Depletion region for diffusion-controlled reactions in a field.

Abstract

The effects of a uniform field on the distribution of mobile particles in the presence of a fixed trap are analyzed. Even small fields are shown to have a drastic effect on magnitudes, such as the reaction rate and the mean distance 〈L〉 from the trap to its nearest neighbor. If the field points towards the trap, a steady state is reached exponentially fast. In this steady state there is a depletion hole next to the trap, whose size depends on the ratio between the drift velocity V and the diffusion coefficient D. If c is the initial concentration of mobile particles, the long-time reaction rate is simply c\ensuremath{\Vert}V\ensuremath{\Vert}. If the field points away from the trap, the maximum of the probability distribution function for the distance to a nearest neighbor moves away from the trap with constant speed (if V=0, 〈L〉\ensuremath{\sim}${\mathit{t}}^{1/4}$) and the reaction rate decays as ${\mathit{t}}^{\mathrm{\ensuremath{-}}3/2}$exp(-${\mathit{V}}^{2}$t/4D), where t is time.