IOP PUBLISHING NANOTECHNOLOGY

Nanotechnology 18 (2007) 395703 (7pp) doi:10.1088/0957-4484/18/39/395703

Collapse and stability of single- andmulti-wall carbon nanotubesJ Xiao1, B Liu1,4, Y Huang2,4, J Zuo3, K-C Hwang1 and M-F Yu2

1 Department of Engineering Mechanics, Tsinghua University, Beijing 100084,People’s Republic of China2 Department of Mechanical Science and Engineering, University of Illinois, Urbana,IL 61801, USA3 Department of Materials Science and Engineering, University of Illinois, Urbana, IL 61801,USA

E-mail: liubin@tsinghua.edu.cn and huang9@uiuc.edu

Received 21 May 2007, in final form 30 July 2007Published 4 September 2007Online at stacks.iop.org/Nano/18/395703

AbstractThe collapse and stability of carbon nanotubes (CNTs) have importantimplications for their synthesis and applications. While nanotube collapsehas been observed experimentally, the conditions for the collapse, especiallyits dependence on tube structures, are not clear. We have studied theenergetics of the collapse of single- and multi-wall CNTs via atomisticsimulations. The collapse is governed by the number of walls and the radiusof the inner-most wall. The collapsed structure is energetically favored abouta certain diameter, which is 4.12, 4.96 and 5.76 nm for single-, double- andtriple-wall CNTs, respectively. The CNT chirality also has a strong influenceon the collapsed structure, leading to flat, warped and twisted CNTs,depending on the chiral angle.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Carbon nanotubes (CNTs) [1] possess unique mechanical,thermal and electrical properties, and have many potentialapplications including nanoscale devices. Owing to the smallsize and the considerable out-of-plane deformability [2], smallexternal forces may induce radial deformation of the CNTsand even their collapse [3], which can significantly affect theelectrical properties of CNTs [4–6].

Gao et al [7] employed classical molecular dynamics andmolecular mechanics methods to study the collapse of single-wall carbon nanotubes (SWCNT). Their results suggestedthat SWCNTs with a radius larger than 3 nm will collapse,since the energy of the collapsed state is lower than thatof the circular, uncollapsed state. Liu et al [8] studiedthe formation of fully collapsed SWCNTs with the atomic-scale finite-element method, which is an order-N atomisticsimulation method [9, 10] based on Brenner et al’s recentmany-body interatomic potential for carbon [11]. For thearmchair (n, n) SWCNTs, collapse occurs when n � 30 and

4 Authors to whom any correspondence should be addressed.

the corresponding critical radius (for n = 30) is 2.06 nm,which is smaller than the 3 nm given by Gao et al [7].Figure 1 shows the transmission electron microscopy (TEM)image of a collapsed SWCNT of 2.5 nm in radius obtainedin our experimental study. The SWCNT was synthesized bychemical vapor deposition (CVD). Most of the CVD tubes aresmall tubes with a radius between ∼0.5 and ∼1.5 nm. Large-radius tubes are rare and they are unstable under prolongedTEM observations. The tube shown in figure 1 partiallycollapsed against the supporting Au grid bar used to support thecatalyst for nanotube growth. Its collapse is consistent with Liuet al’s prediction [8], since the radius of the collapsed SWCNT(2.5 nm) is larger than the critical value (2.06 nm).

Chopra et al [3] observed the fully collapsed multi-wallcarbon nanotube (MWCNT) on a substrate, and the collapsedMWCNT was twisted. A continuum model [3, 12] wasdeveloped to estimate the critical radius of collapse, but thecontinuum model could not explain the twist of the collapsedMWCNT. The collapsed MWCNTs may also exist in differentforms [13]. Figure 2 shows the TEM images of three typesof collapsed MWCNT [13]: a doubly supported, twisted andfully collapsed MWCNT (figure 2(a)), a straight and fully

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Nanotechnology 18 (2007) 395703 J Xiao et al

Figure 1. A collapsed single-wall carbon nanotube of 2.5 nm inradius. The tube was synthesized by chemical vapor deposition. Thenanotube was imaged by TEM and the position of the collapsed wallis indicated by the arrow. The collapse was against a Au TEM gridbar, which was used to support the catalyst for nanotube growth.

collapsed MWCNT (figure 2(b)), and a cantilevered, twistedand fully collapsed MWCNT (figure 2(c)). The van der Waalsinteraction related to the lattice registry is directly responsiblefor these different forms of MWCNT collapse [13]. Thecontinuum model (e.g. Chopra et al 1995; Benedict et al 1998)does not account for the effect of lattice registry and thereforecannot explain Yu et al’s experiments [13]. The atomisticsimulation methods are necessary to study the lattice registrydependent collapse of MWCNTs.

