Electronic and nuclear quantum effects on the ice XI/ice Ih phase transition

We study the isotope effect on the temperature of the proton order/disorder phase transition between ice XI and ice Ih, using the quasiharmonic approximation combined with ab initio density functional theory calculations. We show that this method is accurate enough to obtain a phase-transition temperature difference between light ice (${\mathrm{H}}_2\mathrm{O}$) and heavy ice (${\mathrm{D}}_2\mathrm{O}$) of 6 K as compared to the experimental value of 4 K. More importantly, we are able to explain the origin of the isotope effect on the much-debated large temperature difference observed in the phase transition. The source of the difference is directly linked to the physics behind the anomalous isotope effect on the volume of hexagonal ice that was recently explained in Pamuk et al. [Phys. Rev. Lett. 108, 193003 (2012)]. These results indicate that the same physics might be behind the isotope effects in transition temperatures between other ice phases.

Figure 1

Unit cell of the ferroelectric proton-ordered ice XI structure. The four molecules in the unit cell are labeled with star symbols next to them, and $a$ and $c$ lattice vectors are shown. The image on the left is the side view of the $x-z$ plane; the image on the right is the top view of the $x-y$ plane.

Figure 2

Antiferroelectric proton-ordered ice aXI structure. The eight molecules in the unit cell are labeled with star symbols next to them, and $a, b$, and $c$ lattice vectors are shown. The image on the left is the side view of the $x-z$ plane; the image on the right is the top view of the $x-y$ plane.

Figure 3

Proton-disordered ice Ih structure. The image on the left is the side view of the $x-z$ plane; the image on the right is the top view of the $x-y$ plane.

Figure 4

Relative free energy per molecule including the quantum zero-point effects as a function of temperature. The lines show DFT results with the ${\text{vdW-DF}}^{\mathrm{PBE}}$ functional and the dashed lines are the results with the TTM3-F model. DFT correctly predicts the most stable phase as ice XI for low temperatures, with an energy difference of $\sim 4.5\mathrm{meV}$. For low temperatures, the TTM3-F model predicts ice Ih as the stable phase with $\sim 1\mathrm{meV}$ energy difference at zero temperature; and a separation of energy at higher temperatures, making the prediction correct for high temperatures only. The results of the TTM3-F model for ice XI and ice aXI represented by black and red dashed lines, respectively, are almost indistinguishable in this scale.

Figure 5

(Top) Free energy difference per molecule between ice Ih and ice XI calculated with ${\text{vdW-DF}}^{\mathrm{PBE}}$ functional in the region of the phase transition. (Bottom) Contributions to this free energy difference by each term in Eq. (4). (Left) Frozen lattice electronic term. (Middle) Zero-point vibrational energy. (Right) Remaining terms. All the energies on the bottom plots have been shifted to allow them to be compared in the same energy scale.

Figure 6

Vibrational density of states for ${\mathrm{H}}_2\mathrm{O}$ for proton-ordered ice XI and disordered Ih structures, as obtained with ${\text{vdW-DF}}^{\mathrm{PBE}}$ functional. Average Grüneisen constants of the different modes are given in color code. The inset above zooms into the stretching modes and shows the redshift, while the inset below zooms into the librational modes and shows the blueshift in ice XI with respect to ice Ih.