Modeling and Simulation in Polymers

Modeling and Simulation in Polymers......Page 1
Contents......Page 7
Preface......Page 17
List of Contributors......Page 21
1.1 Introduction and Historical Perspective......Page 25
1.2 Governing Equations and Polymer Modeling......Page 30
1.3 Numerical Methods for DNS......Page 34
1.3.1.1 The Semi-Implicit/Explicit Scheme......Page 35
1.3.1.2 The Fully Implicit Scheme......Page 37
1.3.2 The Positive Definiteness of the Conformation Tensor......Page 39
1.4.1 Drag Reduction Evaluation......Page 41
1.4.2 Effects of Flow and Rheological Parameters......Page 43
1.4.3 Effects of Numerical Parameters......Page 50
1.5 Conclusions and Thoughts on Future Work......Page 53
References......Page 55
2.1 Introduction......Page 61
2.2 Polymer Clay Nanocomposites and Coarse-Grained Models......Page 64
2.2.2 Methods and Timescales......Page 66
2.2.2.2 Discrete Lattice Approach......Page 67
2.2.3 Coarse-Grained Sheet......Page 68
2.2.3.1 Conformation and Dynamics of a Sheet......Page 71
2.2.4 Coarse-Grained Studies of Nanocomposites......Page 74
2.2.4.1 Probing Exfoliation and Dispersion......Page 75
2.2.5.1 Solvent Particles......Page 76
2.2.5.2 Polymer Matrix......Page 79
2.2.6 Conclusions and Outlook......Page 84
2.3 All-Atom Models for Interfaces and Application to Clay Minerals......Page 85
2.3.1 Force Fields for Inorganic Components......Page 86
2.3.1.1 Atomic Charges......Page 88
2.3.1.2 Lennard-Jones Parameters......Page 89
2.3.1.3 Bonded Parameters......Page 91
2.3.2 Self-Assembly of Alkylammonium Ions on Montmorillonite: Structural and Surface Properties at the Molecular Level......Page 92
2.3.3 Relationship Between Packing Density and Thermal Transitions of Alkyl Chains on Layered Silicate and Metal Surfaces......Page 102
2.4 Interfacial Thermal Properties of Cross-Linked Polymer–CNT Nanocomposites......Page 103
2.4.1 Model Building......Page 105
2.4.2 Thermal Conductivity......Page 107
References......Page 110
3.1 Introduction......Page 117
3.2.1 Ideal Living Polymerization......Page 119
3.2.3 Chain Transfer to Solvent......Page 121
3.2.4 Multifunctional Initiators......Page 126
3.2.5 Chain Transfer to Polymer......Page 129
3.2.6 Chain Transfer to Monomer......Page 133
3.3 Continuous Polymerization......Page 135
3.3.1 MWD of Living Polymers Formed in CSTR......Page 137
3.3.2 Chain Transfer to Solvent......Page 140
3.3.3 Chain Transfer to Monomer......Page 142
3.3.4 Chain Transfer to Polymer......Page 144
3.4 Conclusions......Page 147
References......Page 149
4.1.1 Polymer Processing......Page 151
4.1.2 Historical Notes on Computations......Page 152
4.2.2 Constitutive Equations......Page 154
4.2.3 Dimensionless Groups......Page 158
4.2.4 Boundary Conditions......Page 162
4.3 Method of Solution......Page 164
4.4.1.1 Flow Inside the Extruder......Page 167
4.4.1.2 Flow in an Extruder Die (Contraction Flow)......Page 170
4.4.1.3 Flow Outside the Extruder – Extrudate Swell......Page 173
4.4.1.4 Coextrusion Flows......Page 174
4.4.1.5 Extrusion Die Design......Page 177
4.4.2 Postextrusion Operations......Page 178
4.4.2.1 Calendering......Page 179
4.4.2.2 Roll Coating......Page 181
4.4.2.3 Wire Coating......Page 186
4.4.2.4 Fiber Spinning......Page 187
4.4.2.5 Film Casting......Page 193
4.4.2.6 Film Blowing......Page 197
4.4.3.1 Blow Molding......Page 200
4.4.3.2 Thermoforming......Page 202
4.4.3.3 Injection Molding......Page 205
4.5 Conclusions......Page 209
4.6 Current Trends and Future Challenges......Page 211
References......Page 212
5.1 Minimal, Coarse-Grained Models, and Universality......Page 221
5.2.1 Hubbard–Stratonovich Transformation: Field-Theoretic Reformulation of the Particle-Based Partition Function......Page 225
5.2.2 Mean Field Approximation......Page 230
5.2.3 Role of Compressibility and Local Correlations of the Fluid of Segments......Page 234
5.3.1 Standard Model for Compressible Multicomponent Polymer Melts and Self-Consistent Field Techniques......Page 235
5.3.2.1 Partial Enumeration Schemes......Page 237
