Folding of carbon nanotubes and graphene nanoribbons into a shape that looks like a tennis racket is considered. An elastic continuum model is utilized in two types of analysis. The first is called an “adhesion model,” in which the adjacent sides of the racket handle are assumed to be straight and bonded together with constant or no separation. The nanotube or nanoribbon is represented as an elastica. This model has been treated in the literature, but new analytical results are derived here, involving the geometry, work of adhesion, and bending and adhesion energies. Expressions are determined for (i) the length for which the total energy is the same as for the straight unstrained equilibrium configuration and (ii) for the minimum length for existence of a stable racket equilibrium shape. The second type of analysis uses the Lennard-Jones potential to model the attractive (van der Waals) and repulsive forces between the two sides of the racket. A nanoribbon is investigated, and the derivative of the interatomic potential is integrated along the length and across the width. Numerical solutions of the integro-differential equations are obtained with a new technique utilizing the finite difference method and minimization of the squares of the resulting algebraic equations. The results are presented for two cases with different flexural rigidities. The separation between the two sides of the handle decreases in the direction of the racket head (loop), and the handle experiences internal compression under the external attractive and repulsive forces. For the adhesion model, the dimensions of the head are proportional to the square root of the flexural rigidity, and this relationship is approximately satisfied in the numerical results based on the Lennard-Jones model.

References

1.
Reich
,
S.
,
Thomsen
,
C.
, and
Maultzsch
,
J.
, 2004,
Carbon Nanotubes: Basic Concepts and Physical Properties
,
Wiley-VCH, Weinheim
,
Germany
.
2.
Terrones
,
M.
,
Botello-Méndez
,
A. R.
,
Campos-Delgado
,
J.
,
López-Urías
,
F.
,
Vega-Cantú
,
Y. I.
,
Rodríguez-Macías
,
F. J.
,
Elías
,
A. L.
,
Muñoz-Sandoval
,
E.
,
Cano-Márquez
,
A. G.
,
Charlier
,
J.-C.
, and
Terrones
,
H.
, 2010, “
Graphene and Graphite Nanoribbons: Morphology, Properties, Synthesis, Defects and Applications
,”
Nano Today
,
5
, pp.
351
372
.
3.
Bottega
,
W. J.
, 1991, “
Peeling and Bond-Point Propagation in a Self-Adhered Elastica
,”
Q. J. Mech. Appl. Math.
,
44
, pp.
17
33
.
4.
Gent
,
A. N.
, 1994, “
Buckles in Adhering Elastic Films and a Test Method for Adhesion Based on the Elastica
,”
J. Adhes. Sci. Technol.
,
8
, pp.
807
819
.
5.
Cohen
,
A. E.
, and
Mahadevan
,
L.
, 2003, “
Kinks, Rings, and Rackets in Filamentous Structures
,”
Proc. Natl. Acad. Sci. U.S.A.
,
100
, pp.
12141
12146
.
6.
Schnurr
,
B.
,
Gittes
,
F.
, and
MacKintosh
,
F. C.
, 2002, “
Metastable Intermediates in the Condensation of Semiflexible Polymers
,”
Phys. Rev. E
,
65
, p.
061904
.
7.
Buehler
,
M. J.
, 2006, “
Mesoscale Modeling of Mechanics of Carbon Nanotubes: Self-Assembly, Self-Folding, and Fracture
,”
J. Mater. Res.
,
21
, pp.
2855
2869
.
8.
Buehler
,
M. J.
,
Kong
,
Y.
,
Gao
,
H.
, and
Huang
,
Y.
, 2006, “
Self-Folding and Unfolding of Carbon Nanotubes
,”
ASME J. Eng. Mater. Technol.
,
128
, pp.
3
10
.
9.
Zhou
,
W.
,
Huang
,
Y.
,
Liu
,
B.
,
Hwang
,
K. C.
,
Zuo
,
J. M.
,
Buehler
,
M. J.
, and
Gao
,
H.
, 2007, “
Self-Folding of Single- and Multiwall Carbon Nanotubes
,”
Appl. Phys. Lett.
,
90
, p.
073107
.
10.
Cranford
,
S.
,
Sen
,
D.
, and
Buehler
,
M. J.
, 2009, “
Meso-Origami: Folding Multilayer Graphene Sheets
,”
Appl. Phys. Lett.
,
95
, p.
123121
.
11.
