# Physical Chemistry of Polymer Solutions

## Theoretical Background

**By**

- K. Kamide
- T. Dobashi

This book is mainly concerned with building a narrow but secure ladder which polymer chemists or engineers can climb from the primary level to an advanced level without great difficulty (but by no means easily, either).

This book describes some fundamentally important topics, carefully chosen, covering subjects from thermodynamics to molecular weight and its distribution effects. For help in self-education the book adopts a "Questions and Answers" format. The mathematical derivation of each equation is shown in detail. For further reading, some original references are also given.

Numerous physical properties of polymer solutions are known to be significantly different from those of low molecular weight solutions. The most probable explanation of this obvious discrepancy is the large molar volume ratio of solute to solvent together with the large number of consecutive segments that constitute each single molecule of the polymer chains present as solute. Thorough understanding of the physical chemistry of polymer solutions requires some prior mathematical background in its students. In the original literature, detailed mathematical derivations of the equations are universally omitted for the sake of space-saving and simplicity. In textbooks of polymer science only extremely rough schemes of the theories and then the final equations are shown. As a consequence, the student cannot learn, unaided, the details of the theory in which he or she is interested from the existing textbooks; however, without a full understanding of the theory, one cannot analyze actual experimental data to obtain more basic and realistic physical quantities. In particular, if one intends to apply the theories in industry, accurate understanding and ability to modify the theory are essential.

**Audience**

For students at universities and researchers, who are studying the physical chemistry of polymer solutions. Also as a reference text for technologists intending to apply the physical chemistry of polymer solutions to industrial practice and to educators teaching this or related subjects.

