Density-functional study of the thermodynamic properties and the pressure–temperature phase diagram of Ti

Abstract

The phonon spectra of $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\omega}$ Ti were studied using the supercell approach. The lattice vibrational energy was calculated in the quasiharmonic approximation using both first-principles phonon density of state and Debye model. The thermal electronic contribution to the free energy was evaluated from the integration over the electronic density of states. The Helmholtz energy was thus obtained by combining them with the 0 K total energy calculated within the framework of all-electron projector-augmented-wave method. The thermodynamic properties of $\ensuremath{\alpha}$ and $\ensuremath{\omega}$ Ti calculated by phonon and Debye model are very close to each other. The predicted enthalpy, entropy, bulk modulus, thermal expansion coefficient and heat capacity of $\ensuremath{\alpha}$ are in good agreement with experiments. By comparing with the experimental enthalpy of $\ensuremath{\beta}$, we found that the 0 K total energy calculated from bcc Ti is incorrect. This problem can be solved by shifting the total energy of $\ensuremath{\beta}$ down by $8\text{ }\text{kJ}\text{ }{\text{mol}}^{\ensuremath{-}1}$ to match the experimental value. With the Gibbs energy calculated from the Debye model as a function of pressure and temperature, the phase transformation conditions of $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\omega}$, $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\beta}$, and $\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\omega}$ were identified. The predicted transition temperature between $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ at ambient pressure and the triple point are close to experiments. It was found that the entropy plays an important role in the $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\alpha}$ and $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\beta}$ transitions, and the thermal electronic contribution to the Gibbs energy cannot be neglected for studying the $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\beta}$ transition. Our calculations also showed that zero-point energy is crucial to predict the transition pressure of Ti at low temperatures.