# Development of a CALPHAD Thermodynamic Database for Pu-U-Fe-Ga Alloys
## 2 Thermodynamic Modeling  
### CALPHAD Method  
Computational Thermodynamics in the form of the CALCulation of PHAse Diagrams (CALPHAD) method provides self-consistent Gibbs energy functions, for a given phase and/or structure [8], which are used to calculate equilibrium conditions of multicomponent systems. The model parameters are optimized using critically selected thermochemical and constitutive data as input. For the pure elements, the data from the SGTE [9] database is used and is expressed via the following temperature (T in K) dependent function, for element, x in a given phase, \(\phi\):  
\[{}^{0}G_{\mathrm{x}}^{\phi}=G^{\phi}-H_{\mathrm{x}}^{\mathrm{SER}}=a+bT+cT\ln(T )\ +\ dT^{2}\ +eT^{3}\ +fT^{-1}\ +g_{n}T^{n} \tag{1}\]  
In the above expression, \(H_{\mathrm{x}}^{\mathrm{SER}}\) is the molar enthalpy of component x in its standard element reference (SER) state at 298.15 K and 10\({}^{5}\) Pa. In the case of the elements described here, the standard states are \(\alpha\)-U (orthorhombic), \(\alpha\)-Fe (body centered cubic), \(\alpha\)-Ga (orthorhombic) and \(\alpha\)-Pu (simple monoclinic), where crystallographic structures are detailed in Table 1. The following phases: \(\alpha\)-U, \(\alpha\)-Pu, \(\beta\)-Pu, \(\gamma\)-Pu and \(\delta^{\prime}\)-Pu are treated as fixed compositions (neglecting solubility) and represented using (Equation (1)). The liquid, bcc (\(\alpha\)-Fe, \(\delta\)-Fe, \(\varepsilon\)-Pu, \(\gamma\)-U), fcc (\(\gamma\)-Fe, \(\delta\)-Pu) and \(\beta\)-U are all treated as solution phases and are modelled as substitutional solutions yielding the following molar Gibbs energy expression for a given phase \(\phi\):  
\[G_{\mathrm{m}}^{\phi}=\sum_{\mathrm{i}}x_{\mathrm{i}}{}^{0}G_{\mathrm{i}}^{ \phi}+RT\sum_{\mathrm{i}}x_{\mathrm{i}}\ln(x_{\mathrm{i}})\ +\ ^{\mathrm{ phys}}G_{\mathrm{m}}^{\phi}\ +\ ^{\mathrm{E}}G_{\mathrm{m}}^{\phi} \tag{2}\]  
where the first term represents the mechanical mixing of the end-members (\({}^{0}G_{\mathrm{i}}^{\phi}\) are the Gibbs energies of the pure elements in the structural state \(\phi\)), the second term represents the contribution due to the ideal entropy of mixing, the third includes physical models such as magnetic contribution and the fourth term (\({}^{\mathrm{ex}}G_{\mathrm{m}}^{\phi}\)) represents the excess molar Gibbs energy. For phases with order-disorder transformations (e.g., bcc_A2/B2), the partitioning model is used, which means the ordering is described as an addition to the Gibbs energy of the disordered phase so that the molar Gibbs energy is described as:  
\[G_{\mathrm{m}}^{\mathrm{bcc}}=G_{\mathrm{m}}^{\mathrm{dis}}\ +\ \Delta G_{ \mathrm{m}}^{\mathrm{ord}} \tag{3}\]  
where \(G_{\mathrm{m}}^{\mathrm{dis}}\) is the contribution to the Gibbs energy based on composition expressed by the disordered solution phases formulae (independent of the ordering state of the phase), and \(\Delta G_{\mathrm{m}}^{\mathrm{ord}}\) is the additional Gibbs energy due to ordering, i.e., configuration (contribution due to the long-range ordering, and must be zero in the when the phase is disordered):  
\[\Delta G_{\mathrm{m}}^{\mathrm{ord}}=G_{\mathrm{m}}^{\mathrm{ord}}\big{(}y_{ \mathrm{i}}\big{)}-G_{\mathrm{m}}^{\mathrm{ord}}\big{(}y_{\mathrm{i}}=x_{ \mathrm{i}}\big{)} \tag{4}\]  
where \(y_{\mathrm{i}}\) is the mole fraction of i per sublattice, also denoted site fraction, and \(x_{\mathrm{i}}\) the mole fraction. The parameter describing the ordering, \(G_{\mathrm{m}}^{\mathrm{ord}}\), is first calculated with the original site fractions, y, which describe the ordering. The site fractions are then set equal to the mole fraction, x, which means that each constituent has the same site fraction in all sublattices (i.e., disordered states), and the value of the expression is calculated again. The difference is the contribution to the Gibbs energy due to ordering. If the phase was originally disordered, the two terms are equal and the difference is zero (no driving for ordering).