melt rheology and its role in plastics processing by dealy.pdf

7/21/2019 Melt Rheology and Its Role in Plastics Processing By Dealy.pdf

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MELT

RHEOLOGY

AND ITS ROLE

IN PLASTICS

PROCESSING

THEORY AND

APPLICATIONS


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MELT

RHEOLOGY

AND

ITS ROLE

IN PLASTICS

PROCESSING

THEORY AND APPLICATIONS

JOHN M. DEALY

Department of Chemical Engineering

McGill University

Montreal, Canada

and

KURT F. WISSBRUN

Hoechst Celanese Research Division

Summit, New Jersey

ImiirI

VAN

NOSTRAND

REINHOLD

~

______ New

York


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Copyright 1990 by Van Nostrand Reinhold

Library

of

Congress Catalog

Card

Number 89-29215

ISBN-13 :978-1-4615-9740-7

DOl:10.1007/978-1-4615-9738-4

e-ISBN-13:978-1-4615-9738-4

All rights reserved. Certain portions

of

this work 1990 by

Van Nostrand Reinhold.

No part of

this work covered by the copyright

hereon may be reproduced or used in any form or by any

means-graphic,

electronic,

or

mechanical, including photocopying, recording, taping,

or

information storage and retrieval

systems-without

written permission

of

the publisher.

Softcover reprint

of

the hardcover 1st edition 1990

Van Nostrand Reinhold

115 Fifth Avenue

New York, New York 10003

Van Nostrand Reinhold International Company Limited

11

New Fetter Lane

London

EC4P

4EE, England

Van Nostrand Reinhold

480 La

Trobe Street

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Nelson

Canada

1120 Birchmount Road

Scarborough, Ontario MIK 5G4, Canada

16

15

14 13 12

11

10 9 8 7 6 5 4 3 2 1

Library

of

Congress Cataloging-in-Publication Data

Dealy,

John

M.

Melt rheology

and

its role in plastics processing: theory

and

applications/John

M. Dealy

and

Kurt F. Wissbrun.

p. cm.

Includes bibliographical references.

1. Plastics-Testing.

II. Title.

TA455.P5D28 1989

668.4'042-dc20

2. Rheology. I. Wissbrun, Kurt F.

89-29215

CIP


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Preface

This book

is

designed to fulfill a dual role. On the one hand it

provides a description

of

the rheological behavior

of

molten poly

mers. On the other, it presents the role of rheology in melt

processing operations. The account

of

rheology emphasises the

underlying principles and presents results, but not detailed deriva

tions

of

equations. The processing operations are described qualita

tively, and wherever possible the role of rheology is discussed

quantitatively. Little emphasis

is

given to non-rheological aspects

of

processes, for example, the design

of

machinery.

The audience for which the book

is

intended

is

also dual in

nature. It includes scientists and engineers whose work in the

plastics industry requires some knowledge

of

aspects

of

rheology.

Examples are the polymer synthetic chemist who

is

concerned with

how a change in molecular weight will affect the melt viscosity and

the extrusion engineer who needs to know the effects of a change in

molecular weight distribution that might result from thermal degra

dation.

The audience also includes post-graduate students in polymer

science and engineering who wish to acquire a more extensive

background in rheology and perhaps become specialists in this area.

Especially for the latter audience, references are given to more

detailed accounts

of

specialized topics, such as constitutive relations

and process simulations. Thus, the book could serve as a textbook

for a graduate level course in polymer rheology, and it has been

used for this purpose.

The structure

of the book is as follows. Chapter 1 is an introduc

tion to rheology and to polymers for readers entering the field for

the first time. The reader is assumed to be familiar with the

mathematics and chemistry that are taught in undergraduate engi

neering and physical science programs.

Chapters 2 through 6 are a treatment

of

rheological behavior that

includes the well established areas

of

steady shear and linear

v


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vi PREFACE

viscoelasticity. There is, in addition, an extensive discussion of

nonlinear viscoelasticity effects, which often play an important role

in melt processing operations. Chapters 7 through 9 are devoted to

the experimental methods used to measure the properties that have

been defined, using both the traditional flows and some special

types of deformation.

The dependence of the parameters of the rheological relations

upon the composition and structure of the polymeric materials is

the subject of Chapters

10

through 13. The description is most

extensive for stable, homogeneous, isotropic molten polymers, and

less so for more complex systems. Chapters

14

through

17

summa

rize what is known about the role

of

rheology in the most important

melt processing operations. Finally, we close with a chapter whose

aim is to provide guidelines, often by example, of how to apply the

information in this book and in the literature to solve problems in

applied rheology.

This volume is not an exhaustive monograph on all aspects of

polymer rheology. However, we have included all the material that

we believe

is

likely to be of direct use to those working in the

plastics industry.

The

reference lists are not intended to be exhaus

tive, but all the work that we believe

is

central to the themes of the

book has been cited.

We have adhered to the Society of Rheology official nomencla

ture wherever possible. Also, we have used index rather than dyadic

notation for tensor quantities, because we felt this would be more

easily understood

by

readers seeing tensor notation for the first

time.

JMD

wishes to acknowledge the support and encouragement

of

McGill University for providing a working environment conducive

to a major writing project. He also wishes to recognize the col

leagues and research students who have played a vital role in the

development

of

his understanding

of

polymer rheology and its

applications. In addition, JMD wishes to express his appreciation to

the University

of

Wisconsin, especially to R. B. Bird and A. S.

Lodge, for their professional hospitality during the time when he

got his

part of

the writing well launched.

KFW wishes to acknowledge the management of Hoechst

Celanese for their permission to participate in this book.

He

also


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PREFACE vii

wishes to thank his many colleagues at Hoechst Celanese, in partic

ular H. M. Yoon, and his colleagues at the University

of

Delaware,

most especially

A.

B.

Metzner, for their contributions to his experi

ence and knowledge of the fields discussed in this book. Others to

whom appreciation

is due

include W. W. Graessley, F.

N.

Cogswell,

D. Pearson, M. Doi, and G. Fuller.

Several people read one or more chapters of the manuscript and

made many helpful suggestions for improvement. These include

H. M.

Laun,

1.

E.

L.

Roovers,

H.

C. Booij, G. A. Campbell,

S.

1.

Kurtz, and 1. V. Lawler. Their contributions are gratefully acknowl

edged. Finally, we wish to thank

Hanser

Publishers, particularly Dr.

Edmund

Immergut, for permission to reproduce some material

from our

chapter

in

the

Blow Molding Handbook.

J. M. Dealy

K.

F.

Wissbrun


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Contents

Preface

1. INTRODUCTION TO RHEOLOGY

1.1

What

is

Rheology?

1.2 Why Rheological Properties are Important

1.3 Stress as a Measure of Force

1.4 Strain as a Measure of Deformation

1.4.1 Strain Measures for Simple Extension

1.4.2

Shear

Strain

1.5 Rheological

Phenomena

1.5.1 Elasticity; Hooke's Law

1.5.2 Viscosity

1.5.3 Viscoelasticity

1.5.4 Structural Time Dependency

1.5.5 Plasticity and Yield Stress

1.6 Why Polymeric Liquids

are

Non-Newtonian

1.6.1 Polymer Solutions

1.6.2 Molten Plastics

1.7 A

Word About

Tensors

1.7.1 Vectors

1.7.2

What

is a Tensor?

