Skip to main content
SpringerLink
Log in

Microscopic models of hardness

  • Theory of Hardness and Superhard Materials
  • Production, Structure, Properties
  • Published:
Journal of Superhard Materials Aims and scope Submit manuscript

Abstract

Recent developments in the field of microscopic hardness models have been reviewed. In these models, the theoretical hardness is described as a function of the bond density and bond strength. The bond strength may be characterized by energy gap, reference potential, electron-holding energy or Gibbs free energy, and different expressions of bond strength may lead to different hardness models. In particular, the hardness model based on the chemical bond theory of complex crystals has been introduced in detail. The examples of the hardness calculations of typical crystals, such as spinel Si3N4, stishovite SiO2, B12O2, ReB2, OsB2, RuB2, and PtN2, are presented. These microscopic models of hardness would play an important role in search for new hard materials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. McColm, I.J., Ceramic Hardness, New York: Plenum Press, 1990.

    Google Scholar 

  2. Handbook of Ceramic Hard Materials, vols. 1, 2, Riedel, R., Ed., Weinheim: WILEY-VCH Verlag, Germany, 2000.

    Google Scholar 

  3. Shaw, M. C., The Fundamental Basis of the Hardness Test. Chapter 1, in The Science of Hardness Testing and its Research Applications, Westbrook, J.H. and Conrad H., Eds., Ohio, USA: American Society for Metals, Metals Park, 1973.

    Google Scholar 

  4. Szymanski, A. and Szymanski, J.M., Hardness Estimation of Minerals Rocks and Ceramic Materials, Amsterdam: Elsevier, 1989.

    Google Scholar 

  5. Glazov, V.M. and Vigdorovich, V.N., Mikrotverdoct’ metallov (Microhardness of Metals), Moscow: Metallurgiya, 1969.

    Google Scholar 

  6. Gilman, J.J., Why Silicon is Hard, Science, 1993, vol. 261, no. 5127, pp. 1436–1439.

    Article  CAS  Google Scholar 

  7. Veprek, S. The Search for Novel Superhard Materials, Vac. Sci. Technol. A, 1999, vol. 17, no. 5, pp. 2401–2420.

    Article  CAS  Google Scholar 

  8. Levine, J.B., Tolbert, S.H., and Kaner, R.B., Advancements in the Search for Superhard Ultra-Incompressible Metal Borides, Adv. Funct. Mater., 2009, vol. 19, pp. 3519–3533.

    Article  CAS  Google Scholar 

  9. Oganov, A.R., Chen, J., Gatti, C., et al., Ionic High Pressure Form of Elemental Boron, Nature, 2009, vol. 457, no. 7231, pp. 863–867.

    Article  CAS  Google Scholar 

  10. Oganov, A.R. and Solozhenko, V.L., Boron: a Hunt for Superhard Polymorphs, J. Superhard Mater., 2009, vol. 31, no. 5, pp. 285–291.

    Article  Google Scholar 

  11. Solozhenko, V.L., Andrault, D., Fiquet, G., et al., Synthesis of Superhard Cubic BC2N, Appl. Phys. Lett., 2001, vol. 78, no. 10, pp. 1385–1387.

    Article  CAS  Google Scholar 

  12. Irifune, T., Kurio, A., Sakamoto, S., et al., Ultrahard Polycrystalline Diamond from Graphite, Nature, 2003, vol. 421, no. 6923, pp. 599–600.

    Article  CAS  Google Scholar 

  13. Wang, Z., Zhao, Y., Tait, K., et al., Quenchable Superhard Carbon Phase Synthesized by Cold Compression of Carbon Nanotubes, in Proc. Nat. Acad. Sci., U.S.A, 2004, vol. 101, no. 38, pp. 13699–13702.

    Article  CAS  Google Scholar 

  14. Dubrovinskaia, N., Dubrovinsky, L., Crichton, W., et al., Aggregated Diamond Nanorods, the Densest and Least Compressible Form of Diamond, Appl. Phys. Lett., 2005, vol. 87, no. 8, pp. 083106 1–083106 3.

