RHEOLOGY AND PROCESSING OF POLYMERIC MATERIALS Polymer Processing

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Author: Chang Dae Han

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Rheology and Processing of Polymeric Materials Volume 2

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RHEOLOGY AND PROCESSING OF POLYMERIC MATERIALS Volume 2 Polymer Processing

Chang Dae Han Department of Polymer Engineering The University of Akron

2007

Oxford University Press, Inc., publishes works that further Oxford University’s objective of excellence in research, scholarship, and education. Oxford New York Auckland Cape Town Dar es Salaam Delhi Hong Kong Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto

Karachi

With ofﬁces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam

Copyright © 2007 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 www.oup.com Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Han, Chang Dae. Rheology and processing of polymeric materials/Chang Dae Han. v. cm. Contents: v. 1 Polymer rheology—; v. 2 Polymer processing— Includes bibliographical references and index. ISBN: 978-0-19-518782-3 (vol. 1); 978-0-19-518783-0 (vol. 2) 1. Polymers–Rheology. 1. Title. QC189.5.H36 2006 620.1 920423—dc22

2005036608

9 8 7 6 5 4 3 2 1 Printed in the United States of America on acid-free paper

In Memory of My Parents

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Preface

In the past, a number of textbooks and research monographs dealing with polymer rheology and polymer processing have been published. In the books that dealt with rheology, the authors, with a few exceptions, put emphasis on the continuum description of homogeneous polymeric ﬂuids, while many industrially important polymeric ﬂuids are heterogeneous, multicomponent, and/or multiphase in nature. The continuum theory, though very useful in many instances, cannot describe the effects of molecular parameters on the rheological behavior of polymeric ﬂuids. On the other hand, the currently held molecular theory deals almost exclusively with homogenous polymeric ﬂuids, while there are many industrially important polymeric ﬂuids (e.g., block copolymers, liquid-crystalline polymers, and thermoplastic polyurethanes) that are composed of more than one component exhibiting complex morphologies during ﬂow. In the books that dealt with polymer processing, most of the authors placed emphasis on showing how to solve the equations of momentum and heat transport during the ﬂow of homogeneous thermoplastic polymers in a relatively simple ﬂow geometry. In industrial polymer processing operations, more often than not, multicomponent and/or multiphase heterogeneous polymeric materials are used. Such materials include microphase-separated block copolymers, liquid-crystalline polymers having mesophase, immiscible polymer blends, highly ﬁlled polymers, organoclay nanocomposites, and thermoplastic foams. Thus an understanding of the rheology of homogeneous (neat) thermoplastic polymers is of little help to control various processing operations of heterogeneous polymeric materials. For this, one must understand the rheological behavior of each of those heterogeneous polymeric materials. There is another very important class of polymeric materials, which are referred to as thermosets. Such materials have been used for the past several decades for the fabrication of various products. Processing of thermosets requires an understanding of the rheological behavior during processing, during which low-molecular-weight oligomers (e.g., unsaturated polyester, urethanes, epoxy resins) having the molecular

viii

PREFACE

weight of the order of a few thousands undergo chemical reactions ultimately giving rise to cross-linked networks. Thus, a better understanding of chemorheology is vitally important to control the processing of thermosets. There are some books that dealt with the chemorheology of thermosets, or processing of some thermosets. But, very few, if any, dealt with the processing of thermosets with chemorheology in a systematic fashion. The preceding observations have motivated me to prepare this two-volume research monograph. Volume 1 aims to present the recent developments in polymer rheology placing emphasis on the rheological behavior of structured polymeric ﬂuids. In so doing, I ﬁrst present the fundamental principles of the rheology of polymeric ﬂuids: (1) the kinematics and stresses of deformable bodies, (2) the continuum theory for the viscoelasticity of ﬂexible homogeneous polymeric liquids, (3) the molecular theory for the viscoelasticity of ﬂexible homogeneous polymeric liquids, and (4) experimental methods for measurement of the rheological properties of polymeric liquids. The materials presented are intended to set a stage for the subsequent chapters by introducing the basic concepts and principles of rheology, from both phenomenological and molecular perspectives, of structurally simple ﬂexible and homogeneous polymeric liquids. Next, I present the rheological behavior of various polymeric materials. Since there are so many polymeric materials, I had to make a conscious, though somewhat arbitrary, decision on the selection of the polymeric materials to be covered in this volume. Admittedly, the selection has been made on the basis of my research activities during the past three decades, since I am quite familiar with the subjects covered. Speciﬁcally, the various polymeric materials considered in Volume 1 range from rheologically simple, ﬂexible thermoplastic homopolymers to rheologically complex polymeric materials including (1) block copolymers, (2) liquid-crystalline polymers, (3) thermoplastic polyurethanes, (4) immiscible polymer blends, (5) particulate-ﬁlled polymers, organoclay nanocomposites, and ﬁber-reinforced thermoplastic composites, (6) molten polymers with solubilized gaseous component. Also, chemorheology is included in Volume 1. Volume 2 aims to present the fundamental principles related to polymer processing operations. In presenting the materials in this volume, again, the objective is not to provide the recipes that necessarily guarantee better product quality. Rather, I put emphasis on presenting fundamental approaches to effectively analyze processing problems. Polymer processing operations require combined knowledge of polymer rheology, polymer solution thermodynamics, mass transfer, heat transfer, and equipment design. Speciﬁcally, in Volume 2, I present the fundamental aspects of several processing operations (plasticating single-screw extrusion, wire-coating extrusion, ﬁber spinning, tabular ﬁlm blowing, injection molding, coextrusion, and foam extrusion) of thermoplastic polymers and three processing operations (reaction injection molding, pultrusion, and compression molding) of thermosets. In Volume 2, I have reused some materials presented in Volume 1. In the preparation of these volumes I have tried to present the fundamental concepts and/or principles associated with the rheology and processing of the various polymeric materials selected and I have tried to avoid presenting technological recipes. In so doing, I have pointed out an urgent need for further experimental and theoretical investigations. I sincerely hope that the materials in this monograph will not only encourage further experimental investigations but also stimulate future development of theory. I wish

PREFACE

ix

to point out that I have tried not to cite articles appearing in conference proceedings and patents unless absolutely essential, because they did not go through rigorous peer review processes. Much of the material presented in this monograph is based on my research activities with very capable graduate students at Polytechnic University from 1967 to 1992 and at the University of Akron from 1993 to 2005. Without their participation and dedication to the various research projects that I initiated, the completion of this monograph would not have been possible. I would like to acknowledge with gratitude that Professor Jin Kon Kim at Pohang University of Science and Technology in the Republic of Korea read the draft of Chapters 4, 6, 7, and 8 of Volume 1 and made very valuable comments and suggestions for improvement. Professor Ralph H. Colby at Pennsylvania State University read the draft of Chapter 7 of Volume 1 and made helpful comments and suggestions, for which I am very grateful. Professor Anthony J. McHugh at Lehigh University read the draft of Chapter 6 of Volume 2 and made many useful comments, for which I am very grateful. It is my special privilege to acknowledge wonderful collaboration I had with Professor Takeji Hashimoto at Kyoto University in Japan for the past 18 years on phase transitions and phase behavior of block copolymers. The collaboration has enabled me to add luster to Chapter 8 of Volume 1. The collaboration was very genuine and highly professional. Such a long collaboration was made possible by mutual respect and admiration. Chang Dae Han The University of Akron Akron, Ohio June, 2006

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Contents

Remarks on Volume 2, xvii

Part I Processing of Thermoplastic Polymers 1 Flow of Polymeric Liquid in Complex Geometry, 3 1.1 Introduction, 3 1.2 Flow through a Rectangular Channel, 4 1.2.1 Flow Patterns in a Rectangular Channel, 4 1.2.2 Extrudate Swell from a Rectangular Channel, 6 1.2.3 Analysis of Flow through a Rectangular Channel, 6 1.3 Flow in the Entrance Region of a Slit Die, 20 1.4 Flow through a Converging or Tapered Channel, 25 1.5 Exit Region Flow, 32 1.6 Flow through a Channel Having Small Side Holes or Slots, 35 1.7 Analysis of Flow in a Coat-Hanger Die, 40 1.7.1 Analysis of Flow in the Manifold, 42 1.7.2 Analysis of Flow in the Coat-Hanger Section, 45 1.8 Summary, 48 Problems, 49 Notes, 53 References, 54

xii

CONTENTS

2 Plasticating Single-Screw Extrusion, 56 2.1 Introduction, 56 2.2 Performance of Plasticating Single-Screw Extruders for Semicrystalline Polymers, 57 2.2.1 Analysis of the Solid-Conveying Section, 59 2.2.2 Analysis of the Melting Section, 60 2.2.3 Analysis of Melt-Conveying Section, 67 2.2.4 Comparison of Prediction with Experiment, 68 2.3 Performance of Fluted Mixing Heads in a Plasticating Single-Screw Extruder, 85 2.3.1 Analysis of the Flow through the Maddock Mixing Head, 86 2.3.2 Comparison of Prediction with Experiment, 90 2.4 Performance of Plasticating Barrier-Screw Extruders, 98 2.4.1 Stability of the Solid Bed in a Plasticating Barrier-Screw Extruder, 102 2.4.2 Analysis of the Performance of Plasticating Barrier-Screw Extruders, 107 2.4.3 Comparison of Prediction with Experiment, 111 2.5 Performance of Plasticating Single-Screw Extruders for Amorphous Polymers, 114 2.5.1 The Concept of Critical Flow Temperature, 115 2.5.2 Analysis of Plasticating Extrusion of Amorphous Polymers, 116 2.5.3 Comparison of Prediction with Experiment, 119 2.6 Summary, 128 Notes, 129 References, 130 3 Morphology Evolution in Immiscible Polymer Blends during Compounding, 132 3.1 Introduction, 132 3.2 Morphology Evolution in Immiscible Polymer Blend during Compounding in an Internal Mixer, 134 3.2.1 Morphology Evolution in Blends Consisting of Two Semicrystalline Polymers, 137 3.2.2 Morphology Evolution in Blends Consisting of Two Amorphous Polymers, 140 3.2.3 Morphology Evolution in Blends Consisting of an Amorphous Polymer and a Semicrystalline Polymer, 144 3.3 Morphology Evolution in Immiscible Polymer Blend during Compounding in a Twin-Screw Extruder, 154 3.3.1 Morphology Evolution in Blends Consisting of Two Amorphous Polymers, 156 3.3.2 Morphology Evolution in Blends Consisting of an Amorphous Polymer and a Semicrystalline Polymer, 161

CONTENTS

3.4 Stability of Co-Continuous Morphology during Compounding, 169 3.5 Summary, 174 Appendix: Theoretical Interpretation of Figure 3.34, 177 Notes, 179 References, 179 4 Compatibilization of Two Immiscible Homopolymers, 181 4.1 Introduction, 181 4.2 Experimental Observations of Compatibilization of Two Immiscible Homopolymers Using a Block Copolymer, 186 4.2.1 A/B/(A-block-B) Ternary Blends, 187 4.2.2 A/B/(A-block-C) Ternary Blends, 194 4.2.3 A/B/(C-block-D) Ternary Blends, 210 4.3 Reactive Compatibilization of Two Immiscible Polymers, 224 4.4 Summary, 229 Notes, 231 References, 232 5 Wire-Coating Extrusion, 235 5.1 5.2 5.3 5.4

Introduction, 235 Analysis of Wire-Coating Extrusion, 236 Experimental Observations, 245 Summary, 253 Problems, 255 Notes, 256 References, 256

6 Fiber Spinning, 257 6.1 Introduction, 257 6.2 Fiber Spinning Processes, 258 6.2.1 Melt Spinning Process, 258 6.2.2 Wet Spinning Process, 260 6.2.3 Dry Spinning Process, 262 6.2.4 Other Fiber Spinning Processes, 263 6.3 High-Speed Melt Spinning, 268 6.3.1 Experimental Observations of High-Speed Melt Spinning, 269 6.3.2 Modeling of High-Speed Melt Spinning, 273 6.3.3 Model Prediction and Comparison with Experiment, 284 6.4 Spinnability, 294 6.5 Summary, 296 Problems, 297 Notes, 300 References, 302

xiii

xiv

CONTENTS

7 Tubular Film Blowing, 305 7.1 Introduction, 305 7.2 Processing Characteristics of Tubular Film Blowing, 307 7.2.1 Kinematics and Stress Field in Tubular Film Blowing, 308 7.2.2 Tensile Stresses at the Freeze Line and Processing–Property Relationships in Tubular Film Blowing, 311 7.3 Analysis of Tubular Film Blowing Including Extrudate Swell, 317 7.3.1 Force Balance Equation, 319 7.3.2 Energy Balance Equation, 322 7.3.3 Viscoelastic Constitutive Equation, 323 7.3.4 Analysis of Tubular Film Blowing in the Extrudate Swell Region, 326 7.3.5 Analysis of Tubular Film Blowing in the Stretching Region, 329 7.3.6 Model Predictions and Comparison with Experiment, 330 7.4 Tubular Film Blowability, 341 7.5 Summary, 346 Problems, 348 Notes, 348 References, 349 8 Injection Molding, 351 8.1 Introduction, 351 8.2 Flow of Molten Polymer through a Runner, 354 8.3 Injection Molding of Amorphous Polymers, 358 8.3.1 Flow Patterns during Mold Filling, 358 8.3.2 Governing System Equations for Mold Filling of Amorphous Polymers, 363 8.3.3 Molecular Orientation during Mold Filling and Residual Stress in Injection Molded Articles, 366 8.4 Injection Molding of Semicrystalline Polymers, 370 8.4.1 Crystallization during Injection Molding, 370 8.4.2 Governing System Equations for Injection Molding of Semicrystalline Polymers, 372 8.4.3 Morphology of Injected-Molded Semicrystalline Polymers, 373 8.5 Summary, 375 Notes, 376 References, 376 9 Coextrusion, 379 9.1 Introduction, 379 9.2 Coextrusion Die Systems, 382 9.2.1 Feedblock Die System for Flat-Film or Sheet Coextrusion, 382 9.2.2 Multimanifold Die System for Flat-Film or Sheet Coextrusion, 383

CONTENTS

9.2.3 9.2.4 9.3

9.4 9.5

xv

Feedblock Die System for Blown-Film Coextrusion, 384 Rotating-Mandrel Die System for Blown-Film Coextrusion, 386 Polymer–Polymer Interdiffusion across the Initially Sharp and Flat Interface, 388 9.3.1 Polymer–Polymer Interdiffusion under Static Conditions, 389 9.3.2 Polymer–Polymer Interdiffusion in the Shear Flow Field, 400 Nonisothermal Coextrusion, 407 Summary, 417 Appendix: Derivation of Equation (9.36), 418 Problems, 419 Notes, 421 References, 421

10 Foam Extrusion, 424 10.1 Introduction, 424 10.2 Solubility and Diffusivity of Gases in a Molten Polymer, 425 10.2.1 Solubility of Gases in a Molten Polymer, 425 10.2.2 Diffusivity of Gases in a Molten Polymer, 433 10.3 Bubble Nucleation in Polymeric Liquids, 443 10.3.1 Experimental Observations of Bubble Nucleation, 446 10.3.2 Theoretical Considerations of Bubble Nucleation in Polymer Solutions, 462 10.4 Foam Extrusion, 468 10.4.1 Processing–Property–Morphology Relationships in Proﬁle Foam Extrusion, 469 10.4.2 Processing–Property Relationships in Sheet Foam Extrusion, 482 10.5 Summary, 487 Problems, 488 Notes, 489 References, 489

Part II Processing of Thermosets 11 Reaction Injection Molding, 495 11.1 Introduction, 495 11.2 Analysis of Reaction Injection Molding, 497 11.2.1 Main Flow, 498 11.2.2 Front Flow, 501

xvi

CONTENTS

11.2.3 Cure Stage, 502 11.2.4 Chemorheological Model, 503 11.3 Conversion and Temperature Proﬁles during Mold Filling, 503 11.4 Summary, 512 Problems, 513 Notes, 514 References, 515 12 Pultrusion of Thermoset/Fiber Composites, 517 12.1 Introduction, 517 12.2 Effect of Mixed Initiators on the Cure Kinetics of Unsaturated Polyester, 519 12.3 Cure Kinetics of Unsaturated Polyester/Fiber Composite, 525 12.4 Analysis of the Pultrusion of Thermoset/Fiber Composite, 528 12.4.1 General System Equations, 528 12.4.2 System Equations with an Empirical Kinetic Model, 530 12.4.3 System Equations with a Mechanistic Kinetic Model, 531 12.5 Conversion and Temperature Proﬁles in a Pultrusion Die, 531 12.6 Summary, 540 Problems, 541 References, 542 13 Compression Molding of Thermoset/Fiber Composites, 544 13.1 13.2 13.3 13.4

Introduction, 544 Thickening Behavior of Unsaturated Polyester, 547 Effect of Pressure on the Curing of Unsaturated Polyester, 552 Analysis of Compression Molding of Unsaturated Polyester/Glass Fiber Composite, 561 13.5 Time Evolution of Temperature during Compression Molding of Unsaturated Polyester/Glass Fiber Composite, 564 13.6 Summary, 568 References, 569 Author Index, 571 Subject Index, 578

Remarks on Volume 2

This volume consists of two parts. Part I has ten chapters presenting fundamental principles associated with the processing of thermoplastic polymers. A thermoplastic polymer, when heated, is transformed into a liquid, which can then readily be transported through a shaping device (e.g., extrusion die or mold cavity), and then cooled down to a solid, rendering speciﬁc mechanical/physical properties. Barring thermal degradation and/or chemical reaction, a thermoplastic polymer can be regenerated by heating and cooling repeatedly. Since the processing of thermoplastic polymers in the molten state invariably involves ﬂow, a successful processing operation requires a good understanding of their rheological behavior, which we have discussed in Part II of Volume 1. Since there are so many different polymer processing operations practiced in industry, I had to make a conscious, though somewhat arbitrary, decision on the selection of the polymer processing operations to be covered in this volume. Admittedly, the selection has been made on the basis of my research activities during the past three decades. Speciﬁcally, Chapter 1 presents the ﬂow of polymeric liquids in complex geometries. In this chapter, we consider the ﬂow of a viscoelastic ﬂuid through a rectangular channel, through a converging channel (entrance ﬂow), and through a channel having small side holes. Chapter 2 presents plasticating single-screw extrusion. This chapter describes the principles associated with the design of screws for single-screw extruders. Chapter 3 presents the morphology evolution in immiscible polymer blends during compounding. Chapter 4 presents the compatibilization of two immiscible homopolymers, in which the principles associated with the selection of a block copolymer to compatibilize a pair of immiscible homopolymers are presented. Chapter 5 presents wire coating extrusion, placing emphasis on the principles of die design. Chapter 6 presents ﬁber spinning, with a detailed discussion of high-speed melt spinning as reported in the 1980s and 1990s. Chapter 7 presents tubular ﬁlm blowing, in which an analysis of tubular ﬁlm blowing including extrudate swell region is discussed. xvii

xviii

REMARKS ON VOLUME 2

Chapter 8 presents the fundamentals of injection molding, placing emphasis on the necessity of developing a mathematical model based on realistic initial and boundary conditions under normal injection speeds of industrial practice, in which the shear rates in the runner usually exceed a few thousand reciprocal seconds. A realistic modeling of the mold-ﬁlling process must include the analysis of the ﬂow of a viscoelastic polymer melt through the runner as an integral part of the analysis of the mold ﬁlling process, because the inlet conditions for the equations of motion and energy for mold ﬁlling must come from the solutions of the equations of motion and energy for the runner. Chapter 9 presents coextrusion, placing emphasis on the importance of polymer–polymer interdiffusion during coextrusion. In the 1970s and 1980s, the fundamental aspects of coextrusion were extensively discussed, while in the 1990s the industry continued to improve machinery. Chapter 10 presents the fundamentals of foam extrusion, placing emphasis on the importance of a good understanding of the solubility and diffusivity of gaseous component or volatile liquid in a molten polymer and, also, the importance of a better understanding of the phenomenon of bubble nucleation in a molten polymer. Part II has three chapters presenting the fundamental principles associated with the processing of thermosets. Thermosets are as important as thermoplastic polymers in the fabrication of various polymeric products of industrial importance. There are many thermosets; to name only a few, epoxy resin, unsaturated polyester resin, and urethane resin. Processing of thermosets is accompanied by exothermic chemical reactions which generate heat. Therefore an analysis of processing of thermosets must include the heat transfer and chemical reaction kinetics, in addition to momentum transfer. Chapter 11 presents reaction injection molding, Chapter 12 presents pultrusion of thermoset/ﬁber composites, and Chapter 13 presents compression molding of thermoset/ﬁber composites. In these three chapters, emphasis is placed on the modeling of seemingly complicated processing operations. Each chapter can be expanded considerably by including the mechanical properties of the fabricated products in terms of material and processing variables, and also the design of processing equipment. Such an expansion of each chapter would require considerable space, which was not available to this volume. Thus, an approach is taken to highlight the modeling aspects by using the chemorheological models presented in Chapter 14 of Volume 1. C.D.H.

Part I

Processing of Thermoplastic Polymers

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1

Flow of Polymeric Liquid in Complex Geometry

1.1

Introduction

The ﬂow geometry encountered in many polymer processing operations of industrial importance is often far more complex than that in cylindrical or slit dies. As will be shown in the following chapters, the industry manufactures polymeric products using very complex ﬂow geometries. For instance, the ﬁber industry produces “shaped ﬁbers,” which have cross sections that are noncircular. What is most intriguing in the production of shaped ﬁbers is that a desired ﬁber shape is often produced by spinneret holes whose cross-sectional shape is quite different from that of the ﬁnal ﬁber produced. Hence, an important question may be raised as to how one can determine, from a sound theoretical basis, the cross-sectional shape of spinneret holes that will produce a ﬁber with a desired cross-sectional shape. In extrusion and injection molding, a polymeric liquid invariably passes through a large cross section before entering into a small cross section, and such a ﬂow is referred to as “entrance ﬂow.” The entrance ﬂow of polymeric liquids, due to their viscoelastic nature, is quite different from that of Newtonian liquids. Similarly, the ﬂow behavior of viscoelastic polymeric liquids near the exit of a die, commonly referred to as “exit ﬂow,” is quite different from that of Newtonian liquids. A better understanding of the unique characteristics of both entrance and exit ﬂows of viscoelastic polymeric ﬂuids is essential for successful design of extrusion dies and molds, as well as to solve difﬁcult technical problems related to a particular processing operation. Before presenting speciﬁc polymer processing operations in following chapters, in this chapter we consider the ﬂow of polymeric liquids through complex geometry: (1) fully developed ﬂow through a rectangular channel with uniform channel depth; (2) fully developed ﬂow through a rectangular channel with a moving channel wall; (3) ﬂow through a rectangular channel with varying channel depth; (4) ﬂow in the entrance region of a rectangular die having constant cross section; (5) ﬂow through 3

4

PROCESSING OF THERMOPLASTIC POLYMERS

a tapered die; (6) ﬂow in the exit region of a cylindrical or slit die; (7) ﬂow through a slit die having side holes; and (8) ﬂow through a coat-hanger die. These ﬂow geometries are encountered in many polymer processing operations. The primary objective of this chapter is to present the unique ﬂow characteristics of viscoelastic polymeric liquids in complex geometries of practical industrial importance.

1.2

Flow through a Rectangular Channel

The ﬂow of polymeric liquids through a rectangular channel having constant cross section or varying cross section is much more complex than the ﬂow through a capillary or slit die considered in Chapter 5 of Volume 1. The complexity arises from both the viscoelastic nature of polymeric ﬂuids and the two-dimensional nature of a rectangular channel. In this section, we present some unique features of ﬂow of polymeric liquids through a rectangular channel. 1.2.1

Flow Patterns in a Rectangular Channel

In the past, using perturbation methods, some investigators (Ericksen 1956; Green and Rivlin 1956; Langlois and Rivlin 1963; Rivlin 1964; Wheeler and Wissler 1966) predicted transverse circulating (secondary) ﬂow patterns in each of the four quadrants of the rectangle, as schematically shown in Figure 1.1, when a viscoelastic ﬂuid ﬂows through a rectangular channel. For instance, using the Rivlin–Ericksen constitutive equation1 (Rivlin and Ericksen 1964), Langlois and Rivlin (1963) found that it required a fourth-order ﬂuid to yield secondary ﬂow, with the second-order ﬂuid only affecting the pressure ﬁeld and the third-order ﬂuid only distorting the normal Newtonian velocity proﬁle. To obtain a streamline pattern of secondary ﬂow in a rectangular channel, one must solve all three components of the equations of motion, whereas in the absence of secondary ﬂow, only the axial component of the equations of motion must

Figure 1.1 Schematic showing secondary ﬂow of viscoelastic ﬂuids in a duct of rectangular cross section.

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

5

be solved. The usually complex form of the constitutive equations for viscoelastic ﬂuids complicates the solution of the equations of motion when secondary ﬂow is to be considered. Wheeler and Wissler (1966) solved the equations of motion numerically for ﬂow through a square duct by considering the Reiner–Rivlin constitutive equation (Reiner 1945; Rivlin 1948), and obtained the streamlines for the secondary ﬂow. They found that the transverse components of velocity are about 1% of the axial component of velocity when the Reynolds number is as large as 100, and that the axial velocity proﬁles computed for a Reynolds number of 26.38 are virtually indistinguishable from those computed when secondary ﬂow was neglected. It should be mentioned that typical values of the Reynolds number in polymer melt ﬂow, owing to very high viscosities, lie below 0.001. The experimental evidence is mixed. Some investigators (Giesekus 1965; Semjonow 1965) report that secondary ﬂows have been observed, and others (Han 1976; Wheeler and Wissler 1965) report that they have not seen evidence of secondary ﬂows. Figure 1.2 gives a micrograph of the cross section of an extrudate, which was obtained in the ﬂow of a blend of polypropylene (PP) and polystyrene (PS) through a rectangular channel having an aspect ratio of 2. In this ﬁgure, the dark areas represent the PS phase and the bright areas represent the PP phase, obtained by etching out the PS phase with the aid of xylene as solvent before the photograph was taken using an optical microscope (Han 1976). Well-characterized rectilinear ﬂow patterns are observed in Figure 1.2, which is at variance with some of the theoretical predictions depicted schematically in Figure 1.1. We can thus conclude that the occurrence of secondary ﬂow is not a general phenomenon that would occur in the ﬂow of every viscoelastic ﬂuid in a rectangular channel. The apparent absence of secondary ﬂow patterns in the rectangular channel having an aspect ratio of 2, shown in Figure 1.2, may be attributable to the extremely slow motion that is characteristic of polymer melt ﬂow having a very low Reynolds number (i.e., below 0.001). Figure 1.3 gives a micrograph of the cross section of an extrudate, which was obtained in the ﬂow of a blend of PP and PS through a rectangular channel having an aspect ratio of 6. It is of interest to observe in this ﬁgure that the ﬂow patterns are only rectilinear at the region that corresponds to an aspect ratio of about 4, and different ﬂow patterns set in at a region near the edge of the long side of the cross section. At present, the physical origin of the ﬂow patterns given in Figure 1.3 is not clear.

Figure 1.2 Micrograph of the extrudate cross section showing the ﬂow patterns of a blend of polypropylene (the bright areas) and polystyrene (the dark areas) that was extruded through a rectangular channel having an aspect ratio of 2. (Reprinted from Han, Rheology in Polymer Processing, Chapter 6. Copyright © 1976, with permission from Elsevier.)

6

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 1.3 Micrograph of the extrudate cross section showing the ﬂow patterns of a blend of polypropylene (the bright areas) and polystyrene (the dark areas) that was extruded through a rectangular channel having an aspect ratio of 6. (Reprinted from Han, Rheology in Polymer Processing, Chapter 6. Copyright © 1976, with permission from Elsevier.)

1.2.2

Extrudate Swell from a Rectangular Channel

In Chapter 1 of Volume 1, we showed that upon exiting from a circular die, the extrudate of a viscoelastic molten polymer gives rise to a circular cross section that is larger than the cross section of the die. Figure 1.4a gives a photograph of the extrudate cross section of a high-density polyethylene (HDPE), extruded through a rectangular die having an aspect ration of 6, and Figure 1.4b gives a schematic depicting the extent that extrudate swell increases with volumetric ﬂow rate. It is seen that the extrudate swell is more pronounced on the long side of a rectangular channel than on the short side, and a maximum swell occurs at the center of the long side. The physical origin of nonuniform extrudate swell from a rectangular channel can be found from measurements of wall normal stresses along a rectangular channel, similar to those for a cylindrical or slit die, as described in Chapter 5 of Volume 1. Figure 1.5 gives a schematic diagram of a rectangular channel, along which pressure transducers are mounted on both long and short sides of the rectangle. Figure 1.6 gives the proﬁles of wall normal stress for an HDPE at 180 ◦ C ﬂowing through the rectangular channel. It can be seen in Figure 1.6 that the wall normal stresses measured along the centerline of the long side of the rectangle are greater than those measured along the centerline of the short side. Figure 1.7 shows plots of the exit pressure (PExit ) versus volumetric ﬂow for HDPE melt, showing that (1) the PExit at the center of the long side of the rectangle is greater than that at the center of the short side, consistent with the experimental observation that extrudate swell is more pronounced at the center of the long side than at the center of the short side (see Figure 1.4), and (2) at any given position in the die, the PExit increases with volumetric ﬂow rate, consistent with the experimental observation that the extrudate swells more as the ﬂow rate is increased. It can be concluded, therefore, that the nonuniform distribution of extrudate swell given in Figures 1.2–1.4 is correlated to the nonuniform distribution of wall normal stresses of the polymer melt at the exit of the rectangular die. 1.2.3

Analysis of Flow through a Rectangular Channel

Let us consider the situation, as schematically shown in Figure 1.8, where a polymeric ﬂuid at temperature To enters into a rectangular channel with constant cross section

Figure 1.4 (a) Photograph of extrudate cross section of an HDPE extruded at 180 ◦ C through a

rectangular channel having an aspect ratio of 6, and (b) schematic diagram showing the extrudate swell as affected by volumetric ﬂow rate. (Reprinted from Han, Rheology in Polymer Processing, Chapter 6. Copyright © 1976, with permission from Elsevier.)

Figure 1.5 Schematic of the layout of pressure transducer tap holes in the rectangular channel

having an aspect ratio of 6. (Reprinted from Han, Rheology in Polymer Processing, Chapter 5. Copyright © 1976, with permission from Elsevier.)

7

Figure 1.6 Proﬁles of wall normal stress of an HDPE melt at 180 ◦ C

along the centerlines of the long side (open symbols) and short side (ﬁlled symbols) of the rectangular channel (aspect ratio of 6) at various volumetric ﬂow rates (cm3 /min): (, 䊉) 97.1, (, ) 72.6, (, ) 65.7, (7, ) 47.6, (, ) 36.6, and (3, ◆) 26.5. (Reprinted from Han, Rheology in Polymer Processing, Chapter 6. Copyright © 1976, with permission from Elsevier.)

Figure 1.7 Plots of exit pressure

versus volumetric ﬂow rate for an HDPE melt at 180 ◦ C in a rectangular channel having an aspect ratio of 6: () at center of long side, and () at center of short side. (Reprinted from Han, Rheology in Polymer Processing, Chapter 6. Copyright © 1976, with permission from Elsevier.)

8

9

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY Figure 1.8 Schematic showing ﬂow through a rectangular channel with constant cross section.

whose walls are kept at temperature Tw . In order to make the solution of the system of equations tractable, the following assumptions are made: (1) the velocity vz in the axial direction (z) depends on y and x only; (2) there exists a cross channel velocity vx in the transverse direction (x); (3) the magnitude of the velocity in the y-direction is negligibly small (i.e., vy ∼ = 0), as compared with that of vz and vx ; (4) the pressure gradient in the y-direction, ∂p/∂y, is equal to zero; (5) the conductive heat transfer in the z-direction is negligibly small, as compared with that in the y- and x-directions; (6) the convective heat transfer in the x- and y-directions are much smaller than that in the z-direction. For simplicity, let us consider the situation where the viscous effect is predominant over the elastic effect and no secondary ﬂow exists. We then have the following system of equations. 1. Momentum balance equations: ∂vx ∂ ∂p + =0 η − ∂x ∂y ∂y ∂v ∂v ∂ ∂ ∂p + η z + η z =0 − ∂z ∂x ∂x ∂y ∂y

(1.1) (1.2)

2. Energy balance equation: ∂T =k ρcp vz ∂z

∂ 2T ∂ 2T + 2 ∂x ∂y 2

+η

∂vx ∂y

2

+

∂vz ∂x

2

+

∂vz ∂y

2 (1.3)

in which η is the viscosity, which depends on both temperature and the rate of deformation, ρ is the density, cp is the speciﬁc heat, and k is the thermal conductivity. Let us assume that η follows a power-law model, which is then expressed by2 η = mo exp(−bT )

∂vx ∂y

2 +

∂vz ∂x

2 +

∂vz ∂y

2 (n−1)/2 (1.4)

10

PROCESSING OF THERMOPLASTIC POLYMERS

where mo is the preexponential factor, b is a constant, and n is the power-law index. Equations (1.1)–(1.4) must be solved, under the appropriate boundary conditions, for the velocities vx and vz and temperature T. Next, we consider a number of different situations. 1.2.3.1 Analysis of Isothermal Flow through a Rectangular Channel with Constant Cross Section For isothermal ﬂow in a rectangular channel with constant cross section, where crosschannel ﬂow is assumed to be negligible (i.e., vx = 0), we only have to solve Eq. (1.2) with the following expression for η: η = ko

∂vz ∂x

2 +

∂vz ∂y

2 (n−1)/2 (1.5)

with the boundary conditions vz = 0 at x = 0 and x = W , and also at y = 0 and y = H (see Figure 1.8 for the coordinates chosen). In the ﬂow under consideration, the velocity ﬁeld is given by vz = f (x, y) and vx = vy = 0. Figure 1.9a gives the contours of constant velocity (i.e., isovels), Figure 1.9b gives the contours of constant velocity gradient, and Figure 1.10 gives the three-dimensional plots of the axial velocity proﬁles vz (x, y) for fully developed isothermal ﬂow of a lowdensity polyethylene (LDPE) melt at 200 ◦ C in a rectangular channel with W/H = 2. These ﬁgures were obtained by numerically solving Eq. (1.2) with the aid of Eq. (1.5) using the following numerical values for the rheological parameters: n = 0.5 and ko = 2.739 × 103 Pa·s0.5 . The contours of constant velocity and constant velocity gradient, respectively, for fully developed isothermal ﬂow of the same power-law ﬂuid are given in Figure 1.11 in a rectangular channel with W/H = 4, in Figure 1.12 in a rectangular channel with W/H = 6, and in Figure 1.13 in a rectangular channel with W/H = 10. It should be mentioned that when the aspect ratio (W/H) of a rectangular channel becomes very large (i.e., larger than 10), the ﬂow through such a channel can be treated as slit ﬂow. In Chapter 5 of Volume 1 we discussed slit ﬂow as a means of measuring the viscometric ﬂow properties of polymeric ﬂuids. With reference to Figure 5.16 in Volume 1, we have pointed out that the long side of the cross section of a slit die must be sufﬁciently large, such that the isovels in the die cross section would not have curvature at the location where the pressure transducer is ﬂush-mounted. A close examination of the contours of the isovels in the ﬂow channels with W/H = 6 (Figure 1.12) and W/H = 10 (Figure 1.13) clearly reveals that the rectangular channel with W/H = 6 would not be suitable for use as a slit die, while the rectangular channel with W/H = 10 may be suitable. 1.2.3.2 Analysis of Isothermal Flow through a Rectangular Channel with Sliding Upper Plate Let us consider the ﬂow through a rectangular channel with the upper plate moving at a constant velocity Vb in the direction that forms an angle θ with the channel axis z, as shown schematically in Figure 1.14. This type of ﬂow is encountered in the

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

11

Figure 1.9 (a) Contours of constant velocity (m/s) and (b) contours of constant velocity gradients (s−1 ) in isothermal ﬂow of an LDPE melt at 200 ◦ C through a rectangular channel having an

aspect ratio of 2.

melt-conveying section of a plasticating single-screw extruder, as will be discussed in greater detail in the next chapter. Due to the sliding motion of the upper plate in the direction forming an angle θ with respect to the z-axis, there is a cross-channel (i.e., transverse) ﬂow in addition to the down-channel ﬂow, as will be elaborated on below. The ﬂow under consideration can be analyzed by Eqs. (1.1) and (1.2), with the expression for η of the form η = ko

∂vx ∂y

2 +

∂vz ∂x

2 +

∂vz ∂y

2 (n−1)/2 (1.6)

Figure 1.10 Three-dimensional plots of velocity proﬁles in the isothermal ﬂow of an LDPE melt at 200 ◦ C through a rectangular channel having an aspect ratio of 2.

Figure 1.11 (a) Contours of constant velocity (m/s) and (b) contours of constant velocity gradient (s−1 ) in the isothermal ﬂow of an LDPE melt at 200 ◦ C through a rectangular channel having

an aspect ratio of 4. 12

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

13

Figure 1.12 (a) Contours of constant velocity (m/s) and (b) contours of constant velocity gradient (s−1 ) in the isothermal ﬂow of an LDPE melt at 200 ◦ C through a rectangular channel having

an aspect ratio of 6.

under the boundary conditions (1) vz = 0 and vx = 0 at y = 0, (2) vz = Vbz and vx = −Vbx at y = H, and (3) vz = 0 at x = 0 and x = W . Figure 1.15a gives contours of constant velocity, Figure 1.15b gives contours of constant velocity gradient, and Figure 1.16 gives three-dimensional plots of the axial velocity vz (x, y) for isothermal fully developed ﬂow of an LDPE melt at 200 ◦ C in the rectangular channel with W/H = 4 and θ = 17.7◦ , in which the upper plate moves at Vb = 0.478 m/s and dp/dz = 1.28 × 106 Pa/m. The power-law constants appearing in Eq. (1.6) for the LDPE melt at 200 ◦ C are n = 0.5 and ko = 2.739 × 103 Pa·s0.5 . Notice the differences in the contours of isovels and velocity gradients between the ﬂow through a rectangular channel with sliding upper plate (Figure 1.15) and the rectilinear ﬂow through the stationary parallel plates (Figure 1.11); speciﬁcally, in Figure 1.15a

Figure 1.13 (a) Contours of constant velocity (m/s) and (b) contours of constant velocity gradient (s−1 ) in the isothermal ﬂow of an LDPE melt at 200 ◦ C through a rectangular channel having

an aspect ratio of 10.

Figure 1.14 Schematic showing the ﬂow through a rectangular channel with a sliding upper

plate.

Figure 1.15 (a) Contours of constant velocity (m/s) and (b) contours of constant velocity gradient (s−1 ) in the isothermal ﬂow of an LDPE melt at 200 ◦ C through a rectangular channel having

an aspect ratio of 4, where the upper plate moves with a velocity of 0.479 m/s in the direction forming an angle of 17.7◦ with respect to the channel axis.

14

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

15

Figure 1.16 Threedimensional plots of velocity proﬁles in the isothermal ﬂow of an LDPE melt at 200 ◦ C through a rectangular channel having an aspect ratio of 4, where the upper plate moves with a velocity of 0.479 m/s in the direction forming an angle of 17.7◦ with respect to the channel axis.

the velocity vz becomes negative near the bottom of the channel (i.e., as y approaches zero), indicating that there is “back ﬂow” in the ﬂow channel (compare it with the schematic given in Figure 1.14). The back ﬂow is caused by the combined effects of drag ﬂow and back pressure, which arises when the ﬂow is restricted at the exit of the channel. When the upper plate is stationary, there is no back ﬂow, thus the velocity vz is positive everywhere in the ﬂow channel (see Figure 1.11a). The presence of back ﬂow in the ﬂow through a rectangular channel with sliding upper plate is seen more clearly in Figure 1.16, as compared with the ﬂow through a stationary rectangular channel given in Figure 1.10. 1.2.3.3 Analysis of Nonisothermal Flow through a Rectangular Channel with Sliding Upper Plate Let us consider nonisothermal ﬂow through a rectangular channel with a sliding upper plate, where a polymer at temperature To enters into a rectangular channel whose walls are kept at temperature Tw and the upper plate slides at a velocity Vb in the direction forming an angle θ with the channel axis (see Figure 1.14). For the situation under consideration, it is reasonable to assume that the heat conduction in the x-direction is much smaller than that in the y-direction, simplifying Eq. (1.3) to ∂ 2T ∂T =k 2 +η ρcp vz ∂z ∂y

∂vx ∂y

2 +

∂vz ∂x

2 +

∂vz ∂y

2 (1.7)

16

PROCESSING OF THERMOPLASTIC POLYMERS

This situation can be analyzed by numerically solving Eqs. (1.1), (1.2), and (1.7), with the aid of Eq. (1.4), subjected to the boundary conditions: (1) vz = 0, vx = 0, and T = Tw at y = 0, (2) vz = Vbz , vx = −Vbx , and T = Tw at y = H, (3) vz = 0 and T = Tw at x = 0, and (4) vz = 0 and T = Tw at x = W . It should be mentioned that the numerical solution of Eq. (1.7) becomes unconditionally unstable when vz is negative (see Figure 1.16 for situations where negative vz occurs). One can overcome this numerical instability by replacing the “laboratory” coordinate system with a coordinate system “ﬂoating along” a streamline, which was ﬁrst suggested by McKelvey (1962) and later used by other investigators (Elbirli and Lindt 1984; Han 1988; Tadmor and Klein 1970). More speciﬁcally, the term ρcp vz (∂T /∂z) appearing on the left side of Eq. (1.7) is replaced by ρcp (∂T /∂tR ) and the resulting equation is rewritten in ﬁnite difference form and solved numerically. Note that tR represents the residence time of the ﬂuid particle along a streamline, deﬁned as tR (y) = L[va (y)], where L is the distance along the channel axis and [va (y)] is the average velocity of the ﬂuid particle in the axial direction, which can be determined from the expression [va (y)] = va (y)tf +va (yc )(1−tf ), in which va (y) and va (yc ) are the axial velocities at the position y and its “complementary” position yc , respectively, and tf is the fractional time that a ﬂuid particle spends in the upper ﬂow ﬁeld. Figure 1.17 shows schematically the streamline and complementary position yc in the cross section of the rectangular channel with sliding upper plate. The axial velocity va (y) can be calculated from (see Figure 1.17): va (y) = −vx (y) cos θ + vx (y) sin θ

(1.8)

and the fractional time tf from (McKelvey 1962): tf =

1 1 + vx (y)/vx (yc )

(1.9)

Figure 1.17 Schematic showing the velocity component vx and streamlines in the cross section

of a rectangular ﬂow channel with sliding upper plate.

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

17

The complementary position yc (see Figure 1.17) corresponding to the position y can be determined by satisfying the expression

yc 0

H

vx (y) dy = −

yc

vx (y) dy

(1.10)

Figure 1.18 gives three-dimensional plots of temperature proﬁles for an LDPE melt ﬂowing through a rectangular channel with W/H = 4 and θ = 17.7◦ at three different dimensionless axial positions, z/H = 4, z/H = 20, and z/H = 100, in which the upper plate moves at Vb = 0.478 m/s and dp/dz = 1.28 × 106 Pa/m. To obtain the temperature proﬁles by numerically solving Eqs. (1.1), (1.2), and (1.7), with the aid of Eq. (1.4), the following rheological parameters and thermal properties of LDPE melt were used: mo = 4.754 × 105 Pa·s0.5 , b = 0.0109 K−1 , n = 0.5, ρ = 77.9 kg/m3 , cp = 2.595 × 103 J/(kg K), and k = 0.182 W/(m·K). It can be seen in Figure 1.18 that the melt temperature has a maximum near the sliding upper plate and that the temperature continues to increase in the down-channel direction, which is due to the heat generated by viscous shear heating. In general, molten polymers are very viscous and thus generate considerable amounts of heat during ﬂow by viscous dissipation.

Figure 1.18 Three-dimensional plots of temperature proﬁles, in the presence of viscous shear heating, of an LDPE melt with an inlet temperature of 200 ◦ C through a rectangular channel having an aspect ratio of 4, where the upper plate moves with a velocity of 0.479 m/s in the direction forming an angle of 17.7◦ with respect to the channel axis, at three different dimensionless axial positions: (a) z/H = 4, (b) z/H = 20, and (c) z/H = 100.

18

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 1.18—(Cont’d)

Figure 1.18 indicates that the heat generated by viscous dissipation is actually carried by the LDPE melt in the axial direction, giving rise to a steady rise in bulk mean temperature (i.e., developing temperature ﬁeld), rather than being conducted away through the upper plate. This is due to the very poor thermal conductivity of the LDPE melt. For comparison, Figure 1.19 gives three-dimensional plots of temperature proﬁles for an LDPE melt at z/H = 100 when the convective heat transfer term, ρcp vz (∂T/∂z), appearing on the left-hand side of Eq. (1.7) is neglected. A comparison of Figure 1.19 with Figure 1.18 reveals that the predicted maximum temperature without the convective heat transfer term in the energy equation is about 30 ◦ C higher than that with the convective heat transfer term. The difference between the two situations suggests that in the presence of signiﬁcant viscous shear heating (i.e., when dealing with polymer melts under the ﬂow conditions of industrial importance), an analysis of ﬂow of a very viscous molten polymer through a rectangular channel must include the convective heat transfer term in the energy equation to accurately predict the temperature proﬁles. Figure 1.20 gives three-dimensional plots of velocity proﬁles vz (x, y) at z/H = 100 when neglecting the convective heat transfer term in the energy equation (i.e., the velocity proﬁles are obtained from the numerical solution of Eqs. (1.1), (1.2), and (1.7)), and neglecting the convective heat transfer term. In the presence of signiﬁcant viscous shear heating, the velocities, vx and vz , appearing in Eqs. (1.1), (1.2), and (1.7) are affected by the temperature through the viscosity η that is deﬁned by Eq. (1.4). A comparison of Figure 1.20 with Figure 1.16 indicates that, when compared with the isothermal ﬂow situation, there is a lesser degree of back ﬂow in the ﬂow channel when a signiﬁcant amount of heat is generated due to viscous shear heating. This observation indicates that the energy equation must be included in the system of

Figure 1.19 Three-dimensional plots of temperature proﬁles, in the presence of viscous shear heating, of an LDPE melt with the inlet temperature of 200 ◦ C through a rectangular channel having an aspect ratio of 4, where the upper plate moves with a velocity of 0.479 m/s in the direction forming an angle of 17.7◦ with respect to the channel axis, at a dimensionless axial position, z/H = 100. Here, the temperature proﬁles are obtained from numerical solution of Eqs. (1.1), (1.2), and (1.7), and neglecting the convective heat transfer term.

Figure 1.20 Three-dimensional plots of velocity proﬁles, in the presence of viscous shear heating, of non-isothermal ﬂow of an LDPE melt through a rectangular channel with an aspect ratio of 4, where the upper plate moves with a velocity of 0.479 m/s in the direction forming an angle of 17.7◦ with respect to the channel axis. Here, the velocity proﬁles are obtained from numerical solution of Eqs. (1.1), (1.2), and (1.7), and neglecting the convective heat transfer term.

19

20

PROCESSING OF THERMOPLASTIC POLYMERS

equations to accurately predict the velocity proﬁles of a very viscous molten polymer ﬂowing through a rectangular channel under the processing conditions of industrial importance.

1.3

Flow in the Entrance Region of a Slit Die

When ﬂuid enters a tube from a large reservoir, the velocity proﬁle becomes fully developed at a certain distance, beyond which it has attained either the classical parabolic form characteristic of Newtonian ﬂuids or the ﬂatter than parabolic form corresponding to non-Newtonian ﬂuids. The criterion for determining fully developed ﬂow in viscoelastic polymeric liquids was discussed extensively in the 1970s (Han and Charles 1970; Han et al. 1970). Today, it is well established that when dealing with viscoelastic ﬂuids, to claim that ﬂow is fully developed inside the tube (or slit die) upon passing the tube entrance, it is necessary, but not sufﬁcient, to establish a constant pressure gradient in a cylindrical tube (or slit die), while such criterion is both necessary and sufﬁcient to claim that ﬂow is fully developed for Newtonian ﬂuids. Conditions other than the constant pressure gradient in the ﬂow of viscoelastic polymeric liquids have been suggested (Han et al. 1970), as will be elaborated on later in this section. This is a very important subject because, once ﬂow is fully developed, one can write momentum balance equations to derive various expressions relating the ﬂuid properties to ﬂow variables (ﬂow rate, pressure gradient, etc.). In Chapter 5 of Volume 1 we considered fully developed ﬂow in order to derive various expressions that allow one to calculate viscometric ﬂow properties of molten polymers ﬂowing through a capillary or slit die. One noticeable difference between viscoelastic polymeric liquids and Newtonian liquids is shown by the pressure drop (PEnt ) at the entrance of a cylindrical tube, as shown in Figure 5.11 for Newtonian ﬂuid (Indopol H1900) and Figure 5.12 for a viscoelastic HDPE melt in Volume 1. It is seen in these ﬁgures that very large PEnt occurs in the HDPE melt, whereas negligibly small PEnt occurs in Indopol H1900. The observed difference in PEnt between the HDPE melt and Indopol H1900 has little to do with the viscosity of the two liquids. Another remarkable viscoelastic nature of polymeric liquids is manifested by the circulatory (secondary) motion at the corners of the reservoir section preceding a cylindrical tube or slit die. Figure 1.21 shows the ﬂow patterns of an HDPE melt at 180 ◦ C in the entrance region of a slit die, where indeed strong circulatory ﬂow patterns are observed at the corners of the reservoir section as the bulk of the HDPE melt ﬂows, forming a natural streamline angle, into the slit die. In the 1980s through 1990s, extensive studies were reported, both experimental (Cable and Boger 1978; Drexler and Han 1973; Nguyen and Boger 1979; White and Kondo 1978/1979) and computational using the ﬁnite element methods (Crochet and Bezy 1979; Kajiwara et al. 1991; Luo and Mitsoulis 1990; Park and Mitsoulis 1992; Viriyayuthakorn and Coswell 1980), on the ﬂow patterns of viscoelastic ﬂuids in the entrance region of capillary and slit dies. One can analyze the velocity distributions of a polymeric liquid in the reservoir section of a slit die using streak photography, for instance. Other experimental techniques, such as doppler velocimetry, can also be employed. To facilitate our presentation here, let us consider the schematic diagram given in Figure 1.22a,

Figure 1.21 Photograph of the ﬂow patterns of an HDPE melt at 180 ◦ C in the entrance region of a slit die, where strong circulatory ﬂow patterns are observed at the corners of the reservoir section as the bulk of the HDPE melt ﬂows into the slit die.

Figure 1.22 (a) Schematic diagram of the entrance region of a slit die superposed by a cylindrical coordinate system and (b) velocity proﬁles of a PS melt at 200 ◦ C in the entrance region of a slit die at different positions from the vertex (r = 0) in the r-direction (see the schematic in part (a)): () at r = 0.685 cm, () at r = 0.565 cm, and () at r = 0.425 cm. (Reprinted from Drexler and Han, Journal of Applied Polymer Science 17:2355. Copyright © 1973, with permission from John Wiley & Sons.)

21

22

PROCESSING OF THERMOPLASTIC POLYMERS

where a cylindrical coordinate system is superposed over the entrance section. Here, we conﬁne our interest in the velocity proﬁles to only inside the angle 2θ , through which a polymeric liquid ﬂows continuously into the slit-die section. Figure 1.22b gives velocity proﬁles of PS melt at 200 ◦ C, which were obtained from streak photography. It is seen that the velocities along the centerline are faster than those away from the centerline and that the PS melt ﬂows faster as it approaches the die entrance. One can also analyze the stress distributions of a polymeric liquid in the entrance region of a slit die using ﬂow birefringence, as schematically shown in Figure 1.23, where the apparatus consists of (1) the optical system, (2) the ﬂow test cell, and (3) the polymer melt feed system. The main components of the optical system are a light source, interference ﬁlter, diffusion screen, polarizer, quarter wave plates, analyzer, and a camera. The rationale behind the use of the optical system to investigate the stress distribution within a polymeric liquid lies in the well-established optical principles (Durelli and Riley 1965; Frocht 1941; Hendry 1966) that when polarized light enters an optically anisotropic medium, the beam separates into two plane-polarized components in the direction of the principal stresses. When the two components emerge from the medium, they have a certain relative path retardation. Further, under certain conditions, extinction of the emerging beam of light occurs, giving rise, when the entire ﬁeld is viewed, to isoclinic fringe patterns when 2θ = N π and to isochromatic fringe patterns when α/2 = Nπ, where θ denotes the direction of principal stresses, α is the angular difference (or retardation) of the emerging beam of light, and N is an integer. When the direction of a light path is made to coincide with the direction of one of the

Figure 1.23 Schematic showing the apparatus for ﬂow birefringence. (Reprinted from Han and

Drexler, Journal of Applied Polymer Science 17:2329. Copyright © 1973, with permission from John Wiley & Sons.)

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

23

principal stresses, the retardation (or fringe order) R can be related to the difference in the other two principal stresses, σ , by n = Cσ

(1.11)

where n = Rλ/d with d being the thickness of the medium and λ being the wavelength. In Eq. (1.11), C is called the stress optical coefﬁcient measured in brewsters (1 brewster = 10−12 cm2 /dyn) and n is the magnitude of the birefringence. Equation (1.11) indicates that n is a function only of the difference of two principal stresses in the plane perpendicular to the axis of the light propagation. Another important relationship in optical photoelasticity is that the orientation of the optical axes (θ opt ) is identical with the orientation of the principal stress axes (θstress ): (θopt ) = (θstress ) = θ

(1.12)

Together, Eqs. (1.11) and (1.12) are called the “stress optical laws.” Thus, measurements of birefringence enable one, via the stress optical laws, to determine the principal stress differences, which can be transformed from the principal coordinates to the Cartesian coordinates by rotation. This transformation then relates shear stress, σxy , and ﬁrst normal stress differences, σxx − σyy , to principal stresses by (Frocht 1941; Hendry 1966) σxy = (σ/2) sin 2θstress

(1.13)

σxx − σyy = σ cos 2θstress

(1.14)

Using Eqs. (1.11) and (1.12), Eqs. (1.13) and (1.14) can be rewritten as σxy = FN sin 2θ

(1.15)

σxx − σyy = 2F cos 2θ

(1.16)

where F = λ/2Cd. Note that θ is to be determined from the isoclinic fringe patterns and N from the isochromatic fringe patterns. However, in order to calculate σxy and σxx − σyy from Eqs. (1.15) and (1.16), the stress optical coefﬁcient C must be known for the ﬂuid being investigated. For a perfectly elastic material, C can be calculated theoretically (Treloar 1958). However, such a theoretical calculation is not applicable to polymeric liquid because it is not a perfectly elastic material. Under such circumstances, C can be determined from measurements of σxy , N, and θ in the well-deﬁned ﬂow ﬁeld, such as fully developed ﬂow in the slit die, using Eq. (1.15). Lodge (1955) appears to have been the ﬁrst to suggest that Eqs. (1.11) and (1.12) may be extended to polymeric liquids. Subsequently, many research groups (Adamse et al. 1968; Funatsu and Mori 1968; Han and Drexler 1973a, 1973b; Philippoff, 1956, 1957, 1961; Wales 1969; Wales and Janeschitz-Kriegl 1967) have applied Eqs. (1.11) and (1.12) to investigate the stress distributions of polymeric solutions or melts in steady-state uniform shear ﬂow, in fully developed ﬂow, or in the entrance-region ﬂow of a slit die.

24

PROCESSING OF THERMOPLASTIC POLYMERS Figure 1.24 Photographs of

(a) isochromatic fringe patterns and (b) isoclinic fringe patterns at θ = 40◦ , where θ denotes the direction of the principal stresses, for an HDPE melt at 200 ◦ C and a volumetric ﬂow rate of 17.3 cm3 /min ﬂowing through a reservoir section followed by a slit die. (Reprinted from Han and Drexler, Journal of Applied Polymer Science 17:2329. Copyright © 1973, with permission from John Wiley & Sons.)

Figure 1.24 gives photographs of (a) isochromatic fringe patterns and (b) isoclinic fringe patterns of an HDPE melt at 200 ◦ C in the entrance region of a slit die. Figure 1.25a gives calculated shear stress proﬁles and Figure 1.25b gives calculated ﬁrst normal stress difference proﬁles in the entrance region of a slit die using Eqs. (1.15) and (1.16). Han and Drexler (1973a) determined the stress optical coefﬁcient C for various molten polymers from the measurements of wall normal stresses along the axis of a slit die, which enabled them to determine the shear stress σxy (see Chapter 5 of Volume 1) and from the measurements of the number of isochromatic fringes N and the isoclinic angles θ in the fully developed region of a slit die. A photograph of typical isochromatic fringe patterns in the fully developed region of a slit die is given in Figure 5.17 in Volume 1. The calculated values of C are 1.23 × 10−9 Pa−1 for an HDPE melt, 4.95 × 10−9 Pa−1 for PS, and 0.605 × 10−9 Pa−1 for PP. It is seen in Figure 1.25 that ﬂow birefringence is a very powerful experimental technique for determining stress distributions of polymeric liquid in the entrance region of a slit die. Note that ﬂow birefringence can be used to determine stress distributions in a ﬂow geometry that is much more complicated than the entrance region of a slit die. However, this experimental technique has limitations. As the ﬂow rate increases, the number of isochromatic fringes also increases. Thus, the distinction between the fringes becomes increasingly difﬁcult to see as the ﬂow rate increases, and eventually one ends up with the situation where the counting the number of fringes becomes

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

25

Figure 1.25 (a) Shear stress proﬁles in the entrance region of a slit die for an HDPE melt at 200 ◦ C with a volumetric ﬂow rate of 2.96 cm3 /min: () σxy = 5.72 × 104 Pa, () σxy = 9.15 × 104 Pa, () σxy = 1.03 × 105 Pa, () σxy = 1.37 × 105 Pa, (䊉) σxy = 4.57 × 104 Pa, () σxy = 2.28 × 104 Pa, () σxy = 1.14 × 104 Pa. (b) First normal stress difference proﬁles: () σxx − σyy = −5.72 × 105 Pa, () σxx − σyy = −4.58 × 105 Pa, () σxx − σyy = −2.29 × 105 Pa, () σxx − σyy = −1.72 × 105 Pa, (3) σxx − σyy = −1.26 × 105 Pa, (䊉) σxx − σyy = 0 Pa, () σxx − σyy = 6.86 × 104 Pa, () σxx − σyy = 1.03 × 105 Pa, () σxx − σyy = 1.72 × 105 Pa. (Reprinted from Han and Drexler, Journal of Applied Polymer

Science 17:2329. Copyright © 1973, with permission from John Wiley & Sons.)

virtually impossible. Accordingly, ﬂow birefringence is limited to relatively low shear rates. Next, the ﬂuid under test must be transparent, and thus ﬂow birefringence cannot be used to determine stress distributions in such polymeric systems as immiscible polymer blends, which are invariably translucent, and ﬁlled polymers.

1.4

Flow through a Converging or Tapered Channel

Since a polymer melt circulating at the corners of the reservoir section (see Figure 1.21) may undergo thermal degradation during extrusion, it is best to design a die to have a conical entrance in the reservoir section. Figure 1.26 gives a streak photograph showing the ﬂow patterns of PS melt at 200 ◦ C in a converging channel having a half-angle of 30◦ . No secondary ﬂow is observed in Figure 1.26 because the angle (60◦ ) of the converging channel is apparently smaller than the natural streamline angle of the PS melt ﬂowing through the die. Thus, the presence of secondary ﬂow can be eliminated by proper die design. In Figure 1.26, the bright streaklines represent tracer particles suspended in the molten PS, the streaklines at the centerline are longer than those

26

PROCESSING OF THERMOPLASTIC POLYMERS Figure 1.26 Streak photograph of PS melt at 200 ◦ C ﬂowing through a converging channel having a half-angle of 30◦

followed by a slit-die section. (Reprinted from Han and Drexler, Journal of Applied Polymer Science 17:2369. Copyright © 1973, with permission from John Wiley & Sons.)

away from the centerline, indicating that particles at the center travel faster than those away from the centerline, and the streaklines near the entrance of the slit-die section are longer than those in the upstream, indicating that the ﬂuid accelerates as it enters the die entrance. Figure 1.27 gives a photograph of isochromatic fringe patterns of a PS melt at 200 ◦ C ﬂowing through a converging channel. It is seen in Figure 1.27 that the number of isochromatic fringes are larger at the corner of the die compared with other areas of the die, indicating that the levels of stress are greater at the corner than other areas. Figure 1.28a gives calculated shear stress proﬁles and Figure 1.28b calculated ﬁrst normal stress difference proﬁles in the entrance region of a converging channel using Eqs. (1.15) and (1.16). Figure 1.29 gives the stress distributions of polybutadiene at 25 ◦ C along the centerline of the reservoir section followed by a slit die. Note that ﬂow along the centerline of a converging channel can be regarded as elongational ﬂow and thus the stress σyy (0, z) in Figure 1.29 can be regarded as the extensional stress. It is seen that σyy (0, z) ﬁrst increases and then decreases, going through a maximum just inside the straight tube.

Figure 1.27 Photograph of isochromatic

fringe patterns in the converging channel having a half-angle of 30◦ followed by a slit-die section for a PS melt at 200 ◦ C with a volumetric ﬂow rate of 17.3 cm3 /min. (Reprinted from Han and Drexler, Journal of Applied Polymer Science 17:2369. Copyright © 1973, with permission from John Wiley & Sons.)

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

27

Figure 1.28 (a) Shear stress proﬁles in the entrance region of a converging ﬂow channel for a PS melt at 200 ◦ C with a volumetric ﬂow rate of 5.36 cm3 /min: () σxy = 0.14×104 Pa, () σxy = 0.28 × 104 Pa, () σxy = 0.49 × 104 Pa, () σxy = 0.70 × 105 Pa, (3) σxy = 0.82 × 104 Pa, (䊉) σxy = 1.12 × 104 Pa, () σxy = 1.40 × 104 Pa, () σxy = 1.54 × 104 Pa. (b) First normal stress difference: () σxx − σyy = −5.72 × 105 Pa, () σxx − σyy = −4.58 × 105 Pa, () σxx − σyy = −2.29 × 105 Pa, () σxx − σyy = −1.72 × 105 Pa, (3) σxx − σyy = −1.26 × 105 Pa, (䊉) σxx − σyy = 0 Pa, () σxx − σyy = 6.86 × 104 Pa. (Reprinted from Han

and Drexler, Journal of Applied Polymer Science 17:2369. Copyright © 1973, with permission from John Wiley & Sons.)

Figure 1.30 gives photographs of isochromatic fringe patterns of a PS melt at 200 ◦ C ﬂowing through a tapered channel3 with various converging angles. It can be seen that the stress distributions of the ﬂuid in the ﬂow channel are greatly inﬂuenced by the converging angle. Figure 1.31 gives calculated ﬁrst normal stress difference proﬁles in the tapered die with an angle of 60◦ using Eq. (1.16). A comparison of Figure 1.31 with Figure 1.28b shows that ﬁrst normal stress difference proﬁles are much more complicated in a tapered die than in a converging die in that very large concentrations of ﬁrst normal stress difference exist just before the ﬂuid exits the die. Let us now consider the schematic diagram given in Figure 1.32, where streamlines emanate radially from the point of intersection of the two nonparallel plates (i.e., the vertex) and therefore the cylindrical coordinate system may be chosen to describe the ﬂow. For the geometry chosen, at any place in the tapered channel we have Trr (r, θ ) = −p(r, θ ) + σrr (r, θ )

(1.17)

Tθθ (r, θ ) = −p(r, θ ) + σθθ (r, θ )

(1.18)

in which Trr (r, θ ) and Tθθ (r, θ ) are the total radial (r-directed) normal stress and the total angular (θ-directed) normal stress, respectively, p is the isotropic pressure, and σrr (r, θ ) and σθθ (r, θ ) are the deviatoric normal stress components. Suppose that

28

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 1.29 Distribution of extensional stress, σyy (0, z), of a polybutadiene at 25 ◦ C along the

centerline (y = 0, z) of the reservoir section followed by a slit die. (a) Die entrance angle of 30◦ at various wall shear stresses (Pa):4 (1) 0.51 × 105 , (2) 1.18 × 105 , (4) 2.22 × 105 , and (5) 2.40 × 105 . (b) Die entrance angle of 45◦ at various wall shear stresses (Pa): (1) 0.58 × 105 , (2) 1.16 × 105 , (3) 1.70 × 105 , (4) 1.99 × 105 , (5) 2.29 × 105 , and (6) 2.4 × 105 . (c) Die entrance angle of 180◦ at various wall shear stresses (Pa): (1) 0.43 × 105 , (2) 0.83 × 105 , (3) 1.08 × 105 , (4) 1.33 × 105 , (5) 1.67 × 105 , (6) 2.08 × 105 , and (7) 2.22 × 105 . (Reprinted from Brizitsky et al., Journal of Applied Polymer Science 22:751. Copyright © 1978, with permission from John Wiley & Sons.)

pressure transducers are mounted at the wall along the r-axis and wall normal stresses are measured at the channel wall. The wall normal stress measured is the θ-directed total normal stress at the channel wall Tθθ (r, α), that is Tθθ (r, α) = −p(r, α) + σθθ (r, α)

(1.19)

in which α is the half-angle of the channel. It is worth pointing out that the concept of pressure gradient, commonly used in the determination of wall shear stress for fully developed ﬂow, does not give us the same convenience for the converging ﬂow ﬁeld. This is because in a converging ﬂow ﬁeld the deviatoric stress component also depends on the direction of ﬂow. Thus, from Eq. (1.18) one has

∂Tθθ ∂r

=− α

∂p ∂r

+ α

∂σθθ ∂r

(1.20) α

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29

Figure 1.30 Photographs of isochromatic fringe patterns of a PS melt at 200 ◦ C ﬂowing through a tapered channel with various angles: (a) 30◦ , (b) 45◦ , (c) 90◦ , and (d) 150◦ . (Reprinted from Yoo

and Han, Journal of Rheology 25:115. Copyright © 1981, with permission from the Society of Rheology.)

Equation (1.20) indicates that, in general, measurement of wall normal stress (Tθθ )α alone is not sufﬁcient to deﬁne the pressure gradient (−∂p/∂r)α in a converging ﬂow ﬁeld unless the gradient of the deviatoric stress at the wall (−∂σθθ /∂r)α is zero. Figure 1.33 gives experimentally measured wall normal stress distributions Tθθ (r, α) along the tapered channel wall for an HDPE melt. It is seen that as the melt ﬂows into the die exit, wall normal stresses perpendicular to the channel wall go through a minimum and then increase very rapidly as the melt approaches the die exit. Figure 1.34 gives theoretically calculated total wall normal stress proﬁles Tθθ (r, α = 30◦ ), which were obtained, via the Coleman–Noll second ﬂuid

30

PROCESSING OF THERMOPLASTIC POLYMERS Figure 1.31 First normal stress difference proﬁles of polystyrene at 200 ◦ C ﬂowing through a tapered die with an angle of 60◦ at a volumetric ﬂow rate of 1.5 cm3 /min: () σxx − σyy = 1.19 × 104 Pa, () σxx − σyy = 2.85 × 104 Pa, () σxx − σyy = 3.95 × 104 Pa, () σxx − σyy = 3.46 × 104 Pa, (3) σxx − σyy = 1.52 × 104 Pa, ( 7) σxx − σyy = 1.02 × 104 Pa. (Reprinted from

Yoo and Han, Journal of Rheology 25:115. Copyright © 1981, with permission from the Society of Rheology.)

(see Chapter 3 in Volume 1), from the following expression (Yoo and Han 1981):

2 ν f (α) η0 f (α) 1 1 1 1 + − − Tθθ (r, α) − Tθθ (r0 , α) = 2 2 r2 r0 2 r4 r0 4

(1.21)

in which r0 denotes the vertex of the converging channel (see Figure 1.32). Note that Eq. (1.21) contains f (α) and f (α), which are related to the function f (θ) given by

cos 2θ − cos 2α f (θ ) = f (0) 1 − cos 2α

Figure 1.32 Schematic showing a

converging ﬂow channel over which a cylindrical coordinate is superposed.

(1.22)

31

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

Figure 1.33 Distribution of wall normal stress in a converging channel having a half-angle of 30◦ for an HDPE melt at 200 ◦ C for various volumetric ﬂow rates (cm3 /min): () 18.4, () 47.4, and () 65.7. (Reprinted from Yoo and Han, Journal of Rheology 25:115. Copyright © 1981, with permission from the Society of Rheology.)

deﬁned by f (θ) in turn is related to the volumetric ﬂow rate per channel width Q = −2 Q

α 0

2α cos 2α − sin 2α rvz (r, θ ) dθ = f (0) 1 − cos 2α

(1.23)

where f (0) is the value of f (θ ) at θ = 0 (i.e., along the centerline of the converging channel; see Figure 1.32).

Figure 1.34 Theoretical predictions of the distributions of wall normal stress for an HDPE melt at 200 ◦ C ﬂowing through a tapered channel with a converging angle of 30◦ , at various volumetric ﬂow rates: (1) Q = 65.7 cm3 /min with Tθθ (r0 = 3.1 cm, α = 15◦ ) = 7.46 × 106 Pa, (2) Q = 47.4 cm3 /min with Tθθ (r0 = 3.1 cm, α = 15◦ ) = 6.25 × 106 Pa, and (3) Q = 18.4 cm3 /min with Tθθ (r0 = 3.1 cm, α = 15◦ ) = 3.3 × 106 Pa. (Reprinted from Yoo and Han, Journal of Rheology 25:115. Copyright © 1981, with permission from the Society of Rheology.)

32

PROCESSING OF THERMOPLASTIC POLYMERS

In obtaining Figure 1.34, the following numerical values of the material parameters appearing in the Coleman–Noll second-order ﬂuid were used: η0 = 1.14 × 105 Pa·s, β = 3.20 × 104 Pa·s2 , and ν = −1.78 × 104 Pa·s2 for an HDPE melt at 200 ◦ C. It is interesting to observe in Figure 1.34 that the wall normal stress Tθθ (r, α) predicted from Eq. (1.21) goes through a minimum and then increases, corroborating the essential features of the experimental results (Figure 1.33). Quantitative agreement between the theoretical prediction and the experimental results should not be expected for several reasons. First, the Coleman–Noll second-order ﬂuid model is not expected to describe well the “rapid motion” of viscoelastic ﬂuids. Note that the ﬂow in a tapered channel is considered to be a “rapid ﬂow.” Second, the HDPE melt employed in the experiment (Figure 1.33) exhibits shear-thinning behavior at the conditions under which the experimental results were obtained. Therefore, the use of the zero-shear viscosity η0 in the theoretical calculation of Tθθ (r, α) must have overestimated the true values of melt viscosity encountered under the actual ﬂow conditions. However, the use of the Coleman–Noll second-order ﬂuid model has yielded analytical expressions that enable one to offer a theoretical interpretation of the experiment results presented in Figure 1.33. Numerical computation is required when using more sophisticated constitutive equations. It is worth noting that a Newtonian ﬂuid, ν = 0 in Eq. (1.21), cannot exhibit a minimum in the distribution Tθθ (r, α) along the tapered channel. We can thus conclude that the increasing trend of Tθθ (r, α) near the exit plane of a tapered channel, predicted from Eq. (1.21), is due to the elastic component of the ﬂuid behavior.

1.5

Exit Region Flow

It has long been recognized that the ﬂow behavior of viscoelastic ﬂuids in the exit region (here referred to as “exit region ﬂow”) of a tube is quite different from that of Newtonian ﬂuids. Two aspects of exit region ﬂow of viscoelastic ﬂuids are of fundamental and practical importance. One aspect is swelling of the extrudate upon exiting from the die (see Figure 1.4 in Volume 1), and the other is the extent of ﬂow disturbance in a region very close to the die exit. In the 1960s and 1970s, experimental investigations of the extrudate swell of polymer melts from a cylindrical tube were extensively reported in the literature (Arai and Aoyama 1963; Bagley et al. 1963; Guillet et al. 1965; Han 1976; Han and Kim 1971; McLuckie and Rogers 1969; Mendelson and Finger 1973; Nakajima and Shida 1966; Rogers 1970; Sekiguchi 1969). In a preceding section, we discussed the extrudate swell of molten polymers upon exiting from a rectangular die (see Figures 1.2–1.4). In the 1970s through 1980s, some effort was made to calculate the extent of extrudate swell of viscoelastic polymeric ﬂuids using ﬁnite element methods (Chang et al. 1979; Crochet and Keunings 1980; Delvaux and Crochet 1990; Reddy and Tanner 1978). In Chapter 5 of Volume 1, we discussed the extent of ﬂow disturbance in the exit region of a cylindrical or slit die. In this section, we present additional information as to the extent that the geometry of a die can increase the extrudate swell of a viscoelastic polymer melt from a tapered die (having no straight die land) compared with that from a slit die, and we then discuss the stress distributions within the polymer melt just before and upon leaving the die. Such a comparison sheds light on the unique features of viscoelastic polymeric liquids.

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

33

Figure 1.35 Photograph of isochromatic fringe patterns of a PS melt at 200 ◦ C ﬂowing through a converging channel with a half-angle of 30◦ . (Reprinted from Han, Rheologica Acta 14:173.

Copyright © 1975, with permission from Springer.)

Figure 1.35 shows a photograph of the isochromatic fringe patterns of PS melt in the exit region of a tapered die. Extremely large extrudate swell is seen in Figure 1.35. Moreover, signiﬁcant amounts of stresses exist in the extrudate just outside the die. Figure 1.36 gives calculated shear stress distributions inside the die and also in the extrudate very close to the die exit plane. The signiﬁcance of Figures 1.35 and 1.36 lies in that the stresses of the PS melt inside the die did not have sufﬁcient time to relax before leaving the die, and thus a considerable amount of stress still remains in the extrudate upon exiting the die. Notice in Figure 1.36 that the level of shear stress in the extrudate is almost as large as that inside the die. This is attributed to the fact that

Figure 1.36 Shear stress proﬁles inside a tapered die with a half-angle of 30◦ and also in the extrudate for a PS melt at 200 ◦ C with a volumetric ﬂow rate of 1.5 cm3 /min: () σxy = 0.68 × 104 Pa,

() σxy = 1.76 × 104 Pa, () σxy = 2.34 × 104 Pa, () σxy = 3.78 × 104 Pa,

(3) σxy = 4.78 × 104 Pa, (䊉) σxy = 0.92 × 104 Pa,

() σxy = 1.46 × 104 Pa, () σxy = 2.06 × 104 Pa. (Reprinted from Han, Rheologica Acta 14:173. Copyright © 1975, with permission from Springer.)

34

PROCESSING OF THERMOPLASTIC POLYMERS

much of the elastic energy stored in the PS melt during the ﬂow through the tapered die did not dissipate fast enough, such that a large amount of energy still remains in the ﬂuid immediately after it exits the die. Such an observation is supported by both the measured and calculated wall normal stress distributions in a tapered die, as shown in Figures 1.33 and 1.34. Figure 1.37 gives a photograph of the isochromatic fringe patterns of PS melt in the exit region of a slit die, where the dark area in the exit plane is due to the shadow cast by the die as the photograph was taken. In Figure 1.37, we observe that when the PS melt leaves the exit plane, it carries with it a large amount of stress that has not relaxed inside the die; the level of stress in the extrudate close to the exit plane is almost as great as that just inside the die. This means that the rate of stress relaxation was too slow to allow for complete relaxation before the PS melt left the exit plane. Notice that very little extrudate swell is seen in Figure 1.37 when compared with Figure 1.35. We have already pointed out the practical limitation of the ﬂow birefringence technique in investigating the stress distributions of viscoelastic ﬂuids in a conﬁned geometry, namely, the number of isochromatic fringes (N in Eq. (1.15)) increases very rapidly as the ﬂow rate increases, making discernment of isochromatic fringe patterns virtually impossible. This is evident in Figure 1.37 in that the isochromatic fringe patterns inside the die are barely distinguishable when the shear rate is only 4.55 s−1 . For instance, when shear rate is increased to, say, 10 s−1 , we would not be able to discern isochromatic fringe patterns inside the die or in the extrudate. This is precisely the limitation of the ﬂow birefringence technique. It is not difﬁcult to surmise from Figure 1.37 that the exit effect (i.e., disturbance of stresses before leaving the die exit plane) would decrease as shear rate increases. This is precisely the rationale behind the use of the “exit pressure method” presented in Chapter 5 of Volume 1, where the exit pressure method is recommended only when wall shear stress is sufﬁciently large, say larger than approximately 25 kPa (see Table 5.1 in Volume 1). Referring to Figure 1.37,

◦ Figure 1.37 Photograph of isochromatic fringe patterns of a PS melt at 200 C with a shear −1 rate of 4.55 s in the exit region of a slit die having an opening (height) of 2 mm. (Reprinted

from Han and Drexler, Transactions of the Society of Rheology 17:659. Copyright © 1973, with permission from the Society of Rheology.)

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

35

the wall shear stress corresponding to a shear rate of 4.55 s−1 for the PS at 200 ◦ C is 10.23 Pa, and yet the amount of residual stress in the extrudate is very large. If there were no or only little residual stress remaining in the extrudate, we would not be able to observe any isochromatic fringe patterns in Figure 1.37.

1.6

Flow through a Channel Having Small Side Holes or Slots

In the processing of polymeric materials, one may encounter a situation where a cylindrical tube or a slit die has a small hole(s) or transverse slot(s) or where a slit die has a transverse slot(s). Under such situations, a polymeric ﬂuid passes by the small hole or transverse slot while ﬂowing through a cylindrical tube or slit die. It is then very important to understand the velocity and stress distributions in the neighborhood of the small hole or transverse slot because the ﬂow in the neighborhood of the side hole or transverse slot may no longer be assumed to be fully developed. To illustrate the situation described here, let us consider a slit ﬂow channel shown schematically in Figure 1.38, where two different sizes of transverse slots are placed opposite to each other in the downstream side, far away from the entrance region. Han and Yoo (1980) investigated ﬂow patterns and stress distributions in a slit ﬂow channel, similar to that shown in Figure 1.38, when a polymer melt was forced to ﬂow through the channel. They used ﬂow birefringence to obtain information on the stress distributions in the neighborhood of the transverse slots and in the ﬂow channel with the aid of a polariscope. With the polariscope removed, they were able to observe ﬂow patterns in the neighborhood of the transverse slots. Figure 1.39 shows photographs of isochromatic fringe patterns of an HDPE melt at 200 ◦ C in the neighborhood of two transverse slots, placed opposite to each other. Figure 1.39a shows the isochromatic fringe patterns in both the upstream and downstream of the transverse slots, in which we observe that the ﬂow is fully developed. To facilitate visualization of the isochromatic fringe patterns in the neighborhood of the transverse slots, an enlarged photograph is shown in Figure 1.39b. It is seen that the shear stress is not symmetric about a plane through the center of the transverse slot and complex distributions of shear stress exist in the neighborhood of two transverse slots. Others have also made similar observations (Arai and Hatta 1980). Further, the ﬂow patterns are not symmetric about a plane through the center of the transverse slots, and circulatory ﬂow patterns are present inside the transverse slot. Figure 1.40 shows

Figure 1.38 Schematic diagram of a slit die having two transverse slots located on opposite sides.

36

PROCESSING OF THERMOPLASTIC POLYMERS Figure 1.39 Photographs of isochromatic fringe patterns of an HDPE melt at 200 ◦

C (a) in both the fully developed region of the slit die and in the neighborhood of transverse slots, and (b) only in the neighborhood of transverse slots with higher magniﬁcation. (Reprinted from Han and Yoo, Journal of Rheology 24:55. Copyright © 1980, with permission from the Society of Rheology.)

the presence of circulatory ﬂow patterns of an aqueous solution of polyacrylamide (a viscoelastic ﬂuid) (Figures 1.40a and 1.40b) and glycerin (a Newtonian ﬂuid) (Figure 1.40c) inside a transverse slot placed in the side wall of a slit die. Other researchers have also made similar experimental observations (Cochrane et al. 1981; Hou et al. 1977). It should be mentioned that ﬁnite element calculations (Jackson and Finlayson 1982; Kajiwara et al. 1991) indeed predict the circulatory ﬂow patterns inside a small side hole, conﬁrming the experimental results presented in Figure 1.40. Careful examination of the ﬂow birefringence patterns, shown in Figure 1.39, in the neighborhood of and inside the transverse slot reveals the physical origin of stress disturbances in that region. To help understand the physical situation, let us look at 1 denotes the entrance plane and the schematic diagram given in Figure 1.41, where

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

37

Figure 1.40 Secondary ﬂow patterns at 25 ◦ C inside a transverse slot placed in the side wall of a slit die for (a) 0.6 wt % aqueous solution of polyacrylamide ﬂowing at γ˙ = 79 s−1 , σ = 9.8 Pa, and NRe = 2.3, (b) 4.0 wt % aqueous solution of polyacrylamide ﬂowing at γ˙ = 676 s−1 , σ = 243 Pa, and NRe = 2.9, and (c) glycerin ﬂowing at γ˙ = 660 s−1 , σ = 475 Pa, and NRe = 5.16, where NRe refers to Reynolds number. (Reprinted from Han and Yoo, Journal of

Rheology 24:55. Copyright © 1980, with permission from the Society of Rheology.)

2 denotes the exit plane of the transverse slot. Referring to Figures 1.39 and 1.41, the 1 is very similar to that in the isochromatic fringe pattern in the neighborhood of plane

exit region (see Figure 1.35), and the isochromatic fringe pattern in the neighborhood 2 is very similar to that in the entrance region (see Figure 1.24). Thus, the of plane streamlines in the neighborhood of a transverse slot are drawn in Figure 1.41, indicating that the ﬂow passing through a transverse slot in a slit die includes both exit and entrance ﬂows. The photographs of ﬂow patterns given in Figure 1.40 further support such schematic streamlines. Namely, in the ﬂow of a viscoelastic polymer solution, a 1 (exit ﬂow) than in the streamline is pushed up more in the neighborhood of plane 2 (entrance ﬂow). However, in the ﬂow of glycerin (which neighborhood of plane 1 is a Newtonian ﬂuid), a streamline is pushed less in the neighborhood of plane

38

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 1.41 Schematic showing asymmetric streamlines in the neighborhood of a transverse slot

1 is equivalent to placed in the wall of a slit die, where the ﬂow in the neighborhood of location 2 is equivalent to the entrance ﬂow. the exit ﬂow, and the ﬂow in the neighborhood of location (Reprinted from Han and Yoo, Journal of Rheology 24:55. Copyright © 1980, with permission from the Society of Rheology.)

2 . The streamline patterns in the neighborhood of than in the neighborhood of plane 1 may be explained in terms of the angle that an extrudate makes at the die plane 2 may be exit (Figure 1.35), and the streamline patterns in the neighborhood of plane explained by the converging streamline angle at the die entrance (Figure 1.21). From a practical point of view, the above situation is encountered when pressure transducers are mounted on a solid wall through a small hole, which connects the die wall surface to the tip of the traducer. Such a small hole is referred to as a “pressuretap hole” or simply a “pressure hole.” However, formation of a pressure hole can be avoided when the die wall surface is ﬂat. In the 1960s and 1970s, some research groups (Brindley and Broadbent 1973; Broadbent and Lodge 1971; Broadbent et al. 1968; Han and Kim 1973; Kaye et al. 1968; Novotny and Eckert 1973; Prichard 1970) reported that the pressure p* measured in the presence of a pressure hole was lower than the pressure p measured without a pressure hole. Under such circumstances, pH = p∗ − p becomes negative, and the pH is referred to as “hole pressure error.” Some research groups made attempts, theoretically (Higashitani and Pritchard 1972; Tanner and Pipkin 1969) and experimentally (Baird 1975, 1976; Higashitani and Lodge 1975), to capitalize on hole pressure error to determine the ﬁrst normal stress difference (N1 ) of viscoelastic ﬂuids in steady-state shear ﬂow. Speciﬁcally, assuming that the ﬂow is so slow that the second-order approximate constitutive equations are valid, for uniform shear ﬂow, Tanner and Pipkin (1969) showed that pH is negative and 25% of N1 , that is

pH = −0.25N1

(1.24)

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39

Figure 1.42 Schematic showing idealized streamline patterns in the neighborhood of a pressure hole. pH = p ∗ − p < 0 for certain ﬂuids in the absence of secondary ﬂow in the pressure hole. (Reprinted from Han and Yoo, Journal of Rheology 24:55. Copyright © 1980, with permission from the Society of Rheology.)

Higashitani and Prichard (1972) obtained the following expression for pH : pH = − 12

σw 0

(σ11 − σ22 ) dσ12 σ12

(1.25)

for a transverse slot in a slit die, where σ11 and σ22 are the normal stresses in the fully developed region of the slit die, σ12 is the shear stress in the fully developed region of the slit die, and σw is the shear stress at the die wall. In the derivation of Eq. (1.25), Higashitani and Prichard made the following assumptions: (1) the streamline pattern is symmetric about a plane through the center of the hole, as schematically shown in Figure 1.42, (2) the shear stress is also symmetric about the centerline of the hole, (3) the motion at the centerline of the hole (or within some neighborhood of the centerline of the hole) is that of unidirectional shear ﬂow, that is, there is no secondary ﬂow (circulating ﬂow pattern) in the hole, and (4) the gradients of axial normal stresses and shear stress with respect to the ﬂow direction vanish at the centerline of the hole. Equation (1.25) can be rewritten as N1 = −2σw

dpH dσw

(1.26)

indicating that ﬁrst normal stress difference N1 in steady-state shear ﬂow can be determined from the measurements of hole pressure error pH as a function of wall shear stress σw . However, as can be seen clearly in Figures 1.39 and 1.40, all the assumptions made in the development of Eqs. (1.25) and (1.26) are not valid. Thus under such circumstances, Eq. (1.25) or Eq. (1.26) are of no rheological signiﬁcance. A suggestion has been made to use the following empirical expression (Baird 1976): pH = −cN1

(1.27)

where c is an empirical constant. Such a suggestion cannot be taken seriously because there is no way of knowing what the value of c might be in the ﬁrst place. It has amply been demonstrated experimentally that pH varies with the diameter (dH ) of

40

PROCESSING OF THERMOPLASTIC POLYMERS

circular pressure hole relative to the channel height (h), the concentration of polymer solutions (φ), and Reynolds number (NRe ) (Han and Kim 1973; Higashitani and Lodge 1975). It is then clear that the value of c in Eq. (1.27) is expected to vary with dH , φ, and NRe , even if such a simple linear relationship between pH and N1 ever exists. However, there is no theoretical basis for pH to be proportional to N1 in the ﬂow of viscoelastic ﬂuids through a cylindrical tube with pressure hole or a slit die with transverse slot. More seriously, the determination of N1 , which is a viscometric ﬂow property, cannot possibly be determined in a complex ﬂow ﬁeld where signiﬁcant exit and entrance effects exist, as we have pointed out. Therefore, it can be concluded that any attempt (Baird 1976; Lodge 1998) to determine N1 from the measurements of pH using Eq. (1.26) or Eq. (1.27) cannot be justiﬁed. Further, it is worth exploring the following view. pH of viscoelastic ﬂuids would depend on the Deborah number NDe , deﬁned by NDe = λ/tf , where λ is the char for a acteristic time of the ﬂuid and tf is the ﬂight time deﬁned by tf = dH /V for a circular pressure hole with V being the characteristic velocity, or tf = W/V transverse slot with W being the width of the slot. According to this view, for a given polymer melt or polymer solution, pH will approach zero when the ﬂuid behaves like an elastic solid; that is, when NDe becomes very large. This situation will occur when the ﬂuid is highly elastic (i.e., very high values of λ), so that the ratio λ/tf is very becomes extremely small, thus large, or when the ﬂuid ﬂows so fast that tf = W/V also making λ/tf very large. It should be pointed out that measurable value of pH can exist only when the ﬂuid relaxes its stress sufﬁciently while passing the distance W for a transverse slot or the distance dH for a circular pressure hole. When the velocity of a ﬂuid is so fast that little time is allowed for the stress to relax while passing the distance W or dH , there would be no measurable pH . No doubt there would be no measurable pH when W or dH approaches zero, since extremely small values of W or dH will make tf very small, which in turn will make NDe very large.

1.7

Analysis of Flow in a Coat-Hanger Die

The coat-hanger die is perhaps one of the most widely used dies in the extrusion operation. This is because it is relatively simple in design and yet versatile from a practical point of view. Figure 1.43 shows both the side view and the plane view of a coat-hanger die. However, the design of a coat-hanger die is not as simple as it may ﬁrst appear. In the past, numerous investigators reported on the design of coathanger dies with varying sophistication; some (Matsubara 1979, 1980, 1983; Liu et al. 1994) obtained analytical expressions based on several simplifying assumptions, while others (Arpin et al. 1992; Vergnes et al. 1984) employed ﬁnite difference methods to solve a system of equations, and still others (Pittman and Sander 1996; Pittman et al. 1995; Puissant et al. 1994; Wang 1991; Wen and Liu 1995) employed ﬁnite element methods to solve a system of equations. Granted that ﬁnite element methods are the best approach for handling the geometry of a coat-hanger die as close to the real situation as possible, but there is very little that we can discuss here if we choose numerical methods to solve a system of equations. Thus, instead of presenting numerical methods to solve a system of equations, here we adopt a more simpliﬁed approach, enabling us

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

41

Figure 1.43 Schematic of a coat-hanger die: (a) side view and (b) plane view.

to derive algebraic expressions that show the underlying principles in the design of a coat-hanger die. In the design of a coat-hanger die, one must consider three separate regions: (1) the manifold, (2) the coat-hanger section, and (3) slit-die section, as shown schematically in Figure 1.43. Let us consider the ﬂow through the manifold and through the coat-hanger section, as shown schematically in Figure 1.44, namely, (a) the variation of the diameter along the axis of the manifold and the plane view of the coat-hanger section, (b) the side view (opening) of the coat-hanger section at various positions (A –A, B –B, C –C, D –D, E –E) across the die width, and (c) the cross-sectional view of the die opening at the end of the coat-hanger section (from A to F). The basic requirement for the design of an acceptable coat-hanger die is to have a uniform linear velocity or ﬂow rate across the die width (from A to F). In order to meet this requirement, qualitatively speaking, the ideal design of a coat-hanger die must have the following features. Referring to Figure 1.44, the cross-sectional area of the manifold must decrease along the direction from A to F due to the leakage ﬂow to the coat-hanger section (Figure 1.44a). Since the machine directional distance decreases as we move from the center (A –A) to the edge (F –F) of the coat-hanger section, the opening (ﬂow path) of the coat-hanger section must increase as we move from the center (A –A) to the edge (F –F) (Figure 1.44b). This then makes the proﬁle of the cross section at the end of the coat-hanger section (A–F) look like the schematic given in Figure 1.44c. It should be pointed out that some investigators (Liu et al. 1994; Matsubara 1979, 1980, 1983) have assumed a constant

42

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 1.44 Schematic showing (a) the ﬂow distribution in the manifold of a coat-hanger die,

(b) openings of the coat-hanger section at various positions along the die width, and (c) crosssectional view of the die opening at the end of coat-hanger section.

opening (ﬂow path) of the coat-hanger section. Such an assumption results in a serious consequence. Brieﬂy stated, a constant opening of the coat-hanger section cannot give rise to a uniform linear velocity or ﬂow rate across the center (A –A) to the edge (F –F) of the coat-hanger section. Before writing down the expressions relevant to the design of a coat-hanger die, let us consider the schematic given in Figure 1.45, describing the coordinates and deﬁning the variables necessary to prepare the various expressions, and let us make the following assumptions: (1) isothermal ﬂow prevails, (2) a truncated power-law model is adequate to describe the ﬂow behavior of a polymer melt, (3) the rectangular coordinates (x, y, z) are chosen inside the manifold, as shown in Figure 1.45, (4) the manifold has a circular cross section and its diameter decreases linearly in the x-direction (i.e., a tapered cylindrical tube), (5) ﬂow is fully developed inside the manifold and it forms a parabolic velocity proﬁle inside the manifold, (6) leakage ﬂow occurs from the manifold to the coat-hanger section (see Figure 1.44a), and (7) the entrance and exit effects are negligible. 1.7.1

Analysis of Flow in the Manifold

For ﬂow of a truncated power-law ﬂuid through a cylindrical tube with constant cross section, we have the following expression for the velocity vx : n vx = n+1

1 2K

dp − dx

1/n

r (n+1)/n R (n+1)/n 1 − R

(1.28)

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

43

Figure 1.45 Schematic showing (a) the manifold and coat-hanger section of a coat-hanger die and (b) the various quantities describing the manifold of a coat-hanger die.

in which n is the power-law index, K is the power-law consistency, −dp/dx is pressure gradient inside the manifold, and R is the radius of a cylindrical tube. Then, the average velocity v¯x is given by n v¯x = 3n + 1

1 2K

dp − dx

1/n R (n+1)/n

(1.29)

and the volumetric ﬂow rate per unit die width, Q(x), is given by nπ Q(x) = 3n + 1

1 2K

dp − dx

1/n R (3n+1)/n

(1.30)

Note that at low shear rates, where Newtonian ﬂow prevails, n = 1 and K = η0 (Newtonian viscosity) in Eqs. (1.28)–(1.30). By keeping in mind that the manifold is assumed to be a tapered cylindrical tube, using Eqs. (1.28) and (1.29) we approximate the average velocity (v¯x ) by n v¯x = 3n + 1

1 2K

dp − dx

1/n (R − kx)(n+1)/n

(1.31)

44

PROCESSING OF THERMOPLASTIC POLYMERS

and the volumetric ﬂow rate (Q) inside the manifold by nπ Q(x) = 3n + 1

1 2K

1/n dp (R − kx)(3n+1)/n − dx

(1.32)

where, according to Figure 1.45, we have θ = θB + θC with θB = tan−1 [(H0 − HE )/L] and θC = tan−1 [R(1 − α)/L sec θB ], and k = tan θC = R(1 − α)/L sec θB with α being the ratio of the manifold radius at the edge (y = L) to the manifold radius at center (y = 0). Let the total volumetric ﬂow rate entering the manifold be 2Q0 with half of the ﬂuid going to the left manifold and half into the right manifold. Note the ﬂow rate in the coat-hanger section is Q0 /L, with L being the half width of the coating-hanger section. Here, we consider only the right manifold, as shown schematically in Figure 1.45. For uniform ﬂow across the width of the coat-hanger section, the following relationship must be satisﬁed: Q(y) = Q0 −

Q0 L

1−

1 y L

(1.33)

Note in Eq. (1.33) that at the center (at y = 0) Q(0) = Q0 , and at the edge (at y = L) Q(L) = Q0 /L, and the second term describes the leakage ﬂow from the manifold to the coat-hanger section. From Eq. (1.33), we have dQ = dy

Q0 L

1 −1 L

(1.34)

which is constant. From Figure 1.45b we have the relationships dy/dx = cos θ and y = x sec θ. Thus, changing the variable x with y, Eq. (1.32) can be rewritten in terms of the variable y as 1/n nπ 1 dp 1/n Q(y) = (cos θ ) (R − ky sec θ)(3n+1)/n − 3n + 1 2K dy

(1.35)

from which we can calculate −(∂p/∂y)y=0 since Q(0) = Q0 . Thus, from Eq. (1.35) we have π dQ = dy 3n + 1

cos θ 2K

1/n

(R − ky sec θ )

(3n+1)/n

dp − dy

(1−n)/n

d2 p − 2 dy

Q 1 dp 1/n (2n+1)/n −1 (R − ky sec θ) −k(sec θ )(3n + 1) − = 0 dy L L (1.36)

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

45

The last term of Eq. (1.36) follows from Eq. (1.34). Equation (1.36) can be rewritten as dp 1/n dp (1−n)/n d2 p (1+2n)/n (3n+1)/n − − − 2 = ξ +β(R −λy) (R −λy) dy dy dy (1.37) Q0 1 π cos θ 1/n −1 where λ = k/cos θ, β = k(3n + 1)/cos θ, ξ = L L (3n + 1) 2K Now, Eq. (1.37) must be solved numerically in order to calculate the pressure pm (y) in the manifold (see Figure 1.45a), which will later be used to calculate the pressure pE at the end of the coat-hanger section (see Figure 1.45a). Equation (1.37) must be solved, subjected to the initial conditions of p = p0 (pressure at the beginning of the manifold with negligible entrance effect) and dp/dy = c1 at y = 0. Note that c1 can be determined from Eq. (1.35) using Q(0) = Q0 . 1.7.2

Analysis of Flow in the Coat-Hanger Section

The coat-hanger section consists of two regions: a triangular coat-hanger section and a small slit section, as schematically shown in Figure 1.45a. The average velocity v¯z for a power-law ﬂuid ﬂowing in a slit die with a constant channel opening h is given by n v¯z = 2n + 1

1 K

dp − dz

1/n (n+1)/n h 2

(1.38)

and apparent shear rate is given by γ˙app = 6Q0 /Lh2 (see Chapter 5 of Volume 1). However, for the situation under consideration here, h varies with distance y along the die width (see Figure 1.44c), while the volumetric ﬂow rate across the die width for uniform ﬂow is given by Q0 /L, where L is the half width of the die (see Figure 1.45a). Therefore, we have v¯z h(y) = Q0 /L, thus n 2n + 1

1 K

1/n Q h(y) (n+1)/n dp h(y) = 0 − dz 2 L

(1.39)

or

h(y) = 2

(n+1)/n

2n + 1 n

Q0 L

K

1/n

dp − dz

−1/n n/(2n+1) (1.40)

which allows one to determine the opening proﬁle of the coat-hanger section across the die width. We can now derive an expression that will allow us to calculate the pressure pE at the end of the coat-hanger section in the machine direction. Let us assume that the axial

46

PROCESSING OF THERMOPLASTIC POLYMERS

pressure gradient, −dp/dz, in the machine direction is constant (i.e., fully developed ﬂow is assumed here). We then have −

p (y) − pE dp = m dz H (y)

(1.41)

where pm (y) is the pressure in the manifold that varies with position y, and H(y) is the distance between the manifold and the end of the slit region of the coat-hanger section (see Figure 1.45a), which is given by H (y) = H0 − [(H0 − HE )/L] y, with H0 being the length of the entire coat-hanger section including the slit region at center (y = 0) and HE being the length of the slit region of the coat-hanger section. Substitution of Eq. (1.41) into (1.40) gives h(y) = 2(n+1)/n

2n + 1 n

Q0 L

H (y)K pm (y) − pE

1/n n/(2n+1) (1.42)

where H(y) is given by H (y) = H0 −

H0 − HE L

(1.43)

y

Thus, pE can be determined from substitution of Eq. (1.39) into (1.41):5 pE = pm (L) − 2

n+1

HE K

1 hE

2n+1

Q0 L

n

2n + 1 n

n (1.44)

where hE = h(L), the die opening at y = L, and HE = H(y = L). Note that values of H0 , HE , and hE are given once the die geometry is speciﬁed, and pm (L) is calculated from the solution of Eq. (1.37). Hence, the calculation procedures for determining the proﬁles of the die opening in the coat-hanger section are as follows. (1) Solve Eq. (1.37) numerically to obtain pressure in the manifold, pm (y), subjected to the initial conditions p = p0 (the inlet pressure to the manifold that is given) at y = 0, and (−dp/dy)y=0 at y = 0 from Eq. (1.35), and by setting Q(y) = Q0 at y = 0. (2) Calculate pE from Eq. (1.44) using the value of pm (L) calculated in the previous step and speciﬁed values of HE , h(L), L, Q0 , K, and n. Note that pE < pm (L) must be satisﬁed. Otherwise, the size of the manifold must be varied until the inequality pE < pm (L) is satisﬁed. (3) Calculate the proﬁles of the die opening h(y) at various positions y from Eq. (1.42), for which H(y) is calculated from Eq. (1.43), and pm (y) and pE are calculated in the previous steps. Figure 1.46 gives the calculated proﬁles of the die opening h(y) of the coat-hanger die for different values of α when an LDPE is extruded at 200 ◦ C. It can be seen in Figure 1.46 that die opening h(y) increases with y, the extent of which increases with decreasing α. In the above calculation, we ﬁnd pE = 4.42 MPa (641 psi) for α = 0.8, pE = 3.62 MPa (525 psi) for α = 0.7, and pE = 2.55 MPa (371 psi) for α = 0.6. This means that a slit-die section (see Figure 1.45) must be added to the coat-hanger section

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

47

Figure 1.46 Computed proﬁles of the opening of the coat-hanger section along the die width

for values of α of (1) α = 0.8, (2) α = 0.7, (3) α = 0.6 in the extrusion of an LDPE at 200 ◦ C in a coat-hanger die having the dimensions R = 0.8 cm, 2L = 100 cm, H0 = 15 cm, HE = 0.5 cm, hE = 0.3 cm, at a mass ﬂow rate (2Q0 ) of 150 kg/h and with the entrance pressure (p0 ) of 10.34 MPa (1,500 psi). The numerical values of the parameters appearing in the truncated power-law model are η0 = 7.46 × 103 Pa·s, K = 6.85 × 103 Pa·s0.44 , n = 0.44, and γ˙c = 0.86 s−1 .

to accommodate the additional pressure drop from the coat-hanger section to the exit plane of the slit-die section. The length of the slit-die section HS can be calculated from the expression (p)S =

2HS hE

K

2n + 1 3n

n

6Q0 LhE

n (1.45)

where (p)S is the pressure drop that would occur in the slit-die section, which is the same as pressure pE at the end of the coat-hanger section. The above observations indicate that whenever the extrusion temperature or ﬂow rate is changed, a new coat-hanger die must be used in order to achieve uniform ﬂow distribution across the die, because different processing conditions require different proﬁles of the die opening in the coat-hanger/slit-die section. The use of so many dies is untenable from a practical point view because it is too costly to keep many dies in the ﬁrst place and frequent interruption of production for the reason of changing the die is not desirable. Here comes engineering ingenuity! The industry has long practiced the adjustment of the die opening by mechanical means, in particular by deﬂecting the die body with the aid of a choker bar, as schematically shown in Figure 1.43a. More often than not, a so-called “lip adjustment screw” is used to provide additional adjustment of the die opening, as schematically shown in Figure 1.43a. Such a practice reduces the operating costs of extrusion considerably. Needless to say, the use of choker bars is successful within practical limitations.

48

PROCESSING OF THERMOPLASTIC POLYMERS

and axial velocity averaged over the die Figure 1.47 Computed ﬂow rate per unit width (Q) width (v¯z ) at the die exit (a) without considering the deﬂection of the die land by choker bar and (b) with consideration of the deﬂection of the die land by choker bar. (Reprinted from Pittman and Sander, Polymer Engineering and Science 36:1982. Copyright © 1996, with permission from the Society of Rheology.)

Pittman et al. (1995, 1996) used ﬁnite element analysis to solve a system of equations for the ﬂow in a coat-hanger die with and without considering the deﬂection of a die body by a mechanical means. Figure 1.47 gives the simulated results, showing that and average velocity (v¯ ) in a coat-hanger the average ﬂow rate per unit die width (Q) z die are not uniform without deﬂection of a die body (part (a)), compared with the situations where a die body is deﬂected (part (b)). Such results from numerical computation underscore the validity of the basic principles we have expounded (Figure 1.46) in the design of coat-hanger dies.

1.8

Summary

In this chapter, we have described the ﬂow of viscoelastic ﬂuids in complex ﬂow geometries. Since one can conceive of many different ﬂow geometries other than those considered in this chapter, it is virtually impossible to show ﬂow behavior in all conceivable complex ﬂow geometries. However, there are some common features in the ﬂow through all complex geometries, such as, ﬂow in the entrance region, ﬂow in the

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

49

exit region, and ﬂow passing by a cavity (or hole). Hence, in this chapter we have attempted to present some unique features of viscoelastic ﬂuids when subjected to ﬂow in such common complex ﬂow geometries. The entrance ﬂow of viscoelastic ﬂuids was discussed extensively, both experimentally and computationally, in the 1970s and 1980s. The details of predicted entrance ﬂow of viscoelastic ﬂuids via ﬁnite element methods depend very much on the choice of constitutive equations. It is well documented that use of differential-type constitutive equations leads to numerical instability at large values of NDe , while use of integral-type constitutive equations can alleviate the problem to some extent. However, little has been reported on ﬁnite element computations of entrance ﬂow at the very large values (10 or 20) of NDe often encountered in industrial extrusion operations. This is a subject that requires further effort in the future. The same comment can be made for the ﬁnite element computations of extrudate swell of viscoelastic ﬂuids. A very important issue when dealing with complex ﬂow which is very often encountered in the processing of thermoplastic polymers in industry is nonisothermal ﬂow. The majority of the investigations reported to date in the literature have dealt with isothermal ﬂow. A case in point is the injection molding of thermoplastic polymers (e.g., PS, PP, and HDPE), which are highly viscoelastic in the molten state. Speciﬁcally, a highly viscoelastic ﬂuid is injected into a cold mold cavity and so heat transfer is inevitable. Thus, a proper description of injection molding requires numerical solutions of both momentum and heat transfer equations. The complexity of the problem is compounded by the fact that a viscoelastic ﬂuid at very high shear rates (much higher than 1,000 s−1 ), upon leaving the gate of a mold, begins to relax within the cold mold cavity. Further, due to very high shear rates encountered in the runner, the very viscous molten polymer experiences shear heating, which then increases the temperature of the melt. This observation suggests that one must ﬁrst solve nonisothermal ﬂow problems in the runner in order to describe accurately the ﬂow before the polymer melt reaches the gate of a mold cavity. Very few studies, if any, have addressed this problem properly. This problem is discussed in more detail in Chapter 8. Another very important issue that must be addressed in solving complex ﬂow problems of industrial importance is the incorporation of phase transition into the momentum and heat transfer equations. This is because all polymer melt processing operations encounter phase transition from the molten state to the solid state. The system equations to be solved become very complicated when dealing with crystallizable polymer. In the past, some efforts have been made to address this issue. Among those efforts, the studies of McHugh and coworkers took a very sound approach to modeling melt spinning (Doufas et al. 2000) and tubular ﬁlm blowing (Doufas and McHugh 2001) processes. More efforts in this direction would be highly desirable.

Problems Problem 1.1

For isothermal ﬂow of a Newtonian ﬂuid with viscosity η0 between two parallel plates with sliding upper plate (see Figure 1.14), where the plate width W is much

50

PROCESSING OF THERMOPLASTIC POLYMERS

larger than the opening H between the two plates (W H ), a simpliﬁcation of Eqs. (1.1) and (1.2) can be made to −

∂ 2v ∂p + η0 2x = 0 ∂x ∂y

(1P.1)

−

∂ 2v ∂p + η0 2z = 0 ∂z ∂y

(1P.2)

Verify that the axial velocity vz and the transverse velocity vx , respectively, are given by vz 1 y + = Vbz H 2η0

∂p y − yH 1− ∂z H

vx y = Vbz H

3y 2− H

(1P.3)

(1P.4)

and the volumetric ﬂow rate Q is expressed by V HW WH 3 + Q= b 2 12η0

∂p − ∂z

(1P.5)

Problem 1.2

Consider the ﬂow of PS through two nonparallel plates, as shown schematically in Figure 1.48, where the channel depths H1 and H2 are very small compared with the channel width W (i.e., H1 W and H2 W ). Under such situations, the dependence of vz on x can be neglected, simplifying Eqs. (1.1)–(1.3) further to −

∂ ∂p + ∂x ∂y

∂ ∂p + − ∂z ∂y

ρcp vz

∂v η x =0 ∂y

∂v η z ∂y

(1P.6)

=0

(1P.7)

∂vz 2 ∂T ∂ 2T = km 2 + η ∂z ∂y ∂y

(1P.8)

Assume that the polymer follows a truncated power-law model: η(γ˙ , T ) =

⎧ ⎨k0 exp(−bT )

for γ˙ ≤ γ˙o

⎩k exp(−bT )(γ˙ /γ˙ )n−1 0 o

for γ˙ > γ˙o

(1P.9)

and the dimensions of the ﬂow channel are H1 = 0.35 cm, H2 = 0.25 cm, W = 6 cm, and L = 5 cm, and the ﬂow conditions are (1) the inlet melt temperature of 180 ◦ C,

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

51

Figure 1.48 Schematic of two non-parallel plates with the upper plane moving in the z-direction,

in which the channel width W is much larger than the channel depths, H1 and H2 .

(2) the mass ﬂow rate of 40 kg/h with the density of 0.76 g/cm3 , and (3) the temperature of both plates is kept at 200 ◦ C. Predict the following: (a) the pressure proﬁle along the z-axis, (b) the axial velocity proﬁle vz (y), (c) the transverse velocity proﬁle vx (y), (d) the temperature proﬁle T (y), and (e) the viscosity proﬁle. Use the following numerical values of the parameters appearing in Eq. (1P.9): ko = 7.269 × 1013 Pa·s, b = 4.886 × 10−2 K−1 , γ˙o = 1.2 s−1 , and n = 0.33. Problem 1.3

Consider the ﬂow of a molten polymer through narrow parallel plates (i.e., height H is much small than the width W ) where the upper plate moves at a constant speed in the direction forming an angle θ with the axis of the stationary lower plate. Assuming that the ﬂow condition is chosen such that an isothermal condition prevails and that the molten polymer follows a power-law model (see Eq. (6.1) in Volume 1), prepare plots of dimensionless ﬂow rate Q versus dimensionless pressure drop p for different values of power-law index n, namely n = 0.1, 0.2, 0.4, 0.5, 0.6, WH and 0.8, for θ = 17.7◦ . Note that Q and p are deﬁned by Q = Q/V n and p = (H/V ) (Hp/KL), respectively, where Q is the volumetric ﬂow rate, is the velocity of the moving upper plate, W is the channel width, H is the V channel height, L is the length of the channel, p is the pressure drop, n is the power-law index, and K is the power-law consistency. Also prepare plots of Q versus p for ﬂow through two stationary parallel plates (i.e., θ = 0). Discuss how the angle θ affects such plots. Verify that for Newtonian ﬂuids (i.e., n = 1), one has the following relationship: Q = 1/2 − (1/12)p . Problem 1.4

As shown in Figure 1.43, a coat-hanger die consists of three parts: the manifold, the coat-hanger section, and the slit-die section. Since a rigorous analysis of ﬂow in the manifold and coat-hanger sections is not possible without numerical calculations, you are asked to consider only the slit-die section. Calculate the pressure drop in the slit-die section when an LDPE is extruded at 200 ◦ C with a throughput of 100 kg/h. Assume that the LDPE follows the power-law model, deﬁned by Eq. (6.1) in Volume 1, with K = 7.6×103 Pa·s0.5 and n = 0.5 at 200 ◦ C, and the dimensions

52

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 1.49 Schematic of a metal-insert mold.

of the slit-die section are: the width of the slit-die section is 100 cm, the opening of the slit-die section is 2 mm, and the length of the slit-die section is 5 cm. In the calculation, you may neglect the entrance and exit effects. Problem 1.5

In polymer processing, one encounters a situation where a polymer melt is extruded through a die having a metal insert, as schematically shown in Figure 1.49. Consider that a PP having the density of 0.76 g/cm3 at 220 ◦ C is extruded at a ﬂow rate of 4 kg/h through the die shown in Figure 1.49. Assuming that the PP melt follows a truncated power-law model deﬁned by Eq. (1P.9), calculate the velocity and shear stress distributions inside the ﬂow channel using a ﬁnite element method. You may use commercial software6 to solve this problem. The numerical values of the parameters appearing in the truncated power-law model (see Eq. (1P.9)) for the PP are as follows: k0 = 1.571 × 109 Pa·s, b = 2.558 × 10−2 K−1 , γ˙o = 0.38 s−1 , and n = 0.39. Problem 1.6

The tubular ﬁlm blowing operation (see Chapter 7) uses an annular die, through which a molten polymer is extruded. Figure 1.50 shows schematics of three different tubular ﬁlm-blowing dies, each having a different entrance angle. It can easily be surmised that the geometry of the ﬂow channel in a tubular ﬁlm-blowing die has a profound inﬂuence on the velocity and stress distributions inside the annular die. Consider that an LDPE with a density of 0.716 g/cm3 at 200 ◦ C is extruded at a ﬂow rate of 5 kg/h through each of the three annular dies shown in Figure 1.50. Assuming that the LDPE melt follows a truncated power-law model deﬁned by Eq. (1P.9), calculate the velocity and shear stress distributions inside the ﬂow channel of the each tubular ﬁlm-blowing die shown in Figure 1.50 using a ﬁnite element method. You may use commercial software6 to solve this problem. The numerical values of the parameters appearing in the truncated power-law model for the LDPE melt are: k0 = 7.467 × 108 Pa·s, b = 2.507 × 10−2 K−1 , γ˙o = 0.30 s−1 , and n = 0.53.

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

53

Figure 1.50 Schematics of three tubular ﬁlm-blowing dies.

Notes 1. See Chapter 3 of Volume 1. 2. The Arrhenius form of the temperature dependence of viscosity is approximated by the empirical expression exp(−bT ), which gives a much more stable numerical solution than the Arrhenius form. The approximation is valid over a narrow range of temperature variations. 3. A tapered die has no slit die section ahead, thus it looks like a die consisting of two nonparallel plates. 4. The authors (Brizitski et al. 1978) of this article did not provide the value of σyy (0, z) for curve 3 in Figure 1.29a. Further, it appears that Figure 12 of this article has typographical errors in the values of σyy (0, z) for curves 1 and 2. Thus, the values of σyy (0, z) for curves 1 and 2 given in Figure 1.29a were guestimated by this author. 5. Note from Eq. (1.41) that pE = pm (L) − HE (−∂p/∂z)|y=L , where HE = H (y = L) and (−∂p/∂z)|y=L is given by Eq. (1.39) with h(y = L) = hE . 6. Commercial software available from Fluent.

54

PROCESSING OF THERMOPLASTIC POLYMERS

References Adamse JWC, Janeschitz-Kriegl H, den Otter JL, Wales JLS (1968). J. Polym. Sci. A-2 6:871. Arai T, Aoyama H (1963). Trans. Soc. Rheol. 7:333. Arai T, Hatta H (1980). Nihon Reoroji Gakkaishi 8:67; ibid. 8:110. Arpin B, Laﬂeur PG, Bergnes B (1992). Polym. Eng. Sci. 32:206. Bagley EB, Storey SH, West DC (1963). J. Appl. Polym. Sci. 7:1661. Baird DG (1975). Trans. Soc. Rheol. 19:147. Baird DG (1976). J. Appl. Polym. Sci. 20:3155. Brindley G, Broadbent JM (1973). Rheol. Acta 12:48. Brizitsky VI, Vinogradov GV, Isayev AI, Podolsky YY (1978). J. Appl. Polym. Sci. 22:751. Broadbent JM, Kaye A, Lodge AS, Vale DG (1968). Nature 217:55. Broadbent JM, Lodge AS (1971). Rheol. Acta 10:557. Cable PJ, Boger DV (1978). AIChE J. 24:869. Chang PW, Patten TW, Finlayson BA (1979). Computer and Fluids 7:285. Cochrane T, Walter K, Webster MF (1981). Phil. Trans. Roy. Soc. (London) A. 301:163. Crochet MJ, Bezy M (1979). J. Non-Newtonian Fluid Mech. 5:201. Crochet MJ, Keunings R (1980). J. Non-Newtonian Fluid Mech. 7:199. Delvaux V, Crochet MJ (1990). J. Rheol. Acta 29:1. Doufas AK, McHugh AJ (2001). J. Rheol. 45:1085. Doufas AK, McHugh AJ, Miller C (2000). J. Non-Newtonian Fluid Mech. 92:27. Drexler LH, Han CD (1973). J. Appl. Polym. Sci. 17:2355. Durelli AJ, Riley WF (1965). Introduction to Photomechanics, Prentice-Hall, Englewood Cliffs, New Jersey. Elbirli B, Lindt JT (1984). Polym. Eng. Sci. 24:482. Ericksen JL (1956). Quart. Appl. Math. 14:318. Frocht MM (1941). Photoelasticity, Vol 1, John Wiley & Sons, New York. Funatsu K, Mori Y (1968). Chem. High Polym. (Japan) 25:337. Giesekus H (1965). Rheol. Acta 4:85. Green AE, Rivlin RS (1956). Quart. Appl. Math. 14:229. Guillet JE, Combs RL, Slonaker DF, Weems DA, Coover HW (1965). J. Appl. Polym. Sci. 9:757. Han CD (1975). Rheol. Acta 14:173. Han CD (1976). Rheology in Polymer Processing, Academic Press, New York. Han CD (1988). Polym. Eng. Sci. 28:1227. Han CD, Charles M (1970). AIChE J. 16:499. Han CD, Charles M, Philippoff W (1970). J. Rheol. 14:393. Han CD, Drexler LH (1973a). J. Appl. Polym. Sci. 17:2329. Han CD, Drexler LH (1973b). J. Appl. Polym. Sci. 17:2369. Han CD, Drexler LH (1973c). Trans. Soc. Rheol. 17:659. Han CD, Kim KU (1971). Polym. Eng. Sci. 11:395. Han CD, Kim KU (1973). Trans. Soc. Rheol. 17:151. Han CD, Yoo HJ (1980). J. Rheol. 24:55. Hendry AW (1966). Photoelastic Analysis, Pergamon Press, New York. Higashitani K, Lodge AS (1975). Trans. Soc. Rheol. 19:307. Higashitani K, Pritchard WG (1972). Trans. Soc. Rheol. 16:687. Hou TH, Tong PP, De Vargas L (1977). Rheol. Acta 16:544. Jackson NR, Finlayson BA (1982). J. Non-Newtonian Fluid Mech. 10:55. Kajiwara T, Kuwano Y, Funatsu K (1991). Rheol. J. (Japan) 19:32. Kaye A, Lodge AS, Vale DG (1968). Rheol. Acta 7:368. Langlois WE, Rivlin RS (1963). Rend. Math. 22:169.

FLOW OF POLYMERIC LIQUID IN COMPLEX GEOMETRY

Liu TJ, Liu LD, Tsou JD (1994). Polym. Eng. Sci. 34:541. Lodge AS (1955). Nature 176:838. Lodge AS (1998). In Rheological Measurement, 2nd ed, Collyer AA and Clegg DW (eds), Chapman & Hall, London, Chapter 10. Luo XL, Mitsoulis E (1990). J. Rheol. 34:309. Matsubara Y (1979). Polym. Eng. Sci. 19:169. Matsubara Y (1980). Polym. Eng. Sci. 20:212, 716. Matsubara Y (1983). Polym. Eng. Sci. 23:17. McKelvey JM (1962). Polymer Processing, John Wiley & Sons, New York, p 319. McLuckie C, Rogers MG (1969). J. Appl. Polym. Sci. 13:1049. Mendelson RA, Finger FL (1973). J. Appl. Polym. Sci. 17:797. Nakajima N, Shida M (1966). Trans. Soc. Rheol. 10:299. Novotny EJ, Eckert RF (1973). Trans. Soc. Rheol. 17:227. Nguyen H, Boger DV (1979). J. Non-Newtonian Fluid Mech. 5:353. Park HJ, Mitsoulis E (1992). J. Non-Newtonian Fluid Mech. 42:301. Philippoff W (1956). J. Appl. Phys. 27:984. Philippoff W (1957). Trans. Soc. Rheol. 1:95. Philippoff W (1961). Trans. Soc. Rheol. 5:163. Pittman JF, Sander R, Schuler W, Pick H, Martin G, Stannek W (1995). Inter. Polym. Processing 10:137. Pittman JF, Sander R (1996). Polym. Eng. Sci. 36:1982. Prichard WG (1970). Rheol. Acta 9:200. Puissant R, Demay Y, Vergnes B, Agassant JF (1994). Polym. Eng. Sci. 34:201. Reddy KR, Tanner RI (1978). J. Rheol. 22:661. Reiner M (1945). J. Amer. J. Math. 67:350. Rivlin RS (1948). Proc. Roy. Soc. A193:260. Rivlin RS (1964). In Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, Reiner M, Abir D (eds). Macmillan, New York, p 668. Rivlin RS, Ericksen JL (1964). J. Ration. Mech. Anal. 4:323. Rogers MG (1970). J. Appl. Polym. Sci. 14:1679. Sekiguchi M (1969). Chem. High Polym. (Japan) 26:721. Semjonow VV (1965). Rheol. Acta 6:171. Tadmor Z, Klein I (1970). Engineering Principles of Plasticating Extrusion, Van Nostrand Reinhold, New York Tanner RI, Pipkin AC (1969). Trans. Soc. Rheol. 13:471. Treloar LRG (1958). The Physics of Rubber Elasticity, 2nd ed, Oxford University Press, London. Vergnes B, Sallilard P, Agassant JF (1984). Polym. Eng. Sci. 24:980. Viriyayuthakorn M, Coswell B (1980). J. Non-Newtonian Fluid Mech. 6:245. Wales JLS (1969). Rheol. Acta 8:38. Wales JLS, Janeschitz-Kriegl H (1967). J. Polym. Sci. A-2 5:781. Wang Y (1991). Polym. Eng. Sci. 31:204. Wen SH, Liu TJ (1995). Polym. Eng. Sci. 35:759. Wheeler JA, Wissler EH (1965). AIChE J. 11:207. Wheeler JA, Wissler EH (1966). Trans. Soc. Rheol. 10:353. White JL, Kondo A (1977/1978). J. Non-Newtonian Fluid Mech. 3:41. Yoo HJ, Han CD (1981). J. Rheol. 25:115.

55

2

Plasticating Single-Screw Extrusion

2.1

Introduction

There are two types of extruder: (1) single-screw extruders and (2) twin-screw extruders. The single-screw extruder is one of the most important pieces of equipment in the processing of thermoplastic polymers. Accordingly, during the past three decades, many attempts have been made to analyze the performance of single-screw extruders using different degrees of mathematical sophistication (Cox and Fenner 1980; Donovan 1971; Edmondson and Fenner 1975; Elbirli et al. 1983, 1984; Halmos et al. 1978; Han et al. 1991a, 1991b, 1996; Lee and Han 1990; Lindt 1976; Lindt and Elbirli 1985; Shapiro et al. 1976; Tadmor 1966; Tadmor and Klein 1970; Tadmor et al. 1967). There are two types of single-screw extruders: (a) plasticating and (b) melt-conveying. The plasticating single-screw extruder conveys a solid polymer from the feed section to the melting section, where most of the melting (or softening) occurs, and then transports the melted or softened polymer to a shaping device (e.g., dies and molds). The meltconveying extruder does not include a melting section; it simply transports an already softened polymer to a shaping device (e.g., rubber extruder). Single-screw extruders are used for various purposes, such as melting and pumping, compounding with an additive(s) or ﬁller, cooling and mixing, removing residual monomers or solvents in polymer (i.e., polymer devolatilization), and cross-linking reactions. Single-screw extruders are simple to operate and relatively inexpensive as compared with twin-screw extruders. However, there are situations where a single-screw extruder cannot function as effectively as a twin-screw extruder. In the design of plasticating single-screw extruders, one needs information on (1) the physical and thermal properties of polymers (e.g., friction coefﬁcient between the solid polymer and barrel wall, thermal conductivity of polymer, speciﬁc heat as a function of temperature, melting point of polymer, and heat of fusion of polymer) and (2) rheological properties of polymers as functions of shear rate and temperature. Due to the complexity involved in the design of extruders, it is highly desirable for one to establish relationships between material variables and processing variables. In this 56

PLASTICATING SINGLE-SCREW EXTRUSION

57

regard, it is important for one to be able to evaluate the performance of plasticating extruders. For this, one needs a mathematical model that simulates plasticating singlescrew extruders. Once a mathematical model is established, it will help predict power requirement, throughput, pressure distributions along the extruder axis, temperature distributions inside the screw channel, the extent of temperature rise due to viscous shear heating, residence time distribution in the extruder, and optimum operating conditions. In this chapter, we discuss the performance of different types of single-screw extruders.

2.2

Performance of Plasticating Single-Screw Extruders for Semicrystalline Polymers

Figure 2.1 shows a photograph of a plasticating single-screw extruder, which has three geometrical sections: (1) the feed section, with a deep uniform channel, (2) the transition (tapered) section, with a uniformly decreasing channel depth, and (3) the metering section, with shallow uniform channel. One can also divide the extruder, as depicted schematically in Figure 2.2, into three sections from the point of view of function: (1) solid-conveying section, (2) melting section, and (3) melt-conveying section. The locations of these sections along the screw axis depend on operating conditions and therefore they usually do not coincide with those of the geometrical sections. Of the

Figure 2.1 A plasticating single-screw extruder. (Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.)

58

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 2.2 Schematic showing the functions of a plasticating single-screw extruder: (a) solidconveying section, (b) melting section, and (c) melt-conveying section. (Reprinted from Lee and Han, Polymer Engineering and Science 30:665. Copyright © 1990, with permission from the Society of Plastics Engineers.)

three sections, the melting section has received the most attention by researchers. This is understandable because in the melting section one must deal not only with the phase transition of a polymer from the solid state to ﬂuid state, but also with the ﬂow of rheologically complex molten polymers in a complicated ﬂow geometry. It must be emphasized that an evaluation of the performance and/or a design of plasticating single-screw extruders is of little practical signiﬁcance without a proper analysis of both the thermal and the ﬂow phenomena occurring in the melting section of the extruder. The heart of a plasticating extruder is the screw, the design of which varies with the type of polymer to be extruded; that is, whether the polymer can be regarded as soft or rigid, or whether the polymer is semicrystalline or amorphous, whether the polymer has a high or low melting temperature, or whether the polymer is sensitive to thermal degradation or not. Figure 2.3 gives photographs of some typical plasticating single screw designs, which are in wide use in the plastics fabrication industry. The design of screws must, therefore, be based on a sound theoretical understanding of the various physical processes underway during extrusion. Suggestions that the performance of a plasticating single-screw extruder can be analyzed with consideration of only the meltconveying section are believed to be grossly misleading. Among the three functional sections in the extruder, the melting section is the most important, and of course it is very complex to analyze. In this section, we ﬁrst present currently held theories of plasticating extrusion of semicrystalline polymers, followed by an analysis of the extrusion of amorphous polymers. Finally, we compare theoretical predictions with experimental results.

PLASTICATING SINGLE-SCREW EXTRUSION

59

Figure 2.3 Typical single screws: (a) standard metering screw, (b) metering screw with the

Maddock mixing head, and (c) barrier screw with the Maddock mixing head. (Reprinted with permission from Davis-Standard Corporation.)

2.2.1

Analysis of the Solid-Conveying Section

In the solid-conveying section, the pressure builds up as polymer pellets or powders are transported by the rotating motions of the screw. At the same time, the temperature of the polymer, especially near the barrel wall, increases due to the heat conducted from the barrel wall and the heat generated by the friction between the polymer and the barrel wall. The heat generated due to the frictional force will help form thin melt ﬁlms at and near the barrel wall. We present the Darnell–Mol analysis (Darnell and Mol 1956) to determine the temperature proﬁles along the screw axis and the position at which melting begins. Subsequently, this information will be used as initial conditions for the analysis of the melting section. The pressure proﬁle along the screw axis, P (z), in the solid-conveying section may be described by P (z) = P0 exp

B1 − A1 Kz A2 Kz + B2

z

(2.1)

where P0 is the pressure at the inlet of the extruder (i.e., at z = 0), and A1 , A2 , B1 , B2 , and Kz are parameters that depend on the screw geometry, coefﬁcients of friction between the polymer and the barrel wall and between the polymer and the screw surface, and pressures acting on the screw, barrel, and ﬂights. The heat qb generated by friction on the barrel surface along the screw axis z is related to the pressure P by qb = Pf b WVb

sin θ b sin(θ b + φ)

(2.2)

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PROCESSING OF THERMOPLASTIC POLYMERS

where W is the channel width, fb is the friction coefﬁcient, Vb is the barrel velocity, θb is the ﬂight helix angle, and φ is the angle of movement of the outer surface of the solid plug, which is deﬁned by tan φ =

Vb ρs π(Db − H )H 1 − 4mT tan θ b

−1 (2.3)

where mT is the total mass ﬂow rate, ρs is the density of solid polymer, Db is the barrel diameter, and H is the channel height. Note that the volumetric ﬂow rate Qs of the solid plug can be written as Qs = π2 NHDb (D b − H )H

tan φ tan θb tan φ + tan θb

(2.4)

where the ﬂight width is assumed to be negligibly small compared with the channel width and N is the screw speed. The heat generated at the barrel surface, given by Eq. (2.2), is dissipated into the solid plug and conducted away through the barrel walls. This can be expressed by qb = −ks

∂Ts ∂y

+ kb y=0

∂Tb ∂y

(2.5) y=0

where ks is the thermal conductivity of the solid plug, kb is the thermal conductivity of the barrel, Ts is the temperature of the solid plug, and Tb is the temperature of the barrel. If we assume that the temperature gradient in the barrel, ∂Tb /∂y, is constant and that the temperature proﬁle of the solid plug in the y-direction, which is perpendicular to the down-channel direction z, can be described by one-dimensional heat conduction, Eq. (2.5) can be rewritten as y/ks qb + Ts (1, n) + kb /ks (y/b) Tb (b, n) Ts (0, n) = 1 + kb /ks (y/b)

(2.6)

where Ts (0, n) and Ts (1, n) are the temperatures of the solid plug at y = 0 (i.e., at the barrel wall) and y, respectively, at a down-channel distance z = nz, n being an integer, and Tb (b, n) is the temperature of the barrel outer wall, where b is the thickness of the barrel wall, at a down-channel distance z = nz. Note that values of Ts (1, n) can be determined from solutions of the heat conduction equation. It can be seen that the temperature proﬁle Ts (0, n) along the screw axis, which depends on the amount of heat generated qb due to pressure rise, can be calculated from Eq. (2.6). 2.2.2

Analysis of the Melting Section

Maddock (1959) was the ﬁrst to have performed a very signiﬁcant experiment in order to understand the melting process in a plasticating single-screw extruder, which formed the basis of a proposed melting mechanism, which is today referred to as the “Maddock melting mechanism.” The Maddock melting mechanism states that the solid particles

PLASTICATING SINGLE-SCREW EXTRUSION

61

in contact with the hot barrel surface partially melt and form a thin melt ﬁlm over the barrel surface. This melt ﬁlm is dragged by the barrel surface, meeting the leading edge of the advancing ﬂight, and is then mixed with previously melted material, forming a melt pool, while the width of the solid bed gradually decreases. The melting process comes to an end when the solid bed disappears completely. Based on the Maddock melting mechanism, Tadmor and coworkers (Tadmor 1966; Tadmor et al. 1967; Tadmor and Klein 1970) developed a comprehensive mathematical model for the melting behavior of semicrystalline polymers in a single-screw extruder. Since then, many researchers either extended or modiﬁed the Tadmor analysis by relaxing one or more of the assumptions that Tadmor made. Speciﬁcally, Donovan (1971) introduced into the Tadmor model an empirical parameter, which describes the acceleration of the solid bed, and was able to predict a rise in the solid-bed temperature along the screw axis. Edmondson and Fenner (1975) also included acceleration of the solid bed in their melting model and concluded that break-up of the solid bed occurs when the model predicts rapid bed acceleration. As shown schematically in Figure 2.4, an idealized cross section of the melting section of the unwound screw channel, which depicts the Maddock melting mechanism, can be divided into ﬁve zones (Shapiro et al. 1976): (A) solid bed, (B) melt pool, (C) a thin melt ﬁlm between the barrel surface and the solid bed, (D) a thin melt ﬁlm between the screw surface and the solid bed, and (E) a thin melt ﬁlm between the screw ﬂight and the solid bed. In Figure 2.4, the barrel surface is at the top, moving at constant velocity Vb with velocity components Vbx and Vbz in the cross-channel and down-channel directions, respectively. The screw surface is at the bottom, and the screw ﬂights are at the two sides. Pearson and coworkers (Halmos et al. 1978; Shapiro et al. 1976) solved the momentum and energy equations by using the assumptions,

Figure 2.4 Schematic of the idealized cross section in the melting section of an unwound screw

channel, describing the Maddock melting mechanism. (Reprinted from Lee and Han, Polymer Engineering and Science 30:665. Copyright © 1990, with permission from the Society of Plastics Engineers.)

62

PROCESSING OF THERMOPLASTIC POLYMERS

among others, that (1) the solid bed is freely deformable and (2) the melt ﬁlms in zones D and E in Figure 2.4 have no cross-channel velocity component, thus no melt circulation around the solid bed. It was Lindt and coworkers (Elbirli et al. 1984; Lindt and Elbirli 1985) who ﬁrst relaxed these assumptions and solved all of the conservation equations for the ﬁve-zone model.1 By numerically solving two-dimensional momentum and energy equations in the melt pool as well as in thin melt ﬁlms, Lindt and Elbirli (1985) predicted the axial pressure and solid bed width proﬁles along the extruder axis. In their analysis, however, Lindt and Elbirli assumed a priori whether the solid bed was rigid or freely deformable; speciﬁcally, the solid-bed velocity was determined from a mass balance equation. Lee and Han (1990) introduced the concept of solid-bed deformation into the Lindt–Elbirli analysis. This concept does not assume a priori whether the solid bed is rigid or freely deformable. The solution of the force balance on the surfaces of the solid bed, which is surrounded by thin melt ﬁlms and a melt pool, with the assumption that the solid bed in the bulk state follows a linear stress–strain relationship, gives us information as to whether the solid bed deforms under a given extrusion condition and/or with a given screw geometry. We now present the Lee-Han analysis for plasticating single-screw extruders, which is based on the following assumptions: (1) the ﬂows are fully developed, (2) the lubrication approximation is applicable, (3) the Maddock melting mechanism is valid (see Figure 2.4), (4) the screw channel curvature does not affect the melting process appreciably, (5) the solid–liquid interfaces are smooth, (6) the solid bed is homogeneous and isotropic, (7) the polymer has a sharp melting point, and (8) the ﬂow through ﬂight clearance is negligible. (a) Zone A (Solid Bed)

The energy equation in the solid bed may be

described by ∂T ∂T ∂ 2T ρs cps Vsy + Vsz = ks 2 ∂y ∂z ∂y

(2.7)

where ρs is the density, cps is the speciﬁc heat, and ks is the conductivity of solid polymer, Vsy is the solid-bed velocity in the y-direction, Vsz is the solid-bed velocity in the down-channel direction, and the left-hand side represents convective heat transfer associated with the movement of the solid bed in the y- and z-directions. The solution of Eq. (2.7) must satisfy the following boundary conditions: T = To (y) at z = 0

(2.8a)

T = Tm and Vsy = −Vsy2 at y = 0

(2.8b)

T = Tm and Vsy = Vsy1

(2.8c)

at y = Hs

in which Tm is the melting point of solid polymer and Hs is the solid-bed height. Note that the temperature distribution within the solid bed cannot be symmetric, since solid/melt interfaces AC and AD (see Figure 2.4) are exposed to different thermal environments associated with the different heat conduction and dissipation rates in the

63

PLASTICATING SINGLE-SCREW EXTRUSION

respective melt ﬁlms. Therefore, in specifying the boundary conditions, Eq. (2.8), we have divided zone A into two different directions, Vsy2 describing the solid-bed velocity moving downward (in the negative direction) to the screw surface and Vsy1 describing the solid-bed velocity moving upward to the barrel surface. This means that there exists a position y = a ∗ at which Vsy = 0 (i.e., (∂Ts /∂y)|y=a ∗ = 0). It is important to notice that the following energy equations must hold at the two interfaces. 1. At solid/melt interface AC (i.e., between the solid bed and upper melt ﬁlm): km

∂T ∂T − k = ρs Vsy1 λ s ∂y melt ∂y solid

(2.9a)

2. At solid/melt interface AD (i.e., between the solid bed and lower melt ﬁlm): ks

∂T ∂T − k = ρs Vsy2 λ m ∂y solid ∂y melt

(2.9b)

where km is the thermal conductivity of polymer melt and λ is the heat of fusion of solid polymer. The down-channel solid-bed velocity Vsz can be determined by solving the force balance equations at the surfaces of the solid bed. In other words, the solution of the system of equations can tell us whether the solid bed will deform under a given set of operating conditions for a speciﬁc screw geometry. Referring to the schematic given in Figure 2.5, the forces acting on the surfaces of the solid bed in the down-channel

Figure 2.5 Schematic of the solid bed in the melting section of an unwound screw channel,

describing the various forces exerted on the surfaces of the solid bed. (Reprinted from Lee and Han, Polymer Engineering and Science 30:665. Copyright © 1990, with permission from the Society of Plastics Engineers.)

64

PROCESSING OF THERMOPLASTIC POLYMERS

direction may be written as (Lee and Han 1990) F1 = Ws zτyz |AD

(2.10a)

F2 = Hs zτxz |AE

(2.10b)

F3 = Ws zτyz |AC

(2.10c)

F4 = Hs zτxz |AB

(2.10d)

F5 − F6 = Ws Hs z(−∂p/∂z)

(2.10e)

where τxz |AB is the shear stress acting on the solid/melt interface AB in the z-direction, and the meanings of other shear stress components are self-explanatory. We postulate here that when the sum of all forces deﬁned in Eq. (2.10) is not equal to zero, the solid bed will deform and accelerate in the z direction. Under this postulation, we assume that a linear relationship between stress and strain can describe the rheology of the solid bed, so that

Fi = Eapp ε¯ W s Hs

(2.11)

where Ws and Hs are the width and height, respectively, of the solid bed, which vary with z, and Eapp is the “apparent” modulus and ε¯ is average strain of the solid bed. Note in Eq. (2.11) that Fi represents the sum of forces of the solid bed, consisting of shear stress imposed by the surrounding thin ﬁlms and melt pool, and the hydrostatic pressure differences between two faces perpendicular to the ﬂights, and that Eapp is by (Lee and Han 1990) related to the average modulus of the polymer E Eapp = fm E

(2.12)

where fm is a parameter whose value is a function of the size and shape of the polymer particles being fed into the extruder and the operating conditions (i.e., pressure and temperature). Basically, Eq. (2.11) represents the constitutive equation for the solid bed, in which the apparent modulus describes the rheological properties of the solid bed in the “bulk” state. In view of the fact that the temperature of the solid bed varies with positions y and z (see Eq. (2.7)), we must consider the dependence of the modulus of the polymer on temperature. For this, the following expression may be used (van Krevelen 1976):

E(Tr ) log E T (y)

Tm /Tr − Tm /T (y) = 1.15 Tm /Tr − 1

(2.13)

where E(Tr ) is the modulus of the polymer at a reference temperature Tr , and Tm is the melting point of the polymer, and E(T (y)) is the modulus of the polymer at temperature T, which in turn varies with position y. Since the solution of Eq. (2.7)

PLASTICATING SINGLE-SCREW EXTRUSION

65

provides the dependence of T on y and z, at a given value of z, the average modulus of the polymer can be calculated from = E

Hs 0

Hs E T (y) dy 0 dy

(2.14)

Since the solid bed consists of a very large number of pellets or powder particulates, the modulus of the individual pellets or powder particulates calculated from Eq. (2.14) must be related to the apparent modulus of the solid bed in the bulk state. At present, there is no theory that suggests how one can relate the average modulus of the polymer to the apparent modulus of the solid bed in the bulk state. Thus, we introduce an empirical relationship, Eq. (2.12), and calculate values of the average strain ε¯ of the solid bed by substituting values of Eapp , via Eqs. (2.12)–(2.14), into Eq. (2.11). In so doing, fm is regarded as an adjustable parameter, so as to ﬁt predicted pressure proﬁles along the extruder axis to experimental results. The down-channel solid-bed velocity Vsz nz at position z = nz is related to the solid-bed velocity Vsz (n−1)z at position z = zo + (n − 1)z, where zo is the position at which melting begins, by Vsz nz = 1 + ε¯ n Vsz (n−1)z

(2.15)

where ε¯ n is the average strain of the solid bed at position nz. Since the strain (i.e., the extent of deformation) of the solid bed will vary along the z-direction, Eq. (2.15) can be rewritten as N 1 + ε¯ n Vsz 0 Vsz nz =

(2.16)

n=1

where N is the total number of increments in the z-direction, such that melting is complete at z = Nz, and Vsz 0 is the down-channel solid-bed velocity at the beginning of the melting zone (at z = zo ), which can be estimated from a mass balance Vsz = mT /ρs Ho W

(2.17)

where mT is the total mass ﬂow rate, ρs is the density of polymer in the solid state, Ho is the channel height in the feed section, and W is the channel width. It should be pointed out that in their analysis Pearson and coworkers (Halmos et al. 1978; Shapiro et al. 1976) assumed that the sum of the forces acting on the surfaces of the solid bed in the down-channel direction, deﬁned in Eq. (2.11), was equal to zero (i.e., Fi = 0). In so doing, they calculated the axial pressure gradient −∂p/∂z from the momentum equation around the solid bed and then calculated the solid-bed velocity from a mass balance equation under the assumption of a freely deformable solid bed. The signiﬁcance of Eq. (2.11) lies in that the rheology of the solid-bed deformation was introduced into the force balance around the solid bed. Furthermore, the Lee–Han melting model calculates the solid-bed velocity using Eq. (2.16), instead of using a mass balance under the assumption of a freely deformable solid bed.

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PROCESSING OF THERMOPLASTIC POLYMERS

(b) Zone B (Melt Pool)

The momentum equation may be written as

∂vx η ∂y ∂v ∂v ∂p ∂ ∂ η z + η z = ∂x ∂x ∂y ∂y ∂z

∂ ∂p = ∂x ∂y

(2.18) (2.19)

where η is the viscosity of polymer melt, and the energy equation may be written as ∂T = km ρm cpm vz ∂z

∂ 2T ∂ 2T + 2 ∂x ∂y 2

+η

∂vx ∂y

2

+

∂vz ∂x

2

+

∂vz ∂y

2 (2.20)

where ρm is the density, cpm is the speciﬁc heat, and km is the thermal conductivity of polymer melt. For the general situations under consideration, where the molten polymer is assumed to follow a power-law model, the viscosity η appearing in Eqs. (2.18)–(2.20) is expressed by η = mo exp(−bT )

∂vx ∂y

2

+

∂vz ∂x

2

+

∂vz ∂y

2 (n−1)/2 (2.21)

where mo is the preexponential factor, b is a constant, and n is the power-law index. The solution of Eqs. (2.18)–(2.20) must satisfy the following boundary conditions: vz = 0,

vx = 0,

T = Tc

at y = 0

(2.22a)

vz = Vbz ,

vx = −Vbx ,

T = Tb

at y = H

(2.22b)

vz = 0,

T = Tc

at x = 0

(2.22c)

vz = Vsz ,

T = Tm

at x = WB

(2.22d)

where Tc is the screw temperature, Tb is the barrel inner wall temperature, and WB is the melt-pool width. (c) Zones C, D, and E (Melt Films) Zone C represents a thin melt ﬁlm, which is the primary source for the supply of molten polymer to the melt pool, zone D is a thin melt ﬁlm between the solid bed and the screw root, and zone E is a thin melt ﬁlm between the solid bed and the screw ﬂight. Under the assumptions that ﬂows are fully developed in the down- and cross-channel directions and that the melt ﬁlms circulate around the solid bed, the x- and z-components of the force balance may be written as

∂ ∂p = ∂x ∂y ∂ ∂p = ∂z ∂y

η η

∂vx ∂y ∂vz ∂y

(2.23) (2.24)

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PLASTICATING SINGLE-SCREW EXTRUSION

and the energy equation is given by ∂T ∂ 2T ρm cpm vz = km 2 + η ∂z ∂y

∂vx ∂y

2 +

∂vz ∂y

2 (2.25)

Note that the viscosity of the melt η appearing in Eqs. (2.23)–(2.25) is given by η = mo exp(−bT )

∂vz ∂y

2 +

∂vx ∂y

2 (n−1)/2 (2.26)

The boundary conditions for zone C are vz = Vsz ,

vx = 0,

T = Tm

at y = 0

(2.27a)

vz = Vbz ,

vx = −Vbx ,

T = Tb

at y = HC

(2.27b)

where HC is the melt ﬁlm thickness in zone C, and the boundary conditions for zones D and E are vz = 0,

vx = 0,

T = Tc

at y = 0

(2.28a)

vz = Vsz ,

vx = 0,

T = Tm

at y = HD

(2.28b)

where HD is the melt ﬁlm thickness in zone D. 2.2.3

Analysis of Melt-Conveying Section

Equations (2.23)–(2.25) also apply to the description of melt ﬂow in the metering section of single-screw extruders, under the following boundary conditions: vz = 0,

vx = 0,

T = Tc

at y = 0

(2.29a)

vz = Vbz ,

vx = −Vbx , T = Tb

at y = H

(2.29b)

Note that in solving Eqs. (2.23)–(2.25) for melt ﬂow in the metering section, the following relationships must be satisﬁed:

H 0

vx dy = 0;

W

H 0

vz dy = Q

(2.30)

where H is the channel height, W is the channel width, and Q is the volumetric ﬂow rate.

