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 Superhard materials are ultra-incompressible materials with high electron density and high bond covalency. As a result of their unique properties, these materials are of great interest in many industrial areas including but not limited to abrasives, polishing and cutting tools, wear-resistant and protection coatings and machining. Superhard materials are defined as such by the Vickers Hardness Test which is designed to measure the technical hardness of a material. A material classified as superhard under the Vickers Test exceeds 40 Gpa (gigapascals). By this convention, diamond is the hardest known material to date with a Vickers Hardness in the range of 70-150 GPa. Diamond demonstrates both high thermal conductivity and insulative properties and much attention has been put into finding practical applications of this material. However, diamond has several limitations for mass industrial application, including its high cost of synthesis, high pressure and high temperature (HPHT) synthetic requirements and oxidation at high temperatures. In addition diamond has demonstrated an inability to effectively cut ferrous materials including steel due to the formation of iron carbide during the cutting process. While diamond is the "gold standard" of hardness, its other molecular properties make it undesirable for many mass industrial applications. The search for new superhard materials is driven by both the prospect of understanding and utilizing micro- and macroscopic properties of these unique materials as well as addressing the technical importance of superhard materials for industrial purposes. Recent research of superhard materials has been focusing on compounds that may be used as an alternative to diamond. Superhard materials can be generally classified into two categories: intrinsic compounds and extrinsic compounds. The intrinsic group includes diamond, cubic boron nitride (c-BN, HV ≈ 48 GPa), carbonitrides, ternary compounds (such as B-N-C triangles), etc. which possess an innate hardness. Conversely, extrinsic materials are those that have superhardness and other mechanical properties that are determined by their microstructure.

Details and Mechanics of Hardness
The definition of hardness is now majorly accepted as follows: 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. Any dislocations or deformations within the material must be extremely small in order for the material to pass the Vickers test and therefore be classified as superhard. Ideally superhard materials should be isotropic. This greatly reduces structural deformations that can lower the strength of the material. But in reality defects can actually strengthen some covalent structures. The details of these strengthening effects will be discussed later. Traditionally HPHT conditions are used to synthesize materials with little impurities. Recent superhard material syntheses focus on using lower energetic methods and lower cost materials which are still able to give materials with few deformations.

The process of defining materials as superhard has progressed greatly. Hardness was first defined as the ability of one material to scratch another. This ability was measured by the nonlinear Mohs scale but quickly found to be complicated by a number of factors including bulk behaviors of materials at different temperatures. I.e. a cold material may have a much higher Mohs hardness than the same material at warmer temperatures. Measuring the mechanical hardness of materials changed to using a material indenter, giving bulk moduli readings. The Brinell, Rockwell, Vickers and Knoop scales use this measure of hardness. As stated in this page previously, the Vickers scale is the most definitive test for superhard materials today. However, there remain controversies with the classification of materials as superhard in terms of the weight load necessary for resistance tests. Today three major components (bulk moduli, shear moduli, and elasticity) remain key factors in the superhard classification process.

The incompressibility of a material is quantified by bulk modulus, the resistance of a solid to volume compression under hydrostatic stress. The bulk modulus equation is as follows:

Eq. 1  B = -Vdp/dV

where V is the volume, p is the 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 (B) was the first major test of hardness and originally shown to be correlated with the molar volume (Vm) and cohesive energy (Ec) as shown below:

Eq. 2  BEc/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. In the early 2000’s 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 plane of shear and direction of shear. The shear modulus is defined as ratio of shear stress to sheer strain:

Eq. 3   G = stress/strain=F/A/(dx/L)

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 in the following manner:

Eq. 4   G = 3/2B(1-2v)/(1+v)

where B is the bulk modulus, G is the shear modulus and v is the Poisson’s ratio.

The typical value for the Poisson’s ratio is 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 that this material does not have the strength of say, ReB2 or diamond.

It is important to note that one must take into account all of these properties in order to classify a material as hard. While it can be said that a hard material must have a high bulk modulus, a hard bulk modulus does not directly classify a material as hard. Elastic characteristics must be account for as well. In fact, shear modulus provides a better correlation with hardness than bulk modulus. Covalent materials generally have high bond-bending force constants, fixed lattice points, have highly directional bonds and have high shear moduli. Ionic materials have lower bond-bending force constants and lower shear moduli. Therefore covalent materials by nature are more likely to give superhard structures.

