melt rheology and its role in plastics processing by dealy.pdf
TRANSCRIPT
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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
Melbourne, Victoria 3000, Australia
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
4.1 Introduction 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
5.1 Introduction 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
6.1 Introduction 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
7.1 Introduction 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
11.1 Introduction 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
16.1 Introduction 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
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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
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INTRODUCTION TO RHEOLOGY 5
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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6 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
--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
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INTRODUCTION TO RHEOLOGY 7
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-
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8 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
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
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INTRODUCTION TO RHEOLOGY 9
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.
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10 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
~ 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)
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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
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12 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
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.
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INTRODUCTION TO RHEOLOGY 13
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.
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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
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INTRODUCTION TO RHEOLOGY 15
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
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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.
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m
m
w
II:
I
m
II:
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18 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
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:
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INTRODUCTION TO RHEOLOGY 19
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
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INTRODUCTION TO RHEOLOGY 21
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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22 MELT RHEOLOGY AND ITS ROLE
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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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INTRODUCTION TO RHEOLOGY 23
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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24 MELT RHEOLOGY AND ITS ROLE
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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
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INTRODUCTION TO RHEOLOGY 25
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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26 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
> _ _ _ _ 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:
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INTRODUCTION TO RHEOLOGY 27
- + - - - ~ 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-
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28 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
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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INTRODUCTION TO RHEOLOGY 29
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."
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30 MELT RHEOLOGY AND ITS ROLE IN PLASTICS PROCESSING
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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INTRODUCTION TO RHEOLOGY 31
In the case of simple extension, there
is
only one rheologically
significant feature of the stress field, because there are