Mechanics of carbon nanotubes

[+] Author and Article Information
Dong Qian

Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd, Evanston, IL 60208d-qian@northwestern.edu, g-wagner@northwestern.edu, w-liu@northwestern.edu

Gregory J Wagner

Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd, Evanston, IL 60208d-qian@northwestern.edu, g-wagner@northwestern.edu, w-liu@northwestern.edu

Wing Kam Liu

Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd, Evanston, IL 60208d-qian@northwestern.edu, g-wagner@northwestern.edu, w-liu@northwestern.edu

Min-Feng Yu

Zyvex Corporation, Advanced Technologies Group, 1321 North Plano Rd, Richardson, TX 75081mfyu@zyvex.com

Rodney S Ruoff

Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd, Evanston, IL 60208 r-ruoff@northwestern.edu

Appl. Mech. Rev 55(6), 495-533 (Oct 16, 2002) (39 pages) doi:10.1115/1.1490129 History: Online October 16, 2002
Copyright © 2002 by ASME
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Basic hexagonal bonding structure for one graphite layer (the graphene sheet); carbon nuclei shown as filled circles, out-of-plane π-bonds represented as delocalized (dotted line), and σ-bonds connect the C nuclei in-plane.
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Definition of roll-up vector as linear combinations of base vectors a and b
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Examples of zigzag, chiral, and arm-chair nanotubes and their caps corresponding to different types of fullerenes (Reprinted from 13 with permission from Elsevier Science.)
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Upper left: High resolution transmission electronic microscopy (HRTEM) image of an individual MWCNT. The parallel fringes have ∼0.34 nm separation between them and correspond to individual layers of the coaxial cylindrical geometry. Bottom left: HRTEM image showing isolated SWCNT as well as bundles of such tubes covered with amorphous carbon. The isolated tubes shown are approximately 1.2 nm in diameter. Top right: HRTEM image showing the tip structure of a closed MWCNT. The fringe (layer) separation is again 0.34 nm. Top right: The tip structure of a conical end. Bottom right: The image of a MWCNT showing the geometric changes due to the presence of five and seven membered rings (position indicated in the image by P for pentagon and H for heptagon) in the lattice. Note that the defects in all the neighboring shells are conformal (from 25).
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HRTEM image of a MWCNT. Note the presence of anomalously large interfringe spacings indicated by arrows (from 26)
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Eight stress versus strain curves obtained from the tensile-loading experiments on individual SWCNT bundles. The values of the nominal stress are calculated using the cross-sectional area of the perimeter SWCNTs assuming a thickness of 0.34 nm (from 28). The strain is the engineering strain. Same for Fig. 7.
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Plot of stress versus strain curves for 5 individual MWCNTs (Reprinted with permission from 29. Copyright 2000 American Association for the Advancement of Science.)
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Scanning electron microscope (SEM) images of electric field induced resonance of an individual MWCNT at its fundamental resonance frequency (a) and at its second order harmonic (b)
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(a) Buckling of SWCNT under bending load, (b) Buckling of SWCNT under torsional load
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Left: HRTEM image of two adjacent MWCNTs a and b69. Nanotube a has 10 fringes and Nanotube b has 12 fringes. The average interlayer spacing for inner layers and outer layers belonging to the MWCNT a are 0.338 nm and 0.345 nm; respectively. For MWCNT b, these are 0.343 nm and 0.351 nm, respectively. This 0.07 and 0.08 nm differences are due to the compressive force acting in the contact region, and the deformation from perfectly cylindrical shells occurring in both inner and outer portions of each MWCNT. Right, from top to bottom: (a) Calculated deformation resulting from van der Waals forces between two double layered nanotubes. (b) Projected atom density from (a). The projected atom density is clearly higher in the contact region, in agreement with the experimental observation of the much darker fringes in the contact region as compared to the outer portions of MWCNTs a and b. (c) Calculated deformation for adjacent single layer nanotubes (from 69).
