self-folding and unfolding of carbon nanotubes

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Markus J. BuehlerCalifornia Institute of Technology,

Division of Chemistry and ChemicalEngineering,

Pasadena, CA 91125

Yong Kong

Huajian Gao1

e-mail: [email protected]

Max Planck Institute for Metals Research,Heisenbergstrasse 3, D-70569

Stuttgart, Germany

Yonggang HuangDepartment of Mechanical and Industrial

Engineering,University of Illinois at Urbana-

Champaign,1206 West Green Street, Urbana, IL

Self-Folding and Unfolding ofCarbon NanotubesCarbon nanotubes (CNTs) constitute a prominent example of nanomaterials. In moststudies on mechanical properties, the effort was concentrated on CNTs with relativelysmall aspect ratio of length to diameters. In contrast, CNTs with aspect ratios of severalhundred can be produced with today’s experimental techniques. We report atomistic-continuum studies of single-wall carbon nanotubes with very large aspect ratios subject tocompressive loading. It was recently shown that these long tubes display significantlydifferent mechanical behavior than tubes with smaller aspect ratios (Buehler, M. J., Kong,Y., and Guo, H., 2004, ASME J. Eng. Mater. Technol. 126, pp. 245–249). We distinguishthree different classes of mechanical response to compressive loading. While the defor-mation mechanism is characterized by buckling of thin shells in nanotubes with smallaspect ratios, it is replaced by a rodlike buckling mode above a critical aspect ratio,analogous to the Euler theory in continuum mechanics. For very large aspect ratios, ananotube is found to behave like a wire that can be deformed in a very flexible manner tovarious shapes. In this paper, we focus on the properties of such wirelike CNTs. Usingatomistic simulations carried out over a several-nanoseconds time span, we observe thatwirelike CNTs behave similarly to flexible macromolecules. Our modeling reveals thatthey can form thermodynamically stable self-folded structures, where different parts of theCNTs attract each other through weak van der Waals (vdW) forces. This self-folded CNTrepresents a novel structure not described in the literature. There exists a critical lengthfor self-folding of CNTs that depends on the elastic properties of the tube. We observe thatCNTs fold below a critical temperature and unfold above another critical temperature.Surprisingly, we observe that self-folded CNTs with very large aspect ratios never unfolduntil they evaporate. The folding-unfolding transition can be explained by entropic driv-ing forces that dominate over the elastic energy at elevated temperature. These mecha-nisms are reminiscent of the dynamics of biomolecules, such as proteins. The differentstable states of CNTs are finally summarized in a schematic phase diagram of CNTs.�DOI: 10.1115/1.1857938�

Keywords: Carbon Nanotubes, Self-Folding, Folding, van der Waals Interaction, Mo-lecular Dynamics, Mechanical Properties, Biomolecules

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1 IntroductionThe science of materials has always been of great interest to

human beings. Over thousands of years, the knowledge about ma-terials has paved the way for our modern technologies. Materialseven served as eponyms for civilization eras, as, for example, theStone Age or Bronze Age. At this moment, we are at the cross-roads to a new era where humans, for the first time, start creatingtechnologies at the scale of single atoms. Such nanotechnologycould revolutionize the way we live, learn, and organize our livesin the next decades. Since the effects of single atoms can domi-nate the materials behavior at nanoscale, the atomistic viewpointis becoming a very critical aspect not only for scientists but alsofor engineers. In addition, computer modeling will become moreand more important in the development of new technologies.

Since their discovery in 1991 by Iijama �1�, carbon nanotubes�CNTs� have received tremendous attention from various branchesof science. Extensive research studies have been carried out viavarieties of experimental, theoretical, and computer simulation ap-proaches. One of the striking features of CNTs is their enormouspotential for use in nanoscale devices. Due to their very interest-ing mechanical, optical, and electrical properties, CNTs could be-come one of the best nanomaterials to be established as one of thecornerstones for tomorrow’s technology.

CNTs are found in various configurations and geometries �2�.Essentially, they consist of a graphite sheet rolled along a certain

1Corresponding author.Manuscript received May 30, 2004; revision received October 25, 2004. Review

conducted by: M. Zhou.

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orientation into a tube. If the tube consists of a single graphitelayer, then it is referred to as a single-wall nanotube �SWNT�. Ifthere is more than one layer in one tube, then the tube is referredto as a multiwall nanotube �MWNT�. SWNTs are usually found inbundles of nearly parallel tubules, but individual SWNTs aresometimes found as well. SWNTs are known be extremely flex-ible. They can be easily bent into arcs with very small curvatureson the order of several nanometers or below. CNTs are consideredthe strongest and most flexible molecular material because of theC-C covalent bonding and seamless hexagonal network architec-ture. Van der Waals �vdW� forces, known to play a critical role innanoscale engineering, dominate the interlayer and intertubularinteractions between carbon nanotubes.

