Aggregation According to Classical Kinetics--From Nucleation to Coarsening

TL;DR

This paper analyzes the Lifshitz-Slyozov model for particle aggregation, revealing a growth phase before coarsening, and characterizing the evolution of cluster sizes and distributions during these stages.

Contribution

It introduces an effective initial condition based on nucleation exhaustion, and provides a detailed analysis of growth and coarsening phases using asymptotic and numerical methods.

Findings

Identification of a growth era before coarsening.

Characterization of cluster size evolution during growth and coarsening.

Confirmation that the long-time limit is the discontinuous LS similarity solution.

Abstract

We solve the standard Lifshitz-Slyozov (LS) model with conservation of total particles in the limit of small super-saturation. The new element is an effective initial condition that follows from the initial exhaustion of nucleation as described in a previous paper [Farjoun and Neu, Phys. Rev. E 78]. The effective initial condition is characterized by a narrow distribution of cluster-sizes, all much larger than critical. In the subsequent solution, one of the LS similarity solutions emerges as the long-time limit, as expected. But our solution tells more. In particular, there is a "growth" era prior to what is usually called "coarsening". During "growth" the clusters (all of nearly the same size much larger than critical) eventually exhaust the super-saturation (the exhaustion of nucleation in the previous era results from only a small decrease in super-saturation). This allows the critical size to catch up to the clusters, and the traditional "coarsening" begins: Subcritical clusters dissolve and fuel the growth of the remaining super-critical clusters. Our analysis tracks the evolution of cluster sizes during growth and coarsening by complimentary use of asymptotic and numerical methods. We establish characteristic times and cluster sizes associated with growth and coarsening from physical parameters and the initial super-saturation. The emerging distribution is discontinuous at the largest cluster size, and our model selects the discontinuous LS similarity solution as the long time limit. There are strong indications that the smooth similarity solution proposed in the original LS paper emerges on a, yet longer, late-coarsening time-scale.