# The Role Played by Computation in Understanding Hard Materials

## Abstract

**:**

## 1. Introduction

_{2}tetrahdrea [15,16,17,18]. Under pressure, a superhard phase of SiO

_{2}named stishovite is created, consisting of octahedral SiO

_{2}units [16,19]. However, this material has not been stabilized and as such rapidly reverts back to a quartz related amorphous structure [20]. There is also a very high pressure transformation of stishovite into a CaCl

_{2}form [21] with some related enhancement of elastic properties. Thereafter enhancement of the coordination has required metals which involve d states. In this way the near nine-fold coordinated TiO

_{2}cotunnite phase has now been suggested to be the hardest known oxide [22]. However, synthesis of this material needs very high pressures (in excess of 50 GPa) while similar cotunnite phases of ZrO

_{2}or HfO

_{2}are synthesized at lower pressures between 12–18 GPa [23,24,25] and could also show superhard features.

## 2. Computational Approaches

_{tot}, and from variation of this with a unit cell distortion the elastic constants (c

_{ij}) and thereafter an effective Voigt isotropic bulk (B

_{v}) and shear elastic moduli (G

_{v}) through an expression of the form:

_{11}, c

_{12}and c

_{33}, the stability criteria is [41]:

_{11}− c

_{12}) > 0; c

_{44}> 0; (c

_{11}+ 2c

_{12}) > 0

## 3. Boron Carbon Nitrogen Structures

_{4}C, B

_{6}O or MgAlB

_{4}etc. [44]. Some of the latter materials are established for application or are under research for further improvement [45]. Other structures based on B–C or B–C–N are under intense investigation for their potential to rival the properties of diamond because of chemical differences and to produce new application. Of importance to the synthesis of these materials are the precursor phases. As with diamond, graphitic structures are considered to be the most likely possibilities here. However, in this respect, it is noted that there are differences in the relative energetics of the diamond-graphite systems as compared with the cubic BN-hexagonal BN system. In the case of the diamond-graphite system, the energy of diamond lies above that of graphite [46] whereas for the c-BN-hBN system it is the reverse [47,48].

#### 3.1. Graphitic Precursor Structures

_{n}graphitic structures have now been obtained and the frequencies are somewhat consistent with theory [49] as shown in Figure 1. The theory assumed flat graphitic related BC

_{n}sheets as indicated in Figure 2, but it likely that these will slightly buckle and this could account for the very marginal differences between theory and experiment.

**Figure 3.**Various graphitic structures considered for hexagonal BC

_{2}N. Structures (

**a**) and (

**c**) follow stacking as for graphite, structures (

**b**) and (

**d**) as for h-BN. The shading of spheres is white B, black C and grey N.

_{2}N structures (b) and (d) have the lowest energy, consistent with a sheet phase separation in this material. This phase separation could be somewhat disappointing from the superhard synthesis of a crystalline diamond-like superhard phase of BC

_{2}N. In fact, suggested crystalline structures of diamond-like BC

_{2}N have been studied using ab-initio approaches [54] with several forms having large elastic moduli again consistent to that being found. But the superhardness of such a material is likely to relate mainly to the number of C-C bonds in the material. Calculations have also suggested that large amounts of B in diamond would lead to graphitization [55].

**Table 1.**Cell structure and calculated total energy of hexagonal BC

_{2}N structures. Experimental values from [53].

a (A) | c (A) | E_{tot} (eV/atom) | |
---|---|---|---|

h-BC_{2}N (a) | 2.50 | 5.74 | −8.456 |

h-BC_{2}N (b) | 2.47 | 6.41 | −9.880 |

h-BC_{2}N (c) | 2.50 | 5.89 | −8.440 |

h-BC_{2}N (d) | 2.47 | 6.82 | −9.868 |

Expt. | 2.42 | 7.25 |

_{n}and BC

_{2}N structures appear to be quite similar around P = 18 GPa and around 2000 K. Although the structures are metastable relative to diamond/graphite or phases of BN, this is a difficulty that will be overcome [56].

#### 3.2. Superhard Structures

_{n}[10] or BC

_{2}N [54] structure has been used. Relative to diamond and graphite, all of these structures are metastable from the point of view of the relative energies and this will be a problem to overcome in a satisfactory synthesis. The essential conclusion that is emerging in the case of the superhard BC structures is that the number of C-C bonds will dictate the ultimate strength of the material [57]. Although there is still some uncertainty as to how much boron can actually be incorporated into a diamond structure, recent calculations suggest that a very heavily B-doped material will easily graphitize [55].

