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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
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Abstract

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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 

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