5.3.2.2 Monte Carlo Sampling of the Single-Chain Partition Function and Self-Consistent Brownian Dynamics......Page 238
5.3.3.1 Single-Chain-in-Mean-Field Simulations......Page 241
5.3.3.2 Minimal, Particle-Based, Coarse-Grained Model: Discretization of Space and Molecular Contour......Page 243
5.3.3.3 Monte Carlo Simulations and Advantages of Soft Coarse-Grained Models......Page 244
5.3.3.4 Comparison Between Monte Carlo and SCMF Simulations: Quasi-Instantaneous Field Approximation......Page 245
5.3.4 Off-Lattice, Soft, Coarse-Grained Models......Page 249
5.4.1 Crystallization in Hard Condensed Matter Versus Self-Assembly of Soft Matter......Page 251
5.4.2 Field-Theoretic Reference State: The Einstein Crystal of Grid-Based Fields......Page 252
5.4.3.1 How to Turn a Disordered Melt into a Microphase-Separated Morphology Without Passing Through a First-Order Transition?......Page 253
5.4.3.2 Thermodynamic Integration Versus Expanded Ensemble and Replica-Exchange Monte Carlo Simulation......Page 256
5.4.3.3 Selected Applications......Page 259
5.4.4 Simultaneous Calculation of Pressure and Chemical Potential in Soft, Off-Lattice Models......Page 262
5.5 Outlook......Page 263
References......Page 266
6.1 Introduction......Page 271
6.2.1 Simulation Method......Page 275
6.2.2 Degree of Ionization......Page 277
6.2.3 Size and Shape of the Polyelectrolyte......Page 279
6.2.4 Effect of Salt Concentration on Degree of Ionization......Page 280
6.2.6 Dependence of Degree of Ionization on Polymer Density......Page 283
6.2.7.1 Theoretical Background......Page 286
6.2.7.2 Dependence of Radius of Gyration on Salt with Monovalent Counterions......Page 288
6.2.7.3 Bridging Effect by Divalent Counterions......Page 289
6.3 The Variational Theory......Page 290
6.3.1 Free Energy......Page 293
6.3.2.1 Salt-Free Solutions......Page 299
6.3.2.2 Divalent Salt and Overcharging......Page 302
6.3.4 Competitive Adsorption of Divalent Salts......Page 303
6.3.6 Effect of Monomer Concentration and Chain Length......Page 306
6.3.7 Free energy Profile......Page 308
6.3.8 Diagram of Charged States: Divalent Salt......Page 311
6.3.9 Effect of Ion-Pair Correlations......Page 314
6.3.10 Collapse in a Poor Solvent......Page 315
6.3.11 Bridging Effect: Divalent Salt......Page 319
6.3.12 Role of Chain Stiffness: The Rodlike Chain Limit......Page 323
6.4 The Self-Consistent Field Theory......Page 325
6.4.1 Extension of Edward.s Formulation......Page 327
6.4.2.1 Transformation Using Functional Integral Identities......Page 333
6.4.2.2 Hubbard–Stratonovich Transformation......Page 334
6.4.4 Saddle-Point Approximation......Page 336
6.4.5 Numerical Techniques......Page 338
6.4.5.1 Finite Difference Methods......Page 339
6.4.5.2 Spectral Method: Method of Basis Functions......Page 340
6.4.5.3 Pseudospectral Method......Page 342
6.4.6 Fluctuations Around the Saddle Point......Page 344
6.5 Comparison of Theories: SCFT and Variational Formalism......Page 346
6.5.1 Self-Consistent Field Theory for Single Chain......Page 347
6.5.2 Variational Formalism......Page 349
6.5.3 Numerical Techniques......Page 351
6.5.4 Degree of Ionization......Page 352
6.5.5 Term-by-Term Comparison of Free Energy: SCFT and Variational Formalism......Page 354
References......Page 363
7.1.1 Challenges in Polymer Dynamics Under Flow......Page 367
7.1.2 Modeling Polymer Dynamics Beyond Equilibrium......Page 368
7.1.3 Challenges in Standard Simulations of Polymers in Flow......Page 370
7.2 Coarse-Grained Variables and Models......Page 371
7.2.1 Beads and Superatoms......Page 372
7.2.2 Uncrossable Chains of Blobs......Page 374
7.2.3 Primitive Paths......Page 375
7.2.4 Other Single-Chain Simulation Approaches to Polymer Melts: Slip-Link and Dual Slip-Link Models......Page 377
7.2.5 Entire Molecules......Page 378
7.2.6 Conformation Tensor......Page 379
7.3.1 The Need for and Benefits of Consistent Coarse-Graining Schemes......Page 381
7.3.2 Different Levels of Description and the Choice of Relevant Variables......Page 382
