Analytic Model for U-Nb Liquidus and U-6Nb Melting Curve

Abstract

Uranium–niobium (U-Nb) alloys, used in a variety of industrial and energy applications that require high density, ductility, and good corrosion resistance, comprise a highly complex, multiphasic system with a phase diagram well established through decades of extensive experimental and theoretical research. They are also one of the best candidates for a metallic fuel alloy with high-temperature strength sufficient to support the core, acceptable nuclear properties, good fabricability, and compatibility with usable coolant media. The key factor determining the performance and safety of a metallic fuel such as U-Nb is its operational limits in the application environment, which are closely related to material’s structure and thermodynamic stability. They are in turn closely related to the ambient (zero-pressure) melting point $\left(T_m\right);$ thus, $T_m$ is an important engineering parameter. However, the current knowledge of $T_m$ of the U-Nb system is limited, as the only experimental study of its Nb-rich portion dates back to 1958. In addition, it has not yet been adequately modeled based on general thermodynamics principles or using an equation-of-state approach. In this study, we present a theoretical model for the melting curve (liquidus) of a mixture, and apply it to U-Nb, which is considered as a mixture of pure U and pure Nb. The model uses the known melting curves of pure constituents as an input and predicts the melting curve of their mixture. It has only one free parameter, which must be determined independently. The ambient liquidus of U-Nb predicted by the model appears to be in good agreement with the available experimental data. We calculate the melting curve (the pressure dependence of $T_m$) of pure U using ab initio quantum molecular dynamics (QMD), the knowledge of which is required for obtaining the model parameters for U. We also generalize the new model to nonzero pressure and consider the melting curve of U-6 wt.% Nb (U-6Nb) alloy as an example. The melting curve of U-6Nb alloy predicted by the model appears to be in good agreement with the ab initio melting curve obtained from our QMD simulations. We suggest that the U-18Nb alloy can be considered as a proxy for protactinium (Pa) and demonstrate that the melting curves of U-18Nb and Pa are in good agreement with each other.

1. Introduction

The results of numerous irradiation studies which have been performed on metallic fuels indicate that there is a need for a metallic fuel alloy with high-temperature $\left(T\right)$ strength sufficient to support the core, acceptable nuclear properties, good fabricability, and compatibility with usable coolant media. In an effort to improve ductility and corrosion resistance, elements like titanium, zirconium, niobium, and molybdenum have been used to alloy uranium (U) [1,2,3]. However, among those, only niobium (Nb) does not form intermediate (stoichiometric) compounds with U; hence, Nb has been traditionally used more than the other three. Also, the addition of Nb significantly increases the oxidation resistance of U, because Nb is distributed uniformly in the U matrix [4]. Studies of the early 1960s of the phase diagram of the U-Nb system indicated that this system was a logical choice for a high-T fuel [5]. However, the original idea of the requirement for strength at elevated Ts, which automatically removed uranium-rich alloys from consideration and concentrated on the niobium-rich alloys instead, e.g., the 20 weight percent (wt.%) uranium–80 wt.% niobium (U-80Nb) alloy considered in [5], was later on reconsidered in favor of the use of the U-rich alloys as a high-T fuel. Specifically, U-Nb alloys in the vicinity of the monotectoid composition of 6 wt.% Nb (U-6Nb), namely, within the range from 5 wt.% (U-5Nb) to 8 wt.% (U-8Nb), exhibit the shape memory effect [1,2,6], which depends on strain conditions and $T.$ Specifically, the U-6Nb alloy can recover up to 7% strain in tension by heating the material back into the high-T $\gamma$ phase [6].

Another aspect of alloying U with Nb is the extension of the T range of the stability of its high-T body-centered cubic (bcc) structure, referred to as the $\gamma$ phase, which is stable at (in K) $1049\le T\le T_m=1408$ relative to that of the low-T orthorhombic (oS4) structure, referred to as the $\alpha$ phase, which is stable in the 0–941 K range [7]. The intermediate tetragonal (tP30) structure of U, which is referred to as the $\beta$ phase, is stable between 941 and 1049 K. From the material processing standpoint (machining, etc.), $\gamma$-U is much more preferable to $\alpha$ because of the brittle nature of the latter and its tendency to form uranium oxides. In fact, U-6Nb is stable in the bcc structure, also referred to as the $\gamma$ phase, already at $T\stackrel{>}{\sim }600$ K. However, as the alloy is cooled down rapidly to room $T,$ it undergoes a two-stage transformation process: at $\sim 570$ K, the material exhibits the distorted cubic (i.e., tetragonal) structure referred to as the ${\gamma }^0$ phase. At a lower T of ∼370–450 K, the alloy undergoes further transformation into the metastable monoclinic ${\alpha }^{″}$ structure. These transformations are discussed in detail in Refs. [8,9]. With a Nb concentration of ∼60 to 100 wt.%, the U-Nb system forms a continuous solid solution of the bcc $\left(\gamma \right)$ structure in the entire T range [10].

Of all the physical properties of a material, melting behavior is a fundamental property closely related to its structure and thermodynamic stability, and therefore has always been a crucial research subject. The ambient (zero-$P,$ where P stands for pressure) melting point $\left(T_m\right)$ is also an important engineering parameter, as it defines the operational limits of a material in its application environment. It becomes critical in nuclear engineering where the thermo-mechanical stability of a nuclear fuel element is a key factor determining fuel performance and safety, and in this respect, the U-Nb system is no exception.

At high $T,$ U and Nb form a continuous solid solution without any stoichiometric compound. The ambient phase diagram of the U-Nb system is summarized in Refs. [10,11]. The high-T Nb-rich portion of this phase diagram is based on the experimental results of Rogers et al. of the late 1950s [12,13], the accuracy of which has not been assessed. These old results have not been updated since then. The high-T U-rich portion of the U-Nb phase diagram can be considered as reliable, since the results of several experimental studies that it is based on, including those of Drotning of 1982 [14], as well as the more recent unpublished LANL MST-6/Sigma data from Thoma [11], are consistent with each other within experimental uncertainties. Nonetheless, taking into account a potential uncertainty associated with the high-T Nb-rich portion of the U-Nb phase diagram, the knowledge of the (ambient) melting behavior of the U-Nb system remains incomplete. In particular, it has not yet been adequately modeled based on general thermodynamics principles or using an equation-of-state (EOS) approach. The present article directly addresses this deficiency.

