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.
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References
Gibbs, J.W.: The Scientific Papers of J. Willard Gibbs, (vol. 1). Longmans, Green and Company, New York (1906)
Cammarata, R.C., Sieradzki, K.: Surface and interface stresses. Ann. Rev. Mater. Sci. 24, 215–234 (1994)
Sun, C.Q.: Thermo-mechanical behavior of low-dimensional systems: the local bond average approach. Prog. Mater. Sci. 54, 179–307 (2009)
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)
Gao, W., Yu, S.W., Huang, G.Y.: Finite element characterization of the size-dependent mechanical behavior in nanosystems. Nanotechnology 17, 1118–1122 (2006)
Huang, Z.P., Wang, J.: A theory of hyperelasticity of multi-phase media with surface/interface energy effect. Acta Mech. 182, 195–210 (2006)
Wu, H.A.: Molecular dynamics study on mechanics of metal nanowire. Mech. Res. Commun. 33, 9–16 (2006)
Zhou, L.G., Huang, H.: Are surfaces elastically softer or stiffer? Appl. Phys. Lett. 84, 1940–1942 (2004)
Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)
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)
Ru, C.Q.: Size effect of dissipative surface stress on quality factor of microbeams. Appl. Phys. Lett. 94, 051905 (2009)
Shenoy, V.B.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B. 71, 094104 (2005)
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)
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)
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)
Nanda, K.K., Maisels, A., Kruis, F.E., et al.: Higher surface energy of free nanoparticles. Phys. Rev. Lett. 91, 106102 (2003)
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)
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)
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)
Ouyang, G., Tan, X., Yang, G.W.: Thermodynamic model of the surface energy of nanocrystals. Phys. Rev. B. 74, 195408 (2006)
Lu, H.M., Jiang, Q.: Size-dependent surface energies of nanocrystals. J. Phys. Chem. B. 108, 5617–5619 (2004)
Yao, Y., Wei, Y., Chen, S.: Size effect of the surface energy density of nanoparticles. Surf. Sci. 636, 19–24 (2015)
Wang, Y., Weissmüller, J., Duan, H.L.: Mechanics of corrugated surfaces. J. Mech. Phys. Solids. 58, 1552–1556 (2010)
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)
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)
Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995)
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)
Voter, A.F., Chen, S.P.: Accurate interatomic potentials for Ni, Al and Ni\(_3\)Al. MRS Proc. 82, 175 (1986)
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)
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)
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)
Wei, Y., Chen, S.: Size-dependent surface energy density of spherical face-centered-cubic metallic nanoparticles. J. Nanosci. Nanotechnol. 15, 9457–9463 (2015)
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This project was supported by the National Natural Science Foundation of China (Grants 11272249 and 11525209).
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Wang, J., Bian, J., Niu, X. et al. A universal method to calculate the surface energy density of spherical surfaces in crystals. Acta Mech. Sin. 33, 77–82 (2017). https://doi.org/10.1007/s10409-016-0605-z
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DOI: https://doi.org/10.1007/s10409-016-0605-z