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PROCESSING OF THERMOPLASTIC POLYMERS

2.2.4

Comparison of Prediction with Experiment

Let us consider the screw conﬁguration summarized in Table 2.1 and the two extrusion conditions designated as examples 1 and 2. Comparison of predictions of axial pressure proﬁle with experimental results is given in Figure 2.6 for example 1. The physical and thermal properties of a low-density polyethylene (LDPE) used for the predictions given in Figure 2.6 are summarized in Table 2.2. In Figure 2.6 curve (1) is computed with the assumption of a rigid solid bed (see Eq. (2.17)) and curves (2)–(4) are obtained with different values of fm , deﬁned by Eq. (2.12). Note that the smaller the value of fm , the greater the extent of solid-bed deformation in the bulk state, where fm = 1 means that the solid bed has the same modulus as the LDPE itself. It can be seen in Figure 2.6 that when compared with experimental results, the predicted axial pressure proﬁles are not sensitive to the value of fm chosen, leading us to conclude that fm = 0.1 gives the best ﬁt to experimental results. However, as the throughput is increased by about threefold and the screw speed is increased to 100 rpm (example 2 in Table 2.1), the predicted axial pressure proﬁles are seen to become very sensitive to the values of fm chosen, as can be seen in Figure 2.7. It appears that fm = 0.01 gives the best ﬁt of predicted proﬁles to experimental results. Notice that the screw used in Figures 2.6 and 2.7 has very long feed and tapered sections. Such a screw was popular in the 1960s. However, most modern commercial plasticating extruders use relatively short feed and transition (tapered) sections. Table 2.1 Geometrical conﬁguration of the single-screw extruder and operating conditions

(a) Extruder Conﬁguration Extruder diameter (mm) = 63.5 Extruder length (L/D) = 26 Length of the feed section (L/D) = 12.4 Length of the tapered section (L/D) = 9.5 Length of the metering section (L/D) = 4.5 Channel depth of the feed section (mm) = 9.39 Channel depth of the metering section (mm) = 3.23 Flight pitch angle (◦ ) = 17.7 (b) Operating Conditions Used for Extruding LDPE Example 1: Experiment no. 2 described in Table 5.1 of Tadmor and Klein (1970) Screw speed (rpm) = 40 Throughput (kg/h) = 32.6 Barrel temperature (◦ C) = 232 Pressure proﬁle data in Figure 8.6 of Tadmor and Klein (1970) Solid bed width proﬁle data in Figure 5.40 of Tadmor and Klein (1970) Example 2: Experiment no. 13 described in Table 5.1 of Tadmor and Klein (1970) Screw speed (rpm) = 100 Throughput (kg/h) = 99.8 Barrel temperature (◦ C) = 204 Pressure proﬁle data given in Figure 8.4 of Tadmor and Klein (1970) Solid bed width proﬁle data given in Figure 5.38 of Tadmor and Klein (1970) Reprinted from Lee and Han, Polymer Engineering and Science 30:665. Copyright © 1990, with permission from the Society of Plastics Engineers.

Figure 2.6 Comparison of theoretically predicted axial pressure proﬁles (solid line) with experi-

mental results (symbol 䊉) in the extrusion of an LDPE (example 1). Table 2.1 gives the extruder conﬁguration and operating conditions employed for experiment and prediction. Curve (1) is based on the assumption of a rigid solid bed and the other curves are based on different values of fm : curve (2) with fm = 1.0, curve (3) with fm = 0.1, and curve (4) with fm = 0.01. (Reprinted from Lee and Han, Polymer Engineering and Science 30:665. Copyright © 1990, with permission from the Society of Plastics Engineers.)

Table 2.2 Physical and thermal properties of LDPE

(a) Properties in the Molten State Power-law index for viscosity, η Power-law consistency, mo Temperature coefﬁcient, b Thermal conductivity, km Density, ρm Speciﬁc heat, cpm

0.345 5.6 × 104 Pa·s0.345 0.01 1/K 0.182 W/(m K) 810 kg/m3 2604 J/(kg K)

(b) Properties in the Solid State 0.335 W/(m K) 915 kg/m3 2772 J/(kg K) 110 ◦ C 1.3 × 105 J/kg

Thermal conductivity, ks Density, ρs Speciﬁc heat, cps Melting point, Tm Heat of fusion, λ

Reprinted from Lee and Han, Polymer Engineering and Science 30:665. Copyright © 1990, with permission from the Society of Plastics Engineers.

69

70

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 2.7 Comparison of theoretically predicted axial pressure proﬁles (solid line) with experimental results (symbol 䊉) in the extrusion of an LDPE (example 2). Table 2.1 gives the extruder conﬁguration and operating conditions employed for experiment and prediction. Curve (1) is based on the assumption of a rigid solid bed and the other curves are based on different values of fm : curve (2) with fm = 1.0, curve (3) with fm = 0.1, and curve (4) with fm = 0.01. (Reprinted from Lee and Han, Polymer Engineering and Science 30:665. Copyright © 1990, with permission from the Society of Plastics Engineers.)

One such screw conﬁguration is summarized in Table 2.3. Note that the diameter of this extruder is the same as that given in Table 2.1. Also given in Table 2.3 are two extrusion conditions, designated as examples 3 and 4. Figure 2.8 gives a comparison of predictions of axial pressure proﬁles with experimental results for example 3. We observe from Figure 2.8 that fm = 0.01 gives the best ﬁt of predicted pressure proﬁles to experimental results. Note that the throughput employed in example 3 is almost Table 2.3 Geometrical conﬁguration of the single-screw extruder and operating conditions

(a) Extruder Conﬁgurations Extruder diameter (mm) = 63.5 Extruder length (L/D) = 24.35 Length of the feed section (L/D) = 3.0 Length of the transition section (L/D) = 4.0 Length of the metering section (L/D) = 17.35 Channel width (mm) = 54.14 Channel depth of the feed section (mm) = 8.31 Channel depth of the metering section (mm) = 2.72 Flight width (mm) = 6.35 Flight clearance (mm) = 0.08 Flight helix angle (◦ ) = 17.7 (continued)

Table 2.3—(Cont’d)

(b) Operating Conditions Used for Extruding LDPE Example 3: Run no. 6 in Table 4 of Han et al. (1990) Screw speed (rpm) = 50 Throughput (kg/h) = 32.3 Barrel temperature (◦ C) = 162 Head pressure (MPa) = 6.92 Pressure proﬁle data in Figure 2.8 Example 4: Run no. 8 in Table 4 of Han et al. (1990) Screw speed (rpm) = 50 Throughput (kg/h) = 29.6 Barrel temperature (◦ C) = 162 Head pressure (MPa) = 20.67 Pressure proﬁle data in Figure 2.9 Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.

Figure 2.8 Comparison of theoretically predicted axial pressure proﬁles (solid line) with experi-

mental results (symbol 䊉) in the extrusion of an LDPE (example 3). Table 2.3 gives the extruder conﬁguration and operating conditions employed for experiment and prediction. Curve (1) is based on the assumption of a rigid solid bed and the other curves are based on different values of fm : curve (2) with fm = 1.0, curve (3) with fm = 0.1, curve (4) with fm = 0.01, and curve (5) with fm = 0.001. (Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.)

71

72

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 2.9 Comparison of theoretically predicted axial pressure proﬁles (solid line) with experimental results (symbol ●) in the extrusion of an LDPE (example 4). Table 2.3 gives the extruder conﬁguration and operating conditions employed for experiment and prediction. Curve (1) is based on the assumption of a rigid solid bed and the other curves are based on different values of fm : curve (2) with fm = 1.0, curve (3) with fm = 0.1, curve (4) with fm = 0.01, and curve (5) with fm = 0.001. (Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.)

the same as that in example 1, but the screw speed in example 3 is 50 rpm whereas the screw speed in example 1 is 40 rpm. However, the large differences in the lengths of the feed and tapered sections between the two screws mean that the axial pressure proﬁles in the two extruders are seen to be quite different. As the head pressure of the extruder (i.e., pressure at the extruder exit) is increased from 6.92 MPa (1,000 psi) to 20.67 MPa (3,000 psi), as can be seen in Figure 2.9 for example 4, the axial pressure proﬁle is seen to increase steadily, which is quite different from what is observed in Figure 2.8. Note that head pressure is dictated by the design of a die attached to the extruder. The steady increase in axial pressure proﬁle observed in Figure 2.9 is due to the back pressure caused by the die attached to the extruder. It should be noted that as the head pressure increases, the ﬂow in the screw channel is dominated by drag ﬂow, consequently deformation of the solid bed will become greater. This is clearly shown to be the case by Figure 2.9, in which fm = 0.001, which is much smaller than the value observed in Figure 2.8, appears to give the best ﬁt of predicted pressure proﬁles to experimental results. Figures 2.10 and 2.11 give comparison of predictions of axial pressure proﬁles with experimental results for an LDPE in a screw whose conﬁgurations are described in Table 2.3. Note that the predicted pressure proﬁles given in Figures 2.10 and 2.11 are based on the value of fm = 0.01. In view of the fact that several assumptions were

PLASTICATING SINGLE-SCREW EXTRUSION

73

Figure 2.10 Comparison of axial pressure proﬁles between experimental results (symbols) and

theoretical predictions (solid line) in the extrusion of an LDPE at a screw speed of 50 rpm for different head pressures (P ) and throughputs (Q): curve (1) and () for P = 3.35 MPa and Q = 33.2 kg/h, curve (2) and () for P = 6.92 MPa and Q = 32.3 kg/h, curve (3) and () for P = 13.78 MPa and Q = 30.9 kg/h, and curve (4) and () for P = 20.67 MPa and Q = 29.6 kg/h. All the theoretical predictions are based on fm = 0.01 and Table 2.3 gives the extruder conﬁgurations employed for experiment and prediction. (Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.)

made in the melting model, the agreement between experimental results and theoretical predictions, displayed in Figures 2.10 and 2.11, is judged to be reasonable and very encouraging. Predicted proﬁles of reduced solid-bed velocity, Vsz /Vbz , along the extruder axis are given in Figure 2.12 for example 1, in Figure 2.13 for example 2, in Figure 2.14 for example 3, and in Figure 2.15 for example 4, in which Vsz is the solid-bed velocity along the z-direction and Vbz is the barrel velocity in the down-channel direction. It can be seen in these ﬁgures that the smaller the value of fm , the greater the solid-bed velocity, and that the solid-bed velocities in examples 3 and 4 for screws with short feed and short tapered sections, are greater than those in examples 1 and 2, for screws with long feed and long tapered sections. Several research groups (Bruker and Balch 1989; Cox and Fenner 1980; Donovan 1971; Edmondson and Fenner 1975; Fukase et al. 1982; Zhu and Chen 1991) reported measurements of the solid-bed velocity in single-screw extrusion. Figure 2.16 gives the experimental results obtained by Zhu and Chen (1991), who employed a tracer particle technique to measure the solid-bed velocities of LDPE, high-density polyethylene (HDPE), and polypropylene (PP) in a single-screw extruder.2 It can be seen in

74

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 2.11 Comparison of axial pressure proﬁles between experimental results (symbols) and

theoretical predictions (solid line) in the extrusion of an LDPE at a screw speed of 100 rpm for different head pressures (P ) and throughputs (Q): curve (1) and () for P = 4.27 MPa and Q = 66.8 kg/h, curve (2) and () for P = 6.82 MPa and Q = 66.2 kg/h, curve (3) and () for P = 13.78 MPa and Q = 63.4 kg/h, and curve (4) and () for P = 20.67 MPa and Q = 60.9 kg/h. All the theoretical predictions are based on fm = 0.01 and Table 2.3 gives the extruder conﬁgurations employed for experiment and prediction. (Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.)

Figure 2.16 that the solid bed accelerates slowly in the early part of the melting zone and then accelerates rapidly, supporting the theoretical predictions given in Figures 2.14 and 2.15. Note that the location inside the extruder at which the solid bed begins to accelerate depends on the screw conﬁguration. To illustrate the point, Figure 2.17 gives a schematic of two types of unwound screw conﬁgurations: (a) a screw with long feed and tapered sections and a very short metering section, and (b) a screw with short feed and tapered sections and a very long metering section. For the screw depicted in Figure 2.17a, the solid-bed acceleration would be rather small, as predicted in Figure 2.12, whereas for the screw depicted in Figure 2.17b, the solid-bed acceleration is expected to be rather large, as predicted in Figures 2.14 and 2.15. Most of the screws currently used in the plastics industry have the conﬁguration depicted in Figure 2.17b, and thus considerable solid-bed acceleration is expected to occur in the melting section of these screws. Figure 2.18 gives a photograph of the LDPE sample obtained from a “screw pushout” experiment for example 3. Such an experimental technique appears to have ﬁrst been suggested by Maddock (1959). Note that the dark areas in the photograph

Figure 2.12 Theoretically predicted reduced solid-bed velocity, Vsz /Vbz , versus distance L/D along the extruder axis in the extrusion of an LDPE (example 1): curve (1) for a rigid solid bed, curve (2) for fm = 1.0, curve (3) for fm = 0.1, and curve (4) for fm = 0.01. Table 2.1 gives the extruder conﬁguration and operating conditions employed for prediction. (Reprinted from Lee and Han, Polymer Engineering and Science 30:665. Copyright © 1990, with permission from the Society of Plastics Engineers.)

Figure 2.13 Theoretically predicted reduced solid-bed velocity, Vsz /Vbz , versus distance L/D along the extruder axis in the extrusion of an LDPE (example 2): curve (1) for a rigid solid bed, curve (2) for fm = 1.0, curve (3) for fm = 0.1, and curve (4) for fm = 0.01. Table 2.1 gives the extruder conﬁguration and operating conditions employed for prediction. (Reprinted from Lee and Han, Polymer Engineering and Science 30:665. Copyright © 1990, with permission from the Society of Plastics Engineers.)

75

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PROCESSING OF THERMOPLASTIC POLYMERS

Figure 2.14 Theoretically predicted reduced solid-bed velocity, Vsz /Vbz , versus distance L/D along the extruder axis in the extrusion of an LDPE (example 3): curve (1) for a rigid solid bed, curve (2) for fm = 1.0, curve (3) for fm = 0.1, curve (4) for fm = 0.01, and curve (5) for fm = 0.001. Table 2.3 gives the extruder conﬁguration and operating conditions employed for prediction. (Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.)

given in Figure 2.18 represent the pigment added as tracer. In the screw pushout experiment, samples were cut at various positions along the extruder axis and the distributions of the pigment in the cross section of the screw channel were recorded (Han et al. 1990). The results of the solid-bed proﬁles for the LDPE are given in Figure 2.19. It can be seen in Figure 2.19 that solid-bed breakup occurred at an L/D of about 6. Figure 2.20 gives the solid bed width proﬁles along the extruder axis for example 1, together with predictions for different values of fm . Note in Figure 2.20 that a breakup of the solid bed occurred at an L/D of about 17. Similar plots are given in Figure 2.21 for example 2, in which a breakup of the solid bed occurred at an L/D of about 10.5. Notice that the axial position at which a breakup of the solid bed occurred in example 2 is much shorter than that in example 1. This can be understood in view of the fact that the solid-bed acceleration in the tapered section of the extruder is much greater in example 2 than in example 1 (see Figures 2.12 and 2.13). Figure 2.22 gives a comparison of predictions of the solid bed width proﬁle along the extruder axis for example 3. Note in Figure 2.22 that the solid-bed breakup actually occurred at an L/D of about 5.5, which corresponds with the tapered section of the screw channel (see Table 2.3 for the geometrical conﬁgurations of the screw employed). In the case of a constant solid-bed velocity, the computation of solid bed width proﬁles had to be stopped at an L/D of about 5 (Lee and Han 1990), because, as shown in Figure 2.8,

Figure 2.15 Theoretically predicted reduced solid-bed velocity, Vsz /Vbz , versus distance L/D along the extruder axis in the extrusion of an LDPE (example 4): curve (1) for a rigid solid bed, curve (2) for fm = 1.0, curve (3) for fm = 0.1, curve (4) for fm = 0.01, and curve (5) for fm = 0.001. Table 2.3 gives the extruder conﬁguration and operating conditions employed for prediction. (Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.)

Figure 2.16 Experimentally measured reduced solid-bed velocity, Vsz /Vbz , versus distance L/D along the extruder axis in the extrusion of () LDPE, () HDPE, and () PP. (Reprinted from Zhu and Chen, Polymer Engineering and Science 31:1113. Copyright © 1991, with permission from the Society of Plastics Engineers.)

77

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PROCESSING OF THERMOPLASTIC POLYMERS

Figure 2.17 Comparison of unwound screw channel between (a) a screw with long feed and

tapered sections and (b) a screw with short feed and tapered sections.

the axial pressure decreases rapidly, which is physically not realistic. Note that the predictions of solid-bed width given in Figures 2.20–2.22 assumed that there was no solid-bed breakup during extrusion. At present there is no criterion for determining the onset of solid-bed breakup in the development of a mathematical model for simulating the plasticating extrusion process. This is one of the most challenging problems yet to be resolved in the modeling efforts. When it is assumed that the melt ﬁlms in zones D and E do not circulate around the solid bed (see Figure 2.4), the x-component velocity vx in Eqs. (2.23) and (2.25) can be neglected, and one must then solve the z-component force balance: ∂ ∂p = ∂z ∂y

∂v η z ∂y

(2.31)

Figure 2.18 Photograph of an LDPE sample obtained from a screw pushout experiment at extru-

sion conditions identical with example 3 given in Table 2.3. (Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.)

PLASTICATING SINGLE-SCREW EXTRUSION

79

Figure 2.19 Solid bed width proﬁles of an LDPE in the screw channel obtained from the screw

pushout experiment at extrusion conditions identical to example 3 given in Table 2.3. The numbers on each of the schematics represent the position along the extruder axis (L/D): (1) 2.0, (2) 2.5, (3) 3.0, (4) 3.5, (5) 4.0, (6) 4.5, (7) 5.0, (8) 5.5, (9) 6.0, (10) 6.5, (11) 7.0, (12) 7.5, (13) 8.0, (14) 8.5, (15) 9.0, and (16) 9.5. (Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.)

and the energy equation ∂vz 2 ∂T ∂ 2T = km 2 + η ρm cpm vz ∂z ∂y ∂y

(2.32)

where η = mo exp(−bT )

∂vz ∂y

n−1 (2.33)

Figure 2.20 Comparison of theoretically predicted solid bed width proﬁles (solid lines) with

experimental results (symbols) in the extrusion of an LDPE (example 1). Table 2.1 gives the extruder conﬁguration and operating conditions employed for experiment and prediction. Curve (1) is based on the assumption of a rigid solid bed and the other curves are based on different values of fm : curve (2) with fm = 1.0, curve (3) with fm = 0.1, and curve (4) with fm = 0.01. (Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.)

Figure 2.21 Comparison of theoretically predicted solid bed width proﬁles (solid lines) with

experimental results (symbols) in the extrusion of an LDPE (example 2). Table 2.1 gives the extruder conﬁguration and operating conditions employed for experiment and prediction. Curve (1) is based on the assumption of a rigid solid bed and the other curves are based on different values of fm : curve (2) with fm = 1.0, curve (3) with fm = 0.1, and curve (4) with fm = 0.01. (Reprinted from Lee and Han, Polymer Engineering and Science 30:665. Copyright © 1990, with permission from the Society of Plastics Engineers.) 80

PLASTICATING SINGLE-SCREW EXTRUSION

81

Figure 2.22 Comparison of theoretically predicted solid bed width proﬁles (solid line) with

experimental results (symbol 䊉) in the extrusion of an LDPE (example 3). Table 2.3 gives the extruder conﬁguration and operating conditions employed for experiment and prediction. Curve (1) is based on the assumption of a rigid solid bed and the other curves are based on different values of fm : curve (2) with fm = 1.0, curve (3) with fm = 0.1, curve (4) with fm = 0.01, and curve (5) with fm = 0.001. (Reprinted from Han et al., Polymer Engineering and Science 30:1557. Copyright © 1990, with permission from the Society of Plastics Engineers.)

subject to the following boundary conditions: vz = 0;

T = Tc

at y = 0

(2.34a)

vz = Vsz ;

T = Tm

at y = HD

(2.34b)

Figure 2.23 gives predicted melt-ﬁlm thickness in zone D along the extruder axis, with and without cross-channel velocity. It can be seen in Figure 2.23 that when the cross-channel velocity is included, the predicted melt-ﬁlm thickness in zone D initially increases and then levels off at a certain axial position along the extruder, but when the cross-channel velocity is neglected, the predicted melt-ﬁlm thickness increases without bound at a certain axial position along the extruder, which is physically unacceptable. When the melt-ﬁlm thickness in zone D grows without bound, the calculation of solid bed width proﬁles (also axial pressure proﬁles) is not possible because the continuous increase of melt-ﬁlm thickness in zone D makes the solid-bed height unacceptably small, which then causes the solid-bed width to become larger than the channel width in order to satisfy the material balance. Thus, we can conclude that the inclusion of

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Figure 2.23 Theoretically predicted lower ﬁlm thickness along the extruder axis in the extrusion of an LDPE (example 3): curve (1) with cross-channel velocity in the momentum and energy equations, and curve (2) without cross-channel velocity in the momentum and energy equations. Predictions are based on fm = 0.01, and Table 2.3 gives the extruder conﬁguration and operating conditions employed for prediction.

a cross-channel velocity for melt ﬁlms in zones D and E is absolutely essential for successful predictions of axial pressure and solid bed width proﬁles in the melting zone.3 Indeed Bruker and Balch (1989) presented experimental evidence supporting the existence of circulating ﬂow patterns in melt ﬁlms surrounding the solid bed. Predicted contours of constant velocity (isovels) for example 3 considered above in the cross-section of melt pool at L/D = 4 are given in Figure 2.24, in which the numerical values represent the axial velocity in the melt pool in meters per second (m/s). It can be seen in Figure 2.24 that the velocities of the melt are very low at positions far away from the barrel wall (i.e., near the root of the screw surface and near the screw ﬂight). Figure 2.25 gives the axial velocity proﬁles vz (x, y) in the melt pool of LDPE (1) at the root of the screw, (2) at the screw ﬂight, and (3) at the solid bed/melt pool interface, at the position of L/D = 10 along the screw axis, where one half of the screw channel is ﬁlled with the melt (i.e., X/W = 0.5) and the aspect ratio of the melt pool (Wm /Hm ) is 2.77. Predicted contours of constant temperature (isotherms) in the cross section of the melt pool at L/D = 4 are given in Figure 2.26 for example 3, in which numerical values represent the temperature in the melt pool in Celsius. It is of interest to note in Figure 2.26 that the temperatures in the upper-left corner of the melt pool are about 8 ◦ C higher than the temperature (160 ◦ C) of the barrel inner surface. This is attributed

Figure 2.24 Theoretically predicted contours of constant melt-pool velocity (isovels) at L/D = 4 in the extrusion of an LDPE (example 3), in which the numbers inside the ﬁgure represent velocity in m/s. Predictions are based on fm = 0.01 with convective heat transfer, and Table 2.3 gives the extruder conﬁguration and operating conditions employed for prediction.

Figure 2.25 Three-dimensional plots of axial velocity proﬁles vz (x, y) in the melt pool at L/D = 10, X/W = 0.5, and Wm /Hm = 2.77 in the extrusion of an LDPE (example 3).

83

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PROCESSING OF THERMOPLASTIC POLYMERS

Figure 2.26 Theoretically predicted contours of constant melt-pool temperature (isotherms) at L/D = 4 in the extrusion of an LDPE (example 3), in which the numbers inside the ﬁgure represent temperature in Celsius. Predictions are based on fm = 0.01 by including convective heat transfer, and Table 2.3 gives the extruder conﬁguration and operating conditions employed for prediction.

to the heat generated by viscous shear heating. Of course, the amount of heat generated by viscous shear heating depends on the viscosity of the polymer and the operating conditions (i.e., screw speed and throughput). Figure 2.27 gives contours of constant temperature (isotherms) in the cross section of melt pool at L/D = 4, which are obtained by neglecting the convective heat transfer

Figure 2.27 Theoretically predicted contours of constant melt-pool temperature (isotherms) at L/D = 4 in the extrusion of an LDPE (example 3), in which the numbers inside the ﬁgure represent temperature in Celsius. Predictions are based on fm = 0.01 by neglecting the convective heat transfer term in the energy equation, Eq. (2.20), and Table 2.3 gives the extruder conﬁguration and operating conditions employed for prediction.

PLASTICATING SINGLE-SCREW EXTRUSION

85

term in the energy equation, Eq. (2.20). Comparison of Figure 2.27 with Figure 2.26 leads us to make the following interesting observations: (1) in the regions away from the melt pool/solid bed interface, the predicted temperature distribution in the melt pool is more or less symmetric when convective heat transfer is neglected, whereas it is highly asymmetric when it is included, and (2) the predicted maximum temperature in the melt pool is much higher when convective heat transfer is neglected, compared with that when it is included. It should be pointed out that neglecting the convective heat transfer terms in the energy equation is tantamount to the assumption that the temperature ﬁeld is fully developed in the z-direction. This assumption requires that the heat generated by viscous shear heating inside the melt pool be conducted away through the barrel wall. However, due to the very poor thermal conductivity of molten polymers, the heat generated by viscous shear heating inside the melt pool, instead of being conducted away through the barrel wall, is actually carried by the melt in the z-direction (i.e., along the extruder axis), giving rise to a developing temperature ﬁeld. More importantly, the temperature ﬁeld in the melting section cannot possibly be fully developed because melting continues along the extruder axis.

2.3

Performance of Fluted Mixing Heads in a Plasticating Single-Screw Extruder

It has long been recognized that the metering screws for plasticating extruders are not effective for mixing. In an attempt to improve the mixing capabilities of single-screw extruders, some investigators (LeRoy 1969; Gregory and Street 1968) suggested that a barrier-type mixing section, today generally referred to as a “ﬂuted mixing device,” be placed near the end of a metering screw. Maddock (1967) offered an explanation as to how such a ﬂuted mixing device might help to improve the mixing capability of metering screws. Figure 2.28 shows a photograph of one such ﬂuted mixing device, known as the “Maddock mixing head.” Figure 2.29 gives schematics of the side and cross-sectional

Figure 2.28 A Maddock mixing head. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

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Figure 2.29 Schematic of the Maddock mixing head: (a) side view and (b) cross-sectional view.

(Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

views of the Maddock mixing head, showing the rationale behind its design. Brieﬂy stated, in reference to Figure 2.29, the Maddock mixing head consists of four pairs of deeply grooved inlet and outlet channels, a barrier ﬂight (referred to as the “mixing land”), and a wiping ﬂight (referred to as the “wiping land”). The rationale behind its design lies in that when a stream of molten polymer from the metering section of a single screw enters the inlet channel of the mixing head, it ﬂows over the barrier ﬂight into the outlet channel by the rotation of the screw. Since the space between the barrier ﬂight and the inner wall of the barrel is very small compared with the depth of the inlet channel, any pellets that are still unmelted when they reach the mixing head can be prevented from crossing over into the outlet channel and will therefore have the chance to be melted completely while ﬂowing through the inlet channel of the mixing head. 2.3.1

Analysis of the Flow through the Maddock Mixing Head

In order to facilitate the formulation of the system of equations that describes the ﬂow through a Maddock mixing head, a schematic of the plane and side views of the mixing head is given in Figure 2.30, where the directions of melt ﬂow from the entrance to the exit of the mixing head are indicated by arrows. It can be seen in Figure 2.30 that the ﬂow from the screw channel of the metering section is divided primarily into two directions: (1) in the z-direction by pressure-driven ﬂow in both the inlet and outlet channels, and (2) in the x-direction by drag ﬂow through the clearance between the barrier ﬂight and the barrel. Figure 2.31 gives a schematic of the cross-sectional view of the Maddock mixing head, in which the clearance δ2 of the wiping ﬂight is small compared with the clearance δ1 of the barrier ﬂight. Using Cartesian coordinates and making the usual assumptions for isothermal ﬂow of very viscous molten polymers at steady state, we have the continuity equation ∂v ∂vx + z =0 ∂x ∂z

(2.35)

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87

Figure 2.30 Schematic of the Maddock mixing head: (a) plane view, (b) depth proﬁle along the

centerline of the inlet channel, and (c) depth proﬁle along the centerline of the outlet channel. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

and x- and z-components of the force balance −

∂p ∂σxy + =0 ∂x ∂y

(2.36)

−

∂p ∂σzy + =0 ∂z ∂y

(2.37)

For Newtonian ﬂuids, the solution of Eqs. (2.36) and (2.37), subject to the boundary conditions vx = 0;

vz = 0

at y = 0

(2.38a)

vx = Vb ;

vz = 0

at y = H

(2.38b)

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Figure 2.31 Schematic depicting the cross-sectional view of the Maddock mixing head.

(Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

becomes y 2 H 2 ∂p y y − vx = Vb − H 2η ∂x H H y 2 H 2 ∂p y − vz = − 2η ∂z H H

(2.39) (2.40)

Then, the ﬂow rates in the x- and z-directions, qx and qz , respectively, can be calculated by qx =

1 H 3 ∂p Vb H − 2 12η ∂x

qz = −

H 3 ∂p 12η ∂z

(2.41) (2.42)

Using ﬁnite analysis network (FAN) method, Eqs. (2.41) and (2.42) can be rewritten in the discretized form (Han et al. 1991a) H (i + 1, j ) + H (i, j ) H (i + 1, j ) + H (i, j ) 3 1 1 Vb − 2 2 12η 2 p(i + 1, j ) − p(i, j ) (2.43) × x H (i, j + 1) + H (i, j ) 3 p(i, j + 1) − p(i, j ) 1 qz (i, j ) = − (2.44) 12η 2 z

qx (i, j ) =

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PLASTICATING SINGLE-SCREW EXTRUSION

where x and z are the mesh sizes in the x- and z-directions, respectively. Notice that the use of Eqs. (2.43) and (2.44) enable one to calculate the pressure p(i, j ) by satisfying the mass balance

qx (i − 1, j ) − qx (i, j ) z + qz (i, j − 1) − qz (i, j ) x = 0

(2.45)

That is, by combining Eqs. (2.43)–(2.45) we obtain ⎡

z ⎣ −

Vb 2

Vb 2

H (i + 1, j ) + H (i, j ) 2

H (i, j ) + H (i − 1, j ) 2

−

1 12η

1 − 12η

H (i + 1, j ) + H (i, j ) 3 p(i + 1, j ) − p(i, j ) 2 x ⎤ H (i, j ) + H (i − 1, j ) 3 p(i, j ) − p(i − 1, j ) ⎦ 2 x

H (i, j + 1) + H (i, j ) 3 p(i, j + 1) − p(i, j ) 2 z H (i, j ) + H (i, j − 1) 3 p(i, j ) − p(i, j − 1) 1 =0 − 12η 2 z − x

1 12η

(2.46)

from which pressure distributions p(i, j ) in the mixing head can be calculated with known values of viscosity η and channel depth H (i, j ). When pressure distributions are known, the ﬂow rates qx and qz can be calculated using Eqs. (2.41) and (2.42). In so doing, we require that the sum of the mass ﬂow rates in the z-direction be equal to the total mass ﬂow rate G: G = ρm

W 0

qz (x, z) dx = ρm

M #

(2.47)

qz (i, j )

i=1

where ρm is the density of the ﬂuid, W is the distance from the beginning of the wiping ﬂight to the end of the outlet channel in the cross-channel direction (see Figure 2.30b), and M is the number of discretized meshes in the x-direction. The following mass balance must be satisﬁed along the path CDEFG designated in Figure 2.30a: G/ρm =

Lc

+

[qx ]BF − [qx ]WF dz + W [qz ]L2 dx + W [qz ]L1 dx

Wf

i

[qz ]L2 dx

o

(2.48)

where [qx ]BF and [qx ]WF are the ﬂow rates in the x-direction at the barrier ﬂight and wiping ﬂight, respectively, and Lc , L1 , L2 , Wi , Wo , and Wf are deﬁned in Figure 2.30a. In order to compute the pressure distributions p(i, j ) using Eq. (2.46), one must ﬁrst calculate the channel depth H (i, j ) and viscosity η of the ﬂuid. For power-law ﬂuids, instead of a constant value of viscosity, a shear-dependent viscosity η can be calculated from η = ko exp(−bT )(Vb /H )n−1

(2.49)

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PROCESSING OF THERMOPLASTIC POLYMERS

where Vb is the barrel velocity in the x-direction. Note that Vb can be calculated using Vb = πDN, where N is the screw speed. Strictly speaking, the use of Eq. (2.49) is not valid because the dependence of viscosity on the z-directional shear rate is neglected. However, the dependence of viscosity on the z-directional shear rate (i.e., the effect of pressure-driven ﬂow) is negligibly small compared with the dependence of viscosity on the x-directional shear rate; that is, the effect of drag ﬂow (Han et al. 1991a). 2.3.2

Comparison of Prediction with Experiment

Using specially designed screws, as schematically shown in Figure 2.32, Han et al. (1991a) measured pressure variations across the Maddock mixing head in the downchannel direction with the aid of two pressure transducers, one placed near the entrance and the other near the exit of the mixing head (Screw #1 in Figure 2.32). They also measured pressure variations across the barrier and wiping ﬂights in the cross-channel direction with the aid of a pressure transducer placed in the middle of the mixing head (Screw #2 in Figure 2.32). The geometrical conﬁgurations of the Maddock mixing heads employed in their study are given in Figure 2.33, and the dimensions of the two Maddock mixing heads are given in Table 2.4. Figure 2.34 gives the pressure proﬁles for an LDPE along the extruder axis at a screw speed of 75 rpm and at three different throughput rates, which were obtained using screw #1 depicted in Figure 2.32. Table 2.5 gives a summary of the extrusion conditions employed and pressure difference P across the mixing head #1 for other experimental runs, for which the set point of the barrel temperature along the entire extruder axis was 162 ◦ C for all experimental runs. It can be seen in Figure 2.34 and, also in Table 2.5, that the pressure may drop (P = P8 − P7 < 0) or rise (P =P8 − P7 > 0) across the Maddock mixing head, depending upon the extrusion conditions. A close examination of the values of P given in Table 2.5 reveals that at a given

Figure 2.32 Schematic

depicting the positions of pressure transducers mounted in the Maddock mixing head. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

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91

Figure 2.33 Geometrical conﬁgurations and dimensions of the Maddock mixing head.

(Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

screw speed, P changes from a negative value (i.e., pressure drop) to a positive value (i.e., pressure rise) as the throughput rate decreases. Figure 2.35 gives the recordings of pressure variation for an LDPE for three screw speeds (25, 50, and 75 rpm) at a head pressure of 20.67 MPa (3,000 psi), and Figure 2.36 gives the recordings of pressure variation for an LDPE for three head pressures (6.89, 13.78, and 20.67 MPa) for mixing head #2 at a screw speed of 50 rpm. Table 2.6 gives a summary of the extrusion conditions employed and pressure difference P across mixing head #2 for other experimental runs, for which the set point of the barrel temperature along the entire extruder axis was 162 ◦ C for all experimental runs. Referring to Figures 2.35 and 2.36, symbols A, B, C, and D represent pressure variations in the inlet channel, in the outlet channel, in the clearance of the barrier ﬂight, and in the

Table 2.4 Dimensions of the Maddock mixing heads

Mixing Head #1 Length of the mixing head (mm) Length of the channel (mm) Width of the channel (mm) Clearance of the barrier ﬂight (mm) Clearance of the wiper ﬂight (mm) Maximum radius of the channel (mm) Number of inlet channels Number of outlet channels

212.73 149.23 9.35 0.53 0.08 14.88 3 3

Mixing Head #2 212.73 149.23 9.35 0.66 0.08 14.88 3 3

Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.

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Figure 2.34 Experimental pressure proﬁles for an LDPE along the extruder axis with mixing head #1: () run #31, () run #32, and () run #33. Table 2.5 gives the extrusion conditions

employed. The solid lines are drawn through the data points for a visual aid. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

clearance of the wiping ﬂight, respectively, as shown schematically in Figure 2.31 (see Table 2.4 for the geometrical conﬁgurations of the Maddock mixing head used in the experiment). The following observations can be made on the measured pressure variations given in Figures 2.35 and 2.36: P across the mixing head becomes negative (i.e., pressure drops) when the pressure either decreases or increases slowly across the

Table 2.5 Summary of the extrusion conditions employed for mixing head #1 and comparison of predicted pressure drop across Maddock mixing head with experiment

Run No. 28 29 30 31 32 33 34 35 36

Screw Speed (rpm)

Throughput (kg/h)

Head Pressurea (MPa)

Measured P b (MPa)

Predicted P (MPa)

50 50 50 75 75 75 100 100 100

41.95 38.77 35.59 64.33 60.34 56.21 86.81 81.81 77.14

6.89 13.78 20.67 6.89 13.78 20.67 6.89 13.78 20.67

−0.27 0.28 1.03 −0.43 0.14 0.62 −0.89 −0.28 0.34

−0.12 −0.01 0.15 −0.18 −0.05 0.06 −0.21 −0.11 −0.02

a

The pressure measured at the extruder exit. b Pressure difference across the Maddock mixing head: the minus sign denotes pressure drop and the plus sign denotes pressure rise. Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.

Figure 2.35 Experimental pressure variations for an LDPE in the cross-channel direction of mixing head #2 at a head pressure of 20.67 MPa (3,000 psi): (a) run #45, (b) run #42, and (c) run #47. Table 2.6 gives the extrusion conditions employed. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

Table 2.6 Summary of the extrusion conditions employed for mixing head #2 and comparison of predicted pressure drop across Maddock mixing head

Run # 40 41 42 43 44 45 46 47 48 49

Screw Speed (rpm)

Throughput (kg/h)

Head Pressurea (MPa)

Measured P b (MPa)

Predicted P (MPa)

50 50 50 50 25 25 75 75 100 100

54.39 48.53 42.04 35.37 27.42 16.34 77.59 64.29 99.88 85.44

6.89 13.78 20.67 27.56 6.89 20.78 6.89 20.67 6.89 20.67

−0.55 0.14 0.82 1.72 −0.55 1.58 −0.41 0.62 −0.34 0.90

−0.19 −0.06 0.09 0.36 −0.13 0.33 −0.19 0.07 −0.13 0.08

a

The pressure measured at the extruder exit. b Pressure difference across the Maddock mixing head: the minus sign denotes pressure drop and the plus sign denotes pressure rise. Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.

93

94

PROCESSING OF THERMOPLASTIC POLYMERS Figure 2.36 Experimental

pressure variations for an LDPE in the cross-channel direction of mixing head #2 at a screw speed of 50 rpm: (a) run #40, (b) run #41, and (c) run #42. Table 2.6 gives the extrusion conditions employed. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

barrier ﬂight (region C in reference to Figure 2.31), and P across the mixing head becomes positive (i.e., pressure rises) when the pressure increases rapidly across the barrier ﬂight. Figure 2.37 gives predictions of pressure variations for an LDPE in the downchannel direction of the Maddock mixing head, and Figure 2.38 gives predictions of pressure variations in the cross-channel direction at a position in the middle of the outlet channel, at a head pressure of 20.67 MPa and two different screw speeds, 25 and 75 rpm, which are identical extrusion conditions used to generate the results given in Figures 2.35a and 2.35c. Note that for two sets of extrusion conditions, the numerical values of n = 0.5, ko = 4.754 × 105 Pa·s0.5 , b = 0.01093 and T = 443 K were used to calculate viscosities. Comparison of Figure 2.38 with Figure 2.35 indicates a good agreement between the predicted and experimentally measured pressure variations in the cross-channel direction. It can be seen in Figure 2.37 that a pressure increase (P > 0) across the mixing head is predicted, which is also in agreement with experimental results (see run #45 and run #47 in Table 2.6). Figure 2.39 gives predictions of pressure variations for an LDPE in the downchannel direction of the Maddock mixing head, and Figure 2.40 gives predictions of pressure variations in the cross-channel direction at a position in the middle of the outlet channel, at a screw speed of 50 rpm and two different head pressures, 6.89 and 20.67 MPa, which are identical extrusion conditions used to generate the results

PLASTICATING SINGLE-SCREW EXTRUSION

95

Figure 2.37 Predicted pressure variations for an LDPE in the down-channel direction of mixing

head #2 at a head pressure of 20.67 MPa: (a) run #45 and (b) run #47. Table 2.6 gives the extrusion conditions employed for prediction. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

given in Figures 2.36a and 2.36c. It can be seen in Figure 2.39 that the theory predicts a pressure decrease (P < 0) across the mixing head at a head pressure of 6.89 MPa and a pressure increase (P > 0) across the mixing head at a head pressure of 20.67 MPa, both at a screw speed of 50 rpm. Again, the theory predicts correctly the trend observed experimentally (see run #40 and run #42 in Table 2.6). We observe further that the shape of predicted pressure variations in the cross-channel direction is very similar to that observed experimentally (compare Figure 2.40 with Figure 2.36). Note in Figure 2.39b that at a head pressure of 20.67 MPa there is a rapid increase in pressure near the entrance of the outlet channel and, also, near the exit of the inlet channel. Notice in Figure 2.30 that the depth proﬁle in the down-channel direction is the narrowest near the entrance of the outlet channel and, also, near the exit of the inlet channel. In view of the fact that throughput is decreased from 54.39 to 42.04 kg/h as head pressure is increased from 6.89 to 20.67 MPa (see run #40 and run #42 in Table 2.6), we conclude that as the head pressure increases, drag ﬂow becomes predominant over

Figure 2.38 Predicted pressure

variations for an LDPE in the cross-channel direction of mixing head #2 at a head pressure of 20.67 MPa: (a) run #45 and (b) run #47. Table 2.6 gives the extrusion conditions employed for prediction. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

Figure 2.39 Predicted pressure variations for an LDPE in the down-channel direction of mixing

head #2 at a screw speed of 50 rpm: (a) run #40 and (b) run #42. Table 2.6 gives the extrusion conditions employed for prediction. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.) 96

PLASTICATING SINGLE-SCREW EXTRUSION

97

Figure 2.40 Predicted pressure variations for an LDPE in the cross-channel direction of mixing head #2 at a screw speed of 50 rpm: (a) run #40 and (b) run #42. Table 2.6 gives the extrusion conditions employed for prediction. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

pressure-driven ﬂow. This then leads us to conclude that the positive P observed in Figure 2.39b at a head pressure of 20.67 MPa is attributable to the predominant role of drag ﬂow over pressure-driven ﬂow. It should be noted that the analysis, which is based on the FAN method, enables one to predict pressure distributions along the entire length of the Maddock mixing head, including the shaded areas shown in Figure 2.30a. Notice in Figure 2.39b that the pressure increases rapidly near the entrance of the outlet channel and, also, near the exit of the inlet channel. This seems to suggest that there might be circulatory ﬂow and perhaps backﬂow (i.e., reverse ﬂow) in the area where pressure buildup occurs rapidly in the down-channel direction. As can be seen in Figure 2.41, indeed we observe circulatory ﬂow in the area where the inlet channel ends (see the left side of the shaded area designated CD in Figure 2.41a) and, also, in the area where the outlet channel begins (see the right side of the shaded area designated EF in Figure 2.41b). Notice in Figure 2.41 that the shaded area CD represents the region where the depth proﬁle is the narrowest along the centerline of the inlet channel (see Figure 2.30b), and the shaded area EF represents the region where the depth proﬁle is the narrowest along the centerline of the outlet channel (see Figure 2.30c). The presence of backﬂow inside the inlet and outlet channels of a Maddock mixing head under certain extrusion conditions, which is predicted in the theory presented above, has indeed been conﬁrmed by experiment. Figure 2.42 is a photograph of a piece of plastic (from the beginning of the outlet channel), which was obtained from a screw pushout experiment conducted at a screw speed of 100 rpm and a head pressure of 6.89 MPa (see run #34 in Table 2.5). According to Han et al. (1991a), in the screw pushout experiments a red pigment tracer was introduced into the LDPE before extrusion. It is quite clear in Figure 2.42 that there is backﬂow at the beginning of the outlet channel, as expected, based on the theoretical predictions (see right-hand side of EF in Figure 2.41).

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Figure 2.41 Predicted streamlines for an LDPE in the entire mixing head #2, using the same

extrusion conditions as run #42, where circulatory ﬂow patterns are seen near the exit of the inlet channel (on the right side of the top ﬁgure) and near the entrance of the outlet channel (on the left side of the bottom ﬁgure). Table 2.6 gives the extrusion conditions employed for prediction. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.) Figure 2.42 A photograph showing the

presence of backﬂow inside the outlet channel of a Maddock mixing head. (Reprinted from Han et al., Polymer Engineering and Science 31:818. Copyright © 1991, with permission from the Society of Plastics Engineers.)

2.4

Performance of Plasticating Barrier-Screw Extruders

One of the most troublesome aspects in the use of plasticating metering screw extruders, which has long been recognized, is ﬂuctuations of ﬂow rate in the extruder, commonly referred to as “surging.” The cause of surging has been debated for a long time. Today, it is widely accepted that the breakup of the solid bed in the melting section of a plasticating extruder is the primary source of the surging. Consequently, some clever ideas have emerged to prevent the breakup of the solid bed during extrusion. These ideas are documented primarily in patents (Barr 1972; Dray and Lawrence 1972; Ingen Housz 1980; Kim 1975; Maillefer 1959; Wheeler 1982). The basic idea in these patents is to design a plasticating screw in such a way that the solid bed can be separated from the melt pool. Figure 2.43 gives a photograph of a barrier screw. (However, in fact several different types of barrier screws are currently in use in the plastics industry.) Figure 2.44 gives

Figure 2.43 A photograph of a Davis-Standard barrier screw. (Reprinted from Han et al., Polymer

Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

Figure 2.44 Schematic of three different types of barrier screw: (a) the Barr barrier screw, (b) the

Davis-Standard barrier screw, and (c) the Kim barrier screw. (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

99

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Figure 2.45 Schematic showing the side view of three different types of barrier screws:

(a) the Barr barrier screw, (b) the Davis-Standard barrier screw, and (c) the Kim barrier screw. (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

schematics of the side view of three different types of barrier screw suggested in patents, namely, (a) the Barr barrier screw (Barr 1972), (b) the Davis-Standard barrier screw (Wheeler 1982), and (c) the Kim barrier screw (Kim 1975). It can be seen in Figure 2.44 that the solid and melt channels are separated by a secondary (barrier) ﬂight. The major differences between the three barrier screws lie in the proﬁles (the width and height) of the melt and solid channels along the extruder axis, as shown schematically in Figure 2.45. Speciﬁcally, as schematically shown in Figure 2.46, the Barr barrier screw has the following features: (1) beyond the initial few turns, the width of solid channel is constant but the height of the solid channel decreases, approaching zero at the end of the melting section, (2) both the width and height of the melt channel increase along the screw axis, (3) the total widths of solid and melt channels are kept constant along the screw axis, (4) within the distance ZM from the feed hopper the helix angle of the barrier ﬂights is greater than that of the primary ﬂights, but beyond ZM the helix angle of the barrier ﬂights is the same as that of the primary ﬂights, and (5) the helix angle of the primary ﬂights is kept constant along the screw axis. The volumetric ratio of melt channel to solid channel along the screw axis must be determined by properly adjusting channel heights, such that the melting rate in the solid channel can be balanced by the pumping rate into the melt channel. As schematically shown in Figure 2.47, in the Davis-Standard barrier screw the heights of both solid and melt channels vary in the same way as those in the Barr barrier screw, but it has the following special features: (1) the helix angles of both

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101

Figure 2.46 Schematic showing (a) the solid bed height proﬁle, (b) the top view of the screw

channel, and (c) the melt channel height proﬁle in the Barr barrier screw.

primary and barrier ﬂights vary along the screw axis, (2) the helix angle of the barrier ﬂights is greater than that of the primary ﬂights, and (3) there are transition zones between the feed section and the melting section, and between the melting section and the melt-conveying section of the screw. As schematically shown in Figure 2.48, the Kim barrier screw has the following features: (1) the width of the solid channel is constant while the width of the melt channel increases along the screw axis, (2) the height of the solid channel decreases

Figure 2.47 Schematic showing (a) the solid bed height proﬁle, (b) the top view of the screw

channel, and (c) the melt channel height proﬁle in the Davis-Standard barrier screw.

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PROCESSING OF THERMOPLASTIC POLYMERS

Figure 2.48 Schematic showing (a) the solid bed height proﬁle, (b) the top view of the screw

channel, and (c) the melt channel height proﬁle in the Kim barrier screw.

while the height of the melt channel increases along the screw axis, and (3) the helix angles of both primary and barrier ﬂights increase gradually along the screw axis. 2.4.1

Stability of the Solid Bed in a Plasticating Barrier-Screw Extruder

Figure 2.49 gives a schematic of the cross-section of an unwound Davis-Standard barrier screw. Figure 2.50 gives a schematic showing the positions at which nine pressure transducers were mounted along the extruder axis of (a) a metering screw and (b) a Davis-Standard barrier screw. Table 2.7 describes the dimensions of the barrier

Figure 2.49 Schematic of the cross section of an unwound Davis-Standard barrier screw.

(Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

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Figure 2.50 Schematic showing the position of pressure transducers mounted along the extruder

axis of (a) metering screw and (b) Davis-Standard barrier screw, employed in an experimental study. (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

screw employed, and Table 2.8 gives a summary of the extrusion conditions used for an LDPE in which the set point of the barrel temperature along the entire extruder axis was at 162 ◦ C for all experimental runs (Han et al. 1991). Figure 2.51 gives the traces of pressure signals during the extrusion of an LDPE, recorded from the transducers mounted on the barrel along the axis of the barrier-screw extruder (see Figure 2.49 and Table 2.7 for the geometrical conﬁguration of the screw), Table 2.7 Geometrical conﬁgurations of the Davis-Standard barrier-screw extruder

Extruder diameter (mm) = 63.5 Extruder length (L/D) = 24.35 Length of the feed section (L/D) = 5.0 Length of the ﬁrst transition section (L/D) = 1.0 Length of the melting section (L/D) = 10.5 Length of the second transition section (L/D) = 1.4 Length of the metering section (L/D) = 16.35 Channel depth at the beginning of the solid channel (mm) = 12.19 Channel depth at the end of the solid channel (mm) = 0.254 Channel depth at the beginning of the melt channel (mm) = 3.81 Channel depth at the end of the melt channel (mm) = 4.826 Primary ﬂight width (mm) = 6.35 Primary ﬂight clearance (mm) = 0.076 Barrier ﬂight width (mm) = 3.67 Barrier ﬂight clearance (mm) = 0.245 Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.

104

PROCESSING OF THERMOPLASTIC POLYMERS Table 2.8 The extrusion conditions employed in the extrusion of an LDPE in a Davis-Standard barrier-screw extruder

Run No. 18 19 20 21 22 23 24 25 26

Screw Speed (rpm)

Throughput (kg/h)

Head Pressurea (MPa)

50 50 50 75 75 75 100 100 100

56.8 50.0 41.9 81.4 74.7 67.0 105.0 97.4 89.6

6.89 13.78 20.67 6.72 13.95 20.67 6.79 13.67 20.56

a The pressure measured at the extruder exit.

Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.

Figure 2.51 Traces of pressure signals for an LDPE recorded at various positions along the extruder barrel using a Davis-Standard barrier screw. Run #25 in Table 2.8 describes the extrusion conditions employed. (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

at a screw speed of 100 rpm and a head pressure of 13.67 MPa (run #25 in Table 2.8). For comparison, Figure 2.52 gives the traces of recorded pressure signals along the metering screw extruder at the identical screw speed and head pressure. It is quite apparent in Figures 2.51 and 2.52 that the pressure patterns at each position are much steadies in the barrier screw as compared with those in the metering screw. Figure 2.53 gives magniﬁed pressure signals recorded at positions p2 , p3 , and p4 in both the barrier-screw and metering-screw extruders. Notice that in the metering screw extruder, p3 is located 0.4D from the end of the transition section of the screw and

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105

Figure 2.52 Traces of pressure signals for an LDPE recorded at various positions along the extruder barrel using a metering screw. The extrusion conditions employed are: screw speed of 100 rpm, throughput of 63.4 kg/h, and head pressure of 13.68 MPa. Table 2.3 gives the screw conﬁgurations employed. (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

p4 is 2.4D into the metering section (see Figure 2.52). It is quite evident in Figure 2.53 that pressure instability begins to develop in the vicinity of p3 in the metering screw, whereas no pressure instability develops in the barrier screw. Notice in Figure 2.53b that in the vicinity of p3 of the metering screw it is difﬁcult to detect from the pressure signal the presence of either the solid bed or the melt pool, but it can be observed that solid-bed breakup did occur. Thus, efﬁcient melting has ended by the time the material reached position p4 . However, from the pressure signals given in Figure 2.53a, there is no indication of solid-bed breakup. Notice that in the barrier-screw extruder p3 is located 0.6D after the formation of the melt channel is completed (see Figure 2.51 and Table 2.7). Notice further in Figure 2.53 that both screws show similar pressure signals patterns at p2 . This is just after the start of the transition zone in the case of the metering screw (see Figure 2.52) and before the ﬁrst barrier ﬂight in the case of the barrier screw (see Figure 2.51). No instability of pressure had developed at this point. It should be pointed that the pressure instability observed with the metering screw is not caused by the short feed-transition design (3D feed section and 4D transition section) (see Table 2.3). It has been reported that for designs utilizing a longer feed section and/or a longer transition section, solid-bed breakup is more catastrophic and so pressure variations are more severe. Figure 2.54 gives a further magniﬁcation of the pressure signals recorded at p3 in both the barrier screw and the metering screw, at a screw speed of 100 rpm and a head pressure of 13.67 MPa. In the case of the barrier screw, the locations of the solid channel, barrier ﬂight, melt channel, and primary ﬂight can be identiﬁed by the recorded pressure signals. But this is not possible, due to solid-bed breakup, in the case

106

PROCESSING OF THERMOPLASTIC POLYMERS Figure 2.53 Comparison of

pressure signals for an LDPE at p2 , p3 , and p4 : (a) in a Davis-Standard barrier screw, and (b) in a metering screw. (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

of metering screw. Under virtually identical extrusion conditions, the barrier screw had a throughput of 97.4 kg/h, while the metering screw’s throughput was only 63.4 kg/h. Thus, the barrier screw produced a 54% increase in throughput, supporting the claim that generally a well-designed barrier screw has superior performance to that of a metering screw for the same operating conditions. Figure 2.55 gives the results of a screw pushout experiment for an LDPE, describing the proﬁles of the solid-bed width at various positions along the extruder axis, in which the dark areas represent the molten LDPE mixed with a pigment. Notice in Figure 2.49 that the barrier section begins at an L/D of about 5 and the melt channel depth begins to increase at an L/D of about 6. This observation helps us to understand the presence of a small melt pool in the screw channel for an L/D of less than 5 (see #1 through #6 in Figure 2.55), where there is no barrier ﬂight. The following observations are worth noting in Figure 2.55: (1) at values of L/D from 6.3 (#8) to 7.7 (#10), a small melt pool is present on the left side of the solid channel that is in contact with the barrier ﬂight, (2) at an L/D of about 7.7 (#10), the size of the melt pool in the solid channel begins to decrease, and ﬁnally the melt pool disappears completely at an L/D of about 10.5, and (3) even when there is no melt pool in the solid channel, there exist thin melt ﬁlms surrounding the solid bed.

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107

Figure 2.54 Magniﬁed pressure signals for an LDPE recorded at p3 : (a) in a Davis-Standard barrier screw and (b) in a metering screw. (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

In reference to Figure 2.55, the fact that during extrusion there was a small melt pool in the solid channel at values of L/D from 6.3 (#8) to 9.8 (#11) indicates to us that under the particular extrusion conditions the melting rate was not balanced by the pumping rate. When a melt pool builds up in the solid channel for whatever reason, this will make the solid-bed width smaller, thus decreasing the contact area between the upper surface of the solid bed and the inner surface of the barrel. In view of the fact that the majority of melting occurs from the frictional heat generated between the barrel surface and the upper surface of the solid bed, a decrease in the contact area between the upper surface of the solid bed and the barrel inner surface will result in a decrease in the melting rate. This certainly is not a desirable situation in operating barrier-screw extruders. Therefore, a buildup of melt pool in the solid channel must be avoided in order to have good performance in barrier-screw extruders. 2.4.2

Analysis of the Performance of Plasticating Barrier-Screw Extruders

In the past, some serious attempts were made to simulate the performance of barrierscrew extruders, using varying degrees of mathematical sophistication (Amellal and Elbirli 1988; Han et al. 1991b; Ingen Housz and Meijer 1981). We present here a mathematical analysis of the performance of barrier-screw extruders by extending the analysis presented in Section 2.2.2 for metering screw extruders. In order to facilitate the presentation of our analysis, a schematic of an idealized cross section of the “unwound” screw channel, which consists of six zones (A through F), is given in Figure 2.56,

Figure 2.55 Melt ﬁlm proﬁles in the solid bed channel that were obtained from a screw pushout

experiment using a Davis-Standard barrier screw at different values of L/D: (1) 2.7, (2) 3.2, (3) 3.7, (4) 4.2, (5) 4.7, (6) 5.2, (7) 5.7, (8) 6.3, (9) 7.0, (10) 7.7, (11) 9.8, and (12) 10.5. Run #24 in Table 2.8 describes the extrusion conditions employed. (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

Figure 2.56 Schematic of the cross-section of an idealized unwound barrier screw. (Reprinted

from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

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PLASTICATING SINGLE-SCREW EXTRUSION

109

where the barrel surface is at the top, moving at constant velocity Vb with velocity components Vbx and Vbz in the cross-channel and down-channel directions, respectively. The screw surface is at the bottom, and the barrier ﬂight is placed between the melt channel (zone B) and the solid channel, which consists of the solid bed (zone A) and thin melt ﬁlms (zones C, D, E1 , and E2 ) surrounding the solid bed. It should be mentioned that the melt ﬁlm thickness E1 , between the barrier ﬂight and the solid bed, can be different from the melt ﬁlm thickness E2 , between the primary ﬂight and the solid bed. Figure 2.56 is similar to Figure 2.4 for a metering screw extruder, but differs in the following aspects: (1) there is a barrier ﬂight separating the solid bed and the melt pool and (2) the melt ﬁlm in the barrier ﬂight clearance (zone F) must be treated separately from the upper melt ﬁlm (zone C) above the solid bed. Nevertheless, the mathematical analysis of the barrier-screw extruder can be performed by following the approach already described in Section 2.2 for metering screw extruders, with some modiﬁcations, as summarized below. (a) Zone A (Solid Bed) The energy equation for the solid bed is exactly the same as Eq. (2.7). However, since the solid bed is not in direct contact with the melt pool in the barrier screw, it is reasonable to assume that the solid bed is freely deformable. Then, the down-channel solid-bed velocity at a given position along the screw axis, Vsz z , can be expressed by

Vsz z = mA z /ρs Ws Hs

(2.50)

where mA z is the mass ﬂow rate in zone A at axial position z. (b) Zone B (Melt Channel) The approach used to analyze the ﬂow in the melt channel of a barrier-screw extruder is essentially the same as that used in the analysis of metering-screw extruders, except that since the melt pool is not in direct contact with the solid bed, the boundary conditions must be different for the two situations. Equations (2.18)–(2.20) must be solved under the following boundary conditions:

vz = 0,

vx = 0,

T = Tc

at y = 0

(2.51a)

vz = Vbz ,

vx = −Vbx ,

T = Tb

at y = H

(2.51b)

vz = 0,

T = Tc

at x = 0

(2.51c)

vz = 0,

T = Tc

at x = WB

(2.51d)

where WB is the melt-pool width. (c) Zone C, D, E1 , and E2 (Melt Films in the Solid Channel) Zone C represents the upper melt ﬁlm, which is the primary source for the supply of molten polymer to the melt channel. Notice that the thickness of the melt ﬁlm in zone C is very small as compared with the height of the barrier ﬂight. The system of equations for zones C, D, E1 , and E2 is the same as for metering-screw extruders, Eqs. (2.23)–(2.25). The boundary conditions for zone C are given by Eq. (2.27), and the boundary conditions for zones D, E1 , and E2 are given by Eq. (2.28).

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(d) Zone F (Barrier Flight Clearance) In an ideal situation, where the design of barrier ﬂights is perfect, all the polymer melted in the solid channel (zone A) will be transported to the melt channel (zone B) through the barrier ﬂight clearance HF . However, when HF is too small, the melted polymer that is not transported to the melt channel will remain in the solid channel, thus increasing the thicknesses of melt ﬁlms in zones C, D, E1 , and E2 and eventually breaking up the solid bed. Conversely, when HF is too large, the rate of ﬂow from zone C to the melt channel (zone B) will exceed the melting rate in the solid channel, thus raising the height of solid bed and consequently increasing the chances of unmelted solid pellets (or powders) ﬂowing over the barrier ﬂight into the melt channel. This means that the barrier ﬂight clearance HF plays an important role in controlling the rate of ﬂow of melted polymer from the solid channel into the melt channel. Therefore, for zone F, the momentum and energy equations, Eqs. (2.23)–(2.25) must be solved with the boundary conditions:

vz = 0,

vx = 0,

T = Tc

at y = 0

(2.52a)

vz = Vbz ,

vx = −Vbx ,

T = Tb

at y = HF

(2.52b)

where HF is the clearance of the barrier ﬂight in zone F. Note that boundary conditions, Eq. (2.52), are not the same as the boundary conditions given by Eq. (2.27) for zone C. In solving the system of differential equations for six different zones for the barrierscrew extruder, it is important to understand that the mass balance for each zone must be satisﬁed, as described below. The mass ﬂow rate of unmelted polymer in the downchannel direction, m ˙ Az , is determined by the melting rate at solid/melt interfaces AC, RAC , and the melting rate at the solid/melt interface ADE, RADE : ˙ Az z − RAC − RADE m ˙ Az z+ = m

(2.53)

The mass ﬂow rate in the melt channel m ˙ Bz in the down-channel direction is determined by the total throughput G, the mass ﬂow rate in the solid bed m ˙ Az , the ﬂow rate at the ˙ DEz , and the ﬂow rate over upper melt ﬁlm m ˙ Cz , the ﬂow rate at the lower melt ﬁlm m the barrier ﬂight along the down-channel direction m ˙ Fz , that is m ˙ Bz = G − m ˙ Az − m ˙ Cz − m ˙ DEz − m ˙ Fz

(2.54)

where m ˙ Fz = Ws ρm

HF 0

vz (F) (y) dy

(2.55)

where vz (F) (y) is the velocity of melt in the down-channel direction. direction is The mass ﬂow rate in the upper melt ﬁlm m ˙ Cz in the down-channel determined by the melting rate at the solid/melt interface AC, RAC z , and the pumping rate m ˙ Cx of the melt from the upper melt ﬁlm to the melt channel, that is ˙ Cz z + RAC z − m ˙ Cx z (2.56) m ˙ Cz z+z = m

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111

where

HC m ˙ Cz z = Ws ρm 0 vz (C) (y) dy RAC z = ρs Vsyl Ws z

(2.57)

m ˙ Cx z = z ρm

(2.59)

HC 0

vx (C) (y) dy

(2.58)

The mass ﬂow rate in the lower melt ﬁlm m ˙ DEz in the down-channel direction is determined by the melting rate RADE at solid/melt interface ADE, mass ﬂow rate ˙ Cx in the m ˙ DEx in the cross-channel direction, mass ﬂow rate at the upper melt ﬁlm m cross-channel direction, and mass ﬂow rate through the barrier ﬂight clearance m ˙ Fx in the cross-channel direction, that is ˙ DEz z + RADE − m ˙ DEx z + m ˙ Cx z − m ˙ Fx z m ˙ DEz z+z = m

(2.60)

HDE m ˙ DEz z = Ws ρm 0 vz (DE) (y) dy

(2.61)

where

HDE m ˙ DEx z = zρm 0 vx (DE) (y) dy m ˙ Fx z = z ρm RADE

HF

vx (F) (y) dy = z ρs vsy2 z 2Hs + Ws 0

(2.62) (2.63) (2.64)

in which vsy2 is the solid-bed velocity in the y-direction at interface AD. The meltconveying section of the barrier-screw extruder can be analyzed in exactly the same way as for the metering-screw extruders discussed in Section 2.2.3. 2.4.3

Comparison of Prediction with Experiment

Figure 2.57 gives a comparison of predicted and experimental axial pressure proﬁles for an LDPE, the operating conditions of which are given in Table 2.8 (run #21, run #22, and run #23). It can be seen in Figure 2.57 that at a given screw speed, the predicted axial pressure proﬁles are in reasonable agreement with experimental results at L/D ratios of up to about 16, but theory overpredicts axial pressure proﬁles at L/D ratios greater than about 16, especially at low head pressures. Among the several possible reasons that might have contributed to the poor agreement between axial pressure proﬁles from theory and experiment at L/D ratios greater than 16, the ﬁnite difference method employed might have generated too crude an approximation of the complex geometrical conﬁgurations of the barrier screw near the end of the melting section. Notice in Figure 2.49 that at an L/D ratio of about 16, a rapid transition occurs from

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Figure 2.57 Comparison of axial pressure proﬁles from experimental results (symbols) and

theoretical predictions (solid line) for an LDPE at various extrusion conditions: curve (1) and () for run #21, curve (2) and () for run #22, and curve (3) and () for run #23. Table 2.8 gives the extrusion conditions employed for experiment and prediction. (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

the barrier section to the melt-conveying section. Under such circumstances, the ﬁnite element method must be used because the ﬁnite difference method cannot describe the ﬂow geometry accurately, leading to poor pressure proﬁle predictions along the extruder axis. Figure 2.58 gives predictions of upper melt-ﬁlm thickness (zone C), and Figure 2.59 gives predictions of lower melt-ﬁlm thickness (zone D), at various extrusion conditions (see Table 2.8). It can be seen in Figures 2.58 and 2.59 that at a screw speed of 50 rpm, the head pressure inﬂuences greatly the thickness proﬁles of both upper and lower melt ﬁlms along the extruder axis, but as the screw speed is increased to 75 rpm, the head pressure is seen to have little inﬂuence on the thickness of upper and lower melt ﬁlms. From the point of view of stable extrusion operations, it is desirable to have as little sensitivity of melt-ﬁlm thickness in the solid channel to extrusion conditions as possible. This then suggests that, in reference to Figures 2.58 and 2.59, at a given screw speed, one would expect to have a stable extrusion operation at low head pressures. Figure 2.60 gives prediction of reduced solid-bed velocity, Vsz /Vbz , along the extruder axis under various extrusion conditions (see Table 2.8), where Vsz is the solidbed velocity at position z and Vbz is the z-component barrel velocity. It can be seen in Figure 2.60 that at a ﬁxed screw speed, the solid-bed velocity decreases with increasing head pressure. This can be explained by the fact that at a ﬁxed screw speed, the solidbed velocity decreases with increasing head pressure, which in turn is explained by the

PLASTICATING SINGLE-SCREW EXTRUSION

113

Figure 2.58 Plots of predicted upper melt-ﬁlm thickness for an LDPE versus distance along the

extruder axis. Table 2.8 gives the extrusion conditions employed for prediction. (a) At a screw speed of 50 rpm for different head pressures: curve (1) for 6.89 MPa (run #18), curve (2) for 13.78 MPa (run #19), and curve (3) for 20.67 MPa (run #20). (b) At a screw speed of 75 rpm for different head pressures: curve (1) for 6.72 MPa (run #21), curve (2) for 13.95 MPa (run #22), and curve (3) for 20.67 MPa (run #23). (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

fact that at a ﬁxed screw speed, throughput decreases with increasing head pressure. We can now explain why in Figures 2.58 and 2.59 we observe an increase in melt-ﬁlm thickness with increasing head pressure. As the head pressure increases, polymer pellets (or powders) will stay longer in the melting section, and consequently the chances for more melting to occur will increase. However, for a given design of barrier ﬂight and screw speed, the mass ﬂow rate from the solid channel to the melt channel will remain the same. As a result, the amounts of polymer melted in the solid channel and thus the thickness of upper and lower melt ﬁlms will increase because the pumping rate of molten polymer into the melt channel is not balanced by the melting rate in the solid channel. We have shown in Section 2.4.2d that the barrier ﬂight clearance HF (zone F in Figure 2.56) plays an important role in determining the performance of barrier-screw extruders. Speciﬁcally, too small a value of HF will make the ﬂow of molten polymer from the upper melt ﬁlm (zone C) to the melt channel (zone B) difﬁcult and thus will cause the molten polymer to accumulate in the solid channel. Under such circumstances, the pumping rate will become less than the melting rate. This will then create a melt pool in the solid channel, consequently inducing solid-bed breakup, which defeats the

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Figure 2.59 Plots of predicted lower melt-ﬁlm thickness versus distance along the extruder

axis. Table 2.8 gives the extrusion conditions employed for prediction. (a) At a screw speed of 50 rpm for different head pressures: curve (1) for 6.89 MPa (run #18), curve (2) for 13.78 MPa (run #19), and curve (3) for 20.67 MPa (run #20). (b) At a screw speed of 75 rpm for different head pressures: curve (1) for 6.72 MPa (run #21), curve (2) for 13.95 MPa (run #22), and curve (3) for 20.67 MPa (run #23). (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

purpose of having barrier ﬂights. Conversely, too large a value of HF will make the pumping rate faster than the melting rate, thus depleting the molten polymer in the solid channel rather quickly. Under such circumstances, the solid bed may lose coherence and the chances of melted pellets ﬂowing over the barrier ﬂights (thus into the melt channel) may increase. This certainly is not a desirable situation when trying to achieve a stable extrusion operation.

2.5

Performance of Plasticating Single-Screw Extruders for Amorphous Polymers

So far, we have discussed plasticating extrusion of semicrystalline polymers, which have a sharp melting point. When dealing with the ﬂow of an amorphous polymer in the screw channel, the concept of “melting model” applied to semicrystalline polymer becomes very dubious. This is because there are no the sharp interfaces between the solid bed and the melt pool, and between the solid bed and the thin melt ﬁlms surrounding the solid bed, as in the situations for semicrystalline polymers (see Figure 2.4).

PLASTICATING SINGLE-SCREW EXTRUSION

115

Figure 2.60 Plots of reduced solid-bed velocity versus distance along the extruder axis. Table 2.8

gives the extrusion conditions employed for prediction. (a) At a screw speed of 50 rpm for different head pressures: curve (1) for 6.89 MPa (run #18), curve (2) for 13.78 MPa (run #19), and curve (3) for 20.67 MPa (run #20). (b) At a screw speed of 75 rpm for different head pressures: curve (1) for 6.72 MPa (run #21), curve (2) for 13.95 MPa (run #22), and curve (3) for 20.67 MPa (run #23). (Reprinted from Han et al., Polymer Engineering and Science 31:831. Copyright © 1991, with permission from the Society of Plastics Engineers.)

Thus, the concept of the melting model usually used for semicrystalline polymers must be modiﬁed before we can develop a mathematical model for the ﬂow of amorphous polymers in the screw channel. 2.5.1

The Concept of Critical Flow Temperature

Let us ﬁrst introduce the concept of “critical ﬂow temperature” (Tcf ) to an amorphous polymer in the screw channel (Han et al. 1996), and then modify the Lee–Han melting model presented in Section 2.2 to describe the plasticating extrusion of semicrystalline polymers. As shown schematically in Figure 2.61, Tcf is a temperature that lies above the glass transition temperature (Tg ) of an amorphous polymer, such that the polymer may be regarded as a “rubberlike” solid at temperatures between Tg and Tcf . Thus, Tcf plays the role of a de facto melting point for amorphous polymers, allowing us to deﬁne the interface between the solid bed and the melt pool, and the interface

116

PROCESSING OF THERMOPLASTIC POLYMERS Figure 2.61 Schematic showing the

deﬁnition of critical ﬂow temperature (Tcf ) for an amorphous polymer. Tcf is a temperature that lies above the glass transition temperature (Tg ), such that the polymer may be regarded as a “rubberlike” solid at temperatures between Tg and Tcf . (Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.)

between the solid bed and the thin melt ﬁlms surrounding the solid bed, in the screw channel. This means that an amorphous polymer may be regarded as “liquid” at temperatures above Tcf . In the analysis of plasticating extrusion of amorphous polymers, the following assumptions may be used: (1) at temperatures between Tg and Tcf , the temperature dependence of the modulus of the bulk solid bed may be described by the Williams–Landel–Ferry (WLF) (see Chapter 6 of Volume 1) equation, (2) at temperatures between Tcf and Tg + 100 ◦ C, the temperature dependence of the viscosity of the amorphous polymer may be described by the WLF equation, and (3) at temperatures above Tg + 100 ◦ C, the temperature dependence of the viscosity of the amorphous polymer may be described by the Arrhenius relationship. Granted that there is no theoretical guideline as to how the value of Tcf may be determined. In the following section, we show how the value of Tcf can be determined when solving the system of equations. 2.5.2

Analysis of Plasticating Extrusion of Amorphous Polymers

(a) Solid Bed (Zone A) The energy equation, Eq. (2.7), must be solved for the solid bed using the boundary conditions

T = Tz0 (y) at z = 0

(2.65a)

T = Tcf

and Vsy = −Vsy2

at y = 0

(2.65b)

T = Tcf

and Vsy = Vsy1

at y = Hs

(2.65c)

where Tz0 (y) is the temperature proﬁle in the y-direction at z = 0, and Tcf is the temperature at the interface between the solid bed and the lower melt ﬁlm (i.e., at y = 0),

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117

and between the solid bed and the upper melt ﬁlm (i.e., at y = Hs ). Note that following energy equations must be satisﬁed at the two interfaces (see Figure 2.4): (1) At interface AC between the solid bed and the upper melt ﬁlm: km

∂T ∂T − ks = ρs cps Vsy1 (Tcf − Tavg ) ∂y melt ∂y solid

(2.66a)

(2) At interface AD between the solid bed and the lower melt ﬁlm: ks

∂T ∂T − km = −ρs cps Vsy2 (Tcf − Tavg ) ∂y solid ∂y melt

(2.66b)

where Tavg is the average temperature of the solid bed, which varies along the extruder axis as the amorphous polymer continues to soften. Therefore, Tavg must be determined by solving Eq. (2.7) using the boundary conditions given by Eq. (2.66). Notice that Eq. (2.66) is different from its counterpart, Eq. (2.9), for semicrystalline polymers. The bulk density of the solid bed, ρB , for an amorphous polymer may be deﬁned by ρB = ρs (1 − εb )fsd + ρm (1 − fsd )

(2.67)

where ρs is the density of the polymer at T < Tg , ρm is the density of the polymer at T Tg , εb is the void fraction of the solid bed, and fsd is the fraction of the solid bed at T < Tg . The density of the solid bed deﬁned by Eq. (2.67) consists of two parts: (1) one that can be regarded as representing the glassy state at T < Tg , and (2) one that can be regarded as a rubberlike solid at Tg T < Tcf . Below Tg , each of these polymers may not undergo deformation under usual extrusion conditions. The values of ρB can be calculated using the expression (Hyun and Spalding 1990) ρmax − ρB = exp(FP ) ρmax − ρmin

(2.68)

where P is the pressure, ρmax is the maximum bulk density at inﬁnitely high pressure, ρmin the minimum bulk density at zero pressure, and F is the bulk density coefﬁcient function, which varies with temperature as F = b0 + b1 T + b2 T 2 + b3 /(Tg − T )

(2.69)

where T is expressed in Celsius. The numerical values for the coefﬁcients appearing in Eq. (2.69) for polystyrene (PS) and polycarbonate (PC) are given in Table 2.9. The temperature dependence of the modulus of an amorphous polymer must be calculated as follows: (1) when the temperature of the solid bed is lower than the Tg of the polymer, the polymer is regarded as a glassy solid and thus it has a constant value of modulus, Es , and (2) when the temperature lies between Tg and Tcf , the polymer

118

PROCESSING OF THERMOPLASTIC POLYMERS Table 2.9 Parameters appearing in the expression for the bulk density of the solid bed

Parameters ρmin (kg/m3 ) ρmax (kg/m3 ) b0 (1/MPa) b1 (1/MPa ◦ C) b2 (1/MPa (◦ C)2 ) b3 (◦ C/MPa) Tg (◦ C)

PS

PC

610 790 1.43 × 10−2 −2.84 × 10−3 3.97 × 10−5 −3.35 100

725 1,100 1.325 × 10−2 −1.133 × 10−3 9.739 × 10−6 −1.916 148

Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.

is regarded as a rubberlike solid and its modulus E(T ) varies with temperature T , following the WLF expression (Ferry 1980):

E(Tr ) log E T (y)

=

−C1 (T − Tr ) (C2 + T − Tr )

(2.70)

E(Tr ) is the modulus of the polymer at a reference temperature Tr , and C1 and C2 are constants characteristic of the polymer. The average modulus of the polymer can be calculated from Eq. (2.14). Then, the solid-bed velocity in the z-direction Vsz can be determined. (b) Melt Pool (Zone B) The momentum and energy equations, Eqs. (2.18)– (2.20), must be solved using the boundary conditions

vz = 0,

vx = 0,

T = Tc

at y = 0

(2.71a)

vz = Vbz ,

vx = −Vbx ,

T = Tb

at y = H

(2.71b)

vz = 0,

T = Tc

at x = 0

(2.71c)

vz = Vsz ,

T = Tcf

at x = WB

(2.71d)

However, the viscosity deﬁned by Eq. (2.21) must be estimated as follows: (1) For Tcf T Tg + 100 ◦ C: η0 (T ) = η0 (Tcf )

aT (T ) aT (Tcf )

(2.72)

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119

where η0 (Tcf ) is the Newtonian viscosity at temperature Tcf , and aT is a shift factor deﬁned by the WLF expression (Ferry 1980) −C1 T − Tr log aT (T ) = C2 + T − Tr

(2.73)

where Tr is a reference temperature. (2) For T > Tg + 100 ◦ C: η0 (T ) = ko exp(−bT )

(2.74)

(c) Melt Films Surrounding the Solid Bed (Zones C, D and E) The momentum and energy equations, Eqs. (2.23)–(2.25), must be solved for zone C using the boundary conditions

vz = Vsz ,

vx = 0,

T = Tcf

at y = 0

(2.75a)

vz = Vbz ,

vx = −Vbx ,

T = Tb

at y = HC

(2.75b)

and for zones D and E using the boundary conditions vz = 0,

vx = 0,

T = Tc

at y = 0

(2.76a)

vz = Vsz ,

vx = 0,

T = Tcf

at y = HD

(2.76b)

(d) Melt-Conveying Section The analysis of this section is the same as for semicrystalline polymers, as discussed in Section 2.2.

2.5.3

Comparison of Prediction with Experiment

Han et al. (1996) numerically solved a system of equations to predict the pressure proﬁles along the extruder axis in the extrusion of PS and PC by adjusting the numerical values of Tcf and fm until predicted pressure proﬁles agreed reasonably well with measured ones. Table 2.10 gives the dimensions of the metering screw simulated. In the numerical simulations presented below, the expression (see Chapter 6 of Volume 1) η(T , γ˙ ) =

η0 (T )

for γ˙ < γ˙0

η0 (T )(γ /γ˙0 )n−1

for γ˙ ≥ γ˙0

(2.77)

will be used for the rheological properties of PS and PC for temperatures T > Tg + 100 ◦ C, where γ˙0 is the critical shear rate at which the viscosity starts to deviate from Newtonian behavior, and η0 (T ) is temperature-dependent zero-shear viscosity given by Eq. (2.74). For temperatures Tcf T Tg + 100 ◦ C, Eqs. (2.72) and (2.73) will be

120

PROCESSING OF THERMOPLASTIC POLYMERS Table 2.10 Geometrical conﬁgurations of the DavisStandard metering screw simulated for PS and PC

Screw diameter (mm) Extruder length (L/D) Length of the feed section (L/D) Length of the tapered section (L/D) Length of the metering section (L/D) Channel depth in the feed section (mm) Channel depth in the metering section (mm) Channel width (mm) Flight width (mm) Flight clearance (mm) Flight helix angle (◦ )

63.5 24.4 6.0 6.4 12.0 8.31 2.72 54.14 6.35 0.08 17.7

Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.

used to calculate η0 (T ). γ˙0 is virtually constant (0.07 s−1 ) over the temperature range tested for PC, but it varies with temperature for PS, which can be represented by γ˙0 = 3.111 × 10−10 exp(0.0458T )

(2.78)

Table 2.11 gives the physical and rheological parameters for the PC and PS employed in the numerical simulation.4 Figure 2.62 gives a comparison of predicted pressure proﬁles for PS with measured ones under the extrusion conditions (run #2A) given in Table 2.12, for which fm = 0.01 and Tcf = 155 ◦ C were used. In Figure 2.62, curve (1) represents the prediction based on the barrel temperature proﬁle, which was preset during experiment, of 177 ◦ C for zone 1 (L/D = 1–6) and 218 ◦ C for zones 2–4 (L/D = 6–24), and curve (2) represents the prediction using the barrel temperature proﬁle speciﬁed in Table 2.13. As can be seen in Figure 2.62, we ﬁnd that the set points of the barrel temperature proﬁle, curve (1), used during the experiment give rather poor agreement between predicted pressure proﬁle and the measured one. Therefore, the barrel temperature proﬁle was adjusted until the best ﬁt could be made between predicted pressure proﬁle, curve (2), and the measured one. Note that during the experiment there is no way of knowing the actual melt temperature at the inner side of the extruder barrel (at y = H ). This is because, due to viscous shear heating, the true melt temperature might be much higher than the set point. The choices for the values for fm and Tcf are not arbitrary; that is, outside a very narrow range of the values of fm and Tcf , predicted pressure proﬁles would never agree with measured ones. Figure 2.63 gives predicted pressure proﬁles for PS along the extruder axis for different values of fm , namely, 0.01, 0.1, and 1.0, with otherwise the same extrusion conditions as in Figure 2.62. It can be seen in Figure 2.63 that the choice of the value of fm larger than 0.01 gives rise to a predicted pressure proﬁle far below the measured one.

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121

Table 2.11 Physical properties and rheological parameters of PS and PC

PS

PC

1030 978 0.117 0.261 1218 1923 100

1200 1100 0.192 0.264 1180 1420 145

ko × 10−14 (Pa·s) b × 102 (1/K) n (dimensionless) C1

68.63 5.728 0.36 13.7

C2

50.0

0.758 4.264 0.75 10.2a 3.1b 73.26a 64.43b 1.1 × 109 1.1 × 108

(a) Physical Properties Density of solid polymer, ρs (kg/m3 ) Density of molten polymer, ρm (kg/m3 ) Thermal conductivity of solid polymer, ks (W/(m K)) Thermal conductivity of softened polymer, km (W/(m K)) Speciﬁc heat of solid polymer, cps (J/(kg K)) Speciﬁc heat of softened polymer, cpm (J/(kg K)) Glass transition temperature (◦ C) (b) Rheological Parameters

E(Tr ) (Pa) Emin (Pa) a

3.2 × 109 3.0 × 105

◦ For Tg T Tcf . b For Tcf < T < Tg + 100 C.

Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.

It was not possible to bring the predicted pressure proﬁle closer to the measured one by adjusting the other parameters involved in the system of equations (Han et al. 1996). Figure 2.64 gives predicted pressure proﬁles for PS along the extruder axis for three different values of Tcf , namely, 150, 155, and 160 ◦ C, with otherwise the same extrusion conditions as in Figure 2.62. It can be seen in Figure 2.64 that the choice of the value of Tcf of 160 ◦ C gives rise to a predicted pressure proﬁle that lies far below the measured one, while the choice of the value of Tcf of 150 ◦ C gives rise to a pressure increase in the screw channel that is too steep. It was not possible to bring the predicted pressure proﬁle closer to the measured one by adjusting the other parameters involved in the system of equations, and when the value of Tcf was set at much lower than 150 ◦ C, the computation could not proceed due to numerical instability (Han et al. 1996). Figure 2.65 gives a comparison of predicted pressure proﬁles with measured ones for PC for run #A46 and run #A48 (see Table 2.12 for the extrusion conditions), for which fm = 0.1 and Tcf = 200 ◦ C were used. In the numerical simulation, the barrel temperature proﬁle given in Table 2.13 and the physical and rheological properties given in Table 2.11 were employed.

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PROCESSING OF THERMOPLASTIC POLYMERS

Figure 2.62 Axial pressure proﬁles in the extrusion of PS, comparing experimental results

() with theoretical predictions. Curve (1) shows the predicted axial proﬁle for the preset barrel temperatures, 177 ◦ C for zone 1 (L/D = 1–6) and 218 ◦ C for zones 2–4 (L/D = 6–24), used in the experiment, and curve (2) shows the predicted axial pressure proﬁles for the barrel temperatures given in Table 2.13. The barrel temperature proﬁle given in Table 2.13 was chosen, such that the best ﬁt of the predicted axial proﬁles to the experimental ones could be obtained. In the theoretical predictions of the axial pressure proﬁles, the values of fm = 0.01 and Tcf = 155 ◦ C were used, and the extrusion conditions employed are given in Table 2.12. (Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.)

The results presented above lead us to conclude that there are only very narrow ranges of parameters fm and Tcf , that give rise to predicted pressure proﬁles that agree reasonably well with measured ones. In other words, fm and Tcf should not be regarded as being parameters that can be arbitrarily adjusted to ﬁt predicted pressure proﬁles to measured ones. This can be explained by the fact that fm represents the extent of solid-bed deformation and Tcf represents a temperature that can be regarded as being equivalent to a melting point in a semicrystalline polymer. Figure 2.66 gives predicted reduced solid-bed velocity (Vsz /Vbz ) along the extruder axis in the extrusion of PS for three different values of fm , 0.01, 0.1, and 1.0, under otherwise identical extrusion conditions to those used in Figure 2.62. It can be seen in Figure 2.66 that for fm = 0.1 and 1.0, the solid bed has a constant velocity along the extruder axis, while for fm = 0.01, the solid bed initially increases slowly and then accelerates very rapidly along the extruder axis. In other words, the simulation results indicate that when there is no or little deformation of the solid bed, the solid

Table 2.12 Extrusion conditions employed for PS and PC

Screw Speed (rpm)

Throughput (kg/h)

Head Pressurea (MPa)

(a) PS A1 A2 A3 A4 A5 A6 A7 A8 A9

50 50 50 75 75 75 100 100 100

46.76 44.49 40.86 68.55 64.92 61.74 91.25 85.81 81.27

6.89 13.78 20.67 6.89 13.78 20.67 6.89 13.78 20.67

(b) PC A46 A47 A48

25 25 25

26.33 25.88 25.42

17.06 24.12 31.01

Run Number

a Head pressure refers to the pressure measured at p . 9

Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.

Figure 2.63 Effect of fm on the predicted axial pressure proﬁles in the extrusion of PS: curve (1) for fm = 0.01, curve (2) for fm = 0.1, and curve (3) for fm = 1. In the numerical simulation, we used Tcf = 155 ◦ C, the extrusion conditions given in Table 2.12, and the barrel temperatures given in Table 2.13. (Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.)

123

124

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 2.64 Effect of Tcf on the predicted axial pressure proﬁles in the extrusion of PS: curve (1) for Tcf = 150 ◦ C, curve (2) for Tcf = 155 ◦ C, and curve (3) for Tcf = 160 ◦ C. fm = 0.01 was

used in the numerical simulation and the extrusion conditions employed are given in Table 2.12 and the barrel temperatures employed are given in Table 2.13. (Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.)

bed moves along the extruder axis with a constant velocity. Figure 2.67 gives predicted Vsz /Vbz for along the extruder axis in the extrusion of PS for three different values of Tcf , 150, 155, and 160 ◦ C, under otherwise identical extrusion conditions to those used in Figure 2.62. It can be seen in Figure 2.67 that for Tcf = 150 ◦ C, the solid bed accelerates very rapidly from the beginning, approaching inﬁnity at an L/D ratio of about 6. This implicitly suggests that softening of PS from the solid state at T < Tg and then from the rubberlike solid at Tg ≤ T < Tcf is completed at an L/D ratio of about 6, which simply cannot be true. However, it can be seen in Figure 2.67 that for Tcf = 155 and 160 ◦ C, the solid-bed velocity increases initially very slowly and then accelerates rapidly at an L/D ratio of about 8. This means that there exists an optimum value of Tcf , which predicts with reasonable accuracy measured axial pressure proﬁles of PS in a metering screw. Figure 2.68 gives the experimentally determined solid-bed velocity proﬁles reported by Bruker and Balch (1989), who employed a ﬂow visualization technique in the extrusion of PC in a Davis-Standard metering screw, which had a diameter of 63.5 mm, a feed section of L/D = 5, a tapered section of L/D = 8, and a metering section of L/D = 11. In Figure 2.68, the solid line represents a constant solid-bed velocity and the

PLASTICATING SINGLE-SCREW EXTRUSION

125

Figure 2.65 Comparison of experimentally measured axial pressure proﬁles and prediction for

PC in a metering screw: () run #A46 and () run #A48. The predictions (solid lines) were made with fm = 0.1 and Tcf = 200 ◦ C. The extrusion conditions employed are given in Table 2.12 and the barrel temperatures employed are given in Table 2.13. (Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.)

broken line represents the solid-bed velocity that was estimated with the assumption that it is proportional to channel depth ratio Ho /H (z), where Ho is the depth at the beginning of the tapered section and H (z) is the channel depth at position z in the tapered section. Experimental results (Bruker and Balch 1989; Zhu and Chen 1991) support the predictions given in Figures 2.66 and 2.67, indicating clearly that the assumption of a constant solid-bed velocity, which is tantamount to the assumption Table 2.13 The extruder barrel temperatures used in the numerical simulation

(a) For PS L/D T (◦ C)

3 180

6 190

9 200

12 220

15 225

18 230

21 235

24 240

3 220

6 260

9 280

12 290

15 320

18 320

21 320

24 320

(b) For PC L/D T (◦ C)

Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.

Figure 2.66 Predicted reduced solid-bed velocity, Vsz /Vbz , along the extruder axis in the extru-

sion of PS for different values of fm : curve (1) for fm = 0.01, curves (2) and (3) for fm = 0.1 and fm = 1, respectively. In the numerical simulation, Tcf = 155 ◦ C was used. The extrusion conditions (run #A2) employed are given in Table 2.12 and the barrel temperatures are given in Table 2.13. (Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.)

Figure 2.67 Predicted reduced solid-bed velocity, Vsz /Vbz , along the extruder axis in the extrusion of PS for different values of Tcf : curve (1) for Tcf = 150 ◦ C, curve (2) for Tcf = 155 ◦ C, and curve (3) for Tcf = 160 ◦ C. In the numerical simulation, fm = 0.01 was used. The extrusion

conditions (run #A2) employed are given in Table 2.12 and the barrel temperatures are given in Table 2.13. (Reprinted from Han et al., Polymer Engineering and Science 36:1360. Copyright © 1996, with permission from the Society of Plastics Engineers.) 126

PLASTICATING SINGLE-SCREW EXTRUSION

127

Figure 2.68 Experimentally measured reduced solid-bed velocity (), Vsz /Vbz , versus distance L/D along the extruder axis in the extrusion of PC, where the solid line represents constant solid bed velocity and the broken line represents the solid bed velocity that was estimated with the assumption that it is proportional to the channel depth ratio Ho /H (z), where Ho is the channel depth at the beginning of the tapered section and H (z) is the channel depth at position z in the tapered section. (Reprinted from Bruker and Balch, Polymer Engineering and Science 29:258. Copyright © 1989, with permission from the Society of Plastics Engineers.)

that the solid bed does not deform during softening in the tapered section of the screw channel, is not justiﬁed. In this section, we have shown that there exists a temperature, referred to as “critical ﬂow temperature,” Tcf , which may be regarded as an effective softening temperature at which an amorphous polymer can be regarded as liquid in the development of a mathematical model for plasticating single-screw extrusion. We have shown with the aid of experimentally measured pressure proﬁles along the extruder axis that the Tcf for PS is about 155 ◦ C and the Tcf for PC is about 200 ◦ C. This observation suggests that Tcf is about 55 ◦ C above the glass transition temperature of an amorphous polymer. Whether or not this can be used as a general guideline for amorphous polymers in general remains to be seen, which requires further investigation. The Tg of an amorphous polymer cannot be regarded as being equal to the melting temperature (Tm ) of a semicrystalline polymer because the viscosities of an amorphous polymer at or near its Tg are too high to ﬂow like a semicrystalline polymer above its Tm . Speciﬁcally, owing to the excessively large values of viscosity at or near the Tg of an amorphous polymer, one will not be able to ﬁnd numerical solution for the

128

PROCESSING OF THERMOPLASTIC POLYMERS

energy equation, Eq. (2.20), because the numerical solutions will become unbounded when Tcf in the boundary condition, Eq. (2.76b), is replaced by Tg .

2.6

Summary

In this chapter, we have described the fundamental principles of plasticating extrusion in a single-screw extruder for semicrystalline and amorphous polymers. Emphasis was placed on the importance of a clear understanding of melting mechanisms inside a screw when semicrystalline polymers are extruded. The concept of critical ﬂow temperature (Tcf ) was introduced in order to properly model the ﬂow of amorphous polymers inside a screw. Though empirical, the concept of Tcf has been proven to be very useful for understanding the extrusion characteristics of amorphous polymers, which do not have a sharp melting point. We have analyzed the performance of plasticating single-screw extruders with and without mixing elements and the performance of barrier-screw extruders. The idea put forth in the melting model presented in this chapter does not assume a priori whether the solid bed is rigid or freely deformable. The solution of the force balance on the surfaces of the solid bed, which is surrounded by thin melt ﬁlms and a melt pool, with the assumption that the solid bed in the bulk state follows a linear stress–strain relationship, gives us information as to whether or not the solid bed deforms under given extrusion conditions and/or for a given screw geometry. Predictions of axial pressure proﬁle and solid-bed width along the extruder axis have been compared with experimental results. We have shown experimental evidence that breakup of the solid bed occurs in the tapered section of the screw channel, where melting occurs. At present, there is no theoretical criterion (or criteria) established that enables one to predict the breakup of the solid bed during extrusion. When the breakup of the solid bed occurs, the solution of the momentum and energy equations has no physical signiﬁcance. This is perhaps the weakest point in all the efforts spent so far on developing various melting models for plasticating extrusion. Serious effort is needed to establish a criterion (or criteria) for the onset of solid-bed breakup, which could then be incorporated into efforts towards developing mathematical models for melting behavior in a screw channel. We have shown that under normal operating conditions of industrial practice, the extent of solid-bed deformation (thus solid-bed acceleration) is very large in a screw, which has a short feed section (L/D = 4–5), a short tapered section (L/D = 3–5), and a long metering section (L/D = 14–17). Currently, such screw designs are very widely used in many industrial extrusion operations, but it is not clear how the acceleration of the solid bed is related to the breakup of the solid bed. This subject must be addressed in future research. In presenting the melting model based on the premise of solid-bed deformation, we have assumed a linear stress–strain relationship for the solid bed in the bulk state and introduced a parameter fm relating the average modulus of a polymer in the solid state to the apparent modulus of the solid bed in the bulk state. In the absence of experimental data for the apparent modulus of the solid bed in

PLASTICATING SINGLE-SCREW EXTRUSION

129

the bulk state, we have adjusted the value of fm until predicted axial pressure proﬁles agreed reasonably well with experimental results. What is needed in the future is the development of an experimental technique that can enable one to determine the apparent modulus of the solid bed in the bulk state as a function of temperature and pressure. This is essential for successful evaluation of the performance of plasticating single-screw extruders. In other words, like measurement of the friction coefﬁcient between polymer pellets (or powders) and the barrel surface, measurement of the rheological behavior of the solid bed in the bulk state is urgently needed in order to be able to successfully predict the performance of plasticating single-screw extruders. We have also shown that the inclusion of cross-channel velocity in thin melt ﬁlms, which surround the solid bed, is essential for realistic predictions of axial pressure proﬁles in plasticating single-screw extruders. When the cross-channel velocity in thin melt ﬁlms is neglected, the thickness of the melt ﬁlm between the solid bed and the root of the screw (i.e., lower melt ﬁlm) increases without bound, making the numerical integration of the system of equations unstable. Unbounded growth of lower melt-ﬁlm thickness is not physically acceptable. The inclusion of the convective heat transfer term in the energy equation for the melt pool and thin melt ﬁlms gives rise to more realistic predictions of melt temperature in the screw channel, although predictions of axial pressure and solid bed width proﬁles may not be greatly affected by neglecting the contribution of convective heat transfer. In this chapter, we have shown how to analyze the performance of plasticating single-screw extruders for amorphous polymers by modifying the Lee–Han analysis (1990) for semicrystalline polymers. In doing so, we introduced the concept of “critical ﬂow temperature” (Tcf ), which can be regarded as being a temperature at which plastication (or softening) of an amorphous polymer begins (similar to melting of a semicrystalline polymer), yielding a “liquid-state” polymer melt. We postulated that an amorphous polymer may be regarded as being a rubberlike solid for temperatures between its Tg and Tcf , and as being a liquid for temperatures between its Tcf and Tg + 100 ◦ C. The use of the concept of “critical ﬂow temperature” enables one to deﬁne the melt pool for an amorphous polymer in the screw channel. We have shown how to determine the Tcf of an amorphous polymer with the aid of experimentally measured proﬁles of pressure along the extruder axis. It has been found that Tcf is 155 ◦ C for PS and 200 ◦ C for PC, suggesting Tcf ≈ Tg + 55 ◦ C.

Notes 1. It should be pointed that the Tadmor melting model (1966) deals with one-dimensional ﬂow in the upper melt ﬁlm (zone C in Figure 2.4) and neglects the presence of zones D and E in Figure 2.4. Moreover, this model does not solve a system of equations (two-dimensional momentum and energy equations) for the melt pool (zone B in Figure 2.4), and thus it cannot predict the pressure proﬁles along the axis of an extruder. It is fair to say that the Tadmor melting model is useful only for describing the early part of the melting zone.

130

PROCESSING OF THERMOPLASTIC POLYMERS

2. In their paper, Zhu and Chen (1991) did not present information on the lengths of feed section, or melting section, or metering section. It appears, however, from Figure 2.16 that the melting section in their screw begins at an L/D of about 7 and ends at an L/D of about 15. 3. The Tadmor melting model neglects the presence of the cross-channel velocity around the solid bed. Thus, the prediction of the lower melt ﬁlm thickness would not be possible using the Tadmor melting model. 4. Emin in Table 2.11 denotes the minimum value of the temperature-dependent shear modulus E(T ) of a polymer in a rubberlike solid state, deﬁned by Eq. (2.70); in other words, Emin is the shear modulus of PS or PC at Tcf (see Figure 2.61), above which the polymer may be regarded as liquid.

References Amellal K, Elbirli B (1988). Polym. Eng. Sci. 28:311. Barr RA (1972). U.S. Patent 3698541. Bruker I, Balch GS (1989). Polym. Eng. Sci. 29:258. Cox APD, Fenner RT (1980). Polym. Eng. Sci. 20:562. Darnell WH, Mol EAJ (1956). SPE J. 12:21. Donovan RC (1971). Polym. Eng. Sci. 11:247. Dray RF, Lawrence DL (1972). U.S. Patent 3650652. Edmondson IR, Fenner RT (1975). Polymer 16:49. Elbirli E, Lindt JT, Gottgetreu SR, Baba SM (1983). Polym. Eng. Sci. 23:86. Elbirli E, Lindt JT, Gottgetreu SR, Baba SM (1984). Polym. Eng. Sci. 24:988. Ferry JD (1980). Viscoelastic Properties of Polymers, 3rd ed, John Wiley & Sons, New York. Fukase H, Kunio T, Shinyo S, Nomura A (1982). Polym. Eng. Sci. 22:578. Gregory RB, Street LF (1968). U.S. Patent 3411179. Halmos AJ, Pearson JRA, Trottnow R (1978). Polymer 19:1199. Han CD, Lee KY, Wheeler NC (1990). Polym. Eng. Sci. 30:1557. Han CD, Lee KY, Wheeler NC (1991a). Polym. Eng. Sci. 31:818. Han CD, Lee KY, Wheeler NC (1991b). Polym. Eng. Sci. 31:831. Han CD, Lee KY, Wheeler NC (1996). Polym. Eng. Sci. 36:1360. Hyun KS, Spalding MA (1990). Polym. Eng. Sci. 30:571. Ingen Housz JF (1980). U.S. Patent 4218146. Ingen Housz JF, Meijer HEH (1981). Polym. Eng. Sci. 21:352. Kim HT (1975). U.S. Patent 3867079. Lee KY, Han CD (1990). Polym. Eng. Sci. 30:665. LeRoy G (1969). U.S. Patent 3486192. Lindt JT (1976). Polym. Eng. Sci. 16:284. Lindt JT, Elbirli E (1985). Polym. Eng. Sci. 25:412. Maddock BH (1959). SPE J. 15:383. Maddock BH (1967). SPE J. 23:23. Maillefer C (1959). Swiss Patent 363149. Shapiro J, Halmos AL, Pearson JRA (1976). Polymer 17:905. Tadmor Z (1966). Polym. Eng. Sci. 6:185.

PLASTICATING SINGLE-SCREW EXTRUSION

Tadmor Z, Duvdevani I, Klein I (1967). Polym. Eng. Sci. 7:198. Tadmor Z, Klein I (1970). Engineering Principles of Plasticating Extrusion, Van Nostrand Reinhold, New York. van Krevelen DW (1976). Properties of Polymers, 2nd ed, Elsevier, Amsterdam. Wheeler, NC (1982). U.S. Patent 4341474. Zhu F, Chen L (1991). Polym. Eng. Sci. 31:1113.

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3

Morphology Evolution in Immiscible Polymer Blends during Compounding

3.1

Introduction

Polymer researchers have had a long-standing interest in understanding the evolution of blend morphology when two (or more) incompatible homopolymers or copolymers are melt blended in mixing equipment. In industry, melt blending is conducted using either an internal (batch) mixer (e.g., a Banbury mixer or a Brabender mixer) or a continuous mixer (e.g., a twin-screw extruder or a Buss kneader). There are many factors that control the evolution of blend morphology during compounding, the ﬁve primary ones being (1) blend composition, (2) rheological properties (e.g., viscosity ratio) of the constituent components, (3) mixing temperature, which in turn affects the rheological properties of the constituent components, (4) the duration of mixing in a batch mixer or residence time in a continuous mixer, and (5) rotor speed in a batch mixer or screw speed in a continuous mixer (i.e., local shear rate or shear stress). When two immiscible polymers are compounded in mixing equipment, two types of blend morphology are often observed: dispersed morphology and co-continuous morphology. Numerous investigators have reported on blend morphology of immiscible polymers, and there are too many papers to cite them all here. Some investigators (Han 1976, 1981; Han and Kim 1975; Han and Yu 1972; Nelson et al. 1977; van Oene 1978) examined blend morphology to explain the seemingly very complicated rheological behavior of two-phase polymer blends, and others (Favis and Therrien 1991; He et al. 1997; Ho et al. 1990; Miles and Zurek 1988; Scott and Macosko 1995; Shih 1995; Sundararaj et al. 1992, 1996) investigated blend morphology as affected by processing conditions. Today, it is fairly well understood from experimental studies under what conditions a dispersed morphology or a co-continuous morphology may be formed, and whether a co-continuous morphology is stable, giving rise to an equilibrium morphology, or whether it is an unstable intermediate morphology that eventually is transformed into a dispersed morphology (Lee and Han 1999a, 1999b, 2000). 132

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Let us consider the morphology evolution in an immiscible blend consisting of two semicrystalline polymers, A and B, in a compounding machine, and let us assume that the melting point (Tm,A ) of polymer A is lower than the melting point (Tm,B ) of polymer B. Under such circumstances, polymer A will melt ﬁrst, forming the continuous phase in which the pellets (or powders) of polymer B will be suspended until Tm,B is reached, at which point the binary mixture will form an emulsion. As the temperature is increased further above Tm,B (i.e., Tm,A < Tm,B < T, with T being the processing temperature), the evolution of blend morphology will depend on the viscosity ratio of the two polymers and blend composition. When the viscosity (ηA ) of polymer A is lower than the viscosity (ηB ) of polymer B and polymer A is the major component, most likely the blend will form a dispersed morphology having polymer A as the continuous phase and polymer B as the discrete phase (i.e., drops). The drops can be elongated and/or broken into smaller drops, depending upon the intensity of mixing and the viscosity ratio of the two polymers. An interesting question, however, can be raised. Would it be possible for polymer A to form the discrete phase and polymer B to form the continuous phase when polymer A is the minor component while maintaining the relationship ηA < ηB ? In order to understand the morphology evolution in an immiscible blend consisting of a semicrystalline polymer and an amorphous polymer in mixing equipment, one must ﬁrst deﬁne the temperature that can be regarded as being the effective “melting point” of the amorphous polymer. From a thermodynamic point of view, an amorphous polymer can be regarded as “liquid” at temperatures above its Tg . In practice, however, the viscosity of an amorphous polymer at temperatures slightly above its Tg is very high; the polymer hardly ﬂows until it reaches a certain temperature much higher than Tg . Thus, the Tg of an amorphous polymer cannot be regarded as being a temperature that is equivalent to the melting temperature (Tm ) of a semicrystalline polymer. At such a temperature, when deﬁned properly, an amorphous polymer can be regarded as liquid. Shih (1995) recognized the importance of this problem when melt blending a semicrystalline high-density polyethylene (HDPE) and an amorphous ethylene-propylene-diene terpolymer (EPDM). In his study, however, Shih did not elaborate on the temperature at which EPDM actually began to ﬂow as a liquid during mixing with HDPE. In a study on the plasticating extrusion of amorphous polymer in a single-screw extruder, Han et al. (1996) introduced the concept of “critical ﬂow temperature” (Tcf ) for amorphous polymers (see Figure 2.61 in Chapter 2). According to this concept, an amorphous polymer may be regarded as a “rubberlike” solid at Tg < T < Tcf and as a “liquid” at T ≥ Tcf , that is, an amorphous polymer may be considered to ﬂow at T ≥ Tcf . In this regard, the Tcf of an amorphous polymer is de facto equivalent to the “melting point” of a semicrystalline polymer. Han et al. (1996) have suggested that Tcf ≈ Tg + 55 ◦ C. In this chapter, we present morphology evolution in immiscible polymer blend during compounding in terms of (1) the difference(s) in Tcf or Tm of the constituent components, (2) blend composition, (3) the viscosity ratio of the constituent components, (4) mixing temperature, (5) the intensity of mixing (rotor speed in a batch mixer or shear rate in a twin-screw extruder), and (6) the duration of mixing in an internal mixer or the residence time in a twin-screw extruder. The purpose of this chapter is to present the fundamental concepts associated with the morphology evolution in immiscible polymer blends during compounding. In this chapter, no attempt is made

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to relate the evolution of morphology of immiscible polymer blends to their physical or mechanical properties in the solid state, which is beyond the limited scope of this chapter. In Chapter 11 of Volume 1 we have discussed the rheology of immiscible polymer blends.

3.2

Morphology Evolution in Immiscible Polymer Blend during Compounding in an Internal Mixer

In this section, we present experimental observations of morphology evolution in immiscible polymer blends in an internal mixer investigated by Lee and Han (1999b). In the preparation of blends, they selected six commercial homopolymers: poly(methyl methacrylate) (PMMA), polystyrene (PS), polycarbonate (PC), polypropylene (PP), HDPE, and nylon 6. Table 3.1 gives values of Tg or Tm for the six homopolymers selected for preparing blends. Table 3.2 gives values of the difference between the Tcf s of two amorphous polymers, values of the difference between the Tcf of an amorphous polymer and the Tm of a semicrystalline polymer, and the value of the difference between the Tm s of two semicrystalline polymers for the ﬁve polymer pairs selected.

Table 3.1 Six homopolymers selected by Lee and Han to investigate morphology evolution in immiscible polymer blends during compounding

Sample Code

Manufacturer

Morphology

PMMA PS PC PP HDPE Nylon 6

Rohm & Haas (Plexiglas V825) Dow Chemical (STYRON 615PR) Dow Chemical Exxon Chemical (Escorene 1052) Dow Chemical (HF-1030 INSITE) AlliedSignal (Capron 8202)

amorphous amorphous amorphous crystalline crystalline crystalline

Tg (◦ C ) or T m (◦ C ) 118 98 146 165 125 221

Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.

Table 3.2 Values of Tcf , |Tcf − Tm |, or Tm of the ﬁve polymer blend pairs employed

Sample Code

Morphology

PMMA/PS PC/PS PS/HDPE PS/PP Nylon 6/HDPE

amorphous/amorphous amorphous/amorphous amorphous/crystalline amorphous/crystalline crystalline/crystalline

a

Tcf , |Tcf − Tm |, or Tm (◦ C ) 15a,b 45c 30a 10a 96

◦ ◦ ◦ Tcf of PS ≈ 155 C. b Tcf of PMMA ≈ 170 C. c Tcf of PC ≈ 200 C.

Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.

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Figure 3.1 (a) Photograph of the side view of a Brabender internal mixer, (b) photograph of

the plane view of a Brabender internal mixer showing the conﬁgurations of two rotors, and (c) simulation results of the ﬂow patterns in a Brabender internal mixer using the CFX-3FD multiphase solver. (See color plate 1.)

Figure 3.1 gives a photograph of the side view of a commercial internal mixer (Brabender), a photograph of the plane view of the Brabender internal mixer showing the conﬁgurations of two rotors, and simulation results of the ﬂow patterns in a Brabender internal mixer using the CFX-3FD multiphase solver.1 Figure 3.2a gives a schematic of the Brabender internal mixer and Figure 3.2b shows the postulated evolution of blend morphology when two semicrystalline polymers, A and B, are melt

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Figure 3.2 (a) Schematic of an internal mixer showing the blade and bowl, in which materials are mixed, and (b) schematic of how the melting and morphology evolution take place when two semicrystalline polymers are heated in an internal mixer, where the melting point of polymer A (Tm,A ) is assumed to be lower than that of polymer B (Tm,B ).

blended in the internal mixer, where the melting point (Tm,A ) of polymer A is assumed to be lower than the melting point (Tm,B ) of polymer B. In reference to Figure 3.2b, (1) when the processing temperature (T) lies between Tm,A and Tm,B (Tm,A < T < Tm,B ), the pellets of polymer A (the bright areas) melt ﬁrst, forming the matrix phase (the shaded areas) in which the pellets of polymer B (the dark areas) are suspended, and (2) when the processing temperature is increased to become Tm,A < Tm,B < T, the pellets of polymer B melt, forming drops that are stretched due to the motion of the two rotors. As will be shown below, such simple morphology evolution may not always be observed, depending upon many factors including the viscosities of the constituent components, blend composition, and so on. In this section, emphasis is placed on identifying the governing factors that control morphology evolution in immiscible polymer blends.

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3.2.1

137

Morphology Evolution in Blends Consisting of Two Semicrystalline Polymers

Figure 3.3a gives logarithmic plots of shear viscosity (η) versus shear rate (γ˙ ) for nylon 6 and HDPE at 240, 250, and 260 ◦ C for γ˙ = 0.001−1,000 s−1 , where values of η at γ˙ = 0.001−10 s−1 were obtained using a cone-and-plate rheometer and values of η at γ˙ = 10−1,000 s−1 were obtained using a plunger-type capillary rheometer (Instron rheometer) with a length-to-diameter ratio of 28.5, thus allowing end corrections to be neglected. Figure 3.3b shows the dependence of viscosity ratio, ηnylon 6 /ηHDPE , on temperature at γ˙ = 54.5 s−1 , showing that the ratio ηnylon 6 /ηHDPE decreases with increasing temperature. Note that γ˙ = 54.5 s−1 represents the maximum shear rate at a rotor speed of 50 rpm in the Brabender internal mixer employed. The value of γ˙ = 54.5 s−1 was determined using the expression γ˙ = Ω/δ = π DN/δ, where Ω is

Figure 3.3 (a) The shear-rate dependence of viscosity for nylon 6 (open symbols) and HDPE (ﬁlled symbols) at various temperatures (◦ C): (, 䊉) 240, (, ) 250, and (, ) 260. (b) The temperature dependence of viscosity ratio, ηnylon 6 /ηHDPE , at γ˙ = 54.5 s−1 .

(Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

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the angular speed of the rotor tip, δ is the gap opening between the rotor tip and the mixing chamber, D is the diameter of the inner or outer rotor, and N is the rotor speed in rpm. The outer diameter of the rotor was 37.5 mm with δ = 1.8 mm, and the inner diameter of the rotor was 12 mm with δ = 27.3 mm. Thus, the value of γ˙ = 54.5 s−1 , determined from the above expression, approximately represents the intensity of mixing in the Brabender internal mixer at a rotor speed of 50 rpm. The simplistic approach adopted here is not rigorous, simply relating the intensity of mixing to shear rate and thus enabling us to estimate the ratio ηnylon 6 /ηHDPE . Figure 3.4 shows images from scanning electron microscopy (SEM) for 30/70, 50/50, and 70/30 nylon 6/HDPE blends, where 30/70, 50/50, and 70/30 refer to the

Figure 3.4 SEM images showing the effect of the duration of mixing (2, 5, or 10 min) on the morphology evolution in 30/70, 50/50, and 70/30 nylon 6/HDPE blends during compounding at a rotor speed of 50 rpm and at 240 ◦ C in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

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weight percent of the component polymers, which were prepared at 240 ◦ C at a rotor speed of 50 rpm for 2, 5, and 10 min, where the dark areas represent the nylon 6 phase and the bright areas represent the HDPE phase. Note that since the Tm of nylon 6 is 221 ◦ C and the Tm of HDPE is 125 ◦ C, the HDPE melts ﬁrst, forming the matrix phase in which nylon 6 pellets are suspended until the temperature reaches 221 ◦ C. Upon melting at 221 ◦ C, the nylon 6 forms the discrete phase. Two possibilities then exist: either nylon 6 remains as the discrete phase and is dispersed in the HDPE matrix, or the discrete phase of nylon 6 transforms into the continuous phase, thus phase inversion takes place. In Figures 3.4a–c, we observe that (1) after 2 min of mixing the 30/70 nylon 6/ HDPE blend already formed a well-established dispersed morphology, in which the nylon 6 formed drops and was dispersed in the HDPE matrix, and (2) the same morphology persisted when the mixing time was extended to 10 min. Note that the HDPE has a higher viscosity than the nylon 6 (see Figure 3.3). Thus, we can conclude that the blend ratio determined the state of dispersion in the 30/70 nylon 6/HDPE blend. If the blend ratio were to determine an equilibrium state of dispersion in the 70/30 nylon 6/HDPE blend, we would expect that the major component, nylon 6, would form the continuous phase and the minor component, HDPE, would form the discrete phase. This means that phase inversion must take place after a sufﬁciently long duration of mixing because the HDPE melts ﬁrst to form the continuous phase while the nylon 6 remains in pellet form until the melt blending temperature reaches the Tm (221 ◦ C) of nylon 6. Indeed, SEM images of the 70/30 nylon 6/HDPE blend demonstrate this to be the case, as can be seen in Figures 3.4g–i. Speciﬁcally, during the initial 2 min of mixing, nylon 6 formed the discrete phase (dark holes in Figure 3.4g) and the HDPE formed the continuous phase. As the mixing continued for 5 min, we observe a co-continuous morphology consisting of interconnected structures of nylon 6 and HDPE (Figure 3.4h). As the mixing continued for 10 min, we observe a dispersed morphology in which HDPE is dispersed in the continuous nylon 6 phase (Figure 3.4i). In Figures 3.4d–f, we observe a morphology evolution in 50/50 nylon 6/HDPE blend, which is very similar to that in the 70/30 nylon 6/HDPE blend discussed above (Figures 3.4g–i). The blend composition being symmetric in the 50/50 nylon 6/HDPE blend, we conclude that the viscosity ratio determines the state of dispersion in that the more viscous HDPE forms the discrete phase and the less viscous nylon 6 forms the continuous phase. What is of great interest in Figure 3.4 is that the co-continuous morphology is a transitory morphological state, through which one mode of dispersed morphology is transformed into another mode of dispersed morphology. The above observation leads us to conclude that the co-continuous morphology observed in 50/50 and 70/30 nylon 6/ HDPE blends is not an equilibrium morphology. If the melt blending experiment had not been allowed to continue for a sufﬁciently long period (say, 10 min), we might have erroneously concluded that 50/50 and 70/30 nylon 6/HDPE blends have a co-continuous morphology. This observation suggests that melt blending be continued for a sufﬁciently long period in order to observe an equilibrium phase morphology in immiscible polymer blends.

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3.2.2

Morphology Evolution in Blends Consisting of Two Amorphous Polymers

Figure 3.5a gives log η versus log γ˙ plots for PMMA and PS at 160, 170, 180, 200, 220, and 240 ◦ C for γ˙ = 0.001−1,000 s−1 , showing that the viscosity of PMMA is much higher than that of PS over the entire range of temperatures and shear rates tested, especially at low γ˙ . The dependence of viscosity ratio, ηPMMA /ηPS , on temperature at γ˙ = 54.5 s−1 is presented in Figure 3.5b, showing that the ratio ηPMMA /ηPS is very high at 160 ◦ C, decreases rapidly as the temperature increases from 160 to about 190 ◦ C, and then tends to level off at 220 ◦ C and higher. Figure 3.6 gives transmission electron microscopy (TEM) images for 70/30, 50/50, and 30/70 PMMA/PS blends, which were prepared at a rotor speed of 50 rpm at 160 ◦ C for 5 and 30 min of mixing, where the dark areas represent the PS phase and

Figure 3.5 The shear-rate dependence of viscosity for PMMA (open symbols) and PS (ﬁlled symbols) at various temperatures ( ◦ C): (, 䊉) 160, (, ) 170, (, ) 180, (, ) 200,

(9, ) 220, and (3, 䉬) 240. (b) The temperature dependence of viscosity ratio, ηPMMA /ηPS , at γ˙ = 54.5 s−1 . (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

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Figure 3.6 TEM images showing the effect of the duration of mixing (5 min versus 30 min) on

the morphology evolution in 70/30, 50/50, and 30/70 PMMA/PS blends during compounding at a rotor speed of 50 rpm and at 160 ◦ C in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

the bright areas represent the PMMA phase. It is of interest to observe in Figure 3.6 a co-continuous morphology in all three blend compositions, regardless of whether they were melt blended for 5 or 30 min. Note that the melt blending temperature employed, 160 ◦ C, was close to the Tcf (155 ◦ C) of PS and below the Tcf (170 ◦ C) of PMMA.2 This observation suggests that during melt blending at 160 ◦ C, the PS barely functioned as a “liquid” and the PMMA functioned as a “rubberlike solid” (see Figure 2.61). Notice from Figure 3.5 that at 160 ◦ C, ηPMMA /ηPS ≈ 2.2×103 at γ˙ = 54.5 s−1 , indicating that there could hardly have been any meaningful mixing between PMMA and PS at 160 ◦ C. This consideration can now explain why in Figure 3.6 a co-continuous morphology is

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observed in all three blend compositions, regardless of whether each blend was mixed for 5 or 30 min. Figure 3.7 gives TEM images of 70/30, 60/40, 55/45, 50/50, and 30/70 PMMA/PS blends, which were prepared at a rotor speed of 50 rpm at 200 ◦ C for 5 and 30 min of mixing. Note that the melt blending temperature employed, 200 ◦ C, is higher than the Tcf s of both PMMA and PS, and that at 200 ◦ C ηPMMA /ηPS ≈ 12 at γ˙ = 54.5 s−1 (see Figure 3.5). The following observations are worth noting in Figure 3.7. For 5 min mixing, the 70/30 PMMA/PS blend has a well-developed dispersed morphology in which PS forms the discrete phase and PMMA forms the continuous phase. Conversely, for the same duration of mixing, the 30/70 PMMA/PS blend has a well-developed dispersed morphology in which PMMA forms the discrete phase and PS forms the continuous phase. As mixing was extended to 30 min, the morphology of both 70/30 and 30/70 PMMA/PS blends remained more or less the same. When two immiscible liquids are mixed we expect, according to the minimum energy dissipation principle, that the more viscous component will form the discrete phase and the less viscous component will form the continuous phase. Accordingly, it is reasonable to expect that in both 70/30 and 30/70 PMMA/PS blends the PMMA would form the discrete phase and the PS would form the continuous phase, because the PMMA is more viscous than the PS (see Figure 3.5). However, from Figure 3.7 we observe that in the 70/30 PMMA/PS blend the major component, PMMA, though more viscous, forms the continuous phase (matrix) and the minor component, PS, forms the discrete phase (drops), contrary to the expectation from the minimum energy dissipation principle. Thus, we conclude that in the 70/30 PMMA/PS blend, the blend ratio plays a dominant role over the viscosity ratio in determining the state of dispersion. Interestingly, in Figure 3.7 we observe that in the 50/50 PMMA/PS blend the more viscous PMMA forms the discrete phase and the less viscous PS forms the continuous phase, suggesting that the viscosity ratio of the constituent components determines the state of dispersion for an equal blend composition. However, we observe a co-continuous morphology in the 60/40 PMMA/PS blend even after melt blending for 30 min at 200 ◦ C. In order to ascertain whether or not the co-continuous morphology observed in 60/40 PMMA/PS blend could be regarded as being an equilibrium morphology, the melt blending temperature was increased further from 200 to 220 ◦ C and to 240 ◦ C, and also the rotor speed was increased from 50 to 150 rpm. In Figure 3.8, we observe that increasing melt blending temperature from 200 to 220 ◦ C and increasing rotor speed from 50 to 150 rpm produced a well-developed dispersed morphology in 60/40 PMMA/PS blend, in which the minor component, PS, forms the discrete phase and the major component, PMMA, forms the continuous phase, in spite of the fact that the PMMA is more viscous than the PS. Also, increasing melt blending temperature from 220 to 240 ◦ C at a rotor speed of 50 rpm produced a well-developed dispersed morphology in the 60/40 PMMA/PS blend, in which the minor component PS forms the discrete phase and the major component PMMA forms the continuous phase. From the above observations, we can conclude that the co-continuous morphology observed in Figure 3.7 for the PMMA/PS blend is not an equilibrium morphology but a transitory morphological state before the blend transforms into a dispersed morphology, which can be achieved either by increasing melt blending temperature or by increasing the rotor speed (i.e., by increasing the intensity of mixing).

Figure 3.7 TEM images showing the effect of the duration of mixing (5 min versus 30 min) on

the morphology evolution in 70/30, 60/40, 55/45, 50/50, and 30/70 PMMA/PS blends during compounding at a rotor speed of 50 rpm and at 200 ◦ C in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

143

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Figure 3.8 TEM images showing the effect of the duration of mixing (5 min versus 30 min), rotor speed (50 rpm versus 150 rpm), and melt blending temperature (220 ◦ C versus 240 ◦ C)

on the morphology evolution in a 60/40 PMMA/PS blend during compounding in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

3.2.3

Morphology Evolution in Blends Consisting of an Amorphous Polymer and a Semicrystalline Polymer

In this section, we show morphology evolution in an amorphous polymer, PS, and a semicrystalline polymer, HDPE or PP, during melt blending in a Brabender

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Figure 3.9 (a) The shear-rate dependence of viscosity for HDPE (open symbols) and PS (ﬁlled symbols) at various temperatures (◦ C): (, 䊉) 160, (, ) 180, (, ) 200, and (, ) 220. (b) The temperature dependence of viscosity ratio, ηPS /ηHDPE , at γ˙ = 54.5 s−1 . (Reprinted

from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

internal mixer. Figure 3.9a gives log η versus log γ˙ plots for PS and HDPE at 160, 180, 200, and 220 ◦ C for γ˙ = 0.001−1,000 s−1 , showing that the viscosity of PS is higher at 160 and 180 ◦ C, but lower at 200 and 220 ◦ C, than that of HDPE over a wide range of γ˙ , and that both PS and HDPE exhibit strong shear-thinning behavior. The temperature dependence of viscosity ratio, ηPS /ηHDPE , at γ˙ = 54.5 s−1 is given in Figure 3.9b, showing that the ratio ηPS /ηHDPE decreases from about 1.2 at 160 ◦ C, passing through 1.0 at about 185 ◦ C and then decreases further to 0.4 at 240 ◦ C. This information will be very useful for interpreting the morphology evolution in PS/HDPE blends. Figure 3.10 gives SEM images of 30/70 and 70/30 PS/HDPE blends, which were prepared at a rotor speed of 50 rpm for 30 min of mixing at 150, 160, and 220 ◦ C, where the dark areas represent the PS phase and the bright areas represent the HDPE phase. The following observations are worth noting on the morphology evolution in the 30/70 PS/HDPE blend given in Figure 3.10. At a mixing temperature of 150 ◦ C, the HDPE (the major component) having the Tm of 125 ◦ C forms the continuous phase, because

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Figure 3.10 SEM images showing the effect of melt blending temperature (150, 160, and 220 ◦ C)

on the morphology evolution in 30/70 and 70/30 PS/HDPE blends during compounding at a rotor speed of 50 rpm for 30 min in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

it already melted, while the PS (the minor component) shows a very irregularly-shaped discrete phase, indicating that the softening of the PS was not complete. This can be interpreted as being a situation where the mixing temperature employed (150 ◦ C) is slightly below the Tcf (≈155 ◦ C) of PS. At a mixing temperature of 160 ◦ C, the discrete PS phase becomes quite different from that at 150 ◦ C, indicating that the discrete phase PS already began to ﬂow at 160 ◦ C. At a mixing temperature of 220 ◦ C, we observe a well-developed dispersed morphology. Note that at 220 ◦ C, the major component, HDPE, is more viscous than the minor component, PS (see Figure 3.9), indicating that the blend ratio is dominant over the viscosity ratio in determining morphology evolution in the 30/70 PS/HDPE blend.

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The following observations are worth noting on the morphology evolution in the 70/30 PS/HDPE blend given in Figure 3.10. At a mixing temperature of 150 ◦ C, we observe a co-continuous morphology in which interconnected structures of HDPE appear to be suspended in the yet unsoftend PS phase. The 70/30 PS/HDPE blend still shows a co-continuous morphology after mixing at 160 ◦ C for 30 min. As the mixing temperature is increased to 220 ◦ C, we observe a well-developed dispersed morphology in which the minor component, HDPE, having higher viscosity forms the discrete phase and the major component, PS, having lower viscosity forms the continuous phase. Figure 3.11 gives SEM images of 50/50 PS/HDPE blends, which were prepared at a rotor speed of 50 rpm for 30 min mixing at 150, 160, 180, 200, 220, and 240 ◦ C. It is clear that at 150 ◦ C there is hardly any mixing, but at 180 ◦ C we observe a dispersed

Figure 3.11 SEM images showing the effect of melt blending temperature (150, 160, 180, 200, 220, and 240 ◦ C) on the morphology evolution in a 50/50 PS/HDPE blend during compounding

at a rotor speed of 50 rpm for 30 min in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

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morphology in which the PS forms the discrete phase and the HDPE forms the continuous phase. Interestingly enough, at 200 ◦ C we observe a co-continuous morphology and at 240 ◦ C we observe a breakdown of interconnected structures of HDPE that existed at 220 ◦ C, giving rise to a dispersed morphology in which the discrete phase of HDPE, though not so well developed, is dispersed in the continuous PS phase. In other words, we observe a “phase inversion” taking place in the 50/50 PS/HDPE blend as the mixing temperature is increased from 180 to 240 ◦ C, passing through a co-continuous morphology at an intermediate temperature. We explain the occurrence of the phase inversion observed in the 50/50 PS/HDPE blend as follows. In reference to Figure 3.9, the viscosity of PS is higher than that of HDPE at temperatures below about 185 ◦ C, and thus the ratio ηPS /ηHDPE is slightly higher than 1 at 180 ◦ C, but it decreases rapidly with increasing temperature and ηPS /ηHDPE ≈ 0.4 at 240 ◦ C. The blend composition being equal in the 50/50 PS/HDPE blend, we conclude from Figure 3.11 that the viscosity ratio determines the state of dispersion in 50/50 PS/HDPE blend in that the more viscous HDPE forms the discrete phase and the less viscous PS forms the continuous phase at T ≥ 240 ◦ C. Figure 3.12 gives SEM images of 45/55 and 55/45 PS/HDPE blends, which were prepared at a rotor speed of 50 rpm for 5 min and 30 min of mixing at 220 ◦ C, showing

Figure 3.12 SEM images showing the effect of the duration of mixing (5 min versus 30 min) on

the morphology evolution in 45/55 and 55/45 PS/HDPE blends during compounding at a rotor speed of 50 rpm and at 220 ◦ C in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

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Figure 3.13 SEM images showing the effect of the duration of mixing (5 min versus 25 min) on

the morphology evolution in 45/55 and 55/45 PS/HDPE blends during compounding at a rotor speed of 50 rpm and at 240 ◦ C in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

that a co-continuous morphology persists even after 30 min of mixing in both blends. However, as can be seen in Figure 3.13, as the melt blending temperature is increased from 220 to 240 ◦ C and mixing is continued for 25 min, both 45/55 and 55/45 PS/HDPE blends form a dispersed morphology in which the major component forms the continuous phase and the minor component forms the discrete phase. A transformation from a co-continuous morphology into a dispersed morphology in 45/55 PS/HDPE is also realized, as can be seen in Figure 3.14, when the rotor speed is increased from 50 to 150 rpm (compare with the upper panels of Figure 3.12). In Figure 3.14, we observe a clear picture about the morphology evolution in 45/55 PS/HDPE as the duration of mixing is increased from 5 to 30 min, leading us to conclude that a co-continuous morphology is not an equilibrium morphology. Figure 3.15a gives log η versus log γ˙ plots for PS and PP at 190, 200, 220, and 240 ◦ C for γ˙ = 0.001−1,000 s−1 , showing that the viscosity of PS is higher at temperatures below about 184 ◦ C and lower at higher temperatures than that of PP over a wide range of γ˙ tested, and that both PS and PP exhibit strong shear-thinning behavior. Figure 3.15b shows the temperature dependence of viscosity ratio, ηPS /ηPP , at γ˙ = 54.5 s−1 . Note that the inequality Tcf ,PS < Tm,PP holds for PS/PP blends, while the inequality Tcf ,PS > Tm,HDPE holds for PS/HDPE blends.

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Figure 3.14 SEM images showing the effect of the duration of mixing (5, 10, 15, and 30 min)

on the morphology evolution in a 45/55 PS/HDPE blend during compounding at a rotor speed of 150 rpm and 220 ◦ C in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

SEM images of a 45/55 PS/PP blend are given in Figure 3.16 for a rotor speed of 10 rpm, in Figure 3.17 for a rotor speed of 50 rpm, in Figure 3.18 for a rotor speed of 100 rpm, and in Figure 3.19 for a rotor speed of 150 rpm. In Figures 3.16–3.19 we observe that a co-continuous morphology eventually transforms into a dispersed morphology when mixing is carried out for a sufﬁciently long period: (1) after 90 min mixing at a rotor speed of 10 rpm (Figure 3.16), (2) after 60 min mixing at a rotor speed of 50 rpm (Figure 3.17), (3) after 45 min mixing at a rotor speed of 100 rpm (Figure 3.18), and (4) after 30 min mixing at a rotor speed of 150 rpm (Figure 3.19). That is, the higher the rotor speed (i.e., the greater the intensity of mixing), the shorter the mixing period required for a transformation from a co-continuous morphology to a dispersed morphology to occur. Figure 3.20a gives variations of torque with mixing period during the compounding of the 45/55 PS/PP blend at 220 ◦ C at four different rotor speeds: 10, 50, 100, and 150 rpm. It can be seen in Figure 3.20a that the torque goes through a maximum within a few minutes after melt blending begins and then decreases rapidly, giving rise to a more or less constant value for the remaining period of mixing (30 min). It should be mentioned that the peak value of torque observed in Figure 3.20a is due to the initial mixing of two polymers in the solid state and thus the torque decreases rapidly once the melting or softening of the polymers begins. It is worth noting in Figure 3.20a that

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Figure 3.15 (a) The shear-rate dependence of viscosity for PS (open symbols) and PP (ﬁlled symbols) at various temperatures (◦ C): (, 䊉) 190, (, ) 200, (, ) 220, and (, ) 240. (b) The temperature dependence of viscosity ratio, ηPS /ηPP , at γ˙ = 54.5 s−1 . (Reprinted from

Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

the values of torque are relatively insensitive to the rotor speed (10–150 rpm) of a Brabender internal mixer. However, as can be seen in Figure 3.20b, the mechanical energy applied to the blend by the Brabender internal mixer is very sensitive to the rotor speed, that is, the mechanical energy increases with increasing rotor speed from 10 to 150 rpm: the higher the rotor speed of the mixing equipment, the greater is the mechanical energy applied to the blend. We should mention that values of the mechanical energy plotted in Figure 3.20b were read off from the Brabender mixer. Figure 3.20b seems to explain the experimental results given in Figures 3.16–3.19, that is, the reason why a shorter mixing period was required to transform a co-continuous morphology into a dispersed morphology as the rotor speed was increased from 10 to 150 rpm. It appears from Figure 3.20 that there exists a critical value of mechanical energy that is required to transform a co-continuous morphology into a dispersed morphology. The critical value of mechanical energy will depend on the characteristics of the pair of polymers being melt blended and the blend ratio.

Figure 3.16 SEM images showing the effect of the duration of mixing (5, 40, 60, and 90 min)

on the morphology evolution in a 45/55 PS/PP blend during compounding at a rotor speed of 10 rpm and at 220 ◦ C in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

Figure 3.17 SEM images showing the effect of the duration of mixing (5, 30, 45, and 60 min)

on the morphology evolution in a 45/55 PS/PP blend during compounding at a rotor speed of 50 rpm and at 220 ◦ C in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.) 152

Figure 3.18 SEM images showing the effect of the duration of mixing (5, 20, 30, and 45 min)

on the morphology evolution in a 45/55 PS/PP blend during compounding at a rotor speed of 100 rpm and at 220 ◦ C in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

Figure 3.19 SEM images showing the effect of the duration of mixing (5, 10, 15, and 30 min)

on the morphology evolution in a 45/55 PS/PP blend during compounding at a rotor speed of 150 rpm and at 220 ◦ C in a Brabender mixer. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.) 153

154

PROCESSING OF THERMOPLASTIC POLYMERS Figure 3.20 (a) The torque and (b) the

mechanical energy, recorded on a Brabender mixer during compounding of a 45/55 PS/PP blend at 220 ◦ C for various rotor speeds (rpm): () 10, () 50, () 100, and () 150. (Reprinted from Lee and Han, Polymer 40:6277. Copyright © 1999, with permission from Elsevier.)

3.3

Morphology Evolution in Immiscible Polymer Blend during Compounding in a Twin-Screw Extruder

A twin-screw extruder has two screws, which rotate either in the same direction (referred to as co-rotating screws) or in opposite directions (referred to as counterrotating screws), in contrast to a single-screw extruder. The twin screws consist of basically two sections: kneading elements and mixing chambers. Depending on the design, which will vary according to the speciﬁc application, a twin-screw extruder might have two to four kneading elements and three to ﬁve mixing chambers. Most of the melting occurs in the ﬁrst kneading element and molten polymers are mixed in the mixing chambers. Figure 3.21 shows (a) a model depicting the kneading elements of a twin-screw, (b) kneading elements and mixing chambers of a co-rotating modular twin screws, and (c) an entire section of a pair of modular co-rotating twin screws. The details of the principles for designing twin-screw extruders and their functions (Janssen 1978; Martelli 1983; White 1990) are beyond the scope of this chapter. Figure 3.22a gives a schematic of a twin-screw extruder and, for illustration purposes, Figure 3.22b shows schematically the morphology evolution in two immiscible crystalline polymers, polymer A and polymer B, where the melting point (Tm,A ) of

Figure 3.21 (a) A model depicting the kneading elements of a twin-screw extruder, (b) kneading

elements and mixing chamber of modular co-rotating twin screws, and (c) the entire section of a pair of modular co-rotating twin screws.

Figure 3.22 (a) Schematic of a twin-screw extruder, in which a pair of immiscible polymers

are extruded under a preset temperature proﬁle along the extruder axis. (b) Schematic of the reduction of dispersed droplets along the screw axis, where polymer A (the shaded areas) is assumed to have a lower melting point than polymer B (the dark areas). 155

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polymer A is assumed to be lower than the melting point (Tm,B ) of polymer B. Figure 3.22b shows the situation where a gradual reduction in drop size, without “phase inversion,” takes place as the two molten polymers move along the extruder axis. As will be shown in the following section, the morphology evolution in two immiscible polymers can be much more complicated than the very simple picture depicted in Figure 3.22b.

3.3.1

Morphology Evolution in Blends Consisting of Two Amorphous Polymers

Figure 3.23 gives TEM images showing the morphology evolution in a 70/30 PMMA/PS blend along the axis of a twin-screw extruder and a schematic showing barrel temperature proﬁle and positions where blend specimens were taken after screw pullout. In Figure 3.23, the dark areas represent the PS phase and the bright areas represent the PMMA phase. The following observations are worth noting on the morphology evolution displayed in Figure 3.23. At the front end of the ﬁrst kneading block (position A), we observe a co-continuous morphology. Although the barrel temperature at position A was set at 160 ◦ C, we are quite certain that owing to viscous shear heating the real temperature of the blend must have been higher than 160 ◦ C. In view of the fact that the Tcf of PS is 155 ◦ C and the Tcf of PMMA is 170 ◦ C, it is reasonable to speculate that the PMMA in the 70/30 PMMA/PS blend barely begins to ﬂow at position A. At the exit of the ﬁrst kneading block (position B), we observe some breakdown of interconnected structures of PMMA that are dispersed in the PS phase. At position C, where the barrel temperature was set at 200 ◦ C, we clearly observe a dispersed morphology in which the discrete PMMA phase is dispersed in the continuous PS phase. It is interesting to note that the PMMA phase, which is broken down at position B, apparently undergoes coalescence, forming a continuous phase, at position C; that is, a phase inversion takes place inside the extruder. This might have resulted from a slowdown of melt ﬂow at position C, which consists of screw elements forming mixing chambers, while mixing was very intense at the ﬁrst kneading block preceding position C. At the second kneading block (position D), we observe a ﬁbrillation of the discrete PS phase, giving rise to very long, threadlike drops. At position E, where the barrel temperature was set at 210–220 ◦ C, we observe, once again, evidence of drop coalescence which, as pointed out above, might have resulted from a slowdown of melt ﬂow in the mixing chamber past the second kneading block. At position G, past the third kneading block, we observe both breakup and coalescence of PS drops. Breakup of drops at position G (with the barrel temperature set at 240 ◦ C) is much easier than at position E (with the barrel temperature set at 210 ◦ C) because the viscosities of both PS and PMMA decrease with increasing temperature and the viscosity ratio, ηPMMA /ηPS is higher than 10 (see Figure 3.5). According to the literature (Grace 1982; Karam and Bellinger 1968; Torza et al. 1972), drop breakup becomes easier when the viscosity ratio of drop to matrix lies between 0.1 and 1. According to the minimum energy dissipation principle, it is reasonable to expect that in the 70/30 PMMA/PS mixture the PMMA forms the discrete phase and the PS forms the continuous phase. However, from Figure 3.23 we observe that the major component, PMMA, though more viscous, forms the continuous phase (matrix) and

Figure 3.23 The morphology evolution in a 70/30 PMMA/PS blend during compounding in a twin-screw extruder: (A) at the front end of the ﬁrst kneading block (160 ◦ C), (B) at the exit of the ﬁrst kneading block (160 ◦ C), (C) between the ﬁrst and second kneading blocks (200 ◦ C), (D) at the front end of the second kneading block (200 ◦ C), (E) between the second and third kneading blocks (220 ◦ C), (F) at the exit of the third kneading block (230 ◦ C), (G) between the third kneading block and the die (240 ◦ C), and (H) extrudate. (Reprinted from Lee and Han, Polymer 41:1799. Copyright © 2000, with permission from Elsevier.)

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the minor component, PS, forms the discrete phase (drops), contrary to the expectation from the minimum energy dissipation principle. Thus, we conclude that in the 70/30 PMMA/PS blend, the blend composition ratio plays a predominant role over the viscosity ratio in determining the state of dispersion. This conclusion is consistent with that made with reference to Figure 3.7. Figure 3.24 gives TEM images showing the morphology evolution in a 50/50 PMMA/PS blend along the axis of a twin-screw extruder, and a schematic showing the barrel temperature proﬁle and positions where blend specimens were taken after screw pullout. The following observations are worth noting on the morphology evolution displayed in Figure 3.24. At the front end of the ﬁrst kneading block (position A), where the barrel temperature was set at 160 ◦ C, we observe a mixture of small drops and very large domains of PMMA suspended in the PS matrix. Note that at 160 ◦ C (although the actual melt temperature was probably higher), which is slightly lower than the Tcf of PMMA, the mobility of the PMMA in the 50/50 PMMA/PS blend would have been extremely low. Note in Figure 3.5 that at 160 ◦ C the viscosity of the PMMA is about 2,000 times that of the PS. Therefore, adequate mixing between the PMMA and PS in the 50/50 PS/PMMA blend at 160 ◦ C would have been very difﬁcult. At the exit of the ﬁrst kneading block (position B) we observe a breakdown of large PMMA domains, though some are still present, dispersed in the PS phase. At position C, where the barrel temperature was set at 200 ◦ C, we observe coalescence of PMMA drops, the mechanism for which was presented above. In the second kneading block (position D), we observe an elongation of PMMA drops, and at position E, where the barrel temperature is set at 210–220 ◦ C, we observe evidence of drop coalescence. At position G, where the barrel temperature is set at 240 ◦ C, we observe a dispersed morphology, which becomes much clearer in the extrudate (position H). In the 50/50 PMMA/PS blend, we observe that the more viscous PMMA forms the discrete phase and the less viscous PS forms the continuous phase. Thus, we conclude that in this blend composition, the viscosity ratio played a dominant role over the blend ratio in determining the state of dispersion. This conclusion is consistent with that made with reference to Figure 3.7. Figure 3.25 gives TEM images showing the morphology evolution in a 30/70 PMMA/PS blend along the axis of a twin-screw extruder, and a schematic showing barrel temperature proﬁle and positions where blend specimens were taken after screw pullout. The following observations are worth noting on the morphology evolution displayed in Figure 3.25. At the front end of the ﬁrst kneading block (position A), we observe a mixture of small drops (the bright areas) and very large domains suspended in the PS matrix (the dark areas). Being the minor component in the 30/70 PMMA/PS blend, it might have been easier for the PMMA here (as compared with the PMMA in the other blend compositions considered) to form the discrete phase in the environment of the less viscous PS phase, which forms the continuous phase. At the exit of the ﬁrst kneading block (position B), we observe a breakdown of large PMMA domains as well as coalescence of very small PMMA drops dispersed in the PS matrix. Along the remainder of the extruder axis, we observe a dispersed morphology in which PMMA drops are dispersed in the PS matrix. This observation is consistent with the expectation from the minimum energy dissipation principle, stating that the less viscous component would form a continuous phase and the more viscous component would form a discrete phase. After all, the PMMA is the minor component in the 30/70 PMMA/PS blend.