Thus, the clarification of the behavior of the ambient $T_m$ of U-Nb alloy as a function of the Nb content requires further study. Here, we present a theoretical model for the melting curve (liquidus) of a mixture and apply it to the U-Nb system, which is being considered as a mixture of pure U and Nb. To this end, we calculate the melting curve (the P dependence of $T_m$) of U using the so-called Z method implemented with quantum molecular dynamics (QMD) simulations with the first-principles-based code VASP. We also consider the generalization of this model to a nonzero P and derive the melting curve of U-6Nb as an example.

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 $A_{1-x}{B}_x,0\le x\le 1,$ and no stoichiometric $A_n{B}_m$ (both m and n are integers $\ge 1$) 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 $\left(P\right)$ dependence of the melting point: $T_m=T_m\left(P\right).$ This form can be either a simple polynomial fit, e.g., quadratic or cubic, or a more sophisticated Simon–Glatzel (SG) form $T_m\left(P\right)=T_m\left(0\right){(1+P/a)}^b,$ where $T_m\left(0\right)$ is the ambient $T_m$ and $a,b=\mathrm{const},$ 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 ℓ 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

$$P_1(\rho ,T)=P_1(\rho ,0)+{\alpha }_1{B}_{T,1}T,P_2(\rho ,T)=P_2(\rho ,0)+{\alpha }_2{B}_{T,2}T,P(\rho ,T)=P(\rho ,0)+\alpha B_{T}T,$$

(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

$$P_1({\rho }_1,0)=P_2({\rho }_2,0)$$

(2)

and

$${\displaystyle \frac{1-x}{{\rho }_1}}+{\displaystyle \frac{x}{{\rho }_2}}={\displaystyle \frac{1}{\rho }},$$

(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. The thermodynamic formulation based on the “specific” Gibbs energy function $G=(1-x)G_1+x{G}_2$ [15] leads to the relations

$${\displaystyle \frac{1}{\rho B_T}}={\displaystyle \frac{1-x}{{\rho }_1{B}_{T,1}}}+{\displaystyle \frac{x}{{\rho }_2{B}_{T,2}}}$$

(4)

and

$${\displaystyle \frac{\alpha }{\rho }}={\displaystyle \frac{(1-x){\alpha }_1}{{\rho }_1}}+{\displaystyle \frac{x{\alpha }_2}{{\rho }_2}}$$

(5)

which, when divided by each other, result in [15]

$$\alpha B_T=(1-x){\alpha }_1{B}_{T,1}+x{\alpha }_2{B}_{T,2}+(1-x)x\left({\alpha }_1{B}_{T,1}-{\alpha }_2{B}_{T,2}\right)\left({\displaystyle \frac{\rho B_T}{{\rho }_1{B}_{T,1}}}-{\displaystyle \frac{\rho B_T}{{\rho }_2{B}_{T,2}}}\right).$$

(6)

We now consider a binary mixture at temperature equilibrium at the liquidus point $(P,T_{\ell })$ and assume that each of its constituents is described by the thermal EOS of the form (1); therefore, ($P_{\ell ,1}\equiv P_1({\rho }_1,T_{\ell }),P_{\ell ,2}\equiv P_2({\rho }_2,T_{\ell }),P_{\ell }\equiv P(\rho ,T_{\ell })$)

$$\alpha B_T{T}_{\ell }=P_{\ell }-P,{\alpha }_1{B}_{T,1}{T}_{\ell }=P_{\ell ,1}-P,{\alpha }_2{B}_{T,2}{T}_{\ell }=P_{\ell ,2}-P.$$

Then, the use of these three relations in Equation (6) multiplied by $T_{\ell }$ leads to $(P$ cancels out)

$$P_{\ell }=(1-x)P_{\ell ,1}+x{P}_{\ell ,2}+(1-x)xC\left(P_{\ell ,1}-P_{\ell ,2}\right),$$

(7)

where $C\equiv \rho B_T/\left({\rho }_1{B}_{T,1}\right)-\rho B_T/\left({\rho }_2{B}_{T,2}\right).$ Equation (7) is our formula for the liquidus of a binary mixture. Since both $P_{\ell ,1}=P_{\ell ,1}\left(T\right)$ and $P_{\ell ,2}=P_{\ell ,2}\left(T\right)$ are assumed to be available, provided that the value of C is known, this formula gives the value of the melting P of a mixture at any given melting temperature $T_{\ell }^{*},$ via $P_{\ell ,1}^{*}=P_{\ell ,1}\left(T_{\ell }^{*}\right)$ and $P_{\ell ,2}^{*}=P_{\ell ,2}\left(T_{\ell }^{*}\right).$

To assign a physical meaning to $C,$ we associate it with the strength of an “effective” interaction between the two constituents of a binary mixture. In this sense, C is an effective “coupling” parameter for a pair of constituents of a mixture. In the case of an N-component mixture, where $N>2,$ a set of $C_{ij}$ for each of the $N(N-1)/2$i-j pairs of constituents should be introduced, and Equation (7) should be modified accordingly. Among the properties of these $C_{ij}$s are (i) their absolute values, which are of order 0.1–1 and (ii) that they satisfy the antisymmetry condition $C_{ij}=-C_{ji};$ in our case of a binary mixture, this condition manifests itself in the invariance of Equation (7) under the simultaneous permutation $1\leftrightarrow 2,x\leftrightarrow 1-x,$ and $C\leftrightarrow -C.$ The derivation of the analog of Equation (7) for an N-component mixture, $N>2,$ goes beyond the scope of this work and will be presented elsewhere.

2.1. Liquidus of a Mixture at $P=0$ and Small P

At low P, both $T_{\ell ,1}\left(P\right)$ and $T_{\ell ,2}\left(P\right)$ can be approximated by simple linear forms:

$$T_{\ell ,1}\left(P\right)=T_{\ell ,1}\left(0\right)+T_{\ell ,1}^{\prime }·P,T_{\ell ,2}\left(P\right)=T_{\ell ,2}\left(0\right)+T_{\ell ,2}^{\prime }·P,$$

or $P_{\ell ,1}\left(T\right)=(T-T_{\ell ,1}\left(0\right))/T_{\ell ,1}^{\prime }\left(0\right),P_{\ell ,2}\left(T\right)=(T-T_{\ell ,2}\left(0\right))/T_{\ell ,2}^{\prime }\left(0\right),$ where $T_{\ell ,1}\left(0\right),T_{\ell ,2}\left(0\right),T_{\ell ,1}^{\prime }\left(0\right)$, and $T_{\ell ,2}^{\prime }\left(0\right)$ are the corresponding ambient melting points and the initial slopes of the melting curves. Using these expressions in Equation (7) leads to

