Continuum electrostatics for ionic solutions with non-uniform ionic sizes

This work concerns electrostatic properties of an ionic solution with multiple ionic species of possibly different ionic sizes. Such properties are described by the minimization of an electrostatic free-energy functional of ionic concentrations. Bounds are obtained for ionic concentrations with low electrostatic free energies. Such bounds are used to show that there exists a unique set of equilibrium ionic concentrations that minimizes the free-energy functional. The equilibrium ionic concentrations are found to depend sorely on the equilibrium electrostatic potential, resembling the classical Boltzmann distributions that relate the equilibrium ionic concentrations to the equilibrium electrostatic potential. Unless all the ionic and solvent molecular sizes are assumed to be the same, explicit formulae of such dependence are, however, not available in general. It is nevertheless proved that in equilibrium the ionic charge density is a decreasing function of the electrostatic potential. This determines a variational principle with a convex functional for the electrostatic potential.