Inhomogeneous two-species annihilation in the steady state

Abstract

The authors investigate steady-state geometrical properties of a reaction interface in the two-species annihilation process, A+B to 0, when a flux j of A and B particles is injected at opposite extremities of a finite domain. By balancing the input flux with the number of reactions, they determine that the width w of the reaction zone scales as j-1/3 in the large flux limit, and that the concentration in this zone is proportional to j2/3. This same behaviour is deduced from the solution to the reaction-diffusion equation. In the low flux limit, the concentration is almost independent of position and is proportional to j. In the latter case, the local reaction rate reaches maximum at the edges of the system rather than at the midpoint. When the two species approach at a finite velocity, there exists a critical velocity, above which the reactants essentially pass through each other. Results similar to those in one dimension are found in two- and three-dimensional radial geometries. Finally, they apply the quasistatic approximation to their steady-state solution to recover the known time dependence for the reaction zone width for the case of initially separated components with no external input.