Hardbound, 676 Pages

Published: October 2000

Imprint: Elsevier

ISBN: 978-0-444-89430-4

## Contents

**Preface. Glossary. Fundamentals of thermodynamics.**Internal energy, free energy and enthalpy. Partial molar quantities. Gibbs-Duhem relation. Mixing volume change and mixing entropy. Gibbs condition for two-phase equilibrium. Heat of mixing.**Ideal and non-athermal solutions.**Ideal solution. Molar quantities in mixing. Entropy of mixing for ideal solution. Raoult's law. Boiling point elevation and freezing point depression (I). Boiling point elevation and freezing point depression (II). Membrane osmometry. van 't Hoff's equation. Empirical determination of osmotic pressure. Empirical determination of number-average molecular weight. Non-ideal solution (I). Non-ideal solution (II). Mixing in non-ideal solution. Real solution. Vapor pressure osmometry (I). Vapor pressure osmometry (II). Vapor pressure osmometry (III). Vapor pressure osmometry (IV). Vapor pressure osmometry (V). Vapor pressure osmometry (VI). Vapor pressure osmometry (VII). Vapor pressure osmometry (VIII).**Lattice Theory.**Lattice theory for low molecular weight solution. Bragg-Williams approximation. Free energy of mixing for random mixing. Free energy of mixing for athermal solution. Flory's theory (I): 0th approximation theory for polymer solution. Flory's theory (II): entropy of polymer solution. Flory's theory (III): entropy of mixing for polymer solution. Flory's theory (IV): partial molar entropy of mixing of solvent andpolymer. Flory's theory (V): van Laar-Scatchard approximation. Flory's theory (VI): Gibbs free energy of mixing of polymer solution. Flory's theory (VII): chemical potential of solvent for non-athermal random mixing polymer solution. Flory's theory (VIII): chemical potential of polymer for non-athermal random mixing polymer solution. Flory's theory (IX): Gibbs-Duhem relation for polymer solution. Flory's theory (X): assumptions in Flory's 0th approximation theory. Thermodynamic interaction parameter &khgr;. Concentration dependence of &khgr;. Virial coefficient at &thgr; point. Determination of &khgr; from cloud point curve. Chemical potential of polymer in multicomponent polymer solution. Huggins' free energy correction parameter g. Gibbs free energy of mixing for ternary system.**Phase Equilibria.**Stability of thermodynamic system (I). Stability of thermodynamic system (II). Stability of thermodynamic system (III). Stability of thermodynamic system (IV). Criteria for stable equilibrium for binary mixture. Gibbs free energy surface and phase diagram for binary mixture. Criteria for stable equilibrium for ternary mixture. Gibbs free energy on composition triangle. Critical condition for ternary mixture. Critical condition for (r+1)-component system. Critical condition for ideal solution and regular solution. Critical condition for Flory-Huggins solution. Range of critical temperature and critical composition. Determination of &thgr; and &psgr; (I): Shultz-Flory plot. Determination of &thgr; and &psgr; (II): application to experimental data. Determination of &thgr; and &psgr; (III). Chemical potential in Flory-Huggins solution. Mean molar Gibbs free energy of regular solution. Mean volume Gibbs free energy and critical condition for Flory-Huggins solution. Critical condition for homologous polymer solution. Critical parameters for homologous polymer solutions with concentration-independent &khgr;. Chemical potential for polydisperse polymer in single solvent (P/S) with concentration-dependent &khgr;. Critical condition for polydisperse polymer in single solvent (P/S) with concentration-dependent &khgr;. Determination of cloud point curve for polydisperse polymer in single solvent (P/S) with concentration-dependent &khgr;. Effect of molecular weight distribution on critical concentration. Experimental method for determining Flory's &thgr; condition. Experimental method for determining &thgr; and &psgr;. Experimental method for determining &kgr;0. Critical condition in terms of g. Relationship between g, &thgr; and &psgr;. Slope of spinodal curve. Phase equilibria of polymer blend (P1/P2) (I): Gibbs free energy of mixing per unit volume for monodisperse polymer / monodisperse polymer. Flory-Huggins free energy for multicomponent solution. Phase equilibria of polymer blend (P1/P2) (II): Critical parameters formonodisperse polymer / monodisperse polymer. Phase equilibria of polymer blend (P1/P2) (III): chemical potential for polydisperse polymer / polydisperse polymer. Phase equilibria of polymer blend (P1/P2) (IV): critical condition for polydisperse polymer / polydisperse polymer. Phase equilibria of polymer blend (P1/P2) (V): critical condition for polydisperse polymer / polydisperse polymer. Phase equilibria of polymer blend (P1/P2) (VI): critical parameters for polydisperse polymer / polydisperse polymer. Second-order derivatives of Gibbs free energy for ternary mixturesP2/S1/S0, P2/P1/S0 and P2/P1/P0. Spinodal condition for quasi-ternary system polydisperse polymer /polydisperse polymer / solvent (P2/P1/S0). Neutral equilibrium condition for quasi-ternary system