Physical Chemistry of Polymer Solutions: Theoretical Background

Cover Page......Page 1
Title: Physical Chemistry of Polymer Solutions......Page 4
ISBN 0444894306......Page 5
Preface......Page 8
2. Ideal and Non-Athermal Solutions......Page 14
4. Phase Equilibria......Page 15
5. Colligative Properties and Virial Coefficients of Polymer Solutions......Page 17
6. Statistical Mechanics and Excluded Volume of Polymer Chains......Page 18
7. Light Scattering......Page 19
8. Hydrodynamic Properties......Page 20
9. Molecular Weight and Molecular Weight Distribution......Page 22
Glossary......Page 24
1-1 Internal energy, free energy and enthalpy......Page 30
1-2 Partial molar quantities......Page 32
1-3 Gibbs-Duhem relation......Page 34
1-4 Mixing volume change and mixing entropy......Page 36
1-5 Gibbs condition for two-phase equilibrium......Page 37
1-6 Heat of mixing......Page 38
2-1 Ideal solution......Page 40
2-2 Molar quantities in mixing......Page 42
2-4 Raoult's law......Page 45
2-5 Boiling point elevation and freezing point depression (I)......Page 48
2-6 Boiling point elevation and freezing point depression (II)......Page 49
2-7 Membrane osmometry......Page 51
2-8 van't Hoff's equation......Page 52
2-9-b Empirical determination of number-average molecular weight......Page 55
2-10 Non-ideal solution (I)......Page 56
2-11 Non-ideal solution (II)......Page 57
2-12 Mixing in non-ideal solution......Page 58
2-13 Real solution......Page 61
2-14 Vapor pressure osmometry (I)......Page 63
2-15 Vapor pressure osmometry (II)......Page 64
2-16 Vapor pressure osmometry (III)......Page 66
2-17 Vapor pressure osmometry (IV)......Page 71
2-18 Vapor pressure osmometry (V)......Page 72
2-19-a Vapor pressure osmometry (VI)......Page 76
2-20 Vapor pressure osmometry (VIII)......Page 77
3-1 Lattice theory for low molecular weight solution......Page 79
3-2 Bragg-Williams approximation......Page 81
3-3 Free energy of mixing for random mixing......Page 82
3-4 Free energy of mixing for athermal solution......Page 84
3-5 Flory's theory (I): 0th approximation theory for polymer solution......Page 86
3-6 Flory's theory (II): Entropy of polymer solution......Page 89
3-7 Flory's theory (III): Entropy of mixing for polymer solution......Page 90
3-8 Flory's theory (IV): Partial molar entropy of mixing of solvent and polymer......Page 92
3-9 Flory's theory (V): van Laar-Scatchard approximation......Page 93
3-10 Flory's theory (VI): Gibbs free energy of mixing of polymer solution......Page 94
3-11 Flory's theory (VII): Chemical potential of solvent for non-athermal random mixing polymer solution......Page 95
3-12 Flory's theory (VIII): Chemical potential of polymer for non-athermal random mixing polymer solution......Page 96
3-13 Flory's theory (IX): Gibbs-Duhem relation for polymer solution......Page 97
3-14 Flory's theory (X): Assumptions in Flory's 0th approximation theory......Page 99
3-15 Thermodynamic interaction parameter X......Page 107
3-16-a Concentration dependence of X......Page 110
3-16-b Virial coefficient at theta point......Page 111
3-16-c Determination of x from cloud point curve......Page 113
3-17 Chemical potential of polymer in multicomponent polymer solution......Page 117
3-18 Huggins' free energy correction parameter g......Page 118
3-19 Gibbs free energy of mixing for ternary system......Page 119
4-1 Stability of thermodynamic system (I)......Page 123
4-2 Stability of thermodynamic system (II)......Page 124
4-3 Stability of thermodynamic system (III)......Page 126
4-4 Stability of thermodynamic system (IV)......Page 128
4-5 Criteria for stable equilibrium for binary mixture......Page 131
4-6 Gibbs free energy surface and phase diagram for binary mixture......Page 134
4-7 Criteria for stable equilibrium for ternary mixture......Page 138
4-8 Gibbs free energy on composition triangle......Page 144
4-9 Critical condition for ternary mixture......Page 145
4-10 Critical condition for (r+1 )-component system......Page 147
4-11 Critical condition for ideal solution and regular solution......Page 151
4-12 Critical condition for Flory-Huggins solution......Page 153
4-13 Range of critical temperature and critical composition......Page 156
4-14 Determination of theta and psi (I): Shultz-Flory plot......Page 157
4-15 Determination of theta and psi (II): Application to experimental data......Page 158
4-16 Determination of theta and psi (III)......Page 159
4-18 Mean molar Gibbs free energy of regular solution......Page 160
4-19 Mean volume Gibbs free energy and critical condition for Flory-Huggins solution......Page 162
4-20 Critical condition for homologous polymer solution......Page 165
4-21 -a Critical parameters for homologous polymer solutions with concentrationindependent x......Page 175
