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Asymptotic analysis of the Poisson-Boltzmann equation describing electrokinetics in porous media

We consider the Poisson-Boltzmann equation in a periodic cell, representative of a porous medium. It is a model for the electrostatic distribution of $N$ chemical species diluted in a liquid at rest, occupying the pore space with charged solid boundaries. We study the asymptotic behavior of its solution depending on a parameter $β$ which is the square of the ratio between a characteristic pore length and the Debye length. For small $β$ we identify the limit problem which is still a nonlinear Poisson equation involving only one species with maximal valence, opposite to the average of the given surface charge density. This result justifies the {\it Donnan effect}, observing that the ions for which the charge is the one of the solid phase are expelled from the pores. For large $β$ we prove that the solution behaves like a boundary layer near the pore walls and is constant far away in the bulk. Our analysis is valid for Neumann boundary conditions (namely for imposed surface charge densities) and establishes rigorously that solid interfaces are uncoupled from the bulk fluid, so that the simplified additive theories, such as the one of the popular Derjaguin, Landau, Verwey and Overbeek (DLVO) approach, can be used. We show that the asymptotic behavior is completely different in the case of Dirichlet boundary conditions (namely for imposed surface potential).