Acta Mechanica Sinica

, Volume 33, Issue 1, pp 77–82 | Cite as

A universal method to calculate the surface energy density of spherical surfaces in crystals

Research Paper
  • 203 Downloads

Abstract

Surface energy plays an important role in the mechanical performance of nanomaterials; however, determining the surface energy density of curved surfaces remains a challenge. In this paper, we conduct atomic simulations to calculate the surface energy density of spherical surfaces in various crystalline metals. It is found that the average surface energy density of spherical surfaces remains almost constant once its radius exceeds 5 nm. Then, using a geometrical analysis and the scaling law, we develop an analytical approach to estimate the surface energy density of spherical surfaces through that of planar surfaces. The theoretical prediction agrees well with the direct atomic simulations, and thus provides a simple and general method to calculate the surface energy density in crystals.

Keywords

Surface energy density Nanoparticle Nanocavity 

References

  1. 1.
    Gibbs, J.W.: The Scientific Papers of J. Willard Gibbs, (vol. 1). Longmans, Green and Company, New York (1906)MATHGoogle Scholar
  2. 2.
    Cammarata, R.C., Sieradzki, K.: Surface and interface stresses. Ann. Rev. Mater. Sci. 24, 215–234 (1994)CrossRefGoogle Scholar
  3. 3.
    Sun, C.Q.: Thermo-mechanical behavior of low-dimensional systems: the local bond average approach. Prog. Mater. Sci. 54, 179–307 (2009)CrossRefGoogle Scholar
  4. 4.
    Duan, H.L., Wang, J., Huang, Z.P., et al.: Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J. Mech. Phys. Solids. 53, 1574–1596 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gao, W., Yu, S.W., Huang, G.Y.: Finite element characterization of the size-dependent mechanical behavior in nanosystems. Nanotechnology 17, 1118–1122 (2006)CrossRefGoogle Scholar
  6. 6.
    Huang, Z.P., Wang, J.: A theory of hyperelasticity of multi-phase media with surface/interface energy effect. Acta Mech. 182, 195–210 (2006)CrossRefMATHGoogle Scholar
  7. 7.
    Wu, H.A.: Molecular dynamics study on mechanics of metal nanowire. Mech. Res. Commun. 33, 9–16 (2006)CrossRefMATHGoogle Scholar
  8. 8.
    Zhou, L.G., Huang, H.: Are surfaces elastically softer or stiffer? Appl. Phys. Lett. 84, 1940–1942 (2004)CrossRefGoogle Scholar
  9. 9.
    Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)Google Scholar
  10. 10.
    Wang, G.F., Feng, X.Q.: Timoshenko beam model for buckling and vibration of nanowires with surface effects. J. Phys. D 42, 155411–155415 (2009)CrossRefGoogle Scholar
  11. 11.
    Ru, C.Q.: Size effect of dissipative surface stress on quality factor of microbeams. Appl. Phys. Lett. 94, 051905 (2009)CrossRefGoogle Scholar
  12. 12.
    Shenoy, V.B.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B. 71, 094104 (2005)CrossRefGoogle Scholar
  13. 13.
    Cuenot, S., Fretigny, C., Champagne, S.D., et al.: Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B. 69, 165410 (2004)CrossRefGoogle Scholar
  14. 14.
    Mi, C., Jun, S., Kouris, D.A., et al.: Atomistic calculations of interface elastic properties in noncoherent metallic bilayers. Phys. Rev. B. 77, 075425 (2008)CrossRefGoogle Scholar
  15. 15.
    Jo, M., Choi, Y., Koo, Y., et al.: Scaling behavior of the surface energy in face-centered cubic metals. Comput. Mater. Sci. 92, 166–171 (2014)CrossRefGoogle Scholar
  16. 16.
    Nanda, K.K., Maisels, A., Kruis, F.E., et al.: Higher surface energy of free nanoparticles. Phys. Rev. Lett. 91, 106102 (2003)CrossRefGoogle Scholar
  17. 17.
    Naicker, P.K., Cummings, P.T., Zhang, H., et al.: Characterization of titanium dioxide nanoparticles using molecular dynamics simulations. J. Phys. Chem. B. 109, 15243–15249 (2005)CrossRefGoogle Scholar
  18. 18.
    Bian, J.J., Wang, G.F., Feng, X.Q.: Atomistic calculations of surface energy of spherical copper surfaces. Acta Mech. Solida. Sin. 25, 557–561 (2012)CrossRefGoogle Scholar
  19. 19.
    Sun, X.Y., Qi, Y.Z., Ouyang, W., et al.: Energy corrugation in atomic-scale friction on graphite revisited by molecular dynamics simulations. Acta Mech. Sin. 32, 604–610 (2016)Google Scholar
  20. 20.
    Ouyang, G., Tan, X., Yang, G.W.: Thermodynamic model of the surface energy of nanocrystals. Phys. Rev. B. 74, 195408 (2006)CrossRefGoogle Scholar
  21. 21.
    Lu, H.M., Jiang, Q.: Size-dependent surface energies of nanocrystals. J. Phys. Chem. B. 108, 5617–5619 (2004)CrossRefGoogle Scholar
  22. 22.
    Yao, Y., Wei, Y., Chen, S.: Size effect of the surface energy density of nanoparticles. Surf. Sci. 636, 19–24 (2015)CrossRefGoogle Scholar
  23. 23.
    Wang, Y., Weissmüller, J., Duan, H.L.: Mechanics of corrugated surfaces. J. Mech. Phys. Solids. 58, 1552–1556 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Daw, M.S., Baskes, M.I.: Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B. 29, 6443 (1984)CrossRefGoogle Scholar
  25. 25.
    Mishin, Y., Mehl, M., Papaconstantopoulos, D., et al.: Structural stability and lattice defects in copper: ab initio, tight-binding, and embedded-atom calculations. Phys. Rev. B. 63, 224106 (2001)CrossRefGoogle Scholar
  26. 26.
    Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995)CrossRefMATHGoogle Scholar
  27. 27.
    Foiles, S., Baskes, M., Daw, M.: Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys. Phys. Rev. B. 33, 7983 (1986)CrossRefGoogle Scholar
  28. 28.
    Voter, A.F., Chen, S.P.: Accurate interatomic potentials for Ni, Al and Ni\(_3\)Al. MRS Proc. 82, 175 (1986)CrossRefGoogle Scholar
  29. 29.
    Adams, J., Foiles, S., Wolfer, W.: Self-diffusion and impurity diffusion of fee metals using the five-frequency model and the embedded atom method. J. Mater. Res. 4, 102–112 (1989)CrossRefGoogle Scholar
  30. 30.
    Mendelev, M., Han, S., Srolovitz, D., et al.: Development of new interatomic potentials appropriate for crystalline and liquid iron. Philos. Magn. 83, 3977–3994 (2003)CrossRefGoogle Scholar
  31. 31.
    Zhou, X.W., Wadley, H.N.G., Johnson, R.A., et al.: Atomic scale structure of sputtered metal multilayers. Acta Mater. 49, 4005–4015 (2001)CrossRefGoogle Scholar
  32. 32.
    Wei, Y., Chen, S.: Size-dependent surface energy density of spherical face-centered-cubic metallic nanoparticles. J. Nanosci. Nanotechnol. 15, 9457–9463 (2015)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Engineering Mechanics, SVLXi’an Jiaotong UniversityXi’anChina
  2. 2.CASM and Department of Mechanical and Biomedical EngineeringCity University of Hong KongHong KongChina

Personalised recommendations