Interfacial territory covered by surface-mediated diffusion

We consider a minimal model of heterogeneous catalysis in which a molecule performs surface-mediated diffusion inside a confining domain whose boundary contains catalytic sites. We explicitly take into account the combination of surface and bulk diffusion, and we obtain exact results for the mean and variance of the territory covered on the boundary by the particle before its exit in the case of a two-dimensional spherical domain. Depending on the relative positions of the entrance and exit points, very different behaviors with respect to the mean adsorption time of the molecule on the surface are found. We also determine both exact lower and upper bounds and an approximate expression of the probability of reacting with catalytic sites before exiting the domain. These results provide a quantitative measure of the efficiency of an idealized catalyst.