Localization-delocalization transition of a reaction-diffusion front near a semipermeable wall

Abstract

The $A+B\ensuremath{\rightarrow}C$ reaction-diffusion process is studied in a system where the reagents are separated by a semipermeable wall. We use reaction-diffusion equations to describe the process, and to derive a scaling description for the long-time behavior of the reaction front. Furthermore, we show that a critical localization-delocalization transition takes place as a control parameter which depends on the initial densities and on the diffusion constants is varied. The transition is between a reaction front of finite width that is localized at the wall and a front which is detached and moves away from the wall. At the critical point, the reaction front remains at the wall but its width diverges with time (as ${t}^{1/6}$ in the mean-field approximation). Below two dimensions, the fluctuations play an important role and the critical exponents have no longer their mean-field values.