Equation of state and phase diagram of water at ultrahigh pressures as in planetary interiors

We present QMD simulations of water in the ultra-high-pressure regime up to conditions typical for the deep interior of Jupiter and Saturn. We calculate the equation of state and the Hugoniot curve and study the structural properties via pair correlation functions and self-diffusion coefficients. In the ultradense superionic phase, we find a continuous transition in the protonic structure. With rising density, the mobile protons stay with increasing probability at the octahedral sites while leaving the ice X positions to the same degree unoccupied. Water forms a fluid dense plasma at the conditions of Jupiter’s core (i.e., $20000\text{K}$, 50 Mbar, $11\text{g}/{\text{cm}}^3$), while it may be superionic in the core of Saturn. We expect a substantial amount of superionic water inside Neptune.

Figure 1

(Color) QMD pressure isotherms (solid lines) in comparison with the SESAME 7150 EOS (Ref. 26) (dashed lines) for temperatures of 1000 K (black), 4000 K (red), 6000 K (green), and 10000 K (blue). The cyan line is a 300 K QMD isotherm to compare with DAC measurements of Sugimura et al. (Ref. 27) represented by the dots. The inset shows the 8000 K (black) and 24000 K (red) isotherms for higher densities.

Figure 2

(Color) Principal Hugoniot curves based on the QMD simulations in comparison with the SESAME 7150 predictions and experiments of Mitchell and Nellis (Ref. 28), Volkov et al. (Ref. 29), Celliers et al. (Ref. 30) and Podurets et al. (Ref. 31). Orange squares are calculations with the 8e-potential. At high temperatures, the system is a disordered plasma, reducing the need for large simulation cells: the results for 16 and 54 molecules converge.

Figure 3

(Color) Phase diagram of warm and ultradense water for temperatures above 3000 K. Each colored point represents a QMD simulation in equilibrium. The plasma-to-superionic phase boundary (black dashed line), the principal Hugoniot curve (black solid line), and two Neptune isentropes are also shown. The electronic conductivities are taken from Ref. 6.

Figure 4

(Color online) Radial oxygen-oxygen pair correlation functions at 16000 K. The picture includes also an ideal bcc lattice at $20\text{g}/{\text{cm}}^3$ $\left(L_{box}=5.76\text{Å}\right)$ to illustrate the appearance of a bcc-like short-range order already in the plasma.

Figure 5

(Color online) Protonic pair correlation functions with octahedral sites $g_{H,oct}$ and ice X positions $g_{H,X}$ in the bcc oxygen lattice at 6000 K. For comparison, we plot the same correlation functions of ice X at $4\text{g}/{\text{cm}}^3$ and 1000 K (dotted lines). Only the symmetric positions are occupied in ice X.

Figure 6

Mean square displacements for protons in superionic water at 6000 K. The simulation box contained 54 molecules.

Figure 7

Mean square displacements for protons and oxygen ions in water plasma at 16000 K and $20\text{g}/{\text{cm}}^3$.

Figure 8

Thermodynamic functions and oxygen diffusion during the formation of the superionic phase at 8000 K and $7\text{g}/{\text{cm}}^3$. The simulation cell contains 54 molecules and the initial configuration was a fluid at 12000 K and $7\text{g}/{\text{cm}}^3$. The system remains a fluid until there are simultaneous changes in energy, pressure and diffusion after 1000 fs. This particular simulation rapidly changed phase, a sign that 8000 K is deep in the superionic phase. Other points required several tens of ps long simulations before the phase transitions were observed. In the majority of our calculations the temperature difference between the initial configurations and the simulations was 2000 K.

Figure 9

(Color online) The $3\text{g}/{\text{cm}}^3$ isochore showing discontinuities in the thermodynamic functions. For this example, the transition temperature was determined to an accuracy of 200 K. This required simulation times of 15 ps. The symbol sizes represent the statistical error of the pressure and energy, see Sec. 1.

Figure 10

(Color online) Mean square displacements for the oxygen ions at the $3\text{g}/{\text{cm}}^3$ isochore. The oxygen movement is diffusive at 4500 K and higher temperatures. At 4300 K or lower temperatures they can only vibrate around their bcc lattice positions and, thus, show no diffusive movement.

Figure 11

(Color online) Phase diagram of warm and ultradense water for temperatures above 3000 K. Each colored point represents a QMD simulation in equilibrium. The principal Hugoniot curve (black solid line) is also shown. The electronic conductivities are taken from Ref. 6. The arrows represent the slope of the phase boundary calculated by the Clausius-Clapeyron equation above $4\text{g}/{\text{cm}}^3$.