Preface ................................................... xiii
1 Statistical properties of polymer chains ..................... 1
1.1 Conformation of polymers ................................ 1
1.1.1 Internal coordinates of a polymer chain and its
hindered rotation ................................ 1
1.1.2 Coarse-grained models of polymer chains .......... 3
1.2 The ideal chain ......................................... 5
1.2.1 Single-chain partition function .................. 5
1.2.2 Tension-elongation curve ......................... 8
1.2.3 Distribution of the end-to-end vector ........... 10
1.3 Fundamental properties of a Gaussian chain ............. 11
1.4 Effect of internal rotation and stiff chains ........... 13
1.4.1 Characteristic ratio ............................ 13
1.4.2 Persistence length and the stiff chain .......... 15
1.5 Excluded-volume effect ................................. 16
1.6 Scaling laws and the temperature blob model ............ 19
1.7 Coil-globule transition of a polymer chain in a poor
solvent ................................................ 21
1.8 Coil-helix transition .................................. 23
1.9 Hydration of polymer chains ............................ 33
1.9.1 Statistical models of hydrated polymer chains ... 33
1.9.2 Models of the globules and hydrated coils ....... 38
1.9.3 Competitive hydrogen bonds in mixed solvents .... 39
References .................................................. 44
2 Polymer solutions ........................................... 46
2.1 Thermodynamics of phase equilibria ..................... 46
2.1.1 Gibbs' phase rule and phase diagrams ............ 46
2.1.2 Stability of a phase ............................ 48
2.1.3 Liquid-liquid separation by a semipermeable
membrane ........................................ 52
2.1.4 Spontaneous liquid-liquid phase separation ...... 55
2.2 Characteristic properties of polymer solutions ......... 57
2.2.1 Vapor pressure and osmotic pressure ............. 58
2.2.2 Viscosity ....................................... 61
2.2.3 Diffusion of a polymer chain .................... 65
2.3 Lattice theory of polymer solutions .................... 69
2.3.1 The free energy of mixing ....................... 69
2.3.2 Properties of polymer solutions predicted by
Flory-Huggins lattice theory .................... 74
2.3.3 Extension to many-component polymer solutions
and blends ...................................... 79
2.3.4 Refinement beyond the simple mean field
approximation ................................... 81
2.4 Scaling laws of polymer solutions ...................... 87
2.4.1 Overlap concentration ........................... 87
2.4.2 Correlation length .............................. 89
2.4.3 Radius of gyration .............................. 90
2.4.4 Osmotic pressure ................................ 91
2.4.5 Phase equilibria (reduced equation of states) ... 92
2.4.6 Molecular motion ................................ 94
References .................................................. 95
3 Classical theory of gelation ................................ 97
3.1 What is a gel? ......................................... 97
3.1.1 Definition of a gel ............................. 97
3.1.2 Classification of gels .......................... 97
3.1.3 Structure of gels and their characterization .... 98
3.1.4 Examples of gels ............................... 100
3.2 Classical theory of gelation .......................... 103
3.2.1 Random branching ............................... 104
3.2.2 Polycondensation ............................... 106
3.2.3 Polydisperse functional monomers ............... 111
3.2.4 Cross-linking of prepolymers ................... 113
3.3 Gelation in binary mixtures ........................... 114
3.3.1 Finding the gel point using the branching
coefficient .................................... 114
3.3.2 Molecular weight distribution function of the
binary mixtures R{Aƒ}/R{Bg} .................... 116
3.3.3 Polydisperse binary mixture R{Aƒ}/R{Bg} ........ 118
3.3.4 Gels with multiple junctions ................... 119
3.A Moments of the Stockmayer distribution function ........ 121
3.B Cascade theory of gelation ............................. 122
References ................................................. 127
4 Elasticity of polymer networks ............................. 128
4.1 Thermodynamics of rubber elasticity ................... 128
4.1.1 Energetic elasticity and entropie elasticity ... 128
4.1.2 Thermoelastic inversion ........................ 131
4.1.3 Gough-Joule effect ............................. 131
4.2 Affine network theory ................................. 133
4.2.1 Local structure of cross-linked rubbers ........ 133
4.2.2 Affine network theory .......................... 134
4.2.3 Elastically effective chains ................... 139
4.2.4 Simple description of thermoelastic
inversion ...................................... 141
4.3 Phantom network theory ................................ 142
4.3.1 Micronetworks of tree form ..................... 143
4.3.2 Fluctuation theorem and the elastic free
energy ......................................... 145
4.4 Swelling experiments .................................. 146
4.5 Volume transition of gels ............................. 150
4.5.1 Free swelling .................................. 153
4.5.2 Swelling under uniaxial elongation ............. 154
4.6 Networks made up of nonlinear chains .................. 156
References ................................................. 159
5 Associating polymer solutions and thermoreversible
gelation ................................................... 160
5.1 Historical survey of the study of associating
solutions ............................................. 160
5.2 Statistical thermodynamics of associating polymers .... 161
5.2.1 Pregel regime .................................. 167
5.2.2 Sol-gel transition and postgel regime .......... 168
5.3 Renormalization of the interaction parameters ......... 168
5.4 Phase separation, stability limit, and other
solution properties ................................... 169
5.5 Scattering function of associating polymer mixtures ... 170
5.A Renormalization of the interaction parameters ......... 173
5.B Scattering function in RPA ............................ 175