Diamond
Diamond is an allotrope of carbon where the crystal packing is arranged in a modified version of face-centered cubic (fcc) known as diamond lattice. It is a very valuable material due to its highly unique physical and chemical properties. Besides being known for its hardness (115 GPa) and incompressibility, it is also targeted for its potential in optical and electrical applications. Unfortunately, natural diamond is not applicable in mainstream industry as it has a very low natural abundance due to the intricate process required for its formation within the earth’s surface. Because of this, synthetic diamond became a major research focus following the discovery that it is made of pure carbon.

Synthetic Diamond
The high pressure synthesis of diamond in 1953 in Sweden and in 1954 in the USA, made possible by the development of new apparatus and techniques, became a milestone in synthesis of artificial superhard materials. The synthesis clearly showed the potential of high-pressure applications for industrial purposes and stimulated growing interest in the field. Four years after the first synthesis of artificial diamond, cubic boron nitride cBN was obtained that was found to be second hard phase.

Synthetic diamond can exist as a single, continuous crystal or as small, interconnected polycrystals. The inherent spatial separation of these subunits causes the formation of grains, which are visible by the unaided eye due to the light absorption and scattering properties of the material.

The hardness of synthetic diamond is very dependent on the relative purity of the crystal itself (70-150 GPa). The more perfect the crystal structure, the harder the diamond becomes. It has recently been reported that single-crystal diamonds and HPHT nanocrystalline diamonds are capable of being harder than natural diamond. This is due to careful control of crystal formation and exclusion of impurities.

Historically, it was thought that synthetic diamond needed to be structurally perfect to be considered useful. This is because diamond was mainly preferred for its aesthetic qualities--small flaws in structure and composition gave crystalline grains and changes in color. Although this is true, the properties associated with these small changes has led to interesting new potential applications of synthetic diamond. Impurities can be evenly spaced throughout the entire lattice or can be grouped into regions called inclusions. These inclusions can change the properties in a macroscopic region of the crystal.

All diamonds have a high optical dispersion due to their inherent structure. It is possible to change diamond from an electrical insulator to a conductor by adding boron into the lattice. Some current research is focusing on turning diamond into a superconductor. The addition of nitrogen into the lattice will cause dislocations and imperfections that will not only show up as a yellow color, but will cause an increase in compressive stress of the lattice, which could potentially make it harder and tougher.

Despite being an electrical insulator, diamond has a very high thermal conductivity due to the highly covalent nature of the crystal. It has the highest thermal conductivity of any pure solid. These effects can be enhanced by the synthetic creation process. As an example, single synthetic diamond crystals that are enriched with carbon-12 have the highest known thermal conductivity at room temperature of any known material at 30 Watt/cm-Kelvin (W/cm-K). To compare, copper has a thermal conductivity of approximately 6 W/cm-K, which is five times lower.

History
Cubic boron nitride or c-BN, is the second strongest and hardest material (H ~ 48 GPa). c-BN was first synthesized in 1957 by Robert H. Wentorf, a physical chemist (of General Electric), shortly after the synthesis of diamond The general process for c-BN synthesis is the dissolution of hexagonal boron nitride (h-BN) in a solvent-catalyst (SC), usually alkali or alkaline-earth metals and their nitrides, followed by spontaneous nucleation of c-BN under HPHT conditions. The yield of c-BN is lower and substantially slower compared to diamond's synthetic route due to the complicated intermediate steps. Its insolubility in iron and other metal alloys makes it more useful for some industrial applications than diamond.



Pure cubic boron nitride is transparent or slightly amber. Different colors can be produced depending on defects or an excess of boron (less than 1%). Defects can be produced by doping solvent-catalysts (i.e. Li, Ca, or Mg nitrides) with Al, B, Ti, or Si. This induces a change in the morphology and color of c-BN crystals. The result is darker and larger (500 μm) crystals with better shapes and a higher yield.

Structure & Properties
Cubic boron nitride adopts a sphalerite crystal structure. The structure can be considered by replacing every two carbon atoms in diamond with one boron atom and one nitrogen atom. The short B-N (1.57 Å) bond is close to the C-C bond length (1.54 Å), creating strong covalent bonding between atoms in the same fashion as diamond. Although c-BN is superhard, the slight decrease in covalency for B-N bonds compared to C-C bonds reduces the hardness from ~100 GPa down to 48 GPa. As diamond is less stable than graphite, c-BN is less stable than h-BN, but the conversion rate between those forms is negligible at room temperature.

