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. C 2015 AIP Publishing LLC. [http://dx

Bulk-mediated surface transport in the presence of bias

“…This paper extends the formalism developed for unbiased bulk-mediated surface diffusion in Ref. 35 to the case where the particle has arbitrary drift velocities on the surface and in the bulk. This is done in Secs.…”

“…Recently we proposed an alternative approach to this problem 35 that focuses on the cumulative time spent by the particle on the surface. The approach exploits the fact that the free-particle propagators on the surface and in the bulk are Gaussian.…”

“…One of the advantages of the formalism developed in Ref. 35 for the case of unbiased bulk-mediated surface diffusion is that it can be readily extended to the case where the particle is allowed to have arbitrary drift velocities in the bulk and on the surface. One can easily imagine that the particle has a drift velocity in the bulk, for example, due to the flow of the bulk liquid.…”

An encounter-based approach for restricted diffusion with a gradient drift

“…For instance, one can consider the effect of local potentials near the boundary (such as, e.g., electrostatic repulsion or attractive interactions) onto the statistics of boundary encounters. To some extent, such an approach can provide a microscopic model for so-called intermittent diffusions, in which a particle alternates diffusions in the bulk and on the surface [59][60][61][62][63][64][65][66][67][68]. One can also consider other surface reaction mechanisms [8] and investigate how the drift may affect them.…”

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