polydispersepolymer / polydisperse polymer/ solvent (P2/P1/S0). Chemical potential of mixing for (r+1)-component Flory-Huggins solution. Critical condition for quasi-ternary system polydisperse polymer in mixed solvent (P3/S2/S1). Symmetry of critical condition. Fractionation (I): coexistence curve of polymer solution. Fractionation (II). Fractionation (III): partition coefficient. Fractionation (IV): Weight of polymer partitioned in each phase. Fractionation (V): characteristic specific value for the degree of polymerization na. Fractionation (VI): fractionation efficiency. Fractionation (VII): molecular weight distribution of polymers remaining in concentrated phase. Fractionation (VIII): effect of fraction size. Fractionation (IX): effect of overall concentration on efficiency. Fractionation (X): shape of molecular weight distribution. fractionation (XI): successive precipitation fractionation and successive solution fractionation.**Colligative Properties and Virial Coefficients of Polymer Solutions.**Osmotic pressure (I): vapor pressure and osmotic pressure of polymer solution. Osmotic pressure (II): virial expansion. Osmotic pressure (III). Second virial coefficient (I): internal energy and entropy terms. Second virial coefficient (II): van der Waals equation. Flory temperature for van der Waals equation. Partition function for semi-grand canonical ensemble (I). Partition function for semi-grand canonical ensemble (II). N-body distribution function. Osmotic pressure (IV): cluster integrals. Osmotic pressure (V): relationship for second and third virial coefficients with cluster integrals. Second virial coefficient (III): relationship with pair segment potential. Second virial coefficient (IV). Second virial coefficient (V): polymer segment with rigid sphere potential. Second virial coefficient (VI): comparison of Flory lattice theory with imperfect gas theory. Second virial coefficient (VII): mean force potential. Second virial coefficient (VIII): temperature dependence. Second virial coefficient (IX): ideal solution. Second virial coefficient (X): rigid sphere solution. Second virial coefficient (XI): molecular weight dependence. Two-body cluster integral. Second virial coefficient (XII): various polymer solutions. Second virial coefficient (XIII): rod-like molecule. Second virial coefficient (XIV): chain molecule with n sequential rigid rod segments. Relationship between second virial coefficient and excess chemical potential. Third virial coefficient of rigid sphere solution. Relationship between second and third virial coefficients.**Statistical Mechanics and Excluded Volume of Polymer Chains.**Probability density distribution for Gaussian chain. Distribution function of end-to-end distance of random chain (I). Distribution function of end-to-end distance of random chain (II). Elastic force of Gaussian chain (I). Elastic force of Gaussian chain (II). Mean square end-to-end distance of Gaussian chain. End-to-end distance for chain molecule with internal rotation (I). End-to-end distance for chain molecule with internal rotation (II). End-to-end distance for chain molecule with internal rotation (III): Oka's equation. Distribution function of end-to-end distance of polymer chain. Bresler-Frenkel's equation. Mean square radius of gyration. End-to-end distance of partial chain. Distribution function of separation between segments and the center of gravity. Excluded volume effect (I). Excluded volume effect (II). Excluded volume effect (III). Increase in free energy by swelling. &agr;5-law. Segment density at the origin. Mean internal energy. Relationship between &agr; and Z (I). &agr;3-law. Relationship between &agr; and Z (II). Relationship between &agr; and Z (III). Relationship between &agr; and Z (IV). Relationship between &agr; and Z (V). Relationship between &agr; and Z (VI): Kurata-Stockmayer-Roig's equation. Relationship between &agr; and Z (VII). Determination of Flory constant K. Determination of Z. &agr;s and &psgr; comparison between experiment and theory.**Light Scattering.**Rayleigh's equation for scattered light intensity. Total scattered light intensity. Turbidity (I). Rayleigh ratio. Scattering from large particles. Particle scattering factor. Guinier plot. Determination of the shape of particles from P(&thgr;) (I). Determination of the shape of particles from P(&thgr;) (II). Determination of the shape of particles from P(&thgr;) (III). Determination of polarizability &agr;. Scattering from small particles. Scattering from polymer solution. Effect of molecular weight distribution of polymer chains on P(&thgr;). Zimm plot. Particle scattering factor for polymers with Schulz-Zimm molecular weight distribution (I). Particle scattering factor for polymers with Schulz-Zimm molecular weight distribution (II). Light scattering arising from concentration fluctuation. Relationship between concentration fluctuation and chemical potential. Light scattering arising from copolymer (I). Light scattering arising from copolymer (II). Light scattering arising from optically anisotropic particles (I). Light scattering arising from optically anisotropic particles (II). Fluctuation theory of light scattering. Turbidity (II). Light scattering arising from polymer solution with molecular weight distribution. Osmotic pressure of polymer solution with molecular weight distribution. Light scattering arising from the system polymer in mixed solvent (P2/S1/S0).