The liquid phases in nuclear fuels are important to model so that the phase equilibria can be completely assessed through comparison of experimental and computed phase diagrams. The availability of solidus and liquidus information also provides necessary boundaries for modeling the solid-state behavior. Finally, safety analysis requirements with regard to the potential onset of melting will benefit from accu­rate representations of the complex liquids.

Ideal, regular/subregular, or Bragg-Williams formulations are not very successful in representing metal and especially oxide liquids where there are strong interactions between constituents. The CEF model is designed for fixed lattice sites, and thus it too will not handle liquids. The issues for these complex liquids involve the short-range ordering that generally occurs and its effect on the form of the Gibbs free energy expressions. One approach to dealing with the issue of these strong interactions is the modified associate species method.

The modified associate species technique for crystalline materials was discussed to an extent in Section 1.17.4.2. Its application to, for example, oxide melts has been more broadly covered recently by Besmann and Spear39 with much of the original development by Hastie and coworkers.40-43 The approach assumes that the liquid can be modeled by an ideal solution of end-member species together with intermediate species. The modified term refers to the fact that an ideal solution cannot represent a miscibility gap in the liquid as that requires repulsive (positive) interaction energy terms. Thus, when a mis­cibility gap needs to be included, interaction energies between appropriate associate species are added to the formulation.

In the associate species approach, the system standard Gibbs free energy is simply the sum of the constituent end-member and associate free energies, for example, A, B, and A2B, where inclusion of the A2B associate is found to reproduce the behavior well,

G° = XaGa + XbGb + Xa2bGA2b [16]

Consequently, ideal mixing among end members and associates generates the entropy contribution

Gid = RT (Xa ln Xa + Xb ln Xb + Xa2b ln Xa2b) [17]

Should a nonideal term providing positive interac­tion energies be needed to properly address a misci­bility gap, it would be added into the total Gibbs free energy as in eqn [10]. For example, for an interaction between A and A2B in the Redlich-Kister-Muggianu formulation the excess term is expressed as

Gex = XaXa2bELk, A:A2B(XA — Xa2b)* [18]

k

The associate species are typically selected from the stoichiometry of intermediate crystalline phases, but others as needed can be added to accurately reflect the phase equilibria even when no stable crys­talline phases of that stoichiometry exist. Gibbs free energies for these species can be derived from fits to the phase equilibria and other data following the CALPHAD method with first estimates gener­ated from crystalline phases of the same stoichiome­try or weighted sums of existing phases when no stoichiometric phase exists. The application of the method for the liquid phase in the Na2O-Al2O3 is seen in the computed phase diagram in Figure 8. For this system, the associate species required to represent the liquid were only Na2O, NaAlO2, (1/3) Na2Al4O7, and Al2O3. In nuclear fuel systems, Chevalier et al44 applied an associate species approach using the components O, U, and O2U, although it deviated from the associate species approach in using binary interaction parameters in a Redlich-Kister-Muggianu form. The computed

phase diagram showing agreement with the liquidus/ solidus data is seen in Figure 9.

The use of the modified associate species model with ternary and higher order systems can require the use of ternary or possibly quaternary associates. Another issue with the modified associate species approach is that in the case of a highly ordered solution which requires an overwhelming content of an associate compared to an end-member, the rela­tions do not follow what should be Raoult’s law for dilute solutions. At the other extreme, it is also appar­ent that in the case of essentially zero concentration of associates, the relationships do not default to an ideal solution as one would expect.