Variational approach for electrolyte solutions: From dielectric interfaces to charged nanopores

A variational theory is developed to study electrolyte solutions, composed of interacting pointlike ions in a solvent, in the presence of dielectric discontinuities and charges at the boundaries. Three important and nonlinear electrostatic effects induced by these interfaces are taken into account: surface charge induced electrostatic field, solvation energies due to the ionic cloud, and image-charge repulsion. Our variational equations thus go beyond the mean-field theory, or weak coupling limit, where thermal fluctuations overcome electrostatic correlations, and allows one to reach the opposite strong coupling limit, where electrostatic interactions induced by interfaces dominate. The influence of salt concentration, ion valency, dielectric jumps, and surface charge is studied in two geometries. (i) A single neutral dielectric interface (e.g., air-water or electrolyte-membrane) with an asymmetric electrolyte. A charge separation and thus an electrostatic field get established due to the different image-charge repulsions for coions and counterions. Both charge distributions and surface tension are computed and compared to previous approximate calculations. For symmetric electrolyte solutions close to a charged surface, two zones are characterized. In the first one, in contact with the surface and with size proportional to the logarithm of the coupling parameter, strong image forces and strong coupling impose a total ion exclusion, while in the second zone the mean-field approach applies. (ii) A symmetric electrolyte confined between two dielectric interfaces as a simple model of ion rejection from nanopores in membranes. The competition between image-charge repulsion and attraction of counterions by the membrane charge is studied. For small surface charge, the counterion partition coefficient decreases with increasing pore size up to a critical pore size, contrary to neutral membranes. For larger pore sizes, the whole system behaves like a neutral pore. For strong coupling and small pore size, coion exclusion is total and the counterion partition coefficient is solely determined by global electroneutrality. A quantitative comparison is made with a previous approach, where image and surface charge effects were smeared out in the pore. It is shown that the variational method allows one to go beyond the constant Donnan potential approximation, with deviations stronger at high ion concentrations or small pore sizes. The prediction of the variational method is also compared with MC simulations and good agreement is observed.