$$T_{\ell }\left(0\right)={\displaystyle \frac{\frac{(1-x)T_{\ell ,1}\left(0\right)}{T_{\ell ,1}^{\prime }\left(0\right)}+\frac{x{T}_{\ell ,2}\left(0\right)}{T_{\ell ,2}^{\prime }\left(0\right)}+(1-x)xC\left(\frac{T_{\ell ,1}\left(0\right)}{T_{\ell ,1}^{\prime }\left(0\right)}-\frac{T_{\ell ,2}\left(0\right)}{T_{\ell ,2}^{\prime }\left(0\right)}\right)}{\frac{1-x}{T_{\ell ,1}^{\prime }\left(0\right)}+\frac{x}{T_{\ell ,2}^{\prime }\left(0\right)}+(1-x)xC\left(\frac{1}{T_{\ell ,1}^{\prime }\left(0\right)}-\frac{1}{T_{\ell ,2}^{\prime }\left(0\right)}\right)}}$$

(8)

and

$${\displaystyle \frac{1}{T_{\ell }^{\prime }\left(0\right)}}={\displaystyle \frac{1-x}{T_{\ell ,1}^{\prime }\left(0\right)}}+{\displaystyle \frac{x}{T_{\ell ,2}^{\prime }\left(0\right)}}+(1-x)xC\left({\displaystyle \frac{1}{T_{\ell ,1}^{\prime }\left(0\right)}}-{\displaystyle \frac{1}{T_{\ell ,2}^{\prime }\left(0\right)}}\right),$$

(9)

which are the expressions for the ambient melting point and the initial slope of the melting curve of a mixture, respectively. Hence, the analytic form of the liquidus of a mixture at low P is

$$T_{\ell }\left(P\right)=T_{\ell }\left(0\right)+T_{\ell }^{\prime }\left(0\right)·P,$$

where $T_{\ell }\left(0\right)$ and $T_{\ell }^{\prime }\left(0\right)$ are determined, respectively, by Equations (8) and (9).

The example of the Mo-W system considered below demonstrates how the value of C is determined in practice.

2.2. Example: The Mo-W System

As an example of the practical use of our new analytic model, let us consider a mixture of molybdenum and tungsten which, according to [18], form a continuous solid solution without any eutectics. According to studies in the literature [19], the melting curves of Mo and W are, respectively, $T_m^{\mathrm{Mo}}\left(P\right)=2896{(1+P/36.6)}^{0.43}$ and $T_m^{\mathrm{W}}\left(P\right)=3695{(1+P/41.8)}^{0.50}.$ Hence, in Equation (8), $T_{\ell ,1}\left(0\right)=2896,T_{\ell ,1}^{\prime }\left(0\right)=34.0,$ and $T_{\ell ,2}\left(0\right)=3695,T_{\ell ,2}^{\prime }\left(0\right)=44.2.$ Then, the best fit of the form (8) with the above parameters to the experimental data of Ref. [18] brings up $C=-0.4.$ In Figure 1, the resulting $T_{\ell }\left(x\right)$ is compared to the liquidus of ${\mathrm{Mo}}_{1-x}{\mathrm{W}}_x$ from experiments.

In this case of Mo-W, a variation in C by as much as 60%, $C=-0.4\pm 0.24,$ causes a shift in the model liquidus within $\sim 50$ K, or $\sim 1.5$% only. Specifically, the largest deviations from the $C=-0.4$ liquidus are +49.7 K at $x=0.564$ with $C=-0.64,$ and −51.4 K at $x=0.563$ with $C=-0.16,$ compared to $T_{\ell }=3392.0$ K at $x=0.57$ with $C=-0.4.$ Thus, the exact knowledge of the value of C may not be really necessary for the model to produce the liquidus of a mixture in good agreement with experiment.

3. Application to the Uranium–Niobium System

We now apply Formula (8) to the uranium–niobium system. Note that, as the previous example of the Mo-W system shows, in a general case, the value of C can be determined from fitting $T_m\left(0\right)$ of the functional form (8) to the experimental and/or theoretical data on the liquidus of a mixture. In the case of U-Nb, we will determine the value of C by fitting to the experimental data on the U-rich portion of the U-Nb liquidus of Ref. [14]. We consider U as a host and Nb as a dopant. The other parameter values required for the application of Equation (8) to U-Nb, which we use for the determination of the values of C, are $T_{\ell ,1}\left(0\right),$ $T_{\ell ,1}^{\prime }\left(0\right),$ and $T_{\ell ,2}\left(0\right),T_{\ell ,2}^{\prime }\left(0\right).$ Given that the latter pair of values follows directly from the published melting curve of Nb of Ref. [20],

$$T_{\ell }^{\mathrm{Nb}}\left(P\right)=2750{\left(1+{\displaystyle \frac{P}{22.6}}\right)}^{0.30},$$

(10)

the following can be deduced:

$$T_{\ell ,2}\left(0\right)=2750\mathrm{K},T_{\ell ,2}^{\prime }\left(0\right)=36.5\mathrm{K}/\mathrm{GPa}.$$

(11)

However, while the value of the ambient $T_m$ of U is known with certainty, $1407\pm 2$ K [21], its $P=0$ value of $d{T}_m\left(P\right)/dP$ has been ambiguous. According to Fokin [22], its value can be as low as ∼30 and as high as ∼180 K/GPa, a difference of a factor of 6. Some typical literature values are (in K/GPa) ≈31 from the best fit to the low-P experimental data of [23], ≈33 from the best fit to the experimental data to ∼50 GPa of Ref. [24], ≈50 from the fit to the data of [24] with the fixed value of $T_m\left(0\right)=1408$ K, and 40.5 [25] from the melting curve of U obtained by the Lindemann approach using the Grüneisen parameter given by the analytic model of Burakovsky and Preston [26,27]. Based on these three references alone, the ambient value of $d{T}_m\left(P\right)/dP$ of U lies in the interval ∼30–50 K/GPa and cannot be chosen unambiguously. In order to proceed, we need to re-evaluate this value more accurately using the most contemporary research tools available. Such tools are QMD simulations using the Z method.

