Equation of state modeling with pseudoatom molecular dynamics

Using a modified version of the pseudoatom molecular-dynamics approach, the silicon and oxygen equations of state were generated and then employed to construct the equation of state of silicon dioxide. The results are supported by the close agreement with ab initio simulations of the silicon pressure and experimental shock Hugoniot of silicon dioxide. Ion thermal contributions to thermodynamic functions provided by the PAMD simulations are compared to their counterparts obtained with the one-component plasma and charged-hard-sphere approximations.

Figure 1

Isotherms of the silicon mean ion charge at temperatures $T=10, {10}^2$, and ${10}^3$ eV calculated by using various approximations in the context of the Liberman model.

Figure 2

Present-work radial screening electron density for aluminum at $\rho =2.7$ g/${\mathrm{cm}}^3, T=2$ eV calculated with quantum-mechanical description of electrons and various definitions of free-electron densities—(33) and (34) (blue solid curve) and (35) and (36) (red solid curve)—as compared to the calculations of Ref. [21] with free-electron densities (33) and (34) (magenta dashed curve) and their semiclassical counterparts similar to (35) and (36) (green dashed curve).

Figure 3

Ion-ion pair distribution functions for aluminum at $\rho =2.7$ g/${\mathrm{cm}}^3, T=2$ eV, calculated with PAMD using various definitions of the screening electron density providing the pseudoatom mean ion charge $Z_0=Z_{*}$ (blue curve) and $Z_0=Z_{\text{wkb}}$ (red curve), as compared to QMD simulations of Ref. [21] (brown circles).

Figure 4

Ion-ion pair distribution functions for aluminum at a density $\rho =6.3$ g/${\mathrm{cm}}^3$ and various temperatures obtained with PAMD utilizing the definition of the screening electron density providing the pseudoatom mean ion charge $Z_0=Z_{*}$. Solid and dashed curves corresponding to the simulations of the present work and of Ref. [51], respectively, coincide almost perfectly.

Figure 5

Isochores of the pressure-to-temperature ratio for normal density silicon ($\rho ={\rho }_0=2.329$ g/${\mathrm{cm}}^3$) calculated by using the PAMD approach and AA models with various approximations for the ion thermal contribution to the EOS as compared to the ab initio simulation data of Ref. [7]. Red solid and blue dashed curves correspond to the PAMD calculations utilizing the screening electron density definitions providing the pseudoatom mean ion charge $Z_0=Z_{\text{wkb}}$ and $Z_0=Z_{*}$, respectively; dark green and light green solid curves correspond respectively to the NWS and Liberman AA-model calculations with the ideal-gas approximation for the ion thermal contribution; magenta dashed curve—to the NWS calculation with the OCP approximation for the ion thermal contribution utilizing $Z_0=Z_{\text{wkb}}$; orange dashed-dotted, cyan dash-dotted, and yellow dash-dotted curves—to the NWS calculations with the CHS approximation for the ion thermal contribution utilizing $Z_0=Z_{\text{wkb}}, Z_0=Z_{*}$, and $Z_0=Z_{\text{WS}}$, respectively. Squares and circles represent the QMD and PIMC simulation data [7], respectively.

Figure 6

Isochores of the pressure-to-temperature ratio for silicon at $\rho =2{\rho }_0$. The colors, lines, and symbols have the same meanings as in Fig. 5.

Figure 7

Isochores of the pressure-to-temperature ratio for silicon at $\rho =4{\rho }_0$. The colors, lines, and symbols have the same meanings as in Fig. 5.

Figure 8

Isochores of the pressure-to-temperature ratio for silicon at $\rho =5{\rho }_0$. The colors, lines, and symbols have the same meanings as in Fig. 5.

Figure 9

Isochores of the pressure-to-temperature ratio for silicon at $\rho =7{\rho }_0$. The colors, lines, and symbols have the same meanings as in Fig. 5.

Figure 10

Ratios of the OCP corrections to the NWS-model internal energy $\mathrm{\Delta }{E}_i^{\text{OCP}}$ evaluated at various temperatures $T$ (eV) of silicon and $Z_0=Z_{\text{wkb}}$ to their PAMD counterparts $\mathrm{\Delta }{E}_i^{\text{PAMD}}$ found with the screening electron density definition providing $Z_0=Z_{\text{wkb}}$.

