Two-dimensional polymer configuration via mean-field theory

Abstract

We consider determining the configurational properties of a neutral polymer in two dimensions (2D) via self-consistent mean-field methods. By suitably scaling the problem we recover the Flory result for polymers under the excluded volume interaction, i.e. RN ∼ N3/4, where RN is the mean scaling length of a polymer which consists of (N + 1) monomers. If we let x denote the scaled distance from one end of the polymer to a point in space we find that there exists a point y*, where the scaled polymer density fN(x), decays rapidly to zero. Physically the existence of such a point is expected since the polymer has a finite length. For y* - x > O(N-1/3) we find fN(x) ∼ 1/2x[fN(x)-fN(y*)]1/2 while for x-y* > O(N-1/3) we obtain fN(x) ∼ o(1). We discuss the consequence of these results on the validity of the asymptotic methods used.