Thermodynamics of the Zr-O system from first-principles calculations

We investigate the electronic and thermodynamic properties of Zr and its oxides from first principles to elucidate phase stability in the Zr-O system. Hexagonally close-packed Zr is unusual in its ability to dissolve very high concentrations of oxygen over its interstitial octahedral sites, forming a variety of ordered suboxides that undergo both first-order and second-order phase transitions upon heating. We perform a first-principles, statistical-mechanical analysis of finite temperature phase stability of ZrO${}_x$ using a cluster expansion Hamiltonian and Monte Carlo calculations. This analysis predicts the existence of 0-K ground-state oxygen orderings at composition ZrO${}_{1/6}$, ZrO${}_{2/9}$, ZrO${}_{1/3}$, ZrO${}_{4/9}$, and ZrO${}_{1/2}$ along with evidence of an infinite sequence of ground-state suboxide orderings at intermediate oxygen concentrations consisting of different stackings of empty, $\frac{1}{3}$-filled and $\frac{2}{3}$-filled two-dimensional oxygen layers. We also predict the stability of a previously uncharacterized Zr-monoxide phase, which we label ${\delta }^{\prime }$-ZrO due to its crystallographic relation to $\delta$-TiO. The ${\delta }^{\prime }$-ZrO structure is equivalent to the high-pressure $\omega$-Zr phase but has interstitial oxygen ordering. Finally, as part of the technical implementation of our statistical mechanical study, we introduce a new algorithm to parametrize the coefficients of a cluster expansion Hamiltonian and apply a $k$-space analysis to rigorously track order-disorder phenomena at finite temperature.

Figure 1

The highly converged calculated (a) formation energy and (b) volume of various crystal structures, using the Zr pseudopotential with reference valence configuration 4$s^2$, 4$p^6$, 5$s^1$, 4$d^3$ and energy cutoff 600 eV. The reference structures are $\alpha$-Zr (without any oxygen) and $\alpha$-ZrO${}_2$. The structures labeled $\alpha$-Zr w/O are those generated based on an hcp Zr lattice, but may not be hcp after relaxation. Other notable structures are also indicated. The convex hull is shown with a black line and circles indicating the ground states.

Figure 2

The structure of the major ground-state hcp ordered phases. The gray and white circles indicate Zr and O, respectively. In the plane view, large and small gray circles represent Zr on different [0001] planes and empty squares indicate unoccupied octahedral interstitial sites. The cluster, $\Delta$, corresponding to $\sqrt{3}a\times \sqrt{3}a$ triangle triplets is indicated.

Figure 3

The DOS of the major ground-state hcp ordered phases projected onto (a) Zr and (b) O atomic orbitals, with Fermi level set to 0.

Figure 4

ECI for hcp Zr with octahedral O, with “$\times$” indicating ECI with value 0. Tabulated values of the ECI are included in Ref. 30.

Figure 5

The structure of the (a) ground-state ${\delta }^{\prime }$-ZrO phase and (b) mechanically unstable $\delta$-ZrO phase, viewed along the $c$ axis. Gray atoms are Zr and white atoms are O.

Figure 6

The band structure and DOS of the (a) mechanically unstable $\delta$-ZrO phase and (b) ground-state ${\delta }^{\prime }$-ZrO phase, with Fermi level set to 0.

Figure 7

The calculated phase diagram of hcp ZrO${}_x$. Colored areas indicate single phase regions, and uncolored areas indicate two-phase regions. Solid lines indicate first-order phase boundaries, and dashed lines indicate higher-order transitions. The region between ZrO${}_{1/6}$ and ZrO${}_{1/3}$ is shaded the same color as solid solution $\alpha$-Zr since it has no long-range order parallel to $\mathit{c}$.

Figure 8

Oxygen chemical potential for the Zr-O system. ${\mu }_{\mathrm{O}}$ is referenced to the two-phase region with Zr${}_2$O${}_3$ and $\alpha$-ZrO${}_2$. Regions with increasing ${\mu }_{\mathrm{O}}$ correspond to single phase regions, while ${\mu }_{\mathrm{O}}$ remains constant in two-phase regions.

Figure 9

The values of the simulated diffraction intensities $I_{\mathit{k}}$ for (a) ${\mathit{k}}_A=[0,0,1]$ and (b) ${\mathit{k}}_B=[\frac{1}{3},\frac{1}{3},1]$, obtained from GCMC calculations with decreasing ${\mu }_{\mathrm{O}}$ (decreasing $x$), over the range $200\text{K}\le T\le 2500$ K.

Figure 10

Analysis of the ZrO${}_{1/2}$ disordering process with increasing temperature, obtained from a series of GCMC calculations with ${\mu }_{\mathrm{O}}$ corresponding to the maximum concentration in Fig. 9 and increasing $T$ in $12\times 12\times 12$ supercells. $x_l$ is the O concentration in basal layer $l$ of the simulation cell.

Figure 11

Analysis of the ZrO${}_{1/2}$ disordering process with decreasing $x$ at constant $T=1000$ K, obtained from a series of GCMC calculations in $12\times 12\times 12$ supercells. $x_l$ is the O concentration in basal layer $l$ of the simulation cell.

Figure 12

Schematic of the microstructure observed experimentally during Zr oxidation, along with the variation of ${\mu }_{\mathrm{O}}$ with distance from the surface.