×

zbMATH — the first resource for mathematics

Instabilities of core-shell heterostructured cylinders due to diffusions and epitaxy: Spheroidization and blossom of nanowires. (English) Zbl 1162.74317
Summary: The morphological instabilities of core-shell heterostructures consisting of an epitaxially stressed cylinder embedded in a finite shell are investigated. The contributions of the surface diffusion, interface diffusion and volume diffusion, and a combination of these processes to the mass transport along the surface of the shell and the interface between the cylinder and shell are examined, respectively. As the driving forces for the instabilities, the capillary terms and the mismatch strain energy are also taken into account. The governing evolution equations of the surface and interface are established in terms of a linear instability analysis of the longitudinal and radial variations of the surface and the interface positions. The critical conditions for the zero and maximum fluctuations of the surface and interface in the radial and longitudinal directions are given. For a core-shell cylinder of nickel, it is demonstrated that at a small size, the contribution of the surface/interface diffusion to the morphological evolution is larger than that of the volume diffusion, even at an elevated temperature. It is shown that the analysis of the instabilities of the longitudinal and radial fluctuations can be used to predict the spheroidization and blossom of core-shell nanowires.

MSC:
74A50 Structured surfaces and interfaces, coexistent phases
74H55 Stability of dynamical problems in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Asaro, R.J.; Tiller, W.A., Interface morphology development during stress-corrosion cracking. I. via surface diffusion, Metall. trans., 3, 1789-1796, (1972)
[2] Cahn, J.W., On spinodal decomposition, Acta metall., 9, 795-801, (1961)
[3] Cahn, J.W.; Larche, F., Surface stress and the chemical-equilibrium of small crystals. II. solid particles embedded in a solid matrix, Acta metall., 30, 51-56, (1982)
[4] Colin, J., Morphological instabilities of stressed axi-symmetrical structures embedded in a matrix: volume diffusion approach, Acta mater., 52, 4985-4995, (2004)
[5] Colin, J., Morphological instability of stressed spherical particles growing by diffusion in a matrix, Phys. rev. B, 71, 165403, (2005)
[6] Crank, J., The mathematics of diffusion, (1975), Clarendon Press Oxford · Zbl 0071.41401
[7] Duan, H.L.; Wang, J.; Huang, Z.P.; Karihaloo, B.L., Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress, J. mech. phys. solids, 53, 1574-1596, (2005) · Zbl 1120.74718
[8] Fratzl, P.; Penrose, O.; Lebowitz, J.L., Modeling of phase separation in alloys with coherent elastic misfit, J. stat. phys., 95, 1429-1503, (1999) · Zbl 0952.74052
[9] Fried, E.; Gurtin, M.E., A unified treatment of evolving interfaces accounting for small deformations and atomic transport with emphasis on grain-boundaries and epitaxy, Adv. appl. mech., 40, 1-177, (2004)
[10] Gao, H.J., Some general-properties of stress-driven surface evolution in a heteroepitaxial thin-film structure, J. mech. phys. solids, 42, 741-772, (1994) · Zbl 0800.73369
[11] Gao, H.J.; Nix, W.D., Surface roughening of heteroepitaxial thin films, Annual rev. mater. sci., 29, 173-209, (1999)
[12] Grilhé, J., Study of roughness formation induced by homogeneous stress at the free surfaces of solids, Acta metall., 41, 909-913, (1993)
[13] Grinfeld, M.A., Thermodynamics methods in the theory of heterogeneous systems, (1993), Longman Sussex · Zbl 0843.73040
[14] Gurtin, M.E.; Weissmüller, J.; Larche, F., A general theory of curved deformable interfaces in solids at equilibrium, Philos. mag. A, 78, 1093-1109, (1998)
[15] Herring, C., Effect of change of scale on sintering phenomena, J. appl. phys., 21, 301-303, (1950)
[16] Johnson, W.C.; Voorhees, P.W., Interfacial stress, interfacial energy, and phase equilibria in binary alloys, J. stat. phys., 95, 1281-1309, (1999) · Zbl 0952.74005
[17] Kirill, D.J.; Davis, S.H.; Miksis, M.J.; Voorhees, P.W., Morphological instability of a whisker, Proc. R. soc. London A, 455, 3825-3844, (1999) · Zbl 0942.74006
[18] Kolb, F.M.; Hofmeister, H.; Zacharias, M.; Gösele, U., On the morphological instability of silicon/silicon dioxide nanowires, Appl. phys. A, 80, 1405-1408, (2005)