The stability and collapse of carbon nanotubes, includingMWCNTs, are studied in this paper to determine the criticalradius of nanotube collapse and its dependence on structure.The results show different collapse modes, which can explainYu et al’s experiments [13].

2. Methodology

The computational method used in this study is the atomic-scale finite-element method (AFEM) [9, 10], which modelseach atom as a node, and is an order-N atomistic simulationmethod based on the recent many-body interatomic potentialfor carbon [11]. The method accurately describes the twotypes of atomistic interactions between carbon atoms: theinteractions between carbon atoms from the same nanotubewall are modeled using the recent many-body interatomicpotential [11], while the van der Waals interactions betweencarbon atoms from different nanotube walls are modeled with aLennard-Jones potential for carbon, v(r) = 4ε[(σ

r )12 − (σr )6],

where ε = 2.39 meV and σ = 0.342 nm [14]. The total

energy in the system is the sum of the energy in all covalentbonds within the same tube and the energy due to van derWaals interactions between pairs of atoms from different CNTwalls. The minimization of the total energy then gives theatom positions at equilibrium. Similar to the continuum finite-element method, AFEM uses both the first- and second-orderderivatives of the system energy in the energy minimization.The method is therefore more effective and robust than thewidely used conjugate gradient method.

The approach for investigating the CNT collapse consistsof two steps. First, a circular CNT is collapsed by anexternal force. The external force is then removed, and a newequilibrium configuration is obtained via energy minimization.The collapsed CNT is stable if its system energy is lower thanthat of the circular CNT, which is called the stable collapse(figure 3). The collapsed CNT is metastable (figure 3) if theenergy of the new equilibrium configuration is higher than thatof the circular CNT. For a small CNT radius, the collapsedCNT is not stable and it springs back to the circular CNT(figure 3).

The ends of the CNT are open, and the dangling bondsdo not undergo any reconstruction in the simulation, even aftercollapse. This is because the equilibrium distance due to thevan der Waals interactions between the collapsed CNT wallsis around 0.34 nm, which is much larger than the covalentbond length (0.14 nm). Six degrees of freedom of the CNTare constrained to eliminate the rigid body motion. Theconvergence is verified by comparing the results for differentlengths of CNTs.

3. Results

3.1. Stability map of multi-wall carbon nanotubes

We use (n1, m1)(n2, m2) · · · (nl, ml) to denote the chirality of aMWCNT, where (ni , mi) represent the chirality of the i th wall(i = 1, 2, . . . , l), with the first being the inner-most wall andthe last being the outer-most wall. Figure 3 shows a stabilitymap for the armchair (n1, n1)(n2, n2) · · · (nl , nl) MWCNTs at

Figure 2. The transmission election microscopy images of three types of collapsed multi-wall carbon nanotube (MWCNT) [13]: (a) a doublysupported, twisted, and fully collapsed MWCNT; (b) a purely flat and fully collapsed MWCNT; and (c) a cantilevered, twisted, and fullycollapsed MWCNT.

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SWCNT

DWCNT

(15,15) (30,30) (n,n)

CircularStable

CollapseMetastable

Full CollapseStable

CircularStable

CollapseMetastable

Full CollapseStable

CollapseMetastable

Full CollapseStable

CircularStable

TWCNT

(15,15) (36,36) (n,n)

(n,n)(42,42)(16,16)

Figure 3. A stability map for the deformation of armchair carbonnanotubes (CNTs). For double-wall and triple-wall CNTs, (n, n)represent the chirality of the inner-most tube in this map.

room temperature, where ni+1 = ni + 5.5 There are threedifferent energetically favorable structures at room temperaturefor single-, double-, and triple-wall CNTs:

(i) for the single-wall armchair CNTs (n = m) [8], theuncollapsed CNTs with a circular cross section are stablefor n1 < 15, the partially collapsed or fully collapsedCNTs are metastable6 for 15 � n1 < 30, and the fullycollapsed CNTs are stable for n1 � 30;

(ii) for the double-wall armchair CNTs, the uncollapsed CNTswith a circular cross section are stable for n1 < 15, thepartially collapsed or fully collapsed CNTs are metastablefor 15 � n1 < 36, and the fully collapsed CNTs are stablefor n1 � 36; and

(iii) for the triple-wall armchair CNTs, these three differentregimes become n1 < 16, 16 � n1 < 42, and n1 � 42,respectively.

