xanthos and S. K. Dey, “In-line Measurement of Gas Solubility in Polyethylene Terephthalate Melts During Foam Extrusion,” Symp. on Porous, Cellular and Microcellular, ASME Intern. Mech. Eng. Congr. and Exp., MD-Vol. 82, 75 (1998). 49. G. Boone, Proc. Foam Conference 96, LCM Public Relations, pp. 145–157, Somerset, NJ, December 10–12 (1996). 50. V. Tan, Polymer Processing Institute, Newark, NJ, private communication (1998). 51. T. M. Cheung, C. L. Davis and J. E. Prince, US 4,891,631 (1991). 52. Y. Kitamori, “PET Foams and Applications,” Proc. of Thermoplastics Foams Technical Conference, sponsored by Industrial Technology Research Institute, Taipei, Taiwan (1995). 53. M. T. Huggard, US 4,462,947 (1984) and 4,466,933 (1984). 54. Y. Ahn, G. Ivanov, T. Shutov and H. Al-Ghatta, in “Cellular and Microcellular Materials,” Proc. ASME Intern. Mech. Eng. Cong. and Exp., MD-Vol 53, p. 1 (1994).

© 2000 by CRC Press LLC