[19] Kukta, R.V.; Freund, L.B., Minimum energy configuration of epitaxial material clusters on a lattice-mismatched substrate, J. mech. phys. solids, 45, 1835-1860, (1997)
[20] Lauhon, L.J.; Gudiksen, M.S.; Wang, D.; Lieber, C.M., Epitaxial core – shell and core – multishell nanowire heterostructures, Nature, 420, 57-61, (2002)
[21] Lauhon, L.J.; Gudiksen, M.S.; Lieber, C.M., Semiconductor nanowire heterostructures, Philos. trans. R. soc. London A, 362, 1247-1260, (2004)
[22] Leo, P.H.; Sekerka, R.F., The effect of surface stress on crystal melt and crystal equilibrium, Acta metall., 37, 3119-3138, (1989)
[23] Leo, P.H.; Lowengrub, J.S.; Jou, H.J., A diffuse interface model for microstructural evolution in elastically stressed solids, Acta mater., 46, 2113-2130, (1998)
[24] Lewis, N.S., Toward cost-effective solar energy use, Science, 315, 798-801, (2007)
[25] Lu, W.; Suo, Z., Dynamics of nanoscale pattern formation of an epitaxial monolayer, J. mech. phys. solids, 49, 1937-1950, (2001) · Zbl 0998.74006
[26] McCarty, K.F.; Nobel, J.A.; Bartelt, N.C., Vacancies in solids and the stability of surface morphology, Nature, 412, 622-625, (2001)
[27] Mullins, W.W., Capillarity-induced surface morphologies, Interface sci., 9, 9-20, (2001)
[28] Mullins, W.W.; Sekerka, R.F., Morphological stability of a particle growing by diffusion or heat flow, J. appl. phys., 34, 323-329, (1963)
[29] Nichols, F.A.; Mullins, W.W., Surface (interface) and volume diffusion contributions to morphological changes driven by capillarity, Trans. metall. soc. AIME, 233, 1840-1848, (1965)
[30] Peng, H.Y.; Pan, Z.W.; Xu, L.; Fan, X.H.; Wang, N.; Lee, C.S.; Lee, S.T., Temperature dependence of si nanowire morphology, Adv. mater., 13, 317-320, (2001)
[31] Rockenberger, J.; Troger, L.; Rogach, A.L.; Tischer, M.; Grundmann, M.; Eychmuller, A.; Weller, H., The contribution of particle core and surface to strain, disorder and vibrations in thiolcapped cdte nanocrystals, J. chem. phys., 108, 7807-7815, (1998)
[32] Savina, T.V.; Voorhees, P.W.; Davis, S.H., The effect of surface stress and wetting layers on morphological instability in epitaxially strained films, J. appl. phys., 96, 3127-3133, (2004)
[33] Shklyaev, O.E.; Miksis, M.J.; Voorhees, P.W., Equilibrium shapes of strained islands with non-zero contact angles, J. mech. phys. solids, 54, 2111-2135, (2006) · Zbl 1120.74619
[34] Spencer, B.J.; Tersoff, J., Equilibrium shapes and properties of epitaxially strained islands, Phys. rev. lett., 79, 4858-4861, (1997)
[35] Spencer, B.J.; Voorhees, P.W.; Davis, S.H., Morphological instability in epitaxially strained dislocation-free solid films, Phys. rev. lett., 67, 3696-3699, (1991)
[36] Thornton, K.; Ågren, J.; Voorhees, P.W., Modelling the evolution of phase boundaries in solids at the meso- and nano-scales, Acta mater., 51, 5675-5710, (2003)
[37] Tian, B.Z.; Zheng, X.L.; Kempa, T.J.; Fang, Y.; Yu, N.F.; Yu, G.H.; Huang, J.L.; Lieber, C.M., Coaxial silicon nanowires as solar cells and nanoelectronic power sources, Nature, 449, 885-890, (2007)
[38] Timoshenko, S.P.; Goodier, J.N., Theory of elasticity, (1951), McGraw-Hill New York · Zbl 0045.26402
[39] Weissmüller, J.; Cahn, J.W., Mean stresses in microstructures due to interface stresses: a generalization of a capillary equation for solids, Acta mater., 45, 1899-1906, (1997)
[40] Weissmüller, J.; Viswanath, R.N.; Kramer, D.; Zimmer, P.; Wurschum, R.; Gleiter, H., Charge-induced reversible strain in a metal, Science, 300, 312-315, (2003)
[41] Xia, L.; Bower, A.F.; Suo, Z.; Shih, C.F., A finite element analysis of the motion and evolution of voids due to strain and electromigration induced surface diffusion, J. mech. phys. solids, 45, 1473-1493, (1997) · Zbl 0974.74566
[42] Yang, F.Q., On the interface instability of a cylindrical fiber embedded in a matrix, Scr. mater., 49, 571-575, (2003)
[43] Yang, W.; Wang, W.; Suo, Z., Cavity and dislocation in instability under electric current, J. mech. phys. solids, 42, 897-911, (1994)
[44] Zhang, Y.W.; Bower, A.F.; Xia, L.; Shih, C.F., Three dimensional finite element analysis of the evolution of voids and thin films by strain and electromigration induced surface diffusion, J. mech. phys. solids, 47, 173-199, (1999) · Zbl 0964.74073
[45] Zhang, Y.; Wang, L.W.; Mascarenhas, A., “quantum coaxial cables” for solar energy harvesting, Nano lett., 7, 1264-1269, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.