5 The equilibrium distance between CNT walls is 0.34 nm, based on which the(n1, n1)(n2, n2) · · · (nl , nl ) armchair MWCNTs approximately satisfy ni+1 =ni + 5.6 Metastable means that the collapsed CNT corresponds to a local energyminimum which is still larger than the energy for the uncollapsed CNT, suchthat the CNT may resume the uncollapsed state under disturbance.

Figure 4. (a) A schematic diagram of a collapsed single-wall carbonnanotube (CNT); (b) the collapsed armchair CNT achieves ABstacking via a translation normal to the axial (z) direction; (c) thecollapsed zigzag CNT achieves AB stacking via translations normaland parallel to the axial (z) direction; (d) the collapsed chiral CNTachieves AB stacking via translations normal and parallel to the axial(z) direction, plus rotation.

The collapse of CNTs decreases the interwall binding energyat the expense of increasing the deformation energy. As thenumber of CNT walls increases, the CNT collapse requiresa large decrease in the interwall binding energy in order tocompensate the increase in deformation energy. Since thespacing of (circular or collapsed) CNT walls is always around0.34 nm, the change in the intertube binding is much smallerthan that associated with the collapse of the inner-most wall.This requires an increase in the critical radius of the inner-mostwall governing the stable, fully collapsed CNTs, from 2.06 nmfor SWCNTs to 2.48 nm and 2.88 nm for double- and triple-wall CNTs, respectively.

We have also studied the collapse of other (non-armchair)MWCNTs7. For zigzag double-wall CNT (n1, 0)(n2, 0), thecritical radius of the inner-wall governing the fully collapsedCNT is 2.44 nm, which is very close to its counterpart(2.48 nm) for armchair double-wall CNTs, and suggests thatthe collapse of the CNT is mainly controlled by its inner-wall radius (and the number of walls). In fact, we havestudied double-wall CNTs with an inner armchair (n1, n1)

and an outer zigzag (n2, 0) or chiral (n2, m2)(m2 �= n2 �= 0)

CNTs, as well as with an inner zigzag (n1, 0) or chiral(n1, m1)(m1 �= n1 �= 0) and outer armchair (n2, n2) CNTs.The critical inner-wall radius for a fully collapsed double-wallCNT is all around 2.48 nm.

Yu et al [15] observed the collapse of double- and triple-wall CNTs. For double-wall CNTs, the inner radii of collapsednanotubes are 5 and 5.5 nm, which are definitely in the range

7 The chirality of each wall is chosen to keep the equilibrium distance betweenCNT walls approximately 0.34 nm.

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-0.0360

-0.0362

-0.0364

-0.0366

-0.0368

-0.0370

-0.0372

ener

gy/a

tom

(eV

)

withlattice

registry

withlattice

registry

(62,0)

(60,20)(41,41)

(41,39)

without significantlattice registry

α

α

Self Lattice Registry

Cross Lattice Registry

0 5 10 15 20 25 30 35 40 45 50 55 60

(a)

(b)

-0.03613

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-0.03637

-0.03645

-0.03653

-0.03660

-0.03668

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eV/atom

Figure 5. (a) The system energy versus the alignment angle α for two parallel graphenes shown in the inset [8]; (b) the contour plot of energylevels for two parallel graphenes subject to translation.

of stable collapse (>2.48 nm). For triple-wall CNTs, the innerradii of collapsed nanotubes are 2, 2.5, 3.5, 5 and 5.5 nm. Herethe last three are in the range of stable collapse, and the firsttwo are in the range of metastable collapse.

3.2. Collapse mode of single- and double-wall carbonnanotubes

Even though CNT collapse is governed by its radius (and thenumber of walls), the collapse mode depends strongly on theCNT charality. We first summarize the results for SWCNT [8],which pave the way for understanding the collapse of double-wall CNTs.

3.2.1. Single-wall carbon nanotubes. Figure 4(a) shows acollapsed CNT, with z along the CNT axis and x normal tothe collapse direction. Except around the edges, the interactionbetween the collapsed CNT walls is essentially the same as

that between two parallel graphenes, for which the minimalenergy is around the AB stacking. For a collapsed armchairCNT (figure 4(b)), the AB stacking can be reached via atranslation along the x direction such that the collapsed CNTwalls remain planar (except near edges). For a collapsedzigzag CNT (figure 4(c)), the AB stacking can be reachedvia two translations along the x and z (axial) directions,which lead to warping of the collapsed CNT walls. The ABstacking can never be reached for a collapsed chiral CNTunless the collapsed CNT walls undergo rotations, as shownin figure 4(d), and to be further discussed in the following.

