Finite Volume Kolmogorov-Johnson-Mehl-Avrami Theory

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

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