3.1. Melting Simulations of $\gamma$-U

Here, we present the results of our QMD simulations of the melting of U, considered as its bcc modification $\gamma$-U, which were carried out using the Z method implemented with VASP. The Z method is a procedure used to calculate the melting curves of materials by modeling a solid system at different initial total energies. The idea of the method was put forward by Belonoshko et al. in Ref. [28]. A solid system is modeled in an NVE ensemble (N is the number ot atoms, and V and E are the volume and total energy of the system, respectively). In each simulation, E is controlled by the corresponding choice of the initial $T.$ Essentially, the Z method simulates a solid in a superheated limit. The solid will melt spontaneously if the simulation is long enough, and T will drop to $T_m$ (at $P=P_m$). The shape of the P-T trajectory mapped out by the final states of simulations with different initial Ts resembles the letter Z, which is the origin of the method’s name. The Z method has been used in numerous studies of melting curves, e.g., of Zn [29], Sn [30], and Pt [17], just to name a few. The practical implementation of the Z method can be found in Refs. [28,31,32]. The typical time evolution of P and T in a QMD run is shown, with an example of Pa considered in what follows.

In our simulations of the melting of U, its electronic structure was represented by [Xe ${\mathrm{4f}}^{14}{\mathrm{5d}}^{10}$] ${\mathrm{5f}}^3{\mathrm{6s}}^2{\mathrm{6p}}^6{\mathrm{6d}}^1{\mathrm{7s}}^2$ (14 valence electrons). For the electron exchange and correlation functionals, the Perdew–Burke–Ernzerhof (PBE) implementation of generalized gradient approximation (GGA) was used, as suggested in Ref. [33].

A large 432-atom $(6\times 6\times 6)$ supercell with a single $\Gamma$-point was used to avoid potential size effects associated with the mechanical instability of bcc at low T and its high-T stabilization. As our (unpublished) study of Ti, Zr, and Hf, which all melt from bcc and have bcc mechanically unstable at low T but stabilized at high $T,$ just like in U, shows, the critical supercell sizes needed to avoid the size effects related to bcc stabilization are $686(7\times 7\times 7)$ for both Ti and Zr but 432 for Hf. Thus, the critical supercell size seems to decrease with the atomic number; hence, we assume that a 432-atom supercell is adequate for the simulations of U. We considered six different cases of the $\gamma$-U unit cells, which correspond to six different U densities and six different melting points on the U melting curve. In each of the six cases, the corresponding supercell was subject to a set of initial temperatures $\left(T_{\mathrm{init}}\right)$ separated by an increment of 150 K in the first case and 250 K in the remaining five cases, and run with QMD in the $NVE$ ensemble, for a total of up to 15,000 time steps of 2 fs each, i.e., up to 30 ps, to determine the corresponding values of $P_m$ and $T_m.$ Uncertainty of the value of $T_m$ intrinsic to the Z method is therefore 75 K in the first case and 125 K in the remaining five cases, half of the corresponding increment in $T_{\mathrm{init}}$ [32], which constitutes 3–9% of $T_m$ (the largest uncertainty of 9% is for the first datapoint, and the lowest one of 3% for the last one). The uncertainties of $P_m$ are negligibly small, within 0.5 GPa. The results of our melting simulations are summarized in

Table 1. The six ab initio melting points of $\gamma$-U, of the form $(P_m,T_m\pm \Delta T_m),$ obtained from the Z method implemented with VASP, along with the values of lattice constants and densities at the corresponding $T_m$.

Lattice Constant (Å)	Density (${\mathrm{g/cm}}^3$)	${\mathit{P}}_{\mathit{m}}$ (GPa)	${\mathit{T}}_{\mathit{m}}$ (K)	$\mathbf{\Delta }{\mathit{T}}_{\mathit{m}}$ (K)
3.70	15.607	−9.5	810	75
3.50	18.438	6.1	1690	125
3.35	21.027	29.7	2540	125
3.25	23.028	55.5	3260	125
3.20	24.125	72.8	3670	125
3.15	25.292	93.9	4120	125

.

Fitting the SG functional form to the six melting datapoints in Table 1 results in

$$T_{\ell }^{\mathrm{U}}\left(P\right)=1408{\left(1+{\displaystyle \frac{P}{14.9}}\right)}^{0.54},$$

(12)

which is the melting curve of U in the interval of P that corresponds to the six datapoints in Table 1. Therefore, $(1408·0.54/14.9=51.03),$

$$T_{\ell ,1}\left(0\right)=1408\mathrm{K},T_{\ell ,1}^{\prime }\left(0\right)=51.0\mathrm{K}/\mathrm{GPa}.$$

(13)

Interestingly, the numerical values of the parameters of the best fit to the experimental data of [24], with the fixed value of $T_m\left(0\right)=1408$ K, $T_m^{\mathrm{U}}\left(P\right)=1408{(1+P/13.0)}^{0.46}$ (for which $T_{\ell ,1}^{\prime }\left(0\right)=49.8$ K/GPa), are very close to those of Equation (12).

3.2. The $P=0$ Value of $d{T}_m\left(P\right)/dP$ for U from the Clausius–Clapeyron Formula

To validate the value of the initial slope of the U melting curve, 51 K/GPa, given by Equation (13), we calculate this value independently, using the Clausius–Clapeyron (CC) formula, which is valid at the first-order phase transition point:

$${\displaystyle \frac{d{T}_{\mathrm{tr}}\left(P\right)}{dP}}{|}_{P_{\mathrm{tr}}}={\displaystyle \frac{\Delta V_{\mathrm{tr}}}{\Delta S_{\mathrm{tr}}}}=T_{\mathrm{tr}}{\displaystyle \frac{\Delta V_{\mathrm{tr}}}{\Delta H_{\mathrm{tr}}}},$$

where $\Delta V_{\mathrm{tr}},\Delta S_{\mathrm{tr}}$, and $\Delta H_{\mathrm{tr}}=T_{\mathrm{tr}}\Delta S_{\mathrm{tr}}$ are, respectively, the volume, entropy, and enthalpy changes across the transition point $(P_{\mathrm{tr}},T_{\mathrm{tr}}).$ In our case of the melting of $\gamma$-U, the corresponding values of $\Delta V_{\mathrm{m}}$ and $\Delta H_{\mathrm{m}}$ are, respectively, $0.3624{\mathrm{cm}}^3$/mol [34] or $0.2874{\mathrm{cm}}^3$/mol [35], and $8.47\pm 1.0$ kJ/mol [21]. Note that the estimate of Ref. [21] does not take into account the value $\Delta H_{\mathrm{m}}=9.142$ kJ/mol of Ref. [36], which was later on corrected by Hackenberg [11]: 9.190 kJ/mol. With this value, the estimate of Ref. [21] would have been $8.6\pm 0.8$ kJ/mol, which we adopt as the value of $\Delta H_{\mathrm{m}}$ for the melting of U. Thus,

$$\Delta V_{\mathrm{m}}=0.325\pm 0.04{\mathrm{g}/\mathrm{cm}}^3,\Delta H_{\mathrm{m}}=8.6\pm 0.8\mathrm{kJ}/\mathrm{mol}.$$

(14)

Therefore, the CC formula gives $d{T}_{\mathrm{m}}{\left(P\right)/dP|}_{P=0}=53.2\pm 8.2$ K/GPa, which our value of 51.0 K/GPa in Equation (13) is in full agreement with.

