Charge renormalization and other exact coupling corrections to the dipolar effective interaction in an electrolyte...

The aim of the paper is to study the renormalizations of the charge and screening length that appear in the large-distance behavior of the effective pairwise interaction w(alphaalpha') between two charges e(alpha) and e(alpha') in a dilute electrolyte solution, both along a dielectric wall and in the bulk. The electrolyte is described by the so-called primitive model in the framework of classical statistical mechanics and the electrostatic response of the wall is characterized by its dielectric constant. In a previous paper [Phys. Rev. E 68, 022133 (2003)] a graphic reorganization of resummed Mayer diagrammatics has been devised in order to exhibit the general structure of the 1/y3 leading tail of w(alphaalpha') (x,x',y) for two charges located at distances x and x' from the wall and separated by a distance y along the wall. When all species have the same closest approach distance b to the wall, the coefficient of the 1/y3 tail is the product Dalpha(x)Dalpha'(x') of two effective dipoles. Here we use the same graphic reorganization in order to systematically investigate the exponential large-distance behavior of w(alphaalpha') in the bulk. (We show that the reorganization also enables one to derive the basic screening rules in both cases.) Then, in a regime of high dilution and weak coupling, the exact analytical corrections to the leading tail of w(alphaalpha'), both in the bulk or along the wall, are calculated at first order in the coupling parameter epsilon and in the limit where b becomes negligible with respect to the Debye screening length. (Epsilon is proportional to the so-called plasma parameter.) The structure of corrections to the terms of order epsilon is exhibited, and the scaling regime for the validity of the Debye limit is specified. In the vicinity of the wall, we use the density profiles calculated previously [J. Stat. Phys. 105, 211 (2001)] up to order epsilon and a method devised [J. Stat. Phys. 105, 245 (2001)] for the determination of the corresponding correction in the auxiliary screened potential, which also appears in the linear-response theory. The first coupling correction to the effective dipole Dalpha(x) is a function (not a mere exponential decay) determined by the nonuniformity of the density profiles as well as by three- and four-body screened interactions in w(alphaalpha'). Though the effective screening length (beyond the Debye value) in the direction perpendicular to the wall is the same as in the bulk, the bare solvated charges are not renormalized by the same quantity as in the bulk, because of combined steric and electrostatic effects induced by the wall.