    Article  CAS  Google Scholar 

  15. Dubrovinsky, L.S., Dubrovinskaia, N.A, Swamy, V., et al., The Hardest Known Oxide, Nature, 2001, vol. 410

  16. Brazhkin, V., Dubrovinskaia, N., Nicol, M., et al., What Does “Harder than Diamond” Mean?, Nature Materials, 2004, vol. 3, no. 2, pp. 576–577.

    Article  CAS  Google Scholar 

  17. Dubrovinskaia, N., Solozhenko, V.L., Miyajima, N., et al., Superhard Nanocomposite of Dense Polymorphs of Boron Nitride: Noncarbon Material Has Reached Diamond Hardness, Appl. Phys. Lett., 2007, vol. 90, no. 10, pp. 101912 1–101912 3.

    Article  CAS  Google Scholar 

  18. Liu, A.Y. and Cohen, M.L., Prediction of New Low Compressibility Solids, Science, 1989, vol. 245, no. 4920, pp. 841–842.

    Article  CAS  Google Scholar 

  19. Teter, D.M. and Hemley, R.J., Low Compressibility Carbon Nitrides, ibid., 1996, vol. 271, no. 5245, pp. 53–55.

    Article  CAS  Google Scholar 

  20. Kroll, P., Hafnium Nitride with Thorium Phosphide Structure: Physical Properties and an Assessment of the Hf-N, Zr-N, and Ti-N Phase Diagrams at High Pressures and Temperatures, Phys. Rev. Lett., 2003, vol. 90, no. 12, pp. 125501 1–125501 4.

    Article  CAS  Google Scholar 

  21. Shiyou Chen, Gong, X.G., and Wei, Su-Huai, Superhard Pseudocubic BC2N Superlattices, ibid., 2007, vol. 98, no. 1, pp. 015502 1–015502 4.

    Article  CAS  Google Scholar 

  22. Li, Q., Ma, Ya., Oganov, A.R., et al., Superhard Monoclinic Polymorph of Carbon, ibid., 2009, vol. 102, no. 17, pp. 175506 1–175506 4.

    Google Scholar 

  23. Wang, H., Li, Q., Li, Y., et al., Ultra-Incompressible Phases of Tungsten Dinitride Predicted from First Principles, Phys. Rev. B, 2009, vol. 79, no. 13, pp. 132109 1–132109 6.

    Article  CAS  Google Scholar 

  24. Li., Q., Wang, M., Oganov, A.R., et al., Rhombohedral Superhard Structure of BC2N, J. Appl. Phys., vol. 105, no. 5, pp. 053514 1–053514 4.

  25. Zhang, M., Wang, M., Cui, T., et al., Electronic Structure, Phase Stability, and Hardness of the Osmium Borides, Carbides, Nitrides, and Oxides: First-Principles Calculations, J. Phys. Chem. Solids, 2008, vol. 69, no. 8, pp. 2096–2102.

    Article  CAS  Google Scholar 

  26. Li, Y., Wang, H., Li, Q., et al., Twofold Coordinated Ground-State and Eightfold High-Pressure Phases of Heavy Transition Metal Nitrides MN2 (M=Os, Ir, Ru and Rh), Inorganic Chemistry, 2009, in press.

  27. Haines, J., Leger, J. M., and Bocquillon, G., Synthesis and Design of Superhard Materials, Annu. Rev. Mater. Res., 2001, vol. 31, pp. 1–23.

    Article  CAS  Google Scholar 

  28. Hong, S., Jhi, S.H., Roundy, D., et al., Structural Forms of Cubic BC2N, Phys. Rev. B, 2001, vol. 64, no. 9, p. 094108.

    Article  CAS  Google Scholar 

  29. He, J.L., Guo, L.C., Wu, E., et al., First-Principles Study of B2CN Crystals Deduced from the Diamond Structure, J. Physics-Condensed Matter, 2004, vol. 16, no. 46, pp. 8131–8138.

    Article  CAS  Google Scholar 

  30. Wang, Y.X., Masao, A., Taizo, S., Fan, Z., Ab initio Study of Monoclinic Iridium Nitride as a High Bulk Modulus Compound, Phys. Rev. B, 2007, vol. 75, no. 10, pp. 104110 1–104110 6.