1.8

The

Stress Tensor

1.9 A Strain Tensor for Infinitesimal Deformations

1.10 The Newtonian Fluid

1.11 The Basic Equations of Fluid Mechanics

1.11.1 The Continuity Equation

1.11.2 Cauchy's Equation

1.11.3 The Navier-Stokes Equation

References

2. LINEAR VISCOELASTICITY

2.1 Introduction

2.2 The Relaxation Modulus

v

1

1

3

3

6

7

9

10

10

11

13

16

18

19

19

20

22

23

23

25

31

36

37

38

39

40

41

42

42

43

ix


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x CONTENTS

2.3 The Boltzmann Superposition Principle 44

2.4 Relaxation Modulus of Molten Polymers

48

2.5 Empirical Equations for the Relaxation Modulus

51

2.5.1

The

Generalized Maxwell Model 52

2.5.2 Power Laws and an Exponential Function

53

2.6

The

Relaxation Spectrum 54

2.7

Creep

and Creep Recovery;

The

Compliance 55

2.8 Small Amplitude Oscillatory Shear 60

2.8.1 The Complex Modulus and the Complex

Viscosity 61

2.8.2 Complex Modulus of Typical Molten Polymers 66

2.8.3 Quantitative Relationships between

G*(w)

and

MWD 68

2.8.4

The

Storage and Loss Compliances 69

2.9 Determination of Maxwell Model Parameters 70

2.10 Start-Up and Cessation

of

Steady Simple Shear and

Extension 72

2.11 Molecular Theories: Prediction of Linear Behavior 74

2.11.1 The Modified Rouse Model for Unentangled

Melts 74

2.11.1.1

The

Rouse Model for Dilute Solutions 74

2.11.1.2

The

Bueche Modification of the Rouse

Theory 75

2.11.1.3

The

Bueche-Ferry Law 79

2.11.2 Molecular Theories for Entangled Melts 79

2.11.2.1 Evidence for the Existence of

Entanglements 79

2.11.2.2

The

Nature of Entanglement Coupling 80

2.11.2.3 Reptation

81

2.11.2.4 The Doi-Edwards Theory 82

2.11.2.5

The

Curtiss-Bird Model 85

2.11.2.6 Limitations of Reptation Models 86

2.12 Time-Temperature Superposition 86

2.13 Linear Behavior of Several Polymers 94

References

100

3. INTRODUCTION TO NONLINEAR VlSCOEIASTICITY 103

3.1 Introduction 103

3.2 Nonlinear Phenomena

105


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CONTENTS xi

3.3

Theories of Nonlinear Behavior

106

3.4

Finite Measures of Strain

108

3.4.1 The Cauchy Tensor and the Finger Tensor

109

3.4.2 Strain Tensors

110

3.4.3 Reference Configurations

112

3.4.4 Scalar Invariants of the Finger Tensor 113

3.5

The Rubberlike Liquid

114

3.5.1 A Theory of Finite Linear Viscoelasticity

115

3.5.2 Lodge's Network Theory and the Convected

Maxwell Model

117

3.5.3

Behavior of the Rubberlike Liquid in Simple

Shear Flows 118

3.5.3.1 Rubberlike Liquid in Step Shear Strain

119

3.5.3.2

Rubberlike Liquid in Steady Simple

Shear

119

3.5.3.3

Rubberlike Liquid in Oscillatory Shear

121

3.5.3.4

Constrained Recoil of Rubberlike

Liquid

122

3.5.3.5

The Stress Ratio

(N1/u)

and the

Recoverable Shear

122

3.5.4 The Rubberlike Liquid in Simple Extension

123

3.5.5 Comments on the Rubberlike Liquid Model

126

3.6

The BKZ Equation 127

3.7

Wagner's Equation and the Damping Function 128

3.7.1 Strain Dependent Memory Function 128

3.7.2 Determination of the Damping Function

131

3.7.3 Separable Stress Relaxation Behavior

132

3.7.4 Damping Function Equations for Polymeric

Liquids

134

3.7.4.1 Damping Function for Shear Flows 134

3.7.4.2

Damping Function for Simple Extension

138

3.7.4.3

Universal Damping Functions 139

3.7.5

Interpretation of the Damping Function in Terms

of Entanglements 141

3.7.5.1 The Irreversibility Assumption

142

3.7.6 Comments on the Use of the Damping Function

144

3.8

Molecular Models for Nonlinear Viscoelasticity

146

3.8.1 The Doi-Edwards Constitutive Equation

148

3.9 Strong Flows; The Tendency to Stretch and Align

Molecules

150

References

151


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xii CONTENTS

4.

STEADY SIMPLE SHEAR FLOW AND THE VISCOMETRIC

FUNCTIONS 153


4.2 Steady Simple Shear Flow 153

4.3 Viscometric Flow 155

4.4 Wall Slip and Edge Effects 158

4.5 The Viscosity of Molten Polymers 158

4.5.1 Dependence

of

Viscosity on Shear

Rate

159

4.5.2 Dependence

of

Viscosity

on

Temperature 169

4.6 The First Normal Stress Difference 170

4.7 Empirical Relationships Involving Viscometric

Functions 173

4.7.1

The

Cox-Merz Rules

173

4.7.2

The

Gleissle Mirror Relations 175

4.7.3 Other Relationships 176

References 176

5. TRANSIENT SHEAR FLOWS USED TO STUDY

NONLINEAR VISCOELASTICITY 179


5.2 Step Shear Strain 181

5.2.1 Finite Rise Time 181

5.2.2

The

Nonlinear Shear Stress Relaxation Modulus 183

5.2.3 Time-Temperature Superposition 188

5.2.4 Strain-Dependent Spectrum and Maxwell

Parameters 188

5.2.5 Normal Stress Differences for Single-Step Shear

Strain 190

5.2.6 Multistep Strain Tests

191

5.3 Flows Involving Steady Simple Shear 194

5.3.1 Start-Up Flow 194

5.3.2 Cessation

of

Steady Simple Shear 199

5.3.3 Interrupted Shear 203

5.3.4 Reduction in Shear Rate 205

5.4 Nonlinear Creep 206

5.4.1 Time-Temperature Superposition of Creep

Data

209

5.5 Recoil and Recoverable Shear 210

5.5.1 Creep Recovery 210

5.5.1.1 Time-Temperature Superposition;

Creep

Recovery

213


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CONTENTS xiii

5.5.2 Recoil During Start-Up Flow 214

5.5.3 Recoverable Shear Following Steady Simple

Shear

215

5.6 Superposed Deformations 217

5.6.1 Superposed Steady and Oscillatory Shear 218

5.6.2 Step Strain with Superposed Deformations 219

5.7 Large Amplitude Oscillatory Shear 219

5.8 Exponential Shear; A Strong Flow 225

5.9 Usefulness of Transient Shear Tests 228

References 228

6. EXTENSIONAL FLOW PROPERTIES AND

THEIR

MEASUREMENT 231


6.2 Extensional Flows 232

6.3 Simple Extension 237

6.3.1 Material Functions for Simple Extension 238

6.3.2 Experimental Methods

241

6.3.3 Experimental Observations for LDPE 249

6.3.4 Experimental Observations for Linear Polymers 258

6.4 Biaxial Extension 260

6.5 Planar Extension 263

6.6

Other

Extensional Flows 265

References 266

7. ROTATIONAL AND SLIDING SURFACE RHEOMETERS 269


7.2 Sources

of Error

for Drag Flow Rheometers 270

7.2.1 Instrument Compliance 270

7.2.2 Viscous Heating 274

7.2.3 End and Edge Effects 275

7.2.4 Shear Wave Propagation 275

7.3 Cone-Plate Flow Rheometers 277

7.3.1 Basic Equations for Cone-Plate Rheometers 278

7.3.2 Sources of Error for Cone-Plate Rheometers 279

7.3.3 Measurement of the First Normal Stress

Difference 281

7.4 Parallel Disk Rheometers 283

7.5 Eccentric Rotating Disks 284


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xiv CONTENTS

7.6 Concentric Cylinder Rheometers 285

7.7 Controlled Stress Rotational Rheometers 286

7.8 Torque Rheometers 287

7.9 Sliding Plate Rheometers 287

7.9.1 Basic Equations for Sliding Plate Rheometers 288

7.9.2 End and Edge Effects for Sliding Plate

Rheometers 289

7.9.3 Sliding Plate Melt Rheometers 290

7.9.4

The

Shear Stress Transducer 292

7.10 Sliding Cylinder Rheometers 294

References 294

8. FLOW IN CAPILLARIES, SLITS AND DIES 298

8.1

Introduction 298

8.2 Flow in a Round Tube 298

8.2.1 Shear Stress Distribution 298

8.2.2 Shear Rate for a Newtonian Fluid 299

8.2.3 Shear Rate for a Power Law Fluid 301

8.2.4

The

Rabinowitch Correction 303

8.2.5

The

Schiimmer Approximation 304

8.2.6 Wall Slip in Capillary Flow 305

8.3 Flow in a Slit 307

8.3.1 Basic Equations for Shear Stress and Shear Rate 307

8.3.2 Use of a Slit Rheometer to Determine Nt 309

8.3.2.1 Determination of Nt from the Hole

Pressure 310

8.3.2.2 Determination of Nt from the Exit

Pressure 313

8.4 Pressure Drop in Irregular Cross Sections 317

8.5 Entrance Effects 317

8.5.1 Experimental Observations 318

8.5.2 Entrance Pressure

Drop-the

Bagley End

Correction 319

8.5.3 Rheological Significance of the Entrance

Pressure Drop 323

8.6 Capillary Rheometers 324

8.7 Flow in Converging Channels 329

8.7.1

The

Lubrication Approximation 329

8.7.2 Industrial Die Design 332

8.8 Extrudate Swell 332

8.9 Extrudate Distortion 336


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CONTENTS

xv

8.9.1 Surface Melt Fracture-Sharkskin 337

8.9.2 Oscillatory Flow in Linear Polymers 338

8.9.3 Gross Melt Fracture 339

8.9.4 Role of Slip in Melt Fracture 340

8.9.5 Gross Melt Fracture Without Oscillations

341

References 341

9.