Figure 3.24 The morphology evolution in a 50/50 PMMA/PS blend during compounding in a twin-screw extruder: (A) at the front end of the ﬁrst kneading block (160 ◦ C), (B) at the exit of the ﬁrst kneading block (160 ◦ C), (C) between the ﬁrst and second kneading blocks (200 ◦ C), (D) at the front end of the second kneading block (200 ◦ C), (E) between the second and third kneading blocks (220 ◦ C), (F) at the exit of the third kneading block (230 ◦ C), (G) between the third kneading block and the die (240 ◦ C), and (H) extrudate. (Reprinted from Lee and Han, Polymer 41:1799. Copyright © 2000, with permission from Elsevier.)

Figure 3.25 The morphology evolution in a 30/70 PMMA/PS blend during compounding in a twin-screw extruder: (A) at the front end of the ﬁrst kneading block (160 ◦ C), (B) at the exit of the ﬁrst kneading block (160 ◦ C), (C) between the ﬁrst and second kneading blocks (200 ◦ C), (D) at the front end of the second kneading block (200 ◦ C), (E) between the second and third kneading blocks (220 ◦ C), (F) at exit of the third kneading block (230 ◦ C), (G) between the third kneading block and the die (240 ◦ C), and (H) extrudate. (Reprinted from Lee and Han, Polymer 41:1799. Copyright

© 2000, with permission from Elsevier.)

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3.3.2

161

Morphology Evolution in Blends Consisting of an Amorphous Polymer and a Semicrystalline Polymer

Figure 3.26 gives SEM images showing the morphology evolution in a 70/30 PS/HDPE blend along the axis of a twin-screw extruder, and a schematic showing barrel temperature proﬁle and positions where blend specimens were taken after screw pullout. The following observations are worth noting on the morphology evolution displayed in Figure 3.26. At the front end of the ﬁrst kneading block (position A), where the barrel temperature was set at 160 ◦ C, we observe a poorly developed morphology which, again, is due to the rather low barrel temperature. However, at the exit of the ﬁrst kneading block (position B), we clearly observe a dispersed morphology in which the PS drops (the dark areas) are dispersed in the HDPE matrix (the bright areas). The same dispersed morphology persists at position C, where the barrel temperature is set at 200 ◦ C, and at the second kneading block (position D). However, at position E, where the barrel temperature is set at 210–220 ◦ C, we observe a co-continuous morphology, and at the third kneading block (position F), where the barrel temperature is set at 230 ◦ C, we observe some breakdown of interconnected structures. Finally, at position G, where the barrel temperature is set at 240 ◦ C, we observe a dispersed morphology in which the more viscous, minor component HDPE forms drops and the less viscous, major component PS forms the continuous phase. We conclude that a phase inversion took place inside the extruder, while the 70/30 PS/HDPE blend was melt blended. Figure 3.27 gives SEM images showing the morphology evolution in a 50/50 PS/HDPE blend along the axis of a twin-screw extruder, and a schematic showing barrel temperature proﬁle and positions where blend specimens were taken after screw pullout. The following observations are worth noting on the morphology evolution displayed in Figure 3.27. At the ﬁrst kneading block (position A), where the barrel temperature is set at 160 ◦ C, we observe a dispersed morphology in which the discrete phase of PS having irregular shapes is dispersed in the continuous phase of HDPE. Owing to the low barrel temperature of 160 ◦ C, which is slightly above the Tcf (≈155 ◦ C) of PS, the state of dispersion is rather poor. Interestingly enough, however, at position C, where the barrel temperature is set at 200 ◦ C, we observe a co-continuous morphology with interconnected structures of PS and HDPE. At the second and third kneading blocks (positions E and F), we still observe a co-continuous morphology. The same morphology persists along the rest of the extruder axis. However, in the extrudate (position H), we observe a dispersed morphology in which the HDPE forms drops dispersed in the PS matrix. It is of great interest to observe in Figure 3.27 that a phase inversion took place from one mode of dispersed morphology to another inside the extruder, while the 50/50 PS/HDPE blend was extruded. Figure 3.28 gives SEM images showing the morphology evolution in a 30/70 PS/HDPE blend along the extruder axis, and a schematic showing barrel temperature proﬁle and positions where blend specimens were taken after screw pullout. The following observations are worth noting on the morphology evolution displayed in Figure 3.28. At the ﬁrst kneading block (positions A and B), where the barrel temperature was set at 160 ◦ C, we observe a dispersed morphology in which PS domains are dispersed in the HDPE matrix. It should be remembered that before reaching the Tcf of PS, the HDPE had already melted and formed the continuous phase. At position C,

Figure 3.26 The morphology evolution in a 70/30 PS/HDPE blend during compounding in a twin-screw extruder: (A) at the front end of the ﬁrst kneading block (160 ◦ C), (B) at the exit of the ﬁrst kneading block (160 ◦ C), (C) between the ﬁrst and second kneading blocks (200 ◦ C), (D) at the front end of the second kneading block (210 ◦ C), (E) between the second and third kneading blocks (220 ◦ C), (F) at the exit of the third kneading block (230 ◦ C), (G) between the third kneading block and the die (240 ◦ C), and (H) extrudate. (Reprinted from Lee and Han, Polymer 41:1799. Copyright © 2000, with permission from Elsevier.)

Figure 3.27 The morphology evolution in a 50/50 PS/HDPE blend during compounding in a twin-screw extruder: (A) at the front end of the ﬁrst kneading block (160 ◦ C), (B) at the exit of the ﬁrst kneading block (160 ◦ C), (C) between the ﬁrst and second kneading blocks (200 ◦ C), (D) at the front end of the second kneading block (210 ◦ C), (E) between the second and third kneading blocks (220 ◦ C), (F) at the exit of the third kneading block ◦ (230 ◦

C), (G) between the third kneading block and the die (240 C), and (H) extrudate. (Reprinted from Lee and Han, Polymer 41:1799. Copyright © 2000, with permission from Elsevier.)

Figure 3.28 The morphology evolution in a 30/70 PS/HDPE blend during compounding in a twin-screw extruder: (A) at the front end of the ﬁrst kneading block (160 ◦ C), (B) at the exit of the ﬁrst kneading block (160 ◦ C), (C) between the ﬁrst and second kneading blocks (200 ◦ C), (D) at the front end of the second kneading block (210 ◦ C), (E) between the second and third kneading blocks (220 ◦ C), (F) at the exit of the third kneading block (230 ◦ C), (G) between the third kneading block and the die (240 ◦ C), and (H) extrudate. (Reprinted from Lee and Han, Polymer 41:1799. Copyright © 2000, with permission from Elsevier.)

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where the barrel temperature is set at 200 ◦ C, we observe a slightly improved blend morphology, and in the second kneading block (position D), we clearly observe a much improved dispersed morphology in which PS drops are dispersed in the HDPE matrix. Along the remainder of the extruder axis, we observe little change in blend morphology. Based on the above observations, we conclude that in the 30/70 PS/HDPE blend, blend ratio determined the state of dispersion. Figure 3.29 gives SEM images showing the morphology evolution in a 70/30 PS/PP blend along the axis of a twin-screw extruder, and a schematic showing barrel temperature proﬁle and positions where blend specimens were taken after screw pullout. The following observations are worth noting on the morphology evolution displayed in Figure 3.29. In the ﬁrst kneading block (positions A and B), where the barrel temperature was set at 160 ◦ C, we observe many interconnected thin strands of PP (the bright areas) dispersed in the PS matrix (the dark areas). At position C, where the barrel temperature is set at 200 ◦ C, we observe a mixture of a co-continuous morphology and a dispersed morphology. However, in the second kneading block (position D), we clearly observe a co-continuous morphology. At position E, where the barrel temperature is set at 210–220 ◦ C, we observe a breakdown of interconnected structures of PP, yielding broken drops of irregular shape dispersed in the PS matrix. Interestingly, in the third kneading block (position F), where the barrel temperature is set at 230 ◦ C, we observe elongated PP drops dispersed in the PS matrix, and the drops become spherical at position G, where the barrel temperature is set at 240 ◦ C. The extrudate (position H) has a dispersed morphology in which PP drops are dispersed in the PS matrix. Again, we observe that the more viscous, minor component, PP, forms drops and the less viscous, major component, PS, forms the continuous phase. Figure 3.30 gives SEM images showing the morphology evolution in a 50/50 PS/PP blend along the axis of a twin-screw extruder, and a schematic showing barrel temperature proﬁle and positions where blend specimens were taken after screw pullout. The following observations are worth noting on the morphology evolution displayed in Figure 3.30. At the front end of the ﬁrst kneading block (position A), where the barrel temperature was set at 160 ◦ C, we observe a very poorly developed blend morphology, which is attributed to the low barrel temperature. However, at the exit of the ﬁrst kneading block (position B), we observe a dispersed morphology in which the discrete phase of PS is dispersed in the PP matrix. We observe a dispersed morphology at position C, where the barrel temperature is set at 200 ◦ C, and at the second kneading block (position D). At position E, where the barrel temperature is set at 210–220 ◦ C, we begin to observe a co-continuous morphology with interconnected structures of PS and PP. The same morphology persists along the rest of the extruder axis. However, as can be seen in Figure 3.31, when the barrel temperature is increased to 220 ◦ C at position C, to 250–260 ◦ C at position E, and to 260 ◦ C at positions F and G, before the 50/50 PS/PP blend leaves the extruder we observe a transformation taking place from a co-continuous morphology to a dispersed morphology, in which the more viscous component, PP, forms the discrete phase dispersed in the less viscous PS (see Figure 3.15 for the ηPS /ηPP ratio as a function of temperature). The above observation leads us to conclude that for the symmetric blend ratio, the viscosity ratio plays the dominant role in determining the mode of dispersion, consistent with the observations made above for the PMMA/PS and PS/HDPE blends. Comparison of Figure 3.30 with Figure 3.31 demonstrates clearly that under the right processing conditions,

Figure 3.29 The morphology evolution in a 70/30 PS/PP blend during compounding in a twin-screw extruder: (A) at the front end of the ﬁrst kneading block (160 ◦ C), (B) at the exit of the ﬁrst kneading block (160 ◦ C), (C) between the ﬁrst and second kneading blocks (200 ◦ C), (D) at the front end of the second kneading block (210 ◦ C), (E) between the second and third kneading blocks (220 ◦ C), (F) at the exit of the third kneading block (230 ◦ C), (G) between the third kneading block and the die (240 ◦ C), and (H) extrudate. (Reprinted from Lee and Han, Polymer 41:1799. Copyright © 2000, with permission from Elsevier.)

Figure 3.30 The morphology evolution in a 50/50 PS/PP blend during compounding in a twin-screw extruder: (A) at the front end of the ﬁrst kneading block (160 ◦ C), (B) at the exit of the ﬁrst kneading block (160 ◦ C), (C) between the ﬁrst and second kneading blocks (200 ◦ C), (D) at the front end of the second kneading block (210 ◦ C), (E) between the second and third kneading blocks (220 ◦ C); (F) at the exit of the third kneading block (230 ◦ C), (G) between the third kneading block and the die (240 ◦ C), and (H) extrudate. (Reprinted from Lee and Han, Polymer 41:1799. Copyright © 2000, with permission from Elsevier.)

Figure 3.31 The morphology evolution in a 50/50 PS/PP blend during compounding in a twin-screw extruder: (A) at the front end of the ﬁrst kneading block (200 ◦ C), (B) at the exit of the ﬁrst kneading block (200 ◦ C), (C) between the ﬁrst and second kneading blocks (220 ◦ C), (D) at the front end of the second kneading block (250 ◦ C), (E) between the second and third kneading blocks (260 ◦ C), (F) at the exit of the third kneading block (260 ◦ C), (G) between the third kneading block and the die (260 ◦ C), and (H) extrudate. (Reprinted from Lee and Han, Polymer 41:1799. Copyright © 2000, with permission from Elsevier.)

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a co-continuous morphology can be transformed into a dispersed morphology, suggesting that a co-continuous morphology is not a stable morphology.

3.4

Stability of Co-Continuous Morphology during Compounding

There have been several reports in the literature on the stability of co-continuous morphology. Speciﬁcally, Miles and Zurek (1988) ﬁrst observed that in a 53/47 PS/PMMA blend that had been melt blended in a Brabender internal mixer at 200 ◦ C at a rotor speed of 20 rpm (equivalent to a shear rate of about 90 s−1 ), the co-continuous morphology transformed into a dispersed morphology when the melt-blended specimen was extruded through a capillary rheometer at a shear rate of 27 s−1 . In investigating the formation of a co-continuous morphology in nylon 6/poly(ethersulfone) and poly(butylene terephthalate)/polystyrene blend systems, He et al. (1997) observed that the range of blend ratios over which a co-continuous morphology formed became narrower as the duration of mixing was increased. They then speculated that after a sufﬁciently long period of mixing a co-continuous morphology would have disappeared. Lee and Han (1999b; 2000) reached essentially the same conclusion when investigating the morphology evolution in several different blend systems in a Brabender internal mixer and in a twin-screw extruder, the results of which are summarized in the previous sections of this chapter. Namely, a co-continuous morphology may not be an equilibrium structure and it can be made to disappear at a sufﬁciently high melt blending temperature or after a sufﬁciently long period of mixing. We will now examine in depth the stability of co-continuous morphology from the point of view of the kinetics of phase separation. Figure 3.32 gives TEM images of rapidly precipitated PMMA/PS blends, in which the bright areas represent the PMMA phase and the dark areas represent the PS phase. In Figure 3.32 we observe a modulated co-continuous morphology in all three blend compositions. This can easily be understood when we recognize the fact that, thermodynamically speaking, the phase separation mechanism involved with rapid precipitation (via composition quenching) is very similar to that involved with spinodal decomposition (via temperature quenching). In accordance with the Cahn theory of spinodal decomposition (Cahn 1961, 1968), the modulated co-continuous morphology observed in Figure 3.32 may be regarded as being formed by the superposition of the sine waves of thermal composition ﬂuctuations. Figure 3.33 shows a schematic representation of a three-dimensional, periodically modulated co-continuous structure, having the domain spacing Λ(t), for PMMA/PS blends obtained by rapid precipitation. From the measurements of the distance between the centers of two adjacent modulated structures, Lee and Han (1999a) calculated Λ(t) for two PMMA/PS blend systems having different molecular weights using the TEM images obtained for the respective blends, and the results are summarized in Figure 3.34. It is seen in Figure 3.34 that the values of Λ(t) for equal blend compositions (50/50 PMMA/PS blends) are larger than those for the unequal blend compositions (30/70 and 70/30 PMMA/PS blends). This can be explained using Cahn’s linearized theory (see Appendix). Figure 3.35 gives TEM images showing how, during isothermal annealing at 170 ◦ C, the morphology of a rapidly precipitated 70/30 PMMA/PS blend specimen evolved with

Figure 3.32 TEM images of as-precipitated PMMA/PS blends: (a) 70/30 PMMA/PS, (b) 50/50 PMMA/PS, and (c) 30/70 PMMA/PS. The images show co-continuous morphology irrespective of blend composition. (Reprinted from Lee and Han, Polymer 40:2521. Copyright © 1999, with permission from Elsevier.) Figure 3.33 Three-dimensional

schematic representation of the periodically modulated co-continuous structure, formed via spinodal decomposition, and wavelength Λ(t) of PMMA/PS blends. (Reprinted from Lee and Han, Polymer 40:2521. Copyright © 1999, with permission from Elsevier.)

170

Figure 3.34 Plots of Λ(t) versus blend composition for PMMA/PS blends, formed via spinodal decomposition, having different molecular weights: () Mw = 7.9 × 104 and Mw /Mn = 1.56 for PMMA and Mw = 2.1 × 105 and Mw /Mn = 2.0 for PS, and () Mw = 4.1 × 104 and Mw /Mn = 2.2 for PMMA and Mw = 1.1 × 105 and Mw /Mn = 1.09 for PS. (Reprinted

from Lee and Han, Polymer 40:2521. Copyright © 1999, with permission from Elsevier.)

Figure 3.35 TEM images of a rapidly precipitated 70/30 PMMA/PS blend after being annealed at 170 ◦ C for: (a) 5 min, (b) 15 min, (c) 30 min, and (d) 2 h. (Reprinted from Lee and Han,

Polymer 40:2521. Copyright © 1999, with permission from Elsevier.) 171

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time, from 5 min to 2 h. In Figure 3.35, we observe that (1) annealing at 170 ◦ C for 5 min was not long enough for us to observe a discernible change in blend morphology, (2) prolonging the annealing period from 5 to 15 min enables us to observe that a modulated co-continuous morphology (see Figure 3.32a) evolved into a dispersed morphology, and (3) prolonging the annealing period to 30 min and to 2 h helps achieve a well-developed dispersed morphology. Notice in Figure 3.35 that the minor component, PS (the dark areas), forms the discrete phase and the major component, PMMA (the bright areas), forms the continuous phase. Figure 3.36 gives TEM images showing how, during isothermal annealing at 170 ◦ C, the morphology of a rapidly precipitated 30/70 PMMA/PS blend specimen evolved with time from 30 min to 6 h. In Figure 3.36a, we observe clearly that the modulated co-continuous morphology of 30/70 PMMA/PS blend (Figure 3.32c), which was formed upon rapid precipitation of a homogeneous solution, did not evolve into a welldeveloped dispersed morphology after annealing for 30 min at 170 ◦ C, in contrast to the situations with 70/30 PMMA/PS blend (Figure 3.35c). This observation indicates that the rate of morphology development is much slower in 30/70 PMMA/PS blend than in 70/30 PMMA/PS blend. As annealing continues to 6 h, in Figure 3.36c we observe that the minor component, PMMA (the bright areas), forms the discrete phase and the major component, PS (the dark areas), forms the continuous phase. Figure 3.37 gives TEM images showing how, during isothermal annealing at 170 ◦ C, the morphology of a rapidly precipitated 50/50 PMMA/PS blend specimen evolved with

Figure 3.36 TEM images of a rapidly precipitated 30/70 PMMA/PS blend after being annealed at 170 ◦ C for (a) 30 min, (b) 2 h, and (c) 6 h. (Reprinted from Lee and Han, Polymer 40:2521. Copyright © 1999, with permission from Elsevier.)

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Figure 3.37 TEM images of a rapidly precipitated 50/50 PMMA/PS blend after being annealed at 170 ◦ C for (a) 30 min, (b) 2 h, (c) 6 h, and (d) 12 h. (Reprinted from Lee and Han, Polymer 40:2521. Copyright © 1999, with permission from Elsevier.)

time from 30 min to 12 h. It is not clear in Figure 3.37a how the morphology evolved during the annealing period of 30 min. As annealing continued to 2 h, as shown in Figure 3.37b, the morphology of 50/50 PMMA/PS blend evolved into more or less a “dual mode” of dispersed morphology, that is, there is one region where PS forms the discrete phase dispersed in the PMMA matrix and there is another region where PMMA forms the discrete phase dispersed in the PS matrix. In Figure 3.37d, when annealing continued to 12 h at 170 ◦ C, we observe a well-developed dual mode of dispersed morphology which started from a modulated co-continuous morphology (see Figure 3.32b). A number of research groups (Inoue et al. 1985; McMaster 1975; Snyder et al. 1983; Strobl 1985) investigated, via light scattering, the kinetics of phase separation of binary polymer blends and reported that Cahn’s linearized theory describes well the initial stage of spinodal decomposition. Hashimoto and coworkers (Hashimoto et al. 1986, 1992; Nakai et al. 1996; Takenaka et al. 1989, 1990, 1992; Takeno and Hashimoto 1997) conducted, via time-resolved light scattering, an extensive investigation of late stages of spinodal decomposition of polymer blends, which were obtained by either rapid quenching or solvent casting. The time evolution of the rapidly precipitated blend morphology during isothermal annealing, as presented in Figure 3.35–3.37, can be regarded as being equivalent to late stages of spinodal decomposition. A theoretical investigation of the late stages of spinodal decomposition was carried out by Siggia (1979), who showed that the hydrodynamic effects are very important, and that the domain size (d) grows according to two mechanisms: (1) d ∝ (kB T/η0 )1/3 t 1/3

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in the early stage, which is governed by hydrodynamic effects and diffusion, where kB T represents thermal energy with kB being the Boltzmann constant and η0 is the zero-shear viscosity, and (2) d ∝ (γ /η0 )t in the long-term coarsening, which is driven by the interfacial tension γ at a rate controlled by the viscosity η0 . Using Siggia’s analysis (1979), Lee and Han (1999a) explained their experimental results (presented in Figures 3.35–3.37), namely, the time evolution of a modulated co-continuous morphology into a dispersed morphology during isothermal annealing at 170 ◦ C. Although there was no bulk ﬂow in the rapidly precipitated blend specimen during isothermal annealing, according to Siggia’s analysis, the zero-shear viscosity must have played an important role in controlling the growth rate of domain d(t). Note that the diffusion coefﬁcient (Dc ) (or mobility) is related to the η0 , which in turn depends on the molecular weight (M) of the constituent components. Thus, we have d(t) ∝ Dc ∝

γ γ ∝ 3.4 η0 M

(3.1)

in which the polymers under consideration are assumed to be entangled. Referring to the PMMA/PS blends (Figures 3.35–3.37) at 170 ◦ C, we have the following relationships, η0,PMMA /η0,PS = 11.3 with η0,PMMA = 1.30 × 105 Pa·s and η0,PS = 1.15 × 104 Pa·s. We can now explain why the rate of evolution of the dispersed morphology in a 70/30 PMMA/PS blend (see Figure 3.35), in which the minor component, PS, forms the discrete phase dispersed in the major component, PMMA, is faster than that in 30/70 PMMA/PS blend (see Figure 3.36), in which the minor component, PMMA, forms the discrete phase dispersed in the major component, PS. Having explained above that the zero-shear viscosity played an important role in determining the state of dispersion during isothermal annealing of a rapidly precipitated PMMA/PS blend, let us revisit the state of dispersion in asymmetric PMMA/PS blend compositions. Speciﬁcally, in Figure 3.35 we observe that the major component, PMMA, forms the continuous phase and the minor component, PS, forms the discrete phase, and in Figure 3.36 we observe that the major component, PS, forms the continuous phase and the minor component, PMMA, forms the discrete phase, regardless of the viscosity ratio of the constituent components. The above observation leads us to conclude that the blend composition played a dominant role over the viscosity ratio in determining the state of dispersion during isothermal annealing of a rapidly precipitated PMMA/PS blend. From the point of view of the minimum energy dissipation principle, which is applicable to bulk ﬂow of a two-phase liquid, we expect that the less viscous component will form the continuous phase and the more viscous component will form the discrete phase. Having realized the fact that isothermal annealing does not involve bulk ﬂow, we can easily surmise that the minimum energy dissipation principle would not be applicable to the situation where a blend specimen was subjected to isothermal annealing under quiescent condition.

3.5

Summary

In this chapter, we have presented experimental observations of morphology evolution in immiscible polymer blends during compounding in an internal mixer or in a

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twin-screw extruder. Emphasis was placed on a better understanding of morphology evolution during compounding in relation to the rheological properties of the constituent polymers, blend composition, and processing conditions. We have pointed out that although from a thermodynamic point of view an amorphous polymer may be regarded as “liquid” at temperatures above its Tg , from a rheological point of view an amorphous polymer may be regarded as “liquid” when the temperature is approximately 50 ◦ C above its Tg . Such a temperature, termed “critical ﬂow temperature,” has helped us to understand the morphology evolution of polymer blends consisting of two amorphous polymers (e.g., PMMA and PS) or consisting of an amorphous polymer and a semicrystalline polymer (e.g., PS and HDPE, or PS and PP). We have shown that when two immiscible polymers are compounded in a twin-screw extruder, an equilibrium morphology of the binary blends depends, among many factors, on (1) the melt blending temperature relative to the Tm of a semicrystalline polymer and the Tcf of amorphous polymer, (2) the screw speed (the intensity of mixing, thus the residence time), (3) duration of mixing (the number of kneading elements and mixing chamber), (4) the viscosity ratio of the constituent components, and (5) the blend composition. We have demonstrated that a co-continuous morphology may be formed, irrespective of blend composition, when the melt blending temperature is lower than the Tcf of the amorphous constituent component(s). However, a co-continuous morphology may be transformed into a dispersed morphology when the melt blending temperature is much higher than the Tcf . In such a situation, the mode of dispersed blend morphology depends on blend composition and viscosity ratio, provided that a sufﬁciently long mixing period is allowed. When the duration of mixing is not sufﬁciently long, one may end up with a co-continuous morphology, which certainly is not an equilibrium morphology. We have shown further that when a sufﬁciently long period is allowed for melt blending or a proper processing condition (e.g., sufﬁciently high temperature) is chosen, a co-continuous morphology may be transformed into a dispersed morphology. The experimental results presented in this chapter may be summarized as given schematically in Figure 3.38, where an immiscible blend consisting of two semicrystalline polymers is considered. When dealing with an immiscible blend consisting of two amorphous polymers or consisting of an amorphous polymer and a semicrystalline polymer, Tm in Figure 3.38 should be replaced by Tcf for the amorphous polymer(s). With reference to Figure 3.38, the Tm (or Tcf ) of the constituent components plays an important role in the morphology evolution in an immiscible polymer blend. When the melt blending temperature T lies between the Tm or Tcf of the constituent components (say Tm,A < T < Tm,B or Tcf ,A < T < Tcf ,B ), component A ﬁrst forms the matrix phase in which component B, still in the solid state, is suspended, forming a suspension. When T > Tm,B > Tm,A or T > Tcf ,B > Tcf ,A , initially a dispersed two-phase liquid forms having drops of component B dispersed in the matrix phase of component A. At this temperature, if the viscosity of component B is lower than that of component A (i.e., ηB < ηA ) and/or component B is the major component (φB > φA ), phase inversion may take place, giving rise to the matrix phase of component B and the discrete phase of component A. When phase inversion takes place, the two-phase mixture must go through the transitory morphological state, a co-continuous phase; that is, a co-continuous morphology is a transitory morphological state between two stable modes of dispersed morphology: (1) morphology I, in which component B forms the

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Figure 3.38 Schematic showing the evolution of blend morphology during compounding of two

immiscible polymers in an internal mixer or in a twin-screw extruder, where the melting point of polymer A is assumed to be lower than that of polymer B. (Reprinted from Lee and Han, Polymer 41:1799. Copyright © 2000, with permission from Elsevier.)

discrete phase dispersed in component A, and (2) morphology II, in which component A forms the discrete phase dispersed in component B. Breakup and coalescence of the discrete phase may take place under certain processing conditions during the compounding of two immiscible polymers. The geometrical conﬁgurations of mixing equipment would play an important role in the extent of drop breakup and/or coalescence. Speciﬁcally, drop breakup may occur more readily in the converging section of a ﬂow channel, where an extensional ﬂow is dominant (Chin and Han 1980), as discussed in Chapter 11 of Volume 1. Conversely, coalescence of drops may occur more readily in the diverging section of a ﬂow channel, where stresses relax and thus elongated drops recoil. Coalescence is a physical phenomenon that is associated with a kinetic process (Elmendorp and van der Vegt 1986; Fortenly and Zivny 1995; Roland and Böhm 1984). Theoretical treatment of coalescence of the discrete phase (drops) during compounding of two immiscible polymers has been discussed little in the literature and so requires greater attention in the future. From the point of view of the minimum energy dissipation principle in channel ﬂow of two immiscible liquids, the component having the lower viscosity is expected to form the continuous phase, wetting the channel wall where the shear stress is greatest. Following this line of logic, we expect that in the 70/30 PMMA/PS blend

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considered in Figure 3.7, for example, the more viscous component, PMMA, would form the discrete phase and the less viscous PS would form the continuous phase. But, the experimental results given in Figure 3.7 show the opposite in that PMMA forms the continuous phase and PS forms the discrete phase. The experimental results presented in this chapter indicate that for asymmetric blend compositions, regardless of viscosity ratio of the constituent components, blend ratio determines the state of dispersion, that is, the minor component forms the discrete phase and the major component forms the continuous phase. In this chapter, we have not discussed the role, if any, of ﬂuid elasticity in the development of morphology when two immiscible polymers are melt blended. Van Oene (1972) advanced a theory suggesting that the second normal stress difference (N2 ) may play an important role in determining the morphology of immiscible polymer blends. Since then, however, little experimental evidence has been reported to either support or question van Oene’s argument because measurement of N2 for polymer melts is difﬁcult, especially at high shear rates. We emphasize that great challenges are ahead of us in the development of a comprehensive theory, combined with sophisticated computational methods, that will enable us to predict morphology evolution in immiscible polymer blends in commercial operations. In the past, some efforts were spent on modeling the ﬂow of two immiscible polymers in a twin-screw extruder using a single-phase ﬂow approach. Such efforts are meaningless without considering the two-phase nature of the ﬂow, that is, without considering morphology evolution in a twin-screw extruder, because the bulk viscosity of the blend depends, among many factors, on blend morphology. As alluded to in Chapter 11 of Volume 1, any effort to model the morphology evolution in immiscible polymer blends must deal with two-phase ﬂow, which cannot and should not be substituted by single-phase ﬂow. That is to say, any theoretical attempt to describe the ﬂow of two immiscible polymers in an internal mixer or in a twin-screw extruder must include morphology evolution during compounding, suggesting that the mixing of two immiscible polymers with proper moving boundary conditions at the phase interface must be included. In Chapter 11 of Volume 1 we discussed the deformation of a single drop in the entrance region of a cylindrical tube. Such an approach can be extended to many drops in complex ﬂow geometry in order to predict the shapes of many drops. In addition to the deformation of drops, breakup and coalescence of drops must also be included in the modeling effort. Further, heat transfer must be included in the formulation of system equations, because the compounding of two immiscible polymers in a twin-screw extruder is accompanied by a gradual increase in temperature along the extruder axis.

Appendix: Theoretical Interpretation of Figure 3.34 The time evolution of the phase-separated structures of polymer mixtures obtained by spinodal decomposition may be divided into three stages: (1) the early stage, (2) the late stage, and (3) the ﬁnal stage. The early stage of spinodal decomposition was interpreted using Cahn’s theory (Cahn 1961, 1968), which enables one to obtain the following expression for the composition φ A in an inhomogeneous mixture consisting

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PROCESSING OF THERMOPLASTIC POLYMERS

of components A and B: 2 ∂ f (φA ) 2 ∂φA 4 =M ∇ φA − 2κ∇ φA ∂t ∂φA 2

(3A.1)

where f(φA ) is the free energy density of homogeneous material of composition φA , κ is the gradient energy coefﬁcient arising from the effects of localized composition gradient, and M is the mobility deﬁned by the ratio of diffusion ﬂux (JB or JA ) to the gradient of chemical potential, µA − µB : JB = −JA = M∇(µA − µB )

(3A.2)

Equation (3A.1) is very similar to the conventional diffusion equation with diffusion coefﬁcient D if we deﬁne Dc = M|∂ 2 f/∂φA 2 | in Eq. (3A.1). Note that Dc = D and Dc is called the cooperative (or apparent) diffusion coefﬁcient. The solution of Eq. (3A.1) may be written as (Cahn 1961, 1968)

φA ∝ exp R(β)t

(3A.3)

R(β) = −Dc β 2 − 2Mκβ 4

(3A.4)

where

in which β is the wavenumber deﬁned by β = 2π/Λ, with Λ being the wavelength or domain spacing. According to Cahn’s linearized theory, the growth rate of concentration ﬂuctuations in terms of the volume fraction of the total polymer, φp (t), in a ternary system consisting of a pair of immiscible polymers and a solvent is given by (Inoue et al. 1985) φp (t) − φp (0) ∝ exp(Rm t)

(3A.5)

where φp (0) is the volume fraction of the total polymer in the initially homogeneous solution and Rm is the maximum rate constant of concentration ﬂuctuation growth at βm , which is the value of β at {∂R(β)/∂β}|β=βm = 0, with βm = 2π/Λm , where Λm is the maximum wavelength. Λm and Rm , respectively, are related to the polymer concentrations φp by (Inoue et al. 1985) Λm ∝ (φp − φps )−1/2 Rm ∝ M

(3A.6)

χAB φ (φ − φPs )2 NA P P

(3A.7)

where φps is the polymer concentration at the point of spinodal decomposition and it is given by φps

NA = 2χAB

1 1 + NA θ A N B θB

(3A.8)

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179

in which χAB is the Flory–Huggins interaction parameter, NA and NB are degrees of polymerization for polymers A and B, respectively, θA = φA /(φA + φB ) is the volume fraction of polymer A, θB = φB /(φA + φB ) is the volume fraction of polymer B, and φp = φA + φB = 1 − φS , with φS being the volume fraction of solvent in the ternary mixture. The following observation can be made from Eq. (3A.6) and (3A.8). For given molecular weights of the component polymers (i.e., for ﬁxed values of NA and NB ), from Eq. (3A.8) we have a minimum value of φps at θA = θB = 0.5, thus from Eq. (3A.6) we have the longest periodic distance of modulated structure. This observation now explains the experimental results given in Figure 3.34, the largest value of Λ being for the 50/50 PMMA/PS blends. For a given blend composition (i.e., for given values of θA and θB ), it follows from Eq. (3A.8) that φps will decrease with increasing molecular weight (NA or NB ) of polymers A and B, indicating that, in accordance with Eq. (3A.6), the higher the molecular weights of the component polymers, the larger the value of Λm will be. This observation again explains the experimental results in Figure 3.34.

Notes 1. CFX-3FD is a commercial software distributed by Computational Fluid Dynamics Services, AEA Technology Engineering Softwares. 2. In Chapter 2, we introduced the concept of critical ﬂow temperature (Tcf ) for glassy polymers. On the basis of PS and PC, it has been found that Tcf is 50–55 ◦ C above the glassy transition temperature (Tg ) of a glassy polymer.

References Cahn JW (1961). Acta Metall. 9:795. Cahn JW (1968). Trans. Met. Soc. 242:166. Chin HB, Han CD (1980). J. Rheol. 24:1. Elmendorp JJ, van der Vegt AK (1986). Polym. Eng. Sci. 26:1332. Favis BD, Therrien D (1991). Polymer 32:1474. Fortenly I, Zivny A (1995). Polymer 36:4113. Grace H (1982). Chem. Eng. Comm. 14:225. Han CD (1976). Rheology in Polymer Processing, Academic Press, New York, Chap 7. Han CD (1981). Multiphase Flow in Polymer Processing, Academic Press, New York, Chap 4. Han CD, Kim YW (1975). Trans. Soc. Rheol. 19:245. Han CD, Lee KY, Wheeler NC (1996). Polym. Eng. Sci. 36:1360. Han CD, Yu TC (1972). Polym. Eng. Sci. 12:81. Hashimoto T, Itakura M, Hasegawa H (1986). J. Chem. Phys. 85:6118. Hashimoto T, Takenaka M, Izumitani T (1992). J. Chem. Phys. 97:679. He J, Bu W, Zeng J (1997). Polymer 38:6347. Ho RM, Wu CH, Su AC (1990). Polym. Eng. Sci. 30:511. Inoue T, Ougizawa T, Yasuda O, Miyasaka K (1985). Macromolecules 18:57. Janssen LPBM (1978). Twin Screw Extrusion, Elsevier, Amsterdam.

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Karam H, Bellinger JC (1968). Ind. Eng. Chem. Fundam. 7:576. Lee JK, Han CD (1999a). Polymer 40:2521. Lee JK, Han CD (1999b). Polymer 40:6277. Lee JK, Han CD (2000). Polymer 41:1799. Martelli FG (1983). Twin-Screw Extruders, Van Nostrand Reihold, New York. McMaster LP (1975). In Copolymers, Polyblends, and Composites, Platzer NAJ (ed), Adv. Chem. Series no 142, American Chemical Society, Washington, DC, p 43. Miles IS, Zurek A (1988). Polym. Eng. Sci. 28:796. Nakai A, Shiwaku T, Wang W, Hasegawa H, Hashimoto T (1996). Macromolecules 29:5990. Nelson CJ, Avgeropoulos GN, Weisser FC, Böhm GGA (1977). Angew. Makromol. Chem. 60/61:49. Roland CM, Böhm GGA (1984). J. Polym. Sci., Polym. Phys. Ed. 22:79. Scott CE, Macosko CW (1995). Polymer 36:461. Shih CK (1995). Polym. Eng. Sci. 35:1688. Siggia ED (1979). Phys. Rev. A 20:595. Snyder HL, Meakin P, Reich S (1983). Macromolecules 16:757. Strobl GR (1985). Macromolecules 18:558. Sundararaj U, Macosko CW, Rolando RJ, Chan HT (1992). Polym. Eng. Sci. 32:1814. Sundararaj U, Macosko CW, Shi CK (1996). Polym. Eng. Sci. 36:1769. Takenaka M, Izumitani T, Hashimoto T (1990). J. Chem. Phys. 92:4566. Takenaka M, Izumitani T, Hashimoto T (1992). J. Chem. Phys. 97:6855. Takenaka M, Tanaka K, Hashimoto T (1989). In Multiphase Macromolecular Systems, Culbertson BM (ed), Plenum, New York, p 363. Takeno H, Hashimoto T (1997). J. Chem. Phys. 107:1634. Torza S, Cox RC, Mason SG (1972). J. Colloid Interface Sci. 38:395. van Oene H (1972). J. Colloid Interface Sci. 40:448. van Oene H (1978). In Polymer Blends, Vol 1, Paul DR, Newman S (eds), Academic Press, New York, p 295. White JL (1990). Twin Screw Extrusion, Hanser, Munich.

4

Compatibilization of Two Immiscible Homopolymers

4.1

Introduction

More often than not, the mechanical properties (e.g., impact and tensile properties) of immiscible polymer blends are very poor owing to the lack of adhesion between the constituent components, which originates from strong repulsive thermodynamic (segmental) interactions. Therefore, in the past, a great deal of effort (Barlow and Paul 1984; Fayt and Teyssie 1989; Fayt et al. 1981, 1987, 1989; Gupta and Purwar 1985; Ouhadi et al. 1986a; Park et al. 1992; Schwarz et al. 1988, 1989; Srinivasan and Gupta 1994; Traugott et al. 1983) has been made to improve the mechanical properties of two immiscible polymers by adding a third component (e.g., a block copolymer). In this chapter, we conﬁne our attention primarily to the situations where a nonreactive third component is added to two immiscible homopolymers in order to improve their mechanical properties. A polymer blend consisting of two immiscible homopolymers (say, A and B) has a very narrow interface, as schematically shown in Figure 4.1, because they have strong repulsive segmental interactions giving rise to a positive value of the Flory–Huggins interaction parameter (χ ), i.e., χAB > 0. Helfand and Tagami (1971, 1972) derived the following expression relating the interfacial thickness d of a pair of immiscible homopolymers of inﬁnite molecular weight to χ : d = 2b/(6χ )1/2

(4.1)

where the Kuhn length b is assumed to be the same for both components. They also derived an expression for the interfacial tension γ between two immiscible homopolymers: γ = (χ /6)1/2 bρo kB T 181

(4.2)

182

PROCESSING OF THERMOPLASTIC POLYMERS Figure 4.1 Schematic showing the presence of a narrow interface between two homopolymers, A and B, where repulsive segmental interactions are assumed to exist.

in terms of χ , where kB is the Boltzmann constant, T is the absolute temperature, and ρo is the reference density (the inverse of monomeric volume of a reference component). Equation (4.1) indicates that the interfacial thickness between two immiscible homopolymers will be larger when the extent of repulsive segmental interactions is less, and Eq. (4.2) indicates that the interfacial tension between two immiscible homopolymers will be lower when the extent of repulsive segmental interactions is less. However, Eqs. (4.1) and (4.2) have practical limitations in that d will become exceedingly large as the value of χ becomes exceedingly small which is physically unacceptable. Practically speaking, the interfacial thickness of a pair of immiscible homopolymers is expected be less than 1 nm, certainly less than the radius of gyration of the constituent components. Nevertheless, Eqs. (4.1) and (4.2) reveal how the extent of repulsive segmental interactions (i.e., the degree of immiscibility) of a pair of immiscible homopolymers is related to the interfacial thickness and interfacial tension. Broseta et al. (1990) extended the Helfand–Tagami analysis to ﬁnite molecular weight of polymers and concluded that for ﬁnite molecular weights, the interfaces are broader and the interfacial tensions smaller than predicted by the assumption of inﬁnite molecular weights of the constituent polymers. They noted that in polydisperse systems, there is some accumulation of small chains at the interface, lowering the interfacial tension. In the literature, in some instances the terms “emulsiﬁcation” and “compatibilization” have been used interchangeably when the addition of a third component decreased the size of domains dispersed in the matrix of an immiscible polymer blend. In this chapter, we make a distinction between an “emulsifying agent” and a “compatibilizing agent.” We regard a third component as being an emulsifying agent (or a dispersing agent) when it only decreases the size of domains dispersed in the matrix of an immiscible polymer blend without increasing the interfacial thickness. Under this deﬁnition of emulsifying agent, the added third component acts merely as a “surfactant,” decreasing the interfacial tension of the two homopolymers being melt blended, which in turn decreases the domain size of the dispersed phase. Conversely, we regard a third component as being a “compatibilizing agent” when it not only decreases the domain size of the dispersed phase, but also forms an “interphase,” the width of which being greater than the radius of gyration of the constituent components. The formation of such an interphase is possible only when an added third component, C, has attractive segmental interactions with the two immiscible

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183

Figure 4.2 Schematic showing the formation of an “interphase” between two immiscible homopolymers, A and B, in the presence of a third homopolymer, C, where attractive segmental interactions between homopolymers A and C and between homopolymers B and C are assumed to exist.

homopolymers, A and B, and thus all three components coexist inside the interphase, as schematically shown in Figure 4.2. Under this deﬁnition of compatibilizing agent, the formation of a sufﬁciently wide interphase is possible only when attractive segmental interactions exist between components A and C and between components B and C. We hasten to point out that the interphase of such a ternary blend (having χAB > 0, χAC < 0, and χBC < 0) will grow continuously, after a sufﬁciently long time of melt blending, ultimately giving rise to a homogeneously mixed phase in which all three components coexist. In practice, however, melt blending is done over a relatively short period (say less than 5 min) in an internal mixer or in a twin-screw extruder (see Chapter 3) and thus a ﬁnite thickness of interphase is expected in such a ternary blend. Therefore, from a practical point of view, one must consider the time dependence of the formation of an interphase during melt blending. Since it is very difﬁcult, in general, to ﬁnd or synthesize a homopolymer C that can meet the thermodynamic requirements χAC < 0 and χBC < 0 when melt blended with two immiscible homopolymers A and B, it seems natural to look for a C-block-D copolymer with attractive segmental interactions between homopolymer A and block C of the copolymer (χAC < 0) and between homopolymer B and block D of the copolymer (χBD < 0). Figure 4.3 gives a schematic describing the “dynamic” formation of an interphase between homopolymer A and block C of the copolymer, and an interphase between homopolymer B and block D of the copolymer. Here, the word “dynamic” is used because the width of the interphase depicted in Figure 4.3 (the shaded areas) would change with the duration of mixing. The premise of the deﬁnition of compatibilizing agent given above is that a signiﬁcant improvement in mechanical properties of a polymer blend can occur only in the presence of a sufﬁciently wide interphase. This implies that an effective compatibilizing agent is also an effective emulsifying agent, but the converse is not true; that is, a third component that decreases the interfacial tension between two immiscible homopolymers and thus decreases the domain size of the dispersed phase does not necessarily form a sufﬁciently wide interphase. The distinction made above between emulsifying agent and compatibilizing agent is very important from both theoretical and practical points of view in that a compatibilizing agent signiﬁcantly improves the mechanical properties of an immiscible polymer blend, while an emulsifying agent may not. Figure 4.4 gives, for illustration,

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Figure 4.3 Schematic showing the formation of an “interphase” between two immiscible homopolymers, A and B, in the presence of a C-block-D copolymer, where attractive segmental interactions between the homopolymer A and the block C of the copolymer and between the homopolymer B and the block D of the copolymer are assumed to exist. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

a schematic showing the role that an emulsifying agent and a compatibilizing agent, respectively, play in improving the mechanical properties of two immiscible polymers. With reference to Figure 4.4, curve (1) describes the composition dependence of a mechanical property without an emulsifying agent, curve (2) describes the composition dependence of a mechanical property with an emulsifying agent, curve (3) describes the composition dependence of a mechanical property with a compatibilizing agent having weak attractive interactions, and curve (4) describes the composition dependence of a mechanical property with strong attractive interactions. Figure 4.4 is given only to

Figure 4.4 Schematic showing the

mechanical properties of two immiscible homopolymers with and without an emulsifying agent or compatibilizing agent, where curve (1) represents the mechanical properties of two immiscible homopolymers, curve (2) represents the mechanical properties of two immiscible homopolymers in the presence of an emulsifying agent, curve (3) represents the mechanical properties of two immiscible homopolymers in the presence of a mild compatibilizing agent, and curve (4) represents the mechanical properties of two immiscible homopolymers with a strong compatibilizing agent.

COMPATIBILIZATION OF TWO IMMISCIBLE HOMOPOLYMERS

185

illustrate qualitatively the different roles that emulsifying agents and compatibilizing agents play and thus that variations in the shapes of the curves given in Figure 4.4 can exist. Thus, within the context of the deﬁnitions given above, “emulsifying agent” is not synonymous with “compatibilizing agent.” Some research groups (Anastasiadis et al. 1989; Cho et al. 2000) observed a decrease in interfacial tension, while other research groups (Fayt et al. 1982, 1986, 1987; Macosko et al. 1996; Ouhadi et al. 1986b) observed a decrease in the domain size, when a block copolymer was added to two immiscible homopolymers. Still other research groups (Barlow and Paul 1984; Gupta and Purwar 1985; Park et al. 1992; Schwarz et al. 1988, 1989; Traugott et al. 1983) observed a marginal improvement in mechanical properties of two immiscible homopolymers when a block copolymer was added and then concluded, without identifying the formation of an interphase, that the block copolymer played the role of a compatibilizing agent. According to the deﬁnition of compatibilizing agent we have given here, the block copolymers used in the studies cited did not function and could not have functioned as effective compatibilizing agents, the reason for which will be made clear in the next section of this chapter. Based on equilibrium statistical thermodynamics, some theoretical studies (Leibler 1988; Noolandi 1991; Noolandi and Hong 1982, 1984; Shull and Kramer 1990) have predicted a decrease in interfacial tension when an A-block-B copolymer is mixed with two immiscible homopolymers, A and B, while others (Vilgis and Noolandi 1990) made the same prediction when a C-block-D copolymer is mixed with a pair of immiscible homopolymers A and B. The former mixture is much simpler to treat theoretically than the latter, because only one χ value is needed when an A-block-B copolymer is added to a pair of immiscible homopolymers A and B, while six χ values are needed when a C-block-D copolymer is added to a pair of immiscible homopolymers A and B. But, those theoretical studies used positive values of χ in their calculations of the concentration proﬁles at the interface. Leibler (1988) argued that a reduction in interfacial tension between two immiscible homopolymers, in the presence of a block copolymer, originates from the equilibrium adsorption at the interface. Conversely, Noolandi (1991) argued that the main contribution to the reduction in interfacial tension is the enthalpic orientational effect when a block copolymer is present in a mixture of two immiscible homopolymers. In those statistical thermodynamic theories we have cited, equilibrium situations were considered. However, melt blending of a pair of immiscible homopolymers with a block copolymer in mixing equipment (e.g., in a twin-screw extruder) would not attain an equilibrium state because the duration of mixing (say 5 min of residence time in a twin-screw extruder) is too short to achieve an equilibrium morphology. Moreover, in the compatibilization of two immiscible homopolymers using a block copolymer, even when the segmental interactions between a homopolymer and a block copolymer are attractive (i.e., negative χ value), the molecular weights of the constituent components would also play an important role, via melt viscosity, in determining the rate of formation of an interphase. This is because the higher the viscosities of the polymers to be melt blended, the slower will be the rate of polymer–polymer interdiffusion and thus a longer period of mixing will be required to achieve an equilibrium morphology. In other words, during melt blending, a block copolymer will act as an effective compatibilizing agent only when the molecular weights of block chains and the corresponding homopolymers lie within a certain range, because the rate of interdiffusion between

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component polymers depends on molecular weight. Molecular viscoelasticity theory (Doi and Edwards 1986) suggests that the center-of-mass diffusion coefﬁcient (Do ) of a polymer is inversely proportional to molecular weight M (i.e., Do ∼ 1/M) for M < Mc and Do ∼ 1/M 2 for M > Mc , where Mc is a critical molecular weight (see Chapter 4 in Volume 1). This means that when M > Mc , the polymer–polymer interdiffusion would be extremely slow. However, from the point of view of mechanical properties, it is highly desirable to have M > Mc , so that the chains would not be easily pulled out from the interphase when an external force is applied. The rate of formation of an interphase, during melt blending, between two immiscible homopolymers in the presence of a block copolymer would depend not only on meeting the thermodynamic requirements, but also on the melt blending temperature in relation to the order–disorder transition temperature (TODT ) of the block copolymer (Chun and Han 1999, 2000). This subject will be elaborated on in the next section. In this chapter, we ﬁrst present compatibilization of two immiscible polymers using a block copolymer. We will then present very brieﬂy reactive compatibilization of two immiscible polymers using a reactive third component. There have been many reactive polymeric compatibilizing agents reported in the literature (Datta and Lohse 1996). Since different reactive compatibilizing agents may lead to different reaction paths, it seems impossible to present the reaction mechanisms for all conceivable ternary blends in the presence of a reactive compatibilizing agent. Without a comprehensive theory, an exhaustive discussion of such subject is neither possible nor useful. Thus, we will present a few speciﬁc examples of reactive compatibilization of two immiscible polymers.

4.2

Experimental Observations of Compatibilization of Two Immiscible Homopolymers Using a Block Copolymer

Block copolymer, when properly designed in terms of chemical structure, architecture, and molecular weight, can be used as an effective compatibilizing agent for a pair of immiscible homopolymers. In this section, we present experimental observations of compatibilization of two homopolymers, A and B, using (1) A-block-B copolymer, (2) A-block-C copolymer, or (3) C-block-D copolymer. Needless to say, the block length ratio of a block copolymer and the molecular weight of a block of the given block copolymer relative to the molecular weight of corresponding homopolymer would play very important roles in determining the effectiveness of compatibilization. In this section, emphasis is placed on pointing out that not only are the thermodynamic requirements for compatibilization, as described in the preceding section, crucial factors determining the effectiveness of compatibilization, but so too is the melt blending temperature relative to the TODT of the block copolymer. Speciﬁcally, we demonstrate that thermodynamic requirements alone are not sufﬁcient to form an interphase between two immiscible homopolymers in the presence of a block copolymer, unless melt blending of all three components is carried out at a temperature above the TODT of the block copolymer. We will illustrate the situations in which an added block copolymer played the role of an emulsifying agent instead of a compatibilizing agent.

COMPATIBILIZATION OF TWO IMMISCIBLE HOMOPOLYMERS

4.2.1

187

A/B/(A-block-B) Ternary Blends

A number of research groups have investigated experimentally the A/B/(A-block-B) ternary system; some research groups (Adedeji et al. 1996; Macosko et al. 1996) investigated the effectiveness of an A-block-B copolymer in the compatibilization of two immiscible homopolymers, A and B, while other research groups (Dai et al. 1992; Russell et al. 1991a, 1991b; Shull et al. 1990) investigated segregation of an A-block-B copolymer to interfaces between two immiscible homopolymers, A and B. And, still others (Brown et al. 1993) investigated the effects of an A-block-B copolymer on adhesion between two immiscible homopolymers, A and B, on the interface toughness using fracture test, and on the organization of the diblock copolymer using secondary ion mass spectroscopy. An interpretation of the experimental results for the A/B/(A-block-B) ternary system is relatively simple compared with that for A/B/(A-block-C) and A/B/ (C-block-D) ternary systems because only one interaction parameter χAB is involved in the A/B/(A-block-B) ternary system. In this section, we will consider three different combinations of the A/B/(A-block-B) ternary blends in terms of the chemical structures of the constituent components. 4.2.1.1 PS/PMMA/(PS-block-PMMA) Ternary Blends or Three-Layer Films Macosko et al. (1996) investigated the morphology of polystyrene (PS)/poly(methyl methacrylate) (PMMA)/(PS-block-PMMA) ternary blends that were prepared by melt blending PMMA (forming the minor component) with PS (forming the major component) at 180 ◦ C with varying amounts (1–5 wt %) of PS-block-PMMA copolymer. They observed a signiﬁcant reduction in the domain size of PMMA in the presence of as little as 1 wt % of a PS-block-PMMA copolymer, and that the molecular weight of added block copolymer greatly inﬂuenced the size of the dispersed PMMA phase as well as the stability of the dispersion. They observed the coalescence of the dispersed PMMA phase during annealing at 195 ◦ C for 20 min and concluded that the principal role of added block copolymer was to control the morphology by preventing coalescence. No attempt was made to identify the location of added block copolymer in the ternary blend. Adedeji et al. (1996) investigated the morphology of PS/PMMA/(PS-block-PMMA) ternary blends that were prepared by solvent casting. In their study, the molecular weights of homopolymers, PS and PMMA, and also of PS-block-PMMA copolymer, were varied to obtain different microstructures of the ternary blend. No attempt was made to identify the location of added block copolymer in the ternary blend. Russell et al. (1991a, 1991b) investigated segment density distribution of homopolymer and copolymer chains at the interface of a thin ﬁlm of PS/(PS-blockPMMA)/PMMA layers using neutron reﬂectivity. For the investigation, selective deuteration of either homopolymer chains or either block of the copolymer was conducted in order to have the contrast necessary to determine the individual segment density distribution across the interface or to determine the width of homopolymer interfaces in the presence of a symmetric PS-block-PMMA copolymer using thin three-layer ﬁlm specimens, in which a PS-block-PMMA copolymer was sandwiched between PS and PMMA ﬁlms, prepared by spin coating. The three-layer PS/(PS-blockPMMA)/PMMM ﬁlms prepared were annealed at 170 ◦ C for 5–10 days before neutron

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Table 4.1 Molecular parameters of the PS-block-PMMA copolymers employed by various investigators for preparing ternary blends with two immiscible homopolymers or random copolymers

Sample Code

Mw 5.8 × 104 8.9 × 104

SM55 SM85 SM160 SM780 B65 B283 B680 B1170 P(S-d-b-MMA)a P(S-b-MMA-d)a P(S-d-b-MMA-d)a a

10.8 × 104 96.7 × 104 6.6 × 104 28.3 × 104 68.0 × 104 117.0 × 104 10.1 × 104 12.1 × 104 11.5 × 104

PS (wt frac) 0.58 0.55 0.50 0.40 0.50 0.43 0.67 0.75 0.52 0.46 0.49

References Macosko et al. (1996) Macosko et al. (1996) Macosko et al. (1996) Macosko et al. (1996) Adedeji et al. (1996) Adedeji et al. (1996) Adedeji et al. (1996) Adedeji et al. (1996) Russell et al. (1991a) Russell et al. (1991a) Russell et al. (1991a)

d denotes deuterated block.

reﬂectivity measurements. Shull et al. (1990) observed that the addition of a PS-blockPMMA copolymer broadened the interface between PS and PMMA homopolymers by 50% (from 5 nm to 7.5 nm) and concluded that the broadening of the interface was due to a signiﬁcant penetration of the homopolymer into the interfacial region (“interphase”). They observed further that the junction points of the block copolymer were localized to a region around the midpoint of the interface over a distance comparable with the interfacial width observed for the neat block copolymers. The molecular weights of the PS-block-PMMA copolymers employed in these studies are summarized in Table 4.1. Since a block copolymer forms microdomains at temperatures below its TODT , and the viscosity of a block copolymer in the ordered state (at T < TODT ) would be orders of magnitude higher than that in the disordered state (at T > TODT ), it would be instructive to estimate the TODT of each of the PS-block-PMMA copolymers given in Table 4.1. Using the Leibler theory (1980) together with the expression χPS/PMMA = (0.028 ± 0.002) + (3.9 ± 0.6)/T

(4.3)

which was reported by Russell et al. (1990) who measured the small-angle neutron scattering from compositionally symmetric PS-block-PMMA copolymers at temperatures above TODT , we have estimated values of TODT as summarized in Table 4.2. According to Russell et al., Eq. (4.3) was obtained by varying χ to produce the best ﬁt, with the aid of the Leibler theory, to the experimental scattering proﬁles at different temperatures. It can be seen in Table 4.2 that the values of TODT estimated are very sensitive to the molecular weight of PS-block-PMMA copolymer, leading us to conclude that the values of TODT of the PS-block-PMMA copolymers listed in Table 4.1 are exceedingly high compared with either the melt blending temperature employed (Macosko et al. 1996) or annealing temperature employed (Russell et al. 1991a, 1991b).

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189

Table 4.2 TODT of compositionally symmetric PS-block-PMMA copolymers predicted from the Leibler theory

Mw 2.0 × 104 2.4 × 104 2.8 × 104

TODT (◦ C)a

Mw

TODT (◦ C)a

− 122 − 42 100

3.2 × 104 3.4 × 104 3.6 × 104

416 774 1544

a Equation (4.3) was used to predict T

ODT

.

Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.

This observation indicates that the PS-block-PMMA copolymers employed by the various research groups cited were in a microphase-separated state. It is then reasonable to speculate that those PS-block-PMMA copolymers at 180–195 ◦ C employed for melt blending (Macosko et al. 1996) or at 170 ◦ C employed for annealing (Russell et al. 1991a, 1991b) must have had exceedingly low mobility owing to the very high viscosity at such temperatures, which were much below the TODT of the respective block copolymers. Under such circumstances, it seems reasonable to ask how such microphase-separated PS-block-PMMA copolymers could have functioned effectively as emulsifying agents, let alone as compatibilizing agents, at T TODT . The evolution of the width of interphase in a mixture consisting of a block copolymer and two immiscible homopolymers is a kinetic process involving the mechanism of polymer interdiffusion, which in turn depends on the molecular weights of the constituent components, the interaction parameter χ , and the blend composition (Kramer et al. 1984). Let us look at the schematic given in Figure 4.5, in which a PS-blockPMMA copolymer is placed between two homopolymers, PS and PMMA. We have the following three situations in reference to Figure 4.5. (a) When the processing temperature is lower than the TODT of PS-block-PMMA copolymer, little interdiffusion is expected to take place between the homopolymer PS and the PS segments of the copolymer, and between the homopolymer PMMA and the PMMA segments of the copolymer. Under such situations, the molecular weight of the polymers are expected to play little role in the compatibilization of the immiscible PS/PMMA pair, the reason for which will be elaborated on later. (b) When the processing temperature is higher than the TODT of PS-block-PMMA copolymer and the molecular weight of PS-block-PMMA copolymer is lower than the molecular weights of the two homopolymers, PS and PMMA, the short copolymer chains are expected to be stretched out towards the homopolymer sides, while the homopolymer chains would not move towards the copolymer side. (c) When the processing temperature is higher than the TODT of PS-block-PMMA copolymer and the molecular weight of PS-block-PMMA copolymer is higher than the molecular weights of the two homopolymers, PS and PMMA, the long copolymer chains would stretch towards the homopolymer sides, while the homopolymer chains can move towards the block copolymer side. Since there are neither attractive nor repulsive segmental interactions between the homopolymer PS and the PS blocks of copolymer, and between the homopolymer PMMA and the PMMA blocks of copolymer, the PS–PMMA joints of the block copolymer are expected to be more or less stationary. The above considerations may explain the experimental observations of

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Figure 4.5 Schematic showing the distribution of components in an A/B/(A-block-B) ternary blend: (a) in a specimen that was melt blended at a temperature below the TODT of the block copolymer, (b) in a specimen that was melt blended at a temperature above the TODT of the block copolymer and when the molecular weight of each homopolymer was higher than the molecular weight of the corresponding block in the copolymer, and (c) in a specimen that was melt blended at a temperature above the TODT of the block copolymer and when the molecular weight of each homopolymer was lower than the molecular weight of the corresponding block in the copolymer.

Russell et al. (1991b), who reported that the junction points of the PS-block-PMMA copolymer, as determined from neutron reﬂectivity, were localized around the midpoint of the interface when the block copolymer was placed between the homopolymer PS and PMMA. Thin three-layer PS/(PS-block-PMMA)/PMMA ﬁlms employed in their experiments were annealed at 170 ◦ C, whereas the TODT of their PS-block-PMMA copolymers must have been much higher than the annealing temperature. As discussed in Chapter 3, when PS and PMMA were melt blended in an internal mixer or in a twin-screw extruder, the minor component formed the discrete phase and the major component formed the continuous phase, irrespective of viscosity ratio. Therefore, the following fundamental question may be raised: Where will a small amount (say, 5 wt %) of added PS-block-PMMA copolymer go when it is melt blended

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191

with 30 wt % PS and 70 wt % PMMA or when it is melt blended with 70 wt % PS and 30 wt % PMMA? Will the PS-block-PMMA copolymer always surround, during melt blending for say 5 min, the surface of the discrete phase (drops) in the PS/PMMA blend? On the basis of the discussion presented above, the location of the added block copolymer in a PS/PMMA/(PS-block-PMMA) ternary blend would depend very much on the melt blending temperature relative to the TODT of the PS-block-PMMA copolymer. It is quite possible that the PS-block-PMMA copolymer may form its own domains in the matrix of PS or PMMA, depending on blend composition, when the melt blending temperature is lower than the TODT of the PS-block-PMMA copolymer (i.e., at T < TODT ). Even when a PS-block-PMMA copolymer is melt blended with PS and PMMA at a temperature above its TODT by choosing a relatively low molecular weight block copolymer, there is no assurance that the added block copolymer will necessarily surround, during melt blending for say 5 min, the surface of the dispersed drops of PS or PMMA because there are no attractive segmental interactions between the homopolymer and the block copolymer. We will return to this subject when dealing with A/B/(A-block-C) and A/B/(C-block-D) ternary blends, in which attractive segmental interactions exist between the homopolymer and the block copolymer. 4.2.1.2 PS/PI/(PS-block-PI) and PS/PB/(PS-block-PB) Ternary Blends Figure 4.6a gives a transmission electron microscopy (TEM) image for as-precipitated 63/27/10 PS/polyisoprene (PI)/polystyrene-block-polyisoprene (SI diblock) copolymer (SI-9/9) ternary blend, in which 63/27/10 refers to the weight percent of the component polymers, and Figure 4.6b gives a TEM image for as-precipitated 63/27/10 PS/polybutadiene (PB)/polystyrene-block-polybutadiene (SB diblock) copolymer (SB9/8) ternary blend, where the dark areas represent the PI or PB phase and the bright areas represent the PS phase. The molecular weights of SI-9/9 and SB-9/8 are given in Table 4.3. Owing to the low molecular weights of the SI and SB diblock copolymers employed, which were in the liquid state at room temperature, melt blending was not possible in preparing these ternary blends (Chun and Han 1999).

Figure 4.6 (a) TEM image of as-precipitated 63/27/10 PS/PI/(SI-9/9) ternary blend. (b) TEM

image of as-precipitated 63/27/10 PS/PB/(SB-9/8) ternary blend.

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Table 4.3 Molecular characteristics of SB and SI diblock copolymers

Sample Code SI-9/9a SB-9/8b SB-14/11c SB-13/12d

Mn

Mw /Mn

wt % PS

Mw of PS

Mw of PI or PB

18,000 16,700 24,000 23,000

1.10 1.04 1.06 1.09

51.0 52.8 53.9 53.9

9,400 9,200 13,700 13,500

9,000 8,200 11,700 12,200

a

PI block has 94% 1,4-addition. b PB blocks has 89% 1,2-addition. Member osmometry was employed to determined numberaverage molecular weight (Mn ) and gel permeation chromatography was used to determine polydispersity Mw /Mn . The weight percent of PS in each block copolymer was determined using nuclear magnetic resonance spectroscopy. c PB block has 9% 1,2-addition. d PB block has 44% 1,2-addition.

Reprinted from Chun and Han, Macromolecules 32:4030. Copyright © 1999, with permission from the American Chemical Society.

Therefore, a homogeneous solution of two homopolymers and a block copolymer, which were dissolved in toluene, was rapidly precipitated by pouring into methanol. This rapid precipitation may be termed “composition quenching” (Lee and Han 1999; Yang and Han 1996), and gives rise to a morphology very similar to that often obtained from spinodal decomposition by rapid quenching. Owing to the low magniﬁcation, SI or SB diblock copolymers are not discernible in Figure 4.6, in which we observe a co-continuous morphology very similar to that observed during rapid precipitation of two immiscible homopolymers (see Chapter 3). Figure 4.7a gives a TEM image of as-precipitated 63/27/10 PS/PI/(SI-9/9) ternary blend after annealing for 12 h at 110 ◦ C (below the TODT of SI-9/9), and Figure 4.7b gives a TEM image of as-precipitated 63/27/10 PS/PB/(SB-9/8) ternary blend after annealing for 12 h at 120 ◦ C (below the TODT of SB-9/8). The TODT of SI-9/9 is 133 ◦ C and the TODT of SB-9/8 is 123 ◦ C. In Figure 4.7 we observe that both 63/27/10 PS/PI/(SI-9/9) and 63/27/10 PS/PB/(SB-9/8) ternary blends form a co-continuous morphology, consistent with the experimental observations presented in Chapter 3 for

Figure 4.7 (a) TEM image of as-precipitated 63/27/10 PS/PI/(SI-9/9) ternary blend after annealing at 110 ◦ C (below the TODT of SI-9/9) for 12 h. (b) TEM image of as-precipitated 63/27/10 PS/PB/(SB-9/8) ternary blend after annealing at 120 ◦ C (below the TODT of SB-9/8) for 12 h.

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193

binary blends of PS and PMMA. A close look at Figure 4.7a reveals that there are many very small dark areas, which represent the macrophase-separated SI diblock copolymer, as well as the large dark areas that represent the homopolymer PI. Since the annealing temperature (110 ◦ C) was below the TODT of SI-9/9, and further the annealing temperature was not far above the glass transition temperature (Tg ) of the PS block of SI-9/9, apparently the diffusion of the block copolymer SI-9/9 towards the interface between the discrete phase of PI and the continuous phase of PS might have been extremely slow (i.e., apparently the annealing at 110 ◦ C for 12 h was not sufﬁciently long). Figure 4.8 gives TEM images of 63/27/10 PS/PI/(SI-9/9) ternary blends and 63/27/10 PS/PB/(SB-9/8) ternary blends, each annealed for 12 h at 160 ◦ C, well above the TODT of the respective SI and SB diblock copolymers. Upon annealing at an elevated temperature for a sufﬁciently long period, a rapidly precipitated blend undergoes a transformation from a co-continuous morphology into a dispersed morphology. It is interesting to observe in Figure 4.8 that the SI block copolymer has diffused into the PI drop and that the SB diblock copolymer has diffused into the PB drop, while maintaining lamellar microdomain structure. There is little evidence to indicate that the block copolymer, SI-9/9 or SB-9/8, is distributed uniformly on the surface of the drop that is dispersed in the PS matrix. These observations lead us to conclude that the block copolymer, SI-9/9 or SB-9/8, did not function as an effective emulsifying agent, let alone as an effective compatibilizing agent. What is interesting here is that an A-block-B copolymer in A/B/(A-block-B) ternary blends may not function as an effective compatibilizing agent, even when the as-precipitated specimen was annealed at temperature above the TODT of the block copolymer. This conclusion is consistent with the studies of Adedeji et al. (1997a, 1997b), who demonstrated that the effective emulsiﬁcation of two immiscible homopolymers or random copolymers using a diblock

Figure 4.8 (a) TEM image of as-precipitated 63/27/10 PS/PI/(SI-9/9) ternary blend after annealing at 160 ◦ C (above the TODT of SI-9/9) for 12 h, where the block copolymer maintains

lamellar microdomain structure inside a PI droplet. (b) TEM image of as-precipitated 63/27/10 PS/PB/(SB-9/8) ternary blend after annealing at 160 ◦ C (above the TODT of SB-9/8) for 12 h, where the block copolymer maintains lamellar microdomain structure inside a PB droplet. (Reprinted from Chun and Han, Macromolecules 32:4030. Copyright © 1999, with permission from the American Chemical Society.)

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copolymer requires attractive segmental interactions (i.e., negative χ values) between the homopolymer or random copolymer and the corresponding block of the copolymer. Anastasiadis et al. (1989) observed a decrease in interfacial tension between homopolymer PS and homopolymer 1,2-polybutadiene (1,2-PB) in the presence of PS-block-1,2-PB copolymer at 145 ◦ C, and they concluded that the block copolymer played the role of a compatibilizing agent. The number-average molecular weight (Mn ) of PS-block-1,2-PB copolymer was 1.86 × 104 , with Mw /Mn = 1.06 and 49 wt % PS block, and the block copolymer was preblended with a PS having Mn = 2.2 × 103 . A ﬂuid drop of this mixture was subsequently blended with the 1,2-PB phase, having Mn = 7.8 × 103 , which formed the continuous phase. Using the following expression for the interaction parameter α (Han et al. 1998): α = −0.2388 × 10−2 + 1.355/T − 0.0186φPS /T

(4.4)

where α is related to χ by α = Vref χ with Vref being the molar reference volume, T is the absolute temperature, and the theory of Helfand and Wasserman (1982), we estimate TODT to be 156 ◦ C for the PS-block-1,2-PB copolymer. Considering the uncertainties associated with Eq. (4.4) and with the Helfand–Wasserman theory, it is reasonable to conclude that the interfacial tension measurement by Anastasiadis et al. might have been conducted at a temperature very close to the disordered state. However, in the absence of a TEM image showing the location of the added block copolymer in the ternary mixture and in the absence of a measurement of interfacial thickness, we cannot conclude whether the added PS-block-1,2-PB copolymer acted as a compatibilizing agent and formed an “interphase” with signiﬁcant thickness or acted merely as an emulsifying agent lowering the interfacial tension. 4.2.2

A/B/(A-block-C) Ternary Blends

4.2.2.1 PS/LDPE/(PS-block-PEB) and PS/HDPE/(PS-block-PEB) Ternary Blends Fayt et al. (1981, 1982, 1986, 1987, 1989) synthesized a series of PS-blockpoly(ethylene-co-1-butene) (PS-block-PEB) copolymers to emulsify or compatibilize PS/low-density polyethylene (LDPE) blends or PS/high-density polyethylene (HDPE) blends. For brevity, PS-block-PEB copolymer will be referred to as SEB diblock copolymer. The rationale behind such efforts lies in that the PEB block might be miscible, during melt blending, with LDPE in a PS/LDPE binary blend and with HDPE in a PS/HDPE binary blend. Note that PEB block is obtained from the hydrogenation of 1,2-addition of the PB block and the hydrogenation of 1,4-addition of the PB block of SB diblock copolymer. The PB block of an SB diblock copolymer usually has about 30% 1,2-addition and about 70% 1,4-addition when using tert-butyllithium as an initiator, while it can have a very high vinyl content (80–94% 1,2-addition) when using a specially designed initiator (Halasa et al. 1981). Later in this chapter we will show that the microstructure of the PB block in SB diblock copolymer plays very important role in the compatibilization of certain immiscible blends. Therefore, the miscibility between LDPE or HDPE with SEB diblock copolymer depends on the microstructure of the PEB block of the copolymer.

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195

The effective segmental interaction parameter for a binary blend consisting of homopolymer, A, and a random copolymer, C-ran-D copolymer, can be estimated from the following expression (ten Brinke et al. 1983): χA/(C-ran-D) = yχA/D + (1 − y)χA/C − y(1 − y)χC/D

(4.5)

where y is the volume fraction of D in the C-ran-D copolymer, χA/D is the interaction parameter for the A/D pair, χA/C is the interaction parameter for the A/C pair, and χC/D is the interaction parameter for the C/D pair. Thus, the interaction parameter for the PE (LDPE or HDPE)/PEB pair can be estimated from χPE/PEB = y 2 χPE/(PB-1)

(4.6)

We have the following expression (Chun and Han 2000): χPE/PB-1 = −5.05 × 10−2 + 39.5/T

(4.7)

Therefore, at the melt blending temperature of 210 ◦ C employed in the studies of Fayt et al. (1986), from Eqs. (4.5) and (4.6) we have χPE/PEB = 2.52 × 10−2 for y = 0.9 and χPE/PEB = 0.28 × 10−2 for y = 0.3. It is then clear that the larger the amount of PB-1 in the PEB block of an SEB diblock copolymer, the less the miscibility between the homopolymer LDPE or HDPE and the PEB block of the copolymer. The ideal situation would be to have an SB diblock copolymer without 1,2-addition at all, which will then give rise to, after hydrogenation, an SEB diblock copolymer consisting of PS and PE blocks. In practice, it is virtually impossible to synthesize an SB diblock copolymer without 1,2-addition at all. Therefore, we can conclude that PS/LDPE/SEB and PS/HDPE/SEB ternary blends, belonging to the A/B/(A-block-C) ternary system, may not be the ideal combinations from the point of view of miscibility between the homopolymer, LDPE or HDPE, and the PEB block of the copolymer as long as the SEB diblock copolymer contains PB-1. Thus, the SEB diblock copolymers, in the presence of repulsive segmental interactions between the LDPE or HDPE and the PEB block of the copolymer, could not have played the role of an effective compatibilizing agent for PS/LDPE and PS/HDPE blends. Indeed, Fayt et al. (1981, 1982, 1986, 1987, 1989) concluded that their SEB diblock copolymers played the role of an emulsifying agent for PS/LDPE and PS/HDPE blends, and noted that SEB diblock copolymers acted as efﬁcient emulsifying agents only when the molecular weight of block sequences was comparable with or higher than the molecular weight of the corresponding homopolymer. There is little doubt that the SEB diblock copolymer did not play the role of an emulsifying agent for PS/LDPE and PS/HDPE blends. However, as discussed above with reference to PS/PMMA/ (PS-block-PMMA) ternary blends, the TODT of SEB diblock copolymer would also play an important role in the effectiveness of emulsiﬁcation of the PS/LDPE and PS/HDPE blends. Using information on the molecular weights, given in Table 4.4, of the SEB diblock copolymers employed in the studies of Fayt et al. (1986) we have estimated the

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Table 4.4 Molecular parameters of the SEB and SEBS block copolymers employed by various investigators for preparing ternary blends with two immiscible homopolymers

Sample Code

Mw

PS (wt frac)

Predicted TODT (◦ C)a

6.1 × 104c 28.9 × 104c 16.3 × 104c 8.4 × 104c

0.43 0.50 0.49 0.50

373 439 426 397

Barlow and Paul (1984) Barlow and Paul (1984) Fayt et al. (1986) Fayt et al. (1986)

8.7 × 104d

0.29

316

Gupta and Purwar (1985) Schwarz et al. (1989) Setz et al. (1996) Srinivasan and Gupta (1994)

References

(a) SEB SE-1b SE-3b SE-2b SE-5b (b) SEBS Kraton G1650

a Equation (4.8) was used to predict T

from the Helfand–Wasserman theory. b The amount of 1-butene in the PEB ODT block was not provided by the investigators. c Since the investigators provided values of Mn only values of Mw given here

were estimated by assuming Mw /Mn = 1.05. d Taken from the paper by Setz et al. (1996) by assuming Mw /Mn = 1.04.

Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.

values of TODT for the SEB block copolymers with the aid of the Helfand–Wasserman theory (1982),1 for which the following expression (Han et al. 1998) was used: αPS/PEB = −0.1678 × 10−2 + 1.2536/T − 0.0648φPS /T

(4.8)

A summary of predicted values of TODT is also given in Table 4.4. Note that Eq. (4.8) was obtained via cloud point measurements for fairly low molecular weights of PS and PEB: Mw = 1.5 × 103 for PS and Mw = 3.9 × 103 for PEB with 84% 1-butene (Han et al. 1998). In view of the fact that α may depend on the molecular weight of the constituent components, the estimated TODT values given in Table 4.4 may be regarded as being very conservative values, because the molecular weights of the SEB diblock copolymers employed (see Table 4.4) by Fayt et al. (1986) are far greater than those used to obtain Eq. (4.8). In Table 4.4, we observe that predicted values of TODT are much higher than the melt blending temperatures employed (e.g., 210 ◦ C) in their experiments. Such high values of the predicted TODT are attributable to the high molecular weight of the SEB diblock copolymers. This observation indicates that the SEB diblock copolymers could not have formed an interphase during melt blending, and thus could not have played the role of an effective compatibilizing agent, because the LDPE or HDPE and the PEB block of the copolymer have repulsive segmental interactions. However, such block copolymers might have played the role of emulsifying agent to reduce the domain size.

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197

4.2.2.2 PS/PP/(PS-block-PEB-block-PS) Ternary Blends Polystyrene-block-poly(ethylene-co-1-butene)-block-polystyrene (PS-block-PEB-blockPS) copolymer, which is commercially available (Kraton G1600 series, Shell Development Company), has been used extensively to compatibilize PS/LDPE, PS/HDPE, PS/polypropylene (PP), poly(phenylene oxide) (PPO)/HDPE or PPO/LDPE blends by several research groups (Barlow and Paul 1984; Gupta and Purwar 1985; Heck et al. 1997; Park et al. 1992; Schwarz et al. 1988, 1989; Setz et al. 1996; Srinivasan and Gupta 1994; Traugott et al. 1983). For brevity, we will refer to PS-block-PEB-block-PS copolymer as SEBS triblock copolymer. An improvement in some mechanical properties of immiscible binary blends was observed with the presence of an SEBS triblock copolymer, leading to the conclusion that the SEBS triblock copolymer acted as a compatibilizing agent. Figure 4.9 shows the dependence of ultimate tensile strength (Sb ) and modulus (Gm ) on volume fraction of PP (φPP ) in PS/PP/Kraton G1650 ternary blends2 with 10 vol % of Kraton G1650. Note that the PS/PP/Kraton G1650 ternary blends were prepared by mixing at 200 ◦ C for 5 min in an internal mixer at a rotor speed of 60 rpm. The specimens for tensile property measurement were prepared by compression molding the meltblended polymer in a hot press at 200 ◦ C (Chun and Han 1999). For comparison,

Figure 4.9 Plots of (a) ultimate tensile strength versus volume fraction of PP and (b) modulus versus volume fraction of PP for (PS-220)/PP/Kraton G1650 ternary blends (䊉) and (PS-220)/PP binary blends (). (Reprinted from Chun and Han, Macromolecules 32:4030. Copyright © 1999, with permission from the American Chemical Society.)

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the dependence of Sb and Gm on φPP for PS/PP binary blends is also given in Figure 4.9. The following observations are worth noting in Figure 4.9: (1) the ultimate tensile strength of neat PP is 36 MPa, while the ultimate tensile strength of PS/Kraton G1650 (10 vol %) binary blend is 22 MPa, indicating that Kraton G1650 and PS are not miscible, (2) the addition of Kraton G1650 (10 vol %) increased the ultimate tensile strength of the 65/35 PS/PP binary blend from about 11 to about 18 MPa, but improved little the ultimate tensile strength of the PS/PP binary blends having the volume fraction of PP greater than 0.5, and (3) the addition of Kraton G1650 (10 vol %) did virtually nothing to improve the modulus of the PS/PP binary blends over the entire blend compositions. The above observations indicate that Kraton G1650 did not act as a compatibilizing agent to the immiscible PS/PP binary blends. It is clear from Figure 4.9 that values of Sb and Gm of the PS/PP/Kraton G1650 ternary blends are lower than the linear additive rule, leading us to conclude that there was little interfacial adhesion between the block copolymer Kraton G1650 and homopolymers PS and PP. TEM images of the 45/45/10 PS/PP/Kraton G1650 ternary blend are given in Figure 4.10, in which the numbers represent volume percentages, the bright areas represent the PP phase, the medium dark areas represent PS phase stained by ruthenium tetroxide (RuO4 ), and the very dark areas represent the block copolymer, Kraton G1650. Notice in Figure 4.10a that Kraton G1650, like PS, forms separate domains that are dispersed in the PP matrix. In Figure 4.10b, which is a higher magniﬁcation image of Figure 4.10a, we observe that a smaller drop of Kraton G1650, with a hexagonally packed cylindrical microdomain structure, is attached on the surface of the much larger drop of PS; we see no evidence of uniform distribution of Kraton G1650 at the interface between the PS drop and the PP matrix. This observation now explains why the tensile properties of the PS/PP/Kraton G1650 ternary blends are improved little or marginally over those of the PS/PP binary blends (see Figure 4.9). Figure 4.11a gives log G versus log G plots for Kraton G1650 at temperatures ranging from 190 to 310 ◦ C, showing that the TODT of Kraton G1650 is much higher than 310 ◦ C, the highest experimental temperature employed (see Chapter 8 of Volume 1 for the rheological criteria used). Figure 4.11b gives the temperature dependence of storage modulus G for Kraton G1650, which was obtained from dynamic temperature sweep experiments under isochronal conditions (at ω = 0.01 rad/s). From Figure 4.11b we also conclude that the TODT of Kraton G1650 is much higher than 310 ◦ C, the highest experimental temperature employed (see Chapter 8 of Volume 1 for rheological criteria used), consistent with the conclusion drawn from the log G versus log G plots. It is clear that the TODT of Kraton G1650 is indeed very high compared with the melt blending temperature (200 ◦ C) employed in the experiment that produced Figure 4.9. Although the TODT of Kraton G1650 cannot be determined by experiment, we can estimate it using mean ﬁeld theory. Using the Helfand–Wasserman theory (1982) we have estimated the TODT of Kraton G1650, the molecular weight of which is given in Table 4.4, to be approximately 316 ◦ C with the aid of Eq. (4.8). The estimated value of TODT = 316 ◦ C conﬁrms the experimental results (Figure 4.11), which suggest that the TODT of Kraton G1650 would be much higher than 316 ◦ C. The estimated TODT = 316 ◦ C for Kraton G1650 might be a very conservative value because Eq. (4.8) is based on the low molecular weights (Mw = 1.5 × 103 for PS and Mw = 3.9 × 103 for PEB) compared with the molecular weight of Kraton G1650 (Mw = 2.4 × 104 for PS block and Mw = 6.3 × 104 for PEB block).

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199

Figure 4.10 TEM images of 45/45/10 PS/PP/Kraton G1650 blend at (a) low magniﬁcation and (b) high magniﬁcation. The bright areas represent the PP phase, the medium dark areas represent PS droplets, and the very dark areas represent Kraton G1650. (Reprinted from Chun and Han, Macromolecules 32:4030. Copyright © 1999, with permission from the American Chemical Society.)

We have the following expressions (Chun and Han 2000): χPP/(PB-1) = −7.98 × 10−2 + 3.83/T

(4.9)

χPP/PE = −2.45 × 10−2 + 16.63/T

(4.10)

and

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Figure 4.11 (a) Temperature dependence of log G versus log G plots for Kraton G1650 obtained from dynamic frequency sweep experiments during heating at various temperatures (◦ C): () 190, () 210, () 230, (∇) 250, (䊉) 270, () 290, and () 310. (b) Temperature dependence of dynamic storage modulus G for Kraton G1650 obtained from dynamic temperature sweep experiments under isochronal conditions during heating. (Reprinted from Chun and Han, Macromolecules 32:4030. Copyright © 1999, with permission from the American Chemical Society.)

Therefore, for Kraton G1650 having 30 wt % PB-1 in the PEB block, the substitution of Eqs. (4.7), (4.9), and (4.10) into Eq. (4.5) for y = 0.3 gives χPP/PEB = −3.044 × 10−2 + 4.495/T

(4.11)

From Eq. (4.11) we have χPP/PEB = −0.209 × 10−2 at 200 ◦ C, suggesting that the homopolymer PP and the PEB block of the copolymer have attractive segmental interactions. From this observation we can conclude that SEB diblock copolymer or SEBS triblock copolymer, having about 30 wt % or more of 1-butene in the PEB block of

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201

the copolymer, can function as an effective compatibilizing agent for PS/PP blends if the TODT of the block copolymer is lower than the melt blending temperature. 4.2.2.3 PS/PI/(PS-block-PB) Ternary Blends Let us now consider PS/PI/(PS-block-PB) ternary blends, in which SB diblock copolymers have the following microstructures: (1) SB-14/11 having 9% 1,2-addition, (2) SB-13/12 having 44% 1,2-addition, and (3) SB-9/8 having 89% 1,2-addition in the PB block. Figure 4.12 gives TEM images of three as-precipitated ternary blends: (a) 63/27/10 PS/PI/(SB-14/11), (b) 63/27/10 PS/PI(SB-13/12), and (c) 63/27/10 PS/PI/(SB-9/8), in which the numbers represent weight percentages.3 The molecular characteristics of the three SB diblock copolymers are given in Table 4.3. The micrographs in Figure 4.12 are be very useful for interpreting the morphology of the model ternary blends after annealing under isothermal conditions. Note in Figure 4.12 that the bright areas represent the homopolymer PS and the light dark areas represent the homopolymer PI.

Figure 4.12 (a) TEM image of as-precipitated 63/27/10 PS/PI/(SB-14/11) ternary blend.

(b) TEM image of as-precipitated 63/27/10 PS/PI/(SB-13/12) ternary blend. (c) TEM image of as-precipitated 63/27/10 PS/PI/(SB-9/8) ternary blend. (Reprinted from Chun and Han, Macromolecules 32:4030. Copyright © 1999, with permission from the American Chemical Society.)

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PROCESSING OF THERMOPLASTIC POLYMERS Figure 4.13 Temperature dependence of dynamic storage modulus obtained from dynamic temperature sweep experiments under isochronal conditions during heating for () SB-14/11, () SB-13/12, and () SB-9/8. (Reprinted from Chun and Han, Macromolecules 32:4030. Copyright © 1999, with permission from the American Chemical Society.)

Due to the low magniﬁcation in Figure 4.12, it is not possible to identify the SB diblock copolymer in the ternary blends. Figure 4.13 gives the results of isochronal dynamic temperature sweep experiments at ω = 0.01 rad/s for SB-14/11, SB-13/12, and SB-9/8, in which we can identify a critical temperature at which the value of G drops precipitously for each block copolymer. According to a rheological criterion presented in Chapter 8 of Volume 1 from Figure 4.13 we determine TODT ≈ 174 ◦ C for SB-14/11, TODT ≈ 193 ◦ C for SB-13/12, and TODT ≈ 123 ◦ C for SB-9/8. Figure 4.14a gives a TEM image of the 63/27/10 PS/PI/(SB-14/11) blend annealed at 190 ◦ C (above the TODT of SB-14/11) for 12 h, in which we observe a large PI200 drop that is uniformly covered by the block copolymer SB-14/11, and also smaller drops of macrophase-separated SB-14/11 drops. Figure 4.14b gives a TEM image of the 63/27/10 PS/PI/(SB-14/11) blend annealed at 160 ◦ C (below the TODT of SB-14/11) for 12 h, in which we observe part of the added block copolymer SB-14/11 (having lamellar microdomains) unevenly placed at the surface of a PI drop. Notice the difference in the thickness of SB-14/11 layer, at the same magniﬁcation, on the surface of a PI drop in two situations: one annealed at T > TODT of SB-14/11 and the other annealed at T < TODT of SB-14/11. This observation clearly demonstrates the importance of the TODT of a block copolymer in the compatibilization of PS/PI binary blends. A comparison of Figures 4.14a and 4.14b with Figure 4.12a shows that during isothermal annealing, the co-continuous morphology of the as-precipitated 63/27/10 PS/PI/(SB14/11) blend transformed into a dispersed morphology, and that after annealing the minor component PI forms drops, which are then dispersed in the major component PS. This observation is consistent with that made in Chapter 3. Note that the value of χ for a PI/PB pair (where the PI has 96% 1,4-addition and the PB has 9% 1,2-addition) is estimated to be 0.15 × 10−2 at 190 ◦ C and 0.13 × 10−2 at 160 ◦ C (Chun and Han 1999). Owing to the very small values of χ , though positive, the PI/PB pair may be regarded as being very close to an athermal system. This now may explain the reason why the block copolymer SB-14/11 covered the entire surface of the PI-200 drop (see Figure 4.14a) when the annealing temperature was higher than

Figure 4.14 TEM images of as-precipitated ternary blends after annealing at temperatures above or below the TODT of the SB diblock copolymer: 63/27/10 PS/PI/(SB-14/11) ternary blend after annealing for 12 h at (a) 190 ◦ C (T > TODT ) and (b) 160 ◦ C (T < TODT ); 63/27/10 PS/PI/(SB-13/12) ternary blend after annealing for 12 h at (c) 207 ◦ C (T > TODT ) and (d) 177 ◦ C (T < TODT ); 63/27/10 PS/PI/(SB-9/8) ternary blend after annealing for 12 h at (e) 160 ◦ C (T > TODT ) and (f) 110 ◦ C (T < TODT ). (Reprinted from Chun and Han, Macromolecules 32:4030. Copyright © 1999, with permission from the American Chemical Society.)

203

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PROCESSING OF THERMOPLASTIC POLYMERS

the TODT of SB-14/11. Conversely, when the annealing temperature was lower than the TODT of SB-14/11, neither PS nor PI chains would mix with the lamellar microdomains of PB block in SB-14/11. Figure 4.14c gives a TEM image of the 63/27/10 PS/PI/(SB-13/12) blend annealed at 207 ◦ C (above the TODT of SB-13/12) for 12 h, in which we observe a large PI drop having the block copolymer SB-13/12 with microdomain morphology gradually changing from lamellae near the surface of the drop to spheres deep inside the drop. This observation indicates that during the annealing at T > TODT of SB-13/12 for 12 h, solubilization occurred between the homopolymer PI and the block copolymer SB13/12, which was not the case in the 63/27/10 PS/PI(SB-14/11) blend (see Figure 4.14a). The PB block of SB-13/12 has 44% 1,2-addition, while the PB block of SB-14/11 has 9% 1,2-addition (see Table 4.3). The observed difference in morphology between the two ternary blends, 63/27/10 PS/PI/(SB-14/11) and 63/27/10 PS/PI/(SB-13/12), is attributable to the difference in miscibility between the PI/(SB-14/11) pair and the PI/(SB-13/12) pair. Speciﬁcally, the (1,4-PI)/(1,4-PB) pair has repulsive segmental interactions (positive values of χ ) (Takeno and Hashimoto 1997; Thudium and Han 1996), while the (1,4-PI)/(1,2-PB) pair has attractive segmental interactions (negative values of χ ) (Thudium and Han 1996). Therefore, as the amount of 1,2-addition in the PB block of an SB diblock is increased, better miscibility is expected between 1,4-PI and the SB diblock copolymer. This observation now explains the observed difference in morphology between the 63/27/10 PS/PI/(SB-14/11) ternary blend and the 63/27/10 PS/PI/(SB-13/12) blend, because the PB block of SB-13/12 has a greater amount of 1,2-addition than the PB block of SB-14/11. Figure 4.14d gives a TEM image of the 63/27/10 PS/PI/(SB-13/12) blend annealed at 177 ◦ C (below the TODT of SB-13/12) for 12 h, in which we observe smaller drops of block copolymer SB-13/12, maintaining lamellar microdomain structure, inside a larger PI drop. Observe the totally different morphologies inside the PI drops in Figures 4.14c and 4.14d. This observation can easily be understood from the point of view of the annealing temperatures employed (207 ◦ C versus 177 ◦ C) relative to the TODT (193 ◦ C) of the block copolymer SB-13/12. That is, when the 63/27/10 PS/PI/(SB-13/12) blend was annealed at 177 ◦ C (below the TODT of SB-13/12), the miscibility between the homopolymer PI and the microphase-separated PB block of the copolymer would be extremely poor, thus giving rise to the morphology displayed in Figure 4.14d. However, when the same blend was annealed at 207 ◦ C (above the TODT of SB-13/12), the miscibility between the homopolymer PI and the ﬂexible chains of the PB block of the copolymer would be immensely enhanced, giving rise to the morphology displayed in Figure 4.14c. Again, the above observation demonstrates how important is the choice of annealing (or processing) temperature, relative to the TODT of a block copolymer, in the compatibilization of a pair of immiscible homopolymers. A comparison of Figures 4.14c and 4.14d with Figure 4.12b reveals that during isothermal annealing, the co-continuous morphology of the as-precipitated 63/27/10 PS/PI/(SB-13/12) blend transformed into a dispersed morphology with the minor PI phase dispersed in the continuous PS phase. The value of χ for a PI/PB pair (where the PI has 96% 1,4-addition and the PB has 44% 1,2-addition in SB-13/12) is estimated to be 0.9 × 10−3 at 207 ◦ C and 0.5×10−3 at 177 ◦ C (Chun and Han 1999, 2000; Thudium and Han 1996). Note that the magnitude of these χ values is smaller than that for the PI/PB (where the pair PI has 96% 1,4addition and the PB has 9% 1,2-addition in SB-14/11) given above, indicating that the

COMPATIBILIZATION OF TWO IMMISCIBLE HOMOPOLYMERS

205

miscibility between 1,4-PI and PB is enhanced as the amount of 1,2-addition in PB is increased. Owing to the very small values of χ, though positive, the PI/PB pair having 96% 1,4-addition in PI and 44% 1,2-addition in PB may be regarded as being very close to an athermal system. This now explains the reason why the block copolymer SB13/12 entered into the PI drop and then solubilized at least in part in the PI phase, giving rise to spherical microdomains deep inside the PI drop and lamellar microdomains near the surface of the PI drop. In other words, it is clear that the miscibility between the homopolymer PI and the block copolymer SB-13/12 is greatly enhanced compared with that between the homopolymer PI and the block copolymer SB-14/11 (compare Figure 4.14a with Figure 4.14c) when the annealing temperature was higher than the TODT of the respective block copolymer. This is attributable to the smaller χ value for the pair of the homopolymer PI and the PB block of SB-13/12 compared with that for the pair of the homopolymer PI and the PB block of SB-14/11. However, when the annealing temperature was lower than the TODT of SB-13/12, little mixing between the block copolymer SB-13/12 and the homopolymer PI or little solubilization of the block copolymer SB-13/12 in the homopolymer PI occurred (see Figure 4.14d). This observation points out once again that the TODT of a block copolymer plays a dominant role in the compatibilization of two immiscible homopolymers. Figure 4.14e gives a TEM image of the 63/27/10 PS/PI/(SB-9/8) ternary blend annealed at 160 ◦ C (above the TODT of SB-9/8) for 12 h, in which we observe a PI drop having a uniform distribution of spherical microdomains throughout and smaller drops of macrophase-separated SB-9/8 dispersed in the PS matrix. A comparison of Figure 4.14e with Figure 4.12c shows that during isothermal annealing, the cocontinuous morphology of the as-precipitated 63/27/10 PS/PI/(SB-9/8) ternary blend transformed into a dispersed morphology with the minor PI phase forming drops dispersed in the major PS phase forming the continuous phase. Notice the difference in morphology between the 63/27/10 PS/PI/(SB-9/8) ternary blend (Figure 4.14e) and the 63/27/10 PS/PI/(SB-13/12) ternary blend (Figure 4.14c). The observed difference in morphology between the two ternary blends is attributable to the difference in the microstructures of PB block of the respective block copolymers: 44% 1,2-addition in SB-13/12 and 89% 1,2-addition in SB-9/8. That is, during annealing at a temperature above the TODT of the respective block copolymer, a much faster mixing (or solubilization) occurs between the ﬂexible chains of the PB block of SB-9/8 and the ﬂexible chains of homopolymer PI, giving rise to spherical microdomain structure, than between the ﬂexible chains of the PB block of SB-13/12 and the ﬂexible chains of homopolymer PI. The value of χ for a PI/PB pair (where the PI has 96% 1,4-addition and the PB has 89% 1,2-addition) is estimated to be −0.17 × 10−2 at 160 ◦ C and −0.28 × 10−2 at 110 ◦ C (Chun and Han 1999, 2000; Thudium and Han 1996), and thus the pair of homopolymer PI and the PB block of SB-9/8 has attractive segmental interactions. This observation explains why the block copolymer SB-9/8 is solubilized in the PI drop, giving rise to a morphological transformation from lamellae to spheres (Figure 4.14e) when the annealing temperature was higher than the TODT of SB-9/8. In other words, whereas the chains of homopolymer PI would not mix with the PB block of SB-9/8 at T < TODT of the block copolymer, they would mix at T > TODT of the block copolymer, swelling the PB block chains to increase the average distance of the neighboring junctions of the block copolymers at the interface. The increase of the junction

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distance would have decreased the PS microdomain thickness in order to keep the density of PS lamellae constant. Owing to the attractive segmental interactions between the homopolymer PI and the PB block of SB-9/8 at T > TODT , a morphological transition occurred from lamellar to spherical microdomain structure. The spherical PS microdomains inside the PI drops have lost long-range spatial order and so are randomly distributed in the PI phase. Figure 4.14f gives a TEM image of the 63/27/10 PS/PI/(SB-9/8) ternary blend annealed at 110 ◦ C (below the TODT of SB-9/8) for 12 h, in which we observe irregularly shaped PI drops dispersed in the PS matrix and the block copolymer SB-9/8 appears to be distributed unevenly on the surface of PI drop. The observations made in Figures 4.14e and 4.14f are consistent with those made above with reference to Figures 4.14a and 4.14b, and also Figures 4.14c and 4.14d, insofar as the role that the TODT of a block copolymer plays in the compatibilization of a pair of immiscible homopolymers, PS and PI. 4.2.2.4 PS/PI/(PαMS-block-PI) Ternary Blends Let us consider another model A/B/(A-block-C) ternary system, PS/PI/(PαMS-blockPI) ternary blends. For brevity, PαMS-block-PI copolymers will be referred to as MSPI diblock copolymers. The molecular characteristics of two MSPI diblock copolymers are given in Table 4.5. Figure 4.15 shows the temperature dependence of G for MSPI-23 and MSPI-25, which was obtained, during heating, from isochronal dynamic temperature sweep experiments at ω = 0.01 rad/s. From Figure 4.15 we determine the TODT of MSPI-23 to be about 175 ◦ C and the TODT of MSPI-25 to be much higher than 250 ◦ C. In Figure 4.15 we observe that G for MSPI-25 initially decreases rapidly with increasing temperature and then at a slower rate until reaching 250 ◦ C, at which G starts to increase. It is believed that a sudden increase of G at 250 ◦ C for MSPI-25 is due to the occurrence of cross-linking reactions of PI blocks of MSPI-25. Also given in Figure 4.15 are TEM images of the two block copolymers, MSPI-23 and MSPI-25, each having lamellar microdomain structure. This is expected, because both block copolymers have almost equal volume fractions (Table 4.5). Notice in Figure 4.15 that the Table 4.5 Molecular characteristics of model diblock copolymers

Sample Code MSPI-23c MSPI-25c SEB-15d SEB-19e SB-17f

Mn a

Mw /Mn b

wt % PS or P αMS

TODT (◦ C)

23,000 25,000 15,000 19,000 17,000

1.04 1.08 1.12 1.04 1.04

51.1 50.4 52.7 52.8 53.0

175 >250 125 258 123

a

Membrane osmometry was employed to determine the number-average molecular weight (Mn ). b GPC was used to determine polydispersity index, Mw /Mn . c PαMS-block-PI copolymer having 59% 3,4-addition in PI block. d PS-block-PEB copolymer having 94% 1-butene in PEB block. e PS-block-PEB copolymer having 90% 1-butene in PEB block. f PS-block-PB copolymer having 89% 1,2-addition in PB block.

Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.

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207

Figure 4.15 Variation of G with temperature during isochronal dynamic temperature sweep

experiments at ω = 0.01 rad/s for MSPI-23 () and MSPI-25 (∇). The insets are TEM images of MSPI-23 and MSPI-25. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

TODT of MSPI diblock copolymer is extremely sensitive to molecular weight. Specifically, an increase in molecular weight (Mn ) from 2.3 × 104 (MSPI-23) to 2.5 × 104 (MSPI-25) increased TODT by more than 75 ◦ C. The above observations indicate that there is a very narrow range of molecular weight over which the TODT of MSPI diblock copolymer can lie between the Tg (140 ◦ C) and the thermal degradation/cross-linking temperature (about 220 ◦ C) of PI block. Figure 4.16a gives a TEM image of a 63/27/10 PS/PI/(MSPI-23) ternary blend3 prepared by melt blending for 5 min at 200 ◦ C, higher than the TODT (175 ◦ C) of MSPI-23 (see Figure 4.15). The TEM image in Figure 4.16a shows highly uneven distributions of MSPI-23 on the surface of the discrete phase of homopolymer PI. Interestingly, however, the TEM image given in Figure 4.16b shows that when the melt-blended specimen was annealed at 200 ◦ C for 12 h, MSPI-23 diffused into the PI drop, giving rise to an equilibrium morphology. Figure 4.17a gives a TEM image of a 63/27/10 PS/PI/(MSPI-25) ternary blend3 prepared by melt blending for 5 min at 200 ◦ C, which is lower than the TODT (>250 ◦ C) of MSPI-25 (see Figure 4.15). The TEM image in Figure 4.17a shows highly uneven distributions of MSPI-25 in the PS/PI blend. The TEM image in Figure 4.17b shows that when the melt-blended specimen was annealed at 200 ◦ C for 12 h, PI aggregates break up, forming smaller drops, and MSPI-25 is distributed unevenly at the PI/PS interface as well as between two PI drops. Further, the TEM image in Figure 4.17b shows the presence of MSPI-25 drops dispersed in the PS matrix. Again, the TODT of diblock

208

PROCESSING OF THERMOPLASTIC POLYMERS Figure 4.16 (a) TEM image of the

63/27/10 PS/PI/(MSPI-23) ternary blend prepared by melt blending for 5 min at 200 ◦ C, which is above the TODT (175 ◦ C) of MSPI-23. The block copolymer is distributed nonuniformly on the surface of a PI droplet. (b) TEM image of the melt-blended 63/27/10 PS/PI/(MSPI-23) ternary blend after annealing for 12 h at 200 ◦ C. The lamella-forming block copolymer diffused into the PI droplet, giving rise to an equilibrium morphology.

copolymer plays an important role in determining the locations (or distributions) of the added copolymer in the mixtures of two immiscible homopolymers. Three interaction parameters, αPS/PαMS , αPS/PI , and αPαMS/PI , are associated with the 63/27/10 PS/PI/MSPI ternary blend. We have the following information (Chun and Han 2000): αPS/PI = −0.3188 × 10−2 + 1.6585/T − 0.02018φPS/PI /T

(4.12)

αPαMS/PS = −0.0028 × 10−2 + 0.0319/T − 0.0009φPαMS/PS /T

(4.13)

αPαMS/PI = −0.1735 × 10−2 + 1.1735/T − 0.0929φPαMS/PI /T

(4.14)

where, for instance, φPαMS/PI is the volume fraction of PαMS in the PαMS/PI pair in the PS/PI/(PαMS-block-PI) ternary blend: φPαMS/PI = PαMS /(PαMS +PI ), with PαMS being the volume fraction of PαMS in a given PS/PI/(PαMS-block-PI) ternary blend. For the 63/27/10 PS/PI/(MSPI-23) ternary blend at 200 ◦ C we have χPαMS/PS = 0.447 × 10−2 , χPS/PI = 0.271 × 10−1 , and χPαMS/PI = 0.725 × 10−1 . For the calculations

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209

Figure 4.17 (a) TEM image of the 63/27/10 PS/PI/(MSPI-25) ternary blend prepared by melt blending for 5 min at 200 ◦ C (below the TODT of MSPI-25). The block copolymer forms separate domains, which are dispersed in the PS matrix. (b) TEM image of the melt-blended 63/27/10 PS/PI/(MSPI-25) ternary blend after annealing for 12 h at 200 ◦ C. The block copolymer is located between two PI droplets dispersed in the PS matrix, and also forms separate domains, which are dispersed in the PS matrix.

of χ given above, χ = Vref α was used, where Vref is a reference volume deﬁned by Vref = [(MA vA )(MB vB )]1/2 with MA and MB being the molecular weights of monomer units for components A and B, respectively, and vA and vB being the speciﬁc volumes of components A and B, respectively. For this, the following expressions for speciﬁc volume were used: vPS = 0.9217 + 5.412 × 10−4 t + 1.687 × 10−4 t 2 vPI = 1.0771 + 7.22 × 10

−4

t + 2.46 × 10

vPαMS = 0.87 + 5.08 × 10−4 t

−4 2

t

(4.15) (4.16) (4.17)

where t is the temperature in Celsius. Thus, it is reasonable to conclude that the poor distribution of MSPI-23 at the PS/PI interface during melt blending at 200 ◦ C for 5 min, observed in the TEM image in Figure 4.16a, is due to the repulsive segmental interactions between the homopolymer PI and the PαMS block of the copolymer. However, when the 63/27/10 PS/PI/(MSPI23) ternary blend was annealed at 200 ◦ C for 12 h, apparently χPαMS/PS = 0.447×10−2

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was sufﬁciently small for MSPI-23 to diffuse into PI drops, giving rise to the equilibrium morphology given in the TEM image in Figure 4.16b.

4.2.3

A/B/(C-block-D) Ternary Blends

Several research groups (Auschra et al. 1993a, 1993b; Chun and Han 2000; Jo et al. 1991; Kim et al. 1998, 1999; Ouhadi et al. 1986a) have investigated the compatibilization of two immiscible homopolymers, A and B, with a C-block-D copolymer. Speciﬁcally, Ouhadi et al. (1986a) prepared poly(2,6-dimethyl-1,4-phenylene oxide) (PPO)/poly(vinylidene ﬂuoride) (PVDF)/(PS-block-PMMA) ternary blends at 220 ◦ C for 5 min on a two-roll mill. Since PPO and PS are known to be miscible (Aurelio de Araujo et al. 1988; Fried et al. 1978; Maconnachie et al. 1984a, 1984b; Weeks et al. 1977) and PVDF and PMMA are also known to be miscible (Kwei et al. 1977; Nishi and Wang 1975; Noland et al. 1971; Paul and Altamirano 1975; Wang and Nishi 1977; Wendorff 1980), it sufﬁces to state that PPO/PVDF/(PS-block-PMMA) ternary blends meet with the thermodynamic requirements for compatibilization. Jo et al. (1991) prepared PPO/poly(styrene-ran-acrylonitrile) (PSAN)/(PS-block-PMMA) ternary blends at 270−290 ◦ C using an internal mixer. Auschra et al. (1993a, 1993b) also prepared PPO/PSAN/(PS-block-PMMA) ternary blends by coprecipitation from dilute solution and subsequently compression molding at 240 ◦ C for 45 min. Since PPO and PS are known to be miscible and PSAN and PMMA are also known to be miscible (Bernstein et al. 1977; Kruse et al. 1976; McBriety et al. 1978; McMaster 1975; Naito et al. 1978; Stein et al. 1974), PPO/PSAN/(PS-block-PMMA) ternary blends meet with the thermodynamic requirements for compatibilization. Kim et al. (1998, 1999) prepared poly(cyclohexyl methacrylate) (PCHMA)/PSAN/(PS-block-PMMA) ternary blends by ﬁrst casting from solutions of PSAN, PCHMA, and PS-block-PMMA copolymer followed by drying at 70 ◦ C for 24 h. The dry-mixed powders were then molded to form disks at 200 ◦ C under vacuum. The disk samples prepared were sheared in parallel plate geometry. PCHMA and PS have attractive interactions (Kim et al. 1999), while PSAN and PMMA are known to have attractive segmental interactions (Maconnachie et al. 1984a). Hence, PCHMA/PSAN/(PS-block-PMMA) ternary blends also meet with the thermodynamic requirements for compatibilization. That is, all ternary blend systems referred to above satisfy the thermodynamic requirements depicted in Figure 4.3. However, none of the investigators cited here presented experimental evidence showing that an “interphase” was formed between two immiscible homopolymers in the presence of PS-block-PMMA copolymer during melt blending or during isothermal annealing of solvent-cast specimens. The absence of an interphase in those ternary blends may be attributable, as discussed above with reference to PS/PMMA/(PS-block-PMMA) ternary blends, to the high molecular weights of the PS-block-PMMA copolymers employed. Speciﬁcally, Ouhadi et al. (1986a) used a PS-block-PMMA copolymer having Mw = 1.25 × 105 and about 50 wt % of each block, Jo et al. (1991) used PS-block-PMMA copolymer having Mw = 9.2 × 104 and 57 mol % of PS block, Auschra et al. (1993a, 1993b) used several PS-block-PMMA copolymers having Mw ranging from 1.7 × 104 to 2.58 × 105 and 40–48 wt % of PS block, and Kim et al. (1998, 1999) used two PS-block-PMMA copolymers, one having Mw = 3.34×105 and 48 wt % of PS block and the other having

COMPATIBILIZATION OF TWO IMMISCIBLE HOMOPOLYMERS

211

Mw = 2.99 × 105 and 52.8 wt % of PS block. As can be surmised from Table 4.2, such high-molecular-weight PS-block-PMMA copolymers must have had very high values of TODT , which far exceeded the melt blending or annealing temperatures employed in the experiments. This observation indicates that the satisfaction of the thermodynamic requirements alone is not sufﬁcient for a block copolymer to function as an effective compatibilizing agent for two immiscible homopolymers. 4.2.3.1 PMMA/PP/(PαMS-block-PI) Ternary Blends Figure 4.18a gives a TEM image of a 63/27/10 PMMA/PP/(MSPI-23) ternary blend prepared by melt blending at 200 ◦ C for 5 min. Note that the TODT of MSPI-23 is about 175 ◦ C (Figure 4.15), and thus the melt blending temperature employed is higher than the TODT of MSPI-23. In Figure 4.18a, we observe that MSPI-23 is not located at the PMMA/PP interface, but rather it formed separate domains. Figure 4.18b gives a TEM image of a 63/27/10 PMMA/PP/(MSPI-23) ternary blend that was annealed at 200 ◦ C for 12 h after being melt blended at 200 ◦ C for 5 min. In Figure 4.18b, we can hardly observe any distribution of MSPI-23 at the PMMA/PP interface even

Figure 4.18 (a) TEM image of the 63/27/10 PMMA/PP/(MSPI-23) ternary blend prepared by melt blending for 5 min at 200 ◦ C (above the TODT of MSPI-23). The block copolymer forms a separate phase and is dispersed in the PMMA matrix. (b) TEM image of the melt-blended 63/27/10 PMMA/PP/ (MSPI-23) ternary blend after annealing for 12 h at 200 ◦ C (above the TODT of MSPI-23). There is no evidence of the presence of the block copolymer at the interface. Note that all six values of χ are positive: χPMMA/PαMA > 0, χPMMA/PP > 0, χPMMA/PI > 0, χPP/PαMS > 0, χPP/PI > 0, and χPαMS/PI > 0. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

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after being annealed for 12 h for at 200 ◦ C, which is above the TODT of MSPI-23. These observations should not be surprising, because all six interaction parameters associated with this ternary blend are positive: χPMMA/PαMA > 0, χPMMA/PP > 0, χPMMA/PI > 0, χPP/PαMS > 0, χPP/PI > 0, and χPαMS/PI > 0. This observation clearly indicates that the compatibilization of two immiscible homopolymers, A and B, with a C-block-D copolymer would not occur even when the melt blending temperature is higher than the TODT of the block copolymer, unless the thermodynamic requirements are met (see Figure 4.3). In developing an equilibrium mean ﬁeld theory for predicting the polymer component proﬁles at the interface in A/B/(C-block-D) ternary blends, Vilgis and Noolandi (1990) used positive values for all six interaction parameters: χA/B > 0, χC/D > 0, χA/C > 0, χA/D > 0, χB/C > 0, and χB/D > 0. On the basis of the experimental results presented in Figure 4.18, it does not seem possible that under such a situation the C-block-D copolymer can ever function as an effective compatibilizing agent for the two immiscible homopolymers, A and B, because there are no attractive segmental interactions between homopolymer A and block C of copolymer, and between homopolymer B and block D of copolymer, or between homopolymer A and block D of copolymer, and between homopolymer B and block C of copolymer. 4.2.3.2 PS/PB/(PαMS-block-PI) Ternary Blends Let us now consider a model PS/PB/(PαMS-block-PI) ternary blend4 having attractive segmental interactions between the homopolymer PB and the PI block of copolymer. Figure 4.19 gives TEM images of a 63/27/10 PS/PB/(MSPI-23) ternary blend prepared by melt blending for 5 min at 200 ◦ C, which is higher than the TODT (175 ◦ C) of the block copolymer MSPI-23. Part (b) of Figure 4.19 is at a higher magniﬁcation than part (a). It can be seen in Figure 4.19 that the block copolymer MSPI-23 with a layer thickness of about 30 nm uniformly covers the entire surface of the PB drop dispersed in the PS matrix. This is quite a contrast to Figure 4.18a. In Figure 4.20, we observe that when the melt-blended specimen was annealed at 200 ◦ C for 12 h, the block copolymer MSPI-23, which had uniformly been distributed on the surface of the PB drop during melt blending, diffused into the PB drop during isothermal annealing for 12 h at 200 ◦ C, giving rise to an equilibrium morphology that represents a homogeneously mixed PB/(MSPI-23) blend. Figure 4.21 gives a TEM image of a rapidly precipitated 63/27/10 PS/PB/(MSPI-23) ternary blend,5 which was annealed for 12 h at 200 ◦ C (above the TODT of MSPI-23). In Figure 4.21, we observe that during isothermal annealing MSPI-23 diffused into PB drop, giving rise to an equilibrium morphology. There is a great similarity between Figure 4.20 and Figure 4.21. Six interaction parameters are associated with the 63/27/10 PS/PB/(MSPI-23) ternary blend. We have the following expressions (Chun and Han 2000): αPαMS/PB = −0.5069 × 10−2 + 2.693/T − 0.1368φPαMS/PB /T

(4.18)

αPS/PB = −0.1699 × 10−2 + 1.090/T + 0.0351φPS/PB /T

(4.19)

χPI/PB = 0.2359 × 10−2 − 5.76/T + 2.22φPI/PB /T

(4.20)

Figure 4.19 TEM images of the 63/27/10 PS/PB/(MSPI-23) ternary blend prepared by melt blending for 5 min at 200 ◦ C (above the TODT of MSPI-23). The TEM image in part (b) is a magniﬁcation of part of the dispersed drop shown in part (a) and shows a layer of the block copolymer about 30 nm thick located at the PS/PB interface. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

Figure 4.20 TEM image of a melt-blended 63/27/10 PS/PB/(MSPI-23) ternary blend after annealing for 12 h at 200 ◦ C (above the TODT of MSPI-23). The molecular weight and polydispersity index of the two homopolymers, PS and PB, employed in this ternary blend are: the PS has Mn = 1.1 × 105 and Mw /Mn = 2.01, and the PB has Mn = 7.1 × 104 and Mw /Mn = 1.11. The dark areas represent a PB drop in which the homopolymer PB and the block copolymer are mixed homogeneously at a segmental level. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

213

214

PROCESSING OF THERMOPLASTIC POLYMERS Figure 4.21 TEM image of an

as-precipitated 63/27/10 PS/PB/(MSPI-23) ternary blend after annealing for 12 h at 200 ◦ C (above the TODT of MSPI-23). The molecular weight and polydispersity index of the two homopolymers, PS and PB, employed in this ternary blend are much lower than those employed in Figure 4.20. Namely, the PS has Mn = 1.0 × 104 and Mw /Mn = 1.02, and the PB has Mn = 1.1 × 104 and Mw /Mn = 1.09. The dark areas represent a PB drop in which the homopolymer PB and the block copolymer are mixed homogenously at a segmental level.

where φPαMS/PB , for instance, is the volume fraction of PαMS in the PαMS/PB pair in the PS/PB/(PαMS-block-PI) ternary blend: φPαMS/PB = PαMS /(PαMS + PB ), with PαMS being the volume fraction of PαMS in a given PS/PB/(PαMS-block-PI) ternary blend. For the 63/27/10 PS/PB/(MSPI-23) ternary blend at 200 ◦ C we have χPαMS/PS = 0.447 × 10−2 , χPS/PI = 0.271 × 10−1 , χPαMS/PI = 0.725 × 10−1 , χPαMS/PB = 0.681 × 10−1 , χPS/PB = 0.701 × 10−1 , and χPI/PB = −0.601 × 10−2 . Thus, we have very weak attractive segmental interactions (χPI/PB = −0.601 × 10−2 ) between the PI and PB pair and very weak repulsive segmental interactions (χPαMS/PS = 0.447 × 10−2 ) between the PS and PαMS pair. Note that PB in the PS/PB/(PαMS-block-PI) ternary blend has 90% 1,2-addition5 and that PI and 3,4-PB are not miscible, while PI and 1,2-PB are miscible. In order to explain the morphologies, given in Figures 4.19 and 4.20, of the 63/27/10 PS/PB/(MSPI-23) ternary blend, let us look at Figure 4.22. Figure 4.22a shows schematically the distribution of component polymers after melt blending at 200 ◦ C for 5 min, where the lightly shaded areas represent the chains of homopolymer PB and the chains of PI block of PαMS-block-PI copolymer (MSPI-23), and the darker areas represent a newly formed interphase consisting of the chains of homopolymer PB and the chains of PI block of MSPI-23. The block copolymer chains in the middle of Figure 4.22a depict a thin layer of MSPI-23. During melt blending at 200 ◦ C (>TODT of MSPI-23) for 5 min in an internal mixer, the homogeneous molecules of MSPI-23 covered the entire surface of PB drops and formed a block copolymer ﬁlm with a thickness of about 30 nm at the PS/PB interface (see Figure 4.19). This was made possible because attractive segmental interactions (χPI/PB < 0) existed between homopolymer PB and PI block of MSPI-23. When the ternary blend was cooled down to room temperature after melt blending, microphase separation of MSPI-23 would have occurred at the PS/PB interface, giving rise to alternating layers of lamellar microdomain structure, as shown in the TEM image in Figure 4.19. Presumably, an interphase consisting of the chains of homopolymer PS and the chains of PαMS block of MSPI-23 might have been formed during melt blending. Such an interphase cannot be discerned in the TEM

COMPATIBILIZATION OF TWO IMMISCIBLE HOMOPOLYMERS

215

Figure 4.22 Schematic showing the distributions of components in the 63/27/10 PS/PB/(MSPI-23) ternary blend (a) after melt blending at 200 ◦ C for 5 min and (b) after annealing at 200 ◦ C for 12 h of the melt-blended specimen. ( ) PI block chain, (——–) PαMS block chain, (· · · · · · ) homopolymer PB chain, (- - - -) homopolymer PS chain. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

image in Figure 4.19 because osmium tetroxide (OsO4 ) did not stain the homopolymer PS and the PαMS block of MSPI-23. Figure 4.22b shows schematically the distribution of component polymers after the melt-blended 63/27/10 PS/PB/(MSPI-23) specimen was annealed at 200 ◦ C for 12 h, where the dark areas on the left side represent a homogeneous mixture consisting of homopolymer PB and MSPI-23, which was formed during isothermal annealing at 200 ◦ C for 12 h. During isothermal annealing, all of the MSPI-23 molecules, which had been distributed uniformly on the surface of PB drops during melt blending, diffused into the PB drop, forming an equilibrium morphology, owing to the attractive segmental interactions (χPI/PB = −0.601 × 10−2 ) existing between the homopolymer PB and the PI block of MSPI-23. Owing to the repulsive segmental interactions (χPαMS/PS = 0.447 × 10−2 ), though very weak, between the homopolymer PS and the PαMS block of MSPI-23, all MSPI-23 molecules were dragged into PB drops during the isothermal annealing, suggesting that the attractive segmental interactions between the homopolymer PB and the PI block of MSPI-23 played the predominant role in the formation of the equilibrium morphology depicted schematically in Figure 4.22b. Figure 4.23a gives a TEM image of a 63/27/10 PS/PB/(MSPI-25) ternary blend prepared by melt blending at 200 ◦ C for 5 min. Note that the melt blending temperature (200 ◦ C) employed is lower than the TODT (250 ◦ C) of MSPI-25 (see Figure 4.15). The TEM image given in Figure 4.23a shows that MSPI-25 and PB formed separate domains and dispersed in the PS matrix. It is clear that MSPI-25 is not uniformly

216

PROCESSING OF THERMOPLASTIC POLYMERS Figure 4.23 (a) TEM image of the

63/27/10 PS/PB/(MSPI-25) ternary blend prepared by melt blending for 5 min at 200 ◦ C (below the TODT of block copolymer). The very dark areas represent the PB phase, and the lamella-forming block copolymer forms a separate domain. (b) TEM image of the melt-blended 63/27/10 PS/PB/(MSPI-25) ternary blend after annealing for 12 h at 200 ◦ C (below the TODT of block copolymer). (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

distributed on the surface of the PB drop. Figure 4.23b gives a TEM image of a 63/27/10 PS/PB/(MSPI-25) ternary blend that was annealed at 200 ◦ C for 12 h after being melt blended at 200 ◦ C for 5 min. Note that the annealing temperature employed is below the TODT (250 ◦ C) of MSPI-25. It can be seen in the TEM image of Figure 4.23b that there is no evidence that MSPI-25 diffused into the PB drop. From the observations made in the TEM images in Figures 4.19, 4.20, and 4.23, we can conclude that the TODT of PαMS-block-PI copolymer, in relation to melt blending temperature, plays the decisive role in determining whether or not the block copolymer can be distributed uniformly at the PS/PB interface during melt blending at 200 ◦ C for 5 min. This can be explained by the differences in viscosity between MSPI-23 and MSPI-25 at 200 ◦ C. Figure 4.24 gives temperature dependence of complex viscosity |η∗ | curves for MSPI-23 and MSPI-25, which were obtained from isochronal dynamic temperature sweep experiments at ω = 0.01 rad/s. It can be seen in Figure 4.24 that at 200 ◦ C, the viscosity of MSPI-23 is exceedingly low and the viscosity of MSPI-25 is very high (2 × 104 Pa · s). Note that the molecular weight of MSPI-25 is only about 2,000 higher than that of MSPI-23 (see Table 4.5) and therefore such large difference in viscosity between MSPI-23 and MSPI-25 cannot possibly be due to the difference in molecular weight. Rather, it is due to the difference in the morphological state of the two block copolymers at 200 ◦ C (MSPI-23 in the disordered state and MSPI-25 in the microphase-separated state). Thus, we conclude that the observed difference

COMPATIBILIZATION OF TWO IMMISCIBLE HOMOPOLYMERS

217

Figure 4.24 Temperature dependence of complex viscosity |η∗ | for MSPI-23 () and MSPI-25 () at an angular frequency ω of 0.01 rad/s. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

in the distributions (or locations) of the block copolymer (compare Figure 4.19 with Figure 4.23) in the dispersed mixtures of PS and PB during melt blending at 200 ◦ C is attributable to the exceedingly large difference in melt viscosity between MSPI-23 and MSPI-25. That is, at 200 ◦ C the viscosity of MSPI-25 was so high that it had very limited mobility during melt blending, while the viscosity of MSPI-23 was so low that it could easily ﬂow during melt blending and thus spread over the entire surface of PB drops. The above observations have profound implications, conceptually and practically, in that the attractive segmental interactions alone are not sufﬁcient, though necessary, for a block copolymer to function as an effective compatibilizing agent for two immiscible homopolymers. 4.2.3.3 PαMS/PI/(PS-block-PB) Ternary Blends Figure 4.25 gives a TEM image of the rapidly precipitated 63/27/10 PαMS/PI/ (SB-17) ternary blend,6 which was annealed at 190 ◦ C for 12 h. Table 4.5 gives also the molecular characteristics of SB diblock copolymer, SB-17, which has the TODT of about 122 ◦ C and lamellar microdomain structure, as given in Figure 4.26. Since the annealing temperature employed (190 ◦ C) was above the TODT (122 ◦ C) of the block copolymer SB-17, during isothermal annealing the block copolymer diffused into the PI drop, forming a homogeneously mixed phase consisting of the homopolymer PI and the block copolymer SB-17. Note that attractive segmental interactions exist between the homopolymer 1,4-PI and the 1,2-PB block of copolymer, while very weak repulsive segmental interactions exist between the homopolymer PαMS and the PS block of copolymer. With regard to the chemical structures of two homopolymers and block copolymer, the 63/27/10 PαMS/PI/(SB-17) ternary blend is a reverse situation of the 63/27/10 PS/PB/(MSPI-23) ternary blend considered in Figure 4.21.

218

PROCESSING OF THERMOPLASTIC POLYMERS Figure 4.25 TEM image of

as-precipitated 63/27/10 PαMS/ PI/(SB-17) ternary blend after annealing for 12 h at 190 ◦ C (above the TODT of block copolymer), showing that the block copolymer diffused into a PI droplet and formed a homogeneously mixed phase consisting of the homopolymer PI and block copolymer SB-17. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

Figure 4.26 Variation of G

with temperature during an isochronal dynamic temperature sweep experiment at ω = 0.01 rad/s for SB-17 (). The inset is a TEM image of SB-17. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

Nevertheless, the values of the six interaction parameters remain almost the same even though the annealing temperatures employed in the two blends differ by 10 ◦ C. Therefore, it is not surprising to observe that the equilibrium morphology of the 63/27/10 PαMS/PI/(SB-17) ternary blend is very similar to that of the 63/27/10 PS/PB/(MSPI-23) ternary blend. 4.2.3.4 PPO/PP/(PS-block-PEB) Ternary Blends Let us consider another model PPO/PP/(PS-block-PEB) ternary bend.7 The molecular characteristics of the SEB diblock copolymers are given in Table 4.5, and Figure 4.27

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Figure 4.27 Variation of G with temperature during the isochronal dynamic temperature sweep experiment at ω = 0.01 rad/s for SEB-15 () and SEB-19 (). The insets are TEM images of SEB-15 and SEB-19. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

shows the temperature dependence of G for SEB-15 and SEB-19, obtained during heating from isochronal dynamic temperature sweep experiments at ω = 0.01 rad/s. From Figure 4.27, we determine TODT ≈ 125 ◦ C for SEB-15 and TODT ≈ 258 ◦ C for SEB-19. Note that both SEB-15 and SEB-19 are nearly symmetric diblock copolymers (see Table 4.5), and thus they form lamellar microdomain structures, as shown in Figure 4.27. Figure 4.28a gives a TEM image of the 63/27/10 PPO/PP/(SEB-15) ternary blend prepared by melt blending for 5 min at 200 ◦ C, which is above the TODT (125 ◦ C) of SEB-15. In Figure 4.28a the bright areas represent the PP phase, the gray areas in the matrix represent the PPO phase, and the dark areas inside the PPO matrix represent SEB-15 drops. Note that the major component (PPO) forms the continuous phase and that the minor component (PP) forms the discrete phase (i.e., drops) during melt blending. In Figure 4.28a we observe that an interphase is formed and is distributed uniformly at the PPO/PP interface. Figure 4.28b gives a TEM image of the melt-blended 63/27/10 PPO/PP/(SEB-15) which was subsequently annealed for 12 h at 200 ◦ C, which is higher than the TODT of SEB-15, showing that the interphase broadened during the isothermal annealing.

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PROCESSING OF THERMOPLASTIC POLYMERS Figure 4.28 (a) TEM image of a 63/27/10

PPO/PP/(SEB-15) ternary blend prepared by melt blending for 5 min at 200 ◦ C (above the TODT of block copolymer). (b) TEM image of the melt-blended 63/27/10 PPO/PP/(SEB-15) ternary blend after annealing for 12 h at 200 ◦ C. In both TEM images the block copolymer (the dark layer) is located at the PPO/PP interface. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

We have the following expressions (Chun and Han 2000): χPPO/PS = 0.121 − 77.9/T

(4.21)

χPP/PEB = −0.58 × 10−2 + 1.54/T

(4.22)

αPS/PEB = −0.6868 × 10−2 + 3.386/T + 0.0288φPS /T

(4.23)

and χPP/PS > 0, χPPO/PP > 0, and χPPO/PEB > 0. The miscibility between the homopolymer PP and SEB diblock copolymer depends on the microstructure of PEB of copolymer. As given in Table 4.5, the PEB block of SEB-15 has 94% 1-butene. At 200 ◦ C, we have αPPO/PS = −0.43 × 10−1 , αPP/PEB = −0.2 × 10−2 , and αPS/PEB = 0.3 × 10−1 . It is then clear that at 200 ◦ C the homopolymer PP and the PEB block of copolymer have attractive segmental interactions (are miscible on the segmental level), and also the homopolymer PPO and the PS block of copolymer have attractive segmental interactions (are miscible on the segmental level). Thus, it is reasonable to speculate that, during melt blending, the chains of PEB block of copolymer were stretched preferentially to the surface of the PP phase and the chains of PS block

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221

of copolymer were stretched preferentially to the surface of the PPO phase. This observation now explains why an interphase was formed during melt blending at 200 ◦ C for 5 min in the 63/27/10 PPO/PP/(SEB-15) ternary blend (see Figure 4.28). We can explain further the morphology, given in Figure 4.28, of the 63/27/10 PPO/PP/(SEB-15) ternary blend using the schematic given in Figure 4.29. Figure 4.29a shows schematically the distribution of component polymers after melt blending at 200 ◦ C for 5 min, where the light shaded areas represent the chains of homopolymer PPO stained by RuO4 , and the dark areas represent a newly formed interphase that consists of the homopolymer PPO and the PS block of SEB-15. Note that both PPO and PS phases are stained by RuO4 . The melt blending temperature employed is above the TODT of SEB-15. We are certain that an interphase consisting of the chains of homopolymer PP and the chains of PEB block of SEB-15 must have been formed during melt blending at 200 ◦ C for 5 min because they have attractive segmental interactions, as will be shown below. But, such an interphase cannot be discerned in the TEM image of Figure 4.28 because RuO4 stains neither the homopolymer PP nor the PEB block of SEB-15. Therefore, on the left side of the dark areas in Figure 4.29a, a vertical broken line is drawn to indicate an interphase, which consists of the chains of homopolymer PP and the chains of PEB block of SEB-15. Figure 4.29b shows schematically the distribution of component polymers after the melt-blended 63/27/10 PPO/PP/(SEB-15) was annealed at 200 ◦ C for 12 h. During isothermal annealing at 200 ◦ C, higher than the TODT of SEB-15, the ﬂexible chains of PS block continued to

Figure 4.29 Schematic showing the distributions of components in the 63/27/10 PPO/PP/(SEB-15) ternary blend (a) after melt blending at 200 ◦ C for 5 min and (b) after annealing at 200 ◦ C for 12 h of the melt-blended specimen. ( ) PS block chain, (——–) PEB block chain, (· · · · · · ) homopolymer PPO chain, and (- - - -) homopolymer PP chain. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

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stretch toward the PPO phase, and the ﬂexible chains of PEB block continued to stretch toward the PP phase. The reason for this is that attractive segmental interactions existed between the homopolymer PPO and the PS block of block copolymer (αPPO/PS < 0) and between the homopolymer PP and the PEB block of copolymer (αPP/PEB < 0). Hence, the thickness of the interphase on both sides was broadened during the isothermal annealing, as indicated in Figure 4.29b. Since the chains of each block of SEB-15 were stretched in opposite directions, the overall SEB-15 chains could not have been dragged preferentially to a particular homopolymer. Thus, the position of the newly formed interphase (or the junctions of the SEB-15 diblock copolymer) stayed more or less in the middle of the PPO and PP phases. Figure 4.30a gives a TEM image of a 63/27/10 PPO/PP/(SEB-19) ternary blend prepared by melt blending for 5 min at 200 ◦ C, which is below the TODT (258 ◦ C) of SEB-19 (see Figure 4.27). The TEM image in Figure 4.30a shows a very sharp interface between the PP and PPO phases, with little evidence of the formation of an interphase. Notice in Figure 4.30a that the block copolymer SEB-19 (the dark areas) formed separate domains dispersed in the PPO phase (the gray areas). Figure 4.30b gives a TEM image of a 63/27/10 PPO/PP/(SEB-19) ternary blend that was annealed

Figure 4.30 (a) TEM image of the

63/27/10 PPO/PP/(SEB-19) ternary blend prepared by melt blending for 5 min at 200 ◦ C (below the TODT of block copolymer). There is no evidence of the presence of the block copolymer at the PPO/PP interface, and the block copolymer forms a separated phase (the very dark areas) and is dispersed in the PPO matrix (the gray areas). (b) TEM image of the melt-blended 63/27/10 PPO/PP/(SEB-19) ternary blend after annealing for 12 h at 200 ◦ C (below the TODT of block copolymer). The morphology of the ternary mixture is more or less the same as that before isothermal annealing. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

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Figure 4.31 Temperature dependence of complex viscosity |η∗ | for SEB-15 () and SEB-19 ()

as determined by isochronal dynamic temperature sweep experiment at an angular frequency ω of 0.01 rad/s. (Reprinted from Chun and Han, Macromolecules 33:3409. Copyright © 2000, with permission from the American Chemical Society.)

for 12 h at 200 ◦ C after being melt blended for 5 min at 200 ◦ C (below the TODT of SEB-19). In Figure 4.30b, we observe little change in blend morphology after a prolonged isothermal annealing at 200 ◦ C. We can now explain the differences in blend morphology observed in Figures 4.28 and 4.30 by the differences in the viscosities of SEB-15 and SEB-19 at 200 ◦ C. Figure 4.31 shows the temperature dependence of |η∗ | for SEB-15 and SEB-19, from which we observe that at 200 ◦ C the viscosity of SEB-15 is exceedingly low while the viscosity of SEB-19 is very high (8 × 103 Pa·s). Note that the molecular weight of SEB-19 is only about 4,000 higher than that of SEB-15 (Table 4.5) and therefore such a large difference in viscosity between SEB-15 and SEB-19 is not likely to be due to the difference in molecular weight. Rather, it is due to the difference in the morphological state of the two block copolymers: under the condition of melt blending, SEB-15 is in the disordered state and SEB-19 is in the microphase-separated state. Thus, we conclude that the observed difference in the distributions (or locations) of block copolymer in the respective ternary blends (compare Figure 4.28 with Figure 4.30) is attributable to the difference in viscosity between SEB-15 and SEB-19. The microstructure of PEB block of PS-block-PEB copolymer played a very important role in determining the morphology of PPO/PP/(PS-block-PEB) ternary blends, and thus in the compatibilization of PPO/PP binary blends. This is because the extent of miscibility between homopolymer PP and PEB block of PS-block-PEB depends on the microstructure of the PEB block. Note that the PEB blocks of SEB-15 and SEB-19 have very high 1-butene contents (Table 4.5). If the PEB block of copolymer has very low 1-butene content, which then would be very close to polyethylene, binary mixtures of such PEB block of copolymer and homopolymer PP would not be miscible.

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PROCESSING OF THERMOPLASTIC POLYMERS

Reactive Compatibilization of Two Immiscible Polymers

Another method of compatibilization of two immiscible polymers is to use a reactive third polymer which, when added, may undergo chemical reactions with one or both of the two polymers. Usually, it is necessary that this reactive polymer be miscible with one of the blend components and reactive with the other blend component. Figure 4.32 shows schematically the dual roles that reactive compatibilizing agents are expected to play. One can then easily surmise that a reactive compatibilizing agent must have two components; thus, graft polymers, random copolymers, and functionalized block copolymers have been used. The basic principles involved with “reactive compatibilization” are different from those involved with “reactive blending.” A number of research groups have investigated reactive compatibilization of two immiscible polymers, but there are too many papers to cite them all here. It is readily apparent that there are only a limited number of chemical structures that can fulﬁll the requirements for being a reactive compatibilizing agent. The following ternary blends have been investigated: (1) maleic anhydride (MA) grafted SEBS block copolymer (SEBS-graft-MA) with PPO/polyetherimide-modiﬁed epoxy networks (Girard-Reydet 1999), (2) MA grafted polypropylene (PP-graft-MA) with PP/LDPE binary blends (Tselios et al. 1998) or with PP/nylon 6 binary blends (Cartier and Hu 1999), (3) glycidyl methacrylate (GMA) grafted polypropylene (PP-graftGMA) with PP/nylon 10 blends (Zhang et al. 1997) or with PP/PET binary blends (Champagne et al. 1999), (4) copolymers of styrene and maleic anhydride poly(Sstat-MA) with nylon 6/ABS binary blends (Triacca et al. 1991), with PPO/nylon 6 binary blends (Chiang and Chang 1997), PMMA/nylon 6 binary blends (Dedecker and Groeninckx 1998a, 1998b), or with nylon 12/PS binary blends (Dedecker et al. 1998), and (5) random copolymers of styrene and glycidyl methacrylate poly(S-ranGMA) with PS/poly(butylene terephthalate) (PBT) binary blends (Jeon and Kim 1988). Other types of reactive compatibilizing agents have also been used (Kim et al. 1997; Majumdar et al. 1994a, 1994b, 1994c, 1994d, 1997; Park et al. 1997; Pietransanta et al. 1999; Wildes et al. 1999; Zhang and Yin 1997). During melt blending, a reactive compatibilizing agent is expected to form a graft polymer. The graft copolymer formed in situ is located preferentially at the interface between two immiscible polymers to act as an effective compatibilizing agent. The chemistry of the reactions depends on the chemical structures of the reactive compatibilizer and the homopolymers to be melt blended. In this section, instead of

Figure 4.32 Schematic showing the roles of a

reactive compatibilizing agent when melt blended with two immiscible homopolymers.

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225

presenting in detail how each of the reactive compatibilizing agents reported in the literature functions, we consider two examples to illustrate how reactive compatibilization works. Let us ﬁrst consider the use of SEBS-graft-MA polymer to compatibilize two immiscible polymers. In the preceding section, we have shown that commercial SEBS triblock copolymer (e.g., Kraton G Series) cannot be used as an effective compatibilizing agent (see Figure 4.10). However, SEBS-graft-MA polymer can be used as an effective compatibilizing agent for certain pairs of polymers if the maleic anhydride grafted onto the PEB midblock of copolymer can have chemical reactions with one of the blend components and the PS endblock of copolymer is miscible with the other blend component. One such pair of polymers to be melt blended is PPO/nylon 6 (or nylon 6,6), because PPO and PS are miscible (as discussed in the preceding section) and the carboxyl groups in maleic anhydride can react with the amine end groups of nylon 6 (or nylon 6,6). Figure 4.33 gives reaction schemes between SEBS-graft-MA polymer and nylon 6. Needless to say, when a functionalized block copolymer is used, the TODT of the block copolymer is not an issue because the chemical reactions determine the effectiveness of compatibilization. Let us now consider another example, compatibilization of PPO/nylon 6 binary blends with poly(styrene-stat-maleic anhydride) having 8% maleic anhydride as reactive compatibilizing agent. For brevity, poly(styrene-stat-maleic anhydride) will be referred to as SMA. Figure 4.34 gives reaction schemes between SMA and nylon 6.

Figure 4.33 Reaction scheme between SEBS-graft-MA polymer and nylon 6.

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Figure 4.34 Reaction scheme between SMA and nylon 6.

Figure 4.35 gives plots of unnotched Izod impact of PPO/nylon 6/SMA ternary blend versus the amount of SMA in PPO/nylon 6 binary blends with varying compositions. In Figure 4.35 we observe that the impact strength of the ternary blends, especially the 70/30/x PPO/nylon 6/SMA ternary blends, increases considerably as the amount (x) of SMA increases. Similar observation can be made in Figure 4.36, describing the Figure 4.35 Effect of the

concentration of SMA in PPO/nylon 6/SMA ternary blends on unnotched Izod impact: (a) 30/70 PPO/nylon 6 with varying amounts of SMA (b) 50/50 PPO/nylon 6 with varying amounts of SMA, and (c) 70/30 PPO/nylon 6 with varying amounts of SMA. (Reprinted from Chiang and Chang, Polymer 38:4807. Copyright © 1997, with permission from Elsevier.)

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Figure 4.36 Effect of the concentration

of SMA in PPO/nylon 6/SMA ternary blends on tensile strength. (a) 30/70 PPO/nylon 6 with varying amounts of SMA, (b) 50/50 PPO/nylon 6 with varying amounts of SMA, and (c) 70/30 PPO/nylon 6 with varying amounts of SMA. (Reprinted from Chiang and Chang, Polymer 38:4807. Copyright © 1997, with permission from Elsevier.)

effect of added amount of SMA on the tensile strength of 70/30/xPPO/nylon 6/SMA ternary blends. It is interesting to observe in Figures 4.35 and 4.36 that both unnotched impact strength and tensile strength increase very rapidly initially and then tend to level off as the amount of SMA increases further. The above observation seems to suggest that there exists an optimum range of added compatibizing agent, beyond which its effectiveness does not increase. Using Fourier transform infrared (FTIR) spectroscopy, Chiang and Chang (1997) conﬁrmed the formation of SMA-graft-nylon 6 in situ during melt blending. They offered the following compatibilization mechanism. Since SMA is thermodynamically miscible with PPO, it tends to be dissolved into PPO during the melt blending of PPO and nylon 6. A fraction of the dissolved SMA would have the opportunity to make contact and react with nylon 6 to form the desirable SMA-graft-nylon 6 molecules at the interface, depending on the amount of SMA employed for melt blending. Most of the SMA-graft-nylon 6 molecules formed in situ at the interface may be only lightly grafted, one or two grafts per SMA main chain, because the rest of the SMA segment is miscible with PPO and tends to mix intimately with the PPO phase. They found that the concentration of reactive group (MA) was an important factor for optimal compatibilization; too high an MA concentration in SMA produced an excessively grafted comblike copolymer, while too low an MA concentration, although having the advantage of not producing the excessive graft copolymer, tended to produce smaller numbers of the desirable SMA-graft-nylon 6 and left a greater fraction of the unreacted free SMA in the PPO phase. Thus, the key factor in determining the efﬁciency of a reactive compatibilization is the extent to which the added reactive compatibilizing agent turns into lightly grafted polymers anchored along the interface, functioning as an effective compatibilizing agent by reducing the interfacial tension and enhancing

228

PROCESSING OF THERMOPLASTIC POLYMERS Figure 4.37 Effect of the amount of

MA in SMA on the interfacial layer thickness of the nylon 12/SMA bilayer system. (Reprinted from Dedecker et al., Polymer 39:5001. Copyright © 1998, with permission from Elsevier.)

the interfacial adhesion. They observed that the domain size of the dispersed phase was reduced with increasing amount of SMA. Since SMA is a copolymer, its composition might also play an important role in determining its effectiveness when melt blended with a pair of immiscible polymers. Using ellipsometry, Dedecker et al. (1998) measured the interfacial layer thickness (i.e., the width of interphase) of bilayer specimens consisting of nylon 12 and SMA with varying amounts of MA, the results of which are given in Figure 4.37. The width of interphase was measured after a specimen was annealed at 210 ◦ C for 30 min. It is interesting to observe in Figure 4.37 that the width of interphase initially increases with increasing amounts of MA in SMA, going through a maximum followed by a decline with further increase of MA content in SMA. They noted that each grade of SMA employed in their study had a slight variation in molecular weight, which however does not seem to explain completely the existence of an optimum content of MA in the SMA employed. Reactive compatibilization has another advantage in that it helps minimize drop coalescence during melt blending (Sundararaj and Macosko 1995). In other words, lowering the interfacial tension in the presence of an interfacial agent residing at the interface between the drop phase and the matrix helps to suppress the coalescence of the discrete phase (drops) formed during compounding in mixing equipment (see Chapter 3). The subject of drop coalescence has not been discussed much in the literature (Fortenly and Zivny 1995a, 1995b, 1998) compared with the extensive studies on drop breakup (see Chapter 11 of Volume 1). A better understanding of the mechanism(s) of drop coalescence is essential for the control of blend morphology during compounding. There is experimental evidence (Elmendorp and van Der Vegt 1986; Roland and Böhm 1984), suggesting that shear ﬂow may actually induce coalescence, although shear ﬂow is necessary to achieve ﬁner dispersion. This means that there is a competition (or balance) between breakup and coalescence of the dispersed phase during compounding in mixing equipment, which would depend on the concentration of added compatibilizing agent and processing conditions (the intensity of

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mixing and mixing temperature). This subject requires more intense research effort, both experimental and theoretical, in the future.

