# Analytic Model for U-Nb Liquidus and U-6 Nb Melting Curve
## 4 Generalization of the Theoretical Model to Nonzero \(P\): Melting Curve of U-6 Nb as an Example
### The \(P=0\) Value of \(dT_{m}(P)/dP\) for U-6 Nb from the Clausius-Clapeyron Formula  
Here, we again use the CC formula to calculate the initial slope of the U-6 Nb melting curve independently for comparison with that in Equation (15). The density of liquid U-6 Nb at its ambient \(T_{m}\) can be determined from the experimental data of [PERSON] [14]: \(15.57\pm 0.08\) g/cm\({}^{3}\); the corresponding \(V_{\rm liq}\) is \(13.98\pm 0.07\) cm\({}^{3}\)/mol. There are no experimental data on the density of solid U-6 Nb at its ambient \(T_{m}\). The formula for the molar volume of U-6 Nb as a function of \(T\) in Ref. [37] is based on the empirical relation \(\alpha(X)=(1-X)\alpha_{\rm U}+X\alpha_{\rm Nb}\), which is in disagreement with the thermodynamically based Formula (5). Hence, the value of \(\rho(\)U-6 Nb\()\) of \(\simeq 16.2\) g/cm\({}^{3}\) and the corresponding \(V_{\rm sol}\simeq 13.43\) cm\({}^{3}\)/mol that Ref. [37] suggests may not be accurate enough. We carried out the QMD simulations of the EOS of both solid and liquid U-6 Nb at a fixed \(T\) of 1600 K, virtually the ambient \(T_{m}\) of U-6 Nb (of 1577.6 K). We obtained several values of \(P\) that correspond to several fixed volumes of the 250-atom (\(5\times 5\times 5\)) computational supercell (ordered for solid and disordered for liquid), and then determined the size of the supercell that corresponds to \(P=0\) (and \(T=1600\) K) by interpolating between all the values. We found solid and liquid densities of, respectively, 16.235 and 15.708 g/cm\({}^{3}\), and the corresponding \(V_{\rm sol}\) and \(V_{\rm liq}\) of, respectively, 13.405 and 13.855 cm\({}^{3}\)/mol. Hence, \(\Delta V_{m}\equiv V_{\rm liq}-V_{\rm sol}=0.45\) cm\({}^{3}\)/mol. Note that our value of the solid density is consistent with that from [37], while the difference between our value of the liquid density and that of [14] is within 1%. Essentially, the same values of the solid density at the ambient \(T_{m}\) result from interpolating either between the \(\rho=\rho(P_{m})\) values in Table 3 as \(\rho(P_{m}=0)\), or between its \(\rho=\rho(T_{m})\) values as \(\rho(T_{m}=1577.6~{}K):16.230\) and 16.251 g/cm\({}^{3}\), respectively (the difference between the latter and our 16.235 is 0.1%).  
As noted by [PERSON] [11], because the U-6 Nb alloy melts over a range of \(T\), its true enthalpy of melting cannot be defined in a thermodynamically rigorous manner. Instead, two variants of this quantity can be considered: (i) the change in enthalpy that transpires  
Figure 3: The melting curves of (i) U-6 Nb (orange) given by the new analytic model with \(C=-0.6\) fixed (Equation (7)), (ii) pure U (blue, Equation (12)) as the best fit to the six QMD melting datapoints in Table 1 shown as blue bullets, and (iii) pure Nb (green, Equation (10)). The QMD melting datapoints of U-6 Nb in Table 3 are shown as orange squares, for comparison with Equation (7). Also shown are the melting curve of U-18 Nb (gray, Equation (19)), the experimental ambient melting point of Pa (exp. datum), and the QMD melting datapoint of Pa (gray square with error bars), which are all discussed in what follows.  
over the entire melting range, and (ii) the enthalpy difference between the extrapolated single-phase solid and liquid lines of \(H=H(T)\) at a single \(T\) halfway between the liquidus and solidus \(T\)s. The corresponding two values are 16.36 and 12.62 kJ/mol. We therefore adopt \(\Delta H_{m}=14.49\pm 1.87\) kJ/mol as the enthalpy of melting of U-6 Nb. Then, via the SS formula, the initial slope of the U-6 Nb melting curve is \(dT_{m}/dP=T_{m}\cdot\Delta V_{m}/\Delta H_{m}=1577.6\cdot 0.45/(14.49\pm 1.87)=48.99 \pm 6.32\) K/GPa. Thus, the initial slope of the U-6 Nb melting curve is \(\approx\)49 K/GPa, in essential agreement with both Equations (15) and (18).