5.С Spinodal condition in RPA ............................. 177
References ................................................. 178
6 Nongelling associating polymers ............................ 180
6.1 Dimer formation as associated block-copolymers ........ 180
6.2 Linear association and ring formation ................. 186
6.3 Side-chain association ................................ 189
6.4 Hydration in aqueous polymer solutions and closed-
loop miscibility gaps ................................. 197
6.5 Cooperative hydration in solutions of temperature-
responsive polymers ................................... 200
6.6 Hydrogen-bonded liquid-crystalline supramolecules ..... 207
6.7 Polymeric micellization ............................... 212
References ................................................. 219
7 Thermoreversible gelation .................................. 222
7.1 Models of thermoreversible gelation ................... 222
7.2 Application of the classical theory of gelation ....... 224
7.2.1 Pregel regime .................................. 226
7.2.2 The gel point .................................. 227
7.2.3 Postgel regime ................................. 228
7.2.4 Phase diagrams of thermoreversible gels ........ 232
7.3 Thermodynamics of sol-gel transition as compared
with Bose-Einstein condensation ............................ 233
7.4 Thermoreversible gels with multiple cross-linking ..... 235
7.4.1 Multiple association ........................... 235
7.4.2 Distribution function of multiple trees ........ 237
7.4.3 The average molecular weight and the
condition for the gel point .................... 240
7.4.4 Solution properties of thermoreversible gels
with multiple junctions ........................ 242
7.4.5 Simple models of junction multiplicity ......... 243
References ................................................. 245
8 Structure of polymer networks .............................. 247
8.1 Local structure of the networks-cross-linking
regions ............................................... 247
8.2 Global structure of the networks - elastically
effective chains and elastic modulus .................. 250
8.2.1 Fundamental parameters of the network
topology ....................................... 250
8.2.2 Structure parameters of multiplty cross-
linked gels .................................... 252
8.2.3 The number of elastically effective chains ..... 258
8.3 Percolation model ..................................... 262
8.3.1 Percolation threshold .......................... 262
8.3.2 Distribution function of clusters .............. 265
8.3.3 Percolation in one dimension ................... 266
8.3.4 Site percolation on the Bethe lattice .......... 268
8.4 Self-similarity and scaling laws ...................... 269
8.4.1 Static scaling laws ............................ 269
8.4.2 Viscoelastic scaling laws ...................... 273
8.5 Percolation in continuum media ........................ 276
8.5.1 Critical volume fraction of percolation ........ 276
8.5.2 Gelation of sticky hard spheres (Baxter's
problem) ....................................... 277
References ................................................. 279
9 Rheology of thermoreversible gels .......................... 281
9.1 Networks with temporal junctions ...................... 281
9.1.1 Models of transient networks ................... 282
9.1.2 Equilibrium solutions .......................... 286
9.1.3 Stress-strain relation ......................... 289
9.1.4 Integral form of the equation .................. 290
9.1.5 Generalization of the model .................... 292
9.2 Linear response of transient networks ................. 292
9.2.1 The Green-Tobolsky limit ....................... 295
9.2.2 Exponential dissociation rate .................. 296
9.2.3 Power-law dissociation rate .................... 297
9.2.4 Coupling to the tension ........................ 298
9.3 Stationary flows ...................................... 299
9.3.1 GT limit and quadratic P ....................... 300
9.3.2 Coupling to the tension ........................ 302
9.3.3 Expansion in powers of the shear rate .......... 303
9.3.4 Elongational flows ............................. 305
9.4 Time-dependent flows .................................. 309
9.4.1 Transient flows of Gaussian networks in the
GT limit ....................................... 309
9.4.2 Start-up shear flows with tension-
dissociation coupling .......................... 311
9.4.3 Nonlinear stress relaxation .................... 316
9.A Expansion in powers of the shear rate and time ........ 321
9.B Solvable model of the quadratic dissociation rate ..... 322
9.B.1 Start-up and stationary flows .................. 323
9.B.2 Stress relaxation .............................. 328
References ................................................. 329
10 Some important thermoreversible gels ....................... 331
10.1 Polymer-surfactant interaction ........................ 331
10.1.1 Modification of the gel point by surfactants ... 333
10.1.2 Surfactant binding isotherms ................... 335
10.1.3 CMC of the surfactant molecules ................ 336
10.1.4 High-frequency elastic modulus ................. 338
10.2 Loop-bridge transition ................................ 339
10.3 Competing hydration and gelation ...................... 345
10.3.1 Models of competitive hydration and gelation ... 345
10.3.2 Degree of hydration and the gel point .......... 349
10.4 Coexisting hydration and gelation ..................... 352
10.5 Thermoreversible gelation driven by polymer
conformational change ................................. 359
10.5.1 Models of conformational transition ............ 361
10.5.2 Theory of gelation with conformation change .... 363
10.5.3 Simple models of excitation .................... 367
10.6 Thermoreversible gelation driven by the coil-helix
transition of polymers ................................ 370
10.6.1 Models of helix association .................... 372
10.6.2 Multiple helices ............................... 374
10.6.3 Multiple association of single helices ......... 378
References ................................................. 379
Index ...................................................... 383
|
