Phase field modeling of shallow cells during directional solidification of a binary alloy

We report some computational results obtained by using the phase field model for binary alloy solidification. We study directional solidification and concentrate on the transition between a planar interface and steady shallow cells near the onset of morphological instability at low growth speeds. The model is formulated on the principle of local positive entropy production. This gives rise to a set of partial differential equations for the temperature, composition and the phase fields. We make a frozen temperature approximation and compute steady-state nearly periodic cellular crystal–melt interfaces in two dimensions. The cell shapes can be characterized by a small number of Fourier coefficients and become nearly sinusoidal as stability is approached. Wavelength selection by tip splitting or coarsening is also observed.