Quantum molecular dynamics of warm dense iron and a five-phase equation of state

Through quantum molecular dynamics (QMD), utilizing both Kohn-Sham (orbital-based) and orbital-free density functional theory, we calculate the equation of state of warm dense iron in the density range $7–30{\mathrm{g}/\mathrm{cm}}^3$ and temperatures from 1 to 100 eV. A critical examination of the iron pseudopotential is made, from which we find a significant improvement at high pressure to the previous QMD calculations of Wang et al. [Phys. Rev. E 89, 023101 (2014)]. Our results also significantly extend the ranges of density and temperature that were attempted in that prior work. We calculate the shock Hugoniot and find very good agreement with experimental results to pressures over 20 TPa. These results are then incorporated with previous studies to generate a five-phase equation of state for iron.

Figure 1

Zero temperature calculation for bcc iron. Overlap of the cutoff radius, and the lack of sufficient number of valence states, causes the VASP 8-electron pseudopotential to be valid only for densities less than $18{\mathrm{g}/\mathrm{cm}}^3$.

Figure 2

Upper panel: Pressure isotherms at $T=1,5,10,15$ eV from the current work compared with the fit from Wang et al. [7]. The pseudopotential used in Ref. [7] is insufficient at the higher densities it is employed at and those results deviate from ours starting below 17.$5{\mathrm{g}/\mathrm{cm}}^3$. Lower panel: High temperature pressure isotherms at $T=10,15,20,50,100$ eV. Curves at 10 eV and above are from the orbital-free QMD.

Figure 3

Shock Hugoniot of iron with ${\rho }_0=7.874{\mathrm{g}/\mathrm{cm}}^3$. Our QMD results (triangles) are shown along with our $U_s\left(u_p\right)$ fit, and are seen to be in good agreement with available experimental results [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. The SESAME 2140 Hugoniot begins to soften relative to our results and experiment from just below the shock melt completion at about 12.$5{\mathrm{g}/\mathrm{cm}}^3$ and 270 GPa. The uncertainty in the experiments is mostly conveyed by the scatter in the data, except at the highest densities, above $24{\mathrm{g}/\mathrm{cm}}^3$, where the uncertainty in density is $1{\mathrm{g}/\mathrm{cm}}^3$ or more.

Figure 4

Temperature along the iron shock Hugoniot. Our calculated Hugoniot temperatures and current fit, are shown along with the SESAME 2140 temperatures and the melt curve of Bouchet et al. [6]. Also marked is the experimental shock melt completion at 270 GPa.

Figure 5

Total heat capacity of iron from experiment [49, 50] is shown with comparison to the alpha phase EOS with and without the $F_{\text{mag}}$ contribution.

Figure 6

Isobaric results for thermal expansion [52, 53] and entropy [50] at 1 atm for iron. Transitions from alpha to gamma to delta to liquid for the EOS can be seen to be in good agreement with experiment.

Figure 7

Diamond anvil cell compression at 300 K, shown with the multiphase EOS 300 K isotherm (solid line) which transitions from alpha to epsilon phase at 13 GPa. The individual phases are shown with dashed lines.

Figure 8

Iron Hugoniot for ${\rho }_0=7.874$ g/cc. Here the current EOS and two previous EOSs are compared to the QMD fit. The current EOS gives the best agreement up to 50 Mbar, while the previous Kerley EOS (SES 2150) diverges at 10 Mbar.

Figure 9

Pressure isotherms from the EOS (solid lines) are in good agreement with QMD results (triangles) for $T=1,4,8$ eV. The SESAME 2150 EOS (dashed line) isotherms are shown to be overly stiff with temperature and pressure increase. The Hugoniot path is also shown.

Figure 10

Hugoniot for gamma iron shocked from ambient pressure and 1573 K.

Figure 11

Phase diagram for iron from the current EOS. The five phases including the liquid phase are shown along with the priciple Hugoniot, ${\rho }_0=7.874{\mathrm{g}/\mathrm{cm}}^3$.