7.3.3.2 Irreversibility and Dissipation Through Coarse Graining......Page 384
7.4.1 GENERIC Coarse-Graining Applied to Unentangled Melts: Foundations......Page 387
7.4.2 Thermodynamically Guided Atomistic Monte Carlo Methodology for Generating Realistic Shear Flows......Page 389
7.4.3.1 Systematic Timescale Bridging Algorithm......Page 393
7.4.3.3 Results......Page 395
7.5 Conclusions and Perspectives......Page 396
References......Page 398
8.1 Introduction......Page 409
8.2 Nonlinear Finite Element Analysis......Page 410
8.3 Incompressibility Conditions......Page 413
8.4 Solution Strategy......Page 417
8.5 Treatment of Contact Constraints......Page 418
8.6 Tire Modeling......Page 421
References......Page 427
9.1.1 Lessons from Nature......Page 429
9.1.2 A Historical Overview......Page 430
9.2.1 Theory of Elasticity of an Elastic Rod......Page 432
9.2.2 Hydrodynamic Friction of a Filament: Resistive Force Theory......Page 434
9.2.3 Hydrodynamic Friction of a Filament: Method of Hydrodynamic Interaction......Page 436
9.3.1 Details of the Modeling......Page 438
9.3.2 Microscopic Artificial Swimmer......Page 440
9.3.3 Fluid Transport......Page 444
9.3.3.1 Two-Dimensional Stroke......Page 445
9.3.3.2 Three-Dimensional Stroke......Page 449
References......Page 451
10.1.1 Equilibrium and Metastable States: Supercooled Liquids......Page 457
10.1.2 Common Folklore......Page 458
10.1.3 Systems Being Considered......Page 459
10.1.4 Long-Time Stability......Page 460
10.1.5.1 Cell Model......Page 461
10.1.5.2 Communal Entropy, Free Energy, and Lattice Models......Page 464
10.1.6 Fundamental Postulate: Stationary Limit......Page 465
10.1.7 Thermodynamics of Metastability......Page 467
10.1.8 Scope of the Review......Page 468
10.2.2 Entropy Extension in the Gap......Page 470
10.2.3 Gibbs–Di Marzio Theory......Page 471
10.3.1 Experimentally Observed Glassy State......Page 475
10.3.2 Glass Phenomenology......Page 476
10.3.3 Fragility......Page 477
10.3.4 Ideal Glass Transition as r → 0......Page 478
10.3.5 Kauzmann Paradox and Thermodynamics......Page 480
10.3.6 Entropy Crisis and Ideal Glass Transition......Page 481
10.4.1 Communal Entropy, Confinement, and Ideal Glass......Page 483
10.4.2 Partitioning of Zτ(T, V)......Page 487
10.5.1 Thermodynamic Theory of Adam and Gibbs......Page 488
10.5.2 Free Volume Theory......Page 489
10.5.3 Mode Coupling Theory......Page 490
10.6 Progigine–Defay Ratio Π and the Significance of Entropy......Page 491
10.7.1 Canonical Partition Function......Page 493
10.7.3 Order Parameter and Classification of Microstates......Page 494
10.8.2 Restricted and Extended Restricted PF\'s......Page 495
10.9 Three Useful Theorems......Page 496
10.10.1 Polymer Model and Classification of Configurations......Page 499
10.10.2 Exact Calculation......Page 501
10.11 Glass Transition in a Binary Mixture......Page 504
10.12.1 Singular Free Energy......Page 508
10.12.2 Order Parameter......Page 509
10.12.3 Relevance for Experiments......Page 510
10.13 Conclusions......Page 512
Appendix 10.A: Classical Statistical Mechanics......Page 514
Appendix 10.B: Negative Entropy......Page 515
References......Page 516
11.1.1 Low Molecular Weight and Polymeric Liquid Crystals......Page 521
11.1.2 Molecular and Continuum Theories of LCP......Page 522
11.1.3 Soft Deformation Modes in LCP......Page 524
11.1.4 Specific Problems in LCP Theories......Page 526
11.1.5 Experimental Effects in Flows of LCP......Page 527
11.2 General Equations and Simulation Procedures......Page 528
11.3 LCP and their Parameters Established in Simulations......Page 532
11.4.1 Simulations of Steady Shearing Flows......Page 535
11.4.2 Simulations of Transient Start-Up Shear Flows......Page 538
11.4.3 Simulations of Relaxation after Cessation of Steady Flow......Page 542
11.4.4 On the Time-Temperature Superposition in Weakly Viscoelastic Nematodynamics......Page 545
11.5 Conclusions and Discussions......Page 546
References......Page 548
Index......Page 551