Competing reaction processes on a lattice as a paradigm for catalyst deactivation

Abstract

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 = 0 and = 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 ≈ 1 and s ≈ 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 ≈ 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.

Heterogeneous continuous-time random walks

“…One can also easily treat partially absorbing boundaries [52][53][54][55][56][57][58][59][60] by allowing nonzero leakage probability from the sink site x * . If a particle can disappear or loose its activity during diffusion, FPT problems for such "mortal" walkers [61][62][63][64][65][66][67][68] can be treated by introducing two sink sites, x * 1 and x * 2 , that represent an absorbing boundary and a reactive bulk. Using the exchange time distributions ψ xx * 2 (t) depending on x, one can model space-dependent bulk reaction rates.…”

Random walks on lattices. Influence of competing reaction centers on diffusion-controlled processes

“…In the present work we study the above model in Euclidean dimension d = 3 and compare our results with previous results for d = 2 [5]. In this context, we also consider the "two-walker" problem; that is, instead of assuming that the reaction center is static, we allow for the possibility that it is a moving target.…”

“…In Ref. [5], periodic boundary conditions were chosen, implying that the lattice becomes translationally invariant. Here, we stick to this choice, since we wish to keep our setting as simple as 4 possible by assessing the influence of finite size effects only (for a discussion of boundary effects, see Ref.…”

“…The case where a perfect trap at one lattice site (s = 1) coexists with a non-zero probability of absorption (0 < s < 1) at all other lattice sites (also termed background trapping in what follows) lends itself particularly well to the analysis of finite size effects [5]. Although the above model is of interest for a variety of different situations, a possible interpretation of non-zero background trapping is site deactivation due to partial poisoning of a catalyst.…”

“…Recently, there has been a surge of interest in such problems [10][11][12][13][14][15][16][17][18][19][20][21][22] motivated by a large number of applications, including photosynthetic trapping [5], oocyte fertilization [13], game theory [5], radioactive decay [5,18], target deactivation problems [14] or foraging behavior [21].…”