A generic model for long self-avoiding chain molecules

Abstract

A generic model, which applies rigorously to finite chains in the absence of inter-segmental interactions, and to subchains of infinite chains with practically arbitrary interactions, furnishes a simple recurrence relation. Self-avoiding walks on the diamond lattice (SAW-D) form the paradigmatic example for polymer science. The generic model as a whole is solved (in terms of parameters) for configurational statistics, asymptotically (large number n of segments) and on stated plausible conjectures, by classical methods, i.e. without postulating a singularity of power law form. The appropriate generalisation of the power law form of scaling theory emerges very simply, e.g. in terms of Kummer's hypergeometric function. Extensive computations on the SAW-D model lead to two correction terms to the crude scaling form. In this way, some arguments in the literature on asymptotic theories can be settled. Several examples illustrate the danger of mistaking non-asymptotic experimental results for those desired in the asymptotic range. Thus, contrary to his own conclusions and those of Fleming (1979) Brun's data (1977) data for freely hinged hard-sphere chains are here reconciled with the 'universal' exponent gamma of Le Guillou and Zinn-Justin (1977). For the SAW-D model subject always to further refinements (which will never produce a final answer), the present experimental estimate for gamma is also shown to be about 1.2, in line with conjectural estimates from many forms of modern theory.