Bulk-mediated surface transport in the presence of bias

Abstract

Surface transport, when the particle is allowed to leave the surface, travel in the bulk for some time, and then return to the surface, is referred to as bulk-mediated surface transport. Recently, we proposed a formalism that significantly simplifies analysis of bulk-mediated surface diffusion [A. M. Berezhkovskii, L. Dagdug, and S. M. Bezrukov, J. Chem. Phys. 143, 084103 (2015)]. Here this formalism is extended to bulk-mediated surface transport in the presence of bias, i.e., when the particle has arbitrary drift velocities on the surface and in the bulk. A key advantage of our approach is that the transport problem reduces to that of a two-state problem of the particle transitions between the surface and the bulk. The latter can be solved with relative ease. The formalism is used to find the Laplace transforms of the first two moments of the particle displacement over the surface in time t at arbitrary values of the particle drift velocities and diffusivities on the surface and in the bulk. This allows us to analyze in detail the time dependence of the effective drift velocity of the particle on the surface, which can be highly nontrivial.

FIG. 1.

Schematic representation of the model geometry analyzed in the paper.

FIG. 2.

The time dependences of the effective drift velocity v_eff(t) scaled by the particle drift velocity in the bulk, v_b, with v_s = 0, for several values of L, L = 0.5, 1, 5, 10, and ∞, and k = κ = 1. Different panels show the dependences in different ranges of time. The black dashed lines in panels (b) and (c) represent the long-time asymptotic values of the corresponding v_eff(t) at finite L. When L = ∞, v_eff(t) vanishes as t → ∞; its long-time asymptotic behavior is given by Eq. (5.3), ${\left.{\mathit{v}}_{\mathrm{eff}}\left(t\right)\right|}_{t\to \infty }={\mathit{v}}_b\sqrt{{𝜃}_{\infty }∕\left(\pi t\right)}$.

FIG. 3.

The time dependences of the effective drift velocity v_eff(t) scaled by the particle drift velocity on the surface, v_s, with ${\mathit{v}}_b∕{\mathit{v}}_s=\pm 5$, for several values of L, L = 0.5, 1, 5, 10, and ∞, and k = κ = 1. Different panels show the dependences in different ranges of time. The black dashed lines in panels (b) and (c) represent the long-time asymptotic values of the corresponding v_eff(t) at finite L. When L = ∞, v_eff(t) vanishes as t → ∞; its long-time asymptotic behavior is given by Eq. (5.3), ${\left.{\mathit{v}}_{\mathrm{eff}}\left(t\right)\right|}_{t\to \infty }={\mathit{v}}_b\sqrt{{𝜃}_{\infty }∕\left(\pi t\right)}$.