RHEO-OPTICS

AND

MOLECUlAR

ORIENTATION 345

9.1 Basic Concepts-Interaction of Light and Matter 345

9.1.1 Refractive Index and Polarization 346

9.1.2 Absorption and Scattering 347

9.1.3 Anisotropic Media; Birefringence and Dichroism 349

9.2 Measurement of Birefringence 353

9.3 Birefringence and Stress 358

9.3.1 Stress-Optical Relation 358

9.3.2 Application of Birefringence Measurements 362

References 363

10. EFFECTS

OF MOLECUlAR

STRUCTURE 365

10.1

Introduction and Qualitative Overview of Molecular

Theory 365

10.2 Molecular Weight Dependence of Zero Shear Viscosity 368

10.3 Compliance and First Normal Stress Difference 370

10.4 Shear Rate Dependence of Viscosity 374

10.5 Temperature and Pressure Dependence

381

10.5.1 Temperature Dependence of Viscosity

381

10.5.2 Pressure Dependence of Viscosity 384

10.6 Effects

of

Long Chain Branching 386

References 389

11. RHEOWGY

OF

MULTIPHASE SYSTEMS 390


11.2 Effect of Rigid Fillers 390

11.2.1 Viscosity 392

11.2.2 Elasticity 400

11.3 Deformable Multiphase Systems (Blends, Block

Polymers)

401

11.3.1 Deformation

of

Disperse Phases and Relation to

Morphology 403


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xvi CONTENTS

11.3.2 Rheology of Immiscible Polymer Blends 406

11.3.3 Phase-Separated Block and Graft Copolymers 407

References 408

12. CHEMORHEOLOGY OF REACTING SYSTEMS 410

12.1

Introduction 410

12.2 Nature of the Curing Reaction

411

12.3 Experimental Methods for Monitoring Curing Reactions

413

12.3.1 Dielectric Analysis

417

12.4 Viscosity of the Pre-gel Liquid

418

12.5 The Gel Point and Beyond 419

References

421

13. RHEOLOGY OF THERMOTROPIC LIQUID CRYSTAL

POLYMERS 424

13.1

Introduction 424

13.2 Rheology of Low Molecular Weight Liquid Crystals 426

13.3 Rheology of Aromatic Thermotropic Polyesters

431

13.4 Relation

of

Rheology to Processing of Liquid Crystal

Polymers 437

References

439

14. ROLE OF RHEOLOGY IN EXTRUSION 441

14.1

Introduction

441

14.1.1 Functions of Extruders 442

14.1.2 Types of Extruders 443

14.1.3 Screw Extruder Zones 444

14.2 Analysis of Single Screw Extruder Operation

446

14.2.1 Approximate Analysis of Melt Conveying Zone

446

14.2.2 Coupling Melt Conveying to Die Flow 454

14.2.3 Effects of Simplifying Approximations

459

14.2.3.1 Geometric Factors 459

14.2.3.2 Leakage Flow 460

14.2.3.3 Non-Newtonian Viscosity 462

14.2.3.4 Non-Isothermal Flow 467

14.2.4 Solids Conveying and Melting Zones 470

14.2.4.1 Feeding and Solids Conveying 470

14.2.4.2 Melting Zone 472

14.2.5 Scale-Up and Simulation 476


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CONTENTS xvii

14.2.5.1 Scale-Up 476

14.2.5.2 Simulation 477

14.3 Mixing, Devolatilization and Twin Screw Extruders 480

14.3.1 Mixing 480

14.3.2 Devolatilization 484

14.3.3 Twin Screw Extruders

485

References 489

15. ROLE

OF

RHEOLOGY IN INJECTION MOLDING 490

15.1 Introduction

491

15.2 Melt Flow in Runners and Gates 492

15.3 Flow in the Mold Cavity 494

15.4 Laboratory Evaluation of Molding Resins 500

15.4.1 Physical Property Measurement 501

15.4.2 Moldability Tests 502

15.5 Formulation and Selection of Molding Resins 506

References 507

16. ROLE

OF

RHEOLOGY IN BLOW MOLDING 509


16.2 Flow in the Die 510

16.3 Parison Swell 512

16.4 Parison Sag 519

16.4.1 Pleating 521

16.5 Parison Inflation 521

16.6 Blow Molding of Engineering Resins 522

16.7 Stretch Blow Molding 523

16.8 Measurement of Resin Processability 524

16.8.1 Resin Selection Tests 524

16.8.2 Quality Control Tests 528

References 529

17. ROLE

OF

RHEOLOGY IN FILM BLOWING AND SHEET

EXTRUSION 530

17.1

The

Film Blowing Process 531

17.1.1 Description of the Process 531

17.1.2 Criteria for Successful Processing 533

17.1.3 Principal Problems Arising in Film Blowing 534

17.1.4 Resins Used for Blown Film 534


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xviii CONTENTS

17.2 Flow in the Extruder and Die; Extrudate Swell 536

17.3 Melt Flow in the Bubble 540

17.3.1 Forces Acting

on

the Bubble

541

17.3.1.1 Viscous Stress in the Molten Region of

the Bubble

17.3.1.2 Aerodynamic Forces

17.3.2 Bubble Shape

17.3.3 Drawability

17.4 Bubble Stability

17.5 Sheet Extrusion

References

18. ON-LINE MEASUREMENT

OF

RHEOLOGICAL

PROPERTIES 557

18.1

Introduction 557

18.2 Types of On-Line Rheometers for Melts 558

18.2.1 On-Line Capillary Rheometers for Melts 558

18.2.2 Rotational On-Line Rheometers for Melts 560

18.2.3 In-Line Melt Rheometers 562

18.3 Specific Applications

of

Process Rheometers 563

References 565

19. INDUSTRIAL USE

OF

RHEOMETERS 567

19.1 Factors Affecting Test and Instrument Selection 567

19.1.1 Purposes of Rheological Testing 568

19.1.2 Material Limitations on Test Selection 569

19.1.3 Instruments

571

19.2 Screening and Characterization 573

19.2.1 Advantages and Disadvantages of Rheological

T h ~

5n

19.2.2 Other Information Useful for Screening 574

19.2.3 Stability 577

19.2.3.1 Stability Measurement 578

19.2.3.2 Use of Stability Data 580

19.2.4 Temperature and Frequency Dependence 582

19.2.4.1 Measurement Tactics 582

19.2.4.2 Interpretation

of

Results 583

19.3 Resin Selection and Optimization and Process Problem

Solving 585


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CONTENTS xix

19.4 Rheological Quality Control Tests 595

References 599

APPENDIX

A:

MEASURES

OF

STRAIN FOR LARGE

DEFORMATIONS

601

APPENDIX

B:

MOLECULAR WEIGHT DISTRIBUTION

AND

DETERMINATION

OF

MOLECULAR WEIGHT

AVERAGES 607

APPENDIX C: THE INTRINSIC VISCOSIlY

AND

THE

INHERENT VISCOSIlY 613

APPENDIX D: THE GLASS TRANSITION TEMPERATURE 617

APPENDIX E: MANUFACTURERS

OF

MELT RHEOMETERS

AND

RELATED EQUIPMENT 622

NOMENCLATURE 630

AUTHOR INDEX 639

SUBJECT INDEX 649


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MELT RHEOLOGY

AND ITS ROLE

IN PLASTICS

PROCESSING

THEORY AND APPLICATIONS


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Chapter 1

Introduction to Rheology

1.1

WHAT

IS

RHEOLOGY?

I t is anticipated that many readers will have little previous knowl

edge about rheology but will wish to find out how it can be useful to

them in solving practical problems involving the

flow of

molten

plastics. For this reason, it

is

our intention to supply sufficient basic

information about rheology to enable the reader to understand and

make use

of

the methods described. With this in mind, we begin at

the beginning, with a definition of rheology.

Rheology is the science that deals with the way materials deform

when forces are applied to them. The term is most commonly

applied to the study of liquids and liquid-like materials such as

paint, catsup, oil well drilling mud, blood, polymer solutions and

molten plastics, i.e., materials that flow, although rheology also

includes the study of the deformation of solids such as occurs in

metal forming and the stretching of rubber.

The two key words in the above definition of rheology are

deformation

and

force.

To learn anything about the rheological

properties

of

a material, we must either measure the deformation

resulting from a given force or measure the force required to

produce a given deformation. For example, let us say you wish to

evaluate the relative merits

of

several foam rubber pillows. Instinc

tively, you would squeeze (deform) the various products offered,

noting the force required to deform the samples.

A pillow that required a high force to compress would be consid

ered

"hard,"

and you probably wouldn't buy it, because it would be

painful to sleep on. On the other hand, if it required too little force

(too "soft") it would not provide adequate support for your weary

head. Foam rubber is a lightly crosslinked elastomer, and in squeez-


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INTRODUCTION TO RHEOLOGY 3

at the present time, except for very simple materials such as

Newtonian fluids. In the case of more complex materials, one can at

least develop relationships showing how specific rheological proper

ties such as the viscosity and the relaxation modulus are influenced

by

molecular structure, composition, temperature and pressure.

Molten plastics are rheologically complex materials that can

exhibit both viscous

flow

and elastic recoil. A truly general constitu

tive equation has not been developed for these materials, and our

present knowledge of their rheological behavior is largely empirical.

This complicates the description and measurement

of

their rheolog

ical properties but makes polymer rheology an interesting and

challenging field

of

study.

1.2 WHY RHEOLOGICAL PROPERTIES ARE IMPORTANT

The forces that develop when a lubricant

is

subjected to a high-speed

shearing deformation are obviously

of

central importance to me

chanical engineers. The rheological property of interest in this

application

is

the viscosity. The stiffness

of

a steel beam used to

construct a building is of great importance to civil engineers, and

the relevant property here

is

the modulus

of

elasticity.

The viscoelastic properties

of

molten polymers are

of

importance

to plastics engineers, because it

is

these properties that govern

flow

behavior whenever plastics are processed in the molten state.

For

example, in

order

to optimize the design

of

an extruder, the

viscosity must

be

known

as

a function

of

temperature and shear

rate. In injection molding, the same information is necessary in

order to design the mold in such a

way

that the melt will completely

fill it in every shot. In blow molding, the processes

of

parison sag

and swell are governed entirely

by

the rheological properties

of

the

melt.

1.3 STRESS AS A MEASURE OF FORCE

I t

was emphasized in Section

1.1

that force and deformation are the

key

words in the definition

of

rheology. In order to describe the

rheological behavior of a material in a quantitative way, i.e., to

define rheological material constants (such

as

the viscosity

of

a

Newtonian fluid)

or

material functions (such as the relaxation

modulus of a rubber), it

is

necessary to establish clearly defined and


23/682

4 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING

quantitative measures

of

force and deformation. Furthermore, it

is

necessary to define these measures in such a

way

that they describe

the state

of

the material

of

interest without detailed reference to

the procedure used to make the rheological measurement.

For

example, in the case

of

the evaluation

of

the pillows that was

described in Section 1.1, one

way

to quantify the test results would

be to place the pillow on a table, place a flat board on top, and

measure the distance between the board and the table both before

and after a weight

of

a certain mass was placed on top

of

the board,

as shown in Figure

1-1. I f

our objective

is

simply to compare several

pillows

of

the same size, it would be sufficient to simply list the

amount

of

compression, in centimeters, caused

by

a weight having a

mass

of

1

kg.

Figure

1-1.

Setup for testing pillows.

However, if our objective

is

to make a quantitative determination

of

the elastic properties

of

the foam rubber, the reporting

of

the

test results

is

awkward. We must report the size and shape

of

the

sample (the pillow), the mass

of

the weight applied, and the amount

of

compression.

I f

one wishes to compare the behavior

of

this foam

with that

of

a second foam, the second material must be tested in

exactly the same way as the first.

I t

would be advantageous to be

able to describe the elastic behavior

of

the rubber using physical

quantities which are defined

so

that they describe the state

of

the

material under test, without reference to the details

of

the test

procedure.

First let's look at force. Two types of force can act on a fluid

element. A "body force" acts directly on the mass

of

the element as

the result

of

a force field. Usually only gravity need be considered,

but a magnetic field can also generate a body force. A surface force

is

the result of the contact of a fluid element with a solid wall or

with the surrounding fluid elements.

I t is

the surface forces that are

of interest in rheology. The force exerted

by

a weight sitting

on

top


24/682


F

Figure

1-2.

Uniaxial (simple) extension.

of a pillow

is

an example of a surface force. The fact that the

ultimate cause of the surface force acting on the pillow

is

the body

force acting on the weight is not of rheological importance, as the

compressive force on the sample could equally well be supplied

by

means of a testing machine, and the observed relationship between

force and deformation would be the same.

Placing a weight having a mass of 1

kg

on a small pillow will

cause more compression than placing it on a larger pillow. From

the point of view of the material, it is obviously not the total force

that is important. In fact, since the deforming force acts on the

upper surface of the sample, it

is

found that if the force

is

divided

by

the area of the surface we obtain a quantity suitable for describ

ing the properties of the material. We call this quantity the "stress."

In

general, then, the stress is calculated by dividing the force

by

the area over which it acts. In the case of a test like the one with

the pillow which involves squeezing, we call this the compressive

stress. A more common type of test method for elastic materials

involves stretching rather than compressing, as shown in Figure 1-2.

Again, the stress

is

the force divided

by

the cross sectional area of

the sample, and in this case we call it a "tensile stress." We will use

the symbol (FE for this quantity.

(FE =

cross sectional area

stretching force

(1-1)


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--l

f:j.X \- ; - WETTED AREA ~ A

is,,-

F

I - - .

I I

Figure 1-3. Simple shear (2 plates, gap

~ h).

Compressive and tensile stresses are the two types of "normal

stress," so called because the direction of the force is normal

(perpendicular) to the surface on which it acts. In addition to

normal stresses, we can have a "shear stress"; in this case, the

direction of the force is tangential to the surface on which it acts, as

shown in Figure

1-3.

This figure shows the deformation called

"simple shear," in which the sample

is

contained between two fiat

plates with a fixed spacing, h, between them. The upper plate

moves in a direction parallel to itself while the lower plate is

stationary. The shear stress

is

the shear force divided by the

tangential area. We will use the symbol

(T ,

with no subscript, to

refer to the shear stress in a simple shear deformation.

shear force

T=

- - - - - - - - - - - -

tangential area

(1-2)

1.4 STRAIN AS A MEASURE OF DEFORMATION

In the previous section,

we

stated that shear stresses and normal

stresses are useful measures of the forces that act to deform a

material. Now we need a quantitative measure of deformation that

is rheologically significant. The description

of

deformation in terms

of

strain

is

more complex than the description of force in terms of

stress, and there are many alternative, rheologically significant,

measures of strain. While we

will consider this question further in

Chapter 3, we

will

define here only those measures that are useful


26/682


in the description of deformations commonly used to make rheolog

ical measurements, namely simple shear and simple extension.

In

Section 1.9 a strain measure for small deformations that

is

not

restricted to describing simple shear or simple extension will be

defined.

The thing that complicates the definition of a measure of strain

is

that it is necessary to refer to two states of a material element.

