Rheology and Processing of Polymeric Materials, Vol. 2 Polymer Processing

Rheology and Processing of Polymeric Materials Volume 2

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RHEOLOGY AND PROCESSING OF POLYMERIC MATERIALS Volume 2 Polymer Processing

Chang Dae HanDepartment of Polymer Engineering The University of Akron

2007

Oxford University Press, Inc., publishes works that further Oxford Universitys 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

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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. PolymersRheology. 1. Title. QC189.5.H36 2006 620.1 920423dc22

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


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 uids, while many industrially important polymeric uids 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 uids. On the other hand, the currently held molecular theory deals almost exclusively with homogenous polymeric uids, while there are many industrially important polymeric uids (e.g., block copolymers, liquid-crystalline polymers, and thermoplastic polyurethanes) that are composed of more than one component exhibiting complex morphologies during ow. 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 ow of homogeneous thermoplastic polymers in a relatively simple ow 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 lled 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 uids. In so doing, I rst present the fundamental principles of the rheology of polymeric uids: (1) the kinematics and stresses of deformable bodies, (2) the continuum theory for the viscoelasticity of exible homogeneous polymeric liquids, (3) the molecular theory for the viscoelasticity of exible 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 exible 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. Specically, the various polymeric materials considered in Volume 1 range from rheologically simple, exible thermoplastic homopolymers to rheologically complex polymeric materials including (1) block copolymers, (2) liquid-crystalline polymers, (3) thermoplastic polyurethanes, (4) immiscible polymer blends, (5) particulate-lled polymers, organoclay nanocomposites, and ber-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. Specically, in Volume 2, I present the fundamental aspects of several processing operations (plasticating single-screw extrusion, wire-coating extrusion, ber spinning, tabular lm 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


Contents

Remarks on Volume 2, xvii

Part I Processing of Thermoplastic Polymers1 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

xiii

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

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 ProcessingProperty 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

xv

9.2.3 9.2.4 9.3

9.4 9.5

Feedblock Die System for Blown-Film Coextrusion, 384 Rotating-Mandrel Die System for Blown-Film Coextrusion, 386 PolymerPolymer Interdiffusion across the Initially Sharp and Flat Interface, 388 9.3.1 PolymerPolymer Interdiffusion under Static Conditions, 389 9.3.2 PolymerPolymer 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 ProcessingPropertyMorphology Relationships in Prole Foam Extrusion, 469 10.4.2 ProcessingProperty Relationships in Sheet Foam Extrusion, 482 10.5 Summary, 487 Problems, 488 Notes, 489 References, 489

Part II Processing of Thermosets11 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 Proles 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 Proles 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 specic 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 ow, 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. Specically, Chapter 1 presents the ow of polymeric liquids in complex geometries. In this chapter, we consider the ow of a viscoelastic uid through a rectangular channel, through a converging channel (entrance ow), 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 ber spinning, with a detailed discussion of high-speed melt spinning as reported in the 1980s and 1990s. Chapter 7 presents tubular lm blowing, in which an analysis of tubular lm 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-lling process must include the analysis of the ow of a viscoelastic polymer melt through the runner as an integral part of the analysis of the mold lling process, because the inlet conditions for the equations of motion and energy for mold lling 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 polymerpolymer 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/ber composites, and Chapter 13 presents compression molding of thermoset/ber 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