    Article  CAS  Google Scholar 

  31. Gilman, J.J., Electronic Basis of Hardness and Phase Transformations, J. Phys. D: Appl. Phys., 2008, vol. 41, no. 7, pp. 074020–074024.

    Article  CAS  Google Scholar 

  32. Cohen, M.L., Calculation of Bulk Moduli of Diamond and Zinc-Blende Solids, Phys. Rev. B, 1985, vol. 32, no. 12, pp. 7988–7991.

    Article  CAS  Google Scholar 

  33. Zhang, S., Li, H., Li, H., et al., Calculation of the Bulk Modulus of Simple and Complex Crystals with the Chemical Bond Method, J. Phys. Chem. B, 2007, vol. 111, no. 6, pp. 1304–1309.

    Article  CAS  Google Scholar 

  34. Sung, C.M. and Song, M., Carbon Nitride and Other Speculative Superhard Materials, Mater. Chem. Phys., 1996, vol. 43, no. 1, pp. 1–18.

    Article  CAS  Google Scholar 

  35. Brazhkin, V., Lyapin, A., and Hemley, R.J., Harder than Diamond: Dreams and Reality, Philosophical Magazine A, 2002, vol. 82, no. 2, pp. 231–253.

    CAS  Google Scholar 

  36. Ceder, G., Predicting Properties from Scratch, Science, 1998, vol. 280, no. 5366, pp. 1099–1100.

    Article  CAS  Google Scholar 

  37. Teter, D.M., Computational Alchemy: the Search for New Superhard Materials, Mater. Res. Soc. Bull., 1998, vol. 23, no. 1, pp. 22–27.

    CAS  Google Scholar 

  38. Gilman, J.J., Hardness—a Strength Microprobe, Chapter 4 of The Science of Hardness Testing and Its Research Applications, Westbrook, J.H. and Conrad, H., Eds., Ohio, USA: American Society for Metals, Metals Park, 1973.

    Google Scholar 

  39. Nye, J.F., Physical Properties of Crystals, London: Oxford University Press, 1957.

    Google Scholar 

  40. Hill, A., The Elastic Behavior of a Crystalline Aggregate, Proc. Phys. Soc., 1952, vol. 65, pp. 349–354.

    Article  Google Scholar 

  41. Gao, F.M., He, J.L., Wu, E.D., et al., Hardness of Covalent Crystals, Phys. Rev. Lett., 2003, vol. 91, no. 1, pp. 015502 1–015502 4.

    Article  CAS  Google Scholar 

  42. Gao, F.M., Hardness Estimation of Complex Oxide Materials, Phys. Rev. B, 2004, vol. 69, no. 9, pp. 094113 1–094113 6.

    Article  CAS  Google Scholar 

  43. Šimunek, A. and Vackar, J., Hardness of Covalent and Ionic Crystals: First-Principle Calculations, Phys. Rev. Lett., 2006, vol. 96, no. 8, pp. 085501 1–085501 4.

    Article  CAS  Google Scholar 

  44. Šimunek, A., How to Estimate Hardness of Crystals on a Pocket Calculator, Phys. Rev. B, 2007, vol. 75, no. 17, pp. 172108 1–172108 4.

    Article  CAS  Google Scholar 

  45. Li, K., Wang, X., Zhang, K., and Xue, D., Electronegativity Identification of Novel Superhard Materials, Phys. Rev. Lett., 2008, vol. 100, no. 23, pp. 235504 1–235504 4.

    Google Scholar 

  46. Li, K. and Xue, D., Hardness of Materials: Studies at Levels from Atoms to Crystals, Chinese Science Bulletin, 2009, vol. 54, no. 1, pp. 131–136.

    Article  CAS  Google Scholar 

  47. Mukhanov, V.A., Kurakevych, O.O., and Solozhenko, V.L., The Interrelation between the Hardness and Compressibility of Substances and Their Structure and Thermodynamic Properties, J. Superhard Materials, 2008, vol. 30, no. 6, pp. 368–378.

