Survival in a nanoforest of absorbing pillars

TL;DR

This paper provides an exact analytical solution for the survival probability of a diffusing particle in a nanoforest of absorbing pillars, highlighting the influence of geometric parameters and exploring asymptotic behaviors.

Contribution

It introduces a mode matching method to solve the modified Helmholtz equation in a cylindrical domain with a pillar, offering new insights into the first-passage time distribution.

Findings

Exact Laplace transform of survival probability obtained

Geometric parameters significantly affect first-passage times

Capacitance approximation fails for principal eigenvalue

Abstract

We investigate the survival probability of a particle diffusing between two parallel reflecting planes toward a periodic array of absorbing pillars. We approximate the periodic cell of this system by a cylindrical tube containing a single pillar. Using a mode matching method, we obtain an exact solution of the modified Helmholtz equation in this domain that determines the Laplace transform of the survival probability and the associated distribution of first-passage times. This solution reveals the respective roles of several geometric parameters: the height and radius of the pillar, the inter-pillar distance, and the distance between confining planes. This model allows us to explore different asymptotic regimes in the probability density of the first-passage time. In the practically relevant case of a large distance between confining planes, we argue that the mean first-passage time is much larger than the typical time and thus uninformative. We also illustrate the failure of the capacitance approximation for the principal eigenvalue of the Laplace operator. Some practical implications and future perspectives are discussed.