High-entropy high-hardness metal carbides discovered by entropy descriptors

High-entropy materials have attracted considerable interest due to the combination of useful properties and promising applications. Predicting their formation remains the major hindrance to the discovery of new systems. Here we propose a descriptor—entropy forming ability—for addressing synthesizability from first principles. The formalism, based on the energy distribution spectrum of randomized calculations, captures the accessibility of equally-sampled states near the ground state and quantifies configurational disorder capable of stabilizing high-entropy homogeneous phases. The methodology is applied to disordered refractory 5-metal carbides—promising candidates for high-hardness applications. The descriptor correctly predicts the ease with which compositions can be experimentally synthesized as rock-salt high-entropy homogeneous phases, validating the ansatz, and in some cases, going beyond intuition. Several of these materials exhibit hardness up to 50% higher than rule of mixtures estimations. The entropy descriptor method has the potential to accelerate the search for high-entropy systems by rationally combining first principles with experimental synthesis and characterization.

MoNbTaVWC 5−x was synthesized using both hexagonal WC and W 2 C precursors, to determine their impact on the homogeneity of the final sample. The x-ray diffraction spectrum for the sample synthesized using WC is displayed in the top panel of Figure 37, while the spectrum for the sample prepared with W 2 C is shown in the bottom panel. Both spectra feature sharp peaks at similar values of 2θ, indicating that the choice of precursor has little effect on the structure of the high-entropy material. Supplementary Figure 38. Mechanical properties. (a) Load-displacement curves for 40 indents for HfNbTaTiZrC5 at a maximum load of 50mN. This curve provides the three necessary parameters -the peak load: Pmax, the depth at peak load: δmax, and the initial unloading contact stiffness: κ, (indicated by arrows for an indent) -for obtaining Vickers hardness: HV, and elastic modulus: Comparison of calculated and measured HV for 6 single-phase 5-metal carbides. All 5-metal carbides have a higher hardness than expected from their respective ROM predictions (indicated by the dotted upward arrows), whereas the calculated HV (green circles) are consistent with the ROM for the AFLOW-AEL results (blue diamonds). The measured HV for the 6 rock-salt structure binary carbide samples along with the calculated HV for rock-salt MoC and WC are also plotted. Error bars represent the standard deviations for a series of 40 indents.
The experimentally measured load-displacement indentation curves for the hardness measurements for the HfNbTaTiZrC 5 sample are shown in Figure 38(a). Each curve provides the three parameters necessary to obtain the Vickers hardness H V and elastic modulus E [2, 3]: the peak load: P max , the depth at peak load: δ max , and the initial unloading contact stiffness: κ. The measured and calculated H V for 6 5-metal carbides along with 8 rock-salt binary carbides are plotted in Figure 38(b). The calculated H V are obtained from the thermally averaged bulk (B) and shear (G) moduli, weighted according to the Boltzmann distribution at a temperature of 2200 • C (the experimental sintering temperature), using the model of Chen et al. [4]. The experimentally measured hardness for all 5-metal carbides exceeds the rule of mixture (ROM) predictions, whereas the calculated values are consistent with the ROM for the AFLOW-AEL (Automatic Elasticity Library) results. The atomic disorder is not accounted for in the AFLOW-AEL calculations of the AFLOW-POCC ordered-configurations, suggesting that the measured enhancement in H V is disorder-driven. For HfNbTaTiZrC 5 , H V is about 50% higher than the ROM estimate. Note that the ROM predictions for Mo and/or W containing 5-metal carbides are obtained from the AFLOW-AEL calculations, since MoC and WC do not stabilize in a rock-salt phase at ambient temperature.
The values of B and G for each of the 49 configurations of 6 5-metal carbides are listed in Supplementary Table 3. These elastic moduli are calculated using the Voigt-Reuss-Hill (VRH) average within the AEL module [5] of the AFLOW framework.
Supplementary Table 3. B and G for each configuration of 6 5-metal carbides, as calculated using AFLOW-AEL. Units: B, G in GPa. conf.
Supplementary The PARTCAR (the geometry input file for the AFLOW partial occupation (AFLOW-POCC) algorithm [6]) and initial POSCARs (the VASP [7] input file for the atomic geometry) for all 49 configurations of HfNbTaTiZrC 5 are presented below. Starting with the rock-salt crystal structure (spacegroup: F m3m, #225; Pearson symbol: cF8; AFLOW Prototype: AB cF8 225 a b [8]) as the input parent lattice, the AFLOW-POCC algorithm generates a set of 49 distinct configurations, each containing one atom of each of the 5 metals, along with 5 carbon atoms. This is the minimum cell size necessary to accurately reproduce the required stoichiometry: C atom with full occupancy at the anionic lattice site and 5 different refractory metal elements with a 0.2 occupancy probability for each at the cationic lattice site. The degeneracy for each configuration g i is given by DG in the header of each POSCAR. For HfNbTaTiZrC 5 , all configurations have g i = 10, except for the 49 th where g i = 120. Each of the other 5-metal carbides also have the same 49 distinct configurations, with the same number of anions and cations in the unit cell, but with a different set of 5 refractory metal elements at the cationic sites. Note that the numerical designation of the configurations can vary from system to system. The total degeneracy ( i g i = 600) is same for all 5-metal systems.