Skip to main content
SpringerLink
Log in

On using many-particle interatomic potentials to compute elastic properties of graphene and diamond

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

The elastic properties of diatomic crystals are considered. An approach is proposed that permits calculating the elastic characteristics of crystals by using the interatomic interaction parameters specified as many-particle potentials, i.e., potentials that take into account the effect of the environment on the diatomic interaction. The many-particle interaction is given in the general form obtained in the framework of linear elastic deformation. It is shown that, by expanding in series in small deformation parameters, a group of nonlinear potentials frequently used to model covalent structures can be reduced to this general form. An example of graphene and diamond lattices is used to determine how adequately these potentials describe the elastic characteristics of crystals.

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. K. S. Novoselov, A. K. Geim, S. V. Morozov, et al., “Electric Field Effect in Atomically Thin Carbon Films, Science 306(5696), 666–669 (2004).

    Article  ADS  Google Scholar 

  2. K. S. Novoselov, A. K. Geim, S. V. Morozov, et al., “Two-Dimensional Gas of Massless Dirac Fermions in Graphene,” Nature 438, 197–200 (2005).

    Article  ADS  Google Scholar 

  3. M. Segal, “Wait for a Breakthrough in a Year,” Publication in Internet. URL: http://www.nanometer.ru/2009/10/27/12566498911870_157791.html (Date 27.08.2010).

  4. E. A. Belenkov, V. V. Ivanovskaya, and A. L. Ivanovskii, Nanodiamonds and Relative Carbon Materials. Computer Simulation (Ekaterinburg, UrO RAN, 2008) [in Russian].

    Google Scholar 

  5. J. C. Bowman and J. A. Krumhansl, “The Low-Temperature Specific Heat of Graphite,” J. Phys. Chem. Solids 6(4), 367–379 (1958).

    Article  ADS  Google Scholar 

  6. O. L. Blakslee, D. G. Proctor, E. J. Seldin, et al., “Elastic Constants of Compression-Annealed Pyrolytic Graphite,” J. Appl. Phys. 41(8), 3373–3382 (1970).

    Article  ADS  Google Scholar 

  7. E. J. Seldin and C. W. Nezbeda, “Elastic Constants and Electron-Microscope Observations of Neutron-Irradiated Compression-Annealed Pyrolytic and Single-Crystal Graphite,” J. Appl. Phys. 41(8), 3389–3400 (1970).

    Article  ADS  Google Scholar 

  8. R. Nicklow, N. Eakabayashi, and H. G. Smith, “Lattice Dynamics of Pyrolytic Graphite,” Phys. Rev. B 5, 4951–4962 (1972).

    Article  ADS  Google Scholar 

  9. W. B. Gauster and I. J. Fritz, “Pressure and Temperature Dependencies of the Elastic Constants of Compression-Annealed Pyrolytic Graphite,” J. Appl. Phys. 45(8), 3309–3314 (1974).

    Article  ADS  Google Scholar 

  10. M. Grimsditch, “Shear Elastic Modulus of Graphite,” J. Phys. C: Solid State Phys. 16, L143–L144 (1983).

    Article  ADS  Google Scholar 

  11. A. Bosak, M. Krisch, M. Mohr, et al., “Elasticity of Single-Crystalline Graphite: Inelastic X-Ray Scattering Study,” Phys. Rev. B 75, 153408(4) (2007).

    ADS  Google Scholar 

  12. I. W. Frank, D. N. Tannenbaim, A. M. Van der Zande, and P. L. McEuen, “Mechanical Properties of Suspended Graphene Sheets,” J. Vac. Sci. Tech. 25(6), 2558–2561 (2007).

    Article  Google Scholar 

  13. M. Poot and S. J. Van der Zant, “Nanomechanical Properties of Few-Layer Graphene Membranes,” Appl. Phys. Lett. 92, 063111 (2008).

    Article  ADS  Google Scholar 

  14. C. Lee, X. Wei, J. W. Kysar, and J. Hone, “Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene,” Science 321, 385–388 (2009).

