# High-Temperature Thermodynamics Modeling of Graphite
## 3 Results  
By combining our calculated electronic and lattice dynamics contributions, we were able to construct the Gibbs free energy (or Helmoltz free energy at zero pressure), see Equation (1). This free energy is a fundamental quantity in thermodynamics, but it is typically not directly measured. Thus, rather than validating our ab initio free energy explicitly with experiments, we compared our results with other theories and modeling that have been developed for graphite. Figure 3 shows our first-principles free energy together with quasi-harmonic (QH) theory, CALPHAD [32], and an equation of state (EOS) model [33]. We calculated the quasi-harmonic free energy in Figure 3 from the GIBBS2 package [34, 35, 36, 37] using the energies determined by FPLMTO calculations of graphite. For all FPLMTO calculations and QH modeling, we made the same approximation to keep the axial c/a ratio constant at an experimental value. Although QH modeling for graphite has been researched, we were not able to find any free-energy data in the literature. Notably, in Figure 3, the CALPHAD and EOS models are in excellent agreement, which was expected considering that both are thermodynamically consistent representations of the experimental data. Furthermore, the figure shows a good agreement between our first-principles anharmonic theory and the two empirical models. Therefore, we conclude that the anharmonic approximation is adequate even at very high temperatures of graphite. The QH treatment, on the other hand, appears reasonable up to approximately 1000 K, but after that, it becomes increasingly inaccurate. This is certainly not unexpected; we found similar behaviors in our thermodynamic study of actinide monocarbides and monointirides [16]. Figure 2: DFT (VASP) phonon dispersions for graphite (solid line). Data shown as blue circles, red squares, and black circles represent results from room-temperature experiments [23, 26, 27]. Next, we focused on the specific heat at constant pressure. Analogous to the free energy, we obtained \(C_{p}\) as the sum of an electronic and a lattice contribution, see Equation (2). Contrary to the actinide systems, where the density of the actinide 5f states is quite large at excitation energies [16], the electronic contribution for graphite is actually very small because there are very few energy states available for electronic excitations; see the discussion of the graphite electronic density of states in Section 4. Although small, \(C_{p}{}^{d}\) is still included in the total specific heat shown in Figure 4, but on the scale of the figure, the contribution is not clearly visible. In addition to our anharmonic theory, Figure 4 shows experimental data [8,38,39], EOS models [33,40], results from quasi-harmonic theory [7] and CALPHAD [32]. The QH model [7] assumes the same fixed c/a value as our own modeling. The EOS models attempt to reproduce the most reliable experimental data for carbon. [PERSON] and [PERSON] [33] represented the zero-temperature EOS with density and temperature as the independent variables and a Birch-Murnaghan equation, while the lattice vibration terms were formulated using conventional [PERSON] and [PERSON] models. Contrarily, [PERSON] and [PERSON] [40] proposed an EOS that combines a representation where pressure and temperature are the independent variables within an accurate model modified by Murnaghan, which is able to match all known experimental data for carbon. They included thermal effects through the  
Figure 4: Experimental (red squares) [38], (blue diamonds) [8], and (purple triangles) [39] specific heats. Anharmonic (black line), quasi-harmonic (QH) (red line) [7], Sandia EOS model (open black squares) [33], [PERSON] et al. EOS model (open blue circles) [40], and CALPHAD (black dashed line) [32] specific heats. Figure 3: Anharmonic (black line), quasi-harmonic (QH) (red line), CALPHAD [32] (dashed line) and Sandia EOS model [33] (open squares) free energies for graphite.