Figure 11

Ratios of the OCP corrections to the NWS-model pressure $\mathrm{\Delta }{P}_i^{\text{OCP}}$ evaluated at various temperatures of silicon and $Z_0=Z_{\text{wkb}}$ to their PAMD counterparts $\mathrm{\Delta }{P}_i^{\text{PAMD}}$ found with the screening electron density definition providing $Z_0=Z_{\text{wkb}}$. The colors and lines have the same meanings as in Fig. 10.

Figure 12

Ratios of the CHS corrections to the NWS-model pressure $\mathrm{\Delta }{P}_i^{\text{CHS}}$ evaluated at various temperatures of silicon and $Z_0=Z_{\text{wkb}}$ to their PAMD counterparts $\mathrm{\Delta }{P}_i^{\text{PAMD}}$ found with the screening electron density definition providing $Z_0=Z_{\text{wkb}}$. The colors and lines have the same meanings as in Fig. 10.

Figure 13

Ratios of the CHS corrections to the NWS-model pressure $\mathrm{\Delta }{P}_i^{\text{CHS}}$ evaluated at various temperatures of silicon and $Z_0=Z_{*}$ to their PAMD counterparts $\mathrm{\Delta }{P}_i^{\text{PAMD}}$ found with the screening electron density definition providing $Z_0=Z_{\text{wkb}}$. The colors and lines have the same meanings as in Fig. 10.

Figure 14

Ratios of the CHS corrections to the NWS-model pressure $\mathrm{\Delta }{P}_i^{\text{CHS}}$ evaluated at various temperatures of silicon and $Z_0=Z_{\text{WS}}$ to their PAMD counterparts $\mathrm{\Delta }{P}_i^{\text{PAMD}}$ found with the screening electron density definition providing $Z_0=Z_{\text{wkb}}$. The colors and lines have the same meanings as in Fig. 10.

Figure 15

The OCP pressure-to-internal energy-corrections ratios ${\left(\mathrm{\Delta }{P}_i/\mathrm{\Delta }{E}_i\right)}^{\text{OCP}}$ evaluated using the NWS model at various temperatures of silicon and $Z_0=Z_{\text{wkb}}$ as divided by their PAMD counterparts ${\left(\mathrm{\Delta }{P}_i/\mathrm{\Delta }{E}_i\right)}^{\text{PAMD}}$ found with the screening electron density definition providing $Z_0=Z_{\text{wkb}}$. The colors and lines have the same meanings as in Fig. 10.

Figure 16

The CHS pressure-to-internal energy-corrections ratios ${\left(\mathrm{\Delta }{P}_i/\mathrm{\Delta }{E}_i\right)}^{\text{CHS}}$ evaluated using the NWS model at various temperatures of silicon and $Z_0=Z_{\text{wkb}}$ as divided by their PAMD counterparts ${\left(\mathrm{\Delta }{P}_i/\mathrm{\Delta }{E}_i\right)}^{\text{PAMD}}$ found with the screening electron density definition providing $Z_0=Z_{\text{wkb}}$. The colors and lines have the same meanings as in Fig. 10.

Figure 17

The CHS pressure-to-internal energy-corrections ratios ${\left(\mathrm{\Delta }{P}_i/\mathrm{\Delta }{E}_i\right)}^{\text{CHS}}$ evaluated using the NWS model at various temperatures of silicon and $Z_0=Z_{*}$ as divided by their PAMD counterparts ${\left(\mathrm{\Delta }{P}_i/\mathrm{\Delta }{E}_i\right)}^{\text{PAMD}}$ found with the screening electron density definition providing $Z_0=Z_{\text{wkb}}$. The colors and lines have the same meanings as in Fig. 10.

Figure 18

The CHS pressure-to-internal energy-corrections ratios ${\left(\mathrm{\Delta }{P}_i/\mathrm{\Delta }{E}_i\right)}^{\text{CHS}}$ evaluated using the NWS model at various temperatures of silicon and $Z_0=Z_{\text{WS}}$ as divided by their PAMD counterparts ${\left(\mathrm{\Delta }{P}_i/\mathrm{\Delta }{E}_i\right)}^{\text{PAMD}}$ found with the screening electron density definition providing $Z_0=Z_{\text{wkb}}$. The colors and lines have the same meanings as in Fig. 10.

Figure 19

Ratios of the PAMD corrections to the NWS-model internal energy $\mathrm{\Delta }{E}_i^{\text{PAMD}}$ found at various temperatures of silicon with the screening electron density definition providing $Z_0=Z_{\text{wkb}}$ to their OCP counterparts $\mathrm{\Delta }{E}_i^{\text{OCP}}$ evaluated with $Z_0=Z_{\text{wkb}}$. Solid curves correspond to the total PAMD corrections and dashed curves—to corrections to the electrostatic energy only (violet solid and dashed curves almost coincide with each other).