SWNTs have already found applications in technological de-vices. They are, for instance, used as sensors to detect proteinbinding �3�. Other research has revealed that DNA can be sponta-neously encapsulated into a SWNT if the tube radius exceeds acritical value �4�. All these mechanisms could potentially be inte-grated in nanodevices, as for example in lab-on-a-chip technology.A proper understanding of the mechanics of SWNT is crucial toengineer novel nanoscale devices. A joint treatment using classicalmechanics method �e.g., continuum finite element method� in con-junction with studies on the atomic scale will be important forsuch type of novel engineering in the 21st century.

In nanoscale devices, large stresses can occur due to thermal orlattice mismatch between different materials. Therefore, the reli-ability of many devices depends critically on the understanding of

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their response to mechanical loading. From this point of view, themechanical properties of SWNTs would play an important role inCNT-based nanodevices.

Yacobsen et al. �5� investigated the behavior of single, free-standing SWNTs under compressive loading using classical, mo-lecular dynamics �MD� with empirical potentials. The longest tubeconsidered was 6 nm with a diameter of 1 nm. The authors devel-oped a continuum shell model to describe the buckling modes ofthe CNTs. Ozaki and co-workers �6� investigated SWNTs underaxial compression using tight-binding �TB� MD methods and sys-tem sizes up to a few thousand atoms. In their studies, the lengthof the nanotubes was limited to about 14 nm with a diameter of0.67 nm. A ripple shell buckling was observed once the SWNTwas put in compressive loading. The details of the ripple buckling�e.g., the ripple wavelength� were found to be strongly dependenton the temperature, and the stress under large strain and zerotemperature depends on the helicity. SWNTs under tensile andcompressive loading were studied by Dereli and Ozdogan �7� us-ing a TB-MD scheme in an attempt to obtain the stress-straincurve, theoretical strength, and Poisson ratio for SWNTs. The au-thors modeled CNTs with about 400 atoms, featuring a totallength of 20 layers, corresponding to a few nanometers. Ru �8�recently considered buckling of a double-walled carbon nanotubeembedded in an elastic medium under compressive loading usinga double-shell continuum mechanics model. The main finding wasthat critical buckling strain for MWNTs may be reduced comparedto SWNTs, indicating that MWNT could even be more susceptibleto axial buckling than SWNTs. Other research focused on themechanical properties of CNTs filled with small molecules. Theauthors in �9� investigated compression of CNTs filled with nano-particles and molecules �e.g., C60 , NH4). The longest tube con-sidered had a length around 20 nm. Research has also been carriedout to investigate the elastic properties of CNTs. As recentlyshown by Hod and Rabani �10�, SWNTs can be bent into closed-ring structures �‘‘nanorings’’�. The authors in �10� used theTersoff-Brenner potential in a classical MD scheme to study theelastic properties of the CNT ring structure. There are also studiesfocused on the interaction between different CNTs. It was shownthat when SWNTs are formed into bundles due to vdW interaction�11�, their cross-sectional shape can change significantly. Thechange in shape can modify the flexural rigidity and promotebending or other forms of elastic deformation �11�. The shapechange can also affect the electrical conductivity of CNTs. Failuremechanisms such as fracture nucleation in SWNTs under tensionhave been discussed using a combined continuum-atomistic ap-proach �12�.

Significant research has been carried out to understand the be-haviors of SWNT under mechanical loading. However, most stud-ies in the literature are directed toward short tubes with lengthbelow 20 nm and diameter in the range of 1–2 nm. The reason ofstudying nanotubes with rather small aspect ratios can at leastpartly be attributed to limited computational resource in the past.Experiments, on the other hand, have shown that SWNTs cangrow to lengths above 700 nm with a diameter of 0.9 nm, result-ing in an aspect ratio as large as about 800 �2�. In fact, recentreports by Wu and colleagues suggest that CNTs can be grown aslong as 20 cm �13�. Such long SWNTs could potentially be uti-lized as nanowires or means to transport biological or other mol-ecules. On the other hand, today’s supercomputer technology andparallel computing algorithms have allowed MD simulations to beperformed on system sizes up to 1 billion atoms and time scalesup to nanoseconds �14�.

