First-principles multiphase equation of state of carbon under extreme conditions

We describe the construction of a multiphase equation of state for carbon at extreme pressures based on ab initio electronic structure calculations of two solid phases (diamond and BC8) and the liquid. Solid-phase free energies are built from knowledge of the cold curves and phonon calculations, together with direct ab initio molecular-dynamics calculations of the equation of state, which are used to extract anharmonic corrections to the phonon free energy. The liquid free energy is constructed based on results from molecular-dynamics calculations and constraints determined from previously calculated melting curves, assuming a simple solidlike free-energy model. The resulting equation of state is extended to extreme densities and temperatures with a Thomas Fermi-based free-energy model. Comparisons to available experimental results are discussed.

Figure 1

(Color online) Several carbon phase diagrams proposed in the last decade, some of which include the BC8 phase. The bold line indicates phase boundaries used in this work for the construction of the liquid EOS.

Figure 2

(Color online) Vinet equation fits to the cold curves for diamond and BC8.

Figure 3

(Color online) Phonon density of states of diamond for two different volumes: (a) 5.57 and (b) $2.32{\text{Å}}^3/\text{atom}$, together with the double-Debye model PDOS for those same volumes. Note that in (a) the model virtually reduces to that of a single-Debye model.

Figure 4

(Color online) Phonon density of states of BC8 for two different volumes: (a) 3.19 and (b) $1.72{\text{Å}}^3/\text{atom}$, together with the double-Debye model PDOS for those same volumes.

Figure 5

(Color online) Debye temperatures ${\theta }_{\text{A}}$ and ${\theta }_{\text{B}}$ for diamond and BC8 phases at each volume (dots), together with fits assuming Grüneisen parameters which vary linearly with $V$ (lines).

Figure 6

(Color online) Phonon moments ${\theta }_0$ and ${\theta }_1$ for diamond and BC8 phases as computed directly from the PDOS (points), together with results from our double-Debye model (lines), assuming Grüneisen parameters which vary linearly with $V$.

Figure 7

(Color online) Diamond energy vs temperature obtained from averaging over constant-$T$ constant-$V$ molecular-dynamics trajectories (dots) and from quasiharmonic solid model ($\text{cold}+\text{quasiharmonic}$ ion-thermal) (lines). The anharmonic term (not shown) is added later in order to bring the solid equation of state into agreement with the MD points. The diamonds indicate solidus temperature at the appropriate density, computed using our resulting multiphase EOS model.

Figure 8

(Color online) Diamond pressure vs temperature obtained from averaging over constant-$T$ constant-$V$ molecular-dynamics trajectories. In this case the quasiharmonic solid model (lines) reproduces the molecular-dynamics points with no added correction due to volume-independent anharmonic term (see text). The diamonds indicate solidus temperature at the appropriate density, computed using our resulting multiphase EOS model.

Figure 9

(Color online) Electronic density of states of BC8 crystal for a volume of $1.72{\text{Å}}^3/\text{atom}$, corresponding to a pressure of 3685 GPa. The Fermi level is at 43.5 eV (vertical line), where the density of states is quite small.

Figure 10

(Color online) Density of states at the Fermi energy, $D\left(E_F\right)$, as a function of volume for BC8 phase. For low pressure this is effectively zero due to the presence of a finite electronic gap. The solid line is a smooth fit to the results (see text).

Figure 11

(Color online) Liquid energy vs temperature obtained from averaging over constant-$T$ constant-$V$ molecular-dynamics trajectories (dots) and from liquid model (lines). Each curve corresponds to an isochore. From the bottom to the top of the figure, the volumes represented are 5.67, 4.93, 4.50, 4.00, 2.85, 2.31, and $1.98{\text{Å}}^3/\text{atom}$. The diamonds indicate liquidus temperature at the appropriate density, computed using our resulting multiphase EOS model.

Figure 12

(Color online) Liquid pressure vs temperature obtained from averaging over constant-$T$ constant-$V$ molecular-dynamics trajectories and from liquid model (lines). Each curve corresponds to an isochore. From the bottom to the top of the figure, the volumes represented are 5.67, 4.93, 4.50, 4.00, 2.85, 2.31, and $1.98{\text{Å}}^3/\text{atom}$. The diamonds indicate liquidus temperature at the appropriate density, computed using our resulting multiphase EOS model.

Figure 13

(Color online) Phase diagram for carbon as obtained from the free energies described in this work (solid black) compared with melting curve (red error bars) of the two-phase simulations (extracted from Ref. 27). The diamond-BC8 phase line can be compared to the one resulting from neglecting anharmonic terms in both phases (dotted blue); this curve is shown to illustrate the effect of anharmonicity in the solid-solid-phase line. The slightly larger anharmonic term in the BC8 phase results in the entropy of that phase being slightly greater than the entropy of diamond, causing BC8 to be slightly favored as $T$ is increased. This moves the triple-point to lower $P$ and lower $T$ when compared with that resulting from harmonic-only free energies.

Figure 14

(Color online) Hugoniot curves of carbon in $\left(P,T\right)$-space as calculated from our EOS model, initial condition in the diamond state, and extensions to BC8 and liquid phase. The full line (red) is the diamond primary Hugoniot (calculated with an initial state at ambient condition). Note that primary Hugoniot does not reach BC8 region of stability. The dashed lines indicate secondary Hugoniot with a second shock started at 200, 300, and 400 GPa. The blue crosses are the data of Ref. 45. The black lines are the coexistence and melting lines.

Figure 15

Comparison of multi-phase and QEOS carbon isochores at the boundaries of the multiphase region.

Figure 16

The EOS table generated $\left(\rho ,T\right)$ grid used in the hydrodynamic simulations. The multiphase region (as constructed in Secs. 2, 3) is shaded, while the rest is described by the QEOS model.