Cubic boron nitride is insoluble in iron, nickel, and related alloys at high temperatures, but it binds well with metals due to formation of interlayers of metal borides and nitrides. It is also insoluble in most acids, but is soluble in alkaline molten salts and nitrides, such as LiOH, KOH, NaOH-Na2CO3, NaNO3 which are used to etch c-BN. Because of its stability with heat and metals, c-BN surpasses diamond in mechanical applications. The thermal conductivity of BN is among the highest of all electric insulators. In addition, c-BN consists of only light elements and has low X-ray absorptivity, capable of reducing the X-ray absorption background.

Research and Development
With great chemical and mechanical robustness, c-BN has wide applications as abrasives, cutting tools, and even one of the popular X-ray membranes. Several hundred tons of of c-BN amounts is produced worldwide each year. . By modification, “Borazon”, a US brand name, is used in industrial applications to shape tools, as it can withstand temperatures greater than 2000 °C. Cubic boron nitride-coated grinding wheels, referred to as Borazon wheels, are routinely used in the machining of hard ferrous metals, cast irons, and nickel-base and cobalt-base superalloys. Other brand names, such as "Elbor" and "Cubonite", are marketed by Russian vendors.

New approaches in research focus on improving c-BN pressure capabilities of the devices used for c-BN synthesis. At present, the capabilities for the production of c-BN are restricted to pressures of about 6GPa. Increasing the pressure limit will permit synthesis of larger single crystals than from the present catalytic synthesis. However, the use of solvents under supercritical conditions for c-BN synthesis has been shown to reduce pressure requirements. The high cost of c-BN still limits its application, which motivates the search for other superhard materials.

Carbonitride
The structure of carbon nitride (C3N4) was first proposed by Liu and Cohen in 1985. This compound, isostructural with Si3N4, was predicted to be harder than diamond in 1989. The calculated bond length was 1.47 Å, 5% shorter than the C-C bond length in diamond. However, despite two decades pursuing this compound, no synthetic sample of C3N4 has ever validated the hardness predictions.

Later calculations indicated that the shear modulus is 60% of that of diamond, and carbon nitride is less hard than c-BN. This discrepancy comes from the difficulty in material synthesis and C3N4's instability. Carbon nitride is only stable at a pressure that is higher than that of the graphite-to-diamond transformation. The synthesis conditions would require extremely high pressures because carbon is four- and six-fold coordinated. In addition, C3N4 would pose problems of carbide formation if they were to be used to machine ferrous metals. Although publications have reported preparation of C3N4 at lower pressure than stated, synthetic C3N4 was not proved superhard. Thus, there remains a need for a scalable, alternative route to produce new superhard phases that combine high hardness with chemical inertness and low-cost synthesis to yield practical benefits.

Boron Carbon Nitride
The similar atomic sizes of B, C, and N, as well as the similar structures of carbon and boron nitride polymorphs, suggest that it might be possible to synthesize diamond-like phase containing all three elements. It is also possible to make compounds containing B-C-O, B-O-N, or B-C-O-N under high pressure, but their synthesis would expect to require a complex chemistry and in addition, their elastic properties would be inferior to that of diamond.

History
Beginning in 1990, a great interest has been put in studying the possibility to synthesize dense ternary phase in the B-C-N system. Dense B–C–N ternary phases are expected to be thermally and chemically more stable than diamond, and harder than c-BN, and would therefore be excellent materials for high speed cutting and polishing of ferrous alloys. These characteristic properties are attributed to the diamond-like structure combined with the sp3 σ-bonds among carbon and the hetero atoms. However, a superhard phase c-BC2N with diamond-like structure was synthesized by Solozhenko et al until 2001.

Synthesis
The first reported BCxNy films were synthesized by Badzian et al. using the deposition of the ternary by thermal CVD in 1972. However, data on the attempted synthesis of B-C-N dense phases reported by different authors have been contradictory. It is unclear whether the synthesis products are diamond-like solid solutions between carbon and boron nitride or just mechanical mixtures of highly dispersed diamond and c-BN. Until 2001, a diamond-like-structured c-BC2N with was synthesized at pressures >18 GPa and temperatures >2200 K by a direct solid-state phase transition of graphite-like (BN)0.48C0.52. The Vickers (HV) and Knoop (HK) hardnesses of c-BC2N, are intermediate between diamond and c-BN, making the new phase the second hardest known material. Ternary B–C–N phases can also be made using shock-compression synthesis. It was suggested that the great potential of the B–C–N triangle can be extended further when ternary and quanternary compounds with silicon are included.

