Understanding Failure Strain and Hardness in Solid Solutions and High-Entropy Superhard Metal Dodecaborides

A range of intrinsically hard metal dodecaboride binary, ternary, and quinary solid solutions containing Y, Zr, Gd, Hf, and/or Ho were investigated using radial X-ray diffraction under nonhydrostatic compression up to 60 GPa to understand how metal composition influences boron cage structures and therefore controls hardness. Differential strain was measured to study the deformation mechanisms of these materials. Y_0.23Zr_0.53Gd_0.24B₁₂, which has the highest Vickers hardness (H_V = 46.9 ± 2.4 GPa at 0.49 N load), was also found to show the highest differential strain in all lattice planes studied, and hardness was found to correlate with differential strain across a range of materials. Differential strain did not correlate in any simple way with material composition, but it did correlate with the density of atom packing within the unit cell if samples were refined using a noncubic unit cell. Specifically, materials with the highest hardness showed anomalously compact atomic packing, with metal atoms of complementary sizes contained in the smallest volume in the rigid boron cage network. This atomic packing hypothesis was then tested by applying it to a series of quinary high-entropy borides (HEBs). The HEBs were found to have a wide range of plateau differential strain values, and all values correlated very well with the proposed atomic packing hypothesis, where higher differential strain/hardness correlates with more compact structures. Finally, the bulk modulus was calculated for all dodecaborides. All dodecaborides have high bulk moduli (K₀ > 170 GPa), but the first derivative with respect to pressure (K₀′) of most samples is well below 4, indicating that significant compression is possible in these materials before repulsive interactions start to raise the modulus. This structural flexibility is consistent with the compactness hypothesis.