3.3. The U-Nb Liquidus

We now apply Formula (8) to the U-Nb system, as the values of two $T_{\ell }$s and two $T_{\ell }^{\prime }$s are available from Equations (11) and (13). As noted before, in the case of U-Nb, we determine the value of C from fitting to the most reliable experimental data of Ref. [14] on the U-rich portion of the U-Nb liquidus. The fitting brings up the value of $C=-0.6.$ Therefore, as $\left|C\right|<1,$ there is no local extremum of the U-Nb liquidus (see Ref. [15] for more detail); i.e., such local extrema are at the two endpoints.

The liquidus curve of the U-Nb system is determined by Equation (8) with $C=-0.6$ and the values of two $T_{\ell }$s and two $T_{\ell }^{\prime }$s from (11) and (13). It is shown in Figure 2, along with the experimental data of Refs. [13,14] and the best fit by Hackenberg et al. [11] to the two sets of the experimental data shown in Figure 2. We note that the Calphad-type modeling of the U-Nb system by Turchi [37] results in the U-Nb liquidus being very close to that of Hackenberg shown in Figure 2. However, such a Calphad-type modeling requires the adjustment of the values of too many free parameters, which makes its practical use more complicated and more time-consuming.

Again, just like in the Mo-W case considered above, we determined that for the U-Nb system, a variation in C by as much as 50%, i.e., $C=-0.6\pm 0.3,$ causes a shift in the model’s liquidus within $\sim 100$ K, or $\sim 4$%. Specifically, the largest deviations from the $C=-0.6$ liquidus are +93.6 K at $x=0.414$ with $C=-0.9,$ and −89.2 K at $x=0.412$ with $C=-0.3,$ compared to $T_{\ell }=2265.0$ K at $x=0.413$ with $C=-0.6.$ Thus, the exact knowledge of the value of C may not be really necessary to obtain the liquidus of a mixture in good agreement with experiment.

For the specific case of U-6Nb considered below, the new analytic model gives, via Equations (8) and (9), respectively,

$$T_{\ell }^{\mathrm{U}-6\mathrm{Nb}}\left(0\right)=1577.6\mathrm{K},T_{\ell }^{\prime \mathrm{U}-6\mathrm{Nb}}\left(0\right)=49.167\mathrm{K}/\mathrm{GPa}.$$

(15)

4. Generalization of the Theoretical Model to Nonzero $\mathit{P}$: Melting Curve of U-6Nb as an Example

Formula (8), applied above to the U-Nb system, represents the particular $P=0$ case of the more general Formula (7) that predicts the liquidus of the mixture at any $P\ne 0$ if both $P_{\ell ,1}$ and $P_{\ell ,2}$ (as functions of T), as well as the value of C as a function of P, are all available. Note that if the melting curve of a constituent is given by the SG form $T\left(P\right)=T\left(0\right){(1+P/a)}^b,$ $P_{\ell }$ is the analytic function of $T:P_{\ell }=a\left({[T/T\left(0\right)]}^{1/b}-1\right).$ As regards $C,$ we assume that C is a very weak function of $P.$ In other words, C can be (approximately) considered as a constant over some interval of $P.$ In general, this interval of P should not be expected to be very wide; its (effective) width must be determined by the differences in the values of the EOS parameters of the constituents such as $\alpha ,B_T,$ etc.

As an example, consider the application of Equation (7) to U-6Nb, where the two $P_{\ell }$s are determined by the corresponding SG forms (10) and (12), and $C(\approx \mathrm{const})=-0.6.$ With $x=0.06$ (6 wt.% Nb), we numerically solve Equation (7) at the seven Ts between 1600 (close to $T_{\ell }^{\mathrm{U}-6\mathrm{Nb}}\left(0\right)=1577.6$ K) and 4000 K, with an increment of 400 K. The seven $(P_{\ell },T_{\ell })$ points are listed in

Table 2. The eight $(P_{\ell },T_{\ell })$ melting points of U-6Nb given by Equation (7) with $C=-0.6,$ and $P_{\ell }^{\mathrm{Nb}}$ and $P_{\ell }^{\mathrm{U}}$ from Equations (10) and (12), respectively.

$P_{\ell }$ (GPa)	0	0.43	9.23	19.9	32.6	47.3	64.1	83.2
$T_{\ell }$ (K)	1577.6	1600	2000	2400	2800	3200	3600	4000

along with the ambient melting point.

The best fit of the SG form to the eight datapoints in Table 2 results in the model melting curve of U-6Nb:

$$T_{\ell }^{\mathrm{U}-6\mathrm{Nb}}\left(P\right)=1577.6{\left(1+{\displaystyle \frac{P}{16.73}}\right)}^{0.5214},$$

(16)

for which $T_{\ell }^{\prime \mathrm{U}-6\mathrm{Nb}}\left(0\right)=1577.6·0.5214/16.73=49.167$ K/GPa, in agreement with (15).

