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Restricted Three Body Problems at the Nanoscale

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Abstract

In this paper, we investigate some of the classical restricted three body problems at the nanoscale, such as the circular planar restricted problem for three C60 fullerenes, and a carbon atom and two C60 fullerenes. We model the van der Waals forces between the fullerenes by the Lennard–Jones potential. In particular, the pairwise potential energies between the carbon atoms on the fullerenes are approximated by the continuous approach, so that the total molecular energy between two fullerenes can be determined analytically. Since we assume that such interactions between the molecules occur at sufficiently large distance, the classical three body problems analysis is legitimate to determine the collective angular velocity of the two and three C60 fullerenes at the nanoscale. We find that the maximum collective angular velocity of the two and three fullerenes systems reach the terahertz range and we also determine the stationary points and the points which have maximum velocity for the carbon atom, for the carbon atom and the two fullerenes system.

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References

  1. Jones D.E.H.: Hollow molecules. New Sci. 32, 245 (1966)

    Google Scholar 

  2. Kroto H.W., Heath J.R., O’Brien S.C., Curl R.F., Smalley R.E.: C60: Buckminsterfullerene. Nature 318, 162–163 (1985)

    Article  ADS  Google Scholar 

  3. Cumings J., Zettl A.: Low-friction nanoscale linear bearing realized from multiwall carbon nanotubes. Science 289, 602 (2000)

    Article  ADS  Google Scholar 

  4. Zheng Q., Jiang Q.: Multiwalled carbon nanotubes as gigahertz oscillators. Phys. Rev. Lett. 88, 045503 (2002)

    Article  ADS  Google Scholar 

  5. Legoas S.B., Coluci V.R., Braga S.F., Coura P.Z., Dantas S.O., Galvao D.S: Molecular-dynamics simulations of carbon nanotubes as gigahertz oscillators. Phys. Rev. Lett. 90, 055504 (2003)

    Article  ADS  Google Scholar 

  6. Rivera J.L., McCabe C., Cumming P.T.: Oscillatory behavior of double nanotubes under extension: a simple nanoscale damped spring. Nano Lett. 3, 1001 (2003)

    Article  ADS  Google Scholar 

  7. Rivera J.L., McCabe C., Cumming P.T.: The oscillatory damped behaviour of incommensurate double-walled carbon nanotubes. Nanotechnology 16, 186 (2005)

    Article  ADS  Google Scholar 

  8. Baowan D., Hill J.M.: Accurate expressions for the force distribution for double-walled carbon nanotubes. Z. Angew. Math. Phys. 16, 186 (2005)

    Google Scholar 

  9. Cox B.J., Thamwattana N., Hill J.M.: Mechanics of atoms and fullerenes in single-walled carbon nanotubes. I. Acceptance and suction energies. Proc. R. Soc. Lond. A 463, 461 (2007)

    Article  MATH  ADS  Google Scholar 

  10. Cox B.J., Thamwattana N., Hill J.M.: Mechanics of atoms and fullerenes in single-walled carbon nanotubes. II. Oscillatory behaviour. Proc. R. Soc. Lond. A 463, 477 (2007)

    Article  MATH  ADS  Google Scholar 

  11. Cox B.J., Thamwattana N., Hill J.M.: Mechanics of nanotubes oscillating in carbon nanotube bundles. Proc. R. Soc. Lond. A 464, 691 (2008)

    Article  MATH  ADS  Google Scholar 

  12. Cox B.J., Thamwattana N., Hill J.M.: Mechanics of fullerenes oscillating in carbon nanotube bundles. J. Phys. A: Math. Theor. 40, 13197 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Hilder T.A., Hill J.M.: Orbiting nanosectors inside carbon nanotori. Micro Nano Lett. 2, 50 (2007)

    Article  Google Scholar 

  14. Hilder T.A., Hill J.M.: Orbiting atoms and C60 fullerenes inside carbon nanotori. J. Appl. Phys. 101, 064319 (2007)

    Article  ADS  Google Scholar 

  15. Chan, Y., Cox, G.M., Hill, J.M.: A carbon atom orbiting around the outside of a carbon nanotube. In: Proceedings of International Conference on Nanoscience and Nanotechnology, pp. 152–155, February 2008. ICONN 2008 (2008)

  16. Chan, Y., Thamwattana, N., Cox, G.M., Hill, J.M.: Mechanics of nanoscale orbiting systems. J. Math. Chem. doi:10.1007/s10910-008-9516-y (2009)

  17. Goldstein H., Poole C., Safko J.: Classical Mechanics. Addison Wesley, Reading (2002)

    Google Scholar 

  18. Christian M.: The Three-Body Problem. Elsevier, Amsterdam (1990)

    MATH  Google Scholar 

  19. Pollard H.: Celestial Mechanics. The Mathematical Association of America, USA (1976)

    MATH  Google Scholar 

  20. Moulton F.R.: An Introduction to Celestial Mechanics. Dover, New York (1970)

    Google Scholar 

  21. Freundlich E.F.: Celestial Mechanics. Pergamon Press, London (1958)

    MATH  Google Scholar 

  22. Sterne T.E.: An Introduction to Celestial Mechanics. Interscience Publishers, Inc., New York (1960)

    MATH  Google Scholar 

  23. Mccuskey S.W.: Introduction to Celestial Mechanics. Addison-Wesley, London (1963)

    MATH  Google Scholar 

  24. Jones J.E.L.: The determination of molecular fields: from the variation of the viscosity of a gas with temperature. Proc. R. Soc. 106A, 441 (1924)

    ADS  Google Scholar 

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Authors and Affiliations

  1. Nanomechanics Group, School of Mathematicsand Applied Statistics, University of Wollongong, Wollongong, NSW, 2522, Australia

    Yue Chan, Ngamta Thamwattana & James M. Hill

Authors
  1. Yue Chan
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  2. Ngamta Thamwattana
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  3. James M. Hill
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Correspondence to Yue Chan.

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Chan, Y., Thamwattana, N. & Hill, J.M. Restricted Three Body Problems at the Nanoscale. Few-Body Syst 46, 239–247 (2009). https://doi.org/10.1007/s00601-009-0070-3

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  • DOI: https://doi.org/10.1007/s00601-009-0070-3

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