Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T20:43:58.915Z Has data issue: false hasContentIssue false
  • English
  • Français

DETERMINATION OF JOIN REGIONS BETWEEN CARBON NANOSTRUCTURES USING VARIATIONAL CALCULUS

Part of: Chemistry Geometry and physics Optimality conditions

Published online by Cambridge University Press:  04 September 2013

D. BAOWAN ,
B. J. COX  and
J. M. HILL
D. BAOWAN
Affiliation:
Department of Mathematics, Faculty of Science, Mahidol University, Rama VI Rd., Bangkok 10400, Thailand email duangkamon.bao@mahidol.ac.th
B. J. COX*
Affiliation:
Nanomechanics Group, School of Mathematical Sciences, The University of Adelaide, South Australia 5005, Australia email jim.hill@adelaide.edu.au
J. M. HILL
Affiliation:
Nanomechanics Group, School of Mathematical Sciences, The University of Adelaide, South Australia 5005, Australia email jim.hill@adelaide.edu.au
*
✉barry.cox@adelaide.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We review the work of the present authors to employ variational calculus to formulate continuous models for the connections between various carbon nanostructures. In formulating such a variational principle, there is some evidence that carbon nanotubes deform as in perfect elasticity, and rather like the elastica, and therefore we seek to minimize the elastic energy. The calculus of variations is utilized to minimize the curvature subject to a length constraint, to obtain an Euler–Lagrange equation, which determines the connection between two carbon nanostructures. Moreover, a numerical solution is proposed to determine the geometric parameters for the connected structures. Throughout this review, we assume that the defects on the nanostructures are axially symmetric and that the into-the-plane curvature is small in comparison to that in the two-dimensional plane, so that the problems can be considered in the two-dimensional plane. Since the curvature can be both positive and negative, depending on the gap between the two nanostructures, two distinct cases are examined, which are subsequently shown to smoothly connect to each other.

Keywords

variational calculusnanostructurescarbon nanotubesC60 fullerenesgeometry

MSC classification

Secondary: 49K15: Problems involving ordinary differential equations 51P05: Geometry and physics (should also be assigned at least one other classification number from Sections 70--86) 92E10: Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
Type
Research Article
Information
The ANZIAM Journal , Volume 54 , Issue 4 , April 2013 , pp. 221 - 247
Copyright
Copyright ©2013 Australian Mathematical Society 

