Models for polymeric and anisotropic liquids

Description

Part I of this monograph is concerned with the theoretical, analytical as well as numerical prediction of field-induced dynamics and structure for simple models describing soft matter. It presents selected results and demonstrates ranges of applications for the methods described in Part II. Special emphasis is placed on the finitely extendable nonlinear elastic (FENE) chain models for polymeric liquids, their dynamical and rheological behavior and the description of their inherently anisotropic material properties via deterministic and stochastic approaches. A number of representative examples are given on how simple (but high-dimensional) models can, and have been implemented in order to enable the analysis of the microscopic origins of the dynamical behavior of polymeric materials. These examples are shown to provide us with a number of routes for developing and establishing low-dimensional models devoted to the prediction of a reduced number of significant material properties. Concerning the types of complex fluids, we cover the range from flexible polymers in melts and solutions, wormlike micelles, actin filaments, rigid and semiflexible molecules in flow-induced anisotropic, and also liquid crystalline phases. Fokker-Planck equations and molecular and brownian dynamics computer simulation methods are involved to formulate and analyze the model fluids.

Part II allows the reader to redo simulations and motivates for further investigation of polymeric and anisotropic fluids. It contains computational recipes for devising simulation methods and codes, including Monte Carlo, molecular and brownian dynamics (written in Mathwork's Matlab, thus allowing for simple visualization and animation). A special chapter on isotropic and irreducible tensors allows for comfortable conversion between stochastic differential equations, tensorial balances, and equations for coefficients including the testing of closure approximations. We explicitely derive coupled equations for alignment tensors for arbitrary tensor fields suitable for nth order approximations strictly valid close to equilibrium, and also highly anisotropic states.

Table of contents

Part I Illustrations & Applications

1 Simple Models for Polymeric and Anisotropic Liquids . . . . . . . . . . . . . . .  3
1.1 Section-by-Section Summary. . . . . . . . . . . . . . . . . . . . . . . . . . .  7

2 Dumbbell Model for Dilute and Semi-Dilute Solutions . . . . . . . . . . . . . . . 13
2.1 FENE-PMF Dumbbell in Finitely Diluted Solution . . . . . . . . . . . . . . . .  15
2.2 Introducing a Mean Field Potential . . . . . . . . . . . . . . . . . . . . . .  16
2.3 Relaxation Equation for the Tensor of Gyration . . . . . . . . . . . . . . . .  16
2.4 Symmetry Adapted Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . .  18
2.5 Stress Tensor and Material Functions . . . . . . . . . . . . . . . . . . . . .  21
2.6 Reduced Description of Kinetic Models . . . . . . . . . . . . . . . . . . . . . 23

3 Chain Model for Dilute Solutions . . . . . . . . . . . . . . . . . . . . . . . .  25
3.1 Hydrodynamic Interaction . . . . . . . . . . . . . . . . . . . . . . .  . . . . 25
3.2 Long Chain Limit, Cholesky Decomposition . . . . . . . . . . . . . . . . . . .  27
3.3 NEBD Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Universal Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  29

4 Chain Model for Concentrated Solutions and Melts . . . . . . . . . . . . . . . .  33
4.1 NEMD Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .  34
4.2 Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Lennard–Jones (LJ) Units . . . . . . . . . . . . . . . . . . . . . . . . . . .  35
4.4 Flow Curve and Dynamical Crossover for Polymer Melts . . . . . . . . . . . . .  35
4.5 Characteristic Lengths and Times . . . . . . . . . . . . . . . . . . . . . .  . 36
4.6 Linear Stress-Optic Rule (SOR) and Failures . . . . . . . . . . . . . . . . . . 39
4.7 Nonlinear Stress-Optic-Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.8 Stress-Optic Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . .  44
4.9 Interpretation of Dimensionless Simulation Numbers. . . . . . . . . . . . . . . 48

5 Chain Models for Transient and Semiflexible Structures . . . . . . . . . . . . .  49
5.1 Conformational Statistics of Wormlike Chains (WLC) . . . . . . . . . . . . . .  49
5.1.1 Functional Integrals for WLCs . . . . . . . . . . . . . . . . . . . . . . . . 50
5.1.2 Properties of WLCs . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . 51
5.2 FENE-C Wormlike Micelles . . . . . . . . . . . . . . . . . . . . . . . . . . .  53
5.2.1 Flow-Induced Orientation and Degradation . . . . . . . . . . . . . . . . . .  54
5.2.2 Length Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.3 FENE-C Theory vs Simulation, Rheology, Flow Alignment . . . . . . . . . . . . 57
5.3 FENE-B Semiflexible Chains . . . . . . . . . . . . . . . . . . . . . . . . . .  61
5.3.1 Actin Filaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4 FENE-B Liquid Crystalline Polymers . . . . . . . . . . . . . . . . . . . . . .  66
5.4.1 Static Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.5 FENE-CB Transient Semiflexible Networks, Ring Formation . . . . . . . . . . . . 72

