Transformation kinetics for nucleation on random planes and lines

Birth and growth processes are known in materials science as nucleation and growth processes. In crystalline materials nucleation almost always takes place in an internal crystalline defect. These defects are classified according to their dimensionality: point, line or planar defects. Therefore, investigating nucleation on sets of dimensionality lower than the set in which the transformation takes place is of paramount importance. Cahn (1956) in a classical work derived expressions for transformation kinetics when nucleation took place on random planes and on random straight lines. He used these expressions to describe nucleation in polycrystalline materials. He considered that nucleation on grain faces could be treated as nucleation on random planes and, likewise, nucleation on grain edges could be treated as nucleation on random lines. The present work revisits and generalizes Cahn’s treatment of nucleation on planes and lines. First a general expression for the case of nucleation on lower dimensional sets is obtained. After that general expressions for nucleation on random planes and random lines are given. This paper provides the mathematical basis for the development of more specific expressions to be used in practical applications. Although this work has been done bearing applications to materials science in mind the results obtained here may be applied to birth and growth processes in any field of science.