    Article  Google Scholar 

  48. Pauling, L., The Nature of the Chemical Bond, Ithaca, New York: Cornell Univ. Press, 1960.

    Google Scholar 

  49. Shanker, J. and Verma, M.P., The Fractional Ionic Character of Alkali and Silver Halide Crystals, Pramana, 1973, vol. 1, no. 6, pp. 243–246.

    Article  CAS  Google Scholar 

  50. Hidaka, T., Pauling’s Ionicity and Phillips’ Ionicity, J. Phys. Soc. Japan, 1978, vol. 44, no. 4, pp. 1204–1207.

    Article  CAS  Google Scholar 

  51. Phillips, J.C., Ionicity of the Chemical Bond in Crystals, Rev. Mod. Phys., 1970, vol. 42, no. 3, pp. 317–356.

    Article  CAS  Google Scholar 

  52. Van Vechten, J.A., Quantum Dielectric Theory of Electronegativity in Covalent Systems, Phys. Rev. B, 1969, vol. 182, no. 3, pp. 891–905.

    Article  Google Scholar 

  53. Van Vechten, J.A., Quantum Dielectric Theory of Electronegativity in Covalent systems. II. Ionization Potentials and Interband Transition Energies, ibid., 1969, vol. 187, no. 3, pp. 1007–1020.

    Article  Google Scholar 

  54. Levine, B.F., Bond Susceptibilities and Ionicities in Complex Crystal Structures, J. Chem. Phys., 1973, vol. 59, no. 3, pp. 1463–1485.

    Article  CAS  Google Scholar 

  55. Levine, B.F., Bond-Charge Calculation of Nonlinear Optical Susceptibilities for Various Crystal Structures, Phys. Rev. B, vol. 7, no. 6, pp. 2600–2626.

  56. Zhang, S.Y., Investigation of Chemical Bonds on Complex Crystals, Chin. J. Chem. Phys., 1991, vol. 4, pp. 109–115.

    Google Scholar 

  57. Zhang, S.Y., Gao, F.M., Wu, C.X., Chemical Bond Properties of Rare Earth Ions in Crystals, J. Alloys Compd., 1998, nos. 275–277, pp. 835–837.

  58. Gao, F.M. and Zhang, S.Y., Investigation of Covalency and Spectrum Shifts in 3d Transition Metal Compounds, Chin. J. Inorg. Chem., 2000, vol. 16, no. 5, pp. 751–756.

    CAS  Google Scholar 

  59. Gao, F.M., Li, D.C., and Zhang, S.Y., Mossbauer Spectroscopy and Chemical Bonds in BaFe12O19 Hexaferrite, J. Phys.-Cond. Matter., 2003, vol. 15, no. 29, pp. 5079–5084.

    Article  CAS  Google Scholar 

  60. Krell, A., Vickers Hardness and Microfracture of Single and Polycrystalline Al2O3, Kristall und Technik, 1980, vol. 15, no. 12, pp. 1467–1474.

    Article  CAS  Google Scholar 

  61. Luo, S.N., Swadenerb, J.G., Ma, C., and Tschauner, O., Examining Crystallographic Orientation Dependence of Hardness of Silica Stishovite, Physica B, 2007, vol. 399, no. 6, pp. 138–142.

    Article  CAS  Google Scholar 

  62. Brazhkin, V.V., Grimsditch, M., Guedes, I., et al., Elastic Moduli and the Mechanical Properties of Stishovite Single Crystals, Phys. Uspekhi, 2002, vol. 45, no. 4, pp. 447–448.

    Article  Google Scholar 

  63. Zerr, A., Miehe, G., Serghiou, G., et al., Synthesis of Cubic Silicon Nitride, Nature, 1999, vol. 400, no. 6742, pp. 340–342.

    Article  CAS  Google Scholar 

  64. Mo, S.D., Ouyang, L., Ching, W.Y., et al., Interesting Physical Properties of the New Spinel Phase of Si3N4 and C3N4, Phys. Rev. Lett., 1999, vol. 83, no. 24, pp. 5046–5049.