4.4

Summary

In this chapter, we have made a distinction between the role of an emulsifying agent and the role of a compatibilizing agent when either of them is melt blended with a pair of immiscible polymers. An emulsifying agent is expected to lower the interfacial tension between two immiscible polymers and thus produce smaller domain sizes of the dispersed phase, without necessarily forming an interphase. In this regard, an emulsifying agent can be viewed as the same as a surfactant for low-molecular-weight liquids. Thus, an emulsifying agent is not expected to signiﬁcantly improve the mechanical properties of an immiscible polymer blend. Conversely, an effective compatibilizing agent is expected to signiﬁcantly improve the mechanical properties of an immiscible blend by forming an interphase, thus creating strong interfacial adhesion. We have shown that two requirements must be met for a nonreactive block copolymer to function as an effective compatibilizing agent for a pair of immiscible homopolymers or random copolymers. They are: (1) attractive segmental interactions between a homopolymer and corresponding block of copolymer must be present, and (2) the block copolymer must be designed such that its TODT is lower than the intended melt blending temperature. We have pointed out that melt blending of a pair of immiscible homopolymers or random copolymers with a nonreactive block copolymer for a relatively short time (not longer than say 5 min) in conventional mixing equipment would not give rise to an equilibrium morphology, and that the molecular weight of polymers and the melt blending temperature, which in turn control the viscosities of the polymers, would play signiﬁcant roles in determining the effectiveness of a block copolymer as a compatibilizing agent. The higher the viscosities of the polymers, the slower will be the rate of polymer–polymer interdiffusion, and thus a longer period of mixing will be required to achieve the desired blend morphology. One cannot expect the formation of an interphase between a homopolymer and corresponding block of nonreactive copolymer during melt blending unless they have attractive segmental interactions. We have presented TEM images showing the formation of an interphase in two model ternary blend systems: (1) 63/27/10 PS/PB/(PαMS-block-PI) blend (see Figure 4.19) and (2) 63/27/10 PPO/PP/(PS-blockPEB) blend (see Figure 4.28). The formation of an interphase in those blends was made possible by the attractive segmental interactions (negative χ ) existing (1) between the homopolymer PB (having high 1,2-addition) and the PI block (having high 1,4addition) of copolymer in the 63/27/10 PS/PB/(PαMS-block-PI) blend, and (2) between the homopolymer PPO and the PS block of copolymer, and between the homopolymer PP and the PEB block (having high 1-butene content) of copolymer in the 63/27/10 PPO/PP/(PS-block-PEB) blend. Also, we have presented TEM images showing no formation of an interphase in the 63/27/10 PMMA/PP/(PαMS-block-PI) ternary blend, in which only positive segmental interactions exist in all six pairs of components (i.e., all six χ’s associated with the blend are positive). Most importantly, we have shown that even when attractive segmental interactions exist between a homopolymer and corresponding block of nonreactive copolymer,

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little or no signiﬁcant interphase is formed when the melt blending temperature is lower than the TODT of block copolymer. The absence of an interphase under such circumstances is attributed to the very high viscosity of the microphase-separated block copolymers. We have shown that the viscosity of a block copolymer in an ordered state (i.e., at T < TODT ) is a few orders of magnitude higher than that in the disordered state (i.e., at T > TODT ), and that melt blending in practical polymer processing operations (e.g., in a twin-screw extruder) would not exceed 5 min. In this regard, it is fair to state that equilibrium morphology in a ternary blend consisting of two homopolymers and a nonreactive block copolymer cannot be expected in practical melt blending operations. At present there are equilibrium mean ﬁeld theories available for predicting the equilibrium composition proﬁles for A/B/(A-block-B) ternary blends (Leibler 1988; Noolandi 1991; Noolandi and Hong 1982, 1984; Shull and Kramer 1990) and A/B/ (C-block-D) ternary blends (Vilgis and Noolandi 1990). In this chapter, we have shown that C-block-D copolymer is much more effective than A-block-B copolymer in compatibilizing two immiscible homopolymers, A and B. The experimental results presented in this chapter suggest that in the use of C-block-D copolymer to compatibilize two immiscible homopolymers, A and B, the segmental interactions between homopolymer A and block C of the copolymer, and between homopolymer B and block D of the copolymer, or between homopolymer A and block D of the copolymer, and between homopolymer B and block C of the copolymer, must be attractive (i.e., χAC < 0 and χBD < 0, or χAD < 0 and χBC < 0), so that the block chains can segregate preferentially to the interface between two immiscible homopolymers, A and B, thereby creating an interphase that greatly promotes adhesion between the two homopolymers. Since practical melt blending operations would not exceed, say, 5 min in a twinscrew extruder, equilibrium morphology will never be achieved. This then suggests that we need to develop a transient theory, so that the time evolution of interphase in A/B/(C-block-D) ternary blend during melt blending can be predicted for (1) χAC < 0, χBD < 0, χAB > 0, χCD > 0, χAD > 0, and χBC > 0, or (2) χAD < 0, χBC < 0, χAB > 0, χCD > 0, χAC > 0, and χBD > 0. Further, the prediction of an optimal range of molecular weight of a nonreactive block copolymer relative to the molecular weights of two homopolymers to be melt blended is very important for achieving a sufﬁciently wide interphase in A/B/(C-block-D) ternary blends. The adhesion sufﬁcient for obtaining the desired mechanical properties of a two-phase polymer blend, in the presence of an effective compatibilizing agent, can be realized only when the interphase is sufﬁciently wide. At present, we do not have a comprehensive theory enabling us to predict, during melt blending, the time evolution of interphase in A/B/(C-block-D) ternary blends as functions of the molecular weights of two homopolymers, A and B, the molecular weight and block length ratio of C-block-D copolymer, blend composition, and the interaction parameters, which in turn depend on temperature. When such a theory becomes available, one could minimize very time-consuming and costly, sometimes unnecessary, experiments. In this chapter, we have presented the basic principles associated with selecting or synthesizing a reactive compatibilizing agent (see Figure 4.32), and then described a few reactive compatibilizing agents that have been used by different research groups. It goes without saying that a reactive compatibilizing agent is far more effective

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231

than a nonreactive compatibilizing agent in forming an interphase. In the situations where a functionalized block copolymer is used as compatibilizing agent, the TODT of the block copolymer will not become an issue for the choice of melt blending temperature because most likely chemical reaction(s) would take place at an elevated temperature. When dealing with a reactive compatibilizing agent during melt blending, the chemical reaction(s) between the reactive (or functional) groups in a compatibilizing agent and the functional groups in one of the two homopolymers or random copolymers to be melt blended would generate an in situ graft polymer, which then is located at the interface and plays dual roles: the role of an interfacial (or emulsifying) agent and the role of adhesive layer. The graft polymer formed in situ not only helps to reduce the size of the dispersed phase, but also helps to suppress coalescence of the dispersed particles. Above all, the graft polymer formed in situ helps improve the mechanical properties of the blend signiﬁcantly. The most serious problem, however, is that there are no universal reactive compatibilizing agents that will work with any pair of polymers. For a speciﬁc pair of immiscible polymers, a particular reactive compatibilizing agent must be designed and synthesized.

Notes 1. The Leibler theory (1980) tends to predict the TODT of lamellar-forming block copolymers higher than does the Helfand–Wasserman theory (1982), while both theories predict comparable values of TODT of sphere-forming block copolymers. For the details of this ﬁnding, see the papers by Han et al. (1990, 1995). 2. In these blends, the PS has Mn = 1.96 × 105 determined from membrane osmometry (MO) and Mw /Mn = 1.07 determined from gel permeation chromatography (GPC), and PP has Mn = 3.39 × 105 determined from MO and Mw /Mn = 5.88 determined from high-temperature GPC. 3. In these blends, the PS has Mn = 1.96 × 105 determined from MO and Mw /Mn = 2.23 determined from GPC, and the PI has Mn = 1.96 × 105 determined from MO and Mw /Mn = 1.07 determined from GPC. 4. In this blend, the PS has Mn = 1.1 × 105 determined from MO and Mw /Mn = 2.03 determined from GPC, and the PB has Mn = 7.1 × 104 determined from MO and Mw /Mn = 1.11 determined from GPC. 5. In this blend, the PS has Mn = 1.0 × 104 determined from MO and Mw /Mn = 1.02 determined from GPC, and the PB contains 90% 1,2-addition determined from NMR, and has Mn = 1.12 × 104 determined from MO and Mw /Mn = 1.09 determined from GPC. 6. In this blend, the PαMS has Mn = 8.69 × 104 determined from MO and Mw /Mn = 1.12 determined from GPC, and the PI has Mn = 1.96 × 105 determined from MO and Mw /Mn = 1.07 determined from GPC. 7. In this blend, the PPO is an experimental polymer having Mw = 0.82 × 104 and Mw /Mn = 1.24 determined from GPC, and the PP is a commercial polymer having Mw = 3.39 × 105 and Mw /Mn = 5.88 determined from high-temperature GPC.

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References Adedeji A, Hudson SD, Jamieson AM (1997a). Polymer 38:737. Adedeji A, Jamieson AM, Hudson SD (1996). Macromol. Chem. Phys. 197:2521. Adedeji A, Jamieson AM, Hudson SD (1997b). Polymer 38:2753. Anastasiadis SH, Gancarz I, Koberstein JT (1989). Macromolecules 22:1449. Aurelio de Araujo M, Stadler R, Cantow HJ (1988). Polymer 29:2235. Auschra C, Stadler R, Voigt-Martin IG (1993a). Polymer 34:2081. Auschra C, Stadler R, Voigt-Martin IG (1993b). Polymer 34:2094. Barlow JW, Paul DR (1984). Polym. Eng. Sci. 24:525. Bernstein RE, Cruz CA, Paul DR, Barlow JW (1977). Macromolecules 10:681. Broseta D, Fredrickson GH, Helfand E, Leibler L (1990). Macromolecules 23:132. Brown HR, Char K, Deline VR, Green PF (1993). Macromolecules 26:4155. Cartier H, Hu GH (1999). Polym. Eng. Sci. 39:996. Champagne MF, Hunealut MA, Roux C, Peyrel W (1999). Polym. Eng. Sci. 39:976. Chiang CR, Chang FC (1997). Polymer 38:4807. Cho D, Hu W, Koberstein JT, Lingelser JP, Gallot T (2000). Macromolecules 33:5245. Chun SB, Han CD (1999). Macromolecules 32:4030. Chun SB, Han CD (2000). Macromolecules 33:3409. Dai KH, Kramer EJ, Shull KR (1992). Macromolecules 25:220. Datta S, Lohse DJ (1996). Polymeric Compatibilizers, Hanser, Munich. Dedecker K, Groeninckx G (1998a). Polymer 39:4985. Dedecker K, Groeninckx G (1998b). Polymer 39:4993. Dedecker K, Groeninckx G, Inoue T (1998). Polymer 39:5001. Doi M, Edwards SF (1986). The Theory of Polymer Dynamics, Oxford University Press, Oxford, England. Elmendorp JJ, van Der Vegt AK (1986). Polym. Eng. Sci. 26:1332. Fayt R, Jérôme R, Teyssie Ph (1981). J. Polym. Sci., Polym. Lett. Ed. 19:79. Fayt R, Jérôme R, Teyssie Ph (1982). J. Polym. Sci., Polym. Phys. Ed. 20:2209. Fayt R, Jérôme R, Teyssie Ph (1986). Makromol. Chem. 187:837. Fayt R, Jérôme R, Teyssie Ph (1987). Polym. Eng. Sci. 27:328. Fayt R, Jérôme R, Teyssie Ph (1989). J. Polym. Sci., Polym. Phys. Ed. 27:775. Fayt R, Teyssie Ph (1989). Polym. Eng. Sci. 29:538. Fortenly I, Zivny A (1995a). Polymer 36:4113. Fortenly I, Zivny A (1995b). Polym. Eng. Sci. 35:1872. Fortenly I, Kovar J (1998). Polym. Compos. 9:11924. Fried JR, Karasz FE, MacKnight WJ (1978). Macromolecules 11:150. Girard-Reydet E, Sautereau H, Pascault TP (1999). Polymer 40:1677. Gupta AK, Purwar SN (1985). J. Appl. Polym. Sci. 30:1799. Halasa AF, Lohr DF, Hall JE (1981). J. Polym. Sci., Polym. Chem. Ed. 19:1357. Han CD, Baek DM, Kim JK (1990). Macromolecules 23:561. Han CD, Baek DM, Kim JK, Ogawa T, Sakamoto N, Hashimoto T (1995). Macromolecules 28:5043. Han CD, Chun SB, Hahn FS, Harper SQ, Savickas PJ, Meunier DJ, Li L, Yalcin T (1998). Macromolecules 31:394. Heck B, Arends P, Ganter M, Kressler J, Stühn B (1997). Macromolecules 30:4559. Helfand H, Tagami Y (1971). J. Polym. Sci., Polym. Lett. Ed. 9:741. Helfand H, Tagami Y (1972). J. Chem. Phys. 56:3592. Helfand E, Wasserman ZR (1982). In Development in Block Copolymers, Goodman I (ed), Applied Science, New York, Chap 4. Jeon HK, Kim JK (1988). Macromolecules 31:9273.

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Jo WH, Kim H, Baek DH (1991). Macromolecules 24:2231. Kim JR, Jamieson AM, Hudson S, Manas-Zloczower I, Ishida H (1998). Macromolecules 31:5383. Kim JR, Jamieson AM, Hudson S, Manas-Zloczower I, Ishida H (1999). Macromolecules 32:4582. Kim S, Kim JK, Park CE (1997). Polymer 38:1809. Kramer EJ, Green PF, Palmstrom C (1984). Polymer 25:473. Kruse WA, Kirste RG, Haas J, Schmitt BJ, Stein DJ (1976). J. Makromol. Chem. 177:1145. Kwei TK, Frisch HL, Radigan W, Vogel S (1977). Macromolecules 10:157. Lee JK, Han CD (1999). Polymer 40:2521. Leibler L (1980). Macromolecules 13:1602. Leibler L (1988). Makromol. Chem. Macromol. Symp. 16:1. Maconnachie A, Kambour RP, Bopp RC (1984a). Polymer 25:357. Maconnachie A, Kambour RP, White DM, Rostami S, Walsh DJ (1984b) Macromolecules 17:2645. Macosko CW, Guegan P, Khandpur AK, Nakayama A, Marechal P, Inoue T (1996). Macromolecules 29:5590. McBriety VJ, Douglass DC, Kwei TK (1978). Macromolecules 11:1265. McMaster LP (1975). In Copolymers, Polyblends, and Composites, Platzer NAJ (ed), Adv. Chem. Series no 142, American Chemical Society, Washington, DC, p 43. Majumdar B, Keskkula H, Paul DR (1994a). Polymer 35:3164. Majumdar B, Keskkula H, Paul DR (1994b). Polymer 35:5453. Majumdar B, Keskkula H, Paul DR (1994c). Polymer 35:5468. Majumdar B, Keskkula H, Paul DR, Harvery NG (1994d). Polymer 35:4263. Majumdar B, Paul DR, Oshinski AJ (1997). Polymer 38:1787. Naito K, Johnson RE, Allara KL, Kwei TK (1978). Macromolecules 11:1260. Nishi T, Wang TT (1975). Macromolecules 8:909. Noolandi J (1991). Makromol. Chem. Rapid Commun. 12:517. Noolandi J, Hong KM (1982). Macromolecules 15:482. Noolandi J, Hong KM (1984). Macromolecules 17:1531. Noland S, Hsu HHC, Saxon R, Schmitt JM (1971). In Multicomponent Polymer Systems, Platzer NAJ (ed), Adv. Chem. Series no. 99, American Chemical Society, Washington, DC, p 99. Ouhadi T, Fayt R, Jérôme R, Teyssie Ph (1986a). J. Polym. Sci., Polym. Phys. Ed. 24:973. Ouhadi T, Fayt R, Jérôme R, Teyssie Ph (1986b). Polym. Commun. 27:212. Park I, Barlow JW, Paul DR (1992). J. Appl. Polym. Sci. 45:1313. Park KY, Park SH, Suh KD (1997). J. Appl. Polym. Sci. 66:2183. Paul DR, Altamirano JO (1975). In Copolymers, Polyblends and Composites, Platzer NAJ (ed), Adv. Chem. Series no. 142, American Chemical Society, Washington, DC, p 371. Pietransanta Y, Robin JJ, Torres N, Boutevin B (1999). Macromol. Chem. Phys. 200:142. Roland CM, Böhm GGA (1984). J. Polym. Sci., Polym. Phys. Ed. 22:79. Russell TP, Anastasiadis SH, Menelle A, Felcher GP, Satija SK (1991a). Macromolecules 24:1575. Russell TP, Hjelm RP, Seeger PA (1990). Macromolecules 23:890. Russell TP, Menelle A, Hamilton WA, Smith GS, Satija SK, Majkrzak CF (1991b). Macromolecules 24:5721. Schwarz MC, Barlow JW, Paul DR (1988). J. Appl. Polym. Sci. 35:2053. Schwarz MC, Barlow JW, Paul DR (1989). J. Appl. Polym. Sci. 37:403. Setz S, Stricker F, Kressler J, Duschek T, Mülhaupt R (1996). J. Appl. Polym. Sci. 59:1117. Shull KR, Kramer EJ (1990). Macromolecules 23:4769. Shull KR, Kramer EJ, Hadziioannou G, Tang W (1990). Macromolecules 23:4780.

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Srinivasan KR, Gupta AK (1994). J. Appl. Polym. Sci. 53:1. Stein DJ, Jung RH, Illers KH, Hendus H (1974). Angew. Makromol. Chem. 36:89. Sundararaj U, Macosko CW (1995). Macromolecules 28:2647. Takeno H, Hashimoto T (1997). J. Chem. Phys. 107:1634. ten Brinke G, Karasz FE, MacKnight WJ (1983). Macromolecules 16:1827. Thudium RN, Han CC (1996). Macromolecules 29:2143. Tselios Ch, Bikiaris D, Maslis V, Panayiotou C (1998). Polymer 39:6807. Traugott TD, Barlow JW, Paul DR (1983). J. Appl. Polym. Sci. 28:2947. Triacca VJ, Ziaee S, Balow JW, Keskkula H, Paul DR (1991). Polymer 32:1401. Vilgis TA, Noolandi J (1990). Macromolecules 23:2941. Wang TT, Nishi T (1977). Macromolecules 10:142. Weeks NE, Karasz FE, MacKnight WJ (1977). J. Appl. Phys. 48:4068. Wendorff JH (1980). J. Polym. Sci., Polym. Lett. Ed. 18:439. Wildes GS, Harada T, Keskkula H, Paul DR, Janarthananm V, Padwa AR (1999) Polymer 40:3069. Yang K, Han CD (1996). Polymer 37:5795. Zhang XM, Yin Z, Na T, Yin JH (1997). Polymer 38:5905. Zhang XM, Yin JH (1997). Polym. Eng. Sci. 37:197.

5

Wire-Coating Extrusion

5.1

Introduction

The process of coating a wire with a polymeric material is basically an extrusion operation in which either the molten polymer is extruded continuously over an axially moving wire or the wire is pulled through the extruded molten polymer. As schematically shown in Figure 5.1, the typical wire-coating unit consists of a pay-off device, a wire preheater, an extruder equipped with a cross-head die, a cooling trough, and a take-off device. Various control and measuring instruments are utilized in the commercial line (Griff 1962). The two basic wire-coating dies are pressure-type dies and tubing-type dies, as shown schematically in Figure 5.2. The tubing-type dies are annular in cross-section. The ﬂow geometry outside the tubing die is important from the point of view of obtaining a coating with better mechanical and electrical properties and surface smoothness. However, little effort has been spent on studying this particular aspect of the process. The pressure-type wire-coating die is an annulus, the side surface of which is the wire to be coated, moving at a constant speed. The ﬂow through this type of die is analogous to the ﬂow through an annulus formed by coaxial cylinders with the inner cylinder moving in the axial direction. In the past, analysis of wire-coating extrusion for pressure-type die has been carried out for Newtonian and power-law ﬂuids (Bagley and Storey 1963; Bernhardt 1962; Carley et al. 1979; Han and Rao 1978; McKelvey 1963). Like in the ﬁlm coextrusion process presented in Chapter 9, in wire-coating coextrusion two different polymers may be concentrically coated on the wire in a single step (LeNir 1974). Tough abrasive-resistant nylon, for example, can be coated over a much less expensive polyethylene core, or one can have a thin coat of color compound over unpigmented insulator, thus taking advantage of the different properties of two components at a reduced cost. Considerable savings in the cost of processing can be achieved by applying two coats in a single step. In this process, two different polymer melts from separate extruders are brought into a single cross-head die, where they are

235

236

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 5.1 Schematic of the layout of a wire-coating extrusion unit.

made to ﬂow in a concentric annular manner over an axially moving wire, as shown schematically in Figure 5.3. Only a few studies (Basu 1981; Heng and Mitsoulis 1989; Han 1981; Han and Rao 1980) have reported on wire-coating coextrusion. In this chapter, we present an analysis of wire-coating extrusion followed by some experimental observations. There are very few experimental results reported on wirecoating extrusion. After presenting a system of equations describing nonisothermal single-layer wire-coating extrusion, we will consider an isothermal situation for a power-law ﬂuid, because analytical expressions for the shear stress and velocity proﬁles can be obtained, giving us clear ideas about the role that certain processing variables play in determining the performance of the process. We will not present two-layer wire-coating coextrusion because very little new development on wirecoating coextrusion has been reported since the publication of the monograph of Han (1981).

5.2

Analysis of Wire-Coating Extrusion

Here, we present a system of equations necessary for predicting the velocity and stress distributions of a molten polymer in a pressure-type wire-coating die, as shown schematically in Figure 5.4. We make the following assumptions: (1) the ﬂow is in steady state; (2) the polymer melt, following a power-law model, ﬂows through a sufﬁciently long cylindrical die in which a wire moves along the centerline at a constant speed; (3) the ﬂow is laminar and the velocity in the radial direction is negligibly small compared with that in the axial direction; (4) inertial effect is negligibly small compared with viscous effect, which is reasonable owing to the very high viscosity

Figure 5.2 Schematic of wire-coating dies: (a) pressure-type die, and (b) tubing-type die.

WIRE-COATING EXTRUSION

237

Figure 5.3 Schematic of a die for two-layer wire-coating coextrusion, in which two separate

melt streams are fed to the die.

of polymer melt; (5) the heat conduction in the ﬂow direction is negligibly small compared with that in the radial direction; (6) the melt density, speciﬁc heat, and thermal conductivity are independent of temperature; and (7) the gravitational effect is negligible. Under the assumptions given above, using the cylindrical coordinates (r, θ , z) the z-component of the momentum balance equation for the ﬂuid is written as −

∂p 1 ∂ + (rσrz ) = 0 ∂z r ∂r

Figure 5.4 Schematic of the ﬂow geometry of a pressure-type wire-coating die.

(5.1)

238

PROCESSING OF THERMOPLASTIC POLYMERS

in which −∂p/∂z is the axial pressure gradient and σrz is the rzth component (shear stress component) of the deviatoric stress tensor. The energy balance equation for the ﬂuid is written as k ∂ ∂T ρcp vz = ∂z r ∂r

∂T r ∂r

∂vz +η ∂r

2 (5.2)

where ρ is the density, cp is the speciﬁc heat, k is the thermal conductivity, and η is the viscosity. The second term on the right-hand side of Eq. (5.2) describes viscous shear heating. Note that Eqs. (5.1) and (5.2) are identical to Eqs. (5.92) and (5.93) given in Chapter 5 of Volume 1. For a speciﬁed expression for η, Eqs. (5.1) and (5.2) for nonisothermal operation must be solved under the boundary conditions: (i) at r = κR0 and 0 ≤ z ≤ L, vz = Vwire and T = Twire

(5.3a)

(ii) at r = R0 and 0 ≤ z ≤ L, vz = 0 and T = Twire

(5.3b)

for the constant temperature at the wire surface and die wall, where κ is the dimensionless radius (Ri /R0 ) of the wire (0 < κ < 1), R0 is the radius of the circular tube (i.e., the inner radius of the wire-coating die), and Ri (= κR0 ) is the radius of the wire, and Vwire is the wire speed and Twire is the temperature of the wire surface. For the adiabatic condition at the wire surface and die wall, the boundary conditions are given by (i) at r = κR0 and 0 ≤ z ≤ L, vz = Vwire and (∂T /∂r) r=κR0 = 0 (ii) at r = R0 and 0 ≤ z ≤ L, vz = 0 and (∂T /∂r) r=R0 = 0

(5.4a) (5.4b)

Here, we consider an isothermal operation, for which we only have to solve Eq. (5.1). We do this, because the consideration of isothermal operation allows us to obtain analytical expressions for the velocity, shear stress, and the volumetric ﬂow rate, providing insight into the wire-coating extrusion process, while the numerical solution of Eqs. (5.1) and (5.2) does not. Granted that the consideration of isothermal operation cannot describe the effect of viscous shear heating on the velocity, shear stress, and temperature distributions inside the die.1 Integrating Eq. (5.1) we get σrz = − 12 ςr + c1 /r

(5.5)

in which ς = −∂p/∂z and c1 is an integration constant. At r = λR0 , with λ being the dimensionless radial position at which the ﬂuid velocity goes through a maximum, we have σrz = 0

at r = λR0

(5.6)

239

WIRE-COATING EXTRUSION

Use of Eq. (5.6) in Eq. (5.5) gives ⎧ 2 2 ⎪ ⎪1R ζ λ − ξ ⎪ ⎨2 0 ξ σrz = 2 ⎪ ξ − λ2 ⎪ 1 ⎪ ⎩ 2 R0 ζ ξ

for κ ≤ ξ < λ (5.7) for λ < ξ ≤ 1

in which ξ is the dimensionless radial position deﬁned by ξ = r/R0 . In order to obtain the velocity proﬁle, one needs to choose a rheological model to solve Eq. (5.7). For the analysis here, we choose a power-law model: σrz = η(γ˙ )

dvz dr

(5.8)

where dv n−1 η(γ˙ ) = K z dr

(5.9)

in which K and n are material constants (see Chapter 6 of Volume 1). Use of Eq. (5.8) in (5.7) and integration of the resulting expression with the aid of the boundary conditions given by Eq. (5.3), under the isothermal condition, give ⎧ 1/n ⎪ R0 ζ 1/n ξ λ2 − s 2 ⎪ ⎪ ds ⎪ ⎨Vwire + R0 2K s κ vz (ξ ) = 1/n ⎪ ⎪ R0 ζ 1/n 1 s 2 − λ2 ⎪ ⎪ ds ⎩R0 2K s ξ

for κ ≤ ξ < λ (5.10) for λ < ξ ≤ 1

At ξ = λ, it follows from Eq. (5.10) that Vwire + R0

R0 ζ 2K

λ λ2

1/n κ

− s2 s

1/n ds = R0

R0 ζ 2K

1 s2

1/n λ

− λ2 s

1/n ds (5.11)

This is the expression to be used to determine the value of λ, which depends on the wire speed Vwire , dimensionless wire radius κ, the radius R0 of the wire-coating die, the pressure gradient ζ , and the material constants K and n. For the determination of λ from Eq. (5.11), one must resort to a trial-and-error procedure, using some kind of numerical scheme. The volumetric ﬂow rate Q may be obtained from Q = 2πR0 2

1 κ

vz (ξ ) ξ dξ

(5.12)

240

PROCESSING OF THERMOPLASTIC POLYMERS

Use of Eq. (5.10) in Eq. (5.12) gives Q = Vwire πR0 (λ − κ ) + πR0 2

2

2

3

R0 ζ 2K

1/n

1 κ

1 λ2 − s 2 n +1 s 1/n

ds

(5.13)

in which the ﬁrst term on the right-hand side represents the volumetric ﬂow rate due to drag ﬂow (QD ) and the second term on the right-hand side represents the volumetric ﬂow rate due to pressure-driven ﬂow (Qp ). It should be pointed out that for a power-law ﬂuid, the contribution from drag ﬂow and the contribution from pressure-driven ﬂow are not linearly additive because the value of λ appearing in Eq. (5.13) depends on the wire speed Vwire . Further, as will be shown in the next section, the experimental results show that the pressure gradient ζ appearing in the second term on the right-hand side of Eq. (5.13) is decreased as the wire speed is increased. It can be shown, however, that for a Newtonian ﬂuid the contribution from drag ﬂow and the contribution from pressure-driven ﬂow are linearly additive. Figure 5.5 gives calculated velocity proﬁles, and Figure 5.6 gives calculated shear stress proﬁles, of a low-density polyethylene (LDPE) at 200 ◦ C in a pressure-type wire-coating die for three different wire speeds, where the die dimensions employed are R0 = 0.154 cm and Ri = 0.064 cm, and the rheological parameters of the LDPE employed are K = 0.30 × 104 Pa·s0.45 and n = 0.45. The values of pressure gradient

Figure 5.5 Calculated velocity proﬁles of an LDPE at 200 ◦ C with K = 0.3 × 104 Pa·s0.45 and

n = 0.45 in a pressure-type die with R0 = 0.154 cm and Ri = 0.064 cm at a melt ﬂow rate Q of 56 cm3 /min: curve (1) with Vwire = 3.23 m/min, −∂p/∂z = 156.1 MPa/m, and λ = 0.655, curve (2) with Vwire = 7.48 m/min, −∂p/∂z = 133.0 MPa/m, and λ = 0.620, and curve (3) with Vwire = 11.39 m/min, −∂p/∂z = 103.2 MPa/m, and λ = 0.520. (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

WIRE-COATING EXTRUSION

241

Figure 5.6 Calculated shear stress proﬁles of an LDPE at 200 ◦ C with K = 0.3 × 104 Pa·s0.45

and n = 0.45 in a pressure-type die with R0 = 0.154 cm and Ri = 0.064 cm at a melt ﬂow rate Q of 56 cm3 /min: curve (1) with Vwire = 3.23 m/min, −∂p/∂z = 156.1 MPa/m, and λ = 0.655, curve (2) with Vwire = 7.48 m/min, −∂p/∂z = 133.0 MPa/m, and λ = 0.620, and curve (3) with Vwire = 11.39 m/min, −∂p/∂z = 103.2 MPa/m, and λ = 0.520. (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

−∂p/∂z employed for the calculation were obtained from experiment (Han and Rao 1978). It is seen in Figure 5.5 that as the wire speed increases, the location of the maximum ﬂuid velocity moves toward the wire surface, giving rise to smaller values of λ. It is seen in Figure 5.6 that the shear stress at the wire surface can be greater or less than the shear stress at the die wall, depending on the wire speed. This can be readily seen when one examines Eq. (5.7) carefully. Figure 5.7 gives plots of λ versus draw-down ratio (DR ) for different values of κ (the ratio Ri /R0 ) at a ﬁxed value of n = 0.5, in which DR is deﬁned by

DR =

R0 2 − R i 2 (Ri + h)2 − Ri 2

(5.14)

where h is the thickness of the coating. Note that Figure 5.7 is obtained using Eq. (5.7). It is seen in Figure 5.7 that there are two regions: one region where the shear stress at the wire σwire is greater than the shear stress at the die wall σdie , that is, σwire > σdie , and another region where σwire < σdie . At the wire surface, namely where ξ = κ, we have σwire = 12 R0 ζ (λ2 − κ 2 )/κ

(5.15)

242

PROCESSING OF THERMOPLASTIC POLYMERS Figure 5.7 Plots of λ versus DR for a power-law ﬂuid with n = 0.5, describing the region where σwire < σdie and σwire > σdie . (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

and at the die wall, where ξ = 1, we have σwire = 12 R0 ζ (1 − λ2 )

(5.16)

The condition at which σwire = σdie is satisﬁed yields λ = κ 1/2

for 0 < κ < 1

(5.17)

The practical signiﬁcance of Figure 5.7 lies in that one can judiciously choose the extrusion conditions (e.g., DR ) so as to prevent the breakage of wire inside the die, as might occur in the region where σwire > σdie . Figure 5.8 gives calculated velocity proﬁles of an LDPE in a pressure-type die for (1) drag ﬂow alone at Vwire = 14.7 m/min and QD = 26.9 cm3 /min, (2) pressure-driven ﬂow alone at −∂p/∂z = 168.3 MPa/m and QP = 61.3 cm3 /min, (3) the sum of drag and pressure ﬂows at QD + QP = 88.2 cm3 /min, and (4) exact analysis of combined drag and pressure-driven ﬂows at Vwire = 14.7 m/min, −∂p/∂z = 168.3 MPa/m, and Q = 99.5 cm3 /min. Referring to Figure 5.8, curve (1) is obtained from the expression vz (ξ ) = Vwire

ξ (n−1)/n − 1 κ (n−1)/n − 1

for n = 1

(5.18)

WIRE-COATING EXTRUSION

243

Figure 5.8 Comparison of an approximate analysis with the exact analysis of wire-coating extru-

sion in a pressure-type die: curve (1) represents drag ﬂow alone, with Vwire = 14.72 m/min and QD = 26.9 cm3 /min, curve (2) represents pressure-driven ﬂow alone, with −∂p/∂z = 168.3 MPa/m and QP = 61.3 cm3 /min, curve (3) represents the sum of curves (1) and (2) with QD + QP = 88.2 cm3 /min, and curve (4) represents the exact analysis of combined drag and pressure-driven ﬂows, with Vwire = 14.72 m/min, −∂p/∂z = 168.3 MPa/m, Q = 99.5 cm3 /min. (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

curve (2) from the expression ⎧ 1/n ⎪ R0 ζ 1/n ξ λ2 − s 2 ⎪ ⎪ ds ⎪ ⎨R0 2K s κ vz (ξ ) = ⎪ R ζ 1/n 1 s 2 − λ2 1/n ⎪ ⎪ 0 ⎪ ds ⎩R0 2K s ξ

for κ ≤ ξ < λ (5.19) for λ < ξ ≤ 1

curve (3) from the sum of Eqs. (5.18) and (5.19), and curve (4) from Eq. (5.10). It is clearly seen in Figure 5.8 that for a power-law ﬂuid, one cannot simply add the solutions for drag ﬂow and pressure-driven ﬂow in order to obtain the solution for combined drag and pressure-driven ﬂows. This is attributed to the nonlinear characteristics of the ﬂow properties given by Eq. (5.9). For the purpose of either designing a wire-coating die or running the wire-coating process, one would be interested in having a sort of operating guide that would give important relationships, for instance, between the pressure gradient and volumetric ﬂow rate (or wire speed). Figure 5.9 gives plots of dimensionless pressure gradient ΠP versus DR for a power-law ﬂuid with n = 0.5 with κ as a parameter, in which ΠP is

244

PROCESSING OF THERMOPLASTIC POLYMERS Figure 5.9 Plots of ΠP versus DR for a power-law ﬂuid for n = 0.5 and κ as a parameter. (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

deﬁned by ΠP =

R0 Vwire

R0 ζ 2K

(5.20)

If we deﬁne the dimensionless ﬂow rate ΠQ by ΠQ =

π(R0

2

Q − Ri 2 )Vwire

(5.21)

use of Eq. (5.21) in (5.14) yields DR =

ρs 1 ρm ΠQ

(5.22)

in which ρs and ρm are densities of the polymer in the solid and molten states, respectively. In view of the fact that the DR is proportional to wire speed Vwire and inversely proportional to volumetric ﬂow of the melt, the plots given in Figure 5.9 are of practical value, both for determining the dimensions of a pressure-type wire-coating die and for predicting the processing conditions (e.g., pressure gradient and wire speed).

WIRE-COATING EXTRUSION

245

Figure 5.10 Plots of 2(1−κ)n+1 (ΠP )n versus DR for a power-law ﬂuid for n = 0.5 and n = 1.0

with κ as a parameter. (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

Figure 5.10 gives another dimensionless plot, 2(1 − κ)n+1 (ΠP )n versus DR , with κ as a parameter, for two different values of n. It is seen in Figure 5.10 that the choice of the dimensionless variable 2(1 − κ)n+1 (ΠP )n over ΠP suppresses considerably the plots of ΠP versus DR given in Figure 5.9. Again, Figure 5.10 is very useful for the design of a pressure-type die or to predict processing conditions for wire-coating extrusion.

5.3

Experimental Observations

There are very few experimental studies reported in the literature on wire-coating extrusion. Han and Rao (1978) constructed a laboratory-scale wire-coating extrusion unit with a pressure-type die, shown schematically in Figure 5.11. In their experiments, they employed an LDPE, a high-density polyethylene (HDPE), and a thermoplastic rubber (TPR), and measured wall normal stresses on the downstream side of a pressuretype die using three pressure transducers, P1, P2, and P3, as shown schematically in Figure 5.12.

Figure 5.11 Schematic of a wire-coating extrusion die system.

Figure 5.12 Schematic of the pressure-type die employed. (Reprinted from Han and Rao,

Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

246

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Figure 5.13 Plots of melt viscosity η versus shear rate γ˙ at 220 ◦ C for LDPE (), HDPE (), and TPR (). (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

Figure 5.13 gives logarithmic plots of melt viscosity η versus shear rate γ˙ at 220 ◦ C for the LDPE, HDPE, and TPR employed, and Table 5.1 gives a summary of the power-law constants of the three polymers. Figure 5.14 gives axial wall normal stress proﬁles2 for the LDPE at 180 ◦ C and at Vwire = 7.46 m/min for three different ﬂow rates, showing that the axial wall normal stress increases as the melt ﬂow rate increases and that the slope of the wall normal stress is constant in the region where wall normal stresses were measured. This experimental observation is very similar to that presented in Chapter 5 of Volume 1 for ﬂow in a cylindrical tube or in a slit die. According to the theoretical interpretation presented in Chapter 5 of Volume 1, the constant slope of wall normal stress observed in Figure 5.14 indicates that ﬂow is fully developed and thus the slope of wall normal stress is equal to the pressure gradient, that is, ∂Trr /∂z = −∂p/∂z.3 Figure 5.15 gives plots of −∂p/∂z versus ﬂow rate for LDPE at 180 ◦ C with wire speed as a parameter. Similar plots are given in Figure 5.16 for HDPE and in Figure 5.17 for TPR. It is interesting to observe in Figures 5.15–5.17 that pressure gradient decreases as the wire speed increases, indicating that drag ﬂow becomes

Table 5.1 Power-law constants of LDPE, HDPE, and TPR

Polymer LDPE HDPE TPR

Temp. ( ◦ C)

K (Pa·sn )

n

220 220 220

0.30 × 104 1.01 × 104 1.16 × 104

0.45 0.43 0.33

Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.

248

PROCESSING OF THERMOPLASTIC POLYMERS Figure 5.14 Axial wall normal stress proﬁles for LDPE at 180 ◦

C and at a wire speed of 7.46 m/min for various melt ﬂow rates (cm3 /min): () 94.6, () 70.6, and () 40. (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

increasingly important as the wire speed increases. In the preceding section, we have shown that both drag ﬂow and pressure-driven ﬂow prevail in a pressure-type die. Figure 5.18 gives plots of experimentally measured dimensionless pressure gradient ΠP versus draw down ratio DR for HDPE and TPR. Note that ΠP is deﬁned by Eq. (5.20). In Figure 5.18 we observe that the ΠP is decreased as the DR is increased and the shape of the plots is very similar to that predicted from theoretical considerations given in Figure 5.9. It is interesting to observe in Figure 5.18 that ΠP versus DR plots are independent of wire speed and ﬂow rate. Note that, according to Eq. (5.22), DR is inversely proportional to ΠQ deﬁned by Eq. (5.21). Therefore, Figure 5.18 may be

Figure 5.15 Plots of −∂p/∂z versus Q for LDPE at 180 ◦ C at various wire speeds (m/min): () 2.46, () 3.90, and () 7.46. (Reprinted from Han and Rao, Polymer Engineering and

Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

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Figure 5.16 Plots of −∂p/∂z versus Q for HDPE at 220 ◦ C at various wire speeds (m/min): () 2.46, () 3.90, and () 7.46. (Reprinted from Han and Rao, Polymer Engineering and

Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

conveniently used to estimate the extent of pressure gradient reduction when either the wire speed or melt ﬂow rate vary. Referring to Figure 5.14, the exit pressure PExit increases with increasing melt ﬂow rate, consistent with the observation made in Chapter 5 of Volume 1 for ﬂow through capillary and slit dies. It is interesting to observe in Figure 5.14 that the PExit decreases Figure 5.17 Plots of −∂p/∂z versus Q for TPR at 220 ◦ C at various wire speeds (m/min): () 3.81, () 7.56, and () 9.75. (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

250

PROCESSING OF THERMOPLASTIC POLYMERS Figure 5.18 Experimentally

determined plots of ΠP versus DR . Curve (1) TPR at 220 ◦ C for various wire speeds (m/min): () 3.81, () 7.56, and () 9.75. Curve (2) HDPE at 220 ◦ C for various wire speeds (m/min): (䊉) 2.46, () 3.90, and () 7.46. (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

with increasing wire speed. Since PExit may be regarded as being the amount of elastic (recoverable) energy stored in the melt at the die exit,4 the decrease of PExit with increasing wire speed indicates that the amount of recoverable elastic energy stored in the melt becomes less as the wire speed increases. Since the wire is pulled, the force on the wire is transmitted inside the die and thus a decrease in PExit is expected as the wire speed increases. Needless to say, the force pulling the wire affects both the viscosity and shear stress distributions of molten polymer inside the die and hence the degree of orientation of macromolecules there. Figure 5.19 gives logarithmic plots of PExit versus Q for LDPE at 180 ◦ C and at two different wire speeds, 2.46 and 7.55 m/min, and at 240 ◦ C and at a wire speed of 7.55 m/min. In Figure 5.19 we observe that PExit is decreased as the wire speed is increased and also as the melt temperature is increased. It is very interesting to observe in Figure 5.20, however, that when PExit is plotted against pressure gradient −∂p/∂z, instead of Q, in logarithmic coordinates, such plots give rise to a single correlation, which is independent of wire speed and temperature. According to Eq. (5.7), −∂p/∂z is proportional to shear stress σrz . Thus, the logarithmic plot of PExit versus −∂p/∂z in wire-coating extrusion is equivalent to the logarithmic plot of PExit versus σ in capillary ﬂow or slit ﬂow considered in Chapter 5 of Volume 1. Then, we can conclude that the temperature independence of the logarithmic plot of PExit versus

Figure 5.19 Plots of PExit

versus Q for LDPE: () at 180 ◦ C and Vwire = 2.46 m/min, () at 180 ◦ C and Vwire = 7.55 m/min, and (䊉) at 240 ◦ C and Vwire = 7.55 m/min.

Figure 5.20 Plots of PExit versus −∂p/∂z for LDPE: () at 180 ◦ C and Vwire = 2.46 m/min, () at 180 ◦ C and Vwire = 7.55 m/min, and (䊉) at 240 ◦ C and Vwire = 7.55 m/min.

251

252

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 5.21 Plots of PExit versus DR for LDPE at 180 ◦ C for two different wire speeds (m/min): () 2.46 and () 7.55. (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

−∂p/∂z observed in Figure 5.20 has the same physical origin as that of the logarithmic plots of PExit versus σ in capillary ﬂow or slit ﬂow considered in Chapter 5 of Volume 1. Figure 5.21 gives plots of PExit versus DR for LDPE at 180 ◦ C. Similar plots are given in Figure 5.22 for HDPE at 220 ◦ C and in Figure 5.23 for TPR at 220 ◦ C. What Figure 5.22 Plots of PExit versus DR for HDPE at 220 ◦ C for various wire

speeds (m/min): () 2.46. () 3.90, and () 7.46. (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

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Figure 5.23 Plots of PExit versus DR ◦

for TPR at 220 C for various wire speeds (m/min): () 3.81, () 7.56, and () 9.75. (Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.)

is common in all three ﬁgures is that PExit is decreased as the wire speed is increased. In other words, pulling of the wire greatly inﬂuences the amount of elastic energy stored in the melt at the die exit plane. Such an observation has signiﬁcant rheological implications in that the severity of melt fracture on the surface of wire-coated extrudate would be much less than that on the extrudate surface in the conventional extrusion operation. This is because the occurrence of melt fracture is commonly ascribed to the elastic properties of molten polymer.

5.4

Summary

In this chapter, we have presented an analysis of wire-coating extrusion in a pressuretype die, followed by some experimental observations. Very few experimental studies on high-speed wire-coating operations have been reported in the literature. This is understandable because the operation of a high-speed wire-coating process is not practical at an academic institution. From a theoretical point of view, wire-coating extrusion is a relatively simple polymer processing operation. Table 5.2 gives a comparison of experimental and theoretically predicted volumetric ﬂow rates in a pressure-type die, in which Eq. (5.13) was used to calculate the volumetric ﬂow rate with information on the pressure gradient obtained from experiment. It is seen that the calculated volumetric ﬂow rates are in reasonable agreement with the experimentally observed ones. It should be pointed out that the experimental results summarized in Table 5.2 were obtained at low wire speeds and at low melt ﬂow rates because the experiments were conducted in an academic institution. Thus, the simulation of the isothermal analysis of wire-coating extrusion presented in this chapter is justiﬁed for comparison with such experimental results.

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Table 5.2 Summary of measured and calculated ﬂow rates in wire-coating extrusion in a pressure-type die

Polymer LDPE

HDPE

TPR

Vwire (m/min)

−∂p/∂z (MPa/m)

Measured Q (cm3 /min)

Calculated Q (cm3 /min)

3.90 3.90 7.56 7.56 3.90 3.90 7.56 7.56 3.90 3.90 7.56 7.56

103.1 192.3 128.2 188.4 293.2 519.4 365.9 545.6 251.1 321.4 262.3 306.1

34.4 84.2 46.6 90.7 34.5 77.8 54.6 95.9 51.8 79.8 61.3 79.0

31.9 90.5 50.3 95.9 29.0 81.1 61.2 97.2 49.8 78.8 66.1 78.9

Note: The radius of the die (R0 ) is 0.158 cm and the radius of the wire (Ri ) is 0.064 cm. Reprinted from Han and Rao, Polymer Engineering and Science 18:1019. Copyright © 1978, with permission from the Society of Plastics Engineers.

Numerical solutions of Eqs. (5.1) and (5.2) are needed to simulate nonisothermal wire-coating extrusion for very high wire speeds or high melt ﬂow rates. In this chapter, we have not presented the numerical solutions of nonisothermal analysis of wire-coating extrusion for two reasons. The ﬁrst reason is that no experimental results for very high wire speeds or high ﬂow rates are available for comparison with numerical solutions. The second reason is that the consequence of nonisothermal analysis is very obvious in that it will predict a temperature rise near the die wall. The extent of temperature rise due to viscous shear heating will become greater as the wire speed and/or ﬂow rate is increased. Figure 5.24 gives a schematic showing (a) drag ﬂow, (b) pressure-driven ﬂow, and (c) combined drag/pressure-driven ﬂow in a pressure-type die. It has been demonstrated in this chapter that for a power-law ﬂuid, one must not add the solutions of drag ﬂow and pressure-driven ﬂow, obtained separately, to determine combined drag and pressuredriven ﬂow. This is attributed to the nonlinear characteristics of the ﬂow properties given, for instance, by Eq. (5.9). Needless to say, the same would be true whenever a nonlinear relationship between viscosity and shear rate is employed in solving a system of equations, Eqs. (5.1) and (5.2). The isothermal analysis presented in this chapter gives a clear insight into the intricate relationships between the rheological properties of the material and the processing variables. In this regard, Figures 5.9 and 5.10 would be very useful for determining the dimensions of a pressure-type die or for predicting the processing conditions (e.g., pressure gradient and wire speed) when an isothermal analysis is justiﬁed.

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255

Figure 5.24 Schematic showing the velocity proﬁles in (a) drag ﬂow, (b) pressure-driven ﬂow,

and (c) combined drag/pressure-driven ﬂow.

Problems Problem 5.1

Numerically solve Eqs. (5.1) and (5.2) for a truncated power-law model under the boundary conditions given by Eq. (5.3). Prepare the temperature, velocity, and viscosity proﬁles inside a pressure-type die using the following numerical values of various quantities: (1) die and wire dimensions: R0 = 0.154 cm and Ri = 0.064 cm, and (2) processing conditions: −∂p/∂z = 35.25 MPa/m, Vwire = 7.6 m/min, and Q = 96 cm3 /min. Use the truncated power-law model deﬁned by Eq. (1P.1) in Chapter 1 and the following numerical values for the parameters appearing in Eq. (1P.1): ko = 8.1 × 1013 Pa.s, b = 5.811 × 10−2 K −1 , γ˙0 = 1.0 s−1 , and n = 0.50, and the following numerical values for the parameters appearing in Eq. (5.2): ρ = 810 kg/m3 , cp = 2,640 J/kg, and k = 0.182 W/(m K). Problem 5.2

Keep all parameters other than the wire speed considered in Problem 5.1. Now, increase the wire speed to 50 m/min, 100 m/min, and 300 m/min and plot the temperature distribution inside the die for each wire speed. You will observe an overshoot of temperature near the die wall, which is due to the viscous shear heating of the polymer melt, as the wire speed is increased. Problem 5.3

Verify that the total volumetric ﬂow rate from combined drag and pressure-driven ﬂows is linearly additive for a Newtonian ﬂuid, as given by

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PROCESSING OF THERMOPLASTIC POLYMERS

Q = Vwire π(κR0 )

2

πR0 4 ζ (1/κ)2 − 1 (1 − κ 2 )2 4 −1 + 1−κ − (5P.1) 2 ln(1/κ) 8η0 ln(1/κ)

Notes 1. You are encouraged to write a computer code to obtain numerical solutions of Eqs. (5.1) and (5.2) under the boundary conditions given by Eq. (5.3). You will be able to observe the effect of viscous shear heating on the velocity and temperature distributions inside the die as the wire speed is increased. Namely, a temperature overshoot will be observed near the die wall, the extent of which will increase with increasing wire speed and extrusion rate of polymer. 2. See Chapter 5 of Volume 1. 3. See Chapter 5 of Volume 1. 4. See Chapter 5 of Volume 1.

References Bagley EB, Storey SH (1963). Wire and Wire Products 38:1104. Basu S (1981). Polym. Eng. Sci. 21:1128. Bernhardt EC (ed) (1962). Processing of Thermoplastic Materials, Reinhold Publishing, New York, p 269. Carley JF, Endo T, Krantz WB (1979). Polym. Eng. Sci. 19:1178. Griff AL (1962). Plastics Extrusion Technology, Reinhold Publishing, New York, p 125. Heng FL, Mitsoulis E (1989). Intern. Polymer Processing 4:44. Han CD (1981). Multiphase Flow in Polymer Processing, Academic Press, New York, Chap 7. Han CD, Rao D (1978). Polym. Eng. Sci. 18:1019. Han CD, Rao D (1980). Polym. Eng. Sci. 20:128. LeNir V (1974). Wire J. 7:59. McKelvey JM (1963). Polymer Processing, Wiley, New York, p 111.

6

Fiber Spinning

6.1

Introduction

Fiber spinning is one of the oldest polymer processing operations that have contributed signiﬁcantly to our society, especially after the commercialization of polyamide (nylon) synthetic ﬁbers in the 1940s by DuPont Company. Subsequent commercialization of poly(ethylene terephthalate) (PET) and polyacrylonitrile ﬁbers in the 1950s made the synthetic ﬁber industry very prosperous. For a given ﬁber-forming polymer, different spinning techniques can produce ﬁbers possessing markedly different physical and/or mechanical properties. Thus, the ﬁber industry made continuous efforts through the 1960s and 1970s to modify existing processes and develop new ones. One very important breakthrough from such efforts emerged in the late 1970s, enabling one to melt spin at exceedingly high take-up speeds, widely known today as “high-speed melt spinning.” While the ﬁber manufacturers carefully guarded their spinning techniques, the commercial developments were documented in numerous patents. Beginning in the early 1960s, some fundamental studies on ﬁber spinning were reported in the open literature, and they are summarized in the three-volume monograph edited by Mark et al. (1967). An understanding of ﬁber spinning requires knowledge of momentum, energy, and/or mass transport. In addition, knowledge of macromolecular behavior under deformation (i.e., stretching) is also necessary for understanding such complicated problems as molecular orientation under stretching, crystallization kinetics under cooling, and ﬁber morphology as affected by spinning conditions. In the late 1950s, and the early 1960s, Ziabicki and coworkers (Ziabicki 1959, 1961; Ziabicki and Kedzierska 1959, 1960a, 1960b, 1962a, 1962b) made seminal contributions to a fundamental understanding of ﬁber-spinning processes, and their efforts were summarized in Ziabicki’s monograph (1976a). In the 1970s, a new class of synthetic ﬁbers, known as “high-modulus wholly aromatic ﬁbers,” was developed (Bair and Morgan 1972; Daniels et al. 1971; Frazer 1972; Kwolek 1971; Logullo 1971; Morgan et al. 1974) and subsequently commercialized 257

258

PROCESSING OF THERMOPLASTIC POLYMERS

with the trade name of Kevlar by DuPont (Kwolek 1971). The chemical structure of such synthetic ﬁbers consists of rigid rodlike molecules that orient easily along the stretching direction during spinning, giving rise to high modulus in the spun ﬁbers. The chemical structure and mechanical properties of the wholly aromatic ﬁbers are well documented in the monograph edited by Black and Preston (1973). Some ﬁberforming wholly aromatic polymers were found to exhibit liquid crystallinity in the solution state, which led to intensive research efforts to investigate liquid-crystalline polymers. For example, Kevlar ﬁber is based on structures of poly(p-benzamide) or poly(p-phenylene terephthalamide) type (Kwolek et al. 1977; Morgan 1977). Because of the high melting temperatures of these polymers, indicative of a high degree of chain rigidity, Kevlar ﬁbers are spun using solutions via air-gap spinning (which will be brieﬂy described in the next section). In Chapter 9 of Volume 1, we have discussed the rheology of liquid-crystalline polymers having rigid chains. It is fair to state that the ﬁber industry is a mature industry, in the sense that currently there is little fundamental investigation of ﬁber spinning, particularly from the point of view of rheology, as compared with the activities of the 1960s and 1970s. Fundamental research activities before the emergence of high-speed melt spinning have been reviewed by Denn (1983). In this chapter, we ﬁrst present various ﬁber spinning processes from a historical perspective generally, and then high-speed melt spinning as summarized in the monograph edited by Ziabicki and Kawai (1985). Within the spirit of the major thrust of this volume, this chapter puts emphasis primarily on the analysis (mathematical modeling) of high-speed melt spinning, based on a paper by Doufas et al. (2000a). Discussion of the morphology and mechanical properties of synthetic ﬁbers is outside the scope of this chapter.

6.2

Fiber Spinning Processes

In this section, we brieﬂy describe three conventional spinning process, (1) melt spinning, (2) wet spinning, and (3) dry spinning. We then describe other spinning processes, including air-gap spinning, shaped-ﬁber spinning, and conjugate ﬁber spinning processes. 6.2.1

Melt Spinning Process

In melt spinning, the bulk polymer is melted and extruded through a spinneret and the liquid threadlines solidify while passing through a cooling medium, as schematically shown in Figure 6.1. Needless to say, commercial spinnerets have many holes, while for simplicity Figure 6.1 shows only a single ﬁlament. Referring to Figure 6.1, a molten polymer, which was in the shear ﬂow ﬁeld inside the spinneret hole, relaxes its stress upon exiting the spinneret hole, thus giving rise to disorientation of polymer chains in the short distance from the spinneret. The molten threadline is then stretched by the force exerted by the take-up device positioned beneath the spinneret. During stretching, the molten threadline undergoes, in the presence of quench air being blown across the threadline, phase transformation into semicrystalline phase and

FIBER SPINNING

259

Figure 6.1 Schematic showing the deformation of a single ﬁlament in the melt spinning process.

eventually solidiﬁes. As will be presented in the next section, a molten threadline may undergo phase transformation into semicrystalline phase even at a temperature above the equilibrium melting point of the polymer when the take-up speed exceeds a certain critical value, the phenomenon known as “ﬂow-induced (or stress-induced) crystallization.” However, one may argue that crystallization can occur above the equilibrium melting point even without ﬂow being present. The argument is based on a theoretical analysis (Ziabicki 1976a) showing that use of the Avrami rate equation and ﬁtting of quiescent transformation rate data to a Gaussian curve predict the presence of very small but still nonzero amounts of crystal above the equilibrium melting point. This is an empirical consequence of the Avrami ﬁt, nonetheless indicating that some nuclei or crystalline embryos can exist. However, the amount is so small as to have no discernible effect on the system and thus, for all intents and purposes, one can neglect it when solving a system of equations from the point of view of modeling. Commercial ﬁbers such as nylon and polyester ﬁbers are melt spun. An important requirement for a polymer to be melt spun is that it should not degrade when softened by heat. Hence, a polymer that is degradable at the desired spinning temperature is certainly not suitable for melt spinning. Sometimes, the problem of thermal degradation can be avoided by adding a heat stabilizer or plasticizer (some plasticizers are lowmolecular weight substances, which can reduce the melting point of a polymer to below its thermal degradation point). The melt spinning process has two advantages over the wet spinning process, which will be described in the next section, in that high throughputs and high take-up speeds can be achieved, and solvent recovery is not an issue. The high throughput is made possible by the relatively low drag force acting on the ﬁlament while it passes through the cooling gas medium. The take-up speed can be very high (up to say 8,000–9,000 m/min in high-speed spinning), depending on the type of polymer and its rheological properties and melt temperature. Note however that the magnitude of drag force increases as the take-up speed is increased. Since the viscosity of a melt is much higher than

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PROCESSING OF THERMOPLASTIC POLYMERS

that of a solution for wet spinning, the stretch ratio, deﬁned by the ratio of the velocity of the ﬁlament at the take-up device to the average velocity of the melt at the exit of a spinneret hole, is very high in melt spinning. Therefore, to a certain extent, one can control the physical properties of the ﬁnished ﬁbers by judiciously choosing an optimum value of stretch ratio, because the stretch ratio can signiﬁcantly affect the molecular orientation and crystallinity in the ﬁnished ﬁber. As will be presented below, a high stretch ratio may cause ﬂow-induced (or stress-induced) crystallization in the spinline. In the 1960s and 1970s, numerous investigators (Denn et al. 1975; Fisher and Denn 1977; Gagon and Denn 1981; George 1982; Henson et al. 1988; Kase and Matsuo 1965, 1967; Lamonte and Han 1972; Ziabicki 1976b) reported on the analysis (mathematical modeling) of low-speed melt spinning. Some investigators (Bankar et al. 1977a; George 1982; Lamonte and Han 1972; Minoshima et al. 1980) reported on experimental studies of low-speed melt spinning. Here, low-speed melt spinning is considered to be a spinning process in which ﬂow-induced crystallization does not occur (if it occurs, it is very limited). Thus, in low-speed melt spinning the crystallization in the spinline is associated strictly with the cooling by quench air. Conversely, in high-speed melt spinning the stretching of a ﬁlament may induce crystallization and thus the effect of stretching (or tensile stress) of a ﬁlament must be incorporated into the crystallization of a ﬁber-forming polymer. The subject of structure formation during melt spinning has been investigated by numerous research groups (Bankar et al. 1977b; Dees and Spruiell 1974; George et al. 1983; Katayama and Yoon 1985; Nedalla et al. 1977; White et al. 1974). Notwithstanding with the fact that the dynamics of melt spinning greatly inﬂuence the structure formation (morphology) and hence the mechanical properties of the ﬁnished ﬁbers, the review of this subject is not presented here for the reasons that it would require a very large amount of space and the main thrust of this chapter is to show how the information presented in Chapters 2, 3, and 9 of Volume 1 may be used in simulations of the high-speed melt spinning process. 6.2.2

Wet Spinning Process

Figure 6.2 shows schematically the wet spinning process, where a spin dope is pumped into the spinneret and the ﬁlaments thus produced in the spinning bath are passed

Figure 6.2 Schematic of the wet spinning process.

FIBER SPINNING

261

through coagulating baths to yield solid ﬁlaments. The solvent originally added to the spin dope is removed in coagulating baths by the mechanism of counter-diffusion. Sometimes, a chemical reaction also takes place in the coagulating bath (e.g., Viscose ﬁber). In addition, the wet spinning process is economically less attractive than the melt spinning process because an additional cost is involved in the removal of the solvent from the ﬁlaments spun. In general, more than one coagulating bath is needed to remove the solvent, and sometimes the steps involved with aftertreatment (washing and drying) are used for further stretching the ﬁlaments. However, the wet spinning process gives low throughput and low stretch ratios, mainly because of the large force acting on the ﬁlaments while they pass through the liquid coagulating medium. The details of the wet spinning process are described in the literature (Siclari 1967; Ziabicki 1976c). In commercial wet spinning, a series of complex simultaneous operations is performed on and within each spun ﬁlament. These are: extrusion in the spinneret hole, ﬁber elongation, molecular orientation, and coagulation. In addition, counter-diffusion of solvent and nonsolvent occurs between the solidiﬁed skin and the ﬂuid core, and between the coagulated ﬁlament skin and the coagulating bath. Ignoring crystallization and heat transfer, a thorough investigation of this system would involve equations expressing mass transfer, the motion of threads being stretched, and the rheological behavior of the liquid thread in the elongational ﬂow ﬁeld. Furthermore, the physical constants that are needed to solve such a system of equations are very difﬁcult to determine under such complex conditions. Spinning under maximum tension, and therefore under maximum tensile stress, may improve the tensile properties of the ﬁnished ﬁber by increasing molecular orientation. Three factors that could affect spinnability and the mechanism of ﬁber breakage in the wet spinning process may be isolated: (1) coagulating bath concentration, (2) system temperature, and (3) stretch ratio. Coagulating bath concentration is important because this factor actually determines the rate of coagulation, skin formation, take-up speed, maximum stretch ratio, and tensile stress. Temperature is important primarily because it affects the shear viscosity of the spinning solution inside the spinneret hole and the elongational viscosity of the spinline under stretching. Its main inﬂuence is on tensile stress and stretch ratio, rather than on coagulation rate. Stretch ratio is listed independently because it is actually a measure of the ﬁlament residence time in the coagulating bath; the higher the stretch ratio, the shorter the ﬁlament residence time. Even under optimum conditions of temperature and coagulating bath concentration, the stretch ratio may be considered to be an independent variable. Commercial wet spinning for acrylic ﬁbers is performed under conditions of moderate bath concentration and at low temperatures (25–40 ◦ C). These conditions are probably close to optimum, since solution viscosity, stretch ratio, and coagulation rate are balanced so as to allow near-maximum tensions, near-maximum stretch ratio, and probably the minimum chance of thread breakage. At extremely low bath concentration, which brings rapid coagulation, the maximum tension attainable and the maximum stretch ratio are both limited, probably by the breakage of the ﬁlament “skin,” since at a very low coagulating bath concentration the rapid formation of a solid skin inhibits further coagulation of the ﬂuid core. The resultant breakage can be caused by “slippage” between the solid annular skin and the ﬂuid core. At somewhat higher coagulating bath concentrations, slower skin formation allows more thorough hardening of the

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PROCESSING OF THERMOPLASTIC POLYMERS

entire ﬁber, and an optimum ﬁlament strength dependent on coagulation rate may be achievable. Breakage of the “slowly hardened” ﬁlament would then occur if a critical tensile stress is exceeded, and at this point the hardened portion would separate completely from the ﬂuid (at or in the spinneret) over the entire cross section of the ﬁlament. It follows, therefore, that there is an optimum coagulation rate that depends on coagulating bath temperature and concentration, thus maximizing the combination of ﬂuid and skin strength of the partially hardened ﬁber. Temperature may play a dual role here, since the effect of temperature on the spin dope viscosity appears to be more important than its effect on coagulation rate. Thus, there are only a few studies (Han and Segal 1970; Ziabicki 1976c) that have reported on the analysis of wet spinning. A rigorous analysis of the wet spinning process is much more complicated than an analysis of melt spinning because wet spinning involves mass transfer, as we have described. 6.2.3

Dry Spinning Process

Figure 6.3 shows schematically the dry spinning process, where a spin dope is pumped into the spinneret, which is placed in an enclosed chamber (the drying tower) for solvent recovery. Upon exiting the spinneret, the solvent in the spin dope in the moving ﬁlament evaporates inside the drying tower, to which an inert gas at an elevated temperature is supplied. The inert gas leaving the drying tower carries most, if not all, of the solvent from the spin dope (Corbiere 1967; Ziabicki 1976c). Dry spinning is usually recommended when the polymer has no ﬁnite melting point or is easily degraded when heated. The dry spinning process calls for a solvent that has a low boiling point and a low heat of vaporization. In general, a nonpolar solvent is preferred to a polar one, and, in addition, it should satisfy the following requirements: ease of recovery, thermal stability, inertness, nontoxicity, minimum tendency to form

Figure 6.3 Schematic of the dry

spinning process.

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electrostatic charges, and free from explosion hazard. Examples of dry spun ﬁbers are acrylic ﬁber and polyvinyl chloride ﬁber. From the point of view of solvent recovery, concentrated solutions are preferred for dry spinning. However, in practice the preparation of concentrated spin dope is very difﬁcult because of the solubility limit and its difﬁculty in handling. However, as in melt spinning, the dry spinning process can give a high stretch rate and hence a high production rate. Dry spinning is advantageous over wet spinning in that higher throughputs and higher take-up speeds possible compared with those in wet spinning. From the point of view of process simulation, dry spinning is much more complicated than wet spinning in that heat and mass transfers occur simultaneously, and thus dry spinning has received the least attention of researchers from a fundamental point of view. This is mainly because an analysis of the dry spinning process requires the solution of the momentum, energy, and mass transfer equations. In the energy balance equation, a term describing the latent heat of solvent vaporization, in addition to a term describing the convective heat transfer, must be included. In the mass transfer equation, the mechanisms of mass transfer must be included, describing (1) ﬂash vaporization, (2) diffusion within the threadline, and (3) convective mass transfer from the threadline surface to the surrounding gas medium. As one may surmise, the experimental determination of the physical constants associated with mass and heat transfers (e.g., the diffusion coefﬁcient and the heat and mass transfer coefﬁcients) is very difﬁcult. The lack of experimental data is the major reason as to why so few simulation studies on the dry spinning process have been carried out. In the 1960s and early 1970s, some investigators (Fok and Griskey 1966; Ohzawa et al. 1969; Ohzawa and Nagano 1970) reported on experimental and theoretical studies of dry spinning. Since then, very few fundamental studies on dry spinning have been reported until recently (Gou and McHugh 2004a, 2004b, 2004c). Much more experimental work is needed to gain a better understanding of the complicated interactions; for instance, between the momentum and mass transfer, and between mass and heat transfers. Note also that the elongational viscosity in dry spinning should be represented in terms of temperature and concentration, as well as axial velocity gradient. This is because both the temperature and concentration change continuously as the ﬁlament solidiﬁes in the spinline. Since only a small number of fundamental studies on dry spinning have been reported, it is very difﬁcult to present here the fundamental aspects of dry spinning at an intelligent level. 6.2.4

Other Fiber Spinning Processes

Over the years, the ﬁber industry has developed processes other than those described above to produce synthetic ﬁbers. In this section, we describe those processes that have met with great commercial success. 6.2.4.1 Air-Gap Spinning Figure 6.4 shows schematically the air-gap spinning process, where a spin dope is pumped into the spinneret and the ﬁlaments pass through a short distance of air gap (more precisely, through a short, enclosed drying chamber), which is very similar to

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Figure 6.4 Schematic of the air-gap spinning process.

the dry spinning process, and then into a spinning bath, like the coagulating bath in the wet spinning process. Because the bulk of the solvent in the spin dope is expected to evaporate in the air gap, the removal of the remaining solvent from the ﬁlaments should not require a long, multistage coagulating bath. This method of ﬁber spinning was apparently developed for producing Kevlar ﬁbers by DuPont (Kwolek 1971). The extent of improvement in the mechanical properties of ﬁbers spun from air-gap spinning over those spun from dry spinning alone or wet spinning alone has not been reported. Such information is a well-guarded secret by the industry. It is, however, clear that the air-gap spinning process is a combination of dry spinning and wet spinning. Kevlar ﬁbers are made from a spin dope consisting of poly(p-benzamide) or poly(p-phenylene terephthalamide) type, exhibiting liquid crystallinity in the solution state. It is quite possible that air-gap spinning of Kevlar spin dope might enable one to have a better control of chain orientation along the ﬁber axis than can be achieved by dry spinning alone or wet-spinning alone. 6.2.4.2 Shaped-Fiber Spinning The ﬁber industry has long practiced the production of synthetic ﬁbers having a variety of cross sections other than circular. Such ﬁbers are referred to as “shaped ﬁbers” (Buckley and Philipps 1969; Forney et al. 1966). What is most intriguing in making shaped ﬁbers is that a desired cross-sectional shape of ﬁber can be produced from spinneret holes whose shape is quite different from that of the ﬁber itself. There are many variables that may play an important role in determining the ﬁnal shape of a ﬁber’s cross section. For instance, wet-spun ﬁbers of an elliptical shape, having various aspect ratios, can be produced from the same rectangular spinneret hole by judiciously choosing spinning variables, such as stretch ratio or coagulating bath concentration

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(Han and Park 1973). In addition, equally as important as the spinning variables, if not more, are the rheological properties of the spin dope itself. Because of the complexity of the problem, a rigorous analysis of the processes involved in producing shaped ﬁbers has not been reported in the literature. Therefore, the development of shaped ﬁbers appears to have largely depended on trial and error, and, understandably, much of the technology in this ﬁeld has been kept as proprietary information by the ﬁber manufacturers. Han (1971) made some interesting experimental observations, which enabled him to explain why, for instance, an elliptical ﬁber can be produced from a rectangular spinneret hole. He believed that the nonuniform distribution of wall normal stresses along the long and short sides of the rectangular spinneret holes can give rise to nonuniform extrudate swelling upon its exiting from the die, which leads to an elliptical ﬁber cross section. In Chapter 1 we presented experimental results for wall normal stress distributions in a high-density polyethtylene (HDPE) melt ﬂowing through a channel having a rectangular cross section (see Figure 1.6), where wall normal stresses measured along the long side of the rectangular channel were greater than those measured at the center of the short side of the rectangular channel. Han and Park (1973) investigated shaped ﬁbers using both wet spinning and melt spinning; in particular, they studied (1) the effect of shape and size of a spinneret hole on the shape of extruded ﬁlaments, and (2) the effect of spinning conditions, such as stretch ratio and coagulating bath concentration, on the shape of ﬁbers spun. Figure 6.5 shows the shapes of two spinneret holes: (a) a trilobal hole and (b) a round hole with lugs. Also given in Figure 6.5 are pictures of the cross sections of melt-spun polystyrene ﬁbers: (c) from a trilobal spinneret hole and (d) from a round hole with lugs. It is seen in Figure 6.5 that the cross-sectional shape of the melt-spun ﬁbers resembles very much the cross-sectional shape of the spinneret holes from which ﬁbers were spun. Figure 6.6 gives pictures of the cross-sectional shape of wet-spun acrylic ﬁbers: (a) from a trilobal spinneret hole and (b) from a round hole with lugs. There are three variables that can affect the cross-sectional shape of the ﬁber when

Figure 6.5 (a) Shape of a spinneret with a trilobal hole, (b) shape of a round spinneret hole with lugs, (c) the cross section of a polystyrene ﬁber that was melt-spun from a spinneret with a trilobal hole at an apparent stretch ratio of 30.5, and (d) the cross section of a polystyrene ﬁber that was melt-spun from a round spinneret hole with lugs at an apparent stretch ratio of 40.5. (Reprinted from Han and Park, Journal of Applied Polymer Science 17:187. Copyright © 1973, with permission from John Wiley & Sons.)

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Figure 6.6 Cross sections of acrylic ﬁbers that were wet-spun from a spinneret with (a) trilobal hole at an apparent stretch ratio of 5.6 and (b) round spinneret hole with lugs at an apparent stretch ratio of 10.5, both with a bath concentration of 0 wt % NaSCN (i.e., in a water coagulating bath) and at room temperature. (Reprinted from Han and Park, Journal of Applied Polymer Science 17:187. Copyright © 1973, with permission from John Wiley & Sons.)

it is spun through noncircular spinneret holes (Han and Park (1973)): (1) stretch ratio, (2) coagulating bath concentration (thus interfacial tension) in wet spinning, and (3) mass throughput. Regarding the effect of stretch ratio, they observed that the cross-sectional shape of the wet-spun acrylic ﬁbers, which were obtained through rectangular spinneret holes having an aspect ratio of 5, had almost a circular cross section at a low stretch ratio, while the cross-sectional shape of the ﬁbers more closely resembled the cross-sectional shape of the spinneret hole as the stretch ratio increased. Regarding the role of interfacial tension, Han and Park (1973) concluded that in wet spinning, the interfacial tension between the liquid-state threadline that is only partially coagulated and the coagulating bath solution can play a very important role in determining the ﬁnal cross-sectional shape of wet-spun ﬁber; the interfacial tension tends to make the initially noncircular cross section of the wet-spun threadline, upon exiting the spinneret hole, circular when the applied stretch ratio is low. However, as the stretch ratio is increased, the applied tension may be transmitted to the liquid threadline just outside the spinneret face. If this happens, the applied tension becomes dominant over the interfacial tension, causing the cross-sectional shape of the wet-spun ﬁber to resemble the cross-sectional shape of the spinneret hole. 6.2.4.3 Conjugate-Fiber Spinning The ﬁber industry has long produced ﬁbers consisting of two components, which are referred to as “bicomponent” or “conjugate” ﬁbers (Hicks et al. 1960, 1967; Sisson and Morehead 1953, 1960; Ziabicki 1976d). As the word conjugate implies, these ﬁbers are produced by pumping two feed streams side by side into a common spinneret hole, as shown schematically in Figure 6.7. The two streams meet at the entrance of

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Figure 6.7 Schematic of a spinneret into which two separate streams (melts for melt spinning or spin dopes for wet spinning) are pumped, producing a conjugate (two-component) ﬁber.

the spinneret hole and ﬂow side by side through the spinneret hole. Historically, the ﬁber industry developed conjugate ﬁbers to mimic natural wools. These ﬁbers have a crimped (curly) characteristic and do not lose their shape even after washing and drying many times. Figure 6.8 gives a photograph of conjugate ﬁbers of polypropylene (PP) and poly(ethylene terephthalate) (PET), showing a crimping characteristic (Han 1981). The crimpy characteristic results from the different thermal expansion coefﬁcients of the individual components, leading to the buckling of the ﬁlament upon exiting from the spinneret, while it is being either cooled or coagulated along the length of the spinline. In producing conjugate ﬁbers, the distribution of the two components (i.e., interfacial shape) in the ﬁber cross section is of paramount importance in controlling the amplitudes and frequencies of crimps in the ﬁnished ﬁber. For instance, when two components are fed side by side at the inlet of a spinneret with circular cross section, it is highly desirable to maintain the same interface in the ﬁnished ﬁber in order to have

Figure 6.8 Photographs showing the crimped characteristics of a conjugate ﬁber of PP and PET.

(Reprinted from Han, Multiphase Flow In Polymer Processing, Chapter 7. Copyright © 1981, with permission from Elsevier.)

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PROCESSING OF THERMOPLASTIC POLYMERS Figure 6.9 Schematic showing the deformation of the

interface when two separate feed streams meet at the inlet of a spinneret and then ﬂow side by side inside a spinneret with a round cross section. (Reprinted from Han, Multiphase Flow In Polymer Processing, Chapter 7. Copyright © 1981, with permission from Elsevier.)

crimped characteristic. However, as schematically shown in Figure 6.9, the interface of two components will deform (or shift), depending on the rheological properties of the individual components; namely, while ﬂowing side by side inside a spinneret hole, the less viscous component tends to surround the more viscous component. Such interfacial deformation is not desirable from the point of view of producing conjugate ﬁbers. Several research groups have investigated, experimentally (Han 1973, 1975; Han and Kim 1976; Lee and White 1974; Southern and Ballman 1973, 1975) and theoretically (Everage 1973; MacLean 1973; Khan and Han 1976), the effect of rheological properties (viscosity and elasticity) of the constituent components on the interface deformation inside a die (or spinneret) when two polymers ﬂow side by side. The fundamentals of coextrusion from the point of view of rheology are well documented in a monograph by Han (1981).

6.3

High-Speed Melt Spinning

Beginning in the late 1970s, high-speed melt spinning attracted the attention of the ﬁber industry for two obvious reasons: it certainly increased production rate, and it most likely enhanced the tensile properties of ﬁbers due to a higher degree of orientation of macromolecules during stretching. The spinning speed that could be achieved with high-speed melt spinning of nylon and PET is reported to be as high as 9,000 m/min (Ziabicki and Kawai 1985). Owing to the very high stretch rates applied to the spinline during high-speed melt spinning, some very unusual physical phenomena, which were absent in low-speed melt spinning, have been observed experimentally (Doufas and McHugh 2001a; Doufas et al. 2000b; Haberkorn et al. 1990; Ishizuka and Koyama 1985; Matsui 1985; Shimizu et al. 1985; Vassilatos et al. 1985; Zieminski and Spruiell 1988). Such experimental observations stimulated the modeling efforts on high-speed melt spinning during the past two decades (Doufas 2002; Doufas et al. 2000a; Doufas and McHugh 2001b; Kim and Kim 2000; Kulkarni and Beris 1998; Zieminski and Spruiell 1988). Since the experimentally observed unusual phenomena are intimately related to the rheological behavior of the ﬁber-forming polymers, in this section we present ﬁrst some typical experimental observations reported in the literature and then an analysis of high-speed melt spinning, associated primarily with its rheological aspects. For the reasons given in the introduction to this chapter, we will not address the problems associated with the

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structure development, the orientation of macromolecular chains, and the mechanical properties of the ﬁnished ﬁbers produced from high-speed melt spinning. 6.3.1

Experimental Observations of High-Speed Melt Spinning

There are primarily two physical phenomena reported in the literature that distinguish high-speed melt spinning from low-speed melt spinning; they are (1) a sudden decrease in ﬁlament diameter, often referred to as “necklike deformation,” at a certain position along the spinline before solidiﬁcation begins, and (2) ﬂow-induced (or stress-induced) crystallization in the spinline in the high-speed melt spinning of notably nylon and PET. In this section, we present some representative experimental results reported in the literature as they relate to the unusual rheological behavior observed in high-speed melt spinning. Figure 6.10 shows the variations of ﬁlament diameter (D) in terms of denier along the spinline distance (z) in the melt spinning of PET for different values of take-up speed (VL ), mass throughput (Q), and melt extrusion temperature (T ). Denier is the weight in grams of a 9,000 m length of a ﬁber or yarn, and thus denier is related to the ﬁlament diameter. Notice in Figure 6.10 that the ﬁlament denier increases with mass throughput. What is of great interest in Figure 6.10 is that the ﬁlament diameter suddenly decreases at z ≈ 50 cm as the take-up speed is increased from 5,000 to 6,000 m/min, giving rise to “necklike deformation” of the ﬁlament diameter. Figure 6.11 gives the proﬁles of ﬁlament diameter D, temperature T, axial velocity gradient dvz /dz, tensile stress Tzz , and birefringence n along the spinline distance z in the high-speed melt spinning of PET at a take-up speed of 6,000 m/min. The following

Figure 6.10 Proﬁles of ﬁlament diameter

(in denier) along the spinline distance from spinneret in the melt spinning of PET under different spinning conditions: () at a take-up speed of 4,000 m/min, mass throughput of 0.33 g/min, and extrusion temperature of 295 ◦ C, () at a take-up speed of 4,000 m/min, mass throughput of 1.14 g/min, and extrusion temperature of 280 ◦ C, () at a take-up speed of 5,000 m/min, mass throughput of 1.35 g/min, and extrusion temperature of 290 ◦ C, and () at a take-up speed of 6,000 m/min, mass throughput of 2.55 g/min, and extrusion temperature of 295 ◦ C. Spinneret hole diameter employed was 0.03 cm. (Reprinted from Matsui, High-Speed Fiber Spinning, Ziabicki and Kawai (eds), p. 137. Copyright © 1985, with permission from John Wiley & Sons.)

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Figure 6.11 Proﬁles of ﬁlament diameter D (), axial velocity gradient dvz /dz (), tensile stress Tzz (), birefringence (), and temperature (3) along the spinline distance from spinneret in the melt spinning of PET at a take-up speed of 6,000 m/min, mass throughput of 8.4 g/min, and extrusion temperature of 295 ◦ C. The spinneret hole diameter employed was 0.24 cm. (Reprinted from Ishizuka and Koyama, High-Speed Fiber Spinning, Ziabicki and Kawai (eds), p. 151. Copyright © 1985, with permission from John Wiley & Sons.)

observations are worth noting in Figure 6.11. (1) The dvz /dz ﬁrst increases, goes through a maximum, and then decreases along the spinline distance z. Interestingly, the position at which a maximum in dvz /dz occurs coincides with the position at which necklike deformation of the ﬁlament diameter occurs. (2) The ﬁlament temperature also shows a rapid decrease at the spinline position where the value of ﬁlament diameter suddenly decreases. (3) The tensile stress Tzz begins to increase very rapidly as dvz /dz increases very rapidly and then Tzz increases very slowly along the spinline even though dvz /dz, after going through a maximum, rapidly decreases. (4) The birefringence n increases very rapidly along the spinline z and then levels off as dvz /dz ﬁrst increases very rapidly, goes through a maximum, and then decreases rapidly. The above observations indicate that some unusual physical phenomena take place in the spinline where necklike deformation of the ﬁlament diameter occurs. We will elaborate on this further after we have presented an analysis of high-speed melt spinning. Figure 6.12 gives the proﬁles of ﬁlament diameter in the melt spinning of an HDPE having very broad molecular weight distribution (MWD). It is seen in Figure 6.12 that the HDPE also exhibits necklike deformation of the ﬁlament diameter as the take-up speed exceeds a certain critical value. Notice that the values of take-up speed applied to generate Figure 6.12 are rather low (1, 4, and 8 m/min) compared with those practiced in industrial high-speed melt spinning of PET and nylon. Nevertheless, the melt spinning of a very broad MWD HDPE exhibits the same feature of necklike deformation of the ﬁlament diameter as that observed in high-speed melt spinning of PET (compare Figure 6.12 with Figure 6.10). Figure 6.13 shows the effect of take-up speed on the axial velocity gradient (dvz /dz) along the spinline in the melt spinning of a very broad MWD HDPE, showing that a maximum in dvz /dz increases very rapidly with increasing take-up speed. Note that

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Figure 6.12 Proﬁles of ﬁlament diameter along the spinline distance from spinneret in melt spinning of an HDPE with broad molecular weight distribution at various take-up speeds (m/min): () 1, () 4, and () 8. (Reprinted from Ishizuka and Koyama, High-Speed Fiber Spinning, Ziabicki and Kawai (eds), p. 151. Copyright © 1985, with permission from John Wiley & Sons.)

Figure 6.13 was obtained under the identical spinning conditions as produced the ﬁlament diameter proﬁles presented in Figure 6.12. Notice in Figure 6.13 that dvz /dz varies with z and thus the melt spinning operation does not give rise to steady-state (constant) elongation rate, in contrast to the controlled rheological experiment that gives rise to steady-state elongation rate discussed in Chapter 5 of Volume 1. Thus, in Figure 6.13 Proﬁles of apparent elongation rate dvz /dz along the spinline distance from spinneret in the melt spinning of an HDPE with broad molecular weight distribution at various take-up speeds (m/min): () 1, () 4, and () 8. (Reprinted from Ishizuka and Koyama, High-Speed Fiber Spinning, Ziabicki and Kawai (eds), p. 151. Copyright © 1985, with permission from John Wiley & Sons.)

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melt spinning we do not have steady-state elongational ﬂow. This distinction is very important to the analysis of the melt spinning process, as will be elaborated on later in this chapter. Therefore, to distinguish a constant elongation rate that can be realized under controlled elongational ﬂow experiments, throughout this chapter the dvz /dz encountered in high-speed melt spinning is referred to as “apparent” elongation rate (˙εapp ). There is experimental evidence (Han and Lamonte 1972), suggesting that dvz /dz along the spinline is not constant, even when a molten polymer is spun through a short isothermal chamber attached beneath the spinneret. After all, in melt spinning one is not interested in controlling elongation rate along the spinline. It is then fair to state that the situation of constant elongation rate is not encountered in practical melt spinning operations. Figure 6.14 gives the proﬁles of “apparent” elongational viscosity (ηE,app ) along the spinline distance z for a very broad MWD HDPE at three different take-up speeds, where values of ηE,app were calculated from the deﬁnition ηE,app = T zz /(dvz /dz), where the tensile stress Tzz is deﬁned by Tzz = F/A with F being tensile force and A being ﬁlament cross-sectional area. In Figure 6.14, we observe that ηE,app ﬁrst decreases, goes through a minimum, and then increases very rapidly, the value of the minimum in ηE,app being greater with increasing take-up speed. Note that melt spinning is conducted under nonisothermal conditions and thus the variations in ηE,app along the spinline distance, given in Figure 6.14, is greatly inﬂuenced by cooling; cooling is expected to increase ηE,app along the spinline distance. Interestingly enough, however, Figure 6.14 shows that along the spinline distance, ηE,app initially decreases away from the spinneret and then begins to increase very rapidly. What is of great interest in Figure 6.14 is that the spinline position at which ηE,app goes through a minimum roughly corresponds to the position where ε˙ app goes through a maximum (see Figure 6.13),

Figure 6.14 Proﬁles of apparent

elongational viscosity ηE,app along the spinline distance from the spinneret in the melt spinning of an HDPE with broad molecular weight distribution at various take-up speeds (m/min): () 1, () 4, and () 8. (Reprinted from Ishizuka and Koyama, High-Speed Fiber Spinning, Ziabicki and Kawai (eds), p. 151. Copyright © 1985, with permission from John Wiley & Sons.)

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and also roughly corresponds to the position where necklike deformation of the ﬁlament diameter occurs (see Figure 6.12). We will return to this subject after presenting an analysis of the high-speed melt spinning process.

6.3.2

Modeling of High-Speed Melt Spinning

Flow-induced (or stress-induced) crystallization in the spinline, which occurs only in high-speed melt spinning, began to attract the attention of researchers in the late 1970s. Kulkarni and Beris (1998) appear to be the ﬁrst who developed a structure-based phenomenological model for isothermal melt spinning by including stress-induced crystallization in the spinline. The primary objective of their study was to explain the necklike deformation of the ﬁlament diameter in high-speed melt spinning. In their study, Kulkarni and Beris employed phenomenological constitutive equations to describe both the amorphous and semicrystalline phases, and neglected air drag effects in addition to ﬁlament cooling. Doufas et al. (2000a) developed a very comprehensive structure-based model to simulate both low- and high-speed melt spinning, which included the effects of stress-induced crystallization, viscoelasticity of both the amorphous and semicrystalline phases, ﬁlament cooling, air drag, inertia, surface tension, and gravity. In so doing, they employed a modiﬁed version of the single-mode Giesekus constitutive equation (Giesekus 1982) (see Chapter 3 of Volume 1) to describe the rheology of the amorphous phase prior to the onset of crystallization and a rigid rod model (Bird et al. 1987) to describe, after the onset of ﬂow-induced crystallization, the rheology and orientation of the semicrystalline phase in the spinline. Their model explains successfully many important features of the experimental observations reported hitherto when model predictions are compared with experiment (Haberkorn et al. 1990; Ishizuka and Koyama 1985; Matsui 1985; Shimizu et al. 1985; Vassilatos et al. 1985; Zieminski and Spruiell 1988). Since the analysis of high-speed melt spinning by Doufas et al. (2000a) seems to be the most comprehensive among all the studies reported to date, we will present their analysis and then compare its predictions with experiment. To help facilitate our presentation, let us look at the schematic given in Figure 6.15, describing the variation in ﬁlament diameter beneath the spinneret in high-speed melt spinning. There are three issues, among others, that make a rigorous analysis of melt spinning very complicated. The ﬁrst issue has to do with the extrudate swell of a viscoelastic melt upon exiting a spinneret. Needless to say, for all intents and purposes under practical spinning conditions, the extrudate swell can be neglected when dealing with nylon and PET because the ﬁber-spinning grades of nylon and PET follow nearly Newtonian behavior. However, the ﬁber-spinning grades of PP and HDPE exhibit strong viscoelastic behavior, giving rise to a signiﬁcant degree of extrudate swell upon exiting a spinneret. For a given polymer, the exact location at which the maximum extrudate swell occurs and the extent of extrudate swell depend on the deformation history of the melt inside the spinneret hole. This then suggests that one must solve the momentum balance equation inside the spinneret hole followed by the extrudate swell region outside the spinneret. A rigorous analysis of the ﬂow of a molten threadline in the extrudate swell region is very difﬁcult owing to as-yet undetermined boundary conditions, compounded by the usually complicated expressions for the constitutive equations of

274

PROCESSING OF THERMOPLASTIC POLYMERS Figure 6.15 Schematic of the high-speed

melt spinning process, where extrudate swell and necklike deformation are exaggerated for the purpose of illustration. In the schematic, z = 0 represents the position where the initial conditions are speciﬁed for the solution of the system equations described in the text, and z = L is the position of the take-up roll. (Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.)

viscoelastic ﬂuids. In the analysis presented below, we neglect the extrudate swell region, as indicated in the schematic shown in Figure 6.15. The second issue has to do with the stretching and reorientation of polymer chains before solidiﬁcation begins. Referring to Figure 6.15, the polymer chains, which were once oriented inside the spinneret passages, begin to disorient, upon exiting the spinneret, over a very short distance. Then, below the extrudate swell region the orientation of the polymer chains starts again under the inﬂuence of stretching by the take-up device. It may be supposed that chain orientation ceases at a point where solidiﬁcation of the molten threadline has progressed to such an extent that the long polymer chains lose their mobility. In Figure 6.15, necklike deformation is indicated schematically. One must then formulate a system of equations, such that necklike deformation can be predicted as part of the solution of the system equations. The third issue has to do with being able to adequately describe stress-induced crystallization in the spinline during high-speed melt spinning. Note that the rate of cooling and the rate of stretching inﬂuence both the rate of crystallization and the degree of crystallization. It is not difﬁcult to surmise that crystallization would greatly affect the elongational viscosity of the spinline during melt spinning. Some research groups (Katayama and Yoon 1985; Patel et al. 1991; Ziabicki 1988; Zieminski and Spruiell 1988) incorporated, on an ad hoc basis, the rate of crystallization into the variation in viscosity (strictly speaking, elongational viscosity) along the spinline in high-speed melt spinning. Since cooling also affects the viscosity (η) of the spinline, they employed an empirical expression of the form (Ziabicki 1988) η (T , X) = f (T ) [1 − X]n

(6.1)

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where f (T) is a function describing the temperature dependence of η, n is an empirical constant characteristic of a polymer, and X = φ/φ∞ represents the degree of transformation from the molten state to the semicrystalline state and is therefore bounded by 0 and 1, although it does not necessarily have to go to 1, and where φ∞ represents the ﬁnal degree of crystallinity of the transformed semicrystalline phase and is generally less than 1. For crystallization under quiescent, nonisothermal conditions, X is often expressed by the generalized Avrami equation (Nakamura et al. 1973): m t K (T )dt X(t) = 1 − exp − 0 av

(6.2)

where Kav (T ) is the Avrami constant under quiescent (no ﬂow) conditions and m is the Avrami index determined in the isothermal experiments. Since an increase of take-up speed in high-speed melt spinning introduces extra stresses in the spinline, crystallization in high-speed melt spinning will occur in the spinline at a position closer to the spinneret, at a rate that will be higher than in low-speed melt spinning. Ziabicki (1988) advocated that an increase in crystallization rate under stress can reach many orders of magnitude, which in turn may lead to extremely high elongation rates and solidiﬁcation of the polymer within a narrow range of the spinline. He speculated that necklike deformation in the spinline might possibly be caused by the gradient of apparent elongational viscosity along the spinline. 6.3.2.1 Mass Balance, Momentum Balance, and Energy Equation for One-Dimensional Steady-State Melt Spinning In the past, numerous research groups (Denn et al. 1975; Fisher and Denn 1977; Gagon and Denn 1981; George 1982; Henson et al. 1988; Kase and Matsuo 1965, 1967; Lamonte and Han 1972; Ziabicki 1976b) set up and solved, often numerically, a system of equations describing the melt spinning process. Here, we present the mass balance, momentum balance, and energy equations in cylindrical coordinates, which were used by Doufas (2002) and Doufas et al. (2000a), describing the one-dimensional steady-state melt spinning process. (a) Mass Balance

In melt spinning the mass is conserved and it is

expressed by W = ρπD 2 vz /4

(6.3)

where W is mass ﬂow rate, ρ is the polymer melt density that is assumed to be constant, D is ﬁlament diameter that decreases along the axial direction z, and vz is the axial velocity that increases in the z-direction. (b) Momentum Balance

W

dvz = dz

The momentum equation can be written as

% & d A τzz − τrr dz

dD 1 − πBµa (vz − vd ) + ρgA + πς 2 dz

(6.4)

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where A is the cross-sectional area of the ﬁlament, (τzz − τrr ) is the tensile stress1 (see Chapter 3 of Volume 1) at an arbitrary position z in the ﬁlament, B is a dimensionless variable (Bingham number) related to the quench air drag force,2 µa is the viscosity of the quench air, vd is the downward velocity of quench air, g is the gravitational force, and ς is the surface tension of the ﬁlament. On the right-hand side of Eq. (6.4), the ﬁrst term describes the average tensile force at the axial position z, the second term represents the air drag between the moving ﬁlament and the quench air, the third term describes the force due to gravity, and the fourth term represents the surface tension effect of the moving ﬁlament whose diameter decreases along the z-direction. (c) Energy Equation In writing down the energy equation given below, it is assumed that the conductive and radiant heat transfers are negligibly small compared with the convective heat transfer:

ρcp vz

dv 4 dT dφ = − h(T − Ta ) + (τzz − τrr ) z + ρHf vz dz D dz dz

(6.5)

where cp is the heat capacity, h is the heat transfer coefﬁcient, T is the ﬁlament temperature, Ta is the quench air temperature, Hf is the heat of crystallization per unit mass, and φ is the average absolute degree of crystallinity of the system (mass fraction of crystals) at the axial position z. On the right-hand side of Eq. (6.5), the ﬁrst term describes the convective heat transfer between the ﬁlament and the quench air, the second term describes the viscous dissipation, and the third term describes the release of heat generated from the crystallization taking place in the ﬁlament. 6.3.2.2 Microstructural Model for the Amorphous Phase Prior to the Onset of Crystallization Doufas et al. (2000a) chose a modiﬁed version of the single-mode Giesekus model (1982) to describe the rheological behavior of a ﬁber-forming polymer melt prior to the onset of crystallization. In Chapter 3 of Volume 1 we described the Giesekus model that introduces the conﬁguration tensor c, deﬁned by Eq. (3.25). In modifying the single-mode Giesekus model, following Kulkarni and Beris (1998), Doufas et al. (2000a) introduced the conformation tensor c = rr to describe the stretching and orientation of the ﬂexible polymer chain. Here the bracket in rr denotes the averaging of the quantity rr with respect to the distribution function of the polymer melt (see Chapter 4 of Volume 1), where the vector r denotes the end-to-end distance of a ﬂexible polymer chain. Note that the conformation tensor c introduced by Doufas et al. is different from the conﬁguration tensor c in the Giesekus model in that the conformation tensor c has the dimension of length squared, while the conﬁguration tensor c is a strain tensor that is dimensionless. This distinction between the two is very important for the correct interpretation of the simulation results presented below. The second-order conformation tensor c = rr can be considered to represent the second moment of the end-to-end vector r of the polymer chain. The concept of relating the conformation tensor c to the stress tensor σ was introduced by Bird et al. (1987), who developed the kinetic theory for the dynamics of macromolecules.

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By recognizing the fact that in reality polymer chains cannot be stretched indeﬁnitely, Doufas et al. (2000a) employed the following form of a modiﬁed expression of Eq. (3.26) given in Chapter 3 of Volume 1, which is due to Peterlin (1966):3 K0 K0 Ec · Ec − δ (1 − α)δ + α kB T kB T

kB T 1 ᒁc =− ᒁt λa,0 (T ) K0

(6.6)

where ᒁ/ᒁt is the upper convected derivative (see Chapter 2 in Volume 1), λa,0 (T ) is a temperature-dependent characteristic (Hookean) relaxation time of the amorphous melt phase, T is the absolute temperature, kB is the Boltzmann constant, K0 is the Hookean spring constant, deﬁned by (Bird et al. 1987) K0 = 3kB T /Nb2

(6.7)

with N being the number of ﬂexible statistical links in a single polymer chain and b being the length of a statistical link, α is the molecular (Giesekus) mobility parameter that lies in the range 0 ≤ α ≤ 1 (see Chapter 3 of Volume 1), and E is the nonlinear spring force factor introduced by Peterlin (1966)4

E(r/Lc ) =

⎧ 3 2 ⎪ ⎨1 + 5 (r/Lc ) + . . . ) c = 1 ⎪ 3(r/Lc )

⎩ 3(r/Lc ) 1 − (r/Lc )

L−1 (r/L

r/Lc → 0 r/Lc → 1

(6.8)

where L−1 is the inverse Langevin function, deﬁned by % &−1 L−1 (x) = (coth x − x −1 )

(6.9)

with Lc being the contour length Nb. Note that E in Eq. (6.6) was introduced to describe ﬁnite extensibility of the Hookean spring (Peterlin 1966). Other forms of ﬁnite extensibility of the Hookean spring have been suggested in the literature (Bird et al. 1987). Hence, the solution of Eq. (6.6) describes the evolution of the stretching of the polymer chains under a given ﬂow condition. Once the solution of Eq. (6.6) for c is available,5 one can calculate the components of the extra stress tensor τ for the amorphous (melt) phase prior to the onset of crystallization from τ = nK0 Ec − nkB T δ

(6.10)

in which n is the number of chains in unit volume in accordance with the rigid rod model (Bird et al. 1987), and nkB T may be considered to represent the melt shear modulus. Note that the conformation tensor c appearing in Eq. (6.10) has the dimension of length squares and Eq. (6.10) enables one to calculate the tensile stress from Tzz = τzz − τrr in uniaxial elongational ﬂow (see Chapter 3 of Volume 1). The salient feature of the approach presented above lies in that microstructure parameters, via the tensor c, are connected to the stress tensor τ. This is very important because the stress tensor alone, when using the strictly continuum-based constitutive equations

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PROCESSING OF THERMOPLASTIC POLYMERS

(see Chapter 3 of Volume 1), does not provide information on the stretching of the polymer chains in the spinline. Speciﬁcally, the solution of Eq. (6.6) enables one to understand in terms of the zz- and rr-components of the conformation tensor c how, prior to the onset of crystallization, the amorphous melt chains stretch in the spinline, as will be shown below. Several research groups (Doufas 2002; Doufas et al. 2000a; Kim and Kim 2000; Kulkarni and Beris 1998; Zieminski and Spruiell 1988) have reported on the development of mathematical models to simulate high-speed melt spinning including crystallization. Some studies were based on a phenomenological approach in that the rheological behavior (e.g., tensile stress or elongational viscosity) of the spinline was coupled, on an ad hoc basis, with the structure parameters associated with the semicrystalline phase of a ﬁber-forming polymer. In high-speed melt spinning, since the combination of high tensile stress and temperature gradient along the spinline induces crystallization, it is highly desirable for one to be able to describe the orientation of the semicrystalline phase in ﬂow, while coupling between the structural variables and stress is realized through a set of governing equations. To that end, the approach taken by Doufas et al. (2000a) is very appealing. They used the elastic dumbbell model, Eq. (6.10), to describe the rheological behavior, after the onset of crystallization, of the semicrystalline phase of a ﬁber-forming polymer. They used the schematic given in Figure 6.16 to describe their microstructural model for the semicrystalline phase after the onset of crystallization in high-speed melt spinning. Referring to Figure 6.16, a ﬁnite number N of statistical segments are available for crystallization. Thus, after the onset of crystallization, the ﬁber-forming polymer consists of trapped amorphous and crystalline regions, forming a semicrystalline phase, and the remaining untransformed molten phase (described by the Giesekus equation). The molten phase keeps decreasing, until a point is reached where crystallization is complete. The untransformed molten phase is described by Eq. (6.6) with λa,0 (T ) replaced by a temperature- and crystallinity-dependent relaxation time λa (T , X) in the form

Figure 6.16 Schematic showing the amorphous phase in a crystallizable polymer melt rep-

resented by a modiﬁed Giesekus model and, after ﬂow-induced (or stress-induced) phase transformation during stretching or cooling, the resultant semicrystalline phase represented by the rigid-rod model. In the schematic, N denotes the number of statistical segments that are available for crystallization, X denotes the degree of phase transformation, T denotes temperature, ∇v denotes the velocity gradient, c = rr denotes the conformation tensor with r being chain extension, and S = uu − (1/3)δ with u being the unit vector. (Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.)

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(Doufas et al. 2000a) λa (T , X) = λa,0 (T )(1 − X)2

(6.11)

where X is the degree of phase transformation. In the limit X → 1, λa (T , X) reduces to zero due to complete consumption of the N ﬂexible segments. The nonlinear force factor E appearing in Eq. (6.6) should be modiﬁed with a factor of 1 − X in the denominator to account for the loss of untransformed molten phase during crystallization. Then, the contribution of the untransformed molten phase to the extra stress tensor τa can be written as E c − Gδ (6.12) τa = nK 1−X with K = 3kB T /N(1 − X)b2

(6.13)

which corrects the Hookean spring constant for the remaining untransformed chains in the semicrystalline phase. Since both K and τa increase indeﬁnitely as X → 1 (zero statistical links in the untransformed molten phase), the calculations of K from Eq. (6.13) and τa from Eq. (6.12) must be terminated for the value of X slightly less than 1 (say 0.997). Doufas et al. (2000a) described the semicrystalline phase using a rigid-rod model (Bird et al. 1987). Speciﬁcally, in describing the semicrystalline phase, they employed the traceless orientational tensor S = uu − (1/3)δ, where u is the unit vector, to describe the conformational state of the semicrystalline region in the spinline. In Chapter 9 of Volume 1, we introduced the tensor S to describe the orientation of rigid, rodlike liquid-crystalline polymer chains. Doufas et al. employed the following expression:6 β ᒁS =− S + 13 (∇v + (∇v)T ) − 2 (∇v)T : uuuu ᒁt λsc (X, T )

(6.14)

to describe the evolution of the tensor S, where λsc (X, T ) is the temperature- and crystallinity-dependent orientation relaxation time of the semicrystalline phase and β is a molecular anisotropic drag parameter, similar to the molecular mobility parameter α appearing in Eq. (6.6), which lies in the range 0 ≤ β ≤ 1. Here, a decrease in β corresponds to an increase in the ratio of resistance encountered perpendicular to the rigid rod to that encountered along the rod axis. In other words, the semicrystalline phase after the onset of crystallization may be regarded as rigid rods suspended in a medium. In the absence of ﬁrst principles that allow one to determine the dependence of λsc (X, T ) on X and T, the following form was assumed (Doufas et al. 2000a): λsc (X, T ) = λsc,0 (T ) exp(F X) ≈ ελa,0 (T ) exp(F X)

(6.15)

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PROCESSING OF THERMOPLASTIC POLYMERS

where λsc,0 is the value of λsc (X → 0, T ) in the limiting situation of X → 0, ε and F are model parameters that need to be determined from experimental data, and λa,0 (T ) is the temperature-dependent relaxation time of the amorphous phase. Since experimental determination of λsc,0 is rather difﬁcult, Doufas et al. assumed that λsc,0 may be approximated by λsc,0 ≈ ελa,0 , with 0 < ε < 1. They rationalized this approximation with an argument that for an inﬁnitesimally small value of X (at X → 0), λsc (X, T ) should not be made to be zero in order to maintain a smooth continuity as the singlephase amorphous polymer melt begins to transform into a two-phase semicrystalline phase. That is, Eq. (6.15) at X → 0 yields λsc (X → 0, T ) ≈ ελa,0 (T ), a ﬁnite value that will then avoid the untenable situation where the right-hand side of Eq. (6.14) would become inﬁnite. In other words, λsc (X → 0, T ) ≈ ελa,0 (T ) will be a ﬁnite value for 0 < ε < 1. It is seen from Eq. (6.15) that λsc (X, T ) increases as the crystallization progresses and attains a maximum value in the limit X → 1. Once the solution of Eq. (6.14) for S is available, one can calculate the components of the extra stress tensor τsc of semicrystalline phase from (Bird et al. 1987):7 τsc = 3nkB T

' ( S + 2λsc (∇v)T : uuuu

(6.16)

in which the ﬁrst term on the right-hand side represents the elastic (entropic) contribution to the semicrystalline stress and the second term on the right-hand side represents the viscous contribution to the semicrystalline stress. Note that in Eq. (6.16) use was made of S = uu − (1/3)δ. Equations (6.14)–(6.16) describe the rheological behavior of the semicrystalline phase. The salient feature of the approach presented above lies in that the orientational order parameter, via the tensor S, is related to the stress tensor τsc . This is very important because the stress tensor alone, when using strictly continuum-based constitutive equations (see Chapter 3 of Volume 1), does not provide information on the orientation of the semicrystalline phase in the spinline. However, the evaluation of the stress tensor τsc from Eq. (6.16) requires an evaluation of the fourth-order tensor uuuu. As mentioned in Chapters 9 and 12 of Volume 1, there are several different ways of approximating the fourth-order tensor uuuu. In their study, Doufas et al. (2000a) employed the following approximation (Advani and Tucker 1990): (∇v) : uuuu ∼ = (1 − w) T

1 15

∇v + (∇v)

T

+

1 7

%

(∇v)T : S δ

& + S · ∇v + (∇v)T + ∇v + (∇v)T ·S + w (∇v)T : S (S + 13 δ)

(6.17)

with w = 1 − 27 det(S + 13 δ)

(6.18)

In other words, the Doufas–McHugh model presented here is a two-phase model with two separate but linked constitutive equations for each phase. It is this feature along

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with the associated separate relaxation times for the two equations that gives the desired and needed behavior and that was missing in all earlier models. As mentioned above, in high-speed melt spinning, stress-induced crystallization is very important. Thus, the stress and the rate of crystallization must be coupled. At present, no ﬁrst principles, upon which such a relationship between the two can be made, have been identiﬁed and thus one resorts to empiricism. In their study, Doufas et al. (2000a) employed a differential form of Eq. (6.2): (m−1)/m 1 DX = mKav (T )(1 − X) ln exp(ξ trτ/G) (6.19) Dt 1−X to represent stress-induced crystallization, by multiplying the factor exp(ξ tr τ/G) with ξ being a temperature-independent dimensionless model parameter, G being the shear modulus, and τ being the total extra stress tensor given by τ = τa + τsc

(6.20)

In Eq. (6.19) D/Dt denotes the substantial derivative, m is an empirical constant, often referred to as the Avrami index, and Kav (T ) is the temperature-dependent Avrami constant under quiescent conditions. Note that the right-hand side of Eq. (6.19) will become inﬁnite as X → 1. Thus, a value of X slightly less than 1 (say 0.997) must be chosen for the integration of Eq. (6.19) to converge. In integrating Eq. (6.19), Doufas et al. (2000a) employed the following expression for Kav (T ): −3

Kav (T ) = 1.47 × 10

4πNu 3φ∞,is

1/3

T − 141 2 exp − 47.44

(6.21)

in which Nu denotes the number density of nuclei initially present within the melt in the spinneret, which was taken to be 1010 per cm3 , φ∞,is denotes the ultimate isotropic crystallinity, which was taken to be 0.5, T has the unit of Celsius, and Kav has the unit of reciprocal second (s−1 ). It is seen from Eqs. (6.19) and (6.20) that the crystallinity is coupled both with the amorphous region of the semicrystalline phase through the conformation tensor c, the time evolution of which is described by kB T 1 ᒁc =− ᒁt λa (X, T ) K

K E E K c · c − δ (6.22) (1 − α) δ + α kB T 1 − X kB T 1 − X

which follows from Eq. (6.6), and with the crystalline region of the semicrystalline phase through the orientational tensor S, the time evolution of which is described by Eq. (6.14). Note that the tensor c in the evolution equation, Eq. (6.22), is associated with the stress tensor τa in the untransformed molten phase deﬁned by Eq. (6.12), and the tensor S is associated with the stress tensor τsc in the semicrystalline region deﬁned by Eq. (6.16). Again, the value of X slightly less than 1 (say 0.997) must be chosen in order for the integration of Eq. (6.22) to converge. Note that the Doufas–McHugh model described above arbitrarily assumes that crystallization is zero until the point in the spinline where the ﬁber temperature drops to

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PROCESSING OF THERMOPLASTIC POLYMERS

the equilibrium melting point. Up to that point, the model uses only the amorphous melt phase constitutive equation (Giesekus). Once temperature drops below the equilibrium melt point, the second constitutive equation for the semicrystalline phase (i.e., rigid rods) along with the Avrami kinetic equation is introduced so as to make the total stress to be the sum of the amorphous and semicrystalline phase stresses (i.e., the Giesekus and rigid-rod constitutive equations, respectively). This approach causes a discontinuity in the computation because of the need to arbitrarily match velocity gradients in the two phases (i.e., constitutive equations) at that point. The problem of discontinuity in the computation remains to be resolved in future investigation. 6.3.2.3 Dimensionless System Equations To predict the proﬁles of ﬁlament diameter, axial velocity, temperature, and degree of phase transformation along the spinline, one must numerically solve the following dimensionless system equations (Doufas et al. 2000a). (a) Momentum Balance Equation

∗ d (τzz∗ − τrr∗ )/vz∗ dvz∗ D3 ∗ ∗ −3/2 dvz D1 ∗ = − D (v − v ) + − D (v ) 2 z r 4 z dz dz∗ vz∗ dz∗

(6.23)

(b) Energy Balance Equation

∗ ∗ dv ∗ τzz − τrr dT ∗ dX z ∗ −1/2 ∗ = −D 5 (vz ) (T − Tr ) + D6 + D7 ∗ ∗ ∗ ∗ dz vz dz dz

(6.24)

with (m−1)/m 1 dX 1 = ∗ mK ∗ (1 − X) ln exp(ξ tr τ∗ ) dz∗ vz 1−X

(6.25)

(c) Stress Tensor Prior to the Onset of Crystallization

τ∗ = Ec∗ − δ

(6.26)

(d) Stress Tensor After the Onset of Crystallization

τ∗ =

E c∗ − δ + 3S + 6NDe,sc (∇ ∗ vz∗ )T : uuuu 1−X

(6.27)

in which NDe,sc is the Deborah number in the semicrystalline phase deﬁned by NDe,sc = v0 λsc /L with v0 being the average axial velocity at z = 0 and L being the spinline length, and λsc is deﬁned by Eq. (6.15).

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(e) Evolution Equation for c*zz and c*rr

E ∗ czz −1 1−X ∗ dv ∗ crr E E 1−X z ∗ ∗ (1 − α) + α c c −1 =− ∗ ∗ − ∗ vz dz vz NDe,a 1 − X rr 1 − X rr

∗ dczz c∗ dv ∗ 1−X = 2 zz∗ z∗ − ∗ ∗ dz vz dz vz NDe,a ∗ dcrr dz∗

(1 − α) + α

E c∗ 1 − X zz

(6.28) (6.29)

in which NDe,a is the Deborah number in the amorphous phase deﬁned by NDe,a = v0 λa /L with v0 being the average axial velocity at z = 0 and L being the spinline length, and λa is deﬁned by Eq. (6.11). Prior to the onset of crystallization, Eqs. (6.28) and (6.29) with X = 0 must be solved. Note that the temperature-dependent relaxation time of the amorphous melt phase, λa,0 , can be calculated using the relationship λa,0 (T ) = η0 (T )/G, where η0 (T ) is temperature-dependent zero-shear viscosity and G is melt shear modulus. It has been observed that the temperature dependence of shear modulus is very weak compared with that of shear viscosity, and thus, for all intents and purposes, a constant value of G is acceptable within a reasonably narrow range of temperatures. After the onset of crystallization (for 0 < X < 1) Eqs. (6.28) and (6.29) must be solved. (f) Evolution Equation for Szz

Szz dvz∗ dSzz β 2 1 dvz∗ 1−w = 2 − S + −2 ∗ zz ∗ ∗ ∗ ∗ ∗ ∗ dz vz dz vz NDe,sc 3 vz dz vz ∗ w 1 dvz − 3 ∗ Szz Szz + vz 3 dz∗

2 11 + Szz 15 14

dvz∗ dz∗ (6.30)

The dimensionless variables appearing in Eqs. (6.23)–(6.30) are deﬁned as follows: z∗ = z/L, with L being the spinline length, vz∗ = vz /v0 , vr = vd /v0 , ∇ ∗ = ∇/L, T ∗ = T /T0 , with T0 being the temperature at z = 0, τ∗ = τ/G, c∗ = cK0 /kB T , Tr = Ta /T0 , D1 = ρv0 2 /G, D2 = πµa BLρv0 2 /GW , D3 = gρL/G, D4 = (πς 2 ρv0 /4W G2 )1/2 , D5 = (4πL2 h2 /ρcp 2 v0 W )1/2 , D6 = G/ρcp T0 , and D7 = Hf φ∞ /cp T0 . The system equations presented above contain several parameters (α, β, ε, F, m, ξ , Nu , φ∞,is ) that must be determined either from controlled experiments or by curve ﬁtting to experimental results. In view of the complexity of the problem dealt with, such an approach may be unavoidable. Nevertheless, the salient feature of the modeling approach presented above lies in that a structure-based approach has been employed throughout the entire analysis, including the phase transition from the amorphous melt phase prior to the onset of crystallization to the semicrystalline phase after the onset of crystallization and to the solidiﬁcation. Consequently, the freeze point along the spinline can be calculated as part of the numerical solution of the system equations by considering the rheological responses of both amorphous and crystalline phases simultaneously.

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PROCESSING OF THERMOPLASTIC POLYMERS Table 6.1 Physical and rheological properties of nylon 66 melt used in the simulation

Properties

Value

Density, ρ Zero-shear viscosity, η0 at 280 ◦ C Melt shear modulus, G Mobility parameter, α Number of statistical links per chain, N Melting temperature, Tm0 Glass transition temperature, Tg cs1 cs2 cs3 ca1 ca2 ca3 Hf (0) Surface tension, ζ

0.98 g/cm3 163 Pa·s 1.1 ×105 Pa 0.4 200 265 ◦ C 45 ◦ C 1.26 J/(g ◦ C) 0.83 × 10−2 J/(g 0.00 J/(g (◦ C)3 ) 2.09 J/(g ◦ C) 1.95 × 10−3 J/(g 0.00 J/(g (◦ C)3 ) 210 J/g 0.036 N/m

(◦ C)2 ) (◦ C)2 )

Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.

6.3.3

Model Prediction and Comparison with Experiment

In this section, we present some representative predictions of the analysis described above and then compare them with experiment.8 As can be seen in Eq. (6.23)–(6.30), there are many physical and rheological parameters associated with the system equations. Table 6.1 gives numerical values of the physical and rheological properties employed, Table 6.2 gives numerical values of the processing conditions employed, and Table 6.3 gives numerical values of the model parameters that appear in the system

Table 6.2 Processing conditions for melt spinning of nylon 66 used in the simulation

Processing Parameter

Value

Take-up speed, vL Capillary tube diameter, D0 Mass throughput, W Temperature at the exit of spinneret, T0 Spinline length, L Cross velocity of quench air, vc Downward velocity of quench air, vd Quench air temperature, Ta

400–6,500 m/min 0.025 cm 1.5–3.6 g/(min-capillary) 297 ◦ C 135 cm 19.8 m/min 0 m/min 24 ◦ C

Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.

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Table 6.3 Parameters used for the model calculations

Processing Parameter F ξ ε β φ∞ m a

Equation

Value

6.15 6.25 6.15 6.14 6.14a 6.25

60 0.06 0.001 0.5 0.45 1.0

φ∞ appears in the deﬁnition of D7 = Hf φ∞ /cp T0

Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.

equations. Here, we present some expressions that explain the origins of some of the parameters appearing in Tables 6.1–6.3. (1) Zero-Shear Viscosity of Nylon 66 Melt η0 (T ) 1.35 × 104 (280 − t) ◦ η0 (T ) = η0 (280 C) exp 1099.2(t + 273.2)

(6.31)

where η0 (280 ◦ C) = 163 Pa·s, t is temperature in Celsius, and the relaxation time λa,0 of the amorphous phase before the onset of crystallization can be calculated using λa,0 (T ) = η0 (T )/G, with G = 1.1 × 105 Pa for nylon 66. (2) Heat Capacity of the Melt-Spun Fiber cp cp = cs Xφ∞ + ca (1 − Xφ∞ )

(6.32)

where cs is the heat capacity of the crystalline region, ca is the heat capacity of the amorphous region, X is the degree of transformation, and φ∞ is the degree of crystallinity within the semicrystalline phase. The temperature dependence of cs and ca is expressed as cs (T ) = cs1 + cs2 T + cs3 T 2

(6.33)

ca (T ) = ca1 + ca2 T + ca3 T 2

(6.34)

(3) Heat of Fusion Hf (T ) Hf (T ) = Hf (0) + (ca1 − cs1 )T + (ca2 − cs2 )T 2/2 + (ca3 − cs3 )T 3/3 (6.35) where Hf (0) is a reference heat of fusion taken equal to 210 J/g for nylon 66. (4) Isotropic Crystallization Kinetics 2/3 1 DX = 3Kav (T ) ln (1 − X) Dt 1−X where Kav (T ) is given by Eq. (6.21).

(6.36)

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(5) Physical Properties of Quenched Air at 1 atm: ρa = 1.352/Tf for density (g/cm3 ), µa = 1.446 × 10−6 Tf 1.5/ Tf + 113.9 for viscosity (Pa·s), and ka = 4.49 × 10−7 Ta 0.866 for thermal conductivity (W/m ◦ C), in which Ta in Celsius is the quench air temperature and Tf in Kelvin is the temperature deﬁned as the arithmetic mean of the ﬁlament temperature and quench air temperature. Figure 6.17 shows the effect of take-up speed on the predicted axial velocity (vz ) proﬁle along the spinline distance z for a mass ﬂow rate of 2.2 g/min through a round spinneret hole. The following observations are worth noting in Figure 6.17: (1) at a take-up speed of 1,000 m/min, vz increases gradually and then levels off along the spinline distance z, (2) as the take-up speed is increased to 4,500 m/min, vz increases very quickly and then levels off at z ≈ 70 cm, and (3) as the take-up speed is increased further to 6,500 m/min, vz increases much more rapidly and then levels off at z ≈ 50 cm, showing that the freeze point moves closer to the spinneret (i.e., towards the spinneret face) as the take-up speed increases. These observations seem to indicate that ﬂowinduced (or stress-induced) crystallization takes place in the spinline at high take-up speeds. What is signiﬁcant in Figure 6.17 is that the freeze point is predicted naturally as part of the solution of the system equations in the modeling approach we have presented. Such prediction has been made possible by the incorporation of a consistent molecule-based microstructural approach adopted for both before and after the onset of crystallization. In other words, the Doufas–McHugh model presented above does not specify the freeze point along the spinline distance in terms of the melting point or glass transition temperature of a polymer; that is, it is a two-phase model that distinguishes itself from other models presented in the literature. Figure 6.18 shows the effect of take-up speed on the predicted ﬁlament diameter (D) proﬁle along the spinline distance z for a mass ﬂow rate of 2.2 g/min through a round spinneret hole. Necklike deformation of the ﬁlament diameter can be observed

Figure 6.17 Predicted proﬁles of axial velocity vz at a constant mass ﬂow rate along the spinline distance z from the spinneret for three take-up speeds (m/min): (1) 1,000, (2) 4,500, and (3) 6,500. The numerical values of the system parameters and processing conditions employed for the simulation are summarized in Tables 6.1–6.3. (Reprinted from Doufas et al., Journal of NonNewtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.)

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Figure 6.18 Predicted proﬁles of ﬁlament diameter D at a constant mass ﬂow rate along the spin-

line at a distance z from the spinneret for three take-up speeds (m/min): (1) 1,000, (2) 4,500, and (3) 6,500. The numerical values of the system parameters and processing conditions employed for the simulation are summarized in Tables 6.1–6.3. (Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.)

in Figure 6.18 when the take-up speed is increased above 4,000 m/min, very similar to the experimental observations given in Figures 6.10 and 6.12 and also by other research groups (Haberkorn et al. 1990; Vassilatos et al. 1985). Notice in Figure 6.18 that necklike deformation is not predicted at a take-up speed of 1,000 m/min. It seems appropriate to mention at this juncture that necklike deformation of the ﬁlament diameter was not predicted by Spruiell and coworkers (Patel et al. 1991; Zieminski and Spruiell 1988), who employed Newtonian ﬂuids in modeling high-speed melt spinning including crystallization. Based on such an experimental observation, Doufas et al. (2000a) concluded that their prediction of necklike deformation of the ﬁlament diameter in high-speed melt spinning (see Figure 6.18) was made possible because viscoelasticity together with stress-induced crystallization was considered in their analysis. ∗ Figure 6.19 gives typical predicted proﬁles of dimensionless axial component czz ∗ and radial component crr of the conformation tensor along the spinline z from the spinneret.9 Figure 6.19 is presented to merely demonstrate that indeed the conformation tensor describes the stretching of the amorphous polymer chain along the spinline. It should be emphasized that the conformation tensor c* represents the untransformed phase in the two-phase model, which is to distinguish the fact that the rigid rodlike phase is semicrystalline (i.e., it also contains trapped amorphous material that is part ∗ initially remains more or of the transformed phase). It is seen in Figure 6.19 that czz less constant below the spinneret, down to the spinline, distance of z ≈ 20 cm, and then increases rapidly, going through a maximum at z ≈ 60 cm, and then decreases ∗ also initially remains more or less very rapidly to virtually zero at z ≈ 75 cm. crr constant down to the spinline distance of z ≈ 20 cm, decreases slowly, going through ∗ exhibits a maximum), and then decreases a slight minimum at z ≈ 60 cm (where czz ∗ at some distance very rapidly to virtually zero at z ≈ 75 cm. Note that the drop of czz below the spinneret indicates the decrease of the contribution of the untransformed ∗ and c∗ phase stress relative to the semicrystalline stress. The predicted proﬁles of czz rr given in Figure 6.19 represent the chain extension in the untransformed molten phase.

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∗ and radial compoFigure 6.19 Typical predicted proﬁles of dimensionless axial component czz ∗ nent crr of the conformation tensor along the spinline z from the spinneret for a take-up speed of 5,500 m/min and W = 2 g/min.9 The numerical values of the system parameters and processing

conditions employed for the simulation are summarized in Tables 6.1–6.3.

Figure 6.20 shows predicted variations in the fractional amorphous chain extension (e) along the spinline distance z for three different take-up speeds, where e was calculated from 1 e= 1−X

tr c∗ 3N

1/2 (6.37)

Note that Eq. (6.37) is different from Eq. (6N.2) by a factor 1/(1 − X ).10 It can be seen from Figure 6.20 that the amorphous chain extension increases along the spinline distance and then levels off at a certain position, and that the percent amorphous chain extension increases with increasing take-up speed, indicative of the onset of stress-induced crystallization in the spinline. The leveling off of the amorphous chain

Figure 6.20 Typical predicted proﬁles of amorphous chain extension e, deﬁned by Eq. (6.37),

at three take-up speeds (m/min): (1) 1,000, (2) 2,000, and (3) 3,000. The numerical values of the system parameters and processing conditions employed for the simulation can be found from Tables I and II in the original paper. (Reprinted from Doufas and McHugh, Journal of Rheology 45:403. Copyright © 2001, with permission from the Society of Rheology.)

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extension at a certain spinline position observed in Figure 6.20 is consistent with the leveling off of the axial velocity vz at a certain spinline position observed in Figure 6.17. Figure 6.21 shows the effect of take-up speed on the predicted tensile stress (Tzz ) proﬁle and apparent elongational viscosity (ηE,app ) proﬁle along the spinline distance z for a mass ﬂow rate of 2.2 g/min through a round spinneret hole. Note in Figure 6.21 that Tzz is proportional to the difference τzz − τrr (see Eq. (3.95) in Chapter 3 of Volume 1 for the deﬁnition of the tensile stress in uniaxial elongation ﬂow) and ηE,app was calculated from ηE,app = (τzz − τrr )/(dvz /dz) (see Chapter 3 of Volume 1 for the ∗ − c∗ ) is meaningdeﬁnition). It should be noted that the difference czz − crr (or czz rr ∗ ∗ ful, but the individual components czz and crr (or czz and crr ) are not, in describing the physical quantities of interest. Note from Eq. (6.10) that τzz − τrr ∝ czz − crr . The following observations are worth noting in Figure 6.21: (1) at a take-up speed of 1,000 m/min, Tzz increases gradually and then levels off, while ηE,app steadily increases along the spinline distance z, (2) as the take-up speed is increased to 4,500 m/min, Tzz increases rapidly and then levels off at z ≈ 70 cm, while ηE,app initially increases very slowly along the spinline distance z, exhibiting a plateau region followed by a slight dip, and then increases rapidly at z ≈ 70 cm, and (3) as the takeup speed is increased further to 6,500 m/min, Tzz increases much more quickly and then shows a slightly positive slope as a result of the air drag, while ηE,app increases, going through a maximum and then a minimum, before increasing asymptotically at z ≈ 50 cm. The predicted Tzz proﬁles given in Figure 6.21 at high take-up speeds are very similar to the experimental observations presented in Figure 6.11, and the predicted ηE,app proﬁles given in Figure 6.21 at high take-up speeds are very similar to the experimental observations presented in Figure 6.14. Comparison of Figure 6.21 with Figure 6.18 seems to indicate that strain softening in ηE,app and necklike deformation of the ﬁlament diameter are closely related. Doufas et al. (2000a) attributed

Figure 6.21 Predicted proﬁles of tensile stress Tzz (solid lines) and apparent elongational viscosity ηE,app (broken lines) at a constant mass ﬂow rate along the spinline distance z from the

spinneret for three take-up speeds (m/min): (1) 1,000, (2) 4,500, and (3) 6,500. The numerical values of the system parameters and processing conditions employed for the simulation are summarized in Tables 6.1–6.3. (Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.)

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the sharp decrease in ηE,app (which is often referred to as strain softening) at a take-up speed of 6,500 m/min observed in Figure 6.21 to the nonlinear viscoelasticity that produces an apparent strain softening (see, for instance, Figure 3.12 given in Chapter 3 of Volume 1 for a strain-softening behavior in accordance with the Giesekus constitutive equation). In other words, they concluded that the combination of the onset of a rapid increase in the semicrystalline phase relaxation time and the nonlinear viscoelastic effects is responsible for the formation and stabilization of the necklike deformation of the ﬁlament diameter. Figure 6.22 shows the effect of take-up speed on the predicted temperature (T) proﬁle along the spinline distance z for a mass ﬂow rate of 2.2 g/min through a round spinneret hole. It is seen in Figure 6.22 that (1) at a take-up speed of 1,000 m/min, the temperature decreases gradually along the spinline distance z, (2) at a take-up speed of 4,500 m/min, the temperature ﬁrst decreases along the spinline distance and then suddenly increases at z ≈ 65 cm, followed by a steady decrease along the spinline distance, and (3) at a take-up speed of 6,500 m/min, the temperature ﬁrst decreases along the spinline and then rapidly increases at z ≈ 50 cm, followed by a steady decrease along the spinline distance. It is worth noting that the spinline distance z at which a rapid increase in ηE,app occurs (see Figure 6.21) roughly corresponds to the position z at which a rapid increase in temperature occurs (see Figure 6.22). Also note in Figure 6.21 that the sharp increase in ηE,app (which is often referred to as strain hardening) is due to the cooling effect. In the literature, a question was raised as to whether or not necklike deformation of the ﬁlament diameter arises from stress-induced crystallization in the spinline. Figure 6.23 shows the predicted proﬁles of ηE,app and temperature T along the spinline distance z at a take-up speed of 5,500 m/min. It is seen in Figure 6.23 that the decrease of ηE,app occurs before the temperature rise due to stress-induced crystallization. Note that the decrease of ηE,app and necklike deformation of the ﬁlament diameter occur

Figure 6.22 Predicted proﬁles of temperature T at a constant mass ﬂow rate along the spinline distance z from the spinneret for three take-up speeds (m/min): (1) 1,000, (2) 4,500, and (3) 6,500. The numerical values of the system parameters and processing conditions employed for the simulation are summarized in Tables 6.1–6.3. (Reprinted from Doufas et al., Journal of NonNewtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.)

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Figure 6.23 Predicted proﬁles of (1) apparent elongational viscosity ηE,app and (2) temperature T at a constant mass ﬂow rate along the spinline distance z from the spinneret for a take-up speed of 5,500 m/min, showing that the decrease of ηE,app occurs before the temperature rise due to ﬂow-induced crystallization. The numerical values of the system parameters and processing conditions employed for the simulation are summarized in Tables 6.1–6.3. (Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.)

at roughly the same spinline distance from the spinneret (see Figures 6.18 and 6.21). On the basis of Figure 6.23, Doufas et al. (2000a) dismissed the view that necklike deformation of the ﬁlament diameter arises from the onset of stress-induced crystallization. Instead, they advocated the nonlinear viscoelastic behavior in the spinline for the occurrence of necklike deformation in high-speed melt spinning. In this case, they used NDe,a = λa ε˙ . Speciﬁcally, they argued that NDe,a can be as large as 50 in the highspeed melt spinning of nylon, because the axial velocity gradient (apparent elongation rate ε˙ ) in the spinline during high-speed spinning can be as high as 1,000–5,000 s−1 , while the relaxation time λa of nylon is very small. Note that the melt-spinning grade of nylon may be considered to be a Newtonian ﬂuid when subjected to shear ﬂow inside a spinneret. Large values of NDe,a in the high-speed melt spinning of nylon are due to the nonlinear viscoelasticity, and therefore the necklike deformation of the nylon ﬁlament diameter observed in high-speed melt spinning is attributed to the nonlinear viscoelasticity and not to stress-induced crystallization (Doufas et al. 2000a). Figure 6.24 shows the effect of take-up speed on the predicted proﬁles of the degree of transformation X along the spinline distance z. It is seen in Figure 6.24 that ﬂow enhances crystallization by many orders of magnitude relative to isotropic crystallization in that the crystallization rate increases very rapidly as the take-up speed is increased from 1,000 to 6,500 m/min, conﬁrming experimental observations (Haberkorn et al. 1990). Notice further that the completion of phase transformation is realized closer to the spinneret as the take-up speed is increased. The onset of crystallization results in a temperature rise of the ﬁlament owing to the heat of crystallization. Some research groups (Kulkarni and Beris 1998; Shimizu et al. 1985; Ziabicki 1988) have associated necklike deformation of the ﬁlament diameter in the spinline with the onset of crystallization. However, Haberkorn et al. (1990) measured both the temperature and ﬁlament diameter proﬁles simultaneously during the high-speed melt

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Figure 6.24 Predicted proﬁles of the degree of phase transformation X at a constant mass ﬂow

rate along the spinline distance z from spinneret for different take-up speeds: (1) under isotropic (quiescent) conditions, (2) 1,000 m/min, (3) 4,500 m/min, and (4) 6,500 m/min. The numerical values of the system parameters and processing conditions employed for the simulation are summarized in Tables 6.1–6.3. (Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.)

spinning of PET, and they indeed observed that the temperature started to increase at the end of necklike deformation in the spinline. They also observed that crystallization took place at the end of the necklike deformation in the spinline. They concluded that necklike deformation of the ﬁlament diameter in the spinline is not primarily caused by the onset of the crystalline solidiﬁcation of the ﬁlament, although necklike deformation is somewhat related to crystallization, and that crystallization in the spinline does not occur before the end of the necklike deformation. Such experimental observations seem to be supported by the simulation results presented here. Figure 6.25 shows predicted tensile stress proﬁles along the spinline distance z from the amorphous phase (τ zz − τrr )a , which was calculated from Eq. (6.12), and from the semicrystalline phase (τ zz − τrr )sc , which was calculated from Eq. (6.16). It is seen in Figure 6.25 that up to approximately 60 cm below the spinneret the tensile stress from the amorphous phase dominates, and that the tensile stress from the semicrystalline phase starts to increase rapidly at z ≈ 50 cm and dominates from the position where the freeze point sets in. After the freeze point, as the crystallization proceeds, the tensile stress from semicrystalline phase takes over that from the amorphous phase, and the total tensile stress increases very slowly as the crystallization continues in the spinline. Also shown in Figure 6.25 is the predicted proﬁle of the degree of crystallinity along the spinline, indicating that crystallization increases very slowly from z ≈ 40 cm to z ≈ 60 cm, and then increases very rapidly at the freeze point. The very slow increase of crystallinity at spinline distances between 40 and 60 cm is attributable to stress-induced crystallization, while the spinline is still predominantly in the amorphous state. Figure 6.25 further indicates that most of the crystallization takes place after the freeze point. Figure 6.26 gives predictions for the effect of molecular mobility parameter α on the apparent elongational viscosity ηE,app along the spinline distance z. It is seen in Figure 6.26 that α = 0.1 predicts strictly strain hardening, while α = 1 predicts strain

Figure 6.25 Predicted proﬁles of tensile stresses Tzz and the degree of crystallinity (– – – –) at a constant mass ﬂow rate along the spinline distance z from the spinneret for a take-up speed of 5,500 m/min, where the total tensile stress (— · —) consists of the contributions from the amorphous phase (—) and semicrystalline phase (- - - - -). The numerical values of the system parameters and processing conditions employed for the simulation are summarized in Tables 6.1–6.3. (Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:27. Copyright © 2000, with permission from Elsevier.)

Figure 6.26 Predicted proﬁles of apparent elongational viscosity ηE,app at a constant mass ﬂow rate along the spinline distance z from spinneret for two different values of the mobility parameter α: (1) α = 0.1 and (2) α = 1. The processing conditions employed for the experiment are: take-up speed (vz ) = 5,700 m/min, mass throughput (W ) = 2.22 g/min, melt feed temperature (T0 ) = 290 ◦ C, spinline length (L) = 160 cm, and quench air velocity (vc ) = 5,333 cm/min. The numerical values of the system parameters employed for the simulation can be found from Table 3 in the original paper. (Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:81. Copyright © 2000, with permission from Elsevier.)

293

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Figure 6.27 Comparison of experimentally determined proﬁles of axial velocity vz (䊉) at a constant mass ﬂow rate along the spinline distance z from spinneret with model predictions with two different values of the mobility parameter α: (1) α = 0.1, and (2) α = 1. The processing conditions employed for the experiment are: take-up speed (vz ) = 5,700 m/min, mass throughput (W ) = 2.22 g/min, melt feed temperature T0 = 290 ◦ C, spinline length (L) = 160 cm, and quench air velocity (vc ) = 5, 333 cm/min. The numerical values of the system parameters employed for the simulation can be found from Table 3 in the original paper. (Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:81. Copyright © 2000, with permission from Elsevier.)

softening, which is compatible with high-speed spinning (see Figure 6.14). Figure 6.27 shows the predicted effect of α on the axial velocity vz along the spinline distance z. Also given in Figure 6.27 is the experimentally determined proﬁle of vz for high-speed spinning of nylon. It appears from Figure 6.27 that the predicted proﬁle of vz with α = 1 agrees better with experiment, which is consistent with the observation made above in reference to Figure 6.26. Note that the parameter α was introduced into the Giesekus constitutive equation (see Chapter 3 of Volume 1) and hence into Eq. (6.6). But, the determination of α requires independent experiment. Figure 6.28 compares experimentally determined proﬁles of axial velocity vz and birefringence n along the spinline distance z, together with prediction in the highspeed spinning of nylon. It is seen in Figure 6.28 that n begins to increase very rapidly at z ≈ 60 cm and attains a plateau region at z ≈ 80 cm, that is, n increases sharply once vz levels off. Spruiell and coworkers (Patel et al. 1991; Zieminski and Spruiell 1988) reported similar experimental results. This observation indicates that the rapid increase in n reﬂects the fast crystallization rates that occur after the freeze point, and that most of the crystallization takes place after the freeze point. These observations are consistent with that made above in reference to Figure 6.25.

6.4

Spinnability

The most fundamental question one must ask, when discussing ﬁber spinning, is: What makes only certain polymers (e.g., nylon; PET) melt spinnable and able to resist breakage of spinline at high take-up speeds? Note that other ﬁber-forming

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Figure 6.28 Comparison of experimentally determined proﬁles of axial velocity vz (䊉) and birefringence n () at a constant mass ﬂow rate along the spinline distance z from the spinneret with model predictions (solid lines). The processing conditions employed for the experiment are: take-up speed vz = 5,100 m/min, mass throughput (W ) = 1.94 g/min, melt feed temperature T0 = 279.9 ◦ C, spinline length (L) = 200 cm, and quench air velocity (vc ) = 2, 294 cm/min. The numerical values of the system parameters employed for the simulation can be found from Table 3 in the original paper. (Reprinted from Doufas et al., Journal of Non-Newtonian Fluid Mechanics 92:81. Copyright © 2000, with permission from Elsevier.)

polymers, such as PP and HDPE, cannot withstand very high take-up speeds (say up to 8,000 m/min). Other related questions that may be asked include the following. (1) Is there a simple experimental method available that can guide us in the synthesis of new polymers or in modiﬁcation of existing polymers such that they can be used for ﬁber spinning? (2) Is the spinnability of a polymer associated strictly with its chemical structure alone? (3) If not, how do processing conditions affect the spinnability of a polymer? It seems that no clear answers to the above questions can be found in the literature, even though the synthetic ﬁber industry has existed for over half a century. It was once thought by some researchers that measurement of elongational viscosity might provide some clues as to whether or not a polymer may be melt spinnable. It appears that the rationale behind such a thought was based on experimental observation that the lower the elongational viscosity of a polymer, the better its spinnability. At this juncture, one must deﬁne “spinnability.” It appears from the literature that no simple deﬁnition with which everyone can agree of spinnability is available. In the absence of a universally accepted deﬁnition of spinnability, for the sake of this discussion let us deﬁne spinnability by the maximum draw-down ratio (or stretch ratio), which is the ratio of the maximum take-up speed (VL,max ), at which the spinline breaks down, to the average velocity of the melt inside a spinneret hole (V0 ). It is then not difﬁcult to surmise that the ratio VL,max /V0 depends, among many other things, on (1) the rheological properties of the melt, (2) the hole size of the spinneret, and (3) the cooling rate of the spinline. A rheological property of a polymer that is relevant to ﬁber spinning is elongational viscosity. In ﬁber spinning, the axial velocity gradient dvz /dz is not constant along the spinline distance z (see Figures 6.11 and 6.13), indicating that melt spinning is operated at uncontrolled elongation rates. Consequently, the melt spinning operation

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gives rise to apparent elongational viscosity ηE,app under the inﬂuence of cooling (see Figure 6.14). It is then very doubtful that the steady-state elongational viscosity determined at a constant elongation rate under isothermal conditions, which was discussed in Chapter 5 of Volume 1, would ever be helpful to predict (or understand) the spinnability of a polymer. It is clear from the simulation results presented in this chapter that ηE,app is a derived quantity that can be obtained from the solution of system equations, and that it is not an intrinsic rheological property that can be related directly to the chemical structure alone of a ﬁber-forming polymer. No doubt, however, that ηE,app would depend on the molecular weight and molecular weight distribution of a polymer. There is experimental evidence (Han and Lamonte 1972; Minoshima et al. 1980) suggesting that the narrower the molecular weight distribution, the lower the ηE,app and the greater the ratio VL,max /V0 and thus the better the spinnability. These observations seem to suggest that there may be intricate relationships between molecular weight distribution, apparent elongational viscosity, and spinnability. However, more systematic investigations are needed to establish, at least experimentally, a ﬁrm scientiﬁc basis upon which spinnability can be better understood.

6.5

Summary

In this chapter, we have presented the theoretical analysis of high-speed melt spinning by Doufas and McHugh (2000a, 2000b). Their analysis is based on a structurebased two-phase model that includes stress-induced crystallization in high-speed melt spinning. Due to the very complex nature of the problem dealt with, their analysis requires independent evaluations of several model parameters, which seems inevitable when modeling such a complex processing operation as high-speed melt spinning. The analysis captures the essential features of high-speed melt spinning. Fundamental research on ﬁber spinning was very active in the 1960s and 1970s, and then continued in the 1980s upon the emergence of high-speed melt spinning. One of the research interests in ﬁber spinning in the 1960s and 1970s was the “draw resonance” phenomenon, where oscillatory ﬂuctuations developed in the ﬁlament diameter along the spinline. Many experimental and theoretical studies on draw resonance in lowspeed melt spinning were reported in the same period. Many papers (Chang and Denn 1979; Fischer and Denn 1976, 1977; Freeman and Coplan 1964; Han et al. 1972; Kase and Matsuo 1966; Pearson and Matovic 1966) dealt with draw resonance in low-speed melt spinning, but we ﬁnd very few publications on draw resonance associated with high-speed melt spinning, indicating that draw resonance is not an important issue in high-speed melt spinning. As a matter of fact, there is experimental and theoretical evidence indicating that draw resonance disappears as the draw-down ratio in melt spinning exceeds a certain critical value. It is fair to state that for more than a decade since the late 1980s, fundamental research activities on ﬁber spinning have been very limited in number, especially in academia. This may be due in part to the fact that the ﬁber industry has matured. However, this does not mean that there are few problems to be investigated in ﬁber spinning. In fact, there are some challenging theoretical problems yet to be solved. One such problem is an analysis of melt spinning of viscoelastic ﬂuids, including extrudate swell below the spinneret. Note that PP and HDPE are melt-spun commercially to meet with

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the demands of certain markets and that these polymers exhibit considerable extrudate swell under normal extrusion conditions. In view of the fact that the length-to-diameter (L/D) ratio of spinneret holes in industrial melt spinning is rather small (varying from 1 to 4) and that the extrudate swell depends very much on the deformation history (thus L/D ratio), there is little chance for PP or HDPE to attain fully developed ﬂow at the exit plane of the spinneret. This means that a prediction of extrudate swell below the spinneret must consider the deformation history of a PP or HDPE melt inside the spinneret hole. The occurrence of extrudate swell beneath the spinneret means that the macromolecules sheared (and thus oriented) inside the spinneret hole will disorient during extrudate swell, as schematically shown in Figure 6.1. However, the extent of extrudate swell (or the extent of macromolecular disorientation) may be affected by the tension transmitted to the spinline from the take-up device. Therefore, a connection must be made between the extent of extrudate swell and the processing conditions. To date, this problem has received little attention from researchers, although it is granted that the problem is a difﬁcult one to solve. As described in the beginning of this chapter, relatively few modeling efforts on wet and dry spinning processes have been reported in the literature. One of the many barriers to a successful mathematical modeling effort on wet and dry spinning processes is the lack of information on mass transfer coefﬁcients between the moving ﬁlament and the surrounding environment (the coagulating bath in wet spinning and the drying tower in dry spinning). Carefully planned and controlled experiments could be devised to obtain such fundamental information, which might then stimulate others to simulate wet and drying spinning processes. Another challenging problem is a simulation of the ﬁber spinning process for producing shaped ﬁbers. Since the cross section of a spinneret hole for producing shaped ﬁbers can be almost any shape, a simulation of ﬂow of a polymeric liquid through such complex geometry requires use of ﬁnite element methods. This problem can be handled without much difﬁculty as the powerful commercial software now available is able to generate meshes for an almost any arbitrary shape of spinneret cross section. We emphasize the importance of processing–morphology–property relationships in ﬁber spinning, although this subject is outside the scope of this chapter. Mathematical modeling (or simulation) efforts on various ﬁber spinning processes play important but limited roles in producing successful ﬁbers, and such efforts cannot provide one with all the answers to produce ﬁbers of desired mechanical properties. One should keep in mind that end-users will buy ﬁbers only when the ﬁbers meet the expectations of enduse properties. In this regard, the ﬁnal goal of mathematical simulation of ﬁber spinning should be to predict the tensile properties and/or modulus of ﬁbers in terms of the molecular characteristics and rheological properties of polymers and processing variables.