In

other words, it is not possible to specify the strain of a material

element unless we specify at the same time the reference state

relative to which the strain

is

measured. In the case of an elastic

material that cannot

flow,

for example a crosslinked rubber, this

is

straightforward, because there is a unique, easily identifiable, un

strained state that a material element will always return to when

ever deforming stresses are not acting.

For materials that

flow,

i.e., fluids, such a unique reference state

does not exist. In the case of a well-controlled experiment, however,

in which a simple homogeneous deformation is imposed on a

sample initially at rest and free

of

all deforming stresses, this initial

condition provides a meaningful reference state with respect to

which strain can be defined. We will make use of this fact in the

next two sections to define strain measures for simple extension and

simple shear.

1 4.1 Strain Measures for Simple Extension

Consider the simple extension test illustrated in Figure

1-4.

Let Lo

be the length of the sample prior to the application of a tensile

stress and

L

the length after deformation has occurred. A simple

measure of the deformation

is

the quantity (L - Lo). However, this

quantity

is

meaningful only in terms of a specific sample, whereas

we desire a measure of deformation that describes the state of a

material element. We can easily form such a quantity by dividing

this length difference

by

the initial length to obtain the "linear

strain" for simple extension.

(l-3)

For a uniform deformation, every material element of the sample

experiences this same strain. For example, if the initial length at

time to of a material element measured in the direction of stretch-


27/682


F

8X,(t)

Figure 1-4. Quantities used to describe simple extension.

ing

is

oX

1

(t

O

) '

and the length at a later time, t, after deformation

has occurred

is

OX/f),

the linear strain

of

the material element

is:

(1-4)

This measure of deformation has some convenient features. I t

is

independent of sample size, and it

is

zero

in

the unstressed, initial

state.

However, it is not the only measure of deformation that has these

properties. Another is the Hencky strain, which

is

defined as

follows

in

terms

of

the length of a material element.

(1-5)

For a sample with initial length Lo undergoing uniform strain the

Hencky strain can also be expressed as:

(1-6)

For materials that

flow,

e.g., molten plastics, this quantity

is

more

useful than the linear strain. In fact, the linear and Hencky strains


28/682


become equivalent in the limit

of

very small strains. This can be

demonstrated for simple extension

by

noting that

e = In{1

+ S)

and

that the first term of the series expansion of

In{1

+

S)

is

S.

The Hencky strain

rate

is

also a useful quantity for describing

rheological phenomena in simple extension. This

is

defined in

Equation

1-7

in terms of the length,

L,

of

the sample.

i: =

d In(L)/dt

(1-7)

We note that the initial length does not enter into the Hencky

strain rate but does enter into the linear strain rate

dS

/

dt.

1.4.2 Shear Strain

Now consider simple shear, which

is

the type of deformation most

often used to make rheological measurements on fluids. Referring

to Figure

1-3,

an obvious choice

of

a strain measure

is

the displace

ment of the moving plate,

ax,

divided

by

the distance between the

plates,

h.

'Y

=

aX/h

(1-8)

Referring to the two material particles shown in Figure 1-5 rather

than to the entire sample,

we

can define the

shear strain, 'Y,

for the

fluid element located at

(Xl ' X

2

, X

3

)

as

(1-9)

where

aX

I

is

the distance, measured in the Xl direction, between

two neighboring material particles that are separated

by

a distance

Figure 1-5. Two material particles in simple shear.


29/682


~ X 2 in the

x

2

direction. In the absence

of

edge effects, i.e., for a

uniform deformation, every fluid element will undergo the same

strain, and the local shear strain will everywhere be equal to the

overall sample strain:

(1-10)

And the shear rate is simply the rate of change of the shear strain

with time:

1 dX V

Y=hdt=-;;

(1-11)

where V is the velocity

of

the moving plate.

1.5 RHEOLOGICAL PHENOMENA

In this section we will examine the general types of rheological

behavior that can be exhibited by materials. These are elasticity,

viscosity, viscoelasticity, structural time dependency, and plasticity.

Although we will use simple extension and simple shear behavior in

this section to describe rheological phenomena, it is important to

remember

that

for rheologically complex materials such as poly

meric liquids,

the

behavior observed in these simple deformations

does not tell

the

whole rheological story.

1.5.1

Elasticity: Hooke's Law

Elasticity

is

a type

of

behavior in which a deformed material returns

to its original shape whenever a deforming stress is removed. This

implies that a deforming stress is necessary to produce and main

tain any deviation in shape from the original (unstressed) shape,

i.e., to produce strain.

The

simplest type of elastic behavior is that

in which the stress required to produce a given amount

of

deforma

tion is directly proportional to the strain associated with that

deformation.

For

example, in simple extension this can be ex

pressed as:

(1-12)


30/682

INTRODUCTION TO RHEOLOGY

11

The constant

of

proportionality, E, is called Young's modulus. This

relationship is called Hooke's law. The corresponding form of

Hooke's law for simple shear

is:

er

= Gy (1-l3)

where G is the shear modulus or modulus of rigidity. We note that

in a purely elastic material like this, all the work done to deform

the material

is

stored as elastic energy and can be recovered when

the material

is

permitted to return to its equilibrium configuration.

Another way

of

describing elastic behavior

is

to specify the strain

that results from the application of a specific stress. For a Hookean

material we have, for simple shear:

y = Jer (1-14)

where J is the shear compliance. Obviously, for a material following

Hooke's

law:

J = I /G

( 1-15)

1.5.2 Viscosity

Viscosity

is

a property of a material that involves resistance to

continuous deformation. Unlike elasticity, the stress is not related

to the amount of deformation but to the rate of deformation. Thus

it is a property peculiar to materials that flow rather than to solid

materials. We will consider first the simplest type

of

rheological

behavior for a material that can

flow.

For simple shear this type of

behavior

is

described

by

a linear relationship between the shear

stress and the shear rate:

er

= 'Y1Y

(1-16)

where

'Y1

is the viscosity. A material that behaves in this way is

called a Newtonian fluid.

For a Newtonian fluid, the viscosity

is

a "material constant,"

in that it does not depend on the rate or amount of strain. Single

phase liquids containing only

low

molecular weight compounds are


31/682


1)0

>

r-

OO

o

o

(f)

'>

SHEAR RATE (1)

Figure

1-6.

Viscosity-shear rate curve for a typical molten polymer.

Newtonian for all practical purposes. For multiphase systems, for

example suspensions and emulsions, and for polymeric liquids, the

relationship between stress and strain rate is no longer linear and

cannot be described in terms of a single constant. It is still conve

nient, however, to present the results of a steady simple shear

experiment in terms of a viscosity function 1](

y)

defined as follows:

(1-17)

where a is the shear stress and y is the shear rate. A typical

viscosity-shear rate curve for a molten polymer

is

shown in Figure

1-6.

The important features of this curve are listed below.

1. At sufficiently low shear rates, the viscosity approaches a

limiting constant value

1]0

called the zero shear viscosity.

2. The viscosity decreases monotonically as the shear rate

is

increased. This type of behavior is called shear thinning. (An

older terminology is "pseudoplastic.")

3. At sufficiently high shear rates the viscosity might be expected

to level off again, although a high-shear-rate limiting value is

not observed in melts, because viscous heating and polymer

degradation usually make it impossible to carry out experi

ments at sufficiently high shear rates. Specific forms for the

viscosity function are presented in Chapter 4.


32/682


1 5.3 Viscoelasticity

Polymeric materials, including solutions, melts, and crosslinked

elastomers, exhibit both viscous resistance to deformation and

elasticity. In the case

of

a vulcanized (crosslinked) rubber, flow

is

not possible, and the material has a unique configuration that it will

return to in the absence

of

deforming stresses. However, the viscous

resistance to deformation makes itself felt

by

delaying the response

of

the rubber to a change in stress. To illustrate this point, consider

the phenomenological analog

of

a viscoelastic rubber shown in

Figure 1-7. This mechanical assembly consists

of

a spring in parallel

with a dashpot. The force in the spring

is

assumed to be propor

tional to its elongation, and the force in the dashpot

is

assumed to

be proportional to its rate

of

elongation. Thus, the spring

is

a linear

elastic element, in which the force

is

proportional to the extension,

X,

and the dashpot

is

a linear viscous element in which the force

is

proportional to the rate of change of

X.

