HOMOGENIZATION OF MESOSCOPIC THEORIES: EFFECTIVE PROPERTIES OF MODEL MEMBRANES

A new mathematical framework is introduced for modeling diffusion in nanoporous materials or on surfaces exhibiting heterogeneity in properties over large length scales while retaining molecular scale information typically captured by molecular simulations only. This framework entails first the use of newly developed mesoscopic equations derived rigorously from underlying master equations by coarse-graining statistical mechanics techniques. Homogenization techniques are then employed to derive the leading-order effective mesoscopic models that are subsequently solved by spectral methods. These solutions are also compared to direct numerical simulations for selected two-dimensional model membranes with defects, when attractive adsorbate-adsorbate interactions affect particle diffusion. It has been found that not only the density but also the dispersion of defects significantly alters the macroscopic behavior in terms of fluxes and concentration patterns, especially when phase transitions can occur. It is also shown that homogenization techniques could potentially offer a promising alternative to direct numerical simulations, when complex, large-scale heterogeneities are present. Implications for various applications, including heterogeneous catalysts, materials growth, and separations using membranes, are also discussed. * Corresponding author