4.1. Melting Simulations of U-6Nb

As a consistency check for Equation (16), we carried out QMD melting simulations of U-6Nb using the Z method implemented with VASP. As discussed above, the addition of Nb enhances the thermal stability of $\gamma$-U. Hence, a smaller bcc supercell can be used for the QMD simulations of the melting of U-6Nb. Specifically, for these simulations, we used a 250-atom $(5\times 5\times 5)$ U-6Nb bcc supercell. Since for Nb, 6 wt.% corresponds to ≈14 atomic % [3], the 35 atoms of U chosen at random were substituted with 35 atoms of Nb; thus, the supercell consisted of 215 atoms of U and 35 atoms of Nb. As before, the electronic structure of U was represented by [Xe ${\mathrm{4f}}^{14}{\mathrm{5d}}^{10}$] ${\mathrm{5f}}^3{\mathrm{6s}}^2{\mathrm{6p}}^6{\mathrm{6d}}^1{\mathrm{7s}}^2$ (14 valence electrons), while that of Nb by [Ar ${\mathrm{3d}}^{10}$] ${\mathrm{4s}}^2{\mathrm{4p}}^6{\mathrm{4d}}^4{\mathrm{5s}}^1$ (13 valence electrons). The PBE implementation of GGA was used for both U and Nb. Just as for $\gamma$-U, we considered six different cases of the U-6Nb unit cells, which correspond to six different densities and six different melting points on the U-6Nb melting curve. In each of the six cases, the corresponding supercell was subject to a set of initial Ts separated by an increment of 150 K in the first case and 250 K in the remaining five cases, and run with QMD in the $NVE$ ensemble, for a total of up to 20,000 time steps of 1.5 fs each, i.e., up to 30 ps. The results of our melting simulations are summarized in

Table 3. The six ab initio melting points of U-6Nb, of the form $(P_m,T_m\pm \Delta T_m),$ obtained from the Z method implemented with VASP, along with the values of lattice constants and densities at the corresponding $T_m$.

Lattice Constant (Å)	Density (${\mathrm{g/cm}}^3$)	${\mathit{P}}_{\mathit{m}}$ (GPa)	${\mathit{T}}_{\mathit{\ell }}$ (K)	$\mathbf{\Delta }{\mathit{T}}_{\mathit{\ell }}$ (K)
3.70	14.269	−11.0	860	75
3.50	16.858	4.3	1780	125
3.35	19.225	26.5	2550	125
3.25	21.055	50.5	3170	125
3.20	22.057	66.6	3520	125
3.15	23.125	86.1	3900	125

.

Just as in the case of pure $\gamma$-U considered above, the uncertainties of the values of $T_m$ are within 3% for the last datapoint and 9% for the first one, and uncertainties of $P_m$ are negligibly small, within 0.5 GPa.

Fitting the SG form to the six melting datapoints in Table 2 results in

$$T_{\ell }^{\mathrm{U}-6\mathrm{Nb}}\left(P\right)=1563.6{\left(1+{\displaystyle \frac{P}{15.0}}\right)}^{0.48},$$

(17)

which is the ab initio melting curve of U-6Nb in the interval of P that corresponds to the six dataponts in Table 3, and for which $T_{\ell }^{\prime \mathrm{U}-6\mathrm{Nb}}\left(0\right)\approx 50.0$ K/GPa. If the value of $T_{\ell }\left(0\right)$ is fixed at 1577.6 K, in agreement with (15), the fitting brings up

$$T_{\ell }^{\mathrm{U}-6\mathrm{Nb}}\left(P\right)=1577.6{\left(1+{\displaystyle \frac{P}{15.4}}\right)}^{0.48},$$

(18)

which is only slightly different from Equation (17). The value of the initial slope of this melting curve (in K/GPa), $1577.6·0.48/15.4=49.172,$ virtually coincides with that in (15).

The melting curve of U-6Nb given by the new analytic model, Equation (7), with fixed $C=-0.6$, is shown in Figure 3. The melting curves of both pure U and pure Nb are also shown in Figure 3 for comparison.

As Figure 3 clearly demonstrates, the agreement between the new model generalized to nonzero P (with a fixed value of C) and the ab initio results is very good. This justifies the assumption that keeping the value of C fixed over some range of P is a good approximation. In our case, keeping $C=-0.6$ fixed seems to work for pressures up to at least ∼50 GPa, and the model curve and QMD results start to deviate from each other at higher $P,$ but the deviation is rather small, ∼300 K at ∼90 GPa, which constitutes ∼7.5%. Possible ways to improve the accuracy of the model in regard to its melting curve predictions are discussed below in the concluding remarks.

4.2. The $P=0$ Value of $d{T}_m\left(P\right)/dP$ for U-6Nb from the Clausius–Clapeyron Formula

Here, we again use the CC formula to calculate the initial slope of the U-6Nb melting curve independently for comparison with that in Equation (15). The density of liquid U-6Nb at its ambient $T_m$ can be determined from the experimental data of Drotning [14]: $15.57\pm 0.08{\mathrm{g/cm}}^3;$ the corresponding $V_{\mathrm{liq}}$ is $13.98\pm 0.07{\mathrm{cm}}^3$/mol. There are no experimental data on the density of solid U-6Nb at its ambient $T_m.$ The formula for the molar volume of U-6Nb as a function of T in Ref. [37] is based on the empirical relation $\alpha \left(X\right)=(1-X){\alpha }_{\mathrm{U}}+X{\alpha }_{\mathrm{Nb}}$, which is in disagreement with the thermodynamically based Formula (5). Hence, the value of $\rho$(U-6Nb) of $\simeq 16.2{\mathrm{g/cm}}^3$ and the corresponding $V_{\mathrm{sol}}\simeq 13.43{\mathrm{cm}}^3$/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-6Nb at a fixed T of 1600 K, virtually the ambient $T_m$ of U-6Nb (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 $15.708 {\mathrm{g/cm}}^3,$ and the corresponding $V_{\mathrm{sol}}$ and $V_{\mathrm{liq}}$ of, respectively, 13.405 and $13.855 {\mathrm{cm}}^3$/mol. Hence, $\Delta V_m\equiv V_{\mathrm{liq}}-V_{\mathrm{sol}}=0.45{\mathrm{cm}}^3$/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 $\rho =\rho \left(P_m\right)$ values in Table 3 as $\rho (P_m=0),$ or between its $\rho =\rho \left(T_m\right)$ values as $\rho (T_m=1577.6\mathrm{K}):$ 16.230 and $16.251 {\mathrm{g/cm}}^3,$ respectively (the difference between the latter and our 16.235 is 0.1%).

As noted by Hackenberg [11], because the U-6Nb 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 over the entire melting range, and (ii) the enthalpy difference between the extrapolated single-phase solid and liquid lines of $H=H\left(T\right)$ at a single T halfway between the liquidus and solidus Ts. 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-6Nb. Then, via the SS formula, the initial slope of the U-6Nb melting curve is $d{T}_m/dP=T_m·\Delta V_m/\Delta H_m=1577.6·0.45/(14.49\pm 1.87)=48.99\pm 6.32$ K/GPa. Thus, the initial slope of the U-6Nb melting curve is ≈49 K/GPa, in essential agreement with both Equations (15) and (18).

