Effect of hydrostatic pressure on the quantum paraelectric state of dipolar coupled water molecular network

We measure the real part ${\epsilon }^{\prime }$ of the dielectric permittivity of beryl crystals with heavy water molecules ${\mathrm{D}}_2\mathrm{O}$ confined in nanosized cages formed by an ionic crystal lattice. The experiments are performed at a frequency of 1 MHz in the temperature interval from 300 down to 4 K under different hydrostatic pressures up to $P=6.3$ GPa. At high temperatures, a Curie-Weiss-like increase of ${\epsilon }^{\prime }\left(T\right)$ is observed upon cooling. Application of pressure leads to flattening of ${\epsilon }^{\prime }\left(T\right)$ at low temperatures due to quantum effects, i.e., tunneling of deuterium atoms in the hexagonal localizing potential. Analyzing the temperature behavior of ${\epsilon }^{\prime }$ with the Barrett expression allows us to obtain pressure dependencies of the quantum temperature $T_1$, the Curie-Weiss temperature $T_C$, and the Barrett constant $C$. The increase of $T_1$ observed up to 4 GPa is associated with an enhanced azimuthal tunneling of the confined water molecules through the barriers of the potential. For $P>4$ GPa, $T_1\left(P\right)$ levels off since the barriers disappear. Any further pressure increase does not affect the tunneling rate because of the absence of a barrier. The behavior is modeled by solving the Schrödinger equation for the water molecule in the azimuthal potential numerically. Small negative values of $T_C\approx -10$ K obtained for $P<4$ GPa indicate the antiferroelectric ordering tendency of the water dipoles localized in the crystalline nanochannels. For higher pressure, a strong decrease of $T_C$ toward negative values is observed that would correspond to the enhanced interdipole coupling strength, which is however hard to explain in the present case, and thus calls for additional theoretical and experimental studies.

Figure 1

(a) Beryl (${\mathrm{Be}}_3{\mathrm{Al}}_2{\mathrm{Si}}_6{\mathrm{O}}_{18}$) structure viewed along the $c$ axis. Honeycomb six-membered Be, Al, and ${\mathrm{SiO}}_4$ tetrahedra enclose a water molecule (oxygen is gray, deuterium is red). The color coding of the atoms is given through the legend. The blue arrow indicates the dipole moment of the water molecule. (b) Separated water molecule within the ionic cage. The cage is in diameter of 5.1 Å with narrow bottlenecks of 2.8 Å. The water molecule encaged is subjected to a six-well localizing potential (green belt). (c) Schematic view of water molecules in nanosized cages of the hexagonal beryl crystal lattice. Cages are organized in channels aligned along the crystallographic $c$ axis. Characteristic sizes are shown: interchannel distance (9.2 Å) and intercage distance along the channel (4.6 Å). Hydrogen bonding between water molecules is negated, and only dipole-dipole interaction is present.

Figure 2

Water molecule dipole configuration in (a) the $ab$ plane and (b) the $bc$ plane in beryl at zero temperature obtained by Monte Carlo simulation. (a) In the $ab$ plane, the ferroelectric alignment is favored between adjacent water dipoles, but exceptions can be found, and no long-range order is developed. (b) The water dipoles have a zero component along the $c$ axis. The vertical component of the dipole in this figure is the component in the $a$ axis, such that the spatial configuration, in the $bc$ plane, of the orientation of the dipole in the $ab$ plane is displayed. We see the perfect antiferroelectric order along the $c$ axis for the nearest dipoles.

Figure 3

Inverse dielectric susceptibility ${\chi }^{-1}$ of water molecule dipoles in the $ab$ plane of beryl. Lines and symbols correspond to $\chi$ with the electric field polarization directed along $a$ (red) and $b$ (blue) Cartesian axes, respectively, obtained by Monte Carlo simulations. Experimental $\chi$ (green square) is also extracted from ${\epsilon }^{\prime }$ by subtracting the high-temperature limit $\chi ={\epsilon }^{\prime }-{\epsilon }_{\infty }^{\prime }$. The simulated $\chi$ will match with the experiment data by multiplying by a factor of $\alpha =$ 2.5. The discrepancy is due to approximation in the model. The strong growth of ${\chi }^{-1}$ below $\approx 5$ K is because of the absence of quantum effects in Monte Carlo simulations. Dashed lines are extrapolations of Curie-Weiss fitting above (purple) and below (orange) the kink feature at 20 K. Both extrapolations project a negative $T_C$.

Figure 4

Temperature dependencies of the real part of the permittivity ${\epsilon }^{\prime }$ of beryl crystal filled with heavy water molecules confined inside nanosized cages within the crystal lattice. The measurements are performed at a frequency of 1 MHz and at different hydrostatic pressures as indicated. Solid lines show results of least-square fits according to the Barrett expression in Eq. (2), as described in the text. The inset presents the corresponding dependencies of the inverse permittivity ${({\epsilon }^{\prime }-{\epsilon }_{\infty }^{\prime })}^{-1}$, where ${\epsilon }_{\infty }^{\prime }\approx 7.3$ is the high-temperature dielectric constant independent of pressure. The colors in the inset correspond to those in the main panel. Straight line extrapolation toward zero values of ${({\epsilon }^{\prime }-{\epsilon }_{\infty }^{\prime })}^{-1}$ indicates negative critical temperature $T_C\approx -10$ K.

Figure 5

Dependence on hydrostatic pressure of (a) quantum temperature $T_1$, (b) Curie-Weiss temperature $T_C$, and (c) Barrett constant $C$ obtained by fitting the experimental data presented in Fig. 4 with the Barrett expression [Eq. (2)] in the temperature range 4–20 K and with the Curie-Weiss expression [Eq. (1)] in the temperature range 20–50 K. Dashed lines are guides for the eye. Dark blue solid line in (a) corresponds to the water molecule ground state energy splitting in the azimuthal potential (see text).

Figure 6

Calculated (see text) energy levels of a water molecule in the azimuthal potential (a) at zero pressure and (b) at $P=4$ GPa. The thick blue curve is the potential energy, dashed lines are the energy levels, and solid lines are the corresponding wave functions.