Revisiting Cahn’s expression for grain boundary nucleated transformations using rigorous mathematical methods

It is well known that grain boundaries are the preferred location for nucleation. Those nucleation sites are not located uniformly randomly in space as assumed by the KJMA theory. As the grains are polyhedra, they comprise faces, edges, and vertices, and nucleation may take place in all these geometrical features. In his pioneering paper, John W. Cahn proposed modeling nucleation and growth on each of these geometrical faces, making some approximations. One of his assumptions was that the grain faces were approximated by "random planes." In this paper, we clarify the meaning of Cahn's random planes by revisiting his derivation using modern methods of stochastic geometry. We illustrate our result with a computer simulation. In summary, this paper provides a mathematical explanation of when and why Cahn's equation, a well-known model, can be used to approximate the volume fraction of phase transformations nucleated on grain faces. We consider both the site-saturated and the constant nucleation rate cases.