| The phase field model provides a promising theoretical technique for computational studies of crystal growth. Instead of assuming that the interface between the solid and the liquid has zero thickness, as in the sharp interface model, we use a model assuming a diffuse interface. An auxiliary variable, the phase field, is used to identify the phases. The phase field variable takes on constant values in bulk phases and changes continuously across a thin region, the solid-liquid interface. As a result, there is no need for explicit interface tracking. Based on the principles of irreversible thermodynamics, dynamical governing equations are obtained for the temperature field, the phase field and the concentration field (in the case of alloy growth). First we examine a phase field model for a pure material in which the internal energy density is used to identify the phase. The resulting fourth order equation is solved for some one-dimensional cases. Then, we concentrate on our main topic, the directional growth of a binary alloy. Morphological instability occurs and a steady-state cellular interface results. The cellular shape of the interface is studied with Fourier analysis. Transitions between a stable planar interface and shallow cells, wavelength selection, domain size effects, and transitions between shallow cells and deep cells are investigated. |
