First-principles equation-of-state table of silicon and its effects on high-energy-density plasma simulations

Using density-functional theory–based molecular-dynamics simulations, we have investigated the equation of state for silicon in a wide range of plasma density and temperature conditions of $\rho =0.001–500\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$ and $T=2000–{10}^8\mathrm{K}$. With these calculations, we have established a first-principles equation-of-state (FPEOS) table of silicon for high-energy-density (HED) plasma simulations. When compared with the widely used SESAME-EOS model (Table 3810), we find that the FPEOS-predicted Hugoniot is ∼20% softer; for off-Hugoniot plasma conditions, the pressure and internal energy in FPEOS are lower than those of SESAME EOS for temperatures above $T$ ≈ 1–10 eV (depending on density), while the former becomes higher in the low-$T$ regime. The pressure difference between FPEOS and SESAME 3810 can reach to ∼50%, especially in the warm-dense-matter regime. Implementing the FPEOS table of silicon into our hydrocodes, we have studied its effects on Si-target implosions. When compared with the one-dimensional radiation-hydrodynamics simulation using the SESAME 3810 EOS model, the FPEOS simulation showed that (1) the shock speed in silicon is ∼10% slower; (2) the peak density of an in-flight Si shell during implosion is ∼20% higher than the SESAME 3810 simulation; (3) the maximum density reached in the FPEOS simulation is ∼40% higher at the peak compression; and (4) the final areal density and neutron yield are, respectively, ∼30% and ∼70% higher predicted by FPEOS versus the traditional simulation using SESAME 3810. All of these features can be attributed to the larger compressibility of silicon predicted by FPEOS. These results indicate that an accurate EOS table, like the FPEOS presented here, could be essential for the precise design of targets for HED experiments.

Figure 1

The pressure as a function of silicon plasma temperature for all densities $(\rho =0.001–500\mathrm{g}/\mathrm{c}{\mathrm{m}}^3)$ scanned by our first-principles (KSMD+OFMD) calculations.

Figure 2

(a) The shock Hugoniot of silicon predicted by FPEOS (blue solid line) is compared to the EOS-model SESAME 3810 (red dashed line), a recent KSMD study (green dashed line) [28], and available experiments (various symbols) by Pavlovskii [16], Gust and Royce [17], and Goto et al. [18] (b) A comparison of heat capacity calculated from FPEOS and SESAME 3810 along the principal Hugoniot. The diamond phase silicon $({\rho }_0=2.329\mathrm{g}/\mathrm{c}{\mathrm{m}}^3)$ is chosen as the initial state for the Hugoniot calculations.

Figure 3

The off-Hugoniot equation-of-state comparisons between FPEOS and SESAME 3810. The (a) pressures and (b) internal energies are plotted as functions of temperature for a silicon density of $\rho =5\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$.

Figure 4

Same as Fig. 3 except for a different silicon density of $\rho =10\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$.

Figure 5

The pressure comparisons between FPEOS and SESAME 3810 for higher densities of silicon plasmas: (a) $\rho =50\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$ and (b) $\rho =500\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$.

Figure 6

The percentage difference of pressure and internal energy between FPEOS and SESAME 3810 as a function of silicon density for the isotherms of $T=15625\mathrm{K}$ (a,b) and $T=125000\mathrm{K}$ (c,d).

Figure 7

The laser pulse shape and target dimensions for implosion simulations to test the silicon-EOS effects. The capsule consists of a 40-μm-thick silicon shell (${\rho }_0=2.329\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$) filled with 3 atm of DT gas. The initial target radius $R=1200\mu \mathrm{m}$. The total laser energy is 800 kJ with 8-ns pulse duration, available at NIF-type facilities.

Figure 8

Comparisons of density and electron temperature profiles predicted by the two lilac simulations using FPEOS (blue solid lines) and SESAME 3810 (red dashed line) EOS models. The snapshot was taken at $t=0.9\mathrm{ns}$, when the first shock was still propagating in the silicon layer.

Figure 9

Same as Fig. 8 but for different implosion times: (a) $t=5.4\mathrm{ns}$ (in flight of the imploding shell) and (b) $t=7.9\mathrm{ns}$ (the end of shell acceleration).

Figure 10

Comparisons of density and ion temperature profiles predicted by the two lilac simulations using FPEOS (blue solid lines) and SESAME 3810 (red dashed line) EOS models. The snapshot is shown for the instant at their peak neutron production ($t\sim 9.0\mathrm{ns}$).

Figure 11

Comparisons of (a) the areal density ρR and (b) the total neutron yield as functions of time, for the two lilac simulations using FPEOS (blue solid lines) and SESAME 3810 (red dashed line) EOS models.