Boundary homogenization for partially reactive patches

TL;DR

This paper derives an effective trapping rate for particles interacting with surfaces containing small reactive patches, using advanced mathematical techniques and validating results with kinetic Monte Carlo simulations.

Contribution

It provides a new analytical formula for the trapping rate of partially reactive patches, combining asymptotic analysis, homogenization theory, and probabilistic algorithms.

Findings

Derived explicit formulas for trapping rates.

Validated formulas with kinetic Monte Carlo simulations.

Compared results to previous heuristic approximations.

Abstract

A wide variety of physical, chemical, and biological processes involve diffusive particles interacting with surfaces containing reactive patches. The theory of boundary homogenization seeks to encapsulate the effective reactivity of such a patchy surface by a single trapping rate parameter. In this paper, we derive the trapping rate for partially reactive patches occupying a small fraction of a surface. We use matched asymptotic analysis, double perturbation expansions, and homogenization theory to derive formulas for the trapping rate in terms of the far-field behavior of solutions to certain partial differential equations (PDEs). We then develop kinetic Monte Carlo (KMC) algorithms to rapidly compute these far-field behaviors. These KMC algorithms depend on probabilistic representations of PDE solutions, including using the theory of Brownian local time. We confirm our results by comparing to KMC simulations of the full stochastic system. We further compare our results to prior heuristic approximations.