THE TIME CONE METHOD FOR NUCLEATION AND GROWTH KINETICS ON
A FINITE DOMAIN.
John W. Cahn
Materials Sci. and Eng. Lab.
Gaithersburg, MD 20899
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.