Crespi
,
V. H.
, 2009, “
Nanotechnology: Soggy Origami
,”
Nature (London)
,
462
, pp.
858
859
.
12.
Ke
,
C.
,
Zheng
,
M.
,
Zhou
,
G.
,
Cui
,
W.
,
Pugno
,
N.
, and
Miles
,
R. N.
, 2010, “
Mechanical Peeling of Free-Standing Single-Walled Carbon-Nanotube Bundles
,”
Small
,
6
, pp.
438
445
.
13.
Mikata
,
Y.
, 2010, “
Approximate Solutions for a Self-Folding Problem of Carbon Nanotubes
,”
ASME J. Eng. Mater. Technol.
,
132
, p.
011013
.
14.
Xu
,
Z.
, and
Buehler
,
M. J.
, 2010, “
Geometry Controls Conformation of Graphene Sheets: Membranes, Ribbons, and Scrolls
,”
ACS Nano
,
4
, pp.
3869
3876
.
15.
Zhang
,
J.
,
Xiao
,
J.
,
Meng
,
X.
,
Monroe
,
C.
,
Huang
,
Y.
, and
Zuo
,
J.-M.
, 2010, “
Free Folding of Suspended Graphene Sheets by Random Mechanical Stimulation
,”
Phys. Rev. Lett.
,
104
, p.
166805
.
16.
Lu
,
W.
, and
Chou
,
T.-W.
, 2011, “
Analysis of the Entanglements in Carbon Nanotube Fibers Using a Self-Folded Nanotube Model
,”
J. Mech. Phys. Solids
,
59
, pp.
511
524
.
17.
Paparcone
,
R.
,
Cranford
,
S. W.
, and
Buehler
,
M. J.
, 2011, “
Self-Folding and Aggregation of Amyloid Nanofibrils
,”
Nanoscale
,
3
, pp.
1748
1755
.
18.
Gao
,
G.
,
Çagin
,
T.
, and
Goddard
,
W. A.
, 1998, “
Energetics, Structure, Mechanical and Vibrational Properties of Single-Walled Carbon Nanotubes
,”
Nanotechnology
,
9
, pp.
184
191
.
19.
Pantano
,
A.
,
Parks
,
D. M.
, and
Boyce
,
M. C.
, 2004, “
Mechanics of Deformation of Single- and Multi-Wall Carbon Nanotubes
,”
J. Mech. Phys. Solids
,
52
, pp.
789
821
.
20.
Tang
,
T.
,
Jagota
,
A.
,
Hui
,
C.-Y.
, and
Glassmaker
,
N. J.
, 2005, “
Collapse of Single-Walled Carbon Nanotubes
,”
J. Appl. Phys.
,
97
, p.
074310
.
21.
Zhang
,
S.
,
Khare
,
R.
,
Belytschko
,
T.
,
Hsia
,
K. J.
,
Mielke
,
S. L.
, and
Schatz
,
G. C.
, 2006, “
Transition States and Minimum Energy Pathways for the Collapse of Carbon Nanotubes
,”
Phys. Rev. B
,
73
, p.
075423
.
22.
Tang
,
T.
, and
Glassmaker
,
N. J.
, 2010, “
On the Inextensible Elastica Model for the Collapse of Nanotubes
,”
Math. Mech. Solids
,
15
, pp.
591
606
.
23.
Mockensturm
,
E.
,
Mahdavi
,
A.
, and
Crespi
,
V.
, 2005, “
Van der Waals’ Elastica
,”
Proceedings of the ASME 2005 International Mechanical Engineering Congress and Exposition
,
Orlando, FL
, IMECE2005-82991.
24.
Pugno
,
N. M.
, 2010, “
The Design of Self-Collapsed Super-Strong Nanotube Bundles
,”
J. Mech. Phys. Solids
,
58
, pp.
1397
1410
.
25.
Lu
,
W.
,
Chou
,
T.-W.
, and
Kim
,
B.-S.
, 2011, “
Radial Deformation and Its Related Energy Variations of Single-Walled Carbon Nanotubes
,”
Phys. Rev. B
,
83
, p.
134113
.
26.
Oyharcabal
,
X.
, and
Frisch
,
T.
, 2005, “
Peeling off an Elastica From a Smooth Attractive Substrate
,”
Phys. Rev. E
,
71
, p.
036611
.
27.