More recently, it has been found by large-scale computer simu-lation that CNTs undergo a shell-rod-wire transition with increas-ing aspect ratio �see Fig. 1� �15�. The wirelike CNTs behave simi-larly as a macromolecule, as is illustrated in Fig. 2. CNTs withvery large aspect ratios will not only be bent by application ofexternal forces, but also by thermal fluctuations, similar in mac-romolecules. It was further suggested in �15� that due to vdW

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interaction between different parts of CNTs, such structures couldundergo a self-folding mechanism where different parts of CNTsattract each other and form ‘‘self-bundles’’ of CNTs. Given theCNT is sufficiently flexible �or the aspect ratio is large enough�,the gain in potential energy due to formation of vdW bonds can belarger than the elastic energy necessary to bend the CNT. Underthese conditions, CNTs can form energetically stable self-foldedstructures similar to tennis rackets.

Here we concentrate on the dynamical properties of such wire-like CNTs, with a particular focus on self-folding of the tubes andthe properties of self-folded tubes at different temperatures. Purelyenergetical considerations based on continuum mechanics theories�corresponding to 0 K� suggest that the self-folded state of CNTsis a stable configuration. Considering the free energy of the sys-tem (F�U�TS), where U is the potential energy, T the tempera-ture, and S entropy�, it is suggested that at higher temperaturesentropy dominates over potential energy. Does this lead to unfold-ing of CNTs, similar as known in the dynamics of many biologicalmolecules? In which temperature regime are folded CNTs stable?What is the atomic dynamics of folding and unfolding?

In the remainder of the paper, we first give a brief introductioninto the numerical method and the interatomic potentials. We thendescribe the self-folded state of CNTs and provide several numeri-cal results that support the notion of a self-folded, thermodynami-cally stable state of CNTs. In particular, we show by atomistic

Fig. 1 Shell-rod wire transition of CNTs, „a… represents shell-like behavior, „b… behavior of CNTs as a rod, and „c… shows theCNT that behaves similarly as a wire. The properties of suchwirelike CNTs are in the focus of this paper.

Fig. 2 Wirelike behavior of CNTs. Upon application of com-pressive loading at the ends of the tube as discussed in †14‡,the armchair-carbon nanotube forms a helical structure.

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computer simulation that self-folded CNTs are stable at low tem-peratures and unfold at elevated temperatures. Discussion is alsodevoted to folded CNTs that never unfold until the temperature ishigher than the melting point and the CNT starts to evaporate. Wefinally draw a schematic phase diagram of CNTs summarizing thedifferent modes of deformation, ranging from shell-like, rodlike towirelike behavior that leads to the self-folded state. We discussimplications of our findings on the properties of CNTs and give anoutlook to future research work that could be carried out based onour results.

2 Simulation Method and Atomistic ModelOur simulation tool is classical molecular dynamics �MD� �16�.

MD predicts the motion of a large number of atoms by numeri-cally integrating the equations of motion governed by interatomicinteraction. Normally, it is necessary to rely on classical MD tosimulate system sizes above a few thousand atoms and time scaleson the order of nanoseconds, as such system sizes and time scalesare still far beyond the capabilities of quantum mechanics basedmethods.

Interatomic potentials are the core of classical MD methods.During the last decades, numerous potentials describing atomicinteraction in various materials with different levels of accuracyhave been proposed, each having unique problems and strengths.Approaches range from quantum-mechanics-based treatments�e.g., tight-binding potentials� to multibody potentials. For co-valently bonded materials, such as carbon or silicon, bond-ordermultibody potentials have been developed �e.g., Tersoff potentialor Stillinger-Weber �17,18��. These multibody potentials capturenot only pairwise interactions, but also additional contributionsfrom the local geometric configuration of the neighboring atoms.

In the present work, we use the Tersoff potential �17� to de-scribe the interatomic bonding of C-C atoms. The Tersoff poten-tial has proven to be a reliable empirical potential to describe thebonding inside carbon nanotubes �bond length of C-C bonds is1.42 Å�. For the van der Waals interaction between nonbondingatoms, we use Lennard-Jones potential �16� with parameters 3.49Å and 0.002 4 eV. The time step is chosen to be on the order of afemtosecond. Whereas the vdW interaction features a larger cutoffof 20 Å, the Tersoff potential has a cutoff of 2.1 Å. The fact thatthe LJ potential is long range dictates a fairly large cell divisionfor neighbor search in the molecular-dynamics algorithm, whichsignificantly slows down the simulation. Another important pointis that effective parallelization of this problem based on domaindecomposition is difficult to achieve, since the mass distribution is

Fig. 3 Decomposition of the CNT into virtual atom types.Types 1 and 4 interact via vdW interactions „LJ potential…. Thetype 3 is being used to pin the CNT during application of exter-nal forces in order to bring ends of the CNT into contact „seeFig. 4 and associated discussion in the text…, and types 6 and 7are utilized to apply external forces. All virtual atom types in-teract via the Tersoff potential with themselves and others.

