Dedication ...................................................... v
Preface ...................................................... xvii
Chapter 1 Monomeric Liquid Dynamics ............................ 1
1.1 Introduction ............................................... 1
1.2 Molecular Liquids .......................................... 3
1.3 Liquid State Transport Coefficients ........................ 7
1.3.1 Viscosity .......................................... 11
The Arrhenius Equation ............................. 13
The Fulcher-Tammann and Williams-Landel-Ferry
Equations .......................................... 13
Viscosity-Temperature Fits ......................... 16
1.3.2 Diffusion .......................................... 19
Diffusion Equations ................................ 19
Tracer and Self Diffusion .......................... 21
1.3.3 Diffusion Measurement Methods ...................... 24
1.4 The Viscosity-Diffusion Relationship ...................... 26
1.4.1 A Random Walk Approach ............................. 27
1.4.2 The Stokes-Einstein Equation ....................... 28
The Langevin Equation .............................. 28
Velocity and Displacement .......................... 30
Equipartition and Mean Displacement ................ 30
1.4.3 Orientational Diffusion and Relaxation ............. 34
1.4.4 Experimental Comparisons ........................... 35
1.5 Viscoelasticity in Monomeric Liquids ...................... 39
1.5.1 Modulus ............................................ 41
1.5.2 Relaxation Time .................................... 42
1.5.3 Common Theoretical Forms ........................... 45
1.6 The Class Temperature ..................................... 46
1.6.1 Supercooling and Vitrification ..................... 46
1.6.2 Structural Mixtures ................................ 47
1.6.3 Class Temperature Detection ........................ 50
Specific Volume .................................... 50
Specific Heat ...................................... 51
1.6.4 Effects on Viscosity ............................... 52
1.6.5 History Dependence ................................. 53
1.7 Analogies and Parallels ................................... 56
1.7.1 Hard Spheres ....................................... 56
1.7.2 Colloidal Spheres .................................. 59
1.7.3 Monomeric Spheres .................................. 61
1.8 Properties Near the Glass Temperature ..................... 62
1.8.1 The Strong-Fragile Classification .................. 62
1.8.2 Relaxation Time Distribution ....................... 65
Mechanical Response ................................ 66
Electromagnetic Response ........................... 66
1.8.3 Stokes-Einstein Departures ......................... 69
Self Diffusion ..................................... 70
Tracer Diffusion ................................... 71
Related Phenomena .................................. 72
1.8.4 Liquid Heterogeneity ............................... 74
A Two-Microstate Interpretation .................... 75
A More Detailed Picture ............................ 76
1.9 Theories about Vitrification ......................... 78
1.10 Free Volume .......................................... 82
Appendix 1.1 Listing and Characteristics of
Polymer Species .................................... 87
References ......................................... 93
Chapter 2 Linear Viscoelasticity ............................. 101
2.1 Polymeric Viscoelasticity ................................ 101
2.2 Linear Viscoelastic Functions ............................ 103
2.2.1 Stress Relaxation ................................. 104
2.2.2 Creep and Creep Recovery .......................... 106
2.2.3 Steady-State Oscillatory Response ................. 109
2.2.4 Experimental Aspects .............................. 115
Stress Relaxation ................................. 115
Creep and Recoil .................................. 116
Dynamic Compliance and Modulus .................... 116
2.3 Linear Response Properties ............................... 116
2.3.1 Limits of Linear Response ......................... 117
2.3.2 Boltzmann Superposition ........................... 118
2.3.3 Relationships between Response Functions .......... 119
Kramers-Kronig Relations .......................... 121
Limiting Functions ................................ 123
2.3.4 Complex Notation .................................. 124
2.3.5 Viscoelastic Models ............................... 124
Maxwell ........................................... 126
Burgers ........................................... 128
Davidson-Cole ..................................... 129
Barlow-Erginsav-Lamb .............................. 129
2.3.6 Auxiliary Response Functions ...................... 130
2.3.7 Linear Viscoelastic Parameters .................... 132
2.3.8 Extensional Deformation ........................... 135
2.4 Temperature Dependence ................................... 136
2.4.1 Time-Temperature Superposition .................... 136
2.4.2 Temperature Shift Factors ......................... 142
2.4.3 Thermorheological Complexity ...................... 144
2.5 Flow-Related Properties .................................. 148
2.5.1 The Terminal Dispersion ........................... 148
2.5.2 Plateau Modulus ................................... 151
2.5.3 Recoverable Compliance ............................ 156
2.5.4 Zero-Shear Viscosity .............................. 158
2.5.5 Commentary ........................................ 163
2.6 Molecular Origin of Polymeric Response ................... 164
2.6.1 Conformational Rearrangement ...................... 165
2.6.2 Relaxation of Stress .............................. 168
