Ion diffusion at charged interfaces

Abstract

We investigate the statistical-mechanical basis and the numerical accuracy of the Smoluchowski-Poisson-Boltzmann (SPB) approximation for describing ion diffusion in non-uniform electrolytes. The many-particle generalized Smoluchowski equation is formally reduced to a hierarchy of coupled nparticle equations. A closure relation, called the Instantaneous Relaxation Approximation (IRA), is used to decouple the equation for the one-particle self-propagator. Introducing also a mean field approximation (MFA), we recover the SPB equation. The accuracy of the IRA and MFA is quantitatively assessed for a model system consisting of two parallel uniformly charged plates with an intervening solution containing point ions in a dielectric medium. This is done by comparing diffusion propagators, survival probabilities and mean first passage times obtained by (1) solving the manyparticle generalized Smoluchowski equation by the stochastic dynamics simulation technique (2) numerically solving the one-particle Smoluchowski equation with the exact (simulated) equilibrium potential of mean force, and (3) analytically solving the SPB equation. The IRA is found to be a useful approximation, whereas the MFA can lead to substantial error for systems with strong Coulomb coupling, as in the case of polyvalent counterions. Provided with a realistic potential of mean force, the one-particle Smoluchowski equation thus yields an accurate description of ion diffusion in nonuniform electrolytes.