First-principles equation of state database for warm dense matter computation

We put together a first-principles equation of state (FPEOS) database for matter at extreme conditions by combining results from path integral Monte Carlo and density functional molecular dynamics simulations of the elements H, He, B, C, N, O, Ne, Na, Mg, Al, and Si as well as the compounds $\mathrm{LiF}, {\mathrm{B}}_4\mathrm{C}, \mathrm{BN}, {\mathrm{CH}}_4, {\mathrm{CH}}_2, {\mathrm{C}}_2{\mathrm{H}}_3, \mathrm{CH}, {\mathrm{C}}_2\mathrm{H}, \mathrm{MgO}, \mathrm{and} {\mathrm{MgSiO}}_3$. For all these materials, we provide the pressure and internal energy over a density-temperature range from $\sim 0.5$ to 50 g ${\mathrm{cm}}^{-3}$ and from $\sim {10}^4$ to ${10}^9$ K, which are based on $\sim 5000$ different first-principles simulations. We compute isobars, adiabats, and shock Hugoniot curves in the regime of $\mathrm{L}$- and $\mathrm{K}$-shell ionization. Invoking the linear mixing approximation, we study the properties of mixtures at high density and temperature. We derive the Hugoniot curves for water and alumina as well as for carbon-oxygen, helium-neon, and CH-silicon mixtures. We predict the maximal shock compression ratios of ${\mathrm{H}}_2\mathrm{O},$ ${\mathrm{H}}_2{\mathrm{O}}_2,$ ${\mathrm{Al}}_2{\mathrm{O}}_3, \mathrm{CO}, \mathrm{and} {\mathrm{CO}}_2$ to be 4.61, 4.64, 4.64, 4.89, and 4.83, respectively. Finally we use the FPEOS database to determine the points of maximum shock compression for all available binary mixtures. We identify mixtures that reach higher shock compression ratios than their end members. We discuss trends common to all mixtures in pressure-temperature and particle-shock velocity spaces. In the Supplemental Material, we provide all FPEOS tables as well as computer codes for interpolation, Hugoniot calculations, and plots of various thermodynamic functions.

Figure 1

Density-temperature-pressure conditions of our first-principles simulations of helium [77]. Isobars, isentropes, and shock Hugoniot curves are shown. In the upper panel, we include the interior conditions of Jupiter [32, 44] and main-sequence stars of different masses [13] as well as the highest-pressure conditions reached in recent shock wave experiments on CH [122], ${\mathrm{B}}_4\mathrm{C}$ [91], MgO [123], BN [92], ${\mathrm{MgSiO}}_3$ [124], and ${\mathrm{CO}}_2$ [125]. The corresponding temperatures were not measured but derived from simulations.

Figure 2

Shock Hugoniot curves for helium at different initial densities (${\rho }_0=0.1235$ g ${\mathrm{cm}}^{-3}$) are shown in compression-temperature space. Relativistic and radiation effects have also been introduced.

Figure 3

Internal energy of hot, dense helium is shown for a collection of isochores. Their densities are given in the captions in units of g ${\mathrm{cm}}^{-3}$. For clarity, we shifted the individual curves in the upper panel by $\mathrm{\Delta }$ and removed $E_0$, the energy of a noninteracting Fermi gas of electrons and classical nuclei. In the lower panel, we show that the results from our first-principles simulation converge to prediction from the Debye plasma models in the limit of high temperature where screening effects are the dominant type of the interaction.

Figure 4

Shock Hugoniot curves of different mixtures of plastic (CH) and silicon. The silicon Hugoniot curve exhibits two compression maxima corresponding to the ionization of $\mathrm{K}$- and $\mathrm{L}$-shell electrons. Conversely, the CH curve shows only one maximum because the states of $\mathrm{L}$-shell electrons in carbon have merged with the conduction bands [93]. We predict that the two-maxima signature is preserved as long as the CH:Si concentration does not exceed a 2:1 ratio. With increasing CH concentration, the peak compression ratio decreases, as is expected from two end-member curves.