    Article  CAS  Google Scholar 

  65. Sekine, T. and Mitsuhashi, T., High-Temperature Metastability of Cubic Spinel Si3N4, Appl. Phys. Lett., 2001, vol. 79, no. 17, pp. 2719 1–2719 3.

    Article  CAS  Google Scholar 

  66. Gao, F.M., Xu, R., and Liu, K., Origin of Hardness in Nitride Spinel Materials, Phys. Rev. B, 2005, vol. 71, no. 5, pp. 052103 1–052103 4.

    Article  CAS  Google Scholar 

  67. Soignard, E., Somayazulu, M., Dong, J., et al., High Pressure-High Temperature Synthesis and Elasticity of the Cubic Nitride Spinel γ-Si3N4, J. Phys. C, 2001, vol. 13(22), pp. 557–563.

    CAS  Google Scholar 

  68. Jiang, J.Z., Kragh, F., Frost, D.J., et al., Hardness and Thermal Stability of Cubic Silicon Nitride, J. Phys.: Condens. Matter., 2001, vol. 13, no. 22, p. L515–L520.

    Article  CAS  Google Scholar 

  69. Zerr, A., Kempf, M., Schwarz, M., et al., Elastic Moduli and Hardness of Cubic Silicon Nitride, J. Am. Ceram. Soc., 2002, vol. 85, no. 1, pp. 86–90.

    Article  CAS  Google Scholar 

  70. Chung, H.Y., Weinberger, M.B., Levine, J.B., et al., Synthesis of Ultra-Incompressible Superhard Rhenium Diboride at Ambient Pressure, Science, 2007, vol. 316, no. 5823, pp. 436–439.

    Article  CAS  Google Scholar 

  71. Qin, J., He, D., Wang, J., et al., Is Rhenium Diboride a Superhard Material?, Adv. Mater., 2008, vol. 20, no. 24, pp. 4780–4783.

    Article  CAS  Google Scholar 

  72. Cumberland, R.W., Weinberger, M.B., Gilman, J.J., et al., Osmium Diboride, an Ultra-Incompressible Hard Material, J. Amer. Chem. Soc., 2005, vol. 127, no. 20, pp. 7264–7265.

    Article  CAS  Google Scholar 

  73. Crowhurst, J.C., Goncharov, A.F., Sadigh, B., et al., Synthesis and Characterization of the Nitrides of Platinum and Iridium, Science, 2006, vol. 311, no. 5765, pp. 1275–1278.

    Article  CAS  Google Scholar 

  74. Gu, Q., Krauss, G., and Steurer, W., Transition Metal Borides: Superhard versus Ultra-Incompressible, Adv. Mater., 2008, vol. 20, pp. 3620–3626.

    Article  CAS  Google Scholar 

  75. Emin, D., Aselage, T.L., Switendick, A.C., et al., Boron-Rich Solids, New York: AIP, 1990.

    Google Scholar 

  76. Hubert, H., Devouard, B., Garvie, L.A.J., et al., Icosahedral Packing of B12 Icosahedra in Boron Suboxide (B6O), Nature, 1998, vol. 391, pp. 376–378.

    Article  Google Scholar 

  77. Gao, F.M., Hou, L., and He, Y.H., Origin of Superhardness in Icosahedral B-12 Materials, J. Phys. Chem. B, 2004, vol. 108, no. 35, pp. 13069–13073.

    Article  CAS  Google Scholar 

  78. Gao, F.M., Qin, X.J., Wang, L.Q., et al., Prediction of New Superhard Boron-Rich Compounds, J. Phys. Chem. B, 2005, vol. 109, no. 31, pp. 14892–14895.

    Article  CAS  Google Scholar 

  79. Olofsson, M. and Lundstrom, T., Synthesis and Structure of Nonstoichiometric B6O, J. Alloys. Compounds, 1997, vol. 257, nos. 1–2, pp. 91–95.

    Article  CAS  Google Scholar 

  80. Kurakevych, O.O. and Solozhenko, V.L., Rhombohedral Boron Subnitride, B13N2, by X-ray Powder Diffraction, Acta Crystallogr. Sect. C, 2007, vol. 63, pp. i80–i82.

