Modeling of vibrational and configurational degrees of freedom in hexagonal and cubic tungsten carbide at high temperatures

Transition metal carbide is a class of materials characterized by high hardness, high melting points, and low chemical reactivity. It is widely used in industrial applications involving exposure to elevated temperatures, aggressive media, and heavy loads, and is thus of technological and industrial importance. In this paper the high-temperature thermodynamic properties of tungsten carbide, WC, is studied. At most temperatures below melting, WC assumes a hexagonal structure with essentially no vacancies ($\delta$-WC). Only at very high temperatures (around 3000 K), WC crystallizes in the cubic rocksalt structure ($\gamma$-WC), which is more common for the transition metal carbides and in the case for WC can contain up to 40% carbon vacancies. At lower temperatures, $\gamma$-WC can, however, form as thin interfacial structures or nanoparticles. Hence, the thermodynamic properties of both $\gamma$-WC and $\delta$-WC are of relevance. Here, we conduct a first-principles density-functional theory based computational investigation of the $\gamma$-WC and $\delta$-WC phases, which requires modeling of high carbon vacancy concentrations at high temperatures. The configurational degrees of freedom are modeled with an alloy cluster expansion and sampled through Monte Carlo simulations. To account for the dynamic instability of the cubic $\gamma$-WC phase at low temperatures, the vibrational degrees of freedom are treated using effective harmonic models constructed from $ab$ initio molecular dynamics simulations. Finally, we obtain a part of the W-C phase diagram in reasonably quantitative agreement with experimental data.

Figure 1

Free energy per atom of the stoichiometric ($y=1)$ $\gamma$-WC and $\delta$-WC phases. The dotted line represents the free energy from electronic excitations. The dashed line represents the vibrational free energy $(\mathrm{EHM}+\mathrm{FEP})$ excluding thermal expansion. The solid line represents the total free energy $G_{\alpha }(y=1,T)$, which includes electronic and vibrational contributions as well as thermal expansion. For $\delta$-WC the total free energy is shifted by the volume-relaxed total energy of $\delta$-WC. For the cubic phase, all free energies are shifted with the relaxed total-energy difference between the two WC phases, i.e., $E_{\gamma }^0-E_{\delta }^0$.

Figure 2

EDOS per atom for the $\delta$-WC and $\gamma$-WC phases. The energy $\varepsilon$ is given relative to the Fermi energy ${\varepsilon }_{\text{F}}$. The dashed lines correspond to the Fermi distribution at 3000 K for the $\delta$-WC and $\gamma$-WC phases.

Figure 3

(a) $\delta$-WC phonon dispersion at 0 K (HA) and 3000 K (EHM). (b) VDOS for $\delta$-WC and $\gamma$-WC at 3000 K (EHM). The dashed line shows $-f_{\text{HA}}(\omega ,T)$ at $T=3000\mathrm{K}$ [see Eq. (12)], which indicates the magnitude of the contribution to the vibrational free energy. (c) $\gamma$-WC phonon dispersion at 0 K (HA), 1000 K (EHM), and 3000 K (EHM). Imaginary frequencies are shown as negative.

Figure 4

Potential energy surface for the unstable mode at the X point for stoichiometric $\gamma$-WC computed with DFT and using harmonic models corresponding to 0 K (HA), 1000 K (EHM), and 3000 K (EHM). The relaxed volume at 0 K without zero-point motion effects is used at all temperatures.

Figure 5

Formation free energy per atom of the $\gamma$-WC and $\delta$-WC phases relative bcc-W and graphite, computed using DFT and the PBE functional. The dashed line presents the experimental formation free energy given in Ref. [62] for the hexagonal phase.

Figure 6

Effective cluster interactions for the carbon-vacancy cluster expansion for $\gamma$-WC.

Figure 7

Configurational contribution to heat capacity $C_V$ as a function of temperature and composition. The heat capacity becomes maximal at the order-disorder transitions.

Figure 8

Mixing free energy per atom for the $\gamma \text{−}{\mathrm{WC}}_y$ phase as function of the concentration $y=N_{\text{C}}/N_{\text{W}}$. (a) The total mixing free energy at 0, 2000, and 2800 K, including and excluding the effect of thermal expansion. (b) Electronic, configurational, and vibrational contributions to the mixing free energy at 2000 K, excluding the effect of thermal expansion. The configurational contribution is compared with the ideal mixing model. The vibrational free energy is shown for both ground states (GS) and high-temperature representative structures (RS).

Figure 9

(a) VDOS for the ground states of $\gamma$-WC at $y=1$ and 0.5 with EHMs corresponding to 2000 K. (b) VDOS for the ground state (GS) and one representative structure (RS) from a MC simulation at a high temperature for composition $y=0.5$ with HAs.

Figure 10

VDOS for the ground states of $\gamma$-WC at $y=1$, 0.75, and 0.5 within the HA corresponding to 0 K.

Figure 11

C vacancy formation energy in the $\delta$-WC phase using HA in equilibrium with graphite (left axis, green line) and the corresponding vacancy concentration (right axis, red line).

Figure 12

Mixing free energy per atom for the $\delta$-WC and $\gamma$-WC phases as a function of C concentration at 2500 K. The convex hull is indicated by a dashed black line. $\delta$ indicates $\delta$-WC, $\gamma$ indicates $\gamma$-WC, W indicates bcc-W, and C indicates graphite.

Figure 13

Binary W-C phase diagram constructed from the free-energy landscape calculated in this work. Here, $\delta$ indicates $\delta$-WC, $\gamma$ indicates $\gamma$-WC, W indicates bcc-W, and C indicates graphite. The dashed lines indicate transitions with high uncertainty.

Figure 14

Free energy for graphite and bcc tungsten as a function of temperature.