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Deformability of a MWCNT deposited on a patterned silicon wafer as visualized with tapping-mode AFM operated far below mechanical resonance of a cantilever at different set points. The height in this and all subsequent images was coded in gray scale, with darker tones corresponding to lower features. (a) Large-area view of a MWCNT bent upon deposition into a hairpin shape. (b)–(e) Height profiles taken along the thin marked line in (a) from images acquired at different set-point (S/S0) values: (b) 1.0; (c) 0.7; (d) 0.5; (e) 1.0. (f)–(i) Three-dimensional images of the curved region of the MWCNT acquired at the corresponding set-point values as in (b)–(e). (from 80).
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A freestanding twisted MWCNT ribbon. (a) A TEM image of this ribbon anchored on one end by a carbon support film on a lacy carbon grid. Arrows point to the twists in the ribbon. (b) and (c) 8 resolved fringes along both edges of the ribbon imaged near the anchor point. (d) A schematic depicting the AB stacking between armchair CNT shells (the two layers are: the layer having brighter background and black lattices versus the layer having darker background and white lattices). The AB stacking can be achieved by just shifting the layer positions along the x direction that is perpendicular to the long axis of the MWCNT. (e) A schematic depicting the lattice alignment between the zigzag CNT shells by allowing the relative shifting of the layers along x direction. The AB stacking is not possible and only AA stacking or other stacking (as shown in the schematic) is possible (from 83).
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SEM image of a tensile loaded SWCNT bundle between an AFM tip and a SWCNT buckytube paper sample (from 28)
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Tensile loading of individual MWCNTs. (a) An SEM image of a MWCNT attached between two AFM tips. (b) Large magnification image of the indicated region in (a) showing the MWCNT between the AFM tips (Reprinted with premission from 29. Copyright 2000 American Association for the Advancement of Science.)
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The “5-7-7-5” dislocation evolves as either a crack (brittle cleavage), or as a couple of dislocations gliding away along the spiral slip plane (plastic yield). In the latter case, the change of the nanotube chirality is reflected by a stepwise change of diameter and by corresponding variations of electrical properties (Reprinted from 99 with permission from Elsevier Science).
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The forces involved in the shell-sliding experiment can be described by Fa=Fs+Fi=πdτL(t)+Fi, where Fa is the applied pulling force as a function of time, τ is the shear strength, L is the contact length, d the shell diameter, and Fi is a diameter dependent force originating from both surface tension and edge effects. SEM images showing the sword-in-sheath breaking mechanism of MWCNTs: (a) A MWCNT attached between AFM tips under no tensile load, (b) The same MWCNT after being tensile loaded to break. Notice the apparent overall length change of the MWCNT fragments after break compared to the initial length and the curling of the top MWCNT fragment in (b) (from 11).
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TEM images of vibrating single-walled nanotubes. Inserted with each micrograph is the simulated image corresponding to the best least-square fit for the adjusted length L and tip vibration amplitude σ. The tick marks in each micrograph indicate the section of the nanotube shank that was fitted. The nanotube length, diameter W, tip amplitude and the estimated Young’s modulus E, are (a) L=36.8 nm, σ=0.33 nm, W=1.50 nm,E=1.33±0.2 TPa; (b) L=24.3 nm, σ=0.18 nm, W=1.52 nm,E=1.20±0.2 TPa; and (c) L=23.4 nm, σ=50.30 nm, W=1.12 nm,E=1.02±0.3 TPa (from 46).
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Electric field driven resonance of MWCNT: (a) In the absence of a potential, the nanotube tip (L=6.25 mm,D=14.5 nm) vibrated slightly because of thermal effects; (b) Resonant excitation of the fundamental mode of vibration (f1=530 kHz; (c) Resonant excitation of the second harmonic (f2=3.01 MHz). For this nanotube, a value of Eb=0.21 TPa was fit to the standard continuum beam mechanics formula (Reprinted with permission from 44. Copyright 1999 American Association for Advancement of Science.)
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Bending and buckling of MWCNTs: (a) An original straight MWCNT, (b) The MWCNT is bent upwards all the way back onto itself, (c) The same MWCNT is bent all the way back onto itself in the other direction (Reprinted by permission from Nature66. Copyright 1997 Macmillan Publishers Ltd.)