In order to understand the rotation of the collapsed chiralCNTs, Liu et al [8] studied the energy between two parallelgraphenes subject to van der Waals interactions. Let α denotetheir alignment angle. Without losing generality, α is between0◦ and 60◦, where α = 0◦ and 60◦ correspond to AA stackingand AB stacking, respectively. Figure 5(a) shows the energybetween two graphenes versus the alignment angle α, which

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clearly has a maximum at α = 0◦ and a minimum at α =60◦. This suggests that the graphenes with AA stacking tendto rotate away from this angle α = 0◦, and that with ABstacking are stable. However, there are many local minimafor α between these two limits, and the energy curve is fairlyflat for α between 10◦ and 50◦. Therefore, for α in this range(10◦–50◦), the graphenes may undergo small rotation to reachthe local minimum, instead of large rotation towards the globalminimum at α = 60◦.

In contrast to figure 5(a), which focuses on the effect ofrotation, figure 5(b) shows the contour plot of energy levelsfor two parallel graphenes subject to translation. The centercorresponds to AA stacking, while the vertices correspond toAB stacking. Each point on the contour gives the energy levelcorresponding to the relative position of the two graphenesvia translation in this direction from the center. It is clearlyobserved that there are no energy minima between AA stackingand AB stacking such that the local energy minima cannot beachieved via translation only.

Figure 6 illustrates the relation between the alignmentangle α for two graphenes in figure 5(a) and the chiralangle θ of a collapsed SWCNT. Let Ch be the chiral vector(perpendicular to the z axis) [16], and t represents the directionnormal to the carbon bond on the top wall of the collapsedCNT, as shown in figure 6. The angle between Ch and t isdenoted by θ (figure 6) [16]. The vector t′ represents thedirection normal to the carbon bond on the bottom wall of thecollapsed CNT (figure 6). The alignment angle α in figure 5(a)is the acute angle between t and t′, which gives α = 2θ .Therefore, α = 0◦ and 60◦ in figure 5(a) correspond to thecollapsed zigzag and armchair CNTs, respectively.

Based on figures 5(a) and 6, the collapsed SWCNTdisplays two possible configurations:

(i) The alignment angle α is between 10◦ and 50◦ (chiral) orequals 60◦ (armchair). The rotation is small (α: 10◦–50◦)or non-existent (α = 60◦, translation only) such that thecollapsed CNT walls are essentially flat. This is markedby ‘without significant lattice registry’ in figure 5(a).

(ii) The alignment angle α is less than 10◦ (zigzag or chiral)or larger than 50◦ (chiral). The rotation is large (α < 10◦or α > 50◦) or the CNT warps (α = 0◦) such thatthe collapsed CNT walls are twisted or warped. This ismarked by ‘with lattice registry’ in figure 5(a).

3.2.2. Double-wall carbon nanotubes. For a collapseddouble-wall CNT, the van der Waals interactions are significantbetween neighbor walls, i.e. between two collapsed walls ofthe inner CNT and between one inner CNT wall and one outerCNT wall, which are called the self-lattice registry and cross-lattice registry (figure 5(a)), respectively. The alignment anglefor two collapsed walls of the inner CNT is denoted by αself

and that between the inner and outer CNT walls is denoted byαcross. Here αcross is related to chiral angles of the inner andouter walls by αcross = |θin − θout|, which also ranges from 0◦to 60◦. Also based on figure 5(a), the inner and outer CNTwalls are non-flat when αcross is less than 10◦ or more than 50◦(except 60◦).

Figure 7 shows four different collapse modes of double-wall CNTs for different combinations of αself and αcross.

Figure 6. A schematic diagram of the alignment angle and chiralangle of a collapsed single-wall carbon nanotube.

(i) No lattice registry (αself, αcross: 10◦–50◦ or 60◦). Thereis no rotation between two collapsed walls of the inner CNTnor between the inner and outer CNT walls, as shown infigure 7(a) for a (37, 37)(60, 20) CNT, which gives αself = 60◦and αcross = 15◦. The collapsed double-wall CNT appears tobe perfectly flat, which is consistent with the flat, collapsedCNT observed in experiment shown in figure 2(b).