History
Research has been focused on moving away from carbon based systems and investigating metal borides. The goal of this class of materials is to be easily synthesized in large quantities under ambient conditions. This is an important advantage due to the high temperature and high pressure (HPHT) requirements of the previously addressed synthetic diamond and cubic boron nitrides. A few examples of these metal borides include RuB2, OsB2, and ReB2. The density of states of these materials reveal their metallic nature, but the extensive covalent bonding between the B-B and M-B lead to their high hardness. Metals such as Osmium, Rhenium, Tungsten, etc. are desirable due to the high electron density, small atomic radius, high bulk modulus, and highly controlled directional bonding with boron. The M-B bond contributes to this due to the overlapping of the transition metal d states and boron p states.

Osmium Diboride
‎Because OsB2 has a high bulk modulus (395 GPa) it is targeted to be a good candidate as a superhard material. The maximum Vickers hardness that it is measured to possess is 37 GPa. So while it is said to be classified as superhard in the past, we now know that the properties of OsB2 still need to be optimized in order to try bring it up to the classification of a superhard material.

Synthesis and Material Structure
A common way to synthesize Osmium Diboride (OsB2) is by a solid-state metathesis reaction containing a 2:3 mixture of OsCl3:MgB2. After the MgCl2 product is washed away, X-ray diffraction indicates products of OsB2, OsB, and Os. Heating this product at 1000 °C for three days produces pure OsB2 crystalline product. OsB2 is an orthorhombic structure (Pmmn space group) and consists of two planes of osmium atoms separated by a non-planar layer of hexagonal boron atoms with lattice parameters of a = 4.684 Å, b = 2.872 Å, c = 4.096 Å. The b direction of the crystal is the most compressible and the c direction is the least compressible. This can be explained by its orthorhombic structure. When looking at the boron and osmium atoms in the a and b directions, they are arranged in a way that is offset from one another. Therefore, when they are compressed they are not pushed right up against one another. Electrostatic repulsion is the force that maximizes the materials incompressibility and so in this case the electrostatic repulsion is not taken full advantage of. When compressed in the c direction, the osmium and boron atoms are almost directly in line with one another and the electrostatic repulsion is therefore high causing direction c to be the least compressible. These results imply that if boron is more evenly distributed throughout the lattice, there could potentially be higher incompressibilities. Electron backscattering diffraction coupled with hardness measurements quantitatively defines the structure. This technique reveals that in the (010) plane, the 100 is 54% harder than that of the 001 direction. This is seen by looking at how long the indentation is along a certain direction (related to the indentations made with a Vickers hardness test). Along with the alignment of the atoms, this is also due to the short covalent B-B (1.80 Å) bonds in the 100 direction, which are absent in the 001 direction (B-B = 4.10 Å).

Current and Future Applications
Just as cubic boron nitrides are a model after the carbon analogue, there will be a new generation of superhard materials generated from the structure of MB2. It is also a possibility to produce a superhard material using a combination of metals such as Os1-x¬RuxB2, of whose chemical properties may be studied further in depth in the future. Many applications of superhard materials require that they be applied as a thin film or coating. This is an advantage for a material containing a relatively expensive metal where cutting down on the amount needed will improve the cost effectiveness. These films will be useful if they possess a good combination of physical and mechanical properties, such as chemical stability, thermal stability, oxidative resistance, high fracture toughness, good adhesion to the substrate, and high hardness. The importance of these coatings for machine applications is demonstrated by the fact that 40% of all cutting tools are coated with a wear resistant film. If the productivity of expensive automated machines could be increased, companies could save on high costs currently needed for environmentally unfriendly coolants. Under certain conditions, these superhard materials are shown to increase in hardness when synthesized as a film compared to a bulk material. The discovery of testing the properties of these superhard films is very recent and is a pathway to improving further upon these composite materials. A second improvement involves increasing the number of boron atoms attached to a metal atom. This induces more covalent bonding and strengthen the lattice. A metal currently be looked upon to make such bonds is tungsten in WB4. Advantages of this complex include the lower metal content reduce the overall cost of production and the higher content of boron lower the density of the compound and in turn lower the weight of the material.