Appendix 2.1 Fourier and Laplace Transforms ....... 174
References ........................................ 175
Chapter 3 Stress Response to Shear Deformations .............. 179
3.1 Introduction ............................................. 179
3.1.1 Contact Forces, Traction, and Stress .............. 180
3.1.2 Elements of Matrix Algebra ........................ 189
3.1.3 Reference Frame ................................... 194
3.2 Conservation of Mass and Momentum ........................ 197
3.2.1 Equation of Continuity ............................ 197
3.2.2 Equation of Motion ................................ 198
3.2.3 Rheological Considerations ........................ 201
Constitutive Equations ............................ 201
Pressure .......................................... 201
Inertial Effects .................................. 202
3.3 Simple Shear Deformation ................................. 202
3.3.1 Symmetry Considerations ........................... 204
3.3.2 Equilibrium Stress for Solids ..................... 206
3.3.3 Steady-State Stress for Liquids ................... 207
3.4 Polymeric Liquid Response ........................... 209
3.4.1 Steady-State Properties ........................... 210
Viscosity ......................................... 211
First Normal Stress Difference .................... 216
Approximate Relationships ......................... 219
Second Normal Stress Difference ................... 225
3.4.2 Start-up and Shut-down Behavior ................... 227
Start-up from Rest ................................ 228
Relaxation from Steady State ...................... 231
3.4.3 Relaxation from Large Step Strains ................ 233
Lodge-Meissner Relation ........................... 234
Strain-Time Factorability ......................... 234
Normal Stress Ratio ............................... 239
3.5 Time-Temperature Superposition ........................... 239
References ............................................... 243
Chapter 4 Stress-Deformation Relationships ................... 247
4.1 Deformation .............................................. 247
4.1.1 Particle Position ................................. 248
4.1.2 Particle Displacement ............................. 249
4.1.3 Displacement Gradient Tensor ...................... 251
4.1.4 Cauchy Strain Tensor .............................. 254
4.1.5 Finger Strain Tensor .............................. 257
4.1.6 Other Strain Measures ............................. 261
Green and Piola Tensors ........................... 261
Deformation Rate Tensor ........................... 261
Infinitesimal Deformation Tensor .................. 264
4.1.7 Reference Frame Rotation .......................... 266
4.1.8 Scalar Invariants ................................. 267
4.2 Constitutive Considerations .............................. 268
4.2.1 Incompressibility ................................. 269
4.2.2 Reference Frame Indifference ...................... 271
4.3 Elastic Response of Solids ............................... 272
4.3.1 Neo-Hookean Solids ................................ 273
4.3.2 The Rivlin Equation ............................... 276
4.3.3 Mooney-Rivlin Solids .............................. 280
4.3.4 Valanis-Landel Formulation ........................ 280
4.4 Viscoelastic Response of Liquids ......................... 281
4.4.1 The Lodge Model ................................... 283
Simple Shear ...................................... 285
Uniaxial Extension ................................ 288
Extensional Flow Behavior ......................... 291
Assessment ........................................ 300
4.4.2 Consistency Tests ................................. 301
4.5 Strain-Dependent Models .................................. 303
4.5.1 General KBKZ Properties ........................... 304
4.5.2 Simple Shear Histories ............................ 304
4.5.3 Factorization ..................................... 306
4.5.4 Irreversibility ................................... 310
Nonlinear Recoil .................................. 311
Double Step Strain ................................ 314
References ........................................ 316
Chapter 5 Experimental Aspects ............................... 319
5.1 Introduction ............................................. 319
5.2 Hookean Solids and Newtonian Liquids ..................... 325
5.2.1 Simple Shear ...................................... 326
Hookean Solids .................................... 326
Newtonian Liquids ................................. 330
5.2.2 Uniaxial Extension ................................ 331
Hookean Solids .................................... 331
Poisson Ratio ..................................... 334
Newtonian Liquids ................................. 335
5.2.3 Property Measurement .............................. 337
Shear Modulus ..................................... 337
Shear Viscosity ................................... 339
5.2.4 General Comments .................................. 343
5.3 Simple Shear Rheometry ................................... 344
5.3.1 Analysis of Rotational Rheometers .................. 344
Concentric Rotating Plates ........................ 346
Concentric Cylinders .............................. 349
Concentric Cone and Plate ......................... 350
5.3.2 Analysis of Flow-Through Rheometers ............... 354
Capillary Flow .................................... 355
Annular Flow ...................................... 359
Slit Flow ......................................... 361
5.3.3 Weissenberg-Rabinowitsch Procedures ............... 363
Capillary Flow .................................... 363
Slit Flow ......................................... 366
Concentric Plates ................................. 367
