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Cambridge2011An Introduction to Atmospheric Physics
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Contents Preface page 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 2Polymer 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 viii Contents 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 Randombranching 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{Af }/R{Bg} 116 3.3.3 Polydisperse binary mixture R{Af }/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 entropic elasticity 128 4.1.2 Thermoelastic inversion 131 4.1.3 Gough–Joule effect 131 Contents ix 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 Phantomnetwork theory 142 4.3.1 Micronetworks of tree form 143 4.3.2 Fluctuation theoremand 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.C 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 x Contents 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 continuumm edia 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 Equilibriumsolutions 286 9.1.3 Stress–strain relation 289 9.1.4 Integral formof the equation 290 9.1.5 Generalization of the model 292 Contents xi 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 β 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 |
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