Homogenization of a diffusion process in a rarefied binary structure

We study the homogenization of a diffusion process which takes place in a binary structure formed by an ambiental connected phase surrounding a suspension of very small spheres distributed in an $\veps$-periodic network. The asymptotic distribution of the concentration is determined for both phases, as $\veps\to 0$, assuming that the suspension has mass of unity order and vanishing volume. Three cases are distinguished according to the values of a certain limit capacity. When it is positive and finite, the macroscopic system involves a two-concentration system, coupled through a term accounting for the non local effects. In the other two cases, where the capacity is either infinite or going to zero, although the form of the system is much simpler, some peculiar effects still account for the presence of the suspension