Hardness of materials at high temperature and high pressure

Abstract

The intrinsic character of the correlation between hardness and thermodynamic properties of solids has been established. The proposed thermodynamic model of hardness allows one to easily estimate hardness and bulk moduli of known or even hypothetical solids from the data on Gibbs energy of atomization of the elements or on the enthalpy at the melting point. The correctness of this approach is illustrated by an example of the recently synthesized superhard diamond-like BC₅ and orthorhombic modification of boron, γ-B₂₈. The pressure and/or temperature dependences of hardness were calculated for a number of hard and superhard phases, i.e. diamond, cBN, B₆O, B₄C, SiC, Al₂O₃, β-B₂O₃ and β-rh boron. Excellent agreement between experimental and calculated values is observed for temperature dependences of Vickers and Knoop hardness. In addition, the model predicts that some materials can become harder than diamond at pressures in the megabar range.

Keywords:

• superhard materials
• hardness theory
• high pressure
• high temperature

Acknowledgements

The authors are grateful to Agence Nationale de la Recherche for financial support (grant ANR-05-BLAN-0141).

Notes

Notes

1. See 13" for example, in which the linear dependence between the hardness and density has been established for carbon phases.

2. The use of this coefficient allows the hardness of the A^IB^VII (ε = 1/N) and A^IIB^VI (ε = 2/N) compounds, i.e. LiF, NaCl, BeO, ZnS, MgO, etc, to be established.

3. The compounds/phases considered are diamond, Si, Ge, d-Sn, SiC, cBN, wBN, c-BC₂N, α-rh B, β-rh B, B₄C, B₆O;, TiC, Si₃N₄, BeO, TiN, Al₂O₃, quartz, coesite, stishovite, WC, ReB₂, LiF, Al₂SiO₄F₂, KAlSi₃O₈, Ca₅(PO₄)₃F, CaF₂, CaCO₃, BAs, BP, AlN, AlP, AlAs, AlSb, GaN, GaP, GaAs, GaSb, InN, InP, InAs, InSb, ZnS, ZnSe, ZnTe, ZnO.

4. The /NV value has been set to a mean (∼51 GPa) of corresponding values for B₆O and B₄C; β = 0.79.

5. The compounds/phases considered are diamond, Si, Ge, d-Sn, cBN, SiC, B₄C, β-rh B, α-Al₂O₃, quartz, Ga₂O₃, Si₃N₄, GeO₂, Be₃N₂, Al₄C₃, hp-B₂O₃, BeO, TiO₂, LiF, NaCl, MgO, CaO, SrO, BaO, Cr₂O₃, FeO, Fe₂O₃, ZnO, ZnS, CaF₂, CaCO₃.

6. The compounds/phases considered are diamond, Si, Ge, d-Sn (α-Sn), β-Sn, SiC, cBN, c-BC₂N, c-BC₅, BP, TiC, ZrC, WC, TiN, ZrN, SiO₂ (stishovite), BeO, α-Al₂O₃, Al, Ca, Sr, Ba, Li, Na, K, Rb, Cs, Ti, Zr, Hf, Sc, Y, Zn, Cd, Hg (∼−40°C), Cu, Ag, Au, Cr, Mo, W, Ni, Pd, Pt, Co, Rh, Ir, Fe, Ru, Os, V, Nb, Ta, P, As, Sb, Bi, I₂, NaCl, LiF.

7. For diamond and cBN the corresponding temperatures of sublimation are 4300 K and 3300 K, respectively 20.

8. The results are qualitatively similar for any other equation of state having finite value of at high pressure.