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.