5.4 Comments on Shear Rheometry .............................. 369
5.4.1 Flow Instabilities ................................ 370
Edge Instabilities ................................ 370
Entry Region Vortices ............................. 372
Sharkskin, Slip, and Melt Fracture ................ 373
5.4.2 Instrumental Aspects .............................. 377
Stiffness and Response Time ....................... 377
Wall Pressure ..................................... 380
5.4.3 Other Issues ...................................... 382
5.4.4 Summary ........................................... 383
5.5 Extensional Rheometry .................................... 384
5.6 Optical Methods .......................................... 386
5.6.1 Optical Birefringence ............................. 387
5.6.2 Flow Modeling ..................................... 390
References ............................................... 395
Chapter 6 Flexible-Chain Dynamics ............................ 399
6.1 Methods and Forces ....................................... 400
6.2 Elastic Dumbbells ........................................ 403
6.2.1 The Diffusion Equation ............................ 405
6.2.2 External and Internal Distributions ............... 406
6.2.3 Flow-Induced Effects .............................. 409
6.2.4 Stress ............................................ 411
6.2.5 Flow Properties ................................... 415
Relation to the Lodge Model ....................... 416
Association with Observables ...................... 416
6.2.6 Extended Formulation .............................. 418
6.3 The Rouse Model .......................................... 419
6.3.1 Diffusion and Stress .............................. 421
6.3.2 Diffusion Coefficient ............................. 423
6.3.3 Viscoelastic Properties ........................... 426
Matrix Diagonalization ............................ 426
Association with Observables ...................... 427
6.4 The Zimm Model ........................................... 430
6.4.1 Stress and Diffusion .............................. 434
6.4.2 Association with Observables ...................... 435
6.4.3 Discussion ........................................ 437
6.5 Dilute Solution Dynamics ................................. 440
6.5.1 Linear Response ................................... 443
Slow Dynamics ..................................... 443
Excluded Volume Effects ........................... 447
High Frequency Response ........................... 451
6.5.2 Nonlinear Response ................................ 451
Shear Flows ....................................... 451
Stretching Flows .................................. 454
6.6 Dynamics of Short Chains ................................. 460
6.6.1 Dynamic Moduli .................................... 462
6.6.2 Recoverable Compliance ............................ 464
6.6.3 Self Diffusion .................................... 465
6.6.4 Monomeric Unit Friction ........................... 468
Appendix 6.1 Integration Formulas for
Calculating Averages of
Distributions ........................ 472
References ........................................ 474
Chapter 7 Entangled Chain Dynamics ........................... 479
7.1 Entangled Melts .......................................... 480
7.1.1 Polarizability and Polarity Aspects ............... 481
Stress-Optical Rule ............................... 482
Dielectric Methods ................................ 483
7.1.2 The Plateau Modulus ............................... 484
7.1.3 The Shifted Rouse Formulation ..................... 487
7.2 Reptation and the Tube Model ............................. 488
7.2.1 Doi-Edwards Theory ................................ 489
7.2.2 Non-Mechanical Dynamics ........................... 492
7.2.3 Dynamic-Mechanical Properties ..................... 496
Gaussian Networks ................................. 497
Step Strain Response .............................. 498
7.3 Linear Viscoelasticity ................................... 500
7.3.1 Dynamic Properties ................................ 501
7.3.2 Comparisons with Data ............................. 503
7.3.3 Competing Relaxation Mechanisms ................... 509
7.4 Constraint Release ....................................... 510
7.4.1 Tracer Diffusion .................................. 511
7.4.2 Viscoelastic Aspects .............................. 514
Single Component Properties ....................... 514
Tracer Relaxation ................................. 518
7.5 Fluctuations ............................................. 520
7.5.1 Viscosity ......................................... 520
7.5.2 Self Diffusion .................................... 527
7.5.3 Nonuniversality ................................... 531
7.5.4 Simulations ....................................... 533
7.6 Nonlinear Viscoelastic Properties ........................ 535
7.6.1 Step Shear Strain ................................. 536
7.6.2 Step Extensional Strain ........................... 539
7.6.3 Viscometric Functions ............................. 540
7.6.4 Irreversibility ................................... 541
7.7 Independent Alignment Approximation ...................... 542
7.7.1 Shear Deformation ................................. 543
7.7.2 Uniaxial Extension ................................ 546
7.8 Theoretical Additions .................................... 548
7.8.1 Critique .......................................... 548
7.8.2 Commentary ........................................ 550
7.8.3 Convective Constraint Release ..................... 552
Collective Approach ............................... 553
Pairwise Junction Approach ........................ 554
