Exact results for nucleation-and-growth in one dimension

TL;DR

This paper provides exact analytical results for nucleation-and-growth processes in one dimension, including gap and island distributions, and explores effects of accelerated growth mechanisms.

Contribution

It introduces exact solutions for the one-dimensional nucleation-and-growth model and analyzes the impact of accelerated growth on system coverage and critical behavior.

Findings

Exact gap density and island distribution derived

System reaches complete coverage in finite time

Near-critical behavior is robust to nucleation details

Abstract

We study statistical properties of the Kolmogorov-Avrami-Johnson-Mehl nucleation-and-growth model in one dimension. We obtain exact results for the gap density as well as the island distribution. When all nucleation events occur simultaneously, the island distribution has discontinuous derivatives on the rays x_n(t)=nt, n=1,2,3... We introduce an accelerated growth mechanism where the velocity increases linearly with the island size. We solve for the inter-island gap density and show that the system reaches complete coverage in a finite time and that the near-critical behavior of the system is robust, i.e., it is insensitive to details such as the nucleation mechanism.