1

Flow of Polymeric Liquid in Complex Geometry

1.1

Introduction

The ow 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 ow geometries. For instance, the ber industry produces shaped bers, which have cross sections that are noncircular. What is most intriguing in the production of shaped bers is that a desired ber shape is often produced by spinneret holes whose cross-sectional shape is quite different from that of the nal ber 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 ber 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 ow is referred to as entrance ow. The entrance ow of polymeric liquids, due to their viscoelastic nature, is quite different from that of Newtonian liquids. Similarly, the ow behavior of viscoelastic polymeric liquids near the exit of a die, commonly referred to as exit ow, is quite different from that of Newtonian liquids. A better understanding of the unique characteristics of both entrance and exit ows of viscoelastic polymeric uids is essential for successful design of extrusion dies and molds, as well as to solve difcult technical problems related to a particular processing operation. Before presenting specic polymer processing operations in following chapters, in this chapter we consider the ow of polymeric liquids through complex geometry: (1) fully developed ow through a rectangular channel with uniform channel depth; (2) fully developed ow through a rectangular channel with a moving channel wall; (3) ow through a rectangular channel with varying channel depth; (4) ow in the entrance region of a rectangular die having constant cross section; (5) ow through3

4

PROCESSING OF THERMOPLASTIC POLYMERS

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

1.2

Flow through a Rectangular Channel

The ow of polymeric liquids through a rectangular channel having constant cross section or varying cross section is much more complex than the ow through a capillary or slit die considered in Chapter 5 of Volume 1. The complexity arises from both the viscoelastic nature of polymeric uids and the two-dimensional nature of a rectangular channel. In this section, we present some unique features of ow 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) ow patterns in each of the four quadrants of the rectangle, as schematically shown in Figure 1.1, when a viscoelastic uid ows through a rectangular channel. For instance, using the RivlinEricksen constitutive equation1 (Rivlin and Ericksen 1964), Langlois and Rivlin (1963) found that it required a fourth-order uid to yield secondary ow, with the second-order uid only affecting the pressure eld and the third-order uid only distorting the normal Newtonian velocity prole. To obtain a streamline pattern of secondary ow in a rectangular channel, one must solve all three components of the equations of motion, whereas in the absence of secondary ow, only the axial component of the equations of motion must

Figure 1.1 Schematic showing secondary ow of viscoelastic uids 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 uids complicates the solution of the equations of motion when secondary ow is to be considered. Wheeler and Wissler (1966) solved the equations of motion numerically for ow through a square duct by considering the ReinerRivlin constitutive equation (Reiner 1945; Rivlin 1948), and obtained the streamlines for the secondary ow. 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 proles computed for a Reynolds number of 26.38 are virtually indistinguishable from those computed when secondary ow was neglected. It should be mentioned that typical values of the Reynolds number in polymer melt ow, owing to very high viscosities, lie below 0.001. The experimental evidence is mixed. Some investigators (Giesekus 1965; Semjonow 1965) report that secondary ows have been observed, and others (Han 1976; Wheeler and Wissler 1965) report that they have not seen evidence of secondary ows. Figure 1.2 gives a micrograph of the cross section of an extrudate, which was obtained in the ow of a blend of polypropylene (PP) and polystyrene (PS) through a rectangular channel having an aspect ratio of 2. In this gure, 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 ow 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 ow is not a general phenomenon that would occur in the ow of every viscoelastic uid in a rectangular channel. The apparent absence of secondary ow 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 ow 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 ow 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 gure that the ow patterns are only rectilinear at the region that corresponds to an aspect ratio of about 4, and different ow patterns set in at a region near the edge of the long side of the cross section. At present, the physical origin of the ow patterns given in Figure 1.3 is not clear.

Figure 1.2 Micrograph of the extrudate cross section showing the ow 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


Figure 1.3 Micrograph of the extrudate cross section showing the ow 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 ow 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 proles of wall normal stress for an HDPE at 180 C owing 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 ow 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 ow rate, consistent with the experimental observation that the extrudate swells more as the ow rate is increased. It can be concluded, therefore, that the nonuniform distribution of extrudate swell given in Figures 1.21.4 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 uid 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 ow 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 Proles of wall normal stress of an HDPE melt at 180 C

along the centerlines of the long side (open symbols) and short side (lled symbols) of the rectangular channel (aspect ratio of 6) at various volumetric ow rates (cm3 /min): ( , ) 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 ow 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

