Virial expansion for a polymer chain: the two parameter approximation

Abstract

A complete diagrammatic expansion is developed for the Domb-Joyce model of an N-step chain, with an interaction w which varies between 0 and 1. Simple rules are given for obtaining the diagrams. The correspondence between these diagrams and appropriate generating functions permits computation of the coefficients of the series α²_N(w) = 1 + k₁w + k₂w² + . . ., where α²_N(w) is the expansion factor of the mean square end-to-end length of the chain. The dominant term in N of each of the first three k_r is shown to be identical for the three cubic lattices and for the Gaussian continuum model, with the exception of a scale factor h₀. Retention of only this dominant term yields a ‘two-parameter’ expansion equivalent to that of Zimm (1946), Fixman (1955) and others. Diagrams are classed either as ladder or as non-ladder graphs. The ladder graph contributions are summed by using functional relations of Domb & Joyce (1972). The non-ladder contributions for the first three coefficients are computed individually, thereby yielding results for k₁, k₂ and k₃ in terms of the ‘universal’ parameter z = h₀N^1/2w. The terms k₁ and k₂ agree with previous computations for the Gaussian model but k₃ differs slightly.