Coffin
,
D. W.
,
Carlsson
,
L. A.
, and
Pipes
,
R. B.
, 2006, “
On the Separation of Carbon Nanotubes
,”
Compos. Sci. Technol.
,
66
, pp.
1132
1140
.
28.
Sasaki
,
N.
,
Toyoda
,
A.
,
Saitoh
,
H.
,
Itamura
,
N.
,
Ohyama
,
M.
, and
Miura
,
K.
, 2006, “
Theoretical Simulation of Atomic-Scale Peeling of Single-Walled Carbon Nanotube From Graphite Surface
,”
e-J. Surf. Sci. Nanotechnol.
,
4
, pp.
133
137
.
29.
Sasaki
,
N.
,
Toyoda
,
A.
,
Itamura
,
N.
, and
Miura
,
K.
, 2008, “
Simulation of Nanoscale Peeling and Adhesion of Single-Walled Carbon Nanotube on Graphite Surface
,”
e-J. Surf. Sci. Nanotechnol.
,
6
, pp.
72
78
.
30.
Strus
,
M. C.
,
Zalamea
,
L.
,
Raman
,
A.
,
Pipes
,
R. B.
,
Nguyen
,
C. V.
, and
Stach
,
E. A.
, 2008, “
Peeling Force Spectroscopy: Exposing the Adhesive Nanomechanics of One-Dimensional Nanostructures
,”
Nano Lett.
,
8
, pp.
544
550
.
31.
Ishikawa
,
M.
,
Harada
,
R.
,
Sasaki
,
N.
, and
Miura
,
K.
, 2009, “
Adhesion and Peeling Forces of Carbon Nanotubes on a Substrate
,”
Phys. Rev. B
,
80
, p.
193406
.
32.
Strus
,
M. C.
, and
Raman
,
A.
, 2009, “
Identification of Multiple Oscillation States of Carbon Nanotube Tipped Cantilevers Interacting With Surfaces in Dynamic Atomic Force Microscopy
,”
Phys. Rev. B
,
80
, p.
224105
.
33.
Strus
,
M. C.
,
Cano
,
C. I.
,
Pipes
,
R. B.
,
Nguyen
,
C. V.
, and
Raman
,
A.
, 2009, “
Interfacial Energy Between Carbon Nanotubes and Polymers Measured From Nanoscale Peel Tests in the Atomic Force Microscope
,”
Compos. Sci. Technol.
,
69
, pp.
1580
1586
.
34.
Xie
,
H.
, and
Régnier
,
S.
, 2010, “
In Situ Peeling of One-Dimensional Nanostructures Using a Dual-Probe Nanotweezer
,”
Rev. Sci. Instrum.
,
81
, p.
035112
.
35.
Fu
,
Y.-M.
, and
Zhang
,
P.
, 2011, “
Peeling Off Carbon Nanotubes From Rigid Substrates: An Exact Model
,”
J. Adhes. Sci. Technol.
,
25
, pp.
1061
1072
.
36.
Israelachvili
,
J. N.
, 2011,
Intermolecular and Surface Forces
, 3rd ed.,
Elsevier
,
Amsterdam
.
37.
Chen
,
B.
,
Gao
,
M.
,
Zuo
,
J. M.
,
Qu
,
S.
,
Liu
,
B.
, and
Huang
,
Y.
, 2003, “
Binding Energy of Parallel Carbon Nanotubes
,”
Appl. Phys. Lett.
,
83
, pp.
3570
3571
.
38.
Majidi
,
C.
, 2007, “
Remarks on Formulating an Adhesion Problem Using Euler’s Elastica
,”
Mech. Res. Commun.
,
34
, pp.
85
90
.
39.
Santillan
,
S.
,
Virgin
,
L. N.
, and
Plaut
,
R. H.
, 2005, “
Equilibria and Vibration of a Heavy Pinched Loop
,”
J. Sound Vib.
,
288
, pp.
81
90
.
40.
Wang
,
C.-Y.
, 1981, “
Folding of Elastica—Similarity Solutions
,”
ASME J. Appl. Mech.
,
48
, pp.
199
200
.
41.
Wolfram
,
S.
, 1996,
The Mathematica Book
, 3rd ed.,
Cambridge University Press
,
Cambridge, UK
.
42.
Liu
,
B.
,
Yu
,
M.-F.
, and
Huang
,
Y.
