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Explicit solutions for a SWCNT collapse

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Abstract

Self-collapse of carbon nanotubes may prominently affect their electrical properties and holds promise for important applications in the emerging nano-mechanical or electronic systems. Based upon the potential energy functional, we derived the governing equation and transversality boundary condition for a collapsed single-walled carbon nanotube (SWCNT). Considering the inextensible condition of the elastica, some closed-form solutions for the collapsed configuration were obtained in terms of elliptical integrals. These analytical solutions include the critical length of the flat contact segment, critical radii for collapsed and circular shapes of the cross-section, deflections and potential energies of the SWCNTs. Finally, the energy states and the collapsed morphologies of the SWCNTs were presented. These explicit solutions are beneficial for the design of nano-structured materials, and cast a light on enhancing their mechanical, chemical, optical and electronic properties.

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References

  1. Krishnan A., Dujardin E., Ebbesen T.W. et al.: Young’s modulus of single-walled nanotubes. Phys. Rev. B 58, 14013–14019 (1998)

    Article  Google Scholar 

  2. Poncharal P., Wang Z.L., Ugarte D. et al.: Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science 283, 1513–1516 (1999)

    Article  Google Scholar 

  3. Baughman R.H., Zakhidov A.A., de Heer W.A.: Carbon nanotubes: the route toward applications. Science 297, 787–792 (2002)

    Article  Google Scholar 

  4. Fygenson D.K., Marko J.F., Libchaber A.: Mechanics of microtubule-based membrane extension. Phys. Rev. Lett. 79, 4497–4500 (1997)

    Article  Google Scholar 

  5. Cohen A.E., Mahadevan L.: Kinks, rings, and rackets in filamentous structures. P. Natl. Acad. Sci. USA 100, 12141–12146 (2003)

    Article  Google Scholar 

  6. Lee H.S., Yun C.H., Kim H.M. et al.: Persistence length of multiwalled carbon nanotubes with static bending. J. Phys. Chem. C 111, 18882–18887 (2007)

    Article  Google Scholar 

  7. Zheng L.X., O’Connelll M.J., Doorn S.K. et al.: Ultralong single-wall carbon nanotubes. Nat. Mater. 3, 673–676 (2004)

    Article  Google Scholar 

  8. Buehler M.J., Kong Y., Gao H. et al.: Self-folding and unfolding of carbon nanotubes. J. Eng. Mater. Tech. 128, 3–10 (2006)

    Article  Google Scholar 

  9. Zhou W., Huang Y., Liu B. et al.: Self-folding of single- and multiwall carbon nanotubes. Appl. Phys. Lett. 90, 073107 (2007)

    Article  Google Scholar 

  10. Mikata Y.: Approximate solutions for a self-folding problem of carbon nanotubes. J. Eng. Mater. Tech 132, 011013 (2010)

    Article  Google Scholar 

  11. Hertel T., Walkup R., Avouris P.: Deformation of carbon nanotubes by surface van der Waals forces. Phys. Rev. B 58, 13870–13873 (1998)

    Article  Google Scholar 

  12. Tang T., Jagota A., Hui C.: Adhesion between single-walled carbon nanotubes. J. Appl. Phys. 97, 074304 (2005)

    Article  Google Scholar 

  13. Ruoff R.S., Tersoff J., Lorents D.C. et al.: Radial deformation of carbon nanotubes by van der Waals forces. Nature 364, 514–516 (1993)

    Article  Google Scholar 

  14. Chopra N.G., Benedict L.X., Crespi V.H. et al.: Fully collapsed carbon nanotubes. Nature 377, 135–138 (1995)

    Article  Google Scholar 

  15. Martel R., Schmidt T., Shea H.R. et al.: Single- and multi-wall carbon nanotube field-effect transistors. Appl. Phys. Lett. 73, 2447–2449 (1998)

    Article  Google Scholar 

  16. Yu M.F., Dyer M.J., Chen J. et al.: Locked twist in multiwalled carbon-nanotube ribbons. Phys. Rev. B 64, 241403 (2001)

    Article  Google Scholar 

  17. Yu M.F., Kowalewski T., Ruoff R.S.: Investigation of the radial deformability of individual carbon nanotubes under controlled indentation force. Phys. Rev. Lett. 85, 1456–1459 (2000)

    Article  Google Scholar 

  18. Wong E.W., Sheehan P.E., Lieber C.M.: Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277, 1971–1975 (1997)

    Article  Google Scholar 

  19. Ru C.Q.: Effect of van der Waals forces on axial buckling of a doublewall carbon nanotube. J. Appl. Phys. 87, 7227–7231 (2000)

    Article  Google Scholar 

  20. Ru C.Q.: Elastic buckling of singlewalled carbon nanotube ropes under high pressure. Phys. Rev. B 62, 10405–10408 (2000)

