First-Principles Calculations of Free Energies of Unstable Phases: The Case of fcc W

Ab initio molecular dynamics simulations are used to solve the long-standing problem of calculating the free energies of unstable phases, such as fcc W. We find that fcc W is mechanically unstable with respect to long-wavelength shear at all temperatures considered ($T>2500\mathrm{K}$), while the short-wavelength phonon modes are anharmonically stabilized. The calculated fcc-bcc enthalpy and entropy differences at $T=3500\mathrm{K}$ (308 meV and $0.74k_B$ per atom, respectively) agree well with the recent values derived from analysis of experimental data.

Figure 1

Calculated stress difference in Eq. (1) at $T=2500\mathrm{K}$ as a function of tetragonal shear $\gamma$. The solid line represents a polynomial fit to the directly calculated data points using Eq. (2), excluding bcc 32-atom $\gamma =0.85$ and 0.9, and bcc 54-atom $\gamma =0.95$ points.

Figure 2

Calculated $T=2500\mathrm{K}$ free energy and internal electronic energy of W along the Bain path. The free energy was obtained from Eq. (1), while the internal energy was calculated directly (empty squares) and fit to a polynomial obtained by integrating Eq. (2) with respect to $\gamma$ (dashed line).

Figure 3

Calculated fcc-bcc differences in the free energy $\Delta F$ (filled squares), total entropy $\Delta S$ (empty squares), enthalpy $\Delta H$ (filled circles), and electronic entropy $\Delta S_{\mathrm{el}}$ (empty circles) for W.

Figure 4

Comparison of the DFT fcc-bcc energy and entropy differences (empty red squares) with previous results obtained from CALPHAD assessments of thermodynamic data (filled blue symbols).