Thermodynamic stabilization of $\gamma \text{−}\mathrm{U}-\mathrm{Mo}$ alloys: Effect of Mo content and temperature

The $\gamma \text{−}\mathrm{U}-\mathrm{Mo}$ alloys have been studied by means of ab initio molecular-dynamic simulations at 900 K as a function of the Mo concentration (0%, 25%, 50%, 75%, and 100%). Using the temperature-dependent effective potential method with the symmetry-imposed force constant extension, we obtained the vibrational, thermodynamic, and elastic properties of U–Mo, including anharmonicity and disorder. We show that the addition of Mo and temperature effects are responsible for the stabilization of the $\gamma$ phase of U–Mo alloys in the range of 0–25 % of Mo concentration above 900 K. The vibrational entropy is found to be very large compared to its value in other alloys, about half of the configurational entropy counterpart, and to act against the thermodynamic stabilization of the solid solution. The vibrational and thermodynamic properties are strongly impacted by the disorder, due to the large differences between mass and interatomic bonds of uranium and molybdenum. The increase of elastic shear modulus as a function of the Mo concentration also indicates a mechanical stabilization of the solid solution and validates the thermodynamic findings.

Figure 1

Phase diagram of U–Mo, adapted from Ref. [2].

Figure 2

Schematic representation of the different methods used. The circles and the lines represent the atoms and the interatomic force constants (IFCs), respectively. (a) TDEP for an ideal crystal, (b) TDEP-VCA for an alloy with average mass and IFC, (c) SIFC-MD for an alloy with mass disorder and average IFC, and (d) SIFC-MDTD for an alloy with mass disorder and type-dependent IFC. In (c) and (d), to avoid finite-size effects, the IFCs are mapped onto a bigger supercell.

Figure 3

Phonon band structure of molybdenum at 300 K (purple) and 900 K (red). Experimental results are from [50].

Figure 4

Phonon band structure of uranium, computed with TDEP at 900 K and DFPT, respectively.

Figure 5

Evolution of the excess ground-state energy with respect to the molybdenum concentration. The results have been interpolated using cubic spline. The full (dashed) lines are results without (with) relaxation of the SQS, blue (purple) lines are results with (without) inclusion of SOC. FPLMTO-SQS (orange) results are extracted from [9].

Figure 6

Evolution of the temperature-dependent part of the excess free energy $-T\mathrm{\Delta }S$ with respect to molybdenum concentration at 900 K. The green dashed line corresponds to computation with configurational entropy only. On the black lines, the vibrational entropy is added to the configurational entropy, with the three different approximations (VCA, SIFC-MD, and SIFC-MDTD).

Figure 7

Evolution of the excess free energy with respect to molybdenum concentration at 900 K, obtained by adding the excess entropy to the excess ground-state energy, including SOC. The green dashed curve corresponds to computation including configurational entropy only. On the black curves, the vibrational entropy is added to the configurational entropy, with the different approximations. The blue curve is the excess ground-state energy from Fig. 5, including SOC.

Figure 8

Phonon DOS and pDOS of ${\mathrm{U}}_{1-x}{\mathrm{Mo}}_x$ at 900 K with VCA (dashed lines), SIFC-MD (dash-dotted lines), and SIFC-MDTD (solid lines). The shaded green area corresponds to the DOS computed with SIFC-MDTD.

Figure 9

Evolution of the excess vibrational entropy with respect to molybdenum concentration at 900 K, computed within the SIFC-MDTD method. The configurational entropy is plotted as the green dashed curve for comparison. The blue (red) curve corresponds to the uranium (molybdenum) partial vibrational entropy, and the black curve is the total vibrational entropy.

Figure 10

Evolution of elastic constants (a) and moduli (b) with respect to the molybdenum concentration at 900 K. The continuous lines are the elastic properties computed with the type-dependent IFC, while the dashed lines are computed with the type-independent IFC.