5. U-18Nb as a Proxy for Protactinium

Another much less known application of the U-Nb system is the ability of U-18 wt.% Nb (U-18Nb) to serve as a proxy for protactinium. Protactinium (Pa), element 91, falls between thorium and uranium and is one of the more intractable natural radioactive elements. Protactinium is one of the rarest and most expensive naturally occurring elements. The average concentrations of Pa in the Earth’s crust is typically on the order of a few parts per trillion, but may reach up to a few parts per million in some uraninite ore deposits. It is only comparatively recently that a reasonably clear understanding of its physical and chemical properties has begun to emerge. Pa is typically obtained through isolation from spent nuclear fuel, typically uranium materials such as U-Nb, and subsequent purification. However, separating Pa from U-Nb alloy, which, as mentioned above, is a common material in the nuclear fuel cycle, is challenging due to the chemical similarity of Pa and Nb, as the two, along with vanadium and tantalum, share the same vertical group in the periodic table. Because of all this, Pa is extremely difficult to study in in situ experiments, and theoretical approaches should be invoked.

It appears that having the values of density, bulk modulus, and ambient melting temperature all very close to those of Pa, the U-18 wt.% Nb (U-18Nb) alloy is a natural candidate for being a proxy for protactinium. Indeed, the $T=0$ density of Pa is ≈$15.5 {\mathrm{g/cm}}^3$ (≈15.4 at 300 K) [7], its (ambient) bulk modulus is in the interval 100–157 GPa [38], and the three experimental measurements cited in [21], $1848\pm 20,1833\pm 30,$ and $1838\pm 20$ K, give the ambient $T_m$ of Pa of $1840.5\pm 22.5$ K, or 1818–1863 K. For U-18Nb, the corresponding values are as follows: (i) a density of $15.69 {\mathrm{g/cm}}^3,$ from Equation (3), with $x=0.18$ and the literature values of ${\rho }_{\mathrm{U}}=19.14$ and ${\rho }_{\mathrm{Nb}}=8.620;$ (ii) a bulk modulus of 132.9 GPa, from Equation (4), with $x=0.18$ and the literature values of $B_{\mathrm{U}}=120.0$ and $B_{\mathrm{Nb}}=170.5;$ and (iii) the ambient $T_m$ of 1863.0 K from Equation (8). Thus, the two densities agree within 1%, the bulk modulus of U-18Nb is right in the middle of the experimental interval of that of Pa, and the ambient $T_m$ of U-18Nb is at the upper end of the experimental interval of that of Pa. It is therefore quite plausible to consider U-18Nb as a proxy for Pa, at least in regard to its melting curve, i.e., the behavior of $T_m$ as a function of $P.$

We now consider the application of Equation (7) to U-18Nb, where the two $P_{\ell }$s are determined by the corresponding SG forms (10) and (12), and C (≈const) = $-0.6.$ With $x=0.18$ (18 wt.% Nb), we numerically solve Equation (7) at the seven Ts between 2000 (close to $T_{\ell }^{\mathrm{U}-6\mathrm{Nb}}\left(0\right)=1863.0$ K) and 4400 K with an increment of 400 K. The seven $(P_{\ell },T_{\ell })$ points are listed in Table 3 along with the ambient melting point.

The best fit of the SG form to the eight datapoints in

Table 4. The eight $(P_{\ell },T_{\ell })$ melting points of U-18Nb given by Equation (7) with $C=-0.6,$ and $P_{\ell }^{\mathrm{Nb}}$ and $P_{\ell }^{\mathrm{U}}$ from Equations (10) and (12), respectively.

$P_{\ell }$ (GPa)	0	3.01	13.2	25.4	39.9	56.9	76.6	94.2
$T_{\ell }$ (K)	1863.0	2000	2400	2800	3200	3600	4000	4400

results in the model melting curve of U-18Nb:

$$T_{\ell }^{\mathrm{U}-18\mathrm{Nb}}\left(P\right)=1863.0{\left(1+{\displaystyle \frac{P}{19.67}}\right)}^{0.4866},$$

(19)

for which $T_{\ell }^{\prime \mathrm{U}-18\mathrm{Nb}}\left(0\right)=1863.0·0.4866/19.67=46.087$ K/GPa, which virtually coincides with 46.083 given by Equation (9). This melting curve is shown as a gray line in Figure 3.

It is interesting to compare this melting curve of U-18Nb with the results of QMD melting simulations on Pa, which U-18Nb is presumably a proxy for.

Melting Simulations of Pa

The high-T crystal structure of Pa is bcc; hence, the QMD melting simulations were carried out for this structure. These simulations were performed in the same way as described above for $\gamma$-U. The electronic structure of Pa was represented by [Hg] ${\mathrm{5f}}^2{\mathrm{6p}}^6{\mathrm{6d}}^1{\mathrm{7s}}^2$ (11 valence electrons). Just as for $\gamma$-U, the PBE implementation of GGA was employed, and a 432-atom $(6\times 6\times 6)$ supercell with a single $\Gamma$-point was used to avoid potential size effects associated with the mechanical instability of bcc at low $T,$ which is also present in Pa, and its high-T stabilization.

We obtained a single melting point of bcc-Pa with a lattice constant of 3.225 Å (a density of 22.8756$\approx 22.9{\mathrm{g/cm}}^3$). The corresponding supercell was subject to a set of initial temperatures $\left(T_{\mathrm{init}}\right)$ separated by an increment of 625 K and run with QMD in the $NVE$ ensemble, for a total of up to 10,000 time steps of 2 fs each, i.e., up to 20 ps, to determine the corresponding values of $P_m$ and $T_m.$

Figure 4 and Figure 5 demonstrate the time evolution of, respectively, T and P during the QMD runs with three initial temperatures, $T_0.$ The intermediate run with $T_0$ = 11,250 K is the melting run during which the system melts, and simultaneously, its temperature decreases from ∼5800 K for a superheated solid down to ∼4800 K for a liquid, and its pressure increases from ∼96.8 GPa for a superheated solid up to ∼97.3 K for a liquid.

Our QMD simulations result in the melting point of bcc-Pa of (P in GPa, T in K) (97.3, 4790). Since the increment in $T_0$ in our simulations is 625 K, the error bar of $T_m$ is 312.5 K (the error bar of P is of order 0.1 GPa).

At $P=97.3$ GPa, the melting curve of U-18Nb, Equation (19), gives 4435.8 K, within 7.5% of the QMD value of 4790 K. However, if the error bars are taken into account, the model value of 4435.8 K is within 1% of 4477.5 K at the lower error bar, which Figure 3 clearly demonstrates. We therefore conclude that U-18Nb can indeed be considered as a reliable proxy for Pa because its melting curve is a good approximation for that of Pa; in fact, it represents the lower bound on the melting curve of Pa. Most likely, U-18Nb can be used to approximate other physical properties of Pa such as its equation of state, principal Hugoniot, etc.