4-21 -b Chemical potential for polydisperse polymer in single solvent (P/S) with concentration-dependent x......Page 176
4-21 -c Critical condition for polydisperse polymer in single solvent (P/S) with concentration-dependent x......Page 178
4-21 -d Determination of cloud point curve for polydisperse polymer in single solvent (P/S) with concentration-dependent x......Page 180
4-22 Effect of molecular weight distribution on critical concentration......Page 184
4-23-a Experimental method for determining Flory's theta condition......Page 186
4-23-b Experimental method for determining theta and psi......Page 187
4-23-c Experimental method for determining K0......Page 196
4-24 Critical condition in terms of g......Page 201
4-25 Relationship between g, theta and psi......Page 204
4-27 Phase equilibria of polymer blend (P1/P2) (I): Gibbs free energy of mixing per unit volume for monodisperse polymer / monodisperse polymer......Page 205
4-28-a Flory-Huggins free energy for multicomponent solution......Page 206
4-28-b Phase equilibria of polymer blend (P1/P2) (II): Critical parameters for monodisperse polymer / monodisperse polymer......Page 209
4-28-c Phase equilibria of polymer blend (P1/P2) (III): Chemical potential for polydisperse polymer / polydisperse polymer......Page 211
4-28-d Phase equilibria of polymer blend (P1/P2) (IV): Critical condition for polydisperse polymer / polydisperse polymer......Page 212
4-28-f Phase equilibria of polymer blend (P1/P2) (VI): Critical parameters for polydisperse polymer / polydisperse polymer......Page 217
4-28-g Second-order derivatives of Gibbs free energy for ternary mixtures P2/S1/S0, P2/P1/S0 and P2/P1/P0......Page 225
4-29-a Spinodal condition for quasi-ternary system polydisperse polymer / polydisperse polymer / solvent (P2/P1/S0)......Page 226
4-29-b Neutral equilibrium condition for quasi-ternary system polydisperse polymer / polydisperse polymer/ solvent (P2/P1/S0)......Page 230
4-30-a Chemical potential of mixing for (r+1 )-component Flory-Huggins solution......Page 235
4-30-b Critical condition for quasi-ternary system polydisperse polymer in mixed solvent (P3/S2/S1)......Page 239
4-30-c Symmetry of critical condition......Page 245
4-31 Fractionation (I): Coexistence curve of polymer solution......Page 247
4-32 Fractionation (II)......Page 250
4-33 Fractionation (III): Partition coefficient......Page 251
4-34 Fractionation (IV): Weight of polymer partitioned in each phase......Page 252
4-35 Fractionation (V): Characteristic specific value for the degree of polymerization na......Page 253
4-36 Fractionation (VI): Fractionation efficiency......Page 254
4-37 Fractionation (VII): Molecular weight distribution of polymers remaining in concentrated phase......Page 255
4-38 Fractionation (VIII): Effect of fraction size......Page 256
4-41 Fractionation (XI): Successive precipitation fractionation and successive solution fractionation......Page 261
5-1 Osmotic pressure (I): Vapor pressure and osmotic pressure of polymer solution......Page 265
5-2 Osmotic pressure (II): Virial expansion......Page 267
5-3 Osmotic pressure (III)......Page 268
5-4 Second virial coefficient (I): Internal energy and entropy terms......Page 269
5-5 Second virial coefficient (II): van der Waals equation......Page 270
5-7 Partition function for semi-grand canonical ensemble (I)......Page 271
5-8 Partition function for semi-grand canonical ensemble (II)......Page 274
5-9 N-body distribution function......Page 277
5-10 Osmotic pressure (IV): Cluster integrals......Page 280
5-12 Second virial coefficient (III): Relationship with pair segment potential......Page 284
5-13 Second virial coefficient (IV)......Page 286
5-14 Second virial coefficient (V): Polymer segment with rigid sphere potential......Page 288
5-16 Second virial coefficient (VII): Mean force potential......Page 289
5-17 Second virial coefficient (VIII): Temperature dependence......Page 291
5-18 Second virial coefficient (IX): Ideal solution......Page 292
5-19 Second virial coefficient (X): Rigid sphere solution......Page 293
5-20 Second virial coefficient (XI): Molecular weight dependence......Page 294
5-21 Two-body cluster integral......Page 296
5-22 Second virial coefficient (XII): Various polymer solutions......Page 297
5-23 Second virial coefficient (XIII): Rod-like molecule......Page 298
5-24 Second virial coefficient (XIV): Chain molecule with n sequential rigid rod segments......Page 301
5-25 Relationship between second virial coefficient and excess chemical potential......Page 302
5-26 Third virial coefficient of rigid sphere solution......Page 304
5-27 Relationship between second and third virial coefficients......Page 308
