Effective permeability conditions for diffusive transport through impermeable membranes with gaps

TL;DR

This paper derives explicit effective conditions for diffusive transport across membranes with gaps, accounting for microscale geometry and time-dependent effects, validated through numerical simulations and applicable to biological and industrial membranes.

Contribution

It generalizes classical membrane coupling conditions by incorporating microscale geometry and temporal memory effects, providing explicit formulas for permeability.

Findings

Permeability depends mainly on membrane thickness.

Transport exhibits memory effects for long channels with time-varying external concentrations.

Alterations in membrane microstructure influence overall transport efficiency.

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 characterised in general. Motivated by the application of transport across a bacterial membrane, in this paper we use formal multiscale analysis to derive explicit effective coupling conditions for macroscale transport across a two-dimensional impermeable membrane with periodically spaced gaps, and validate these with numerical simulations. We derive analytic expressions for effective macroscale quantities associated with the membrane, such as the permeability, in terms of the microscale geometry. Our results generalise the classic constitutive membrane coupling conditions to a wider range of membrane geometries and time-varying scenarios. Specifically, we demonstrate that if the exterior concentration varies in time, for membranes with long channels, the transport gains a memory property where the coupling conditions depend on the system history. By applying our effective conditions in the context of small molecule transport through gaps in bacterial membranes called porins, we predict that bacterial membrane permeability is primarily dominated by the thickness of the membrane. Furthermore, we predict how alterations to membrane microstructure, for example via changes to porin expression, might affect overall transport, including when external concentrations vary in time. These results will apply to a broad range of physical applications with similar membrane structures, from medical and industrial filtration to carbon capture.