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

9

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 ow exists. We then have the following system of equations. 1. Momentum balance equations: p + z x p + x y + y vx y vz y =0 =0 (1.1) (1.2)

vz x

2. Energy balance equation: T =k z 2T 2T + 2 x y 2 + vx y2

cp vz

+

vz x

2

+

vz y

2

(1.3)

in which is the viscosity, which depends on both temperature and the rate of deformation, is the density, cp is the specic heat, and k is the thermal conductivity. Let us assume that follows a power-law model, which is then expressed by2 vx y2

= mo exp(bT )

+

vz x

2

+

vz y

2 (n1)/2

(1.4)

10


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 ow in a rectangular channel with constant cross section, where crosschannel ow is assumed to be negligible (i.e., vx = 0), we only have to solve Eq. (1.2) with the following expression for : = ko vz x2

+

vz y

2 (n1)/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 ow under consideration, the velocity eld 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 proles vz (x, y) for fully developed isothermal ow of a lowdensity polyethylene (LDPE) melt at 200 C in a rectangular channel with W/H = 2. These gures 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 Pas0.5 . The contours of constant velocity and constant velocity gradient, respectively, for fully developed isothermal ow of the same power-law uid 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 ow through such a channel can be treated as slit ow. In Chapter 5 of Volume 1 we discussed slit ow as a means of measuring the viscometric ow properties of polymeric uids. 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 sufciently large, such that the isovels in the die cross section would not have curvature at the location where the pressure transducer is ush-mounted. A close examination of the contours of the isovels in the ow 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 ow through a rectangular channel with the upper plate moving at a constant velocity Vb in the direction that forms an angle with the channel axis z, as shown schematically in Figure 1.14. This type of ow is encountered in the


11

Figure 1.9 (a) Contours of constant velocity (m/s) and (b) contours of constant velocity gradients (s1 ) in isothermal ow 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 with respect to the z-axis, there is a cross-channel (i.e., transverse) ow in addition to the down-channel ow, as will be elaborated on below. The ow under consideration can be analyzed by Eqs. (1.1) and (1.2), with the expression for of the form = ko vx y2

+

vz x

2

+

vz y

2 (n1)/2

(1.6)

Figure 1.10 Three-dimensional plots of velocity proles in the isothermal ow 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 (s1 ) in the isothermal ow of an LDPE melt at 200 C through a rectangular channel having

an aspect ratio of 4. 12


13


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 ow of an LDPE melt at 200 C in the rectangular channel with W/H = 4 and = 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 Pas0.5 . Notice the differences in the contours of isovels and velocity gradients between the ow through a rectangular channel with sliding upper plate (Figure 1.15) and the rectilinear ow through the stationary parallel plates (Figure 1.11); specically, in Figure 1.15a


an aspect ratio of 10.

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

plate.


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


15

Figure 1.16 Threedimensional plots of velocity proles in the isothermal ow 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 ow in the ow channel (compare it with the schematic given in Figure 1.14). The back ow is caused by the combined effects of drag ow and back pressure, which arises when the ow is restricted at the exit of the channel. When the upper plate is stationary, there is no back ow, thus the velocity vz is positive everywhere in the ow channel (see Figure 1.11a). The presence of back ow in the ow through a rectangular channel with sliding upper plate is seen more clearly in Figure 1.16, as compared with the ow 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 ow 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 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 + z y vx y2

cp vz

+

vz x

2

+

vz y

2

(1.7)

16


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 oating along a streamline, which was rst suggested by McKelvey (1962) and later used by other investigators (Elbirli and Lindt 1984; Han 1988; Tadmor and Klein 1970). More specically, the term cp vz (T /z) appearing on the left side of Eq. (1.7) is replaced by cp (T /tR ) and the resulting equation is rewritten in nite difference form and solved numerically. Note that tR represents the residence time of the uid particle along a streamline, dened as tR (y) = L[va (y)], where L is the distance along the channel axis and [va (y)] is the average velocity of the uid particle in the axial direction, which can be determined from the expression [va (y)] = va (y)tf +va (yc )(1tf ), 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 uid particle spends in the upper ow eld. 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 + vx (y) sin and the fractional time tf from (McKelvey 1962): tf = 1 1 + vx (y)/vx (yc ) (1.9) (1.8)

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

of a rectangular ow channel with sliding upper plate.


17

The complementary position yc (see Figure 1.17) corresponding to the position y can be determined by satisfying the expressionyc 0

vx (y) dy =

H yc

vx (y) dy

(1.10)

Figure 1.18 gives three-dimensional plots of temperature proles for an LDPE melt owing through a rectangular channel with W/H = 4 and = 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 proles 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 Pas0.5 , b = 0.0109 K1 , n = 0.5, = 77.9 kg/m3 , cp = 2.595 103 J/(kg K), and k = 0.182 W/(mK). 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 ow by viscous dissipation.

Figure 1.18 Three-dimensional plots of temperature proles, 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


Figure 1.18(Contd)

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 eld), 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 proles for an LDPE melt at z/H = 100 when the convective heat transfer term, cp 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 signicant viscous shear heating (i.e., when dealing with polymer melts under the ow conditions of industrial importance), an analysis of ow 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 proles. Figure 1.20 gives three-dimensional plots of velocity proles vz (x, y) at z/H = 100 when neglecting the convective heat transfer term in the energy equation (i.e., the velocity proles 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 signicant 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 that is dened by Eq. (1.4). A comparison of Figure 1.20 with Figure 1.16 indicates that, when compared with the isothermal ow situation, there is a lesser degree of back ow in the ow channel when a signicant 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 proles, 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 proles 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 proles, in the presence of viscous shear heating, of non-isothermal ow 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 proles are obtained from numerical solution of Eqs. (1.1), (1.2), and (1.7), and neglecting the convective heat transfer term.

19

20


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

1.3

Flow in the Entrance Region of a Slit Die

When uid enters a tube from a large reservoir, the velocity prole becomes fully developed at a certain distance, beyond which it has attained either the classical parabolic form characteristic of Newtonian uids or the atter than parabolic form corresponding to non-Newtonian uids. The criterion for determining fully developed ow 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 uids, to claim that ow is fully developed inside the tube (or slit die) upon passing the tube entrance, it is necessary, but not sufcient, to establish a constant pressure gradient in a cylindrical tube (or slit die), while such criterion is both necessary and sufcient to claim that ow is fully developed for Newtonian uids. Conditions other than the constant pressure gradient in the ow 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 ow is fully developed, one can write momentum balance equations to derive various expressions relating the uid properties to ow variables (ow rate, pressure gradient, etc.). In Chapter 5 of Volume 1 we considered fully developed ow in order to derive various expressions that allow one to calculate viscometric ow properties of molten polymers owing 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 uid (Indopol H1900) and Figure 5.12 for a viscoelastic HDPE melt in Volume 1. It is seen in these gures 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 ow patterns of an HDPE melt at 180 C in the entrance region of a slit die, where indeed strong circulatory ow patterns are observed at the corners of the reservoir section as the bulk of the HDPE melt ows, 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 nite element methods (Crochet and Bezy 1979; Kajiwara et al. 1991; Luo and Mitsoulis 1990; Park and Mitsoulis 1992; Viriyayuthakorn and Coswell 1980), on the ow patterns of viscoelastic uids 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 ow patterns of an HDPE melt at 180 C in the entrance region of a slit die, where strong circulatory ow patterns are observed at the corners of the reservoir section as the bulk of the HDPE melt ows 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 proles 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


where a cylindrical coordinate system is superposed over the entrance section. Here, we conne our interest in the velocity proles to only inside the angle 2 , through which a polymeric liquid ows continuously into the slit-die section. Figure 1.22b gives velocity proles 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 ows 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 ow birefringence, as schematically shown in Figure 1.23, where the apparatus consists of (1) the optical system, (2) the ow test cell, and (3) the polymer melt feed system. The main components of the optical system are a light source, interference lter, 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 eld is viewed, to isoclinic fringe patterns when 2 = N and to isochromatic fringe patterns when /2 = N, where denotes the direction of principal stresses, 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 ow birefringence. (Reprinted from Han and