_{2}N is not as straightforward as the BC

_{n}system obviously because of the presence of both B-N and C-C bonds. Although various structures have been speculated for this material [58,59,60,61,62,63,64], as yet a precise structure is not determined. However, there is some indication that the relative energies of these structures are quite close.

_{3}and BC

_{2}N. Results of LDA calculations are shown in Table 2—the results suggest that N concentration dramatically affects the relative energy and may possibly affect the transition pressure.

**Table 2.**Local density approximation (LDA) calculated relative energies of precursor graphitic and super-hard diamond-like structures. A positive value indicates that the precursor phase is higher in energy that the diamond-like phase. The estimated calculational uncertainty is 10 meV/atom.

Stoichiometry | hexagonal phase | Diamond-like phase | Energy difference (eV/atom) |
---|---|---|---|

BC_{2}N | h-BC_{2}N | BC_{2}N | −0.397 |

BC_{3} | BC_{3} | BC_{3} | +0.058 |

C | graphite | diamond | −0.003 |

BN | h-BN | c-BN | +0.056 |

## 4. Advanced Nitrides

_{2}[68], InN

_{2}and OsN

_{2}[69]—have successfully been synthesized under extreme conditions of pressure and temperature. There have been several studies of the crystal structure of these nitrides [70,71,72], some pointing to an enhanced bulk modulus of the nitride over the metal with a rock-salt structure being the most stable structure [72] or a pyrite structure [73] and this arises due to strong hybridization between the metal d and N 2p states [74].

_{3}N

_{6}, Ta

_{4}N

_{5}up to Ta

_{3}N

_{5}—three polymorphs of the mononitride TaN and two phases of Ta

_{2}N. High pressure δ-TaN having a NaCl structure and orthorhombic Ta

_{3}N

_{5}have outstanding properties among the Ta-N system and are said to be superconducting.

_{2}S

_{3}structure (space group Pbnm) at high pressure and temperature conditions with the aim of obtaining the high pressure phase. The structure is shown in Figure 4.

_{2}N

_{3}displays high hardness and a unique texture and this makes it a potential superhard material for industrial applications.

_{2}N

_{3}is unstable as one of the elastic moduli (c

_{66}) is less tha zero and contravenes the Born criteria for orthrhombic systems. This is so despite the other moduli—buk or shear are quite large and indicators of a superhard material. However, more recent calculations using LDA are not consistent with this interpretation and, in fact, predict all elastic constants that are consistent with the Born criteria, thus indicating the stability of the orthorhombic structure of Ta

_{2}N

_{3}.

## 5. Strong Metallic Alloys

_{3}Al has proven an interesting material for both ab-initio [81,82] and molecular dynamic simulation [83,84]. Sutton-Chen potentials [85] seem quite good in simulating this phase with calculations of the melting temperatures being in quite good agreement with experimental. Such calculations show the distribution of atoms at various temperatures—a typical result is shown in Figure 5 for the cubic structure of Ni

_{3}Al. As can be seen, initially the cubic lattice has only slightly deformed but at very higher temperatures there is no sign of a cubic lattice, showing that the material essentially has melted. A more quantitative analysis of the behavior is based upon the behavior of the diffusion coefficient of the various component atoms which increases dramatically at the melting temperature.

**Figure 5.**Molecular Dynamics results for the melting of Ni

_{3}Al [83]. Top: T = 0 K; bottom left T = 1000 K and bottom right T = 2000 K.

_{3}Al alloy has been considered as a possible alternative to Ni

_{3}Al as a high temperature strong material but is currently hampered by its weight and high cost. However these factors are expected to be counterbalanced by platinum’s exceptional chemical stability, oxidation resistance, ductility, thermal-shock resistance, and electrical or thermal conductivity, which makes it desirable for the design of next generation of super-alloys [86,87].

## 6. Conclusions

## Acknowledgements

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Lowther, J.E.
The Role Played by Computation in Understanding Hard Materials. *Materials* **2011**, *4*, 1104-1116.
https://doi.org/10.3390/ma4061104

**AMA Style**

Lowther JE.
The Role Played by Computation in Understanding Hard Materials. *Materials*. 2011; 4(6):1104-1116.
https://doi.org/10.3390/ma4061104

**Chicago/Turabian Style**

Lowther, John Edward.
2011. "The Role Played by Computation in Understanding Hard Materials" *Materials* 4, no. 6: 1104-1116.
https://doi.org/10.3390/ma4061104