Polymers with excluded volume in various geometries: Renormalization group methods

Abstract

Renormalization group (RG) methods are generalized to study a single polymer chain with excluded volume in various geometries with different boundary conditions (or polymer–surface interactions) on the limiting surfaces. Methods for the renormalization of these theories are presented and are used to derive the RG equations which dictate the generalized scaling behavior as a function of the several interaction and geometrical parameters. We illustrate the general theory by studying a polymer chain confined between two parallel plates with three different (Neumann, Dirichlet, and periodic) boundary conditions to one-loop order. We show that ε expansions are well behaved as long as the radius of gyration of the chain is smaller than the interplate separation L. The finite size corrections to the full space (bulk) limit are found to be proportional to L−1 for free boundaries, while they are exponentially small for periodic boundary conditions. The presence of several lengths and/or interactions produces interesting crossovers, which we illustrate for a Gaussian polymer chain attached to the exterior surface of a repulsive sphere where full crossover scaling functions are obtained for the partition function and moments of the end-vector distribution function. A new exponent associated with the radius of the sphere is predicted on the basis of scaling arguments which are supported by the RG equations. This work provides the necessary input ingredients for extension of the theory to treat semidilute concentrations.