Extending effective medium theory to finite size systems: Theory and simulation for permeation in mixed-matrix membranes

We present a novel theory for estimation of the effective permeability of pure gases in flat mixed-matrix membranes (MMMs), in which effective medium theory (EMT) is extended to systems with finite filler size and membrane thickness. We introduce an inhomogeneous filler volume fraction profile, which arises due to depletion of the filler in regions adjacent to the membrane ends, into the MMM permeation model. In this way, the effective medium approach (EMA) can still be applied to systems where the dispersant size is not small in comparison to the membrane thickness, and for which a permeability profiles arises in the MMM that is dependent on both filler size and membrane thickness, besides the filler-polymer equilibrium constant. It is found that increase in particle size reduces the effective membrane permeability at fixed membrane thickness, and that the effective membrane permeability increases with increase of the membrane thickness to asymptotically reach the value predicted by existing models. The present theory is validated against detailed simulations of the transport in MMMs, and theoretical predictions are found to be in agreement with those obtained from the exact calculations. Further, comparison of the exact effective permeability at different filler volume fractions is made for different packing configurations, showing variations in dispersant packing structure to have only a very weak effect on MMM performance.