Finite Volume Kolmogorov-Johnson-Mehl-Avrami Theory

TL;DR

This paper extends KJMA theory to finite volumes, deriving a relationship for conversion time involving a scaling function that depends on volume size and expansion dynamics.

Contribution

It introduces a finite-volume KJMA model, deriving a universal scaling function for conversion time across different dimensions and volume sizes.

Findings

Derived a formula for conversion time in finite volumes.

Calculated the scaling function in 1D, 2D, and 3D.

Showed volume size influences nucleation and spinodal decomposition limits.

Abstract

We study Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory of phase conversion in finite volumes. For the conversion time we find the relationship $\tau_{\rm con} = \tau_{\rm nu} [1+f_d(q)]$. Here $d$ is the space dimension, $\tau_{\rm nu}$ the nucleation time in the volume $V$, and $f_d(q)$ a scaling function. Its dimensionless argument is $q=\tau_{\rm ex}/ \tau_{\rm nu}$, where $\tau_{\rm ex}$ is an expansion time, defined to be proportional to the diameter of the volume divided by expansion speed. We calculate $f_d(q)$ in one, two and three dimensions. The often considered limits of phase conversion via either nucleation or spinodal decomposition are found to be volume-size dependent concepts, governed by simple power laws for $f_d(q)$.