    Article  CAS  Google Scholar 

  81. Solozhenko, V.L., Kurakevych, O.O., and Oganov, A.R., On the Hardness of a New Boron Phase, Orthorhombic γ-B28, J. Superhard Materials, 2008, vol. 30, no. 6, pp. 428–429.

    Article  Google Scholar 

  82. Gou, H.Y., Zhang, J.W., and Gao, F.M., First-Principles Calculations of Boron-Rich Compounds of B13N2 and B12C2X (X=Si, Ge), J. Phys.-Cond. Matter., 2008, vol. 20, no. 50, pp. 505211–505216.

    Article  CAS  Google Scholar 

  83. Rama Krishna, M.V. and Friesner, R. A., Quantum Confinement Effects in Semiconductor Clusters, J. Chem. Phys., 1991, vol. 95, no. 11, pp. 8309–8322.

    Article  Google Scholar 

  84. Halperin, W.P., Quantum Size Effects in Metal Particles, Rev. Mod. Phys., 1986, vol. 58, no. 3, pp. 533–606.

    Article  CAS  Google Scholar 

  85. Tse, J.S., Klug, D.D., and Gao, F.M., Hardness of Nanocrystalline Diamonds, Phys. Rev. B, 2006, vol. 73, no. 14, pp. 140102 1–140102 4.

    Article  CAS  Google Scholar 

  86. Gao, F.M., Klug, D.D., and Tse, J.S., Theoretical Study of New Superhard Materials: B4C3, J. Appl. Phys., 2007, vol. 102, no. 8, pp. 084311 1–084311 5.

    Article  CAS  Google Scholar 

  87. Andrievski, R.A., Superhard Materials Based on Nanostructured High-Melting Point Compounds: Achievements and Perspectives, Int. J. Refract. Met. Hard Mater., 2001, vol. 19, nos. 4–6, pp. 447–452.

    Article  CAS  Google Scholar 

  88. Siegel, R.W. and Fougere, G.E., Mechanical Properties of Nanophase Metals, Nanostruct.Mater., 1995, vol. 6, nos. 1–4, pp. 205–216.

    Article  CAS  Google Scholar 

  89. Yamakov, V., Wolf, D., Phillpot, S.R., et al., Deformation-Mechanism Map for Nanocrystalline Metals by Molecular-Dynamics Simulation, Nat. Mater., 2004, vol. 3, no. 1, pp. 43–47.

    Article  CAS  Google Scholar 

  90. Segall, M.D., Shah, R., Pickard, C.J., and Payne, M.C., Population Analysis of Plane-Wave Electronic Structure Calculations of Bulk Materials, Phys. Rev. B, 1996, vol. 54, no. 23, pp. 16317–16320.

    Article  CAS  Google Scholar 

  91. Ching, W.Y. and Xu, Y.N., Nonscalability and Nontransferability in the Electronic Properties of the Y-Al-O system, ibid., 1999, vol. 59, no. 20, pp. 12815–12821.

    Article  CAS  Google Scholar 

  92. Gao, F.M., Theoretical Model of Intrinsic Hardness, ibid., 2006, vol. 73, no. 13, pp. 132104 1–132104 4.

    Article  CAS  Google Scholar 

  93. Mulliken, R.S., Electronic Population Analysis on LCAO-MO Molecular Wave Functions. I, J. Chem. Phys., 1955, vol. 23, no. 10, pp. 1833–1840.

    Article  CAS  Google Scholar 

  94. Gou, H.Y., Hou, L., Zhang, J.W., and Gao, F.M., Pressure-Induced Incompressibility of ReC and Effect of Metallic Bonding on its Hardness, Appl. Phys. Lett., 2008, vol. 92, no. 24, pp. 241901 1–241901 3.

    Article  CAS  Google Scholar 

  95. Zhang, X., Luo, X., Han, J., et al., Electronic Structure, Elasticity and Hardness of Diborides of Zirconium and Hafnium: First-Principles Calculations, Computational Mater. Sci., 2008, vol. 44, pp. 411–421.