6-1 Probability density distribution for Gaussian chain......Page 309
6-2 Distribution function of end-to-end distance of random chain (I)......Page 313
6-3 Distribution function of end-to-end distance of random chain (II)......Page 315
6-4-a Elastic force of Gaussian chain (I)......Page 316
6-4-b Elastic force of Gaussian chain (II)......Page 317
6-5 Mean square end-to-end distance of Gaussian chain......Page 323
6-6-a End-to-end distance for chain molecule with internal rotation (I)......Page 326
6-6-b End-to-end distance for chain molecule with internal rotation (II)......Page 328
6-6-c End-to-end distance for chain molecule with internal rotation (III): Oka's equation......Page 329
6-7 Distribution function of end-to-end distance of polymer chain......Page 333
6-8 Bresler-Frenkel's equation......Page 334
6-9 Mean square radius of gyration......Page 337
6-10 End-to-end distance of partial chain......Page 340
6-11 Distribution function of separation between segments and the center of gravity......Page 344
6-12-a Excluded volume effect (I)......Page 346
6-12-b Excluded volume effect (II)......Page 348
6-12-c Excluded volume effect (III)......Page 352
6-13-a Increase in free energy by swelling......Page 358
6-13-b a5 -law......Page 360
6-13-c Segment density at the origin......Page 365
6-14-a Mean internal energy......Page 366
6-14-b Relationship between a and Z (I)......Page 368
6-15-a a3-law......Page 372
6-15-b Relationship between a and Z (II)......Page 374
6-15-c Relationship between a and Z (III)......Page 376
6-16-a Relationship between a and Z (IV)......Page 383
6-16-b Relationship between a and Z (V)......Page 387
6-17-a Relationship between a and Z (VI): Kurata-Stockmayer-Roig's equation......Page 390
6-17-b Relationship between a and Z (VII)......Page 395
6-18 Determination of Flory constant K......Page 396
6-19 Determination of Z......Page 398
6-20 as and psi; comparison between experiment and theory......Page 401
7-1 Rayleigh's equation for scattered light intensity......Page 406
7-2-a Total scattered light intensity......Page 410
7-2-b Turbidity (I)......Page 411
7-3 Rayleigh ratio......Page 412
7-4 Scattering from large particles......Page 413
7-5 Particle scattering factor......Page 418
7-7-a Determination of the shape of particles from P(theta) (I)......Page 420
7-8 Determination of the shape of particles from P(theta) (III)......Page 426
7-9 Determination of polarizability a......Page 427
7-10-a Scattering from small particles......Page 429
7-10-b Scattering from polymer solution......Page 430
7-11 -a Effect of molecular weight distribution of polymer chains on P(theta)......Page 432
7-11-b Zimm plot......Page 435
7-12-a Particle scattering factor for polymers with Schulz-Zimm molecular weight distribution (I)......Page 436
7-13 Light scattering arising from concentration fluctuation......Page 439
7-14 Relationship between concentration fluctuation and chemical potential......Page 441
7-15-a Light scattering arising from copolymer (I)......Page 445
7-15-b Light scattering arising from copolymer (II)......Page 447
7-16 Light scattering arising from optically anisotropic particles (I)......Page 448
7-17 Light scattering arising from optically anisotropic particles (II)......Page 454
7-18 Fluctuation theory of light scattering......Page 456
7-19 Turbidity (II)......Page 462
7-20 Light scattering arising from polymer solution with molecular weight distribution......Page 465
7-21 Osmotic pressure of polymer solution with molecular weight distribution......Page 467
7-22 Light scattering arising from the system polymer in mixed solvent (P2/S1,/So)......Page 469
8-1 Equation of motion for viscoelastic fluids (I)......Page 472
8-2 Stress-strain relationship......Page 474
8-3 Lame constant......Page 477
8-5 Equation of motion for viscoelastic fluids (II)......Page 481
8-6 Equation of continuity......Page 483
8-7 Navier-Stokes equation and Euler's equation......Page 485
8-9 Couette flow......Page 486
8-10 Equation of motion and equation of continuity for slow steady flow (I)......Page 489
8-11 Equation of motion and equation of continuity for slow steady flow (II)......Page 490
8-13 Oseen tensor......Page 491
8-14 Capillary flow......Page 495
8-15-a Frictional heat and viscosity......Page 498
8-15-b Estimation of volume fraction dependence of viscosity coefficient......Page 499
8-16 Two-dimensional steady shear flow of solution of dumbbell-like molecule (I): Diffusion equation......Page 500
8-17 Two-dimensional steady shear flow of solution of dumbbell-like molecule (II): Force and its corresponding moment acting on molecule......Page 506