Problems Problem 6.1

Consider that PP is being melt spun at 220 ◦ C through a circular spinneret hole with the diameter of 0.1 mm and the length of 1 cm into an isothermal chamber of 10 cm long followed by air cooling. Assume that the ﬂow inside the spinneret is fully developed when the molten PP leaves the spinneret, and that upon exiting the spinneret the PP swells. The extent of extrudate swell of the molten PP upon exiting

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the spinneret depends, among many factors, on the deformation history inside the spinneret, the mass ﬂow rate, and the draw-down ratio, VL /V0 , with V0 being the average velocity of the PP melt leaving the spinneret and VL being the take-up speed at the distance L from spinneret. For very low values of VL /V0 , the tension from the take-up roll is not strong enough to affect the extrudate swell. Under such circumstances, the residual wall normal stress in the PP melt at the spinneret exit plane will relax completely below the spinneret, giving rise to a maximum extrudate swell, before the diameter of the molten threadline begins to decrease under the inﬂuence of the tension from the take-up roll. Conversely, for very high values of VL /V0 , the tension from the take-up roll is so high that there will be little or no extrudate swell beneath the spinneret. For the melt spinning situations described above, one must calculate the extent of extrudate swell below the spinneret in order to be able to predict the proﬁles of the threadline diameter under the inﬂuence of the tension from the take-up roll. (a) Using the constitutive equation

σ=

t

⎧ ⎨G

−∞ ⎩ λ1

exp −

t t

⎫ ) ⎬' ( 1 + cλ1 I2 (t ) C−1 (t ) − I dt dt ⎭ t λ1

(6P.1)

predict the diameter proﬁle of the molten PP threadline inside the isothermal chamber. Referring to Eq. (6P.1), λ1 is the relaxation time, G is the modulus, c is a material constant, I2 is the second invariant of the rate-of-strain tensor d, and C−1 t is the relative Finger strain tensor deﬁned by Eq. (2.73) given in Chapter 2 of Volume 1. Note that λ1 = η0 /G, with η0 being the zero-shear viscosity. (b) Using the constitutive equation Eq. (6P.1) with the temperaturedependent rheological parameters as described below, predict the diameter and temperature proﬁles of the molten PP threadline when the isothermal chamber is removed and the molten threadline is cooled by the ambient air over the entire spinline. Assume the following relationship describing the temperature dependence of λ1 (T ):

λ1 (T ) = λ10 aT (T )

(6P.2)

where λ10 is the relaxation time at temperature T0 and aT (T) is a temperaturedependent shift factor deﬁned by (see Chapter 6 of Volume 1) aT (T ) =

η0 (T )ρ0 T0 η0 (T0 )ρT

(6P.3)

in which η0 (T ) and η0 (T0 ) are the zero-shear viscosities at T and T0 , respectively, and ρ and ρ 0 are the densities at T and T0 , respectively. Over the range of temperatures encountered in melt spinning (i.e., from the melting temperature 175 ◦ C of PP to the extrusion temperature 220 ◦ C) we have ρ0 T0 /ρT ≈ 1 and

FIBER SPINNING

thus Eq. (6P.3) can be expressed by the Arrhenius relation: E 1 1 − aT (T ) = exp R T T0

299

(6P.4)

where E is the viscous ﬂow activation energy and R is the universal gas constant. Further, assume that the temperature dependence of G is so weak that G can be assumed to be constant over the range of temperatures of melt spinning. Table 6.4 gives a summary of the physical, thermal, and rheological properties of PP and the melt spinning conditions. You may use the following expression for the heat Table 6.4 Physical, thermal, and rheological properties of PP melt and spinning conditions for Problem 6.1

(a) Physical and Thermal Properties Density at a reference temperature T0 (220 ◦ C), ρ0 Melting temperature, Tm0 Speciﬁc heat capacity, cp Thermal conductivity, k Heat of crystallization, Hf Degree of crystallinity within the semicrystalline phase, φ∞ Surface tension, ζ Parameter B appearing in Eq. (6.4)

760 kg/cm3 170 ◦ C 2.01 × 103 J/(kg K) 0.181 W/(m K) 8.8 ×102 J/kg 0.6 0.04 N/m 1.0

(b) Rheological Parameters Zero-shear viscosity at a reference temperature T0 (220 ◦ C), η0 Melt shear modulus, G Viscous ﬂow activation energy, E Mobility parameter α appearing in Eq. (6.6) Parameter c appearing in Eq. (6P.1) (c) Spinning Conditions Mass throughput, W Take-up speed, vL Capillary tube diameter, D0 Melt temperature entering the spinneret (reference temperature), T0 Spinline length, L Velocity of quench air, va Viscosity of quench air, µa

Quench air temperature, Ta a

5.24 × 103 Pa·s 2.62 × 104 Pa 4.74 × 104 J/mol 0.5 0.6 5.0 g/min (single hole) 400 m/min 0.05 cm 220 ◦ C 100 cm 10 m/min 1.446 × 10−6 Tf 1.5 /(Tf + 113.9) in Pa·s, with Tf being the ﬁlament temperaturea 25 ◦ C

Tf is deﬁned as the arithmetic mean of the ﬁlament temperature and quench air temperature.

300

PROCESSING OF THERMOPLASTIC POLYMERS

transfer coefﬁcient h appearing in Eq. (6.5), given by Shimizu et al. (1985): h = ka NNu /D

(6P.5)

in which ka is the thermal conductivity (W/(m ◦ C)) of quench air given by, ka = 1.88 × 10−8 Ta0.866 , in which Ta in Celsius is the quench air temperature, NNu is the Nusselt number, deﬁned by % 2 &0.167 NNu = 0.42NRe 1 + 8va /vz

(6P.6)

where NRe is the Reynolds number based on ﬁlament diameter D(z), va is the quench air velocity, and vz is the ﬁlament velocity. You may also use the following expression for the temperature dependence of melt density ρ(T) for a narrow range of temperatures (170 ◦ C < T ≤ 220 ◦ C), ρ(T ) = 1.222 − 2.1 × 10−3 (T − 273)

(6P.7)

having the unit of g/cm3 . Further, you may assume that the PP has a weight-average molecular weight of 2.1 × 105 and a polydispersity index of 5.5. Problem 6.2

Derive Eq. (6.17).

Notes 1. The stress tensor τ appearing in Eq. (6.4) is deﬁned by τ = σ − Gδ with σ being the extra stress tensor in the deﬁnition of the total stress, T = −pI + σ in Cartesian coordinates (see Chapter 2 of Volume 1). Since the stress tensor for incompressible ﬂuids is only deﬁned to within an isotropic constant, either τ or σ can be used to calculate, for instance, the tensile stress from Tzz = τ11 − τ22 = σ11 − σ22 . In Chapter 3 of Volume 1, we presented a single-mode Giesekus constitutive equation in terms of τ, Eq. (3.23), and obtained the time evolution equation for the conﬁguration tensor c, Eq. (3.26), from Eq. (3.23) by assuming σ = Gc, where the strain tensor c is dimensionless. 2. According to Doufas et al. (2000a), the parameter B appearing in Eq. (6.4) is related to a dimensionless quench air drag force per unit length of the ﬁlament, and it could be also viewed as a type of drag coefﬁcient appearing in standard air drag correlations in the literature. A. J. McHugh and A. K. Doufas kindly provided the author with a plot, showing that the magnitude of B is in the order of 1 along the spinline distance. 3. Equation (6.6) was obtained by introducing the nonlinear spring force factor E, due to Peterlin (1966), into Eq. (3.26) given in Chapter 3 of Volume 1. The ratio kB T /K0 outside the bracket and the ratio K0 /kB T inside the bracket are multiplied to maintain the consistency in dimension. 4. Peterlin (1966) introduced the concept of ﬁnite extensibility into the freely jointed bead–spring model having N + 1 beads connected by N elastic springs with the spring length b. Namely, the force (or tension) F preventing the ends of the polymer chain

FIBER SPINNING

301

from stretching inﬁnitely may be described by F=

3kB T Er Lc b

(6N.1)

where kB is the Boltzmann constant, T is the absolute temperature, Lc is the contour length Nb, r is the end-to-end vector of the ﬂexible chains, and E is deﬁned by Eq. (6.8). For E = 1, Eq. (6N.1) reduces to the situation where the chain extends to any length as the applied force is increased (i.e., the force will increase continuously even when the chain is stretched to approaching its extended length (i.e., as Lc → 1), while in reality polymer chains do not exhibit such behavior. Equation (6.8) indicates that the force deﬁned by Eq. (6N.1) will become very large as r/Lc → 1, suggesting that the polymer chain will not be stretched beyond a certain limit (which is a signature of ﬁnite chain extensibility), that is, the chain stiffens as r/Lc → 1. This is illustrated in Figure 6.29. 5. Since the conformation tensor c is deﬁned by c = rr, we have (tr c)1/2 = 1/2 3 rrii . Thus, e deﬁned by i=1

e = (tr c)1/2 /Lc

(6N.2)

may be interpreted as the ratio of the chain extension to the contour length of the completely extended molecule, that is, the fractional extension of the chains. 6. Equation (6.14) is obtained as follows. In the absence of the external potential, Eq. (14.2–11) given in the monograph of Bird et al. (1987) reduces to ' ( 1 1 ᒁuu = δ − uu − 2 (∇v)T : uuuu ᒁt 3λ λ

(6N.3)

Using S = uu − (1/3)δ, the convected derivative of uu appearing on the left-hand side of Eq. (6N.3) can be rewritten as & ᒁuu ᒁS 1 % = − ∇v + (∇v)T ᒁt ᒁt 3

(6N.4)

Figure 6.29 Plots of dimensionless force

F /(3kB T /b) versus dimensionless chain extension r/Lc .

302

PROCESSING OF THERMOPLASTIC POLYMERS

in which use was made of ᒁδ/ᒁt = −2d = −[∇v + (∇v)T ]. Thus, substitution of Eq. (6N.4) into Eq. (6N.3) gives & ' ( 1 1% ᒁS =− S+ ∇v + (∇v)T − 2 (∇v)T : uuuu λ 3 ᒁt

(6N.5)

7. See Eq. (14.6–16) given in the monograph of Bird et al. (1987) for the derivation of Eq. (6.16). See also Chapter 9 of Volume 1, in which we have presented the details of the derivations of the expressions for the stresses of rigid rodlike macromolecules. 8. For details, refer to the paper of Doufas et al. (2000a). 9. This ﬁgure was kindly provided by A. J. McHugh and A. K. Doufas. 10. Use of the deﬁnition of the dimensionless conformation tensor c∗ = cK0 /kB T in Eq. (6.37) gives 1 e= 1−X

tr c 3NkB T /K0

1/2 (6N.6)

The denominator on the right-hand side of Eq. (6N.6) can be rewritten, with the aid of Eq. (6.7), as 3NkB T 3N 2 kB T b2 = (Nb)2 = K0 3kB T

(6N.7)

Substitution of Eq. (6N.7) into (6N.6) reduces to e=

1 (tr c)1/2 1 − X Nb

(6N.8)

which is consistent with Eq. (6N.2) for the amorphous melt phase before the onset of crystallization.

References Advani SG, Tucker CL (1990). J. Rheol. 34:367. Bair TI, Morgan PW (1972). U.S. Patent 3673143. Bankar VG, Spruiell JE, White JL (1977a). J. Appl. Polym. Sci. 21:2135. Bankar VG, Spruiell JE, White JL (1977b). J. Appl. Polym. Sci. 21:2141. Bird RB, Curtiss CF, Armstrong RC, Hassager O (1987). Dynamics of Polymeric Liquids: Kinetic Theory, 2nd ed, Vol 2, John Wiley & Sons, New York. Black BW, Preston J (eds) (1973). High-Modulus Wholly Aromatic Fibers, Marcel Dekker, New York. Buckley RA, Philipps RJ (1969). Chem. Eng. Progr. 65(10):41. Chang JC, Denn MM (1979). J. Non-Newtonian Fluid Mech. 5:369. Corbiere J (1967). In Man-Made Fibers, Mark HF, Atlas SM, Cernia E (eds), Vol 1, John Wiley & Sons, New York, p 133.

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Daniels BK, Peston J, Zaukelies DA (1971). U.S.Patent 3600269. Dees J, Spruiell JE (1974). J. Appl. Polym. Sci. 18:1053. Denn MM (1983). In Computational Analysis of Polymer Processing, Pearson JRA, Richardson SM (eds), Applied Science, New York. Denn MM, Petrie CJS, Avenas P (1975). AIChE J. 21:791. Doufas AK (2002). A Microstructural Model for Flow-Induced Crsytallization with Applications to the Simulation of Polymer Processes, Doctoral Dissertation, University of Illinois, Urbana Champaign, Illinois. Doufas AK, McHugh AJ, Miller C (2000a). J. Non-Newtonian Fluid Mech. 92:27. Doufas AK, McHugh AJ, Miller C, Immaneni A (2000b). J. Non-Newtonian Fluid Mech. 92:81. Doufas AK, McHugh AJ (2001a). J. Rheol. 45:403. Doufas AK, McHugh AJ (2001b). J. Rheol. 45:825. Everage AE (1973). Trans. Soc. Rheol. 17:629. Fischer RJ, Denn MM (1976). AIChE J. 22:236. Fisher RJ, Denn MM (1977). AIChE J. 23:23. Fok SY, Griskey RG (1966). Appl. Sci. Res. 16:141. Forney RC, McCune LK, Pierce NC, Tompson RY (1966). Chem. Eng. Progr. 62(3):89. Frazer AH (1972). U.S. Patents 3536651, 3642707. Freeman HI, Coplan MJ (1964). J. Appl. Polym. Sci. 8:2389. Gagon DK, Denn MM (1981). Polym. Eng. Sci. 21:844. George HH (1982). Polym. Eng. Sci. 22:292. George HH, Holt A, Buckley A (1983). Polym. Eng. Sci. 23:95. Giesekus H (1982). J. Non-Newtonian Fluid Mech. 11:69. Gou Z, McHugh, AJ (2004a). J. Non-Newtonian Fluid Mech. 118:121. Gou Z, McHugh, AJ (2004b). Intern. Polymer Process. 19:244. Gou Z, McHugh, AJ (2004c). Intern. Polymer Process. 19:254. Haberkorn H, Hahn K, Breuer H, Dorrer HD, Matthies P (1990). J. Appl. Polym. Sci. 39:447. Han CD (1971). J. Appl. Polym. Sci. 15:1091. Han CD (1973). J. Appl. Polym. Sci. 17:1289. Han CD (1975). J. Appl. Polym. Sci. 19:1875. Han CD (1981). Multiphase Flow in Polymer Processing, Academic Press, New York, Chap 7. Han CD, Kim YW (1976). J. Appl. Polym. Sci. 20:2609. Han CD, Lamonte RR (1972). Trans. Soc. Rheol. 16:447. Han CD, Lamonte RR, Shah YT (1972). J. Appl. Polym. Sci. 16:3307. Han CD, Park JY (1973). J. Appl. Polym. Sci. 17:187. Han CD, Segal L (1970). J. Appl. Polym. Sci. 14:2999. Henson GM, Cao D, Bechtel SE, Forest MG (1988). J. Rheol. 42:329. Hicks EM, Ryan JF, Taylor RB, Tichenor RL (1960). Textile Res. J. 30:675. Hicks EM, Tippetts EA, Hewett JV, Brand RH (1967). In Man-Made Fibers, Mark HF, Atlas SM, Cernia E (eds), Vol 1, John Wiley & Sons, New York, p 375. Ishizuka O, Koyama K (1985). In High-Speed Fiber Spinning, Ziabicki A, Kawai H (eds), John Wiley & Sons, New York, p 151. Kase S, Matsuo T (1965). J. Polym. Sci., A 3:2541. Kase S, Matsuo T (1967). J. Appl. Polym. Sci. 11:251. Kase S, Matsuo Y (1966). Sen-I Kikai Gakkaishi (J. Japan Text. Mech. Soc.) 19:T63. Katayama K, Yoon MG (1985). In High-Speed Fiber Spinning, Ziabicki A, Kawai H (eds), John Wiley & Sons, New York, p 207. Khan AA, Han CD (1976). Trans. Soc. Rheol. 20:595. Kim JS, Kim SY (2000). J. Appl. Polym. Sci. 76:446. Kulkarni JA, Beris AN (1998). J. Rheol. 42:971. Kwolek SL (1971), U.S. Patents 3600500, 3671542.

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Kwolek SL, Morgan PW, Schaefgen JR, Gurlich LW (1977). Macromolecules 10:1390. Lamonte RR, Han CD (1972). J. Appl. Polym. Sci. 16:3285. Lee BL, White JL (1974). Trans. Soc. Rheol. 18:467. Logullo FM (1971). U.S. Patent 3595951. MacLean DL (1973). Trans. Soc. Rheol. 17:385. Mark HF, Atlas SM, Cernia E (eds) (1967). Man-Made Fibers, John Wiley & Sons, New York. Matsui M (1985). In High-Speed Fiber Spinning, Ziabicki A, Kawai H (eds), John Wiley & Sons, New York, p 137. Minoshima W, White JL, Spruiell JE (1980). J. Appl. Polym. Sci. 25:287. Morgan HS, Preston J, Black WB (1974). U.S. Patent 3796693. Morgan PW (1997). Macromolecules 10:1381. Nakamura K, Katayama K, Amano T (1973). J. Appl. Polym. Sci. 17:1031. Nedalla HP, Henson HM, Spruiell JE, White JL (1977). J. Appl. Polym. Sci. 21:3003. Ohzawa Y, Nagano Y (1970). J. Appl. Polym. Sci. 14:1879. Ohzawa Y, Nagano Y, Matsuo T (1969). In Proc. 5th Int. Congr. Rheol. Vol 4, Onogi S (ed), University Park Press, Baltimore, Maryland. Patel RM, Bhed, JH, Spruiell JE (1991). J. Appl. Polym. Sci. 42:1671. Pearson JRA, Matovic MA (1966). Ind. Eng. Chem. Fundam. 15:31. Peterlin A (1966). Pure Appl. Chem. 12:563. Shimizu I, Okui N, Kikutani T (1985). In High-Speed Fiber Spinning, Ziabicki A, Kawai H (eds), John Wiley & Sons„ New York, p 173. Siclari F (1967). In Man-Made Fibers, Mark HF, Atlas SM, Cernia E (eds), Vol 1, John Wiley & Sons, New York, p 95. Sisson WE, Morehead FF (1953). Textile Res. J. 23:152. Sisson WE, Morehead FF (1960). Textile Res. J. 30:153. Southern JH, Ballman RL (1973). Appl. Polym. Symp. 20:1234. Southern JH, Ballman RL (1975). J. Polym. Sci. Part A-2 13:863. Vassilatos G, Knox BH, Frankfort HRE (1985). In High-Speed Fiber Spinning, Ziabicki A, Kawai H (eds), John Wiley & Sons, New York, p 383. White JL, Dharod KC, Clark ES (1974). J. Appl. Polym. Sci. 18:2539. Ziabicki A (1959). J. Appl. Polym. Sci. 2:24. Ziabicki A (1961). Kolloid Z. 175:14. Ziabicki A (1976a). Fundamentals of Fiber Formation, John Wiley & Sons, New York. Ziabicki A (1976b). In Fundamentals of Fiber Formation, John Wiley & Sons, New York, Chap 3. Ziabicki A (1976c). In Fundamentals of Fiber Formation, John Wiley & Sons, New York, Chap 4. Ziabicki A (1976d). In Fundamentals of Fiber Formation, John Wiley & Sons, New York, Chap 5. Ziabicki A (1988). J. Non-Newtonian Fluid Mech. 30:157. Ziabicki A, Kedzierska K (1959). J. Appl. Polym. Sci. 11:14. Ziabicki A, Kedzierska K (1960a). Kolloid Z. 171:51. Ziabicki A, Kedzierska K (1960b). Kolloid Z. 171:111. Ziabicki A, Kedzierska K (1962a). J. Appl. Polym. Sci. 6:111. Ziabicki A, Kedzierska K (1962b) J. Appl. Polym. Sci. 6:361. Ziabicki A, Kawai H (eds) (1985). High-Speed Fiber Spinning, John Wiley & Sons, New York. Zieminski KF, Spruiell J (1988). J. Appl. Polym. Sci. 35:2223.

7

Tubular Film Blowing

7.1

Introduction

Tubular ﬁlm blowing has long been used to produce biaxially oriented ﬁlms using such thermoplastic polymers as low-density polyethylene (LDPE), high-density polyethylene (HDPE), and polypropylene (PP). Here, LDPE refers to a polymer that is synthesized by free-radical polymerization under high pressure (Fawcett et al. 1937). The discovery of linear low-density polyethylene (LLDPE) in the 1980s via the Unipol process (Beret et al. 1986; Jones et al. 1985), which uses a low-pressure gas-phase process, has led to additions to the family of tubular blown ﬁlms during the past two decades. The discovery of metallocene catalysts (Stevens and Neithamer 1991; Welborn and Ewen 1994) in the 1990s further increased the number of LLDPEs that have been used to produce tubular blown ﬁlms during the last decade. To distinguish LLDPE from LDPE, LLDPE is sometimes referred to as low-pressure low-density polyethylene (LP-LDPE) and LDPE is referred to as high-pressure low-density polyethylene (HP-LDPE) (see Chapter 6 of Volume 1). In this chapter, however, we use the terminologies LDPE and LLDPE. As described in Chapter 6 of Volume 1, LDPE has a high degree of long-chain branching, while LLDPE has short-chain branching with little or no longchain branching. However, the metallocene catalysts apparently allow one to produce LLDPEs having a wide range of side chains, including a certain degree of long-chain branching. The details of the synthetic procedures for producing such a variety of LLDPEs are closely guarded industrial secrets. Biaxially oriented ﬁlm can be strong and tough in all directions in the plane of the ﬁlm. As in ﬁber spinning, the polymer melt exiting from the die ﬂows under a mechanical tension in the direction of ﬂow. However, in the ﬁlm blowing process, the tube of molten polymer is extended in both the transverse and the axial (machine) directions. Therefore, rheologically speaking, the ﬁlm blowing process may be treated from the point of view of biaxial elongational ﬂow, whereas the ﬁber spinning process may be treated from the point of view of uniaxial elongational ﬂow. 305

306

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 7.1 Schematic of the tubular ﬁlm blowing process.

As schematically shown in Figure 7.1, in the tubular ﬁlm blowing process, a thin ﬁlm is produced by means of the extrusion of a polymer melt through an annular die. The molten polymer tube exiting from the die is drawn upward by a take-up device. At the bottom of the die, air is introduced, inﬂating the tube to form a bubble. An air ring is also used to rapidly cool the hot bubble and solidify it at some distance above the die exit. The inﬂated, solidiﬁed tubular bubble is then ﬂattened as it passes through the nip rolls. The nip rolls, driven by a variable-speed motor, provide the axial tension needed to pull the ﬁlm upward, and they form an air-tight seal so that a constant pressure, slightly above atmospheric, is maintained in the inﬂated tubular bubble. The pressure inside the bubble is controlled by adjusting the air supply to the bottom of the die. A fundamental approach to a better understanding of the tubular ﬁlm blowing process started in 1970 when Pearson and Petrie (1970a, 1970b) formulated a system of equations to model the rather complex processing operation for Newtonian ﬂuids. Subsequently, other research groups have used basically the same formalism to model the tubular ﬁlm blowing process for power-law ﬂuids (Han and Park 1975b) or viscoelastic ﬂuids (Cao and Campbell 1990; Gupta et al. 1982; Haw 1984; Luo and Tanner 1985; Petrie 1973, 1975; Wagner 1976b). Also, a number of research groups (Cao and Campbell 1990; Farber and Dealy 1974; Gupta 1980; Gupta et al. 1982; Han and Park 1975a; Haw 1984; Liu et al. 1995a; Wagner 1976a, 1976b) conducted experimental

TUBULAR FILM BLOWING

307

investigations on the effect of processing variables on tubular ﬁlm blowing, and others investigated the effects of processing variables on the crystalline orientation (Aggarwal et al. 1959; Ashizawa et al. 1984; Choi et al. 1980, 1982; Desper 1969; Holmes and Palmer 1958; Kendall 1963; Kwack et al. 1988; Lindenmeyer and Lustig 1965; Lu et al. 2001; Maddams and Preedy 1978a, 1978b, 1978c; Maddams and Vickers 1983; Nagasawa et al. 1973; Shimomura et al. 1982) and mechanical properties (e.g., ultimate tensile strength and tensile modulus) (Han and Kwack 1983; Huck and Clegg 1961; Kaylon and Moy 1988; Kwack and Han 1983) of tubular blown ﬁlms. In this chapter, we present some fundamental aspects of tubular ﬁlm blowing that have been reported since the publication of the monograph by Han (1976) and the review article by Petrie (1983). The major thrust of this chapter is to present a fundamental approach for the simulation of the effects of processing variables on tubular ﬁlm blowing characteristics from ﬂuid mechanics and rheological points of view. Such an approach does not depend on the speciﬁc polymers being or to be used to produce tubular blown ﬁlms, as long as information on the pertinent rheological properties of the polymers is available. Speciﬁcally, in this chapter we ﬁrst present relationships between the tensile stresses in both the machine and transverse directions and processing variables in tubular ﬁlm blowing with the aid of simple force balance equations (without using elaborate momentum and energy balance equations for the moving tubular ﬁlm). Then, we present a more rigorous analysis of the tubular ﬁlm blowing process, including the extrudate swell region outside the tubular ﬁlm die. It is granted that this is not a trivial problem. Since the majority of thermoplastic polymers in use for tubular ﬁlm blowing exhibit strong elastic effects (e.g., extrudate swell), the inclusion of extrudate swell in the modeling of tubular ﬁlm blowing seems to be very appropriate. In this chapter, we do not discuss the effects of processing variables on the chain orientation of macromolecules in biaxial stretching, optical properties, and mechanical properties (tensile properties, puncture resistance, tear resistance, etc.) of tubular blown ﬁlms, although the subjects are of fundamental and practical importance. This is because the presentation of such subjects will require much more space than is available in this book, and meaningful discussion of those subjects will require the descriptions of many different experimental techniques (e.g., X-ray scattering, light scattering, birefringence, transmission electron microscopy, and scanning electron microscopy). Such subjects have been discussed extensively in the literature (Bafna et al. 2001; Butler and Donald 1998; Clark and Garber 1971; Garber and Clark 1970; Gupta et al. 1993; Holmes and Palmer 1958; Huck and Clegg 1961; Keller and Machin 1967; Samuels 1974; Schultz 1974; Sherman 1997).

7.2

Processing Characteristics of Tubular Film Blowing

In this section, we ﬁrst present the kinematics and stress ﬁeld in tubular ﬁlm blowing, because it will help us to understand the unique features of the tubular ﬁlm blowing process, and we then discuss the relationships between processing variables and tensile stress in tubular ﬁlm blowing. This section will serve a very useful purpose in the development of a rigorous analysis of tubular ﬁlm blowing that will be presented in the next section.

308

PROCESSING OF THERMOPLASTIC POLYMERS

7.2.1

Kinematics and Stress Field in Tubular Film Blowing

Consider the coordinates shown in Figure 7.2. We focus our attention in the region where both the bubble diameter a(z) and the ﬁlm thickness h(z) vary with the machine direction z. To express the velocity gradients in terms of a, h and z, we use the theory of thin shells as an approximation (Pearson and Petrie 1970a, 1970b, 1970c). This approximation assumes that the ﬁlm thickness h is small compared with other dimensions of the bubble and its radii of curvature (i.e., h a), permitting one to approximate the curved ﬁlm by a plane ﬁlm. Our task is, then, to relate the cylindrical coordinates (r, θ, z) to the rectangular Cartesian coordinates (ξ1 , ξ2 , ξ3 ) at a point P on the surface of the ﬁlm. In Figure 7.2, ξ1 is in the direction of ﬂow, ξ2 is normal to the ﬁlm surface, and ξ3 is in the transverse (circumferential) direction. Let (v1 , v2 , v3 ) be velocity components in the coordinate directions (ξ1 , ξ2 , ξ3 ). The rate-of-strain tensor d may be written as -d - 11 d = -0 -0

0 d22 0

0 0 d33 -

(7.1)

in which d11 , d22 , and d33 are deﬁned by d11 =

∂v1 ∂v ∂v , d22 = 2 , d33 = 3 ∂ξ1 ∂ξ2 ∂ξ3

(7.2)

Figure 7.2 Coordinate systems for describing the deformation of a tubular bubble.

TUBULAR FILM BLOWING

309

In Eq. (7.1), the ﬁlm is assumed to be almost plane, and consequently the shear components of the rate-of-strain are assumed to be negligible. Note that d22 and d33 may be represented in terms of a and h as functions of z (Pearson and Petrie 1970a): d22 =

1 dh dξ1 1 dh 1 dh = = v1 cos θ h dt h dξ1 dt h dz

(7.3)

d33 =

1 da dξ1 1 da 1 da = v1 cos θ = a dz a dt a dξ1 dt

(7.4)

In order to satisfy the equation of continuity, we require d11 + d22 + d33 = 0

(7.5)

d11 may now be expressed in terms of a and h by d11 = −(d22 + d33 ) = −v1 cos θ

1 dh 1 da + a dz h dz

(7.6)

Since v1 can be expressed in terms of the volumetric ﬂow rate (Q), a, and h by dξ1 Q = v1 = dt 2πah

(7.7)

Eqs. (7.3), (7.4), and (7.6) may be rewritten as d11

Q cos θ =− 2πah Q cos θ 2πah2 Q cos θ = 2πa 2 h

d22 = d33

1 dh 1 da + a dz h dz

(7.8)

dh dz da dz

(7.9) (7.10)

Substituting Eqs. (7.8), (7.9), and (7.10) into (7.1), we obtain: - 1 da 1 dh -− + a dz h dz Q cos θ d = 0 2πah 0 -

0 1 dh h dz 0

0 0 1 da a dz

(7.11)

The total stress component Tij may be given by Tij = −pδij + σij

(7.12)

310

PROCESSING OF THERMOPLASTIC POLYMERS

in which p is the pressure that can be determined within an accuracy of an isotropic term (i.e., p is undetermined hydrostatic pressure), δij is the Kronecker delta, and the σij are extra stresses. Referring to Figure 7.2, the fact that the stress at the free surface is equal to atmospheric pressure gives T22 = 0

(7.13)

p = σ22

(7.14)

T11 = σ11 − σ22

(7.15)

T33 = σ33 − σ22

(7.16)

Using Eq. (7.13) in (7.12), we obtain

Using Eq. (7.14) in (7.12), we obtain

in which T11 is the tensile stress in the machine direction (MD) and T33 is the tensile stress in the transverse direction (TD) (i.e., the hoop stress). Alternatively, we can deﬁne biaxial elongational viscosity ηB in tubular ﬁlm blowing by σ11 − σ22 = ηB (I2 )(d11 − d22 )

(7.17)

σ33 − σ22 = ηB (I2 )(d33 − d22 )

(7.18)

or

where I2 is the second invariant, deﬁned as I2 = d11 2 +d22 2 +d33 2 , of the rate-of-strain tensor d. With the aid of Eqs. (7.8)–(7.10), (7.15) and (7.16), Eqs. (7.17) and (7.18) can be rewritten as ηB =

cos θ − Q2πah

T 11

1 da a dz

+

2 dh h dz

(7.19)

or ηB =

Q cos θ 2πah

T33 1 da a dz

−

1 dh h dz

By combining Eqs. (7.19) and (7.20), we obtain & % 2 dh 1 da T11 h dz + a dz & =% 1 dh T33 − 1 da h dz

(7.20)

(7.21)

a dz

The signiﬁcance of Eq. (7.21) lies in that one can calculate the ratio T11 /T33 along the machine direction z when the proﬁles of ﬁlm thickness h(z) and bubble shape a(z) are

TUBULAR FILM BLOWING

311

available below the freeze line, 0 < z < Z (see Figure 7.1), either from experiment or simulation. In the next section, we will present experimental observations of the tensile stress T11F in the machine direction at the freeze line, the tensile stress T33F in the transverse direction at the freeze line, and also the ratio T11F /T33F as functions of processing variables. 7.2.2

Tensile Stresses at the Freeze Line and Processing–Property Relationships in Tubular Film Blowing

A complete understanding of the effect of processing variables on the mechanical/ optical properties and morphology of tubular blown ﬁlms is indeed a formidable task. Before presenting a more rigorous approach, here we will consider a grossly simpliﬁed approach by neglecting the region where the kinematics of biaxial stretching are involved (i.e., the region between the die exit and the freeze line in Figure 7.1). That is, we will take the steady-state force balances above the freeze line (i.e., in the region between the freeze line and the position at which the axial tension is measured near the take-up roll), and then consider processing–property relationships in tubular ﬁlm blowing of LDPEs. Referring to Figure 7.2, the tensile stress in the MD at the freeze line, T11F , is given by T11F = FZ /2πAH

(7.22)

where A and H are the radius of the tubular bubble and the ﬁlm thickness, respectively, at the freeze line, and FZ is the tensile force at the freeze line Z, as calculated by FZ = FL − 2πAHρs g(L − Z)

(7.23)

where FL is the axial tension measured at a distance L above the die exit, ρs is the density at the solidiﬁed ﬁlm, g is the gravitational acceleration. The tensile stress in the TD at the freeze line, T33F , is given by T33F = Ap/H

(7.24)

where p is the pressure inside the tubular bubble above ambient. From a consideration of the conservation of mass, we have m ˙ = π(ao 2 − ai 2 )vo ρm = 2πρs AHV

(7.25)

where ρm is the density of the polymer in the molten state, vo is the linear average velocity of the melt at the die exit, V is the linear velocity of the tubular bubble at the freeze line (i.e., take-up speed), and ao and ai are the outer and inner radii, respectively, of the die opening.

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PROCESSING OF THERMOPLASTIC POLYMERS

Let us deﬁne blow-up ratio (BUR) and take-up ratio (TUR), respectively, by BUR = A/ao

(7.26)

TUR = V /vo

(7.27)

Substitution of Eqs. (7.26) and (7.27) into (7.25) gives (Han and Kwack 1983) (a 2 − ai 2 ) H = o 2ao

ρm ρs

1 (BUR) (TUR)

(7.28)

It is of interest to observe in Eq. (7.28) that for a given die gap opening and a given polymer, the ﬁlm thickness H is inversely proportional to the product of BUR and TUR. Substitution of Eq. (7.28) into (7.22), with the aid of Eq. (7.23), gives T11F =

FL ρs (TUR) − (L − Z)ρs g C ρm

(7.29)

where C = π(ao 2 − ai 2 ). Substitution of Eq. (7.28) into (7.24), with the aid of (7.26), gives T33F =

1 ρs (p)(BUR)2 (TUR) B ρm

(7.30)

where B = (ao 2 − ai 2 )/2(ao )2 . It is seen that the tensile stresses T11F in MD and T33F in TD at the freeze line can be calculated from measured values of FL , TUR, BUR, p and the freeze-line height Z. The striking feature of Eqs. (7.29) and (7.30) is that measurements of ﬁlm thickness are not needed to determine the tensile stresses in the tubular blown ﬁlm at the freeze line. Notwithstanding the simplicity of the macroscopic force balances considered so far, we will show some interesting observations that can be used to explain experimental results. Figure 7.3 gives experimental results describing the dependence of ﬁlm thickness H on the inverse of the product of BUR and TUR for three LDPEs having different molecular weight and long-chain branching, the values of which are summarized in Table 7.1. It is seen in Figure 7.3 that the experimental data conﬁrm the prediction of Eq. (7.28). The effect of BUR on T11F and T33F is given in Figure 7.4 for an LDPE at various values of TUR with other processing variables ﬁxed: a melt extrusion temperature at 200 ◦ C and a ﬂow rate of cooling air at 2.21 × 103 cm3 /s. It is seen in Figure 7.4 that the values of T11F are larger than the values of T33F but at a ﬁxed TUR T33F increases much faster with BUR than T11F does. This experimental observation can be explained with the theoretical predictions represented by Eqs. (7.29) and (7.30). Note that the axial tension FL appearing in Eq. (7.29) increases with BUR and thus T11F increases with BUR. The effect of TUR on T11F and T33F is given in Figure 7.5 for three LDPEs at a BUR of 3 with other processing variables ﬁxed: a melt extrusion temperature at 200 ◦ C and a ﬂow rate of cooling air at 2.21 × 103 cm3 /s. It is seen in Figure 7.5 that both

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313

Figure 7.3 Plots of ﬁlm thickness H versus 1/(BUR)(TUR) for () LDPE-A, () LDPE-B, and () LDPE-C. (Reprinted from Han and Kwack, Journal of Applied Polymer Science 28:3399. Copyright © 1983, with permission from John Wiley & Sons.)

T11F and T33F increase with TUR at approximately the same rate. This experimental observation also is readily explainable with Eqs. (7.29) and (7.30). In ﬁlm blowing operations, BUR is decreased as TUR is increased, unless p is increased. Therefore, the experimental data obtained at different values of TUR, given in Figure 7.5, were obtained for different values of p. Note that the axial tension FL and the freeze-line height Z also vary with BUR and TUR. If we assume that there is no molecular orientation occurring in the tubular ﬁlm above the freeze line, the values of T11F and T33F represent the MD and TD tensile strengths, respectively, of the as-blown tubular ﬁlm. From measured values of T11F and T33F , one can prepare plots of the ratio T11F /T33F versus BUR with TUR as a parameter, as given in Figure 7.6 for three LDPEs (see Table 7.1 for their molecular characteristics). It is seen in Figure 7.6 that the ratio T11F /T33F decreases with BUR and increases with TUR, and that LDPE-A, having a broad molecular weight distribution (MWD), exhibits higher values of T11F /T33F than the other two polymers over the range of TUR and BUR investigated. Note in Figure 7.6 that the ratio T11F /T33F

Table 7.1 Molecular characteristics of three LDPEs

Sample Code LDPE-A LDPE-B LDPE-C

Density (g/cm3 )

Mw

Mw /Mn

λN a

0.918 0.921 0.923

2.0 × 105 1.4 × 105 1.1 × 105

9.43 6.03 4.18

3.4 2.5 1.6

aλ represents the long-chain branching frequency deﬁned as the numberN average number of branch points per 1,000 carbon atoms.

Reprinted from Han and Kwack, Journal of Applied Polymer Science 28:3399. Copyright ©1983, with permission from John Wiley & Sons.

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PROCESSING OF THERMOPLASTIC POLYMERS

Figure 7.4 Plots of T11F (open symbols) and T33F (ﬁlled symbols) versus BUR for LDPE-A at various TURs: (, 䊉) 3.98, (, ) 9.86, (, ) 15.61, and (, ) 21.44. Other processing conditions are: melt extrusion temperature at 200 ◦ C and cooling air ﬂow rate at 2.21 × 103 cm3 /s.

(Reprinted from Han and Kwack, Journal of Applied Polymer Science 28:3399. Copyright © 1983, with permission from John Wiley & Sons.)

approaches 2 in the limit of high BURs. It is worth pointing out that Eq. (7.21) predicts that the ratio T11F /T33F approaches 2 at the freeze line, where da/dz becomes zero, that is, T11F /T33F = 2 as observed experimentally under certain processing conditions (see Figure 7.6). It has been observed experimentally that values of (1/a)da/dz approach zero for large BURs (e.g., BUR 5). From Eqs. (7.29) and (7.30) we have the following expression for the ratio T11F /T33F (Han and Kwack 1983): T11F = T33F

Bρm g(L − Z) − (p)(TUR) 2π(ao )2 p FL

1 (BUR)2

(7.31)

Equation (7.31) would be very useful for determining the processing conditions (TUR, BUR, and/or p) and/or the die design variables (die radius ao and die gap opening, ao − ai ), such that the tensile strengths in the MD and TD can be optimized. For instance, with the guidance of Eq. (7.31) one can produce tubular blown ﬁlms with

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315

Figure 7.5 Plots of T11F (open symbols) and T33F (ﬁlled symbols) versus TUR for at a BUR of 3 for three LDPEs: (, 䊉) LDPE-A, (, ) LDPE-B, and (, ) LDPE-C. Other processing conditions are melt extrusion temperature at 200 ◦ C and cooling air ﬂow rate at 2.21×103 cm3 /s.

(Reprinted from Han and Kwack, Journal of Applied Polymer Science 28:3399. Copyright © 1983, with permission from John Wiley & Sons.)

balanced axial and hoop stresses (T11F = T33F ) by a judicious selection of processing variables. Without Eq. (7.31) it would be extremely difﬁcult, if not impossible, to determine the combination of processing variables that can optimize the MD and TD tensile strengths of tubular blown ﬁlms. In this regard, the analysis presented above, although much simpliﬁed, can be considered as a very useful processing guideline. It is granted, however, that the simpliﬁed analysis presented above does not provide information on the shape of the blown bubble below the freeze line, the proﬁle of ﬁlm thickness below the freeze line, or the temperature proﬁle of tubular bubble below the freeze line. Such information can be predicted using a more sophisticated analysis, as will be presented in the next section. The axial and hoop stresses greatly inﬂuence the molecular orientation in tubular blown ﬁlms. A number of research groups (Aggarwal 1959; Ashizawa et al. 1984; Choi et al. 1980, 1982; Desper 1969; Han and Kwack 1983; Holmes and Palmer 1958; Kendall 1963; Kwack and Han, 1983; Lindenmeyer and Lustig 1965; Lu et al. 2001; Maddams and Preedy 1978a, 1978b, 1978c; Maddams and Vickers 1983; Nagasawa et al. 1973; Shimomura et al. 1982) reported on the effect of processing variables on the crystalline orientations of tubular blown ﬁlms, while others (Ghaneh-Fard et al.

316

PROCESSING OF THERMOPLASTIC POLYMERS Figure 7.6 Plots of T11F /T33F

versus BUR. (a) LDPE-A at three different TURs: () 4.4, () 9.9, and () 15.6. (b) LDPE-B at three different TURs: (䊉) 4.4, () 9.9, and () 15.6. (c) LDPE-C at three different TURs: (䊎) 4.4, (䊖) 9.9, and, (䊒) 15.6. Other processing conditions are: melt extrusion temperature at 200 ◦ C and cooling air ﬂow rate at 2.21 × 103 cm3 /s. (Reprinted from Han and Kwack, Journal of Applied Polymer Science 28:3399. Copyright © 1983, with permission from John Wiley & Sons.)

1997a, 1997b; Kurtz 1995; Kwack et al. 1988) related the axial and hoop stresses to the mechanical properties of tubular blown ﬁlms. However, at present there is no theory that relates the crystalline orientation in a tubular blown ﬁlm to the axial and hoop stresses for semicrystalline polymers. This is a subject that requires further investigation. The majority of the polymers employed for tubular ﬁlm blowing are semicrystalline polymers, such as LDPE, LLDPE, and HDPE. It is well established that the mechanical properties of semicrystalline polymers are determined by the morphology of the crystalline phase, the size of crystals and the extent of orientation of the crystalline phase, which in turn are determined by the processing conditions. Figure 7.7 shows a schematic of the mechanism of lamellar twisting behavior of crystalline regions of semicrystalline tubular blown ﬁlm with an increase in tensile stress. Referring to Figure 7.7, depending on the stress ratio T11F /T33F at the freeze line, the ﬁbrillar nuclei are distributed along the MD and TD. From the center of each nucleus, lamellae emanate radially, and with twisting, along the growing direction. At a low stress state (0.1 MPa), the mode of lamellar twisting is rather irregular and the crystalline structure is more or less spherillitic. At a high stress state (1 MPa), the growing lamellae have a tendency to be aligned along the ﬁbrillar direction, although they still twist and distort themselves along the growing axis. Figure 7.8 shows schematically the “row structure” of one unit crystal structure as affected by the applied stresses, in which only one disc of lamellae along the row is depicted, for (a) T11F > T33F and (b) T11F = T33F (about 1 MPa). Note that the b-axis is the lamellae-growing direction, and that the c-axis lies perpendicular to the lamellar surface. This structural model agrees well with the X-ray diffraction patterns obtained from the tubular blown ﬁlm specimen (Kwack et al. 1988).

TUBULAR FILM BLOWING

317

Figure 7.7 Schematic of lamellar twisting behavior of crystalline regions of semicrystalline tubular blown ﬁlm with an increase in tensile stress, in which TD denotes transverse direction and ND denotes neutral direction.

Figure 7.9 gives transmission electron microscopy (TEM) images of HDPE tubular blown ﬁlm samples: (a) a cross-sectional view along the MD of an as-blown sample and (b) a cross-sectional view along the MD of a tubular blown ﬁlm that was later subjected to further stretching at room temperature. Note that both TEM images in Figure 7.9 represent the areas close to the edges of the cross section. In Figure 7.9a we observe “row structure” (Keller and Machin 1967) aligned along the MD (i.e., the lamellae in the row are aligned perpendicular to the MD), suggesting that molecular chains are aligned more or less along the MD and the sample must have experienced relatively high stresses (>1 MPa) along the MD. Similar experimental observations have been reported by Lu et al. (2001). In Figure 7.9b, we observe that the distance between the lamellae in the row structure is increased due to the cold stretching applied to the as-blown ﬁlm sample. Garber and Clark (1970) and Clark and Garber (1971) ﬁrst noted the presence of a row structure in tubular blown ﬁlms of poly(oxymethylene).

7.3

Analysis of Tubular Film Blowing Including Extrudate Swell

Figure 7.10 gives two photographs, showing (a) the “wine-glass shape” of a tubular blown bubble of an HDPE and (b) the standard shape of a tubular blown bubble of an LDPE. It is interesting to observe that upon exiting die, the two polymers form different tubular bubble shapes. The origin of the wine-glass shape shown in Figure 7.10a is not well understood. Therefore, in the analysis below we will consider the standard shape of a tubular bubble shown in Figure 7.10b. In this section, we present an analysis simulating nonisothermal tubular ﬁlm blowing, including extrudate swell just

Figure 7.8 Schematic showing the “row structure” as affected by the applied stresses, in which only one disc of lamellae along the row is depicted: (a) T11F > T33F , and (b) T11F = T33F (≈ 1 MPa). MD denotes machine direction, TD denotes transverse direction, and ND denotes neutral direction.

Figure 7.9 TEM images of (a) as-blown ﬁlm of HDPE taken in the MD–ND plane, and

(b) as-blown ﬁlm of HDPE followed by cold stretching at room temperature. MD denotes machine direction and ND denotes neutral direction. 318

TUBULAR FILM BLOWING

319

Figure 7.10 Photographs of (a) a wine-glass shape of HDPE tubular blown bubble and (b) a

standard shape of LDPE tubular blown bubble. (Reprinted from Han and Park, Journal of Applied Polymer Science 19:3257. Copyright © 1975, with permission from John Wiley & Sons.)

outside the die. For this, we will employ an integral-type constitutive equation taking into account the temperature dependence of the material constant involved. The effect of gravity will be included in the force balance equation. In the upward tubular ﬁlm blowing operation, the inclusion of the gravity effect in the force balance equation is very important. Note that the force exerted on the ﬁlm by the take-up device counterbalances the gravity force. We make the following assumptions in writing down the force balance equation (Pearson and Petrie 1970a) and energy balance equation (Han and Park 1975b; Wagner 1976a): (1) the ﬁlm is thin enough so that variations in the ﬂow ﬁeld across it may be neglected; (2) the velocity gradients may be approximated locally by those of a plane ﬁlm being extended biaxially; (3) the effects of surface tension, air drag, and the inertial force are negligible, compared with the axial tension (i.e., the rheological force needed for the deformation of the bubble of molten polymer); (4) heat transfer between the inner surface and the air trapped within the bubble is negligible; (5) heat conduction in the thin ﬁlm is negligible; (6) the cooling of the tubular blown bubble is controlled by radiation and convective heat transfer; (7) heat generation due to the frictional force is negligible; and (8) the heat of crystallization is assumed to be negligible compared with the overall energy transfer involved. Figure 7.11 gives a schematic of the extrudate swell region (zone I) and the stretching region (zone II), where the ﬁlm thickness h(z) decreases along the axial direction z, and Figure 7.12 gives a schematic of the rectangular coordinates (ξ1 , ξ2 , ξ3 ) at a point P on the surface of the tubular ﬁlm. 7.3.1

Force Balance Equation

From the “thin shell” theory (Pearson and Petrie 1970a), the force in the MD per unit length of a tubular bubble (the axial tension), PL , and the force in the TD per unit

320

PROCESSING OF THERMOPLASTIC POLYMERS Figure 7.11 Schematic

depicting the swelling of ﬁlm thickness of a tubular blown bubble upon exiting an annular die, where zone I denotes the extrudate swell region and zone II denotes the stretching region.

length of a tubular bubble (the hoop tension), PH , are balanced by the pressure inside the bubble p: p =

PL P + H − ρgh sin θ RL RH

(7.32)

where RH and RL are the principal radii of curvature of the tubular bubble given by RH = a sec θ

(7.33)

and

RL =

2 3/2 − 1 + da dz d2 a dz2

=−

sec3 θ d2 a dz2

(7.34)

respectively, and PL and PH can be represented by (Pearson and Petrie 1970a, 1970b) PL =

h 0

T11 dξ2 = hT11

(7.35)

TUBULAR FILM BLOWING

321

Figure 7.12 Schematic showing the rectangular coordinates (ξ1 , ξ2 , ξ3 ) at a point on the surface of the tubular ﬁlm, including extrudate swell and stretching regions, where zm denotes the axial position at which the swelling of ﬁlm thickness attains a maximum, ho denotes the die gap opening, ao − ai , h(z) is the thickness of the tubular ﬁlm below the freeze line, a(z) is the bubble radius below the freeze line, and A is the bubble radius above the freeze line. Notice that the ﬁlm thickness in the axial direction increases in the extrudate swell region and decreases in the stretching region.

and

PH =

h 0

T33 dξ2 = hT33

(7.36)

respectively, where T11 is tensile stress in the machine direction and T33 is the tensile stress in the transverse direction (i.e., the hoop stress). Substitution of Eqs. (7.33) and (7.34) into (7.32), with the aid of Eqs. (7.35) and (7.36), gives1

p = −

hT11 dθ dz sec θ

+

hT33 − ρgh sin θ a sec θ

(7.37)

which, after rearrangement, can be rewritten as 1 dθ = dz hT11 cos θ

hT33 − p − ρgh sin θ a sec θ

(7.38)

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PROCESSING OF THERMOPLASTIC POLYMERS

7.3.2

Energy Balance Equation

The energy balance on the tubular ﬁlm may be written as (Han and Park 1975b; Wagner 1976a)

ρcp v1

∂T ∂q =− ∂ξ1 ∂ξ2

(7.39)

where q is the heat ﬂux in the ξ2 direction, cp is the speciﬁc heat capacity, v1 is the velocity in the ξ1 direction, and T is the temperature on the surface of the tubular bubble. conditions (1) q = 0 at Integrating Eq. (7.39) with respect to ξ2 using the boundary ξ2 = 0 (inner surface) and (2) q = U T − Ta + λr ε T 4 − Ta 4 at ξ2 = h (outer surface), where U is the overall heat transfer coefﬁcient, λr is the Stefan–Boltzmann constant, ε is the emissivity of the tubular ﬁlm, and Ta is the ambient temperature, we obtain

dT 1 =− U T − Ta + λr ε T 4 − Ta 4 dξ1 ρcp v1 h

(7.40)

Note that the conservation of the mass ﬂow rate m ˙ can be expressed as m ˙ = 2πρahv1

(7.41)

Substitution of Eq. (7.41) in (7.40), with the aid of d/dξ1 = cos θ d/dz, yields

2πa dT =− U T − Ta + λr ε T 4 − Ta 4 dz cp m ˙ cos θ

(7.42)

The temperature dependence of speciﬁc heat capacity cp (T) and melt density ρ(T) may be expressed as cp0 D1 + D2 T cp (T ) = D1 + D2 T0

(7.43)

ρ0 E1 + E2 T ρ(T ) = E1 + E 2 T 0

(7.44)

and

where cp0 and ρ0 are speciﬁc heat capacity and density, respectively, at the reference temperature T0 and D1 , D2 , E1 , and E2 are constants.

TUBULAR FILM BLOWING

323

The system equations describing the dynamics and heat transfer of a tubular bubble in ﬁlm blowing may be summarized as follows2 (Han and Park 1975b): dz = v1 cos θ dt da = v1 sin θ dt

2πav1 dT 4 4 =− U T − Ta + λ r ε T − T a dt cp m ˙ hT33 v dθ = 1 − p − ρgh sin θ dt hT11 a sec θ

(7.45) (7.46) (7.47) (7.48)

Note that Eq. (7.46) follows from the geometrical relationship da/dz = tan θ (see Figure 7.12). In order to solve Eqs. (7.45)–(7.48) with appropriate boundary conditions, one must have information on T11 and T33 appearing in Eq. (7.48). The full description of T11 and T33 , thus σ11 , σ22 , and σ33 in accordance with Eqs. (7.15) and (7.16), can be obtained from a constitutive equation chosen. 7.3.3

Viscoelastic Constitutive Equation

In the past, some attempts (Cao and Campbell 1990; Gupta et al. 1982; Haw 1984; Luo and Tanner 1985; Wagner 1976b) were made at modeling the nonisothermal tubular ﬁlm blowing process using a viscoelastic constitutive equation. Luo and Tanner (1985) employed the upper convected Maxwell model (see Eq. (3.4) given in Chapter 3 of Volume 1) while others (Cao and Campbell 1990; Gupta et al. 1982) employed an empirically modiﬁed Maxwell model (see Eq. (3.12) given in Chapter 3 of Volume 1). They noted that the numerical solutions of the system equations were highly unstable and thus it was difﬁcult to obtain convergence with arbitrary values of the ﬁlm-blowing parameters. Such numerical instability is frequently encountered when some of the differential-type constitutive equations are coupled with highly nonlinear momentum and energy balance equations. In this regard, the use of integral-type constitutive equations may be preferable to the use of differential-type constitutive equations when modeling the nonisothermal tubular ﬁlm blowing process. It should be mentioned that in those studies (Cao and Campbell 1990; Gupta et al. 1982; Luo and Tanner 1985), the details of the ﬂow at the die exit (the region where extrudate swell takes place) were ignored. In the molten state, the majority of the polymers (e.g., LDPE, LLDPE, HDPE, and PP) being employed in industrial tubular ﬁlm blowing operations exhibit nonlinear viscoelasticity. For example, upon exiting a cylindrical tube, LDPE, LLDPE, HDPE, and PP melts exhibit considerable extrudate swell. It is then not difﬁcult to speculate that upon exiting an annular die, the thickness of a tubular blown ﬁlm would swell. In fact, experimental evidence (Dowd 1972; Gregory 1969; Han and Park 1975a; Scholl 1968; Steffen 1966) exists which supports such a speculation. The extent of extrudate swell just outside a die is expected to inﬂuence the tubular ﬁlm blowing characteristics of viscoelastic molten polymers. Then it is very important to include extrudate swell

324

PROCESSING OF THERMOPLASTIC POLYMERS

when modeling the nonisothermal tubular ﬁlm blowing process that employs highly elastic molten polymers. Wagner (1976b) employed an integral-type constitutive equation to model the nonisothermal tubular ﬁlm blowing process. As described in Chapter 3 of Volume 1, a viscoelastic molten polymer has a memory that depends on the deformation history. Thus, the extent of extrudate swell of a molten polymer, upon exiting an annular die, would depend on the deformation history that the molten polymer experienced inside the die and also the amount of residual stress the molten polymer has at the exit plane. Besides the numerical stability when using an integral-type constitutive equation in place of some of the differential-type constitutive equations, the use of integral-type constitutive equations offers further advantages; for instance, the implementation of temperature-dependent rheological parameters is much easier with an integral-type constitutive equation than with differential-type constitutive equations. In his study, Wagner (1976b) employed an integral-type constitutive equation3 deﬁned by Eq. (3.48) (in Volume 1). As pointed out in Chapter 3 of Volume 1, however, the particular constitutive equation cannot predict shear-rate dependent viscosity in steady-state shear ﬂow. In this section, we present the analysis of Haw (1984), who employed the Meister model (Meister 1971) given by Eq. (3.52) in Chapter 3 of Volume 1:

σ=

t

⎧ ⎪ ⎪ ⎪ N ⎨#

−∞ ⎪ ⎪ ⎪ i=1

⎩

⎡ Gi ⎢ exp⎣− λi

t t

/ 1 + cλi I2 (t ) λi

⎫ ⎤⎪ ⎪ ⎪ ⎬' ( ⎥ C−1 t − I dt dt ⎦ t ⎪ ⎪ ⎪ ⎭

(7.49)

where λi are the relaxation time constants, and Gi are moduli, c is a material constant, I2 is the second invariant of the rate-of-strain tensor d deﬁned by I2 = −1 1 2 is the relative Finger 2 (tr d) − tr (d : d) (see Chapter 2 of Volume 1), and Ct strain tensor deﬁned by4 -v 2 - 1 (t) - v (t )2 - 1 −1 Ct (t ) = - 0 - 0

0 h(t)2 h(t )2

0

0 0 a(t)2 2

(7.50)

a(t )

Note that C−1 t (t = t ) reduces to I for t = t as discussed in Chapter 2 of Volume 1. Due to the choice of a memory function that depends on the second invariant of the rate-of-strain tensor, Eq. (7.49) predicts non-Newtonian viscosity and also reasonable shear-rate dependence of ﬁrst normal stress difference in steady-state shear ﬂow (see Chapter 3 of Volume 1). To model the nonisothermal tubular ﬁlm blowing process, one must be able to describe the temperature-dependent stresses, T11 and T33 , thus σ11 , σ22 , and σ33 . For this, Haw (1984) assumed that the temperature dependence of λi (T ) may be

TUBULAR FILM BLOWING

325

described by λi (T ) = λi0 aT (T )

(7.51)

where λi0 (i = 1, . . . , N) are relaxation time constants at a reference temperature T0 and aT (T ) is a temperature-dependent shift factor deﬁned by (see Chapter 6 of Volume 1) aT (T ) =

η0 (T ) ρ0 T0 η0 T0 ρT

(7.52)

in which η0 (T ) and η0 (T0 ) are the zero-shear viscosities at T and T0 , respectively, and ρ and ρ0 are the densities at T and T0 , respectively. Over the range of temperatures encountered in tubular ﬁlm blowing (i.e., from the melting temperature of a semicrystalline polymer to the extrusion temperature, which is usually below 240 ◦ C) we have ρ0 T0 /ρT ≈ 1 and thus Eq. (7.52) can be expressed by the Arrhenius relation: E 1 1 − (7.53) aT (T ) = exp R T T0 where E is the viscous ﬂow activation energy and R is the universal gas constant. Further, the following approximation is made: Gi = Gi0 T/T0

(7.54)

where Gi0 (i = 1, . . . , N) are shear moduli at temperature T0 . For all intents and purposes, it is reasonable to assume Gi ≈ Gi0 , because shear modulus changes very slowly with temperature and the ratio T /T0 does not deviate much from 1 over the range of temperatures encountered in tubular ﬁlm blowing. Thus, Eq. (7.49) may be rewritten as ⎫ ⎧ ⎡ ⎤⎪ / ⎪ ⎪ ⎪ ⎪# ⎪ t ⎨ t 1+cλ a (t ,T ) I (t ) N ⎬ Gi i0 T 2 ⎢ ⎥ σ= exp − dt ⎣ ⎦ ⎪ λi0 aT (t ,T ) −∞ ⎪ t ⎪ i=1 λi0 aT (t ,T ) ⎪ ⎪ ⎪ ⎩ ⎭ ' ( × C−1 t −I dt t

(7.55)

It is then clear that Eq. (7.55) will enable us to calculate the stresses in the tubular blown ﬁlm in both the extrudate swell zone and the stretching zone. The material parameters appearing in Eq. (7.49) are to be determined with the aid of experimental data. Speciﬁcally, the relaxation time constants λi and shear moduli Gi may be determined by curve ﬁtting linear dynamic viscoelastic data to the following expressions (see Chapter 3 of Volume 1): G (ω) =

N # ω 2 Gi λ i 2 ; 1 + ω2 λi 2 i=1

η (ω) =

N # i=1

Gi λ i 1 + ω 2 λi 2

(7.56)

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PROCESSING OF THERMOPLASTIC POLYMERS

Alternatively, values of λi and Gi may be determined by curve ﬁtting steady-state shear viscosity (η) and ﬁrst normal stress difference (N1 ) data to the following expressions:

η(γ˙ ) =

N #

i=1

7.3.4

Gi λ i 1 + cγ˙ λi

2 ;

σ11 − σ22 =

N # 2Gi λi 2 γ˙ 2 3 i=1 1 + cγ˙ λi

(7.57)

Analysis of Tubular Film Blowing in the Extrudate Swell Region

It is generally accepted that upon exiting a die, extrudate swell occurs due to the relaxation of the residual stress (i.e., the recoverable energy per unit volume) that a molten polymer has at the exit plane of the die. This is a unique characteristic of viscoelastic ﬂuids, as discussed in Chapter 5 of Volume 1. If during ﬂow a ﬂuid dissipated all its energy, which was supplied by an external source (e.g., from a pump or extruder), and retained no recoverable energy at the die exit, there should be no extrudate swell. As discussed in Chapter 5 of Volume 1, the amount of residual stress, which a viscoelastic ﬂuid retains at the die exit, depends on the shear rate inside the die, the temperature of the ﬂuid, die geometry (the length-to-diameter ratio), and the rheological characteristic of the ﬂuid, namely Deborah number. When LDPE, LLDPE, HDPE, or PP are extruded through a tubular ﬁlm die at reasonably high shear rates the ﬁrst normal stress difference of the polymer (say, 300–1,000 s−1 ), we can estimate melt at the exit plane, σ11 − σ22 t=0 (i.e., just before the polymer melt exits the die). In ﬂow through an annular die, the shear across the die opening. Thus, rate varies one must calculate the average value of σ11 − σ22 t=0 over the cross-sectional area of the die opening, which is rather cumbersome. For simplicity, let us assume here that from Eq. (7.57) the ﬁrst normal stress difference of the melt inside the die at T = T0 represents σ11 − σ22 t=0− . Then, from Eq. (7.57) we have

T11

0−

N # 2Gi λi0 2 γ˙ 2 = σ11 − σ22 t=0− = 3 i=1 1 + cγ˙ λi0

(7.58)

Equation (7.58) is valid (1) when the ﬂow inside the annular ﬂow channel is fully developed at the exit plane and (2) when substantial amounts of residual stress remain at the die exit plane. These two conditions are met when the ﬂow channel has a large length-to-gap opening ratio, Lc /(ao − ai ) with Lc being the length of the die, and ao and ai being the outer and inner diameters of the annular die respectively, and when the polymer melt is highly elastic under the ﬂow conditions employed (i.e., at a high Deborah number). This indeed is the case when LDPE, LLDPE, PP, or HDPE are extruded under the usual processing conditions practiced in industry. In the extrudate swell region (zone I in Figure 7.12), where the tubular ﬁlm is not stretched by an external force, we assume that the ﬁrst normal stress difference

TUBULAR FILM BLOWING

327

of the melt, upon exiting the die, starts to relax following the expression

t N #

2Gi λi0 2 γ˙ 2 dt 1 = σ = exp − (t)−σ (t) 3 11 22 R R λi0 0 aT (t , T ) i=1 1+cγ˙ λi0

T11 (t)

(7.59)

which describes the residual stress of the melt being relaxed outside the die. Note that the tubular ﬁlm in the swelling zone will deform and generate stresses. Under such circumstances, we can use Eq. (7.55) to calculate the ﬁrst normal stress difference from

T11 (t) D = σ11 (t) − σ22 (t) D ⎧ ⎫ ⎡ ⎤⎪ / ⎪ ⎪ ⎪ ⎪ t 1 + cλ a (t , T ) I (t ) t ⎪ N ⎬ ⎨# Gi i0 T 2 ⎢ ⎥ exp − dt = ⎣ ⎦ ⎪ λ a (t , T ) λi0 aT (t , T ) −∞ ⎪ t ⎪ ⎪ ⎪ i=1 i0 T ⎪ ⎩ ⎭ ' ( × β1 (t )2 − β2 (t )2 dt

(7.60)

where β1 (t ) = v(t)/v(t ) and β2 (t ) = h(t)/ h(t ) Then the total axial (tensile) stress T11 (t) in the extrudate swell region of a tubular ﬁlm consists of two components, one from the residual stress in the melt, represented by Eq. (7.59), and the other from the rheological force due to the deformation of the melt, represented by Eq. (7.60): T11 (t) = σ11 (t) − σ22 (t) =

t N # 2Gi λi0 2 γ˙ 2 dt 1 exp − 3 λi0 0 aT (t , T ) i=1 1 + cγ˙ λi0 ⎧ ⎫ ⎡ ⎤⎪ / ⎪ ⎪ ⎪ ⎪# ⎪ t 1 + cλ a (t , T ) I (t ) t ⎨ N ⎬ Gi i0 T 2 ⎢ ⎥ exp − dt + ⎣ ⎦ ⎪ λi0 aT (t , T ) λi0 aT (t , T ) −∞ ⎪ t ⎪ ⎪ ⎪ ⎪ ⎩ i=1 ⎭ ' ( × β1(t )2 − β2 (t )2 dt

(7.61)

The total hoop stress T33 (t) of the tubular ﬁlm in the extrudate swell region may be described by T33 (t) = σ33 (t) − σ22 (t) ⎧ ⎫ ⎡ ⎤⎪ / ⎪ ⎪ ⎪ ⎪ t ⎪ t 1 + cλ a (t , T ) I (t ) N ⎨# ⎬ Gi i0 T 2 ⎢ ⎥ = exp − dt ⎣ ⎦ ⎪ λ a (t , T ) λi0 aT (t , T ) −∞ ⎪ t ⎪ ⎪ ⎪ i=1 i0 T ⎪ ⎩ ⎭ ' ( × β3 (t )2 − β2 (t )2 dt where β3 (t ) = a(t)/a(t ).

(7.62)

328

PROCESSING OF THERMOPLASTIC POLYMERS

The swelling of the tubular ﬁlm thickness will occur only when the residual stress given by Eq. (7.58) is greater than the tensile stress exerted by the take-up rolls, which can be calculated from the axial force balance. Referring to the coordinate system deﬁned in Figure 7.12, the axial force balance on the ﬁlm can be written as (Pearson and Petrie 1970a, 1970b) FZ = (2πa cos θ )PL + π(A − a ) p + 2πg 2

Z

2

(ahρ sec θ)dz

(7.63)

z

in which FZ is the tensile force at the freeze line (z = Z), where the diameter and the thickness of a tubular bubble no longer change. Whereas FZ cannot be measured directly, the tensile force at z = L, FL , can be measured. Then, FZ can be calculated from Eq. (7.23). Substituting Eqs. (7.23) and (7.35) into (7.63), we obtain the following expression for the tensile stress of the tubular blown bubble in the stretching zone:

T11 =

1 FL − 2πgρs AH (L − Z) − π A2 − ao 2 p 2πah cos θ t Z − 2πg ahρv1 dt

(7.64)

t

in which Eq. (7.45) was used, and tZ is the ﬂight time for the melt traveling from the die exit (z = 0) to the freeze line (at z = Z). From Eq. (7.64), the tensile stress exerted by the take-up rolls to the molten tubular bubble at the die exit (at z = 0) can be expressed as

T11

0+

=

1 FL − 2πgρs AH(L − Z) − π A2 − ao 2 p 2πao h0 cos θ0 t Z (7.65) (ahρv1 )dt − 2πg 0

If the magnitude of (T11 )0+ given by Eq. (7.65) is less than that of (T11 )0− given by Eq. (7.58), the thickness of the tubular ﬁlm, upon exiting the die, will swell. Therefore, the axial force at time t in the extrudate swell region (zone I in Figure 7.12) consisting of (1) the total force exerted by the residual stress at the die exit plane, (2) the force generated by the deformation of the melt in the extrudate swell region, (3) the gravitational force, and (4) the force exerted by the bubble inﬂation pressure is balanced by the total force exerted by the residual stress being relaxed in the

TUBULAR FILM BLOWING

329

extrudate region:

2πa0 h0 cosθ0

N # 2Gi λi0 2 γ˙ 2 3 i=1 1+cγ˙ λi0

+2πahcosθ

t

⎧ ⎪ ⎪ ⎪ N ⎨#

−∞ ⎪ ⎪ ⎪ i=1

⎩

⎡ Gi ⎢ exp⎣− λi0 aT (t , T )

t

/ 1+cλi0 aT (t , T ) I2 (t )

t

λi0 aT (t , T )

⎫ ⎤⎪ ⎪ ⎪ ⎬ ⎥ dt ⎦ ⎪ ⎪ ⎪ ⎭

t & % × β1 (t )2 −β2 (t )2 dt −2πg ahρv1 dt −π(a 2 −ao 2 )p 0

= 2πahcosθ

t N # 2Gi λi0 2 γ˙ 2 dt 1 exp − 3 λi0 0 aT (t , T ) i=1 1+cγ λi0

(7.66)

in the distance between the die exit (at z = 0) and the position (at z = zm ) at which the thickness of a tubular blown ﬁlm attains a maximum. Thus, the governing equations for the extrudate swell region, Eqs. (7.45)−(7.48), (7.61), (7.62), and (7.66), must be solved simultaneously, subject to the following initial conditions at t = 0: (1) z = 0, (2) a = a0 , (3) h = h0 , (4) T = T0 , (5) θ = θ0 (unknown), (6) T11 = (T11 )0− given by Eq. (7.58), and (7) T33 = 0. The computation must be continued until the stress exerted by the take-up rolls, represented by Eq. (7.65), is balanced by the stress in the melt outside the die, represented by Eq. (7.60), at the position z = zm , the position where the thickness of tubular ﬁlm attains a maximum due to extrudate swell (see Figure 7.12). 7.3.5

Analysis of Tubular Film Blowing in the Stretching Region

If the magnitude of (T11 )0+ given by Eq. (7.65) is greater than that of (T11 )0− given by Eq. (7.58), the tubular blown ﬁlm will be stretched. It is postulated here that the relaxation of the axial stress in the melt, upon exiting the die, makes the thickness of the tubular ﬁlm swell outside the die. When the stress exerted by the take-up rolls becomes greater than the stress in the melt, the tubular ﬁlm will be stretched and the thickness of the tubular ﬁlm will be decreased until the tubular blown bubble reaches the freeze line, z = Z. In the stretching region (zone II in Figure 7.12), the governing equations of the extrudate swell region presented above hold, except for the force balance; namely, in place of Eq. (7.66), Eq. (7.64) becomes the appropriate expression describing the stretching zone. The governing equations, Eqs. (7.45)−(7.48), (7.58), (7.62), and (7.64), must be solved simultaneously subject to the following initial conditions at t = tm : (1) z = zm , (2) a = am , (3) h = hm , (4) T = Tm , (5) θ = θm , (6) T11 = (T11 )m , and (7) T33 = (T33 )m , where subscript m refers to the position at which the thickness of the tubular ﬁlm attains a maximum due to extrudate swell. The computation must be continued until the tubular blown bubble reaches the freeze line at z = Z (see Figure 7.12).

330

PROCESSING OF THERMOPLASTIC POLYMERS

In the analysis presented above, the freeze line is speciﬁed by the position at which a molten tubular blown bubble begins to crystallize, that is, when the calculated temperature of a tubular blown bubble in the stretching zone reaches the melting point of a semicrystalline polymer (e.g., LDPE, LLDPE, HDPE, or PP). It is desirable to determine the position of freeze line as part of the solution of the governing equations, as shown by the analysis of Doufas and McHugh (Doufas and McHugh 2001).

7.3.6

Model Predictions and Comparison with Experiment

To solve the system equations presented above numerically, one must have information on the model parameters, c, aT , λi0 , and Gi , appearing in the constitutive equation, Eq. (7.55). Using the log η versus log γ˙ and log N1 versus log γ˙ plots for an LDPE at three temperatures given in Figure 7.13, Eq. (7.57) was curve-ﬁt to the experimental data using the nonlinear least-squares method (Haw 1984). For this, three values of λi0 at a reference temperature (T0 = 200 ◦ C), λ10 = 1.0 s, λ20 = 0.1 s, and λ30 = 0.01 s, were ﬁrst chosen, and then values of Gi corresponding to the three λi0 values were determined. The value of c = 0.3 was found to give the best curve ﬁt, and the viscous ﬂow activation energy (E) appearing in Eq. (7.53) was determined to be 4.35 × 105 J/mol when using the experimentally determined zero-shear viscosity (η0 ) at 180, 200, and 220 ◦ C (see Figure 7.13) with the aid of Eq. (7.52). The numerical values of Gi determined are: G1 = 3.89 × 103 Pa, G2 = 1.09 × 104 Pa, and G3 = 5.59×104 Pa. The LDPE having these values of λi0 and Gi will be referred to as ﬂuid 1.

Figure 7.13 Plots of log η versus log γ˙ and log N1 versus log γ˙ for an LDPE at three different temperatures (◦ C): 180, () 200, and () 220. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

TUBULAR FILM BLOWING

331

Table 7.2 Values of λi0 and Gi at a reference temperature T0 for four viscoelastic molten polymers employed for the numerical solution of the governing equations

Sample Code

λi0 (s)

Gi (Pa)

Fluid 1

λ10 = 1.0 λ20 = 0.1 λ30 = 0.01

G1 = 3.89 × 103 G2 = 1.09 × 104 G3 = 5.59 × 104

Fluid 2

λ10 = 0.2 λ20 = 0.02 λ30 = 0.002

G1 = 1.95 × 104 G2 = 5.45 × 104 G3 = 2.79 × 105

Fluid 3

λ10 = 5.0 λ20 = 0.5 λ30 = 0.05

G1 = 0.78 × 103 G2 = 0.22 × 104 G3 = 1.12 × 104

Fluid 4

λ10 = 5.0 λ20 = 0.5 λ30 = 0.05

G1 = 3.89 × 103 G2 = 1.09 × 104 G3 = 5.59 × 104

Other parameters: c = 0.3; E = 4.35 × 105 J/mol. Based on the doctoral dissertation of Jungrim Haw, 1984.

In order to investigate the effect of the rheological parameters, λi0 and Gi , appearing in Eq. (7.55), on the tubular ﬁlm blowing characteristics, Haw (1984) varied the values of λi0 and Gi for ﬂuid 1 to generate three new hypothetical ﬂuids, ﬂuid 2, ﬂuid 3, and ﬂuid 4. The numerical values thus determined for λi0 and Gi for these ﬂuids, including for ﬂuid 1, are summarized in Table 7.2. Figure 7.14 compares log η versus log γ˙ and log N1 versus log γ˙ plots for ﬂuid 1 with those for ﬂuid 2 (see Table 7.2 for the numerical values of λi0 and Gi ). In Figure 7.14, we observe that the log N1 versus log γ˙ plot for ﬂuid 1 and the log N1 versus log γ˙ plot for ﬂuid 2 cross each other as γ˙ is increased, while the log η versus log γ˙ plot for ﬂuid 2 lies above that for ﬂuid 1 over the entire range of γ˙ considered. Speciﬁcally, at γ˙ = 70 s−1 , the value of N1 of ﬂuid 2 is two times larger than that of ﬂuid 1. Thus, when each of the two polymers is extruded at γ˙ = 70 s−1 through an annular die, at the exit plane of the die ﬂuid 2 is expected to have greater amounts of residual stress, deﬁned by Eq. (7.59), than ﬂuid 1. Under such circumstances, ﬂuid 2, upon exiting the die, is expected to have a larger extrudate swell than ﬂuid 1, as long as (T11 )0+ given by Eq. (7.65) is less than (T11 )0− given by Eq. (7.58). Figure 7.15 gives predicted proﬁles of dimensionless ﬁlm thickness (h/ho ) for ﬂuid 1 and ﬂuid 2. The die dimensions, the speciﬁc heat capacity and density of LDPE, heat transfer parameters, and processing conditions employed for the computations are given in Table 7.3. It is seen in Figure 7.15 that the ﬁlm thickness of the tubular blown bubble, upon exiting the die, initially increases and then decreases as the tubular blown bubble travels upward in the MD. This prediction indicates that under the processing conditions chosen, the tension from the take-up rolls was not sufﬁciently large to be transmitted all the way to the exit plane of the die, and thus the extrudate swelled. Note that the ﬁlm thickness of ﬂuid 2 initially decreases at a slower rate and then later at a faster rate than the ﬁlm thickness of ﬂuid 1. Eventually, ﬂuid 2 produces thinner ﬁlms than ﬂuid 1.

332

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 7.14 Plots of log η versus log γ˙ and log N1 versus log γ˙ for two viscoelastic polymer melts, ﬂuid 1 (solid line) and ﬂuid 2 (dashed line), employed for the simulation of tubular ﬁlm blowing. The numerical values of a set of λi0 (i = 1, 2, 3) and Gi (i = 1, 2, 3) at a reference temperature T0 for the respective ﬂuids are given in Table 7.2. These values were obtained, via the nonlinear least-squares method, by curve ﬁtting Eq. (7.57) to the log η versus log γ˙ and log N1 versus log γ˙ plots. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

Figure 7.16 gives predicted proﬁles of dimensionless maximum ﬁlm thickness (h/ ho )max of two tubular blown bubbles as a function of applied tension FL from the take-up rolls. It can be seen that upon exiting the die, the extrudate swell of ﬂuid 2 is greater than that of ﬂuid 1. It is interesting to observe in Figure 7.16 that the maximum extrudate swell decreases rapidly with increasing FL , indicating that extrudate swell can completely be suppressed when the applied tension from the take-up rolls becomes sufﬁciently large. Figure 7.17 gives the effect of applied tension FL from the take-up rolls on the dimensionless position zm /ao , at which the maximum extrudate swell of tubular blown bubble occurs. It is seen in Figure 7.17 that a larger tension is required to suppress the extrudate swell of ﬂuid 2 than to suppress the extrudate swell of ﬂuid 1. Figure 7.18 compares predicted dimensionless bubble radius (a/ao ) of two tubular blown bubbles, showing that the radius of tubular blown bubble for ﬂuid 2 increases in the MD faster than that for ﬂuid 1. Note that the larger the radius of a tubular blown bubble, the thinner the ﬁlm thickness will be. We then observe that the predicted proﬁles of bubble radius given in Figure 7.18 are consistent with the predicted proﬁles of ﬁlm thickness given in Figure 7.15. Figure 7.19 compares predicted temperature proﬁles of two tubular blown bubbles in the MD, showing that the temperature of ﬂuid 2 decreases faster in the MD than that of ﬂuid 1. Again, this temperature prediction is consistent with the predicted proﬁles of ﬁlm thickness (Figure 7.15), in that the thinner the ﬁlm thickness of a tubular blown bubble, the faster the cooling of the bubble will be.

Figure 7.15 Predicted proﬁles of dimensionless ﬁlm thickness h/ ho of tubular blown bubble in the axial direction for ﬂuid 1 (solid line) and ﬂuid 2 (dashed line). The numerical values of the parameters λi0 and Gi for the respective ﬂuids are given in Table 7.2. The dimensions of the die, physical/thermal properties of the ﬂuids, heat transfer parameters, and processing conditions employed for the numerical computations are given in Table 7.3. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

Table 7.3 Die dimensions, speciﬁc heat capacity and density of LDPE, heat transfer parameters, and processing conditions employed for the numerical solution of the governing equations

(a) Die Dimensions ai = 1.27 cm; ao − ai = 0.078 cm (b) Speciﬁc Heat Capacity of LDPE cp0 = 2.73 × 103 J/(kg K); D1 = 1.45; D2 = 0.271 × 10−2 K−1 (c) Density of LDPE ρ0 = 747 kg/m3 ; E1 = 0.993; E2 = 0.732 × 10−5 K−1 (d) Heat Transfer Parameters U = 69.6 W/(m2 K); ε = 0.3 (e) Processing Conditions m ˙ = 0.99 kg/h; T0 = 200 ◦ C; p = 24.5 Pa; FL = 0.94 N; Qair = 1.19 × 103 cm3 /s; Ta = 27 ◦ C; L = 1.65 m Based on the doctoral dissertation of Jungrim Haw, 1984.

333

Figure 7.16 Predicted proﬁles of dimensionless maximum ﬁlm thickness (h/ ho )max of two tubular blown bubbles as a function of applied tension FL for ﬂuid 1 (solid line) and ﬂuid 2 (dashed line). The numerical values of the parameters λi0 and Gi for the respective ﬂuids are

given in Table 7.2. The dimensions of the die, physical/thermal properties of the ﬂuids, heat transfer parameters, and processing conditions employed for the numerical computations are given in Table 7.3. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

Figure 7.17 Predicted proﬁles of dimensionless axial position (zm /ao ), at which the maximum extrudate swell of tubular blown bubble occurs, as a function of applied tension FL for ﬂuid 1 (solid line) and ﬂuid 2 (dashed line). The numerical values of the parameters λi0 and Gi for the respective ﬂuids are given in Table 7.2. The dimensions of the die, physical/thermal properties of the ﬂuids, heat transfer parameters, and processing conditions employed for the numerical computations are given in Table 7.3. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

334

Figure 7.18 Predicted proﬁles of dimensionless bubble radius a/ao of tubular blown bubble

along the axial direction for ﬂuid 1 (solid line) and ﬂuid 2 (dashed line). The numerical values of the parameters λi0 and Gi for the respective ﬂuids are given in Table 7.2. The dimensions of the die, physical/thermal properties of the ﬂuids, heat transfer parameters, and processing conditions employed for the numerical computations are given in Table 7.3. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

Figure 7.19 Predicted temperature proﬁles of two tubular blown bubbles in the axial direction

for ﬂuid 1 (solid line) and ﬂuid 2 (dotted line). The numerical values of the parameters λi0 and Gi for the respective ﬂuids are given in Table 7.2. The dimensions of the die, physical/thermal properties of the ﬂuids, heat transfer parameters, and processing conditions employed for the numerical computations are given in Table 7.3. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.) 335

336

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 7.20 Plots of log η versus log γ˙ and log N1 versus log γ˙ for two viscoelastic polymer melts, ﬂuid 1 (solid line) and ﬂuid 3 (dashed line), employed in the simulation of tubular ﬁlm blowing. The numerical values of a set of λi0 (i = 1, 2, 3) and Gi (i = 1, 2, 3) at a reference temperature T0 for the respective ﬂuids are given in Table 7.3. These values were obtained, via nonlinear least-squares methods, by curve-ﬁtting Eq. (7.57) to the log η versus log γ˙ and log N1 versus log γ˙ plots. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

Figure 7.20 compares log η versus log γ˙ and log N1 versus log γ˙ plots for ﬂuid 1 with those for ﬂuid 3 (see Table 7.2 for the numerical values of λi0 and Gi ). In Figure 7.20, we observe that the log η versus log γ˙ plot for ﬂuid 3 lies below that for ﬂuid 1 over the entire range of γ˙ considered, while the N1 of ﬂuid 3 is greater at low γ˙ , but less at high γ˙ than that of ﬂuid 1. Notice in Figure 7.20 that at γ˙ = 70 s−1 for instance, the value of N1 for ﬂuid 3 is about one-half the value of N1 for ﬂuid 1. Therefore, it is expected that ﬂuid 3, upon exiting the die at γ˙ = 70 s−1 , will have less extrudate swell than ﬂuid 1. This indeed is borne out to be the case, as can be seen in Figure 7.21, which shows predicted proﬁles of dimensionless maximum ﬁlm thickness (h/ ho )max , where ho is the die opening (ao − ai ), of two tubular blown bubbles as a function of applied tension FL . Since the extrudate swell of ﬂuid 3 is less than that of ﬂuid 1, the maximum extrudate swell of ﬂuid 3 is expected to occur at a shorter distance from the die exit than the maximum extrudate swell of ﬂuid 1. This expectation is again borne out to be the case, as can be seen in Figure 7.22, showing predicted proﬁles of zm /ao of two tubular blown bubbles as a function of applied tension FL . Figure 7.23 compares log η versus log γ˙ and log N1 versus log γ˙ plots for ﬂuid 1 with those for ﬂuid 4 (see Table 7.2 for the numerical values of λi0 and Gi ). In Figure 7.23, we observe that the log N1 versus log γ˙ plot for ﬂuid 4 lies above that for ﬂuid 1 over the entire range of γ˙ considered, while at γ˙ < 150 s−1 the viscosity of ﬂuid 4 is higher than the viscosity of ﬂuid 1, but at γ˙ > 150 s−1 the viscosity

Figure 7.21 Predicted proﬁles of dimensionless maximum ﬁlm thickness (h/ ho )max of tubular blown bubble as a function of applied tension FL for ﬂuid 1 (solid line) and ﬂuid 3 (dashed line). The numerical values of the parameters λi0 and Gi for the respective ﬂuids are given in Table 7.2. The dimensions of the die, physical/thermal properties of the ﬂuids, heat transfer parameters, and processing conditions employed for the numerical computations are given in Table 7.3. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

Figure 7.22 Predicted proﬁles of dimensionless axial position (zm /ao ) at which the maximum extrudate swell of tubular blown bubble occurs, as a function of applied FL for ﬂuid 1 (solid line) and ﬂuid 3 (dashed line). The numerical values of the parameters λi0 and Gi for the respective ﬂuids are given in Table 7.2. The dimensions of the die, physical/thermal properties of the ﬂuids, heat transfer parameters, and processing conditions employed for the numerical computations are given in Table 7.3. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

337

338

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 7.23 Plots of log η versus log γ˙ and log N1 versus log γ˙ for two viscoelastic polymer melts, ﬂuid 1 (solid line) and ﬂuid 4 (dashed line), employed in the simulation of tubular ﬁlm blowing. The numerical values of a set of λi0 (i = 1, 2, 3) and Gi (i = 1, 2, 3) at a reference temperature T0 for the respective ﬂuids are given in Table 7.2. These values were obtained, via nonlinear least-squares methods, by curve-ﬁtting Eq. (7.57) to the log η versus log γ˙ and log N1 versus log γ˙ plots. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

of ﬂuid 4 is lower than the viscosity of ﬂuid 1. Speciﬁcally, at γ˙ = 70 s−1 the value of N1 for ﬂuid 4 is about three times greater than that for ﬂuid 1. Under such circumstances, ﬂuid 4, upon exiting the die, is expected to have a larger extrudate swell than ﬂuid 1, as long as (T11 )0+ given by Eq. (7.65) is less than (T11 )0− given by Eq. (7.58). Figure 7.24 gives predicted proﬁles of dimensionless maximum ﬁlm thickness (h/ ho )max of two tubular blown bubbles as a function of applied tension FL , showing that the extrudate swell of ﬂuid 4 is much greater than that of ﬂuid 1. Owing to the very large values of N1 for ﬂuid 4 (see Figure 7.23), at the exit plane of the die a larger residual stress is expected in ﬂuid 4 than in ﬂuid 1. Under such circumstances, it is reasonable to speculate that a very large applied tension from the take-up rolls is needed to suppress the extrudate swell of a tubular blown bubble. In practice, however, an excessively large tension from the take-up rolls may break the tubular melt ﬁlm near the die exit, suggesting that extrudate swell is inevitable in the tubular ﬁlm blowing of very highly elastic molten polymers. Figure 7.25 shows the effect of applied tension FL from the take-up rolls on the dimensionless position zm /ao of two tubular blown bubbles. It is seen in Figure 7.25 that initially (near the die exit) the value of zm /ao of ﬂuid 4 decreases rapidly with increasing FL , and then very slowly with a further increase in FL from 4 to 12 N, while the value of zm /ao for ﬂuid 1 decreases rapidly with increasing FL , approaching zero when FL ≈ 5 N . In other words, the distance

Figure 7.24 Predicted proﬁles of dimensionless maximum ﬁlm thickness (h/ ho )max of tubular blown bubble as a function of applied tension FL for ﬂuid 1 (solid line) and ﬂuid 4 (dashed line). The numerical values of the parameters λi0 and Gi for the respective ﬂuids are given in Table 7.2. The dimensions of the die, physical/thermal properties of the ﬂuids, heat transfer parameters, and processing conditions employed for the numerical computations are given in Table 7.3. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

Figure 7.25 Predicted proﬁles of dimensionless axial position (zm /ao ) at which the maximum extrudate swell of tubular blown bubble occurs as a function of applied tension FL for ﬂuid 1 (solid line) and ﬂuid 4 (dashed line). The numerical values of the parameters λi0 and Gi for the respective ﬂuids are given in Table 7.2. The dimensions of the die, physical/thermal properties of the ﬂuids, heat transfer parameters, and processing conditions employed for the numerical computations are given in Table 7.3. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

339

340

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 7.26 Comparison of predicted dimensionless bubble radius (solid line) with experiment

() in the tubular ﬁlm blowing of an LDPE. The numerical values of the parameters λi0 and Gi for the LDPE (ﬂuid 1) are given in Table 7.2. The dimensions of the die, physical/thermal properties of the ﬂuids, heat transfer parameters, and processing conditions employed for the numerical computations are given in Table 7.3. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

through which the extrudate swells is indeed quite long for the very elastic ﬂuid 4, compared with that for the less elastic ﬂuid 1. Figure 7.26 compares predicted bubble shape with experiment, and Figure 7.27 compares predicted temperature proﬁle with experiment, in the tubular ﬁlm blowing of an LDPE. Within experimental uncertainties, the agreement between prediction and experiment seems reasonable.

Figure 7.27 Comparison of predicted temperature proﬁle (solid line) with experiment () in the tubular ﬁlm blowing of an LDPE. The numerical values of the parameters λi0 and Gi for the LDPE (ﬂuid 1) are given in Table 7.2. The dimensions of the die, physical/thermal properties of the ﬂuids, heat transfer parameters, and processing conditions employed for the numerical computations are given in Table 7.3. (Reprinted from Haw, doctoral dissertation of Jungrim Haw. Copyright © 1984, with permission from Polytechnic University.)

TUBULAR FILM BLOWING

7.4

341

Tubular Film Blowability

In Chapter 6 of Volume 1 we showed that in steady-state shear ﬂow LDPE exhibits a stronger shear-thinning behavior than LLDPE (see Figure 6.35 in Volume 1) and LDPE is more elastic than LLDPE (see Figure 6.36 in Volume 1). We showed that in transient uniaxial elongational ﬂow, LDPE exhibits strong strain-hardening behavior compared with LLDPE (see Figure 6.37 in Volume 1), and speculated that the strong strain-hardening behavior of LDPE limits its drawability, while the less strain-hardening behavior of LLDPE gives rise to good drawability. We then inferred the “tubular ﬁlm blowability” of LDPE from the transient uniaxial elongational ﬂow behavior in relation to bubble stability. Namely, the stronger strain-hardening behavior of LDPE gives rise to better bubble stability than would be achieved from LLDPE. We speculated that the presence of a certain amount of long branching in LDPE may be necessary to have good bubble stability in tubular ﬁlm blowing. In tubular ﬁlm blowing operations, one is always interested in controlling the ultimate ﬁlm thickness. This is understandable because, for a given grade of polymer, the thinner the ﬁlm, the greater the area it can cover and hence the greater the proﬁtability will be. In the tubular ﬁlm industry, one often introduces the term “draw-down ratio” (DDR), deﬁned by the ratio of die opening to ﬁlm thickness: DDR = ao − ai /H

(7.67)

Substitution of Eq. (7.28) into (7.67) yields DDR =

2ao ao + ai

ρm ρs

(BUR)(TUR)

(7.68)

In the same way that we were interested in “spinnability” in ﬁber spinning, as discussed in Chapter 6, we are interested in “ﬁlm blowability” in tubular ﬁlm blowing. Speciﬁcally, we are interested in understanding the scientiﬁc reason(s) why only certain polymers are good for ﬁlm blowing, while others are not. For simplicity, let us deﬁne here “tubular ﬁlm blowability” by the maximum draw-down ratio, (DDR)max , at which the tubular blown bubble breaks by cohesive failure. Note that Eq. (7.68) deﬁnes DDR in terms of BUR and TUR. Let us assume that the limited experimental studies reported to date, claiming that LLDPE gives rise to better mechanical properties than LDPE, are valid. Fundamental questions may still be raised. What is the scientiﬁc basis for such a claim? Is it possible for one to produce LDPE tubular blown ﬁlms having better mechanical properties than LLDPE tubular blown ﬁlms by optimizing processing conditions? This question is reasonable, because processing conditions can greatly inﬂuence the mechanical and optical properties of blown ﬁlms. This is precisely why the question of tubular ﬁlm blowability is very important. As the TUR and/or BUR are increased during a tubular ﬁlm blowing operation, the tubular blown bubble will eventually break when the tension from the take-up rolls exceeds a certain critical value, which the tubular blown bubble can no longer withstand. Such a critical tension, which may be termed “ultimate melt strength,” must be inherent in the molecular characteristics of polymers. Note that as

342

PROCESSING OF THERMOPLASTIC POLYMERS

the TUR and/or BUR are increased, the tubular blown bubble will be stretched, giving rise to thinner ﬁlms. If a polymer melt exhibits strain hardening, the tensile stress of the melt will increase rapidly with increasing stretch (elongation) rate and thus it will soon reach the ultimate melt strength, at which the tubular blown bubble breaks. Conversely, if a polymer melt exhibits strain softening, the tensile stress of the melt will decrease with increasing elongation rate, and thus the tubular blown bubble will be stretched further until the tensile strength of the melt reaches the ultimate melt strength. Let us ﬁrst consider the rheological properties of some LDPEs and LLDPEs and then their ﬁlm blowability. Figure 7.28 gives log η versus log γ˙ and log N1 versus log γ˙ plots for three blown-ﬁlm grade LDPEs, whose molecular characteristics are given in Table 7.1. Figure 7.29 gives log η versus log γ˙ and log N1 and log γ˙ plots for three blown-ﬁlm grade LLDPEs whose molecular characteristics are given in Table 7.4. Comparison of Figure 7.28 with Figure 7.29 indicates that LLDPEs are less shear thinning than LDPEs, and at high γ˙ , shear-rate dependence of N1 is greater for LLDPEs than for LDPEs. In Figure 6.35 given in Chapter 6 of Volume 1 we showed that an LDPE exhibits greater shear-thinning behavior (with a power-law index n of 0.45) than an LLDPE (with n = 0.62). One may be tempted to use information on the steady-state uniaxial elongational viscosity of a polymer to assess its tubular ﬁlm blowability. The idea is very appealing because tubular ﬁlm blowing is associated with elongational ﬂow. As discussed in Chapter 5 of Volume 1, steady-state uniaxial elongational viscosity of a molten polymer can be measured under controlled, isothermal experimental conditions. It is doubtful, however, that such information would be useful to assess tubular ﬁlm blowability for the following reasons. First, tubular ﬁlm blowing is associated with

Figure 7.28 Plots of log η versus log γ˙ and log N1 versus log γ˙ at 200 ◦ C for three LDPEs: (, 䊉) LDPE-A, (, ) LDPE-B, and (, ) LDPE-C, in which open symbols represent data

taken with a cone-and-plate rheometer and ﬁlled symbols represent data taken by the exit pressure method using a capillary rheometer (see Chapter 5 of Volume 1). (Reprinted from Han and Kwack, Journal of Applied Polymer Science 28:3399. Copyright © 1983, with permission from John Wiley & Sons.)

TUBULAR FILM BLOWING

343

Figure 7.29 Plots of log η versus log γ˙ and log N1 versus log γ˙ at 240 ◦ C for three LLDPEs: (, 䊉) LLDPE-A, (, ) LLDPE-B, and (, ) LLDPE-C, in which open symbols represent

data taken with a cone-and-plate rheometer and ﬁlled symbols represent data taken by the exit pressure method using a capillary rheometer (see Chapter 5 of Volume 1). (Reprinted from Kwack and Han, Journal of Applied Polymer Science 28:3419. Copyright © 1983, with permission from John Wiley & Sons.)

nonuniform (unequal) biaxial elongational ﬂow, that is, the elongation rates in the axial and transverse directions are different, d11 = d33 (see Eq. (7.2)). Second, as in melt spinning (discussed in Chapter 6), in tubular ﬁlm blowing the elongation rates in the axial and transverse directions would vary with the direction of stretching (in MD and TD, respectively), that is, no steady-state elongation rates prevail in tubular ﬁlm blowing. Third, tubular ﬁlm blowing is operated under nonisothermal conditions. Very few experimental studies have been reported on the measurements of nonuniform elongational viscosity of molten polymers under nonisothermal conditions. Referring to Tables 7.1 and 7.4, LLDPEs have a relatively narrow MWD compared with LDPEs. It can easily be surmised that a change in molecular characteristics brings

Table 7.4 Molecular characteristics of three LLDPEs

Sample Code LLDPE-A LLDPE-B LLDPE-C

Density (g/cm3 )

Mw

Mw /Mn

0.920 0.921 0.919

2.5 × 105 2.2 × 105 2.8 × 105

4.40 3.95 3.70

Reprinted from Kwack and Han, Journal of Applied Polymer Science 28:3419. Copyright © 1983, with permission from John Wiley & Sons.

344

PROCESSING OF THERMOPLASTIC POLYMERS

about different rheological properties, which can result in different physical, mechanical, and/or optical properties in the tubular blown ﬁlms. It is generally understood that LLDPE ﬁlms have higher tensile strength and elongation, outstanding ﬁlm puncture resistance, greater stiffness, excellent environmental stress crack resistance, and outstanding draw-down characteristics, compared with LDPEs. However, only a few research groups (Guichon et al. 2003; Han and Kwack 1983; Kwack and Han 1983) have reported on the differences in tubular ﬁlm blowing characteristics between LLDPE and LDPE. In the tubular ﬁlm blowing operation, the cooling rate of the tubular blown bubble, upon exiting from the die, greatly inﬂuences both TUR and BUR, thus the tubular ﬁlm blowing characteristics of polymers. Hence, it is of practical interest to determine how the cooling rate of the tubular blown bubble, upon exiting a die, might inﬂuence (DDR)max , at which the tubular blown bubble breaks. Since the tubular blown bubble always breaks in the zone below the freeze line, it will break when the tension from the take-up rolls exceeds the ultimate melt strength of a polymer. Figure 7.30 shows the effect of cooling air ﬂow rate on maximum take-up ratio (TUR)max at a ﬁxed BUR for three LDPEs having different molecular weights and different amounts of long-chain branching (see Table 7.1), and Figure 7.31 shows the effect of cooling air ﬂow rate on (DDR)max for the same LDPEs. It is seen in Figures 7.30 and 7.31 that both (TUR)max and (DDR)max decrease with increasing cooling air ﬂow rate, that is, as the tubular blown bubble cools faster. Faster cooling of a tubular blown bubble would not allow sufﬁcient stretching of the polymer molecules, giving rise to lower tensile strengths of the blown ﬁlms. It is of interest to note in Figures 7.30 and 7.31 that LDPE-C gives higher (TUR)max and (DDR)max than the other two polymers. The above observations seem to suggest that the narrower the MWD and the lower the amount of long-chain branching of LDPE, the greater the tubular ﬁlm blowabiltiy might be.

Figure 7.30 Plots of (TUR)max

versus cooling air ﬂow rate at a BUR of 3.5 for () LDPE-A, () LDPE-B, and () LDPE-C. (Reprinted from Han and Kwack, Journal of Applied Polymer Science 28:3399. Copyright © 1983, with permission from John Wiley & Sons.)

TUBULAR FILM BLOWING

345

Figure 7.31 Plots of (DDR)max versus cooling air ﬂow rate at a BUR of 3.5 for () LEPE-A, () LDPE-B, and () LDPE-C. (Reprinted from Han and Kwack, Journal of Applied Polymer Science 28:3399. Copyright © 1983, with permission from John Wiley & Sons.)

In tubular ﬁlm blowing experiments, Kwack and Han (1983) increased TUR stepwise until the tubular blown bubble broke. Within the limits of their equipment, which was capable of handling (TUR)max of about 80, they could not break the tubular blown bubble of LLDPEs, while they could break the tubular blown bubble of LDPEs at a (TUR)max of about 60. Due to the greater strechability (drawability) of the LLDPEs employed, the tubular blown bubble was stretched continuously without experiencing cohesive failure; that is, the tension from the take-up rolls could not bring about cohesive failure of the tubular blown bubble of the LLDPE. On the basis of these observations, they concluded that the stretching operation caused the stress to build up much faster in the LDPE than in the LLDPE employed, thus reaching the ultimate melt strength of the LDPE at a much lower strain rate compared with the LLDPE. Hence, the LLDPE permitted greater (TUR)max and (DDR)max (hence thinner ﬁlms) than the LDPE. Figure 7.32 gives plots of the ratio T11F /T33F versus BUR for three LDPEs and three LLDPEs. It is seen clearly in Figure 7.32 that under comparable processing conditions, a more uniform tensile strength in both the MD and the TD is achievable with LLDPEs than with LDPEs. It should be pointed out that one does not have to employ identical (comparable) processing conditions to determine whether or not LLDPE can produce superior mechanical properties in tubular blown ﬁlms than LDPE, or vice versa, because LLDPE and LDPE may have different optimum processing conditions, each producing the maximum attainable mechanical properties. Crystallization takes place at the freeze line, where the tubular blown bubble begins to solidify and form

346

PROCESSING OF THERMOPLASTIC POLYMERS Figure 7.32 Comparison of the

plots of T11F /T33F versus BUR for three LDPEs with those for three LLDPEs at various TURs: LDPE-A at three different TURs: () 4.4, () 9.9, and () 15.6; LDPE-B at three different TURs: (䊉) 4.4, () 9.9, and () 15.6; LDPE-C at three different TURs: (䊋) 4.4, (䊕) 9.9, and (䊑) 15.6; LLDPE-A at three different TURs: (䊋) 7.6, (䊕) 11.0, and (䊑) 14.3, LLDPE-B at three different TURs: () 7.6, () 11.0, and () 14.3, and LLDPE-C at three different TURs: (䊎) 7.6, (䊖) 11.0, and (䊒) 14.3. (Reprinted from Kwack and Han, Journal of Applied Polymer Science 28:3419. Copyright © 1983, with permission from John Wiley & Sons.)

a constant bubble diameter under biaxial stretching. Therefore, one must consider the molecular and structural parameters of polymers in order to have a better understanding of structure–processing–property relationships in tubular ﬁlm blowing. The molecular and structural parameters that must be considered include the molecular weight, the MWD, the amount of long-chain branching, the orientation of the amorphous phase, the degree of crystallinity, the distribution of crystalline axis orientation, and the morphology evolution during cooling. Systematic investigation in this area is much needed in the future.

7.5

Summary

Technologically speaking, tubular ﬁlm blowing is an attractive processing technique that, in principle, can be used to produce biaxially stretched ﬁlms with an equal mechanical strength in both the MD and the TD. In this chapter, we ﬁrst presented how processing variables (e.g., blow-up ratio, take-up ratio, and bubble inﬂation pressure) might inﬂuence the tubular ﬁlm blowing characteristics in terms of the tensile stress in the machine and transverse directions, T11F and T33F , at the freeze line. Then, we presented an analysis of tubular ﬁlm blowing, including extrudate swell just outside the die. In doing so, however, we did not include crystallization in the analysis. In view of the fact that the majority of polymers in use for tubular ﬁlm blowing are semicrystalline,

TUBULAR FILM BLOWING

347

a better understanding of crystallization during tubular ﬁlm blowing is very important. It is well understood that the crystalline structure in a tubular blown ﬁlm greatly inﬂuences its mechanical/physical properties. Liu et al. (1995b) included crystallization into a system of equations describing the dynamics of tubular ﬁlm blowing. In that approach, the effect of crystallization was incorporated into the temperature-dependent viscosity that appears in the system equations. Doufas and McHugh (2001) applied their structure-based model (Doufas et al. 1999, 2000), which had been developed to simulate high-speed melt spinning (see Chapter 6), to describe tubular ﬁlm blowing, including ﬂow-induced (stress-induced) crystallization. In their analysis, for simplicity Doufas and McHugh (2001) ignored extrudate swell, while dealing with a viscoelastic ﬂuid and also the curvature of the tubular blown bubble in the machine direction; thus, a “quasi cylindrical” shape of tubular blown bubble was considered. However, to date few experimental studies have been reported on stress-induced crystallization in tubular ﬁlm blowing. In tubular ﬁlm blowing, some polymers have a much greater sensitivity than others to small variations in nip-roll speed or in temperature of the air surrounding the traveling tubular bubble, giving rise to nonuniform bubble diameter and, consequently, nonuniform ﬁlm thickness. Such ﬂow behavior is referred to as “bubble instability” in tubular ﬁlm blowing. In the mid 1970s, Han and coworkers (Han and Park 1975c; Han and Shetty 1977) conducted seminal experimental investigations of bubble instability in tubular ﬁlm blowing of LDPE and HDPE. They observed that LDPE was much more susceptible to bubble instability than HDPE. However, they were able to stabilize the tubular blown bubble by lowering the melt temperature inside the die, and thereby increasing the melt viscosity of LDPE exiting the annular die. A decade later, using several polymers White and coworkers (Kanai and White 1984; Minoshima and White 1986) also conducted experiments on bubble instability in tubular ﬁlm blowing, and observed different types of bubble instability. They interpreted the different types of bubble instability observed in terms of the uniaxial elongational viscosity of the polymers under isothermal conditions. Such an interpretation is not warranted in the rigorous sense because the kinematics in tubular ﬁlm blowing are associated with unequal biaxial elongational ﬂow under nonsiothermal conditions. On the theoretical side, Yeow (1976) carried out a stability analysis of isothermal tubular ﬁlm blowing by considering axisymmetric disturbances in a Newtonian ﬂuid. A decade later, Cain and Denn (1988) performed a linear stability analysis of isothermal and also nonisothermal tubular ﬁlm blowing operations of a Newtonian ﬂuid, and also a viscoelastic ﬂuid represented by the upper convected Maxwell model. They observed some similarities and dissimilarities between the draw resonance in ﬁber spinning and the bubble instability in tubular ﬁlm blowing. They concluded that the additional processing variable that is associated with tubular ﬁlm blowing, the hoop stress balance, and hence bubble inﬂation pressure, made bubble (process) instability predictions not observed in ﬁber spinning. It is, however, granted that a rigorous stability analysis of nonisothermal tubular ﬁlm blowing of viscoelastic ﬂuids is a very difﬁcult and challenging subject to deal with. It is expected that predictions of a theoretical analysis of bubble instability in tubular ﬁlm blowing would depend on the constitutive equations chosen. For more than a decade, few theoretical studies have been reported on bubble instability in tubular ﬁlm blowing. This is a subject that requires further attention in the future.

348

PROCESSING OF THERMOPLASTIC POLYMERS

Problems Problem 7.1

Predict the bubble shape in isothermal tubular ﬁlm blowing using the Meister model given by Eq. (7.49). Problem 7.2

When c = 0 in the Meister model, Eq. (7.49) reduces to σ=

N #G

t

−∞

i=1

i

λi

exp −

t t

dt λi

'

( C−1 t − I dt t

(7P.1)

Predict the bubble shape in nonisothermal tubular ﬁlm blowing using the above constitutive equation.

Notes 1. Use was made of d2 a

dθ dz

(7N.1)

d d = v1 cos θ dt dz

(7N.2)

dz2

= sec2 θ

from da/dz = tan θ (see Figure 7.12). 2. Use was made of

dξ1 dh dξ1 dh dz dh = v1 , (2) = v1 = , and (3) = cos θ dt dt dξ1 dt dξ1 dξ1 d d (see Figure 7.12) or = cos θ . dξ1 dz which follows from: (1)

3. The constitutive equation used by Wagner (1976b) has the form t dt G exp − C−1 t dt σ= t −∞ λ t t λ t t

(7N.3)

where G is the elastic modulus, λ(t) is the relaxation time at time t, and C−1 t (t ) is the relative Finger tensor deﬁned by Eq. (7.50). Wagner used the following relationship

η0 (T ) = λ(T )G

(7N.4)

349

TUBULAR FILM BLOWING

to determine the temperature dependence of λ(T ), in which G was assumed to be independent of temperature and the zero-shear viscosity η0 (T ) was assumed to follow the Arrhenius relationship:

1 E 1 − η0 (T ) = η0 (T ) exp R T T0

(7N.5)

4. The rate-of-strain tensor d corresponding to the tubular ﬁlm blowing process can be written as - 1 dv1 0 0 - v dz - 1 1 dh d = v1 cos θ - 0 0 h dz 1 da (7N.6) 0 - 0 z dz where v1 is deﬁned by Eq. (7.7).

References Aggarwal SL, Tilley GP, Sweeting OJ (1959). J. Appl. Polym. Sci. 1:91. Ashizawa H, Spruiell JE, White JL (1984). Polym. Eng. Sci. 24:1035. Bafna A, Beaucage G, Mirabella F, Skillas G, Sukumaran S (2001). J. Polym. Sci., Polym. Phys. Ed. 39:2923. Beret S, Jones RL, Jones TM, Jenkins JM (1986). U.S. Patent 4588790. Butler MF, Donald AM (1998). J. Appl. Polym. Sci. 67:321. Cain JJ, Denn MM (1988). Polym. Eng. Sci. 28:1527. Cao T, Campbell GA (1990). AIChE J. 36:420. Choi KJ, Spruiell JE, White JL (1980). J. Appl. Polym. Sci. 25:2777. Choi KJ, Spruiell JE, White JL (1982). J. Polym. Sci., Polym. Phys. Ed. 20:27. Clark ES, Garber CA (1971). Intern. J. Polym. Mater. 1:31. Desper CR (1969). J. Appl. Polym. Sci. 13:169. Doufas AK, Dairanieh IS, McHugh AJ (1999). J. Rheol. 43:85. Doufas AK, McHugh AJ (2001). J. Rheol. 45:1085. Doufas AK, McHugh AJ, Miller C (2000). J. Non-Newtonian Fluid Mech. 92:27. Dowd LE (1972). SPE J. 28:22. Farber R, Dealy J (1974). Polym. Eng. Sci. 14:435. Fawcett EW, Gibson RO, Perrin MW, Paton JG, Williams EG (1937). British Patent 471590. Garber CA, Clark ES (1970). J. Macromol. Sci. Phys. B4(3):499. Ghaneh-Fard A, Carreau PJ, Laﬂeur PG (1997a). Polym. Eng. Sci. 37:1148. Ghaneh-Fard A, Carreau PJ, Laﬂeur PG (1997b). Intern. Polym. Processing 12:136. Gregory BH (1969). Plastics 24:165. Guichon O, Séguéla R, David L, Vigier G (2003). J. Polym. Sci., Polym. Phys. Ed. 42:327. Gupta A, Simpson DM, Harrison IR (1993). J. Appl. Polym. Sci. 50:2085. Gupta RK (1980). A New Non-Isothermal Rheological Constitutive Equation and Its Application to Industrial Film Blowing Processes, Doctoral Dissertation, University of Delaware, Newark, Delaware. Gupta RK, Metzner AB, Wissbrun KF (1982). Polym. Eng. Sci. 22:172.

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8

Injection Molding

8.1

Introduction

Injection molding is one of the oldest polymer processing operations used to produce goods from thermoplastic polymers. Today, almost all commercial injection molding machines have a reciprocating single screw for softening (or melting) under heat a thermoplastic polymer, and polymer melt is then injected into an empty mold cavity, as schematically shown in Figure 8.1. In the injection molding operation, the mold is ﬁrst closed and then a predetermined amount of polymer melt from the screw section is injected into an empty mold cavity. Pressure is maintained for some time after the mold cavity has been ﬁlled to permit the build-up of adequate pressure in the mold cavity. Cooling water is circulated through channels in the mold so as to keep the mold cavity walls at a temperature usually between room temperature and the softening (or melting) temperature of the polymer. Thus, the hot polymer begins to cool as it enters the mold cavity. When it is cooled to a state of sufﬁcient rigidity, the mold is opened and the part is removed. Some of the important variables in the operation of an injection molding machine are: (1) pressure applied by the screw, (2) temperature proﬁle of the screw section, (3) mold temperature, (4) the screw forward time, (5) the mold closed time, and (6) the mold open time. Relationships between these variables are very complicated. In general, one would like to know the pressure, temperature, and density of the polymer in the mold cavity as functions of time during and after the mold is ﬁlled. In principle, these quantities can be calculated, via a mathematical model, during the entire period of mold ﬁlling and subsequent cooling when information on the geometry of the mold cavity, the rheological properties of the polymer, the temperature at which the polymer enters the mold cavity, and the mold temperature is available. However, in practice it

351

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PROCESSING OF THERMOPLASTIC POLYMERS

Figure 8.1 Schematic of typical screw-injection molding machine.

is not easy to develop a rigorous theory because of the geometrically complex shapes of mold cavities, the complex nature of mold ﬁlling patterns (i.e., jetting) at normal injection speeds of industrial practice, and the highly viscoelastic nature of polymer melts, which varies with temperature, pressure, and injection rate (i.e., shear rate in the runner). We have already presented in Chapter 2 the principles of plasticating single-screw extrusion. Thus, in this chapter, we present analyses of injection molding operations associated only with the melt ﬂow through the runner followed by injection into a mold cavity. Let us look at a diagram representing the pressure–temperature (or time) relationships during the injection molding cycle, as schematically shown in Figure 8.2. Referring to Figure 8.2, the following sequence of time periods is identiﬁed. (1) Dead time before the polymer starts ﬂowing into the mold. (2) Filling time, during which the material is ﬁlling the cavity. (3) Packing time, during which the pressure increases rapidly and packing occurs (at this stage, melt ﬂow into the cavity is at a very slow rate) accompanied by cooling. The compressibility of the polymer melt (or shrinkage during cooling or crystallization for semicrystalline polymers) allows some ﬂow during pressure build-up. (4) Sealing time, during which cooling causes sealing at the gate of the cavity. (5) Sealed cooling time, during which further cooling causes the pressure remaining in the cavity to decay. (6) Ejection time, required for removing the

Figure 8.2 Sequence of events

during a typical injection molding cycle: (1) dead time, (2) mold ﬁlling, (3) packing, (4) sealing, (5) sealed cooling, and (6) discharge. (Reprinted from Han, Rheology in Polymer Processing, Chapter 11. Copyright © 1976, with permission from Elsevier.)

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molded part. Usually, there is pressure remaining in the cavity just before part ejection, and too high a residual pressure (or residual stress) in the molded article can cause sticking, scoring, or cracking. However, if the mold pressure is too low, sink marks and bubbles may appear on the surface of the ﬁnished article. Successful injection molding operations must produce molded articles that are free from warpage, sinks, bubbles, scoring, cracks, etc. There are variations of the traditional injection molding process, a few of which are: structural foam injection molding, gas-assist injection molding, and reaction injection molding. In Chapter 11 we present the principles of reaction injection molding as part of thermoset processing. Complicating matters further, in addition to neat thermoplastic polymers, other forms of polymers are used in injection molding, such as highly ﬁlled polymers, glass-ﬁber-reinforced polymers, and heterogeneous polymers, including immiscible polymer blends, etc. Owing to the very limited space available in this chapter, we shall not discuss those subjects here. Even when we restrict our attention to the traditional injection molding process, the analysis must be based on all three of the major variables: time, temperature, and pressure (or injection rate). The molding cycle and the three variables are determined by three sets of conditions or variables: (1) machine variables, (2) mold variables, and (3) polymer variables. In the 1950s, a few research groups (Ballman et al. 1959; Spencer and Gilmore 1951) carried out simplistic analyses of the injection molding process. Later, in the period of the 1970s through the 1990s, numerous research groups made efforts toward a better understanding of the injection molding process by applying some carefully chosen experimental techniques and/or by making grossly simpliﬁed assumptions in modeling efforts, but there are too many papers to cite them all here. Later in this chapter, we will cite some speciﬁc studies that are closely related to the topics discussed in this chapter. In this chapter, we do not describe (1) how to produce better molded articles, (2) mold design, (3) operating procedures for injection molding machines, or (4) how to choose speciﬁc polymers for certain molded articles. Such subjects are discussed in the book by Rubin (1972), which describes the general features of injection molding operations. Some of the comparisons made between experiment and model prediction reported in the literature appear to be highly subjective, because the experimental results reported in the literature are strongly dependent upon processing conditions employed (e.g., injection rate, injection pressure, or mold-ﬁll time), the type of polymer employed, and the geometry of the molds employed. Absent in the literature is a systematic study that enables us to make general conclusions. A generalization of the experimental results in the literature is not possible without a very comprehensive mathematical model that includes all the relevant important physical phenomena (e.g., a jetting from the gate into an empty mold cavity, extrudate swell upon injection from the runner, and the stress relaxation of a polymer melt under cooling during mold ﬁlling) associated with the injection molding process. For the reasons delineated above, in this chapter we only present a fundamental approach to a better understanding of the injection molding process. From the point of view of modeling the injection molding process, there are three steps that may be considered, although all three steps are integral parts of the entire process; namely, (1) ﬂow of a viscoelastic polymer melt in the runner, (2) mold ﬁlling in an empty cavity under nonisothermal conditions, and (3) cooling of the polymer melt

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after mold ﬁlling is completed. Considering the complicated viscoelastic behavior of thermoplastic molten polymers in general, one can easily appreciate the difﬁculties inherent in developing a comprehensive mathematical model for the ﬁrst two steps, especially mold ﬁlling. In this chapter, we describe a fundamental approach to modeling the ﬁrst two steps referred to above. For reasons that will become clear soon, an analysis of mold ﬁlling using semicrystalline polymers is much more complicated than that using amorphous polymers. Thus, the two situations must be treated separately. We will ﬁrst consider the mold ﬁlling using amorphous polymers and then the mold ﬁlling using semicrystalline polymers.

8.2

Flow of Molten Polymer through a Runner

Referring to Figure 8.1, the temperature and stresses of a polymer melt at the end of the runner, which is essentially the gate of a mold cavity, provide the initial conditions for the governing system equations that describe the mold-ﬁlling process. Thus, an accurate description of the temperature and stresses of a polymer melt at the gate of a mold cavity is very important to an accurate description of the mold-ﬁlling process. The runner is typically a cylindrical tube that receives melt from the screw section and delivers it to the mold cavity. Although the ﬂow of a viscoelastic polymer melt through the cylindrical runner may seem straightforward, the problem is not as simple as it may seem at a ﬁrst glance for the following reasons. In injection molding in industrial practice, injection rates are very high, giving rise to shear rates in the runner usually that exceed a few thousand reciprocal seconds, depending upon the type of the polymer and melt temperature employed. Under such circumstances, it is very important to consider the possibility of signiﬁcant viscous shear heating of the polymer melt in the runner, and thus the temperature of the polymer melt leaving the runner (i.e., entering the mold cavity) may no longer have a uniform temperature. Being viscoelastic ﬂuids, most of the thermoplastic polymers used for injection molding would have accumulated signiﬁcant amounts of elastic energy (normal stresses) by the time they reach the end of the runner (i.e., at the gate of the mold cavity). Part of the normal stresses will relax during mold ﬁlling under cooling, while the unrelaxed normal stresses will remain as “frozen-in stress” in the molded part. We can easily estimate the extent of viscous heating and the level of ﬁrst normal stress difference of a viscoelastic polymer melt at the end of a runner at normal injection rates. For illustration, let us consider a runner in the shape of a cylindrical tube having a diameter (D) of 5 mm and a length-to-diameter (L/D) ratio of 10, and three commercial polymers: polystyrene (PS), low-density polyethylene (LDPE), and polypropylene (PP). To estimate the extent of viscous shear heating in the runner, we can solve Eqs. (5.92) and (5.93) in Chapter 5 of Volume 1 using the physical and rheological parameters given in Table 8.1 for the three polymers. The values of computed maximum temperature (Tmax ) of PP (Profax 6423), LDPE (NPE 952), and PS (Styron 686) at the end of the runner are summarized in Table 8.2 for ﬁve shear rates, showing that viscous shear heating is signiﬁcant at the end of the runner. Figure 8.3 gives the computed radial temperature distributions of PP (Profax 6423) at the end of the runner (L/D = 10) for three different shear rates (γ˙ ) ranging from 1,000 to 3,000 s−1 , where it is assumed that the polymer melt entering the runner is at

Table 8.1 Physical and rheological parameters for some commercial PS, HDPE, and PP polymers

(a) Physical Properties Polymer PS (Styron 686) LDPE (NPE 952) PP (Profax 6423)

ρ (kg/m3 )

cp (J/kg K)

k (W/m K)

978 716 760

1.92 × 103 2.60 × 103

0.261 0.182 0.181

2.01 × 103

(b) Rheological Parameters Appearing in the Truncated Power-Law Model ko (Pa·s)

Polymer PS (Styron 686) LDPE (NPE 952) PP (Profax 6423)

b (1/T ) 4.48 × 10−2 2.24 × 10−2

7.27 × 1013 4.56 × 108

2.56 × 10−2

1.57 × 109

γ˙0 (s−1 )

n

1.20 0.40 0.38

0.28 0.38 0.39

ρ denotes the density, cp denotes the speciﬁc heat, and k denotes the thermal conductivity.

Table 8.2 Effect of shear rate at normal injection speeds on the maximum temperature (Tmax ) due to viscous heating, shear stress (σ ) and ﬁrst normal stress difference (N1 ) of some ◦ commercial polymers at the end of a cylindrical runner at 200 C

W (kg/h)

γ˙ (s −1 )a

−∂p/∂z (P a/m)b

σ (P a)c

N1 (P a)d

NWe e

Tmax (◦ C)

(a) PP (Profax 6423) 24.5 49.0 73.5 98.0 122.5

1.0 × 103 2.0 × 103 3.0 × 103 4.0 × 103 5.0 × 103

7.73 × 9.57 × 1.12 × 1.19 × 1.26 ×

107 107 108 108 108

9.66 × 104 1.20 × 105 1.40 × 105 1.49 × 105 1.58 × 105

1.27 1.93 2.61 2.94 3.28

× × × × ×

106 106 106 106 106

6.6 8.2 9.3 9.8 10.4

209 214 218 222 224

6.58 × 8.23 × 9.34 × 1.02 × 1.09 ×

107 107 107 108 108

8.32 × 104 1.03 × 105 1.17 × 105 1.27 × 105 1.36 × 105

8.32 1.23 1.53 1.77 1.99

× × × × ×

105 106 106 106 106

5.1 5.9 6.6 6.9 7.3

206 210 213 217 220

3.99 × 4.68 × 5.12 × 5.69 × 6.02 ×

107 107 107 107 107

4.99 × 104 5.85 × 104 6.40 × 104 7.11 × 104 7.53 × 104

2.80 3.54 4.05 4.73 5.15

× × × × ×

105 105 105 105 105

2.8 3.1 3.2 3.3 3.4

203 205 206 207 208

(b) LDPE (NPE 952) 22.4 44.8 67.1 89.5 111.9

1.0 × 103 2.0 × 103 3.0 × 103 4.0 × 103 5.0 × 103

(c) PS (Styron 686) 26.3 52.7 79.0 105.3 131.6

1.0 2.0 3.0 4.0 5.0

× × × × ×

103 103 103 103 103

a

Shear rate is calculated from γ˙ = [(3n + 1)/4n](32Q/πD 3 ) with Q being volumetric ﬂow rate and D being the capillary diameter (0.5 cm). b Pressure gradient −∂p/∂z is obtained from the solution of the governing system equations, Eqs. (5.92) and (5.93) in Volume 1. c Shear stress is calculated from σ = (−∂p/∂z)D/4. d N1 is calculated from Eq. (5.84) in Volume 1 with the numerical values of A and b given in Table 5.3 of Volume 1. e NWe = N1 /2σ .

355

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PROCESSING OF THERMOPLASTIC POLYMERS Figure 8.3 Computed radial

temperature distributions of PP (Profax 6423), which was fed at 200 ◦ C to the inlet of a tubular runner at the end of the runner (D = 0.5 cm and L/D = 10), for three shear rates (s−1 ): (1) 1,000, (2) 2,000, and (3) 3,000. It is assumed that the surface temperature of the tubular runner is kept at 200 ◦ C. Note that r/R = 0 refers to the center of the circular cross-section of a tabular runner.

200 ◦ C and the wall temperature of the runner is kept at 200 ◦ C. It is seen in Figure 8.3 that the temperature near the inner wall of the runner is much higher, due to viscous heating, than the preset wall temperature (200 ◦ C), and that the extent of temperature rise inside the runner increases as the shear rate is increased. It should be mentioned that the L/D ratio of the runner used in industrial mold design is often much larger than 10, which was considered in the above illustration. Thus, values of Tmax in such mold designs would be much larger than those given in Table 8.2. These observations attest to the fact that the assumption of a uniform temperature of the melt stream leaving the runner (entering the mold cavity) is a gross approximation and is far from the reality. Regarding the estimation of ﬁrst normal stress difference at the end of the runner, a rigorous approach requires the solution of the equations of motion for capillary ﬂow of a viscoelastic polymer melt using an appropriate constitutive equation, enabling one to calculate the components of velocity and stresses at the end of the runner. The complexity here is compounded by the fact that the polymer melt does not have a uniform temperature in the runner (see Figure 8.3) and thus the equations of both motion and energy must be solved simultaneously. Few investigators, if any, have ever carried out such an analysis in the runner when they modeled the mold ﬁlling process. It should be emphasized that an analysis of the ﬂow of a viscoelastic polymer melt through the runner is an integral part of the analysis of the mold ﬁlling process, because the inlet conditions for the equations of motion and energy for mold ﬁlling must come from the solutions of the equations of motion and energy for the runner. Next, we will elaborate on the consequence of neglecting the analysis of the ﬂow of a viscoelastic polymer melt in the runner. For the three commercial polymers considered for the estimate of viscous heating (see Table 8.2), let us estimate, without having to solve the equations of motion and energy for the runner, the extent of ﬂuid elasticity (in terms of ﬁrst normal stress difference) at the end of a cylindrical runner having a diameter (D) of 5 mm and an L/D ratio of 10. For simplicity, let us assume that ﬂow is fully developed at the end of the cylindrical runner at isothermal conditions. Under this assumption, using Eq. (5.84) in Chapter 5 of Volume 1 we can estimate ﬁrst normal stress difference (N1 )

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as a function of steady-state shear stress (σ ). Table 8.2 gives also a summary of the calculated values of N1 for ﬁve different value of σ corresponding to the ﬁve γ˙ ranging from 1,000 to 5,000 s−1 in the runner for PP (Profax 6423), LDPE (NPE 952), and PS (Styron 686). Referring to Table 8.2, γ˙ was calculated using the expression γ˙ = [(3n + 1)/4n](32Q/πD 3 ), with Q being the volumetric ﬂow rate and D being the capillary diameter, and σ was calculated using the expression σ = (−∂p/∂z)D/4 with −∂p/∂z being pressure gradient that was obtained from the numerical solution of Eqs. (5.92) and (5.93) in Volume 1. In Chapter 6 of Volume 1, we suggested that the ratio N1 /σ (often referred to as the Weissenberg number, NWe ), not the absolute value of N1 , be used to assess (or determine) how elastic a polymeric ﬂuid is. The very large values of the ratio N1 /σ summarized in Table 8.2 suggest that all three polymers (PP, LDPE, and PS) have very high ﬂuid elasticity at the end of the runner (at the gate of mold cavity), part of which will relax during mold ﬁlling while the rest will remain as frozen-in stress in the molded part. The above simple analyses give us a very good idea about the extent of nonuniform temperature distribution at the end of a runner and the level of normal stresses accumulated in the melt just before entering the gate of an empty mold cavity at normal injection speeds of practical (commercial) interest. Needless to say, the situation will become very different at very low injection speeds, where the extent of viscous heating of a polymer melt in the runner can be negligibly small just before entering the gate of an empty mold cavity. For illustration, Table 8.3 gives a summary of the calculated maximum temperature and N1 for PP (Profax 6423), LDPE (NPE 952), and PS (Styron) at two different shear rates, γ˙ = 10 and 100 s−1 , in the runner. It is seen in Table 8.3 that the temperature rise near the inner wall of the runner is very small at such low shear rates, but values of NWe are still larger than 1. However, such low injection speeds do not occur in any practical (commercial) injection molding operations. In the next two sections, using the analysis presented above, we will discuss the mold-ﬁlling process of amorphous and semicrystalline polymers from the point of view of mathematical modeling. Table 8.3 Maximum temperature (Tmax ) due to viscous heating, shear stress (σ ) and ﬁrst normal stress difference (N1 ) of some commercial polymers at the end of a cylindrical runner at 200 ◦ C at low injection speeds

W (kg/h)

γ˙ (s−1 )

−∂p/∂z (P a/m)

σ (P a)

N1 (P a)

NWe

Tmax (◦ C)

1.44 × 107 3.47 × 107

1.84 × 104 4.32 × 104

5.10 × 104 2.76 × 105

1.4 3.2

200.1 201.3

1.23 × 107 2.92 × 107

1.54 × 104 3.65 × 104

4.50 × 104 2.02 × 105

2.9 5.5

200.1 200.9

1.19 × 107 2.25 × 107

1.49 × 104 2.80 × 104

4.67 × 104 1.19 × 105

1.6 2.2

200.1 200.5

(a) PP (Profax 6423) 0.245 2.450

10 100

(b) LDPE (NPE 952) 0.224 2.240

10 100

(c) PS (Styron 686) 0.263 2.630

10 100

358

8.3 8.3.1

PROCESSING OF THERMOPLASTIC POLYMERS

Injection Molding of Amorphous Polymers Flow Patterns during Mold Filling

In this section, we consider the ﬂow patterns during mold ﬁlling of an amorphous polymer (e.g., PS), which undergoes a phase transformation from the liquid state to the solid state as the injected polymer melt is cooled down during mold ﬁlling followed by packing in the mold cavity. Spencer and Gilmore (1951) reported for the ﬁrst time on a visualization study of mold ﬁlling of PS into a rectangular mold cavity with quartz windows. They recorded the ﬂow patterns as affected by ﬁll time, temperature, and pressure. Later, other investigators (Han 1974; Han and Villamizar 1978; Kamal and Kenig 1972b; Oda et al. 1976; Schmidt 1974; Sleeman and West 1974; White and Dee 1974) also conducted ﬂow visualization studies of mold ﬁlling. Of particular note are the ﬂow visualization studies of Sleeman and West (1974) and White and Dee (1974), who observed a “jetting phenomenon” of melt stream issuing from the runner into an empty rectangular mold cavity with quartz windows. In the study of Sleeman and West (1973), a cylindrical runner having a diameter (D) of 3.175 mm and a length (L) of 102 mm was used. Sleeman and West observed jetting of PS at 220 ◦ C at the injection rates of 0.01 and 0.02 kg/s, which they regarded as being “normal injection rates.” Using the value of n = 0.363 of a power-law index for the PS employed by them, we estimate that the shear rates in the runner are 1,477 s−1 for the injection rate of 0.01 kg/s and 2,944 s−1 for the injection rate of 0.02 kg/s. On the basis of this observation it is fair to state that the typical shear rates in the runner at normal injection rates of commercial injection molding operations would exceed 2,000 s−1 . Thus, we can conclude that the range of shear rates considered in the estimate of the extent of viscous heating and the ﬁrst normal stress difference of PP, LDPE, and PS, the results of which are summarized in table 8.2, lies within normal injection molding conditions. Figure 8.4 shows the ﬂow patterns of LDPE and high-density polyethylene (HDPE) at low injection rates1 into a rectangular mold cavity with quartz windows. It is seen in Figure 8.4 that upon exiting the runner, the melt spreads in an approximately radial manner and then ﬁlls the corners, followed by forward movement to ﬁll the rest of the empty mold cavity. In doing so, the melt front changes from a circular shape to an almost ﬂat proﬁle. The radial ﬂow patterns at the ﬂuid–ﬂuid interface were referred to as “fountain ﬂow” by Rose (1961). Later, some investigators (Broyer et al. 1973; Richardson 1972; Sato and Richardson 1995) analyzed the ﬁlling of a narrow channel with a Newtonian ﬂuid or a viscoelastic ﬂuid on the basis of the Hele-Shaw approximation (Batchelor 1967; Hele-Shaw 1898; Lamb 1932), which treats the advancing ﬂuid front as fountain ﬂow. While Richardson (1972) advocated that Hele-Shaw ﬂow exists for certain cavity geometries, White (1975) argued that Hele-Shaw ﬂow is not a good approximation for computing the velocity ﬁeld in planes perpendicular to narrow mold sections, except in the vicinity of the mold cavity walls. Sato and Richardson (1995) numerically simulated, via ﬁnite element method (FEM), the fountain ﬂow problem in uniform mold ﬁlling of viscoelastic ﬂuids under isothermal conditions. It should be emphasized that a viscoelastic polymer melt issuing from the runner at very high injection rates into an empty mold cavity under nonisothermal conditions has neither a uniform temperature distribution (see Figure 8.3) nor a fully developed velocity proﬁle at the gate. Thus, the assumption of fountain ﬂow of a viscoelastic polymer melt

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359

Figure 8.4 Photographs of isothermal mold ﬁlling patterns of (a) and (b) an LDPE and (c) and (d) an HDPE into an end-gated thin rectangular mold cavity at low injection speeds. (Reprinted from White and Dee, Polymer Engineering and Science 14:212. Copyright © 1974, with permission from the Society of Plastics Engineers.)

into an empty mold cavity for the advancing front during mold ﬁlling at high injection speeds of industrial practice is not justiﬁed. Figure 8.5 shows the ﬂow patterns of an LDPE at high injection rates2 into a rectangular mold cavity with quartz windows. It is seen in Figure 8.5 that upon exiting the runner, a jet of melt strikes the opposite cavity wall and piles up upon itself, later ﬁlling the entire mold cavity. Such a “jetting phenomenon” was independently observed by Sleeman and West (1974) at shear rates of 2,000–3,000 s−1 in the runner. It is then very clear from Figures 8.4 and 8.5 that the mode of mold ﬁlling (radial ﬂow of the advancing melt front or jetting) depends very much on injection speeds. This observation has a very signiﬁcant implication for the formulation of governing system equations for simulating the mold ﬁlling process. To facilitate our discussion here, let us consider two simple geometries of mold cavity that have been considered extensively in the literature: (1) an end-gated thin rectangular channel mold cavity with very large aspect ratio, as schematically shown in Figure 8.6 and (2) a center-gated disk-type mold cavity, as schematically shown in Figure 8.7. Referring to Figure 8.6, a polymer melt, upon the entering the gate at

Figure 8.5 Photograph showing the jetting phenomenon of an LDPE during molding ﬁlling into an end-gated thin rectangular mold cavity at a high injection speed. (Reprinted from White and Dee, Polymer Engineering and Science 14:212. Copyright © 1974, with permission from the Society of Plastics Engineers.)

Figure 8.6 Schematic showing the mold ﬁlling pattern of a molten polymer at high injection speed into an end-gated thin rectangular mold cavity with very large aspect ratio where a “jetting phenomenon” is seen: (a) coordinate system, (b) view in the z–y plane, and (c) view in the z–x plane.

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361

Figure 8.7 Schematic showing the mold ﬁlling pattern of a molten polymer at high injection speed into a center-gated disk-type mold cavity.

“normal” injection rates (at shear rates in the runner above, say, 2,000 s−1 ), will jet into an empty end-gated thin rectangular cavity, and then the jet melt stream strikes the opposite cavity wall and piles up upon itself, later ﬁlling the entire mold cavity as seen from both the z–x plane (see Figure 8.6c). Under such a circumstance, it is very difﬁcult to imagine whether the advancing front will ever form fountain ﬂow during mold ﬁlling. Referring to Figure 8.7, a polymer melt, upon leaving the runner (or entering the gate) at “normal” injection rates, will ﬁrst hit the wall of the center-gated disk (region I) and then the jet stream will travel radially (in the r-direction) to ﬁll the empty cavity (regions II and III). It is very difﬁcult to imagine exactly where in the empty mold cavity the advancing melt front might form fountain ﬂow, although region II in Figure 8.7 shows the presence of fountain ﬂow formed by the advancing melt front. It is quite possible that the advancing melt front might never form fountain ﬂow during mold ﬁlling. Since the hot polymer melt undergoes a phase transformation near the glass transition temperature (Tg ) of the amorphous polymer being injected, a frozen layer will be formed as the polymer melt comes in contact with the cold mold wall, which will grow continuously with time and along the radial direction as the rest of the empty mold cavity is ﬁlled. However, when injection speed is decreased sufﬁciently to the extent (at shear rates in the runner below, say, 100 s−1 ) that the jetting phenomenon would not occur at the gate, the polymer melt ﬂowing from the runner would swell until contacting the cavity wall and ﬁll the rest of the empty cavity, as schematically shown in Figure 8.8 for an end-gated thin rectangular mold cavity with very large aspect ratio, and schematically shown in Figure 8.9 for a center-gated disk-type mold cavity. Under such a circumstance, one must include extrudate swell accompanied by stress relaxation in describing the ﬂow patterns during mold ﬁlling. At very slow injection speeds (at shear rates in the runner below, say, 10 s−1 ) extrudate swell of the polymer melt leaving the runner will become negligible for all intents and purposes. Under such a circumstance, the ﬂow

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PROCESSING OF THERMOPLASTIC POLYMERS

Figure 8.8 Schematic showing the mold ﬁlling pattern of a molten polymer at low injection speed into an end-gated thin rectangular mold cavity with very large aspect ratio, where use of a Hele-Shaw approximation with the advancing melt front forming fountain ﬂow may be appropriate.

patterns during mold ﬁlling may be described by a Hele-Shaw approximation with the advancing melt front forming fountain ﬂow, as schematically shown in Figure 8.10. On the basis of these observations, we can summarize the ﬂow patterns during mold ﬁlling as follows. The mold ﬁlling is truly a very complex process, the complexity of which depends on, among many factors: (1) the geometry of a mold cavity into which a hot molten polymer is injected; (2) injection speed (i.e., shear rate at the end of the runner), which determines whether or not the ﬁlling of an empty mold cavity is accompanied by jetting phenomenon; (3) the extent of normal stresses accumulated in the melt leaving the runner; (4) the rate of stress relaxation in the polymer melt upon leaving the runner and while an empty mold cavity is ﬁlled; and (5) the temperature Figure 8.9 Schematic showing the

mold ﬁlling pattern of a molten polymer at low injection speed into a center-gated disk-type mold cavity, where use of a Hele-Shaw approximation with the advancing melt front forming fountain ﬂow may be appropriate.

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363

Figure 8.10 Schematic showing the advancing melt front forming fountain ﬂow during very low-speed molding ﬁlling of a center-gated disk-type mold cavity.

difference between the melt exiting the gate and the mold wall, which determines both the temperature distribution across the mold cavity and the rate of skin-layer formation near the inner wall of the cold mold. 8.3.2

Governing System Equations for Mold Filling of Amorphous Polymers

During the past three decades, numerous investigators (Berger and Gogos 1973; Chang and Chiou 1994; Chiang et al. 1991; Falman 1993; Gogos et al. 1986; Harry and Parrott 1970; Hétu et al. 1998; Ilinca and Hétu 2001; Isayev and Hieber 1980; Kamal and Kenig 1972a; Kim et al. 1999; Kuo and Kamal 1976; Mavridis et al. 1988; Papathanadiou and Kamal 1993) have carried out analyses, with different degrees of numerical sophistication, of the mold-ﬁlling process by making some simplifying assumptions for mathematical convenience. Some of the most critical assumptions made are the following. The interface between the advancing melt front of a viscoelastic polymer melt and the air may be described by fountain ﬂow. The ﬁlling of a center-gated disk-type mold cavity under such assumptions (see Figure 8.9) and nonisothermal conditions may be described by the following governing equations in cylindrical coordinates: (1) Continuity equation: ∂ρ 1 ∂ + ρrvr = 0 ∂t r ∂r

(8.1)

(2) Momentum equation: −

∂p ∂σzr + =0 ∂r ∂z

(8.2)

364

PROCESSING OF THERMOPLASTIC POLYMERS

(3) Energy equation:

∂T ∂T + vr ρcp (T ) ∂t ∂r

∂vr ∂ ∂T = k(T ) + σzr ∂z ∂z ∂z

(8.3)

in which ρ denotes the density of polymer melt, which may vary with temperature and pressure when the polymer melt is regarded as a compressible ﬂuid, cp (T ) is the speciﬁc heat capacity of polymer melt that may vary with temperature, and k(T ) is the thermal conductivity of polymer melt that may also vary with temperature. In writing down Eqs. (8.1)–(8.3), a number of simplifying assumptions are made: (1) vz = vθ = 0, (2) ∂vr /∂r ∂vr /∂z, (3) the magnitude of normal stresses in the z- and θ -directions (σzz and σθθ , respectively) are negligibly small compared with that of shear stress σzr , and (4) the axial heat conduction is negligible. Referring to Eq. (8.2), the transient term is omitted because the ﬂow rate is constant, and creeping ﬂow is assumed owing to the fact that polymer melts are very viscous. Referring to the right-hand side of Eq. (8.3), the ﬁrst term describes the heat conduction in the shear (z-axis) direction and the second term describes heat generated due to viscous heating. The ﬁlling of an end-gated thin rectangular mold cavity (see Figure 8.8) under nonisothermal conditions may be described by the following governing equations in rectangular coordinates: (1) Continuity equation: ∂ ∂ρ + ρvz = 0 ∂t ∂z

(8.4)

(2) Momentum equation: −

∂p ∂σzy + =0 ∂z ∂y

(8.5)

(3) Energy equation: ρcp (T )

∂T ∂T + vz ∂t ∂z

=

∂ ∂y

k(T )

∂T ∂y

+ σzy

∂vz ∂y

(8.6)

The last term on the right-hand side of Eq. (8.6) describes the viscous heating during mold ﬁlling. In solving the governing system equations given above, Eqs. (8.1)–(8.3) for a centergated disk-type mold cavity or Eqs. (8.4)–(8.6) for an end-gated thin rectangular mold cavity, previous investigators made additional assumptions: (1) the velocity proﬁle at the gate of an empty mold cavity may be described by fully developed Poiseuille ﬂow, and (2) the temperature of the polymer melt leaving the runner is uniform. Such assumptions essentially adopt the ﬂow patterns, for instance, depicted schematically by Figure 8.10 for a center-gated disk-type mold cavity. As pointed out above, such assumptions are not warranted when describing the mold-ﬁlling process at normal injection speeds (at shear rates greater than say 2,000 s−1 in the runner) in industrial

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Figure 8.11 Schematic showing the formation of a frozen layer near the inner wall of a mold cavity during mold ﬁlling into an end-gated thin rectangular mold cavity at very low injection speeds, where a fountain ﬂow is assumed.

practice. Further, on cooling of an amorphous polymer during mold ﬁlling, a frozen layer is formed near the inner wall of the mold cavity, as schematically shown in Figure 8.11 for the simpliﬁed situation where the advancing melt front is assumed to have ‘fountain ﬂow’ for very low injection speeds. Note that the frozen layer grows continuously inwards until the entire mold is ﬁlled. Figure 8.11 is shown only to illustrate the formation of a frozen layer near the inner wall of a mold cavity, although fountain ﬂow may not be observed at normal injection speeds (at shear rates exceeding, say, 2,000 s−1 in the runner). Therefore, an expression describing the continuous growth of a frozen layer during mold ﬁlling must be provided, and it must be solved together with the system of equations, Eqs. (8.1)–(8.3), for a center-gated disk-type mold cavity and, Eqs. (8.4)–(8.6), for an end-gated thin rectangular mold cavity. It cannot be overemphasized that the accumulation of a high level of normal stresses in the polymer melt (see Table 8.2), while ﬂowing through the runner at normal injection speeds, suggests that there would be a very large extrudate swell of polymer melt at the gate of a mold cavity. Extrudate swell is a manifestation of the relaxation of the normal stresses accumulated in a viscoelastic polymer melt as it ﬂows through the runner. Therefore, an accurate description of mold ﬁlling of a viscoelastic molten polymer must include extrudate swell and stress relaxation of the melt having a nonuniform temperature distribution at the gate of the mold cavity (see Figure 8.3). In the past, some investigators (Berger and Gogos 1973; Chiang et al. 1991; Harry and Parrott 1970; Hétu et al. 1998; Ilinca and Hétu 2001) employed inelastic non-Newtonian models (e.g., power-law, Cross, or Carreau model, as discussed in Chapter 6 of Volume 1), while others (Chang and Chiou 1994; Falman 1993; Isayev and Hieber 1980; Kim et al. 1999; Papathanadiou and Kamal 1993) employed viscoelastic constitutive equations to solve Eqs. (8.1)–(8.3) for mold ﬁlling into a center-gated disk-type cavity or Eqs. (8.4)–(8.6) for mold ﬁlling into an end-gated thin rectangular mold cavity, by assuming uniform temperature and fully developed velocity proﬁle of a polymer melt at the gate, and neglecting extrudate swell of the polymer melt leaving the runner. It is then fair to state that the previous studies on mold ﬁlling may be considered to be gross approximations that do not describe the real situations of mold ﬁlling at the normal injection speeds

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of industrial practice. As pointed out above, a rigorous analysis of mold ﬁlling of viscoelastic polymer melt at normal injection speeds (at shear rates exceeding a few thousand reciprocal seconds in the runner) is indeed extremely complicated in light of the jetting phenomenon and extrudate swell accompanied by the stress relaxations of polymer melt during cooling in the empty mold cavity. 8.3.3

Molecular Orientation during Mold Filling and Residual Stress in Injection Molded Articles

The polymer molecules are oriented during mold ﬁlling, and part of this orientation is retained in the molded article as it is cooled. The subject of molecular orientation as a result of the injection molding process has been studied over a long period by a number of investigators (Ballman and Toor 1960; Clark 1967; Cleereman 1967; Dietz et al. 1978; Jackson and Ballman 1960; Janeschitz-Kriegl 1977; Kamal and Tan 1979; Spencer and Gilmore 1950; Wales et al. 1972). Molecular orientation can be measured by a number of methods. Birefringence is often the easiest and most rapid measurement that can be made. In this procedure, the difference in refractive indices in two mutually perpendicular directions is estimated. Since the refractive index (or polarizability) parallel to a polymer chain usually is different from that perpendicular to the chain, birefringence is a sensitive indicator of molecular orientation. Wales et al. (1972) compared, as a measure of the molecular orientation, the birefringence of injection molded articles with the ﬂow birefringence of molten PS during steady, isothermal shear ﬂow. Since the ﬂow birefringence was found to be related to the wall shear stress in isothermal ﬂow (see Chapter 1), Wales et al. concluded that the orientation of injection molded articles was dominated by the shear stresses during the mold-ﬁlling process. Figure 8.12 gives, for illustration, a photograph of birefringence of an injection molded specimen of PS obtained from a thin rectangular mold cavity under isothermal

Figure 8.12 Photograph of birefringence patterns of an injection-molded specimen of PS, which

was obtained from an end-gated thin rectangular mold cavity. (Reprinted from Han, Rheology in Polymer Processing, Chapter 11. Copyright © 1976, with permission from Elsevier.)

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Figure 8.13 Schematic showing the distribution of birefringence, n, in an injection-molded specimen of PS, which was obtained from an end-gated thin rectangular mold cavity.

conditions. The speciﬁcation of the mold and injection-molding conditions employed are described in a paper by Wales et al. (1972). It is seen in Figure 8.12 that there are small optical effects in the center of the section, and sharp peaks near the edges, while at the edges (i.e., the cavity walls), the birefringence is nonzero. Figure 8.13 gives a schematic of the absolute value of birefringence (n) across the thickness of the molded specimen obtained from a rectangular mold cavity. Since birefringence is increased as molecular orientation is increased, molecular orientation is highest at some distance away from the cavity wall and is lowest at the center of the cross section. Some investigators (Isayev and Hieber 1980; Kim et al. 1999) calculated the birefringence n of amorphous polymers during mold ﬁlling using the expression (Oda et al. 1978) n = C(T )(N1 2 + 4σ 2 )1/2

(8.7)

in which C(T ) is the temperature-dependent stress-optical coefﬁcient, N1 is ﬁrst normal stress difference, and σ is shear stress of the viscoelastic polymer melt undergoing cooling during mold ﬁlling. Note that Eq. (8.7) is valid only for shear ﬂow. Since, strictly speaking, mold ﬁlling is not shear ﬂow, even for very slow injection speeds, use of Eq. (8.7) to calculate the n of a viscoelastic polymer melt during mold ﬁlling under nonisothermal conditions must be regarded as being an approximation. Figure 8.14 gives a schematic describing the distribution of velocity gradient (dvz /dy) under the assumption of fountain ﬂow, across the thickness of a thin rectangular mold cavity (see Figure 8.8 for the coordinate system employed). It is very interesting to observe in Figure 8.14 that the distribution of dvz /dy over the cross section of the thin rectangular mold cavity goes through a maximum at some distance away from the cavity wall and a minimum at the center of the cross section, very similar to the birefringence patterns given in Figure 8.12. It is very interesting to observe that the distribution of velocity gradient of the molten PS in the cavity during mold

368

PROCESSING OF THERMOPLASTIC POLYMERS Figure 8.14 Schematic showing the

distribution of velocity gradient dvz /dy during mold ﬁlling under the assumption of fountain ﬂow across the thickness of a thin rectangular mold cavity.

ﬁlling (see Figure 8.14) is very similar to the distribution of ﬂow birefringence (see Figure 8.13) of the polymer chains over the cross section of the mold cavity. Some theoretical studies have reported on the molecular orientation during mold ﬁlling in the injection molding process (Mavridis et al. 1988; Tadmor 1974). Those studies were based on the premise that during mold ﬁlling the advancing melt front assumes a fountain ﬂow pattern, and they did not consider the effect of extrudate swell accompanied by stress relaxation when a viscoelastic polymer melt was injected into an empty mold cavity. As pointed out above, the jetting phenomenon may occur when a polymer melt enters the gate at the normal injection speeds of industrial practice. Thus, the theoretical studies of molecular orientation reported in the literature referred to above are applicable to mold ﬁlling at very slow injection speeds. As would be expected, molecular orientation increases with injection speed (thus injection pressure). But in some cases, an additional increase in injection pressure after the minimum molding pressure has been reached can affect the molecular orientation of the molding in two different ways. That is, an increase in injection pressure increases stress, which in turn causes higher molecular orientation. In some cases, additional pressure causes the cavity to ﬁll more rapidly, allowing more relaxation. The net result of these two compensating phenomena depends on the particular molding conditions used. Figure 8.15 gives a schematic summarizing the effects of injection molding variables on molecular orientation. Note that the relationships displayed in Figure 8.15 do not take into account some of the compensating factors discussed above and that others not mentioned. In the past, a number of studies have reported on the inﬂuence of molecular orientation on the mechanical properties of injection molded specimen. Rigid polymers, which have been oriented by stretching while being heated, have anisotropic mechanical properties. Uniaxially oriented materials have higher values of certain mechanical properties (such as Young’s modulus, tensile strength and elongation at break) in the direction parallel to the orientation than in the direction perpendicular to the orientation. These effects would be expected, since, parallel to the orientation, stresses are exerted largely on the primary bonds of the polymer chains, while in the direction perpendicular to the orientation, forces act to a large extent on the weak secondary bonds between polymer chains. For this reason, in a uniaxially oriented molding, the tensile strength along the line of the ﬂow rises as the degree of orientation rises, while that at right angles to ﬂow decreases, as schematically shown in Figure 8.16. It also demonstrates that molecular orientation has a great effect on tensile strength and elongation at break.

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Figure 8.15 Effect of processing variables on the orientation of injection-molded specimen: (1) mold temperature, (2) cavity thickness, (3) injection pressure, (4) packing time, and (5) runner temperature. (Reprinted from Han, Rheology in Polymer Processing, chapter 11. Copyright © 1976, with permission from Elsevier.)

There are three types of residual stress (or strain) in injection molded articles: (1) those accompanying quenching stresses, (2) frozen-in molecular orientation, and (3) conﬁgurational volume strains. Quenching stresses sometimes relieve themselves by producing bubbles or sink marks in the article, and may be otherwise relieved by annealing. Conﬁgurational volume strains can be relieved only by annealing (which is

Figure 8.16 Effect of orientation on the tensile properties of molded specimen of PS. (Reprinted from Han, Rheology in Polymer Processing, Chapter 11. Copyright © 1976, with permission from Elsevier.)

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PROCESSING OF THERMOPLASTIC POLYMERS

frequently impractical), and are not important in many practical cases. Much of the frozen-in orientation present originates during “packing.” Note that considerable reduction in the amount of frozen-in orientation may be effected by minimizing the packing time. Reduction in the amount of frozen-in orientation reduces the tendency of the moldings to “craze,” improves their dimensional stability on heating, and produces more consistent specimens for mechanical testing. If a material is cooled to the softening point while the stress is still acting, the molecular chains are immobilized in the uncoiled state and we ﬁnd that frozen orientation is present in the molded article. Residual stresses are commonly found in molded articles that have been cooled rapidly through their freezing points or hardening ranges. The magnitude of the residual stresses may be reduced by slowing down the process of cooling during the period in which hardening is taking place throughout the injected material. This annealing procedure decreases the temperature differences present during hardening, and thus reduces the residual stresses. In general, annealing to minimize quenching stresses is impractical during injection molding. However, annealing of the ﬁnished articles may be feasible in some cases.

8.4

Injection Molding of Semicrystalline Polymers

In the injection molding of semicrystalline polymers (e.g., PP or HDPE), the spherulitic structure will be formed in the molded articles. At normal injection speeds, which give rise to shear rates of a few thousand reciprocal seconds in the runner (see Table 8.2), crystallization in a polymer melt may be induced by the ﬂow inside the mold cavity. It is then fair to state that an analysis of the mold-ﬁlling process of a semicrystalline polymer would be much more complicated than that of an amorphous polymer. Some investigators (Guo and Isayev 1999a, 1999b; Isayev et al. 1995; Kamal and Laﬂeur 1986; Kamal and Papathanadiou 1993; Laﬂeur and Kamal 1986; Pantani et al. 2001) have studied the mold ﬁlling of semicrystalline polymers. In this section, we ﬁrst review very brieﬂy the literature describing the crystallization kinetics of a polymer melt under static conditions or under the inﬂuence of ﬂow, and then discuss the governing system equations that include the crystallization of the polymer inside the mold cavity. 8.4.1

Crystallization during Injection Molding

In the injection molding of semicrystalline polymers, two types of crystallization may take place, depending upon the injection speed: (1) quiescent crystallization and (2) ﬂow-induced crystallization. Quiescent crystallization in a mold cavity will take place as soon as the polymer melt leaving the runner is cooled down to its melting point, suggesting that a thin skin layer will ﬁrst be formed near the wall of the cold mold cavity and then the skin layer will grow continuously, via quiescent crystallization, toward the center of the mold cavity during the rest of the mold ﬁlling followed by packing. When the injection speed is reasonably high, as is usually the case for all commercial injection molding operations, combined shear and elongational ﬂows near the thin skin layer may induce crystallization, even when the temperature of the polymer melt is above its melting point. The subject of quiescent crystallization in

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molten polymers has been investigated by numerous research groups over the past ﬁve decades, and there are too many references to cite them all here. However, the crystallization of a polymer is still not completely understood even today. The difﬁculty lies in gaining a clear understanding of the “nucleation phenomenon.” Thus, it sufﬁces to state that any discussion of this subject is beyond the scope of this chapter. Setting aside the issue of nucleation phenomenon, the empirical Avrami equation (Avrami 1939) has long been used to describe the crystallization rate of polymers under quiescent conditions. The differential form of Eq. (6.2) given in Chapter 6 can be written as (Nakamura et al. 1973) (m−1)/m DX 1 = mKav (T )(1 − X) ln Dt 1−X

(8.8)

where D/Dt is the substantial derivative, X is the relative degree of crystallinity in the system deﬁned by X = φ/φ∞ , with φ∞ being the ultimate degree of crystallinity in the semicrystalline phase. Note that X is bound by 0 and 1, although it does not necessarily have to go to 1, m is the Avrami constant, and Kav (T ) is the temperature-dependent rate constant. The half-time of crystallization, t1/2 (the time required for 50% of the crystallization), may be expressed by (Hoffman et al. 1976) KN 1 U∗ 1 exp − = exp − (8.9) t1/2 t1/2 R(T − T∞ ) T Tf o

where U ∗ denotes the activation energy for segment jump rate in polymer, which may be taken as equal to 6,285 J/mol, R is the universal gas constant, T∞ is a hypothetical temperature at which all motion associated with viscous ﬂow stops, which may be taken as Tg − 30 K with Tg being the glass transition temperature of polymer, T = Tmo − T with Tmo being the equilibrium melting temperature of polymer, f accounts for the decrease in latent heat of fusion as the temperature is lowered, which may be taken to be approximately equal to 2T/ T + Tmo , and KN is a constant. Note that values of (t1/2 )o and KN appearing in Eq. (8.9) can be determined from plots of ln(t1/2 )+U ∗ /(R(T −T∞ ) versus 1/(T Tf ). The rate constant Kav (T ) appearing in Eq. (8.8) can be expressed, with the aid of Eq. (8.9), as3 KN U∗ 1 1/m exp − (8.10) Kav (T ) = (ln 2) exp − t1/2 R(T − T∞ ) T Tf o

Equation (8.8) was used together with Eq. (8.10) to investigate the quiescent crystallization of nylon 6 (Patel and Spruiell 1991), and to investigate the quiescent crystallization of poly(ethylene terephthalate) and isotactic polypropylene (Chan and Isayev 1994; Isayev and Catignani 1997). There is a general agreement among researchers that crystallization is enhanced when a polymer solution or crystallizable polymer melt is subjected to shear ﬂow or elongational ﬂow. Over the past three decades, ﬂow-induced crystallization in polymer systems has been an active research area. Speciﬁcally, ﬂow-induced crystallization in polymer solutions or melts under isothermal conditions has been discussed extensively (Miller 1979; McHugh and Blank 1986; McHugh and Spevacek 1987, 1991).

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An accurate description of ﬂow-induced crystallization during mold ﬁlling is a complicated subject. In describing ﬂow-induced crystallization during mold ﬁlling, Isayev et al. (1995) used an expression for the rate of crystallization proposed by Janeschitz-Kriegl and coworkers (Eder and Janeschitz-Kriegl 1988; Eder et al. 1990). However, the expression contains several parameters that require a great deal of effort to determine. An alternative approach might be to use the much simpler expression (Doufas et al. 2000) (m−1)/m DX 1 = mKav (T )(1 − X) ln exp(ξ tr τ/G) Dt 1−X

(8.11)

where ξ is a constant, τ is total extra stress tensor, and G is the shear modulus of the polymer, to describe ﬂow-induced crystallization during mold ﬁlling. Note that Eq. (8.11) was originally suggested by Doufas et al. (2000), who investigated shearinduced crystallization in high-speed melt spinning (see Chapter 6). The salient feature of Eq. (8.11) lies in that it contains only one additional adjustable parameter, ξ , to the modiﬁed Avrami equation, Eq. (8.8). Further discussion of the crystallization mechanisms during injection molding is beyond the scope of this chapter. 8.4.2

Governing System Equations for Injection Molding of Semicrystalline Polymers

Analysis of injection molding of semicrystalline polymer must include the crystallization in the mold cavity during and after the mold ﬁlling. For illustration, let us consider an analysis of mold ﬁlling of a semicrystalline polymer into a center-gated disk-type mold cavity. For this, one must solve Eqs. (8.1), (8.2), and the energy equation that includes the term describing crystallization: ρcp (T )

∂T ∂T + vr ∂t ∂r

=

∂vr DX ∂ ∂T k(T ) + σzr + ρHf ∂z ∂z ∂z Dt

(8.12)

in which Hf is the latent heat of crystallization per unit mass. The last term on the right-hand side of Eq. (8.12) describes the heat release due to crystallization. In solving Eqs. (8.1), (8.2), and (8.12), subjected to proper inlet and boundary conditions, one must specify an expression for DX/Dt, say Eq. (8.8), describing the quiescent crystallization taking place in the stationary region, where the polymer is cooled down below its crystallization temperature, and another expression for DX/Dt, say Eq. (8.11), describing the shear-induced crystallization taking place outside the frozen layer near the wall of the cold mold cavity. For this, one must ﬁrst calculate, using a carefully chosen constitutive equation, the stresses of the polymer melt over the cavity cross section during mold ﬁlling from the solutions of Eqs. (8.1)−(8.3) as a ﬁrst approximation, and then solve Eqs. (8.1), (8.2), and (8.12) with the aid of Eq. (8.8) for quiescent crystallization, and with the aid of Eq. (8.11) for shear-induced crystallization. Note that according to Eq. (8.11), the effect of shear-induced crystallization on mold ﬁlling will be signiﬁcant only in the region where the shear stress exceeds a certain critical value.

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During the mold-ﬁlling stage, the polymer melt crystallizes, forming a frozen layer, near the wall of the cold mold cavity, the temperature of which is far below the crystallization temperature of the polymer. Thus, it is highly unlikely that the frozen layer would ﬂow along with the advancing melt front. The previous studies (Guo and Isayev 1999a, 1999b; Isayev et al. 1995; Laﬂeur and Kamal 1986; Pantani et al. 2001) reporting on the mold ﬁlling of semicrystalline polymers were based on the assumptions of uniform temperature and fully developed velocity proﬁle at the gate of a mold cavity. As pointed out in the preceding section (which discussed the mold ﬁlling of amorphous polymers), numerical solutions of Eqs. (8.1), (8.2) and (8.12) with such unrealistic assumptions are too crude to accurately describe the mold ﬁlling of semicrystalline polymers at normal injection speeds (at shear rates of a few thousand reciprocal seconds in the runner) of commercial injection molding operations. Further, in solving Eqs. (8.1), (8.2), and (8.12), one may have to consider the effect of shear-induced crystallization on the melt viscosity and melt elasticity during mold ﬁlling. This subject requires serious attention in the future.

8.4.3

Morphology of Injected-Molded Semicrystalline Polymers

It should be pointed out that during injection molding, the development of superstructure and orientation in semicrystalline polymers (e.g., PP or HDPE) occurs in a manner quite different from that in amorphous polymers (e.g., PS). The development of morphology during injection molding of semicrystalline polymers can best be understood in terms of the interrelationships between melt rheology, crystallization kinetics, and thermal environments (Clark 1973; Fujiyama and Wakino 1991; Guo et al. 1999; Kamal et al. 1980; Katti and Schultz 1982; Moy and Kamal 1980). Three types of morphology occur (Clark 1973): (1) a skin of high molecular orientation, (2) a less highly oriented intermediate “transcrystalline layer,” and (3) a spherulitic core. The skin represents that portion of the melt crystallizing during the ﬁlling period and has a structure similar to the well-known “shish-kebabs.” The transcrystalline and spherulitic regions result from crystallization under low melt stress, with their relative volumes determined by the rate of heat transfer to the mold wall. Figure 8.17 gives optical micrographs of an injection-molded PP specimen in the thickness direction, which were obtained from a center-gated thin dumbbell-shape mold cavity. The specimens were obtained at thee different injection speeds, which gave rise to shear rates of 50, 190, and 324 s−1 in the runner. It is seen in Figure 8.17 that the thin skin layer near the cavity wall has ﬁne textures and the core region has larger spherulites, and that the size of spherulites increases with increasing injection speed. Similar optical micrographs have been reported by Fujiyama and Wakino (1991), who conducted quantitative analysis of their experimental results in terms of the size and distribution of the crystallites in the thickness and ﬂow directions of the injection-molded PP specimens. We now offer an explanation as to why the crystalline morphology of PP in the thin skin layer beneath the cavity wall, as shown in Figure 8.17, is very different from that in the core region of an injection-molded part of semicrystalline PP. As the ﬂowing melt stream begins to freeze near the cavity wall owing to the difference in temperature between the cavity wall and the ﬂowing melt stream,

Figure 8.17 Polarized optical micrographs of injection-molded PP specimens, which were

obtained from an end-gated thin dumbbell-shaped mold cavity at different injection speeds, represented in terms of shear rate (γ˙ ), in the cylindrical runner. Micrograph (a) describes the entire cross section and micrograph (b) describes the edge area at γ˙ = 50 s−1 in the runner. Micrograph (c) describes the entire cross section and micrograph (d) describes the edge area at γ˙ = 190 s−1 in the runner. Micrograph (e) describes the entire cross section and micrograph (f) describes the edge area at γ˙ = 324 s−1 in the runner. (See color plate 2.) 374

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the melt stream just outside the frozen skin layer becomes subjected to elongational ﬂow, causing the polymer chains to stretch in the ﬂow direction. As the melt stream advances forward to ﬁll the rest of the empty mold cavity, cooling continues and thus the thickness of the frozen skin layer will grow. As this process continues until the mold cavity is completely ﬁlled, the successive layers of stretched molten polymer undergo cooling and thus crystallize when the melt temperature reaches near the melting point. It seems reasonable to speculate that the combined elongational and shear ﬂows just outside the frozen skin layer in the mold cavity may enhance crystallization and thus crystallization may actually take place at a temperature slightly higher than the melting point of the polymer under quiescent conditions. When this occurs, the frozen skin layer would have elongated microstructure, the size of which would be much smaller than in the core region. This is because crystallization in the core region would take place after the mold is completely ﬁlled, that is, during the packing and cooling stages (crystallization under quiescent conditions). Since the thermal conductivity of a molten polymer is very poor compared with that of a metal, the cooling of polymer melt in the core region will take place very slowly, during which period the nucleated crystallites grow in three dimensions, giving rise to spherulites. Further discussion of the kinetics of ﬂow-induced crystallization associated with mold ﬁlling is beyond the scope of this chapter.

8.5

Summary

Injection molding is indeed a very complicated processing operation. A complete understanding of the injection molding operation from a fundamental point of view requires an understanding of interrelationships between material variables, mold design, and morphology development during mold ﬁlling and cooling. When a mold cavity is ﬁlled, the velocity of the polymer melt at the gate of a mold cavity drops drastically and cooling begins. An accurate design of a complicated mold geometry is often a formidable task; it requires sophisticated mathematical models which in turn require information on the temperature-dependent nonlinear viscoelastic behavior of polymers. There are some commercial software packages available, which can be regarded as producing very crude ﬁrst-order approximations of real situations. In the packing stage, the compressibility of a melt would be very important. Therefore, the use of a pressure–volume–temperature (PVT) thermodynamic relationship is necessary to accurately describe the packing stage of injection molding. At normal injection speeds, with shear rates exceeding a few thousand of reciprocal seconds in the runner, the injection pressure would be very high. Under such a circumstance, the effect of pressure (p) on shear viscosity may be important to accurately describe mold ﬁlling, and thus shear viscosity (η) must be expressed in terms of shear rate (γ˙ ), temperature (T ), and pressure ( p), η(γ˙ , T , p), to simulate mold ﬁlling. Thus, some may argue that the use of melt viscosity obtained at low pressure may give misleading predictions for the mold ﬁlling. Practically speaking, the use of pressure-dependent viscosity function η(γ˙ , T , p) to solve the momentum and energy equations would pose a formidable task. Moreover, experimental determination of the pressure-dependent melt viscosity in the shear-thinning regime is a great challenge, as pointed out in Chapters 5 and 13 of Volume 1. An accurate prediction of pressure drops

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in the runner (for the purpose of designing a runner) at normal injection speeds would require information on pressure-dependent shear viscosity, η(γ˙ , T , p). With the limited space available in this chapter, we have made an attempt to describe the fundamental aspects of injection molding, placing emphasis on the development of a mathematical model based on realistic initial and boundary conditions. We have pointed out that an accurate and realistic description of the mold-ﬁlling process from a modeling point of view must ﬁrst simulate the ﬂow of a viscoelastic polymer melt in the runner, which could then provide realistic initial conditions to solve a system of equations for mold ﬁlling. Most, if not all, of the modeling efforts reported in the literature to date were based on the assumption of uniform temperature and fully developed velocity proﬁle at the end of a runner (thus at the gate of a mold cavity) without solving a system of equations for the runner. Such an assumption is not realistic and will not accurately describe the mold ﬁlling process under normal injection speeds. In this chapter, we have shown that at normal injection speeds, signiﬁcant shear heating occurs in the runner, giving rise to nonuniform temperature proﬁle at the end of a runner. Further, at normal injection speeds, at the end of a runner the levels of ﬁrst normal stress difference (thus the accumulated elastic energy per unit volume), or the Weissenberg number, of most of the commercial thermoplastic polymer melts (e.g., PS, PP, or HDPE), used for injection molding is so large that they will give rise to signiﬁcant extrudate swell upon leaving the runner. Thus, a realistic modeling of the mold-ﬁlling process must also take into account the elasticity of the melt at the gate of a mold cavity. We recognize the fact that the suggested approach is a difﬁcult task, but it can be done. We suggest that serious efforts should be made in the future to address the issues raised here.

Notes 1. The authors did not report the values of shear rate employed in their experiments. 2. The authors did not report the values of shear rate employed in their experiments. 3. The Avrami equation, Eq. (6.2) given in Chapter 6, can be rewritten as m t Kav (T )dt ln(1 − X) = −

(8N.1)

0

For X = 12 , Eq. (8N.1) gives Kav (T ) = (ln 2)1/m (1/t1/2 ) Substitution of Eq. (8.9) into the right-hand side of Eq. (8N.2) gives Eq. (8.10).

References Avrami M (1939). J. Chem. Phys. 7:1103; ibid. 8:212; ibid. 9:117. Ballman RL, Shusman T, Toor HL (1959). Ind. Chem. Eng. 51(7): 847.

(8N.2)

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Ballman RL, Toor RL (1960). Modern Plastics 37(10): 113. Batchelor GK (1967). An Introduction to Fluid Mechanics, Cambridge University Press, Oxford, p 222. Berger JO, Gogos CG (1973). Polym. Eng. Sci. 13:102. Broyer E, Tadmor Z, Gutﬁnger C (1973). Israel J. Tech. 11(4):189. Chan TW, Isayev AI (1994). Polym. Eng. Sci. 34:461. Chang RY, Chiou SY (1994). Intern. Polym. Process. 9:365. Chiang HH, Hieber CA, Wang KK (1991). Polym. Eng. Sci. 31:116. Clark ES (1967). SPE J. 23(7):46. Clark ES (1973). Appl. Polym. Symp. 20:325. Cleereman K (1967). SPE J. 23(10):43. Dietz W, White JL, Clark ES (1978). Polym. Eng. Sci. 18:273. Doufas AK, McHugh AJ, Miller C (2000). J. Non-Newtonian Fluid Mech. 92:27. Eder G, Janeschitz-Kriegl H (1988). Colloid Polym. Sci. 266:1087. Eder G, Janeschitz-Kriegl H, Liedauer S (1990). Prog. Polym. Sci. 15:629. Falman AAM (1993). Polym. Eng. Sci. 33:193. Fujiyama M, Wakino T (1991). J. Appl. Polym. Sci. 43:57. Gogos CG, Huang CF, Schmidt LR (1986). Polym. Eng. Sci. 26:1457. Guo X, Isayev AI (1999a). Intern. Polym. Process. 14:377. Guo X, Isayev AI (1999b). Intern. Polym. Process. 14:387. Guo X, Isayev AI, Demiray M (1999). Polym. Eng. Sci. 39:2132. Han CD (1974). J. Appl. Polym. Sci. 18:3581. Han CD (1976). Rheology in Polymer Processing, Academic Press, New York, Chap 11. Han CD, Villamizar C (1978). Polym. Eng. Sci. 18:173. Harry DH, Parrott RG (1970). Polym. Eng. Sci. 10:209. Hele-Shaw HJS (1898). Nature 58:34. Hétu JF, Gao DM, Garcia-Rejon A, Salloum G (1998). Polym. Eng. Sci. 38:223. Hoffman JD, Davis GT, Lauritzen JI (1976). In Treatise on Solid State Chemistry: Crystalline and Noncrystalline Solids, Vol. 3, Hannay NB (ed), Plenum, New York, Chap 7. Ilinca F, Hétu JF (2001). Intern. Polym. Proces. 16:291. Isayev AI, Chan TW, Gmerek M, Shimojo K (1995). J. Appl. Polym. Sci. 55:821. Isayev AI, Catignani BF (1997). Polym. Eng. Sci. 37:1526. Isayev AI, Hieber CA (1980). Rheol. Acta 19:168. Jackson GB, Ballman RL (1960). SPE J. 16(10):1147. Janeschitz-Kriegl H (1977). Rheol. Acta 16:327. Kamal MR, Kaylon DM, Dealy JM (1980). Polym. Eng. Sci. 20:1117. Kamal MR, Kenig S (1972a). Polym. Eng. Sci. 12:294. Kamal MR, Kenig S (1972b). Polym. Eng. Sci. 12:302. Kamal MR, Laﬂeur PG (1986). Polym. Eng. Sci. 26:103. Kamal MR, Papathanadiou TD (1993). Polym. Eng. Sci. 33:410. Kamal MR, Tan V (1979). Polym. Eng. Sci. 19:558. Katti SS, Schultz JM (1982). Polym. Eng. Sci. 22:100. Kim IH, Park SJ, Chung ST, Kwon TH (1999). Polym. Eng. Sci. 39:1930. Kuo Y, Kamal MR (1976). AIChE J. 22:661. Laﬂeur PG, Kamal MR (1986). Polym. Eng. Sci. 26:92. Lamb H (1932). Hydrodynamics, Cambridge University Press, Oxford, p 582. Mavridis H, Hrymak AN, Vlachopoulos J (1988). J. Rheol. 32:639. McHugh AJ, Blank RH (1986). Macromolecules 19:1249. McHugh AJ, Spevacek JA (1987). J. Polym. Sci., Polym. Lett. Ed. 25:105. McHugh AJ, Spevacek JA (1991). J. Polym. Sci., Polym. Phys. Ed. 29:969.

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Miller RL (ed) (1979). Flow Induced Crystallization in Polymer Systems, Gordon Breach, New York. Moy FH, Kamal MR (1980). Polym. Eng. Sci. 20:947. Nakamura K, Katayama K, Amano T (1973). J. Appl. Polym. Sci. 17:1031. Oda K, White JL, Clark ES (1976). Polym. Eng. Sci. 16:585. Oda K, White JL, Clark ES (1978). Polym. Eng. Sci. 18:53. Pantani R, Speranza V, Totomanlio G (2001). Intern. Polym. Process. 16:61. Papathanadiou TD, Kamal MR (1993). Polym. Eng. Sci. 33:400. Patel RM, Spruiell JE (1991). Polym. Eng. Sci. 31:730. Richardson S (1972). J. Fluid Mech. 56:609. Rose W (1961). Nature 191:242. Rubin I (1972). Injection Molding, Interscience, New York. Sato T, Richardson SM (1995). Polym. Eng. Sci. 35:805. Schmidt LR (1974). Polym. Eng. Sci. 14:797. Sleeman MJ, West GH (1973). Brit. Polym. J. 5:91. Sleeman MJ, West GH (1974). Brit. Polym. J. 6:109. Spencer RS, Gilmore GD (1950). Modern Plastics 27(12): 97. Spencer RS, Gilmore GD (1951). J. Colloid Sci. 6:118. Tadmor Z (1974). J. Appl. Polym. Sci. 18:1753. Wales JLS, van Leeuwen IJ, van der Vijgh R (1972). Polym. Eng. Sci. 12:358. White JL (1975). Polym. Eng. Sci. 15:44. White JL, Dee HB (1974). Polym. Eng. Sci. 14:212.

9

Coextrusion

9.1

Introduction

Coextruded products were ﬁrst commercialized in the 1950s by the ﬁber industry, which produced conjugate ﬁbers (Sisson and Morhead 1953; Hicks et al. 1960, 1967). Subsequently, in the 1960s and 1970s, the plastics industry developed coextrusion processes to produce multilayer ﬁlms and sheets by extruding two or more polymers. Schrenk and coworkers (Schrenk 1974; Schrenk and Alfrey 1973; Schrenk et al. 1963) pioneered the concept of a coextrusion die system. However, there are a number of technological problems that must be understood in order to achieve successful coextrusion operations. In the 1970s, a number of research groups devoted their efforts to a better understanding of the fundamental problems associated with the coextrusion processes; namely, (1) interface deformation (i.e., encapsulation of one component by another component) during coextrusion (Everage 1975; Han 1973, 1975; Khan and Han 1976; Lee and White 1974; MacLean 1973; Southern and Ballman 1973, 1975; White and Lee 1975) and (2) interfacial instability during coextrusion (Han and Shetty 1978a; Khan and Han 1977; Schrenk et al. 1978). Those efforts are summarized in two monographs by Han (1976, 1981). Since then, further efforts have been made to investigate interface deformation during coextrusion via ﬁnite element analysis (Karagiannis et al. 1990; Matsunaga et al. 1998; Mavridis et al. 1987; Mitsoulis 1988; Mitsoulis and Heng 1987; Puissant et al. 1994) and to investigate interfacial instability, both experimentally (Han et al. 1984; Wilson and Khomami 1992, 1993) and theoretically (Anturkar et al. 1990; Khomami 1990; Su and Khomami 1992). Coextruded sheet for thermoformed high-barrier containers has become an important business sector for food and beverage packages for meats, baby food, beer, carbonated soft drinks, etc. Such packaging requires improved barrier protection to extend the shelf life of such products in thermoformed barrier containers. It should be mentioned that the coextruded sheets or ﬁlms are of little commercial value unless the component polymers adhere together. This means that the component polymers must be compatible,1 at temperatures ranging from service temperature to the melt 379

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PROCESSING OF THERMOPLASTIC POLYMERS

processing temperature in order to have good adhesion between the layers in the coextruded ﬁlms or sheets. When two polymers to be coextruded are not compatible, one must use a third component as an adhesive (or “tie”) layer between the two polymers. The choice of tie layer depends on the chemical structures of the two polymers to be coextruded. Let us consider the schematic given in Figure 9.1, which shows different combinations of polymer systems that may be coextruded. (a) Polymer A on one layer and pigmented polymer A on the other layer may be coextruded. Such coextruded products may be required when one wishes to have a sheet with a speciﬁc color on the surface. For instance, environmentally resistant polymers (e.g., pigmented PVC) can be coextruded over lower cost base materials (e.g., unpigmented PVC). In obtaining such a coextruded sheet there would be no problem of phase separation between the layers because the amount of pigment added is usually very small, not affecting compatibility between the two layers during coextrusion. Further, in obtaining such a product, the coextrusion process is much more economical than the lamination process of two separately extruded sheets, one without and the other with pigment. (b) Two compatible polymers, A and B, may be coextruded. Such a coextruded product may be required when, for instance, a hard, scratch-resistant layer is needed on a softer material for surface protection. (c) When two polymers, A and B, to be extruded are incompatible, one must coextrude a third component (commonly referred to as tie layer) between the two layers of polymers A and B. (d) Regrind of two compatible polymers A and B can be coextruded in the middle of two layers of polymers A and B. This is very important in commercial applications, because invariably scrap is generated after the coextruded sheets are used for certain applications. (e) Three compatible polymers, A, B, and C, may be coextruded. (f) Three incompatible polymers, A, B, and C, may be coextruded with two separate tie layers. Such coextruded products are commercially available for food packaging as multilayer ﬁlms or high-barrier multilayer rigid containers. For example, poly(ethylene-co-vinyl alcohol) (PEVOH) has a very high gas barrier property and thus it is widely used for coextruding together with low-density polyethylene (LDPE), high-density polyethylene (HDPE), or polypropylene (PP) to produce

Figure 9.1 Schematic showing various combinations of multilayer coextruded ﬁlm or sheet.

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381

multilayer ﬂexible ﬁlm, multilayer sheet, multilayer rigid bottle, or multilayer squeezable bottle. These multilayer materials have excellent oxygen barrier, odor retention, and solvent barrier properties for packaging foods, pharmaceuticals, cosmetics, agricultural chemicals, etc. However, PEVOH/HDPE, PEVOH/LDPE, and PEVOH/PP pairs are not compatible. Hence, a tie layer material must be selected for such purposes, which leads to three-layer or ﬁve-layer coextrusion (see Figures 9.1c and 9.1f). A judicious choice of tie layer is crucial to a successful coextrusion of two or more incompatible polymers. Needless to say, the chemical structure of a tie layer would depend on the chemical structures of the polymers to be coextruded. One may choose functionalized polyoeﬁns as tie layer material, so that it can undergo, during coextrusion, a grafting reaction with PEVOH. There are industrial products (Bynel from duPont de Nemours and Company; ADMER from Mitsui Petrochemical Industries, Ltd; MODIC from Mitsubishi Petrochemical Industries, Inc.) available for such purposes, namely LDPE grafted with maleic anhydride (MA) (LDPE-g-MA), HDPE grafted with MA (HDPE-g-MA), PP grafted with MA (PP-g-MA), or poly(ethylene-co-vinylacetate) (PEVA) grafted with MA (PEVA-g-MA). Hence, one comes up with ﬁve-layer coextruded products: LDPE/(LDPE-g-MA)/PEVOH/(LDPE-g-MA)/LDPE, HDPE/(HDPEg-MA)/PEVOH/(HDPE-g-MA)/HDPE, PP/(PP-g-MA)/PEVOH/(PP-g-MA)/PP, or LDPE/(LDPE-g-MA)/PEVOH/(PP-g-MA/PP)/PP. Another well-known example is the coextruded multilayer ﬁlms for meat packages, for which HDPE and nylon 6 are coextruded. Since HDPE and nylon 6 are not miscible, HDPE-g-MA may be used as a tie layer material, so that grafting reaction can take place between the –NH2 end groups in nylon 6 and the –COOH groups in the HDPE-g-MA, producing HDPE(HDPE-gMA)/nylon 6 three-layer ﬁlms. The applications of HDPE/(HDPE-g-MA)/nylon 6 ﬁlms in the food-packaging industry are based on their low permeability to carbon dioxide, oxygen, and water vapor. The oxygen barrier prevents rapid aging, and the humidity barrier prevents drying of the food. Not infrequently, one encounters a situation where two or more polymers must be fed to the respective extruders at different temperatures. Such a situation arises when polymer A has a rather high melting temperature (e.g., nylon 6), whereas polymer B (e.g., LDPE, HDPE, PP, or PEVA) may undergo thermal degradation at the melting temperature of polymer A. Further, a die temperature may have to be chosen that is different from the incoming melt streamlines. This means that the temperature of the melts inside the die keeps changing during ﬂow through the die, thus nonisothermal ﬂow must be considered inside the coextrusion die. An analysis of nonisothermal coextrusion is of fundamental importance. However, only a few papers (Chin et al. 1984; Sornberger et al. 1986b) have dealt with nonisothermal coextrusion. Another important issue for coextrusion is a proper die design (Han and Shetty 1976; Schrenk 1974; Sornberger et al. 1986a; Yu and Han 1973). The geometry of the coextrusion dies used in industrial practice is much more complex than the ﬂat dies often employed in the theoretical analysis. In this chapter, we present (1) the fundamentals of coextrusion die systems, (2) polymer–polymer interdiffusion during coextrusion, and (3) nonisothermal coextrusion. These topics are sparsely summarized in the literature. When two compatible polymers are coextruded, the extent of polymer–polymer interdiffusion would greatly inﬂuence the interface thickness and hence the extent of adhesion between the two layers. We have chosen not to present interface deformation and interfacial

382

PROCESSING OF THERMOPLASTIC POLYMERS

instability during coextrusion of two immiscible polymers, because they are well documented in two previous monographs by Han (1976, 1981). These subjects are less important when two miscible polymers are coextruded or a tie-layer component is used to coextrude two immiscible polymers.

9.2

Coextrusion Die Systems

Broadly classiﬁed, three types of dies are used in coextrusion: (1) single-manifold die with feedblock for ﬁlm or sheet, (2) multi-manifold die for ﬁlm and sheet, and (3) multilayer blown-ﬁlm die. The key to the design of a coextrusion die is to have uniform ﬂow distribution inside the die, so that uniform layer thickness can be achieved. In this section, we present the coextrusion die systems. 9.2.1

Feedblock Die System for Flat-Film or Sheet Coextrusion

Schrenk (1974) showed that a single-manifold feedblock combined with a coat-hanger die (see Chapter 1) can be used to produce multilayer ﬁlm or sheet, as schematically shown in Figure 9.2. The salient feature of this die system lies in the design of the feedblock, such that two or more polymers are fed through individual feedports that distribute individual layers to their desired location. This coextrusion die system design is based on the premise that the polymer melts, upon exiting the individual feedports, maintain their streamlines as they move forward and spread out towards the coat-hanger die section. Figure 9.3 gives a schematic of the combined feature of single-manifold feedblock and coat-hanger die, where two melt streams leaving the feedports must

Figure 9.2 Schematic of the feedblock arrangement for producing multilayer ﬁlm or sheet using a single-manifold die. (Reprinted from Schrenk, Plastics Engineering 30(3):65. Copyright © 1974, with permission from the Society of Plastics Engineers.)

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Figure 9.3 Schematic of a coat-hanger die with a feedport having two feedslots. (Reprinted from

Schrenk, Plastics Engineering 30(3):65. Copyright © 1974, with permission from the Society of Plastics Engineers.)

spread out to ﬁll the very wide coat-hanger die. In view of the fact that interface deformation (i.e., encapsulation) and interfacial instability take place as two or more melt streams ﬂow through a die (Han 1976, 1981), the extent of which becomes greater as the viscosity difference between the two melt streams increases, the concept of a single-manifold die with feedport is expected to work well only when the viscosity difference between two melt streams is very small. However, the feedblock concept has some advantages in that the same coat-hanger die may be used for different polymer pairs, and so a very large number of layers (say hundreds) can be accommodated by designing a single feedblock. This means that the cost of changing the die system would be rather low. The chances of having many (say, ﬁve) chemically dissimilar polymers with almost the same shear viscosities are very low in practice. Individual polymer melt temperatures at the extruders can be adjusted so that the viscosities of the different melt streams can become similar. However, such an approach can make the situation more complicated in that nonisothermal analysis is called for, which will be discussed later in this chapter. Alternatively, hundreds of alternating layers of two chemically dissimilar polymers with almost the same shear viscosity can be produced using a single-manifold die with a properly designed feedblock. The plastics industry has had some successes in producing such products. 9.2.2

Multimanifold Die System for Flat-Film or Sheet Coextrusion

The multimanifold die system consisting of individual manifolds, extending across the entire width of the die for each layer, and a ﬂat-ﬁlm or sheet-forming die section enables one to coextrude polymers of widely different rheological properties. Such die system, compared with the feedblock die system presented above, would minimize the extent of encapsulation of melt streams during ﬂow. However, the multimanifold die system becomes very expensive in that the entire die system must be designed for each polymer system chosen for coextrusion, and the number of layers to be coexturded would be limited to, say, ﬁve, because the die system would become very bulky to handle, even for a three-layer ﬁlm or sheet.

384

PROCESSING OF THERMOPLASTIC POLYMERS Figure 9.4 Schematic of the side view

of a two-manifold die system for producing coextruded ﬁlm or sheet.

Figure 9.4 gives a schematic of the side view of a two-manifold die system, and Figure 9.5 gives a schematic of the side view of the three-manifold die system. Note that each melt stream supplied from the extruder goes through a separate manifold having the shape of a coat-hanger die (see Figure 9.3) and that the separate melt streams meet together at the inlet of a relatively short die length followed by a die lip. Therefore, each manifold can be designed using the analysis presented in Chapter 1. Again, as presented in Chapter 1, choker bars are used on each manifold to ensure that ﬂow is uniformly distributed before meeting with other melt streams at the inlet of the die land, and the die lip is adjusted further to have uniform thickness of ﬁlm or sheet. We now understand why the multimanifold die system minimizes the extent of encapsulation and also the extent of interfacial instability during ﬂow. It should also be easy to understand now why the multimanifold die system can become very bulky, limiting the use of layers to, say, less than ﬁve from the point of view of mechanical design and handling. 9.2.3

Feedblock Die System for Blown-Film Coextrusion

Multilayer ﬁlms are produced using tubular ﬁlm blowing (the fundamentals of tubular ﬁlm blowing are discussed in Chapter 7). Figure 9.6 gives a schematic of the feedblock tubular blown-ﬁlm die system, where each melt stream is supplied from the extruder and passes through a separate manifold until meeting with other melt streams just before the die lip. Each manifold must be designed such that the ﬂow of the melt stream through the annular channel can be uniform in the circumferential direction.

Figure 9.5 Schematic of the side view of a three-manifold die system for producing coextruded ﬁlm or sheet.

Figure 9.6 Schematic of the front view of a three-manifold tubular ﬁlm-blowing die system for

producing coextruded ﬁlm.

385

386

PROCESSING OF THERMOPLASTIC POLYMERS

Usually, a spiral ﬂow channel is used to help maintain uniform distribution of melt stream inside the mandrel. In the HDPE/(HDPE-g-MA)/nylon 6 three-layer tubular ﬁlm blowing coextrusion, for example, the tie-layer polymer HDPE-g-MA is fed into the middle mandrel. It is clear that a number of layers greater than three would be difﬁcult to operate using the feedblock tubular ﬁlm-blowing die system. 9.2.4

Rotating-Mandrel Die System for Blown-Film Coextrusion

Multilayer tubular ﬁlm blowing can also be produced using a rotating-mandrel die system, as shown schematically in Figure 9.7. Such a die system was ﬁrst suggested by Schrenk and Alfrey (1973) and later used by Han and Shetty (1978b). The principles of operation are as follows. Molten polymers from two extruders are fed separately into a die distribution manifold, while the inner mandrel rotates at a constant speed. One polymer is introduced into the toroidal manifold, while the other polymer is introduced into the annular manifold. The polymers are arranged into alternating layers by connecting feedslots in the feedport ring in such a manner that the individual polymers are at a number of feedslots uniformly spaced around the ring, as schematically shown

Figure 9.7 Schematic of the rotating-mandrel die system for blown-ﬁlm coextrusion: (1) rotating shaft, (2) centering ring, (3) upper die, (4) lower die, (5) threaded ring, (6) bearing retainer, (7) feed ring, (8) thrust bearing, (9) bearing, (10) tinker bearing, and (11) sprocket. (Reprinted from Han and Shetty, Polymer Engineering and Science 18:187. Copyright © 1978, with permission from the Society of Plastics Engineers.)

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Figure 9.8 Schematic showing the details of the feedport for the die described in Figure 9.7.

(Reprinted from Han and Shetty, Polymer Engineering and Science 18:187. Copyright © 1978, with permission from the Society of Plastics Engineers.)

in Figure 9.8. The number of layers to be generated depends, for the given number of feedslots, on the rotational speed of the inner mandrel and feed ratio of the two polymers. Polymer melts, ﬂowing through the annular space, form alternating layers of A and B, as shown schematically in Figure 9.9. Idealized layer patterns generated from the rotating-mandrel tubular ﬁlm-blowing die system are shown schematically in Figure 9.10. The analysis of ﬂow through the rotating-mandrel tubular ﬁlm blowing die is reported by Han and Shetty (1978b). The rotating-mandrel tubular ﬁlm-blowing coextrusion die can generate a large number (even hundreds) of alternating layers of two polymers. The thickness of each layer can be made to be less than the wavelength of white light, such that the multilayer tubular blown-ﬁlm becomes iridescent. Such products are indeed available commercially, and they are used for packaging cosmetics, for instance.

Figure 9.9 Schematic of the layers that may be generated in the blown ﬁlm coextrusion using the rotating-mandrel die system. (Reprinted from Han and Shetty, Polymer Engineering and Science 18:187. Copyright © 1978, with permission from the Society of Plastics Engineers.)

388

PROCESSING OF THERMOPLASTIC POLYMERS Figure 9.10 Schematic of the

idealized layer patterns that may be generated in the blown ﬁlm coextrusion using the rotating-mandrel die system.

9.3

Polymer–Polymer Interdiffusion across the Initially Sharp and Flat Interface

When two layers of compatible (or miscible) molten polymers above their glass transitions temperatures (or above their melting temperatures) are forced to ﬂow side by side through a die, interdiffusion will take place across the initially sharp and ﬂat interface, ultimately leading to a homogeneous phase after a sufﬁciently long time. That is, polymer–polymer interdiffusion is a kinetic process. In the 1980s, polymer–polymer interdiffusion in miscible polymer systems was extensively discussed experimentally (Composto et al. 1988; Fytas 1987; Gilmore et al. 1980; Jones et al. 1986; Kanetakis and Fytas 1987, 1989; Murschall et al. 1986; Wu et al. 1986) and theoretically (Akcasu et al. 1986; Binder 1983; Brochard et al. 1983, 1984; Brochard and de Gennes 1985; Kramer et al. 1984; Sillescu 1984, 1987). Using the concept of reptation of entangled polymer chain dynamics presented in Chapter 4 of Volume 1, some investigators (Adolf et al. 1985; Jud et al. 1981; Kim and Wool 1983; Prager and Tirrell 1981; Wool and O’Connor 1981) considered the polymer–polymer interdiffusion occurring in welding and crack healing problems. All of those studies dealt with the polymer– polymer interdiffusion under static conditions. However, in the coextrusion process, where two miscible polymers are extruded side by side, a better understanding of polymer–polymer interdiffusion in the shear ﬂow ﬁeld is necessary. It can be easily surmised that the rate of interdiffusion across the initially sharp interface between two miscible polymers in the shear ﬂow ﬁeld would be quite different from that under static conditions because the orientation of polymer chains in the shear ﬂow ﬁeld may profoundly inﬂuence the rate of interdiffusion. In this section, we present polymer–polymer interdiffusion for a pair of miscible polymers, ﬁrst under static conditions and then in the shear ﬂow ﬁeld. When in Chapter 7 of Volume 1 we discussed the rheology of miscible polymer blends, we cited pairs of miscible polymers. Since the purpose of this section is to present the fundamentals associated with polymer–polymer diffusion across the initially sharp and ﬂat interface under static conditions or in the shear ﬂow ﬁeld, we will consider, for

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illustration purposes, a pair comprising poly(methyl methacrylate) (PMMA) and poly (vinylidene ﬂuoride) (PVDF). Note that the PMMA/PVDF pair is known to be miscible over the entire range of blend compositions (Nishi and Wang 1975; Noland et al. 1971; Paul and Altamirano 1975; Wendorff 1980).

9.3.1

Polymer–Polymer Interdiffusion under Static Conditions

Let us consider the situation where two semi-inﬁnite sheets of miscible polymers (polymers A and B) are brought together and heated to a temperature well above their glass transition temperatures or melting temperatures in an isothermal environment. The general Fick’s equation for one-dimensional diffusion across the initially sharp and ﬂat interface can be written as ∂φA ∂φA ∂ = D(φA) ∂x ∂x ∂t

(9.1)

where D(φA) is the concentration-dependent interdiffusion coefﬁcient. There are two theories that have been proposed to relate D(φA) to the intrinsic diffusion coefﬁcients of the respective components, DA and DB , by (1) Fast-mode theory (Kramer et al. 1984; Sillescu 1984): D(φA) = (1 − φA )DA + φA DB

(9.2)

(2) Slow-mode theory (Akcasu et al. 1986; Binder 1983; Brochard et al. 1983, 1984; Brochard and de Gennes 1985): 1/D(φA) = (1 − φA )/DA + φA /DB

(9.3)

where φi is the volume fraction of component i (i = A, B), and DA and DB , respectively, are the intrinsic diffusion coefﬁcients deﬁned by % & ∗ 1 − φA /NA + φA /NB + 2 |χ | φA 1 − φA DA = NA DA

(9.4a)

% & DB = NB DB∗ 1 − φA /NA + φA /NB + 2 |χ | φA 1 − φA

(9.4b)

and

∗ and D ∗ are the tracer-diffusion coefﬁcients of components A and B, in which DA B respectively, NA and NB are the degrees of polymerization of components A and B,

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PROCESSING OF THERMOPLASTIC POLYMERS

∗ and D ∗ are related respectively, and χ is the Flory–Huggins interaction parameter. DA B to molecular weights MA and MB by (see Chapter 4 of Volume 1) ∗ DA ∝ MA −2 ; DB∗ ∝ MB −2

(9.5)

∗ DA ∝ MA −1 ; DB∗ ∝ MB −1

(9.6)

in the reptation regime and

in the Rouse regime. Some experimental studies (Composto et al. 1988; Green et al. 1985; Jordan 1988; Kanetakis and Fytas 1989) support Eq. (9.2), while others (Brereton et al. 1987; Kanetakis and Fytas 1987; Murschall et al. 1986) support Eq. (9.3). It can be shown that the values of D(φA ) predicted by Eq. (9.3) will always be lower than those predicted ∗ D ∗ , Eq. (9.2) gives D ∼ D ∗ (dominated by the by Eq. (9.2). Note that for DA B A fast component) and Eq. (9.3) gives D ∼ DB∗ (dominated by the slow component), and ∗ and D ∗ depend on the composition of the that in the most general situations both DA B system. The fundamental difference between Eqs. (9.2) and (9.3) lies in that vacancy ﬂuxes across the interface were included in the derivation of Eq. (9.2) whereas they were neglected in the derivation of Eq. (9.3). Hess and Akcasu (1988) and Kehr et al. (1989) concluded that neither Eq. (9.2) nor (9.3) gives satisfactory description of the interdiffusion of a pair of miscible polymers in the long time limit. Let us consider that an inert marker is placed at the original interface at time zero, t = 0, as schematically shown in Figure 9.11. One would then expect mass transfer to occur, under static conditions, across the interface due to the interdiffusion of longchain macromolecules across the interface, forming a diffuse layer of ﬁnite thickness

Figure 9.11 Schematic showing the direction of one-dimensional interdiffusion ﬂuxes when

two miscible polymers A and B are placed initially (t = 0) with a sharp interface, where the broken vertical line represents the initial interface, the ﬁlled circle (䊉) represents gold marker, jA denotes the mass ﬂux of component A, jB denotes the mass ﬂux of component B, and jv denotes the ﬂux of vacancy. (Reprinted from Wu et al., Journal of Polymer Science, Polymer Physics Edition 24:143. Copyright © 1986, with permission from John Wiley & Sons.)

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that increases with time. The equation of continuity for component A is given by (Wu et al. 1986) ∂(ρwA ) + ∇ · WA = 0 ∂t

(9.7)

where ρ is the density of the mixture, wA is the weight fraction of component A, ∇ is the gradient operator, and WA is the total mass ﬂux, deﬁned by WA = jA − (vA jA − vB jB )ρwA

(9.8)

in which vA and vB are the partial speciﬁc volumes of components A and B, respectively, and jA and jB are the mass ﬂuxes of components A and B, respectively. For one-component mass transfer in the x-direction, jA and jB are given by jA = −ρDA

∂wA ; ∂x

jB = −ρDB

∂wB ∂x

(9.9)

When a mixture is considered to have zero excess volume of mixing, its speciﬁc volume v is given by v = wA vA + wB vB . Note that the speciﬁc volume v of the mixture is related to the density ρ of the mixture by v = 1/ρ and the volume fraction φA is related to the weight fraction wA by φA = wA vA /(wA vA + wB vB ). In the general situation under consideration here, the density ρ varies with both time t and distance x, and thus we have ∂w ∂ρ = −(vA − vB )ρ 2 A ; ∂t ∂t

∂w ∂ρ = −(vA − vB )ρ 2 A ∂x ∂x

(9.10)

Substituting Eqs. (9.8)–(9.10) into (9.7) and after rearranging, one obtains (Kim and Han 1991) ρvB

& ∂ 2 wA ∂wA % = DA −ρwA vA DA −vB DB − ∂t ∂x 2 % & ∂wA 2 ρ vA−vB DA −2ρ vA−vB wA vA DA−vB DB + vA DA−vB DB ∂x ∂wA ∂DA ∂ −ρwA v D −v D (9.11) + ∂x ∂x A A B B ∂x

Substituting Eq. (9.4) into Eq. (9.11), we obtain ∂wA ∂ 2 wA = g1 + g2 ∂τ ∂ξ 2 where

∂wA ∂ξ

2 (9.12)

DB∗ NB NA + 2φA (1 − φA ) |χ | NA (9.13) (1 − φA ) + φA g1 = (1 − wA ) + wA ∗ D A NA NB

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PROCESSING OF THERMOPLASTIC POLYMERS

and g2 =

D∗ N N 1−2vA ρ + 2vA ρ −1 B∗ B 1−φA +φA A +2φA 1−φA |χ |NA D A NA NB ∗ D N N +vA ρ 1−vA ρwA +vB ρwA B∗ B −1+ A +2 1−2φA |χ |NA DA NA NB ∂χ + 2φA 1−φA NA (9.14) ∂φA

In Eq. (9.12), τ and ξ are dimensionless variables deﬁned as ∗ τ = (DA /L2 )t;

ξ = x/L

(9.15)

where L is a characteristic length. When vA DA = vB DB , vA = vB = v = 1/ρ, and DA is independent of x (hence, independent of position), Eq. (9.12) reduces to the well-known expression ∂wA ∂ 2 wA = DA ∂t ∂x 2

(9.16)

When ρA = ρB = ρ and vA = vB , the volume fraction becomes identical to the weight fraction of wA and Eq. (9.12) reduces to ∂φA ∂ = ∂t ∂x

∂φ D A ∂x

(9.17)

where D is deﬁned by Eq. (9.2). The mean-square interfacial layer thickness d 2 can be calculated with the expression (Wu et al. 1986) d2 = 4

∞

−∞

∂wB (x − x0 )2 dx ∂x

∞ −∞

∂wB dx ∂x

(9.18)

where x0 is the initial interfacial position. In order to estimate the mean-square interfacial thickness d 2 under static conditions, as deﬁned by Eq. (9.18), one must solve Eq. (9.12) subject to the following boundary and initial conditions: wA (−∞, t) = 0

for t ≥ 0

(9.19a)

wA (+∞, t) = 1

for t ≥ 0

(9.19b)

wA (ξ, 0) = 0

for −∞ < ξ ≤ 0

(9.19c)

wA (ξ, 0) = 1

for 0 < ξ < ∞

(9.19d)

When experimental data for the tracer-diffusion coefﬁcients are not available, they may o and D o , be replaced by the self-diffusion coefﬁcient2 for the component polymers, DA B

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393

from (Graessley 1980): Dio =

GoNi 135

ρi RT GoNi

2

Roi 2 Mi

Mci 2 Mi η0i (Mci )

(i = A, B)

(9.20)

where GoNi is the plateau modulus for component i, ρi is the density of component i, R is the universal gas constant, T is the absolute temperature, Roi 2 is the unperturbed mean-square end-to-end distance of component i, Mi is the molecular weight of component i, Mci is the critical molecular weight of component i, and η0i (Mci ) is the zero-shear viscosity of component i with the critical molecular weight Mci . Under such circumstances, the predicted values of the interfacial layer thickness must be regarded as approximate. When the value for η0i (Mci ) is not available, the relationship η0i (Mci ) = η0i (Mi )

Mci Mi

3.4 (i = A, B)

(9.21)

may be used, yielding Dio

Go = Ni 135

ρi RT GoNi

2

Roi 2 Mi

Mci η0i (Mi )

Mi Mci

3.4

1 Mi 2

(9.22)

Figure 9.12 gives transmission electron micrographs showing the interdiffusion that initially took place when two ﬂat sheets of PMMA and PVDF were placed side by side. The sample was prepared by sputtering gold particles of about 5 nm on the surface of PVDF in a vacuum chamber (Wu et al. 1986). The sputtered gold covered uniformly about 8% of the entire specimen surface. These gold particles served as markers for tracing the interdiffusion taking place, under static conditions, between the PMMA phase and the PVDF phase after the PMMA/PVDF/PMMA sandwich assembly was placed in a brass frame and heated to 190 ◦ C in a nitrogen atmosphere between heated platens. It can be seen in Figure 9.12 that the inert gold markers migrated either toward the amorphous PMMA phase or toward the crystalline PVDF phase, depending on the ratio of the molecular weights of the PVDF and PMMA used. Note that PVDF begins to crystallize in a mixture of PVDF and PMMA containing 60 wt % or more of PVDF. Since the molecular weight affects the interdiffusion coefﬁcients of the constituent components, as will be shown later in this section, we can conclude that the inert gold markers migrated either toward the PVDF phase or toward the PMMA phase, depending upon the relative magnitudes of the interdiffusion coefﬁcients of the PVDF and PMMA. We can conclude from Figure 9.12 that convective mass transfer occurred between the PVDF and PMMA phases during interdiffusion, which is attributable to the existence of vacancies in the respective phases. o to be 1.05 × Using Eq. (9.22), Kim and Han (1991) calculated the value of DA ◦ −12 2 cm /s for PMMA at 230 C, for which the following values of the various 10 parameters were employed:3 GoN = 6 × 105 Pa, ρ = 1.088 g/cm3 , Ro 2 /M = 0.41 × 10−16 cm2 , η0 = 1.023 × 104 Pa·s at 230 ◦ C, and M = 1.075 × 105 . Since the value for Mc was not available, it was calculated from the relationship Mc = 2Me = 2ρRT/GoN (see Chapter 4 of Volume 1), and obtained Mc = 1.517× 104 at 230 ◦ C.

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PROCESSING OF THERMOPLASTIC POLYMERS

Figure 9.12 Transmission electron micrographs showing the movement of inert gold markers during interdiffusion between PMMA and PVDF at 190 ◦ C under static conditions. (a) The gold markers initially deposited at the sharp interface between the PMMA and PVDF sheets have moved into the amorphous phase 192 min after interdiffusion began, where the amorphous PMMA has Mw = 1.5 × 105 and the semicrystalline PVDF phase has Mw = 5.78 × 105 . (b) The gold markers initially deposited at the sharp interface between the PMMA and PVDF sheets have moved into the semicrystalline phase after interdiffusion began, where the PMMA has Mw = 1.5 × 105 and the PVDF phase has Mw = 1.88 × 105 . (Reprinted from Wu et al., Journal of Polymer Science, Polymer Physics Edition 24:143. Copyright © 1986, with permission from John Wiley & Sons.)

In calculating the value of DBo for PVDF at 230 ◦ C, the following relationship was used (Kim and Han 1991): o (GoN A /η0A )(RoA 2 /MA )(MA /MB )1.4 DA = o DB (GoN B /η0B )(RoB 2 /MB )(McA /McB )0.4

(9.23)

with the aid of ρi RT /GoNi = Mei = Mci /2 (i = A, B) in Eq. (9.22), where GoN A and GoN B are the plateau moduli of components A and B, respectively, η0A and η0B are the zero-shear viscosities of components A and B, respectively, RoA 2 and RoB 2 are the unperturbed mean-square end-to-end distances of components A and B, respectively, MA and MB are the molecular weights of components A and B, respectively, and McA and McB are the critical molecular weights of components A and B, respectively. o /D o = 0.5 and thus D o = 2.1 × 10−12 cm2 /s, for which the It was found that DA B B following values of the various parameters were employed:4 GoN = 4 × 105 Pa,

395

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/

∗t Figure 9.13 Plots of computed interfacial composition proﬁle wB versus u = x/2 DA ∗ 5 −12 2 cm /s for PMMA, and |χ| = 0.3: for MA = 1.08 × 10 for PMMA, DA = 1.05 × 10 ∗ ∗ ∗ /D ∗ = 27.26, (b) M /M (a) MA /MB = 0.1 and DA A B = 0.4 and DA /DB = 1.72, B ∗ /D ∗ = 0.5, and (d) M /M = 2.7 and D ∗ /D ∗ = 0.038. (c) MA /MB = 0.74 and DA A B B B A

Here, component B represents PVDF. The broken line represents the solution of Eq. (9.12) ∗ = D ∗ , N = N , and ρ = ρ = ρ. (Reprinted from Kim and Han, Polymer Engiwith DA A B A B B neering and Science 31:258. Copyright © 1991, with permission from the Society of Plastics Engineers.)

ρ = 1.514 g/cm3 , Ro 2 /M = 0.432 × 10−16 cm2 , η0 = 0.404 × 104 Pa·s at 230 ◦ C, and M = 1.45 × 105 . Since the value of Mc for PVDF was not available, it was calculated from the relationship Mc = 2ρRT/GoN , and Mc = 3.166 × 104 at 230 ◦ C. Figure 9.13 gives composition proﬁles plotted against the dimensionless ) interfacial ∗ t, for four values of M /M , and thus of D ∗ /D ∗ , in which variable, u = x/2 DA A B B A subscript A refers to PMMA and subscript B refers to PVDF. Note that the introduction of the dimensionless variable u permits us to have a single curve for a given pair of PMMA and PVDF. This indicates that the interfacial layer thickness d is proportional to the square root of diffusion time t, that is, d ∝ t 1/2 . The broken straight line in Figure 9.13 represents the expression φB = 1/2 − u/30.0 for −15.0 ≤ u ≤ 15.0

(9.24a)

φB = 1.0

for u ≤ −15.0

(9.24b)

φB = 0.0

for u ≥ 15.0

(9.24c)

∗ = D∗ , which is the solution for a special case of Eq. (9.12); speciﬁcally, for DA B NA = NB and ρA = ρB = ρ. It can be seen in Figure 9.13 that as the molecular weight of PVDF is decreased, the shape of the interfacial composition proﬁle tends to become more asymmetric. This means that the distance penetrated by the PVDF phase into the PMMA phase is greater than the distance penetrated by the PMMA phase into the PVDF phase. Figure 9.14 shows the dependence of the interfacial layer thickness d on the molecular weight of PVDF. It can be seen in Figure 9.14 that d is proportional to MB −1/2 up to a certain value of MB , and then it decreases much more slowly, approaching a limiting value as the value of MB becomes very large. This can be explained upon

396

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 9.14 Plots of computed interfacial layer thickness d versus molecular weight of PVDF, ∗ = 1.05 × 10−12 cm2 /s for PMMA, and |χ| = 0.3, MB , for MA = 1.08 × 105 for PMMA, DA

at various diffusion periods (s): (a) 1, (b) 10, (c) 60, and (d) 600. (Reprinted from Kim and Han, Polymer Engineering and Science 31:258. Copyright © 1991, with permission from the Society of Plastics Engineers.)

examination of the expression D(φA ) ≈

∗ DA

1 − φA + φA

DB∗ ∗ DA

NA NB

2φA 1 − φA |χ | NA

(9.25)

Note that Eq. (9.25) can be obtained from Eq. (9.2) if the entropic term in Eq. (9.4) is neglected for large values of NA and NB . Notice that as NB increases to a sufﬁciently ∗ )(N /N ) negligibly small compared with large value, making the value of (DB∗ /DA B A the value of (1 − φA ), Eq. (9.25) reduces to ∗ D(φA ) ≈ 2DA φA (1 − φA )2 |χ | NA

(9.26)

Equation (9.26) suggests that D(φA ) can become independent of NB for large values and thus of the molecular weight of PVDF. Figure 9.15 gives ) ∗ interfacial composition proﬁles under static conditions plotted t for different values of |χ |, which was assumed to be indeagainst u = x/2 DA pendent of composition. Figure 9.16 ) gives interfacial composition proﬁles under static ∗ |χ | t, for different values of |χ |. It can be conditions plotted against u∗ = x/2 2DA seen in Figure 9.16 that the introduction of the dimensionless variable u∗ yields a master curve, which is independent of |χ |. This means that the interfacial layer thickness d is proportional to the square root of |χ |. This observation indicates that as the degree of compatibility of a pair of polymers increases, the interfacial layer thickness increases accordingly.

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397

/

∗ t for Figure 9.15 Plots of computed interfacial composition proﬁle wB versus u = x/2 DA ∗ = 1.05 × 10−12 cm2 /s for PMMA, MA = 1.08 × 105 for PMMA, MA /MB = 0.74, DA ∗ /D ∗ = 0.5 for different values of |χ|: (a) 0.01, (b) 0.05, (c) 0.15, and (d) 0.30. and DA B

(Reprinted from Kim and Han, Polymer Engineering and Science 31:258. Copyright © 1991, with permission from the Society of Plastics Engineers.)

So far, both PMMA and PVDF have been assumed to be monodisperse. In reality, both PMMA and PVDF used in Figure 9.12 are polydisperse. In order to observe the effect of polydispersity on interdiffusion between PMMA and PVDF, let us consider the situation where the PMMA is monodisperse while the PVDF is polydisperse and assume that the polydispersity is described by the log-normal distribution function. In solving Eq. (9.12) for the interdiffusion between a monodisperse PMMA and a polydisperse PVDF, let us assume that the weight fraction of PVDF, wB , may be

/

∗ |χ |t Figure 9.16 Plots of computed interfacial composition proﬁle wB versus u∗ = x/2 2DA ∗ for MA = 1.08 × 105 for PMMA, MA /MB = 0.74, DA = 1.05 × 10−12 cm2 /s for PMMA, ∗ /D ∗ = 0.5 for different values of |χ |: ( ) 0.3, () 0.15, () 0.05, and () 0.01. and DA 䊉 B

(Reprinted from Kim and Han, Polymer Engineering and Science 31:258. Copyright © 1991, with permission from the Society of Plastics Engineers.)

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PROCESSING OF THERMOPLASTIC POLYMERS

calculated using the expression (Kim and Han 1991) wB =

N #

∗ B,i wB,i (DB,i )

(9.27)

i=1

where B,i is the ith fraction of the molecular weight MB for PVDF and wB,i is the interfacial composition proﬁle, calculated numerically, for the molecular weights MA ∗ for for PMMA and MB,i for PVDF. Note that in Eq. (9.27) one has to compute DB,i each weight fraction of the PVDF having molecular weight MB,i before beginning to solve numerically Eq. (9.12). ) ∗ t, Figure 9.17 gives interfacial composition proﬁles plotted against u = x/2 DA for different values of polydispersity, MwB /MnB , and the constant value of |χ | = 0.3. In Figure 9.17, we observe that the interfacial composition proﬁle is smeared at high values of u as the polydispersity is increased. This means that the interfacial layer thickness increases as the polydispersity is increased, which indeed is predicted, as may be seen in Figure 9.18. This prediction is made by substituting Eq. (9.27) into (9.18), that is, by using the expression (Kim and Han 1991) 4 dpo 2 =

N i=1

B,i N i=1

∞ ∂wB,i −∞

B,i

∂x

∞

−∞

(x − x0 )2 dx

∂wB,i ∂x

(9.28) dx

where dpo denotes the interfacial layer thickness for a polydisperse polymer. As discussed in reference to Figure 9.14 for monodisperse PVDF, the interfacial layer thickness increases as the molecular weight of PVDF decreases. Therefore, we can conclude that an increase in interfacial layer thickness with increasing polydispersity, observed in Figure 9.18, is attributable to the presence of the smaller macromolecules in the polydisperse PVDF.

/

∗ t for Figure 9.17 Plots of computed interfacial composition proﬁle wB versus u = x/2 DA ∗ 5 −12 2 cm /s for PMMA, |χ| = 0.3, and MwB = MA = 1.08 × 10 for PMMA, DA = 1.05 × 10 1.45 × 105 for PVDF, and different values of MwB /MnB : (a) 1.0, (b) 1.4, (c) 2.4, and (d) 3.54.

(Reprinted from Kim and Han, Polymer Engineering and Science 31:258. Copyright © 1991, with permission from the Society of Plastics Engineers.)

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Figure 9.18 Plots of computed interfacial layer thickness d versus diffusion time t for different

values of MwB /MnB for PVDF: (a) 1.0, (b) 1.4, (c) 2.4, and (d) 3.54. Other conditions are the same as in Figure 9.17. The broken lines are based on Eq. (9.31). (Reprinted from Kim and Han, Polymer Engineering and Science 31:258. Copyright © 1991, with permission from the Society of Plastics Engineers.)

However, Eq. (9.28) does not give us insight as to how polydispersity inﬂuences the interfacial layer thickness d. To obtain an analytical expression that enables us to describe how polydispersity inﬂuences d, the following relationship is useful (Kim and Han 1991):

dpo 2 =

N #

⎧ ∞ ⎨4 ∂w B,i

i=1

⎩

2 ⎫ x − x dx ⎬ 0 −∞ ∞ ⎭ ∂wB,i /∂x dx −∞ B,i /∂x

(9.29)

When the relationship (see Figure 9.14 for the case where molecular weight M is less than a certain critical value) di 2 (MB,i ) = K/MB,i

(9.30)

is used in Eq. (9.29), one obtains dpo = dmo (Mw /Mn )1/2

(9.31)

where dmo is the interfacial layer thickness for a monodisperse polymer with molecular weight MwB . Equation (9.31) clearly shows how polydispersity inﬂuences interfacial layer thickness. For comparison, the predictions of interfacial layer thickness based on Eq. (9.31) are also given in Figure 9.18, as denoted by the broken lines. The purpose of having derived Eq. (9.31) is not to replace the exact expression, Eq. (9.29), but to demonstrate in an analytical form how the interfacial layer thickness is affected by polydispersity.

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PROCESSING OF THERMOPLASTIC POLYMERS

9.3.2

Polymer–Polymer Interdiffusion in the Shear Flow Field

Let us turn our attention to polymer–polymer interdiffusion in the shear ﬂow ﬁeld, which is relevant to the coextrusion of two miscible polymers. This situation is quite different from that considered above for static conditions in that in the shear ﬂow ﬁeld the polymer chains are oriented and thus both the rate of polymer–polymer interdiffusion and the interfacial composition proﬁle would be greatly affected by the extent of chain orientation in the ﬂow ﬁeld. This means that the extent of chain orientation must be related to the kinematics of the shear ﬂow ﬁeld. For illustration, let us consider the two-layer coextrusion in a ﬂat-ﬁlm or sheet-forming die, shown schematically in Figure 9.19a. When two polymers A and B have different viscosities, they develop velocity proﬁles in the ﬂow channel, as shown schematically in Figure 9.19b. When the two polymers are miscible, polymer chains will diffuse across the original interface, forming a diffuse interface. In the shear ﬂow ﬁeld, the polymer chains will have average orientation angles in the respective phases, which are greater than 45◦ , as shown schematically in Figure 9.19b. Note that when there is no ﬂow (i.e., in random distribution), the polymer chains can be assumed to have an average orientation angle of 45◦ . During coextrusion, a diffuse interface (i.e., an interphase) is developed by the interdiffusion of two miscible polymers across the original interface. The interphase grows continuously during ﬂow and the rate of formation of an interphase depends on the orientation of polymer chains. A rigorous analysis of the problem at hand is therefore very complicated in that one must relate how the shear ﬂow ﬁeld affects the chain orientation, which in turn affects the rate of polymer–polymer interdiffusion. Thus, here we take a simpliﬁed approach to derive analytical expressions that will enable us to determine the interfacial layer thickness and consequently the adhesive bond strength of coextruded ﬁlm (or sheet) when two miscible polymers ﬂow side by side in the shear ﬂow ﬁeld (Kim and Han 1991). Referring to the schematic given in Figure 9.19a, for simplicity let us assume that the ﬂow is fully developed when two melt streams ﬁrst meet each other inside the die, and that interface migration due to the difference in viscosity is negligible. Then, the shear stress at the interface (σint ) can be calculated by solving the equation of motion in rectangular coordinates (x, y, z): −

∂p ∂σyz + =0 ∂z ∂y

(9.32)

for phases A and B, respectively, subject to the boundary conditions: (σyz )A = (σyz )B = σint at y = δ

(9.33a)

(vz )A = 0 at y = 0

(9.33b)

(vz )B = 0 at y = h

(9.33c)

where σyz is the shear stress, δ is an interface, and (vz )A and (vz )B are the velocities for phases A and B, respectively. Equation (9.33a) assumes that shear stress is continuous at the interface and the interface remains sharp during ﬂow. We are well aware of the

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401

Figure 9.19 (a) Schematic of a two-layer sheet-forming coextrusion die, and (b) schematic

depicting the velocity proﬁles and average orientation of macromolecules in stratiﬁed twocomponent ﬂow in a sheet-forming coextrusion die. (Reprinted from Kim and Han, Polymer Engineering and Science 31:258. Copyright © 1991, with permission from the Society of Plastics Engineers.)

fact that Eq. (9.32) may be valid for two immiscible polymers, but certainly not for two miscible polymers under consideration. As will be shown below, however, we will use an ad hoc approach to estimate the interface thickness during coextrusion of two miscible polymers. In solving Eq. (9.32), let us assume that each of the two miscible polymers in coextrusion follows the truncated power law (see Chapter 6 of Volume 1): σyz =

η0 γ˙ η0 (γ˙ /γ˙0 )n γ˙

for γ˙ ≤ γ˙0 for γ˙ > γ˙0

(9.34)

402

PROCESSING OF THERMOPLASTIC POLYMERS

where η0 is the zero-shear viscosity, γ˙ is shear rate, dvz /dy, and γ˙0 is the critical shear rate at which the viscosity η starts to deviate from η0 . Substituting Eq. (9.34) into (9.32) and then solving the resulting expression, subject to the boundary conditions given by Eq. (9.33), we can obtain the velocity proﬁles and values of σint for a set of given processing conditions. Note that the above analysis can easily be extended to multilayers consisting of more than two miscible polymers. Note also that for a given pair of polymers and extrusion temperature, the value of σint depends on both the total ﬂow rate and the position of the interface. Thus, the interfacial layer thickness and thus adhesive bond strength should depend on the total volumetric ﬂow rate and also on the position of the interface (i.e., the layer thickness ratio of the two polymers). The interfacial layer thickness formed during coextrusion can be predicted with Eq. (9.18) when the effect of shear ﬂow on the interdiffusion coefﬁcient is properly taken into account when solving Eq. (9.12). Apparent intrinsic diffusion coefﬁcients, 1 and D 1 , may be deﬁned by D A B 1 =α D ; D A A A

1 =α D D B B B

(9.35)

where DA and DB are deﬁned by Eq. (9.4), and αA and αB are orientation factors in steady-state shear ﬂow for polymers A and B, respectively, which are deﬁned by (see Appendix) % & o σ ) /2 cos (π/4) + tan−1 (Jeb,i int (i = A, B) (9.36) αi = cos(π/4) o where σint is the shear stress at the interface and Jeb,i is the steady-state shear compliance for component i in a polydisperse polymer. There are several different ways of relating the steady-state shear compliance for o to the steady-state shear compliance for monodisperse polypolydisperse polymers Jeb o mers Je , as suggested in the literature (Graessley 1982; Kurata 1984; Mills 1969; Montfort et al. 1978). Let us use the following relationship (Kurata 1984): o Jeb,i = Jeio (Mw /Mn )3

(9.37)

Note that Jeo for monodisperse polymers can be determined from information on the plateau modulus GoN , by using the relationship (Graessley 1982) Jeo GoN ≈ 3.0 From Eqs. (9.36)–(9.38) we obtain (Kim and Han 1991) ' (2 cos (π/4) + tan−1 3σint (Mwi /Mni )3 /GoNi 2 αi = cos (π/4)

(9.38)

(9.39)

In Eq. (9.39) we observe that αi (i = A, B) depends on the shear stress at the interface σint , the plateau moduli GoNi , and polydispersity Mw /Mn . Note in Eq. (9.39)

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403

that αi (i = A, B) will become unity when there is no ﬂow and will decrease steadily from unity as σint and polydispersity increase. Let us now calculate σint and velocity proﬁles of two melt streams, PMMA and PVDF, which are fed separately into a sheet-forming coextrusion die having the following dimensions (see Figure 9.19a): (a) width (w) of 1 m, (b) die opening (h) of 2 × 10−3 m, and (c) die length (L) of 5 × 10−2 m. Our objective here is ﬁrst to calculate σint for the PMMA and PVDF coextruded under different ﬂow rates in the sheet-forming die and then to calculate the orientation factors, αA and αB , deﬁned by 1 can be estimated using Eq. (9.35). 1 and D Eq. (9.39), so that the values of D A B Figure 9.20 gives the velocity proﬁles of the PMMA/PVDF layers for the situation where the interfacial position lies along the centerline of the die, with increasing volumetric ﬂow rate going from curve (1) to curve (3). The volumetric ﬂow rates used to generate the velocity proﬁles in Figure 9.20 are given in Table 9.1. In this numerical example, the following values of parameters were used: (1) for PMMA, η0 = 1.023 × 104 Pa·s at 230 ◦ C, n = 0.44, γ˙0 = 1.7 s−1 at 230 ◦ C; GoN = 6 × 105 Pa, Mw = 1.075 × 105 , and Mw /Mn = 2.16, and (2) for PVDF, η0 = 0.404 × 104 Pa·s at 230 ◦ C, n = 0.62, γ˙0 = 1.0 s−1 at 230 ◦ C, GoN = 4 × 105 Pa, Mw = 1.45 × 105 , Mw /Mn = 2.31. Also given in Table 9.1 are the computed results of σint , αA , and αB for the nine cases (runs 1–9). It can be seen in Table 9.1 that for runs 1–3, the values of αA and αB decrease little as the volumetric ﬂow rates are increased; speciﬁcally, even as the volumetric ﬂow rate is increased by ﬁvefold. This observation is attributed to the fact that the interfacial position remains almost at the centerline of the die opening,

Figure 9.20 Computed velocity proﬁles in the coextrusion of a PMMA/PVDF system in a sheet-forming coextrusion die. The numbers on the curves represent the run numbers given in Table 9.1. (Reprinted from Kim and Han, Polymer Engineering and Science 31:258. Copyright © 1991, with permission from the Society of Plastics Engineers.)

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PROCESSING OF THERMOPLASTIC POLYMERS

Table 9.1 Orientation factors α A and α B at the interface of PMMA and PVDF in a sheet-forming coextrusion die

Flow Rate (cm3 /min)

Interface Positiona

σint × 10−4 (Pa)

dshear αB

(nm)b

(a) Mw,PMMA = 1.075 × 105 and η0,PMMA = 1.023 × 104 Pa·s at 230 ◦ Cc 1 400 400 0.52 0.746 0.81 2 1200 1200 0.51 1.014 0.74 3 2000 2000 0.51 1.109 0.72 4 3000 1000 0.70 2.561 0.46 5 3500 500 0.81 5.012 0.27 6 3800 200 0.89 7.134 0.19 7 1000 3000 0.34 4.262 0.31 8 500 3500 0.24 5.629 0.24 9 200 3800 0.16 6.498 0.21

0.66 0.56 0.53 0.28 0.15 0.11 0.18 0.13 0.12

1080d 580e 440 325 240 210 265 230 215

(b) Mw,PMMA = 1.50 × 105 and η0,PMMA = 3.175 × 104 Pa·s at 230 ◦ Cf 10 2000 2000 0.57 0.551 0.86 11 3500 500 0.84 4.904 0.27 12 3800 200 0.92 10.300 0.13 13 1000 3000 0.45 8.725 0.16 14 200 3800 0.29 9.555 0.14

0.74 0.15 0.07 0.09 0.08

480 230 160 180 165

Run #

PMMA

PVDF

αA

a

The interface position is deﬁned as δ/ h (see Figure 9.19a). Note that in all cases considered, Mw,PVDF = 1.45 × 105 and ◦ η0,PVDF = 0.404 × 104 Pa·s at 230 C. b The diffusion period (i.e., the average residence time in the die) for runs 3–14 is 1.5 s, whereas it is 7.5 s for run 1, and 2.5 s for run 2. c dstatic for the diffusion period of 1.5 s is 580 nm. d dshear corresponding to the diffusion period of 1.5 s is 480 nm for run 1. e dshear corresponding to the diffusion period of 1.5 s is 450 nm for run 2. f dstatic for the diffusion period of 1.5 s is 560 nm. Reprinted from Kim and Han, Polymer Engineering and Science 31:258. Copyright © 1991, with permission from the Society of Plastics Engineers.

where the shear stress is the lowest in the ﬂow channel. Note in Figure 9.20 that there is a relatively small change in the curvature of the velocity proﬁle at the interface. When the value of η0 of PMMA is increased from 1.023 × 104 to 3.175 × 104 Pa·s by increasing the molecular weight of PMMA from 1.075 × 105 to 1.50 × 105 , while keeping the same value of η0 for PVDF, we observe in Figure 9.21 that the velocity gradient at the interface changes very sharply. Also, the values of αA and αB decrease rapidly when the ratio of volumetric ﬂow rates of PMMA and PVDF is increased, while keeping the total ﬂow rate constant, as may be seen in Table 9.1 (runs 10–14). Note in Figure 9.21 that as the ratio of volumetric ﬂow rates of PMMA and PVDF is increased, the interface position moves toward the PVDF phase. Furthermore, as the ratio of volumetric ﬂow rates of PMMA and PVDF is decreased, the interface position moves toward the PMMA phase, as shown in Figure 9.22. The computed values of αA and αB corresponding to these ﬂow conditions (runs 10–14) are also given in Table 9.1. 1 decrease due to a decrease in the values of α 1 and D Note that as the values of D A B A and αB at a constant contact time, the adhesive bond strength of the coextruded product would be weaker.

Figure 9.21 Computed velocity proﬁles in the coextrusion of a PMMA/PVDF system in a sheet-forming coextrusion die. The numbers on the curves represent the run numbers given in Table 9.1. (Reprinted from Kim and Han, Polymer Engineering and Science 31:258. Copyright © 1991, with permission from the Society of Plastics Engineers.)

Figure 9.22 Computed velocity proﬁles in the coextrusion of a PMMA/PVDF system in a sheet-forming coextrusion die. The numbers on the curves represent the run numbers given in Table 9.1. (Reprinted from Kim and Han, Polymer Engineering and Science 31:258. Copyright © 1991, with permission from the Society of Plastics Engineers.)

405

406

PROCESSING OF THERMOPLASTIC POLYMERS

We can now estimate the interfacial layer thickness under shear ﬂow conditions, 1 and D 1 deﬁned by Eq. (9.35), in place of D dshear , by solving Eq. (9.12) with D A B A and DB , and the results are summarized in Table 9.1. Note that the values of dshear given in Table 9.1 were obtained with χ = −0.3 for the PMMA/PVDF pair. Also given in the footnotes of Table 9.1 are the values of interfacial layer thickness under static conditions, dstatic . Based on the computed values of dshear and dstatic given in Table 9.1 Kim and Han (1991) obtained a correlation with the form ) dshear /dstatic ∼ = αavg

(9.40)

√ where αavg = αA αB . The usefulness of Eq. (9.40), though empirical, lies in that the interfacial layer thickness under shear ﬂow conditions can be estimated within the accuracy of about 10%, without necessarily solving Eq. (9.12), using information on the interfacial layer thickness under static conditions and the orientation factors for components A and B, αA and αB . The following observations can further be made from Table 9.1. As the viscosity (or molecular weight) of PMMA increases, the interfacial layer thickness decreases since the diffusion coefﬁcient decreases with increasing viscosity (or molecular weight). As the interfacial position moves toward the die wall (see Figures 9.21 and 9.22), the interfacial layer thickness decreases rapidly. In practical terms, this observation suggests that, due to the orientation of polymer chains in the shear ﬂow ﬁeld, the interface layer thickness, and hence adhesive bond strength, will decrease as the thickness ratio of the two layers in a coextruded product deviates from unity. In the coextrusion of two miscible polymers, one would like to be able to estimate the adhesive bond strength between the layers. It has been suggested (Wu 1982) that in order to achieve a high adhesive bond strength, polymer chains from each layer must diffuse to form an interfacial layer thickness of at least one entanglement mesh size. There will be virtually no adhesive bond strength when two polymers being coextruded are immiscible. This then suggests that adhesive bond strength would depend on the extent of miscibility (i.e., the value of the interaction parameter χ for a given polymer pair), which in turn controls the interfacial layer thickness. We have already shown that the interfacial layer thickness of coextruded sheet or ﬁlm depends, among many factors, on the rate of interdiffusion and the contact (or residence) time inside the die. Using information given in Table 9.1, we can estimate adhesive bond strength in terms of the number of chains within a given interphase. For example, for run 12 in Table 9.1, we determine the interfacial layer thickness to be about 160 nm in a coextruded PMMA/PVDF sheet under these particular processing conditions. Assuming that the interphase consists of ideal Gaussian chains of PMMA and PVDF, we calculate the radius of gyration (Rg ) to be about 25 nm for both PMMA and PVDF. This means that in the interphase there is a thickness of about 6 chain lengths, suggesting that the residence time of 1.5 s for the particular PMMA/PVDF pair is sufﬁciently long to obtain a strong adhesive bond. The adhesive bond strength under static conditions, Gstatic , is proportional to the square root of the interfacial layer thickness Gstatic ∝

) dstatic

(9.41)

COEXTRUSION

407

Based on the results presented above, however, the adhesive bond strength under shear ﬂow conditions, Gshear , can be related to the diffusion time t, molecular weight M, the orientation factor αavg and the interaction parameter χ by Gshear ∝ t 1/4 M −1/4 αavg 1/4 |χ |1/4

(9.42)

This consideration indicates that the adhesive bond strength Gshear of coextruded sheet (or ﬁlm) decreases with increasing extrusion rate when the ratio of the volumetric ﬂow rates of PMMA and PVDF is kept constant, since the time available for the polymer chains to diffuse will decrease with increasing extrusion rate. It also indicates that Gshear decreases with an increase in the ratio of volumetric ﬂow rates of PMMA and PVDF when the total ﬂow rate is kept constant, since the orientation factor decreases rapidly accordingly.

9.4

Nonisothermal Coextrusion

As mentioned in the introduction to this chapter, nonisothermal coextrusion is practiced in the plastics industry when there is a large difference between the melting temperatures of the component polymers and one of the component polymers may undergo thermal degradation at temperatures above the melting temperature of the other components. For example, when coextruding nylon 6 and PEVA, nylon 6 must be heated to about 250 ◦ C, while PEVA is heated to about 200 ◦ C, because the melting point of nylon is 220–225 ◦ C and PEVA may undergo thermal degradation at temperatures above 220 ◦ C. As mentioned above, PEVA-g-MA must be used as a tie layer because nylon 6 and PEVA are not miscible. The feed temperature of PEVA-g-MA may be chosen as 200 ◦ C. Under such circumstances, the die temperature may be set at 230 ◦ C. This means that the production of PEVA/PEVA-g-MA/nyon 6 three-layer ﬁlms must be analyzed using nonisothermal coextrusion. In this section, we present an analysis of nonisothermal coextrusion in a multimanifold die. For illustration, let us consider a two-layer sheet-forming die. Since the ﬂow in each manifold can be treated as for that in a coat-hanger die (see Chapter 1), here we only consider the ﬂow in the slit die section where two melt streams ﬂow side by side, as schematically shown in Figure 9.23. For simplicity of analysis, we make the following assumptions: (1) the ﬂow is at steady state (i.e., the velocity and temperature do not vary with time at a given position z in the ﬂow channel), (2) the polymer melts considered are incompressible, (3) the thermal conductivity k, speciﬁc heat capacity cp , and density ρ of the polymer melts under consideration have little temperature dependence, (4) the opening of the ﬂow channel is very small compare with its width w (i.e., h w), so the ﬂow may be considered to be unidirectional, (5) the external forces and inertial effects are negligibly small, (6) heat conduction in the ﬂow direction z is negligibly small compared with that in the y direction (see the coordinate system shown in Figure 9.23), (7) convective heat transfer in the y direction is negligibly small compared with that in the z direction, and (8) the rheological

408

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 9.23 Schematic showing the movement of the interface of two melt streams having

different temperatures at the die inlet. (Reprinted from Chin et al., Polymer Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.)

behavior of the polymer melts under consideration may be described by a truncated power-law model deﬁned by η(γ˙ , T ) =

ko exp(−bT ) ko exp(−bT )(γ˙ /γ0 )n−1

for γ˙ ≤ γ˙0 for γ˙ > γ˙0

(9.43)

where γ˙ is shear rate, ko is the preexponential factor, b is a constant, T is the absolute temperature, n is a power-law index, and γ˙0 is the value of shear rate at which η deviates from Newtonian behavior. Note that for small changes in temperature T , the error involved in the use of exp(−bT), instead of the Arrhenius relationship, is negligible. Under these assumptions, the equations of continuity, momentum and energy in rectangular Cartesian coordinates may be expressed by ∂vy ∂y

+

∂vz =0 ∂z

∂ ∂p − + ∂z ∂y

∂v η z ∂y

(9.44) =0

∂vz 2 ∂ 2T ∂T =k 2 +η ρcp vz ∂z ∂y ∂y

(9.45)

(9.46)

To facilitate our analysis, let us introduce the following dimensionless variables: (y*, z*) = (y/h, z/h), (vz∗ , vy∗ ) = (vz /V z , vy /V z ), α ∗ = α/ h, γ˙ ∗ = γ˙ h/V z , ), η∗ = η/η , and θ = b T , where subscript A ∂p∗ /∂z∗ = (∂p/∂z)(h2 /η0,A V z 0,A A refers to phase A (or polymer A), α is the initial interface position (see Figure 9.23), η0 is zero-shear viscosity, and V z is the average velocity deﬁned by V z = Q/ h, with Q being the volumetric ﬂow rate per unit channel width. For convenience, throughout the

COEXTRUSION

409

ensuing analysis, we shall drop the asterisks from the dimensionless variables deﬁned above. Now, Eqs. (9.44)–(9.46) can be rewritten as follows: (1) For phase A: ∂vy,A ∂y − PeA vz,A

∂p ∂z

∂θA ∂z

∂vz,A

+

=0 ∂z ∂vz,A ∂ + ηA =0 ∂y ∂y ∂vz,A 2 ∂ 2 θA = + N a η A A ∂y ∂y 2

(9.47) (9.48) (9.49)

(2) For phase B: ∂vy,B ∂y − PeB vz,B

∂p ∂z

∂θB ∂z

+

∂vz,B

=0 ∂z ∂vz,B ∂ + ηB =0 ∂y ∂y η0,A bA ∂vz,B 2 ∂ 2 θB = + NaB ηB η0,B bB ∂y ∂y 2

(9.50) (9.51) (9.52)

h/k , Pe = ρ c V 2 where PeA = ρA cp,A V z A B B p,B z h/kB , NaA = η0,A bA Vz /kA , 2 /k . Note that in Eqs. (9.47)–(9.52), η and η are and NaB = η0,B bB V z B A B deﬁned by % &nA−1 exp(−θA ) (9.53) ηA = ∂vz,A /∂y γ˙0,A h/V z and

%

ηB = (η0,B /η0,A )

∂vz,B /∂y

&nB−1 γ˙0,B h/V exp(−bB θB /bA ) z

(9.54)

respectively, and ∂pA /∂z = ∂pB /∂z = ∂p/∂z is assumed. Now, Eqs. (9.47)–(9.49) for polymer A and Eqs. (9.50)–(9.52) for polymer B must be solved simultaneously, subject to the following boundary conditions: (1) At the die wall: vz,A = 0 and θA = θw

at y = 0

(9.55a)

vz,B = 0 and θB = θw

at y = 1

(9.55b)

in which θw = bA Tw , with Tw being the die wall temperature. (2) At the phase interface (i.e., at y = α): vz,A = vz,B ; θA = θB ; ηA

∂vz,A ∂y

= ηB

∂vz,B ∂y

; kA

∂θA ∂θ = kB B ∂y ∂y

(9.56)

410

PROCESSING OF THERMOPLASTIC POLYMERS

(3) At the die entrance (i.e., at z = 0): θA = θA,0 ; θB = θB,0

(9.57)

in which θA,0 = bA TA,0 and θB,0 = bB TB,0 , with TA,0 and TB,0 being the inlet melt temperatures for phase A and phase B, respectively. Using the ﬁnite difference method, Chin et al. (1984) solved Eqs. (9.47)–(9.52), subject to the boundary conditions given by Eqs. (9.55)–(9.57), to simulate their experimental study of coextrusion of nylon 6 with CXA 3095,5 which is a maleic anhydride grafted terpolymers. CXA 3095 is a commercial polymer that is used as a tie layer for coextruding nylon 6 with HDPE for meat packaging applications, producing three-layer nylon 6/CXA 3095/HDPE ﬁlm. Note that nylon 6 and HDPE are not miscible, while CXA 3095 is miscible with HDPE and reacts, during coextrusion, with the amine end groups of nylon 6, giving rise to good adhesion. Here, we present the simulation results of Chin et al. (1984) for the following two cases: case (1) for the die temperature being the same as the inlet temperature (240 ◦ C) of nylon 6, while the inlet temperature of CXA 3095 being at 210 ◦ C; and case (2) for the die temperature (230 ◦ C) lying between the inlet temperature (240 ◦ C) of nylon 6 and the inlet temperature (210 ◦ C) of CXA 3095. Table 9.2 gives a summary of the physical and rheological parameters of nylon 6 and CXA 3095 employed for the numerical computation. Case (1) Figure 9.24 shows the development of temperature proﬁle at various axial positions z in the die and Figure 9.25 shows how the temperature at the interface varies along the die axis for the coextrusion of the nylon 6/CXA 3095 pair, where Tw = 240 ◦ C, Tnylon,0 = 240 ◦ C, and TCXA,0 = 210 ◦ C. It is seen that the temperature tends to level off at about z = 2 cm (z/h = 15.75) for the nylon 6/CXA 3095 system. Figure 9.26 shows the development of velocity proﬁle at various axial positions z in the die and Figure 9.27 shows how the interface between the phases varies along the die axis for the nylon 6/CXA 3095 system. It is of interest to note in Figure 9.27 that the interface position of the nylon 6/CXA 3095 system initially dips into the CXA 3095 phase and then bounces back, ﬁnally reaching an equilibrium position. Figure 9.28 shows the variation of pressure gradient, −∂p/∂z, and Figure 9.29 the variation of shear stress at the interface, (σyz )int , as nylon 6 and CXA 3095 ﬂow, Table 9.2 Values of the physical properties and rheological parameters for nylon 6 and CXA 3095 employed for experiment and numerical computation

Polymer nylon 6b CXA 3095

ρ (g/cm3 )a

cp (J/(kg K ))

k (W/(m K ))

n

b (1/K)

0.904 0.750

2.09 × 103 2.07 × 103

0.247 0.415

0.55 0.44

0.026 0.019

a The values of ρ represent melt density. b This is commercial grade Capron 8207, which was available in the 1980s from Allied-Signal Corporation.

Reprinted from Chin et al., Polymer Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.

Figure 9.24 Simulated temperature proﬁles in the coextrusion of the nylon 6/CXA 3095 system

at various positions along the die axis: (1) z = 0, (2) z = 0.032 cm, and (3) z = 0.222 cm. The die temperature is 240 ◦ C, the nylon 6 feedstream temperature is 240 ◦ C, the CXA 3095 feedstream temperature is 210 ◦ C, the volumetric ﬂow rate of nylon 6 is 17.18 cm3 /min, and the volumetric ﬂow rate of CXA 3095 is 26.42 cm3 /min. (Reprinted from Chin et al., Polymer Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.)

Figure 9.25 Simulated interface temperature along the die axis in the coextrusion of the nylon 6/ CXA 3095 system. The die temperature is 240 ◦ C, the nylon 6 feedstream temperature is 240 ◦ C, the CXA 3095 feedstream temperature is 210 ◦ C, the volumetric ﬂow rate of nylon 6 is

17.18 cm3 /min, and the volumetric ﬂow rate of CXA 3095 is 26.42 cm3 /min. (Reprinted from Chin et al., Polymer Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.)

411

Figure 9.26 Simulated velocity proﬁles in the coextrusion of the nylon 6/ CXA 3095 system at

various position along the die axis: (1) z = 0, (2) z = 0.032 cm, and (3) z = 0.422 cm. The die temperature is 240 ◦ C, the nylon 6 feedstream temperature is 240 ◦ C, the CXA 3095 feedstream temperature is 210 ◦ C, the volumetric ﬂow rate of nylon 6 is 17.18 cm3 /min, and the volumetric ﬂow rate of CXA 3095 is 26.42 cm3 /min. (Reprinted from Chin et al., Polymer Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.)

Figure 9.27 Simulated interface position along the die axis in the coextrusion of the nylon 6/CXA 3095 system. The die temperature is 240 ◦ C, the nylon 6 feedstream temperature is 240 ◦ C, and the CXA 3095 feedstream temperature is 210 ◦ C. (Reprinted from Chin et al., Polymer

Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.)

412

Figure 9.28 Simulated proﬁle of pressure gradient −∂p/∂z along the die axis in the coextrusion of the nylon 6/CXA 3095 system. The die temperature is 240 ◦ C, the nylon 6 feedstream temperature is 240 ◦ C, the CXA 3095 feedstream temperature is 210 ◦ C, the volumetric ﬂow rate of nylon 6 is 17.18 cm3 /min, and the volumetric ﬂow rate of CXA 3095 is 26.42 cm3 /min.

(Reprinted from Chin et al., Polymer Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.)

Figure 9.29 Simulated proﬁle of shear stress (σyz )int at the phase interface along the die axis in the coextrusion of the nylon 6/CXA 3095 system. The die temperature is 240 ◦ C, the nylon 6 feedstream temperature is 240 ◦ C, the CXA 3095 feedstream temperature is 210 ◦ C, the volumetric ﬂow rate of nylon 6 is 17.18 cm3 /min, and the volumetric ﬂow rate of CXA 3095 is 26.42 cm3 /min. (Reprinted from Chin et al., Polymer Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.)

413

414

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 9.30 Simulated viscosity proﬁles in the coextrusion of the nylon 6/CXA 3095 system: (1) at the die entrance and (2) at the exit plane of the die. The die temperature is 240 ◦ C, the nylon 6 feedstream temperature is 240 ◦ C, the CXA 3095 feedstream temperature is 210 ◦ C, the volumetric ﬂow rate of nylon 6 is 17.18 cm3 /min, and the volumetric ﬂow rate of CXA 3095 is 26.42 cm3 /min. (Reprinted from Chin et al., Polymer Engineering Reviews 4(4):281.

Copyright © 1984, with permission from Freund Publishing House.)

side by side, through the die. It is of interest to observe in Figure 9.28 that −∂p/∂z initially decreases and then levels off to an equilibrium value as the two polymers ﬂow, side by side, through the die. Note that (σyz )int is related to −∂p/∂z by (σyz )int = (−∂p/∂z) (c − α), with c being the position at which the maximum velocity occurs and α the interface position. In other words, (σyz )int depends not only on the pressure gradient, but also on the interface position, relative to the position where the maximum velocity occurs. Figure 9.30 shows the distribution of viscosity at the entrance (z = 0) and exit plane (z = 6.35 cm) of the die for the nylon 6/CXA 3095 system. It is seen that the viscosity proﬁle of CXA 3095 in the nylon 6/CXA 3095 system has changed dramatically as the two melts ﬂow, side by side, through the die. This is explained by the fact that the temperature of the CXA 3095 feedstream is 210 ◦ C, while the temperature of the nylon 6 feedstream is 240 ◦ C, the same as the die temperature. Case (2) This is the situation in which the die temperature is kept somewhere between the inlet temperatures of the two melt streams. Figure 9.31 shows the development of the temperature proﬁle, and Figure 9.32 the development of the velocity proﬁle, at various axial positions z in the die for the nylon 6/CXA 3095 system, where Tw = 230 ◦ C, Tnylon,0 = 240 ◦ C, and TCXA,0 = 210 ◦ C. Figure 9.33 shows the variation of interface temperature, and Figure 9.34 the variation of interface position, along the die axis for the nylon 6/CXA 3095 system. For the situation under consideration, the CXA 3095 phase is heated, whereas the nylon 6 phase is cooled down, as the two melt streams ﬂow together, side by side, through the die. It is seen in Figures 9.33 and 9.34 that the axial distance at which both the interface position and interface temperature reach an equilibrium is about 3 cm, which is much longer than in case (1), considered above. Therefore, for the particular nylon 6/CXA 3095

Figure 9.31 Simulated temperature proﬁles in the coextrusion of the nylon 6/CXA 3095 system

at various position along the die axis: (1) z = 0, (2) z = 0.064 cm, and (3) z = 0.127 cm. The die temperature is 230 ◦ C, the nylon 6 feedstream temperature is 240 ◦ C, the CXA 3095 feedstream temperature is 210 ◦ C, the volumetric ﬂow rate of nylon 6 is 17.18 cm3 /min, and the volumetric ﬂow rate of CXA 3095 is 26.42 cm3 /min. (Reprinted from Chin et al., Polymer Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.)

Figure 9.32 Simulated velocity proﬁles in the coextrusion of the nylon 6/CXA 3095 system at

various positions along the die axis: (1) z = 0, (2) z = 0.064 cm, and (3) z = 6.36 cm. The die temperature is 230 ◦ C, the nylon 6 feedstream temperature is 240 ◦ C, the CXA 3095 feedstream temperature is 210 ◦ C, the volumetric ﬂow rate of nylon 6 is 17.18 cm3 /min, and the volumetric ﬂow rate of CXA 3095 is 26.42 cm3 /min. (Reprinted from Chin et al., Polymer Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.)

415

Figure 9.33 Simulated interface temperature along the die axis in the coextrusion of the nylon 6/CXA 3095 system. The die temperature is 230 ◦ C, the nylon 6 feedstream temperature is 240 ◦ C, the CXA 3095 feedstream temperature is 210 ◦ C, the volumetric ﬂow rate of nylon 6 is 17.18 cm3 /min, and the volumetric ﬂow rate of CXA 3095 is 26.42 cm3 /min. (Reprinted from

Chin et al., Polymer Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.)

Figure 9.34 Simulated interface position along the die axis in the coextrusion of the nylon 6/CXA 3095 system. The die temperature is 230 ◦ C, the nylon 6 feedstream temperature is 240 ◦ C, the CXA 3095 feedstream temperature is 210 ◦ C, the volumetric ﬂow rate of nylon 6 is 17.18 cm3 /min, and the volumetric ﬂow rate of CXA 3095 is 26.42 cm3 /min. (Reprinted from

Chin et al., Polymer Engineering Reviews 4(4):281. Copyright © 1984, with permission from Freund Publishing House.)

416

COEXTRUSION

417

system considered, we can conclude that, from the point of view of achieving an equilibrium interface position at the shortest distance from the die entrance, it is better to keep the die temperature the same as the higher of the two melt feedstreams. However, in general, other factors, such as the thermal conductivity, speciﬁc heat, and layer thickness, will also play important roles in deciding an optimal die temperature for achieving an equilibrium interface position at the shortest distance from the die entrance.

9.5

Summary

In this chapter, we ﬁrst presented the coextrusion die systems that are of fundamental importance to successful coextrusion operations, and then polymer–polymer interdiffusion during coextrusion, and ﬁnally nonisothermal coextrusion. We pointed out that the adhesion between layers is of utmost importance in order to achieve the desired mechanical and/or physical properties of the coextruded products. This means that the miscibility (or compatibility) of a polymer pair is essential for a successful coextrusion operation. Unlike an immiscible polymer pair, a miscible polymer pair undergoes interdiffusion at an elevated temperature, suggesting that an interphase is created during coextrusion of a miscible polymer pair. We have shown that the rate of formation of an interphase during coextrusion depends, among many factors, on the extent of attractive interactions between the components (i.e., χ parameter), molecular weight and molecular-weight distribution, and the extent of chain orientation at the phase interface. We have presented an expression, though based on an ad hoc basis, that enables one to estimate the time evolution of adhesive bond strength during coextrusion in terms of molecular weight, average chain orientation factor under shear ﬂow, and the χ parameter of a pair of miscible polymers. We have shown further that the shear stress at the interface in a coextrusion die inﬂuences greatly the interfacial position and thus the interfacial layer thickness. In view of the fact that values of the shear stress at the interface depend very much on the viscosity ratio of the polymer pair under consideration, the analysis presented in this chapter points out that the rheological properties of the polymer pair inﬂuence greatly the interfacial layer thickness and consequently adhesive bond strength in coextruded products. The analysis presented in this chapter predicts that the adhesive bond strength increases as the total layer thickness of coextruded sheet (or ﬁlm) increases and as the thickness ratio of two adjacent layers approaches unity. The reason is that, for a given pair of polymers to be coextruded, the processing conditions that give rise to smaller values of interfacial shear stress will make the adhesive bond strength greater. In the future, it would be worth pursuing a rigorous analysis to describe the chain dynamics of macromolecules in the shear ﬂow ﬁeld, where polymer–polymer interdiffusion across the interface takes place during coextrusion. We have pointed out that a third component must be used when coextruding two immiscible polymers, A and B, leading to an A/tie layer/B three-layer ﬁlm or sheet. Without a tie layer, the coextruded ﬁlm or sheet would have little practical value because there would no adhesion between the layers. The chemical structure of tielayer component must be chosen such that it is either miscible with both polymers,

418

PROCESSING OF THERMOPLASTIC POLYMERS

A and B, or it may undergo, during coextrusion, chemical reactions with one or both polymers. In the plastics industry, functionalized polymer is widely used as tie-layer component, so that grafting reactions might occur during coextrusion, between the tie-layer component and one of the two polymers being coextruded. Interface deformation (i.e., encapsulation of one component by the other component) and interfacial instability would be less important to successful coextrusion operations when two miscible polymers are coextruded or when a tie-layer component is used to coextrude two immiscible polymers. This is because an interphase will be created via polymer–polymer interdiffusion when two miscible polymers are coextruded or via chemical reaction when a functionalized tie-layer component is used to coextrude two immiscible polymers. When an interphase is created during coextrusion, it will minimize encapsulation and/or interfacial instability, regardless of the differences in rheological properties of the two polymers being coextruded. More quantitative experimental investigation on this subject is worth pursuing in the future.

Appendix: Derivation of Equation (9.36) According to Janeschitz-Kriegl (1969), the state of stress in steady-state shear ﬂow is related to ﬂow birefringence by p sin 2χ = 2σ12

p cos 2χ = σ11 − σ22

(9A.1) (9A.2)

where p = p| − p|| is the difference of the two principal stresses (p| and p|| ) in the plane of ﬂow, χ is the orientation angle (less than 45◦ ), σ12 is the shear stress, and σ11 − σ22 is the ﬁrst normal stress difference. From Eqs. (9A.1) and (9A.2) we obtain cot 2χ = (σ11 − σ22 )/2σ12

(9A.3)

By invoking the rubber elasticity theory, Janeschitz-Kriegl (1969) obtained the following relationship: o σ12 /vkB T (σ11 − σ22 )/2σ12 = JeR

(9A.4)

where v denotes the number of segments per volume, kB is the Bolztmann constant, T is o is reduced steady-state compliance deﬁned by J o = the absolute temperature, and JeR eR o o vkB T Je with Je being the steady-state compliance of a monodisperse homopolymer. Noting the following relationship (see Chapter 4 of Volume 1): vkB T = ρRT /Me = GoN

(9A.5)

in which ρ is the density (weight of polymer per unit volume), Me is the entanglement molecular weight, and GoN is the plateau modulus, Eq. (9A.4) can be rewritten, with

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the aid of Eqs. (9A.3) and (9A.5), as cot 2χ = Jeo σ12

(9A.6)

o denoting For a polydisperse polymer, under consideration here, Jeo is replaced by Jeb the steady-state compliance of a polydisperse polymer. Since our interest here is the average orientation of the chains of two chemically dissimilar polymers ﬂowing side by side through a sheet-forming die (see the schematic given in Figure 9.19), the shear stress σ12 is replaced by the shear stress at the interface of the two layers σint . Thus, oσ . the right-hand side of Eq. (9A.6) may be replaced by Jeb int Now, we explain the rationale for having chosen in Eq. (9.36) the cosine function to express the average orientation of polymer chains during the coextrusion through a sheet-forming die (see the schematic given in Figure 9.19). The orientation factor αi (i = A, B) deﬁned by Eq. (9.35) denotes the ratio of the average orientation angle of polymer chains under shear ﬂow in the sheet-forming die and the average orientation angle without ﬂow. Referring to the coordinate system shown in Figure 9.19, we take the average orientation angle of the polymer chains with respect to the ﬂow direction (z direction) and thus we replace the cotangent function on the left-hand side of Eq. (9A.6) with the tangent function. Note that the average orientation angle of ◦ the polymer chains without ﬂow is π/4 (45 ), and thus the average orientation angle under shear ﬂow will be larger than π/4. This is the reason why in the numerator of Eq. (9.36) we have two contributions to the orientation angle: the ﬁrst one from the average orientation without ﬂow and the secondone from the orientation of polymer o σ ) /2 that follows from chains due to shear ﬂow, which is expressed by tan−1 (Jeb int Eq. (9A.6).

Problems Problem 9.1

Consider A/B/C three-layer nonisothermal coextrusion in a sheet-forming die having the dimensions of width = 1 m, height = 0.004 m, and length = 0.05 m, and assuming that (1) the rheological behavior of each polymer may be described by the truncated power-law model (see Eq. (9.43)), (2) there is no slippage at the interface between polymer A and polymer B, and between polymer B and polymer C, (3) interface rearrangement during ﬂow is negligible, and (4) interdiffusion between polymer A and polymer B, and interdiffusion between polymer B and polymer C are negligible for the duration of coextrusion, and ﬂow is at steady state. Derive a system of equations for the nonisothermal coextrusion and then solve the system equations numerically, using the physical and rheological properties of the polymers in Table 9.3. Plot the velocity, temperature, and viscosity proﬁles of the A/B/C three layers in the die at 0.005, 0.01, 0.02, 0.03, 0.04, and 0.05 m along the die axis under the following extrusion conditions: (1) volumetric ﬂow rates of 20 cm3 /min for polymer A, 10 cm3 /min for polymer B, and 30 cm3 /min for polymer C, (2) pressure gradient (− ∂p/∂z) of 1.5 × 108 N/m3 , (3) the feedstream temperatures at 200 ◦ C for polymer A, at 210 ◦ C for polymer B, and at 240 ◦ C for polymer C, and (4) the die wall temperature at 220 ◦ C.

420

PROCESSING OF THERMOPLASTIC POLYMERS Table 9.3 Physical, thermal, and rheological properties of three polymers coextruded

Polymer

ρ (g/cm3 )

cp (J/(kg K))

k (W/(m K))

ko (Pa·s)

b (K−1 )

γ˙0 (s−1 )

n

0.750 0.904 0.800

2070 2090 2200

0.415 0.247 0.350

1.04 × 108 1.54 × 109 1.55 × 108

0.0206 0.0271 0.0255

0.50 1.50 1.00

0.4 0.6 0.5

A B C

Problem 9.2

Consider A/B/A three-layer nonisothermal coextrusion in the same sheet-forming die as described in Problem 9.1. Write a system of equations under the same assumptions made in Problem 9.1, and then solve numerically the system equations, using the same physical and rheological properties of the polymers given in Problem 9.1, under the following extrusion conditions: (1) the volumetric ﬂow rates of 10 cm3 /min for the top layer (polymer A), 10 cm3 /min for the middle layer (polymer B), and 30 cm3 /min for the bottom layer (polymer A), (2) pressure gradient (− ∂p/∂z) of 1.2 × 108 N/m3 , (3) the feedstream temperatures at 200 ◦ C for the top layer (polymer A), 230 ◦ C for the middle layer (polymer B), and 200 ◦ C for the bottom layer (polymer A), and (4) the die wall temperature at 220 ◦ C. The same results can be obtained by solving numerically the system equations considered in Problem 9.1, replacing polymer C in the A/B/C three-layer coextrusion system with polymer A. Compare the results obtained from the two approaches. Problem 9.3

Consider A/B two-layer isothermal coextrusion in a sheet-forming die having the dimensions of width = 1 m, height = 0.004 m, and length = 0.05 m. Assume that polymers A and B are miscible with the interaction parameter χ = −0.1 and thus polymer–polymer interdiffusion takes place during coextrusion. Under such circumstances, the polymer–polymer interdiffusion is greatly inﬂuenced by the extent of chain orientation inside the die. However, the extent of chain orientation is affected by the shear stress distribution inside the die. Thus, the momentum equation describing the ﬂow of two polymers A and B inside the die must be solved together with the mass transfer equation describing polymer–polymer interdiffusion, in order to be able to describe the coextrusion of two miscible polymers. Develop a system of equations describing the two-layer coextrusion of miscible polymers A and B, and then calculate shear stress distribution and chain orientation of the two polymers, A and B, inside the die, using the following assumptions: (1) the rheological properties of polymers A and B are the same as those given in Problem 9.1, (2) the volumetric ﬂow rates of 10 cm3 /min for polymer A and 20 cm3 /min for polymer B, (3) pressure gradient (−∂p/∂z) of 0.5 × 108 N/m3 , (4) the feedstream and die temperatures are the same at 200 ◦ C, (5) polymer A has Mw = 0.8×105 , ρ = 1.10 g/cm3 , Ro 2 /M = 0.50 × 10−16 cm2 , GoN = 5 × 105 Pa, and η0 = 0.5 × 105 Pa·s at 200 ◦ C, and (6) polymer B has Mw = 1.2×105 , ρ = 1.0 g/cm3 , Ro 2 /M = 0.65 × 10−16 cm2 , GoN = 6 × 105 Pa, and η0 = 1.0 × 105 Pa·s at 200 ◦ C.

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Notes 1. In this chapter, for simplicity we use “miscibility” and “compatibility” interchangeably. 2. In the absence of available information for tracer-diffusion coefﬁcients Di∗ (i = A, B) it is replaced with self-diffusion coefﬁcients Dio (i = A, B) calculated from Eq. (9.20). Thus, the interfacial composition proﬁles presented here must be regarded as approximate. At present there is no theoretical guideline as to how Di∗ (i = A, B) may be related to Dio (i = A, B). Note that self-diffusion can be regarded as a special case of tracer-diffusion in that, in tracer experiments the tracer-diffusion coefﬁcient of a dilute concentration of labeled A or B chains is measured in a blend of A and B chains that may differ in length and chemical species from the labeled one, while in self-diffusion the diffusing chains and the matrix chains must be chemically identical, except for the label, and of the same degree of polymerization. Equating tracer-diffusion coefﬁcient and self-diffusion coefﬁcient is tantamount to equating the mobility of a labeled PVDF molecule in a PVDF matrix with that of an identically labeled PVDF molecule in a PMMA matrix. In the interface region during the coextrusion of PMMA and PVDF at 230 ◦ C, for instance, the mobility of a labeled PVDF molecule in the PVDF matrix is equal to the mobility of an identically labeled PVDF molecule in the PMMA matrix. Because at 230 ◦ C PVDF is only 46 ◦ C above the melting temperature (which is about 174 ◦ C) of PVDF in the former case, whereas the labeled PVDF is 125 ◦ C above the glass transition temperature (about 105 ◦ C) of the PMMA in the latter case. The labeled PVDF molecule is expected to have a much lower mobility in the PVDF environment as compared with the PMMA environment. 3. The numerical values of GoN = 6 × 105 Pa, ρ = 1.088 g/cm3 and Ro 2 /M = 0.41 × 10−16 cm2 are taken from Wu et al. (1986), and the numerical values of η0 = 1.023× 104 Pa·s at 230 ◦ C and M = 1.075 × 105 for PMMA are taken from Kim and Han (1991). 4. The numerical values of η0 = 0.404 × 104 Pa·s at 230 ◦ C and M = 1.45 × 105 for PVDF are taken from Kim and Han (1991). 5. CXA 3095 (DuPont de Nemours and Company), which was available commercially in the early 1980s, is a physical blend of 81.9 wt % poly(ethylene-stat-vinyl acetate) and 18.1 wt % poly(ethylene-stat-isobutylacrylate-stat-methacrylic acetate), which is grafted with maleic anhydride and contains about 0.1 wt % carboxyl groups as determined from infrared spectroscopy. CXA polymers were later renamed as Bynel.

References Adolf D, Tirrell M, Prager S (1985). J. Polym. Sci., Polym. Phys. Ed. 23:413. Akcasu AZ, Benomouna M, Benoit H (1986). Polymer 27:1935. Anturkar NR, Papanastasiou TC, Wilkes JO (1990). AIChE J. 36:710. Binder K (1983). J. Chem. Phys. 79:6387. Brereton MG, Fischer EW, Fytas G, Murschall U (1987). J. Chem. Phys. 86:5174. Brochard F, de Gennes PG (1985). Europhys. Lett. 1:221. Brochard F, Jouffroy F, Levinson P (1983). Macromolecules 16:1638. Brochard F, Jouffroy F, Levinson P (1984). Macromolecules 17:2925.

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Chin HB, Kim YJ, Han CD (1984). Polym. Eng. Rev. 4(4):281. Composto RJ, Mayer JW, Kramer EJ, White DW (1988). Macromolecules 21:2580. Everage AC (1975). Trans. Soc. Rheol. 19:509. Fytas G (1987). Macromolecules 20:1430. Gilmore PJ, Falabella R, Laurence RL (1980). Macromolecules 13:880. Graessley WW (1980). J. Polym. Sci., Polym. Phys. Ed. 18:27. Graessley WW (1982). Adv. Polym. Sci. 47:102. Green PF, Palmstrom CJ, Mayer JW, Kramer EJ (1985). Macromolecules 18:501. Han CD (1973). J. Appl. Polym. Sci. 17:1289. Han CD (1975). J. Appl. Polym. Sci. 19:1875. Han CD (1976). Rheology in Polymer Processing, Academic Press, New York, Chap 10. Han CD (1981). Multiphase Flow in Polymer Processing, Academic Press, New York, Chaps 7 and 8. Han CD, Kim YJ, Chin HB (1984). Polym. Eng. Rev. 4:177. Han CD, Shetty R (1976). Polym. Eng. Sci. 16:697. Han CD, Shetty R (1978a). Polym. Eng. Sci. 18:180. Han CD, Shetty R (1978b). Polym. Eng. Sci. 18:187. Hess W, Akcasu AZ (1988). J. Phys. France 49:1261. Hicks EM, Ryan JF, Taylor RB, Tichenor RL (1960). Textile Res. J. 30:675. Hicks EM, Tippets EA, Hewett JV, Brand RH (1967). In Man-Made Fibers, Vol 1, Mark H, Atlas SM, Cernia E (eds), Wiley, New York, p 375. Janeschitz-Kriegl H (1969). Adv. Polym. Sci. 6:170. Jones RAL, Klein J, Donald AM (1986). Nature 321:161. Jordan EA, Ball RC, Donald AM, Fetters LJ, Jones RAL, Klein J (1988). Macromolecules 21:235. Jud K, Kausch HH, Williams JG (1981). J. Mat. Sci. 16:204. Kanetakis J, Fytas G (1987). J. Chem. Phys. 87:5048. Kanetakis J, Fytas G (1989). Macromolecules 22:3452. Karagiannis A, Hrymak AN, Vlachopoulos J (1990). Rheol. Acta. 29:71. Kehr KW, Binder K, Reulein SM (1989). Phys. Rev. B 39:4891. Khan AA, Han CD (1976). Trans. Soc. Rheol. 20:595. Khan AA, Han CD (1977). Trans. Soc. Rheol. 21:101. Khomami B (1990). J. Non-Newtonian Fluid Mech. 36:289. Kim JK, Han CD (1991). Polym. Eng. Sci. 31:258. Kim YW, Wool RP (1983). Macromolecules 16:1115. Kramer EJ, Green PF, Palmstrom C (1984). Polymer 25:473. Kurata M (1984). Macromolecules 17:895. Lee BL, White JL (1974). Trans. Soc. Rheol. 18:467. MacLean DL (1973). Trans. Soc. Rheol. 17:385. Matsunaga K, Kajiwara T, Funatsu K (1998). Polym. Eng. Sci. 38:1099. Mavridis H, Hrymak AN, Vlachopoulos J (1987). AIChE J. 33:410. Mills NJ (1969). Eur. Polymer J. 5:675. Mitsoulis E (1988). Adv. Polym. Tech. 8:225. Mitsoulis E, Heng FL (1987). J. Appl. Polym. Sci. 34:1713. Montfort JP, Marin G, Arman J, Monge Ph (1978). Polymer 19:277. Murschall U, Fischer EW, Herkt-Maetzky Ch, Fytas G (1986). J. Polym. Sci., Polym. Lett. Ed. 24:191. Nishi T, Wang TT (1975). Macromolecules 8:909. Noland JS, Hsu NNC, Saxon R, Schmitt JM (1971). In Multicomponent Polymer Systems, Platzer NAJ (ed), Adv. Chem. Series. no 99, American Chemical Society, Washington, DC, p 15.

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Paul DR, Altamirano JO (1975). In Copolymer, Polymer Blends, and Composites, Platzer NAJ (ed), Adv. Chem. Series no 142, American Chemical Society, Washington, DC, p 371. Puissant S, Demay Y, Vergnes B, Agassant JF (1994). Polym. Eng. Sci. 34:201. Prager S, Tirrell M (1981). J. Chem. Phys. 75:5194. Puissant S, Demay Y, Vergnes B, Agassant JF (1994). Polym. Eng. Sci. 34:201. Schrenk WJ (1974). Plastics Eng. 30(3):65. Schrenk WJ, Alfrey T (1973). SPE J. 29:38. Schrenk WJ, Cleereman KJ, Alfrey T (1963). SPE Trans. 19:192. Schrenk WJ, Bradley NL, Alfrey T, Maack H (1978). Polym. Eng. Sci. 18:620. Sillescu H (1984). Macromol. Chem. Rapid Commun. 5:519. Sillescu H (1987). Macromol. Chem. Rapid Commun. 8:393. Sisson WE, Morhead FF (1953). Textile Res. J. 23:152. Sornberger G, Vergnes B, Agassant JF (1986a). Polym. Eng. Sci. 26:455. Sornberger G, Vergnes B, Agassant JF (1986b). Polym. Eng. Sci. 26:682. Southern JH, Ballman RL (1973). Appl. Polym. Symp. 20:1234. Southern JH, Ballman RL (1975). J. Polym. Sci. Part A-2 13:863. Su YY, Khomami B (1992). J. Rheol. 36:357. Wendorff JH (1980). J. Polym. Sci., Polym. Lett. Ed. 18:439. White JL, Lee BL (1975). Trans. Soc. Rheol. 19:457. Wilson GM, Khomami B (1992). J. Non-Newtonian Fluid Mech. 45:355 Wilson GM, Khomami B (1993). J. Rheol. 37:341. Wool RP, O’Connor KM (1981). J. Appl. Phys. 52:5953. Wu S (1982). Polymer Interface and Adhesion, Marcel Dekker, New York. Wu S, Chuang HK, Han CD (1986). J. Polym. Sci., Polym. Phys. Ed. 24:143. Yu TC, Han CD (1973). J. Appl. Polym. Sci. 17:1203.

10

Foam Extrusion

10.1

Introduction

There are two processes used in the production of thermoplastic foams, namely, foam extrusion and structural foam injection molding (Benning 1969; Frisch and Saunders 1973). Foam extrusion, in which either chemical or physical blowing agents are used, is the focus of this chapter. Investigations of foam extrusion have dealt with the type and choice of process equipment (Collins and Brown 1973; Knau and Collins 1974; Senn and Sheneﬁel 1971; Wacehter 1970), the effect of die design (Fehn 1967; Han and Ma 1983b), the effect of blowing agents on foaming characteristics (Burt 1978, 1979; Han and Ma 1983b; Hansen 1962; Ma and Han 1983), and relationships between the foam density, cell geometry, and mechanical properties (Croft 1964; Kanakkanatt 1973; Mehta and Colombo 1976; Meinecke and Clark 1973). Chemical blowing agents are generally low-molecular-weight organic compounds, which decompose at and above a critical temperature and thereby release a gas (or gases), for example, nitrogen, carbon dioxide, or carbon monoxide. Examples of physical blowing agents include nitrogen, carbon dioxide, ﬂuorocarbons (e.g., trichloroﬂuoromethane, dichlorodiﬂuoromethane, and dichlorotetraﬂuoroethane), pentane, etc. They are introduced as a component of the polymer charge or under pressure into the molten polymer in the barrel of the extruder. It is extremely important to control the formation and growth of gas bubbles in order to produce foams of uniform quality (i.e., uniform cell structure). The fundamental questions one may ask in thermoplastic foam processing are: (1) What is the optimal concentration of blowing agent in order to have the minimum number of open cells and thus the best achievable mechanical property? (2) How many bubbles will be nucleated at the instant of nucleation? (3) What is the critical pressure at which bubbles nucleate in a molten polymer? (4) What are the processing–property relationships in foam extrusion and structural foam injection molding? Understandably, the answers to such questions depend, among many factors, on: (1) the solubility of the blowing agent in a molten polymer, (2) the diffusivity of the blowing agent in 424

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425

a molten polymer, (3) the concentration of the blowing agent in the mixture with a molten polymer, (4) the chemical structure of the polymers, (5) the initial pressure of the system, and (6) the equilibrium (or initial) temperature of the system. Some of the questions posed above have been addressed in the literature; namely, bubble nucleation in polymeric liquids (Han and Han 1988, 1990a, 1999b), bubble growth in a molten polymer during foam extrusion (Han et al. 1976; Han and Villamizar 1978; Yang and Han 1985; Yoo and Han 1981) and structural foam injection molding (Han and Yoo 1981; Villamizar and Han 1978). In this chapter, we ﬁrst present methods for estimating the solubility and diffusivity of gases or volatile liquids in a molten polymer, because a better understanding of the thermodynamic and transport properties of mixtures of a molten polymer and blowing agent is of fundamental importance to the control of foam quality in thermoplastic foam processes. Then, we present methods for investigating the phenomenon of bubble nucleation in polymeric liquids under static and ﬂowing conditions. Finally, we discuss processing–property–morphology relationships in proﬁle and sheet foam extrusions. The purpose of this chapter is to present the fundamentals associated with foam extrusion, but not to give recipes for producing speciﬁc foam products. In this chapter, we will not discuss the fundamentals of structural foam injection molding, because this subject has been discussed in the monograph of Han (1981).

10.2

Solubility and Diffusivity of Gases in a Molten Polymer

In this section, we present some useful correlations, which will enable one to estimate the solubility and diffusivity of gases or volatile components in a molten polymer in terms of the molecular parameters and thermodynamic properties of the gas or volatile component. The correlations and computational procedures described in this section will be very useful for estimating the solubility and diffusivity of gases and volatile components in a molten polymer when experimental data are not available. 10.2.1 Solubility of Gases in a Molten Polymer For a given pair of gaseous component (or volatile component) and polymer, the amount of gaseous component that can be dissolved (or solubilized) in a molten polymer is a function of temperature and pressure of the system. In the past, a number of research groups (Atkinson 1977; Bonner et al. 1974; Brockmeier et al. 1972, 1973; Cheng and Bonner 1978; DiPaola-Baranyi and Guillet 1978; Durrill and Griskey 1966; Galin and Rupprecht 1978; Liu and Prausnitz 1976, 1979; Lundberg et al. 1962, 1963; Maloney and Prausnitz 1976a, 1976b; Newman and Prausnitz 1972, 1973, 1974; Patterson et al. 1971; Saeki et al. 1981; Schotte 1982; Stern et al. 1969; Tseng et al. 1985a, 1985b) investigated the solubility of gases or volatile liquids in a molten polymer. Here, we present some correlations and procedures that will enable one to estimate the solubility of gases or volatile components in a molten polymer at elevated temperatures.

426

PROCESSING OF THERMOPLASTIC POLYMERS

According to the Flory–Huggins theory, the activity, a1 , of a solvent (component 1) is given by ln a1 =

ln(p1 /P1s )

1 φ2 + χ φ2 2 = ln(1 − φ2 ) + 1 − r

(10.1)

where p1 is the partial pressure of component 1 (volatile component) in the gas phase, P1s is the vapor pressure of the volatile component at the temperature of the mixture, φ2 is the volume fraction of polymer, r is the ratio of the molar volume of polymer to that of volatile component, and χ is the segmental interaction parameter for the speciﬁc polymer–volatile component system. For concentrated polymer solutions (where φ1 → 0, φ2 → 1.0), Eq. (10.1) reduces to1 p1 = P1s φ1 exp(1 + χ )

(10.2)

p1 = Hc w1

(10.3)

or

where φ1 and w1 , respectively, are the volume and weight fractions of volatile component, and Hc is the weight fraction-based Henry’s constant, deﬁned by Hc = P1s (v1 /v2 ) exp(1+χ ), with v1 , and v2 being the speciﬁc volumes of volatile component and polymer, respectively. From the inverse gas chromatographic (IGC) technique, one can determine the Henry’s constant using the expression (Maloney and Prausnitz 1976a)2 H1 = 22414/Vgo M1 = 22414Kp /M1

(10.4)

where M1 is the molecular weight of component 1 (volatile component), Vgo is the speciﬁc retention volume with units of cm3 of volatile component per gram of polymer at 273.2 K and 1 atm (i.e., at STP per gram of polymer with STP referring to standard temperature and pressure), and Kp is Henry’s constant at approximately 1 atmosphere, given by (Patterson et al. 1971) Kp =

s P1 Ω1∞ P1s M1 1 ) exp (B = − V 1 Vgo 22414 RT 11

(10.5)

where the exponential term is a correction for vapor-phase nonideality and the effect of is molar liquid volume of the pressure on the properties of the volatile component, V 1 volatile component, B11 is the second virial coefﬁcient of the volatile component, R is the universal gas constant, T is the absolute temperature, and Ω1∞ is the weight-fraction inﬁnite-dilution activity coefﬁcient given by 1 ln Ω1∞ = ln(v1 /v2 ) + 1 − +χ r

(10.6)

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427

Since r is very large in mixtures of polymer and low-molecular-weight volatile component, for such mixtures Eq. (10.6) reduces to Ω1∞ = (v1 /v2 ) exp(1 + χ )

(10.7)

One can then calculate χ by substituting Eq. (10.7) into (10.5) once Kp is available from experiment. For ﬁnite concentration, χ can be calculated from (Misovich et al. 1985) χ=

2 (e/Ω1∞ )w2 v1 w1 v1 w1 + 1 − + 1 w1 + (e/Ω1∞ )w2 v2 w2 v2 w2 2 v1 w1 w1 + (v2 /v1 )w2 + + 1 ln v2 w2 w1 + (e/Ω1∞ )w2

(10.8)

where e = 2.718 (the base of the natural logarithm). Note that Eq. (10.8) reduces to (10.7) for inﬁnite dilution (for w1 → 0 and w2 → 1). The experimental data for solubility of gases or volatile liquids reported in the literature, based on gas chromatographic procedures, have been found to correlate to Henry’s law. If this is the case, it is desirable to have generalized equations or correlations that will enable one to predict (or estimate) the solubility for components for which data are not available. Utilizing the solubility data for polystyrene (PS) reported in the literature from high-pressure sorption experiments and gas chromatographic studies, Stiel and Harnish (1976) obtained a single correlation between ln(l/Kp ) and (Tc /T )2 for 109 data points, which obeys the relationship ln (1/Kp ) = −2.338 + 2.706 (Tc /T )2

(10.9)

for (Tc /T )2 > 0.6, where Tc is the critical temperature in Kelvin of the gas. They found that for 109 data points, the average error between the experimental values and values calculated with Eq. (10.9) was 7.53%. Such a correlation would be extremely useful for estimating the solubility of untested gases or volatile solvents in PS at various temperatures. However, using the solubility data in the literature for ﬁve nonpolar n-alkanes, ﬁve slightly polar aromatic solutes, and ﬁve strongly polar solutes, Han (1987) reported that the slope of the ln(1/Kp ) versus (Tc /T )2 plots varies with the type of solute, as shown in Figures 10.1–10.3, although they have a common intercept, −2.028, which is the average value of the intercepts for the 17 solutes considered. Figure 10.4 gives plots of the slope (B) of ln(1/Kp ) versus (Tc /T )2 plots for the 17 organic solutes, showing that values of B for the ﬁve nonpolar n-alkanes and ﬁve slightly polar organic solutes increase linearly with the acentric factor (ω) (Reid et al. 1977), while no clear relationship between B and ω can be established. Notice in Figure 10.4 that at the same value of ω, values of B for the strongly polar solutes are smaller than those of the slightly polar solutes. For a number of gases in low-density polyethylene (LDPE) and polyisobutylene (PIB) at elevated temperatures, Stiel et al. (1985) obtained the following general correlation: ln (l/Kp ) = A + B(Tc /T )2

(10.10)

Figure 10.1 Plots of ln (1/Kp ) versus

(Tc /T )2 for n-alkanes in PS for: () n-dodecane, () n-tetradecane, () n-hexadecane, () n-decane, and (7) n-heptane. Data taken from Galin and Rupprecht (1978). (Reprinted from Han, Journal of Applied Polymer Science 33:2605. Copyright © 1987, with permission from John Wiley & Sons.)

Figure 10.2 Plots of ln (1/Kp ) versus (Tc /T )2 for nonpolar and

slightly polar solutes in PS: () ethylbenzene, () toluene, () benzene, () cyclohexane, (3) chlorobenzene, (7) n-butylbenzene, and ( ) n-butylcyclohexane. (Reprinted from Han, Journal of Applied Polymer Science 33:2605. Copyright © 1987, with permission from John Wiley & Sons.)

428

Figure 10.3 Plots of ln (1/Kp )

versus (Tc /T )2 for strongly polar solutes in PS for: () cyclohexanone, () methyl ethyl ketone, () acetonitrile, () 1,2-dichloroethane, and (3) dioxane. (Reprinted from Han, Journal of Applied Polymer Science 33:2605. Copyright © 1987, with permission from John Wiley & Sons.)

Figure 10.4 Plots of the slope of ln (1/Kp ) versus (Tc /T )2 plots versus acentric factor for

䊋

organic solutes in PS for: () n-heptane, (䊎) n-decane, (䊋) n-dodecane, ( ) n-tetradecane, (䊉) n-hexadecane, ( ) ethylbenzene, (䊑) toluene, () benzene, (䊒) chlorobenzene, () n-butylbenzene, () n-butylcyclohexane, () cyclohexanone, (䊖) n-cyclohexane, (7) methyl ethyl ketone, () 1,2-dichloroethane, () dioxane, and (3) acetone. (Reprinted from Han, Journal of Applied Polymer Science 33:2605. Copyright © 1987, with permission from John Wiley & Sons.) 䊑

429

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PROCESSING OF THERMOPLASTIC POLYMERS

with A = −1.561 and B = 2.057 + 1.438ω for 27 nonpolar solutes in LDPE from the ln (1/Kp ) versus (Tc /T )2 plots given in Figure 10.5. As shown in Figure 10.6, a linear relationship holds between the slope B of Eq. (10.10) and acentric factor ω for 27 nonpolar solutes considered for ω, with the average deviation between the values of B resulting from the experimental data and those calculated from Eq. (10.10) being 1.5%. They also found that the average deviation between the values of 1/Kp obtained from each literature source for 31 nonpolar solutes and the values calculated from Eq. (10.10) was 6.1%. Available solubility data for methane and nitrogen, which have low acentric factors, indicate that the intercept of Eq. (10.10) is the same for these substances as for nonpolar ﬂuids with ω > 0.08, as shown in Figure 10.7. However, the slopes for these molecules are considerably lower than those calculated from Eq. (10.10). For nitrogen, which also has a quadruple moment, the indicated slope is negative.

Figure 10.5 Plots of ln (1/Kp ) versus (Tc /T )2 for 27 nonpolar solutes in LDPE represented by Eq. (10.10). (Reprinted from Stiel et al., Journal of Applied Polymer Science 30:1145. Copyright © 1985, with permission from John Wiley & Sons.)

Figure 10.6 Plots of coefﬁcient B in Eq. (10.10) versus acentric factor ω for 27 nonpolar solutes in LDPE. Key to solutes is given in Figure 10.5. (Reprinted from Stiel et al., Journal of Applied Polymer Science 30:1145. Copyright © 1985, with permission from John Wiley & Sons.)

Figure 10.7 Plots of ln (1/Kp )

versus (Tc /T )2 for eight polar and low-molecular-weight solutes in LDPE represented by Eq. (10.10). (Reprinted from Stiel et al., Journal of Applied Polymer Science 30:1145. Copyright © 1985, with permission from John Wiley & Sons.)

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Large positive deviations in 1/Kp result from Eq. (10.10) for polar ﬂuids, as can be seen in Figure 10.7. However, the values of the slope B for these ﬂuids are lower than those calculated from Eq. (10.10), as shown in Figure 10.8. Therefore, the experimental values of 1/Kp become increasingly lower than the calculated values with increasing Tc /T . In order to accurately correlate the solubility behavior of polar substances, a fourth parameter, which characterizes polarity effects, is required in the relationship for the slope B of Eq. (10.10). Because the values of B increase with the size and decrease with the polarity of the molecule, the slopes are approximately equal to 2.0 for all the polar substances considered. The values of B for several quadrupolar solutes are included in Figure 10.8. The slopes for carbon dioxide, which has a quadrupole moment (B = 1.05), and for sulfur dioxide, which has dipole and quadrupole moment (B = 1.65), appear to be too low compared with those for the other polar ﬂuids. Stiel et al. (1985) found that Eq. (10.10) with A = −1.347 and B = 1.790+1.568ω describes the solubility of nonpolar solutes in polyisobutylene. Once the value of χ is known, one can then calculate the solubility (p1 /P1s or p1 ) of the volatile component in a polymer (with l/r → 0 in Eq. (10.1)) (i.e., the maximum amount of gas that can be dissolved in a molten polymer at any given temperature and pressure) from ln a1 = ln φ1 + φ2 + χ φ2 2

(10.11)

ln a1 ≈ ln (p1 /P1s )

(10.12)

where

Figure 10.8 Plots of slope B of Eq. (10.10) versus acentric factor ω for polar and low-molecular-

weight solutes in LDPE. (Reprinted from Stiel et al., Journal of Applied Polymer Science 30:1145. Copyright © 1985, with permission from John Wiley & Sons.)

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Figure 10.9 Plots of activity a1 versus volume fraction φ1 of ethylbenzene in mixtures with PS at various temperatures (◦ C): () 115.5, (䊉) 130, () 140, () 160, () 170, and () 178. The solid curve is the prediction of the Flory–Huggins theory, which allows us to calculate the interaction parameter at each temperature: χ = 0.205 at 115.5 ◦ C, χ = 0.233 at 130 ◦ C, χ = 0.245 at 140 ◦ C, χ = 0.257 at 160 ◦ C, χ = 0.259 at 170 ◦ C, and χ = 0.258 at 178 ◦ C. (Reprinted from Vrentas et al., Industrial Engineering Chemistry, Product Research and Development 22:326. Copyright © 1983, with permission from the American Chemical Society.)

for an ideal gas, and ln a1 ≈ ln(p1 /P1s ) +

p P1s

v¯ z−1 dp + 1 (P1s − p1 ) p RT

(10.13)

for a nonideal gas, where z is the compressibility factor for the gas phase. Figure 10.9 gives, for illustration, plots of activity a1 versus volume fraction of volatile component φ1 for mixtures of PS and ethylbenzene at various temperatures, in which the solid curve is the prediction based on the Flory–Huggins theory. Figure 10.10 gives plots of calculated partial pressure of ethylbenzene in PS versus weight fraction of ethylbenzene for various temperatures, in which the Flory–Huggins theory was used. Figure 10.11 gives plots of calculated partial pressure of dichlorotetraﬂuoroethane (FC-114) in LDPE versus weight fraction of FC-114 for various temperatures, and Figure 10.12 gives plots of calculated partial pressure of trichloroﬂouromethane (FC-11) in LDPE versus weight fraction of FC-11 for various temperatures. These ﬁgures were obtained by following the procedures we have described: (1) values of Kp were calculated using Eq. (10.10), (2) values of Ω1∞ were calculated using Eq. (10.5), (3) values χ were calculated using Eq. (10.7), and (4) values of p1 were calculated using Eq. (10.12) or Eq. (10.13) for various temperatures. 10.2.2 Diffusivity of Gases in a Molten Polymer When a gas bubble nucleates and then grows, pressure within the bubble will decrease as its radius becomes larger. It will then grow rapidly until much of the excess dissolved

Figure 10.10 Plots of partial pressure p1 versus weight fraction w1 of ethylbenzene

in a mixture with PS at various temperatures (◦ C): (1) 115.5, (2) 130, (3) 140, (4) 160, (5) 170, and (6) 178.