6. Methods

Here is a summary of the theoretical methods used in this work.

• The Z method implemented with VASP for the ab initio calculation of melting curves, presented in detail in Section 3.1.
• Melting simulations of $\gamma$-U using the Z method, presented in Section 3.1.
• Melting simulations of U-6Nb using the Z method, presented in Section 4.1.
• Melting simulations of Pa using the Z method, presented in the Melting Simulations of Pa Section.

7. Concluding Remarks

Here, we briefly summarize the main results of this work.

In this study, we have presented a theoretical model for the melting curve (liquidus) of a mixture and applied it to the uranium–niobium system being considered as a mixture of pure U and pure Nb.

The model is based on several assumptions listed in Section 2, the use of which seems fully justified by the reliability of the results produced by the model. The advantage of our formulation over alternative theoretical approaches that may be found in the literature is that it only requires the knowledge of the melting curves of the constituents of the mixture but not of their EOSs or other thermodynamic parameters, and it only uses one free parameter, $C,$ the value of which can be easily determined based on the available experimental data. As the examples of both the Mo-W system considered in this work and the uranium–plutonium mixed oxide (MOX) system considered in our previous work [15] clearly demonstrate, large variations in the value of $C,$ by as much as 60% in either case, cause a very minor shift in the liquidus of the system within ∼50 K. Thus, as already mentioned above, the exact knowledge of the value of C may not be really necessary to obtain the liquidus of a mixture in good agreement with experiment.

The examples of the application of the model to real mixtures, Mo-W and U-Nb, considered in this work, as well as Si-Ge and MOX, considered in our previous work, clearly demonstrate that the model is reliable and relatively easy to apply in practice, in contrast to more complicated and more time-consuming Calphad calculations.

A comparison of the U-Nb liquidus given by our analytic model to the available experimental results is shown in Figure 2, which demonstrates very good agreement between the model and the available experimental data. Figure 2 represents the current knowledge of the ambient phase diagram of the U-Nb system; this knowledge may be advanced further in subsequent studies on the subject.

The example of U-6Nb of the generalization of our analytic model to nonzero P clearly demonstrates that keeping the value of C a constant over some range of P is a good approximation. In the case of U-6Nb, keeping $C=-0.6$ fixed works well for pressures up to ∼50 GPa, as Figure 3 clearly shows. At higher P, the model curve and QMD results start to deviate from each other, but the deviation is small and so agreement between the two remains good even with further increases in $P.$ In fact, below 100 GPa, the deviation of the model from QMD remains within 10%. Therefore, within the pressure range 0–100 GPa, the model should be considered as quite reliable in its ability to predict the melting curve of U-6Nb, and perhaps the melting curves of ${\mathrm{U}}_{1-x}{\mathrm{Nb}}_x$ with different alloy compositions. Further improvement in agreement between the model and experiment and/or independent theoretical calculations can be achieved via the determination of the more exact physical nature of the coupling parameter C, particularly its possible x dependence and pressure (density) dependence. Below, we present a formula for such a possible x dependence of C, which ensures an almost exact match between the model liquidus and the ambient data. It is very likely that the use of such an x-dependent C will lead to better agreement between the model melting curve and the QMD (as well as the future experimental) results.

We have shown that the U-18Nb alloy can be considered as a proxy for protactinium, an element which is extremely difficult to study experimentally. Specifically, the melting curves of U-18Nb and Pa are very close to each other, as our QMD simulations on the melting point of Pa at ∼100 GPa clearly demonstrate. It appears that the melting curve of U-18Nb can serve as the lower bound on that of Pa. This conclusion can be tested experimentally.

It is also interesting to note that the U-39Nb alloy could have been a proxy for another actinide, the current knowledge of the thermodynamic properties of which is very limited (and the knowledge of the melting curve of which is virtually absent): thorium (Th). Indeed, its cold density and ambient $T_m$ are, respectively, $12.97 {\mathrm{g/cm}}^3$ and 2232.5 K, which are both within 10% of the corresponding 11.824 and 2023 of Th. However, its bulk modulus is ∼3 times as high as that of Th, a fact that disqualifies this alloy from being a good proxy. In this respect, a much better choice would be the U-35V alloy, with a density of $10.98 {\mathrm{g/cm}}^3$ and the ambient $T_m$ of 1877 K (both within 7% of those for Th), and a bulk modulus very close to that of Th.

Real alloys may be different from the ideal mixtures considered in this work, because the presence of impurities and/or porosity may induce additional interactions between the constituents of an alloy and thus alter its melting behavior. Such scenarios can, in principle, be addressed even within the ideal mixing model considered here, after its proper generalization to the case of a multi-component mixture. Indeed, impurities can be considered as additional constituents of an alloy, while porosity can be treated as a mixture with air, as already mentioned before. We plan to carry out a practical analysis of such a real alloy in our subsequent research.

Finally, we note that the present model can be further improved by taking into account more realistic scenarios of non-ideal mixtures, such as those related to the violation of volume additivity because of effective interactions between the constituents of the mixture, as well as a possible dependence of C on $x.$ As we have seen, taking C in Equation (8) to be a constant results in a liquidus of a mixture in very good (or even excellent, in the case of ${\mathrm{Mo}}_{1-x}{\mathrm{W}}_x$) agreement with experiment. Thus, the present model should be expected to predict reliable liquidi of different binary mixtures. However, introducing an x dependence of C may improve agreement with the available experimental data and help address more exotic mixing cases, such as those in which eutectic is present. Specifically, in our case of U-Nb, an x dependence of C of $C\left(x\right)=-1.0+2.0x+0.3x^2-1.5x^3$ is such that $C\left(0\right)=-1.0$ and $C\left(1\right)=-0.2,$ i.e., $C=-0.6\pm 0.4,$ consistent with the $C=-0.6\pm 0.3$ considered in this work. However, with this $C\left(x\right)$, the resulting U-Nb liquidus curve is virtually indistinguishable from that by Hackenberg shown in Figure 2. Hence, introducing an x dependence of C does improve agreement with experiment. The examination of a possible x dependence of C in a general case, as well as the generalization of the present model to mixtures with a number of constituents larger than two (in a general case, an arbitrary number of constituents) will be undertaken in our subsequent studies on this subject.