Definition and Mechanics of Hardness
The hardness of a material is directly related to its incompressibility, elasticity and resistance to change in shape. A superhard material has high shear modulus, high bulk modulus and does not deform plastically. Ideally superhard materials should have a defect-free, isotropic lattice. This greatly reduces structural deformations that can lower the strength of the material. However, defects can actually strengthen some covalent structures. Traditionally, high-pressure and high-temperature (HPHT) conditions have been used to synthesize superhard materials, but recent superhard material syntheses aim at using less energy and lower cost materials.
Historically, hardness was first defined as the ability of one material to scratch another and quantified by an integer (sometimes half-integer) from 0 to 10 on the Mohs scale. This scale was however quickly found too discrete and non-linear. Measuring the mechanical hardness of materials changed to using a nanoindenter (usually made of diamond) and evaluating bulk moduli, and the Brinell, Rockwell, Knoop and Vickers scales have been developed. Whereas the Vickers scale is widely accepted as a most common test, there remain controversies on the weight load to be applied during the test. Bulk moduli, shear moduli, and elasticity are the key factors in the superhard classification process.
| Material | Diamond | c-BC2N | c-BN | OsB2 | B4C | ReB2 |
|---|---|---|---|---|---|---|
| Vickers hardness (GPa) | 115 | 76 | 48 | 37 | 30 | ~20 |
The incompressibility of a material is quantified by the bulk modulus B, which measures the resistance of a solid to volume compression under hydrostatic stress as B = –Vdp/dV. Here V is the volume, p is pressure, and dp/dV is the partial derivative of pressure with respect to the volume. The bulk modulus test uses an indenter tool to form a permanent deformation in a material. The size of the deformation depends on the material’s resistance to the volume compression made by the tool. Elements with small molar volumes and strong interatomic forces usually have high bulk moduli. Bulk moduli was the first major test of hardness and originally shown to be correlated with the molar volume (Vm) and cohesive energy (Ec) as B ~ Ec/Vm Bulk modulus was believed to be a direct measure of a material’s hardness but this no longer remains the dominant school of thought. For example, some alkali and noble metals (Pd, Ag) have anomalously high ratio of the bulk modulus to the Vickers of Brinell hardness. In the early 2000s, a direct relationship between bulk modulus and valence electron density was found as the more electrons were present the greater the repulsions within the structure were. Bulk modulus is still used as a preliminary measure of a material as superhard but it is now known that other properties must be taken into account.
In contrast to bulk modulus, shear modulus measures the resistance to shape change at a constant volume, taking into account the crystalline plane and direction of shear. The shear modulus G is defined as ratio of shear stress to sheer strain: G = stress/strain = F·L/(A·dx), where F is the applied force, A is the area upon which the force acts, dx is the resulting displacement and L is the initial length. The larger the shear modulus, the greater the ability for a material to resist sharing forces. Therefore the shear modulus is a measure of rigidity. Shear modulus is related to bulk modulus as 3/G = 2B·(1–2v)·(1+v), where v is the Poisson’s ratio, which is typically ~0.1 in covalent materials. If a material contains highly directional bonds, the shear modulus will increase and give a low Poisson ratio.
A material is also considered hard if it resists plastic deformation. If a material has short covalent bonds, atomic dislocations that lead to plastic deformation are less likely to occur than in materials with longer, delocalized bonds. If a material contains many delocalized bonds it is likely to be soft. Somewhat related to hardness is another mechanical property fracture toughness, which is a material's ability to resist breakage from forceful impact (note that this concept is distinct from the notion of toughness). A superhard material is not necessarily "supertough". For example, the toughness of diamond is about 7–10 MPa·m1/2, which is high compared to other gemstones and ceramic materials, but poor compared to many metals and alloys – common steels and aluminium alloys have the toughness values at least 5 times higher.
Several properties must be taken into account when evaluating a material as (super)hard. While hard materials have high bulk moduli, a high bulk modulus does not mean a material is hard. Inelastic characteristics must be considered as well, and shear modulus might even provide a better correlation with hardness than bulk modulus. Covalent materials generally have high bond-bending force constants and high shear moduli and are more likely to give superhard structures than, for example, ionic solids.
Read more about this topic: Superhard Materials
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