Exact algorithm ford-dimensional walks on finite and infinite lattices with traps. III. Role of lattice valency in influencing the efficiency of diffusion-controlled reactions

Abstract

In this paper we continue our development of an exact lattice theory of diffusion-controlled reactions. We show how general theorems drawn from the theory of finite Markov processes may be brought to bear on the approach elaborated in our earlier work [Phys. Rev. Lett. 47, 1500 (1981); Phys. Rev. B 26, 4166 (1982)] and, in addition, we exploit well-known diagram and generating-function methods (in particular, those based on the adjacency-walk matrix) to gain further insight into the statistics underlying the processes considered in this paper. We consider reactions in which the diffusing molecule encounters a single reaction center and reacts there, irreversibly, upon first encounter. We also consider the situation where the diffusing particle may, at any of the sites surrounding the reaction center, form an activated complex with a coreactant situated there and, with finite probability, be removed irreversibly from the system. In each case we focus on the problem of reaction efficiency and determine the average number of steps required before a diffusing particle undergoes, eventually, an irreversible reaction. We report extensive new (and exact) results for hexagonal lattices and consider explicitly the role of spatial extent and dimensionality as well as the influence of passive versus active boundary conditions. By comparing the results obtained for hexagonal lattices for $d=2 \mathrm{and} 3$ with those reported earlier for square and/or cubic lattices, quantitative conclusions can be drawn on the role of lattice valency in influencing the efficiency of reactiondiffusion processes. A principal, general conclusion of this study concerns the efficiency of reaction when there exists the possibility of reactant deactivation at the $N\ensuremath{-}1$ sites surrounding the reaction center. We find that a 5% probability of reaction at these adjacent sites effectively erases distinctions between lattices subject to different boundary conditions or characterized by different valencies, i.e., the process becomes kinetically controlled.