Random walk with barriers: Diffusion restricted by permeable membranes

TL;DR

This paper investigates how permeable membranes affect molecular diffusion, revealing a scaling law for the diffusion coefficient over time, which helps characterize microstructure in biological and porous materials.

Contribution

It introduces a theoretical framework for understanding diffusion restricted by permeable membranes using a scattering approach and renormalization group analysis.

Findings

Diffusion coefficient exhibits inverse square root time scaling.

Theoretical predictions align with Monte Carlo simulations.

Method enables quantification of membrane permeability and surface area.

Abstract

Restrictions to molecular motion by barriers (membranes) are ubiquitous in biological tissues, porous media and composite materials. A major challenge is to characterize the microstructure of a material or an organism nondestructively using a bulk transport measurement. Here we demonstrate how the long-range structural correlations introduced by permeable membranes give rise to distinct features of transport. We consider Brownian motion restricted by randomly placed and oriented permeable membranes and focus on the disorder-averaged diffusion propagator using a scattering approach. The renormalization group solution reveals a scaling behavior of the diffusion coefficient for large times, with a characteristically slow inverse square root time dependence. The predicted time dependence of the diffusion coefficient agrees well with Monte Carlo simulations in two dimensions. Our results can be used to identify permeable membranes as restrictions to transport in disordered materials and in biological tissues, and to quantify their permeability and surface area.