# Buckling of Carbon Nanotubes: A State of the Art Review

## Abstract

**:**

## 1. Introduction: Appeal of Nanocarbon Materials

## 2. Background of Nanotube Buckling Research

**Figure 1.**Schematic diagram of buckling of an elastic beam under axial compression: (

**a**) pristine beam; (

**b**) axial compression for a small load; (

**c**) buckling observed beyond a critical load.

**Figure 2.**(

**a**)–(

**f**) Series of TEM images of deformation processes for MWNTs initiated by applying compressive force in the sample direction; (

**g**) Force–displacement diagram. The points indicated by arrows correspond to the TEM images in (

**d**) and (

**e**). Reprinted from Reference [88].

## 3. Resilience and Sensitivity to Buckling

## 4. Axial Compression Buckling

#### 4.1. Shell Buckling or Column Buckling?

**Figure 3.**Axially buckled SWNT pattern deduced from molecular dynamics simulations. [Left] Upper panel: Energy-strain curve of a (10,10) SWNT with a length of 9.6 nm under axial compression. Lower panel: Typical tube geometry (

**a**) before and (

**b**) after buckling, respectively. [Right] Upper panel: Curve of a compressed (10,10) SWNT with 29.5 nm length. Lower panel: Snapshots of the tube (

**a**) before buckling; (

**b**) after column buckling; and (

**c**) undergoing shell buckling. Reprinted from Reference [121].

#### 4.2. Force–Displacement Curve

**Figure 4.**[Top] Force-displacement curve of an MWNT with an aspect ratio of ∼80 under cyclic axial loading. This inset shows a microscopy image; [Bottom] Schematic of the change in the MWNT configuration during the buckling process. The labels correspond to those indicated in the force-displacement curve. Reprinted from Reference [87].

## 5. Radial Compression Buckling

#### 5.1. Uniaxial Collapse of SWNTs

**Figure 5.**Long and short diameters of a (10,10) SWNT as a function of applied hydrostatic pressure. The shape of the cross section at some selected pressures is plotted at the bottom of the figure. Reprinted from Reference [38].

**Figure 6.**Change in the relative volume of the (7,7)@(12,12) DWNT bundle and the corresponding SWNT bundles as a function of hydrostatic pressure. Reprinted from Reference [146].

#### 5.2. Radial Corrugation of MWNTs

**Figure 7.**(

**a**) Cross-sectional views of (

**a**) elliptic $(n=2)$; and (

**b**) corrugated $(n=5)$ deformation modes; The mode index n indicates the wave number of the deformation mode along the circumference; (

**c**) Wall-number dependence of critical pressure ${p}_{c}$. Immediately above ${p}_{c}$, the original circular cross section of MWNTs gets radially corrugated; (

**d**) Stepwise increase in the corrugation mode index n. Reprinted from References [53,61].

**Figure 8.**Cross-sectional views of relaxed MWNTs indexed by (2,8)/(4,16)/…/(2n,8n). The wall numbers n are 5, 10, 15, 20, and 25 from left to right, and all the MWNTs are 20 nm long. Reprinted from Reference [63].

## 6. Bend Buckling of SWNTs

#### 6.1. Kink Formation

**Figure 9.**(

**a**) Kink structure formed in an SWNT with diameters of 1.2 nm under bending. The gap between the tip of the kink and the upper wall is about 0.4 nm; (

**b**) Atomic structure around the kink reproduced by computer simulations. The shaded circles beneath the tube image express the local strain energy at the various atoms, measured relative to a relaxed atom in an infinite graphene sheet. The strain energy scale ranges from 0 to 1.2 eV/atom, from left to right; (

**c**) Total strain energy (in dimensionless units) of an SWNT of diameter ∼1.2 nm as a function of the bending angle up to 120${}^{\circ}$. The dip at ∼30${}^{\circ}$ in the curve is associated with the kink formation. Reprinted from Reference [64].

#### 6.2. Diameter Dependence

**Figure 10.**(

**a**) Relationship between critical bending buckling curvature ${\kappa}_{\mathrm{C}}$ and nanotube diameter d obtained from MD analyses. The tube length is fixed at 24 nm; (

**b**) The length/diameter $(L/d)$ aspect ratio dependence of ${\kappa}_{\mathrm{C}}$ for the tube chiralities of (5,5), (9,0), and (10,10). Reprinted from Reference [167].

#### 6.3. Transient Bending

**Figure 11.**Predicted shape of SWNTs just after buckling, based on MD simulations for (

**a**) a 15.7-nm-long (10, 10) SWNT at the bending angle $\theta ={43}^{\circ}$; and (

**b**) a 23.6-nm-long (30, 30) SWNT at $\theta ={23}^{\circ}$. Note the difference in scale. Reprinted from Reference [68].