, 2004, “
Role of Lattice Registry in the Full Collapse and Twist Formation of Carbon Nanotubes
,”
Phys. Rev. B.
,
70
, p.
161402
(R).
43.
Sasaki
,
N.
,
Saitoh
,
H.
,
Itamura
,
N.
, and
Miura
,
K.
, 2009, “
Analysis of Lateral Orientation of Single-Walled Carbon Nanotube on Graphite
,”
e-J. Surf. Sci. Nanotechnol.
,
7
, pp.
48
52
.
44.
Kitamura
,
N.
,
Oshiyama
,
A.
, and
Sugino
,
O.
, 1998, “
Atomic and Electronic Structures of Deformed Graphite Sheets
,”
J. Phys. Soc. Jpn.
,
67
, pp.
3976
3984
.
45.
Prada
,
E.
,
San-Jose
,
P.
, and
Brey
,
L.
, 2010, “
Zero Landau Level in Folded Graphene Nanoribbons
,”
Phys. Rev. Lett.
,
105
, p.
106802
.
46.
Zhang
,
Z.
,
Liu
,
B.
,
Hwang
,
K.-C.
, and
Gao
,
H.
, 2011, “
Surface-Adsorption-Induced Bending Behaviors of Graphene Nanoribbons
,”
Appl. Phys. Lett.
,
98
, p.
121909
.
47.
Patra
,
N.
,
Wang
,
B.
, and
Král
,
P.
, 2009, “
Nanodroplet Activated and Guided Folding of Graphene Nanostructures
,”
Nano Lett.
,
9
, pp.
3766
3771
.
48.
Lu
,
Q.
,
Arroyo
,
M.
, and
Huang
,
R.
, 2009, “
Elastic Bending Modulus of Monolayer Graphene
,”
J. Phys. D
,
42
, p.
102002
.
49.
Cadelano
,
E.
,
Giordano
,
S.
, and
Colombo
,
L.
, 2010, “
Interplay Between Bending and Stretching in Carbon Nanoribbons
,”
Phys. Rev. B
,
81
, p.
144105
.
50.
Lavin
,
J. G.
,
Subramoney
,
S.
,
Ruoff
,
R. S.
,
Berber
,
S.
, and
Tománek
,
D.
, 2002, “
Scrolls and Nested Tubes in Multiwall Carbon Nanotubes
,”
Carbon
,
40
, pp.
1123
1130
.
51.
Girifalco
,
L. A.
,
Hodak
,
M.
, and
Lee
,
R. S.
, 2000, “
Carbon Nanotubes, Buckyballs, Ropes, and a Universal Graphitic Potential
,”
Phys. Rev. B
,
62
, pp.
13104
13110
.
52.
Buehler
,
M. J.
,
Kong
,
Y.
, and
Gao
,
H.
, 2004, “
Deformation Mechanisms of Very Long Single-Wall Carbon Nanotubes Subject to Compressive Loading
,”
ASME J. Eng. Mater. Technol.
,
126
, pp.
245
249
.
53.
Volkov
,
A. N.
, and
Zhigilei
,
L. V.
, 2010, “
Mesoscopic Interaction Potential for Carbon Nanotubes of Arbitrary Length and Orientation
,”
J. Phys. Chem. C
,
114
, pp.
5513
5531
.
54.
Lu
,
Z.
, and
Dunn
,
M. L.
, 2010, “
Van der Waals Adhesion of Graphene Membranes
,”
J. Appl. Phys.
,
107
, p.
044301
.
55.
Benedict
,
L. X.
,
Chopra
,
N. G.
,
Cohen
,
M. L.
,
Zettl
,
A.
,
Louie
,
S. G.
, and
Crespi
,
V. H.
, 1998, “
Microscopic Determination of the Interlayer Binding Energy in Graphite
,”
Chem. Phys. Lett.
,
286
, pp.
490
496
.
56.
Tang
,
T.
,
Jagota
,
A.
, and
Hui
,
C.-Y.
, 2005, “
Adhesion Between Single-Walled Carbon Nanotubes
,”
J. Appl. Phys.
,
97
, p.
074304
.
57.
Koenig
,
S. P.
,
Boddeti
,
N. G.
,
Dunn
,
M. L.
, and
Bunch
,
J. S.
, 2011, “
Ultrastrong Adhesion of Graphene Membranes
,”
Nat. Nanotechnol.
,
6
, pp.
543
546
.
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