First-principles calculation of phase equilibrium of V-Nb, V-Ta, and Nb-Ta alloys

In this paper, we report the calculated phase diagrams of V-Nb, V-Ta, and Nb-Ta alloys computed by combining the total energies of 40–50 configurations for each system (obtained using density functional theory) with the cluster expansion and Monte Carlo techniques. For V-Nb alloys, the phase diagram computed with conventional cluster expansion shows a miscibility gap with consolute temperature $T_c=1250$ K. Including the constituent strain to the cluster expansion Hamiltonian does not alter the consolute temperature significantly, although it appears to influence the solubility of V- and Nb-rich alloys. The phonon contribution to the free energy lowers $T_c$ to 950 K (about 25$\%$). Our calculations thus predicts an appreciable miscibility gap for V-Nb alloys. For bcc V-Ta alloy, this calculation predicts a miscibility gap with $T_c=1100$ K. For this alloy, both the constituent strain and phonon contributions are found to be significant. The constituent strain increases the miscibility gap while the phonon entropy counteracts the effect of the constituent strain. In V-Ta alloys, an ordering transition occurs at 1583 K from bcc solid solution phase to the V${}_2$Ta Laves phase due to the dominant chemical interaction associated with the relatively large electronegativity difference. Since the current cluster expansion ignores the V${}_2$Ta phase, the associated chemical interaction appears to manifest in making the solid solution phase remain stable down to 1100 K. For the size-matched Nb-Ta alloys, our calculation predicts complete miscibility in agreement with experiment.

Figure 1

The constituent strain energy $\Delta E_{\mathrm{CS}}^{\mathrm{eq}}(x,\widehat{k})$ of V-Nb, V-Ta, and Nb-Ta systems as a function of composition $x$ for several principal interface orientations $\widehat{k}$.

Figure 2

Comparison of the maximum of the constituent strain energy corresponding to principal interface orientations for V-Nb and V-Ta alloys with those of Mo-Ta, Cu-Au, Cu-Ag, and Ni-Au alloys from the literature (Refs. 10 and 22). The constituent strain energy scales with the magnitude of size mismatch of the alloy.

Figure 3

Nearest-neighbor bond stiffness (against stretching or bending) as a function of bond length for V-Nb alloys. Points indicate ab initio data and lines are linear fits representing the transferable force constants relation used in the calculations of the vibrational free energy.

Figure 4

Nearest-neighbor bond stiffness as a function of bond length for V-Ta alloys. Points indicate ab initio data and lines are linear fits used in the calculations of the vibrational free energy.

Figure 5

Nearest-neighbor bond stiffness as a function of bond length for Nb-Ta alloys. Points indicate ab initio data and lines are linear fits used in the calculations of the vibrational free energy.

Figure 6

Effective cluster interactions of V-Nb alloys. (a) is characteristic solely of configuration-dependent chemical interactions. (b) represents cluster expansion that includes both the chemical and constituent strain interactions.

Figure 7

Ground-state search of V-Nb alloys covering all bcc supercells containing up to 12 atoms. (a) is obtained with conventional cluster expansion. (b) is obtained with a Hamiltonian that includes the constituent strain interactions.

Figure 8

Effective cluster interactions of the V-Ta alloys. (a) is characteristic solely of configuration-dependent chemical interactions. (b) represents cluster expansion that includes both the chemical and constituent strain interactions.

Figure 9

Ground-state search of V-Ta alloys covering all bcc supercells containing up to 12 atoms. (a) is obtained with conventional cluster expansion. (b) is obtained with a Hamiltonian that includes the constituent strain to the conventional cluster expansion.

Figure 10

Effective cluster interactions of the Nb-Ta alloys.

Figure 11

Ground-state search of Nb-Ta alloys covering all bcc supercells containing up to 12 atoms.

Figure 12

The calculated phase diagram of V-Nb alloys compared with experiments. The liquidus and solidus are the experimental phase boundaries (Ref. 9). The curves with label CE represent results of the conventional cluster expansion. CS and P represent results obtained by adding the constituent strain and phonon contributions, respectively. The conventional cluster expansion predicts an appreciable miscibility gap. The phonon contribution lowers the miscibility gap temperature by about 24$\%$.

Figure 13

The calculated phase diagram of V-Ta alloys compared with experiments. The solidus, liquidus, and phase boundaries associated with the V${}_2$Ta are the experimental phase boundaries (Ref. 9). The curves with label CE represent the results of conventional cluster expansion. CS and P represent the results obtained by adding the constituent strain and phonon contributions, respectively. The horizontal line at 1073 K is incorporated to indicate that the experimental phase diagram is not available below 1073 K.

Figure 14

The calculated phase diagram of size-matched Nb-Ta alloys compared with experiments. The solidus and liquidus are experimental phase boundaries. The computed phase boundary, with label CE, lies almost on the horizontal axis indicating complete miscibility down to 40 K.