Improved boundary homogenization for a sphere with an absorbing cap of arbitrary size

TL;DR

This paper improves the approximation method for calculating the effective reactivity of a spherical target with an arbitrary-sized absorbing cap, addressing limitations of previous models and validating accuracy across all patch sizes.

Contribution

It identifies limitations of an existing empirical approximation and proposes a simple, improved formula validated against semi-analytical solutions for all patch sizes.

Findings

The original approximation fails at large patch surface fractions.

The proposed amendment provides accurate results across the entire patch size range.

The improved approximation accurately predicts reaction probability and capacitance.

Abstract

Finding accurate approximations for the effective reactivity of a structured spherical target with a circular absorbing patch of arbitrary size is a long-standing problem in chemical physics. In this Communication, we reveal limitations of the empirical approximation proposed in [J. Chem. Phys. 145, 214101 (2016)]. We show that the original approximation fails at large patch surface fractions $\sigma$ and propose a simple amendment. The improved approximation is validated against a semi-analytical solution and is shown to be accurate over the entire range of $\sigma$ from $0$ to $1$. This approximation also determines the probability of reaction on the patch and the capacitance of such a structured target.