Construction of a thermodynamic potential for the water ices VII and X

We employ a combination of density functional theory (DFT), molecular dynamics (MD), and a variety of advanced postprocessing methods to construct an analytic thermodynamic potential (free energy) for ices VII and X. In particular, the temperature-dependent part of the free energy function is constructed using entropy data obtained via the spectrum of vibrational modes from the MD simulations. Conceptional challenges due to the partial absence of stable zero-temperature states and proton disorder are overcome by performing calculations of representative crystalline states combined with a three-stage fitting procedure of data from MD simulations and static DFT calculations. The influence of the exchange-correlation functional is extensively discussed, and a comparison with available experiments is made as well and generally shows good agreement. This work is of significant importance for astrophysical applications, such as the interior modeling of dense icy planets and moons.

Figure 1

Idealized [39] proton-ordered unit cell of ices VII and X, where the lattice constant is labeled $L$. The lengths $l_1$ and $l_2$ indicate the two smallest possible distances of the protons (small spheres) to oxygen nuclei (large spheres), where $l_2\ge l_1$ and $l_1+l_2=\sqrt{3}L/2$.

Figure 2

Oxygen-proton pair-correlation functions of ice VII at $1.75 {\mathrm{g}/\mathrm{cm}}^3$ (upper panel), ice VII* at $2.75 {\mathrm{g}/\mathrm{cm}}^3$ (center panel), and ice X at $3.75 {\mathrm{g}/\mathrm{cm}}^3$ (lower panel). The temperature was 400 K in all cases. Standard pair-correlation functions $g_{OH}\left(r\right)$ from the MD simulations are shown in black. The special pair-correlation functions ${\overline{g}}_{OH}\left(r\right)$ calculated from the time-averaged nuclear coordinates are drawn in red.

Figure 3

Quantum correction term $u_{vc}(\varrho ,T)$ as obtained from Eq. (3). The open circles represent ice VII, the solid circles indicate ice VII*, and the squares show ice X. Small diamonds with colors identical to those of the large symbols (circles and squares) show the residual internal energy data used to fit the nuclear ground-state energy in Sec. 4c. The black line is the respective fit from Eq. (15).

Figure 4

Reduced pressures $p_e\left(\varrho \right)/\varrho$ for different XC functionals.

Figure 5

Entropy calculated from the MD simulations via Eq. (4) for different isotherms. For ice VII the structural entropy from Eq. (10) was added. The open circles represent ice VII, the solid circles indicate ice VII*, and the squares show ice X. The open diamonds are results from the phonon calculations for ice X. Thin lines with the same colors as the symbols show the results from the fit to Eq. $\left(13\right)$.

Figure 6

Nuclear ground-state pressure from Eq. (15) compared with phonon calculations for ice X made with different XC functionals (top). The corresponding nuclear ground-state bulk modulus (bottom).

Figure 7

Isotherms of the internal energy data from the MD simulation, $u_{MD}(\varrho ,T)+u_{vc}(\varrho ,T)$ (circles), compared with the respective results of the fit [Eq. (8); lines].

Figure 8

Entropy (top) and heat capacity (bottom) in ice X at $3.5 {\mathrm{g}/\mathrm{cm}}^3$ (red) and $4 {\mathrm{g}/\mathrm{cm}}^3$ (black) calculated from the fit [Eq. (13); solid lines] and from the phonon calculations (dashed lines).

Figure 9

Reduced pressure $p/\varrho$ derived with Eq. (6) using different XC functionals for $u_e\left(\varrho \right)$ (see Table 1) compared with experiments [8, 12, 15, 23].

Figure 10

Differences in the specific volume derived from Eq. (6) using different XC functionals for $u_e\left(\varrho \right)$ from Table 1. Solid lines were generated with the HSE hybrid functional, and dashed lines are results from the vdW-optB86b XC functional. Squares show the measurements of Fei et al. [9], and the diamonds represent the experimental results of Frank et al. [20]. The data from Sugimura et al. [26] are indicated by circles, and their temperatures are given in the legend.