Fluctuation in electrolyte solutions: The self energy

We address the issue of the self energy of the mobile ions in electrolyte solutions within a general Gaussian renormalized fluctuation theory using a field-theoretic approach. We introduce the Born radii of the ions in the form of a charge distribution allowing for different Born radii between the cations and anions. The model thus automatically yields a theory free of divergences and accounts for the solvation of the ions at the level of continuous dielectric media. In an inhomogeneous dielectric medium, the self energy is in general position dependent and differences in the self energy between cations and anions can give rise to local charge separation in a macroscopically neutral system. Treating the Born radius a as a smallness parameter, we show that the self energy can be split into an O(a(-1)) nonuniversal contribution and an O(a(0)) universal contribution that depends only on the ion concentration, valency, and the spatially varying dielectric constant. For a weakly inhomogeneous dielectric medium, the nonuniversal part of the self energy is shown to have the form of the Born energy with the local dielectric constant. This self energy is incorporated into the Poisson-Boltzmann equation as a simple means of including this local fluctuation effect in a mean-field theory. We illustrate the phenomenon of charge separation by considering cations and anions of difference sizes and valencies in a periodic dielectric medium.