Random walk with barriers

Abstract

Restrictions to molecular motion by barriers (membranes) are ubiquitous in porous media, composite materials and biological tissues. 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 membranes (d - 1 dimensional planes in d dimensions) 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 for any d. Its origin lies in the strong structural fluctuations introduced by the spatially extended random restrictions, representing a novel universality class of the structural disorder. Our results agree well with Monte Carlo simulations in two dimensions. They can be used to identify permeable barriers as restrictions to transport, and to quantify their permeability and surface area.

FIG. 1

A fragment of a two-dimensional patch with randomly placed and oriented membranes (one of the disorder realizations used in the simulations).

FIG. 2

Time-dependent diffusion coefficient D(t) for the two-dimensional random medium of Fig. 1. a, Comparison of the RG solution (6), red, with the Monte Carlo simulations, blue, for the set of decreasing permeabilities, corresponding to ζ = 0.5, 1, 2, 4, 10, 20, 40, 100, 200, 400 (top to bottom). The diffusion time τ_D is marked by blue circles. b, Scaling behavior of D(t). As the strength ζ of the restrictions increases from bottom to top, the numerical curves begin to collapse as a signature of the universal behavior (10). Dashed lines show the ζ → ∞ limits from our RG solution: the “impermeable” limit D(t)/D_∞ = 2d τ_r/t and the scaling limit (10) with $C_2(\infty )=4/\sqrt{\pi }$. c, Parameters of the scaling limit (10), C₂(ζ) (filled circles) and D_∞(ζ) (open circles), determined from the fit of the simulations in b to equation (10), compared with the RG predictions (solid and dashed red lines). Thin solid line is the RG limit $C_2(\infty )=4/\sqrt{\pi }$.

FIG. 3

Effect of a single membrane. a, Meaning of effective membrane thickness: 2ℓ = D₀/κ is the distance by which one should shift the density profile ψ(x) on each side of membrane if one were to heal p the jump discontinuity (1). b, The memory kernel ϕ(t), equation (5), together with its limits $\varphi \simeq 1/\sqrt{\pi \tau t}$ for t ≪ τ, and $\varphi \simeq \sqrt{\tau /4\pi }{t}^{-3/2}$ for t ≫ τ.