7.8.4 Comparisons and Discussion ........................ 556
References ........................................ 560
Chapter 8 Dynamics of Entangled Solutions .................... 567
8.1 Introduction ............................................. 567
8.2 Friction-Insensitive Properties .......................... 569
8.2.1 Plateau Modulus С°N(ø) ............................ 569
Colby-Rubinstein Theory ........................... 571
Analysis of Data .................................. 572
Compliance Support ................................ 576
Entanglement Density Definition ................... 579
8.2.2 Recoverable Compliance J°s(ø, M) .................. 581
8.2.3 Characteristic Shear Rate Ƴo(ø, M) ................ 585
8.2.4 Characteristic Molecular Weight Mc(ø) ............. 589
8.2.5 Characteristic Diffusion Dependence D(ø) .......... 595
8.3 Concentration - Molecular Weight Diagrams ................ 596
8.4 Friction-Dependent Properties ............................ 598
8.4.1 Free Volume Theory ................................ 599
The Williams-Landel-Ferry Form .................... 600
Chain End and Copolymer Effects ................... 602
Diluent Effects ................................... 603
Homologous Series Data ............................ 605
Solutions ......................................... 606
8.4.2 Monomeric Tracer Diffusion ........................ 612
8.4.3 Dilute Solutions .................................. 616
Intrinsic Viscosity ............................... 617
High Frequency Response ........................... 622
Discussion ........................................ 623
References ........................................ 624
Chapter 9 Dynamics of Nonlinear Chains ....................... 629
9.1 Long-Chain Branching ..................................... 629
9.1.1 Types of Structural Nonlinearity .................. 631
9.1.2 Plateau Modulus ................................... 633
9.1.3 Nonlinear Rouse Chains ............................ 635
Linear Chains ..................................... 637
Symmetric Three-Arm Stars ......................... 640
Symmetric f-Arm Stars ............................. 643
Other Architectures ............................... 644
9.2 Linear Dynamics of Entangled Stars ....................... 644
9.2.1 Viscosity ......................................... 644
Diluent Effects ................................... 648
Temperature Effects ............................... 649
9.2.2 Recoverable Compliance ............................ 653
9.2.3 Self Diffusion .................................... 657
9.2.4 Thermorheological Complexity ...................... 661
9.2.5 Exponential Prefactors ............................ 663
9.3 Tube Model for Stars ..................................... 666
9.4 Constraint Release Effects for Stars ..................... 672
9.4.1 Relaxation in Network Environments ................ 672
9.4.2 Tracer Diffusion .................................. 674
9.4.3 Dynamic Dilution Model ............................ 676
9.4.4 Discussion ........................................ 681
Compliance Prediction .................................... 681
Dynamic Dilution Commentary .............................. 681
9.5 Hierarchy: H's, Combs and Networks ....................... 682
9.5.1 Dynamic Hierarchy ................................. 683
Quantitative Aspects ..................................... 685
Terminal Relaxation ...................................... 688
9.5.2 H-Polymers ........................................ 690
9.5.3 Combs ............................................. 693
9.5.4 Dangling Network Strands .......................... 700
9.6 Nonlinear Viscoelasticity ................................ 701
9.6.1 Damping Function .................................. 702
9.6.2 Extensional Behavior .............................. 706
References ............................................... 708
Chapter 10 Polydispersity Effects ............................. 713
10.1 Introduction ............................................. 713
10.2 Macromolecular Heterogeneity ............................. 714
10.2.1 Averages .......................................... 715
10.2.2 Distribution Functions ............................ 716
10.2.3 Distribution Examples ............................. 718
Exponential Distributions ......................... 718
Branching Distributions ........................... 718
10.3 Molecular Weight Mixtures ................................ 719
10.3.1 Rouse and Tube Model Predictions .................. 721
Rouse Mixing Rules ....................................... 721
Doi-Edwards Mixing Rules ................................. 723
10.3.2 Binary Linear-Chain Mixtures ...................... 723
10.3.3 Double Reptation .................................. 734
Binary Mixtures ................................... 738
Continuous Distributions .......................... 739
Commentary ........................................ 746
The Inverse Problem ............................... 746
10.4 Mixtures of Architectures ................................ 748
10.4.1 Binary Linear-Star Mixtures ....................... 748
10.4.2 Continuous Distributions .......................... 750
10.4.3 Nonlinear Viscoelasticity ......................... 757
Shear Thinning .................................... 757
Strain Hardening .................................. 758
Discussion ........................................ 761
References ........................................ 762
Appendix A Symbols ............................................ 767
Appendix В Acronyms and Abbreviations ......................... 775
Subject Index ................................................. 777
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