Note that this assembly

will always return to a unique length, the rest length of the spring,

when no force

is

acting on it. This assembly, called a Voigt body,

is

not intended to be a physical

or

quantitative model for a rubber.

F

Figure 1-7. Voigt body analog of a viscoelastic solid.


33/682

14 MELT RHEOLOGY AND ITS ROLE

IN

PLASTICS PROCESSING

However, the qualitative characteristics of its response to changes

in force are similar in some ways to those exhibited

by

rubbers.

Consider how this assembly would respond to the sudden appli

cation of a tensile load, F. This

is

called a "creep test". The force,

F,

is the sum

of

the force in the spring, KeX, and that in the

dashpot, Kv(dX/dt). Thus:

(1-18)

We note that some

of

the work put into the assembly to deform it is

dissipated in the dashpot, while the remainder is stored elastically

in the spring.

I f

X is initially zero, and the force F is suddenly

applied at time t

= 0,

this differential equation can be solved to

yield:

(1-19)

The important point to note is that the viscous resistance to

elongation introduces a time dependency into the response

of

the

assembly, and that this time dependency is governed by the ratio

(K jK

e

), which has units of time. I f we take the force to be

analogous to the deforming stress in a viscoelastic material, and the

elongation to be analogous to strain,

we

see that a viscoelastic

rubber has a time constant and cannot respond instantaneously to

changes in stress. This is called a "retarded" elastic response. As

the time constant approaches zero, the behavior becomes purely

elastic.

Now we turn to the case of an elastic liquid. To illustrate certain

qualitative features

of

the rheological behavior

of

such a material,

consider the mechanical assembly shown in Figure

1-8.

This assem

bly,

called a Maxwell element, consists of a linear spring in series

with a linear dashpot. Note first that unlike the Voigt body, this

assembly has no unique reference length and will deform indefi

nitely under the influence of an applied force, assuming the dash

pot is infinite in length. This is analogous to the behavior of an

uncrosslinked polymeric material above its glass transition

and

melting temperatures. Such a material will

flow

indefinitely when

subjected to deforming stresses.

Now we examine the force on the Maxwell element when it

is

subjected to a sudden stretching by an amount XO' The force, but


34/682


F

Figure 1-8. Maxwell element analog of a viscoelastic liquid.

not the displacement,

is

the same in both the spring and dash pot.

Thus:

(1-20)

Again we note that some of the work done

is

dissipated in the

dashpot and the remainder is stored in the spring. The total

displacement of the assembly,

Xo, is

the sum

of Xe

and Xv:

(1-21)

Thus:

(1-22)

This ordinary differential equation can be solved to yield

(1-23)

Note that

(Kv/Ke)

is a time constant. The force thus decays or

relaxes exponentially. I f we take the force to be analogous to the

deforming stress in an elastic liquid and the elongation to be


35/682

16 MELT RHEOLOGY

AND

ITS ROLE

IN

PLASTICS PROCESSING

analogous to the strain, this process is analogous to a stress relax

ation experiment. As in the case

of

the viscoelastic rubber, we note

that

the

combination

of

viscous and elastic properties endows

the

material with a characteristic time and makes its response time

dependent. As this "relaxation time" becomes shorter and shorter,

however, it becomes more and more difficult to devise an experi

ment

that

will reveal

the

elastic nature

of

the liquid, and its

behavior appears more and more like that

of

a purely viscous

material.

When we examine the rheological behavior

of

actual polymeric

materials, we find creep and relaxation behavior that

is

qualitatively

like those described above. In particular, the response to a sudden

change in stress or strain is always time dependent, never instanta

neous, and there is both elastic storage

of

energy and viscous

dissipation.

On the other

hand, the creep and relaxation curves

cannot be described by a single exponential function involving a

single characteristic time.

1

As is explained in Section 2.5, however,

practical use can still be made

of

the concept

of

a relaxation time

by describing the viscoelastic behavior

of

real materials in terms

of

a spectrum

of

relaxation times.

1 5.4 Structural Time Dependency

In

our

discussion of the viscosity function, we took the shear stress

to be independent of time at constant shear rate. For a Newtonian

fluid this

is

appropriate, because the stress responds instanta

neously to

the

imposition of a constant shear rate. However, non

Newtonian fluids may not respond instantaneously so that when the

shearing deformation is begun, there is a transient period during

which

the

shear stress varies with time, starting from zero and

finally reaching a steady state value that can be used to calculate

the viscosity by use of Equation 1-17. The origin of this time

dependency may be a flow-induced change in the structure

of

the

fluid, as in the case of a concentrated suspension of solid particles.

For

example, the state of aggregation

of

the suspended particles

can

be

changed significantly

by

shearing. This "structural time

IOther

deficiencies of these simple analogs are that the Voigt body does not exhibit stress

relaxation and the Maxwell element does not exhibit retarded creep.


36/682

m

m

w

II:

I

m

II:


37/682


1.5.5 Plasticity and Yield Stress

The structure in a concentrated suspension can be sufficiently rigid

that it permits the material to withstand a certain level of deform

ing stress without flowing. The maximum stress that can be sus

tained without

flow is

called the "yield stress," and this type

of

behavior

is

called "plasticity." Metals generally exhibit plasticity, as

do semicrystalline polymers at temperatures between their melting

and glass transition temperatures. Highly filled melts are also

thought to have a yield stress, although precise measurement

of

this

property

is

difficult.

The simplest type of plastic behavior

is

that in which the excess

stress, above the yield stress,

is

proportional to the shear rate.

For

simple shear

flow,

this type

of

behavior

is

described

by

Equation

1-24.

(1-24)

Here,

(To is

the yield stress and

7J

p

is

the plastic viscosity. A material

that behaves in this way

is

called a "Bingham plastic." This

is

an

idealized type of behavior that

is

not precisely followed by any real

material, but it

is

sometimes a useful approximation to real behav-

,.....

~

en

en

w

a:

I

en

a:


38/682


ior. Note that Equation 1-24 is not a constitutive equation, as it

does not describe all the components

of

the stress tensor in any

type

of

deformation but only the shear stress in simple shear.

Figure 1-10 compares shear stress versus shear rate curves for a

Bingham plastic, a Newtonian fluid and a shear thinning fluid.

1.6 WHY POLYMERIC LIQUIDS ARE NON-NEWTONIAN

It

is

important in applied polymer science to be able to relate

physical properties, including rheological properties, to molecular

structure. This subject

is

taken up in some detail in Chapters

2,

4

and 10. We

will

mention here only the general mechanisms

by

which polymeric molecules endow liquids with complex rheological

behavior.

First, we should examine the question of why polymer molecules

are elastic. Our physical picture of a polymer molecule is that of a

long chain with many joints allowing relative rotation of adjacent

links. The presence

of

this large number of joints makes the

molecule quite flexible and allows many different configurations of

the molecule. At temperatures above the glass transition tempera

ture a molecule

will

continually change its configuration due to

Brownian motion, but

we

can describe the state of a large number

of molecules in terms of statistical averages. For example, at a given

temperature there will be a unique average value of the end-to-end

distance, R, for the molecules of a polymeric liquid that has been at

rest for a sufficient length

of

time that it is in its equilibrium state.

Deforming the liquid will alter this average length, but if the

deformation is stopped, Brownian motion will tend to return the

average value of

R

to its equilibrium value. This

is

the molecular

origin of the elastic and relaxation phenomena that occur in poly

meric liquids.

1 6.1 Polymer Solutions

We consider first the behavior of a dilute solution in which the

forces acting on the polymer molecule are primarily those due to

the flow of the solvent. This situation is much simpler than that

existing in concentrated solutions and melts, where the rheological

behavior is governed

by

interactions between polymer molecules. In


39/682


40/682


observed that the short term response to rapid deformations of high

molecular weight molten polymers

is

very similar to that

of

a

crosslinked rubber. This has inspired the concept

of

a "temporary

network" that exists in the melt and acts like a rubbery network at

shorter times but whose "junctions" can slip over longer periods

of

time to permit

flow.

