  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 $\beta$ which is the square of the ratio between a
characteristic pore length and the Debye length. For small $\beta$ 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 $\beta$ 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).
