Electrostatic free energy of solvation of an arbitrary charge distribution in the Block-Walker inhomogeneous dielectric

We derive the expressions for the electrostatic free energy and entropy of an arbitrary charge distribution in the dielectric characterized by the distance-dependent Block-Walker (BW) permittivity function epsilon(r) exp(-a ln <epsilon(>)r / r), where a is the solute radius and epsilon(r) is the permittivity of the bulk solvent, This function describes well the effect of dielectric inhomogeneity (e.g., due to nonuniform spatial distribution of dipoles of solvent molecules). As the charge distribution deviates from the center of the solute cavity or as epsilon(r) becomes smaller, the dielectric inhomogeneity gains in importance. The BW function well reproduces the observed free energies and entropies of solvation of univalent ions, without any parametric fittings: its mathematical form leads to appropriate effective radii of solvated ions and produces their sensitive dependence on temperature. We also try to microscopically interpret the BW model by comparing it with the mean spherical approximation (MSA) for the ion-dipolar system and propose the solvent scale BW (SBW) function epsilon(r) exp[-(r(2) ln epsilon(r))/(r - a + r(2))], where r(2) is the radius of the solvent molecule (when r(2) = a, the SBW function is identical with the BW). Although the ion solvation energy for the SBW varies with r(2) more moderately than the MSA, both models provide nearly the same effective radius of an ion, i.e., nearly the same free energy (entropy) of ion solvation.