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

TL;DR

This paper develops a mathematical homogenization framework for reverse osmosis membranes with critically scaled periodic boundary obstacles, revealing how nano-scale features influence macroscopic membrane properties.

Contribution

It extends previous homogenization techniques to model critically scaled nano-structured membranes with chemical reactions, showing the emergence of an effective global semipermeable membrane.

Findings

Homogenized problem exhibits a change in nonlinear constitutive relations.

Critical scale leads to an effective membrane with finite permeability.

Results apply to membranes of arbitrary shape, not just radial symmetry.

Abstract

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 to 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 $\Omega\subset \mathbb R^n$, 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 $\Gamma_1$ 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 $n=2$). Roughly speaking, our result proves that the consideration of the critical case of the scale leads to an homogenized formulation which is equivalent to have a global semipermeable membrane, at the whole part of the boundary $\Gamma_1$, 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.