Figure 5

Shock Hugoniot curves of different mixtures of carbon and oxygen. Like CH in Fig. 4, the carbon Hugoniot curve exhibits only one compression maximum, while oxygen, like silicon, exhibits two separate maxima from its $\mathrm{L}$- and $\mathrm{K}$-shell electrons. Neither CO nor ${\mathrm{CO}}_2$ shows two maxima, which means even a low-carbon concentration such as C:O = 1:2 is sufficient to suppress the $\mathrm{L}$-shell compression maximum that one sees for elemental oxygen. Our simulation results are in very good agreement with the experimental data from Ref. [125].

Figure 6

Shock Hugoniot curves of various mixtures of helium-neon mixtures. The helium curve only exhibits one very pronounced compression maximum because helium does not have an $\mathrm{L}$ shell. The neon curve shows two maxima because of the $\mathrm{L}$- and $\mathrm{K}$-shell ionization like silicon in Fig. 4 and oxygen in Fig. 5. With increasing neon concentration, the compression maxima shift to higher pressure (and temperature) because it takes more energy to ionize electrons from their respective shells. We find that already a low neon concentration of He:Ne = 4:1 is sufficient for the Hugoniot curve to show two compression maxima.

Figure 7

Shock Hugoniot curves of ${\mathrm{Al}}_2{\mathrm{O}}_3$, ${\mathrm{H}}_2\mathrm{O}$, and ${\mathrm{H}}_2{\mathrm{O}}_2$ are compared with those of the elemental constituents. For the lowest pressures, our ${\mathrm{H}}_2\mathrm{O}$ curve converges to the predictions from French et al. [130]. Both theoretical predictions are in better agreement with the reanalyzed experimental results [25] than they are with the original measurements [131]. ${\mathrm{H}}_2\mathrm{O}$ maintains part of the two-peak signature of the oxygen Hugoniot curve; this influence is reduced in ${\mathrm{H}}_2{\mathrm{O}}_2$. ${\mathrm{Al}}_2{\mathrm{O}}_3$ shows only one compression maximum because the compression maxima of aluminum and oxygen are offset in pressure.

Figure 8

The dividing lines between the regimes of thermal and pressure ionization are shown in temperature-density and temperature-pressure spaces.

Figure 9

Conditions of the compression maximum on the shock Hugoniot curves of 21 elements and compounds as well as their mixtures. All materials show compression maxima larger than 4.3 that reflect the effects from electronic excitation. Low-$Z$ materials like helium and helium-rich mixtures already reach their compression maxima for lower temperatures while silicon-rich mixtures require much higher temperatures to excite their $\mathrm{K}$-shell electrons.

Figure 10

Temperature condition of the compression maximum on the shock Hugoniot curves of 21 elements and compounds as well as their mixtures is shown as a function of the average ionic charge $〈Z〉$ of these materials. While generally the compression maximum shifts towards higher temperature with increasing $Z$, the correlation is found to be not very strong.

Figure 11

The particle and shock velocities are shown for the points of maximum compression along shock Hugoniot curves of 21 elements and compounds as well as their mixtures. With good accuracy, the data set can be represented by the linear relationship of $u_s=1.2727u_p-0.8588$ km/s (dashed orange line). The largest positive deviation is seen for a Si:Al mixture, in which case the fit underpredicts $u_s$ by 5.2 km/s, or 1.2%. The largest negative deviation is found for a H:${\mathrm{CH}}_4$ mixture, in which case the fit overpredicts $u_s$ by 5.4 km/s, or 1.9%.

Figure 12

Pressure-temperature conditions for the points of maximum compression along shock Hugoniot curves of 21 elements and compounds as well as their mixtures. As expected, pressure and temperature are highly correlated, which can be represented by the fit $\log (T/\text{K})=0.5744\times \log (P/\text{GPa})+3.7469$. However, specific materials deviate substantially from this fit. The temperature of the maximum compression point of a H:Ne Hugoniot curve is 38% lower than this fit would imply. Conversely, the temperature for B:Al is 49% higher than predicted by the fit.

Figure 13

Conditions of the compression maximum on the shock Hugoniot curves of eight cases for which the mixtures exhibit a higher compression ratio than its end members. The lines emerge because a range of mixing ratios were considered.