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rather inhomogeneous. Therefore we only employ a serial versionof the code and leave parallelization to future work �for instance,parallelization based on OpenMP could be an effective approachwith good load balancing�.

We consider single wall �5,5� armchair nanotubes with aspectratios as large as several hundred. These CNTs feature a diameterof 7.3 Å and a length of several thousand angstroms �the longesttube we consider has a length of 3,670 Å corresponding to anaspect ratio of about 500�. A method of decomposing the CNTinto virtual atom types as depicted in Fig. 3 is used to facilitate thesimulation procedure. This method allows to assume vdW bond-ing only along particular parts of the CNT and to apply forces andother boundary conditions to specific parts of the tube. This isrequired because neighboring atoms must not interact via the LJpotential, as atoms that constitute an actual chemical bond wouldstrongly repel each other. If, on the other hand, the repulsive partwould be neglected, then the vdW forces would lead to strong

Fig. 4 „a… Equilibrium distance of CNTs. The calculated equi-librium distance agrees with molecular statics calculations byHuang †19‡. Subplots „b…–„f… show the results of a computersimulation of formation of bundles of SWNTs. „b… shows theinitial configuration as a square lattice, „c… shows the relaxedstate. It is evident that a triangular lattice has been formed, and„d… shows the relaxed configuration „after a few nanoseconds…of a larger number of CNTs. It can be seen that several ‘‘crystal’’defects have been created, such as grain boundaries between‘‘nanograins’’ of CNTs, and vacancies. „e… and „f… show three-dimensional views on the larger bundle of CNTs depicted in „d….

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attractive forces that eventually yield erroneous formation of newchemical bonds between different parts of the CNT.

We have also computed the equilibrium distance of two straightCNTs interacting via the described vdW interaction. In this calcu-lation, we use two parallel, quite short segments of CNTs withperiodic boundary conditions in the tube axis direction and relaxthe system using an NVE ensemble �on of the tubes has virtualatom type 1, the other 4�. We measure the distance of the center ofthe tubes when the system has reached steady state. The simula-tion result of the two tubes in equilibrium distance is shown inFig. 4�a�, and the calculated distance of the centers of the tube isd�11 Å. This is in agreement with molecular statics calculationsby Huang �19�. The equilibrium spacing will later be comparedwith the distance of the aligned parts in the folded CNTs. Thischeck is important to ensure that no chemical bonding betweenCNTs has occurred between different parts of the tube, and in-stead, purely vdW interactions are present. The binding energy oftwo tubes with the described LJ potential is ��8.65�10–11 J/m�19�.

It is well known that SWNTs often appear as bundles. Here wereport an investigation of bundles of CNTs, with a focus on thetype of structure �lattice� they form under equilibrium. The resultof MD computer simulations of such bundles of CNTs is shown inFigs. 4�b�–4�f�, carried out at a constant temperature around 900K. The plots show the relaxed configuration of a bundle of �5,5�CNTs interacting via a LJ potential, with free boundary conditionsin the direction orthogonal to the tube extension and periodicboundary conditions in the tube directions �corresponding to infi-nitely long tubes�. Figure 4�b� shows the initial configuration as asquare lattice, which quickly reorganizes and relaxes forming atriangular lattice as shown in Fig. 4�c�. Figure 4�d� shows therelaxed state when a larger number of CNTs is used. Interestingly,we observe formation of defects very reminiscent of crystal de-fects, such as grain boundaries and vacancies. We observe that thisstructure is stable on the order of the MD time scale of picosec-onds. Figures 4�e� and 4�f� depict three-dimensional views on thebundle of CNTs. The nearest-neighbor distance between differentCNTs in the triangular lattice is found to be around 11 Å, inagreement with the result shown in Fig. 4�a�.

3 Shell-Rod-Wire Transition of Carbon NanotubesWe briefly review the results reported in �15�, where the me-

chanical response of CNTs due to compressive loading for differ-ent aspect ratio CNTs was investigated. The main concern of thisresearch was how the mechanical response of carbon nanotubesunder compression changes when the aspect ratio is varied.

For nanotubes with very small aspect ratios, the deformationmechanism is characterized by buckling of a cylindrical shellstructure. The results are similar to the deformation patterns ob-served by Yacobsen et al. �5�, although we used CNTs with threetimes larger diameter and length at the same length-to-diameteraspect ratio. This serves as an indication that the CNT bucklingdepends primarily on the aspect ratio. Beyond a critical aspectratio, the shell-buckling behavior of CNTs changes to rod buck-ling described by the Euler theory of continuum mechanics.

Similar Euler buckling in SiO2 nanobeams of high aspect ratiosin compression was recently reported by Carr and Wybourne �20�nanobeams. For extremely large aspect ratios, the CNT is found tobehave like a flexible macromolecule and tends to transform intohelical structures under compressive loading. This suggests ashell-rod-wire transition of the mechanical behavior of carbonnanotubes with increasing aspect ratios.

For a summary over the three different deformation mecha-nisms of SWNTs, we refer the reader to Fig. 1. Whereas the firsttwo classes of deformation �shell and rod� could be effectivelydescribed by continuum mechanics concepts �15�, statistical me-chanics and entropic forces appear to play an important role in thethird class, the wirelike behavior of nanotubes with very largeaspect ratios.


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4 Self-Folding of Carbon NanotubesLarge-aspect ratio CNTs are extremely flexible, and they can be

deformed into almost arbitrary shapes with relatively small ener-getical effort �15�. Also, it is well known that different CNTsattract each other via vdW forces, leading for instance to forma-tion of bundles of CNTs �2�. If different parts of the tube comesufficiently close, these attractive forces should also be presentand can form self-folded structures of CNTs. A possibility of howdifferent parts of highly flexible CNTs can be brought into contactis that thermal fluctuations lead to bending of the tube or bendingresulting from application of external forces for instance by AFMtips. In MD simulations, a method of applying external mechani-cal forces to achieve bending of the tube is necessary due to thetimescale limitation to several nanoseconds. This is because ther-mal fluctuations are not fully explored on that timescale. Furtherdiscussion will be devoted to this point later. Irregardless whetherthe CNT is bent by thermal motion, or by external forces, the mostimportant issue is that at some point in time, different parts of thetube get sufficiently close. Once this happens, the long-range vdWinteractions lead to an attractive force that causes the parts of thetube to align.

By forming vdW ‘‘bonds,’’ the system gains energy. On theother hand, by further bending the CNT into a shape with smallerradius, higher elastic energy is necessary. This suggests that thereexists a tradeoff between the two energy contributions of bindingenergy versus bending energy. This implies that there is a criticallength of CNTs for which such self-folded structures are energeti-cally stable. Recent continuum mechanics analysis carried out byHuang �19� and co-workers allowed calculating the critical lengthbased on finite deformation beam theory, as well as large defor-mation beam theory. The predicted critical length for a �5,5� CNTwith bending stiffness EI�3.2�10–26 Jm is between 2,080 and2,400 Å, depending on whether small- or large-deformation beamtheory is used �details on the modeling will be published else-where �19��. This corresponds to a critical aspect ratio of about

Fig. 5 Application of mechanical force to bring two ends ofthe CNT into contact to initiate the self-alignment folding pro-cess. Once the two ends of the CNT are in contact, the systemis equilibrated for some time „forces are set to zero, and endsare held fixed… until this constraint is released. This must bedone with care to avoid large fluctuations due to release ofpotential energy into kinetic motion, thus hindering the self-folding.


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280. This prediction serves as guidance to our MD simulations,which will be used to check the stability of such a self-foldedstate.

Can a CNT really form self-folded bundles? If yes, how doesthe CNT fold, and what are the conditions? To investigate thesequestions, we create an initial perfectly straight CNT with aspectratio of 340 �thus larger than the critical ratio predicted by con-tinuum theory�, and we bend the tube such that different ends ofthe tube approach each other. For bending, we apply small forcesat the boundaries of the CNT, in regions with virtual atom types 4and 5. During application of forces, we keep the CNT fixed in theregion of virtual atom type 3. The approach is depicted in Fig. 5.Once the two ends of the tube are sufficiently close, vdW bondsstart to form and the CNT self-aligns. This is depicted in Fig. 6, inparticular the first snapshot in the upper-left corner.

Due to the attractive vdW forces between CNTs, the two partsof the tubes start to agglomerate and attach similar to a zipper, asit is shown in the snapshots of Fig. 6. In the case of a CNT withaspect ratio 340, we increase the parameter � in the LJ potential inthis initial folding process, since this has proven to effectivelyaccelerate the folding dynamics because of the stronger attractiveforces between C-atoms. After a large portion of the CNT hasself-aligned, the value for � is reduced to realistic values �and iskept at this value for all other studies reported in the remainder ofthis paper�. Without increasing � in early stages, the CNT withaspect ratio 340 does not self-fold.

However, for very large aspect ratio, tubes that are a few timeslonger than the critical length �around 500�, we observe self-folding when the realistic �small� LJ parameters �as given above�are used during the whole simulation time. As a consequence of

Fig. 6 Self-folding mechanism of CNTs with aspect ratio 340.After two ends of the CNTs are brought into contact by apply-ing small forces as shown in Fig. 4, the CNT self-aligns andforms a tennis-racket-like shape. The reason for this is the vdWinteraction between different parts of the CNT. The self-foldingonly occurs if the CNT is sufficiently long, so that the energygain by forming vdW bonds is larger than the energy necessaryto bend the CNT. In this case, we increase the parameter � toaccelerate the folding dynamics.



the fact that the tube is longer, and the LJ attraction is smaller, thedynamics is slower. The result of this calculation is shown in Fig.7.

We summarize the main findings. Self-folding had to be facili-tated by increasing � during the folding dynamics, if the aspectratio of the tube is close to the theoretical value of the criticallength. �Nevertheless, as soon as sufficiently many vdW bonds arebeing formed, the structure is stable with realistic potential param-eters in agreement with the continuum mechanics prediction.� Incontrast, for tubes with higher aspect ratio, the original values ofthe LJ potential could be used successfully and the tubes started tofold immediately.

One of the possible reasons for these observations is inertiaeffects: We find that care must be taken that the kinetic energy ofdifferent parts of the CNTs is kept moderate. Since the forcesassociated with the vdW bonding are relatively small, the inertiaof the tube and the resulting forces can have significant impactand lead to separation of self-aligned parts. Regions of large ki-netic energy are for instance generated when the CNT is not inelastic equilibrium after the mechanical forces are relieved. Theusage of an energy minimization scheme to drive the system to apotential energy minimum, together with slowly decreasing themagnitude of the applied forces has proven to be helpful in avoid-ing such effects.

The most important result of this section is that, given the as-pect ratio of the CNT is sufficiently large, when two parts of theCNTs are brought into contact the CNT self-folds via a zippermechanism, yielding a tennis-racket-like shape �see Fig. 7�. Thisresult is in consistency with continuum mechanics analyses �19�.

4.1 Thermal Equilibrium, Folding, and Unfolding Due toTemperature Change. In the previous section we discussed for-mation of a self-folded state of CNTs for different aspect ratioCNTs and different choices of the vdW interaction parameter.Here we report computer experiments to study the stability of thisfolded state depending on the temperature. Does a change in tem-perature affect the stability of the folded state?

For that purpose, we run several NVT simulations starting withthe same initial conditions. However, the difference between thesimulations is that we raise the temperature toward different val-ues and then study the dynamics over time spans of several nano-seconds at a constant temperature. The intention of these com-puter experiments is to test the assertion that a change intemperature plays a role in determining the stability of the foldedstate.

An important result is that for the CNTs with aspect ratio of340, the structure never unfolds before the tube starts to evaporate

Fig. 7 Self-folding mechanism of CNTs with aspect ratioaround 500. After two ends of the CNTs are brought into con-tact by applying small forces as shown in Fig. 4, the CNT self-aligns and forms a tennis-racket-like shape due to the vdW in-teraction between different parts of the CNT. Unlike in the studyshown in Fig. 6, here the parameter � is chosen so that thecorrect binding energy of CNTs is achieved during the wholesimulation.


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when the temperature reaches a critical value! Only by loweringthe aspect ratio of the CNT to a value around 200 �correspondinglength is about 1570 Å�, are we able to observe unfolding beforeevaporation. Note that this aspect ratio is comparable to the criti-cal aspect ratio predicted by continuum theory! This structure isstable at finite temperature. To show this, we equilibrate the self-folded CNT with such aspect ratio of around 200 over severalnanoseconds at a temperature of T�232 K. We find that thebonded part of the CNT oscillates around an equilibrium positionduring the simulation time. The results of this calculation areshown in Fig. 8. We also encourage the reader to view an anima-tion of this simulation, which can be downloaded from the web athttp://shasta.mpistuttgart.mpg.de/people/buehler/CNT/fold.avi.

We find that unfolding of this tube occurs at a temperature ofaround T�2,320 K. Due to the temperature change, the thermalfluctuations of the CNT become larger. The ‘‘wilder,’’ and alsomore extensive movements of the wirelike CNT at higher tem-perature cause more vdW bonds to break in the time average, sothat the effectively bonded region gets smaller! Upon a criticaltemperature, the CNT leaves the folded tennis-racket-like stateand unfolds. The vdW bonding is not strong enough to withstandthe large temperature fluctuations. The CNT then returns to anarbitrary shape that is fluctuating in space. The unfolding dynam-ics of such a structure is documented in Fig. 9. As in the previouscase, we encourage the reader to view an animation of the unfold-ing process, which can be downloaded from the web at http://shasta.mpi-stuttgart.mpg.de/people/buehler/CNT/unfold.avi.

