We study the growth of a two-phase finger in eutectic systems, using a boundary-integral formulation. We complement our investigations by a phase-field validation of the stability of the patterns, which were observed experimentally by Akamatsu and Faivre. The deviations from the eutectic temperature and from the eutectic concentration provide two independent control parameters, leading to very different patterns depending on their relative importance.
One of the most common modes of growth of a solid phase from a metastable liquid is the dendritic one where a nearly parabolic front advances at a constant velocity. The possibility of a steady-state growth of a parabolic front was first demonstrated by Ivantsov and was supplemented by the determination of the anisotropy of surface tension as a selection mechanism for the velocity. Recently, selection mechanisms such as the presence of a triple junction or of elastic effects were reported.
Another classical mode of solidification is the lamellar growth in eutectic alloys. The pioneering work of Jackson and Hunt on this topic refers to directional solidification and especially consists of finding the temperature of the solidification front. In opposition to dendritic growth, no unique solution exists for this process. A range of lamellae spacing is stable and bifurcations toward a broad range of oscillatory regimes or tilted patterns have been evidenced. It is worthwhile to mention the recent observation of a three-dimensional spiral dendrite in eutectics.
Despite a large amount of theoretical and experimental studies on eutectics, the problem of dendritic patterns has, to our knowledge, never been addressed on a theoretical level in these systems. However a dendritelike structure called “two-phase fingers” has been observed by Akamatsu and Faivre for an off-eutectic concentration and it is suggested that such a pattern could grow with a constant velocity in directional experiments. This is what we study here in the isothermal case, that is, a two-dimensional solidifying dendrite with one solid phase surrounded by the other one. The framework of the boundary-integral technique is used and we supplement our results with a phase-field calculation showing that the eutectic two-phase finger is a stable mode of growth.