References

Baowan, D., Cox, B. J. and Hill, J. M., “Two least squares analyses of bond lengths and bond angles for the joining of carbon nanotubes to graphenes”, Carbon 45 (2007) 2972–2980; doi:10.1016/j.carbon.2007.09.045.CrossRefGoogle Scholar
Baowan, D., Cox, B. J. and Hill, J. M., “A continuous model for the joining of two fullerenes”, Philos. Mag. 88 (2008) 2953–2964; doi:10.1080/14786430802446682.CrossRefGoogle Scholar
Baowan, D., Cox, B. J. and Hill, J. M., “Discrete and continuous approximations for nanobuds”, Fuller. Nanotubes Carbon Nanostruct. 18 (2010) 160–177; doi:10.1080/15363830903586625.CrossRefGoogle Scholar
Baowan, D., Cox, B. J. and Hill, J. M., “Modelling the joining of nanocones and nanotubes”, J. Math. Chem. 49 (2011) 475–488; doi:10.1007/s10910-010-9753-8.CrossRefGoogle Scholar
Baowan, D., Cox, B. J. and Hill, J. M., “Modeling the join curve between two co-axial carbon nanotubes”, J. Appl. Math. Phys. (ZAMP) 63 (2011) 331–338; doi:10.1007/s00033-011-0140-5.CrossRefGoogle Scholar
Byrd, P. F. and Friedman, M. D., Handbook of elliptic integrals for engineers and scientists, 2nd edn (Springer, Berlin, 1971).CrossRefGoogle Scholar
Charlier, J.-C. and Rignanese, G.-M., “Electronic structure of carbon nanocones”, Phys. Rev. Lett. 86 (2001) 5970–5973; doi:10.1103/PhysRevLett.86.5970.CrossRefGoogle ScholarPubMed
Cox, B. J. and Hill, J. M., “Exact and approximate geometric parameters for carbon nanotubes incorporating curvature”, Carbon 45 (2007) 1453–1462; doi:10.1016/j.carbon.2007.03.028.CrossRefGoogle Scholar
Cox, B. J. and Hill, J. M., “A variational approach to the perpendicular joining of nanotubes to plane sheets”, J. Phys. A: Math. Theor. 41 (2008) 125203; doi:10.1088/1751-8113/41/12/125203.CrossRefGoogle Scholar
Ding, F., Lin, Y., Krasnov, P. O. and Yakobson, B. I., “Nanotube-derived carbon foam for hydrogen sorption”, J. Chem. Phys. 127 (2007) doi:10.1063/1.2790434.CrossRefGoogle ScholarPubMed
Dresselhaus, M. S., Dresselhaus, G. and Eklund, P. C., Science of fullerenes and carbon nanotubes (Academic Press, San Diego, 1995).Google Scholar
Dresselhaus, M. S., Dresselhaus, G. and Saito, R., “Physics of carbon nanotubes”, Carbon 33 (1995) 883–891; doi:10.1016/0008-6223(95)00017-8.CrossRefGoogle Scholar
Ge, M. and Sattler, K., “Observation of fullerene cones”, Chem. Phys. Lett. 220 (1994) 192–196; doi:10.1016/0009-2614(94)00167-7.CrossRefGoogle Scholar
Krishnan, A., Dujardin, E., Treacy, M. M. J., Hugdahl, J., Lynum, S. and Ebbesen, T. W., “Graphitic cones and the nucleation of curved carbon surfaces”, Nature 388 (1997) 451–454; doi:10.1038/41284.CrossRefGoogle Scholar
Nasibulin, A. G. et al. , “A novel hybrid carbon material”, Nature Nanotech. 2 (2007) 156–161; doi:10.1038/nnano.2007.37.CrossRefGoogle ScholarPubMed
Pincak, R. and Osipov, V. A., “Localized electron states near pentagons in variously shaped carbon nanoparticles”, Phys. Lett. A 314 (2003) 315–321; doi:10.1016/S0375-9601(03)00898-3.CrossRefGoogle Scholar
Saito, R., Dresselhaus, G. and Dresselhaus, M. S., “Tunneling conductance of connected carbon nanotubes”, Phys. Rev. B 53 (1996) 2044–2050; doi:10.1103/PhysRevB.53.2044.CrossRefGoogle ScholarPubMed
Saito, R., Dresselhaus, G. and Dresselhaus, M. S., Physical properties of carbon nanotubes (Imperial College Press, London, 1998).CrossRefGoogle Scholar
Sattler, K., “Scanning tunneling microscopy of carbon nanotubes and nanocones”, Carbon 33 (1995) 915–920; doi:10.1016/0008-6223(95)00020-E.CrossRefGoogle Scholar
Ueno, H., Osawa, S., Osawa, E. and Takeuchi, K., “Stone–Wales rearrangement pathways from the hinge-opened $[2+ 2] $ ${\mathrm{C} }_{60} $ dimer to IPR ${\mathrm{C} }_{120} $ fullerenes. Vibrational analysis of intermediates”, Fullerene Sci. Tech. 6 (1996) 319–338.CrossRefGoogle Scholar
Xia, Y., Xing, Y., Tan, C. and Mei, L., “Dimerization and fusion of ${\mathrm{C} }_{60} $ molecules caused by molecular collision”, Phys. Rev. B 53 (1996) 13871–13876; doi:10.1103/PhysRevB.53.13871.CrossRefGoogle Scholar
Xie, Y., Huang, J., Li, B., Liu, Y. and Qian, Y., “A novel peanut-like nanostructure of II–VI semiconductor CdS and ZnS”, Adv. Mater. 12 (2000) 1523–1526; doi:10.1002/1521-4095(200010)12:20<1523::AID-ADMA1523>3.3.CO;2-K.3.0.CO;2-T>CrossRefGoogle Scholar
Zang, J., Treibergs, A., Han, Y. and Liu, F., “Geometric constant defining shape transitions of carbon nanotubes under pressure”, Phys. Rev. Lett. 92 (2004) doi:10.1103/PhysRevLett.92.105501.CrossRefGoogle ScholarPubMed
Zhao, Y., Lin, Y. and Yakobson, B. I., “Fullerene shape transformations via Stone–Wales bond rotations”, Phys. Rev. B 68 (2003) doi:10.1103/PhysRevB.68.233403.CrossRefGoogle Scholar
Zhao, Y., Smalley, R. E. and Yakobson, B. I., “Coalescence of fullerene cages: topology, energetics, and molecular dynamics simulation”, Phys. Rev. B 66 (2002) doi:10.1103/PhysRevB.66.195409.CrossRefGoogle Scholar
Zhao, Y., Yakobson, B. I. and Smalley, R. E., “Dynamic topology of fullerene coalescence”, Phys. Rev. Lett. 88 (2002) doi:10.1103/PhysRevLett.88.185501.CrossRefGoogle ScholarPubMed