Trapping of diffusing particles by spiky absorbers

Abstract

We study trapping of particles diffusing on a flat surface by complex-shaped absorbers formed by periodic absorbing spikes protruding from absorbing circular cores. It is shown that a spiky absorber can be replaced by an equivalent, from the trapping point of view, circular absorber of properly chosen radius. A simple expression for the effective absorber radius in terms of the geometric parameters of the spiky absorber (the number and length of the spikes and the core radius) is derived. To check its accuracy and to establish the range of its applicability, we run Brownian dynamics simulations and obtain the mean lifetimes of particles diffusing inside a reflecting circle with different spiky absorbers placed in its center. These mean lifetimes are then compared with their counterparts given by the theory for equivalent circular absorbers. There is an excellent agreement between the lifetimes obtained by the two methods when the radius of the reflecting circle is sufficiently large.

FIG. 1.

Spiky absorbers: (a) the absorber formed by 4 absorbing spikes of length h protruding from the absorbing core of radius a_c and (b) no-core absorbers formed by 1, 2, and 4 absorbing spikes of length h. Dashed circles are effective circular absorbers.

FIG. 2.

Core radius dependence of the radius a_eff of the effective circular trap for spiky absorbers with n = 1, 2, 4, and 8 spikes (solid lines from bottom to top). Symbols are the values of the $a_{eff}^{sim}/b$ ratio obtained from simulations with R_rb = 10.

FIG. 3.

Absorbing comb-like boundary formed by periodic absorbing spikes of height H protruding from the absorbing base. The spikes are separated by the comb period W. The dashed line is the effective flat absorbing boundary located at distance Δ below the spike tops.