Imperfect Diffusion-Controlled Reactions

TL;DR

This paper explores the mathematical modeling of diffusion-controlled reactions with partial reactivity, emphasizing the stochastic process of partially reflected Brownian motion and its implications for understanding imperfect reactions.

Contribution

It introduces a comprehensive framework for modeling imperfect diffusion-controlled reactions using partially reflected Brownian motion and derives a general propagator representation.

Findings

Derived a general representation of the propagator for imperfect reactions

Linked the propagator to the Dirichlet-to-Neumann operator

Illustrated properties of the model for flat surfaces

Abstract

This chapter aims at emphasizing the crucial role of partial reactivity of a catalytic surface or a target molecule in diffusion-controlled reactions. We discuss various microscopic mechanisms that lead to imperfect reactions, the Robin boundary condition accounting for eventual failed reaction events, and the construction of the underlying stochastic process, the so-called partially reflected Brownian motion. We show that the random path to the reaction event can naturally be separated into the transport step toward the target, and the exploration step near the target surface until reaction. While most studies are focused exclusively on the transport step (describing perfect reactions), the exploration step, consisting is an intricate combination of diffusion-mediated jumps between boundary points, and its consequences for chemical reactions remain poorly understood. We discuss the related mathematical difficulties and recent achievements. In particular, we derive a general representation of the propagator, show its relation to the Dirichlet-to-Neumann operator, and illustrate its properties in the case of a flat surface.