# Analytic Model for U-Nb Liquidus and U-6 Nb Melting Curve
## 2 Theoretical Model for Melting Curve (Liquidus) of a Mixture  
The theoretical model for the melting curve (liquidus) of a two-component mixture was developed in our previous paper [15]. Such a mixture can be a compound, or an alloy, etc. Even a porous material can be considered as a mixture of a regular substance with air. Our approach is based on several assumptions. First, the mixture is of the form \(A_{1-x}B_{x}\), \(0\leq x\leq 1\), and no stoichiometric \(A_{n}B_{m}\) (both \(m\) and \(n\) are integers \(\geq 1\)) compound exists. Second, the ideal mixing occurs; that is, the volume of the mixture is equal to the sum of the volumes of its constituents (we briefly dwell on this in what follows). Third, both of the melting curves of pure constituents are known in the analytic form of the pressure \((P)\) dependence of the melting point: \(T_{m}=T_{m}(P)\). This form can be either a simple polynomial fit, e.g., quadratic or cubic, or a more sophisticated Simon-Glatzel (SG) form \(T_{m}(P)=T_{m}(0)(1+P/a)^{b}\), where \(T_{m}(0)\) is the ambient \(T_{m}\) and \(a,b=\text{const}\), etc. The knowledge of the two melting curves implies that no other information (e.g., EOS) is available on the constituents of the mixture. Finally, both constituents of the mixture are molten and are in a state of thermal equilibrium with each other at a common temperature of \(T_{\ell}\) (in what follows, we use the subscript \(\ell\) instead of \(m\), in order to associate the melting point with the liquidus and to avoid confusion with the solidus). The thermal EOS mixing rules that we also assume to hold are based on the following two assumptions: First, the thermal EOSs of constituents 1 and 2 of a binary mixture and of the mixture itself are of the form  
\[P_{1}(\rho,T)=P_{1}(\rho,0)+\alpha_{1}B_{T,1}\,T,\,\,\,\,P_{2}(\rho,T)=P_{2}( \rho,0)+\alpha_{2}B_{T,2}\,T,\,\,\,\,P(\rho,T)=P(\rho,0)+\alpha B_{T}\,T, \tag{1}\]  
where \(\alpha\) and \(B_{T}\) are, respectively, the corresponding thermal expansion coefficient and the isothermal bulk modulus at temperature \(T\). As our previous studies demonstrate, the thermal EOS of the form (1) holds for many substances; see Refs. [16; 17] for Pt as an example. In fact, along the corresponding melting curves, the thermal EOS of the form (1) is virtually exact in every case considered. And second, at \(T=0\), the two constituents are at pressure equilibrium with each other; that is, the cold (\(T=0\)) counterparts of the two EOSs satisfy the system of equations  
\[P_{1}(\rho_{1},0)\,=\,P_{2}(\rho_{2},0) \tag{2}\]  
and  
\[\frac{1-x}{\rho_{1}}\,+\,\frac{x}{\rho_{2}}\,=\,\frac{1}{\rho}, \tag{3}\]  
where \(\rho_{1}\) and \(\rho_{2}\) are, respectively, the densities of constituents 1 and 2, \(\rho\) is the density of the mixture, and \(x\) is the (fractional) mass percentage of constituent 2 (without any loss of generality, we consider constituent 1 as a host and constituent 2 as a dopant): \(x=M_{2}/(M_{1}+M_{2}),1-x=M_{1}/(M_{1}+M_{2})\). Hence, Equation (3) is equivalent to \(M_{1}/\rho_{1}+M_{2}/\rho_{2}=(M_{1}+M_{2})/\rho\); i.e., the total volume of the mixture is the sum of the volumes of its constituents, which is the case of ideal mixing, one of the assumptions that our theoretical model is based on.