This novel approach to modeling the steady-state solidification of undercooled pure liquids is based upon first principles. Continuum equations are used to describe a volumetrically averaged, coexisting mixture of solid and liquid in the thin phase transformation zone between regions of pure liquid and pure solid. These equations are coupled with a dynamic equilibrium based rate law that describes temperature dependent phase transformation kinetics. The time scale associated with finite rate phase transformation is found to depend on a solidification activation energy, thermal energy, and the enthalpy of fusion. The model leads naturally to an eigenvalue problem for the solidification front propagation speed. In addition, the variation of the volumetrically averaged liquid fraction across the solidification zone is predicted.

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A mathematical model is formulated to describe the effect of finite-rate heat release on solidification in an undercooled pure liquid and to predict the macroscopic propagation speed of a solidification front through the bulk material. The problem is formulated in terms of continuum equations that describe heat and mass transport in a volumetrically averaged mixture of solid and liquid. A relatively thin phase transformation region, called the solidification zone, exists between thicker regions of pure liquid and solid. The solidification zone is examined on a length scale larger than any microstructural detail, yet smaller than macroscopic thermal conduction length scales in the pure liquid and solid regions. The continuum equations used in this 'mesoscale' model contain source terms representing the volumetrically averaged finite-rate phase-transformation process occurring within the solidification zone. Solutions are obtained for a source term of the Arrhenius type, derived from an application of nucleation theory. Front speed variation with degree of undercooling is found to be quantitatively similar to that in relevant experiments in pure metals.