Efficiency of encounter-controlled reaction between diffusing reactants in a finite lattice: topology and boundary effects

The role of dimensionality (Euclidean vs. fractal), spatial extent, boundary effects and system topology on the efficiency of diffusion-reaction processes involving two simultaneously diffusing reactants is analyzed. We present numerically exact values for the mean time to reaction, as gauged by the mean walklength before reactive encounter, obtained via application of the theory of finite Markov processes, and via Monte Carlo simulation. As a general rule, we conclude that for sufficiently large systems, the efficiency of diffusion-reaction processes involving two synchronously diffusing reactants (two-walker case) relative to processes in which one reactant of a pair is anchored at some point in the reaction space (one-walker plus trap case) is higher, and is enhanced the lower the dimensionality of the system. This differential efficiency becomes larger with increasing system size and, for periodic systems, its asymptotic value may depend on the parity of the lattice. Imposing confining boundaries on the system enhances the differential efficiency relative to the periodic case, while decreasing the absolute efficiencies of both two-walker and one-walker plus trap processes. Analytic arguments are presented to provide a rationale for the results obtained. The insights afforded by the analysis to the design of heterogeneous catalyst systems is also discussed.