Renormalization group treatment of excluded volume effects in a polyelectrolyte chain in the weak electrostatic coupling limit

Abstract

We provide the first rigorous treatment of the electrostatic excluded volume for a polyelectrolyte chain which incorporates the effects of salt concentration. Our treatment involves an extension of the t’Hooft–Veltman method of dimensional regularization for polymer excluded volume, developed in the accompanying paper, to the case complicated by the presence of electrostatic interactions. The critical dimensionality for the polyelectrolyte chains with realistic interactions is shown to be four in sharp contrast to previous simplified analyses, which do not consider salt concentration effects explicitly and which lead to a critical dimensionality of six. Our results imply that expansions in ε=4−d (with d the dimensionality of space) can be applied, so the theory reduces to the limit of uncharged polymers with excluded volume when the electrostatic interactions become totally screened. Our renormalization group (RG) treatment indicates the absence of stable fixed points, so there is no simple scaling limit. The range of validity of the perturbation expansion is established on the basis of a RG analysis, and a physical meaning of the weak coupling limit is also determined. The predicted lack of universality for the polyelectrolyte chain is in accord with experimental information. Explicit renormalized expressions are derived for the mean squared end-to-end distance 〈R2〉 to lowest order in both excluded volume and electrostatic coupling constants. These expressions are combined with the solution of the RG equations to provide a generalized scaling representation for 〈R2〉 in terms of three scaling variables. A brief discussion of possible future biological and nonbiological applications is provided.