Scheil Solidification
One approach for modeling solidification processes is the Scheil1 model. The Scheil model assumes that diffusion rates are infinite in the liquid and zero in the solid. In addition, this model assumes that local equilibrium is maintained at the advancing solidification interface, and the ratio of the liquidus composition to the solidus composition, k, remains constant. Lastly, it assumes that there is negligible undercooling due to curvature or kinetics. The Scheil model can be used to describe the microsegregation present in primary phase dendritic growth and directional solidification. The nonequilibrium lever rule, or Scheil equation, can be expressed as:
Insert Equation Here
where C0 is the alloy composition, Cl is the liquid composition, fs is the fraction solid, and k is the partition ratio. Depending on the type of invariant reaction reached, solidification may stop and completely transform the remaining liquid to the invariant reaction product, or solidification will continue in a new phase with a step in the resultant concentration profile.
Phase Equilibria in the Transitional Reaction
Because of an additional degree of freedom possessed by tecause of an additional degree of freedom possessed by ternary alloys, an additional reaction type not found in binary alloys is possible. This reaction is the transitional or U-type reaction and is characterized by the reaction equation:
L + A = B + C.
This reaction occurs in an invariant plane made from the union of two three phase tie triangles separated by a two phase region. This allows an alloy to reach the reaction from three reaction sequences: two three-phase tie-triangles, or one two-phase region. To reach the transitional reaction from a two-phase region the alloy composition must lie at the crossover point between the two-phase tie lines. Otherwise, solidification through a transitional reaction will occur indirectly through a three-phase tie-triangle. Details of the transitional phase equilibria are in Figure 1.
Insert Figure Here
Figure 1. Schematic isothermal sections above, below, and at the invariant temperature and liquidus projection for a ternary transitional or U-type reaction.
At the onset of solidification, the protransitional phase will have a solute distribution as described by the nonequilibrium mass balance. Depending on alloy composition, a three-phase or a two-phapending on alloy composition, a three-phase or a two-phase solidification microstructure is expected prior to reaching the transitional invariant reaction. This means that the solute distribution may occur in either one or two simultaneously growing phases, requiring that assumptions be made about microstructure geometry.
In addition to the equilibrium
phase formation sequence dictated by the phase diagram,
nonequilibrium solidification behavior can add additional
complexity to understanding the resultant microstructures. For
example, if the liquid composition reaches a three-phase
peritectic ridge, the liquid composition will cascade past the
reaction line, rather than
follow the reaction line as in a ternary eutectic reaction.
Without a thorough understanding of the thermodynamics of the
alloy system, predicting solidification paths may be difficult.
At the invariant plane all four phases will be in equilibrium, and therefore all four phases must be in contact. Generally this does not present a problem for nucleation of the product phases, as the alloy composition will determine the sequence of nucleation. However, in a transitional reaction it is not clear how the phases will nucleate for an alloy with a composition at the intersection of the two-phase tie lines. Two phases must nucleate and grow below the invariant temperature, but, itnd grow below the invariant temperature, but, it is unknown if they will grow sequentially or simultaneously.
The Au-Pb-Sn System
To help understand the transitional reaction better it was decided to study solidification microstructures in the Au-Pb-Sn system. The Au-Pb-Sn system is well suited to the study of transitional reactions due to a low melting point transitional reaction occurring at 210 C where:
L + AuSn2 = AuSn4 + (Pb)
This system facilitates analysis because the atomic numbers are far apart on the periodic table providing good atomic number contrast for SEM analysis. In addition, the phase compositions lie far apart on the phase diagram making them easily identifiable by microchemical analysis.
Although the Au-Pb-Sn alloy system has two transitional reactions that feed into a ternary eutectic, the Au-Pb-Sn ternary eutectic reaction has been well studied and is easily identifiable. Thus, determining alloy position within the transitional plane should be straightforward. This micrograph shows a typical microstructure for a transrd. This micrograph shows a typical microstructure for a transitional reaction where the primary phase is consumed by subsequent growth of two product phases.
Proposed Research
A numerical model for ternary transitional alloy solidification will be developed using the Scheil equation as the basis. Incorporation of diffusion will be based on existing analytical models where limited solid state diffusion takes place. In addition to considering diffusion during primary phase solidification, diffusion through the intervening transitional reaction product layers will also be modeled.
Reference
1. Scheil, E., Zeichrift fur Metallkunde, vol. 34, n. 70 (1942)