    Article  Google Scholar 

  21. Wang C.Y., Ru C.Q., Mioduchowskil A.: Elastic buckling of multiwall carbon nanotubes under high pressure. J. Nanosci. Nanotechnol. 3, 199–208 (2003)

    Article  Google Scholar 

  22. Yakobson B.I., Brabec C.J., Bernholc J.: Nanomechanics of carbon tubes: instabilities beyond linear response. Phys. Rev. Lett. 76, 2511–2514 (1996)

    Article  Google Scholar 

  23. Gao G., Cagin T., Goddard W.A. III: Energetics, structure, mechanical and vibrational properties of single-walled carbon nanotubes. Nanotechnology 9, 184–191 (1998)

    Article  Google Scholar 

  24. Pantano A., Parks D.M., Boyce M.C. et al.: Mixed finite element-tight-binding electromechanical analysis of carbon nanotubes. J. Appl. Phys. 96, 6756–6760 (2004)

    Article  Google Scholar 

  25. Tang T., Glassmaker N.J.: On the inextensible elastica model for the collapse of nanotubes. Math. Mech. Solid 15, 591–606 (2010)

    Article  Google Scholar 

  26. Tang T., Jagota A., Hui C.Y. et al.: Collapse of single-walled carbon nanotubes. J. Appl. Phys. 97, 074310 (2005)

    Article  Google Scholar 

  27. Liu B., Yu M.F., Huang Y.: Role of lattice registry in the full collapse and twist formation of carbon nanotubes. Phys. Rev. B 70, 161402 (2004)

    Article  Google Scholar 

  28. Xiao J., Liu B., Huang Y. et al.: Collapse and stability of single- and multi-wall carbon nanotubes. Nanotechnology 18, 395703 (2007)

    Article  Google Scholar 

  29. Love A.E.H.: A treatise on the mathematical theory of elasticity. Cambridge University Press, London (1906)

    Google Scholar 

  30. Bishopp K.E., Drucker D.C.: Large deflections of cantilever beams. Quart. J. Appl. Math. 3, 272–275 (1945)

    Google Scholar 

  31. Liu J.L., Feng X.Q.: Capillary adhesion of microbeams: finite deformation analyses. Chin. Phys. Lett. 24, 2349–2352 (2007)

    Article  MathSciNet  Google Scholar 

  32. Glassmaker N.J., Hui C.Y.: Elastica solution for a nanotube formed by self-adhesion of a folded thin film. J. Appl. Phys. 96, 3429–3444 (2004)

    Article  Google Scholar 

  33. Roman B., Bico J.: Elasto-capillarity: deforming an elastic structure with a liquid droplet. J. Phys. Condens. Matter. 22, 493101 (2010)

    Article  Google Scholar 

  34. Ouyang G., Wang C.X., Yang G.W.: Surface energy of nanostructural materials with negative curvature and related size effects. Chem. Rev. 109, 4221–4247 (2009)

    Article  Google Scholar 

  35. Bormasshenko E., Whyman G.: Variational approcach to wetting problems: calculation of a shape of sessile liquid drop deposited on a solid substrate in external field. Chem. Phys. Lett. 463, 103–105 (2008)

    Article  Google Scholar 

  36. Seifert U., Lipowsky R.: Adhesion of vesicles. Phys. Rev. A 42, 4768–4771 (1990)

    Article  Google Scholar 

  37. Oyharcalbal X., Frisch T.: Peeling off an elastica from a smooth attractive substrate. Phys. Rev. E 71, 036611 (2005)

    Article  Google Scholar 

  38. Py C., Reverdy P., Doppler L. et al.: Capillary origami: spontaneous wrapping of a droplet with an elastic sheet. Phys. Rev. Lett. 98, 156103 (2007)

    Article  Google Scholar 

  39. Zou J., Ji B.H., Feng X.Q. et al.: Self-assembly of single-walled carbon nanotubes into multi-walled carbon nanotubes in water: molecular dynamics simulations. Nano Lett. 6, 430–434 (2006)

    Article  Google Scholar 

  40. Zou J., Ji B.H., Feng X.Q. et al.: Molecular dynamics studies of carbon-water-carbon composite nanotubes. Small 2, 1348–1355 (2006)

    Article  Google Scholar 

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  1. Department of Engineering Mechanics, China University of Petroleum, Qingdao, 266555, China

    Jianlin Liu

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Correspondence to Jianlin Liu.

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Liu, J. Explicit solutions for a SWCNT collapse. Arch Appl Mech 82, 767–776 (2012). https://doi.org/10.1007/s00419-011-0589-x

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  • DOI: https://doi.org/10.1007/s00419-011-0589-x

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