We mobilize both a generating function approach and the theory of finite Markov processes to compute the probability of irreversible absorption of a randomly diffusing species on a lattice with competing reaction centers. We consider an N-site lattice populated by a single deep trap, and $N-1$ partially absorbing traps (absorption probability $0<s<1$). The influence of competing reaction centers on the probability of reaction at a target site (the deep trap) and the mean walk length of the random walker before localization (a measure of the reaction efficiency) are computed for different geometries. Both analytic expressions and numerical results are given for reactive processes on two-dimensional surfaces of Euler characteristic $\Omega =0$ and $\Omega =2$. The results obtained allow a characterization of catalyst deactivation processes on planar surfaces and on catalyst pellets where only a single catalytic site remains fully active (deep trap), the other sites being only partially active as a result of surface poisoning. The central result of our study is that the predicted dependence of the reaction efficiency on system size $N$ and on $s$ is in qualitative accord with previously reported experimental results, notably catalysts exhibiting selective poisoning due to surface sites that have different affinities for chemisorption of the poisoning agent (e.g., acid zeolite catalysts). Deviations from the efficiency of a catalyst with identical sites are quantified, and we find that such deviations display a significant dependence on the topological details of the surface (for fixed values of $N$ and $s$ we find markedly different results for, say, a planar surface and for the polyhedral surface of a catalyst pellet). Our results highlight the importance of surface topology for the efficiency of catalytic conversion processes on inhomogeneous substrates, and in particular for those aimed at industrial applications. From our exact analysis we extract results for the two limiting cases $s\approx 1$ and $s\approx 0$, corresponding respectively to weak and strong catalyst poisoning (decreasing $s$ leads to a monotonic decrease in the efficiency of catalytic conversion). The results for the $s\approx 0$ case are relevant for the dual problem of light-energy conversion via trapping of excitations in the chlorophyll antenna network. Here, decreasing the probability of excitation trapping $s$ at sites other than the target molecule does not result in a decrease of the efficiency as in the catalyst case, but rather in enhanced efficiency of light-energy conversion, which we characterize in terms of $N$ and $s$. The one-dimensional case and its connection with a modified version of the gambler's ruin problem are discussed. Finally, generalizations of our model are described briefly.

• Received 31 July 2014

DOI:https://doi.org/10.1103/PhysRevE.91.022106