History
Rhenium was targeted as a candidate for superhard metal borides because of its desirable physical and chemical characteristics. It has a high electron density, a small atomic radius, and a high bulk modulus. When combined with boron, it is cable of having very ordered and directional bonding with high covalent nature allowing it to be incompressible and potentially very hard A wide array of Rhenium Borides have been investigated including Re3B, Re7B3, Re2B, ReB, Re2B3, Re3B7, Re2B5, ReB3, and ReB2. Each of these materials have their own set of properties and characteristics. Some show promise as superconductors, some have unique elastic and electronic properties. However, the most relevant to the discussion of superhard materials is ReB2.

Synthesis
Rhenium diboride (ReB2) is a refractory compound which was first synthesized in the 1960s. It has been determined to be ultra-incompressible and processing superhardness (hardness ≥ 40GPa). ReB2 has been synthesized by arc melting, zone melting, and by using optical floating zone furnaces. It has a very high melting point approaching 2400 °C. It can be analyzed as a single crystal for its intrinsic properties, but much more is gained from viewing it as a layered crystal because it is anisotropic. An example synthesis of this material is the flux method, which is conducted by placing rhenium metal and amorphous boron in an alumina crucible with excess aluminum. This can be run with a ratio of 1:2:50 for Re:B:Al, with the excess aluminum as a growth medium. The crucible is placed in an alumina tube and inserted into a resistively heated furnace with flowing argon gas. It is heated to 1400 °C for 5 hours, then cooled to 700 °C, and subsequently quickly cooled to room temperature. The aluminum is then dissolved in NaOH, washed with water, and air dried. Each synthesis route has its own drawbacks, and this one gives small inclusions of aluminum incorporated into the crystal lattice.

Material Properties
Rhenium diboride has experimentally determined space groups of both hexagonal (P63mc) and orthorhombic (Cmcm) depending on the phase. Furthermore, the close-packed Re layers alternate with puckered triangular boron layers along the (001) plane. This can be seen above when looking at the example of osmium diboride. The lattice constants of ReB2 were calculated to be a=2.894 Å and c=7.416 Å, which are very close to the experimentally determined values of a=2.897 Å and c=7.472 Å. The correlation between these suggests that both the calculation and experimental results are reasonable. The density of states for ReB2 has the lowest value of N(EF) of all the studied borides, indicating that it possesses the strong covalent bonding behavior which lends to its superhard character.

Due to the anisotropic nature of this material, the hardness can give different values depending on which plane is examined. The (002) plane exhibits a maximum Vickers hardness value of 40.5 GPa, while the perpendicular planes were 6% lower at 38.1 GPa. These values decrease with increased load, settling at around 28 GPa each. The nanoindentation values were found to be 36.4 GPa and 34.0 GPa for the (002) and perpendicular planes respectively. Due to the crystal structure, the (002) plane exhibits the most hardness because it contains the most covalent character. It is important to note that a wide range of hardness values have been reported and this is mainly due to the relative purity of the material. The above values were measured to have a Re:B ratio of approximately 1.00:1.85 instead of the theoretical 1:2 ratio. As purity increases, the material will become harder, and as more boron is lost it will subsequently become less hard. Rhenium diboride also has a reported bulk modulus of 383 GPa and a sheer modulus of 273 GPa.

Rhenium diboride exhibits metallic conductivity which increases as temperature decreases. The room temperature (300K) resistivity has been measured to be about 45µΩ-cm. This is consistent with ReB2 having metal character and exhibiting a nonzero density of states due to the d and p overlap of rhenium and boron respectively. At this point, it is the only superhard material with metallic behavior. It has been determined that it exhibits no superconductivity down to temperatures as low as 2.0K. The material also exhibits relatively high thermal stability. Depending on the heating method, it will maintain its mass up to temperatures of 600-800°C, with any drop being due to loss of absorbed water. A small loss of mass can then be seen at temperatures approaching 1000°C. It performs better when a slower heat ramp is utilized. Part of this small drop at around 1000°C can be explained by the formation of a dull B2O3 coating on the surface as boron is leeched out of the solid, which serves as a protective coating, thereby reducing additional boron loss. This can be easily dissolved by methanol to restore the material to it native shiny state.