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


23

principal stresses, the retardation (or fringe order) R can be related to the difference in the other two principal stresses, , by n=C (1.11)

where n = R/d with d being the thickness of the medium and being the wavelength. In Eq. (1.11), C is called the stress optical coefcient measured in brewsters (1 brewster = 1012 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 ( opt ) is identical with the orientation of the principal stress axes (stress ): (opt ) = (stress ) = (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, xy , and rst normal stress differences, xx yy , to principal stresses by (Frocht 1941; Hendry 1966) xy = ( /2) sin 2stress xx yy = cos 2stress (1.13) (1.14)

Using Eqs. (1.11) and (1.12), Eqs. (1.13) and (1.14) can be rewritten as xy = FN sin 2 xx yy = 2F cos 2 (1.15) (1.16)

where F = /2Cd. Note that is to be determined from the isoclinic fringe patterns and N from the isochromatic fringe patterns. However, in order to calculate xy and xx yy from Eqs. (1.15) and (1.16), the stress optical coefcient C must be known for the uid 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 xy , N, and in the well-dened ow eld, such as fully developed ow in the slit die, using Eq. (1.15). Lodge (1955) appears to have been the rst 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 ow, in fully developed ow, or in the entrance-region ow of a slit die.

24

PROCESSING OF THERMOPLASTIC POLYMERS Figure 1.24 Photographs of

(a) isochromatic fringe patterns and (b) isoclinic fringe patterns at = 40 , where denotes the direction of the principal stresses, for an HDPE melt at 200 C and a volumetric ow rate of 17.3 cm3 /min owing 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 proles and Figure 1.25b gives calculated rst normal stress difference proles in the entrance region of a slit die using Eqs. (1.15) and (1.16). Han and Drexler (1973a) determined the stress optical coefcient 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 xy (see Chapter 5 of Volume 1) and from the measurements of the number of isochromatic fringes N and the isoclinic angles 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 1.23 109 Pa1 for an HDPE melt, 4.95 109 Pa1 for PS, and 0.605 109 Pa1 for PP. It is seen in Figure 1.25 that ow birefringence is a very powerful experimental technique for determining stress distributions of polymeric liquid in the entrance region of a slit die. Note that ow birefringence can be used to determine stress distributions in a ow geometry that is much more complicated than the entrance region of a slit die. However, this experimental technique has limitations. As the ow rate increases, the number of isochromatic fringes also increases. Thus, the distinction between the fringes becomes increasingly difcult to see as the ow rate increases, and eventually one ends up with the situation where the counting the number of fringes becomes


25

Figure 1.25 (a) Shear stress proles in the entrance region of a slit die for an HDPE melt at 200 C with a volumetric ow rate of 2.96 cm3 /min: ( ) xy = 5.72 104 Pa, ( ) xy = 9.15 104 Pa, ( ) xy = 1.03 105 Pa, ( ) xy = 1.37 105 Pa, ( ) xy = 4.57 104 Pa, ( ) xy = 2.28 104 Pa, ( ) xy = 1.14 104 Pa. (b) First normal stress difference proles: ( ) xx yy = 5.72 105 Pa, ( ) xx yy = 4.58 105 Pa, ( ) xx yy = 2.29 105 Pa, ( ) xx yy = 1.72 105 Pa, (3) xx yy = 1.26 105 Pa, ( ) xx yy = 0 Pa, ( ) xx yy = 6.86 104 Pa, ( ) xx yy = 1.03 105 Pa, ( ) xx yy = 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, ow birefringence is limited to relatively low shear rates. Next, the uid under test must be transparent, and thus ow birefringence cannot be used to determine stress distributions in such polymeric systems as immiscible polymer blends, which are invariably translucent, and lled 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 ow patterns of PS melt at 200 C in a converging channel having a half-angle of 30 . No secondary ow 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 owing through the die. Thus, the presence of secondary ow 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 owing 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 uid accelerates as it enters the die entrance. Figure 1.27 gives a photograph of isochromatic fringe patterns of a PS melt at 200 C owing 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 proles and Figure 1.28b calculated rst normal stress difference proles 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 ow along the centerline of a converging channel can be regarded as elongational ow and thus the stress yy (0, z) in Figure 1.29 can be regarded as the extensional stress. It is seen that yy (0, z) rst 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 ow 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.)


