REACTION EFFICIENCY OF DIFFUSION-CONTROLLED PROCESSES ON FINITE, PLANAR ARRAYS

In this paper we investigate the reaction efficiency of diffusion-cont rolled processes on finite, planar arrays having physical or chemical receptors. This problem translates into the statistical-mechanical one of examining the geometrical factors affecting the trapping of a rand om walker on small lattices of dimension d = 2, having N sites and ave rage valency nuBAR. Extensive calculations of the site-specific averag e walk length [n] before trapping, a measure of the efficiency of the underlying diffusion-reaction process, have been carried out on triang ular, square-planar, hexagonal, and Penrose platelets for N = 16 and N = 48. From the variety of distinct lattices considered, and the data generated, three general conclusions can be drawn. First, for fixed N, the smaller the number N(b) of vertices defining the boundary of the finite lattice under consideration, the smaller the value of the (over all) average walk length [nBAR] of the random walker before trapping. Second, for fixed N and fixed N(b), the smaller the value of the (over all) root-mean-square distance (r2BAR)1/2 of the N lattice sites relat ive to the center of the array, the smaller the value of [nBAR]. Third , for fixed {N, N(b), (r2BAR)1/2}, [nBAR] decreases with an increase i n the (overall) average valency nuBAR of lattice sites comprising the array. Thus there are similarities but also real and significant diffe rences in the conclusions drawn here in studying stochastic processes taking place on small, finite lattices of arbitrary shape and those fo und in studying nearest-neighbor random walks on infinite, periodic la ttices of unit cells characterized by a given (N,d,nu). We comment on these and on the possible relevance of this work to one aspect of morp hogenesis, viz., predicting the morphologies assumed by small platelet s when growth is optimized with respect to (chemical or physical) sign al processing at receptor sites.