High pressure-temperature phase diagram and equation of state of titanium

The high pressure-temperature behavior of titanium has been studied with x-ray diffraction in resistively heated and laser-heated diamond anvil cells up to 200 GPa and $\sim 3500$ K. The stability fields of $\alpha \text{−}\mathrm{Ti},\omega \text{−}\mathrm{Ti},\beta \text{−}\mathrm{Ti},\gamma \text{−}\mathrm{Ti}$, and $\delta \text{−}\mathrm{Ti}$ have been determined in this range. $\gamma \text{−}\mathrm{Ti}$ and $\delta \text{−}\mathrm{Ti}$, which had been evidenced earlier under nonhydrostatic compression, are also observed in helium pressure transmitting medium. Equation-of-state parameters are proposed for $\alpha \text{−}\mathrm{Ti}$ and $\omega \text{−}\mathrm{Ti}$ at 300 K, and $\beta \text{−}\mathrm{Ti}$ at high temperature. The stability fields of the $\alpha ,\omega ,\gamma$, and $\delta$ phases are also studied using the projector-augmented wave method based on density-functional theory. Using the relevant core radius to avoid overlapping between atomic spheres, and relaxing cells and atomic positions, we show that all those phases have a stability domain at 0 K. We explain why $\gamma \text{−}\mathrm{Ti}$ and $\delta \text{−}\mathrm{Ti}$ were calculated to be unstable in earlier works. In addition, a new phase, called ${\delta }^{\prime }$-Ti, which is a distortion of $\delta \text{−}\mathrm{Ti}$, is predicted to form between 80 and 120 GPa and below $\simeq 200$ K.

Figure 1

Tentative phase diagram of titanium showing the solid phases reported under high pressure. The gray lines are from Ref. [6]. The stability domains of $\gamma \text{−}\mathrm{Ti}$ and $\delta \text{−}\mathrm{Ti}$ are from Refs. [3] and [4]. The crystal structure unit cells of $\beta \text{−}\mathrm{Ti},\omega \text{−}\mathrm{Ti},\gamma \text{−}\mathrm{Ti}$, and $\delta \text{−}\mathrm{Ti}$ are represented.

Figure 2

Phase boundaries between $\alpha \text{−}\mathrm{Ti},\beta \text{−}\mathrm{Ti}$, and $\omega \text{−}\mathrm{Ti}$ according to the literature [1, 24] and the present study performed under hydrostatic pressurizing conditions (runs 1 and 4). The horizontal bars represent the domain of coexistence of $\alpha \text{−}\mathrm{Ti}$ and $\omega \text{−}\mathrm{Ti}$. The phase boundary (black dashed-double dotted line) is based on the current measurements and the triple point location [1].

Figure 3

XRD patterns of $\alpha \text{−}\mathrm{Ti},\beta \text{−}\mathrm{Ti}$, and $\omega \text{−}\mathrm{Ti}$ recorded during run 5. NaCl (pressure medium) diffraction lines are marked with red dots.

Figure 4

XRD patterns of $\beta \text{−}\mathrm{Ti},\gamma \text{−}\mathrm{Ti}$, and $\delta \text{−}\mathrm{Ti}$ recorded during run 7. KCl (pressure medium) diffraction lines are marked with red dots and rhenium gasket diffraction lines are marked with green asterisks.

Figure 5

$P\text{−}T$ conditions at which XRD data points have been recorded for solid titanium. The temperature has been measured by pyrometry. The pressure has been measured using the pressure gauges specified in Table 1. The symbols' color and shape indicate, respectively, the phase of Ti and run number. The blue dashed lines are tentative phase boundaries based on the current data and literature low pressure studies [1]; they are compared with the boundaries proposed in Ref. [6]. The dots indicate the stable phases according to present DFT calculations (see Sec. 3b).

Figure 6

Calculated enthalpy differences of $\omega ,\gamma$, and $\delta$ phases with respect to the $\beta$ phase as a function of pressure for two PAW atomic datasets: with a large radius (1.2 Å), LCP, on the right panel and with a small radius (1.0 Å), SCP, on the left panel, see text for more details.

Figure 7

Calculated pressure dependence of the lattice parameters, ratios, and internal parameter $y$ of the $\omega ,\gamma ,{\delta }^{\prime },\delta$, and $\beta$ phases, for the SCP atomic dataset $\left(r_{\text{PAW}}=1.0\AA \right)$. The dashed line corresponds to Kutepov et al. [9] results for the $\delta$ phase. The open diamonds correspond to the present experimental ratios for the $\delta$ phase.

Figure 8

Schematic representation of a possible $\gamma$ to ${\delta }^{\prime }$ transition in titanium.

Figure 9

Phonon spectra of $\delta$ (left panel) and ${\delta }^{\prime }$ (center panel) titanium at 300 K (black lines) and 1000 K (red dashed lines) at 110 GPa. The red thick line is the low-frequency optical branch around the $\Gamma$ point calculated at 1000 K. The right panel shows the calculated Gibbs free-energy difference between ${\delta }^{\prime }$ and $\delta$ phases as a function of temperature at 110 GPa.

Figure 10

(a) $P\text{−}V$ points measured at 300 K in helium pressure medium for $\alpha \text{−}\mathrm{Ti},\omega \text{−}\mathrm{Ti},\gamma \text{−}\mathrm{Ti}$, and in KCl pressure medium for $\delta \text{−}\mathrm{Ti}$. Different symbols correspond to different experimental runs. The compression curve of $\omega \text{−}\mathrm{Ti}$ has been fitted with a Rydberg-Vinet EoS (see text). The present DFT-GGA calculations, Ref. [4] measurements, and Ref. [6] model are also plotted. Inset: evolution of the $c/a$ ratio in $\omega \text{−}\mathrm{Ti}$ with pressure; same symbols as in the main graph. (b) Difference between measured and fit volume for $\omega \text{−}\mathrm{Ti}$.

Figure 11

Lattice parameter of $\beta \text{−}\mathrm{Ti}$ vs temperature. The data and the model (see text) are plotted as circles and lines, respectively. Inset: Grüneisen parameter $\gamma$ vs pressure at 300 K, calculated using Eq. (3) with $A=0.0452{\AA }^{-3}$ and $B=0.7$.