Application of Stochastic Geometry to Nucleation and Growth Transformations

In their pioneer work, Johnson-Mehl, Avrami, and Kolmogorov (JMAK) developed well-known analytical expressions to describe the transformation kinetics of nucleation and growth transformations. Their work has had a particularly important influence in recrystallization with the subsequent development of the so-called microstructural path method (MPM). MPM has been extensively applied by Vandermeer and coworkers to analyze recrystallization in a variety of metallic materials. Much of the analytical developments building upon the JMAK theory have retained the core of their original assumptions regarding nucleation and growth. Specifically, JMAK assumed nuclei to be uniformly randomly distributed in space and that the velocity of the moving boundary was the same for every growing grain. Recently, the present authors in a series of papers have introduced a new mathematical methodology involving stochastic geometry and geometric measure theory to deal with nucleation and growth problems. With the help of this new methodology, it has been possible to generalize the original JMAK assumptions. New expressions have been derived to describe transformations originating in a diversity of nucleation sites, such as inhomogeneous nucleation, cluster nucleation, and nucleation on lower dimensional sets. The velocity assumption was also the subject of generalization, and expressions for random velocities could be obtained. Finally, it is possible that two or more transformations may take place simultaneously or sequentially; for example, the evolution of two or more texture components during recrystallization. A new general methodology was developed to deal with this situation. This new and rigorous mathematical methodology has proved to be significantly better than previously used methods. New analytical expressions were derived, which significantly expanded the number of analytical solutions available for nucleation and growth transformations. Moreover, derivation of previous results by the new methodology yielded new insights into their original assumptions and validit