6 Primitive Path Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.1 Doi-Edwards Tube Model and Improvements . . . . . . . . . . . . . . . . . . . . 77
6.2 Refined Tube Model with Anisotropic Flow-Induced Tube Renewal . . . . . . . . . 79
6.2.1 Linear Viscoelasticity of Melts and Concentrated Solutions . . . . . . . . .  80
6.3 Nonlinear Viscoelasticity, Particular Closure . . . . . . . . . . . . . . . . . 84
6.3.1 Example: Refined Tube Model, Stationary Shear Flow . . . . . . . . . . . . .  84
6.3.2 Example: Transient Viscosities for Rigid Polymers . . . . . . . . . . . . . . 85
6.3.3 Example: Doi-Edwards Model as a Special Case . . . . . . . . . . . . . . . .  85
6.4 Nonlinear Viscoelasticity without Closure . . . . . . . . . . . . . . . . . . . 86
6.4.1 Galerkin’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . .  87

7 Elongated Particle Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.1 Director Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2 Structural Theories of Suspensions . . . . . . . . . . . . . . . . . . . . . .  93
7.2.1 Semi-Dilute Suspensions of Elongated Particles . . . . . . . . . . . . . . .  95
7.2.2 Concentrated Suspensions of Rod-Like Polymers . . . . . . . . . . . . . . . . 95
7.3 Uniaxial Fluids, Micro-Macro Correspondence . . . . . . . . . . . . . . . . . . 96
7.3.1 Concentrated Suspensions of Disks, Spheres, Rods . . . . . . . . . . . . . .  97
7.3.2 Example: Tumbling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.3.3 Example: Miesowicz Viscosities . . . . . . . . . . . . . . . . . . . . . . .  98
7.4 Uniaxial Fluids: Decoupling Approximations . . . . . . . . . . . . . . . . . .  99
7.4.1 Decoupling with Correct Tensorial Symmetry . . . . . . . . . . . . . . . . . 102
7.5 Ferrofluids: Dynamics and Rheology . . . . . . . . . . . . . . . . . . . . . . 103
7.6 Liquid Crystals: Periodic and Irregular Dynamics . . . . . . . . . . . . . . . 105
7.6.1 Landau – de Gennes Potential . . . . . . . . . . . . . . . . . . . . . . . . 108
7.6.2 In-Plane and Out-of-Plane States . . . . . . . . . . . . . . . . . . . . . . 109

8 Connection between Different Levels of Description . . . . . . . . . . . . . . . 111
8.1 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.2 Generalized Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . 112
8.3 GENERIC Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  112
8.3.1 Building Block L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.3.2 Building Block M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  115
8.4 Dissipative Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . .  117
8.5 Langevin and Fokker–Planck Equation, Brownian Dynamics . . . . . . . . . . . . 117
8.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.5.2 Interpretation, and Langevin Equation . . . . . . . . . . . . . . . . . . .  118
8.6 Projection Operator Methods . . . . . . . . . . . . . . . . . . . . . . . . .  119
8.7 Stress Tensors: Giesekus – Kramers – GENERIC . . . . . . . . . . . . . . . . . 121
8.8 Generalized Canonical Ensemble and Friction Matrix . . . . . . . . . . . . . . 123
8.9 Beyond-Equilibrium Molecular Dynamics (BEMD). . . . . . . . . . . . . . . . .  124
8.9.1 Multiplostatted Equations . . . . . . . . . . . . . . . . . . . . . . . . .  128
8.9.2 Applicability of BEMD . . . . . . . . . . . . . . . . . . . . . . . . . . .  130
8.9.3 DOLLS/SLLOD Analogy with Multiplostatted Equations . . . . . . . . . . . . . 132
8.10 Examples for Coarse-Graining. . . . . . . . . . . . . . . . . . . . . . . . . 134
8.10.1 From Connected to Primitive Path . . . . . . . . . . . . . . . . . . . . .  134
8.10.2 From Disconnected to Primitive Path . . . . . . . . . . . . . . . . . . . . 136

Part II Theory & Computational Recipes

9 Equilibrium Statistics: Monte Carlo Methods . . . . . . . . . . . . . . . . . .  145
9.1 Expectation Values, Metropolis Monte Carlo . . . . . . . . . . . . . . . . . . 146
9.2 Normalization Constants, Partition Function . . . . . . . . . . . . . . . . .  148
9.2.1 Standard Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.2.2 Direct Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.2.3 Path Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  150
9.3 Density of States Monte Carlo (DSMC) . . . . . . . . . . . . . . . . . . . . . 151
9.4 Quasi Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  153

