Properties of carbon up to 10 million kelvin from Kohn-Sham density functional theory molecular dynamics

Accurately modeling dense plasmas over wide-ranging conditions of pressure and temperature is a grand challenge critically important to our understanding of stellar and planetary physics as well as inertial confinement fusion. In this work, we employ Kohn-Sham density functional theory (DFT) molecular dynamics (MD) to compute the properties of carbon at warm and hot dense matter conditions in the vicinity of the principal Hugoniot. In particular, we calculate the equation of state (EOS), Hugoniot, pair distribution functions, and diffusion coefficients for carbon at densities spanning 8 g/${\mathrm{cm}}^3$ to 16 g/${\mathrm{cm}}^3$ and temperatures ranging from 100 kK to 10 MK using the Spectral Quadrature method. We find that the computed EOS and Hugoniot are in good agreement with path integral Monte Carlo results and the sesame database. Additionally, we calculate the ion-ion structure factor and viscosity for selected points. All results presented are at the level of full Kohn-Sham DFT-MD, free of empirical parameters, average-atom, and orbital-free approximations employed previously at such conditions.

Figure 1

Comparison of the thermal equation of state for carbon based on different methods. Upper panel: Pressure calculated with sqdft (circles) and the planewave Kohn-Sham DFT-MD codes pwscf (triangles) and vasp (squares). PIMC data (diamonds) [10, 28] and the sesame 7831 (colored lines) [60] data are shown for comparison. Lower panels: Pressure difference between sqdft and other approaches using the same color code and symbols as in the upper panel.

Figure 2

Hugoniot curve entirely based on Kohn-Sham DFT-MD by combining sqdft (circles) and pwscf (triangles) as calculated in this work, compared to wide-range equations of state results using sesame 7831 (solid line) [60] and fpeos (dashed line) [28]. The indicated temperatures refer to the sqdft and pwscf points. Inset: Comparison of pressure over compression ratio including pseudoatom models BEMUZE and PAMD as well as the extended FPMD method [2].

Figure 3

Pair distribution function of carbon obtained with sqdft (colored lines) along the 10 g/${\mathrm{cm}}^3$ isochore between 100 kK and 10 MK. For comparison, the pair distribution functions calculated with vasp (filled gray squares) and pwscf (open colored circles) are shown for the two lowest temperatures. Note that the curves above 100 kK are shifted along the $y$ axis to improve visibility.

Figure 4

Static structure factor for 8 (upper panel), 64 (middle panel), and 8000 (lower panel) carbon atoms at 10 g/${\mathrm{cm}}^3$ and 500 kK. The filled blue circles indicate the $k=0$ limit directly obtained from the equation of state results of the 64 atom calculations.

Figure 5

Diffusion coefficients calculated for the considered densities and temperatures. Top panel: Density dependence of the diffusion coefficients obtained with sqdft for various temperatures up to 10 MK. The Hugoniot points are marked by black crosses. Bottom panel: Arrhenius plot to illustrate the temperature behavior of the diffusion coefficient along the 10 g/${\mathrm{cm}}^3$ isochore combining planewave and Spectral Quadrature DFT. For comparison, we show the result by White and Collins [66] as well as the dataset by Grabowski et al. [3]. The solid black line illustrates the best $T^{\alpha }$ fit of our Kohn-Sham DFT-MD data.

Figure 6

Viscosity of carbon at 16 g/${\mathrm{cm}}^3$ and 2 MK calculated for different unit cell sizes containing 216 (top), 64 (middle), and 8 (bottom) atoms. The thin lines represent the individual sqdft simulations whose average is shown as thick line. The final values for each case are given as filled circles with $1\sigma$ error bars.