**Figure 12.**(

**a**) Deformation energy U for a 23.6-nm-long (30,30) SWNT as a function of the bending angle θ. The symbols a–e attached to the curve indicate the points for which the tube shape and cross section at the buckling point are shown in the images (

**b**)–(

**f**) on the right. “TBR" denotes the transient bending regime. Reprinted from Reference [68].

## 7. Bend Buckling of MWNTs

#### 7.1. Emergence of Ripples

**Figure 13.**(

**a**) Under high bending, MWNTs form kinks on the internal (compression) side of the bend; (

**b**)–(

**d**) High-resolution TEM image of a bent nanotube (with a radius of curvature of ∼400 nm), showing the characteristic wavelike distortion. The amplitude of the ripples increases continuously from the center of the tube to the outer layers of the inner arc of the bend. Reprinted from References [66,102].

#### 7.2. Yoshimura Pattern

**Figure 14.**Rippling of a 34-walled carbon nanotube: (

**a**) longitudinal section of the central part of the simulated nanotube; and (

**b**) the morphology of the rippled MWNT reminiscent of the Yoshimura pattern. Highlighted are the ridges and furrows, as well as the trace of the longitudinal section. Reprinted from Reference [174].

**Figure 15.**Energy curves for a bent 34-walled nanotube with respect to the bending curvature. Shown are the strain energy for fictitiously uniform bending (squares), the strain energy for actually rippled deformation (crosses), and the total energy (i.e., sum of the strain energy and the vdW one) for rippled deformation (circles). Reprinted from Reference [174].

## 8. Twist Buckling

#### 8.1. Asymmetric Response of SWNTs

**Figure 16.**Morphological changes for a $(8,3)$ nanotube under torsion. Applied strain and its direction are indicated beneath the diagram; the digit 0.05, for example, corresponds to the strain of 5%, and the sign + (−) indicates right(left)-handed rotation. Under right-handed rotation, the tube buckles at a critical buckling strain ${\gamma}_{\mathrm{cr}}$ = 7.6%, whereas it buckles at ${\gamma}_{\mathrm{cl}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}4.3\%$ under left-handed rotation. Reprinted from Reference [186].

**Figure 17.**Critical buckling shear strains as a function of tube chirality. Some additional data for SWNTs with slightly larger or smaller diameters are also presented for reference. Reprinted from Reference [186].

#### 8.2. Nontrivial Response of MWNTs

**Figure 18.**(

**a**)–(

**c**) Helical buckling of (5,0)@(14,0) DWNTs with lengths: (

**a**) $L=$ 1.095 nm; (

**b**) 4.45 nm; and (

**c**) 6.97 nm. For each tube length, the inner wall shows ordinary buckled patterns, whereas the outer wall exhibits nontrivial buckling modes associated with local rims; (

**d**)–(

**f**) Helical rippling deformation of a 10-walled nanotube (5,5)@⋯@(50,50) of 34 nm in length and 3.4 nm in radius: (

**d**) Longitudinal view; (

**e**) Cross-sectional view; (

**f**) Deformation map, with green for ridges and blue for furrows. Reprinted from References [75,178].

## 9. Universal Scaling Laws Under Bending and Torsion

**Figure 19.**(

**a,d**) Strain energy curves as a function of the bending curvature κ and the twisting angle Θ; (

**b,e**) Data collapse upon appropriate rescaling. The power-law fits with exponents 2 (blue) and $a(<2)$ (red) are shown for illustration. In all four plots, the number of walls increases stepwise from 10 (circles) to 40 (crosses); (

**c**) 40-walled nanotube in pure bending; (

**f**) 35-walled nanotube in torsion. The latter two panels present the deformed shape (top), Gaussian curvature map (middle, with green being zero, red being positive, and blue being negative), and energy density map (bottom, with red being high and blue being low). Reprinted from Reference [176].

## 10. Challenge and Future Directions

#### 10.1. Buckling Effects on Heat Transport

#### 10.2. Role of Defects and Imperfections

#### 10.3. Relevance to Chemical Reaction

## Acknowledgments

## Appendix. Who Discovered Carbon Nanotubes First?

## A1. Iijima’s Nanotube in 1991

**Figure A1.**TEM image of nanotubes and sketches of each nanotube’s cross sections for (

**a**) a five-wall nanotube with a diameter of 6.7 nm; (

**b**) a double-wall nanotube with a diameter of 5.5 nm; and (

**c**) a seven-wall nanotube with a diameter of 6.5 nm, which has the smallest hollow diameter (∼2.2 nm) among the three specimens; Electron diffraction patterns showing (

**d**) the superposition of three sets of $\left\{hk0\right\}$ spots taken from a seven-wall nanotube; and (

**e**) the superposition of four sets of $\left\{hk0\right\}$ spots from a nine-wall nanotube. Reprinted from Reference [3].

## A2. Carbon Nanotubes Prior to 1991

**Figure A2.**The earliest TEM images of carbon nanotubes published in Reference [209].

## A3. Closing Remarks

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Shima, H.
Buckling of Carbon Nanotubes: A State of the Art Review. *Materials* **2012**, *5*, 47-84.
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Shima H.
Buckling of Carbon Nanotubes: A State of the Art Review. *Materials*. 2012; 5(1):47-84.
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2012. "Buckling of Carbon Nanotubes: A State of the Art Review" *Materials* 5, no. 1: 47-84.
https://doi.org/10.3390/ma5010047