Stochastic self-assembly of incommensurate clusters

TL;DR

This paper investigates how incommensurate total mass and maximum cluster size affect stochastic nucleation, revealing significant differences from mean-field models, especially in finite systems with mass ratios that cause broadening of cluster size distribution.

Contribution

It introduces a fully discrete stochastic master equation for nucleation, highlighting the impact of mass incommensurability on cluster distributions, which is not captured by traditional mean-field models.

Findings

Discrepancies between stochastic and mean-field equilibrium concentrations depend on mass divisibility.

Incommensurate mass ratios cause broadening of cluster size distribution.

Classical mean-field approaches fail to accurately describe nucleation in finite, incommensurate systems.

Abstract

We examine the classic problem of homogeneous nucleation and growth by deriving and analyzing a fully discrete stochastic master equation. Upon comparison with results obtained from the corresponding mean-field Becker-D\"{o}ring equations we find striking differences between the two corresponding equilibrium mean cluster concentrations. These discrepancies depend primarily on the divisibility of the total available mass by the maximum allowed cluster size, and the remainder. When such mass incommensurability arises, a single remainder particle can "emulsify" or "disperse" the system by significantly broadening the mean cluster size distribution. This finite-sized broadening effect is periodic in the total mass of the system and can arise even when the system size is asymptotically large, provided the ratio of the total mass to the maximum cluster size is finite. For such finite ratios we show that homogeneous nucleation in the limit of large, closed systems is not accurately described by classical mean-field mass-action approaches.