Figure 10.11 Plots of partial pressure p1 versus weight fraction w1 of FC-114 in a

mixture with LDPE at various temperatures (◦ C): (1) 100, (2) 110, (3) 120, (4) 130, and (5) 140.

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Figure 10.12 Plots of partial pressure p1 versus weight fraction w1 of FC-11 in a

mixture with LDPE at various temperatures (◦ C): (1) 100, (2) 110, (3) 120, (4) 130, (5) 140, (6) 150, (7) 160, and (8) 170.

gas in the contiguous area has been utilized in expanding the polymer. Therefore, bubble growth will be governed by the rate of diffusion of dissolved gas to the polymer–gas interface as well as by the degree of supersaturation and viscosity of the melt. The dynamics of a gas bubble in a molten polymer ﬂowing through a die has been discussed (Han 1981). In this section, we present correlations and procedures that will enable one to estimate the diffusion coefﬁcient of gases or volatile components in a molten polymer at elevated temperatures. Most of the available data on diffusion coefﬁcients of gases or volatile components in molten polymer have been obtained by sorption experiments (Duda and Vrentas 1968; Duda et al. 1978, 1979; Durrill and Griskey 1966; Ju et al. 1981a; Lundberg et al. 1962, 1963; Newitt and Weale 1948). The sorption technique is time-consuming and requires extensive data analysis, but it has the advantage in that it allows one to determine diffusion coefﬁcients at ﬁnite concentrations of gases or volatile components. Inverse gas chromatography (IGC) has also been used widely for the measurement of diffusion coefﬁcients of gases and volatile components in molten polymers (Braun et al. 1976; Edwards and Newman 1977; Gray and Guillet 1973; Hu and Han 1987; Hu et al. 1987; Kong and Hawkes 1976; Senich 1981; Tait and Abushihada 1979). This technique allows rapid determination of values of diffusion coefﬁcients, but it is applicable only to solutes at inﬁnite dilution. The dependence of diffusion coefﬁcient D on temperature T, for a number of gas/polymer systems investigated, was found to follow an Arrhenius expression: D = Do exp(−Ed /RT )

(10.14)

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䊋

Figure 10.13 Plots of log D versus 1/T for a volatile liquid or gas in PS at elevated temperatures: () benzene, () toluene, () ethylbenzene, (3) n-decane, (䊉) benzene (data taken from Pawlisch and Laurence 1983), () toluene (data taken from Pawlisch and Laurence 1983), (䊕) toluene (data taken from Ni 1978), () ethylbenzene (data taken from Pawlisch and Laurence 1983), (䊑) ethylbenzene (data taken from Ni 1978), and ( ) methane (data taken from Lundberg et al. 1963). (Reprinted from Hu et al., Journal of Applied Polymer Science 33:551. Copyright © 1987, with permission from John Wiley & Sons.)

in which Do is the diffusion coefﬁcient at the reference temperature To and Ed is the energy of activation. Using the experimental data in the literature, plots of log D versus 1/T for various solutes in PS are given in Figure 10.13. Table 10.1 gives numerical values of D at various temperatures for PS/benzene, PS/toluene, PS/ethylbenzene, and PS/n-decane mixtures obtained via the IGC method. Note in Figure 10.13 that experimental data from more than one source are given for some organic solutes. It is seen in Figure 10.13 that the slope of the log D versus 1/T plots (i.e., the activation energy for diffusion) varies with temperature, indicating that an Arrhenius expression is not suitable for correlating inﬁnite dilution diffusion coefﬁcients of the four organic solutes and methane in PS in the range of temperatures Tg < T < Tg + 120 ◦ C. Note in Figure 10.13 that the methane data of Lundberg et al. (1963) deviate considerably from the rest of the data points. The free-volume concept has been used to describe the diffusion of low-molecularweight liquids. Cohen and Turnbull (1959) developed a free-volume theory to describe the molecular transport of a liquid consisting of hard spheres, by assuming that the molecular transport occurs by the movement of molecules into voids formed by redistribution of the free volume. Fujita et al. (1960) and Fujita (1961) modiﬁed the free-volume theory for describing solvent–polymer diffusion, which included concentration and temperature dependences, and Fujita (1961) obtained the following expression relating the diffusion coefﬁcient D of a diluent molecule in a polymer to the average fractional free volume f of the system: D = Ad RT exp(−Bd /f )

(10.15)

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Table 10.1 Summary of experimentally determined diffusion coefﬁcients for mixtures of PS and a volatile component via the IGC method

T ( ◦C)

1/TR

103 /(K22 + T − Tg2 ) 1/TRF

log D

ln Dζ

(a) PS/benzene mixture 129 135 150 160 170 181

1.397 1.377 1.328 1.297 1.268 1.237

13.038 12.092 10.235 9.285 8.496 7.770

7.328 6.797 5.763 5.219 4.776 4.368

−7.916 −7.625 −7.214 −7.015 −6.814 −6.638

−21.066 −20.395 −19.452 −18.992 −18.529 −18.125

10.235 9.285 8.873 8.150 7.770 7.536 7.262 6.770

6.056 5.494 5.250 4.822 4.597 4.459 4.297 4.006

−7.304 −7.049 −6.655 −6.560 −6.569 −6.614 −6.500 −6.478

−19.667 −19.080 −18.173 −17.955 −17.975 −18.079 −17.815 −17.765

10.235 8.873 8.150 7.831 7.262 6.770

6.316 5.475 5.029 4.832 4.481 4.178

−7.158 −6.701 −6.530 −6.701 −6.514 −6.464

−19.338 −18.284 −17.891 −18.285 −17.855 −17.738

10.235 9.285 8.873 8.150 7.710 7.479 7.262 6.771

6.321 5.734 5.480 5.033 4.762 4.619 4.485 4.181

−7.741 −7.131 −7.008 −6.796 −6.804 −6.743 −6.682 −6.682

−19.977 −19.293 −19.005 −18.516 −18.546 −18.407 −18.265 −18.093

(b) PS/toluene mixture 150 160 165 175 181 185 190 200

1.398 1.366 1.350 1.320 1.303 1.291 1.277 1.250

(c) PS/ethylbenzene mixture 150 165 175 180 190 200

1.458 1.408 1.377 1.362 1.332 1.304

(d) PS/n-decane mixture 150 160 165 175 182 186 190 200

1.459 1.426 1.409 1.378 1.357 1.344 1.333 1.305

Reprinted from Hu et al., Journal of Applied Polymer Science 33:551. Copyright © 1987, with permission from John Wiley & Sons.

in which Ad is a free volume parameter that depends primarily on the size and shape of the diluent molecule and hence may be independent of temperature and diluent concentration, Bd is a free-volume parameter that presents the minimum hole required for a given diluent molecule to permit a displacement, R is the universal gas constant, T is the temperature. Vrentas and Duda (1977a, 1977b) developed improved free-volume theories for predicting the concentration dependence of diffusion coefﬁcients of solutes in molten

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polymers. Speciﬁcally, they assigned the parameter Ad in Eq. (10.15) to be the preexponential value of the solvent self-diffusion coefﬁcient, and the parameter Bd to be a linear function of the closed-packing volume (i.e., the volume of solvent occupied at zero degree in Kelvin). In a further study, Vrentas and Duda (1977c) made the following assumptions: (1) all thermal expansion coefﬁcients of free volume may be approximated by average values in the temperature range under consideration, (2) the partial speciﬁc volumes of polymer and solvent are independent of concentration, so that the inﬂuence of volume change on mixing on the free volume of the system is considered to be negligible, and (3) the solvent chemical potential in the mixture is given by the Flory–Huggins equation. On this basis, they obtained the following expression for the diffusion coefﬁcient D:

(w V ∗ + w2 ξ V ∗2 D = Do1 (1 − φ1 )2 (1 − 2χ φ1 ) exp − 1 1 VFH /γ

(10.16)

where K K VFH = 11 w1 (K21 + T − Tg1 ) + 12 w2 (K22 + T − Tg2 ) γ γ γ

(10.17)

φ1 =

w1 V o1

w1 V o1 + w2 V o2

Do1 = Do exp(−E/RT )

(10.18)

(10.19)

in which φ1 is the volume fraction of solvent, χ is the interaction parameter for a given polymer–solvent system, wi is the mass fraction of component i, V ∗i is the speciﬁc critical hole volume of component i required for a jump, ζ is the ratio of critical molar volume of solvent jumping unit to critical molar volume of jumping unit of polymer, VFH is the average hole free volume per gram of mixture, γ is an overlap factor for free volume, K11 and K21 are free-volume parameters of solvent, K22 is a free-volume parameter of polymer, Tgi is the glass transition temperature of component i, V oi is the partial speciﬁc volume of component i, Do is a pre-exponential factor, E is the critical energy per mole needed to overcome attraction forces, and R is the universal gas constant. Vrentas and Duda (1977c) have presented procedures for evaluating and predicting the variation of diffusion coefﬁcient D with temperature T and mass fraction wi for a particular polymer–solvent system using Eq. (10.16). At inﬁnitely dilute concentration (i.e., for φ1 = w1 = 0) of volatile component, Eq. (10.16) reduces to

γ V ∗2 ζ /K12 E − ln D = ln Do − RT K22 + T − Tg2

(10.20)

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When the magnitude of the term E/RT in Eq. (10.20) is negligibly small compared with other terms (i.e., E = 0), Eq. (10.20) reduces to

ln D = ln Do −

γ V ∗2 ζ /K12 K22 + T − Tg2

(10.21)

in which K22 and Tg2 (the glass transition temperature) characterize properties of the are properties of the polymer–solvent system. Note polymer, and Do and γ V ∗2 ζ /K 12 that the parameter K22 and γ V ∗2 ζ /K12 are simply related to the Williams–Landel–Ferry (WLF) constants of the polymer (Ferry 1980). Ju et al. (1981a) showed that for a particular polymer–solute system, γ V ∗2 ζ / K12 in Eq. (10.21), which has the dimension of temperature, can be estimated from a linear relationship between values of this constant for the polymer and the molar volume of the solute at zero degree in Kelvin, as estimated by group contribution methods. With the same data as used in Figure 10.13, Hu et al. (1987) prepared plots of log D versus 103 /(K22 +T −Tg2 ) with the aid of Eq. (10.21), as shown in Figure 10.14, for which K22 = 47.7 K and Tg2 = 100 ◦ C were used. The numerical values of log D and 103 /(K22 + T − Tg2 ) are summarized in Table 10.1. For the data taken at temperatures below 178 ◦ C but above Tg2 for PS, Duda et al. (1978, 1979) have observed that such plots show a linear correlation, represented by Eq. (10.21). According to Eq. (10.21), the slopes in the log D versus 103 /(K22 + T − Tg2 ) plots are represented

䊋

Figure 10.14 Plots of log D versus 103 /(K22 + T – Tg2 ) for volatile liquid or gas in PS at elevated temperatures: () benzene, () toluene, () ethylbenzene, (3) n-decane, (䊉) benzene (data taken from Pawlisch and Laurence 1983), () toluene (data taken from Pawlisch and Laurence 1983), (䊕) toluene (data taken from Ni 1978), () ethylbenzene (data taken from Pawlisch and Laurence 1983), (䊑) ethylbenzene (data taken from Ni 1978), and ( ) methane (data taken from Lundberg et al. 1963). (Reprinted from Hu et al., Journal of Applied Polymer Science 33:551. Copyright © 1987, with permission from John Wiley & Sons.)

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by γ V ∗2 ζ / K12 , in which γ represents the polymer–solvent interaction in the jumping process (describing the fact that the same free volume is available for more than one molecule), and V ∗2 ζ /K12 represents an inherent polymer variable. A comparison of Figure 10.14 with Figure 10.13 shows that the introduction of free-volume parameters for PS leads to a better correlation. However, the dependence of D on the structure of solute still remains strong for PS. Note in Figure 10.14 that the methane data of Lundberg et al. (1963) still deviate considerably from the correlation suggested by Eq. (10.21). By introducing the parameter ζ , deﬁned by ζ = M 1/2 /Tc 1/2 Vc 1/3

(10.22)

with M being the molecular weight of the solute, Tc the critical temperature of the solute, and Vc the critical volume of the solute, into Eq. (10.20), Hu et al. (1987) obtained the following generalized Vrentas–Duda free-volume relationship: ln Dζ = a + b/TRF + c/TR

(10.23)

where a = ln Do ζ , b = −γ V ∗2 ξ/K12 Tc , c = −E/RT c , TRF = (K22 + T − Tg2 )/Tc , and TR = T /Tc . Note that the Dζ appearing in Eq. (10.23) may be regarded as being a reduced diffusion coefﬁcient of solute in molten polymer. It can be shown that Dζ /R 1/2 is a dimensionless quantity, with R being the universal gas constant. Figure 10.15 gives plots of ln Dζ versus 1/TRF for mixtures of PS and volatile component. Table 10.1 gives numerical values of ln Dζ and 1/TRF for PS/benzene, PS/toluene, PS/ethylbenzene, and PS/n-decane mixtures. Also, for comparison, plots of ln Dζ versus 1/TRF for poly(vinyl acetate) (PVAc) are given in Figure 10.16. It is seen in Figures 10.15 and 10.16 that two linear regions appear to exist, separated by a certain critical value of 1/TRF , which may be obtained by extrapolating the two linear regions, as indicated by the dotted lines. Note that TRF represents a dimensionless temperature and the slope in the ln Dζ versus 1/TRF plots is represented by the dimensionless parameter b deﬁned by Eq. (10.23). The following observations may be made on the correlations shown in Figures 10.15 and 10.16. Plots of ln Dζ versus 1/TRF give two distinct linear regions, which are separated at a critical value of 1/TRF . As mentioned above, the critical value of 1/TRF may be obtained by extrapolating the two linear regions. In reality, there would be no discontinuity in the slope of ln Dζ . The critical value of 1/TRF appears to be independent of the type of solute and the structure of polymer. The slope b in the ln Dζ versus 1/TRF plots appears to be independent of the type of solute but dependent upon the structure of the polymer. The diffusivity of methane in PS (see Figure 10.15) can be correlated with the same degree of accuracy as other organic solutes, which was not possible with the Vrentas–Duda free-volume correlation (see Figure 10.14). The slope b for PVAc is greater than that for PS, both below and above the critical value of 1/TRF . The critical value of 1/TRF is approximately 4.8. It can therefore be concluded that the use of the reduced variables, ln Dζ and 1/TRF , shown in Figures 10.15 and 10.16, enables us to determine the upper limit of temperature, above which free-volume theories are not entirely applicable for

䊋

Figure 10.15 Plots of ln Dζ versus 1/TRF for volatile liquid or gas in PS at elevated temperatures: () benzene, () toluene, () ethylbenzene, (3) n-decane, (䊉) benzene (data taken from Pawlisch and Laurence 1983), () toluene (data taken from Pawlisch and Laurence 1983), (䊕) toluene (data taken from Ni 1978), () ethylbenzene (data taken from Pawlisch and Laurence 1983), (䊑) ethylbenzene (data taken from Ni 1978), and ( ) methane (data taken from Lundberg et al. 1963). (Reprinted from Hu et al., Journal of Applied Polymer Science 33:551. Copyright © 1987, with permission from John Wiley & Sons.)

Figure 10.16 Plots of ln Dζ versus 1/TRF for volatile liquids in PVAc: () benzene, () toluene, () ethylbenzene, (3) n-decane, (䊉) benzene, and () toluene. (Reprinted from Hu et al., Journal of Applied Polymer Science 33:551. Copyright © 1987, with permission from John Wiley & Sons.)

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correlating the inﬁnite dilution diffusion coefﬁcient of solutes in molten polymers at elevated temperatures. Vrentas and Duda (1977a, 1977b) noted that as the temperature is increased, the speciﬁc hole free volume increases signiﬁcantly, and the energy to overcome attractive forces assumes a more dominant role. The importance of the energy effect can be determined by calculating the fraction of the activation energy for diffusion that is needed to overcome intermolecular forces. Earlier, Macedo and Litovitz (1965) had pointed out the importance of the consideration of energy effects, in addition to free-volume effects, in correlating polymer viscosity data at high temperatures. The suggestion of Macedo and Litovitz was the basis for the inclusion of the term exp(−E/RT ) in Eq. (10.20). Vrentas and Duda (1979) noted that, in reference to Eq. (10.20), the total activation energy ED may be given as ∗ ED = E + (γ Vˆ 2 ξ/K12 )RT 2 /(K22 + T − Tg2 )2

(10.24)

in which the second term represents the contribution from free-volume effects. Ju et al. (1981b) compared experimental data with predictions from the Vrentas–Duda theory, with the aid of Eq. (10.16), for concentrated polymer solutions at temperatures below 180 ◦ C for PS (using the quartz spring sorption balance) and below 65 ◦ C for PVAc (using the Cahn electrobalance). In plotting their data with the aid of Eq. (10.21), the highest temperature available was 178 ◦ C, from the study by Duda et al. (1982) of the PS/ethylbenzene and PS/toluene systems. This temperature still lies in the lower linear region in the ln Dζ versus 1/TRF plots (Hu et al. 1987). Such observations seem to indicate that at higher temperatures, energy effects would be observed, requiring nonzero values of E. Figure 10.17 compares experimental data with theoretical predictions (solid curves) for the diffusion coefﬁcient of ethylbenzene versus weight fraction of ethylbenzene in mixtures with PS at various temperatures, where the experimental data are taken from a paper by Duda et al. (1982). Figure 10.18 compares experimental data with theoretical predictions (solid curves) for the diffusion coefﬁcient of toluene versus weight fraction of toluene in mixtures with PS at various temperatures, where the experimental data are taken from a paper by Duda et al. (1982). In obtaining the solid curves in Figures 10.17 and 10.18, Eq. (10.16) was used together with the numerical values of the free-volume parameters given in Table 1 of a paper by Duda et al. (1982). It can be seen that despite many assumptions made in the development of Eq. (10.16), prediction and experiment are in good agreement. Figure 10.19 gives plots of predicted diffusion coefﬁcient versus weight fraction of FC-11 in mixtures with PS at ﬁve different temperatures (130, 140, 150, 160, and 170 ◦ C), for which Eq. (10.16) was used. The numerical values of the free-volume parameters used to obtain Figure 10.19 are given in Table 10.2. To summarize, in this section we have presented some useful correlations that will enable one to estimate the solubility and diffusivity of gases and volatile components in a molten polymer in terms of the molecular parameters and thermodynamic properties of the gas or volatile component. We also have presented a modiﬁed free-volume theory after Vrentas and Duda, which will enable one to estimate the diffusion coefﬁcient of gases and volatile components in a molten polymer. Such information will be very helpful for successful operation of foam processing.

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Figure 10.17 Plots of diffusion coefﬁcient D versus weight fraction w1 of ethylbenzene in mixtures with PS at various temperatures (◦ C): () 115.5, () 130, () 140, (7) 160, () 170, and (3) 178. The solid curves are the predictions based on the free-volume approach described in the text. The numerical values of free-volume parameters used for the predictions are given in Table 1 of a paper by Duda et al. (1982).

10.3

Bubble Nucleation in Polymeric Liquids

Bubble nucleation in a polymeric liquid is encountered in thermoplastic foam processing and polymer devolatilization. In thermoplastic foam processing, the aim is to create bubbles in a molten polymer (thus, ﬁne cells in solidiﬁed products) by injecting either an inert gas (physical blowing agent) or by adding an organic compound (chemical blowing agent) that generates gas upon thermal decomposition. In foam extrusion, the pressure at the upstream end of the process line is maintained at a sufﬁciently high level to ensure that the injected gas or volatile liquid is dissolved in the molten polymer, but as the pressure decreases downstream in the die, the dissolved gas becomes supersaturated and bubbles are nucleated, which then grow rapidly until the gas pressure in the melt is in equilibrium with that within the bubbles. It is thus quite apparent that the control of bubble nucleation in a polymeric liquid is very important for controling the quality of the products produced from thermoplastic foam processing. Nucleation of a gas bubble in a liquid has been studied for a long time (Becker and Döring 1935; Blander et al. 1971; Farkas 1927; Frenkel 1941; Holden and Katz 1978; Katz and Blander 1973; Porteous and Blander 1975; Volmer and Weber 1926). Nucleation, like ordinary chemical kinetics, involves an activation process, which leads to

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PROCESSING OF THERMOPLASTIC POLYMERS Figure 10.18 Plots of diffusion coefﬁcient D versus weight fraction w1 of toluene in mixtures with PS at various temperatures (◦ C): () 110, () 140, (7) 160, () 170, and () 178. The solid curves are the predictions based on the free-volume approach described in the text. The numerical values of free-volume parameters used for the predictions are given in Table 2 of a paper by Duda et al. (1982).

the formation of unstable intermediate states known as “embryos.” Within a metastable phase, the initial fragments of a new and more stable phase are generated, developing spontaneously into gross fragments of stable phase. Consequently, an investigation of the initial stage of the kinetics of such transformation is needed in order to develop an understanding of the phenomenon of bubble nucleation in a liquid. Classical nucleation theory assumes that gas bubble embryos can be described in terms of the bulk thermodynamic properties, and that the formation of embryos of all sizes (up to the critical size) occurs by a series of intermediate reactions of capturing or losing a molecule. The work required to form a new gas bubble in an isothermal system is the change in Helmholtz free energy. In a homogeneous system, the nucleus for the new phase arises spontaneously because of thermal ﬂuctuations and intermolecular interactions. Such nucleation is termed “homogeneous” and is to be contrasted with “heterogeneous” nucleation, which occurs at an interface between the volatile liquid and another phase it contacts. Heterogeneous nucleation requires much lower degrees of supersaturation. Katz and coworkers (Katz 1970; Katz and Blander 1973; Katz et al. 1966) investigated homogeneous nucleation for a nonideal gas and the predicted critical supersaturation.

Figure 10.19 Plots of predicted diffusion coefﬁcient D versus weight fraction w1 of FC-11 in mixtures with PS at various temperatures (◦ C): (1) 130, (2) 140, (3) 150, (4) 160, and (5) 170. The predictions are based on the free-volume approach described in the text.

Table 10.2 Numerical values of free-volume parameters employed to predict diffusion coefﬁcients of mixtures of PS and FC-11 using Eq. (10.16)

Parameter

PS/(FC-11) System

Vˆ 1∗ (cm3 /g) Vˆ 2∗ (cm3 /g) K11 /γ (cm3 /g K) K12 /γ (cm3 /g K) Tg1 (K) Tg2 (K) K21 (K) K22 (K) ξ Do (cm2 /s) E (J/g mol)

0.504 0.850 5.787 × 10−4 7.186 × 10−3 20.6 373.1 24.5 46.6 0.498 8.925 × 10−5 4.195 × 103

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Some experimental data for homogeneous nucleation in superheated liquids have been reported (Blander and Katz 1975). 10.3.1 Experimental Observations of Bubble Nucleation Although light scattering may be considered as an indirect method for investigating the phenomenon of bubble nucleation, it has long been a powerful tool for the analysis of small particles. For an interpretation of the scattered light intensity versus time curves, a one-to-one correspondence between the extrema of the experimental and theoretical scattering curves must be established. The critical pressure can be determined from the pressure versus time curve with the aid of information on the critical time for bubble nucleation, which can be determined from the scattering curve to be the point at which the light scattering signal begins to appear. Bubble growth can be studied by identifying the number of the extrema in the experimental and theoretical light scattering curves. The size parameters for each extremum can then be obtained from the theoretical scattering curves, while the time scale is read off from the experimental scattering curves. With this information, bubble growth curves can be constructed. In the use of light scattering, a difﬁculty may arise due to the distribution of the scattering data with respect to the particle size. Therefore, it becomes important to consider the uniqueness of a particular solution. For example, in a polydisperse system, there can be more than one size distribution that leads to a particular set of data, at least within experimental uncertainty. This is frequently a serious problem and places severe limitations on the kinds of measurements that can be utilized, as well as on the range of sizes and the width of the size distributions that are amenable to accurate treatment. 10.3.1.1 Bubble Nucleation under Static Conditions Han and Han (1990a) used light scattering to investigate the homogeneous bubble nucleation in a solution consisting of PS and toluene in the temperature range from 150 to 180 ◦ C and in the PS concentration range from 40 to 60 wt %, and in molten polymer containing a blowing agent at an elevated temperature. Figure 10.20 shows a schematic of the layout of the light scattering apparatus employed in their investigation. For the study, they constructed a high-pressure optical cell. Figure 10.21 shows a schematic of the side view of the high-pressure optical cell, where a plate, which supports the light scattering measurement system, is seated on a bearing and can be rotated to any scattering angle ranging from 20 to 160◦ . They used: (1) He–Ne laser as the continuous light source, (2) two photomultipliers to measure the transmitted and scattered light ﬂuxes, (3) a piezoelectric pressure transducer to measure the system pressure, and (4) two storage oscilloscopes to monitor the light scattering ﬂux, transmitted ﬂux, and total pressure, simultaneously. The measurements of light scattering ﬂux and process control of the experiments were performed by means of a microcomputer and a general-purpose data acquisition interface. The bubble nucleation and subsequent growth processes were observed optically. Typical output signals recorded on the oscilloscopes are shown in Figure 10.22a, where, because of the polarity of the photomultipliers, the signals of transmitted and scattered light ﬂuxes are inverted.

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Figure 10.20 Schematic of the layout of the apparatus employed to investigate bubble nucleation using light scattering experiments: (1) high pressure optical cell, (2) He–Ne laser light source, (3) laser exciter, (4) photomultiplier, (5) photomultiplier, (6) high voltage power supply, (7) piezoelectric pressure transducer, (8) pressure transducer chamber, (9) charge ampliﬁer, (10) oscilloscope, (11) oscilloscope, (12) signal processing device and differential ampliﬁer, (13) data acquisition and control system, (14) electronic interface, (15) solenoid valve, (16) metering valve, (17) stop valve, (18) solenoid valve, (19) pressure regulator, and (20) helium gas supply. (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:711. Copyright © 1990, with permission from John Wiley & Sons.)

In Figure 10.22a, the lower curve on the oscillograph represents the pressure variation with time in the optical cell and the upper curve on the oscillograph represents the variation of the scattered and transmitted light ﬂuxes with time. It is seen in Figure 10.22a that the pressure release lasted for about 0.5 s. The total pressure drop can be determined from the pressure versus time curve. The ﬁnal pressure in the system is equal to atmospheric pressure. The ﬁrst peak on the experimental scattering curve at a scattering angle 90◦ given in Figure 10.22b can be identiﬁed as the ﬁrst Mie peak on the theoretical scattering curve. Referring to Figure 10.22, supersaturation of the PS solution in the optical cell was achieved by a fast decrease of the pressure in the cell from the initial pressure P1 ; that is, by opening a solenoid valve, which was connected to the cell, at ambient pressure P∞ . During the expansion period, the partial vapor pressure of toluene Pv (t) can be calculated by (Han and Han 1990a) Pv (t) = Rp P (t) = Rp P exp

%

−t/τp + P∞

&

(10.25)

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Figure 10.21 Schematic of the side view of the optical cell used to investigate bubble nucleation in the light scattering experiments by Han and Han. (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:711. Copyright © 1990, with permission from John Wiley & Sons.)

where Rp describes the expansion ratio deﬁned by Rp = Pv (0)/P1 , with Pv (0) being the partial pressure of toluene in the gas phase at t = 0 and P1 being the system pressure at t = 0, P = P1 − P∞ , with P∞ being ambient pressure, τp is the time constant for a particular initial pressure P1 , and t is the time. Figure 10.23 gives plots of critical pressure for bubble nucleation (Pnc ) versus PS concentration at three temperatures and at the initial equilibrium pressure of 2.86 MPa (400 psig) and the expansion ratio of 28.21. It can be seen in Figure 10.23 that Pnc decreases very rapidly as the PS concentration increases and then tends to level off as the PS concentration increases to about 60 wt %. It is also seen that Pnc decreases with increasing temperature. Figure 10.24 gives plots of Pnc versus temperature for mixtures of PS and a ﬂuorocarbon

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Figure 10.22 (a) Oscillographs of light scattering signal and system pressure for a 60/40 PS/toluene solution at 150 ◦ C, an equilibrium pressure 2.86 MPa, and 90◦ scattering angle. (b) Light scattering signal obtained from the data acquisition system, where Φs denotes scattered light ﬂux. (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:711. Copyright © 1990, with permission from John Wiley & Sons.)

blowing agent, FC-11, at three different FC-11 concentrations and an initial equilibrium pressure of 6.305 MPa (900 psig). It is seen that the value of Pnc increases rapidly with temperature and FC-11 concentration. Figure 10.25 shows how the average bubble population density N, which is the average value of the bubble population density for bubble sizes (radius) from 0.5 to 1.0 µm, varies with the PS concentration in a PS/toluene mixture. It is seen that N increases very rapidly with increasing PS concentration. The bubble population density depends on the rate of bubble nucleation relative to the rate of bubble coalescence, which in turn depends on modal bubble size (or time). This observation leads us to conclude that in the initial stage of bubble nucleation, bubble nucleation rate is greater than bubble coalescence rate, while in the later stages of bubble nucleation, the bubble coalescence rate determines the bubble population density.

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PROCESSING OF THERMOPLASTIC POLYMERS Figure 10.23 Plots of critical pressure for bubble nucleation Pnc versus concentration of PS in PS/toluene mixtures at an equilibrium pressure of 2.86 MPa for three different temperatures (◦ C): () 150, () 170, and () 180. (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:711. Copyright © 1990, with permission from John Wiley & Sons.)

Figure 10.26 gives plots of modal bubble radius3 versus time t − tc , where tc refers to the critical time at which bubble nucleation begins, for the 60/40 PS/toluene mixture at an equilibrium pressure of 2.86 MPa and at two different temperatures, 150 and 170 ◦ C. Notice in Figure 10.26 that two different angles, 60 and 90◦ , give rise to essentially the same correlation. It is seen that the bubble growth rate increases with

Figure 10.24 Plots of critical pressure for bubble nucleation Pnc versus temperature for PS/(FC-11) mixtures at an equilibrium pressure of 6.31 MPa for three different concentrations of FC-11 (wt %): () 5, () 8, and () 10. (Reprinted from Han, doctoral dissertation of James H. Han. Copyright © 1988, with permission from Polytechnic University.)

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Figure 10.25 Plot of average bubble population density N versus concentration of PS in PS/toluene mixture at 150 ◦ C and an equilibrium pressure of 2.86 MPa. (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:711. Copyright © 1990, with permission from John Wiley & Sons.)

temperature. Figure 10.27 gives plots of modal bubble radius versus t −tc for PS/toluene mixture at an equilibrium pressure of 2.86 MPa and at 150 ◦ C for three different PS concentrations. It is seen that polymer concentration has a profound inﬂuence on the bubble growth rate. Figure 10.28 gives plots of modal bubble radius versus t − tc for the 60/40 PS/toluene mixture at 150 ◦ C for two different initial equilibrium pressures, Figure 10.26 Plots of modal bubble radius versus time for the 60/40 PS/toluene mixture at an equilibrium pressure of 2.86 MPa for two different temperatures: () 170 ◦ C (with 90◦ scattering angle), () 150 ◦ C (with 90◦ scattering angle), () 150 ◦ C (with 60◦ scattering angle). (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:711. Copyright © 1990, with permission from John Wiley & Sons.)

Figure 10.27 Plots of modal bubble radius versus time for PS/toluene mixtures at 150 ◦ C at an equilibrium pressure of 2.86 MPa for three different concentrations of PS (wt %): () 40, () 50, and () 60. (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:711. Copyright © 1990, with permission from John Wiley & Sons.)

Figure 10.28 Plots of modal bubble radius versus time for the 60/40 PS/toluene mixture at 150 ◦ C for two different initial equilibrium pressures (MPa): () 2.86 and () 4.24. (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:711. Copyright © 1990, with permission from John Wiley & Sons.)

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2.86 and 4.24 MPa. It is seen that the bubble growth rate increases with increasing initial equilibrium pressure. From Figures 10.26–10.28, Han and Han (1990a) concluded that the true value of critical bubble radius (rc ) could not be detected on the light scattering equipment employed, because the values of modal bubble radius determined varied with temperature, polymer concentration, and the initial equilibrium pressure. However, when plots of modal bubble radius versus t − tc were extrapolated to t − tc = 0, as indicated by the dotted lines in Figures 10.26−10.28, they obtained a more or less constant value (0.2–0.4 µm) of modal bubble radius, which is virtually independent of temperature, polymer concentration, and the initial equilibrium pressure. Thus, they regarded the extrapolated modal bubble radius value (0.2–0.4 µm) as being the critical bubble radius rc . Strictly speaking, the quantity measured in Figures 10.26–10.28 is an “apparent” critical bubble radius. This is because the critical bubble radius was determined by extrapolating the observed bubble radius versus time curves back to the time when bubbles ﬁrst appeared experimentally, (t − tc ). However, bubble nucleation occurred, to some degree, throughout the expansion process (Han and Han 1990a). Note that since the pressure was released over a period of time, the degree of supersaturation changed with time and, therefore, the actual critical bubble radius changed with time. The values of apparent critical bubble radius rc thus obtained are summarized in Table 10.3 for PS/toluene mixtures. In Figures 10.26–10.28, the slope of the r(t) versus t − tc plot at t = tc , (dr(t)/dt)t=t , represents the initial bubble growth rate. It is seen in Figure 10.28 c that (dr(t)/dt)t=t increases as the initial equilibrium pressure P1 is increased from c 2.86 MPa (400 psig) to 4.24 MPa (600 psig). This is attributable to the fact that the larger the value of P1 , the greater the degree of supersaturation S(t). Note that S(t) is the driving force for bubble growth, and thus the greater the value of S(t), the larger the value of (dr(t)/dt)t=t will be. Figure 10.29 gives the effect of PS concentration c on the (dr(t)/dt)t=t for PS/toluene mixture with temperature as parameter, showing c that the (dr(t)/dt)t=t decreases very rapidly as the PS concentration increases but c tends to level off as the PS concentration approaches about 60 wt %. In the theoretical Table 10.3 Summary of critical pressure and critical bubble radius determined from bubble nucleation experiments for PS/toluene mixtures

Polymer (wt %) 60 60 50 50 50 40

Temperature ( ◦C) 150 170 150 170 180 150

Critical Pressure (MPa) 0.271 0.303 0.315 0.381 0.408 0.486

Critical Bubble Radius (µm) 0.24 0.28 0.29 0.31 0.32 0.33

Initial equilibrium pressure at 2.859 MPa (400 psig). Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:711. Copyright © 1990, with permission from John Wiley & Sons.

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PROCESSING OF THERMOPLASTIC POLYMERS Figure 10.29 Plots of initial bubble growth rate versus concentration of PS in PS/toluene mixtures at an equilibrium pressure of 2.86 MPa for three different temperatures (◦ C): () 150, () 170, and (䊉) 180. (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:711. Copyright © 1990, with permission from John Wiley & Sons.)

consideration of bubble dynamics during ﬂow, information on (dr(t)/dt)t=t is very c important since it serves as the initial condition when solving the system equations (Han 1981). In the past, in the absence of such experimental evidence, some theoretical studies (Han and Yoo 1981; Street 1968; Ting 1975; Yang and Yeh 1966; Yoo and Han 1982) assumed (dr(t)/dt)t=t = 0, which now turns out not to be justiﬁed (see c Figures 10.26–10.28). Figure 10.30 gives plots of modal bubble radius versus time t − tc for PS/(FC-11) mixtures at an equilibrium pressure of 6.31 MPa and at 160 ◦ C for three different concentrations of FC-11. It is seen that the bubble growth rate increases as the concentration of FC-11 increases. Figure 10.31 shows the effect of FC-11 concentration in PS/(FC-11) mixtures on the initial growth rate of bubble, (dr(t)/dt)t=t , for three different temc peratures. It is seen that that the (dr(t)/dt)t=t increases very rapidly with both FC-11 c concentration and temperature, very similar to that observed in Figure 10.29 for the PS/toluene mixtures. The values of apparent critical bubble radius rc for PS/(FC-11) mixtures are summarized in Table 10.4. 10.3.1.2 Flow-Induced Bubble Nucleation In foam extrusion, bubble nucleation occurs in a shear ﬂow ﬁeld. Figure 10.32 gives a schematic for the layout of a specially designed apparatus employed in ﬂow-induced bubble nucleation experiments (Han and Han 1988), and Figure 10.33 gives a schematic of the side view of a slit die with glass windows made of fused quartz. Notice in Figure 10.33 that three pressure transducers were mounted on one surface of the slit die along the ﬂow direction, so that the critical wall normal stress could be determined in the ﬂow channel when the location of bubble nucleation was identiﬁed using an He–Ne laser as the light source. In the experiments, a foam extrusion line (shown schematically

Figure 10.30 Plots of modal bubble radius versus time for PS/(FC-11) mixtures at 160 ◦ C and an equilibrium pressure of 6.31 MPa for three different concentrations of FC-11 (wt %): () 5, () 8, and () 10. (Reprinted from Han, doctoral dissertation of James H. Han. Copyright © 1988, with permission from Polytechnic University.)

Figure 10.31 Plots of initial bubble growth rate versus concentration of FC-11 in PS/(FC-11) mixtures at an equilibrium pressure of 6.31 MPa for three different temperatures (◦ C): () 150, () 160, and () 170. (Reprinted from Han, doctoral dissertation of James H. Han. Copyright © 1988, with permission from Polytechnic University.)

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Table 10.4 Summary of critical pressure and critical bubble radius determined from bubble nucleation experiments for PS/(FC-11) mixtures

FC-11 (wt %) 5 5 8 8 10 10 10

Temperature ( ◦C)

Critical Pressure (MPa)

150 160 160 170 150 160 170

0.437 0.635 1.192 2.930 0.865 1.458 5.290

Critical Bubble Radius (µm) 0.24 0.27 0.30 0.34 0.28 0.32 0.35

The initial equilibrium pressure is 6.305 MPa (900 psig). Based on the doctoral dissertation of James H. Han (1988).

Figure 10.32 Schematic of the layout of the apparatus employed in ﬂow-induced bubble nucleation experiments: (1) slit die with glass windows, (2) He–Ne laser light source, (3) laser exciter, (4) photomultiplier, (5) collimator lenses, (6) high-voltage power supply, (7) oscilloscope, (8) signal processing device, (9) differential ampliﬁer, (10) data acquisition device, (11) pressure transducers, (12) pressure measurement device, (13) automatic tracking system, (14) microcomputer, (15) ﬂexible steel tube, and (16) bleed valve. (Reprinted from Han and Han, Polymer Engineering and Science 28:1616. Copyright © 1988, with permission from the Society of Plastics Engineers.)

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Figure 10.33 Schematic of the side view of the slit die with glass windows, used in ﬂow-induced bubble nucleation experiments. (Reprinted from Han and Han, Polymer Engineering and Science 28:1616. Copyright © 1988, with permission from the Society of Plastics Engineers.)

in Figure 13.1 in Volume 1) was used to provide a continuous mixture of polymer melt and ﬂuorocarbon blowing agent under high pressures and at elevated temperatures. An automatic tracking system controlled by a microcomputer was designed for precise determination of the position at which bubbles nucleate in the ﬂow channel, and for the ability to move the control volume of the optical system to any position desired along the ﬂow channel. By moving the control volume within the ﬂow channel, and with the aid of a storage oscilloscope, the position at which a bubble nucleated was determined. The velocity, shear rate, and shear stress in the slit die were calculated by solving the equations of motion. Typical output signals from the oscilloscope are shown in Figure 10.34, in which the scattering signals were obtained at a position where there was no bubble nucleation in the ﬂow of mixture of a molten PS and FC-11 as blowing agent. It can be seen in Figure 10.34a that there was a signiﬁcant level of noise from the optical system, which might have been caused by one or any combination of the following sources: (1) mechanical vibration from the foam process line, (2) velocity ﬂuctuations due to extreme surging, (3) density ﬂuctuations due to incomplete mixing of blowing agent and polymer melt, (4) dirt or dust in the polymer melt, and/or (5) vibration of the optical system during the movement of step motors. Figure 10.34b gives a pulse spectrum at the nucleation site in the ﬂow channel. Figure 10.34c gives a pulse spectrum at a position past the nucleation site, at which point the bubbles had already been nucleated. It is seen that the pulse heights in Figure 10.34c are much greater than those in Figure 10.34b. Figure 10.35 gives the experimentally measured wall normal stress proﬁle along the slit channel (the top panel) and the nucleation sites (the bottom panel) determined by the light scattering technique at the center of the slit die width, for a PS/(FC-11) mixture containing 0.8 wt % FC-11 at a mass ﬂow rate of 0.38 g/s at 180 ◦ C. Similar plots are given in Figure 10.36 for a higher blowing agent concentration, 4.0 wt %, and a higher mass ﬂow rate, 1.7 g/s. It is seen in Figures 10.35 and 10.36 that the pressure

Figure 10.34 Typical output signals from the oscilloscope: (a) output signal at a position in the

ﬂow channel where no bubble nucleation occurred, (b) output signal at the position in the ﬂow channel where bubble nucleation was occurring, (c) signal pattern at the position in the ﬂow channel where the nucleated bubbles were growing. (Reprinted from Han and Han, Polymer Engineering and Science 28:1616. Copyright © 1988, with permission from the Society of Plastics Engineers.)

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Figure 10.35 (a) Experimentally measured wall normal stress proﬁle and (b) nucleating sites in the slit ﬂow channel for a PS/(FC-11) mixture containing 0.8 wt % FC-11 at a mass ﬂow rate of 0.38 g/s at 180 ◦ C. (Reprinted from Han and Han, Polymer Engineering and Science 28:1616. Copyright © 1988, with permission from the Society of Plastics Engineers.)

Figure 10.36 (a) Experimentally measured wall normal stress proﬁle and (b) nucleating sites in the slit ﬂow channel for a PS/(FC-11) mixture containing 4.0 wt % FC-11 at a mass ﬂow rate of 1.70 g/s at 180 ◦ C. (Reprinted from Han and Han, Polymer Engineering and Science 28:1616. Copyright © 1988, with permission from the Society of Plastics Engineers.)

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gradient along the ﬂow direction, −∂p/∂z is constant, and thus the ﬂow of the mixture became fully developed before phase separation (bubble nucleation) occurred. It is interesting to observe that the nucleation site varies with the y direction (perpendicular to the ﬂow direction, z), changing its direction at a certain critical value, y = ync . That is, bubble nucleation occurs near the die wall. In determining the shear stress at y = ync , that is, the critical shear stress σnc for bubble nucleation in the slit die, the equations of motion for steady-state shear ﬂow in a slit die were solved numerically using the truncated power-law model with the aid of the melt viscosity data of PS/(FC-11) mixtures (Han and Han 1988). Figure 10.37 gives a plot of reduced viscosity, η/η, ¯ versus weight fraction of FC-11 in PS/(FC-11) mixtures, where η¯ denotes the melt viscosity of the PS/(FC-11) mixture and η denotes the melt viscosity of neat PS. In Chapter 13 of Volume 1, we have shown that the ratio η/η ¯ is virtually independent of shear rate and temperature over a limited range of shear rates and temperatures. Such an experimental ﬁnding suggests that one can calculate shear rate and temperature dependences of η¯ for the PS/(FC-11) mixtures, via Figure 10.37, only with information on the shear rate and temperature dependences of neat PS. Figure 10.38 gives a plot of σnc versus blowing agent concentration in PS/(FC-11) mixtures ﬂowing through a slit die, where values of σnc were determined from the solution of the equations of motion at experimentally determined ync . It is worth noting in Figure 10.38 that the value of σnc increases with FC-11 concentration in the PS/(FC-11) mixture, suggesting that the nucleation site shifts towards the die wall as the concentration of FC-11 increases. Note that the viscosity of the PS/(FC-11) mixture decreases with FC-11 concentration (see Figure 10.37).

Figure 10.37 Plot of reduced viscosity

η/η ¯ versus concentration of FC-11 in PS/(FC-11) mixtures. (Reprinted from Han and Ma, Journal of Applied Polymer Science 28:851. Copyright © 1983, with permission from John Wiley & Sons.)

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Figure 10.38 Plot of critical shear stress for bubble nucleation σnc versus concentration of FC-11 in PS/(FC-11) mixtures. (Reprinted from Han and Han, Polymer Engineering and Science 28:1616. Copyright © 1988, with permission from the Society of Plastics Engineers.)

Figure 10.39 gives a photograph of a PS/(FC-11) mixture ﬂowing through a slit die, from which we can speculate that bubbles near the die wall of the slit channel might have nucleated earlier than those at the center, corroborating the results obtained from light scattering experiments (see Figures 10.35 and 10.36). The bubble nucleation phenomenon observed in Figures 10.35 and 10.36 suggests that bubble nucleation

Figure 10.39 Photograph of a PS/(FC-11) mixture ﬂowing through a slit die, indicating that bubbles near the wall of the slit channel might have nucleated earlier than those at the center. (Reprinted from Han and Han, Polymer Engineering and Science 28:1616. Copyright © 1988, with permission from the Society of Plastics Engineers.)

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at 0 < y < ync is induced by ﬂow and bubble nucleation at positions ync < y < h (die wall) is induced by shear stress. Note that the bubble near the channel wall might also have been generated by cavitation brought about by the surface roughness of the wall, and also by thermal ﬂuctuations due to the heat transfer between the metal and the PS/(FC-11) mixture. The above observation leads us to conclude that at low FC-11 concentrations, ync moves closer to the center of the ﬂow channel and thus bubble nucleation is induced by ﬂow, while as the FC-11 concentration increases, shear stress seems to be the dominant factor that induces bubble nucleation. 10.3.2 Theoretical Considerations of Bubble Nucleation in Polymer Solutions According to the classical nucleation theory (Blander and Katz 1975), the free-energy change F for the formation of a gas bubble is given by F = Aγ + VG (PL − PG ) + n(µG − µL )

(10.26)

where A is the new surface created, γ is the surface tension, VG is the volume of a spherical gas bubble, PL is the pressure of the liquid medium, PG is the pressure inside the gas bubble, n is the number of gas molecules in the bubble and µG and µL are the chemical potentials of the gas and liquid phases, respectively. For a critical nucleus with radius r under an equilibrium condition, the difference of the chemical potentials, µG − µL , is equal to zero, and PG is equal to the equilibrium vapor pressure PV . Thus Eq. (10.26) reduces to F = 4πr 2 γ + (4/3)πr 3 (PL − PV )

(10.27)

The minimum free energy change F ∗ required for the formation of a critical nucleus is determined by ∂F /∂r = 0, yielding 8πrγ + 4πr 2 (PL − PV ) = 0

(10.28)

Thus, the critical bubble radius rc is determined from rc = 2γ /(PV − PL )

(10.29)

By substituting Eq. (10.29) into Eq. (10.27), F ∗ can be expressed by F ∗ = 16πγ 3 /3(PV − PL )2

(10.30)

and the rate of bubble nucleation (number of bubbles/m3·s) is given by J = MB exp (−F ∗ /kB T )

(10.31)

where M is the number of molecules per unit volume of the metastable phase, B is the frequency factor representing the frequency that gas molecules impinge upon

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the embryo, kB is the Boltzmann constant, T is the absolute temperature, and F ∗ is the minimum free energy change required for the phase transformation. Note that the above theory was developed for a single-component system and is referred to as the “classical nucleation theory.” As shown by Blander and Katz (1975) for bubble nucleation, Eq. (10.31) can be expressed by

2γ J =M πmB

1/2

16πγ 3 exp − 2 3kB T PV − PL

(10.32)

where m is the mass of a gas molecule and B ≈ 2/3 for PL PV and B = 1 for PL = PV . They also modiﬁed the classical nucleation theory for the diffusioncontrolled bubble nucleation in a mixture in which one component is volatile. Han and Han (1990b) used Eq. (10.32) to calculate the rate of bubble nucleation J at critical time tc for the PS/toluene mixtures. Note that PL in Eq. (10.32) is equal to the critical pressure for bubble nucleation, Pnc . Their bubble nucleation experiments indicated that for the PS/toluene mixture at 150 ◦ C, bubble nucleation occurred at a critical pressure PL , which was greater than PV . Under such circumstances, for the PS/toluene mixtures the values of the exponential term in Eq. (10.32) are virtually equal to zero, leading them to conclude that the bubble nucleation rates J were equal to zero, which was unacceptable. Therefore, Eq. (10.32) was not useful for describing bubble nucleation in PS/toluene mixtures. Han and Han (1990b) modiﬁed the classical nucleation theory to describe bubble nucleation in a concentrated polymer solution at elevated temperatures. In the modiﬁcation, they included the following two additional factors. First, the change of free energy due to the presence of macromolecules in solvent was taken into account in the determination of the free energy required for bubble formation in the polymer solution. According to the Flory–Huggins theory, the free energy change of solvent molecules due to the presence of macromolecules in the solution can be expressed by (see Eq. (10.11)) Ft = nkB T [ln φ 1 + φ2 + χ φ2 2 ]

(10.33)

where n is the number of volatile molecules in a critical bubble. Second, while the classical nucleation theory assumes that the critical nuclei are under chemical equilibrium conditions, bubble nucleation in a polymeric solution always occurs under supersaturated conditions, which is not under chemical equilibrium conditions and PG = PV . Therefore, the change in free energy under such conditions must be taken into account. For this, the degree of supersaturation S(t) of the polymer solution during the expansion process may be determined by S(t) = [Co − C(t)]/C∞ (t)

(10.34)

where Co is the initial concentration of the volatile component in a polymer solution at time t = 0, C(t) is the amount of the volatile component consumed per unit volume of the solution at time t, and C∞ (t) is the equilibrium concentration (i.e., the solubility) corresponding to the partial pressure of the volatile component in the vapor phase at time t.

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Han and Han (1990b) proposed the expression Fs = nkB T ln S(t)

(10.35)

to calculate the free energy change due to the presence of supersaturation during the expansion process. Accordingly, the change of free energy Fp∗ for the formation of a critical bubble in polymer solution may be rewritten as Fp∗ = F ∗ − Fs − Ft

(10.36)

where F ∗ is deﬁned by Eq. (10.30), Fs by Eq. (10.35), and Ft by Eq. (10.33). In the classical nucleation theory, the probability of occurrence of nuclei is determined from the minimum work required for bubble formation, and the nucleation rate at steady state is given by Eq. (10.31). Han and Han (1990b) used the following expression to calculate bubble nucleation rate in the polymer solution: J = [M][B] exp(−Fp∗ /nkB T )

(10.37)

where M is the number of solvent molecules per unit volume of the polymer solution and B is the frequency factor. In Eq. (10.37), the probability for the formation of nuclei in a polymer solution is determined by the energy required for each volatile molecule to form a critical nucleus. They proposed the following empirical expression: B = B1 [D(T )/4πrc 2 ] exp(−B2 /T )

(10.38)

where D(T) is the temperature-dependent diffusion coefﬁcient of the volatile molecule in a polymer solution, T is the absolute temperature, and B1 and B2 are constants to be determined by using experimental results of nucleation rate. Due to the consumption of volatile molecules during bubble nucleation and subsequent growth, the rate of bubble nucleation changes during the expansion process. In order to determine the number of bubbles nucleated during the expansion process, the amount of volatile component consumed by the growing nuclei has to be taken into account. As pointed out above, under supersaturated conditions, the pressure PG inside a gas bubble is not equal to the equilibrium vapor pressure PV . The following expression has been proposed to estimate PG during the expansion process (Han and Han 1990b): PG (t)/PV = 1 + Q ln S(t)

(10.39)

where Q is an empirical parameter to be determined. Note that Eq. (10.39) satisﬁes the conditions that PG = PV at S = 1, and PG /PV → ∞ when S → ∞. Since we are concerned with the nucleation stage only, Q may be assumed to be constant during the bubble nucleation process. In order to determine the parameter Q, it may be assumed that the critical nucleus is formed under hydrodynamic equilibrium conditions

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and therefore satisﬁes the hydrodynamic equation for bubble growth in a Newtonian ﬂuid, which is given by (Han and Yoo 1981): ρL [(3/2)˙r 2 + r˙ r¨ ] = PG − 2γ /r − 4η0 (˙r /r) − PL

(10.40)

where the dot above r refers to time derivative, ρL is the density of liquid, and η0 is the zero-shear viscosity. Note that PG and PL are functions of time during the expansion process. Therefore, Q can be determined using Eqs. (10.39) and (10.40) with the aid of the experimental results for bubble nucleation in a polymer solution. If we assume that the bubble growth rate upon nucleation is constant, that is; dr/dt = Z (constant), the bubble growth equation can be expressed as r(t) = rc + Z(t − tc )

(10.41)

where rc is the critical bubble radius and tc is the critical time at which bubbles nucleate. At the critical time tc , PL is equal to the critical pressure for bubble nucleation Pnc , r is equal to rc , and the degree of supersaturation S is equal to Snc . The pressure PG inside the critical bubble at t = tc can be determined from Eq. (10.40): PG (tc ) = (3/2)ρL Z 2 + 2γ /rc + 4η0 (Z/rc ) + Pnc

(10.42)

Also, at t = tc , Eq. (10.39) can be rewritten as PG (tc ) = PV {1 + Q ln Snc }

(10.43)

By combining Eqs. (10.42) and (10.43), Q can be determined by the expression Q=

'

(3/2)ρL Z 2 + 2γ /rc + 4η0 (Z/rc ) + Pnc

PV − 1

(

ln Snc

(10.44)

After the bubble nucleation is cut off, the degree of supersaturation decreases significantly due to the consumption of the volatile component in the solution. Note that bubble growth is a diffusion-controlled process, and also that Henry’s law may be applicable to the determination of the relationship between PG and r using Eq. (10.40). In this case, Eq. (10.40) must be solved numerically with the aid of a mass transfer equation (Han and Yoo 1981). In order to calculate the number of bubbles nucleated during the expansion process, it is necessary to know the amount of volatile component consumed by the growing bubbles. The number of bubbles nucleated at steady state may be calculated from Eq. (10.37), and the growth rate from Eq. (10.41). The number of volatile molecules consumed per unit volume of the solution C(t) as a function of time can be determined by (Han and Han 1990b): C(t) =

t 0

n t − t J t dt

(10.45)

where n(t −t ) is the number of gas molecules inside a bubble at time t which nucleated at t , and J (t ) is the nucleation rate corresponding to the supersaturation at time t .

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Since n(t) is given by n(t) = (4π/3RT )PG (t)r 3 (t)

(10.46)

where R is the universal gas constant, Eq. (10.45) can be rewritten as

t 3 C(t) = (4π/3RT ) 0 PG (t) r t − t J t dt

(10.47)

where r(t − t ) is the radius of a gas bubble at time t which nucleated at t . Thus, the degree of supersaturation S(t) during the expansion process given in Eq. (10.34) can be determined. Han and Han (1990b) used the theory presented above for comparison with experimental results for bubble nucleation in the PS/toluene mixtures. They accomplished this by adjusting the value of B, until a cut-off of supersaturation at critical time tc was obtained, with the aid of information on the temperature-dependent diffusion coefﬁcient D(T) for PS/toluene mixtures presented in Figure 10.18. The frequency factor B for the 50/50 PS/toluene mixture at three temperatures was calculated, yielding B = 4.708 × 1034 [D(T )/4πrc 2 ] exp(−4.23 × 104 /T )

(10.48)

The total number of bubbles nucleated per unit volume may be calculated from N=

t tc

J (t ) dt

(10.49)

There are two sets of variables which control bubble nucleation in a polymer solution: (1) the physical properties of the mixture of a volatile component and a polymer (i.e., the solubility, diffusivity, surface tension, and viscosity) and (2) the process variables (temperature, initial equilibrium pressure, and pressure release rate). In the theories presented above (see Eqs. (10.31) and (10.37)), which were based on the classical nucleation theory, the surface tension was assumed to remain constant during bubble nucleation and subsequent bubble growth. Strictly speaking, such an assumption may be considered to be a gross approximation. Lee and Flumerfelt (1996) applied a thermodynamic approach to investigate bubble nucleation. The approach enabled them to calculate changes in surface tension when a gaseous component was dissolved in a molten polymer. They concluded that the dissolved gas in a molten polymer reduced its surface tension, which in turn had a considerable inﬂuence on bubble nucleation rate. The interested readers are referred to the original paper. Figure 10.40 gives plots of the degree of supersaturation S(t) and bubble nucleation rate J(t) versus time for a 60/40 PS/toluene mixture at 150 ◦ C. It can be seen that both S(t) and J(t) increase with time t before the critical time tc and then decrease rapidly after t = tc . Due to the consumption of the volatile component by both bubble nucleation and subsequent bubble growth, the degree of supersaturation is cut-off for t > tc . The bubble nucleation rate decreases sharply for t > tc , because the concentration

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Figure 10.40 Predicted degree of supersaturation S and bubble nucleation rate J versus time for a 60/40 PS/toluene mixture at 150 ◦ C, where 60/40 refers to the weight percentage of the constituent components. (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:743. Copyright © 1990, with permission from John Wiley & Sons.)

of toluene in the solution and the degree of supersaturation both decrease sharply. It should be mentioned that the bubble nucleation rate at tc , J (tc ), increases with PS concentration (Han and Han 1990b). Figure 10.41 gives a plot of predicted critical nucleation rate J (tc ) versus the reciprocal of absolute temperature for a 50/50 PS/toluene solution. It can be seen that J (tc ) increases rapidly with temperature. Figure 10.42 gives plots of computed N versus time during the expansion process at 150 ◦ C for three different PS/toluene concentrations: 40, 50, and 60 wt % PS. It is seen that N increases very rapidly in the beginning and then tends to level off after a certain time. Note in Figure 10.42 that N increases with polymer concentration, which is consistent with the experimental results presented in Figure 10.25. Table 10.5 gives a summary of the predicted and experimental values of N, showing that the experimentally determined values of N are much smaller than the predicted ones. This may be attributable to the possible coalescence of bubbles that might have occurred during bubble nucleation. Note that the distance between bubbles decreases with increasing N. For large values of N, the coalescence of nuclei would become important. It appears that in order to predict values of N close to the experimentally determined ones, the rate of coalescence would have to be included in any future theoretical development. However, so far, there is no satisfactory theoretical model available for describing the bubble coalescence in polymer solutions. More study is needed to better understand bubble nucleation and coalescence in polymer solutions.

468

PROCESSING OF THERMOPLASTIC POLYMERS Figure 10.41 Bubble nucleation rate at critical time tc versus reciprocal of absolute temperature for a 50/50 PS/toluene mixture, where 50/50 refers to the weight percentage of the constituent components. (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:743. Copyright © 1990, with permission from John Wiley & Sons.)

10.4

Foam Extrusion

Among many polymers, LDPE and PS have enjoyed commercial success via the production of low-density cellular products by extrusion. Historically, ﬂuorocarbons were used as blowing agents in foam extrusion, however, owing to the depletion of the ozone layer, in recent years their use has been discouraged. A schematic layout of the foam extrusion process line is given in Figure 13.1 in Chapter 13 of Volume 1. Due to the difference in their molecular structures, LDPE and PS yield thermoplastic foams with different mechanical properties, which consequently have different end-use applications. For a given chemical structure of polymer, the successful production of foam products depends, among many things, on the following factors: (1) the chemical structure and concentration of the blowing agent, (2) the extrusion melt temperature, (3) the extrusion pressure (hence extrusion rate or apparent shear rate in the die), (4) the type and concentration of nucleating agent, and (5) the geometry of the extrusion die (Han and Ma 1983b). In controlling the quality of low-density cellular products, the following three variables are considered most important: (1) density, (2) cell size and open-cell fraction, and (3) the expansion ratio of extrudate. In this section, we present processing–property–morphology relationships in foam extrusion, with emphasis on

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Figure 10.42 Bubble population density N versus time for the PS/toluene mixtures at 150 ◦ C for different PS concentrations (wt %): (1) 60, (2) 50, and (3) 40. (Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:743. Copyright © 1990, with permission from John Wiley & Sons.)

the effect of die geometry, the type of blowing agent, the type of nucleating agent, and the extrusion temperature. 10.4.1 Processing–Property–Morphology Relationships in Proﬁle Foam Extrusion Let us consider how processing variables affect foam density and cell morphology when a mixture of molten polymer and blowing agent is extruded in a cylindrical die.

Table 10.5 Comparison between the predicted and experimentally measured bubble population density N for PS/toluene mixtures

Temperature ( ◦C) 150 150 150 170 180

Polystyrene (wt %) 60 50 40 50 50

N (number/m3 ) (Experimental)

N (number/m3 ) (Theoretical)

1.733 × 1016 4.374 × 1015 1.481 × 1015 7.480 × 1015 1.229 × 1016

4.000 × 1020 1.676 × 1020 2.531 × 1019 1.676 × 1020 1.405 × 1020

Reprinted from Han and Han, Journal of Polymer Science, Polymer Physics Edition 28:743. Copyright ©1990, with permission from John Wiley & Sons.

470

PROCESSING OF THERMOPLASTIC POLYMERS

In proﬁle foam extrusion, foam density is very important and can depend on, among many factors, bubble nucleation (if any) within the extrusion die and the rate of bubble growth both inside the die and outside the die during cooling. Figures 10.43a and 10.43b give, for illustration, photographs demonstrating the differences in the extrudate swell behavior between a neat LDPE (Rexene 143, Rexene Corporation) and a mixture of the same polymer with a ﬂuorocarbon blowing agent (FC-114) upon exiting from a cylindrical die. It is seen that the extrudate of neat Rexene 143 swells quickly upon exiting from die and then contracts slightly, due to the gravitational force, as it ﬂows downward, whereas the extrudate of the Rexene 143/(FC-114) mixture swells initially due to the recovery of unconstrained elastic strains of the melt and then swells further due to the expansion of gas bubbles in the melt. Figure 10.43c shows schematically the mechanism of extrudate swell of a molten polymer/blowing agent mixture, indicating that gas bubbles grow continuously as the extrudate ﬂows downward. Figure 10.44 gives micrographs (a) near the edge and (b) at the center of the cross section of an extrudate of the 96/4 mixture of PS (Styron 678, Dow Chemical) and FC-12. It is seen that the cell size is much larger at the center than near the edge, which is attributable to the fact that the extrudate, upon exiting from the die, undergoes cooling more rapidly at the extrudate surface than at the center. Hence, the bubbles keep growing at the center of the extrudate, while bubble growth is suppressed near

Figure 10.43 Photographs describing the extrudate swell upon exiting from a cylindrical die (D = 0.315 cm, L/D = 4, entrance angle (α) = 60◦ , and reservoir-to-diameter (DR /D) ratio = 8): (a) neat Rexene 143 extruded at 160 ◦ C and γ˙app = 130 s−1 , (b) 90/10 Rexene 143/(FC-114) mixture with 0.25 wt % talc as nucleating agent extruded at 100 ◦ C and γ˙app = 335 s−1 . (c) Schematic of extrudate swell of a molten polymer/blowing agent mixture. (Reprinted from Han and Ma, Journal of Applied Polymer Science 28:2961. Copyright © 1983, with permission from John Wiley & Sons.)

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Figure 10.44 Micrographs of the cross section of an extrudate of a 96/4 Styron 678/(FC-12) mixture extruded at 150 ◦ C in a cylindrical die (D = 0.315 cm, L/D = 4, DR /D = 8, α = 60◦ ): (a) near the edge of the extrudate cross section and (b) at the center of the extrudate cross section.

the extrudate surface. Note that the extrudate swell due to blowing agent in a molten polymer/blowing agent mixture depends, among many other things, on (1) the type and concentration of the blowing agent, (2) the extrusion rate, (3) the type and concentration of nucleating agent, (4) the die geometry, and (5) the cooling rate of extrudate. Today, it is a well-established fact that the use of nucleating agents is absolutely essential for controlling the cell morphology (namely the number of cells, the cell size, and the cell-size distribution) in thermoplastic foams. Without a nucleating agent, the number of cells is too small, and the cell size is too large to produce low-density foams. However, there is currently no theoretical guidance suggesting the type of nucleating agent to be used for a given combination of polymer and blowing agent. This is understandable because the problem at hand is very complex; it involves heterogeneous nucleation from a mixture of molten polymer and blowing agent at high temperature and high pressure. Evidence indicates that the surface characteristics of a nucleating agent are of paramount importance in controlling the foam quality in the foam extrusion process. Figure 10.45 gives plots of foam density (ρ) versus apparent shear rate (γ˙app ) for the 85/15 Rexene 143/(FC-114) mixture at two die temperatures, 110 ◦ C and 100 ◦ C, with varying talc concentrations. Figure 10.46 shows the effect of talc concentration on open-cell fraction in the extrudate. The following observations are worth noting in

Figure 10.45 Plots of foam density ρ versus apparent shear rate γ˙app for the 85/15 Rexene 143/(FC-114) mixture extruded at (a) 110 ◦ C and (b) 100 ◦ C with various talc concentrations (wt %): () neat Rexene 143, () 0.25, () 0.50, () 0.75, and (7) 1.00. The geometry of the die employed for the extrusion: D = 0.315 cm, L/D = 4, DR /D = 8, and α = 60◦ . (Reprinted from Han and Ma, Journal of Applied Polymer Science 28:2961. Copyright © 1983, with permission from John Wiley & Sons.)

Figure 10.46 Plot of open-cell fraction versus talc concentration for the 85/15 Rexene 143/ (FC-114) mixture extruded at γ˙app = 210 s−1 and 100 ◦ C. The geometry of the die employed for the extrusion: D = 0.315 cm, L/D = 4, DR /D = 8, and α = 60◦ . (Reprinted from Han and Ma, Journal of Applied Polymer Science 28:2961. Copyright © 1983, with permission from John Wiley & Sons.)

472

FOAM EXTRUSION

473

Figures 10.45 and 10.46: (1) the concentration of talc greatly inﬂuences foam density, and a minimum foam density can be achieved only over a certain range of talc concentration, (2) the die temperature also greatly inﬂuences foam density, (3) at both the high talc concentration and high die temperature, the ρ is increased as γ˙app is increased (but the ρ remains constant, while γ˙app is increased, when optimum values of talc concentration and die temperature are chosen), and (4) the open-cell fraction increases with talc concentration. With reference to Figures 10.45 and 10.46, the rapidly increasing trend of ρ at high talc concentrations (0.75 to 1.0 wt %) at 110 ◦ C, as γ˙app is increased, may be attributable to the creation of local “hot spots” (Hansen 1962), which help gas bubbles collapse before the solidiﬁcation of extrudate begins. The hot spots may also give rise to a high open-cell fraction at high talc concentrations. At an optimum die temperature, as may be seen in Figure 10.47, the addition of talc decreases the foam density considerably, which is attributable to the presence of many small gas bubbles in the extrudate. Indeed, Figure 10.48 gives micrographs demonstrating the effect of talc, as nucleating agent, on foam morphology. Figure 10.49 shows the effect of the capillary length-to-diameter ratio (L/D) on extrudate swell ratio (dj /D) and ρ for the 85/15 Rexene 143/(FC-12) mixture with talc as nucleating agent. It is seen that as in the extrusion of neat Rexene 143, the dj /D ratio increases as the L/D ratio decreases. Note that, except for an L/D ratio of 0 (i.e., for a conical die), the dj /D ratio increases very little with γ˙app . This observation is quite different from the situation where Rexene 143 alone is extruded. The effect of L/D ratio on ρ appears to be rather complex, as may be seen in Figure 10.49. It is interesting to observe in Figure 10.49 that the foam density obtained with an L/D ratio of 0 is greater than that obtained with L/D ratios of 2 and 4. Also, the die having an L/D ratio of 2 gives rise to the lowest foam density, as well as, the least open-cell fraction, as shown in Figure 10.50. An explanation as to why the die having an L/D ratio of 0 (i.e., the die having only the converging section) gives rise to higher foam density than the die

Figure 10.47 Plots of foam density ρ versus talc concentration for the 85/15 Rexene 143/ (FC-114) mixture extruded at γ˙app = 200 s−1 for two die temperatures: () 100 ◦ C and () 110 ◦ C. The geometry of the die employed for the extrusion: D = 0.315 cm, L/D = 4, DR /D = 8, and α = 60◦ . (Reprinted from Han and Ma, Journal of Applied Polymer Science 28:2961. Copyright © 1983, with permission from John Wiley & Sons.)

Figure 10.48 Micrographs showing the effect of nucleating agent (talc) on cell size: (a) without talc and (b) with 0.25 wt % talc for the 85/15 Rexene 143/(FC-114) mixture extruded at γ˙app = 200 s−1 and 100 ◦ C. The geometry of the die employed for the extrusion: D = 0.315 cm, L/D = 4, DR /D = 8, and α = 60◦ . (Reprinted from Han and Ma, Journal of Applied Polymer Science 28:2961. Copyright © 1983, with permission from John Wiley & Sons.)

Figure 10.49 Plots of extrudate swell ratio dj /D and foam density ρ versus apparent shear rate γ˙app for the 85/15 Rexene 143/(FC-114) mixture (with 0.25 wt % talc) extruded at 100 ◦ C

through a cylindrical die with varying L/D ratios: () L/D = 0 (conical die), () L/D = 2, () L/D = 4, and () L/D = 8. The other dimensions of the die employed for the extrusion: D = 0.315 cm, DR /D = 8, and α = 60◦ . (Reprinted from Han and Ma, Journal of Applied Polymer Science 28:2961. Copyright © 1983, with permission from John Wiley & Sons.) 474

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Figure 10.50 Plots of open-cell fraction and die pressure versus L/D ratio for the 85/15 Rexene 143/(FC-114) mixture (with 0.25 wt % talc as nucleating agent) extruded at γ˙app = 160 s−1 and 100 ◦ C. (Reprinted from Han and Ma, Journal of Applied Polymer Science 28:2961. Copyright © 1983, with permission from John Wiley & Sons.)

having an L/D ratio of 2, while having greater dj /D ratio, is as follows. An explanation for this seemingly anomalous observation may be given in terms of the die pressure, which was measured during the foam extrusion experiment. Referring to Figure 10.50, the die pressure, which was measured in the upstream end of the reservoir section, decreases as the L/D ratio decreases as expected intuitively. It appears that, under the particular extrusion conditions employed, the die pressure of 2.34 MPa (340 psig) with the die having an L/D = 0 (see Figure 10.50) was not sufﬁciently high to prevent the occurrence of premature foaming inside the die, thus giving rise to a high open-cell fraction. Note in Figure 10.50 that the open-cell fractions for the dies with L/D ratios of 4 and 8 are greater than that for the die with an L/D ratio of 2, in spite of the fact that the die pressure increases with L/D ratio. This is attributable to the fact that although there is less chance of premature foaming occurring in the reservoir section of the dies having L/D ratios of 4 and 8, premature foaming is expected to occur in the capillary section as the Rexene 143/(FC-12) mixture approaches the die exit. Therefore, the larger the L/D ratio of a die, the higher the open-cell fraction will be. These observations suggest that an optimum die geometry is required to obtain extruded foam products having low density. Han and Ma (1983b) reported the effects of L/D ratio (0, 2, 4, and 8), die entrance angle (15◦ , 30◦ , and 60◦ ), and reservoir-to-capillary diameter (DR /D) ratio (2 and 8) on the foam extrusion characteristics of PS. Over the range of die design variables investigated, they observed that only the L/D ratio played a signiﬁcant role in inﬂuencing the foam extrusion characteristics of PS.

476

PROCESSING OF THERMOPLASTIC POLYMERS

Figure 10.51 Plots of extrudate swell ratio dj /D and foam density ρ versus the concentration of a bicomponent nucleating agent system for a 92/4/4 Styron 678/(FC-11)/(FC-12) mixture extruded at γ˙app = 160 s−1 and 130 ◦ C. The geometry of the die employed for the extrusion: D = 0.315 cm, L/D = 2, DR /D = 8, and α = 60◦ . (Reprinted from Ma and Han, Journal of Applied Polymer Science 28:2983. Copyright © 1983, with permission from John Wiley & Sons.)

Figure 10.51 gives plots of dj /D and ρ versus the concentration of a bicomponent nucleating agent system (citric acid/sodium bicarbonate) for a 92/4/4 Styron 678/(FC-11)/(FC-12) mixture. It is seen that, initially, ρ decreases while dj /D remains more or less constant as the concentration of nucleating agent increases, and then ρ levels off. In other words, there exists a minimum amount of nucleating agent (about 0.3 wt % citric acid and 0.375 wt % NaHCO3 ) that may be required for effective foaming of PS. One may surmise that a nucleating agent allows one to achieve ﬁne cell sizes and closed-cell morphology. Figure 10.52 shows micrographs demonstrating the difference in cell size obtained without and with nucleating agent. It is seen that the use of nucleating agent did indeed increase the number of cells and decrease the cell size. Figure 10.53 gives plots of dj /D and ρ versus die temperature for Styron 678/(FC-11)/(FC-12) mixtures with varying concentrations of (FC-11)/(FC-12) blowing agent components. It is seen that the dj /D ratio decreases and ρ increases rapidly when the die temperature passes a certain critical value. Note in Figure 10.53 that below the critical die temperature, ρ decreases and dj /D increases as the blowing agent concentration increases, whereas above the critical die temperature, the opposite trend is observed. In other words, one must control both the blowing agent concentration and the die temperature in order to produce PS foams of acceptable quality. Figure 10.54 gives micrographs of extruded foam obtained under an identical extrusion conditions, except for different blowing agent concentrations. It is seen that the extruded foam with a higher blowing agent concentration has the larger cell sizes. Figure 10.55 shows the effect of the type of ﬂuorocarbon blowing agent on dj /D and ρ at various die temperatures. It is seen that, in the use of FC-11, foam density is very high at 140 ◦ C and decreases rapidly as the die temperature is increased to 150–160 ◦ C

Figure 10.52 Micrographs showing the effects of nucleating agent on cell size for a 96/2/2 Styron 678/(FC-11)/(FC-12) mixture extruded at γ˙app = 160 s−1 and 150 ◦ C (a) without nucleating agent and (b) with nucleating agent, each mixture having 0.3 wt % citric acid and 0.375 wt % NaHCO3 . The geometry of the die employed for the extrusion: D = 0.315 cm, L/D = 2, DR /D = 8, and α = 60◦ . (Reprinted from Ma and Han, Journal of Applied Polymer Science 28:2983. Copyright © 1983, with permission from John Wiley & Sons.)

Figure 10.53 Plots of extrudate swell ratio dj /D and foam density ρ versus die temperature for

mixtures of Styron 678 and varying amounts of 50/50 (FC-11)/(FC-12) blowing agents (wt %): () 2, () 4, and () 8, extruded at γ˙app = 160 s−1 . Each mixture had 0.3 wt % citric acid and 0.375 wt % NaHCO3 . The geometry of the die employed for the extrusion: D = 0.315 cm, L/D = 2, DR /D = 8, and α = 60◦ . (Reprinted from Ma and Han, Journal of Applied Polymer Science 28:2983. Copyright © 1983, with permission from John Wiley & Sons.)

477

478

PROCESSING OF THERMOPLASTIC POLYMERS Figure 10.54 Micrographs of the cross sections of extrudates of mixtures of Styron 678 and 50/50 (FC-11)/(FC-12) blowing agent with varying amounts 50/50 (FC-11)/(FC-12): (a) 2 wt % and (b) 4 wt %. Extruded at 140 ◦ C in a cylindrical die with: D = 0.315 cm, L/D = 4, DR /D = 8, and α = 60◦ .

and, then, increases very rapidly as the die temperature is increased above 160 ◦ C. It should be remembered that FC-11 is a good solvent for PS and, therefore, at low temperatures (say, at 140 ◦ C) gas bubbles of FC-11 come off very slowly from homogeneous mixtures of Styron 678 and FC-11, giving rise to high foam densities. Conversely, FC-12, which has a low boiling point (−29.9 ◦ C) and is a poor solvent for PS, tends to come off easily from mixtures of PS and FC-12 at low temperatures, giving rise to low foam density. The use of a 50/50 (FC-11)/(FC-12) mixture yields a reasonable density of extruded PS foam over the temperature range 130–150 ◦ C. Figure 10.56 gives micrographs of extruded foam obtained with three different types of blowing agents under otherwise identical extrusion conditions. It is seen that the cell sizes in the foam obtained with FC-12 are very large compared with those obtained with FC-11, conﬁrming the assertion that FC-12 would come off easily, due to its poor solubility in Styron 678, from the PS 678/(FC-12) mixture during extrusion. Figure 10.56 suggests that, under such circumstances, a mixture of FC-11 and FC-12 is ideal in terms of controlling the foam density and cell size in extruded PS foam. Figure 10.57 shows that the open-cell fraction increases rapidly with increasing die temperature. Figure 10.58 gives micrographs showing that the cell sizes are much larger in the foam extruded at 160 ◦ C than those extruded at 140 ◦ C. Note that the larger the cell size, the greater will be the chance for bubble collapse. Thus, the micrographs

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Figure 10.55 Plots of extrudate swell ratio dj /D and foam density ρ versus die temperature for

mixtures of Styron 678 and different blowing agents: () 4 wt % FC-11, () 4 wt % FC-12, and () 4 wt % 50/50 (FC-11)/(FC-12) mixture, each extruded at γ˙app = 160 s−1 and 150 ◦ C. Each mixture also contained 0.3 wt % citric acid and 0.375 wt % NaHCO3 as nucleating agent. The geometry of the die employed for the extrusion: D = 0.315 cm, L/D = 2, DR /D = 8, and α = 60◦ . (Reprinted from Ma and Han, Journal of Applied Polymer Science 28:2983. Copyright © 1983, with permission from John Wiley & Sons.)

support the results presented in Figure 10.57 that the open-cell fraction increases with increasing die temperature. Notice further in Figure 10.57 that under identical extrusion conditions, FC-12 gives rise to higher open-cell fraction than either FC-11 alone or a (FC-11)/(FC-12) mixture. This suggests, once again, that the use of (FC-11)/(FC-12) mixtures would allow one to have a better control of the cell morphology than would the use of either FC-11 or FC-12 alone. We now discuss the important roles that the rheological properties of a polymer play in controlling bubble nucleation and bubble growth inside the die. In order to facilitate our discussion here, let us look at Figure 10.59, which shows schematically the wall normal stress distributions along the die axis in a capillary (or slit) die for two different polymers, A and B. In Chapter 5 of Volume 1, we have shown that in fully developed ﬂow in a capillary or slit die, (1) the slope of the wall normal stress is equal to the pressure gradient, −∂p/∂z, (2) the greater the slope of the wall normal stress distribution in a capillary or slit die, the higher the viscosity of the polymer, and (3) the greater the exit pressure, the higher the melt elasticity of the polymer. Then from Figure 10.59a we can conclude that polymer B is more viscous but less elastic than polymer A, while from Figure 10.59b we can conclude that polymer A is more viscous and also more elastic than polymer B. For a given combination of polymer and blowing agent, there exists a critical pressure Pnc at which gas bubbles nucleate, and that once bubbles nucleate inside the die they will keep growing as the mixture of molten polymer and blowing agent approaches the die exit (see Figure 10.39). Therefore, from Figure 10.59a we can conclude that bubbles will nucleate inside the die when a blowing agent is mixed with polymer B and will start to grow inside the die,

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PROCESSING OF THERMOPLASTIC POLYMERS

Figure 10.56 Micrographs of the cross sections of extrudates of mixtures of Styron 678 and different blowing agents: (a) 4 wt % FC-11, (b) 4 wt % FC-12, and (c) 4 wt % 50/50 (FC-11)/(FC-12) mixture, each extruded at γ˙app = 160 s−1 and 150 ◦ C. Each mixture had 0.3 wt % citric acid and 0.375 wt % NaHCO3 , and the geometry of the die employed for the extrusion: D = 0.315 cm, L/D = 2, DR /D = 8, and α = 60◦ .

whereas there would be no bubble nucleation inside the die for polymer A because the exit pressure for polymer A exceeds the Pnc . Conversely, from Figure 10.59b we can conclude that bubble nucleation inside the die will occur in polymer B sooner than in polymer A; thus the cell size in the extrudate will be larger in polymer B than in polymer A. When bubbles are large, they tend to coalesce during ﬂow, giving rise to foams of high density, that is, the foam density increases due to the collapse of cells. However, polymers having high viscosity, while having bubble growth inside the die suppressed, have other features that are not desirable from the standpoint of foam extrusion. The polymers having high melt viscosity experience shear heating at high extrusion rates, especially when the die temperature is very low, as in the foam extrusion process under discussion here, and thus their throughput is limited. The above observations can provide useful guidelines for selecting polymers that would give good foam extrusion characteristics. Speciﬁcally, referring to Figure 10.59a, polymer A, which is less viscous (because it has a smaller pressure gradient)

Figure 10.57 Plots of open-cell fraction versus die temperature for mixtures of Styron 678 and different blowing agents: () 4 wt % FC-11, () 4 wt % FC-12, and () 4 wt % 50/50 (FC-11)/(FC-12) mixture, each extruded at γ˙app = 160 s−1 . Each mixture has 0.3 wt % citric acid and 0.375 wt % NaHCO3 , and the geometry of the die employed for the extrusion: D = 0.315 cm, L/D = 2, DR /D = 8, and α = 60◦ . (Reprinted from Ma and Han, Journal of Applied Polymer Science 28:2983. Copyright © 1983, with permission from John Wiley & Sons.)

Figure 10.58 Micrographs of the cross sections of extrudates of a 92/4/4 Styron 678/(FC-11)/(FC-12) mixture extruded at two different temperatures: (a) 160 ◦ C and (b) 140 ◦ C, at γ˙app = 160 s−1 . Each mixture had 0.3 wt % citric acid and 0.375 wt % NaHCO3 as nucleating agent, and the geometry of the die employed for the extrusion: D = 0.315 cm, L/D = 2, DR /D = 8, and α = 60◦ .

481

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Figure 10.59 (a) Schematic showing the wall normal stress distributions along the die axis, in which polymer A is less viscous and yet more elastic than polymer B. (b) Schematic showing the wall normal stress distributions along the die axis, in which polymer A is more viscous and also more elastic than polymer B. (Reprinted from Han and Ma, Journal of Applied Polymer Science 28:2961. Copyright © 1983, with permission from John Wiley & Sons.)

than polymer B, can completely suppress bubble nucleation inside the die, because the exit pressure is greater than Pnc . Therefore, the choice of polymers for obtaining good foam extrusion characteristics can be made on the basis of the measurements of wall normal stresses in a capillary or slit die, and that the processing conditions must be chosen such that the exit pressure of the mixture of polymer and blowing agent can exceed the critical pressure for bubble nucleation, Pnc . 10.4.2 Processing–Property Relationships in Sheet Foam Extrusion A sheet foam extrusion line is shown schematically in Figure 10.60. It has (1) a feeding system consisting of a single-screw extruder, (2) a blowing agent metering system, (3) two static mixers with hot-oil temperature control units, (4) a tubular die similar to a blown-ﬁlm die, (5) air (cooling) ring, (6) a cooling mandrel, (7) foam sheet, (8) a slitting knife, (9) a stand for slitting knife, (10) a take-off stand, and (11) a winder. A feeding system that delivers mixtures of molten polymer and blowing agent is given schematically in Figure 13.1 in Chapter 13 of Volume 1. Upon exiting the tubular die, the mixture of molten polymer and blowing agent expands considerably in all directions, thus increasing the diameter of the tubular bubble. The inﬂated tubular ﬁlm is then pulled over a water-cooled mandrel, slit into two sheets, and wound onto rolls. A fairly balanced orientation of cells can be achieved by controlling the take-off speed in the machine direction, because the stretching (or orientation) in the transverse direction is determined once the size of the cooling mandrel is chosen. Note that both the blow-up ratio and take-off speed inﬂuence the density and the thickness of the foam sheets produced. The die design, especially the geometry of the ﬂow channel in the vicinity of the die lips, is one of the most important parts of the foam extrusion line.

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Figure 10.60 Schematic of a sheet foam extrusion line: (1) extruder, (2) blowing agent injection port, (3) static mixers, (4) tubular die, (5) air (cooling) ring, (6) cooling mandrel, (7) foam sheet, (8) slitting knife, (9) stand for slitting knife, (10) take-off stand, and (11) winder. (Reprinted from Yang and Han, Journal of Applied Polymer Science 30:3297. Copyright © 1985, with permission from John Wiley & Sons.)

The sheet foam extrusion process has long been used commercially to produce PS and LDPE foam sheets. PS foam sheets are widely used in food packaging applications (e.g., meat and fruit trays and fast-food containers), and LDPE foam sheets are used in packaging applications (e.g., protecting fragile electronic equipment and dishware). The unique feature of the sheet foam extrusion process that employs a tubular-ﬁlm die is that very wide foam sheets, having biaxially oriented cells, can be produced. Conversely, if a ﬂat-ﬁlm die is used, one obtains cells that are oriented uniaxially. Sheet corrugation can occur in both tubular ﬁlm die and ﬂat-ﬁlm die extrusion processes for low-density foam sheet. Figure 10.61 shows the effect of take-off speed on foam density with blowing agent concentration as a parameter. Figure 10.62 does the same with die temperature as a parameter, and Figure 10.63 with the type of blowing agent as a parameter. It is seen in these ﬁgures that the foam density ﬁrst decreases and then increases as the take-off speed is increased. Note that, with other processing variables ﬁxed, an increase in takeoff speed brings about a decrease in the thickness of the foam sheets produced. In sheet foam extrusion, the rate of cooling (i.e., the heat transfer necessary for solidifying the molten polymer) has a profound inﬂuence on foam quality (i.e., cell size and open-cell fraction), namely, the higher foam density observed at low take-off speeds (i.e., thicker foam sheets) may be, in part, due to the fact that the center of the foam sheet could not be cooled fast enough to below the glass transition temperature of the polymer, giving rise to partial cell collapse, which in turn increases foam density. Conversely, the diffusion of ambient air into the cells will help to expand the foam sheet because it enhances the expanding power of the blowing agent. The amount of ambient air that can be diffused into the foam sheet depends on the foam thickness and the time available during which the foam sheet is exposed to cooling air. The latter is determined by the extrusion rate and take-off speed. Note that thicker foams have smaller surface-to-volume ratios available for the ambient air to diffuse into the foam sheet. Therefore, the higher foam density observed at low take-off speeds may also be due to the decreased expanding power of the blowing agent. Figure 10.64 gives plots of foam density versus foam thickness at varying blowing agent concentrations, and Figure 10.65 gives plots of foam density versus

Figure 10.61 Plots of foam density versus take-off speed for Styron 678/(FC-12) mixtures with varying FC-12 concentrations (wt %): () 3, () 4, and () 5. The mixtures had 0.3 wt % citric acid and 0.375 wt % NaHCO3 as nucleating agent, and the foam sheets were extruded at 150 ◦ C and γ˙app = 285 s−1 . (Reprinted from Yang and Han, Journal of Applied Polymer Science 30:3297. Copyright © 1985, with permission from John Wiley & Sons.)

Figure 10.62 Plots of foam density versus take-off speed for a 96/4 Styron 678/(FC-12) mixture extruded at various die temperatures ( ◦ C): () 140, () 150, and () 160. The mixture had 0.3 wt % citric acid and 0.375 wt % NaHCO3 as nucleating agent, and the foam sheets were extruded at γ˙app = 285 s−1 . (Reprinted from Yang and Han, Journal of Applied Polymer Science 30:3297. Copyright © 1985, with permission from John Wiley & Sons.)

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Figure 10.63 Plots of foam density versus take-off speed for Styron 678/FC foam sheets extruded with different types of ﬂuorocarbon blowing agents: () 4 wt % FC-12, () 4 wt % 50/50 (FC-11)/(FC-12) mixture, and () 4 wt % FC-11. The mixtures had 0.3 wt % citric acid and 0.375 wt % NaHCO3 as nucleating agent, and the foam sheets were extruded at 150 ◦ C and γ˙app =

285 s−1 . (Reprinted from Yang and Han, Journal of Applied Polymer Science 30:3297. Copyright © 1985, with permission from John Wiley & Sons.)

foam thickness for different types of ﬂuorocarbon blowing agents. It is clearly seen that the foam density ﬁrst decreases and then increases with increasing thickness of foam sheet. It is seen in Figure 10.65 that the foams obtained with FC-11 have higher densities than those obtained with FC-12. This is because FC-11 is a good solvent for molten PS. Therefore, for instance, at the die temperature of 150 ◦ C, the gas bubbles of FC-11 will come off very slowly from the mixtures of molten PS and FC-11, giving rise to large bubbles and thus high-density foams. Conversely, FC-12 is a poor solvent for PS and has a low boiling point (−29 ◦ C) and therefore gas bubbles of FC-12 will come off quickly from mixtures of PS and FC-12, resulting in low-density foam. It is also

Figure 10.64 Plots of foam density versus foam thickness for the Styron 678/(FC-12) mixtures with varying FC-12 concentrations (wt %): () 3, () 4, and () 5. The mixtures had 0.3 wt % citric acid and 0.375 wt % NaHCO3 as nucleating agent, and the foam sheets were extruded at 150 ◦ C and γ˙app = 285 s−1 . (Reprinted from Yang and Han, Journal of Applied Polymer Science 30:3297. Copyright © 1985, with permission from John Wiley & Sons.)

486

PROCESSING OF THERMOPLASTIC POLYMERS Figure 10.65 Plots of foam density versus foam thickness for the Styron 678/FC foam sheets extruded with different types of ﬂuorocarbon blowing agent: () 4 wt % FC-12, () 4 wt % 50/50 (FC-11)/(FC-12) mixture, and () 4 wt % FC-11. Each of the mixtures had 0.3 wt % citric acid and 0.375 wt % NaHCO3 as nucleating agent, and the foam sheets were extruded at 150 ◦ C and γ˙app = 285 s−1 . (Reprinted from Yang and Han, Journal of Applied Polymer Science 30:3297. Copyright © 1985, with permission from John Wiley & Sons.)

seen in Figure 10.65 that the use of mixtures of FC-11 or FC-12 gives rise to foam densities lying between those when using FC-11 or FC-12 alone. Note that, at the same blowing agent concentration, FC-11 has a lower molar volume for expansion than FC-12. It is worth mentioning that the geometry of cells in a foam also inﬂuences its mechanical properties. It has been found that the compressive stress in the direction of orientation is greater than that of foams of similar structure but having no orientation. Figure 10.66 shows the effect of take-off speed on the machine direction (MD) Figure 10.66 Plots of MD shrinkage versus take-off speed (open symbols) and plots of CD shrinkage versus take-off speed (ﬁlled symbols) for the Styron 678/(FC-12) mixtures extruded with varying FC-12 concentrations (wt %): (, 䊉) 3, (, ) 4, and (, ) 5. The mixtures had 0.3 wt % citric acid and 0.375 wt % NaHCO3 as nucleating agent, and the foam sheets were extruded at 150 ◦ C and γ˙app = 285 s−1 . (Reprinted from Yang and Han, Journal of Applied Polymer Science 30:3297. Copyright © 1985, with permission from John Wiley & Sons.)

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shrinkage and cross direction (CD) shrinkage, with the concentration of blowing agent as a parameter. It is seen that the MD shrinkage increases rapidly as the take-off speed increases, while the CD shrinkage ﬁrst increases very slowly and then levels off at a value of about 50%.

10.5

Summary

In this chapter, we have pointed out the importance of an understanding of the solubility and diffusivity of a blowing agent in a molten polymer in order to control the thermoplastic foam extrusion processes. It is very difﬁcult to comprehend how anyone can select a blowing agent and then design a thermoplastic foam process without ﬁrst having information on the solubility of the blowing agent in the particular polymer chosen. In this regard, the correlation ln (1/Kp ) = A + B(ω)(T /Tc )2 , with ω being the acentric factor of a gaseous component or volatile liquid, described in this chapter will be very useful for estimating the solubility of a blowing agent in a molten polymer. Information on the diffusivity of a blowing agent in a molten polymer is essential for calculating or estimating the rate at which the gas bubbles nucleated will grow during ﬂow. In view of the fact that experimental measurements of the diffusion coefﬁcient of a gaseous component or volatile liquid in a molten polymer are not so easy, the correlation and procedures described in this chapter will be very useful for estimating the diffusion coefﬁcient of a gaseous component or volatile liquid in a molten polymer. We recognize that the estimation of parameters appearing in the free-volume theory of Duda and Vrentas requires some effort, but such effort is far less than the experiments that would otherwise have to be performed. Equation (10.23) will be very useful for estimating the diffusion coefﬁcient of a gaseous component or volatile liquid in a molten polymer (see Figures 10.15 and 10.16) when no such experimental data are available. One of the most fundamental and important physical phenomena in foam processes is bubble nucleation in a molten polymer. In this chapter, we have described bubble nucleation in homogeneous polymeric liquid under static conditions and also during ﬂow in an extrusion die. Invariably, industrial foam processes use nucleating agent(s). Nevertheless, the materials presented in this chapter provide very useful information, namely, in the environment of homogeneous nucleation the diameter of stable nucleating bubbles is about 0.5 µm. There is no question that the size of nucleating embryos would be much smaller than 0.5 µm, but such embryos may not be stable due to thermal ﬂuctuations. Although the experimentally determined value, 0.5 µm in diameter, may not be accurate, such a value would be very helpful for conducting theoretical investigations (numerical calculations) of bubble dynamics encountered in thermoplastic foam processes. It is difﬁcult to anticipate any meaningful progress on bubble dynamics from a theoretical point of view without having information on what might be the reasonable stable bubble size upon nucleation. Further, the experimental observation (see Figures 10.26–10.29) that the initial rate of bubble growth is not equal to zero (i.e., (dr(t)/dt)t=t = 0) should be incorporated into future theoretical study c on bubble growth dynamics. We have presented some fundamental observations on processing–property– morphology relationships in proﬁle foam extrusion. Since there are so many materials

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and processing variables associated with the foam extrusion processes, it is not possible to present generalized processing–property–morphology relationships. Nevertheless, the materials presented in this chapter will be very valuable for gaining an insight into the rather complicated foam extrusion operations.

Problems Problem 10.1

Construct plots of partial pressure of FC-12 versus weight fraction of FC-12 in LDPE/(FC-12) mixtures at 110, 120, 130, and 140 ◦ C. Problem 10.2

Construct plots of partial pressure of FC-12 versus weight fraction of FC-12 in PS/(FC-12) mixtures at 130, 140, 150, and 160 ◦ C. Problem 10.3

Construct plots of partial pressure of CO2 versus weight fraction of CO2 in LDPE/CO2 mixtures at 110, 120, 130, and 140 ◦ C. Problem 10.4

Construct plots of partial pressure of CO2 versus weight fraction of CO2 in PS/CO2 mixtures at 130, 140, 150, and 160 ◦ C. Problem 10.5

Construct plots of diffusion coefﬁcient versus weight fraction of FC-12 for LDPE/(FC-12) mixtures at 110, 120, 130, and 140 ◦ C. Problem 10.6

Construct plots of diffusion coefﬁcient versus weight fraction of FC-12 for PS/(FC-12) mixtures at 130, 140, 150, and 160 ◦ C. Problem 10.7

Construct plots of diffusion coefﬁcient versus weight fraction of FC-11 for PS/(FC-11) mixtures at 130, 140, 150, and 160 ◦ C. Problem 10.8

Construct plots of diffusion coefﬁcient versus weight fraction of CO2 for LDPE/CO2 mixtures at 110, 120, 130, and 140 ◦ C. Problem 10.9

Construct plots of diffusion coefﬁcient versus weight fraction of CO2 for PS/CO2 mixtures at 130, 140, 150, and 160 ◦ C. Problem 10.10

Using the truncated power-law model, calculate the velocity and shear stress distributions in steady-state shear ﬂow of a 90/10 PS/(FC-11) mixture at a volumetric ﬂow rate of 10 cm3 /s in a slit die (die width of 3 cm and die opening of 0.3 cm)

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maintained at 160 ◦ C. For the calculation, you may use the viscosity of the PS/(FC-11) mixture given by Figure 10.37, and assume that the viscosity of neat PS is given by the truncated power-law model with the following numerical values of the parameters: (1) at 200 ◦ C, η0 = 7.36 × 103 Pa·s, γ˙0 = 0.92 s−1 , n = 0.33, and K = 6.93 × 103 Pa·sn , (2) at 210 ◦ C, η0 = 3.92 × 103 Pa·s, γ˙0 = 2.03 s−1 , n = 0.33, and K = 6.30 × 103 Pa·sn , (3) at 220 ◦ C, η0 = 2.14 × 103 Pa·s, γ˙0 = 4.36 s−1 , n = 0.33, and K = 5.75 × 103 Pa·sn .

Notes 1. The volume fraction of volatile component, φ1 , is deﬁned by φ1 = v1 w1 /(v1 w1 + v2 w2 ) ≈ (v1 /v2 )w1 for w1 → 0 and w2 → 1. Thus, ln (1 − φ2 ) = ln φ1 = ln(v1 /v2 )w1 . 2. The partial pressure p1 of solute can be expressed by p1 = P1s γw w1 with P1s being the vapor pressure of pure solute at temperature T , w1 being the weight fraction of solute, and γw being the weight-fraction activity coefﬁcient deﬁned by γw = RT /Vgo P1s M1 , where R is the universal gas constant, T is the absolute temperature, and Vgo is the solubility of solute in cm3 solute per gram of polymer at 271.3 K and at 1 atm. Note that Vgo corresponds to the retention volume that can be determined from gas chromatographic experiments. Thus, the partial pressure p1 of solute can be rewritten as p1 = Hc w1 , where Hc is the weight-fraction-based Henry’s constant, deﬁned by Hc = RT /Vgo M1 = 22,414/Vgo M1 = 22,414Kp /M1 , with 22,414 representing the number of cubic centimeters (STP) per mole, where use is made of 1/Kp = Vgo at atmospheric system pressure. 3. There is a distribution of gas bubbles in the experimental system. Therefore, one must introduce a distribution function in the analysis of light scattering experimental results. Han and Han (1990a) employed a log–normal distribution function to describe bubble size distribution in the optical cell. Thus, the bubble size determined in such an analysis represents the modal bubble size, which is the most probable bubble size.

References Atkinson EB (1977). J. Polym. Sci., Polym. Phys. Ed. 15:795. Becker R, Döring W (1935). Ann. Physik 24:719. Benning CJ (1969). Plastic Foams, Interscience, New York. Blander M, Hengrtenber D, Katz JL (1971). J. Phys. Chem. 7:3613. Blander M, Katz JL (1975). AIChE J. 21:833. Bonner DC, Brockmeier NF, Cheng YI (1974). Ind. Eng. Chem. Process Des. Dev. 13:437. Braun JM, Poos S, Guillet JE (1976). J. Polym. Sci., Polym. Lett. Ed. 14:257. Brockmeier NF, McCoy RW, Meyer JA (1972). Macromolecules 5:464. Brockmeier NF, Carlson RE, McCoy RW (1973). AIChE J. 19:1133. Burt JG (1978). J. Cell. Plast. 14:341. Burt JG (1979). J. Cell. Plast. 15:158. Cheng YI, Bonner DC (1978). J. Polym. Sci., Polym. Phys. Ed. 16:319.

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Cohen MH, Turnbull D (1959). J. Chem. Phys. 31:1164. Collins FH, Brown FP (1973). Plast. Technol. 19(2):37. Croft PW (1964). Brit. Plast. 26(10):47. DiPaola-Baranyi G, Guillet JE (1978). Macromolecules 11, 228. Duda JL, Ni YC, Vrentas JS (1978). J. Appl. Polym. Sci. 22:689. Duda JL, Ni YC, Vrentas JS (1979). J. Appl. Polym. Sci. 23:947 Duda JL, Vrentas JS (1968). J. Polym Sci. A-2 6:675. Duda JL, Vrentas JS, Ju ST, Liu HT (1982). AIChE J. 28:279. Durrill PL, Griskey RG (1966). AIChE J. 12:1147. Edwards TJ, Newman J (1977). Macromolecules 10:609. Farkas L (1927). Z. Physik. Chem. 125:236. Fehn GM (1967). J. Cellular Plast. 3:456. Ferry JD (1980). Viscoelastic Properties of Polymers, 3rd ed, John Wiley & Sons, New York. Frenkel K (1941). Kinetic Theory of Liquids, Dover Publications, New York. Frisch KC, Saunders JH (eds) (1973). Plastic Foams, Marcel Dekker, New York. Fujita H (1961). Fortschr. Hochpolym. Forch. 3:1. Fujita H, Kishimoto A, Matsumoto K (1960). Trans. Faraday Soc. 56:424. Galin M, Rupprecht MC (1978). Polymer 19:506. Gray DG, Guillet JE (1973). Macromolecules 6:223. Han CD (1981). Multiphase Flow in Polymer Processing, Academic Press, New York, Chap 6. Han CD (1987). J. Appl. Polym. Sci. 33:2605. Han CD, Kim YW, Malhotra KD (1976). J. Appl. Polym. Sci. 20:1583. Han CD, Ma CY (1983a). J. Appl. Polym. Sci. 28:851. Han CD, Ma CY (1983b). J. Appl. Polym. Sci. 28:2961. Han CD, Villamizar CA (1978). Polym. Eng. Sci. 18:173. Han CD, Yoo HJ (1981). Polym. Eng. Sci. 21:518. Han JH (1988). Bubble Nucleation in Polymeric Liquids, Doctoral Dissertation, Polytechnic University, Brooklyn, New York. Han JH, Han CD (1988). Polym. Eng. Sci. 28:1616. Han JH, Han CD (1990a). J. Polym. Sci., Polym. Phys. Ed. 28:711. Han JH, Han CD (1990b). J. Polym. Sci., Polym. Phys. Ed. 28:743. Hansen RH (1962). SPE J. 18:77. Holden BS, Katz JL (1978). AIChE J. 24:260. Hu DS, Han CD (1987). J. Appl. Polym. Sci. 34:423. Hu DS, Han CD, Stiel LI (1987). J. Appl. Polym. Sci. 33:551. Ju ST, Duda JL, Vrentas JS (1981a). Ind. Eng. Chem. Prod. Res. Dev. 20:330. Ju ST, Liu HT, Duda JL, Vrentas JS (1981b). J. Appl. Polym. Sci. 26:3735. Kanakkanatt SV (1973). J. Cell. Plast. 1(2):51. Katz JL (1970). J. Stat. Phys. 2:137. Katz JL, Blander M (1973). J. Colloid Interf. Sci. 42:496. Katz JL, Saltsburg H, Reiss H (1966). J. Colloid Interf. Sci. 21:560. Knau, DA, Collins FH (1974). Plast. Eng. 30(2):34. Kong JM, Hawkes SJ (1976). J. Chromatog. Sci. 14:279. Lee JG, Flumerfelt RW (1996). J. Colloid Interface Sci. 184:335. Liu DD, Prausnitz JM (1976). Ind. Eng. Chem. Fundam. 15:330. Liu DD, Prausnitz JM (1979). J. Appl. Polym. Sci. 24:725. Lundberg JL, Wilk MB, Huyett MJ (1962). J. Polym. Sci. 57:275. Lundberg JL, Wilk MB, Huyett MJ (1963). Ind. Eng. Chem. Fundam. 2:37. Ma CY, Han CD (1983). J. Appl. Polym. Sci. 28, 2983. Macedo PB, Litovitz TA (1965). J. Chem. Phys. 42:245. Maloney DP, Prausnitz JM (1976a). AIChE J. 22:74.

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Maloney DP, Prausnitz JM (1976b). Ind. Eng. Chem. Process Des. Dev. 15:216. Mehta BS, Colombo EA (1976). J. Cell. Plast. 12:1. Meinecke EA, Clark RC (1973). Mechanical Properties of Polymeric Foam, Technomic, Westport, Connecticut. Misovich MJ, Grulke EA, Blanks RG (1985). Ind. Eng. Chem. Process Des. Dev. 24:1036. Newitt DM, Weale EW (1948). J. Chem. Soc. (London), 1541. Newman RD, Prausnitz JM (1972). J. Phys. Chem. 76:1492. Newman RD, Prausnitz JM (1973). AIChE J. 19:704. Newman RD, Prausnitz JM (1974). AIChE J. 20:206. Ni YC (1978). Evaluation of Free Volume Theory for Molecular Diffusion in Polymeric Solutions, doctoral dissertation, Pennsylvania State University, University Park, Pennsylvania. Patterson D, Tewari YB, Schreiber HP, Guillet JE (1971). Macromolecules 4:356. Pawlisch CA, Laurence RL (1983). Preprint. Am. Inst. Chem. Eng. Annual Meeting, Washington, DC. Porteous W, Blander M (1975). AIChE J. 21:560. Reid RC, Prausnitz JM, Sherwood TK (1977). The Properties of Gases and Liquids, 3rd ed, McGraw-Hill, New York, Chap 9. Saeki S, Holste JC, Bonner DC (1981). J. Polym. Sci., Polym. Phys. Ed. 19:307. Schotte W (1982). Ind. Eng. Chem. Process Des. Dev. 21:289. Senich GA (1981). Chemtech. 11:360. Senn RK, Sheneﬁel DG (1971). Modern Plastics 48(4):66. Stern SA, Mullhaupt JJ, Bareis PJ (1969). AIChE J. 15:64. Stiel LI, Harnish DF (1976). AIChE J. 22:117. Stiel LI, Chang DK, Chu HH, Han CD (1985). J. Appl. Polym. Sci. 30:1145. Street JR (1968). Trans. Soc. Rheol. 12:103. Tait PJT, Abushihada AM (1979). J. Chromatog. Sci. 17:219. Ting, RY (1975). AIChE J. 21:810. Tseng HS, Lloyd DR, Ward TC (1985a). J. Appl. Polym. Sci. 30:307. Tseng HS, Lloyd DR, Ward TC (1985b). J. Appl. Polym. Sci. 30:1815. Villamizar CA, Han CD (1978). Polym. Eng. Sci. 18:699. Volmer M, Weber A (1926). Z. Physik. Chem. 119:277. Vrentas JS, Duda JL (1977a). J. Polym. Sci., Polym. Phys. Ed. 15:403. Vrentas JS, Duda JL (1977b). J. Polym. Sci., Polym. Phys. Ed. 15:417. Vrentas JS, Duda JL (1977c). J. Appl. Polym. Sci. 21:1715. Vrentas JS, Duda JL (1979). J. Polym. Sci., Polym. Phys. Ed. 17:1086. Vrentas JS, Duda JL, Hsieh ST (1983). Ind. Eng. Chem. Prod. Res. Dev. 22:326. Wacehter CJ (1970). Plast. Design Processing 10(2):9. Yang HH, Han CD (1985). J. Appl. Polym. Sci. 30:3297. Yang WJ, Yeh HC (1966). AIChE J. 12:927. Yoo HJ, Han CD (1981). Polym. Eng. Sci. 21:69. Yoo HJ, Han CD (1982). AIChE J. 28:1002.

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Part II

Processing of Thermosets

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11

Reaction Injection Molding

11.1

Introduction

Reaction injection molding (RIM) is a thermoset processing operation during which the incoming feedstream(s) undergo cure reactions that give rise to a three-dimensional network structure (Becker 1979; Macosko 1989). Different from the operation of injection molding thermoplastic polymers presented in Chapter 8, in RIM operation the component(s) must cure rapidly (say, within 90 seconds) and a ﬁnished product is removed in 1−10 minutes, depending on the chemical systems, the part thickness, and the capabilities of the processing machine. The chief advantages of RIM over the injection molding of thermoplastic polymers are: (1) large parts can be produced at low energy consumption, (2) large parts with varying cross sections with or without inserts can be produced without the problem of sink marks, and (3) lightweight parts, owing to the microcellular structure, can be produced. However, the predominant industrial applications are in the automotive industry; for instance, in the production of automobile fascia. In the 1970s and 1980s, very intensive research activities were reported on a better understanding of the RIM operation. Thermosets must meet with some stringent requirements for RIM operation. These are: (1) viscosities must be fairly low at processing temperature, so that a rapid injection of the feedstreams can be realized; (2) the feedstreams must have sufﬁcient compatibility for efﬁcient mixing by the static impingement mixing technique; (3) cure reaction must be sufﬁciently fast, such that a ﬁnished product can be removed in a very short time after injection is completed; (4) a ﬁnished product must have sufﬁcient stiffness and resiliency at elevated temperatures; and (5) a ﬁnished product must be released easily from the mold surface, etc. It is then clear that not many thermosets meet these requirements. It has been found that urethanes, with proper chemistry of the components, meet with the requirements. For this reason, urethanes have been the most widely used resin for RIM, although other thermosets (e.g., epoxy) have also been used to some extent. 495

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Figure 11.1 Schematic of the RIM process.

Figure 11.1 gives a simpliﬁed schematic of the RIM process using a urethane system; namely, isocynanate and polyol streams, each from a separate storage tank, are pumped into an impingement mixing tank and the well-mixed stream is fed into a mold cavity. It cannot be overemphasized that intimate mixing of isocyanate and polyol streams is extremely important to a successful RIM operation. Hence, in the past, a number of research groups focused their efforts on impingement mixing of isocyanate and polyol streams (Baldyga and Bourne 1983; Kolodziej et al. 1982; Lee et al. 1980; Nguyen and Suh 1986; Tucker and Suh 1980). As schematically shown in Figure 11.2, upon entering the mold cavity the materials follow a series of events, during which the viscosity of the ﬂowing mixture increases during mold ﬁlling due to chemical reaction, the modulus of the material in the mold cavity increases as curing continues after the mold cavity is ﬁlled, and ﬁnally the product is ejected from the

Figure 11.2 Schematic showing the increase in viscosity and modulus during mold ﬁlling, cure

reaction, and post cure in RIM. (Reprinted from Macosko, Fundamentals of Reaction Injection Molding, Chapter 2. Copyright © 1989, with permission from Hanser.)

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497

mold after a sufﬁcient degree of cure is attained. In other words, when the mold cavity is ﬁlled in a reasonably short time (say, within 10–20 s), cure reactions take place to form a three-dimensional network. Two separate liquid streams of the constituent components, each having relatively low viscosities, together with a catalyst(s) are injected via impingement mixing at the inlet of a mold. Then, while chemical reactions continue to yield large molecules with network structure, the mixed stream ﬁlls the mold cavity and subsequently solidiﬁes. Therefore, RIM can be regarded as a process where ﬂuid ﬂow, heat transfer, and chemical reaction are involved simultaneously. Some research groups (Broyer and Macosko 1976; Broyer et al. 1978; Lee and Macosko 1980) have investigated heat transfer problems associated with RIM operation, since the cure reactions are exothermic chemical reactions and generate heat. It is extremely important to get the chemistry of a urethane system correct to ensure that a network is formed, such that the cured product has the desired physical and mechanical properties. The chemistry of urethane systems is well documented (Lenz 1967; Saunders and Frisch 1964), and is beyond the realm of this book. During the cure reaction, the transformation from a liquid state to a solid state takes place inside the mold cavity, thus the modulus of the cured material increases as cure continues. Finally, the molded part is released (i.e., demolding). In the 1980s, some serious efforts (Castro 1980; Castro and Macosko 1982; Domine and Gogos 1980; Kamal and Ryan 1980; Kim 1987; Kim and Kim 1987a, 1987b; Lee and Kim 1988; Manzione 1981) were made to simulate RIM computationally. In this chapter, we present an analysis of the RIM operation, placing emphasis on the importance of relating chemorheology to the heat transfer during the entire RIM operation. However, we do not present mechanical operations of RIM because they are well documented in the literature (Becker 1979; Macosko 1989). Here, we use the chemorheological model for polyurethane systems presented in Chapter 14 of Volume 1.

11.2

Analysis of Reaction Injection Molding

Let us consider a center-gated disk-type mold cavity, as schematically shown in Figure 11.3, in order to present the modeling of RIM. This particular form of mold

Figure 11.3 Schematic of a center-gated disk-type mold cavity simulated for RIM.

498

PROCESSING OF THERMOSETS

cavity has been chosen for two reasons: (1) it is simple enough to demonstrate clearly the modeling principle and (2) it is the same geometry as that considered in Chapter 8 for the analysis of thermoplastic injection molding and thus comparisons, when appropriate, can be made between the two processes. Like the analysis of thermoplastic injection molding considered in Chapter 8, the analysis of RIM must consider two separate ﬂow situations: front ﬂow and mold ﬁlling. However, owing to the chemical reactions taking place in RIM, the system equations for RIM are much more complicated than the system equations for thermoplastic injection molding. In describing the RIM process for a thermoset in a center-gated disk-type mold cavity (Figure 11.3), we make the following assumptions: (1) the density (ρ), speciﬁc heat capacity (cp ), and thermal conductivity (k) of the materials are constant during mold ﬁlling and cure, implying that the ﬂuid is regarded as being incompressible; (2) the rate of chemical reaction is so fast that the molecular diffusion of the components during cure can be neglected; (3) ﬂow rate is constant throughout the entire ﬁlling stage, implying that time-dependent terms in system equations can be neglected; (4) ﬂow rate is sufﬁciently slow to warrant steady-state laminar ﬂow; and (5) ﬂow is fully developed and thus the ﬂuid has a parabolic velocity proﬁle when it enters the mold cavity (i.e., the entrance effect is negligible). 11.2.1 Main Flow Using cylindrical coordinates (r, θ, z), the continuity equation for incompressible ﬂuids is written as 1 ∂ (rur ) = 0 r ∂r

(11.1)

and the momentum balance equation in the radial direction is written as ρ

∂ur ∂u + ur r ∂t ∂r

=−

∂p 1 ∂ + ∂r r ∂r

2ηr

∂ur ∂r

−

2ur η ∂ur ∂ + η ∂z ∂z r2

(11.2)

where ur is the velocity in the radial direction and η is viscosity that varies with temperature and the degree of cure during mold ﬁlling. It can be shown, with the aid of Eq. (11.1), that the second and third terms on the right-hand side of Eq. (11.2) vanish, yielding1

ρur

∂ur ∂u ∂p ∂ =− + η r ∂r ∂r ∂z ∂z

(11.3)

in which the viscosity change in the radial direction is neglected and the transient term appearing in Eq. (11.2) is omitted because the ﬂow rate is constant. For the situation under consideration, where the mold thickness (H) is much smaller than the radius (R) of the disk-type mold cavity (H R in Figure 11.3), inertial effects can be considered to be negligibly small compared with the terms appearing on the right-hand side.2

REACTION INJECTION MOLDING

499

Thus, Eq. (11.3) reduces to

∂p ∂r

∗

∂v ∗ η∗ r∗ ∂z

1 ∂ = ∗ ∗ r ∂z

(11.4)

where vr = rur is introduced and satisﬁes Eq. (11.1) with vr being independent of r, and the following dimensionless variables are introduced: r∗ = r/R, z∗ = z/H, vr∗ = vr /Vr with Vr being average value of vr , (∂p/∂r)∗ = (∂p/∂r)(RH 2 /ηr Vr ), and η∗ = η/ηr , with ηr being reference viscosity.3 Integration of Eq. (11.4) under the boundary condition ∂vr∗ /∂z∗ = 0 at z∗ = 0 gives

∗

∂p ∂r

η∗ ∂vr∗ r ∗ ∂z∗

z∗ =

(11.5)

which, upon integration, gives vr∗ =

∂p ∂r

∗

r∗

z∗

z∗ ∗ dz η∗

1

(11.6)

It can be shown that (∂p/∂r)∗ in Eq. (11.6) can be replaced by4

∂p ∂r

∗

=

r∗

1 0

−1 (z∗ )2 η∗

(11.7)

dz∗

Thus, substitution of Eq. (11.7) into (11.6) gives vr∗ (z∗ , t)

=

1 z∗

z∗ ∗ dz η∗

1 0

(z∗ )2 ∗ dz η∗

(11.8)

which allows one to calculate the velocity proﬁle during mold ﬁlling. Notice in Eq. (11.8) that vr∗ depends on position z∗ and viscosity η∗ , which in turn varies with temperature and the degree of cure, η∗ = f (T, α), as presented in Chapter 14 of Volume 1. Also, the pressure drop p during mold ﬁlling can be calculated by integrating Eq. (11.7), yielding Vη p = r 2r H

rf −r0 R

0

r∗

1 0

dr ∗ (z∗ )2 η∗

dz∗

(11.9)

where rf is the position of the moving front in the mold cavity, r0 is the length of ﬂow front region, which is assumed to be 2H as will be discussed below, and R is the radius

500

PROCESSING OF THERMOSETS

(the characteristic length) of the mold cavity. Thus, p depends on r∗ and η∗ , which in turn vary with temperature and the degree of cure. Equation (11.9) can be rewritten as p =

Qηr 4πH 3

rf −r0 R

r∗

0

1

dr ∗

(z∗ )2 0 η∗

dz∗

(11.10)

where Q = 4πH Vr is used. We have the relationship Rηr Vr ηr = RH 2 2H 2 tf

(11.11)

in which use is made of tf = R 2 /2Vr with tf being the ﬁlling time. Therefore, a speciﬁcation of tf , instead of Q or Vr , as an input variable will be sufﬁcient to calculate p given by Eq. (11.9) or Eq. (11.10). The energy balance equation during mold ﬁlling can be written as ρcp

v ∂T ∂T + r ∂t r ∂r

=k

∂ 2T + HA RA ∂z2

(11.12)

where HR is the heat of reaction due to cure, RA is the rate of formation of the cured resin, and the relationship vr = rur is used. In writing down Eq. (11.12), the following additional assumptions are made: (1) thermal conduction in the r-direction is negligibly small compared with that in the z-direction, which is reasonable because the temperature variation in the r-direction is much smaller than that in the z-direction, and (2) viscous shear heating is negligible, which is justiﬁed for the following considerations. Since the ﬂuid velocity is very slow in the mold cavity compared with that in the runner, viscous shear heating during ﬁlling the mold cavity would be negligibly small. Also, while the ﬂuid velocity in the runner is very high, the residence time in the runner is so short and the viscosity of a thermoset commonly used for RIM is so low that the viscosity rise due to curing reaction in the runner would be negligibly small. What makes the analysis of RIM much more complicated than thermoplastic injection molding is the presence of exothermic chemical reactions during mold ﬁlling, which give rise to a dramatic increase in molecular weight. This means that we need an expression describing the rates of the curing reactions. Note that curing reactions, being exothermic, increase the temperature and molecular weight of the ﬂuid system. The balance equation of reacting species is given by v ∂C ∂CA + r A = RA ∂t r ∂r

(11.13)

where CA is the concentration of the reactive functional group. Deﬁning α = (CA0 − CA )/CA0 , with CA0 being the initial concentration of the reactive functional group, RA is expressed by RA = −CA0

dα dt

(11.14)

REACTION INJECTION MOLDING

501

Thus, use of Eq. (11.14), together with Eq. (14.33) in Volume 1, in Eqs. (11.12) and (11.13) gives the following dimensionless balance equations of energy and reacting species, respectively (Kim 1987) vr∗ ∂θ ∂θ ∂ 2θ + = B + B2 K1 + K2 [Co Na]m (1 − α)n 1 ∗ ∗ ∗ ∗2 ∂t r ∂r ∂z

(11.15)

and v ∗ ∂θ ∂α + r∗ ∗ = B3 K1 + K2 [Co Na]m (1 − α)n ∗ ∂t r ∂r

(11.16)

where t ∗ = tVr /R 2 , r ∗ = r/R, z∗ = z/H, vr∗ = vr /Vr , θ = T /Ti , with Ti being the inlet temperature of the ﬂuid, K1 = k1 /k10 , K2 = k2 /k10 , B1 = ρcp H 2 Vz /kR 2 , B2 = CA0 HR R 2 k10 /ρcp Ti Vr , and B3 = R 2 k10 /Vr . Note that the rate constants, k1 and k2 , are deﬁned by Eq. (14.2) in Chapter 14 of Volume 1 and [Co Na] is the concentration of an accelerator, cobalt naphthenate (see Eq. (14.33) in Volume 1). Equations (11.15) and (11.16), with the aid of Eq. (11.8), must be solved numerically under the following initial and boundary conditions:5 θ = 1.0 and α = 0

at t ∗ ≤ Rr 2 /2R 2

(11.17a)

θ = 1.0 and α = 0

at r ∗ ≤ Rr /R

(11.17b)

∗

∗

at z = 0

(11.17c)

θ = θw

at z∗ = 1

(11.17d)

∂θ/∂z = 0

where Rr is the radius of runner and θw is dimensionless (Tw /Ti ) die wall temperature. Note that the dimensionless viscosity η∗ enters into Eqs. (11.15) and (11.16) through vr∗ deﬁned by Eq. (11.8). 11.2.2 Front Flow As presented in Chapter 8 for thermoplastic injection molding, in RIM the ﬂuid element near the centerline moves faster than the bulk ﬂuid, giving rise to the “fountain effect.” In the fountain ﬂow region, the ﬂuid streamlines no longer have the lamellar velocity proﬁles that form straight pathlines; that is, both the r- and z-directional velocities must be considered in order to describe the balance equations of energy transfer and reacting species during mold ﬁlling. The balance equations of energy transfer and reacting species in the “front-ﬂow” region are written as (Kim 1987) v ∗ ∂θ v ∗ ∂θ ∂ 2θ ∂θ + rf∗ ∗ + B4 zf∗ ∗ = B1 ∗2 + B2 K1 + K2 [Co Na]m (1 − α)n (11.18) ∗ ∂t r ∂r r ∂r ∂z

502

PROCESSING OF THERMOSETS

and ∗ ∗ vzf vrf ∂α ∂θ ∂θ + + B = B3 K1 + K2 [Co Na]m (1 − α)n 4 ∗ ∗ ∗ ∗ ∗ ∂t r ∂r r ∂z

(11.19)

respectively, where B4 = R/H . Note in Eq. (11.18) that the heat conduction in the r-direction is assumed to be negligibly small compared with that in the z-direction. In solving Eq. (11.18) for θ and Eq. (11.19) for α, one must determine ∗ and v ∗ , which represent the dimensionless velocities in the r- and z-directions, vrf zf ∗ and v ∗ is respectively, in the front-ﬂow region. A rigorous determination of vrf zf extremely difﬁcult. Castro (1980) and Castro and Macosko (1982) obtained the ∗ and v ∗ : following approximate expressions for vrf zf % ∗ ∗ ∗ = (vri − 1) 1 − 1.45e−5(1−rff ) sin 0.76 + 2(1 − rff∗ ) vrf & ∗ + 0.53(1 − 5z∗4 )e−5(1−rff ) sin 2(1 − rff∗ ) + 1 ∗ = vzf

z∗

0

∗ vri dz∗ − z∗

(11.20)

%

∗ 7.26e−5(1−rff ) sin 0.76 + 2(1 − rff∗ )

& ∗ −2.90e−5(1−rff ) cos 0.76 + 2(1 − rff∗ ) % ∗ − z∗ (1 − z∗4 ) 2.63e−5(1−rff ) sin 0.76 + 2(1 − rff∗ ) & ∗ −1.05e−5(1−rff ) cos 2(1 − rff∗ )

(11.21)

∗ is the dimensionless velocity at the interface (r ∗ = 0) between the main in which vri ff ﬂow region and the front-ﬂow region, where rff∗ is deﬁned by rff∗ = rf∗ − 1 with rf∗ = rf /2H , in which rf denotes the farthest position in the front-ﬂow region. Here, the length of front-ﬂow region (r0 ) is assumed to be the same as the mold thickness (2H) and thus the main ﬂow prevails in the region Rr ≤ r ≤ rf − 2H, and the front ﬂow prevails in the region rf − 2H ≤ r ≤ rf , where Rr is the radius of the runner (see Figure 11.3).

11.2.3 Cure Stage Once the mold is ﬁlled, the material cures in the mold cavity and the temperature rises due to the exothermic reaction. The balance equations of energy transfer and reacting species can be written as (Kim 1987) ∂θ 1 ∂ = B5 ∗ ∗ ∗ ∂tc r ∂r

∂θ r ∂r ∗ ∗

+ B6

∂ 2θ + B7 K1 + K2 [Co Na]m (1 − α)n ∗2 ∂z

(11.22)

503

REACTION INJECTION MOLDING

and ∂α = B8 K1 + K2 [Co Na]m (1 − α)n ∗ ∂tc

(11.23)

respectively, where tc∗ is the dimensionless cure time deﬁned by tc∗ = t/tc , B5 = ktc /ρcp R 2 , B6 = ktc /ρcp H 2 , B7 = CA0 HR tc k10 /ρcp Ti , and B8 = k10 tc . In the absence of the cobalt naphthenate accelerator, Eq. (11.23) reduces to Eq. (14.1) in Volume 1, with k2 = 0. Notice in Eq. (11.22) that the energy transfer due to the heat conduction in both r- and z-directions is considered. Cure time must be speciﬁed to solve Eqs. (11.22) and (11.23) under the following boundary conditions: ∂θ/∂z∗ = 0

at r ∗ = 0

(11.24a)

∗

∗

at r = 1

(11.24b)

∗

∗

at z = 0

(11.24c)

∗

(11.24d)

∂θ/∂z = 0 ∂θ/∂z = 0 θ = θw

at z = 1

11.2.4 Chemorheological Model In solving Eq. (11.15) and (11.16) with the aid of Eq. (11.8), we must specify an expression for viscosity η(T , α), because vr∗ given by Eq. (11.8) contains the dimensionless viscosity η∗ deﬁned by η∗ = η/ηr , with ηr being reference viscosity, which is chosen to be the viscosity of the feedstream at the inlet temperature Ti . This means that we need an expression for a chemorheological model, η(T, α). Here, we use Eq. (14.30) given in Chapter 14 of Volume 1 as the chemorheological model for polyurethane. Thus, we have6

1 −1 η (θ , α) = exp (Eη /RTi ) θ ∗

αg

a+bα

αg − α

(11.25)

where αg is the degree of cure at a glass transition temperature (Tg ). Note that η1 given in Eq. (14.30) in Volume 1 is related to ηr at Ti by η1 = ηr (Ti ) exp(−Eη /RTi ).

11.3

Conversion and Temperature Proﬁles during Mold Filling

We can now calculate conversion and temperature proﬁles in the mold cavity during the mold ﬁlling and cure stage by numerically solving the system equations presented in the previous section. Besides the physical properties of a thermoset and the parameters appearing in the kinetic expressions and chemorheological models, one must specify the following design and processing variables: (1) the geometry of the mold and the dimensions of the mold cavity, (2) mold ﬁll time (tf ), (3) cure time (tc ), (4) the inlet temperature of the feed streams (Ti ), and (5) the initial mold wall temperature (Tw ).

504

PROCESSING OF THERMOSETS Table 11.1 Summary of the mold geometry and physical properties of polyurethane and epoxy

(a) Mold Geometry Radius of the center-gated disk-type cavity mold (R): 6 cm Radius of mold runner (Rr ): 0.3 cm Half thickness of the mold (H): 0.3 cm (b) Physical Properties of Polyurethane ρ = 1.0 g/cm3 ; cp = 2,220 J/(kg K); k = 0.21 W/(m K) (c) Physical Properties of Epoxy ρ = 1.13 g/cm3 ; cp = 2,100 J/(kg K); k = 0.21 W/(m K)

Note that tf is related to the volumetric ﬂow rate Q. For a center-gated disk-type mold cavity (see Figure 11.3), tf = (πR 2 )(2H )/Q = 2πH R 2 /4πH Vr = R 2 /2Vr , which then enables one to determine injection speed Vr = R 2 /2tf . Below, we only present some representative simulation results to illustrate what the system equations presented above can do in terms of giving a better understanding of RIM operation. In carrying out the computations, we used the dimensions for a center-gated disk-type mold cavity and the physical properties of polyurethane summarized in Table 11.1, the numerical values of the kinetic parameters for polyurethane summarized in Table 11.2, and the numerical values of the parameters appearing in the Eq. (11.25) for polyurethane summarized in Table 11.3. Figure 11.4 shows simulated variations in conversion (α) and temperature (T) proﬁles with time during RIM operation using the diisocyanate/diol system from the center (r∗ = 0) to the end (r∗ = 1) and from the midplane (z∗ = 0) to the inner wall (z∗ = 1) of a center-gated disk-type mold cavity (see Figure 11.3) with R = 6 cm and Rr = 0.3 cm, under the processing conditions: [Co Na] = 0.1 phr, tf (ﬁll time) = 30 s, tc (cure time) = 100 s, Ti (inlet temperature of the feed streams) = 39.5 ◦ C, and Tw (mold wall temperature) = 82 ◦ C. The following observations are worth noting in Figure 11.4. (1) Little cure reaction took place, except very near the mold wall, during mold ﬁlling because α = 0 at t = 30 s and consequently the temperature, except very near the wall, remains at the mold wall temperature 82 ◦ C. (2) At 50 s after mold ﬁlling began, α has increased somewhat everywhere inside the mold cavity, with a value of about 0.2, at the end of the mold cavity and near the mold wall, and consequently the temperature has increased everywhere inside mold cavity to the mold wall temperature Table 11.2 Summary of the kinetic parameters for polyurethane and epoxy

(a) For Polyurethane k10 = 4.83 × 103 s−1 ; E1 = 4.48 × 104 J/mol k20 = 1.74 × 1010 s−1 ; E2 = 7.84 × 104 J/mol m = 0.86; n = 1.3; HR = 259 J/g (b) For Epoxy k10 = 1.67 × 109 s−1 ; E1 = 7.79 × 104 J/mol k20 = 0; E2 = 0; n = 1.64;HR = 214 J/g

REACTION INJECTION MOLDING

505

Table 11.3 Summary of the parameters employed in the chemorheological model for polyurethane and epoxy

(a) For Polyurethane η1 = 1.9 × 10−10 Pa·s; Eη = 5.41 × 104 J/mol αg = 0.694; a = −1.99; b = 6.67 (b) For Epoxy η1 = 1.0 × 10−8 Pa·s; Eη = 5.08 × 104 J/mol αg = 0.894; a = −0.155; b = 5.25

(82 ◦ C), except at the center of the mold cavity. (3) At 70 s after mold ﬁlling began, α has increased everywhere inside the mold cavity to about 0.2, except at the center of the mold cavity where α ≈ 0.1, and consequently the temperature has increased almost uniformly to the mold wall temperature (82 ◦ C). (4) At 100 s after mold ﬁlling began, α has increased to 1.0 almost everywhere, except near the mold wall, which is attributed to the heat generated from the exothermic reaction of the resin, and consequently the temperature at the midplane of the mold cavity has reached a maximum value of about 150 ◦ C. In RIM operation, the maximum temperature rise is due to the exothermic reaction inside the mold cavity and it must be kept below the thermal degradation temperature of cured resin. Figure 11.5 shows simulated variations in conversion and temperature proﬁles with time during RIM operation using the diisocyanate/diol system in the same mold cavity and also under the same processing conditions as used in Figure 11.4, except for that the concentration of cobalt naphthenate accelerator [Co Na] is increased to 0.5 phr. Comparison of Figure 11.5 with Figure 11.4 reveals the following differences between the two situations. (1) At 50 s after mold ﬁlling began, α for [Co Na] = 0.5 phr has increased somewhat everywhere inside mold cavity to about 0.4 (Figure 11.5), while α for [Co Na] = 0.1 phr is about 0.1 everywhere inside the mold cavity, except very near the cavity wall (Figure 11.4). Consequently, the temperature at the end of the cavity (r∗ = 1) along the midplane (z∗ = 0) shows a maximum (200 ◦ C) for [Co Na] = 0.5 phr (Figure 11.5), while the temperature at the same position still shows a minimum (70 ◦ C), which is below the wall temperature (82 ◦ C), for [Co Na] = 0.1 phr (Figure 11.4). This observation suggests that an increase in [Co Na] from 0.1 to 0.5 phr has a profound inﬂuence on both the conversion and temperature proﬁles, accelerating greatly the cure reactions. (2) At 70 s after mold ﬁlling began, α ≈ 1.0 for [Co Na] = 0.5 phr almost everywhere in the mold cavity, except very near the cavity wall, where α ≈ 0.5 (Figure 11.5), while α ≈ 0.2 for [Co Na] = 0.1 phr almost everywhere in the mold cavity (Figure 11.4). Consequently, the temperature along the midplane (z∗ = 0) of the mold cavity goes through a maximum (200 ◦ C) for [Co Na] = 0.5 phr (Figure 11.5), while the temperature for [Co Na] = 0.1 phr is increased barely to the mold wall temperature (82 ◦ C) (Figure 11.4). (3) At 100 s after mold ﬁlling began, α for [Co Na] = 0.5 phr has increased almost uniformly to 1.0, while α ≈ 1.0 for [Co Na] = 0.1 phr almost everywhere inside the mold cavity except near the cavity wall. What is most interesting is the maximum temperature along the midplane (z∗ = 0) of the mold cavity for [Co Na] = 0.5 phr, namely, the temperature has decreased from 200 ◦ C to 140 ◦ C as

Figure 11.4 Simulation results showing the degree of cure α (the left-hand parts) and temperature

(the right-hand parts) at various times during RIM using of diisocyanate/diol system into a disktype mold cavity (see the schematic given in Figure 11.3). R = 6 cm, Rr = 0.3 cm, H = 0.3 cm, [Co Na] = 0.1 phr, tf (ﬁll time) = 30 s, tc (cure time) = 100 s, Ti (inlet temperature) = 39.5 ◦ C, Tw (mold wall temperature) = 82 ◦ C. The physical/thermal properties of polyurethane employed for the simulation are given in Table 11.1, the kinetic parameters employed are given in Table 11.2, and the parameters for the chemorheological model employed are given in Table 11.3.

506

Figure 11.5 Simulation results showing the degree of cure α (the left-hand parts) and temperature

(the right-hand parts) at various times during RIM using diisocyanate/diol system into a disk-type mold cavity (see the schematic given in Figure 11.3). R = 6 cm, Rr = 0.3 cm, H = 0.3 cm, [Co Na] = 0. 5 phr, tf (ﬁll time) = 30 s, tc (cure time) = 100 s, Ti (inlet temperature) = 39.5 ◦ C, Tw (mold wall temperature) = 82 ◦ C. The physical/thermal properties of polyurethane employed for the simulation are given in Table 11.1, the kinetic parameters employed are given in Table 11.2, and the parameters for the chemorheological model employed are given in Table 11.3.

507

508

PROCESSING OF THERMOSETS

the cure continued from 70 to 100 s after mold ﬁlling began (see Figure 11.5). This observation indicates that cooling of the cured material inside the cavity occurred due to the heat conduction from the mold wall that was kept at 82 ◦ C. Conversely, no cooling of the cured material occurred for [Co Na] = 0.1 phr when cure continued from 70 to 100 s after mold ﬁlling began (see Figure 11.4). Figure 11.6 shows simulated variations in conversion and temperature proﬁles with time during RIM operation using the diisocyanate/diol system in the same mold cavity and also under the same processing conditions as used in Figure 11.5, except for tf = 15 s and tc = 60 s. A shorter mold ﬁll time means a faster (higher) injection rate of feed streams, which understandably requires a shorter cure time (or demold time). Comparison of Figure 11.6 with Figure 11.4 reveals the following differences between the two situations. (1) At 40 s after mold ﬁlling began, α ≈ 0.5 uniformly from the center (r∗ = 0) to the end (r∗ = 1) of the mold cavity along the cavity wall and α ≈ 1.0 at the midplane (z∗ = 0) of the mold cavity for tf = 15 s (Figure 11.6), while at 70 s after mold ﬁlling began, α ≈ 0.2 in the mold cavity except very near the cavity wall (Figure 11.4). This observation clearly indicates that the increase in mass ﬂow rate that gave rise to a short mold ﬁll time has accelerated cure reactions in the mold cavity. Notice that Figure 11.6 is based on [Co Na] = 0.5 phr and Figure 11.4 is based on [Co Na] = 0.1 phr. This difference in [Co Na] between the two has accelerated the cure reaction shown in Figure 11.6, as compared with the cure reaction shown in Figure 11.4. The consequence of the accelerated cure reactions is reﬂected on the temperature proﬁles; namely, a maximum temperature rise (200 ◦ C) is observed 40 s after mold ﬁlling began for tf = 15 s (Figure 11.6), while the temperature barely reached mold cavity temperature (82 ◦ C) through heat conduction from the mold wall 70 s after mold ﬁlling began for tf = 30 s (Figure 11.4). (2) At 60 s after mold ﬁlling began, α ≈ 1.0 uniformly everywhere except very near the cavity wall for tf = 15 s (Figure 11.6), while α ≈ 1.0 almost everywhere except near the cavity wall at 100 s after mold ﬁlling began for tf = 30 s (Figure 11.4). The difference between the two situations is manifested in the temperature proﬁles; namely, the maximum temperature rise is 200 ◦ C, 60 s after molding ﬁlling began for tf = 15 s (Figure 11.6), while the maximum temperature rise is 150 ◦ C, 100 s after molding ﬁlling began for tf = 30 s (Figure 11.4). It is then very clear that the injection speed (mass ﬂow rate) has a profound inﬂuence on the conversion and temperature proﬁles in the mold cavity in RIM operations. Figure 11.7 shows simulated variations in conversion and temperature proﬁles with time during RIM operation using the diisocyanate/diol system in the mold cavity of R = 9 cm and H = 0.45. In other words, the radius and thickness of the center-gated disktype cavity simulated in Figure 11.7 are slightly larger than those used in Figure 11.6. Comparison of Figure 11.7 with Figure 11.6 clearly shows that a larger and thicker mold cavity gives rise to a slower rate of conversion and smaller temperature rise in the mold cavity, which is expected intuitively. It is then concluded that the dimension of a mold cavity has a large inﬂuence on the conversion and temperature proﬁles in the mold cavity in RIM operations. Since there are many variables associated with the system equations for RIM, the presentation of the conversion and temperature proﬁles for every conceivable situation would not only be impossible for the reason of limited space available here but also repetitive. An experimental veriﬁcation of the simulated results presented above is difﬁcult because there is no way one can measure conversion and temperature proﬁles

Figure 11.6 Simulation results showing the degree of cure α (the left-hand parts) and temperature

(the right-hand parts) at various times during RIM using of diisocyanate/diol system into a disktype mold cavity (see the schematic given in Figure 11.3). R = 6 cm, Rr = 0.3 cm, H = 0.3 cm, [Co Na] = 0.5 phr, tf (ﬁll time) = 15 s, tc (cure time) = 60 s, Ti (inlet temperature) = 39.5 ◦ C, Tw (mold wall temperature) = 82 ◦ C. The physical/thermal properties of polyurethane employed for the simulation are given in Table 11.1, the kinetic parameters employed are given in Table 11.2, and the parameters for the chemorheological model employed are given in Table 11.3.

509

Figure 11.7 Simulation results showing the degree of cure α (the left-hand parts) and temperature

(the right-hand parts) at various times during RIM using diisocyanate/diol system into a disk-type mold cavity (see the schematic given in Figure 11.3). R = 9 cm, Rr = 0.3 cm, H = 0.45 cm, [Co Na] = 0.5 phr, tf (ﬁll time) = 15 s, tc (cure time) = 60 s, Ti (inlet temperature) = 39.5 ◦ C, Tw (mold wall temperature) = 82 ◦ C. The physical/thermal properties of polyurethane employed for the simulation are given in Table 11.1, the kinetic parameters employed are given in Table 11.2, and the parameters for the chemorheological model employed are given in Table 11.3.

510

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inside a mold cavity during mold ﬁlling and the cure stage. At best, one can measure the inner mold wall temperature by inserting a thermocouple as close as possible to the inner mold wall without disturbing the ﬂow during mold ﬁlling. Manzione and Osinski (1983) immersed a rectangular mold having a 0.48 cm thick mold cavity into an isothermal oil bath kept at 110 ◦ C and measured variations in temperature with time using a thermocouple placed at the centerline of the mold cavity ﬁlled with an epoxy resin and a curing agent. Their experimental results are summarized in Figure 11.8, showing that the centerline temperature ﬁrst increases slowly, followed by a rapid increase going through a maximum, and then decreases with time. The results shown in Figure 11.8 correspond with the variations in centerline temperature with time during cure after the mold ﬁlling is completed. Similar temperature proﬁles, although obtained under a somewhat different experimental setup, were reported earlier by Castro and Macosko (1982). Referring to Figure 11.8, (1) the slow increase in centerline temperature at the beginning is due to the conductive heat transfer from the isothermal oil bath kept at 110 ◦ C to the resin, (2) the rapid increase in centerline temperature after about 30 s is due to the heat generated by the exothermic cure reaction of an epoxy resin after the resin is heated, via conductive heat transfer, to a threshold temperature at which the exothermic reaction can take off, and (3) the decrease in centerline temperature after about 50 s is due to the conductive heat transfer from the cured resin to the isothermal oil bath kept at 110 ◦ C. Such variations in temperature with time are expected in all RIM operations. However, there is no way to measure variations in conversion with time during mold ﬁlling and cure stage in the RIM operation. This is where the process model plays an important role in that it provides information on the progress of conversion, as demonstrated in Figures 11.4−11.7. Since the heat transfer and cure during RIM operation are coupled, one must have some conﬁdence in the predicted conversion when the predicted temperatures seem reasonable, provided that the numerical values employed for the physical properties of resin, kinetic parameters, and chemorheological model are accurate. One must not underestimate the importance of providing accurate numerical values of the various parameters appearing in the simulation model. Needless to say, use of inaccurate numerical values of various system parameters will yield misleading, if not erroneous, predictions. Figure 11.8 Variation in centerline temperature with time during cure of an epoxy in a rectangular mold cavity with a 0.48 cm cavity thickness, which was immersed in an isothermal oil bath at 110 ◦ C, where the horizontal broken line denotes mold wall temperature Tw . (Reprinted from Manzione and Osinski, Polymer Engineering and Science 23:576. Copyright © 1983, with permission from the Society of Plastics Engineers.)

512

11.4

PROCESSING OF THERMOSETS

Summary

In this chapter, we have presented a system of equations to model the RIM process in terms of dimensionless variables (the degree of cure and temperature proﬁles) for a center-gated disk-type mold cavity, which was considered for illustration purposes, and then solved numerically, via the ﬁnite difference method, the system equations to simulate the RIM operation. A more complicated mold cavity could be treated using the ﬁnite element method. The purpose of the analysis presented in this chapter is not to provide a complete picture of the moldability of a thermoset resin system in RIM, but rather to provide a fundamental approach to attack a very complicated reactive processing operation. In the literature, some investigators (Estevez and Castro 1984; Manas-Zloczower and Macosko 1988) constructed “moldability diagrams” for RIM. But, construction of moldability diagrams for RIM is not so simple because a generally accepted deﬁnition of reaction injection moldability does not seem to exist. The basic question is: What are the criteria for reaction injection moldability for producing acceptable properties in the ﬁnal part? There are so many variables (e.g., mixing condition, level of catalyst and/or accelerator, temperature of inlet feed streams, mold ﬁll time, initial mold temperature, and thickness of mold cavity) that determine ﬁnal-part properties. For instance, mixing condition, which is often very difﬁcult to quantify, may greatly inﬂuence ﬁnal part properties, while the inclusion of mixing condition on a moldability diagram is not so easy. One thing that seems clear from the literature (Castro and Macosko 1982; Estevez and Castro 1984) is that “premature gelling” and “short shot” are detrimental to obtaining good (or acceptable) ﬁnal-part properties. Premature gelling, which may be deﬁned as the formation of a three-dimensional network, would be detrimental to completely ﬁlling a mold cavity, resulting in a short shot, because the viscosity of the ﬂowing resin becomes extremely high when a three-dimensional network is formed. In Chapter 14 of Volume 1, we discussed the rheological methods to determine the onset of gelation during cure of a thermoset. One can minimize, if not eliminate completely, premature gelling during mold ﬁlling in RIM by using a lesser amount of catalyst or accelerator, which in turn will decrease the rate of cure, prolonging the overall cure time and thus demold time. This then suggests that an optimum condition be determined on a trial-and-error basis. Upon injection via impingement mixing into a mold cavity, diisocyanate and polyol streams ﬁrst form a linear segmented polyurethane consisting of isocyanate hard segments and polyol soft segments (see Chapter 10 of Volume 1), which then proceeds to form a three-dimensional network via cross-linking reactions. Cross-linking reactions may be accelerated when an excess amount of a diol is used or in the presence of a small amount of a triol or a tertiary amine, which are often used as catalyst. The formation of a cross-linked network is a unique feature in RIM that distinguishes thermosetting polyurethanes from the thermoplastic linear polyurethanes discussed in Chapter 10 in Volume 1. A variety of RIM formulations are used in industry, with most of them being proprietary. Further, from a practical industrial perspective, formulations for RIM can be much more complicated in that, more often than not, particulates are added to produce reinforced reaction injection molded products. Also, other ingredients, such as blowing agents, surfactants, and/or pigments, are added. Modeling of RIM for such complex feed streams is indeed very difﬁcult, if not impossible, and to

REACTION INJECTION MOLDING

513

date little has been reported in the literature dealing with a modeling of such complex RIM formulations. We conclude this chapter by stating that no attempt has been made to relate processing variables to ﬁnal properties of reaction injection molded products. This subject has been reported, although not exhaustively, in the literature (Becker 1979; Macosko 1989), and thus it is felt that there is no reason to reproduce the materials already in the literature. The purpose of this chapter was to present a unifying approach, consistent with the major theme of the remaining two chapters, which will shed some light on the fundamentals of processing thermosets, and not to present an exhaustive collection of materials.

