A Phase-Field Model for Solidification of a Eutectic Alloy

Geoffrey B. McFadden, ACMD
William J. Boettinger, Metallurgy Division
Adam A. Wheeler, Southampton University, UK

Solidification of a eutectic alloy from the melt or liquid phase produces two composite solid phases, in contrast to the simpler case when the liquid phase solidifies into a single solid phase. The two solid phases are characterized by differing solute concentrations; the equilibrium concentrations in each phase are given by the phase diagram for the eutectic system, which gives the relations between temperature and solute concentration for each of the thermodynamic phases in the system. A number of different geometrical arrangements of the two solid phases are possible, depending on the alloy system and on the processing conditions. For example, the solid phases can form in adjacent layers or lamellae, or one phase can form rods that are embedded in the other phase. Both grow continuously from the melt. Since a large number of important binary alloy systems have eutectic phase diagrams, such systems have been studied extensively. It is desirable to be able to predict the geometrical patterns of growth and their associated length scales in order to better understand and possibly control the processing of eutectic materials.

Phase-field methodologies for the solidification of single-phase solids for single-component or pure materials were developed in the early 1980's. In these models an auxiliary variable or order parameter, known as the phase field, is introduced in order to differentiate in a continuous fashion between the liquid and solid phases in the system. In its simplest form, these models include a thermodynamic description of the free energy of the system as a function of the temperature and the phase field. Phase-field models for the solidification of two-component systems having simple phase diagrams were developed in the last several years by the above authors and by researchers elsewhere. These models feature a free energy that depends on the solute concentration in addition to the temperature and the phase field; the free energy is constructed in such as way that the appropriate phase diagrams for the system are recovered. More recently, we have developed models for solidification of eutectic systems. The most general model includes a formulation for the free energy of the system in terms of the temperature and concentration of the system, together with two order parameters rather than a single one. One of the order parameters, phi, is used to indicate whether the system is in a solid or liquid phase, and the other order parameter, psi, is used to indicate which of the two possible solid phases is present. With this free energy it is possible to recover the appropriate eutectic phase diagram for the system.

The resulting phase-field model has been used to study the formation of trijunctions in which both solid phases are in equilibrium with the liquid phase. The angles formed by the inter-phase boundaries should meet at specific angles that satisfy Young's law, representing a balance of surface energies at the trijunction. An example is shown in the figure below, where a temperature gradient is imposed in the vertical direction. The top figure shows the trijunction region, consisting of the liquid phase above the two solid phases, labeled alpha and beta. Contour lines are shown for the solute field. The middle figure shows contour lines for the phase field psi, which determines the solid-solid interface. The lower figure shows contour lines for the phase field phi, which determines the solid-liquid interfaces.

[picture of eutectic]

This work has been described in a paper entitled ``Phase-Field Model for Solidification of a Eutectic Alloy,'' by A. A. Wheeler, G. B. McFadden, and W. J. Boettinger, that has been submitted for publication in the Proceedings of the Royal Society of London, Series A.