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Overview of one approach used to probe mechanical properties of nanorods and nanotubes: (a) SiC nanorods or carbon nanotubes were deposited on a cleaved MoS2 substrate, and then pinned by deposition of a grid of square SiO pads. (b) Optical micrograph of a sample showing the SiO pads and the MoS2 substrate. The scale bar is 8 mm. (c) An AFM image of a 35.3-nm-diameter SiC nanorod protruding from an SiO pad. The scale bar is 500 nm. (d) Schematic of beam bending with an AFM tip. The tip (triangle) moves in the direction of the arrow, and the lateral force is indicated by the red trace at the bottom. (e) Schematic of a pinned beam with a free end. The beam of length L is subjected to a point load P at x=a and to a distributed friction force f (Reprinted with permission from 48. Copyright 1997 American Association for the Advancement of Science.)
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(a) AFM image of a SWCNT bundle adhered to the polished alumina ultrafiltration membrane, with a portion bridging a pore of the membrane; (b) Schematic of the measurement: the AFM is used to apply a load to the nanobeam and to determine directly the resulting deflection. A closed-loop feedback ensured an accurate scanner positioning. Si3N4 cantilevers with force constants of 0.05 and 0.1 N/m were used as tips in the contact mode (from 50).
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Lateral force on SWCNT bundle as a function of AFM tip position. The four symbols represent data from four consecutive lateral force curves on the same rope, showing that this rope is straining elastically with no plastic deformation. Inset: The AFM tip moves along the trench, in the plane of the surface, and displaces the rope as shown (from 93).
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(a) Individual MWCNT is clamped in place and stretched by two opposing AFM tips. (b) Schematic of the tensile loading experiment (Reprinted with permission from 29. Copyright 2000 American Association for the Advancement of Science.)
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Schematic showing the principle of the experiment for the measurement of the tensile strength of SWCNT bundles. The gray cantilever indicates where the cantilever would be if no rope were attached on the AFM tip after its displacement upward to achieve tensile loading.
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(a) An as-grown bamboo section. (b) The same area after the core tubes on the right have been telescoped outward. The line drawings beneath the images are schematic representations to guide the eye (Reprinted with permission from 107. Copyright 2000 American Association for the Advancement of Science.)
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The pair potential and inter-atomic force in a two-atom system
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Comparison of EOS for graphite using different models with experimental data (Reprinted from 213 with permission from Elsevier Science.)
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Multiscale analysis of carbon nanotube
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3D section of a wire and force components in the x-z and y-z planes
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(a) geometric parameters used in relaxed nanotube bundles; (b) configuration of bundled nanotubes after relaxation
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Snapshots of twisting of the SWCNT bundle
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Change in cross-section at the mid-point of the SWNT bundle as a function of twist angle (From a–f, the twisting angles are 30, 60, 90, 120, 150, and 180 degrees, respectively.)
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Transferred load as a function of twisting angle
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Proposed experimental stage for twisting the nanoropes
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Home-built nanomanipulation testing stage, which fits in the palm of the hand and is used in a high resolution scanning electron microscopy. See Yu et al. 112829 for further details.
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Computational modeling of C60 inside nanotubes: (a) The configuration for the problem, (b) The three velocity components history of the C60 as it shuttles through the (10,10) nanotube 20 times (Vz corresponds to the axial direction with an initial value of 0), and (c) Same as (b), but for the case of an (8,8) nanotube. (Reprinted from 213 with permission from Elsevier Science.)
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Before (upper) and after (lower) untangling the nanotubes in the suspension
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Carbon nanotube-based sensor
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CNT deposition in a round gap by AC electrophoresis
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Upper: In situ TEM image of crack in polystyrene film with nanotubes bridging the crack; Lower: arrangement of nanotubes in polymer, good dispersion, random orientation and moderate waviness 337
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Multiscale analysis of nanorope reinforced materials



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