In all the systems discussed heretofore the solidification front was considered to be a mathematical interface of zero thickness endowed with surface properties deemed appropriate to the physics. In this chapter, another approach is taken. The front is allowed to be diffuse, and the fields of interest, such as T and C, are supposed to have well-defined bulk behaviors away from the interfacial region and rapid, though continuous, variations within it. Minimally, one would wish the model to satisfy the laws of thermodynamics, appropriately extended into the nonequilibrium regions, and regain the interfacial properties and jump conditions appropriate to the thin-interface limit when the interfacial thickness approaches zero.

On the one hand one would anticipate that an infinite number of such models likely exists. On the other hand one would anticipate that the thin-interface limit might be taken a number of ways, each giving distinct properties to the front. Nonetheless, conceptually there are two possible virtues of the diffuse-interface approach. If the models are well chosen on the basis of some underlying framework, then there would be a systematic means of generalizing the models to systems such as rapid solidification. In Chapter 6 high rates of solidification were modeled by appending to the standard model variations k = k(Vn), m = m(Vn) with, for example, the equilibrium Gibbs–Thomson undercooling. A systematic generalization could indicate how the relationships for k and m emerge, and what other alterations to the model should simultaneously be included. Call this “model building.”