Figure 20

Principal shock Hugoniots of silicon calculated by using the PAMD approach and AA models with various approximations for the ion thermal contribution to the EOS. Red solid and blue dashed curves correspond to the PAMD calculations with the screening electron density definitions providing $Z_0=Z_{\text{wkb}}$ and $Z_0=Z_{*}$, respectively; dark green and light green solid curves correspond respectively to the NWS and Liberman AA-model calculations with the ideal-gas approximation for the ion thermal contribution; magenta dashed curve—to the NWS calculation with the OCP approximation for the ion thermal contribution utilizing $Z_0=Z_{\text{wkb}}$; orange dashed-dotted, cyan dash-dotted, and yellow dash-dotted curves—to the NWS calculations with the CHS approximation for the ion thermal contribution utilizing $Z_0=Z_{\text{wkb}}, Z_0=Z_{*}$, and $Z_0=Z_{\text{WS}}$, respectively. In Fig. 20, magenta dashed, orange dashed-dotted, cyan dash-dotted, and yellow dash-dotted curves almost coincide with each other.

Figure 21

Principal shock Hugoniots of silicon calculated by using the NWS AA model with the ideal-gas approximation for the ion thermal contribution (dark green solid curve) and the PAMD approach with the screening electron density definition providing $Z_0=Z_{*}$ (blue dashed curve) as compared to the similar PAMD calculations ($Z_0=Z_{*}$) of Ref. [32] (orange dashed curve) and ab initio (QMD $+$ PIMC) simulations of Ref. [7] (brown solid curve).

Figure 22

Isochores of the internal energy-to-temperature ratio for silicon at $\rho =5{\rho }_0=11.645$ g/${\mathrm{cm}}^3$ calculated by using the PAMD approach and AA models with various approximations for the ion thermal contribution to the EOS as compared to the ab initio simulation data of Ref. [3]. Red solid and blue dashed curves correspond to the PAMD calculations with the screening electron density definitions providing $Z_0=Z_{\text{wkb}}$ and $Z_0=Z_{*}$, respectively; dark green and light green solid curves correspond respectively to the NWS and Liberman AA-model calculations with the ideal-gas approximation for the ion thermal contribution; magenta dashed curve—to the NWS calculation with the OCP approximation for the ion thermal contribution utilizing $Z_0=Z_{\text{wkb}}$. Squares and circles represent the QMD and PIMC simulation data [3], respectively.

Figure 23

Isochores of the pressure-to-temperature ratio for normal density aluminum ($\rho ={\rho }_0=2.7$ g/${\mathrm{cm}}^3$) as calculated by using the NWS model with the ideal-gas approximation for plasma ions (present work—green solid curve, Ref. [32]—cyan dashed-dotted curve) and the PAMD approach with the screening electron density definition providing $Z_0=Z_{*}$ (present work—blue solid curve, Ref. [32]—orange dashed-dotted curve). Red dashed curve corresponds to the present-work NWS-model calculation with the ideal-gas approximation for plasma ions and the term $\mathrm{\Delta }P$ (40) of Eq. (39) omitted.

Figure 24

Principal shock Hugoniots of ${\mathrm{SiO}}_2$ calculated by using the PAMD approach and AA models with various approximations for the ion thermal contribution to the EOS. Red solid curve corresponds to the PAMD calculation with the screening electron density definition providing $Z_0=Z_{\text{wkb}}$; dark green and light green solid curves correspond respectively to the NWS and Liberman AA-model calculations with the ideal-gas approximation for the ion thermal contribution; magenta dashed curve—to the NWS calculation with the OCP approximation for the ion thermal contribution utilizing $Z_0=Z_{\text{wkb}}$; orange dashed-dotted, cyan dash-dotted, and yellow dash-dotted curves—to the NWS calculations with the CHS approximation for the ion thermal contribution utilizing $Z_0=Z_{\text{wkb}}, Z_0=Z_{*}$, and $Z_0=Z_{\text{WS}}$, respectively. Dark and light brown symbols represent the Z-machine [60, 61] and NIF [62] experimental data, respectively, and violet squares—the QMD simulations of Ref. [63].

Figure 25

Electron pressure for silicon (blue curves) and oxygen (red curves) at $T=3.16$ and 10 eV calculated with the Liberman (solid curves) and NWS (dashed curves) models.