27

Figure 1.28 (a) Shear stress proles in the entrance region of a converging ow channel for a PS melt at 200 C with a volumetric ow rate of 5.36 cm3 /min: ( ) xy = 0.14104 Pa, ( ) xy = 0.28 104 Pa, ( ) xy = 0.49 104 Pa, ( ) xy = 0.70 105 Pa, (3) xy = 0.82 104 Pa, ( ) xy = 1.12 104 Pa, ( ) xy = 1.40 104 Pa, ( ) xy = 1.54 104 Pa. (b) First normal stress difference: ( ) xx yy = 5.72 105 Pa, ( ) xx yy = 4.58 105 Pa, ( ) xx yy = 2.29 105 Pa, ( ) xx yy = 1.72 105 Pa, (3) xx yy = 1.26 105 Pa, ( ) xx yy = 0 Pa, ( ) xx yy = 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 owing through a tapered channel3 with various converging angles. It can be seen that the stress distributions of the uid in the ow channel are greatly inuenced by the converging angle. Figure 1.31 gives calculated rst normal stress difference proles 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 rst normal stress difference proles are much more complicated in a tapered die than in a converging die in that very large concentrations of rst normal stress difference exist just before the uid 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 ow. For the geometry chosen, at any place in the tapered channel we have Trr (r, ) = p(r, ) + rr (r, ) T (r, ) = p(r, ) + (r, ) (1.17) (1.18)

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

28


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.)

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

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 -directed total normal stress at the channel wall T (r, ), that is T (r, ) = p(r, ) + (r, ) (1.19)

in which 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 ow, does not give us the same convenience for the converging ow eld. This is because in a converging ow eld the deviatoric stress component also depends on the direction of ow. Thus, from Eq. (1.18) one has T r =

p r

+

r

(1.20)


29

Figure 1.30 Photographs of isochromatic fringe patterns of a PS melt at 200 C owing 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 (T ) alone is not sufcient to dene the pressure gradient (p/r) in a converging ow eld unless the gradient of the deviatoric stress at the wall ( /r) is zero. Figure 1.33 gives experimentally measured wall normal stress distributions T (r, ) along the tapered channel wall for an HDPE melt. It is seen that as the melt ows 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 proles T (r, = 30 ), which were obtained, via the ColemanNoll second uid

30

PROCESSING OF THERMOPLASTIC POLYMERS Figure 1.31 First normal stress difference proles of polystyrene at 200 C owing through a tapered die with an angle of 60 at a volumetric ow rate of 1.5 cm3 /min: ( ) xx yy = 1.19 104 Pa, ( ) xx yy = 2.85 104 Pa, ( ) xx yy = 3.95 104 Pa, ( ) xx yy = 3.46 104 Pa, (3) xx yy = 1.52 104 Pa, ( 7) xx yy = 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): f () 0 f () 1 1 2 + 2 2 2 r r02

T (r, ) T (r0 , ) =

1 1 4 4 r r0

(1.21)

in which r0 denotes the vertex of the converging channel (see Figure 1.32). Note that Eq. (1.21) contains f () and f (), which are related to the function f () given by f ( ) = cos 2 cos 2 f (0) 1 cos 2 (1.22)

Figure 1.32 Schematic showing a

converging ow channel over which a cylindrical coordinate is superposed.


