Polymer Viscoelasticity: Basics, Molecular Theories, Experiments and Simulations

441 Pages·2010·3.126 MB·English

by  Yn-Hwang Lin

Polymer
Viscoelasticity
Basics, Molecular Theories,
Experiments and Simulations
2nd
Edition
7786 tp.indd 1 9/9/10 11:27 AM
September6,2010 9:42 WSPC/BookTrimSizefor9inx6in b959-fm FA
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Y n - H w a n g L i n
National Chiao Tung University, Taiwan
Polymer
Viscoelasticity
Basics, Molecular Theories,
Experiments and Simulations
2nd
Edition
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
7786 tp.indd 2 9/9/10 11:27 AM
Published by
World Scientific Publishing Co. Pte. Ltd.
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Library of Congress Cataloging-in-Publication Data
Lin, Y.-H.
Polymer viscoelasticity : basics, molecular theories, experiments, and
simulations / Yn-Hwang Lin. -- 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-981-4313-03-2 (hardcover : alk. paper)
ISBN-10: 981-4313-03-3 (hardcover : alk. paper)
1. Polymers--Viscocity. 2. Viscoelasticity. I. Title.
QD281.P6L56 2010
620.1'9204232--dc22
2010038333
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd.
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Contents
Preface xiii
Preface to the Second Edition xvii
1. Conformation of Polymer Chains 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Probability Distribution Functions, Moments
and Characteristic Functions . . . . . . . . . . . . . . . 2
1.3 A Central Limit Theorem . . . . . . . . . . . . . . . . . 5
1.4 The Freely Jointed Chain Model . . . . . . . . . . . . . 7
1.5 Distribution of the End-to-End Vector . . . . . . . . . . 10
1.6 The Gaussian Chain . . . . . . . . . . . . . . . . . . . . 11
Appendix 1.A — The Dirac Delta Function . . . . . . . . . . . 13
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2. Rubber Elasticity 16
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Entropy and Rubber Elasticity . . . . . . . . . . . . . . 17
2.3 Molecular Theory for Rubber Elasticity . . . . . . . . . 18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3. Polymer Chain Dynamics 26
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 The Smoluchowski Equation . . . . . . . . . . . . . . . . 28
3.3 The Langevin Equation . . . . . . . . . . . . . . . . . . 32
3.4 The Rouse Model . . . . . . . . . . . . . . . . . . . . . . 33
v
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3.5 Diffusion Motion of the Rouse Chain . . . . . . . . . . . 35
3.6 The Rouse Normal Modes of Motion . . . . . . . . . . . 35
Appendix 3.A — Eigenvalues and Eigenvectors
of the Rouse Matrix . . . . . . . . . . . . . . . . . . . . 40
Appendix 3.B — The Langevin Equation of a Particle
in a Harmonic Potential . . . . . . . . . . . . . . . . . . 42
Appendix 3.C — The Continuous Rouse Model . . . . . . . . . 43
Appendix 3.D — Binomial Random Walk . . . . . . . . . . . . 46
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4. Linear Viscoelasticity 51
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Maxwell Equation . . . . . . . . . . . . . . . . . . . . . 52
4.3 Boltzmann’s Superposition Principle . . . . . . . . . . . 57
4.4 Relaxation Modulus . . . . . . . . . . . . . . . . . . . . 58
4.5 Steady-State Shear Flow . . . . . . . . . . . . . . . . . . 60
4.6 Dynamic-Mechanical Spectroscopy . . . . . . . . . . . . 61
4.7 Steady-State Compliance. . . . . . . . . . . . . . . . . . 65
4.8 Creep Compliance . . . . . . . . . . . . . . . . . . . . . 71
Appendix 4.A — The Hopkins–Hamming Method
for the Conversionof G(t) into J(t) . . . . . . . . . . . . 75
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5. Stress and Strain 78
5.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Finite Strain . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 A neo-HookeanMaterial . . . . . . . . . . . . . . . . . . 90
5.4 A Newtonian Fluid . . . . . . . . . . . . . . . . . . . . . 92
Appendix 5.A — Tensor Operations . . . . . . . . . . . . . . . 94
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6. Molecular Theory of Polymer Viscoelasticity —
Elastic Dumbbell Model 98
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2 The Smoluchowski Equation for an Elastic Dumbbell . . 99
6.3 Rheological Constitutive Equation of the Elastic
Dumbbell Model . . . . . . . . . . . . . . . . . . . . . . 104
6.4 Applications of the Constitutive Equation . . . . . . . . 109
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Appendix 6.A — Codeformational (Convected)
Time Derivative . . . . . . . . . . . . . . . . . . . . . . . 113
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7. MolecularTheoryofPolymerViscoelasticity—The
Rouse Model 119
7.1 The Smoluchowski Equation of the Rouse Model . . . . 119
7.2 Rheological Constitutive Equation
of the Rouse Model . . . . . . . . . . . . . . . . . . . . . 127
Appendix 7.A — Eigenvalues and the Inverse
of the Rouse Matrix . . . . . . . . . . . . . . . . . . . . 130
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8. Molecular Theory of Polymer Viscoelasticity —
Entanglement and the Doi–Edwards
(Reptation) Model 133
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.2 The Primitive Chain . . . . . . . . . . . . . . . . . . . . 135
