Hardness and stiffness of materials are different quantities describing properties of solids. Stiffness describes how much it resists being temporarily stretched or compressed, springing back when force is released. In contrast, its hardness is resistance of material to permanent shape change, which happens when atoms are pushed to new locations. The intrinsic hardness of a single crystal can be measured by pressing against its surface with the point of a microscopic diamond pyramid until it stops and measuring the area of the dent.
Since 2005, other researchers have been developing materials of high hardness, like transition metals diborides ReB2 and OsB2. Their hardness rivals that of diamond but they don't require high temperatures and pressures to produce. One striking property of these materials is their extremely direction-dependent hardness. A theory that captures this behavior should enable reliable predicting of hardness in other materials.
In the framework of this research topic methods are being developed for the first-principles calculation of hardness based on knowledge of interatomic chemical bonds. In 2006 and 2007 we have discovered using these methods that larger coordination number, i.e. larger number of close atoms in the chemical bond, causes smaller hardness comparing with materials with lower coordination number. This achievement was in variance with general opinion presented in current literature. However, the methods from 2006 and 2007 did not take into the account directions of the interatomic bonds and could not account for orientation of measured samples toward the external pressure.
New inspiration came in 2009 thanks to an article published in Science Vol.321. The authors Lee, Wei, Kysar and Hone from Columbia University NY, USA, measured elastic properties of grafen and found taht grafen is the most stiff material ever measured. From atomic point of view grafen is two-dimensional crystal where layers are composed of hexagons of carnon atoms. Each carbon has only three closest neighbors. Like with a soap bubble, graphen membrane can be deflected by a force perpendicular to the membrane and the extent of deflection can be measured until the interatomic forces are broken and the bubble bursts. In this experiment interatomic bonds are stretched out by an external force perpendicular to the bonds, like a trampoline under the weight of an athlete. Therefore, the interatomic bonds perpendicular to the external force determine the stiffness of the graphene membrane. An existing teories of hardness were based on idea that hardness is determined by resistence of chemical bonds parallel with the external pressure against their shortening. Based on the article about graphen we came with an idea that the resistence of material against the pressure of the point of the diamond pyramid is caused by the resistence of perpendicular bonds against being stretched out.
New method of calculation of hardness finished by Šimůnek in 2009 prefers transversal bonds to longitudial ones by assigning them a larger weight. The hardness of diborides found experimentally can be reproduces quantitatively and thus the calculations reveal reasons for anisotropy of hardness. With diborides ReB2 and OsB2 the largest hardness occurs in the direction perpendicular to layers where strong chemical bonds between atoms of boron exist.
References
A. Šimůnek, Physical Review B 80, 060103(R)(2009).
A. Šimůnek, Physical Review B 75, 172108 (2007).
A. Šimůnek and J. Vackář, Physical Review Letters 96, 085501 (2006).
Unit cell of ReB2. The bond Re-B are light, the bonds B-B are dark. On the left (a) vertical and horizontal direction runs along unit cell axes c and a, on the right (b) a perspective projection is shown along the axis c. Calculated hardness: along c 50.3, along a 41.8 GPa, perpendicular to a and c 40.0 GPa.
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