A new approach to the problem of bulk-mediated surface diffusion

Abstract

This paper is devoted to bulk-mediated surface diffusion of a particle which can diffuse both on a flat surface and in the bulk layer above the surface. It is assumed that the particle is on the surface initially (at t = 0) and at time t, while in between it may escape from the surface and come back any number of times. We propose a new approach to the problem, which reduces its solution to that of a two-state problem of the particle transitions between the surface and the bulk layer, focusing on the cumulative residence times spent by the particle in the two states. These times are random variables, the sum of which is equal to the total observation time t. The advantage of the proposed approach is that it allows for a simple exact analytical solution for the double Laplace transform of the conditional probability density of the cumulative residence time spent on the surface by the particle observed for time t. This solution is used to find the Laplace transform of the particle mean square displacement and to analyze the peculiarities of its time behavior over the entire range of time. We also establish a relation between the double Laplace transform of the conditional probability density and the Fourier-Laplace transform of the particle propagator over the surface. The proposed approach treats the cases of both finite and infinite bulk layer thicknesses (where bulk-mediated surface diffusion is normal and anomalous at asymptotically long times, respectively) on equal footing.

FIG. 1.

Schematic representation of the geometry analyzed in the paper.

FIG. 2.

The time dependences of the probability P_s(t) of finding the particle on the surface at time t. The dependences are obtained by numerically inverting the Laplace transform in Eq. (3.10) using the Stehfest algorithm. The main panel illustrates variation of P_s(t) as the bulk layer thickness changes from L = 1 to L = ∞, at D_b = k = κ = 1; the values of L are given above the curves. The inset illustrates variation of P_s(t) at L = ∞ as D_b changes from 0.1 to 10, at k = κ = 1; the values of D_b are given above the curves. All the parameter values are given in dimensionless units.

FIG. 3.

The L-dependence of the effective diffusivity, Eq. (5.7), at different values of the diffusivity ratio D_b/D_s = 1, 2, 4, 8, and 16, from bottom to top.

FIG. 4.

The time dependences of the effective diffusivity, D_eff(t), in the absence of the reflecting lid (L = ∞) at different values of D_b, which are given near the curves. The dependences are obtained by numerically inverting the Laplace transform in Eq. (5.21), using the Stehfest algorithm, with D_s = k = κ = 1 (in dimensionless units).

FIG. 5.

The time dependences of the effective diffusivity, D_eff(t), at D_b = 30 (panel (a)) and D_b = 120 (panel (b)) for different values of the bulk layer thickness L, which are given near the curves. The dependences are obtained by numerically inverting the Laplace transform in Eq. (5.21), using the Stehfest algorithm, with D_s = k = κ = 1. All the parameter values are given in dimensionless units.