The increase in covalent bond energy for the collapse of a(37, 37)(60, 20) CNT is 5.323 meV/atom, but the decrease invan der Waals energy due to the collapse of the inner wall aloneis already larger, 5.437 meV/atom. The CNT collapse alsodecreases the van der Waals energy between two CNT wallsby 0.479 meV/atom, which is an order of magnitude smallerthan that for the inner wall collapse. This is because, duringCNT collapse, the CNT wall spacing remains approximately0.34 nm such that the corresponding change of van der Waalsenergy between two CNT walls is not large.

(ii) Self lattice registry dominates (αself: 0◦–10◦ or 50◦–60◦; αcross: 10◦–50◦ or 60◦). The collapse of the inner CNTwall leads to rotation or warping, but there is no rotationbetween the inner and outer CNT walls. The net result is atwisted or warped double-wall CNT, as shown in figure 7(b) fora (62, 0)(41, 41) CNT which gives αself = 0◦ and αcross = 30◦.

The increase in covalent bond energy for the collapseof a (62, 0)(41, 41) CNT is 5.588 meV/atom, which islarger than the decrease in van der Waals energy due to thecollapse of the inner wall alone, 5.256 meV/atom. Thevan der Waals energy between two CNT walls decreases by0.506 meV/atom such that the total van der Waals energydecrease is 5.762 meV/atom, which is larger than the increasein covalent bond energy (5.588 meV/atom).

(iii) Cross lattice registry dominates (αself: 10◦–50◦ or 60◦;αcross: 0◦–10◦ or 50◦–60◦). There is no rotation between twocollapsed walls of the inner CNT, but there is rotation betweenthe inner and outer CNT walls. The net result is a twisteddouble-wall CNT, as shown in figure 7(c) for a (53, 17)(60, 20)CNT which gives αself = 27◦ and αcross = 0.5◦.

The increase in covalent bond energy for the collapse of a(53, 17)(60, 20) CNT is 5.542 meV/atom, which is larger thanboth the decrease in van der Waals energy, 5.263 meV/atom,due to the collapse of the inner wall alone and 0.576 meV/atombetween two CNT walls, but is smaller than their sum.

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Nanotechnology 18 (2007) 395703 J Xiao et al

Figure 7. Four different collapsed double-wall carbon nanotubes.

(iv) Both self and cross lattice registry dominate (αself,αcross: 0◦–10◦ or 50◦–60◦). There is rotation between twocollapsed walls of the inner CNT, as well as between the innerand outer CNT walls. The twist for the double-wall CNT canbe ‘amplified’, as shown in figure 7(d) for a (41, 39)(47, 43)CNT which gives αself = 58.4◦ and αcross = 0.7◦.

The increase in covalent bond energy for the collapse of a(41, 39)(47, 43) CNT is 5.176 meV/atom, which is less thanthe decrease in van der Waals energy due to the collapse of theinner wall alone, 5.598 meV/atom. The decrease in van derWaals energy between two CNT walls is 0.633 meV/atom.

The twisted (or warped) CNTs in figures 7(b)–(d) are onlyshown for a relatively small length (∼7 nm). For much longerCNTs (e.g. 1 μm long in the TEM images in figure 2), thetwist or rotation will be proportionally larger, which leads tothe large twist and rotation observed in figure 2(c).

4. Conclusion and discussion

The collapse of carbon nanotubes (CNTs) is governed bythe number of walls and the radius of the inner-most wall.This radius is 2.06, 2.48 and 2.88 nm for single-, double-and triple-wall CNTs, respectively. The CNT chirality has astrong influence on the collapse mode, leading to flat, warpedand twisted CNTs. The present study explains the differentcollapse modes observed in prior experiments.

The energy difference between the AA and AB stackingsgiven by the Lennard-Jones potential for the vdW interaction

is rather small. This may mean that the warped andtwisted CNT configurations may be thermodynamicallyaccessible. However, our prior study via moleculardynamics simulations [8] shows that the collapsed CNT isthermodynamically stable. It should be pointed out thatthe small energy difference results from the potential used.Other potentials for the van der Waals interactions may givesomewhat different results, but the general trend of latticeregistry in the present study should still hold.

Acknowledgments

The authors acknowledge financial support from the USNational Science Foundation (NSF) through Nano-CEMMS(grant no. DMI 03-28162) at the University of Illinois. Supportfrom the National Natural Science Foundation of China(NSFC) (grant no. 10542001) is also acknowledged.

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