Dynamic Nature of High-Pressure Ice VII

Starting from Shannon’s definition of dynamic entropy, we propose a theory to describe the rare-event-determined dynamic states in condensed matter and their transitions and apply it to high-pressure ice VII. A dynamic intensive quantity named dynamic field, rather than the conventional thermodynamic intensive quantities such as temperature and pressure, is taken as the controlling variable. The dynamic entropy versus dynamic field curve demonstrates two dynamic states in the stability region of ice VII and dynamic ice VII. Their microscopic differences were assigned to the dynamic patterns of proton transfer. This study puts a similar dynamical theory used in earlier studies of glass models on a simpler and more fundamental basis, which could be applied to describe the dynamic states of more realistic condensed matter systems.

Figure 1

(a) Solid, (b) liquid, and (c) the in-between state. (d) The phase diagram of bcc ice. There are well-defined solid and liquid states in the shadowed region, while in the white region the boundaries between ice VII, dynamic ice VII, and SI remain controversial. (e) Density and thermodynamic state function at 500 K. Inset: bcc ice structure. In the bcc skeleton formed by oxygen (red), protons intersperse in the covalent sites (blue) of neighboring oxygen pairs. The equivalent sites (green) can be occupied through transfer motions. The other sites (gray) consist in another hydrogen bonding network.

Figure 2

(a) Schematic of the PES of proton transfer, (b) the realistic trajectory in coordinate space, and (c) the trajectory in component space. $d_{\mathrm{site}}$ is the distance of the proton to its equilibrium position inside each component, in the unit of fractional coordinate of the simulation cell. (d) $p_0(K)$ at different $P$s with $t_{\mathrm{obs}}=2\mathrm{ps}$ (10 000 steps). The inset in (d) shows the $t_{\mathrm{obs}}$ dependence of the percentage of inactive sites for $K(t)=0$. (e) $S_D$ computed from $p_0(K)$. The dashed lines and shadowed region are guides for the eye.

Figure 3

(a) The partition function of the isodynamic ensembles and (b) $⟨k⟩$ as their ensemble averages at different $P$’s and 500 K. (a) and (b) share the legend. All the data are presented with $t_{\mathrm{obs}}=2\mathrm{ps}$ (10 000 steps). (c) The $S_{D,\mathrm{bcc}}$ with different $t_{\mathrm{obs}}$ (in the unit of steps). Inset in (c): the tendency of transition point $s_{\mathrm{tran}}$ toward the long time limit. (d) The tendency of $S_{D,(T,P)}$ on $t_{\mathrm{obs}}$. The dashed lines show theoretical results of Eq. (11).

Figure 4

Schematic of the dynamic constraint. When a single transfer occurs [solid line in (a)], there can be two paths to restore the ice rule: (a) the retrieving motion (dashed line) upon strong constraint and (b) the collective motion upon loosened constraint. (c) and (d) show the PES upon the transfer of H1 and H2 at 10 and 70 GPa, respectively.