First-principles investigation of metal-hydride phase stability: The Ti-H system

Various factors that affect metal-hydride phase stability are investigated from first principles. As a particular example, we consider hydride stability in the Ti-H system, exploring the role of configurational degrees of freedom, zero-point vibrational energy as well as coherency strains. The tetragonal $\gamma \text{−}\mathrm{Ti}\mathrm{H}$ phase is predicted (within generalized gradient approximation) to be unstable relative to hcp Ti ($\alpha$ phase) and the fcc based $\delta \text{−}\mathrm{Ti}{\mathrm{H}}_2$. Zero-point vibrational energy significantly affects the formation energies in this system and makes the $\gamma$ phase even less stable relative to hcp Ti and $\delta \text{−}\mathrm{Ti}{\mathrm{H}}_2$. The effect of stress and strain on the stability of the $\gamma$ phase is also investigated showing that coherency strains between hydride precipitates and the hcp Ti matrix stabilize $\gamma \text{−}\mathrm{Ti}\mathrm{H}$ relative to $\alpha \text{−}\mathrm{Ti}$ and $\delta \text{−}\mathrm{Ti}{\mathrm{H}}_2$. We also find that hydrogen prefers octahedral sites at low hydrogen concentration and tetrahedral sites at high concentration. Both harmonic vibrational as well as electronic origins for the cubic to tetragonal phase transformation of $\mathrm{Ti}{\mathrm{H}}_2$ are investigated, and we argue that anharmonic vibrational degrees of freedom are likely to play an important role in stabilizing cubic $\mathrm{Ti}{\mathrm{H}}_2$.

Figure 1

(Color online) Formation energies of $\mathrm{Ti}{\mathrm{H}}_{3x}$ for fcc and hcp based hydrogen-vacancy configurations with respect to fcc Ti and $\delta \text{−}\mathrm{Ti}{\mathrm{H}}_2$.

Figure 2

Variation of the energy of $\mathrm{Ti}{\mathrm{H}}_{2-z}$ with $c∕a$ ratio (at constant volume) for (a) $\mathrm{Ti}{\mathrm{H}}_2$, (b) $\mathrm{Ti}{\mathrm{H}}_{1.75}$, and (c) $\mathrm{Ti}{\mathrm{H}}_{1.5}$. A $c∕a$ ratio equal to 1 corresponds to the cubic phase.

Figure 3

Effective cluster interactions for the cluster expansion of the hydrogen-vacancy configurational energy over the interstitial sites of fcc Ti. The clusters include an empty and two point clusters (not shown), six pairs, four triplets, and five quadruplets.

Figure 4

Calculated phase diagrams of $\mathrm{Ti}{\mathrm{H}}_{3x}$: (a) considering relative stability over the fcc-Ti host only and (b) considering relative stability over both the hcp and fcc hosts.

Figure 5

Free energies of fcc Ti, $\alpha \text{−}\mathrm{Ti}$ (hcp), $\gamma \text{−}\mathrm{Ti}\mathrm{H}$, and $\delta \text{−}\mathrm{Ti}{\mathrm{H}}_2$ at (a) $300\mathrm{K}$, (b) $450\mathrm{K}$, and (c) $600\mathrm{K}$.

Figure 6

Hydrogen concentration within tetrahedral (solid lines) and octahedral (dashed lines) sites of the fcc-Ti host at (a) $300\mathrm{K}$ and (b) $600\mathrm{K}$.

Figure 7

Formation energies of various configurations of $\mathrm{Ti}{\mathrm{H}}_{3x}$ with (squares) and without (circles) zero-point vibrational energies. Reference states are fcc Ti and $\delta \text{−}\mathrm{Ti}{\mathrm{H}}_2$. The circled energies correspond to configurations with exclusively octahedral occupancy. All other energies in the enlarged plot (bottom) correspond to configurations with exclusively tetrahedral occupancy.

Figure 8

Free energies of $\mathrm{Ti}{\mathrm{H}}_2$ as a function of $c∕a$ ratio, calculated with Fermi broadening of the electron distribution at various temperatures.

Figure 9

Constrained free energies of $\mathrm{Ti}{\mathrm{H}}_2$ as a function of $c∕a$ ratio, calculated within the harmonic approximation using a first-principles parametrized spring model. The free energies at finite temperature are shifted to fit in one plot.

Figure 10

(Color online) (a) The experimentally observed (Ref. 23) habit plane between $\alpha \text{−}\mathrm{Ti}$ (hcp) and coherent $\gamma \text{−}\mathrm{Ti}\mathrm{H}$ precipitates. Empty and black circles are Ti atoms of successive (0001) and (001) planes of the hcp and fcc hosts and green circles are tetrahedrally coordinated hydrogen atoms. (b) Schematic illustration of the orientation of the axis of tension relative to the hcp and fcc hosts used to investigate the role of stress on hydride phase stability.

Figure 11

$0\mathrm{K}$ phase diagrams as a function of hydrogen chemical potential and stress (see Fig. 10 for the orientation of the applied stress relative to the various crystals). (a) Phase diagram in the absence of coherency strains (i.e., equilibrium lattice parameters perpendicular to the axis of tension). (b) Phase diagram accounting for coherency strains that exist across the habit plane of Fig. 10b.