    Article  CAS  Google Scholar 

  96. Gou, H.Y., Hou, L., Zhang, J.W., et al., Theoretical Hardness of PtN2 with Pyrite Structure, Appl. Phys. Lett., 2006, vol. 89, no. 14, pp. 1419101–1419103.

    Article  CAS  Google Scholar 

  97. Gou, H.Y., Hou, L., Zhang, J.W., et al., Cubic Gamma-Be3N2: A Superhard Semiconductor Predicted from First Principles, ibid., 2007, vol. 90, no. 19, pp. 191905 1–191905 3.

    Article  CAS  Google Scholar 

  98. Wang, Z.B., Gao, F.M., Li, N., et al., Density Functional Theory Study of Hexagonal Carbon Phases, J. Phys.-Cond. Matter., 2009, vol. 21, no. 23, pp. 235401–235406.

    Article  CAS  Google Scholar 

  99. Li, C.L., Kuo, J.L., Wang, B.A., et al., A New Layer Compound Nb4SiC3 Predicted from First-Principles Theory, J. Phys. D-App. Phys., 2009, vol. 42, no. 7, pp. 075404–075409.

    Article  CAS  Google Scholar 

  100. Liu, H., Zhu, J., Liu, Y., and Lai, Z., First-Principles Study on the Mechanical Properties of Vanadium Arbides VC and V4C3, Mater. Lett., 2008, vol. 62, nos. 17–18, pp. 3084–3086.

    Article  CAS  Google Scholar 

  101. Chen Z., Gu M., Sun C. Q., et al., Ultrastiff Carbides Uncovered in First Principles, Appl. Phys. Lett., 2007, vol. 91, no. 6, pp. 061905 1–061905 3.

    Google Scholar 

  102. Wang, M., Li, Y., Cui, T., et al., Origin of Hardness in WB4 and Its Implications for ReB4, TaB4, MoB4, TcB4, and OsB4, ibid., 2008, vol. 93, no. 10, pp. 101905 1–101905 3.

    Google Scholar 

  103. Ribeiro, F.J., Tangney, P., Louie, S.G., and Cohen, M.L., Hypothetical Hard Structures of Carbon with Cubic Symmetry, Phys. Rev. B, 2006, vol. 74, no. 17, pp. 172101 1–172101 4.

    Article  CAS  Google Scholar 

  104. Smedskjaer, M.M., Jensen, M., Yue, Y.Z., Theoretical Calculation and Measurement of the Hardness of Diopside, J. Amer. Ceram. Soc., 2008, vol. 91, no. 2, pp. 514–518.

    Article  CAS  Google Scholar 

  105. Gao, L.H. and Gao, F.M., Chemical Bond Properties and Hardness Estimation of Rare Earth Garnets, Mater. Chem. Phys., 2009, vol. 113, no. 1, pp. 145–149.

    Article  CAS  Google Scholar 

  106. Gou, H.Y., Hou, L., Zhang, J.W., et al., First-Principles Study of Low Compressibility Osmium Borides, Appl. Phys. Lett., 2006, vol. 88, no. 22, pp. 221904 1–221904 3.

    Article  CAS  Google Scholar 

  107. Šimunek A., Anisotropy of Hardness from First Principles: The Cases of ReB2 and OsB2, Phys. Rev. B, 2009, vol. 80, no. 6, pp. 060103 1–060103 4.

    Article  CAS  Google Scholar 

  108. Solozhenko, V.L., Kurakevych, O., Andrault, D., et al., Ultimate Metastable Solubility of Boron in Diamond: Synthesis of Superhard Diamond-Like BC5, Phys. Rev. Lett., 2009, vol. 102, pp. 015506 1–015506 4.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Applied Chemistry, Yanshan University, Qinhuangdao, 066004, China

    F. M. Gao & L. H. Gao

Authors
  1. F. M. Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. H. Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © F. Gao, L. Gao, 2010, published in Sverkhtverdye Materialy, 2010, Vol. 32, No. 3, pp. 9–32.

About this article

Cite this article

Gao, F.M., Gao, L.H. Microscopic models of hardness. J. Superhard Mater. 32, 148–166 (2010). https://doi.org/10.3103/S1063457610030020

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1063457610030020

Key words

Search

Navigation