Deposition, diffusion, and nucleation on an interval

TL;DR

This paper models nanoscale film growth via Brownian particles on an interval, revealing a Markovian splitting process and analyzing gap distributions, with implications for understanding thin film deposition.

Contribution

It introduces a continuum model for particle deposition and nucleation on an interval, deriving a splitting density through Brownian motion exit problems and analyzing asymptotic behaviors.

Findings

Convergence to a Markovian splitting process in the sparse limit.

Splitting density governs large-time gap distribution asymptotics.

Asymptotics are independent of deposition rate.

Abstract

Motivated by nanoscale growth of ultra-thin films, we study a model of deposition, on an interval substrate, of particles that perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model studied in the applied literature. We show that the associated interval-splitting process converges in the sparse deposition limit to a Markovian process (in the vein of Brennan and Durrett) governed by a splitting density with a compact Fourier series expansion but, apparently, no simple closed form. We show that the same splitting density governs the fixed deposition rate, large time asymptotics of the normalized gap distribution, so these asymptotics are independent of deposition rate. The splitting density is derived by solving an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson.