The strain rate effect of perfect and defective single-walled carbon nanotubes under axial compression

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Copyright © 2012 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol. 9, 1–7, 2012

The Strain Rate Effect of Perfect and DefectiveSingle-Walled Carbon Nanotubes

Under Axial Compression

Hao Xin1 and Qiang Han1�2�∗1State Key Laboratory of Subtropical Building Science, School of Civil Engineering and Transportation,

South China University of Technology, Guangzhou 510640, P. R. China2College of Architectural and Civil Engineering, Xinjiang University, Urumqi 830047, P. R. China

Different effects of three typical categories of defect on the axial buckling properties of single-walled carbon nanotubes are investigated. Classical molecular dynamics is used to simulate thebuckling behavior of both perfect and defective armchair single-walled carbon nanotubes underaxial compression at different loading rates. Simulation results show that the buckling behavior ofdefective tubes is loading-rate dependent regularly while the mechanical properties of the perfecttube do not change much as the loading rate varies. The in-plane stiffness increases as the tubesare compressed which means the single-walled carbon nanotubes do not deform linearly beforethey buckle. The mechanical properties of single-walled carbon nanotubes are easily influenced bythe defects, but the harmful effects of defects do not simply depend on the size of the defectivearea. The critical buckling loads of single-walled carbon nanotubes decrease sharply due to thedefects, and the influence is strongly related to the buckling modes which specifically differ fromeach other due to the different defect structures. The effective Young’s modulus is also slightlyadversely affected with the effect mainly depending on how many axial zigzag chains of C–C bondsare broken in the defective area.

Keywords: Carbon Nanotubes, Defect, Strain Rate, Buckling, Molecular Dynamics.

1. INTRODUCTION

Since the carbon nanotubes (CNTs)1 were discovered,greater interest has been attracted for their outstandingproperties and wide potential applications in nano-engineering. They are believed to provide the ultimate rein-forcing materials for the development of a new class ofcomposites,2�3 and the applications of these compositesstrongly depend on the mechanical behavior of the CNTs.Nowadays, increasing numbers of studies focus on theirexceptional mechanical properties.Numerous researchers carried out experiments to mea-

sure the effective elastic modulus of CNTs.4–6 Theyreported the effective Young’s modulus of CNTs rangingfrom 0.1 to 1.7 TPa, decreasing as the diameter increased,and the average is about 1�0∼1�2 TPa. Molecular dynam-ics (MD) simulations have provided abundant results forthe understanding of the buckling behavior of CNTs. TheYoung’s modulus of the CNTs was predicted about 1�0∼1�2TPa through various MD methods.7–9 Hu et al.10 proposed

∗Author to whom correspondence should be addressed.

an improved molecular structural mechanics method for thebuckling analysis of CNTs, based on Li and Chou’s model8

and Tersoff-Brenner potential.11 Due to the different meth-ods employed on various CNTs in these researches, thereported data scattered around an average of 1.0 TPa whilethe effective wall thickness of single layer was supposed tobe 0.34 nm.Furthermore, it is quite necessary to study how the

defects in CNTs influence their mechanical properties,which will also be very helpful in realizing CNTs and therelevant composites. Due to the restrictions of CNT manu-facturing, perfect CNTs could be produced with much lesschance than the defective ones. Different configurations ofdefect could exist in the CNTs, i.e., vacancies (vacancy ofsingle atom or of more atoms) and “Stone-Thrower-Wales”(STW) etc.12�13 Hashimoto et al.14 reported observations ofin situ defect formation in single graphite layers with high-resolution TEM. The defect structures can be detected andinvestigated accurately with their method. Chowdhury andOkabe15 investigated the effects of the STW geometricaldefect in the CNTs on the interfacial shear strength (ISS)of CNT reinforced polyethylene composites using MD

J. Comput. Theor. Nanosci. 2012, Vol. 9, No. 3 1546-1955/2012/9/001/007 doi:10.1166/jctn.2012.2033 1

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The Strain Rate Effect of Perfect and Defective Single-Walled Carbon Nanotubes Under Axial Compression Xin and Han

simulations. They found that the STW defect had littleeffect on the ISS when there were no cross-links in theinterface, but it significantly reduced the ISS when cross-links were present in the interface. Buckling properties ofdefective CNTs with vacancy defects were studied withMD method and evident effect of the defects on the car-rying capabilities of CNTs was revealed.16�17 Vali et al.18

investigated the effects of vacancy defects on the bucklingbehavior of single-walled carbon nanotubes (SWCNTs)via a structural mechanics approach.MD simulations are performed in present work to study

the axially compressed buckling behavior of SWCNTs.Both perfect tube and the ones with defects (vacanciesand STW) are studied, and each is simulated at differ-ent loading rates. Different deformations of all the tubesare compared at certain loading rate, and the loading-ratedependence of mechanical properties of different tubesis also investigated. The unique effect of each defect onthe mechanical properties of SWCNTs and the loading-rate dependence of this effect are studied. Especially, thedecisive factor of the buckling properties of defectiveSWCNTs is put forward.

2. MD MODELING

2.1. The Empirical Atomic Potential for CNTs

There are several empirical potentials and relevant para-meters for CNTs.11�19−21 The Morse potential is generallybelieved to accurately model the covalent bonds (‘C–C’bonds in CNTs), and the bond-stretching term of Tersoff-Brenner potential are very close to that of the Morsepotential for compression strains and for tension strains upto 20%.22

The SWCNTs in present paper are modeled with threeterms that respectively describe the Morse potential bondstretching, harmonic potential angle bending, and a peri-odic type of bond torsion. The short-range potential causedby the deformation of CNTs is described as follows,

U�rij ��ijk��ijkl� = Ur+U�+U�

= Kr�1−e−��rij−r0��2+ 12K��ijk−�0�

2

+12K�1+cos�n�ijkl−�0� (1)

where Ur , U� and U� are the potentials of bond stretching,angle bending and bond torsion, respectively, and Kr , K�

and K� are the corresponding force constants. rij repre-sents the distance between any couple of bonded atoms,as �ijk and �ijkl represent all the possible angles of bendingand torsion, and r0, �0 and �0 the corresponding referencegeometry parameters of graphite. � is the steepness of theMorse well. The values19�23�24 of all these parameters arelisted in Table I.

Table I. Parameters for the interactions of carbon atoms in CNTs.

Types ofinteractions Parameters in the equations

Bond stretching Kr = 478�9 KJ/mol, �= 21�867 nm−1,r0 = 0�1418 nm

Angle bending K� = 562�2 KJ/mol, �0 = 120�00�

Bond torsion K� = 25�12 KJ/mol, n= 2, �0 = 180�

L-J �= 0�4396 KJ/mol, � = 0�3851 nm

As to the long-range (or non-bonded) interactions, theLennard-Jones (L-J) potential is adopted to describe thevan der Waals C–C interactions in SWCNTs,

Uvdw�rij �= 4�

[(�

rij

)12

−(�

rij

)6]

(2)

where the parameters � and � are chosen from the univer-sal force field (UFF),25 also listed in Table I.

The whole system of CNTs is energy-minimized andequilibrated before the MD simulation of compression isrun. A steepest descent algorithm is adopted for energyminimization.26�27 And the leapfrog algorithm is used inthe equilibrium MD, with 2 fs as the time step for inte-gral. The axially compressive displacement is imposed onone end of a CNT, with the atoms at the other end fixed.The loading rates are set at different values changing from0�2×10−3 to 1�0×10−2 nm/ps for comparison. The envi-ronmental temperature keeps 0.01 K to avoid the thermalkinetic effect.

2.2. Models of the Perfect and Defective SWCNTs

There are mainly three classes of defects in CNTs, topo-logical defects, defects of hybridization and incompletebonding defects. Two of them are involved in presentwork, STW (one of the topological defects, widely knownas ‘5-7-7-5’ defect) and vacancies (as the ‘incompletebonding defects’). The configurations of them are shownin Figure 1. The defect of single vacancy in CNTs could beformed when an atom and the corresponding three bondsare missing and the defects of bond vacancy (or doublevacancies) could be formed in a similar way. Two differ-ent shapes of double vacancies are studied for comparison.The STW defect can be incorporated in a normal tube withrotation of a C–C bond between two hexagons by 90�. Thisrotation changes four neighboring hexagons into two pen-tagons and two heptagons.28 It causes little change to thediameter and chirality of CNTs and the deformation effectis rather local. And this transformation effectively elon-gates the tube in the strain direction, releasing the excessstrain energy.The nanotubes simulated in present work are all with

the similar length of about 6 nm. And in order to providea general picture of the effect of the defect on the buckling

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Xin and Han The Strain Rate Effect of Perfect and Defective Single-Walled Carbon Nanotubes Under Axial Compression

Fig. 1. Configurations of defects in (7, 7) SWCNTs: ‘Atom’ is for thedefect of single vacancy; ‘Bond-1’ and ‘Bond-2’ for the double vacancies.

behavior, all the defects are located at the same axial posi-tion. It is a random distance (about 1/3 of the tube length)away from the fixed end of the tube.

3. RESULTS AND DISCUSSION

3.1. Effects of the Defects on the ElasticProperties of SWCNTs

The force-strain curves for the perfect and defective (7, 7)armchair SWCNTs are shown in Figures 2–4. Results ofthe simulations indicate that the critical buckling strains(and loads) of the defective tubes are much smaller thanthe perfect tube at different loading rates. The criticalbuckling strain of SWCNTs decreases by nearly 40% dueto the defects.

Fig. 2. Force per atom for SWCNTs at loading rate 0�2×10−3 nm/ps.


It could be noticed that all the straight parts of thecurves almost overlap in Figures 2–4. And if we con-sider each SWCNT an elastic cylindrical shell, the effec-tive elastic modulus should be proportional to the slopeof the linear part of the curve. That means the effectiveelastic modulus of SWCNT does not change much due tothe existence of the defects because.However, the partial enlarged details in Figures 2–4

indicate that the modulus of the defective tubes still differsfrom each other slightly. Equation (1) and the parame-ters in Table I show that the effective elastic modulus ofCNTs is mainly decided by the strength of the C–C bondstretching action and the distribution of the angles betweenthe C–C bonds and the tube axis. Considering the numberof missing C–C bonds of different defects and the anglesbetween those C–C bonds and the tube axis, we can gen-erally conclude that the descending order of the modulusof the defective tubes is as follows: the ‘STW’ tube> the‘Atom’ tube > the tubes with double vacancies (‘Bond-1’and ‘Bond-2’). This result could be observed from the par-tial enlarged details in Figures 2–4 by comparing the slopeof the lines.


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Fig. 5. In-plane stiffness of SWCNTs at different loading rates: ‘L’ forthe lowest value and ‘H ’ for the highest, E and t are the effective elasticmodulus and the wall thickness of SWCNTs.

The different details of the enlarged drawings amongFigures 2–4 also indicate that the modulus of everySWCNT varies a little when the loading rate changes.In order to investigate how the effective elastic modulusof SWCNTs changes at different compressive loading rate,we calculate the in-plane stiffness of each tube at someloading rates. The calculated results show that the tubes areeven not strictly linear elastic, not at any loading rate. Thein-plane stiffness keeps increasing until the tubes buckle.There is a lowest value and a highest value of the in-planestiffness for every tube at each loading rate, all plottedin Figure 5. The lowest value is about 6% smaller thanthe highest value for the defective tubes, and it is about16% for the perfect tube. We also calculate the averagein-plane stiffness of perfect and defective SWCNTs at dif-ferent loading rates (Fig. 6). The average in-plane stiffnessof the defective SWCNTs increases with the increase in

Fig. 6. Average in-plane stiffness of SWCNTs at different loading rates.

Fig. 7. Effective Young’s modulus of SWCNTs at different loadingrates.

the compressive loading rate, but the one of the perfecttube seems stable. Comparing the value for all tubes at anycertain loading rate in Figure 6, we can see that the orderof them are in good agreement with what the enlargeddetails in Figures 2–4 reflect.The effective elastic modulus of SWCNTs is able to

be obtained from the in-plane stiffness if the wall thick-ness is given. Figure 7 displays the effective elasticmodulus of SWCNTs at different loading rates with thewidely accepted wall thickness 0.34 nm. The numericalresults show that the modulus of the perfect tube is about1�169∼1�176 TPa, which is in good agreement with theresults from the former researches.4–11 The modulus couldbe reduced due to the defects by 2.63% (STW) to 5.91%(Bond-1) at low loading rate, and by 1.09% (STW) to2.51% (Bond-1) at higher loading rates. Clearly, the influ-ence of the defects on the effective elastic modulus ofSWCNTs fades while the loading rate increases.Unlike the modulus of the perfect tube which is not

sensitive to the changes of the loading rate, the modulus ofthe defective SWCNTs increases by 2.37% (for the ‘STW’tube) to 4.28% (the ‘Bond-1’ one) with the increase of theloading rate from 0�2× 10−3 nm/ps to 1�0× 10−2 nm/ps.The properties of the other two defective tubes are betweenthe ‘STW’ and the ‘Bond-1’ ones (see Fig. 7). Hence, themore C–C bonds are missing in the defect, the more thedefect influences the modulus of SWCNT and the moresensitive the tube with the same defect is to the loadingrate. The descending order of the influence of the defectsand the sensitivity of the defective tubes is the ‘STW’defect, the ‘Atom,’ the ‘Bond-2’ and the ‘Bond-1,’ and thecorresponding defective tubes.Figure 7 shows that the modulus of the ‘Bond-1’ defec-

tive tube is distinctly smaller than the ‘Atom’ tube whilethe ‘Bond-2’ one is very close to the latter, and that isnot casual but appears at all the simulated loading rates.This could be easily understood when we just investigatehow these vacancy defects make breakage in the armchair


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(a) (b) (c)

Fig. 8. Breakage of the defects in the armchair SWCNTs: (a) ‘Atom’defect, (b) ‘Bond-1’ defect, (c) ‘Bond-2’ defect.

SWCNTs, instead of considering the size of the defectivearea. The size of a defect should be crucial for the bucklingresearches but not substantial for the issues of elastic pro-perties. Figure 8 may visually illustrate this phenomenon.As far as an armchair SWCNT is concerned, the axiallyarranged chains (see the bold sticks in Fig. 8) apparentlyare the main parts to sustain the compressive loads so thatthe effective elastic modulus should have close relation tothe integrity of the chains. However, both of the ‘Atom’and the ‘Bond-2’ defects break only one of the axial zigzagchains of C–C bonds, while the ‘Bond-1’ defect breakstwo. That is why the modulus of the ‘Bond-2’ tube is soclose to the ‘Atom’ tube and they both clearly superior tothe ‘Bond-1’ one.Further more, simulations of axial compression of defec-

tive armchair SWCNTs with multiple vacancies (see Fig. 9)are also carried out, and the in-plane stiffness of the defec-tive tubes are abtained (Fig. 10). Defective SWCNTs withsame number of vacancies could be made based on the‘Bond-1’ defect (Fig. 9(a)) and the ‘Bond-2’ (or ‘Atom’)defect (Fig. 9 (b)), which are named as the ‘Bond-1-x’ and‘Bond-2-x’ series. The defects in ‘Bond-2-x’ tubes breakonly one of the axial zigzag chains of C–C bonds, while

(a) (b)

Fig. 9. Configurations of defective armchair SWCNTs with multiplevacancies: (a) vacancies based on ‘Bond-1’ defect, (b) vacancies basedon ‘Bond-2’/‘Atom’ defect.

Fig. 10. In-plane stiffness of defective SWCNTs with two series ofvacancies.

the ‘Bond-1-x’ defects break two. The in-plane stiffness ofthese two series of defective SWCNTs with the same num-ber of vacancies is compared in Figure 10. It is clear thatthe values of the ‘Bond-2-x’ tubes are always greater thanthe ‘Bond-1-x’ ones. And this result is in good agreementwith what we conclude in the early part.

3.2. Buckling Properties of thePerfect and Defective SWCNTs

Published papers have reported the significant effects ofdefects on the mechanical properties of CNTs, such as onthe tensile strength, the critical buckling load and the criti-cal buckling strain. We investigate the effects of the defectson the axially compressive buckling properties of SWC-NTs, and focus on the difference of the effects at vari-ous loading rates. Critical buckling loads of perfect anddefective SWCNTs at different loading rates are plotted inFigure 11.The influence of the defects on the critical buckling

load of SWCNTs is quite different to that on the effective

Fig. 11. Critical buckling loads of SWCNTs at different loading rates.


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modulus. The critical buckling load is much more sensitivethan the modulus to the defects. It reduces by almost a halfdue to the defects at lower loading rates and by 24%∼40%at higher loading rates (Fig. 11). It is quite similar to mostof ordinary continuum materials, the buckling properties ofwhich are easily influenced due to the uneven distributionof the stress induced by the defects. As far as the effectsof the loading rates are concerned, it is obtained that thecritical buckling load of the perfect tube does not changemuch as the loading rate varies, by 3.0% at most. How-ever, that of the defective tubes is clearly dependent on theloading rate. The critical buckling loads of the defectiveSWCNTs with double vacancies increase the most withthe increase of the loading rate, by 48.20% and 43.12% forthe ‘Bond-1’ and the ‘Bond-2’ tubes, which of the STWand the ‘Atom’ tubes increase by 23.76% and 22.98%.From the comparison above, we could see that the effect

of the defects on the critical buckling load of SWCNTsis less regular than that on the modulus. Our simulationsshow that all the defective tubes buckle locally at thedefective area. And the different structures of the defectsinduce distinct buckling modes of the tubes, which leadto the less regular critical buckling loads of the defectivetubes. Figure 12 shows the enlarge drawing of the loadingcurves around the critical buckling points of the defec-tive SWCNTs, and Figure 13 illustrates different deformedshapes of the defective areas at those critical and typicalpoints. The points ‘a’–‘f’ in Figure 12 and the pictures(a–f) in Figure 13 are with a one-to-one correspondence.It could be noticed that the STW defective tube and

the ‘Atom’ defective tube buckle in a symmetrical mode(see Figs. 13(a and d)). The local buckling only occursat one side of the wall when the other two tubes buckle(see Figs. 13(b and c)). In fact, before the STW tube sym-metrically buckles the defective area becomes deformedinside the wall (see the e point in Fig. 12 and Fig. 13(e)).But the rotated STW bond is still strong enough to sup-port the tube until the other side of the wall buckles. It is

Fig. 12. Enlarge drawing of the load–strain curves of the defectiveSWCNTs.

(a) (b) (c)

(d) (e) (f)

Fig. 13. Buckling shapes of the defective areas.

the similar situation for the ‘Atom’ tube (see the f point inFig. 12 and Fig. 13(f)). However, the firstly sunken sideof the wall of the ‘Atom’ tube is not powerful enoughto resist the deformation rigidly so that the load does notincrease much when the tube is continuously compressedfrom point f to d (see Fig. 12).It seems strange that the critical buckling loads of the

defective tubes with ‘Bond-1’ or ‘Bond-2’ defects aregreater than the tubes with ‘Atom’ defect, though we thinkit is quite reasonable. If we observe the pictures b and cin Figure 13 carefully, we find that ‘Bond-1’ or ‘Bond-2’defects makes 4 atoms, each with a free � bond. Take the


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4 atoms in the Figure 13(b) as an example, one of the 2 topatoms moves inward the wall and the other one outward,and the 2 bottom atoms are in the similar situation, whichwe think be able to make some kind of force balance. Fur-ther more, the top pair and the bottom pair atoms also makea three-dimensional cross, and that should also be helpfulfor the tube to sustain more loads. Situations for the ‘Bond-2’ defective tube is almost the same (Fig. 13(c)), exceptthat the shape of the defective area makes the tube have atrend of distortion (see the top little picture of Fig. 13(c)),which makes it inferior to the ‘Bond-1’ tube. Thus, besidethe category of the defects and the size of the defective area,the specific detail of the deformation at the defective areais another important factor to the investigation of bucklingproperties of defective SWCNTs.

4. CONCLUSIONS

MD method is used to simulate the axially compres-sive buckling behavior of perfect and defective SWCNTs.Mechanical properties of the tubes at different loadingrates are investigated and compared. From these investiga-tions, the following results can be concluded.The critical buckling load of SWCNTs decreases

sharply due to the mentioned defects, and the effectiveYoung’s modulus is also slightly influenced. The influenceof the defects varies at different loading rates.None of the simulated tubes is strictly linear elastic as

widely known before. The in-plane stiffness keeps increas-ing until the tubes buckle.The average effective Young’s modulus of the perfect

tube is about 1.169∼1.176 TPa, which is in good agree-ment with the results from the former researches. The mod-ulus of the perfect tube is not so sensitive to the changesof compressive loading rate, but the modulus of the defec-tive SWCNTs apparently increases with the increase of theloading rate.We think that the way the defects break the axial zigzag

chains of C–C bonds has significant effect on the effectiveelastic modulus of the defective armchair SWCNTs.The critical buckling load of the perfect tube does not

change much as the loading rate varies but that of thedefective tubes are clearly dependent on the loading rate.The effect of the defects on the critical buckling load ofSWCNTs is less regular than on the modulus. The STWdefective tube and the ‘Atom’ one buckle in a symmetricalmode. But when the other two defective tubes buckle thelocal buckling only occurs at one side of the wall. We con-clude that the critical buckling loads of the defective tubesare strongly related to their buckling modes, which spe-cially differ from each other due to the distinct structuresof defects.

As far as the density of the carbon atom or the apparentstructure is concerned, the tube with STW defect appearsmore like a perfect tube than a defective one with defectsof vacancies. However, as to the buckling behavior, theSTW defective tube is very close to the latter.These results also indicate that it is necessary for us to

continuously study the effects of different defects on themechanical properties of CNTs. It may help to understandsome experimental behavior well and truly.

Acknowledgments: The authors wish to acknowl-edge the supports from the Natural Science Foun-dation of Guangdong Province (8151064101000002,10151064101000062).

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Received: 13 March 2011. Accepted: 25 April 2011.


The strain rate effect of perfect and defective single-walled carbon nanotubes under axial compression

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