Equilibrium theory of inhomogeneous polymers.

Format:

Book

Author/Creator:

Fredrickson, Glenn.

Language:

English

Subjects (All):

Polymers--Mathematical models.

Polymers.

Inhomogeneous materials.

Polymer solutions.

Physical Description:

xiv, 437 pages : illustrations ; 23 cm

Place of Publication:

[Place of publication not identified] : Oxford University Press, 2013.

Summary:

The Equilibrium Theory of Inhomogeneous Polymers provides an introduction to the field-theoretic methods and related computer simulation techniques that are used in the design of structured polymer fluids. By such methods, the principles that dictate equilibrium self-assembly in systems ranging from block and graft copolymers, to polyelectrolytes, liquid crystalline polymers, and polymer nanocomposites can be established. Building on an introductory discussion of single-polymer statistical mechanics, the book provides a detailed treatment of analytical and numerical techniques for addressing the conformational properties of polymers subjected to spatially-varying potential fields. This problem is shown to be central to the field-theoretic description of interacting polymeric fluids, and models for a number of important polymer systems are elaborated. Chapter 5 serves to unify and expound the topic of self-consistent field theory, which is a collection of analytical and numerical techniques for obtaining solutions of polymer field theory models in the mean-field approximation. The concluding Chapter 6 provides a discussion of analytical methods for going beyond the mean-field approximation and an introduction to the exciting new field of field-theoretic polymer simulations - the direct numerical simulation of polymer field theory models. No other book brings together in such an instructive fashion the theoretical tools for investigating the structure and thermodynamics of complex polymer systems, including those relevant to soft material nanotechnologies, personal care products, and multiphase plastic materials. Book jacket.

Contents:

1 Introduction 1

1.1 What is a polymer? 1

1.2 Polymeric systems of practical interest 4

1.2.1 Plastic materials 4

1.2.2 Solution formulations 5

1.3 Macrophase and microphase separation 6

1.4 Modern themes 9

1.4.1 Plastics industry trends 9

1.4.2 New synthetic methods 10

1.4.3 Combinatorial methodologies 12

1.4.4 Soft material nanotechnology 13

1.5 Why theory? 14

1.6 Modelling perspective and scales 15

1.6.1 Atomistic perspective 15

1.6.2 Mesoscopic perspective 17

1.6.3 Particles and fields 19

2 Ideal chain models 23

2.1 Real and ideal chains 23

2.2 Freely jointed chain model 26

2.3 Bead-spring models 33

2.4 Continuous Gaussian chain model 37

2.5 Wormlike chain model 44

2.6 Summary 50

3 Single chains in external fields 53

3.1 Partition and distribution functions 53

3.1.1 Discrete Gaussian chain 53

3.1.2 Continuous Gaussian chain 57

3.1.3 Wormlike chain 59

3.1.4 Rodlike polymer 62

3.2 Single-chain averages and operators 62

3.2.1 Density operators 63

3.2.2 Stress operators 70

3.3 Other architectures 71

3.3.1 Branched homopolymers 71

3.3.2 Block and graft copolymers 75

3.4 Approximation schemes 79

3.4.1 Weak inhomogeneity expansion 80

3.4.2 Slow gradient expansion 85

3.4.3 Ground state dominance 87

3.4.4 Other approximations 91

3.5 Boundary conditions 96

3.6 Numerical methods 99

3.6.1 Finite difference method 100

3.6.2 Spectral methods 103

3.6.3 Pseudo-spectral methods 110

3.6.4 Higher dimensions 117

3.6.5 Unit-cell calculations 118

3.6.6 Large-cell calculations 121

3.6.7 Other chain models 124

3.6.8 Other techniques 129

3.7 Summary 129

4 Models of many-chain systems 131

4.1 From particles to fields 131

4.1.1 Monatomic fluid in the canonical (nVT) ensemble 131

4.1.2 Monatomic fluid in the grand canonical (μVT) ensemble 137

4.1.3 Averages and operators for monatomic fluids 139

4.1.4 Averages and operators for the simplified field theories 141

4.2 Neutral polymer solutions 144

4.2.1 Model A: homopolymer in a good solvent 145

4.2.2 Model B: homopolymer solution, explicit solvent 150

4.3 Polymer blends 155

4.3.1 Model C: incompressible homopolymer blend 155

4.3.2 Model D: compressible homopolymer blend 158

4.4 Block and graft copolymers 159

4.4.1 Model E: diblock copolymer melt 159

4.4.2 A₂B graft copolymer melt 161

4.5 Polyelectrolytes 162

4.5.1 Model F: polyelectrolyte solution, implicit solvent 163

4.5.2 Extensions 167

4.6 Liquid crystalline polymers 168

4.6.1 Model G: polymer nematics 168

4.7 Disorder effects 171

4.7.1 Model H: quenched polydispersity 172

4.7.2 Model I: annealed polydispersity, equilibrium polymers 174

4.7.3 Model J: random graft copolymers 179

4.8 Tethered polymer layers 185

4.8.1 Model K: solution polymer brush 185

4.9 Boundary conditions and confinement effects 188

4.9.1 Bulk fluids 188

4.9.2 Confined solutions 190

4.9.3 Confined melts 192

4.9.4 Polymer brushes 194

4.9.5 Irregularly shaped domains 195

4.10 Density functional theory 196

4.10.1 Monatomic fluid 197

4.10.2 Polymer solutions: Model A 198

4.11 Summary 199

5 Self-consistent field theory 203

5.1 The mean-field approximation 203

5.1.1 The mean-field approximation: general considerations 204

5.1.2 Analytic structure of the field theory 205

5.1.3 Multiple solutions 209

5.1.4 Homogeneous saddle points 211

5.2 Further approximations 213

5.2.1 Weak inhomogenities - the RPA 213

5.2.2 Slow gradients 219

5.2.3 Ground state dominance 221

5.2.4 Strong stretching 227

5.3 Numerical methods 230

5.3.1 Nonlinear equation approach 230

5.3.2 Optimization approach 234

5.3.3 Multilevel embedding 243

5.3.4 Metastable states 246

5.3.5 Variable cell shape methods 252

5.3.6 Mean-field dynamics 259

5.4 Applications and examples 263

5.4.1 Unit-cell calculations 263

5.4.2 Large-cell calculations 269

5.5 Summary 276

6 Beyond mean-field theory 281

6.1 Analytical methods 281

6.1.1 Gaussian fluctuations about a homogeneous state: canonical ensemble 282

6.1.2 Gaussian fluctuations about a homogeneous state: grand canonical ensemble 288

6.1.3 Validity of the Gaussian approximation 291

6.1.4 Gaussian fluctuations about an inhomogeneous state 294

6.1.5 Renormalization techniques 297

6.1.6 Other analytical methods 317

6.2 Field-theoretic simulations 323

6.3 Monte Carlo simulations 327

6.3.1 Metropolis Monte Carlo 329

6.3.2 Force-bias Monte Carlo 335

6.3.3 Hybrid Monte Carlo 342

6.4 Complex Langevin simulations 343

6.5 Other numerical techniques 353

6.5.1 Optimal path sampling 354

6.5.2 Partial saddle point approximation 358

6.6 Lattice cutoff and finite size effects 361

6.6.1 Lattice cutoff effects 361

6.6.2 Finite size effects 365

6.7 Numerical renormalization group 366

6.8 Applications 372

6.9 Perspectives 377

6.9.1 Particles or fields? 379

6.9.2 Emerging applications for field-theoretic simulations 381.

ISBN:

0199673799

9780199673797

OCLC:

825181248

Publisher Number:

99954713743