The network

is

sometimes said to arise from

"entanglements" in the melt. However, the modern view

is

that the

rubbery behavior

of

melts

is

due not to an actual looping

or

knotting

of

molecules around each other but simply to the con

straints on their motion resulting from the fact that molecules

cannot cut through each other.

Entanglements occur because

of

the high degree

of

spatial over

lap

of

the molecules. The existence

of

overlap

is

readily demon

strated by considering the measured size of the polymer coils. One

measure of molecular size is the "radius of gyration," R

g

For

linear polyethylene,

Rg

depends on the molecular weight,

M

as

follows:

Rg(cm)

= 4

X

10-

9

X M1/2

For a polyethylene with a molecular weight of 10

6

glmol, the

volume

of

the sphere occupied

by

one molecule

is

therefore about

2.6 X 10-

16

cm

3

The mass of this coil

is

10

6

divided

by

Avogadro's

number, or 1.7

X

10-

18

g. The density of the coil in its occupied

volume

is

thus less than

0.01 g/cm

3

The observed melt density

of

about 0.77 g/cm

3

can only be accounted for if parts

of

many other

coils are present in the volume occupied

by

this coil.

A similar calculation for a polymer with a molecular weight

of

10

4

glmol

gives a density

of

0.1

g/cm

3

,

which

is

considerably closer

to the measured bulk density. This shows that the degree

of

coil

overlap, and therefore the entanglement density increases sharply

with molecular weight.

Rubbery behavior occurs in a melt when the molecular weight

is

above some critical value that varies from one polymer to another.

Above this molecular weight the number

of

entanglements becomes

sufficient to produce strong rubberlike effects.

The

macroscopic effects

of

the strong interactions between poly

mer molecules in a melt include high viscosity and high elastic

recoil, especially just above the melting point or, in the case of an

amorphous polymer, just above the glass transition temperature.

At

the same time, the nature

of

this strong interaction can be altered


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IN

PLASTICS PROCESSING

temporarily

by

deformation so that high molecular weight melts

have highly nonlinear properties; for example, the viscosity is a very

strong function

of

the shear rate.

Polymeric materials are said to have a "memory," in that when

deforming stresses are eliminated, they tend to return to a previous

configuration. Crosslinked polymers have a "perfect memory" in

the sense that since their network is based on permanent chemical

crosslinks they always return to a unique equilibrium configuration,

whereas molten polymers are said to have a "fading memory," since

the entanglement network is not permanent and is altered

by flow

and relaxation processes.

1.7 A WORD ABOUT TENSORS

For those readers who have had little if any experience in the use of

tensor notation, the very word "tensor" probably suggests a mathe

matical system of impenetrable mystery. However, such readers

should have no fear. There

is no mystery While

we

do not claim to

offer here a complete course in tensor analysis, we do present in the

next two brief sections everything you

will

need to know about

tensors in order to describe the rheological properties of polymeric

liquids. After a careful reading of these sections, you too can

impress the uninitiated with your ability to use tensor notation to

describe rheological phenomena.

The concept

of

a tensor was introduced into physics, and thus

into rheology, because it

is

useful; without it, the quantitative

description of many physical phenomena would be hopelessly clumsy

and tedious. Because of this usefulness, most of the literature on

viscoelastic behavior makes some use of tensor notation. This

literature will be inaccessible to a reader having no familiarity with

tensor quantities. Moreover, we will use tensors extensively in

several chapters

of

this book.

In the first section, we stated that rheology involves the relation

ship between deformation (strain) and force (stress) for a material.

It

is

in the quantitative description of the quantities strain and

stress that tensor notation is virtually indispensable. However, be

fore demonstrating this, it

will

be useful to review briefly the

concept of a vector, as this

is

central to an understanding of tensors.


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1.7.1 Vectors

Certain physical quantities, such as force and velocity, are best

specified in terms of vectors, because a vector has a magnitude and

a direction.

One

example

is

the velocity vector,

v.

A vector can be

specified by giving its components,

VI' v

2

, and

v

3

' referring to the

velocities in the three directions, Xl'

x

2

, and

x

3

Generally, we can

refer to a typical velocity component as

Vi'

where i can be 1,

2, or 3.

Note that while the magnitude

is

a physical attribute

of

a vector

that does not depend on the choice

of

a particular coordinate

system, the components

of

the vector do depend

on

the coordinate

system selected to describe the

flow.

There

is

a simple rule that tells

how to use the components of a vector in one coordinate system to

calculate the components of that vector in a second coordinate

system that

is

rotated with respect to the first.

I f vectors are adequate to describe the velocity

of

a body and the

force acting on it, why are they not sufficient for describing rheolog

ical phenomena? The answer

is

that rheology deals not with motion

per

se, but with deformation, and specifically with the relationship

between the deformation

of

a fluid element and the surface forces

exerted on this element

by

the surrounding fluid. Tensors are very

useful in specifying these two types of quantities, and the specific

tensors that are used to represent these quantities are the strain

tensor and the stress tensor.

1.7.2 What is a Tensor?

Like a vector, a tensor can be represented in terms of its compo

nents, and the values

of

these components depend

on

the choice

of

the coordinate system used. Furthermore, there

is

a rule for

using the components

of

a tensor in one coordinate system to

calculate the components

of

that tensor in another coordinate

system, rotated with respect to the first.

The

existence

of

this rule

shows that a tensor has a basic physical significance that transcends

the arbitrary choice

of the coordinate system. However, unlike a

vector, the physical significance

of

a tensor cannot be described in

terms

of

a directed line segment, i.e., in terms

of

a magnitude and a

direction.

In describing rheological behavior associated with a particular

type

of

deformation, we generally select a coordinate system that


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IN

PLASTICS PROCESSING

gives the components a physical significance that

is

easily under

stood. For example, in describing the stretching

of

a rod ("uniaxial

extension") we take the

x

1

direction to be the direction

of

stretch

ing.

Then

the 0"11 component of the stress tensor

is

simply the

tensile stress in the sample.

Whereas a vector has three components, the tensors we will use

have nine. A typical component can be written using two indices,

for example O"ij' where each index can take on one

of

the values

1,

2, or 3 corresponding to the three coordinate directions. To present

the values

of

all the components of a tensor, matrix notation can be

used.

[

0"11

0 . = 0"21

IJ

0"31

(1-25)

Can any nine numbers form the components

of

a tensor? No. These

numbers have a specific mathematical significance, which we

will

find particularly suited to the description

of

deformation and stress.

Specifically, these nine components contain all the information

necessary to transform one vector into another one that has a

certain prescribed relationship with the first. In mathematical lan

guage we say that the tensor "operates on" one vector to yield a

second vector, which contains information taken from both the

original vector and the tensor. For example, we will see that the

strain tensor, i.e. the nine components of the strain tensor, can

be used to operate on the components of the vector describing the

relative position

of

fluid particles within an undeformed fluid ele

ment, to yield the corresponding position vectors after deformation.

Likewise, the stress tensor can be used to operate on the unit

normal vector defining the orientation

of

a surface

of

a fluid

element to yield the surface force vector acting on that element.

Since the vector operated on in both cases

is

an arbitrarily selected

one, we see that the strain tensor actually contains a complete

description

of

the deformation that a fluid element undergoes

during some flow process, while tile stress tensor contains a com

plete description

of

the state

of

stress acting at a point in the fluid

at a particular time.

Our

objective in this book

is

not to solve

flow

problems but only

to describe rheological phenomena. Thus, tensor calculus will not


44/682


be required, and the reader need learn no new mathematics but

only the definitions

of

a

few

quantities.

With regard to notation,

we

will use a bold face symbol to

indicate that it

is

a vector. For example, the velocity vector will be

represented as v. The components

of

a vector will be indicated by

means of a subscript, for example,

Vi'

For the components of a

tensor we will use two subscripts. For example, the components of

the stress tensor will be represented

by

ui j ' In order to minimize the

number

of

new symbols and rules that need be learned, we will not

use dyadic notation or the Einstein summation convention. These

are methods

of

notation that simplify the writing

of

equations

involving tensors, and they are described in the book

by

Aris

[1].

1.8 THE STRESS TENSOR

The deformations that occur in the processing and use

of

materials

are generally more complicated than simple extension and simple

shear and involve a combination

of

these two types

of

deformation.