    Article  ADS  Google Scholar 

  15. S. Bhagavantuam and J. Bhirnuassenachar, “Elastic Constants of Diamond,” Proc. Roy. Soc. London. Ser. A 187(1010), 381–384 (1946).

    Article  ADS  Google Scholar 

  16. R. F. S. Hearmon, “The Elastic Constants of Anisotropic Materials,” Rev. Modern Phys. 18(3), 409–440 (1946).

    Article  ADS  Google Scholar 

  17. E. Prince and W. A. Wooster, “Determination of Elastic Constants of Crystals from Diffuse Reflection of X-Rays. III. Diamond,” Acta Crystallogr. 6(6), 450–454 (1953).

    Article  Google Scholar 

  18. H. J. McSkimin and W. L. Bond, “Elastic Moduli of Diamond,” Phys. Rev. 105, 116–987 (1957).

    Article  ADS  Google Scholar 

  19. H. J. McSkimin and P. Andreatch, “Elastic Moduli of Diamond as a Function of Pressure and Temperature,” J. Appl. Phys. 43(7), 2944–2948 (1972).

    Article  ADS  Google Scholar 

  20. H. F. Markham, National Physical Laboratory Measurements (UK), presented by M. J. P. Musgrave, Diamond Conference, Reading (unpublished, 1965).

  21. M. H. Grimsditch and A. K. Ramdas, “Brillouin Scattering in Diamond,” Phys. Rev. B 11(10), 3139–3148 (1975).

    Article  ADS  Google Scholar 

  22. V. A. Shutilov, Foundations of Ultrasound Physics (Izd-vo LGU, Leningrad, 1980) [in Russian].

    Google Scholar 

  23. J. J. Gilman, “Origins of the Outstanding Mechanical Properties of Diamond,” Mater. Res. Innovat. 6(3), 112–117 (2002).

    Article  Google Scholar 

  24. S. Iijima, “Helical Microtubules of Graphitic Carbon,” Nature 354(6348), 56–58 (1991).

    Article  ADS  Google Scholar 

  25. D. D. Vvedensky, “Multiscale Modeling of Nanostructures,” J. Phys.: Condens. Matter 16(50), R1537–R1576 (2004).

    Article  ADS  Google Scholar 

  26. R. R. Ruoff, D. Qian, and W. K. Liu, “Mechanical Properties of Carbon Nanotubes: Theoretical Predictions and Experimental Measurements,” C. R. Physique 4(10), 993–1008 (2003).

    Article  ADS  Google Scholar 

  27. P. K. Valalvala and G. M. Odegard, “Modelling Techniques for Determination of Mechanical Properties of Polymer Nanocomposites,” Rev. Adv. Mater. Sci. 9, 34–44 (2005).

    Google Scholar 

  28. A. M. Krivtsov, Elastic Properties of One-Atomic and Two-Atomic Crystals (Izd-vo Politekh. Univ., St. Petersburg, 2009) [in Russian].

    Google Scholar 

  29. B. D. Annin, S. N. Korobeinikov, and A. V. Babichev, “Computer Simulation of a Twisted Nanotube Buckling,” Sib. Zh. Industr. Mat. 11(1), 3–22 (2008) [J. Appl. Industr. Math. (Engl. Transl.) 3 (3), 318–333 (2009)].

    MathSciNet  Google Scholar 

  30. R. V. Goldstein and A. V. Chentsov, “Discrete-Continuous Model of a Nanotube,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 57–74 (2005) [Mech. Solids (Engl. Transl.) 40 (4), 45–59 (2005)].

  31. J. Tersoff, “New Empirical Approach for the Structure and Energy of Covalent Systems,” Phys. Rev. B 37(12), 6991–7000 (1988).

    Article  ADS  Google Scholar 

  32. J. Tersoff, “Empirical Interatomic Potential for Carbon, with Applications to Amorphous Carbon,” Phys. Rev. B 61(25), 2879–2882 (1988).

    Article  ADS  Google Scholar 

  33. D.W. Brenner, “Empirical Potential for Hydrocarbons for Use in Simulating the Chemical Vapor Deposition of Diamond Films,” Phys. Rev. B 42(15), 9458–9471 (1990).

