A Definition of Physical Hardness

Each of the elastic constants is a measure of hardness for a particular deformation. A modulus measures the force, or pressure, needed to cause a certain change in shape or volume. But there is one modulus in particular that seems best suited to be a general measure of hardness. This is the bulk modulus, B, which determines the volume change for a sample under hydrostatic pressure. It is defined as 

For an isotropic solid, a force exerted equally in three dimensions will produce a change in volume by a uniform decrease of distances between nearest neighbors. Bond angles will not be changed. The inverse relationship between B and k resembles the relationship between hardness and softness. Certainly one expects a solid which is physically soft to be compressible, and one which is hard to resist compression. 

However, there is an even better reason to single out 5 as a hardness factor. In classical thermodynamics there is a standard equation  

Because of Equation (6.4) Yang et al. have proposed that BVq be called the physical hardness, It has the units of energy, the same as chemical hardness. Its reciprocal is the softness, proportional to k, as desired. Their proposal was strongly reinforced by showing that BVq for a number of substances followed much the same ordering as the Moh hardness of those substances. 

We also see that W and Pt, and a few other noble metals, have H greater than carbon. On this scale they are harder than diamond, which means that H no longer matches the scratch test for hardness. This does not invaUdate as a legitimate scale for hardness. It is well defined, has a thermodynamic basis, and measures the resistance to well-defined changes. But for very hard substances, it is not equivalent to the Moh scale, which measures plastic hardness. Actually B, rather than BVq, matches the scratch test best.  

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