Non-linear boundary condition for non-ideal electrokinetic equations in porous media

This paper studies the partial differential equation describing the charge distribution of an electrolyte in a porous medium. Realistic non-ideal effects are incorporated through the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. The main novelty is the consideration of a non-constant surface charge density on the pore walls. Indeed, a chemical equilibrium reaction is considered on the boundary to represent the dissociation of ionizable sites on the solid walls. The surface charge density is thus given as a non-linear function of the electrostatic potential. Even in the ideal case, the resulting system is a new variant of the famous Poisson-Boltzmann equation, which still has a monotone structure under quantitative assumptions on the physical parameters. In the non-ideal case, the MSA model brings in additional non-linearities which break down the monotone structure of the system. We prove existence, and sometimes uniqueness, of the solution. Some numerical experiments are performed in 2d to compare this model with that for a constant surface charge.