‘‘Order–disorder’’ phenomena in a diffusion-reaction model of interacting dipoles on a surface

Abstract

We study the reaction efficiency of a surficial process in which a diffusing, tumbling dipole A reacts (eventually and irreversibly) with a stationary target dipole B. In contrast to earlier studies of such irreversible diffusion-reaction events (A+B→C), we consider the situation where at each and every site of the space accessible to the diffusing coreactant A, there is also embedded a fixed dipole. To quantify the influence on the reaction efficiency of (angle-averaged, dipole–dipole) potential interactions between the tumbling dipole A and the ensemble of stationary dipoles, we design a lattice-statistical model to describe this problem and use both analytical methods and numerical techniques rooted in the theory of finite Markov processes to work out its consequences. Specifically, we define the reaction space to be an n×n=N square-planar lattice with the target dipole occupying the centrosymmetric site in that space and determine the mean number of steps required before the irreversible event, A+B→C, occurs. Our results reveal two qualitatively-distinct regimes of behavior for this diffusion-reaction process, a low temperature (or strong coupling) regime dominated by nearest-neighbor excursions only, and a high-temperature (or weak-coupling) regime dominated by non-nearest neighbor excursions of the tumbling dipole A, with the transition between these two regimes occurring over a relatively narrow range of interparticle couplings. This behavior has the character of an ‘‘order–disorder’’ transition and is interpreted here in terms of an ‘‘order parameter’’ W related to a generalized Onsager length. The behavior uncovered is studied as a function of system size and of the boundary conditions imposed.