The results of these computer experiments strongly suggest thatthe temperature can govern the stability of the folded state undercertain conditions. This implies that there are at least two neces-sary conditions for folded CNTs to be stable:

1. The aspect ratio needs to be large enough such that the elas-tic energy stored in bending the tube can be compensated by form-ing bonds �see discussion in previous section�.

2. The temperature needs to be low enough such that thermalfluctuations are sufficiently small such that fewer vdW bondsbreak in the time-average, and the CNT remains in its folded state.

Fig. 8 At low temperature, the CNT is thermodynamicallystable and the bonded length oscillates around an equilibriumposition „TÄ232 K…. Similar studies have been carried out withlonger CNTs, where correspondingly the bonded region grows.



However, if the aspect ratio is sufficiently large, the temperaturechange does not lead to unfolding before evaporation of the tube.

The finding that CNTs, once brought into the folded state, neverunfold until they evaporate suggests that the folded state can bequite robust. However, we emphasize that this could, at least par-tially, be affected by the limitation in MD time scale to nanosec-onds as the unfolding of larger aspect ratio CNTs could occur atlater times that are not �yet� accessible to the MD method. Al-though this could change the critical aspect ratio, we expect thatthe main result will prevail even at longer time scales.

How can this observation be understood? A possible explana-tion for this behavior is the decrease in Young’s modulus withincreasing temperature. Since the critical length for a stable self-folded structure increases with Young’s modulus �19�, a decreasein that leads to a decrease in critical length. Indeed, our calcula-tions suggest that at high temperatures, the CNT has a very smallbending stiffness and is extremely flexible. We leave a more quan-titative analysis of this effect to future work.

In summary, we have shown the following. First, we reportedthe simulation of a folded CNT in its thermodynamical equilib-rium at low temperature. An important observation was that thelength of the folded, aligned part fluctuates around an equilibriumposition. We observe that even at moderate aspect ratios, the CNTdoes not unfold even if the temperature is raised beyond the melt-ing point and the CNT starts to evaporate �at least at an MD timescale�. This suggests that the folded state can be quite robust. Inorder to study the unfolding process due to temperature change,we reduced the aspect ratio of the CNTs. In these systems, oncethe temperature was increased to about T�2320 K �still quiteclose to the melting temperature�, the CNT started to unfoldwithin a few tens of picoseconds. The main conclusion is thattemperature change can control the stability of the folded state.The observed unfolding mechanism is reminiscent of the behaviorof other macromolecules such as proteins, although the conceptsof critical length for self-folding and dependence on the geometrymay be considerably different. Nevertheless, the observation offolding-unfolding mechanisms strongly support the viewpointproposed earlier that CNTs behave as macromolecules at veryhigh aspect ratios �15�.

Fig. 9 Unfolding of CNT due to increase in temperature „TÄ2,320 K…. We start from the same initial condition as in thecalculation shown in Fig. 7.



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Fig. 10 Schematic phase diagram of carbon nanotube deformation mechanisms. The plot shows regions where different statesor deformation mechanisms of CNTs govern, as a function of aspect ratio and temperature. For low aspect ratios, the temperaturedependence of the deformation mechanism is negligible. For large aspect ratios, temperature governs if the folded state is stableor not, and for very large aspect ratios the folded state is stable until evaporation of the CNT.

4.2 Schematic Phase Diagram of Carbon Nanotubes. Theresults of the atomistic modeling �together with continuum me-chanics results that are published elsewhere �19�� are used to drawa schematic phase diagram of CNTs. We note that much furtherstudies will be necessary to elucidate the details �and quantify� theexact shape of the diagram. Here the main objective is limited togive a first-order approximation. The diagram also helps to appre-ciate the role of the newly discovered deformation mechanisms incomparison with current understanding of states of CNTs.

The diagram depicted in Fig. 10 shows the different modes ofdeformation of CNTs depending on the aspect ratio and the tem-perature. Note that for very small aspect ratio CNTs �shell- androdlike behavior�, we expect little or no temperature dependence.In contrast, for large aspect ratio CNTs that behave as macromol-ecules, temperature depends on whether the CNT is folded or not.For low temperatures, the CNT will be folded since it is an ener-getically favorable state. However, the lower the temperature theslower the motion of the molecule and the longer it could take inorder for the CNT to fold.

For high temperature, the CNT may tend to unfold. On theother hand, as discussed above, since the elastic properties changewith temperature �as they soften with higher temperature�, theenergetical effort to maintain the folded state is decreasing andthus folded tubes can be stable at temperatures up the meltingtemperature.

