The effect of exclusion on nonlinear reaction diffusion system in inhomogeneous media

We study a minimal model to understand the formation of clusters on surfaces in the presence of surface defects. We consider reaction diffusion model in which atoms undergoes reactions at the defect centers to form clusters. Volume exclusion between particles is introduced through a drift term in the reaction diffusion equation that arises due the repulsive force field produced by the clustering atoms. The volume exclusion terms can be derived from master equation with a concentration dependent hopping rate. Perturbative analysis is performed for both cross-exclusion and self-exclusion one dimensional system. For two dimension numerical analysis is performed. We have found that the clusterization process slows down due to exclusion. As a result the size of the clusters reduces. In this model reaction scheme has algebraic nonlinearity and plausible mechanism is also given.