The Smoluchowski-Poisson-Boltzmann description of ion diffusion at charged interfaces

A theoretical description of ion diffusion in the electric field set up by the double layer in the neighborhood of a charged interface is presented. Such a description is needed for the understanding of diffusion-controlled chemical kinetics and transport of ionic species in a variety of systems of interest in biophysics, electrochemistry, and colloid science. The ion dynamics are taken to be governed by the Smoluchowski diffusion equation and the average electrostatic field is obtained from the nonlinear Poisson-Boltzmann equation. Diffusion in finite regions with partially absorbing boundaries of planar, cylindrical, or spherical geometry is considered. The complete analytical solution of the Smoluchowski-Poisson-Boltzmann equation for counterions between two planar charged interfaces is given. Simple expressions are derived for certain useful integral quantities, viz., mean absorption times and absorption probabilities, in all geometries considered. Finally, lateral counterion diffusion and its consequences for surface re-encounter-enhanced chemoreception is considered.