**Hydrodynamic Properties.**Equation of motion for viscoelastic fluids (I). Stress-strain relationship. Lamé constant. Stress equation. Equation of motion for viscoelastic fluids (II). Equation of continuity. Navier-Stokes equation and Euler's equation. Reynolds number. Couette flow. Equation of motion and equation of continuity for slow steady flow (I). Equation of motion and equation of continuity for slow steady flow (II). Equation of motion and equation of continuity for slow steady flow (III). Oseen tensor. Capillary flow. Frictional heat and viscosity. Estimation of volume fraction dependence of viscosity coefficient. Two-dimensional steady shear flow of solution of dumbbell-like molecule (I): diffusion equation. Two-dimensional steady shear flow of solution of dumbbell-like molecule (II): force and its corresponding moment acting on molecule. Two-dimensional steady shear flow of solution of dumbbell-like molecule (III): probability density. Two-dimensional steady shear flow of solution of dumbbell-like molecule (IV): viscous dissipation. Limiting viscosity number of solution of dumbbell-like molecule (I): estimated from heat dissipation. Limiting viscosity number of solution of dumbbell-like molecule (II): from the ratio of shear stress to shear rate. Definition of solution viscosities. Determination of relative viscosity by viscometer. Huggins' plot and Kraemer's plot (I). Huggins' plot and Kraemer's plot (II). Empirical functional form of the concentration dependence of viscosity. Einstein's viscosity equation: rigid sphere model. Molecular weight dependence of [&eegr;] (I): unpenetrable sphere formed by chain polymer or linear polymer forming sphere. Angular velocity of molecular chain in steady flow. Molecular weight dependence of [&eegr;] (II): free draining random coil molecules. Molecular weight dependence of [&eegr;] (III): Linear polymer Gaussian chain with hydrodynamic interaction (Kirkwood-Riseman theory). Flory constant K (I). Flory constant K (II). Viscosity parameter. Upper limit of the exponent in Mark-Houwink-Sakurada equation. Theoretical relations between two parameters in Mark-Houwink-Sakurada equation (I). Theoretical relations between two parameters in Mark-Houwink-Sakurada equation (II). Flory-Fox-Schaefgen equation. Flory constant K (III). Evaluation of parameters in Mark-Houwink-Sakurada equation by Kurata-Yamakawa theory. Molecular weight dependence of sedimentation coefficient (I). Molecular weight dependence of sedimentation coefficient (II). Molecular weight dependence of diffusion coefficient (I). Molecular weight dependence of diffusion coefficient (II). Two-dimensional steady shear flow of solution of dumbbell-like molecule (V).**Molecular Weight and Molecular Weight Distribution.**Definition of average molecular weight. Schulz polymolecularity index. Average degree of polymerization (I). Average degree of polymerization (II). Condensation polymerization (I). Condensation polymerization (II). Condensation polymerization (III). Condensation polymerization (IV). Molecular weight distribution function (I). Molecular weight distribution function (II). Average degree of polymerization for Schulz-Zimm distribution. Average degree of polymerization for the most probable distribution. Average degree of polymerization for Wesslau distribution. Average degree of polymerization for Lansing-Kraemer distribution. Average degree of polymerization for general log-normal distribution. Average degree of polymerization for Poisson distribution. Molecular weight distribution for equilibrium condensation polymerization. Molecular weight distribution for radical polymerization. Viscosity-average molecular weight. Effect of molecular weight distribution on Mark-Houwink-Sakurada equation (I): Schulz-Zimm type. Effect of molecular weight distribution on Mark-Houwink-Sakuradaequation (II): logarithmic-normal type. Effect of molecular weight distribution on Mark-Houwink-Sakurada equation (III). Effect of molecular weight distribution on Mark-Houwink-Sakurada equation (IV). Effect of molecular weight distribution on viscosity parameter &Fgr;. Effect of molecular weight distribution on Flory-Fox equation. Correction parameter q for viscosity parameter &Fgr;. Effect of molecular weight distribution on parameter qw,z. qw and qw,z for Schulz-Zimm distribution (I). qw and qw,z for Schulz-Zimm distribution (II). Sedimentation coefficient and diffusion coefficient for polydisperse polymer solution (I). Sedimentation coefficient and diffusion coefficient for polydisperse polymer solution (II). Radius of gyration for polymer solution with Schulz-Zimm molecular weight distribution (I). Radius of gyration for polymer solution with Schulz-Zimm molecular weight distribution (II). Chemical potential of mixing for polydisperse polymer solution.**Index.**