Whether it is the nonstoichiometry of fluorite — structure UO2±x or variable composition ortho­rhombic or tetragonal U-Zr alloy fuel, the accurate thermochemical description of these phases has

Diagram illustrating the computer coupling of phase diagrams and thermochemistry approach after Zinkevich.2

been through the use of solution models. Solid and liquid solution modeling from simple highly dilute systems to more complex interstitial and substitu­tional solutions with multiple lattices has been a rich field for some time, yet it is far from fully developed. To accurately describe the energetics of solutions will eventually require bridging atomistic models with the mesoscale treatments currently being used. Recent approaches have begun to deal with defect structures in phases, although only in a very constrained manner which limits clustering and other phenomena. Yet, when they are coupled with accurate data that allow fitting of the model para­meters, the resulting representations have been highly predictive of phase relations and chemistry.

A number of texts provide useful descriptions of solution models from the simple to the complex.12-15 While there are several relatively accurate but rather intricate approaches, such as the cluster variation method, the discussion in this chapter is confined to
simpler models that are easily used in thermochemi­cal equilibrium computational software and thus applicable to large, multicomponent systems of inter­est in nuclear fuels.

The simplest model is the ideal solution where the constituents are assumed to mix randomly with no structural constraints and no interactions (bonding or short — or long-range order). The standard Gibbs free energy and ideal mixing entropy contributions are

G° = X »<G?’

Gid = RT^2"j ln n, [8]

where G is the weighted sum of the standard Gibbs free energy for the constituents in the j phase solution, Gid is the contribution from the entropy increase due to randomly mixing the constituents, which is the configurational entropy, and R is the ideal gas law constant. For an ideal solution, the sum of the two represents the Gibbs free energy of the system.

In cases where there are significant interactions (bonding or repulsive interaction energies) among mixing constituents, an energetic term or terms need to be added to the solution free energy. The inclusion of a simple compositionally weighted excess energy term, Gex, accounts for the additional solution energy for what is historically termed a regular solution

Gex = XX XXEj [9

i=1 j>1

where X is the mole fraction and Ej the interaction energy between the components i and j. The system free energy is thus

G = G° + Gid + Gex [10]