# Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces

## TL;DR

This paper introduces a spectral theoretical framework for modeling diffusion-controlled reactions on heterogeneous catalytic surfaces, simplifying complex boundary conditions and enabling analysis of reaction kinetics with new spectral representations.

## Contribution

It develops a spectral approach that reduces complex Robin boundary problems to Dirichlet problems, allowing for analytical and numerical analysis of heterogeneous surface reactivity.

## Key findings

- Spectral representations of survival probability and reaction times are derived.
- The method simplifies the analysis of heterogeneous reactivity patterns.
- Numerical examples demonstrate the approach's effectiveness for spherical surfaces.

## Abstract

We propose a general theoretical description of chemical reactions occurring on a catalytic surface with heterogeneous reactivity. The propagator of a diffusion-reaction process with eventual absorption on the heterogeneous partially reactive surface is expressed in terms of a much simpler propagator toward a homogeneous perfectly reactive surface. In other words, the original problem with general Robin boundary condition that includes in particular mixed Robin-Neumann condition, is reduced to that with Dirichlet boundary condition. Chemical kinetics on the surface is incorporated as a matrix representation of the surface reactivity in the eigenbasis of the Dirichlet-to-Neumann operator. New spectral representations of important characteristics of diffusion-controlled reactions, such as the survival probability, the distribution of reaction times, and the reaction rate, are deduced. Theoretical and numerical advantages of this spectral approach are illustrated by solving interior and exterior problems for a spherical surface that may describe either an escape from a ball or hitting its surface from outside. The effect of continuously varying or piecewise constant surface reactivity (describing, e.g., many reactive patches) is analyzed.