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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 ow 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.)

f () in turn is related to the volumetric ow rate per channel width Q dened by Q = 20

rvz (r, ) d = f (0)

2 cos 2 sin 2 1 cos 2

(1.23)

where f (0) is the value of f ( ) at = 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 owing through a tapered channel with a converging angle of 30 , at various volumetric ow rates: (1) Q = 65.7 cm3 /min with T (r0 = 3.1 cm, = 15 ) = 7.46 106 Pa, (2) Q = 47.4 cm3 /min with T (r0 = 3.1 cm, = 15 ) = 6.25 106 Pa, and (3) Q = 18.4 cm3 /min with T (r0 = 3.1 cm, = 15 ) = 3.3 106 Pa. (Reprinted from Yoo and Han, Journal of Rheology 25:115. Copyright 1981, with permission from the Society of Rheology.)

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In obtaining Figure 1.34, the following numerical values of the material parameters appearing in the ColemanNoll second-order uid were used: 0 = 1.14 105 Pas, = 3.20 104 Pas2 , and = 1.78 104 Pas2 for an HDPE melt at 200 C. It is interesting to observe in Figure 1.34 that the wall normal stress T (r, ) 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 ColemanNoll second-order uid model is not expected to describe well the rapid motion of viscoelastic uids. Note that the ow in a tapered channel is considered to be a rapid ow. 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 0 in the theoretical calculation of T (r, ) must have overestimated the true values of melt viscosity encountered under the actual ow conditions. However, the use of the ColemanNoll second-order uid 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 uid, = 0 in Eq. (1.21), cannot exhibit a minimum in the distribution T (r, ) along the tapered channel. We can thus conclude that the increasing trend of T (r, ) near the exit plane of a tapered channel, predicted from Eq. (1.21), is due to the elastic component of the uid behavior.

1.5

Exit Region Flow

It has long been recognized that the ow behavior of viscoelastic uids in the exit region (here referred to as exit region ow) of a tube is quite different from that of Newtonian uids. Two aspects of exit region ow of viscoelastic uids 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 ow 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 1.21.4). In the 1970s through 1980s, some effort was made to calculate the extent of extrudate swell of viscoelastic polymeric uids using nite 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 ow 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.


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Figure 1.35 Photograph of isochromatic fringe patterns of a PS melt at 200 C owing 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, signicant 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 signicance of Figures 1.35 and 1.36 lies in that the stresses of the PS melt inside the die did not have sufcient 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 proles 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 ow rate of 1.5 cm3 /min: ( ) xy = 0.68 104 Pa,

( ) xy = 1.76 104 Pa, ( ) xy = 2.34 104 Pa, ( ) xy = 3.78 104 Pa, (3) xy = 4.78 104 Pa, ( ) xy = 0.92 104 Pa, ( ) xy = 1.46 104 Pa, ( ) xy = 2.06 104 Pa. (Reprinted from Han, Rheologica Acta 14:173. Copyright 1975, with permission from Springer.)

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much of the elastic energy stored in the PS melt during the ow through the tapered die did not dissipate fast enough, such that a large amount of energy still remains in the uid 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 ow birefringence technique in investigating the stress distributions of viscoelastic uids in a conned geometry, namely, the number of isochromatic fringes (N in Eq. (1.15)) increases very rapidly as the ow 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 s1 . For instance, when shear rate is increased to, say, 10 s1 , we would not be able to discern isochromatic fringe patterns inside the die or in the extrudate. This is precisely the limitation of the ow birefringence technique. It is not difcult 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 sufciently 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 in the exit region of a slit die having an opening (height) of 2 mm. (Reprinted rate of 4.55 s

