Superionic-Superionic Phase Transitions in Body-Centered Cubic ${\mathrm{H}}_2\mathrm{O}$ Ice

From first-principles molecular dynamics, we investigate the relation between the superionic proton conduction and the behavior of the $\mathrm{O}─\mathrm{H}\cdots \mathrm{O}$ bond (ice ${\mathrm{VII}}^{\prime }$ to ice X transition) in body-centered-cubic (bcc) ${\mathrm{H}}_2\mathrm{O}$ ice between 1300 and 2000 K and up to 300 GPa. We bring evidence that there are three distinct phases in the superionic bcc stability field. A first superionic phase characterized by extremely fast diffusion of highly delocalized protons (denoted ${\mathrm{VII}}^{\prime \prime }$  hereinafter) is stable at low pressures. A first-order transition separates this phase from a superionic ${\mathrm{VII}}^{\prime }$, characterized by a finite degree of localization of protons along the nonsymmetric $\mathrm{O}─\mathrm{H}\cdots \mathrm{O}$ bonds. The transition is identified in structural, energetic, and elastic analysis. Upon further compression a second-order phase transition leads to the superionic ice X with symmetric $\mathrm{O}─\mathrm{H}─\mathrm{O}$ bonds.

Figure 1

(a) Diffusion coefficients of the hydrogen atoms ($D_H$) in solid ice ($D_O=0$) as function of the OO distance. Superionic diffusion regime corresponds to positive $D_H$. The transition pressure to the superionic regime increases with increasing temperature. (b) Variation of the position of H in the $\mathrm{O}─\mathrm{H}\cdots \mathrm{O}$ bond under compression. Black line represents $\mathrm{OH}=\mathrm{OO}/2$, corresponding to ice X. (c) Internal energy ($U$) under compression. The lattice parameter $a$ is related to the OO distance as $a=\mathrm{OO}\times 2/\sqrt{3}$. The abrupt decrease in $D_H$ in the superionic regime corresponds to the jump in the internal energy and to the beginning of the $\mathrm{O}─\mathrm{H}\cdots \mathrm{O}$ bond symmetrization. This marks the phase transition between the superionic ice ${\mathrm{VII}}^{\prime \prime }$ and the superionic ice ${\mathrm{VII}}^{\prime }$.

Figure 2

Behavior of the H and O atoms along the 1300 K isotherm. (a) Representation of the atomic trajectories in one supercell representing the simulation box during a 2.5-ps long run in ice X ($P=300\mathrm{GPa}$), in ice ${\mathrm{VII}}^{\prime }$ (57.3 GPa), in superionic ice ${\mathrm{VII}}^{\prime }$ (34.5 GPa), and in ice ${\mathrm{VII}}^{\prime \prime }$ (34.2 GPa) just after the transition. (b) Projection of the above trajectories along one $\mathrm{O}─\mathrm{H}\cdots \mathrm{O}$ bond. In ice X the H-atom distribution is unimodal and presents a maximum at the center of the $\mathrm{O}─\mathrm{H}\cdots \mathrm{O}$ bond, whereas in ice ${\mathrm{VII}}^{\prime }$ the distribution is bimodal. In ice ${\mathrm{VII}}^{\prime \prime }$, the delocalization of the H atoms is too large to clearly distinguish the $\mathrm{O}─\mathrm{H}\cdots \mathrm{O}$ bond.

Figure 3

Proposed phase diagram obtained from our structural, diffusion, and energetic analysis. Circles indicate the thermodynamic conditions sampled in this study. The colors of the circles correspond to the various phases and/or regimes identified in the phase diagram: light green, ice ${\mathrm{VII}}^{\prime \prime }$; cyan, ice ${\mathrm{VII}}^{\prime }$ in the superionic regime; blue, ice ${\mathrm{VII}}^{\prime }$ in the nondiffusive regime; magenta, ice X in the superionic regime; and white, ice X in the nondiffusive regime. Black dotted lines correspond to the phase transitions lines. The fluid–ice ${\mathrm{VII}}^{\prime \prime }$ and ice ${\mathrm{VII}}^{\prime \prime }$–ice ${\mathrm{VII}}^{\prime }$ transitions are first order. The pale blue background area shows the thermodynamic conditions in which we found the superionic regime. The upper red line is the melting curve observed by Schwager et al. [53], as confirmed by calculations [30]; the lower red line is the melting curve as obtained by spectroscopic measurements [6, 54, 55].

Figure 4

Elastic constants calculated along different isotherms. $C_{11}$, $C_{12}$, and $C_{44}$ are, respectively, represented in blue, red, and green. In the left panels, the stability fields of the different phases are separated with the black vertical lines, and the pale blue area corresponds to the superionic regime. Right panels are zooms of the transition region between the ${\mathrm{VII}}^{\prime \prime }$ phase and ice ${\mathrm{VII}}^{\prime }$. Black arrows show the conditions in which the Born criterion is not respected.