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 α 2 N ( w ) = 1 + k 1 w + k 2 w 2 + . . ., where α 2 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 h 0 . 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 k 1 , k 2 and k 3 in terms of the ‘universal’ parameter z = h 0 N 1/2 w . The terms k 1 and k 2 agree with previous computations for the Gaussian model but k 3 differs slightly.