Finite Volume Kolmogorov-Johnson-Mehl-Avrami Theory

Abstract

We study Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory of phase conversion in finite volumes. For the conversion time we find the relationship ${\tau }_{\mathrm{con}}={\tau }_{\mathrm{nu}}[1+f_d(q)]$. Here $d$ is the space dimension, ${\tau }_{\mathrm{nu}}$ the nucleation time in the volume $V$, and $f_d(q)$ a scaling function. Its dimensionless argument is $q={\tau }_{\mathrm{ex}}/{\tau }_{\mathrm{nu}}$, where ${\tau }_{\mathrm{ex}}$ 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 $f_d(q)$.

Comments: 4 pages, 4 figures. Additions after referee reports: Scaling of the variable q is proven. Additional references are added