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.
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Sheshmani, S., Ashori, A., Fashapoyeh, M.: Wood plastic composite using graphene nanoplatelets. Int. J. Biol. Macromol. 58, 1–6 (2013)
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)
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)
Shahidi, S., Moazzenchi, B.: Carbon nanotube and its applications in textile industry—a review. J. Text. I 109(12), 1653–1666 (2018)
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)
Bianco, A., Kostarelos, K., Prato, M.: Applications of carbon nanotubes in drug delivery. Curr. Opin. Chem. Biol. 9, 674–679 (2005)
Sarapat, P., Hill, J.M., Baowan, D.: A review of geometry, construction and modelling for carbon nanotori. Appl. Sci. 9, 2301 (2019)
Feng, C., Liew, K.M.: A molecular mechanics analysis of the buckling behavior of carbon nanorings under tension. Carbon 64, 033412 (2001)
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)
Hilder, T.A., Hill, J.M.: Oscillating carbon nanotori along carbon nanotubes. Phys. Rev. B 75(12), Art. No. 125415 (2007)
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)
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)
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)
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)
Baowan, D., Cox, B.J., Hill, J.M.: Determination of join regions between carbon nanostructures using variational calculus. ANZIAM J. 54, 221–247 (2013)
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)
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)
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)
Lim, P.H., Bagci, U., Bai, L.: Introducing willmore flow into level set segmentation of spinal vertebrae. IEEE T. Bio-Med. Eng. 60 (2013)
Bui, C., Lleras, V., Pantz, O.: Dynamics of red blood cells in 2D. EDP Sci. 28 (2009)
Willmore, T.J.: Note on embedded surfaces. An. St. Univ. Iasi Mat. 12B, 493–496 (1965)
Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edn. Springer, Berlin (1971)
The authors acknowledge the Development and Promotion of Science and Technology Talents Project (DPST) for financial support.
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Sripaturad, P., Baowan, D. Joining curves between nano-torus and nanotube: mathematical approaches based on energy minimization. Z. Angew. Math. Phys. 72, 20 (2021). https://doi.org/10.1007/s00033-020-01451-0
- Willmore energy
- Calculus of variations
Mathematics Subject Classification