First, we discuss a common feature of single-phase pure metals and amorphous and high-entropy
alloys: the maximum value of hardness corresponding to a valence electron count (VEC)
value of around 6.5-7. This correlation is explained by the coincidence that by subtracting
the number of sp valence electrons (Nsp = 2) from the VEC we obtain the maximal number
of unpaired d electrons, Nd = 4.5-5 in the 3d, 4d, and 5d rows of transition elements.
These unpaired d electrons form orbital overlap bonding, which is stronger than the
isotropic metallic bonds of a delocalized electron cloud. The more unpaired d electrons
there are, the higher the bonding strength. Second, we will discuss the hardness formulas
derived from cohesion energy and shear modulus. We will demonstrate that both types
of formulas originate in the electrostatic energy density of metallic bonds, expressing
a 1/R4 dependence. Finally, we show that only two parameters are sufficient to estimate
hardness: the atomic radius and the cohesion-based valence. In the case of alloys,
our formula gives a lower bound on the hardness only. It is not suitable for calculation
of the hardness increase caused by solid solution, grain size, precipitation, and
phase mixture.