10 Irreducible and Isotropic Cartesian Tensors . . . . . . . . . . . . . . . . . . 155
10.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  155
10.2 Anisotropic (Irreducible) Tensors . . . . . . . . . . . . . . . . . . . . . . 156
10.3 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . .  158
10.4 Isotropic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.4.1 Construction of the Isotropic Tensors . . . . . . . . . . . . . . . . . . . 160
10.4.2 Generalized Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . 160
10.4.3 Generalized Isotropic Tensors . . . . . . . . . . . . . . . . . . . . . . . 161
10.4.4 Implications (Summary) . . . . . . . . . . . . . . . . . . . . . . . . . .  161
10.5 Differential Operations (Tabular Form) . . . . . . . . . . . . . . . . . . .  162
10.6 Nematic Order Parameters . . . . . . . . . . . . . . . . . . . . . . . . . .  163
10.6.1 Uniaxial Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  163
10.6.2 Biaxial Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
10.7 Tensor Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.8 Solutions of the Laplace Equation . . . . . . . . . . . . . . . . . . . . . . 167
10.9 The Reverse Isotropic Tensor Operation . . . . . . . . . . . . . . . . . . .  168
10.10Integrating Irreducible Tensors . . . . . . . . . . . . . . . . . . . . . . . 168

11 Nonequilibrium Dynamics of Anisotropic Fluids . . . . . . . . . . . . . . . .   169
11.1 Orientational Distribution Function . . . . . . . . . . . . . . . . . . . . . 169
11.1.1 Alignment Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
11.1.2 Uniaxial Distribution Function . . . . . . . . . . . . . . . . . . . . . .  170
11.2 Fokker–Planck Equation, Smoluchowski Equation . . . . . . . . . . . . . . . . 170
11.2.1 Spatial Inhomogeneous Distribution . . . . . . . . . . . . . . . . . . . .  172
11.2.2 Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  172
11.2.3 Spatial Homogeneous Distribution, Nth Order Potential . . . . . . . . . . . 173
11.2.4 Examples for Potentials and Applications . . . . . . . . . . . . . . . . .  174
11.3 Coupled Equations of Change for Alignment Tensors . . . . . . . . . . . . . . 174
11.3.1 Dynamical Closures . . . . . . . . . . . . . . . . . . . . . . . . . . . .  176
11.3.2 Equations of Change for Order Parameters . . . . . . . . . . . . . . . . .  176
11.4 Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11.4.1 Brownian Dynamics Simulation . . . . . . . . . . . . . . . . . . . . . . .  179

12 Simple Simulation Algorithms and Sample Applications. . . . . . . . . . . . . . 181
12.1 Index of Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
12.2 Recipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
12.2.1 Random Vectors, Random Paths (2D, 3D) . . . . . . . . . . . . . . . . . . . 182
12.2.2 Periodic and Reflecting Boundary Conditions (nD) . . . . . . . . . . . . .  183
12.2.3 Useful Initial Phase Space Coordinates (nD) . . . . . . . . . . . . . . . . 184
12.2.4 Visualization, Animation & Movies (nD) . . . . . . . . . . . . . . . . . .  184
12.3 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
12.3.1 Standard Monte Carlo Integration (nD) . . . . . . . . . . . . . . . . . . . 185
12.3.2 Ising Model via Metropolis Monte Carlo (2D) . . . . . . . . . . . . . . . . 186
12.4 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  187
12.4.1 Molecular Dynamics of a Lennard–Jones System (nD) . . . . . . . . . . . . . 187
12.4.2 Associating Equilibrium FENE Polymers (2D,3D) . . . . . . . . . . . . . . . 188
12.5 NonEquilibrium Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . 190
12.5.1 NonEquilibrium Molecular Dynamics (nD) . . . . . . . . . . . . . . . . . .  190
12.5.2 Flow through Nanopore (3D) . . . . . . . . . . . . . . . . . . . . . . . .  191
12.6 Brownian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
12.6.1 Brownian Dynamics of a Lennard–Jones System (nD) . . . . . . . . . . . . .  194
12.6.2 Hydrodynamic Interaction via Chebyshev Polynomials (3D) . . . . . . . . . . 194
12.7 Coarse-Graining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
12.7.1 Coarse-Graining Polymer Chains (nD) . . . . . . . . . . . . . . . . . . . . 195

13 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  197
13.1 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   199
14.1 Special Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
14.2 Tensor Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  199
14.3 Upper Case Roman Symbols . . . . . . . . . . . . . . . . . . . . . . . . . .  200
14.4 Lower Case Roman Symbols . . . . . . . . . . . . . . . . . . . . . . . . . .  201
14.5 Greek Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
14.6 Caligraphic Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
14.7 FENE Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
14.8 Gaussian Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  204

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  223

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Corrigenda and Addenda

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