    Article  ADS  Google Scholar 

  34. D.W. Brenner, O. A. Shenderova, J. A. Harrison, et al., “Second-Generation Reactive Empirical Bond Order (REBO) Potential Energy Expression for Hydrocarbons,” J. Phys.: Condens. Matter 14, 783–802 (2002).

    Article  ADS  Google Scholar 

  35. N. L. Allinger, Y. H. Yuh, and J.-H. Lii, “Molecular Mechanics. The MM3 Force Field for Hydrocarbons. 3. The van der Waal’s Potentials and Crystal Data for Aliphatic and Aromatic Hydrocarbons,” J. Am. Chem. Soc. 111(23), 8576–8582 (1989).

    Article  Google Scholar 

  36. D.A. Case, T. E. Cheatham, T. Darden, et al., “The Amber Biomolecular Simulation Programs,” J. Comput. Chem. 26(16), 1668–1688 (2005).

    Article  Google Scholar 

  37. J.W. Ponder and D. A. Case, “Force Fields for Protein Simulations,” Adv. Prot. Chem. 66, 27–85 (2003).

    Article  Google Scholar 

  38. F. Scarpa, S. Adhikari, and A. Srikantha Phani, “Effective Elastic Mechanical Properties of Single Layer Graphene Sheets,” Nanotechnol. 20, 065709 (2009).

    Article  ADS  Google Scholar 

  39. A. Sears and R. C. Batra, “Macroscopic Properties of Carbon Nanotubes from Molecular-Mechanics Simulations,” Phys. Rev. B 69(23), 235406 (2004).

    Article  ADS  Google Scholar 

  40. A. V. Vakhrushev, A. M. Lipanov, and M. V. Suetin, “Modeling of Processes of Hydrogen Adsorption by Nanostructures,” Alternat. Energet. Ekol. 1(45), 22–29 (2007).

    Google Scholar 

  41. A. M. Krivtsov, Deformation and Failure of Solids with Microstructure (Fizmatlit, Moscow, 2007) [in Russian].

    Google Scholar 

  42. P. Zhang, Y. Huang, H. Gao, and K. C. Hwang, “Fracture Nucleation in Single-Wall Carbon Nanotubes under Tension: A Continuum Analysis Incorporating Interatomic Potentials,” Trans. ASME J. Appl. Mech. 69(4), 454–458 (2002).

    Article  ADS  MATH  Google Scholar 

  43. P. Zhang, Y. Huang, P. H. Geubelle, et al., “The Elastic Modulus of Single-Wall Carbon Nanotubes: A Continuum Analysis Incorporating Interatomic Potentials,” Int. J. Solids Struct. 39(13), 3839–3906 (2002).

    Article  Google Scholar 

  44. G. M. Odegrad, T. S. Gates, L. M. Nicholson, and K. E. Wise, “Equivalent-Continuum Modeling of Nano-Structured Materials,” Compos. Sci. Techn. 62(14), 1869–1880 (2002).

    Article  Google Scholar 

  45. R. V. Goldstein, N. M. Osipenko, and A. V. Chentsov, “To Determination of the Strength of Nanodimensional Objects,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 164–181 (2008) [Mech. Solids (Engl. Transl.) 43 (3), 453–469 (2008)].

  46. C. Li and T. W. Chou, “A Structural Mechanics Approach for the Analysis of Carbon Nanotubes,” Int. J. Solids Struct. 40(10), 2487–2499 (2003).

    Article  MATH  Google Scholar 

  47. K. I. Tserpes and P. Papanikos, “Finite Element Modeling of Single-Walled Carbon Nanotubes,” Composites B 36, 468–477 (2005).

    Article  Google Scholar 

  48. E. A. Ivanova, A. M. Krivtsov, N. F. Morozov, and A. D. Firsova, “Inclusion of the Moment Interaction in the Calculation of the Flexural Rigidity of Nanostructures,” Dokl. Ross. Akad. Nauk 391(6), 764–768 (2003) [Dokl. Phys. (Engl. Transl.) 48 (8), 455–458 (2003)].

