A two-scale non-local model of swelling porous media incorporating ion size correlation effects

A new two-scale model is proposed for derivation of the macroscopic modified effective stress principle for swelling porous media saturated by an electrolyte solution containing finite size ions. A non-local pore-scale model is developed within the framework of Statistical Mechanics in conjunction with the thermodynamic approach based on Density Functional Theory leading to a nonlinear integral Fredholm equation of second kind for the ion/nanopore correlation function coupled with Poisson problem for the electric double layer potential. When combined with the fluid equilibrium condition such non-local electrochemical problem gives rise to a constitutive law for the fluid stress tensor in terms of the disjoining pressure which is decomposed into several components of different nature. The homogenization procedure based on formal asymptotic expansions is applied to up-scale the model to the macroscale leading to a two-scale constitutive law for the swelling pressure appearing in the modified effective stress principle with improved accuracy incorporating the deviations from the Gouy–Chapman Poisson–Boltzmann-based theory due to the finite size short-range ion–ion correlation effects. The integro-differential problem posed in a periodic cell is discretized by collocation schemes. Numerical results are obtained for a stratified arrangement of parallel macromolecules showing that the effects of ion–ion correlation forces give rise to anomalous attraction patterns between the particles for divalent ions.