Longitudinal sound velocities, elastic anisotropy, and phase transition of high-pressure cubic ${\mathrm{H}}_2\mathrm{O}$ ice to 82 GPa

Water ice is a molecular solid whose behavior under compression reveals the interplay of covalent bonding in molecules and forces acting between them. This interplay determines high-pressure phase transitions, the elastic and plastic behavior of ${\mathrm{H}}_2\mathrm{O}$ ice, which are the properties needed for modeling the convection and internal structure of the giant planets and moons of the solar system as well as ${\mathrm{H}}_2\mathrm{O}$-rich exoplanets. We investigated experimentally and theoretically elastic properties and phase transitions of cubic ${\mathrm{H}}_2\mathrm{O}$ ice at room temperature and high pressures between 10 and 82 GPa. The time-domain Brillouin scattering (TDBS) technique was used to measure longitudinal sound velocities ($V_L$) in polycrystalline ice samples compressed in a diamond anvil cell. The high spatial resolution of the TDBS technique revealed variations of $V_L$ caused by elastic anisotropy, allowing us to reliably determine the fastest and the slowest sound velocity in a single crystal of cubic ${\mathrm{H}}_2\mathrm{O}$ ice and thus to evaluate existing equations of state. Pressure dependencies of the single-crystal elastic moduli $C_{ij}\left(P\right)$ of cubic ${\mathrm{H}}_2\mathrm{O}$ ice to 82 GPa have been obtained which indicate its hardness and brittleness. These results were compared with ab initio calculations. It is suggested that the transition from molecular ice VII to ionic ice X occurs at much higher pressures than proposed earlier, probably above 80 GPa.

Figure 1

Earlier data on $V_L$($P$) of ice VII and ice X at high pressures. Symbols represent experimental data from classical FDBS measurements: Open green triangles pointing left and right are ${V_L}_{\langle 111\rangle }\left(P\right)$ and ${V_L}_{\langle 100\rangle }\left(P\right)$, respectively, obtained for single crystals of ice VII by Shimizu et al. [28]; open blue triangles pointing up and down are ${V_L}_{\langle 111\rangle }\left(P\right)$ and ${V_L}_{\langle 100\rangle }\left(P\right)$, respectively, calculated from the $C_{ij}\left(P\right)$ dependencies reported by Zha et al. [30]; diamonds and circles are longitudinal sound velocities ${V_L}_{\mathrm{av}}\left(P\right)$ derived from averaged FDBS spectra of polycrystalline ice samples by Polian and Grimsditch [26] and Ahart et al. [29], respectively. The error bar shows the full width at half maximum (FWHM) of the FDBS peak at 77 GPa shown in the latter work. Lines represent theoretical ${V_L}_{\langle 111\rangle }\left(P\right)$ and ${V_L}_{\langle 100\rangle }\left(P\right)$ calculated for ice X: solid lines, earlier results of Journaux et al. [3]; dashed lines, our ab initio calculations. The inset shows orientational distribution of $V_L$ in a single crystal of ice VII calculated using the $C_{ij}$ values at 40 GPa reported by Zha et al. [30]. This distribution is equivalent to the statistical distribution of sound velocities in grains of a texture-free polycrystalline sample along a selected spatial direction and depicts the ideal shape of an averaged FDBS spectrum collected for such a sample. Directions of the fastest and the slowest sound propagation in the cubic ice crystal are $\langle 111\rangle$ and $\langle 100\rangle$, respectively. The star indicates that the $C_{ij}$ values were obtained using the envelope method and are consequently approximate. The distribution has a maximum at 11.64 km/s and its center of mass (corresponds to the most probable velocity) is located at 11.56 km/s. For a texture-free sample with $B$ and $G$ calculated using the same $C_{ij}$ values applying the Voigt approximation we obtained $V_L=11.26\mathrm{km}/\mathrm{s}$. This value deviates from the most probable velocity by 13% of the difference between the fastest and the slowest velocities in a single crystal. Applying the Voigt-Reuss-Hill approximation to determine $G$ of the same polycrystalline sample we obtained $V_L=10.92\mathrm{km}/\mathrm{s}$ which deviates even stronger from the most probable velocity, namely by 27%.

Figure 2

Processing of a TDBS signal collected from an ${\mathrm{H}}_2\mathrm{O}$ ice sample compressed in a DAC to 33 GPa. (a) Raw transient reflectivity signal $S$($t$). (b) Corresponding filtered signal $s$($t$) where the nonoscillating background, caused by a transient heating of the sample, was removed. Here, both amplitude and the phase change with time: $A\left(t\right)cos\left[\varphi \right(t\left)\right]$. (c) The amplitude-variation-free signal $cos\left[\varphi \right(t\left)\right]$ (solid blue line) and normalized root-mean-square error (NRMSE) between the windowed complex exponential $\exp \left[j\varphi \right(t\left)\right]$ and the corresponding extracted complex exponential (dashed black line). The insets show, with a higher magnification, the time segments indicated by the gray background where the signals (blue solid lines) deviate significantly from the best fit by the harmonic function (dashed-dotted lines). These time segments were excluded from further consideration due to a large NRMSE exceeding 10%. (d) Instantaneous signal frequency derived using the synchronous detection technique (solid red line) and the SNR (dashed black line). Gray background in (d) indicates time segments with a poor SNR (smaller than 10 dB) and/or with a large NRMSE (greater than 10%).

