Valence electron concentration as key parameter to control the fracture resistance of refractory high-entropy carbides

Although high-entropy carbides (HECs) have hardness often superior to that of parent compounds, their brittleness—a problem shared with most ceramics—has severely limited their reliability. Refractory HECs in particular are attracting considerable interest due to their unique combination of mechanical and physical properties, tunable over a vast compositional space. Here, combining statistics of crack formation in bulk specimens subject to mild, moderate, and severe nanoindentation loading with ab initio molecular dynamics simulations of alloys under tension, we show that the resistance to fracture of cubic-B1 HECs correlates with their valence electron concentration (VEC). Electronic structure analyses show that VEC ≳ 9.4 electrons per formula unit enhances alloy fracture resistance due to a facile rehybridization of electronic metallic states, which activates transformation plasticity at the yield point. Our work demonstrates a reliable strategy for computationally guided and rule-based (i.e., VEC) engineering of deformation mechanisms in high entropy, solid solution, and doped ceramics.

The PDF file includes: Figs. S1 to S7 Tables S1 to S3 Legends for movies S1 to S4 Other Supplementary Material for this manuscript includes the following:

Surface energies
The surface energies of one low-VEC and one high-VEC HEC composition are calculated by DFT using (i) 504 atoms divided in 6 atomic layers for {001} surfaces and (ii) 480 atoms divided in 6 layers for {110} and {111} surfaces.All structures have a 1:1 metal/carbon ratio.Metal species are randomly arranged on the metal sublattice.Conjugate-gradient energy minimization is iterated to meet an accuracy of 10 -5 eV/supercell while forces on each atom are smaller than 10 -2 eV/Å.Equilibrium structural parameters and bulk energy Ebulk are determined by relaxing the cell shape and atomic positions in 3D-periodic DFT calculations.Thus, a vacuum region of 15 Å is added to separate slab replicas along the surface normal direction, and the atomic positions are relaxed.The surface energies are calculated as Esurf = (Eslab -Ebulk)/(2A), where 2A is the total surface area.
Consistent with DFT results of Ref. ( 27), obtained for binary systems, DFT calculations of the surface energies Esurf of two representative HEC compositions (one low-VEC and one high-VEC) show that Esurf {001} < Esurf {110} < Esurf {111} (Table S1).S2.Summary of theoretical tensile strengths σT, toughness UT, and elongation at fracture δf calculated by AIMD simulations at room temperature for HEC uniaxially strained along the [001] and [110] crystallographic axes.The labels CL, TRIP, and BF, which indicate transformation and fracture mechanisms, stand for brittle "cleavage", "transformation-induced plasticity", and rapid "bond fraying", respectively.S3.Summary of sample chemistry as measured by EDS.The measured atomic percent of each cation in the HECs is in good agreement with the target 20 atom percent.Additionally, the targeted compositions match the measured chemistry.S1).Conversely, stress-activated B1→graphitic-like structural transformations significantly delay fracture in the high-VEC carbide (V,Ta,Cr,Mo,W)C.Chemical bonds have maximum length of 2.6 Å. Sphere color legend: black=C, silver=Ti, cyan=Zr, blue=Hf, ice-blue=V, pink=Nb, red=Ta, violet=Cr, yellow=Mo, orange=W.

Fig. S1 .
Fig. S1.Calculated Cauchy pressure and G/B ratios.(a) Cauchy pressure (CP) and (b) G/B ratio calculated by AIMD at 300 K and plotted as a function of the HEC alloy VEC.Table 2 in the main text reports actual compositions.

Fig
Fig. S3.X-ray diffraction patterns for each of the bulk samples.Only a rocksalt structure B1 phase is observed for each of the HECs.
Fig. S6.Statistics of indents exhibiting fracture.The plots are at constant nanoindenter load (a) and binned penetration depth (b).Gray bars in (a) indicate confidence intervals (CI).

Fig. S7 .
Fig. S7.Energy-resolved electron density of B1 (V,Ta,Cr,Mo,W)C calculated for vertical [001]elongation of 0, 10, and 28%.The lattice planes are (a) (11 ̅ 0) and (b)(100).The energy ranges used for electron-density calculations are [-2.9eV to EF] for the unstrained structure, [-2.3 eV to EF] and [-2.0 eV to EF] for 10 and 28% strained structures, respectively.These energy intervals are delimited by vertical orangedotted and black-dashed lines in Fig.5a,b,cof the main text.The electron densities are normalized by the total number of electrons (586, 529, and 502 e -, for 0, 10, and 28% strain, respectively) included in the corresponding energy ranges.A logarithmic color scale is used to facilitate visualization of strain-induced changes in the electron arrangement.Off-scale electron densities ln(ρ') < -11 or ln(ρ') > -8 [where ρ' denotes a normalized density with units Å -3 ] are colored in red and purple.

Table S1 .
DFT surface energies of representative HEC compositions.

Table 2
in the main text reports actual compositions.