    Google Scholar 

  49. E. A. Ivanova, A. M. Krivtsov, N. F. Morozov, and A. D. Firsova, “Description of Crystal Packing of Particles with Torque Interaction,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 110–127 (2003) [Mech. Solids (Engl. Transl.) 38 (4), 76–88 (2003)].

  50. E. A. Ivanova, A. M. Krivtsov, and H. F. Morozov, “Derivation of Macroscopic Relations of the Elasticity of Complex Crystal Lattices taking into Account the Moment Interactions at the microlevel,” Prikl. Mat. Mekh. 71(4), 595–615 (2007) [J. Appl. Math. Mech. (Engl. Transl.) 71 (4), 543–561 (2007)].

    MathSciNet  MATH  Google Scholar 

  51. I. E. Berinskii, N. G. Dvas, A. M. Krivtsov, et al. Theoretical Mechanics. Elastic and Thermal Properties of Ideal Crystals. Tutorial, Ed. by A. M. Krivtsov (Izd-vo Politekhn. Univ., St. Petersburg, 2009) [in Russian].

    Google Scholar 

  52. A. M. Krivtsov and N. F. Morozov, “Anomalies in Mechanical Characteristics of Nanometer-Size Objects,” Dokl. Ross. Akad. Nauk 381(3), 345–347 (2001) [Dokl. Phys. (Engl. Transl.) 46 (11), 825–827 (2001)].

    Google Scholar 

  53. A. M. Krivtsov and N. F. Morozov, “On Mechanical Characteristics of Nanocrystals,” Fiz. Tverd. Tela 44(12), 2158–2163 (2002) [Phys. Solid State (Engl. Transl.) 44 (12), 2260–2265 (2002)].

    Google Scholar 

  54. B. I. Yakobson, C. J. Brabeck, and J. Bernholc, “Nanomechanics of Carbon Tubes: Instabilities beyond Linear Response,” Phys. Rev. Lett. 76(14), 2511–2514 (1996).

    Article  ADS  Google Scholar 

  55. R. V. Goldstein, V. A. Gorodtsov, and D. S. Lisovenko, “Mesomechanics of Multilayer Carbon Nanotubes and Nanowhiskers,” Fizich. Mezomekh. 11(6), 25–42 (2008).

    Google Scholar 

  56. I. E. Berinskii, A. M. Krivtsov, and A. M. Kudarova, “Two-Parameter Multiparticle Model for Describing the Elastic Characteristics of Graphene,” in Progress in Continuum Mechanics, Collection of Scientific Papers Dedicated to the 70th Anniversary of Academician V. A. Levin (Dal’nauka, Vladivostok, 2009), pp. 67–82 [in Russian].

    Google Scholar 

  57. M. Arroyo and T. Belytschko, “Finite Crystal Elasticity of Carbon Nanotubes Based on the Exponential Cauchy-Born Rule,” Phys. Rev. B 69, 115415 (2004).

    Article  ADS  Google Scholar 

  58. A. Erdemir and J.-M. Martin (Editors), Supserlubricity (Elsevier, Amsterdam, 2007).

    Google Scholar 

  59. C. D. Reddy, S. Rajendran, and K. M. Liew, “Equilibrium Configuration and Continuum Elastic Properties of Finite Sized Graphene,” Nanotechnology 17, 864–870 (2006).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Saint-Petersburg State Polytechnical University, Polytekhnicheskaya 29, St. Petersburg, 195251, Russia

    I. E. Berinskii

  2. Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol’shoy pr-t 61, V.O., St. Petersburg, 199178, Russia

    A. M. Krivtsov

Authors
  1. I. E. Berinskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. M. Krivtsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. E. Berinskii.

Additional information

Original Russian Text © I.E. Berinskii, A.M. Krivtsov, 2010, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2010, No. 6, pp. 60–85.

About this article

Cite this article

Berinskii, I.E., Krivtsov, A.M. On using many-particle interatomic potentials to compute elastic properties of graphene and diamond. Mech. Solids 45, 815–834 (2010). https://doi.org/10.3103/S0025654410060063

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654410060063

Keywords

Search

Navigation