17Nanomechanics

CONTENTSAbstract.........................................................................................17-117.1 Introduction ......................................................................17-117.2 Linear Elastic Properties ...................................................17-417.3 Nonlinear Elasticity and Shell Model ..............................17-517.4 Atomic Relaxation and Failure Mechanisms...................17-817.5 Kinetic Theory of Strength.............................................17-1217.6 Coalescence of Nanotubes as a Reversed Failure..........17-1317.7 Persistence Length, Coils, and Random

FuzzBalls of CNTS ..........................................................17-15Acknowledgments......................................................................17-17References ...................................................................................17-17

Abstract

The general aspects and issues of nanomechanics are discussed and illustrated by an overview of themechanical properties of nanotubes: linear elastic parameters, nonlinear elastic instabilities and buckling,inelastic relaxation, yield strength and fracture mechanisms, and kinetic theory. A discussion of theoreticaland computational studies is supplemented by brief summaries of experimental results for the entirerange of the deformation amplitudes. Atomistic scenarios of coalescence welding and the role of nonco-valent forces (supramolecular interactions) between the nanotubes are also discussed due to their sig-nificance in potential applications.

17.1 Introduction

Nanomechanics is an emerging area of research that deals with the mechanical properties and behaviorof small materials systems. A span of several nanometers in two dimensions (wires, rods, etc.) or threedimensions (clusters, particles, etc.) is a simplistic criterion for a system to be considered nano. Thisformally may exclude the rich and established science of surfaces and interfaces, which are small inonly one dimension. One can also argue that a dislocation is not a nano-object; while it formally meetsthe criterion, it is never isolated from a crystal lattice and as such has long been a subject of researchin solid-state physics. To distinguish itself from the well-established dynamics of molecules, nanome-chanics relies heavily on the heuristics and methods of mechanical engineering and structural mechan-ics. It mainly deals with objects of distinct geometrical shape and function: rods, beams, shells, plates,membranes, etc. At the same time, due to the small length scale, nanomechanics also relies on physics— specifically, interatomic and molecular forces, methods of quantum chemistry, solid-state physics,and statistical mechanics. With these approaches come a variety of numerical and computationalmethods (molecular dynamics, Monte Carlo simulations, classical empirical interatomic potentials,

Boris I. YakobsonRice University

0-8493-1200-0/03/$0.00+$1.50© 2003 by CRC Press LLC 17-1

17

-2

Handbook of Nanoscience, Engineering, and Technology

tight-binding approximation, density functional theory, etc.). This cross-disciplinary aspect makes thisarea both complex and exciting for research and education.

Macroscopic mechanics primarily deals with continuum representation of material, neglecting theunderlying atomic structure that manifests primarily at a smaller scale. In this context it is interesting torealize that a continuum model of a finite object is not self-contained and inevitably leads to a notionof atom as a discrete building block. Indeed, elastic response of continuum is quantified by its moduli,e.g., Young’s modulus, Y (J/m3). A boundary surface of a material piece of finite size L must be associatedwith certain extra energy, surface energy g (J/m2). A combination g/Y is dimensional (m) and constitutesa length not contained within such a finite continuum model, which points to some other inherentparameter of the material, a certain size a. The surface energy is the additional work to “overstretch” ortear apart an elastic continuum. Such work equals to the energy of the formed boundary (two boundaries),that is aY/2 ª 2g, and thus a ª 4g/Y. With typical Y = 50 GPa and g = 1 J/m2, one gets a = 0.1 nm, areasonably accurate atomic size.1,2 The notion of indivisibles has been well known since Democritus, andtherefore a simple mechanical measurement could yield an estimate of atomic size much earlier than themore sophisticated Brownian motion theory. This discussion also shows that for an object of nanometerscale, its grainy, atomistic structure comes inevitably into the picture of its mechanics and that atomisticor hybrid-multiscale methods are necessary.

