Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis 'nano-composite' membranes

One of the main goals of this paper is to extend some of the mathematical techniques of some previous papers by the authors showing that some very useful phenomenological properties which can be observed at the nano-scale can be simulated and justified mathematically by means of some homogenization processes when a certain critical scale is used in the corresponding framework. Here the motivating problem in consideration is formulated in the context of the reverse osmosis. We consider, on a part of the boundary of a domain Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}}, a set of very small periodically distributed semipermeable membranes having an ideal infinite permeability coefficient (which leads to Signorini-type boundary conditions) on a part Γ1{\Gamma_{1}} of the boundary. We also assume that a possible chemical reaction may take place on the membranes. We obtain the rigorous convergence of the problems to a homogenized problem in which there is a change in the constitutive nonlinearities. Changes of this type are the reason for the big success of the nanocomposite materials. Our proof is carried out for membranes not necessarily of radially symmetric shape. The definition of the associated critical scale depends on the dimension of the space (and it is quite peculiar for the special case of n=2{n=2}). Roughly speaking, our result proves that the consideration of the critical case of the scale leads to a homogenized formulation which is equivalent to having a global semipermeable membrane, at the whole part of the boundary Γ1{\Gamma_{1}}, with a “finite permeability coefficient of this virtual membrane”, which is the best we can get, even if the original problem involves a set of membranes of any arbitrary finite permeability coefficients.