Joining curves between nano-torus and nanotube: mathematical approaches based on energy minimization


Due to a variety of applications of nanoscaled materials, several researchers further investigate a joining between two nanostructures as a candidate for new potential applications. Here, the vertically joining between the nanotube with the nano-torus is investigated. Variational calculus is used to predict the joining curve between two nanostructures based on minimizing the elastic energy. Moreover, Willmore energy is also utilized to determine the join region especially for a three-dimensional structure. Since the surface of a catenoid is a minimizer obtained by the Willmore energy function, it is used to join two symmetric nanostructures. We find that these two approaches have less than 10% difference in the positions of the joining curves. These two methods might be used to design other hybrid nano-scaled structures.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    Sheshmani, S., Ashori, A., Fashapoyeh, M.: Wood plastic composite using graphene nanoplatelets. Int. J. Biol. Macromol. 58, 1–6 (2013)

    Article  Google Scholar 

  2. 2.

    Eatemadi, A., Daraee, H., Karimkhanloo, H., Kouhi, M., Zarghami, N., Akbarzadeh, A., Abasi, M., Hanifehpour, Y., Joo, S.W.: Carbon nanotubes: properties, synthesis, purification, and medical applications. Nanoscale Res. Lett. 9, 393 (2014)

    Article  Google Scholar 

  3. 3.

    Kataura, H., Kumazawa, Y., Maniwa, Y., Umezu, I., Suzuki, S., Ohtsuka, Y., Achiba, Y.: Optical properties of single-wall carbon nanotubes. Synth. Met. 103, 2555–2558 (1999)

    Article  Google Scholar 

  4. 4.

    Shahidi, S., Moazzenchi, B.: Carbon nanotube and its applications in textile industry—a review. J. Text. I 109(12), 1653–1666 (2018)

    Article  Google Scholar 

  5. 5.

    Das, R., Ali, M.E., Hamid, S.B.A., Ramakrishna, S., Chowdhury, Z.Z.: Carbon nanotube membranes for water purification: a bright future in water desalination. Desalination 336, 97–109 (2014)

    Article  Google Scholar 

  6. 6.

    Bianco, A., Kostarelos, K., Prato, M.: Applications of carbon nanotubes in drug delivery. Curr. Opin. Chem. Biol. 9, 674–679 (2005)

    Article  Google Scholar 

  7. 7.

    Sarapat, P., Hill, J.M., Baowan, D.: A review of geometry, construction and modelling for carbon nanotori. Appl. Sci. 9, 2301 (2019)

    Article  Google Scholar 

  8. 8.

    Feng, C., Liew, K.M.: A molecular mechanics analysis of the buckling behavior of carbon nanorings under tension. Carbon 64, 033412 (2001)

    Google Scholar 

  9. 9.

    Liu, L., Jayanthi, C.S., Wu, S.Y.: Structural and electronic properties of a carbon nanotorus: effects of delocalized and localized deformations. Phys. Rev. B 64, 033412 (2001)

    Article  Google Scholar 

  10. 10.

    Hilder, T.A., Hill, J.M.: Oscillating carbon nanotori along carbon nanotubes. Phys. Rev. B 75(12), Art. No. 125415 (2007)

    Article  Google Scholar 

  11. 11.

    Woellner, C.F., Botari, T., Perim, E., Galvao, D.S.: Mechanical Properties of Schwarzites—A Fully Atomistic Reactive Molecular Dynamics Investigation, pp. 451–456. Cambridge University Press, Cambridge (2018)

    Google Scholar 

  12. 12.

    Wang, X., Sun, G., Chen, P.: Three-dimensional porous architectures of carbon nanotubes and graphene sheets for energy applications. Front. Energy Res. 2, 1–8 (2014)

    Google Scholar 

  13. 13.

    Zhang, J., Terrones, M., Park, C.R., Mukherjee, R., Monthioux, M., Koratkar, N., Kim, Y.S., Bianco, R.: Carbon science in 2016: status challenges and perspectives. Carbon 98, 708–732 (2016)

    Article  Google Scholar 

  14. 14.

    Lepro, X., Vega-Cantu, Y., Rodriguez-Macias, F.J., Bando, Y., Golberg, D., Terrones, M.: Production and characterization of coaxial nanotube junctions and networks of CNx/CNT. Nano Lett. 7(8)

  15. 15.

    Baowan, D., Cox, B.J., Hill, J.M.: Determination of join regions between carbon nanostructures using variational calculus. ANZIAM J. 54, 221–247 (2013)

    MATH  Google Scholar 

  16. 16.

    Cox, B.J., Hill, J.M.: A variational approach to the perpendicular joining of nanotubes to plane sheets. J. Phys. A: Math. Theor. 41, 125203 (2008)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Sripaturad, P., Alshammari, N.A., Thamwattana, N., McCoy, J.A., Baowan, D.: Willmore energy for joining of carbon nanostructures. Philos. Mag. 98(16), 1511–1524 (2018)

    Article  Google Scholar 

  18. 18.

    Velimirovic, L.S., Ciric, M.S., Cvetkovic, M.D.: Change of the willmore energy under infinitesimal bending of membranes. Comput. Math. Appl. 59, 3679–3686 (2010)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Lim, P.H., Bagci, U., Bai, L.: Introducing willmore flow into level set segmentation of spinal vertebrae. IEEE T. Bio-Med. Eng. 60 (2013)

  20. 20.

    Bui, C., Lleras, V., Pantz, O.: Dynamics of red blood cells in 2D. EDP Sci. 28 (2009)

  21. 21.

    Willmore, T.J.: Note on embedded surfaces. An. St. Univ. Iasi Mat. 12B, 493–496 (1965)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edn. Springer, Berlin (1971)

    Book  Google Scholar 

Download references


The authors acknowledge the Development and Promotion of Science and Technology Talents Project (DPST) for financial support.

Author information



Corresponding author

Correspondence to Duangkamon Baowan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sripaturad, P., Baowan, D. Joining curves between nano-torus and nanotube: mathematical approaches based on energy minimization. Z. Angew. Math. Phys. 72, 20 (2021).

Download citation


  • Nano-torus
  • Nanotube
  • Willmore energy
  • Calculus of variations

Mathematics Subject Classification

  • 03H10
  • 49N99
  • 74G65