For example, consider the deflection of a rubber tire under load or

the

flow of

a molten plastic into a mold. First, the deformation

is

not uniform but varies from one place to another within the

material. I t thus becomes a "field variable," i.e., a quantity that

varies from one point to another and

is

thus a function

of

position.

Secondly, the stress

is

not purely tensile, compressive or shear.

The quantitative specification of the forces acting on a solid

body as a result

of

contact with another body

is

straightforward;

one need only give the components

of

the force vector acting at the

interface. However, the specification

of

the forces acting on the

surface of a fluid element is less obvious, since the orientation of

the surface

is

arbitrary, i.e., it depends on how one defines a fluid

element.

I t

would appear that in order to completely specify the

state of stress on a fluid element one would have to give the

components

of

the stress vector for every possible orientation

of

the surface. Fortunately, this

is

not the case, and we will see that by

specifying only the components

of

the stress

teDSQr,

the state of

stress at a point in a fluid can be completely de,s.cribed. -

For a given, arbitrary, choice

of

a fluid element, the surface stress

vector

is t(o),

where n

is

the unit normal vector for the surface and

specifies its orientation. The (n) superscript on the surface stress


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> _ _ _ _ x,

Figure 1-11. Cubical material element with a typical stress component.

vector indicates that the components of the vector depend on the

orientation of the surface. This vector does not at first seem to

be

a

useful tool for describing the state

of

stress at a point in a fluid

because of the arbitrariness of the choice of the surface orientation.

I t is possible to show, however, that if the stress vectors acting on

each of three mutually perpendicular planes passing through a

point in a fluid are specified, the stress vectors for any other choice

of planes can be calculated

by

means of a simple transformation

rule [1].

I t

is convenient to let these planes be perpendicular to the

coordinate axes. Thus, the unit normal vector for a surface becomes

equal to one of the unit normal vectors for the coordinate system:

and the components

of

the force vector are given by:

t(e.l

=

(T .

]

I ]

where (Tij is the stress tensor.

2

To understand the physical significance of the nine components

of the stress tensor, consider the small cubical element of material

shown in Figure 1-11. The second subscript indicates the direction

of the force and corresponds to the coordinate axis direction.

For

example, the stress component shown in Figure

1-11

acts in the

X l

2The components

of

the surface stress vector for any other surface whose orientation

is

defined by the unit normal vector, n, can be determined as follows:


46/682


- + - - - ~ a l 1

/ ' - - - - - -x ,

Figure 1-12. Several additional stress components.

direction, and the second subscript of this component is thus 1. The

first subscript indicates the face on which the component acts, and

this is specified by reference to the coordinate direction normal to

this face. Thus, the force shown acts on a face normal to the x

2

direction, and the first subscript of this component of the stress is

thus 2. The stress component shown

is

thus

(T21'

To complete our definition

of

the components

of

(Tij ' we need a

sign convention. In this book, we will use the convention generally

used in mechanics, although the reader should be aware that the

opposite convention is used

by

some rheologists [2-4]. We will take

the stress to be positive when it acts in the positive Xj direction, on

a face having the higher value of

Xi'

i.e., the face further from the

origin in the Xi direction. For example, the stress component shown

in Figure

1-11

is positive if the force acts in the direction of the

arrow. This is because it acts in the positive X l direction on a face

having the higher value of

x

2

Figure 1-12 shows several additional

components of the stress.

The set of nine components that is needed to specify completely

the state of stress at a point in a deformable material is an example

of a "second order tensor," and the members of the set are said to

be

the

"components" of the tensor. Since the components

of

the

stress tensor describe the state of stress at a point in the material,

the cubical element shown in" Figure

1-11

must be shrunk to an

infinitesimal size. Thus, the two force vectors shown in Figure

1-13

are acting in opposite directions at the same point. From Newton's

law

of

action and reaction, these two forces must be equal in

magnitude. They are thus both manifestations of the same compo-


47/682


nent

of

the stress tensor, 0"1l' and both have positive values if they

act in the directions indicated. Thus, according to our sign conven

tion, a tensile stress has a positive value.

111

- - - - - - - I ~ 1 1 1

Figure 1-13. Equal, opposite forces at a point are represented

by

the same component

of

the

stress tensor.

The principle

of

conservation

of

angular momentum can be

applied to the infinitesimal material element we have been consid

ering to show that the stress tensor has the following property:

(1-26)

Thus, any two components that have the same subscripts

or

in

dexes, but in reversed order, have the same value. A tensor that has

this property is said to be "symmetric." One result of this property

is that a symmetric tensor has only six independent components

rather than the nine that would be required to completely specify a

nonsymmetric tensor.

To make more concrete our discussion of stress, consider the

simple shearing deformation shown in Figure 1-14. There is a more

or

less universal convention in describing this flow, and it is that the

direction of motion is

Xl '

while the velocity varies in the X

2

direction. To generate this deformation, a force is applied to the

upper plate in the direction shown by the arrow. In the ideal case,

(fully developed

flow

with no edge effects) this force generates a

uniform stress in the sample. Since the force

is

in the

Xl

direction

and acts on a face perpendicular to the X 2 direction, the stress

generated by the force

F

is 0"21' Obviously, this is a shear stress.

Due

to the symmetry of the stress tensor this is equal to 0"12' We

Figure 1-14. Simple shear index convention.


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will use the symbol,

( T ,

with no subscripts, to mean this component

of the stress tensor in simple shear. Thus for simple shear,

(1-27)

where

A

is the area of the sample in contact with the plates. The

other shear stress components are zero:

(1-28)

We can now describe, using matrix notation, the state

of

stress in a

material subjected to simple shear:

(1-29)

Another example of a test that

is

of practical interest in rheology

is simple or uniaxial extension. This test is illustrated in Figure 1-2.

I f we let x

1

be the direction of the applied force, the stress

component resulting from this force will be

(T11'

which is a normal

stress.

I f

it acts in the direction shown, it is a tensile stress. There

are no shear stress components in this case, and the components

of

the stress tensor are as shown below:

[

(Tll

( T .

= 0

I ]

o

o

(1-30)

There is an additional point regarding normal stresses that should

be mentioned here. While all materials are compressible to some

extent, in the case of molten plastics, quite high pressures are

required to produce a significant change in the volume of a sample.

For

this reason, for many purposes these materials can be consid

ered to be incompressible. Now consider what happens when we

subject an incompressible material to a compressive

or

tensile stress

that is equal in all directions, i.e., an isotropic stress or "pressure."


49/682


The

components of the stress tensor in this situation are as shown

below.

(1-31)

A stress of this type

is

said to be "isotropic." The minus signs result

from the fact that pressure is considered positive when it acts to

compress a material, whereas according to our sign convention

compressive stress is a negative quantity. In this situation, the

sample will be totally unaffected

by

the forces associated with this

pressure, i.e., it will not change its size or shape. Thus, such an

isotropic stress field is of no rheological significance. Only when

there are shear stresses acting,

as

in simple shear flow, or when the

normal stress components are different from each other will defor

mation occur in an incompressible material.

Another way of saying this is that if a rheological measurement

on an incompressible material

is

repeated at several different

ambient pressures, for example

by

placing the rheometer in a

hyperbaric chamber, the measurements at various pressures will

yield exactly the same values of all rheological properties.

This means that for an incompressible material a normal compo

nent of stress has no absolute rheological significance. Only

differences between two normal components are of rheological

significance. For example, in the case of simple shear, it is custom

ary to describe the state of stress from a rheological point of view

by specifying the shear stress, a, and the "first and second normal

stress differences."

(1-32)

(1-33)

For Newtonian fluids these two quantities are zero in simple shear,

but in polymeric liquids they generally have nonzero values. One

manifestation

of

the first normal stress difference is observed when

a liquid

is

sheared

by

placing it between two flat parallel disks and

rotating one of the disks. It is found that an elastic liquid exerts a

normal thrust tending to separate the plates, while a Newtonian

fluid exerts no normal thrust on the plates.


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In the case of simple extension, there

is

only one rheologically

significant feature of the stress field, because there are

melt rheology and its role in plastics processing by dealy.pdf

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