Trapping of diffusive particles by rough absorbing surfaces: Boundary smoothing approach

We present analytical results for the first-passage statistics of Brownian particles near a comblike absorbing boundary. Our approach is based on the method of boundary homogenization (or boundary smoothing) when an equivalent flat boundary is introduced to maintain the same diffusion flux as the original rough boundary. By using the conformal invariance of the Laplace equation we derive an analytic expression for the position of an equivalent boundary in terms of its spatial period and amplitude. The main analytical results being initially obtained for the steady state system provide important insights into the statistical characteristics of diffusive transport near rough boundaries (high order moments of the trapping time statistics).

Figure 1

Comblike model of a rough absorbing boundary.

Figure 2

Position of the effective boundary: $\delta =\Delta /H$, as function of the ratio $h=H/W$. The solid line is the theoretical prediction, Eq. (3). Symbols ($\circ$) are the results of the Brownian dynamics simulations. Dashed lines are asymptotic behavior $\delta =h$ for $h\ll 1$ and $\delta =(ln2)/\pi$ and for $h\gg 1$.

Figure 3

Universal PDF of the normalized trapping time of the Brownian particles in the vicinity of the comblike boundary, $P\left(\xi \right)$, $\xi =T/{\tau }_d$: particle simulations (histogram) and theoretical predictions given by Eq. (5) for the equivalent flat boundary (dashed line): $h=0.5$.