Dipolar effective interaction in a fluid of charged spheres near a dielectric plate

Static correlations in a classical fluid of charged spheres at equilibrium are studied in the vicinity of an insulating wall characterized by its dielectric constant. It is well known that the deformations of screening clouds induced by the presence of the wall result into an effective f(alphaalpha('))(x,x('))/y(3) interaction in the pair distribution function between two charges e(alpha) and e(alpha(')) located at distances x and x(') from the wall and separated by a large distance y along the wall. We investigate the structure of f(alphaalpha('))(x,x(')). The method is based on systematic resummations in the Mayer diagrammatics, which are valid both in the bulk and in an inhomogeneous situation. The screened potential phi arising in the formalism happens to coincide with the linearized mean-field approximation for the immersion free energy of two external unit charges. Phi is shown to decay as a repulsive f(phi)(x,x('))/y(3) interaction, whatever the density profiles may be. f(phi)(x,x(')) takes a factorized dipolar structure f(phi)(x,x('))=D(phi)(x)D(phi)(x(')) for distances x and x(') larger than the maximum of the closest approach distances b(alpha)'s to the wall for every species alpha. Moreover, we devise a reorganization of resummed diagrammatics, which is adequate for the determination of the large-distance behavior of correlations, and we prove that, when all species have the same approach distance b to the wall, f(alphaalpha('))(x,x(');b)=D(alpha)(x)D(alpha('))(x(')). In this case, the leading tail of the effective electrostatic interaction between two like charges at the same distance x from a single wall is repulsive. Results are independent of charge magnitudes, of excluded-volume sphere sizes, and of the existence of a surface charge on the wall. It holds whether charges are concentrated at sphere centers or uniformly spread over their surfaces. Comparison is made with an experiment about dilute colloids where the linearized mean-field approximation proves to be relevant. At equilibrium attraction between like charges in confined geometry might arise from purely electrostatic charge-charge interactions only through correlation effects not taken into account in the latter approximation.