As with all superhard materials, it is suited for use with wear-resistant coatings and enhancement of cutting tools. Research is also being done to examine its potential for use in the defense and aerospace industries.

Diboride Controversies
It should also be noted that the hardness properties of metal borides have been called into question by various researchers within this field recently. Although they do not dispute the results that it can exhibit a Vickers hardness rating of over 40 GPa, they do note that this is relative to the experimental load. The hardness ratings over 40 GPa were all measured with an effective load between 0.5-1 N. Other researchers have reported hardness values in the range of 19-17 GPa as a load of 3-49 N was applied. The counter to this argument is that this depends on the relative purity of the samples, the overall sample size, and the use of appropriate load ranges. Regardless, this has led many to believe that superhard materials are not possible outside of the light elements (boron, carbon, nitrogen, and oxygen) due to the presence of metallic and ionic bonds in the transition metal borides. (2008)

Other Rhenium Borides
The theoretical elastic properties calculations of Re2B3, ReB, and Re2B suggest that they also may be classified as superhard materials. More research is necessary to confirm this hypothesis. In light of the recent controversy surrounding the reported hardness values, these might not be as extensively examined as they would have otherwise.

Other Boron Rich Superhard Materials
Other boron-rich compounds include B4C and B6O. Amorphous a-B4C has a hardness of 50 GPa, which is in the range of superhardness. It can be looked at as consisting of boron icosahedra-like crystals embedded in an amorphous medium. However, when studying the crystalline form of B4C, the hardness is only about 30 GPa. This crystalline form has the same stoichiometry as B13C3, which consists of boron icosahedra connected by boron and carbon atoms. Boron suboxide (B6O) has a hardness of about 35 GPa (Hubert 1998). Its structure contains eight B12 icosahedra units, which are sitting at the vertices of a rhombohedral unit cell. There are two oxygen atoms located along the (111) rhombohedral direction. These lighter boron-rich materials have become second to BCNs (boron carbon nitrides) because their properties are not equivalent to this light element doped carbon material. There also may be some future applicable materials to be studied in combining the properties of B6O and c-BN into a mixed B6O–c-BN complex.

Nano Superhard Materials
Nanosuperhard materials fall into the extrinsic category of superhard materials. Because molecular defects affect the superhard properties of bulk materials it is obvious that the microstructure of superhard materials give the materials their unique properties. Focus on synthesizing nano superhard materials is around minimizing microcracks occurring within the structure through grain boundary hardening. The elimination of microcracks can strengthen the material by 3 to 7 times its original strength. Grain boundary strengthening is described by the Hall-Petch equation:

Eq. 5    σc = σo + kgb/√d

Here σC is the critical fracture stress, d the crystallite size, and σo and kgb are constants.

If a material is brittle, the strength of that compound depends mainly on its resistance to forming microcracks. The critical stress which causes the growth of a microcrack of size a0 is given by a general formula,

Eq. 6    σc =kcrack√(2Eγs/πa0)∝1/√d

Here E is the Young’s modulus, kcrack is a constant dependent on the nature and shape of the microcrack and the stress applied and ƴs the surface cohesive energy.

The average hardness of a material decreases with d (crystallite size) decreasing below 10 nm. There have been many mechanisms proposed for grain boundary sliding and hence material softening, but the details are still not understood. Besides grain boundary strengthening, much attention has been put into building microheterostructures, or nanostructures of two materials with very large differences in elastic moduli. Heterostructures were first proposed in 1970. These proposed structures contained such highly ordered thin layers that the layers could not theoretically be separated by mechanical means. These highly ordered heterostrucutres were believed to be stronger than simple mixtures. This theory was confirmed with Al-Cu and Al-Ag structure tests. After the formation of Al-Cu and Al-Ag, a number of researchers followed suit to make multilayer systems including Cu/Ni, TiN/VN, W/WN, Hf/HfN and more. In all cases if the lattice period decreased the hardness factor increased. There are many other proposed equations describing the behavior of nanostructured superhard materials and their dependence upon the structure/layers but they will not be discussed on this page.

One common form of a nanostructured material is aggregated diamond nanorods. Because of high order at the nanomolecular level, nanodiamond is harder than bulk diamond and is currently the hardest (~100-150 GPa), least incompressible material known.