Finite volume Kolmogorov-Johnson-Mehl-Avrami theory

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