8-18 Two-dimensional steady shear flow of solution of dumbbell-like molecule (III): Probability density......Page 508
8-19 Two-dimensional steady shear flow of solution of dumbbell-like molecule (IV): Viscous dissipation......Page 511
8-20 Limiting viscosity number of solution of dumbbell-like molecule (I): Estimated from heat dissipation......Page 513
8-21 Limiting viscosity number of solution of dumbbell-like molecule (II): From the ratio of shear stress to shear rate......Page 514
8-22 Definition of solution viscosities......Page 516
8-23 Determination of relative viscosity by viscometer......Page 517
8-24 Huggins' plot and Kraemer's plot (I)......Page 518
8-25 Huggins' plot and Kraemer's plot (II)......Page 519
8-26 Empirical functional form of the concentration dependence of viscosity......Page 522
8-27 Einstein's viscosity equation: Rigid sphere model......Page 524
8-28 Molecular weight dependence of [eta] (I): Unpenetrable sphere formed by chain polymer or linear polymer forming sphere......Page 528
8-29 Angular velocity of molecular chain in steady flow......Page 530
8-30 Molecular weight dependence of [eta|] (II): Free draining random coil molecules......Page 532
8-31 Molecular weight dependence of [eta] (III): Linear polymer Gaussian chain with hydrodynamic interaction (Kirkwood-Riseman theory)......Page 541
8-32 Flory constant K (I)......Page 553
8-33 Flory constant K (II)......Page 556
8-34 Viscosity parameter (theta)......Page 558
8-35 Upper limit of the exponent in Mark-Houwink-Sakurada equation......Page 561
8-36 Theoretical relations between two parameters in Mark-Houwink-Sakurada equation (I)......Page 563
8-37 Theoretical relations between two parameters in Mark-Houwink-Sakurada equation (II)......Page 565
8-38 Flory-Fox-Schaefgen equation......Page 567
8-40 Evaluation of parameters in Mark-Houwink-Sakurada equation by Kurata-Yamakawa theory......Page 569
8-41 Molecular weight dependence of sedimentation coefficient (I)......Page 572
8-42 Molecular weight dependence of sedimentation coefficient (II)......Page 573
8-44 Molecular weight dependence of diffusion coefficient (II)......Page 574
8-45 Two-dimensional steady shear flow of solution of dumbbell-like molecule (V)......Page 575
9-1 Definition of average molecular weight......Page 591
9-2 Schulz polymolecularity index......Page 594
9-3 Average degree of polymerization (I)......Page 595
9-5-a Condensation polymerization (I)......Page 598
9-5-b Condensation polymerization (II)......Page 600
9-6 Condensation polymerization (III)......Page 604
9-7 Condensation polymerization (IV)......Page 605
9-8 Molecular weight distribution function (I)......Page 606
9-9 Molecular weight distribution function (II)......Page 610
9-10 Average degree of polymerization for Schulz-Zimm distribution......Page 611
9-11 Average degree of polymerization for the most probable distribution......Page 615
9-12 Average degree of polymerization for Wesslau distribution......Page 616
9-13 Average degree of polymerization for Lansing-Kraemer distribution......Page 620
9-14 Average degree of polymerization for general log-normal distribution......Page 623
9-15 Average degree of polymerization for Poisson distribution......Page 627
9-16 Molecular weight distribution for equilibrium condensation polymerization......Page 629
9-17 Molecular weight distribution for radical polymerization......Page 631
9-18 Viscosity-average molecular weight......Page 635
9-19 Effect of molecular weight distribution on Mark-Houwink-Sakurada equation (I): Schulz-Zimm type......Page 638
9-20 Effect of molecular weight distribution on Mark-Houwink-Sakurada equation (II): Logarithmic-normal type......Page 640
9-21 Effect of molecular weight distribution on Mark-Houwink-Sakurada equation (III)......Page 642
9-22 Effect of molecular weight distribution on Mark-Houwink-Sakurada equation (IV)......Page 643
9-23 Effect of molecular weight distribution on viscosity parameter (phi)......Page 645
9-24 Effect of molecular weight distribution on Flory-Fox equation......Page 647
9-25 Correction parameter q for viscosity parameter (phi)......Page 648
9-26 Effect of molecular weight distribution on parameter qw,z......Page 653
9-27 qw and qw,z for Schulz-Zimm distribution (I)......Page 654
9-28 qw and qw,z for Schulz-Zimm distribution (II)......Page 656
9-29 Sedimentation coefficient and diffusion coefficient for polydisperse polymer solution (I)......Page 657
9-30 Sedimentation coefficient and diffusion coefficient for polydisperse polymer solution (II)......Page 659
9-31 Radius of gyration for polymer solution with Schulz-Zimm molecular weight distribution (I)......Page 660
9-33 Chemical potential of mixing for polydisperse polymer solution......Page 663
Index......Page 668