5 Summary and DiscussionThe focus of this paper was on the self-folded state of CNTs.

Compared to earlier studies �15�, here we focused on the foldingand unfolding of CNTs and the dependence of the folded structureon the temperature. The main achievements of this paper can besummarized as follows:

• We were able to demonstrate and describe the dynamics offolding of CNTs. In order to observe the folding of CNTs, we used



mechanical forces to bring two ends of the CNTs sufficiently closeto each other so that the different parts of the CNT align and startform vdW bonds �similar to a zipper mechanism�. In ‘‘real’’ sys-tems, such a state of initial contact could in principle be achievedby sampling of the phase space by thermal fluctuations, althoughfurther �e.g., mesoscopic� studies are yet to be performed toclarify this point.

• We find that CNTs can fold and unfold depending on thetemperature. Atomistic simulations suggest that thermal fluctua-tions cause an increasing number of vdW bonds to break at highertemperature, eventually leading to breaking apart of the tennis-racket-like self-folded state. This is in agreement with the physicalunderstanding of entropic effects starting to dominate over thepurely energetical consideration at low temperature and suggeststhat the CNT essentially behaves quite similarly to a biomolecule.

• We observe that for folded CNTs above a certain aspect ratio,the structure never unfolds until the temperature is so high that theCNT starts to evaporate. This was explained by softening of elas-tic properties with increasing temperature. Therefore, for largeaspect ratio CNTs, the softening of the elastic properties dominateover the increase in thermal fluctuations �that is, the energy nec-essary to bend the CNT becomes smaller and smaller with in-creasing temperature so that the folded state is stable for all tem-peratures�. This suggests that the folded state can be a very robustconfiguration.

• The results of our modeling, including an extrapolation toother phenomena, were summarized in a schematic phase diagramof CNTs. This could motivate and outline further research wherethe details of such a phase diagram could be in the focus of atten-tion. The main conclusion of this diagram is that there is littleimpact of the temperature on relatively small aspect ratio tubes�shell- and rodlike behavior�, whereas in the case of wirelikeCNTs, temperature can play an important role.

Further research in this area could include


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• The temperature dependence of the critical length for self-folding, and a systematic study of change of elastic propertieswith temperature.

• Theoretical analysis of the folded state, possibly based onexisting theories of chain polymers.

• Mesoscopic studies of wirelike CNTs based on the wormlikechain model. This approach could help elucidate the long-timeaspects of the folding process.

• The effect of solvent or other CNTs surrounding molecules.This could significantly change the vdW binding energy �and thusthe critical length for self-folding� or lead to momentum transferby molecular collisions that could either support the foldingmechanism or hinder it.

• Experimental verifications of the self-folded CNTs.

Finally, we note that the present investigation underlines the ne-cessity of using a multidisciplinary and multiscale simulationmethod in understanding the phenomena �concepts from biologi-cal biomolecules, such as folding, play an important role, for in-stance, and the need to create simulation tools that range overlonger time scales is critical to fully understand the phase spacesampling�. Similar considerations may apply to other nanoscalematerials phenomena. The limitations of the MD method becomeobvious, for instance that folding can only be observed with the‘‘help’’ of mechanical forces bringing different parts of the CNTin contact to initiate the process.

Furthermore, we note that despite the similarities of biomol-ecules and large-aspect ratio CNTs, the behavior of CNTs is quitedifferent in some aspects. For instance, the bending stiffness ofbiomolecules is finite at zero temperature �e.g., in theories ofwormlike chain models where it is assumed that the bending stiff-ness vanishes at zero temperature�. The zero bending stiffness isusually assumed because bending the molecules around chemicalbonds is performed with relatively low energetical effort. In con-trast, due to the different structure of CNTs, the bending stiffnessis non-zero at zero temperature! In addition, the bending stiffnessdoes not increase with larger temperature, but in contrast de-creases with higher temperature �as suggested by the results re-ported in this paper�. Such considerations may be important indeveloping mesoscopic models of macromolecules like CNTs, andillustrates that this may be a rather complicated endeavor. Thepresent study is an example for a recent trend in studying nanos-cale materials phenomena �21�.

AcknowledgmentsSome of the simulations were carried out in the Garching Su-

percomputer Center of the Max Planck Society. The MD calcula-tions are preformed using the ITAP IMD code �22,23�, which waskindly provided to us by the group of Prof. Hans Trebin at theInstitute for Theoretical and Applied Physics, University of Stut-tgart, Germany. M.J. Buehler acknowledges discussions on CNTswith Dr. Nicole Grobert at Oxford University, UK.



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self-folding and unfolding of carbon nanotubes

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