The Time Cone Method for Nucleation and Growth Kinetics on a Finite Domain

The Kolmogorov-Johnson-Mehl-Avrami theory is an exact statistical solution for the expected fraction transformed in a nucleation and growth reaction in an infinite specimen, when nucleation is random in the untransformed volume and the radial growth rate after nucleation is constant until impingement. Many of these restrictive assumptions are introduced to facilitate the use of statistics. The introduction of “phantom nuclei” and “extended volumes” are constructs that permit exact estimates of the fraction transformed. An alternative, the time cone method, is presented that does not make use of either of these constructs. The method permits obtaining exact closed form solutions for any specimen that is convex in time and space, and for nucleation rates and growth rates that are both time and position dependent. Certain types of growth anisotropies can be included. The expected fraction transformed is position and time dependent. Expressions for transformation kinetics in simple specimen geometries such as plates and growing films are given, and are shown to reduce to expected formulas in certain limits.