Problems Problem 11.1

Prepare three-dimensional plots of the time required for cure (at least 60% conversion throughout the part) against the mold temperature (Tw ), ranging from 80 to 120 ◦ C, and the initial temperature (Ti ) of the feed streams, ranging from 30 to 70 ◦ C, for RIM operation of urethane with a mold ﬁll time of 30 s. For the computation, use the numerical values for the mold geometry and physical properties given in Table 11.1, the kinetic parameters given in Table 11.2, and the chemorheological parameters given in Table 11.3. Problem 11.2

Prepare three-dimensional plots of the maximum temperature against the mold temperature (Tw ), ranging from 80 to 120 ◦ C, and the initial temperature (Ti ) of the feed streams, ranging from 30 to 70 ◦ C, for RIM operation of urethane with a mold ﬁll time of 30 s. For the computation, use the numerical values for the mold geometry and physical properties given in Table 11.1, the kinetic parameters given in Table 11.2, and the chemorheological parameters given in Table 11.3. Problem 11.3

Prepare three-dimensional plots of the maximum temperature against the mold ﬁll time, ranging from 10 to 50 s, and the initial temperature (Ti ) of the feed streams, ranging from 30 to 70 ◦ C, at a constant mold temperature (Tw ) of 80 ◦ C, for RIM operation of urethane. For the computation, use the numerical values for the mold geometry and physical properties given in Table 11.1, the kinetic parameters given in Table 11.2, and the chemorheological parameters given in Table 11.3. Problem 11.4

Prepare three-dimensional plots of the time required for cure (at least 60% conversion throughout the part) against the mold temperature (Tw ), ranging from 100 to 140 ◦ C, and the initial temperature (Ti ) of the feed streams, ranging from 30 to 70 ◦ C, for RIM operation of epoxy with a mold ﬁll time of 40 s. For the computation, use the numerical values for the mold geometry and physical properties given in Table 11.1, the kinetic parameters given in Table 11.2, and the chemorheological parameters given in Table 11.3.

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Problem 11.5

Prepare three-dimensional plots of the maximum temperature against the mold temperature (Tw ), ranging from 100 to 140 ◦ C, and the initial temperature (Ti ) of the feed streams, ranging from 30 to 70 ◦ C, for RIM operation of epoxy with a mold ﬁll time of 40 s. For the computation, use the numerical values for the mold geometry and physical properties given in Table 11.1, the kinetic parameters given in Table 11.2, and the chemorheological parameters given in Table 11.3. Problem 11.6

Prepare three-dimensional plots of the maximum temperature against the mold ﬁll time, ranging from 20 to 60 s, and the initial temperature (Ti ) of the feed streams, ranging from 30 to 70 ◦ C, at a constant mold temperature (Tw ) of 100 ◦ C, for RIM operation of epoxy. For the computation, use the numerical values for the mold geometry and physical properties given in Table 11.1, the kinetic parameters given in Table 11.2, and the chemorheological parameters given in Table 11.3.

Notes 1. Integration of Eq. (11.1) gives ur + r(∂ur /∂r) = 0. Using this expression, the second and third terms on the right-hand side of Eq. (11.2) can be rewritten as 1 ∂ r ∂r

2ηr

∂ur ∂r

−

2ur η r2

2u η ∂u 1 ∂ 2η −2ηur − 2r = − 2 r r + ur = 0 r ∂r ∂r r r (11N.1)

=

2. Using the dimensionless variables introduced, Eq. (11.3) is rewritten as V −ρ r η0

∗ ∗ 1 ∂ H 2 (v ∗r )2 ∂p ∗ ∂vr η = − + R ∂r r ∗ ∂z∗ ∂z∗ (r ∗ )3

(11N.2)

In typical RIM operations, where the magnitude of ρ, Vr , and η0 are in the order of unity and H/R 1, Eq. (11N.2) reduces to Eq. (11.4). Note that Eq. (11N.2) is obtained using the relationship ∂ur /∂r = −ur /r = −vr /r 2 , which follows from ur + r(∂ur /∂r) = 0 and vr = rur , in Eq. (11.3). 3. Here, ηr may be chosen as the viscosity of the ﬂuid entering the mold at the inlet temperature Ti . 4. Vr is related to the volumetric ﬂow rate Q by

H

Q = 4π

0

rur dz = 4πH Vr

1 0

vr∗ dz∗

(11N.3)

which gives

1 0

vr∗ dz∗ = 1

(11N.4)

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515

because Q = 4πH Vr in the overall mass balance. Integration of Eq. (11N.4) by parts under the boundary condition, vr∗ = 0 at z∗ = 1, gives

−

1 0

z∗

∂vr∗ ∗ dz = 1 ∂z∗

(11N.5)

which can be rewritten, with the aid of Eq. (11.4), as

−

1

r∗ (z∗ )2 ∗ η 0

∂p ∗ ∗ dz = 1 ∂r

(11N.6)

which gives Eq. (11.7). /R 2 = R 2 /2R 2 in which use was made of V = R 2 /2t in the 5. Note that t ∗ = t V r r r r runner under the assumption that there is no chemical reaction in the runner with the radius of Rr . 6. If we assume that the Arrhenius relationship holds, from Eq. (14.30) in Volume 1 we have a+bα α Eη 1 1 η(T , α) g = exp − ηr (Ti ) R T Ti αg − α

(11N.7)

thus η1 = ηr (Ti ) exp(−Eη /RTi ) or ηr = η1 exp(Eη /RTi ).

References Becker WE (ed) (1979). Reaction Injection Molding, Van Nostrand Reinhold, New York. Baldyga J, Bourne JR (1983). Polym. Eng. Sci. 23:556. Broyer E, Macosko CW (1976). AIChE J. 22:258. Broyer E, Macosko CW, Critchﬁeld FE, Lawler LF (1978). Polym. Eng. Sci. 18:382. Castro JM (1980). Mold Filling and Curing Studies for the Polyurethane Reaction Injection Molding Process, Doctoral Dissertation, University of Minnesota, Minneapolis, Minnesota. Castro JM, Macosko CW (1982). AIChE J. 28:250. Domine JD, Gogos CG (1980). Polym. Eng. Sci. 20:847. Estevez SR, Castro JM (1984). Polym. Eng. Sci. 24:428. Kamal MR, Ryan ME (1980). Polym. Eng. Sci. 20:859. Kim DH, Kim SC (1987a). Polym. Compos. 8:208. Kim JH (1987). Reaction Injection Molding Process of Polyurethane-Unsaturated Polyester Blends, Doctoral Dissertation, Korea Advanced Institute of Science and Technology, Taejeon, Korea. Kim JH, Kim SC (1987b). Polym. Eng. Sci. 27:1243. Kolodziej P, Macosko CW, Ranz WE (1982). Polym. Eng. Sci. 22:388. Lee KH, Kim SC (1988). Polym. Eng. Sci. 28:477.

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Lee LJ, Macosko CW (1980). Int. J. Heat Mass Transfer 23:1479. Lee LJ, Ottino JM, Ranz WE, Macosko CW (1980). Polym. Eng. Sci. 20:868 Lenz RW (1967). Organic Chemistry of Synthetic High Polymers, Interscience, New York. Macosko CW (1989). Fundamentals of Reaction Injection Molding, Hanser, Munich. Manas-Zloczower I, Macosko, CW (1988). Polym. Eng. Sci. 28:1219. Manzione LT (1981). Polym. Eng. Sci. 21:1234. Manzione LT, Osinski JS (1983). Polym. Eng. Sci. 23:576. Nguyen LT, Suh NP (1986). Polym. Eng. Sci. 25:799. Saunders JH, Frisch KC (1964). Polyurethanes, Chemistry and Technology, Interscience, New York. Tucker CL, Suh NP (1980). Polym. Eng. Sci. 20:875.

12

Pultrusion of Thermoset/Fiber Composites

12.1

Introduction

Pultrusion of thermoset/ﬁber composites generally consists of pulling continuous rovings and/or continuous glass mats through a resin bath or impregnator and then into preforming ﬁxtures, where the section is partially shaped and excess resin and/or air are removed. Finally, the preformed proﬁles are pulled through heated dies, where the section is cured continuously (Batch 1989; Meyer 1985; Price 1979; Richard 1986). The pultrusion process is one of the most cost-effective continuous processing techniques for producing thermoset composite materials. The laminating resin may be an unsaturated polyester resin, a vinyl ester resin, or an epoxy resin, but the majority of pultruded thermoset products currently use unsaturated polyester resins. The reason for this is that epoxy resins require high heat inputs and have relatively slow gelation, although some effort has been spent on development of new epoxy resin systems that can be pultruded at speeds comparable with unsaturated polyester resin systems (e.g., 0.6–0.9 m/min). Han and coworkers (Han et al. 1986, Han and Chin 1988) formulated and then solved numerically, via the ﬁnite difference method, a system of equations describing the cure kinetics of a thermoset resin and the heat transfer between the resin and the die wall, in order to model the pultrusion process for thermoset/ﬁber composites. Subsequently, other investigators (Batch and Macosko 1993; Chachad et al. 1995; Gorthala et al. 1994a, 1994b; Ma et al. 1986) reported similar studies. Experimental studies (Batch and Macosko 1993; Chachad et al. 1995; Ma et al. 1986; Price 1979; Price and Cupschalk 1984; Roux et al. 1998) on the pultrusion process for thermoset/ﬁber composites have also been reported. Some research groups (Aström and Pipes 1993; Larock et al. 1989; Ma and Chen 1991; Ruan and Liu 1994) have investigated the pultrusion process of ﬁber-reinforced thermoplastic polymers. While there are some similarities

517

518

PROCESSING OF THERMOSETS

between the pultrusion of thermoset/ﬁber composites and ﬁber-reinforced thermoplastic polymers, the most important difference between the two lies in that the former involves chemical reactions during processing, whereas the latter does not. Therefore, the modeling of the pultrusion process of thermoset/ﬁber composites requires information on the chemorheology of the thermoset/ﬁber system under consideration. Conversely, modeling of the pultrusion process of ﬁber-reinforced thermoplastic polymers requires information on the rheological behavior of molten polymers as functions of shear rate and temperature, and on the crystallization kinetics when dealing with semicrystalline thermoplastic polymers. At present, the pultrusion of thermoset/ﬁber composites occupies the bulk of industrial activities, since the mechanical properties of pultruded products from thermoset/ﬁber composites in general are much greater than those from ﬁber-reinforced thermoplastic composites. In this chapter, we consider the pultrusion of thermoset/ﬁber composites. To facilitate our presentation, let us consider the schematic diagram given in Figure 12.1, showing resin-impregnated ﬁbers being pulled at a constant speed through a long cylindrical die, which is heated electrically to maintain a predetermined temperature distribution. As the resin-impregnated ﬁbers enter the die, initially the heat is conducted from the die wall to the material. Due to the low thermal conductivity of the resin and ﬁbers, the center of a given cross section may not reach the die wall temperature until some time after that cross section has entered the die. In other words, the temperature of the material near the die wall initially is higher than that at the center. When the temperature of the material reaches a value at which an initiator (or a catalyst) becomes activated, the curing reaction begins and generates heat due to the exothermic nature of chemical reactions taking place between resin and initiator (or catalyst). As the heat generated by the curing reaction helps accelerate further reactions, like an autocatalytic reaction, the temperature of the material at the center of the cross section becomes higher than that near the die wall. From that point on, the external heat source will no longer be needed. It can be surmised, therefore, that temperature is a most important factor for controlling the pultrusion process and, thus, the quality of pultruded products. It should be noted that a nonuniform distribution of temperature along the cross section of the proﬁle implies a nonuniform distribution of the extent of cross-linking reactions, which in turn results in a nonuniform distribution of molecular weight and, subsequently, of the mechanical properties of pultruded products. To obtain pultruded products with consistent quality, it is essential to develop a strategy for controlling temperature in the

Figure 12.1 Schematic of the pultrusion process for thermoset/ﬁber composites.

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519

pultrusion process. It should be pointed out that control of the die temperature requires information on the curing reactions taking place inside the die. The rate of temperature rise in the material after entering the die depends on the type of resin and initiator (or catalyst) used. The length-to-diameter ratio of a pultrusion die is usually very large, and the pulling speed is relatively slow. Therefore, the use of either too active an initiator or an excessive amount of initiator must be avoided. Under such circumstances, cure reactions will have completed near the die wall ﬁrst, giving rise to a nonuniform distribution of the degree of cure across the cross section and thus poor quality of pultruded products. Conversely, use of either too mild an initiator or an insufﬁcient amount of initiator will give rise to a low degree of cure and thus, again, poor quality of pultruded products. Therefore, a careful balance is required between the reactivity of resin, the reactivity and amount of initiator, the proﬁle of die temperature, the thickness of the part to be pultruded, and also the pulling speed, which determines the residence time of the material in the die. Usually, the volume of ﬁber in the unreacted resin/ﬁber mixture is between 50 and 70%, and the ﬁbers act as a heat sink, by absorbing some of the heat generated by the exothermic reaction of the resin, which helps to control the temperature rise in the material. It is a general practice to add particulate ﬁllers, which also help to control the temperature rise in the material by absorbing heat. One may also add particulates with high thermal conductivity (e.g., carbon black, aluminum powder) in order to achieve a more uniform temperature distribution across the cross section (after the resin-impregnated ﬁbers have entered the die, as well as after the center of the cross section of the material has reached a peak temperature). A variation of the conventional pultrusion process described above has been made by injecting a resin directly into the die inlet, thus eliminating the resin impregnation bath and the shape preformer appearing in the schematic given Figure 12.1. This process is referred to as injection pultrusion. Some research groups (Ding et al. 2000, 2002; Dubé et al. 1995; Kommu et al. 1998; Li et al. 2002; Voorakaranam et al. 1999) have investigated the injection pultrusion process, experimentally and numerically. In this chapter, we ﬁrst present cure kinetics of unsaturated polyester using mixed initiators, and then an analysis of pultrusion of thermoset/ﬁber composites, placing emphasis on the importance of having reliable chemorheological models to simulate the pultrusion process, and then the simulation results for pultrusion of unsaturated polyester/ﬁber and epoxy/ﬁber composites using a cylindrical die. Since the approach for modeling the injection pultrusion process is virtually identical to that for the conventional pultrusion process, except for the ﬂow of a resin at the die inlet region, in this chapter we do not discuss the injection pultrusion process separately.

12.2

Effect of Mixed Initiators on the Cure Kinetics of Unsaturated Polyester

The pultrusion die commonly used in industrial operations is very long (say, 1m) and thus the residence time of a resin/ﬁber composite inside the die is rather long. Therefore, it is a common practice to use mixed initiator systems, having both a lowtemperature initiator and a high-temperature initiator, such that upon entering the die, the resin begins to cure immediately due to the low-temperature initiator and then goes

520

PROCESSING OF THERMOSETS

to reaction completion with the help of the high-temperature initiator. With a hightemperature initiator alone (which may decompose, say, at 90 ◦ C), part of the cure time needed will be lost until the initiator reaches its decomposition temperature because the wall temperature is deﬁned by the predetermined temperature proﬁle along the die axis. Due to the highly exothermic cure reaction, the heat generated locally will give rise to a nonuniform degree of cure in the stock. However, with a low-temperature initiator alone (which may decompose, say, at 50 ◦ C), the cure reaction may not go to completion. After decomposition of the initiator during the initial stages of cure, there may be no free radicals available at high temperatures, say above 90 ◦ C, which would be needed to drive the curing reaction to completion in the remaining cure time available. To demonstrate the necessity of using a mixed initiator system in pultrusion, let us compare the cure rate of a single initiator with that of a mixed initiator system. Figure 12.2 shows variations of the rate of cure, dα/dt, during isothermal cure of a general-purpose unsaturated polyester (OC-E701, Owens-Corning Fiberglas) at 70, 80, and 90 ◦ C, where the initiator PERCADOX 16N (P16N) (Noury Chemical) was used. Note that the detailed procedures employed to obtain Figure 12.2 from differential scanning calorimetry (DSC) experiments are presented in Chapter 14 of Volume 1. It is seen in Figure 12.2 that the peak value of dα/dt occurs sooner when the cure temperature is increased from 70 to 90 ◦ C. Figure 12.3 shows variations of dα/dt during isothermal cure of a general-purpose unsaturated polyester (OC-E701, Owens-Corning Fiberglass) at 90, 100, and 110 ◦ C, where the initiator benzoyl peroxide (BPO) was used. It is seen in Figure 12.3 that the peak value of dα/dt occurs sooner when the cure temperature is increased from 90 to 110 ◦ C. A comparison of Figure 12.2 with Figure 12.3 reveals that the time at which the peak value of dα/dt occurs is approximately the same for OC-E701/P16N at 90 ◦ C and OC-E701/BPO at 110 ◦ C, for OC-E701/P16N at 80 ◦ C

Figure 12.2 Plots of rate of cure

dα/dt versus cure time for OC-E701/P16N = 100/1 (by wt) at three isothermal cure temperatures (◦ C): () 70, () 80, and () 90. (Reprinted from Han and Lee, Journal of Applied Polymer Science 34:793. Copyright © 1987, with permission from John Wiley & Sons.)

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521

Figure 12.3 Plots of rate of cure dα/dt versus cure time for OC-E701/BPO = 100/1 (by wt) at three isothermal cure temperatures (◦ C): () 90, () 100, and () 110. (Reprinted from Han and Lee, Journal of Applied Polymer Science 34:793. Copyright © 1987, with permission from John Wiley & Sons.)

and OC-E701/BPO at 100 ◦ C, and for OC-E701/P16N at 70 ◦ C and OC-E701/BPO at 90 ◦ C. This observation suggests that P16N can be regarded as a low-temperature initiator and BPO as a high-temperature initiator. Figure 12.4 compares variations of dα/dt during isothermal cure at 90 ◦ C for OC-E701/P16N = 100/1 (by wt), OC-E701/BPO = 100/1 (by wt), and OC-E701/P16N/BPO = 100/0.25/0.75 (by wt). In Figure 12.4, we observe that the P16N/BPO mixed initiator system gives rise to the peak value of dα/dt appearing between the two peaks, one for P16N and the other for BPO. In other words, the use of the mixed initiator system, P16N/BPO = 25/75 (by wt), has delayed the cure of OC-E701 in the presence of P16N alone and accelerated the cure of OC-E701 in the presence of BPO alone, suggesting that such a mixed initiator system is effective at controlling the rate of cure of unsaturated polyester. We can describe the above experimental observations using the mechanistic kinetic model presented in Chapter 14 of Volume 1. Figure 12.5 gives temperature dependence of the initiator decomposition rate constant for P16N (kd1 ) and BPO (kd2 ), following the Arrhenius relation. It is seen in Figure 12.5 that the values of kd1 are much larger than those of kd2 , indicating that P16N will decompose at a much lower temperature than BPO, and that BPO will require higher temperatures to decompose. This observation suggests that P16N can be regarded as a low-temperature initiator and BPO as a hightemperature initiator. Below, we demonstrate that the mechanistic kinetic model can be used to predict the effectiveness of a mixed initiator system to cure unsaturated polyester. For the numerical computations, we used the numerical values of the kinetic

Figure 12.4 Plots of rate of cure dα/dt versus cure time at 90 ◦ C

for: () OC-E701/P16N = 100/1 (by wt), () OC-E701/BPO = 100/1 (by wt), and () OC-E701/ P16N/BPO = 100/0.25/0.75 (by wt). (Reprinted from Han and Lee, Journal of Applied Polymer Science 34:793. Copyright © 1987, with permission from John Wiley & Sons.)

Figure 12.5 Plots of initiator rate constant kdi (i = 1, 2) versus 1/T for two initiators: kd1 for P16N and kd2 for BPO. (Reprinted from Han and Chin, Polymer Engineering and Science

28:321. Copyright © 1988, with permission from the Society of Plastics Engineers.) 522

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523

Table 12.1 Parameters for the mechanistic model employed for the simulation of pultrusion using mixed initiators, P16N and BPO

(a) For P16N as Initiator kd10 = 2.205 × 1019 min−1 ; Ed1 = 1.34 × 105 J/mol; f10 = 0.08 (b) For BPO as Initiator kd20 = 2.928 × 1016 min−1 ; Ed2 = 1.29 × 105 J/mol; f20 = 0.18 (c) For Inhibitor kz0 = 3.204 × 1017 min−1 L/M; Ez = 8.73 × 104 J/mol; [Z]0 = 5.09 × 10−4 M/L (d) For Propagation Reaction kp0 = 3.441 × 1014 min−1 L/M; Ep = 8.32 × 104 J/mol; m = 0.8 (e) Heat of Reaction for Unsaturated Polyester HR = 34 J/g

parameters given in Table 12.1 and the initial concentration of unsaturated polyester [M]0 = 0.352 kg/L. Figure 12.6 gives the prediction of dα/dt during cure of an unsaturated polyester with 1 wt % P16N, for which the initial concentration was [I ]0 = 0.256 × 10−1 M/L, at 85, 90, and 100 ◦ C. It is seen that the peak value of dα/dt occurs sooner when the cure temperature is increased from 85 to 100 ◦ C, consistent with the experimental observations shown in Figure 12.2. Figure 12.7 gives the prediction of dα/dt during cure of an unsaturated polyester with 1 wt % BPO, for which the initial concentration was [I ]0 = 0.367 × 10−1 M/L, at 100, 110, and 120 ◦ C. It is seen that the peak

Figure 12.6 Prediction, via a mechanistic kinetic model Eqs. (14.16)–(14.19) given in Chapter 14 of Volume 1, of plots of rate of cure dα/dt versus cure time for resin/P16N = 100/1 (by wt) at three temperatures (◦ C): (1) 85, (2) 90, and (3) 100. The numerical values of the kinetic parameters used for the computation are given in Table 12.1, and the initiator concentration for P16N was [I ]0 = 0.256 × 10−1 M/L.

524

PROCESSING OF THERMOSETS Figure 12.7 Prediction, via a

mechanistic kinetic model Eqs. (14.16)–(14.19) given in Chapter 14 of Volume 1, of plots of rate of cure dα/dt versus cure time for resin/BPO = 100/1 (by wt) at three temperatures (◦ C): (1) 100, (2) 110, and (3) 120. The numerical values of the kinetic parameters used for the computation are given in Table 12.1, and the initiator concentration for BPO was [I ]0 = 0.367 × 10−1 M/L.

value of dα/dt occurs sooner as the cure temperature is increased from 100 to 120 ◦ C, consistent with the experimental observations shown in Figure 12.3. Comparison of Figure 12.7 with Figure 12.6 reveals that at 100 ◦ C, for instance, BPO has a very small value of dα/dt compared with P16N, indicating that BPO is not effective for curing the unsaturated polyester at 100 ◦ C. Conversely, the peak value of dα/dt for BPO becomes very large when the temperature is increased to 120 ◦ C, indicating that BPO would be effective for curing the unsaturated polyester at 120 ◦ C. Figure 12.8 gives the prediction of dα/dt during cure of an unsaturated polyester with a mixed initiator system, 0.25 wt % P16N and 0.75 wt % BPO at 85, 90, and 100 ◦ C, for which the initial concentration was [I1 ]0 = 0.64 × 10−2 M/L for P16N and [I2 ]0 = 0.275 × 10−1 M/L for BPO. Comparison of Figure 12.8 with Figure 12.6 reveals that at 85 ◦ C, the resin/mixed initiator system delays cure reaction until about 2 min after cure began and the peak value of dα/dt occurs at about 6 min after cure began, while the resin/P16N system undergoes cure reaction very soon after cure began and the peak value of dα/dt occurs at about 3.5 min after cure began. This indicates that the resin/mixed initiator system would prolong cure reaction inside the pultrusion die. Conversely, comparison of Figure 12.8 with Figure 12.7 reveals that at 100 ◦ C, the resin/BPO system delays cure reaction until about 8 min after cure began and the peak value of dα/dt occurs at about 14 min after cure began, while the resin/mixed initiator system undergoes cure reaction very soon after cure began and the peak value of dα/dt occurs at about 1 min after cure began. These observations justify use of mixed initiators in pultrusion because the typical residence time in a pultrusion die varies somewhere between 1 and 4 min. More importantly, the temperature of an unsaturated polyester/ﬁber composite increases initially very rapidly owing to highly exothermic reactions and then decreases towards the die exit, as is shown by some examples of die wall temperature presented in the next section.

PULTRUSION OF THERMOSET/FIBER COMPOSITES

525

Figure 12.8 Prediction, via a mechanistic kinetic model Eqs. (14.16)–(14.19) given in Chapter 14 of Volume 1, of plots of rate of cure dα/dt versus cure time for resin/P16N/BPO = 100/0.25/0.75 (by wt) at three temperatures (◦ C): (1) 85, (2) 90, and (3) 100. The numerical values of the kinetic parameters used for the computation are given in Table 12.1, and the initiator concentrations were [I1 ]0 = 0.64 × 10−2 M/L for P16N and [I2 ]0 = 0.275 × 10−1 M/L for BPO.

12.3

Cure Kinetics of Unsaturated Polyester/Fiber Composite

In this section, we present how the cure kinetics of unsaturated polyester are affected by the presence of glass ﬁbers or carbon ﬁbers. This is a very important subject related to the pultrusion process of thermoset/ﬁber composites. Figure 12.9 gives plots of dα/dt during cure of an unsaturated polyester OC-E701/BPO/glass ﬁber system with Figure 12.9 Plots of rate of cure dα/dt versus cure time for OC-E701/BPO/glass ﬁber system with resin/glass ﬁber = 17/83 (by wt) and OC-E701/BPO = 100/1 (by wt) at three isothermal cure temperatures (◦ C): () 90, () 100, and () 110.

526

PROCESSING OF THERMOSETS Figure 12.10 Plots of rate of cure dα/dt versus cure time for epoxy/catalyst/glass ﬁber system with resin/glass ﬁber = 18/82 (by wt) and epoxy/catalyst = 100/3 (by wt) at three isothermal cure temperatures (◦ C): () 90, () 100, and () 110.

resin/glass ﬁber = 17/83 (by wt) and OC-E701/BPO = 100/1 (by wt) at 90, 100, and 110 ◦ C. A comparison of Figure 12.9 with Figure 12.3 reveals that at all three temperatures tested the peak values of dα/dt occur much sooner after cure began, and that they are much greater in the OC-E701/BPO/glass ﬁber system than in the OC-E701/BPO system, indicating that glass ﬁbers accelerate the cure reaction of the OC-E701/BPO system. Figure 12.10 gives plots of dα/dt during cure of an epoxy/catalyst/glass ﬁber system with resin/glass ﬁber = 18/82 (by wt) and epoxy/catalyst = 100/3 (by wt) at 90, 100, and 110 ◦ C. A comparison of Figure 12.10 with Figure 12.9 reveals that at all three temperatures tested the peak values of dα/dt occur much later after cure began, and that they are smaller in the epoxy/catalyst/glass ﬁber system than in the OC-E701/BPO/glass ﬁber system, indicating that glass ﬁbers slow down the cure reaction of the epoxy system. This observation seems to indicate that the interface between thermoset resin and glass ﬁber plays an important role in the cure reaction of thermoset/ﬁber composites. Figure 12.11 gives plots of dα/dt during cure of an epoxy/catalyst/carbon ﬁber system with resin/carbon ﬁber = 18/82 (by wt) and epoxy/catalyst = 100/3 (by wt) at 90, 100, and 110 ◦ C. A comparison of Figure 12.11 with Figure 12.10 reveals that at all three temperatures tested, the peak values of dα/dt occur much sooner after cure began and the peak value of dα/dt at 110 ◦ C is much greater in the epoxy/catalyst/carbon ﬁber system than in the epoxy/catalyst/glass ﬁber system, indicating that the types of ﬁbers employed (owing to the differences in thermal conductivity) for reinforcement also plays an important role in the cure of thermosets/ﬁber composites. Table 12.2 gives a summary of the numerical values of the parameters appearing in the empirical kinetic expressions, Eqs. (14.1) and (14.2), given in Chapter 14 of Volume 1 for the four thermoset/ﬁber composite systems that were considered in Figures 12.9–12.11. These values were obtained using the procedures described in Chapter 14 of Volume 1. In the next sections, we present the simulation results for the pultrusion of unsaturated polyester/P16N/BPO/glass ﬁber system, epoxy/catalyst/glass

Figure 12.11 Plots of rate of cure dα/dt versus cure time for epoxy/catalyst/carbon ﬁber system with resin/carbon ﬁber = 18/82 (by wt) and epoxy/catalyst = 100/3 (by wt) at three isothermal cure temperatures (◦ C): () 90, () 100, and () 110.

Table 12.2 Summary of the kinetic parameters appearing in Eqs. (14.1) and (14.2), given in Chapter 14 of Volume 1, for thermoset/ﬁber composites

(a) Unsaturated Polyestera /BPOb /Glass Fiberc System with Resin/BPO = 100/1 (by wt) and Resin/Glass Fiber = 17/83 (by wt) k10 = 4.575 × 1020 min−1 ; E1 = 1.58 × 105 J/mol; k20 = 2.217 × 107 min−1 ; E2 = 5.23 × 104 J/mol; m = 0.577; n = 1.423; HR = 34 J/g (b) Unsaturated Polyestera /P16Nd /BPOb /Glass Fiberc System with Resin/ P16N/BPO = 100/0.25/0.75 (by wt) and Resin/Glass Fiber = 17/83 (by wt) k10 = 7.224 × 1021 min−1 ; E1 = 1.66 × 105 J/mol; k20 = 6.906 × 1014 min−1 ; E2 = 1.05 × 105 J/mol; m = 0.40; n = 1.60; HR = 34 J/g (c) Epoxye /Catalystf /Glass Fiberc System with Resin/Curing Agent = 100/3 (by wt) and Resin/Fiber = 18/82 (by wt) k10 = 1.165 × 108 min−1 ; E1 = 7.00 × 104 J/mol; k20 = 2.704 × 107 min−1 ; E2 = 5.34 × 104 J/mol; m = 0.93; n = 1.07; HR = 72 J/g (d) Epoxye /Catalystf /Carbon Fiberg System with Resin/Curing Agent = 100/3 (by wt) and Resin/Fiber = 18/82 (by wt) k10 = 1.360 × 1010 min−1 ; E1 = 8.41 × 104 J/mol; k20 = 1.685 × 109 min−1 ; E2 = 6.63 × 104 J/mol; m = 0.91; n = 1.09; HR = 72 J/g a

Low-viscosity, general-purpose isophthalic-based polyester resin (OC-E701), Owens-Corning Fiberglas). b Benzoyl peroxide (Noury Chemical). c PPG-721NT (PPG Company). d PERCADOX P16N (Noury Chemical). e Bisphenol-A/epichlorohydrin epoxy (EPON 9302, Shell Development). f EPON curing agent 9350 (Shell Development). g Thornel T-500 (Union Carbide).

527

528

PROCESSING OF THERMOSETS

ﬁber system, and epoxy/catalyst/carbon ﬁber system, using ﬁrst an empirical kinetic model and then a mechanistic kinetic model.

12.4

Analysis of the Pultrusion of Thermoset/Fiber Composite

In order to show a modeling approach (Han et al. 1986; Han and Chin 1988), here we consider a pultrusion die of cylindrical shape, through which resin-impregnated ﬁbers are pulled (see Figure 12.1). In our analysis, we will not consider the heat transfer occurring in the radio frequency heater in which the resin-impregnated ﬁbers, while traveling, are preheated to the desired temperature. In addition, we make the following assumptions: (1) the process is at steady state, (2) the velocity proﬁle is ﬂat, (3) the heat conduction in the axial direction is negligibly small compared with that in the radial direction, (4) the diffusion of resin during cure is negligible, and (5) the local motion of resin during cure is negligible. 12.4.1 General System Equations The continuity equation, in cylindrical coordinates, is given by vz

∂CA = RA ∂z

(12.1)

where vz is the velocity of the cylindrical rod moving through the die in the pulling direction z, CA is the concentration of the reactive functional group within the pultrusion die, and RA is the rate of formation of cured resin. By deﬁning the degree of cure α as α = (CA0 − CA )/CA0

(12.2)

where CA0 is the initial concentration of the reactive functional group in the resin. In the die entrance, z = 0, we have v ∂CA ∂α dα = vz =− z dt ∂z CA0 ∂z

(12.3)

and substitution of Eq. (12.3) into (12.1) gives CA0

dα = −RA dt

(12.4)

The energy balance equation is given by ρcp vz

1 ∂ ∂T ∂T = r k + RA HR ∂z r ∂r ∂r

(12.5)

where T is the temperature of the cylindrical rod moving through the die, ρ is the bulk density, cp is the bulk speciﬁc heat, k is the bulk thermal conductivity, RA is the rate of

PULTRUSION OF THERMOSET/FIBER COMPOSITES

529

formation of cured resin (i.e., the rate of cure reaction), and HR is the heat of reaction due to cure, which is a negative value owing to the exothermic curing reactions. Note that the material being cured consists of three components, namely, resin, ﬁbers, and cured resin. Therefore, the bulk physical properties of the material must include all three components. In the analysis presented here, we use the following deﬁnitions of bulk physical properties: (a) Bulk Density ρ

wo wo wo ρ o 1 = (1 − α) m + om α m + f ρ ρm ρ m ρp ρf

(12.6)

o is the weight fraction of uncured resin, wo is the weight fraction of ﬁbers, where wm f and the densities of the resin (ρm ), cured product (ρp ), and ﬁbers (ρf ), respectively, are given, by o ρm = ρ m + aT ; ρp = ρpo + bT ; ρf = ρfo + cT

(12.7)

o , ρ o , ρ o , a, b, and c are material constants. The volume fractions of the in which ρm p f resin (φm ), cured product (φp ), and ﬁbers (φf ), respectively, are given by o o (1 − α)ρ/ρm ; φp = αwm ρ/ρp ; φf = wfo ρ/ρf φm = wm

(12.8)

and the weight fractions of the resin (wm ), cured product (wp ), and ﬁbers (wf ), respectively, are given by o o wm = wm (1 − α); wp = αwm ; wf = wfo

(12.9)

(b) Bulk Speciﬁc Heat cp

cp = wm cpm + wp cpp + wf cpf

(12.10)

in which the speciﬁc heat of the resin (cpm ), cured product (cpp ), and ﬁbers (cpf ), respectively, are given by o o o + a1 T ; cpp = cpp + b1 T ; cpf = cpf + c1 T cpm = cpm

(12.11)

o , co , co , a , b , and c are material constants. where cpm pp pf 1 1 1

(c) Bulk Thermal Conductivity k

φp φ φ 1 = m + + f k km kp kf

(12.12)

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PROCESSING OF THERMOSETS

in which the thermal conductivities of the resin (km ), cured product (kp ), and ﬁbers (kf ), respectively are given by o + a2 T ; kp = kpo + b2 T ; kf = kfo + c2 T km = km

(12.13)

In order to solve Eqs. (12.1) and (12.5), a rate expression for cure kinetics must be speciﬁed, because these two equations are related to each other by the rate of cure reaction, RA , which in turn depends on temperature, T . Next, we will consider two different types of rate expressions for the cure of unsaturated polyesters presented in Chapter 14 of Volume 1. 12.4.2 System Equations with an Empirical Kinetic Model In deriving system equations to describe the pultrusion of thermoset/ﬁber composites, we use the empirical kinetic expression presented in Chapter 14 of Volume 1, namely, Eqs. (14.1) and (14.2). Use of Eqs. (12.1)−(12.3) in Eq. (14.1) of Volume 1 gives −RA = CA0 (k1 + k2 α m )(1 − α)n

(12.14)

Substituting Eq. (12.3) into Eq. (14.1) of Volume 1, and Eq. (12.14) into Eq. (12.5), and o , c∗ = c /co , k ∗ = k/k o , B = introducing the dimensionless variables ρ ∗ = ρ/ρm p m p pm 1 o T , Pe = ρ o co v r /k o , K = vz /ro k20 , K1 = k1 /k20 , B2 = k20 HR CA0 ro 2 /km m pm z o m i 2 k2 /k20 , θ = T/Ti , r ∗ = r/ro , and z∗ = z/ro , with ro being the radius of the pultrusion die, we obtain (Han et al. 1986) ∂α = (K1 + K2 α m )(1 − α)n ∂z∗ ∂θ 1 ∂ ∗ ∗ ∂θ ∗ ∗ + B2 (K1 + K2 α m )(1 − α)n ρ cp Pe ∗ = ∗ ∗ r k ∂r r ∂r ∂r ∗ B1

(12.15) (12.16)

Equations (12.15) and (12.16) must be solved simultaneously, subject to the following boundary conditions: (i) at r ∗ = 1 and 0 ≤ z∗ ≤ L∗ , θ = θ (z∗ ) (ii) at r ∗ = 0 and 0 ≤ z∗ ≤ L∗ , ∂θ/∂r ∗ = 0 (iii) at z∗ = 0 and 0 ≤ r ∗ ≤ 1, α = 0 and θ = 1

(12.17a) (12.17b) (12.17c)

where L ∗ is the dimensionless length (L/ro ) of the pultrusion die (see Figure 12.1). Note that K1 and K2 appearing in Eqs. (12.15) and (12.16) are functions of θ . As pointed out in Chapter 14 of Volume 1, there are drawbacks to the use of Eq. (14.1) in that it cannot describe explicitly the effects of the concentration of a curing agent and/or an inhibitor and rate-controlled mechanisms on the rate of cure. As the cure reaction progresses, the molecules become very large and, thus, the diffusion of monomers into growing molecules becomes a rate-controlled step. Such drawbacks can be overcome when using a mechanistic kinetic model.

PULTRUSION OF THERMOSET/FIBER COMPOSITES

531

12.4.3 System Equations with a Mechanistic Kinetic Model Let us consider a mechanistic kinetic model, Eqs. (14.16)−(14.19) given in Chapter 14 of Volume 1, to describe the curing reaction of unsaturated polyester with multiple initiators. Introducing the dimensionless variables [Z ∗ ] = [Z]/[Z]0 , [Ij∗ ] = [Ij ]/[Ij ]0 , [M ∗ ] = [M]/[M]0 , α = ([M]0 − [M])/[M]0 , and [R·∗ ] = [R·]/[R·]m , with [R·]m being the maximum possible concentration of radicals and represented by N [R·]m = 2 [Ij ]0 − [Z]0 , Eq. (12.5) and Eqs. (14.16)−(14.19) from Volume 1 may j =1

be rewritten as (Han et al. 1988)

∂ Z ∗ ∂z∗ = −A Z ∗ [R·∗ ] ∂[Ij∗ ] ∂z∗ = −Bj [Ij∗ ]; j = 1, 2, . . . , N ∂[α] ∂z∗ = C[R·∗ ] (1 − α) N #

∂[R·∗ ] ∂z∗ = Dj [Ij∗ ] − E Z ∗ [R·∗ ]

(12.18) (12.19) (12.20) (12.21)

j =1

ρ ∗ cp∗ P e

∂θ 1 ∂ ∗ ∗ ∂θ = ∗ ∗ r k + F [R·∗ ](1 − α) ∂z∗ r ∂r ∂r ∗

(12.22)

where A = kz ro [R·]m /vz , Bj = kdj ro /vz , C = kp ro [R·]m /vz , Dj = 2kdj fi [Ij ]0 / o T , and Pe = ρ o co v r /k o . vz [R·]m , E = kz ro [Z]0 /vz , F = kp ro 2 [M]0 [R·]m HR /km m p z o m i ∗ ∗ ∗ ∗ ∗ Note that ρ , cp , k , r , and z were deﬁned when writing Eq. (12.16). Note that in deriving Eqs. (12.18)−(12.21) from Eqs. (14.16)−(14.19) from Volume 1, we have made use of the following relationships: d[Z]/dt = vz ∂[Z]/∂z, d[Ij ]/dt = vz ∂[Ij ]/∂z, d[M]/dt = vz ∂[M]/∂z, and d[R·]/dt = vz ∂[R·]/∂z. Equations (12.18)−(12.22) must be solved simultaneously, subject to the boundary conditions given by Eq. (12.17) with a modiﬁcation: at z∗ = 0 and 0 ≤ r ∗ ≤ 1, α = 0, θ = 1, [Z ∗ ]0 = 1, [Ij∗ ]0 = 1, and [R·∗ ] = 0. Note that in writing Eq. (12.21), the termination term is neglected because the contribution of the termination reaction to the cure of unsaturated polyester is negligible (see Chapter 14 in Volume 1).

12.5

Conversion and Temperature Proﬁles in a Pultrusion Die

In this section, we present some representative simulation results, based ﬁrst on an empirical kinetic model and then on a mechanistic kinetic model, using the system equations described in the previous section. In the numerical simulation, we have used the physical properties of resins and ﬁbers given in Table 12.3, and the die dimensions and feed temperature of the thermoset/ﬁber composites given in Table 12.4. Figure 12.12 gives the speciﬁed wall temperature proﬁle along the die axis, which was employed to simulate the pultrusion of unsaturated polyester/glass ﬁber composite. Figure 12.13 gives simulation results, based on the empirical kinetic model, describing

Table 12.3 Physical properties of the resin and ﬁbers employed for simulation

Material Uncured polyester Cured polyester Uncured epoxy Cured epoxy Glass ﬁbers Carbon ﬁbers

ρ (g/cm3 )

cp (J/(kg K))

k (W/(m K))

1.10 1.20 1.13 1.17 2.54 1.79

1,886 1,886 1.08 ×104 + 2.50 T 5.56 × 103 + 0.62 T 843 712

0.17 6.45 ×10−2 + 3.96 × 10−4 T 8.29 × 10−2 + 1.80 × 10−4 T 1.92 × 10−1 + 7.25 × 10−5 T 0.87 8.67

Reprinted from Han et al., Polymer Engineering and Science 26:393. Copyright © 1986, with permission from the Society of Plastics Engineers.

Table 12.4 Die dimensions and feed temperatures employed for simulation

Material Polyester/glass ﬁber Epoxy/glass ﬁber

ro (m)

L (m)

Ti (K)

wf (wt %)

6.35 × 10−3 6.35 × 10−3

1.52 0.89

333 298

83 82

Reprinted from Han et al., Polymer Engineering and Science 26:393. Copyright © 1986, with permission from the Society of Plastics Engineers.

Figure 12.12 Schematic showing the wall temperature proﬁle along the axis of a cylindrical extrusion die that was employed to simulate the pultrusion of unsaturated polyester/glass ﬁber composites.

532

PULTRUSION OF THERMOSET/FIBER COMPOSITES

533

Figure 12.13 Simulation results, based on an empirical kinetic model, Eq. (14.1) given in Chapter 14 of Volume 1, showing the proﬁles of (a) temperature and (b) degree of cure at a pulling speed of 0.61 m/min for unsaturated polyester/P16N/BPO/glass ﬁber system with resin/glass ﬁber = 17/83 (by wt) and resin/P16N/BPO = 100/0.25/0.75 (by wt). The wall temperature proﬁle along the die axis employed for the simulation is given by Figure 12.12 and the numerical values of the kinetic parameters employed for the simulation are given in Table 12.2.

the proﬁles of (a) temperature and (b) degree of cure at a pulling speed of 0.61 m/min for unsaturated polyester resin/P16N/BPO/glass ﬁber system with resin/glass ﬁber = 17/83 (by wt) and resin/P16N/BPO = 100/0.25/0.75 (by wt). The wall temperature proﬁle along the die axis given in Figure 12.12 is used in the numerical simulation. The following observations are worth noting in Figure 12.13. There is a signiﬁcant temperature distribution in the radial direction with the wall temperature, much higher than that at the center. This is attributable to the fact that the die wall temperature is higher than the temperature of the feed stream, since the exothermic reactions taking place near the die wall generate heat, and thus increase the temperature of the resin/ﬁber composite. Consequently, as can be seen in Figure 12.13b, the degree of conversion (α) is higher near the wall than it is at the center of the die. As the cure completes (i.e., α approaching 1), no more heat is generated and thus the temperature rise will cease as the material travels along the die axis. Since the pultruded rod should not be hot when it exits the die, the composite material must be cooled in the latter part of the die (see the imposed wall temperature proﬁle in Figure 12.12). Under the particular processing

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conditions employed for the simulation, the results of which are shown in Figure 12.13, near the die wall (r ∗ ≈ 1), α ≈ 1 at z∗ = 60, and near the center (r ∗ ≈ 0) of the die, α ≈ 1 at z∗ = 180. Referring to Figure 12.13, due to the very low thermal conductivity of both the resin and glass ﬁber (see Table 12.3), the rate of heat transfer in the radial direction (i.e., from the wall to the center of the die) is very slow, and this is reﬂected in the large temperature difference between the region near the die wall and the center of the die. As the temperature is increased to the level at which the initiator begins to decompose, the cure reaction begins. The exothermic cure reaction causes the temperature to increase further as the cure reaction continues. This is the reason why the temperature at the center of the die (i.e., at r ∗ = 0) continues to increase even in the region where the die wall temperature is kept constant. As the die wall temperature decreases near the end of the die, the temperature at and near the center of the die becomes higher than the die wall temperature. This is attributable to the fact that the cure reaction still continues at and near the center of the die, generating heat, and the thermal conductivity of the material is so poor that the center of the die cannot be cooled fast enough. Figure 12.14 gives the speciﬁed wall temperature proﬁle along the die axis that was employed to simulate the pultrusion of unsaturated epoxy resin/glass ﬁber or epoxy resin/carbon composites. Figure 12.15 gives the simulation results, based on the empirical kinetic model, showing the proﬁles of (a) temperature and (b) degree of cure at a pulling speed of 0.43 m/min for epoxy/catalyst/glass ﬁber system with resin/glass ﬁber = 18/82 (by wt) and resin/curing agent = 100/3 (by wt). The following observations are worth noting in Figure 12.15. There is a signiﬁcant temperature distribution in the radial direction, with the wall temperature much higher than the temperature at the center. This is attributable to the fact that the die wall temperature is higher than the temperature of the feed stream, thus the exothermic reaction takes place near the

Figure 12.14 Schematic showing the wall temperature proﬁle along the axis of a cylindrical extrusion die that was employed to simulate the pultrusion of epoxy/glass ﬁber and epoxy/carbon ﬁber composites.

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Figure 12.15 Simulation results, based on the empirical kinetic model, Eq. (14.1) given in Chapter 14 of Volume 1, showing the proﬁles of (a) temperature and (b) degree of cure at a pulling speed of 0.43 m/min for epoxy/curing agent/glass ﬁber system with resin/glass ﬁber = 18/82 (by wt) and resin/curing agent = 100/3 (by wt). The wall temperature proﬁle along the die axis employed for the simulation is given by Figure 12.14 and the numerical values of the kinetic parameters employed for the simulation are given in Table 12.2.

die wall, which then generates heat and thus increases the temperature of the resin/ﬁber composite. Consequently, as can be seen in Figure 12.15b, α is higher near the wall than it is at the center of the die. As the cure completes (i.e., α approaching 1), no more heat is generated and thus the temperature rise will cease as the material travels along the die axis. Since the pultruded rod should not be hot when it exits the die, the composite material must be cooled in the latter part of the die (see the speciﬁed wall temperature proﬁle given by Figure 12.14). It is seen in Figure 12.15 that at the center of the die, cure begins at z∗ ≈ 80, α ≈ 0.7 at the end of the die (z∗ = 140), and the temperature increases rather slowly, whereas near the die wall, α approaches 1 at z∗ ≈ 60 and the temperature rises very rapidly. This is attributed to two factors. One factor is that the heat of reaction (HR ) for the epoxy is very high, twice as high as that for the unsaturated polyester (see Table 12.2), and thus the large amounts of heat

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generated due to the exothermic reaction have raised the temperature of the material and accelerated the cure reaction. The second factor is that the die wall temperature speciﬁed at the front end is much higher in the pultrusion of the epoxy/ﬁber composites compared with that in the pultrusion of the unsaturated polyester/ﬁber composites (compare Figure 12.14 with Figure 12.12). Figure 12.16 gives simulation results, based on the empirical kinetic model, showing the proﬁles of (a) temperature and (b) degree of cure at a pulling speed of 0.17 m/min for epoxy/curing agent/glass ﬁber system with resin/glass ﬁber = 18/82 (by wt) and resin/curing agent = 100/3 (by wt). The speciﬁed wall temperature proﬁle along the die axis employed for the numerical simulation is given in Figure 12.14. It is seen in Figure 12.16 that both temperature and conversion proﬁles are completely different

Figure 12.16 Simulation results, based on the empirical kinetic model, Eq. (14.1) given in Chapter 14 of Volume 1, showing the proﬁles of (a) temperature and (b) degree of cure at a pulling speed of 0.17 m/min for epoxy/curing agent/glass ﬁber system with resin/glass ﬁber = 18/82 (by wt) and resin/curing agent = 100/3 (by wt). The wall temperature proﬁle along the die axis employed for the simulation is given by Figure 12.14 and the numerical values of the kinetic parameters employed for the simulation are given in Table 12.2.

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from those in Figure 12.15, the difference between the two being due to the slower pulling speed in Figure 12.16. At a slow pulling speed, the residence time of the material inside the die increases, thus giving more time for the materials to reach thermal equilibrium. It is seen in Figure 12.16 that at a pulling speed of 0.17 m/min, cure was complete at about half way along the die axis and more uniform temperature distribution across the die was attained. However, a slow pulling speed is not desirable from a production point of view. Figure 12.17 gives simulation results, based on the empirical kinetic model, showing the proﬁles of (a) temperature and (b) degree of cure at a pulling speed of 0.43 m/min for epoxy/curing agent/carbon ﬁber system with resin/carbon ﬁber = 18/82 (by wt) and resin/curing agent = 100/3 (by wt). The speciﬁed wall temperature proﬁle along the

Figure 12.17 Simulation results, based on the empirical kinetic model, Eq. (14.1) given in Chapter 14 of Volume 1, showing the proﬁles of (a) temperature and (b) degree of cure at a pulling speed of 0.43 m/min for epoxy/curing agent/carbon ﬁber system with resin/carbon ﬁber = 18/82 (by wt) and resin/curing agent = 100/3 (by wt). The wall temperature proﬁle along the die axis employed for the simulation is given by Figure 12.14 and the numerical values of the kinetic parameters employed for the simulation are given in Table 12.2.

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die axis employed for the numerical simulation is given in Figure 12.14. Note that Figures 12.17 and 12.15 have the same pulling speed, the difference between the two being the type of ﬁbers used for reinforcement. A comparison of Figure 12.17 with Figure 12.15 reveals that the epoxy/curing agent/carbon ﬁber system undergoes much faster cure and more uniform temperature distribution in the radial direction compared with the epoxy/curing agent/glass ﬁber system. This is attributable to two factors. One factor is that the thermal conductivity of the carbon ﬁbers in the epoxy/curing agent/carbon ﬁber system is an order of magnitude greater than that of the glass ﬁbers in the epoxy/catalyst/glass ﬁber system (see Table 12.3). The other factor is that the rate of cure of epoxy/curing agent/carbon ﬁber system is faster than that of epoxy/curing agent/glass ﬁber system (compare Figure 12.11 with Figure 12.10). These observations indicate that the type of ﬁber employed also plays an important role in determining the performance of the pultrusion operation of thermoset/ﬁber composites. Figure 12.18 gives simulation results, based on the mechanistic kinetic model, showing the proﬁles of (a) temperature and (b) degree of cure at a pulling speed of 0.61 m/min for unsaturated polyester/P16N/BPO/glass ﬁber system with resin/glass

Figure 12.18 Simulation results, based on a mechanistic kinetic model, Eqs. (14.16)–(14.19) given in Chapter 14 of Volume 1, showing the proﬁles of (a) temperature and (b) degree of cure at a pulling speed of 0.61 m/min for unsaturated polyester/P16N/BPO/glass ﬁber system with resin/glass ﬁber = 17/83 (by wt) and resin/P16N/BPO = 100/0.3/0.7 (by wt). The wall temperature proﬁle along the die axis employed for the simulation is given by Figure 12.12 and the numerical values of the kinetic parameters employed for the simulation are given in Table 12.1.

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ﬁber = 17/83 (by wt) and resin/P16N/BPO = 100/0.3/0.7 (by wt). The speciﬁed wall temperature proﬁle along the die axis given in Figure 12.12 was employed for the simulation. The temperature and conversion proﬁles simulated from the mechanistic kinetic model (Figure 12.18) are very similar to those simulated from the empirical kinetic model (Figure 12.13). However, the use of the mechanistic kinetic model provides us with a further insight into the pultrusion process. Speciﬁcally, the proﬁles of initiator concentration inside a pultrusion die can be predicted using the mechanistic kinetic model, as given in Figure 12.19, which would not be possible when using an empirical kinetic model. Figure 12.20 describes the predicted proﬁles of dα/dt at the centerline (r ∗ = 0) along the cylindrical die during the pultrusion of an unsaturated polyester/initiator/glass ﬁber system with resin/glass ﬁber = 17/83 (by wt) and varying resin/initiator systems. It is seen in Figure 12.20 that the rate of cure of the P16N/BPO mixed initiator system lies between the rate of cure of a single initiator P16N and the rate of cure of a single initiator BPO. In obtaining Figures 12.19 and 12.20, the wall temperature proﬁle along the die axis given in Figure 12.12 and the kinetic parameters given Table 12.1 were employed. Thus, it is seen clearly that the use of a mixed initiator system has an advantage over the use of a single initiator in controlling the rate of cure and thus temperature and conversion proﬁles along the axis of a pultrusion die.

Figure 12.19 Simulation results, based on the mechanistic kinetic model, Eqs. (14.16)–(14.19) given in Chapter 14 of Volume 1, showing the proﬁles of initiator concentration [Ij ]/[Ij ]0 at the center (r ∗ = 0) of the die at a pulling speed of 0.61 m/min for unsaturated polyester/initiator/glass ﬁber system with resin/glass ﬁber = 17/83 (by wt) and varying resin/initiator combinations: curve (1) for [I1 ]/[I1 ]0 in resin/P16N = 100/2 (by wt) with [I1 ]0 = 0.0512 M/L for P16N, curve (2) for [I1 ]/[I1 ]0 in resin/BPO = 100/2 (by wt) with [I1 ]0 = 0.0734 M/L for BPO, and curve (3) for [I1 ]/[I1 ]0 in resin/P16N/BPO = 100/1/1 (by wt) with [I1 ]0 = 0.0256 M/L for P16N, and [I2 ]0 = 0.0367 M/L for BPO. The wall temperature proﬁle along the die axis employed for the simulation is given by Figure 12.12 and the kinetic parameters employed for the simulation are given in Table 12.1. (Reprinted from Han and Chin, Polymer Engineering and Science 28:321. Copyright © 1988, with permission from the Society of Plastics Engineers.)

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Figure 12.20 Simulation results, based on the mechanistic kinetic model, Eqs. (14.16)–(14.19) given in Chapter 14 of Volume 1, showing the proﬁles of dα/dt versus z∗ at the center (r ∗ = 0) of the die at a pulling speed of 0.61 m/min for unsaturated polyester/initiator/glass ﬁber system with resin/glass ﬁber = 17/83 (by wt) and varying resin/initiator combinations: curve (1) for resin/P16N = 100/2 (by wt) with [I1 ]0 = 0.0512 M/L for P16N, curve (2) for resin/BPO = 100/2 (by wt) with [I1 ]0 = 0.0734 M/L for BPO, and curve (3) for resin/P16N/BPO = 100/1/1 (by wt) with [I1 ]0 = 0.0256 M/L for P16N and [I2 ]0 = 0.0367 M/L for BPO. The wall temperature proﬁle along the die axis employed for the simulation is given by Figure 12.12 and the kinetic parameters employed for the simulation are given in Table 12.1. (Reprinted from Han and Chin, Polymer Engineering and Science 28:321. Copyright © 1988, with permission from the Society of Plastics Engineers.)

12.6

Summary

In this chapter, we have presented an analysis of heat transfer coupled with cure reaction in the pultrusion of thermoset/ﬁber composites. Emphasis was placed on demonstrating the importance of chemorheology of thermosets in the modeling of pultrusion of thermoset/ﬁber composites. In doing so, we have used the materials presented in Chapter 14 of Volume 1, in which various chemorheological models for thermosets are described. Pultrusion of thermosets can be regarded as being reactive polymer processing in that chemical reactions occur during processing. The pultrusion die can be viewed as a chemical reactor for producing composite materials; in other words, the pultruded stock that emerges from the die is a composite material with a polymeric matrix and ﬁbrous reinforcement. The major thrust of this chapter was to present a fundamental

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541

approach for modeling the pultrusion process for thermoset/ﬁber composites, rather than the details of pultrusion technology. In the analysis presented in this chapter, two important things were left out: one is the pulling force associated with pultrusion (Batch and Macosko 1993; Price and Cupschalk 1984) and the other is the ﬂow of matrix resin through aligned ﬁbers (Batch and Macosko 1993). The pulling force in a pultrusion die is affected by (1) the volume of the resin and ﬁber compacted into and pulled through the die, (2) the die temperature, and (3) the pulling speed. The ﬂow of matrix resin through aligned ﬁbers is important in the early stage of pultrusion (i.e., in the front end of the die), where the cure of resin has not progressed signiﬁcantly. The viscosity of the resin in the impregnating bath (see Figure 12.1) is relatively low and thus there can be signiﬁcant movement or rearrangement of resin through aligned ﬁbers as the resin-impregnated ﬁber bundles move through the front end of the die. At this point, since the degree of cure of resin is very low, one can treat the rearrangement of resin through aligned ﬁber by neglecting cure reactions inside the die. Under such circumstances, the rearrangement of resin through aligned ﬁbers has been dealt with using Darcy’s law, which is often used to describe ﬂuid ﬂow through a porous medium. This problem is analogous to that encountered in compression molding of thermoset/ﬁber composites, which will be presented in the next chapter. For the purpose of demonstrating a fundamental approach to modeling the pultrusion of thermoset/ﬁber composites, in this chapter we purposely have considered pultrusion in a cylindrical die. In many industrial applications, a variety of pultruded shapes other than the circular cross section are produced; examples of products produced in this way include ladders, bus-bar supports, motor-top sticks, cable-support trays, switch actuators, skate boards, paddle shafts, and golf-club shafts (Meyer 1985). The approach presented in this chapter can equally be applied to simulate pultrusion of such products using the ﬁnite element method, which can handle virtually any die geometry. The problem lies only in writing appropriate computer codes for a speciﬁc die geometry and numerically solving a system of equations using the ﬁnite element method.

Problems Problem 12.1

Consider that an unsaturated polyester/P16N/BPO/ﬁber composite is pultruded in each of the dies having the cross sections given in Figure 12.21. Using the physical properties given in Table 12.3 and the kinetic parameters for an empirical kinetic model given in Table 12.2, write ﬁnite element computer codes and solve system equations to predict the temperature proﬁles inside the die. Assume that the die is 1.60 m long and the wall temperature is maintained as given in Figure 12.12. Problem 12.2

Consider that an epoxy/catalyst/carbon ﬁber composite is pultruded in each of the dies given in Figure 12.21. Using the physical properties given in Table 12.3 and the kinetic parameters for an empirical kinetic model given in Table 12.2, write ﬁnite element computer codes and solve system equations to predict the temperature

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Figure 12.21 Irregular cross-sectional shapes of pultrusion dies that are to be simulated

using ﬁnite element method.

proﬁles inside the die. Assume that the die is 0.9 m long and the wall temperature is maintained as given in Figure 12.14. Problem 12.3

Consider that an unsaturated polyester/P16N/BPO ﬁber composite is pultruded in each of the dies given in Figure 12.21. Using the physical properties given in Table 12.3 and the kinetic parameters for a mechanistic kinetic model given in Table 12.1, write ﬁnite element computer codes and solve system equations to predict the temperature proﬁles inside the die. Assume that the die is 1.60 m long and the wall temperature is maintained as given in Figure 12.12.

References Aström BT, Pipes RB (1993). Polym. Compos. 14:173. Batch GL (1989). Crosslinking Free Radical Kinetics and the Pultrusion Processing of Composites, Doctoral Dissertation, University of Minnesota, Minneapolis, Minnesota. Batch GL, Macosko CW (1993). AIChE J. 39:1228. Chachad YR, Roux JA, Vaughan JG (1995). J. Reinf. Plast. Compos. 14:495. Ding Z, Li S, Lee LJ (2002). Polym. Compos. 23:957. Ding Z, Li S, Yang H, Lee LJ, Engelen H, Puckett M (2000). Polym. Compos. 21:762. Dubé MG, Batch GL, Vogel JH, Macosko CW (1995). Polym. Compos. 16:378. Gorthala R, Roux JA, Vaughan JG (1994a). J. Compos. Mater. 28:486. Gorthala R, Roux JA, Vaughan JG, Donti RP (1994b). J. Reinf. Plast. Compos. 13:288. Han CD, Chin HB (1988). Polym. Eng. Sci. 28:321. Han CD, Lee DS (1987). J. Appl. Polym. Sci. 34:793. Han CD, Lee DS, Chin HB (1986). Polym. Eng. Sci. 26:393.

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Kommu S, Khomami B, Kardos JL (1998). Polym. Compos. 19:335. Larock JA, Hahn HT, Evans DJ (1989). J. Thermopl. Comp. Mat. 2:216. Li S, Ding Z, Xu L, Lee LJ, Engelen H (2002). Polym. Compos. 23:947. Ma CC, Chen CH (1991). Polym. Eng. Sci. 31:1086. Ma CCM, Lee KY, Lee YD, Hwang JS (1986). SAMPE J. 22(5):42. Meyer RW (1985). Handbook of Pultrusion Technology, Chapman Hall, New York. Price HL (1979). Curing and Flow of Thermosetting Resins for Composite Material Pultrusion, Doctoral Dissertation, Old Dominion University, Norfolk, Virginia. Price HL, Cupschalk SG (1984). In Polymer Blends and Composites in Multiphase Systems, Han CD (ed)., Adv. Chem. Series no 206, American Chemical Society, Washington, D.C., p 302. Richard RV (1986). Pultrusion Modeling for Die Design and Process Control, Doctoral Dissertation, Tufts University, Medford, MA. Roux JA, Vaughan JG, Shanku R, Arafat ES, Bruce JL, Johnson VR (1998). J. Reinf. Plast. Compos. 17:1557. Ruan Y, Liu J (1994). J. Mater. Proc. Manuf. Sci. 3:91. Voorakaranam S, Joseph B, Kardos JL (1999). J. Compos. Mater. 33:1173.

13

Compression Molding of Thermoset/Fiber Composites

13.1

Introduction

Glass-ﬁber-reinforced thermoset composites have long been used by the plastics industry. Two primary reasons for using glass ﬁbers as reinforcement of thermosets are: (1) to improve the mechanical/physical properties (e.g., tensile modulus, dimensional stability, fatigue endurance, deformation under load, hardness, or abrasion resistance) of the thermosets, and (2) to reduce the cost of production by replacing expensive resins with inexpensive glass ﬁbers. In place of metals, the automotive industry uses glassﬁber-reinforced unsaturated polyester composites. One reason for this substitution is that the weight per unit volume of composite materials is quite low compared with that of metals. This has allowed for considerable reductions in the fuel consumption of automobiles. Another reason is that composite materials are less expensive than metals. The unsaturated polyester premix molding compounds in commercial use are supplied as sheet molding compound (SMC), bulk molding compound (BMC), or thick molding compound (TMC) (Bruins 1976; Parkyn et al. 1967). These molding compounds can be molded in standard compression or transfer molds. The basic challenge in molding unsaturated polyester premix compounds is to get a uniform layer of glass reinforcement in place in the die cavity while the resin ﬁlls the cavity and reaches its gel stage during cure. Temperature, mold closing speed, pressure, and cure time are all functions of the design of the part being produced. The ﬂow of the mixture through the gate(s) can result in variations in strength across the part due to ﬁber orientation during the ﬂow. The precise end-use properties depend on the ﬁber orientation, ﬁber distribution, and ﬁber content in the premix compounds, which are greatly inﬂuenced by the processing conditions. Since the mechanical properties of the molded articles depend strongly upon the orientation of the glass ﬁbers, it is important to understand how to control ﬁber orientation during molding.

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545

Unsaturated polyester accounts for the greater part of all thermosets used in glass-ﬁber-reinforced plastics. Glass-ﬁber-reinforced unsaturated polyesters offer the advantages of a balance of good mechanical, chemical, and electrical properties. Depending upon the application, a number of additives are employed to provide speciﬁc products or end-use properties. For instance, in the preparation of unsaturated polyester molding compounds, such as SMC and BMC, suitable for hot-press matched molding (e.g., compression and transfer moldings), the additives commonly used include inert ﬁllers, low-proﬁle thermoplastic additives, viscosity thickener, and mold release agents. Each additive in a molding compound has a speciﬁc purpose or purposes. Inorganic ﬁllers are used for improving the dimensional stability of the molded parts and serve as a heat sink to achieve better temperature control across a molded part during cure. They are also used to reduce the amount of resin required (hence the cost) and lessen the overall shrinkage. Low-proﬁle thermoplastic additives are used to achieve low shrinkage in molded parts during cure and provide a good surface appearance (see Chapter 14 of Volume 1). Viscosity thickeners are used to increase the viscosity (e.g., up to one million poises) of ﬁber-reinforced unsaturated polyester molding compounds suitable for hot-press matched molding operations. A typical formulation of an unsaturated polyester premix molding compound is: for 100 of resin, 300 phr (parts per hundred resin) of ﬁller, 5 phr of lubricant, 2 phr of catalyst, 75 phr of glass ﬁber. Unsaturated polyesters used in premix molding compounds can vary from rigid to ﬂexible, with viscosities ranging from 3 to 60 Pa·s. The choice of a rigid or ﬂexible resin would be a function of the part and the type of reactive monomer in the resin. For instance, diallyl phthalate-based polyester resins would exhibit signiﬁcantly higher viscosities than styrene-based polyester resins. The two ﬁllers most commonly used are calcium carbonate and clay. The catalyst most commonly used is organic peroxide (e.g., benzoyl peroxide or tert-butyl perbenzoate). Lubricant is used for the purpose of releasing the molded part from the mold, and zinc, calcium, and magnesium stearates are commonly used. The viscosity of the premix compound is a dominant factor, which not only affects the ﬁber orientation, but also controls the processing conditions. The viscosity of the unsaturated polyester molding compound inﬂuences the molding characteristics, and is determined by the level of chemical thickener in the premix and the time-temperature history of the compound. For instance, too low a viscosity can cause the resin to ﬂow ahead of the glass ﬁber and result in resin-rich areas with low mechanical properties. Conversely, too high a viscosity may not properly wet-out the glass ﬁbers. In determining the rheological properties of glass-ﬁber-reinforced thermoset resins, the choice of experimental technique is crucial for obtaining meaningful results. In other words, depending on the size of the glass ﬁbers in a molding compound, the geometry of the rheometer employed can inﬂuence the rheological data obtained, especially for a compound that has a high glass ﬁber content. It is worth pointing out that the conventional rotational rheometers (e.g., the cone-and-plate instrument) are unsuitable for characterizing SMC (or BMC or TMC) since the gap between the moving and stationary members of the instrument is of the same order of magnitude as the diameter of the glass ﬁbers. The conventional capillary viscometer (e.g., the plunger type) is also unsuitable because the converging ﬂow from the barrel into the capillary entrance tends to orient the glass ﬁbers along the ﬂow direction, altering the characteristics of the material. In Chapter 5 of Volume 1, we have described experimental procedures

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for determining the rheological properties of polymeric ﬂuids. Using a squeeze-ﬂow rheometer (in which the test material is squeezed between two parallel discs), some investigators (Lee et al. 1981; Silva-Nieto et al. 1981) have taken rheological measurements of glass-ﬁber-reinforced unsaturated polyester compounds, and reported that an empirical power law was obeyed. They reported that the length-to-diameter ratio (aspect ratio) of the glass ﬁbers signiﬁcantly affected the rheological behavior of the test materials. In the 1970s and 1980s, many studies were reported on squeeze ﬂow (basically, one-dimensional elongational ﬂow) of homogeneous Newtonian, non-Newtonian, or viscoelastic ﬂuids (Brindley et al. 1976; Co and Bird 1977; Covey and Stanmore 1981; Gartling and Phan-Thien 1984; Grimm 1978; Hamza and MacDonald 1981; Lee and Williams 1976; McClelland and Finlayson 1983; Mochimaru 1981; Phan-Thien and Tanner 1983; Shirodkar and Middleman 1982; Zahorski 1978). Although such studies are relevant to mold ﬁlling in compression molding, which also involves squeeze ﬂow, in this chapter we will not elaborate on those studies because an analysis of compression molding of ﬁber-reinforced unsaturated polyester must include cure reactions and the orientation of ﬁbers, which is absent in the squeeze ﬂow of a homogeneous liquid. In the 1980s and 1990s, many studies were reported on ﬁber orientation in compression molding (Advani and Tucker 1990; Barone and Caulk 1986; Barone and Osswald 1988; Castro and Grifﬁth 1989; Jackson et al. 1986; Kau and Hagerman 1987; Lee and Tucker 1987; Lee et al. 1981, 1984, 1991; Silva-Nieto et al. 1980; Tucker and Advani 1994). In those studies, only ﬂow with or without heat transfer was included during mold ﬁlling; cure reactions were not considered. A rigorous analysis of compression molding of ﬁber-reinforced thermoset composites must include combined heat transfer and cure reactions, in addition to the ﬁber orientation during mold ﬁlling. With the very limited space available in this volume, which deals with thirteen different subjects, it is not possible to present many very important aspects (e.g., the orientation of glass ﬁbers and mold design) of the compression molding of thermoset/ﬁber composites. The decision had to be made as to what aspects of compression molding of thermoset/ﬁber composites are to be presented in this chapter. In the processing of unsaturated polyester molding compounds (SMC, BMC, or TMC), it is very important to control their viscosities, commonly referred to as “viscosity thickening.” It is also very important to recognize the fact that pressure has a profound effect on the rate of cure of SMC (or BMC or TMC) during compression molding. These two subjects are fundamental to a better understanding of the performance of unsaturated polyester molding compounds during compression molding, and yet they have had relatively little discussion in the literature. Thus, in this chapter, we ﬁrst present the thickening behavior of unsaturated polyester and the effect of pressure on the cure of unsaturated polyester molding compounds. We then present an analysis of compression molding of thermoset/ﬁber composites into the mold cavity, a simple rectangular mold geometry chosen for illustration purposes, using a mechanistic kinetic model presented in Chapter 14 of Volume 1 together with the heat transfer equation, without including the orientation of glass ﬁbers during mold ﬁlling. As mentioned above, so much has been published on the ﬁber orientation during mold ﬁlling of ﬁber-reinforced thermoset composites that it is not necessary to include it in this chapter. Further, the mechanical design of ﬁber-reinforced thermoset composites is outside the scope of this chapter.

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13.2

Thickening Behavior of Unsaturated Polyester

To produce SMC, BMC, or TMC suitable for various matched molding operations, the viscosity of the materials is deliberately increased up to about 105 Pa·s. It has been suggested in the literature (Burns et al. 1975) that the thickening is a result of chain extension brought about by the condensation reaction between the group II metal oxides and the carboxylic acid groups present in the unsaturated polyesters. Among group II metal oxides and/or hydroxides used as thickeners, magnesium oxide (MgO) is a popular choice, because of its high reactivity and the low concentration required for thickening. The level of chemical thickener in the mixture and the time–temperature history of the compound determine the viscosity of unsaturated polyester molding compounds, and will inﬂuence the molding characteristics. The higher the viscosity at the molding temperature, the more processing tonnage will be required. At some point, complete ﬁll-out of the mold may become a problem and rejects will result. Conversely, too low a viscosity can cause the resin to ﬂow ahead of the glass ﬁbers and result in resin-rich areas with poor mechanical properties. Despite its technological importance, only a few studies (Alvey 1971; Han and Lem 1983; Walton 1976) have reported on the timedependent rheological behavior of unsaturated polyesters during thickening. In this section, we present the fundamentals of thickening behavior of unsaturated polyester and the mechanism(s) of viscosity thickening. Figure 13.1 gives plots of viscosity (η) versus shear rate (γ˙ ) at various periods during thickening of a general-purpose unsaturated polyester (Aropol 7030, Ashland Chemical) when a MgO paste dispersed in styrene monomer was added. The specimen was prepared by adding 1 mole of MgO for each mole of prepolymer, and a mixture of the resin and the viscosity thickener was mixed thoroughly at room temperature with a

Figure 13.1 Logarithmic plots of η versus γ˙ for a mixture of unsaturated polyester (Aropol 7030) and MgO at various thickening periods (h): () 1.3, () 5.3, () 11.1, () 25.9, (䊋) 31.4, (䊕) 34.6, (䊑) 38.8, ( ) 55.3, (䊉) 73.1, () 82.5, () 103, () 124.3, (䊎) 151.5, (䊖) 198.7, (䊒) 272.7, and ( ) 338.1. (Reprinted from Han and Lem, Journal of Applied Polymer Science 28:763. Copyright © 1983, with permission from John Wiley & Sons.)

䊖

䊕

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double axial-ﬂow impeller. The mixture was then transferred into a 500 mL wide-neck Wheaton glass jar, equipped with glass stopper sealed with silicone grease to prevent the evaporation of styrene monomer. The sealed bottle was placed in an oven, maintained at 30 ◦ C for the entire period of thickening, which lasted about three weeks. During thickening, the following measurements were conducted: (1) acid number by titration, (2) the molecular weight by gel permeation chromatography (GPC), and (3) viscoelastic properties using a cone-and-plate rheometer. It is seen in Figure 13.1 that the mixture of unsaturated polyester and thickening agent initially follows Newtonian behavior, and then begins to exhibit shear-thinning behavior as the viscosity increases beyond 100 Pa·s. It is interesting to observe that at about 70 h after thickening began, the viscosity of the solution is of the same order of magnitude as is often observed in the processing of many molten thermoplastic polymers. Figure 13.2 shows variations of zero-shear viscosity (η0 ) of the unsaturated polyester/MgO mixture with thickening period. It is seen that η0 is about 104 Pa·s at about 450 h after thickening began. It should be mentioned that after a viscosity thickener is added to an unsaturated polyester resin during preparation of SMC, BMC, or TMC, the compound is wrapped and stored for a certain period (e.g., 1–2 weeks) before use in compression molding. This means that the viscosity of an SMC is fairly high, though it depends on the duration of storage after the preparation of SMC and the level of thickener used. Figure 13.3 shows variation of the molecular weight of a mixture of Aropol 7030 and MgO during thickening up to 500 h. The number-average molecular weight (Mn ) was determined from the measurement of the acid number during thickening,

Figure 13.2 Plots of log η0 versus thickening period for a mixture of unsaturated polyester (Aropol 7030) and MgO. (Reprinted from Han and Lem, Journal of Applied Polymer Science 28:763. Copyright © 1983, with permission from John Wiley & Sons.)

COMPRESSION MOLDING OF THERMOSET/FIBER COMPOSITES

549

Figure 13.3 Variations of Mn () and Mw () with thickening period for a mixture of unsaturated polyester (Aropol 7030) and MgO. (Reprinted from Han and Lem, Journal of Applied Polymer Science 28:763. Copyright © 1983, with permission from John Wiley & Sons.)

and the weight-average molecular weight (Mw ) was determined from the expression Mw = (1 + P )Mn , with P being the extent of reaction as determined from the titration method. It is seen in Figure 13.3 that over the entire thickening period (about 20 days), Mn increases from 2,850 to 5,580 and Mw increases from 2,850 to 8,310. Table 13.1 gives a summary of Mn determined from the measurement of the acid number, Mw calculated from Mw = (1 + P )Mn , and molecular weight distribution (MWD) (Mw /Mn ) determined from GPC measurements of Aropol 7030 after different periods during thickening. It is seen in Table 13.1 that as thickening progresses, the molecular weight increases and the MWD broadens. Two different mechanisms for thickening behavior in unsaturated polyester have been proposed in the literature: a two-stage reaction (Alvey 1971) and the formation of

Table 13.1 Summary of the molecular weight measurement of a general purpose unsaturated polyester during thickening

Period of Thickening (h)

Mn a

Mw b

Mw /Mn c

0 20 200 480

2,070 2,190 2,570 2,580

5,000 6,660 7,250 8,120

2.4 3.0 2.8 3.3

a Determined from the measurement of the acid number. b Calculated from the expres-

sion Mw = (1 + P )Mn , with P being the extent of reaction as determined from the titration method. c Determined by GPC. Reprinted from Han and Lem, Journal of Applied Polymer Science 28:763. Copyright © 1983, with permission from John Wiley & Sons.

550

PROCESSING OF THERMOSETS

chain extension and chain entanglement (Burns et al. 1975). In the two-stage reaction theory, it is postulated that, ﬁrst, a higher molecular weight salt is formed by the reaction between MgO and carboxylic acid groups on an unsaturated polyester, and then a complex is formed between the salt and carboxylic acid groups of the ester linkages. The formation of a metal complex (MgO in this case) during the thickening reaction may occur at the reactive sites on the carbon atoms adjacent to the carboxylic acid groups. In this theory, the second stage of reaction is considered to be responsible for a large increase in viscosity (i.e., thickening behavior). In the chain extension/chain entanglement theory, it is postulated that the dicarboxyl in the acid groups on an unsaturated polyester chain react with MgO, yielding very high molecular weight species (via condensation polymerization), and thus giving rise to a large increase in viscosity. Burns et al. (1975) reported that a small proportion (0.5– 1.0%) of the unsaturated polyester resin employed contained high-molecular-weight species, with weights in the range of 105 –106 , as determined by GPC, and then postulated that this high-molecular-weight material in the system might have been primarily responsible for the viscosity thickening observed. In explaining the very large increase in viscosity observed, Burns et al. invoked the relationship η0 = KMw 3.4 and postulated that the large increase in viscosity might have arisen from chain entanglement in the unsaturated polyester. However, Han and Lem (1983) did not observe the presence of a high-molecularweight tail in the MWD curves. On the basis of the MWD determined from GPC measurements (see Table 13.1), they argued that the large increase in viscosity observed (see Figure 13.1) could not possibly have been explained simply by the increase in molecular weight. They attributed the extremely large increase in bulk viscosity, observed experimentally, to a strong ionic association in mixtures of unsaturated polyester and MgO paste with the carboxylic acid groups present in unsaturated polyester. In view of the fact that there are at least two carboxylic acid groups along each polymer molecule, a three-dimensional network structure consisting of ionic aggregates may be formed. The size of ionic aggregates is expected to increase as thickening progresses. When subjected to shearing deformation, the temporary three-dimensional network due to ionic association may give rise to high viscosity values. Figure 13.4 gives logarithmic plots of ﬁrst normal stress difference (N1 ) versus γ˙ at various periods during thickening for a mixture of unsaturated polyester and MgO. At about 40 h after thickening began, the mixture of unsaturated polyester and MgO started to exhibit N1 . It is of great interest to note in Figure 13.4 that the mixture of unsaturated polyester and MgO follows second-order ﬂuid behavior (see Chapter 3 of Volume 1) very closely over the entire period of thickening. This observation is of practical importance for modeling the ﬂow problem involved with processing of such mixtures (e.g., SMC or BMC). Figure 13.5 gives logarithmic plots of N1 versus shear stress (σ ) for a mixture of unsaturated polyester and MgO, showing a correlation that is independent of the duration of thickening (i.e., the extent of thickening). Considering the fact that thickening progresses with time, one would not expect to obtain such a correlation if indeed permanent chemical reactions take place, yielding macromolecules of greatly different molecular sizes and/or greatly different degrees of chain branching. Thus, the experimental correlation displayed in Figure 13.5 supports the thickening mechanism proposed above; namely, the thickening behavior involves ionic association

Figure 13.4 Plots of log N1 versus

log γ˙ for a mixture of unsaturated polyester and MgO at various thickening periods (h): (䊑) 38.8, ( ) 55.3, (䊉) 73.1, () 82.5, () 103, () 124.3, (䊎) 151.5, (䊖) 198.7, and (䊒) 272.7. (Reprinted from Han and Lem, Journal of Applied Polymer Science 28:763. Copyright © 1983, with permission from John Wiley & Sons.) 䊖

Figure 13.5 Plots of log N1 versus

log σ for a mixture of unsaturated polyester and MgO at various thickening periods (h): (䊉) 73.1, () 82.5, () 103, () 124.3, (䊎) 151.5, (䊖) 198.7, (䊒) 272.7, and ( ) 338.1. (Reprinted from Han and Lem, Journal of Applied Polymer Science 28:763. Copyright © 1983, with permission from John Wiley & Sons.) 䊕

551

552

PROCESSING OF THERMOSETS

(the formation of ionic aggregation, rather than chain extension/chain branching). In Chapter 6 of Volume 1, we have shown that plots of log N1 versus log σ depend on the degree of chain branching in low-density polyethylenes. In this regard, it is fair to state that the normal stress measurements in steady-state shear ﬂow provide additional important information on the thickening mechanism of unsaturated polyester.

13.3

Effect of Pressure on the Curing of Unsaturated Polyester

In the compression molding process, unsaturated polyester SMC or BMC is cured under elevated pressure, up to about 6.89 MPa (1,000 psi). It has been reported (Oh and Han 1985), as shown in Figure 13.6, that the tensile and impact strengths of unsaturated polyester SMC increase with the applied pressure during compression molding. It is therefore very important to understand how the pressure applied to SMC or BMC in compression molding operation inﬂuences its curing behavior. In this section, we ﬁrst present the effect of pressure on the cure of unsaturated polyester and then determine the kinetic parameters, as functions of applied pressure, of a mechanistic model presented in Chapter 14 of Volume 1. Later in this chapter, we use the mechanistic kinetic model to simulate the compression molding of unsaturated polyester SMC in a thin, disk-type mold cavity.

Figure 13.6 Plots of (a) tensile strength versus applied pressure and (b) impact strength versus

applied pressure during compression molding of an unsaturated polyester SMC at the mold temperature of 143 ◦ C and the cure time of 3 min. (Reprinted from Oh and Han, Polymer Composites 6:13. Copyright © 1985, with permission from the Society of Plastics Engineers.)

COMPRESSION MOLDING OF THERMOSET/FIBER COMPOSITES

553

Figure 13.7 Plots of rate of cure dα/dt versus cure time for a general purpose unsaturated polyester (OC-P340) with 1 wt % TBPB as initiator at 120 ◦ C under various cure pressures (MPa): () 0.10 (1 atm), () 2.07, () 3.45, () 4.83, and (7) 6.21. (Reprinted from Lee and Han, Polymer Composites 8:133. Copyright © 1987, with permission from the Society of Plastics Engineers.)

Kubota (1975) appears to be the ﬁrst to have reported that both the rate of cure and the degree of cure of unsaturated polyester resin increase with an increase in pressure. Lee and Han (1987) also investigated the effect of pressure on the curing behavior of unsaturated polyester resin. Figure 13.7 gives plots of the rate of cure (dα/dt) versus cure time, and Figure 13.8 gives plots of the degree of cure (α) versus cure time, for a general purpose unsaturated polyester (OC-P340, Owens-Corning Fiberglas) cured at 120 ◦ C with tert-butyl perbenzoate (TBPB) as initiator under various levels of pressure. In Chapter 14 of Volume 1 we have described the experimental procedures used to obtain Figures 13.7 and 13.8. It is seen in Figure 13.8 that the ﬁnal value of α ﬁrst increases and then decreases with increasing cure pressure, going through a maximum at a cure pressure of 3.45 MPa (500 psi). Figure 13.9 shows how dα/dt varies with cure time, and Figure 13.10 gives plots of α versus cure time, for the unsaturated polyester at different isothermal temperatures. It is of interest to observe in Figure 13.10 that α increased with cure temperature, irrespective of the cure pressure applied. The polymerization of styrene at high pressure has been studied extensively by a few research groups (Nicholson and Norrish 1956b; Walling 1960), who reported that pressure has a profound inﬂuence on the structure and properties of polymers. The effect of pressure on the rate of reaction may be explained by the transition state of theory of kinetics (Nicholson and Norrish 1956a) V ∗ ∂ ln k =− ∂P RT

(13.1)

Figure 13.8 Plots of degree of cure

α versus cure time for a general purpose unsaturated polyester (OC-P340) with 1 wt % TBPB as initiator at 120 ◦ C under various cure pressures (MPa): () 0.10 (1 atm), () 2.07, () 3.45, () 4.83, and (7) 6.21. (Reprinted from Lee and Han, Polymer Composites 8:133. Copyright © 1987, with permission from the Society of Plastics Engineers.)

Figure 13.9 Plots of rate of cure dα/dt versus cure time for a general purpose unsaturated polyester (OC-P340) with 1 wt % TBPB as initiator at three different isothermal cure temperatures ( ◦ C): (, 䊉) 110, (, ) 120, and (, ) 130. Open symbols are at ambient cure pressure and ﬁlled symbols are at the cure pressure of 3.45 MPa (500 psi). (Reprinted from Lee and Han, Polymer Composites 8:133. Copyright © 1987, with permission from the Society of Plastics Engineers.) 554

COMPRESSION MOLDING OF THERMOSET/FIBER COMPOSITES

555

Figure 13.10 Plots of degree of cure α versus cure time for a general purpose unsaturated polyester (OC-P340) with 1 wt % TBPB as initiator at three different isothermal cure temperatures ( ◦ C): (, 䊉) 110, (, ) 120, and (, ) 130. Open symbols are at atmospheric cure pressure and ﬁlled symbols are at the cure pressure 3.45 MPa (500 psi). (Reprinted from Lee and Han, Polymer Composites 8:133. Copyright © 1987, with permission from the Society of Plastics Engineers.)

where k is the rate constant, V ∗ is the volume change of the activated species, P is the pressure, R is the universal constant, and T is the absolute temperature. Note that V ∗ for the decomposition of an organic peroxide initiator is positive and its rate of decomposition is expected to decrease with an increase in pressure. The effect of pressure on the decomposition rate of benzoyl peroxide is reported by Nicholson and Norrish (1956a), who give the V ∗ for decomposition as 7 cm3 /g mol. Since V ∗ is negative for propagation reactions in the free-radical polymerization of styrene, the application of pressure increases the rate of propagation reaction. Nicholson and Norrish (1956b) determined the rate constant of propagation reactions at various pressures and obtained the value of V ∗ for the propagation step to be 13.3 cm3 /g mol. The rate constant of the termination reaction, kt , in the polymerization of styrene is reported to decrease with an increase in pressure (Evans and Polayni 1935). It has been pointed out that the termination reaction is diffusion controlled and kt varies inversely with the viscosity of monomer. It has been mentioned in the literature (Ferry 1980) that an increase in pressure decreases the free volume in a polymer. Therefore, it is reasonable to expect that, in the curing reaction of unsaturated polyester resin, an increase in pressure will decrease the mobility of the growing molecules and also the free volume of the reactants, and thus limit the degree of cure from the kinetics point of view. After the gel point is reached, the cure reactions become highly diffusion controlled and the mobility of the reactants in the very viscous medium will be greatly retarded. Conversely, from the thermodynamic point of view, the rate of the polymerization reaction is expected to increase with pressure. Since the curing reaction of the unsaturated polyester resin is accompanied

556

PROCESSING OF THERMOSETS

by a decrease in volume, and the changes of both enthalpy H and entropy S are negative, the ceiling temperature of the curing reaction Tc is expected to increase with pressure from the Claypeyron–Clausius equation: dTc V = Tc dP H

(13.2)

where P is pressure and V is the change of volume. Therefore, it can be concluded that an increase in cure pressure has two competing effects on the rate of the curing reaction of unsaturated polyester resin, namely, (1) the rate of the curing reaction will decrease with increasing pressure from the free volume point of view, and (2) the rate of the curing reaction will increase with increasing pressure from the thermodynamic point of view. Referring to Figure 13.7, the appearance of the dα/dt peak is delayed when the curing reactions take place at pressures higher than atmospheric pressure. This is due to the fact that the decomposition of TBPB is retarded under pressure. However, it is quite clear from Figure 13.9 that the temperature has a much greater inﬂuence on the rate of cure than does the pressure. Once the cure reaction begins under a given isothermal condition, as may be seen in Figure 13.11, the ultimate degree of cure αUT ﬁrst increases and then decreases as the cure pressure is increased from atmospheric pressure to 6.21 MPa (900 psi), going through a maximum at the cure pressure of 3.45 MPa (500 psi). This can be explained as follows. When the cure pressure is increased moderately from atmospheric pressure to 3.45 MPa, the thermodynamic effect becomes predominant over the free-volume effect. In other words, a cure pressure below about 3.45 MPa is apparently not high enough to restrict the mobility of growing molecules and thus decreases the free volume of the reacting system signiﬁcantly, to the extent that the free-volume effect becomes predominant over the thermodynamic effect that favors the curing reaction. However, as the cure pressure increased further

Figure 13.11 Plots of ultimate degree of cure αUT versus pressure for a general purpose unsaturated polyester (OC-P340) with 1 wt % TBPB as initiator at three different isothermal cure temperatures ( ◦ C): () 110, () 120, and () 130. (Reprinted from Lee and Han, Polymer Composites 8:133. Copyright © 1987, with permission from the Society of Plastics Engineers.)

COMPRESSION MOLDING OF THERMOSET/FIBER COMPOSITES

557

above about 3.45 MPa, the cure pressure is apparently high enough to decrease the free volume signiﬁcantly, giving rise to a lower ultimate degree of cure than that which can be achieved at pressures below about 3.45 MPa. The experimental results presented in Figures 13.7−13.11 can be analyzed using the mechanistic kinetic model, presented in Chapter 14 of Volume 1, which is based on the free-radical polymerization mechanism, and is given by d[Z]/dt = −kz [Z] [R·]

(13.3)

d[I ]/dt = −kd [I ]

(13.4)

d[M]/dt = −kp [M][R·]

(13.5)

d[R·]/dt = 2f kd [I ] − kz [Z] [R·]

(13.6)

in which [Z] is the inhibitor concentration, [I ] is the concentration of initiator, [M] is the total monomer concentration, [R · ] is the radical concentration, kz is the rate constant of the inhibition reaction, kd is the rate constant of the decomposition reaction of initiator, kp is the rate constant of the propagation reaction, and f is the efﬁciency of initiator. As discussed in Chapter 14 of Volume 1, the termination reaction can be neglected. Equations (13.3)–(13.6) may be solved numerically for [Z],

[I ], [M], and [R

·]

, I = [I ] , M = with the initial conditions, namely, Z = [Z] (0) (0) (0) 0 0

[M]0 , and R·(0) = 0. However, before Eqs. (13.3)–(13.6) are solved, one must specify the rate constants, kz , kd , and kp , each of which is dependent upon temperature. In Chapter 14 of Volume 1 we have suggested the following expressions: kd = kd0 exp(−Ed /RT )

(13.7)

for the decomposition rate constant of TBPB as initiator, kz = kz0 exp(−Ez /RT )

(13.8)

for the inhibition rate constant, and m kp = k˜p0 1 − α/αf

(13.9)

for the propagation rate constant, where k˜p0 = kp0 exp(−Ep /RT ) with kp0 being preexponential factor, Ep the activation energy for propagation reaction, R the universal gas constant, and T the absolute temperature, α = [M]0 − [M] /[M]0 is the degree of conversion at time t, αf = [M]0 − [M]∞ /[M]0 is the ﬁnal degree of conversion, with [M]0 being the initial concentration of monomer and [M]∞ being the monomer concentration after the cure, and m is a constant related to the diffusion-controlled propagation reaction. The initiator efﬁciency f can be expressed by ⎫ 1/2

2 ⎧ ⎬ 4(1 − f0 ) [I]/[I]0 f0 2 1 − (α/αf ) ⎨ − 1 (13.10) 1+ f =

2 ⎭ 2(1 − f0 ) [I]/[I]0 ⎩ f 2 − 1 − (α/α ) 0

f

558

PROCESSING OF THERMOSETS

where f0 is the initial value of the initiator efﬁciency and [I ]0 is the initial concentration of initiator. As was done in Chapter 14 of Volume 1, Eq. (13.5) will be replaced by the expression dα = kp (1 − α)[R·] dt

(13.11)

with the initial condition, α = 0 at t = 0. In order to use the mechanistic kinetic model for simulating the compression molding of ﬁber-reinforced thermoset composites, one needs information on the effects of pressure on the rate constants appearing in Eqs. (13.3)–(13.6). Lee and Han (1987) carried numerical computations using the following numerical values of the parameters: (1) [I ]0 = 5.71 × 10−2 M/L, kd0 = 8.524 × 1015 min−1 L/M, and Ed = 5.45 × 104 J/mol for the initiator TBPB, (2) [Z]0 = 5.09 × 10−4 M/L, kz0 = 3.024 × 1017 min−1 L/M, and Ez = 8.73 × 104 J/mol for the inhibitor. In determining the rate constant kp for the propagation reaction deﬁned by Eq. (13.11), with kp being given by Eq. (13.9), they solved numerically Eq. (13.3)–(13.6), with the aid of Eqs. (13.7)–(13.10), by assuming that the initiator efﬁciency f0 is not affected by pressure and f0 = 0.17 at 110 ◦ C and 120 ◦ C and f0 = 0.20 at 130 ◦ C. Table 13.2 gives a summary of the numerical values of k˜p0 , kp0 , Ep , and m determined. It should be mentioned that the values of kp0 , kt0 , m and n were determined from the computations

Table 13.2 Summary of the parameters associated with the propagation reaction in the mechanistic kinetic model describing the effect of pressure on the cure of a general purpose unsaturated polyester with TBPB as initiator

Pressure (MPa) 0.10

2.07

3.45

4.83

6.21

a

Temperature (◦ C)

k˜p0 (min−1 L/M)

kp0 (min−1 L/M)

m

Ep (kJ/mol)

110 120 130 110 120 130 110 120 130 110 120 130 110 120 130

2.15 × 103 2.82 × 103 5.47 × 103 3.90 × 102 5.83 × 102 9.88 × 102 2.54 × 102 6.04 × 102 9.95 × 102 3.72 × 102 5.92 × 102 8.64 × 102 3.46 × 102 5.06 × 102 6.06 × 102

2.99 × 1011 2.43 × 1011 3.00 × 1011 4.89 × 1010 4.54 × 1010 4.90 × 1010 8.10 × 1012 1.06 × 1013 9.70 × 1012 8.89 × 109 9.18 × 109 8.89 × 109 2.76 × 107 3.03 × 107 2.76 × 107

0.65 0.85 1.45 0.53 0.46 0.49 0.41 0.59 0.88 0.42 0.41 0.73 0.46 0.34 0.58

59.8

59.5

77.2

54.2

36.0

Values of k˜p0 were calculated from the expression k˜p0 = kp0 exp(−Ep /RT ).

Reprinted from Lee and Han, Polymer Composites 8:133. Copyright © 1987, with permission from the Society of Plastics Engineers.

COMPRESSION MOLDING OF THERMOSET/FIBER COMPOSITES

559

Figure 13.12 The dependence of propagation rate constant k˜p0 on cure pressure for a general purpose unsaturated polyester (OC-P340) at various isothermal cure temperatures ( ◦ C): () 110, () 120, and () 130. (Reprinted from Lee and Han, Polymer Composites 8:133. Copyright © 1987, with permission from the Society of Plastics Engineers.)

until the sum of the squares of the differences between the computed values of both dα/dt and α and the experimentally measured ones became a minimum throughout the entire period of cure. Figure 13.12 gives plots of propagation rate constant k˜p0 versus cure pressure at different isothermal cure temperatures. The following observations are worth noting in Figure 13.12 and Table 13.2: (1) at a given cure pressure, k˜p0 increases with cure temperature, indicating that the rate of cure increases with increasing temperature, and (2) at a given temperature, k˜p0 decreases rapidly as the cure pressure is increased from atmospheric pressure to 2.07 MPa (300 psi) and then decreases rather slowly. The decrease of k˜p0 with increasing cure pressure is attributable to the nature of diffusioncontrolled cure reactions of unsaturated polyester resin. Figure 13.13 gives plots of k˜p0 versus the reciprocal of absolute temperature 1/T . Note that Figures 13.12 and 13.13 were prepared with the information given in Table 13.2. The slope of the Arrhenius plots given in Figure 13.13 allows us to determine the activation energy Ep at the cure pressure P . Figure 13.14 displays the dependence of Ep on the cure pressure, showing that Ep goes through a maximum as the cure pressure increases from atmospheric pressure to 6.21 MPa (900 psi). Such seemingly anomalous behavior can be explained by the compressibility of the resin at different cure temperatures. In view of the fact that Ep in Figure 13.14 was obtained from the Arrhenius plot, let us examine the Arrhenius plots given in Figure 13.13. It is seen that the value of k˜p0 decreased as the cure pressure was increased. Also, while the value of k˜p0 at 130 ◦ C is much smaller for the cure pressure 6.21 MPa than for the lower cure pressures,

Figure 13.13 Plots of propagation rate constant k˜p0 versus the reciprocal of absolute temperature 1/T for a general purpose unsaturated polyester (OC-P340) at various cure pressures (MPa): () 0.10 (1 atm), () 2.07, () 3.45, () 4.83, and (7) 6.21. (Reprinted from Lee and Han, Polymer Composites 8:133. Copyright © 1987, with permission from the Society of Plastics Engineers.)

Figure 13.14 The dependence of

activation energy Ep on cure pressure for a general purpose unsaturated polyester (OC-P340). (Reprinted from Lee and Han, Polymer Composites 8:133. Copyright © 1987, with permission from the Society of Plastics Engineers.)

560

COMPRESSION MOLDING OF THERMOSET/FIBER COMPOSITES

561

the value of k˜p0 at 110 ◦ C is larger for the cure pressures of 4.83 and 6.21 MPa than for the cure pressure 3.45 MPa, thus giving rise to an Arrhenius plot of smaller slope. Therefore, we can conclude that the larger decrease in the value of k˜p0 at 130 ◦ C as the cure pressure is increased, compared with those at 110 and 120 ◦ C, is due to the greater compressibility of the resin at 130 ◦ C than at 110 and 120 ◦ C. One can then surmise that the compressibility of the resin will be increased as the cure temperature is increased, because the viscosity of resin will be decreased. The materials presented above should warn those who wish to predict the curing behavior of unsaturated polyester resins, or its molding compounds, at high pressures using information obtained at atmospheric pressure. We will elaborate on the importance of this observation when discussing the simulation results of the compression molding of SMC in the next section.

13.4

Analysis of Compression Molding of Unsaturated Polyester/Glass Fiber Composite

Compression molding of SMC involves squeeze ﬂow of SMC or BMC between two parallel plates preheated to a predetermined temperature, during which ﬁbers orient along the direction perpendicular to the squeezing direction, followed by cure reaction under applied pressure. Therefore, there are three aspects that must be considered in the modeling of the compression molding process of SMC or BMC. They are: (1) the orientation of ﬁbers, (2) the heat transfer between the plates and the charge, and (3) the cure reactions in the charge. Figure 13.15 gives a schematic diagram describing (a) the compression molding process and (b) thermal balance in a compression mold. A comprehensive presentation of the degree of ﬁber orientation during mold ﬁlling in compression molding requires the description of elaborate numerical analysis, which is outside the scope of this chapter. Some investigators (Barone and Caulk 1979; Fan and Lee 1986; Lee 1981; Twu et al. 1993) took a less sophisticated, practical approach to describe the mold ﬁlling of ﬁber-reinforced unsaturated polyester resin without referring to ﬁber orientation, and then compared the predictions with experiment. Since insigniﬁcant cure reactions may occur while an SMC sample is charged and the mold is closed, it seems reasonable to decouple mold charge and subsequent curing reactions. There are some studies (Barone and Caulk 1979; Lee 1981) reporting on the curing behavior of ﬁber-reinforced unsaturated polyester during compression molding, after the mold charge is completed. Speciﬁcally, Barone and Caulk (1979) employed an empirical autocatalytic kinetic model and Lee (1981) employed a mechanistic kinetic model to describe combined heat transfer and curing reaction of ﬁber-reinforced unsaturated polyester during compression molding after the mold charge was completed, using the kinetic parameters obtained at atmospheric pressure. In this section, we present an analysis of combined heat transfer and cure reaction of ﬁber-reinforced unsaturated polyester during compression molding after the mold charge is completed, using the kinetic parameters that were determined in the presence of applied pressure. The geometry to be considered is a rectangular mold cavity, as schematically shown in Figure 13.15. The primary purpose of the analysis presented here is to show a fundamental approach to the seemingly very complicated problem, and not to relate the results of the analysis to the mechanical properties of

562

PROCESSING OF THERMOSETS Figure 13.15 Schematic of (a) a compression mold to which an SMC sample is charged and the mold is still open and (b) the heat conduction from the mold wall to an SMC sample in the mold cavity and the convective heat transfer from the SMC sample in the transverse direction right after the mold is closed.

the cured composites. For the analysis, we employ the mechanistic kinetic model for cure reaction of unsaturated polyester presented in Chapter 14 of Volume 1, with the aid of the kinetic parameters given in Table 13.2, which were obtained in the presence of applied pressure. The importance of the effect of applied pressure on the curing reaction of unsaturated polyester in compression molding cannot be overemphasized, because pressure has a profound inﬂuence on the rate of cure of unsaturated polyester (see Figures 13.7, 13.8, and 13.13), and hence on the degree of cure of ﬁber-reinforced unsaturated polyester SMC. The energy balance equation in a rectangular mold cavity during curing of an SMC may be written by

ρcp

∂ ∂T ∂T = k + RA HR ∂t ∂y ∂y

(13.12)

where T is the temperature of the composites, ρ is the bulk density, cp is the bulk speciﬁc heat, k is the bulk thermal conductivity, RA is the rate of formation of cured resin (i.e., the rate of cure reaction), and HR is the heat of reaction due to cure. Note that the material being cured consists of three components, namely, resin, ﬁbers, and cured resin. Therefore, the bulk physical properties of the material must include all three components. It should be pointed out that it is very important to include the temperature dependence of ρ, cp , and k for the neat resin, glass ﬁbers, and cured resin in the modeling. In the analysis here, we use Eqs. (12.6)–(12.13) given in Chapter 12

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COMPRESSION MOLDING OF THERMOSET/FIBER COMPOSITES

to describe the temperature dependence of ρ, cp , and k for the neat resin, glass ﬁbers, and cured resin, and also the bulk physical properties of the composites. In order to solve Eqs. (13.11) and (13.12), a rate expression for cure kinetics must be speciﬁed, because these two equations are related to each other by the rate of cure reaction, RA , which in turn depends on temperature T . Introducing the dimensionless variables [Z ∗ ] = [Z]/[Z]0 , [I ∗ ] = [I ]/[I ]0 , [M ∗ ] = [M]/[M]0 , and [R·∗ ] = [R·]/[R·]m , with [R·]m being the maximum possible concentration of radicals represented by [R·]m = 2[I ]0 − [Z]0 , Eqs. (13.3)−(13.6) and Eq. (13.12) may be rewritten as ∂[Z ∗ ]/∂t ∗ = −A[Z ∗ ][R·∗ ]

(13.13)

∂[I ∗ ]/∂t ∗ = −B[I ∗ ]

(13.14)

∗

∗

∂[α]/∂t = −C(1 − α)[R· ] ∗

∗

∗

∗

(13.15) ∗

∂[R· ]/∂t = −D[I ] − E[Z ][R· ] ∂θ ∂ ∗ ∗ ∗ ∂θ ρ cp Pe ∗ = ∗ k + F (1 − α)[R·∗ ] ∂t ∂y ∂y ∗

(13.16) (13.17)

where A = kz [R·]m tc , B = kd tc , C = kp [R·]m tc , D = 2kd f [I ]0 tc /[R·]m , E = o co H 2 /k o t , F = k [M] [R·] H H 2 /k o T , ρ ∗ = ρ/ρ o , c∗ = kz [Z]0 tc , Pe = ρm p 0 m R pm m c m i m p o ∗ o , θ = T /T , y ∗ = y/H , with H being half of the part thickness, cp /cpm , k = k/km i and t ∗ = t/tc , with tc being the mold closure time. Equations (13.13)−(13.17) must be solved simultaneously subject to the following initial and boundary conditions: (i) at y∗ = 1 and t ∗ > 1, θ = Tw /Ti

(13.18a)

(ii) at y∗ = 0 and t ∗ > 1, ∂θ/∂y ∗ = 0 ∗

(13.18b)

∗

∗

∗

(iii) at t ≤ 1 and 0 ≤ y ≤ 1, α = 0, θ = 1, [Z ]0 = 1, [I ]0 = 1, and [R·∗ ] = 0 (13.18c) where y ∗ = 0 refers to the center and y ∗ = 1 refers to the wall of the rectangular mold cavity. Since there will be a heat transfer, upon and during the charge of an SMC specimen into the mold cavity, from the mold wall to the ﬁberglass-ﬁlled composite specimen before signiﬁcant curing reactions begin, we will consider the following expression describing the conductive heat transfer: ρ ∗ cp∗ Pe

∂θ ∂ ∗ ∂θ = ∗ k ∂t ∗ ∂y ∂y ∗

(13.19)

which must be solved subject to the following initial and boundary conditions: (i) at y∗ = 1 and t ∗ > 0, θ = Tw /Ti ∗

∗

∗

(13.20a)

(ii) at y = 0 and t > 0, ∂θ/∂y = 0

(13.20b)

(iii) at t ∗ = 0 and 0 ≤ y ∗ ≤ 1, θ = 1

(13.20c)

564

13.5

PROCESSING OF THERMOSETS

Time Evolution of Temperature during Compression Molding of Unsaturated Polyester/Glass Fiber Composite

In this section, we present some representative results simulating the curing and heat transfer during the compression molding of SMC using the system equations presented in the previous section. The purpose of this section is to present an approach that might be useful for gaining a better understanding of a seemingly complex processing operation. As stated in the previous section, a complete description of compression molding of a thermoset/ﬁber composite must address the ﬁber orientation during mold closure after a mold is charged with an SMC sample and then the curing and heat transfer. For convenience, here we assume that the ﬁber orientation during mold closing and the curing/heat transfer after the mold has closed may be decoupled. In Chapter 12 of Volume 1 we have shown that ﬁber orientation during shear ﬂow affects the bulk rheological properties of ﬁber-ﬁlled molten polymers. However, in the compression molding under consideration here, we are not concerned with the rheology of thermoset/ﬁber composite as affected by ﬁber orientation; rather, we are interested primarily in the curing of a thermoset/ﬁber composite after the mold is closed. In this regard, the approach of decoupling of ﬁber orientation during mold closure and curing after the mold has closed, which is adopted here, is justiﬁed for all intents and purposes. Needless to say, the extent of ﬁber orientation greatly affects the mechanical properties of the ﬁnal product of thermoset/ﬁber composites. However, any discussion of the mechanical properties of thermoset/ﬁber composites is beyond the scope of this chapter. For a rectangular mold cavity for which a system of equations has been developed in the previous section, one can solve numerically Eqs. (13.13)–(13.17) subject to Eq. (13.18). Figure 13.16 gives simulated results showing the time evolution of temperature for an SMC sample inside the mold at three different positions: (1) at y ∗ = 0 (center), (2) at y ∗ = 0.5, and (3) y ∗ = 0.9, where y ∗ = y/H , with H being half of the mold height, under the following processing conditions and mold geometry: ◦ (1) mold pressure of 4.83 MPa (700 psi), (2) mold wall temperature T w of 120 C, ◦ (3) feed temperature Ti of 25 C, (4) mold closure time tc of 6 s, (5) 50 wt % glass ﬁber, (6) 2 wt % TBPB as initiator, and (7) H = 0.635 cm. The numerical values of the kinetic parameters employed for the simulation are summarized in Table 13.3. In the numerical simulation, we assumed that propagation reaction begins 2 min after the mold is closed, because the temperature of the material must be increased, via heat conduction from the mold wall, to a certain critical value before a signiﬁcant propagation reaction can actually take place, which in turn depends on the type of initiator used (low-temperature initiator versus high-temperature initiator). In the numerical simulation, we have assumed that TBPB, which is regarded as being a high-temperature initiator (see Chapter 14 of Volume 1), is used. Therefore, referring to Figure 13.16, (1) the zero value of time denotes the instant at which the mold is closed, (2) the temperature increase during the ﬁrst 2 min after the mold is closed is obtained from the solution of Eq. (13.19) subject to Eq. (13.20), and (3) the sharp increase in temperature 2 min after the mold is closed is due to the propagation reaction, which is obtained from the solution of Eqs. (13.13)−(13.17) subject to Eq. (13.18). It is seen in Figure 13.16 that the temperature rise due to the propagation reaction is very large at the center (y ∗ = 0), while the temperature rise near the mold wall (y ∗ = 0.9) is very small. This is

COMPRESSION MOLDING OF THERMOSET/FIBER COMPOSITES

565

Figure 13.16 Simulation results showing the time evolution of temperature, after mold closure, during cure of an SMC sample in a rectangular compression mold (H = 0.635 cm) at three different positions: (1) y ∗ = 0 (center), (2) y ∗ = 0.5, and (3) y ∗ = 0.9. The molding conditions simulated are: mold pressure of 4.83 MPa (700 psi), mold wall temperature (Tw ) of 120 ◦ C, feed temperature (Ti ) of 25 ◦ C, mold closure time (tc ) of 6 s, 50 wt % glass ﬁber, and 2 wt % TBPB as initiator. The numerical values of the kinetic parameters employed for the simulation are given in Table 13.3.

attributable to slow heat dissipation owing to the poor thermal conductivity of the SMC being cured. The choice of 2 min for heat conduction is somewhat arbitrary, but it would not have any signiﬁcant consequence for the simulation results because a shorter period of heat conduction would shift the entire temperature proﬁle in Figure 13.16 towards the left side. It should be mentioned that Lee (1981) determined the time at which propagation reaction begins by calculating the induction time owing to the presence of an inhibitor

Table 13.3 Summary of the kinetic parameters employed in the numerical simulations of compression molding

kd0 = 8.524 × 1015 min−1 L/M for TBPB as initiator Ed = 5.45 × 104 J/mol for TBPB as initiator kz0 = 3.024 × 1017 min−1 L/M for inhibitor Ez = 8.73 × 104 J/mol for inhibitor [Z]0 = 5.09 × 10−4 M/L for inhibitor f0 = 0.17 for TBPB as initiator [M]0 = 1.449 kg/L for 70 wt % resin and 30 wt % ﬁber glass [M]0 = 1.242 kg/L for 60 wt % resin and 40 wt % ﬁber glass [M]0 = 1.035 kg/L for 50 wt % resin and 50 wt % ﬁber glass [I ]0 = 5.716 × 10−2 M/L for 1.0 wt % TBPB as initiator [I ]0 = 8.570 × 10−2 M/L for 1.5 wt % TBPB as initiator [I ]0 = 1.143 × 10−1 M/L for 2.0 wt % TBPB as initiator

566

PROCESSING OF THERMOSETS Figure 13.17 Simulation results showing the effect of initiator (TBPB) concentration (wt %), (1) 1.0, (2) 1.5, and (3) 2.0, on time evolution of temperature at y ∗ = 0 (center), after mold closure, during cure of an SMC sample in a rectangular compression mold (H = 0.635 cm). The compression molding conditions simulated are: mold pressure of 4.83 MPa (700 psi), mold wall temperature (Tw ) of 120 ◦ C, feed temperature (Ti ) of 25 ◦ C, mold closure time (tc ) of 6 s, and 50 wt % glass ﬁber. The numerical values of the kinetic parameters employed for the simulation are given in Table 13.3.

added to the SMC. In such an approach, the solution of the heat conduction equation, Eq. (13.19), is not necessary. However, in the use of commercial unsaturated polyester resins, not only is the amount of an inhibitor added to an unsaturated polyester not known accurately, but also the inhibitor efﬁciency is not known. Figure 13.17 gives simulation results showing the effect of initiator concentration (1.0, 1.5, and 2.0 wt %) on the time evolution of temperature at the center (y ∗ = 0) of the rectangular mold cavity under otherwise identical processing conditions to those used to obtain Figure 13.16. The numerical values of the kinetic parameters employed for the simulation are summarized in Table 13.3. It is seen that the extent of temperature rise due to the propagation reaction increases as the initiator concentration is increased from 1.0 to 2.0 wt %, greatly inﬂuencing the time evolution of temperature of an SMC inside the mold. Figure 13.18 gives simulation results showing the effect of glass ﬁber concentration (40 and 50 wt %) on the time evolution of temperature at the center (y ∗ = 0) of the mold under otherwise the same processing conditions as used to obtain Figure 13.16. The numerical values of the kinetic parameters employed for the simulation are summarized in Table 13.3. It is seen that the extent of temperature rise due to the propagation reaction decreases as the concentration of glass ﬁber in an SMC increases from 40 to 50 wt %. This is attributable to the fact that the glass ﬁber acts as a diluent to the exothermic reaction of unsaturated polyester resin in the SMC. Figure 13.19 gives simulation results showing the effect of mold height (or part thickness) (H = 0.318 and 0.635 cm) on the time evolution of temperature at the

Figure 13.18 Simulation results showing the effect of ﬁber glass concentration (wt %), (1) 40 and (2) 50, on time evolution of temperature at y ∗ = 0 (center), after mold closure, during cure of an SMC sample in a rectangular compression mold (H = 0.635 cm). The compression molding conditions simulated are: mold pressure of 4.83 MPa (700 psi), mold wall temperature (Tw ) of 120 ◦ C, feed temperature (Ti ) of 25 ◦ C, mold closure time (tc ) of 6 s, and 2 wt % TBPB as initiator. The numerical values of the kinetic parameters employed for the simulation are given in Table 13.3.

Figure 13.19 Simulation results describing the effect of mold height (part thickness) (cm), (1) 0.318 and (2) 0.635, on time evolution of temperature at y ∗ = 0 (center), after mold closure, during cure of an SMC sample in a rectangular compression mold. The compression molding conditions simulated are: mold pressure of 4.83 MPa (700 psi), mold wall temperature (Tw ) of 120 ◦ C, feed temperature (Ti ) of 25 ◦ C, mold closure time (tc ) of 6 s, 2 wt % TBPB as initiator, and 50 wt % glass ﬁber. The numerical values of the kinetic parameters employed for the simulation are given in Table 13.3.

567

568

PROCESSING OF THERMOSETS

center (y ∗ = 0) of the rectangular mold cavity under otherwise the same processing conditions as used to obtain Figure 13.16. The numerical values of the kinetic parameters employed for the simulation are summarized in Table 13.3. It is seen that the extent of temperature rise due to the propagation reaction increases as the part thickness increases from 0.318 to 0.635 cm. This is attributable to the fact that a thick SMC part in the mold cannot dissipate the heat, which was generated by the propagation reactions, fast enough through the mold wall.

13.6

Summary

In this chapter, we have presented some fundamental aspects of compression molding of thermoset/ﬁber composites. We have pointed out that in the preparation of SMC, BMC, or TMC, a viscosity thickener is added to increase its viscosity to the level that is needed for various matched molding operations. We have presented experimental results showing that the viscosity of unsaturated polyester increases tremendously during a thickening period, and then offered a thickening mechanism for unsaturated polyester resin. We have presented experimental results showing that an external pressure has a profound inﬂuence on the curing rate, hence the kinetics of curing reaction, of unsaturated polyester. This experimental observation is of great importance for gaining a better understanding of compression molding operation because pressure is invariably applied during compression molding. While we recognize the importance of ﬁber orientation during mold closing, in this chapter we have assumed that the ﬁber orientation during mold closing and the curing/heat transfer after the mold closure can be decoupled. This assumption seems reasonable because the extent of curing reaction during mold closing, owing to relatively low temperature, would be negligibly small. Under such a situation, ﬁber orientation during mold closing in compression molding can be treated in the same way as ﬁber orientation in squeeze ﬂow, which was discussed extensively in the literature in the 1980s and 1990s. Fiber orientation during mold closing was not discussed in this chapter because we already have discussed ﬁber orientation in ﬂow of ﬁberﬁlled molten thermoplastics in Chapter 12 of Volume 1. In this chapter, we have presented a system of equations describing the cure and heat transfer of ﬁber-ﬁlled thermosets after a mold is closed, and then some simulation results describing the effect of processing variables on the time evolution of temperature inside a rectangular mold cavity. There are numerous studies reported in the literature that dealt with the compression molding of thermoset/ﬁber composites, putting emphasis on the effect of processing variables on the mechanical properties of the products. It has been reported that the addition of glass ﬁber to a thermoset (e.g., unsaturated polyester) yields signiﬁcant improvement in physical/mechanical (and/or thermal) properties, and makes the molding of large lightweight parts possible. However, much has yet to be learned about (1) the preparation of thermoset molding compounds having consistent quality, (2) an efﬁcient way of processing such compounds, and (3) the quantitative determination of the relationship of chemical bonds to the strength of the glass ﬁber-reinforced thermoset composites, especially when reinforcement is realized through chemical bonding at

COMPRESSION MOLDING OF THERMOSET/FIBER COMPOSITES

569

interfaces (i.e., the coupling agent/ﬁber interface and the coupling agent/polymer matrix interface). Chemical bonding between the coupling agent and the polymer matrix is necessary to achieve high performance in glass-ﬁber-reinforced composites. However, the chemical composition of a coupling agent is a tightly guarded industrial secret, although silane and titanate coupling agents are widely used. Discussion of this subject is beyond the scope of this chapter.

References Advani SG, Tucker CL (1990). Polym. Compos. 11:164. Alvey FB (1971). J. Polym. Sci. A-1 9:2233. Barone MR, Caulk DA (1979). Int. J. Heat Mass Transfer 22:1021. Barone MR, Caulk DA (1986). J. Appl. Mech. 53:361. Barone MR, Osswald TA (1988). Polym. Compos. 9:158. Brindley G, Davis JM, Walters K (1976). J. Non-Newtonian Fluid Mech. 1:19. Bruins PF (ed) (1976). Unsaturated Polyester Technology, Gordon Breach, New York, 1976. Burns R, Gandhi KS, Hankin AG, Lynsky BM (1975). Plast. Polym. 43:228. Castro JM, Grifﬁth RM (1989). Polym. Eng. Sci. 29:632. Co A, Bird RB (1977). Appl. Sci. Res. 33:385. Covey GH, Stanmore BR (1981). J. Non-Newtonian Fluid Mech. 8:249. Evans MG, Polayni M (1935). Trans. Faraday Soc. 31:185. Fan JD, Lee LJ (1986). Polym. Compos. 7:250. Ferry JD (1980). Viscoelastic Properties of Polymers, 3rd ed, John Wiley & Sons, New York. Gartling DK, Phan-Thien N (1984). J. Non-Newtonian Fluid Mech. 14:147. Grimm RJ (1978). AIChE J. 24:427. Hamza EA, MacDonald DA (1981). J. Fluid Mech. 109:147. Han CD, Lem KW (1983). J. Appl. Polym. Sci. 28:763. Jackson WC, Advani SG, Tucker CL (1986). J. Compos. Mater. 20:539. Kau HT, Hagerman EM (1987). Polym. Compos. 8:176. Kubota H (1975). J. Appl. Polym. Sci. 19:2279. Lee LJ, Marker LF, Griﬁth RM (1981). Polym. Compos. 2:209. Lee DS, Han CD (1987). Polym. Compos. 8:133. Lee CC, Folgar F, Tucker CL (1984). Trans. ASME J. Eng. Ind. 106:114 Lee MCH, Williams MC (1976). J. Non-Newtonian Fluid Mech. 1:323 Lee CC, Tucker CL (1987). J. Non-Newtonian Fluid Mech. 24:245. Lee LJ (1981). Polym. Eng. Sci. 21:483. Lee LJ, Fan JD, Kim J, Im YT (1991). Intern. Polymer Processing 6:61. McClelland MA, Finlayson BS (1983). J. Non-Newtonian Fluid Mech. 13:181. Mochimaru Y (1981). J. Non-Newtonian Fluid Mech. 9:157. Nicholson AE, Norrish RGW (1956a). Disc. Faraday Soc. 22:97. Nicholson AE, Norrish RGW (1956b). Disc. Faraday Soc. 22:104. Oh SY, Han CD (1985). Polym. Compos. 6:13. Phan-Thien N, Tanner RI (1983). J. Fluid Mech. 129:265. Parkyn B, Lamb F, Clifton BV (1967). Polyesters. Vol. 2. Unsaturated Polyesters and Polyester Plasticisers, Elsevier, New York. Shirodkar P, Middleman S (1982). J. Rheol. 26:1. Silva-Nieto RJ, Fisher BC, Birley AW (1980). Polym. Compos. 1:14. Silva-Nieto RJ, Fisher BC, Birley AW (1981). Polym. Eng. Sci. 21:499.

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Tucker CL, Advani SG (1994). In Flow and Rheology in Polymer Composites Manufacturing, Advani SG (ed), Elsevier, Amsterdam, Netherlands, p 147. Twu JT, Hill RR, Wang TJ, Lee LJ (1993). Polym. Compos. 14:503. Walling C (1960). J. Polym. Sci. 48:335. Walton JP (1976). In Unsaturated Polyester Technology, Bruins PF (ed), Gordon Breach, New York, p 109. Zahorski S (1978). J. Non-Newtonian Fluid Mech. 4:217.

Author Index

Abushihada AM, 435 Adamse JWC, 23 Adedeji A, 187, 188 Adolf D, 388 Advani SG, 280, 546 Aggarwal SL, 307, 315 Akcasu AZ, 388, 390 Alfrey T, 379, 386 Altamirano JO, 210 Alvey FB, 547, 549 Amellal K, 107 Anastasiadis SH, 185, 194 Anturkar NR, 379 Aoyama H, 32 Arai T, 32 Arpin B, 40 Ashizawa H, 307, 315 Aström BT, 517 Atkinson EB, 425 Aurelio de Araujo M, 210 Auschra C, 210 Avrami M, 371 Bafna A, 307 Bagley EB, 32, 235 Bair TI, 57 Baird DG, 38, 40 Balch GS, 73, 82, 124, 125 Baldyga J, 496 Ballman RL, 268, 353, 366, 379 Bankar VG, 260

Barlow JW, 181, 185, 196, 197 Barone MR, 546, 561 Barr RA, 98, 100 Basu S, 236 Batch GL, 517, 541 Batchelor GK, 358 Becker R, 443 Becker WE, 495, 497, 513 Bellinger JC, 156 Berger JO, 363, 365 Beris AN, 268, 273, 278, 291 Bernhardt EC, 235 Bernstein RE, 210 Bezy M, 20 Binder K, 388 Bird RB, 273, 277, 279, 280, 301, 302, 546 Black BW, 58 Blander M, 443, 444, 446, 463 Blank RH, 371 Boger DV, 20 Böhm GGA, 176, 228 Bonner DC, 425 Bourne JR, 496 Braun JM, 435 Brindley G, 38, 546 Broadbent JM, 38 Brochard F, 388 Brockmeier NF, 425 Broseta D, 182 Brown FP, 424 Brown HR, 187 571

572

AUTHOR INDEX

Broyer E, 359, 497 Bruins PF, 544 Bruker I, 73, 82, 124, 125 Buckley RA, 264 Burns R, 547, 550 Burt JG, 424 Butler MF, 307 Cable PJ, 20 Cahn JW, 169, 177, 178 Cain JJ, 347 Campbell GA, 306, 323 Cao T, 306, 323 Carley JF, 235 Cartier H, 224 Castro JM, 497, 502, 512, 546 Catignani BF, 371 Caulk DA, 546, 561 Chachad YR, 517 Champagne MF, 224 Chan TW, 371 Chang FC, 224 Chang JC, 296 Chang PW, 32 Chang RY, 363, 365 Charles M, 20 Chen CH, 517 Chen L, 73, 125 Cheng YI, 425 Chiang CR, 224 Chiang HH, 363, 365 Chin HB, 176, 410, 517, 528 Chiou SY, 363, 365 Cho D, 185 Choi KJ, 307, 315 Chun SB, 186, 197, 205, 210, 212, 220 Clark ES, 307, 317, 366, 373 Clark RC, 424 Cleereman K, 366 Clegg PL, 307 Co A, 546 Cochrane T, 36 Cohen MH, 436 Collins FH, 424 Colombo EA, 424 Composto RJ, 388, 390 Coplan MJ, 296 Corbiere J, 262 Coswell B, 20 Covey GH, 546 Cox APD, 56, 73

Crochet MJ, 20, 32 Croft PW, 422 Cupschalk SG, 517, 541 Dai KH, 187 Daniels BK, 57 Darnell WH, 59 de Gennes PG, 388 Dealy J, 306 Dedecker K, 224, 228 Dee HB, 358 Dees J, 260 Delvaux V, 32 Denn MM, 258, 260, 275, 296, 347 Desper CR, 307, 315 Ding Z, 519 DiPaola-Baranyi G, 425 Domine JD, 497 Donald AM, 307 Donovan RC, 56, 61, 73 Döring W, 443 Doufas AK, 258, 268, 273, 275, 278, 279, 280, 281, 282, 291, 296, 302, 330, 347, 372 Dray RF, 98 Drexler LH, 20, 23 Dubé MG, 519 Duda JL, 435, 437, 438, 439, 442 Durelli AJ, 22 Durrill PL, 425, 435 Eckert RF, 38 Eder G, 372 Edmondson IR, 56, 61, 73 Edwards TJ, 435 Elbirli B, 16, 56, 62, 107 Elmendorp JJ, 176, 228 Ericksen JL, 4 Estevez SR, 512 Evans MG, 555 Everage AE, 268, 379 Falman AAM, 365 Fan JD, 561 Farber R, 306 Farkas L, 443 Favis BD, 132 Fayt R, 181, 185, 195, 196 Fehn GM, 424 Fenner RT, 56, 61, 73 Ferry JD, 118, 439, 555

AUTHOR INDEX

Finger FL, 32 Finlayson BS, 36, 546 Fischer RJ, 260, 275, 296 Fok SY, 263 Forney RC, 264 Fortenly I, 176, 228 Frazer AH, 257 Freeman HI, 296 Frenkel K, 443 Fried JR, 210 Frocht MM, 22, 23 Fujita H, 436 Fujiyama M, 373 Fukase H, 73 Funatsu K, 23 Fytas G, 388, 390 Gagon DK, 260, 275 Galin M, 425 Garber CA, 307, 317 Gartling DK, 546 George HH, 260, 275 Ghaneh-Fard A, 315 Giesekus H, 5, 273 Gilmore GD, 348, 353, 366 Gilmore PJ, 388 Girard-Reydet E, 224 Gogos CG, 363, 365, 497 Gorthala R, 517 Gou Z, 263 Grace H, 156 Graessley WW, 393, 402 Gray DG, 435 Green AE, 4 Green PF, 390 Gregory RB, 85 Grifﬁth RM, 546 Grimm RJ, 546 Griskey RG, 263, 425, 435 Groeninckx G, 224 Guichon O, 344 Guillet JE, 32, 425, 435 Guo X, 370, 373 Gupta AK, 181, 185, 196, 197 Gupta RK, 306, 323 Haberkorn H, 268, 273, 287, 291 Hagerman EM, 546 Halmos AJ, 56, 61, 65 Hamza EA, 546 Han CC, 205

573

Han CD, 5, 16, 20, 23, 32, 35, 38, 40, 56, 64, 88, 103, 107, 115, 132, 134, 169, 174, 176, 186, 194, 197, 205, 210, 212, 220, 235, 236, 260, 262, 265, 267, 268, 275, 296, 306, 314, 344, 345, 347, 358, 379, 382, 383, 386, 391, 393, 394, 398, 400, 402, 424, 425, 435, 446, 447, 454, 460, 463, 464, 465, 466, 467, 468, 475, 528, 547, 550, 552, 553 Han JH, 425, 446, 447, 454, 460, 463, 464, 465, 466, 467 Hansen RH, 424 Harnish DF, 427 Harry DH, 365 Hashimoto T, 173 Haw JS, 306, 323, 324, 331 Hawkes SJ, 435 He J, 132, 169 Heck B, 197 Hele-Shaw HJS, 358 Helfand H, 181, 194 Hendry AW, 22, 23 Heng FL, 236 Henson GM, 260, 275 Hess W, 390 Hétu JF, 363, 365 Hicks EM, 266, 379 Hieber CA, 363, 365, 367 Higashitani K, 38, 40 Ho RM, 132 Hoffman JD, 371 Holden BS, 443 Holmes R, 307, 315 Hong KM, 185 Hou TH, 36 Hu DS, 435, 437, 442 Hu GH, 224 Huck ND, 307, 322 Hyun KS, 117 Ilinca F, 363, 365 Ingen Housz JF, 98, 107 Inoue T, 173 Isayev AI, 363, 365, 367, 370–373 Ishizuka O, 268, 273 Jackson GB, 366 Jackson NR, 35 Jackson WC, 546 Janeschitz-Kriegl H, 23, 366, 372, 418

574

AUTHOR INDEX

Jeon HK, 224 Jo WH, 210 Jones RAL, 388 Jordan EA, 390 Ju ST, 435, 439 Jud K, 388 Kajiwara T, 20, 36 Kamal MR, 358, 363, 365, 366, 370, 373, 497 Kanai T, 347 Kanakkanatt SV, 424 Kanetakis J, 388, 390 Karam H, 156 Kase S, 260, 275, 296 Katayama K, 260, 274 Katti SS, 373 Katz JL, 443, 444, 446, 463 Kau HT, 546 Kawai H, 258, 268 Kaye A, 38 Kaylon DM, 307 Kedzierska K, 257 Kehr KW, 390 Keller A, 307 Kendall VG, 307 Kenig S, 358, 363 Keunings R, 32 Khan AA, 268, 379 Khomami B, 379 Kim DH, 497 Kim HT, 98, 100 Kim IH, 363, 365, 367 Kim JH, 497, 501 Kim JK, 224, 391, 393, 394, 398, 400, 402 Kim JR, 210 Kim JS, 268, 278 Kim KU, 32, 38, 40 Kim S, 224 Kim SC, 497 Kim SY, 268, 278 Kim YW, 132, 268, 388 Klein I, 16, 56, 61 Knau, DA, 424 Kolodziej P, 496 Kommu S, 519 Kondo A, 20 Kong JM, 435 Koyama K, 268, 273 Kramer EJ, 185, 388 Kruse WA, 210 Kubota H, 553

Kulkarni JA, 268, 273, 278, 291 Kuo Y, 363 Kurata M, 402 Kurtz SJ, 316 Kwack TH, 307, 314, 316, 344, 345 Kwei TK, 210 Kwolek SL, 257, 258, 264 Laﬂeur PG, 370, 373 Lamb H, 358 Lamonte RR, 260, 275, 296 Langlois WE, 4 Lawrence DL, 98 Lee BL, 268, 379 Lee CC, 546 Lee DS, 553 Lee JK, 132, 134, 169, 174 Lee KH, 497 Lee KY, 56, 64 Lee LJ, 496, 497, 546, 561, 565 Lee MCH, 546 Leibler L, 185 Lem KW, 547, 550 LeRoy G, 85 Li S, 519 Lindenmeyer PH, 307, 315 Lindt JT, 16, 56, 61 Litovitz TA, 442 Liu CC, 306, 347 Liu DD, 425 Liu J, 517 Liu TJ, 40 Lodge AS, 23, 38, 40 Logullo FM, 257 Lu J, 307, 315, 317 Lundberg JL, 425, 435, 440 Luo XL, 20, 306, 323 Lustig S, 307, 315 Ma CC, 517 Ma CCM, 517 Ma CY, 424, 468, 475 MacDonald DA, 546 Macedo PB, 442 Machin MJ, 307 MacLean DL, 268, 379 Maconnachie A, 210 Macosko CW, 132, 185, 187, 189, 228, 495, 497, 502, 512, 513, 517, 541 Maddams WF, 307, 315 Maddock BH, 60, 74, 85

AUTHOR INDEX

Maillefer C, 98 Majumdar B, 224 Maloney DP, 425, 426 Manas-Zloczower I, 512 Manzione LT, 497 Mark HF, 257 Matovic MA, 296 Matsubara Y, 40 Matsui M, 268, 273 Matsuo T, 260, 275, 296 Mavridis H, 363 McBriety VJ, 210 McClelland MA, 546 McHugh AJ, 263, 268, 296, 330, 371 McKelvey JM, 16, 235 McLuckie C, 32 McMaster LP, 173, 210 Mehta BS, 424 Meijer HEH, 98, 100 Meinecke EA, 424 Meister BJ, 324, 348 Mendelson RA, 32 Meyer RW, 517 Middleman S, 546 Miles IS, 132, 169 Miller RL, 371 Mills NJ, 402 Minoshima W, 260, 296, 347 Misovich MJ, 427 Mitsoulis E, 20, 236 Mochimaru Y, 546 Mol EAJ, 59 Montfort JP, 402 Morehead FF, 266, 379 Morgan PW, 257, 258 Mori Y, 23 Moy FH, 307, 373 Murschall U, 388, 390 Nagano Y, 263 Nagasawa T, 307, 315 Naito K, 210 Nakai A, 173 Nakajima N, 32 Nakamura K, 275, 371 Nedalla HP, 260 Nelson CJ, 132 Newitt DM, 435 Newman J, 435 Newman RD, 425 Nguyen H, 20

Nguyen LT, 496 Nicholson AE, 553, 555 Nishi T, 210 Noland S, 210 Noolandi J, 185 Norrish RGW, 553, 555 Novotny EJ, 38 O’Connor KM, 388 Oda K, 358, 367 Oh SY, 552 Ohzawa Y, 263 Osswald TA, 546 Ouhadi T, 181, 185, 210 Palmer RP, 307, 315 Pantani R, 370, 373 Papathanadiou TD, 363, 365, 370 Park HJ, 20 Park I, 181, 185, 197 Park JY, 265, 306, 322, 347 Park KY, 224 Parkyn B, 544 Parrott RG, 365 Patel RM, 274, 287 Patterson D, 425 Paul DR, 181, 185, 196, 210 Pearson JRA, 296, 309, 328 Peterlin A, 277, 300 Petrie CJS, 306, 309, 328, 541 Phan-Thien N, 546 Philippoff W, 23 Philipps RJ, 264 Pietransanta Y, 224 Pipkin AC, 38 Pittman JF, 40 Porteous W, 443 Prager S, 388 Prausnitz JM, 425, 426 Preedy JE, 307, 315 Preston J, 258 Price HL, 517 Prichard WG, 38, 39 Puissant R, 40 Purwar SN, 181, 185, 196, 197 Rao D, 235, 236, 245 Reddy KR, 32 Reid RC, 427 Reiner M, 5 Richard RV, 517

575

576

AUTHOR INDEX

Richardson S, 358 Riley WF, 22 Rivlin RS, 4, 5 Rogers MG, 32 Roland CM, 176, 228 Rose W, 358 Roux JA,517 Ruan Y, 517 Rupprecht MC, 425 Russell TP, 187, 188, 189, 190 Ryan ME, 497 Saeki S, 425 Samuels RJ, 307 Sander R, 40 Sato T, 358 Schmidt LR, 358 Schotte W, 425 Schrenk WJ, 384, 386, 379 Schultz J, 307 Schultz JM, 373 Schwarz MC, 181, 185, 196 Scott CE, 132 Segal L, 262 Sekiguchi M, 32 Semjonow VV, 5 Senich GA, 435 Senn RK, 424 Setz S, 196, 197 Shapiro J, 56, 61, 65 Sheneﬁel DG, 424 Sherman ES, 307 Shetty R, 347, 379, 386 Shida M, 32 Shih CK, 132 Shimizu I, 268, 273, 291, 300 Shimomura Y, 307, 315 Shirodkar P, 546 Shull KR, 185, 188 Siclari F, 261 Siggia ED, 173, 174 Sillescu H, 388 Silva-Nieto RJ, 546 Sisson WE, 266, 379 Sleeman MJ, 358, 359 Snyder HL, 173 Southern JH, 268, 379 Spalding MA, 117 Spencer RS, 353, 358, 366 Spevacek JA, 371 Spruiell JE, 260, 268, 273, 278, 287

Srinivasan KR, 181, 196, 197 Stanmore BR, 546 Stein DJ, 210 Stern SA, 425 Stiel LI, 427, 432 Storey SH, 235 Street JR, 454 Street LF, 85 Strobl GR, 173 Su YY, 379 Suh NP, 496 Sundararaj U, 132, 228 Tadmor Z, 16, 56, 61 Tagami Y, 181 Tait PJT, 435 Takenaka M, 173 Takeno H, 173 Tan V, 366 Tanner RI, 32, 38, 306, 323, 546 Teyssie Ph, 181 Therrien D, 132 Thudium RN, 205 Ting, RY, 454 Tirrell M, 388 Toor RL, 366 Torza S, 156 Traugott TD, 181, 185, 197 Triacca VJ, 224 Tselios Ch, 224 Tseng HS, 425 Tucker CL, 280, 496, 546 Turnbull D, 436 Twu JT, 561 van der Vegt AK, 176, 228 van Oene H, 132 Vassilatos G, 268, 273, 287 Vergnes B, 40 Vickers ME, 307, 315 Vilgis TA, 185 Villamizar C, 358, 425 Viriyayuthakorn M, 20 Volmer M, 443 Voorakaranam S, 519 Vrentas JS, 435, 437, 438, 442 Wacehter CJ, 424 Wagner MH, 306, 322–324, 348 Wakino T, 373 Wales JLS, 23, 366

AUTHOR INDEX

Walling C, 553 Walton JP, 547 Wang TT, 210 Wang Y, 40 Wasserman ZR, 194 Weale EW, 435 Weber A, 443 Weeks NE, 210 Wen SH, 40 West GH, 358, 359 Wheeler JA, 4, 5 Wheeler, NC, 98, 100 White JL, 20, 260, 268, 347, 358, 379 Wildes GS, 224 Williams MC, 546 Wilson GM, 379 Wissler EH, 4, 5 Wool RP, 388 Wu S, 388, 391, 392, 406

Yang HH, 425 Yang WJ, 454 Yeh HC, 454 Yeow YL, 347 Yin JH, 224 Yoo HJ, 35, 425, 454, 465 Yoon MG, 260, 274 Yu TC, 132

Zahorski S, 546 Zhang XM, 224 Zhu F, 73, 125 Ziabicki A, 257–262, 266, 268, 275, 291 Zieminski KF, 268, 273, 278, 287 Zivny A, 176, 228 Zurek A, 132, 169

577

Subject Index

adhesive bond strength, 402, 406 apparent elongational viscosity, 272, 289, 291, 293, 296 apparent intrinsic diffusion coefﬁcient, 402 apparent modulus of solid bed, 64 attractive segmental interaction, 183, 200, 220 average strain of the solid bed, 65 Avrami equation, 275, 372 barrier ﬂight, 100 barrier-screw extruder, 98, 103 biaxial elongational ﬂow, 305 biaxial elongational viscosity, 310 biaxially oriented ﬁlm, 305 block copolymer, 185 blowing agent, 424 blow-up ratio, 312 Brabender internal mixer, 135, 169 bubble growth, 425 bubble growth rate, 451 bubble nucleation, 425, 443, 457, 470 bubble stability in tubular ﬁlm blowing, 347 bulk molding compound, 544 Cahn theory, 169 cell morphology, 467, 471 chemical blowing agent, 424 chemorheological model, 503, 540 chemorheology, 497, 540 circulating ﬂow patterns, 20, 35 classical nucleation theory, 462

coalescence of drops, 176 coat-hanger die, 40 co-continuous morphology, 132, 169, 193 coextruded sheet, 380 coextrusion, 379 Coleman–Noll second-order ﬂuid, 29, 32 compatibilization, 182 compatibilizing agent, 182, 211 compression molding, 544, 558 conjugate-ﬁber spinning, 266 critical bubble radius, 453, 462 critical ﬂow temperature, 115 cure reaction, 495, 500 degree of cure, 500 diffusion-controlled reaction, 557 disordered state, 223 dispersed morphology, 132, 169, 193 draw-down ratio, 241, 341 dry-spinning process, 262 emulsiﬁcation, 182 emulsifying agent, 182, 184 entrance ﬂow, 3, 20 exit ﬂow, 3 exit pressure, 6 extrudate swell, 297, 470, 477 feedblock die system, 382 ﬁber orientation, 544, 546, 564 ﬁber spinnability, 257

578

SUBJECT INDEX

ﬁber-reinforced thermoset composite, 448 ﬁlm blowability, 341 ﬁrst normal stress difference, 23, 38, 550 Flory–Huggins theory, 463 ﬂow birefringence, 22 ﬂow-induced bubble nucleation, 454 ﬂow-induced crystallization, 269, 371 ﬂuted mixing device, 85 foam extrusion, 424 fountain ﬂow, 358 free-volume theory, 437 Giesekus model, 276 heterogeneous nucleation, 471 high-speed melt spinning, 257, 268 high-temperature initiator, 564 hole pressure error, 38 immiscible polymer blends, 132 injection molding, 351 interdiffusion coefﬁcient, 389 interface deformation, 379 interfacial instability, 379 intrinsic diffusion coefﬁcient, 395, 402 jetting phenomenon, 359 Kuhn length, 181 Langevin function, 277 Lee-Han melting model, 65, 115 low-proﬁle thermoset additive, 545 low-temperature initiator, 564 Maddock melting mechanism, 60 Maddock mixing head, 85 mechanistic kinetic model, 531, 538, 557 microphase-separated state, 223 mold ﬁlling, 358, 500 morphology evolution, 132, 154 multimanifold die system, 383 neckline deformation, 269, 287, 291 nonisothermal coextrusion, 407 nucleating agent, 471 nucleation phenomenon, 371 order-disorder transition temperature, 186

579

phase inversion, 156, 161, 175 physical blowing agent, 424 polymer-polymer interdiffusion, 381, 388 power-law model, 9 pressure-tap hole, 38 pultrusion, 517 quiescent crystallization, 370 reaction injection molding, 495 reactive blending, 224 reactive compatibilization, 224 reactive compatibilizing agent, 224 repulsive segment interaction, 196 residual stress, 369 secondary ﬂow, 4 shaped-ﬁber spinning, 265 sheet foam extrusion, 482 sheet molding compound, 544 single-screw extrusion, 56 solid-bed breakup, 76, 105 solid-bed velocity, 73, 112 spinnability, 294 spinodal decomposition, 169, 173 squeeze ﬂow, 546 strain hardening, 341 strain softening, 342 stress optical law, 23 stress-induced crystallization, 273, 281, 288 take-up ratio, 312 thermoset/ﬁber composite, 517, 544 thick molding compound, 544 thickening reaction, 550 tracer diffusion coefﬁcient, 389 tubular ﬁlm blowability, 341 tubular ﬁlm blowing, 305 twin-screw extruder, 154 uniaxial elongational ﬂow, 305 viscosity thickener, 545 Vrentas–Duda free-volume relationship, 440 wall normal stress, 248, 454, 459 wet-spinning process, 260 Williams–Landel–Ferry (WLF) equation, 116 wire-coating extrusion, 235