Evaluating the phase stability of binary titanium alloy Ti-X (X = Mo, Nb, Al, and Zr) using first-principles calculations and a Debye model

To realize bottom-up design of alloys based on theoretical calculations, the thermodynamic stabilities of phases in Ti binary alloys were estimated by a combination of density functional theory calculations for the internal enthalpy energy, the Bragg-Williams approximation for the mixing entropy contribution, the Debye model for the vibrational free energy, and the Sommerfeld model for the electronic excitation entropy. The special quasirandom structure (SQS) model was used to describe the disordered distribution of the alloying element in the solid solution state. We focused on Ti–Mo, Ti–Nb, Ti–Al, and Ti–Zr binary alloys, which have different phases, such as the α phase in the hexagonal close-packed (hcp) structure and the β phase in the body-centered cubic (bcc) structure, depending on the temperature and alloying element fraction. The elastic constants, bulk modulus, and Poisson's ratios were calculated using a strain energy method. Excitations from the vibrational contribution to the quasi-harmonic Debye approximation were added to the 0 K free energy originally derived from ab initio calculations. The effect of temperature up to 1000 K on phase stability was analyzed. Furthermore, to compare phase stabilities, the free energies of formation were calculated using the ground states of the constituent phases as references. The calculated elastic property indicated the mechanical instability of most bcc Ti–Al and bcc Ti–Zr alloys, hcp Ti–Mo and hcp Ti–Nb at high fraction range. The SQS supercell models showed good agreement in elastic constant, bulk modulus, and Poisson's ratio compared to the previous experimental and theoretical results. Free energy results showed that Mo and Nb are β-phase stabilizers, Al is an α-phase stabilizer, and Zr is a neutral element. As the fraction of the alloying element changed, stabilizing or destabilizing effects were observed under different temperatures. Moreover, the linear relationship between the filling of the d band and phase stability was identified in low temperature range. For the β phase, Mo had a stronger stabilizing effect than Nb; both Mo and Nb destabilized the α phase at low temperatures, whereas high temperatures increased the stability of the α phase and the temperature effect became more significant than the element effect. In the examined temperature range, the α phase Ti–Al alloys were stable at all Al fractions, where the thermal effect was negligible. All the α Ti–Zr alloys in this study had similar stabilities to their constituent phases (hcp Ti and hcp Zr) over a wide temperature range.