8.3 Diffusion Motion . . . . . . . . . . . . . . . . . . . . . . 138
8.4 Relaxation Modulus . . . . . . . . . . . . . . . . . . . . 141
8.5 Relaxation of Stress by Reptation . . . . . . . . . . . . . 146
Appendix 8.A — Tension in a Gaussian Chain Between
Two Fixed Points . . . . . . . . . . . . . . . . . . . . . . 150
Appendix 8.B — Equivalent Expressions
for Rubber Elasticity . . . . . . . . . . . . . . . . . . . . 151
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9. MolecularTheoryofPolymerViscoelasticity—The
Extended Reptation Model 153
9.1 Intramolecular Processes . . . . . . . . . . . . . . . . . . 153
9.2 Contour Length Fluctuations of the Primitive Chain . . 156
9.3 Relaxation Processes before t≈Teq . . . . . . . . . . . . 161
9.4 Universality of the G(t) Line Shape in Terms of the
Extended Reptation Theory (ERT) . . . . . . . . . . . . 164
9.5 Zero-Shear Viscosity and Steady-State Compliance . . . 165
9.6 Note: A Clarification of the Term “Transition Region” . 167
Appendix 9.A — Contour Length Fluctuations
of the Primitive Chain . . . . . . . . . . . . . . . . . . . 168
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Appendix 9.B — Rouse Motions in an Entanglement Strand:
The Rouse–Mooney Normal Modes of Motion . . . . . . 171
Appendix 9.C — Eigenvalues and Eigenvectors
of the Rouse–Mooney Matrix . . . . . . . . . . . . . . . 176
Appendix 9.D — Hierarchy and Universality among
the Inherent Characteristic Times in the (Extended)
Slip-Link Model . . . . . . . . . . . . . . . . . . . . . . . 177
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
10. Comparison of the Extended Reptation Theory
(ERT) with Experiments 182
10.1 Effects of the Molecular-Weight Distribution
of the Sample . . . . . . . . . . . . . . . . . . . . . . . . 183
10.2 Analysis of the G(t) Line Shape . . . . . . . . . . . . . . 186
10.3 Zero-Shear Viscosity and Steady-State Compliance . . . 197
10.4 Viscoelasticity and Diffusion . . . . . . . . . . . . . . . . 205
10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Appendix 10.A — The Reason Why G(cid:1)(cid:1)(ω) Should
Be Excluded from the Line-Shape Analysis in Terms
of the Rouse Theory or the ERT If Only the Entropic
Region Is to Be Covered . . . . . . . . . . . . . . . . . . 211
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
11. ERTvs. RouseTheory, Concentration Dependence
and Onset of Entanglement, and Tube Dilation 215
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 215
11.2 Entanglement Region. . . . . . . . . . . . . . . . . . . . 217
11.3 Entanglement-Free Region and Onset of Entanglement . 226
11.4 Tube Dilation . . . . . . . . . . . . . . . . . . . . . . . . 234
Appendix 11.A — Basic Form of the Blending Law
in a Binary Blend . . . . . . . . . . . . . . . . . . . . . . 238
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
12. Molecular Theory of Polymer Viscoelasticity —
Nonlinear Relaxation Modulus of Entangled
Polymers 242
12.1 Chain-Tension Relaxation . . . . . . . . . . . . . . . . . 243
12.2 Comparison of Theory and Experiment. . . . . . . . . . 249
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
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Contents ix
13. Number of Entanglement Strands per
Cubed Entanglement Distance, nt 257
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 257
13.2 Theoretical Basis and Experimental Support for n
t
Being a Universal Constant . . . . . . . . . . . . . . . . 259
13.3 Concentration Dependence of n . . . . . . . . . . . . . 263
t
13.4 Packing of Polymer Chains . . . . . . . . . . . . . . . . 263
13.5 Some Comments . . . . . . . . . . . . . . . . . . . . . . 264
Appendix 13.A — The Rouse Segment vs. the Kuhn
Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
14. Glass Transition-Related Thermorheological
Complexity in Polystyrene Melts 269
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 269
14.2 G(t) Functional Forms . . . . . . . . . . . . . . . . . . . 271
14.3 J(t) Line-Shape Analyses of Entangled Systems . . . . . 273
14.4 Analyses of J(t), Je0 and η0 in an Entanglement-Free
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
14.5 Analyses of the G∗(ω) Spectra of Entanglement-Free
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
14.6 Dynamic Anisotropy in Entangled Systems . . . . . . . 298
14.7 Comparison of the Af and β Values Extracted
G
from the J(t) and G∗(ω) Line Shapes. . . . . . . . . . . 301
14.8 T Defined by the Structural Relaxation Time
g
τ =1,000sec. . . . . . . . . . . . . . . . . . . . . . . . 303
S
14.9 Dependences of τ , s(cid:1) and K(cid:1) on ∆T =T −T . . . . . 305
S g
14.10 Structure as Revealed in G(t) . . . . . . . . . . . . . . . 312
14.11 Frictional Slowdown and Structural Growth . . . . . . . 316
14.12 K Values in the Close Neighborhood of T . . . . . . . . 318
g
14.13 Internal Viscosity and Zero Shear Viscosity . . . . . . . 321
Appendix 14.A — Calculations of the G(cid:1)(ω) and G(cid:1)(cid:1)(ω)
Spectra from a G(t) Functional Form . . . . . . . . . . . 322
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
15. The Basic Mechanism for the Thermorheological
Complexity in Polystyrene Melts 328
15.1 The Basic Mechanism of the Thermorheological
Complexity (TRC) . . . . . . . . . . . . . . . . . . . . . 329