Electrostatic potential inside ionic solutions confined by dielectrics: a variational approach

This paper provides a pedagogical introduction to a variational formulation of electrostatic interactions between free charges close to a dielectric interface. A functional is constructed which, upon minimization, yields a polarization charge density h(r) which satisfies the usual laws of electrostatics. The initial functional of a volume charge density is shown to reduce to a functional of a surface polarization charge density when the interface between dielectrics is sharp. The variational procedure is first shown to provide an elegant systematic solution in cases with very simple geometries, for which analytic solutions are well known. Numerical implementations of the variational procedure using adequate grids are tested against exact results, in the perspective of applying the method to geometrically more complex situations, like ion channels through membranes, which are well beyond analytic treatment. Contact is made with related work on the solvation of biomolecules, based on a functional of the polarization itself. The relative advantages and shortcomings of the two formulations are briefly discussed.