A lattice model for self‐avoiding polymers with controlled length distributions. II. Corrections to Flory–Huggins mean field

We continue the analysis of the lattice spin‐field theory which is introduced in paper I and which formally gives the exact entropy for a set of completely flexible self‐avoiding polymers on a lattice. The model allows for arbitrary chain length distributions and arbitrary polymer volume fractions. Use of random fields produces a field theory from which Flory–Huggins results are recovered in the mean field limit. Here we recast the mean field formulation to allow for the rigorous and systematic evaluation of corrections by means of a cluster expansion. We then calculate corrections to the mean field directly up to fourth order and provide a systematic diagrammatic method for evaluating general order corrections. Our results illustrate the polymer concentration dependence of the corrections to the Flory–Huggins mean field results as well as provide the origins for the entropic contribution to the Flory χ parameter and its concentration dependence. Generalizations to treat rods and semiflexible polymers wil...

[1]  J. D. Cloizeaux,et al.  The Lagrangian theory of polymer solutions at intermediate concentrations , 1975 .

[2]  S. Edwards.  The size of a polymer molecule in a strong solution , 1975 .

[3]  H. C. Andersen,et al.  Roles of Repulsive and Attractive Forces in Liquids: The Optimized Random Phase Approximation , 1972 .

[4]  P. Gennes.  Scaling Concepts in Polymer Physics , 1979 .

[5]  Karl F. Freed,et al.  Renormalization Group Theory of Macromolecules , 1987 .

[6]  J. H. Gibbs,et al.  Chain Stiffness and the Lattice Theory of Polymer Phases , 1958 .

[7]  S. Edwards,et al.  Cross linkage problems of polymers I. The method of second quantization applied to the cross linkage problem of polymers , 1970 .

[8]  M. Huggins.  Solutions of Long Chain Compounds , 1941 .

[9]  C. Itzykson,et al.  An evaluation of the number of Hamiltonian paths , 1985 .

[10]  S. Kirkpatrick.  Percolation and Conduction , 1973 .

[11]  R. Kubo.  GENERALIZED CUMULANT EXPANSION METHOD , 1962 .

[12]  P. Flory,et al.  Stress-Strain Isotherm for Polymer Networks , 1959 .

[13]  F. T. Wall,et al.  Simulation of polymers by self‐avoiding, nonintersecting random chains at various concentrations , 1977 .

[14]  J. D. Cloizeaux.  Polymers in solutions : principles and applications of a direct renormalization method , 1981 .

[15]  K. Freed,et al.  Excluded volume effects for polymers in presence of interacting surfaces: Chain conformation renormalization group , 1985 .

[16]  P. G. de Gennes,et al.  Exponents for the excluded volume problem as derived by the Wilson method , 1972 .

[17]  A. R. Miller,et al.  The number of configurations of a cooperative assembly , 1942, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  E. DiMarzio,et al.  Statistics of Orientation Effects in Linear Polymer Molecules , 1961 .

[19]  T. Morita,et al.  A New Approach to the Theory of Classical Fluids. III: General Treatment of Classical Systems , 1961 .

[20]  J. Nagle.  Statistical mechanics of the melting transition in lattice models of polymers , 1974, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[21]  A. Bellemans,et al.  Concentration dependence of the mean dimension of a polymer chain , 1974 .

[22]  D. Y. Yoon,et al.  Phase transition of cubic-lattice polymer systems , 1984 .

[23]  R. Scott,et al.  The Thermodynamics of High Polymer Solutions. V. Phase Equilibria in the Ternary System: Polymer 1—Polymer 2—Solvent , 1949 .

[24]  P. Flory.  Principles of polymer chemistry , 1953 .

[25]  K. Freed.  New lattice model for interacting, avoiding polymers with controlled length distribution , 1985 .

[26]  R. Fowler,et al.  The number of configurations in an assembly and cooperative phenomena , 1939, Mathematical Proceedings of the Cambridge Philosophical Society.

[27]  R. Fowler,et al.  An attempt to extend the statistical theory of perfect solutions , 1937 .

[28]  D. S. McKenzie.  Polymers and scaling , 1976 .