Figure 3

Experimental and theoretical $V_L$($P$) data for ${\mathrm{H}}_2\mathrm{O}$ ice at high pressures as reported in the present work. Symbols represent the experimental data between 10 and 82 GPa: Solid triangles pointing up and down are our experimental ${V_L}_{\max }\left(P\right)$ and ${V_L}_{\min }\left(P\right)$, respectively, obtained for the window sizes $W$ equal to one (smaller dark-red triangles with error bars) and two-and-half Brillouin oscillations (larger red triangles; error bars are smaller than the symbols). Our theoretical values of ${V_L}_{\langle 111\rangle }\left(P\right)$ and ${V_L}_{\langle 100\rangle }\left(P\right)$ for ice VII and ice X from ab initio calculations are represented by dotted and dashed lines, respectively. Some of the earlier experimental and theoretical results, presented in Fig. 1 in the same style, are shown for comparison.

Figure 4

Pressure dependencies of the single-crystal elastic moduli $C_{ij}\left(P\right)$ of ${\mathrm{H}}_2\mathrm{O}$ ice derived under the assumption that the ice structure is cubic in the entire pressure range. Only experimental TDBS data, obtained for the narrower window (Fig. 3), were used to determine the shown $C_{ij}\left(P\right)$. Solid line represents $B$($P$) reported in Ref. [23]. Solid triangles pointing up and down show $C_{11}\left(P\right)$ and $C_{44}\left(P\right)$, respectively. Solid rhombuses and squares show our $C_{12}\left(P\right)$ and $G$($P$), respectively, where the latter was calculated from $C_{ij}\left(P\right)$ using the Voigt approximation, $G\left(P\right)=\left[C_{11}\left(P\right)-C_{12}\left(P\right)+3C_{44}\left(P\right)\right]/5$. Open squares show $G$($P$) calculated using the Voigt-Reuss-Hill approximation (the experimental uncertainties, not shown for clarity, are the same as for the Voigt approximation).

Figure 5

Experimental and theoretical $V_T$($P$) values for cubic ${\mathrm{H}}_2\mathrm{O}$ ice. Symbols represent experimental data: Solid dark-red triangles pointing up and down show, respectively, our ${V_T}_{\langle 100\rangle }\left(P\right)$ and ${V_T}_{\langle 110\rangle }\left(P\right)$ between 10 and 82 GPa derived using the experimental $C_{ij}\left(P\right)$ depicted in Fig. 4. Open green triangles pointing left and right show ${V_T}_{\langle 100\rangle }\left(P\right)$ and ${V_T}_{\langle 110\rangle }\left(P\right)$, respectively, measured for single crystals of ice VII by Shimizu et al. [28]; open triangles pointing up and down represent the same values calculated from $C_{ij}\left(P\right)$ reported by Zha et al. [30]; circles are ${V_T}_{\mathrm{av}}\left(P\right)$ measured for polycrystalline ice samples by Ahart et al. [29]. Lines represent theoretical values of ${V_T}_{\langle 100\rangle }\left(P\right)$ and ${V_T}_{\langle 110\rangle }\left(P\right)$: Dotted lines, our calculations for ice VII; dashed lines, our calculations for ice X; solid lines, earlier results of Journaux et al. for ice X [3].

Figure 6

Results of present ab initio calculations for the cubic ice phases VII and X between 10 and 100 GPa. (a) Calculated densities of ice VII and ice X are represented by the dotted and dashed red line, respectively, and compared with the experimental EOS (solid line) reported in Ref. [23] which we used to derive $C_{ij}\left(P\right)$ values from the measured ${V_L}_{\max }\left(P\right)$ and ${V_L}_{\min }\left(P\right)$. (b) Elastic moduli of ice VII calculated for the $2\times 2\times 2$ supercell model are represented by thick solid lines. The theoretical $G$($P$) is derived from the $C_{ij}\left(P\right)$ using the Voigt-Reuss-Hill approximation. The calculated values are compared with our experimental results shown in Fig. 4 (symbols connected by thin solid lines). (c) Calculated elastic moduli of ice X represented by thick solid lines. The theoretical $G$($P$) is derived from the shown $C_{ij}\left(P\right)$ using the Voigt-Reuss-Hill approximation. The experimental data points at 82 GPa (see Fig. 4) are shown for comparison. (d) Calculated Gibbs free energy difference between ice VII and ice X at $T=0\mathrm{K}$, shown by connected solid squares.