## 3 CALPHAD Method  
CALPHAD modeling is based on a set of models and adjustable parameters to represent Gibbs energy functions for a given phase as a function of composition, temperature, and pressure. Once a model is selected (depending on the phase), the model parameters are optimized by fitting inputs of critically selected thermochemical and constitutive data from the literature, including experiment and ab initio calculations. The assessed functions are compiled in a database for use in computational thermodynamic predictions across multicomponent materials. The strength of the CALPHAD method relies on the ability of the assessed database to be self-consistent and thus can be applied to predict the thermodynamic behavior of multicomponent systems. To demonstrate the applicability of thermodynamic modeling with respect to fallout glass, we modeled the liquid phase behavior of the \(\mathrm{SiO_{2}}\)-FeO-\(\mathrm{Al_{2}O_{3}}\)-CaO-MgO system using the thermodynamic description from the Thermodynamics of Advanced Fuels-International Database (TAF-ID) and the solid phases models from the constituent binary and ternary systems [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36]. In more detail, the liquid phase for non-metallic systems is described with a two-sublattice ionic liquid model [37; 38] as:  
\[(\mathrm{Al^{3+}},\mathrm{Ca^{2+}},\mathrm{Fe^{2+}},\mathrm{Mg^{3+}},\mathrm{ Si^{4+}})_{\mathrm{P}}(\mathrm{AlO_{2}}^{-},\mathrm{O^{2-}},\mathrm{SiO_{4}}^{-}, \mathrm{FeO_{3/2}},\mathrm{SiO_{2}})_{\mathrm{Q}} \tag{1}\]  
The first sublattice describes the cations of the system and the second considers anions, negatively charged vacancies (\(Va\)), and neutral species, with \(P\) and \(Q\) being the number of sites on each sublattice. The neutral species are introduced to help describe non-metallic liquids. To maintain electroneutrality, \(P\) and \(Q\) vary with composition such that:  
\[P=\sum_{i}v_{i}y_{A_{i}}+Qy_{Va} \tag{2}\]  
\[Q=\sum_{j}v_{j}y_{C_{j}} \tag{3}\]  
where \(v_{i}\) is the valency of ion \(i\) or ion \(j\), and \(y_{i}\) is the site fraction of ion \(i\) or ion \(j\), which is the atomic fraction within each sublattice (\(A\) denotes an anion, \(C\) denotes a cation, and \(B\) is for neutral species). The summation over \(i\) is made over all anions and the summation over \(j\) is made for all cations. The vacancies (\(Va\)) are assumed to hold a charge that is equal to the average charge of the cation sublattice (\(Q\)). The summation of the constituent fractions on each sublattice is unity. The Gibbs energy of the liquid phase described above can be written as:  
\[G^{liq}={}^{ref}G^{liq}+{}^{il}G^{liq}+{}^{ex}G^{liq} \tag{4}\]  
\[{}^{ref}G^{liq}=\sum_{C}\sum_{A}y_{C}y_{A}\ ^{\circ}G^{liq}_{(C)(A)}+Qy_{Va} \sum_{C}y_{C}\ ^{\circ}G^{liq}_{C}+Q\sum_{B}y_{B}\ ^{\circ}G^{liq}_{B} \tag{5}\]  
\[{}^{id}G^{liq}=RT\Bigg{[}P\sum_{C}y_{C}\ln\ y_{C}+Q\Bigg{(}\sum_{A}y_{A}\ln\ y_{A}+y_{Va}\ln\ y_{Va}+\sum_{B}y_{B}\ln y_{B}\ \Bigg{)}\Bigg{]} \tag{6}\]\[ \begin{array}{ll}{}^{ex}G^{liq}=\sum\limits_{C_{1}}\sum\limits_{C_{2}}&\sum \limits_{A}y_{C_{1}}y_{C_{2}}y_{A}\;L_{(C_{1},C_{2})(A)}^{liq}+\sum\limits_{C_{1 }}y_{C_{1}}y_{C_{2}}y_{Va}^{2}\;L_{(C_{1},C_{2})(Va)}^{liq}\\ &+\sum\limits_{C}\sum\limits_{A_{1}}\sum\limits_{A_{2}}y_{C}y_{A_{1}}y_{A_{2}} \;L_{(C)(A_{1},A_{2})}^{liq}+\sum\limits_{C}\sum\limits_{A}y_{C}y_{A}y_{Va}\;L_ {(C)(A,Va)}^{liq}\\ &+\sum\limits_{C}\sum\limits_{A}\sum\limits_{B}y_{C}y_{A}y_{B}\;L_{(C)(A,B)}^{ liq}+\sum\limits_{C}\sum\limits_{B}y_{C}y_{B}y_{Va}\;L_{(C)(Va,B)}^{liq}\\ &+\sum\limits_{B_{1}}\sum\limits_{B_{2}}y_{B_{1}}y_{B_{2}}\;L_{(B_{1},B_{2})}^ {liq}\end{array}