Effective permeability conditions for diffusive transport through impermeable membranes with gaps

Abstract Membranes regulate transport in a wide variety of industrial and biological applications. The microscale geometry of the membrane can significantly affect overall transport through the membrane, but the precise nature of this multiscale coupling is not well characterized in general. Motivated by the application of transport across a bacterial membrane, we use formal multiscale analysis to derive explicit effective coupling conditions for macroscale transport across a two-dimensional (2D) impermeable membrane with periodically spaced gaps, and validate these with numerical simulations. We derive analytic expressions for macroscale membrane quantities, such as the effective permeability, in terms of the microscale geometry. Our results generalize the classic constitutive membrane coupling conditions to a wider range of membrane geometries and time-varying scenarios, e.g. we demonstrate that the unsteady conditions can gain a memory property and depend on the system history. By applying our effective conditions to small-molecule transport through gaps in bacterial membranes called porins, we predict that membrane permeability here is primarily dominated by membrane thickness. Furthermore, we predict how alterations to membrane microstructure, e.g. via changes to porin expression, might affect overall transport. These results apply to other physical applications with similar membrane structures, from medical and industrial filtration to carbon capture.