Electric field effects on the ionic relaxation in semi-infinite adsorbing cells

We analyze the distribution of mobile charges in the absence of external electric fields for semi-infinite electrolytic samples limited by a selectively adsorbing flat surface. Only positive charges are considered mobile for simplicity and to establish the general behavior. The whole scenario is characterized by drift and diffusion phenomena in the bulk, together with selective adsorption of charges at the surface. From a mathematical standpoint, the problem can be addressed within the framework of the Poisson–Nernst–Planck model, in which the continuity equation for the mobile positive charges couples to Poisson’s equation, determining the spatiotemporal distribution of charges throughout the medium. Selective adsorption at the surface is governed by the Langmuir balance equation. Our analyses focus on the mathematical aspects of the problem, where an exact result is derived through integral transform techniques, and on comparisons with alternative solutions obtained by multi-relaxation methods.