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


35

the wall shear stress corresponding to a shear rate of 4.55 s1 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 uid passes by the small hole or transverse slot while owing 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 ow 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 ow 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 ow patterns and stress distributions in a slit ow channel, similar to that shown in Figure 1.38, when a polymer melt was forced to ow through the channel. They used ow birefringence to obtain information on the stress distributions in the neighborhood of the transverse slots and in the ow channel with the aid of a polariscope. With the polariscope removed, they were able to observe ow 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 ow 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 ow patterns are not symmetric about a plane through the center of the transverse slots, and circulatory ow 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.

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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 magnication. (Reprinted from Han and Yoo, Journal of Rheology 24:55. Copyright 1980, with permission from the Society of Rheology.)

the presence of circulatory ow patterns of an aqueous solution of polyacrylamide (a viscoelastic uid) (Figures 1.40a and 1.40b) and glycerin (a Newtonian uid) (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 nite element calculations (Jackson and Finlayson 1982; Kajiwara et al. 1991) indeed predict the circulatory ow patterns inside a small side hole, conrming the experimental results presented in Figure 1.40. Careful examination of the ow 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 the schematic diagram given in Figure 1.41, where 1 denotes the entrance plane and


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Figure 1.40 Secondary ow 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 owing at = 79 s1 , = 9.8 Pa, and NRe = 2.3, (b) 4.0 wt % aqueous solution of polyacrylamide owing at = 676 s1 , = 243 Pa, and NRe = 2.9, and (c) glycerin owing at = 660 s1 , = 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.)

denotes the exit plane of the transverse slot. Referring to Figures 1.39 and 1.41, the isochromatic fringe pattern in the neighborhood of plane 1 is very similar to that in the exit region (see Figure 1.35), and the isochromatic fringe pattern in the neighborhood of plane 2 is very similar to that in the entrance region (see Figure 1.24). Thus, the streamlines in the neighborhood of a transverse slot are drawn in Figure 1.41, indicating that the ow passing through a transverse slot in a slit die includes both exit and entrance ows. The photographs of ow patterns given in Figure 1.40 further support such schematic streamlines. Namely, in the ow of a viscoelastic polymer solution, a streamline is pushed up more in the neighborhood of plane 1 (exit ow) than in the neighborhood of plane 2 (entrance ow). However, in the ow of glycerin (which is a Newtonian uid), a streamline is pushed less in the neighborhood of plane 12

38


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

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

than in the neighborhood of plane 2 . The streamline patterns in the neighborhood of plane 1 may be explained in terms of the angle that an extrudate makes at the die exit (Figure 1.35), and the streamline patterns in the neighborhood of plane 2 may be 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 at. 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 rst normal stress difference (N1 ) of viscoelastic uids in steady-state shear ow. Specically, assuming that the ow is so slow that the second-order approximate constitutive equations are valid, for uniform shear ow, Tanner and Pipkin (1969) showed that pH is negative and 25% of N1 , that is pH = 0.25N1 (1.24)


39

Figure 1.42 Schematic showing idealized streamline patterns in the neighborhood of a pressure hole. pH = p p < 0 for certain uids in the absence of secondary ow 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 = 1 2w 0

(11 22 ) d12 12

(1.25)

for a transverse slot in a slit die, where 11 and 22 are the normal stresses in the fully developed region of the slit die, 12 is the shear stress in the fully developed region of the slit die, and w 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 ow, that is, there is no secondary ow (circulating ow pattern) in the hole, and (4) the gradients of axial normal stresses and shear stress with respect to the ow direction vanish at the centerline of the hole. Equation (1.25) can be rewritten as N1 = 2w dpH dw (1.26)

indicating that rst normal stress difference N1 in steady-state shear ow can be determined from the measurements of hole pressure error pH as a function of wa

Documents

Rheology and Processing of Polymeric Materials, Vol. 2 Polymer Processing