Non-linear boundary condition for non-ideal electrokinetic equations in porous media
Résumé
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 2-d
to compare this model with that for a constant surface charge.
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