Langevin and Fokker-Planck analyses for diffusion-mediated passing of circular and discorectangular species in two-dimensional channels.

The propensity for pairs of diffusing species to pass each other within narrow channels or pores is of basic interest as a first-passage-type problem. It is also of relevance for solution-phase transport in nanoporous materials, and in particular for catalytic conversion reactions where high yield requires that product species can efficiently pass reactant species to exit the pores. We analyze a two-dimensional model with nonoverlapping circular and discorectangular species confined to a rectangular channel, and where passing is mediated by Brownian dynamics in an implicit solvent. For narrower channels where passing is still possible, the discorectangle must align with the channel to pass the circular species. Behavior of the passing propensity, P, can be assessed by strongly damped Langevin simulations, or within an equivalent Fokker-Planck equation (FPE) formalism. The latter corresponds to a diffusion problem in a "higher-dimensional channel" with a constriction. We assess the variation of the passing propensity, P, for a broad range of channel width including its scaling just above the threshold where passing is sterically blocked. Analysis of P versus the rotational diffusion coefficient D_{r} of the discorectangle reveals a significant decrease in P for lower D_{r} for moderate channel width. This prompts a direct analysis of the regime where D_{r}→0, for which the FPE can be reduced to a three-dimensional diffusion problem, precise analysis of which is facilitated by adaptive-mesh finite element methods. The dependence of P on the aspect ratio of the discorectangle is also assessed.