Among the numerous subjects of nanomechanics research (tips, contact junctions, pores, whiskers,etc.), carbon nanotubes (CNTs)3 have earned a special place, receiving much attention. Their molecularlyprecise structure, elongated and hollow shape, effective absence of a surface (which is no different fromthe bulk, at least for the single-walled cylinders, or SWNTs), and superlative covalent bond strength areamong the traits that put CNTs in the focus of nanomechanics. Discussion of numerous other objectsas well as details of the multiscale methods involved in nanomechanics (for example, see recentmonograph4) is far beyond the scope of this chapter.

It is noteworthy that the term resilient was first applied not to nanotubes but to smaller fullerene cages,in the study of high-energy collisions of C60, C70, and C84 bouncing from a solid wall of H-terminateddiamond. The absence of any fragmentation or other irreversible atomic rearrangement in the reboundingcages was somewhat surprising and indicated the ability of fullerenes to sustain great elastic distortion.The very same property of resilience becomes more significant in the case of carbon nanotubes, becausetheir elongated shape, with the aspect ratio close to a thousand, makes the mechanical properties especiallyinteresting and important due to potential structural applications. An accurate simulation (with realisticinteratomic and van der Waals forces) in Figure 17.15 vividly illustrates the appeal of CNTs as a nano-mechanical object: well-defined cylindrical shape, compliance to external forces, and an expected typeof response qualitatively analogous to a common macroscopic behavior puts these objects among molec-ular chemical physics, elasticity theory, and mechanical engineering.

The utility of nanotubes as elements in nanoscale devices or composite materials remains a powerfulmotivation for research in this area. While the feasibility of the practical realization of these applicationsis currently unknown, another incentive comes from the fundamental materials physics. There is aninteresting duality in the nanotubes. CNTs possess, simultaneously, molecular size and morphology aswell as sufficient translational symmetry to perform as very small (nano-) crystals with well-definedprimitive cells, surfaces, possibility of transport, etc. Moreover, in many respects they can be studied aswell defined engineering structures; and many properties can be discussed in traditional terms of moduli,stiffness or compliance, or geometric size and shape. The mesoscopic dimensions (a nanometer diameter),combined with the regular, almost translation-invariant morphology along the micrometer lengths(unlike other polymers, usually coiled), make nanotubes unique and attractive objects of study, includingthe study of mechanical properties and fracture in particular.

Indeed, fracture of materials is a complex phenomenon whose theory generally requires a multiscaledescription involving microscopic, mesoscopic, and macroscopic modeling. Numerous traditionalapproaches are based on a macroscopic continuum picture that provides an appropriate model exceptat the region of actual failure, where a detailed atomistic description (involving chemical bond breaking)is needed. Nanotubes, due to their relative simplicity and atomically precise morphology, offer the

Nanomechanics

17

-3

opportunity of addressing the validity of different macroscopic and microscopic models of fracture andmechanical response. Contrary to crystalline solids, where the structure and evolution of ever-presentsurfaces, grain-boundaries, and dislocations under applied stress determine the plasticity and fractureof the material, nanotubes possess simpler structure while still able to show rich mechanical behaviorwithin elastic or inelastic brittle or ductile domain. This second, theoretical–heuristic value of nanotuberesearch supplements their import due to anticipated practical applications. A morphological similarityof fullerenes and nanotubes to their macroscopic counterparts, like geodesic domes and towers, makesit compelling to test the laws and intuition of macromechanics in a scale ten orders of magnitude smaller.

Section 17.2 discusses theoretical linear elasticity and results for the elastic moduli, compared whereverpossible with the experimental data. The nonlinear elastic behavior, buckling instabilities, and shell modelare presented in Section 17.3, with mention of experimental evidence parallel to theoretical results. Yieldand failure mechanisms in tensile load are presented in Section 17.4, with emphasis on the combineddislocation theory and computational approach. More recent results of kinetic theory of fracture andstrength evaluation in application to CNTs are briefly presented in Section 17.5. Fast molecular tensiontests are recalled in the context of kinetic theory. Section 17.6 presents some of the most recent resultson CNT “welding,” a process essentially the reverse of fracture. In Section 17.7 we also briefly discussthe large-scale mechanical deformation of nanotubes caused by their attraction to each other, and therelation between nanomechanics and statistical persistence length of CNT in thermodynamic suspension.Throughout the discussion we do not attempt to provide a comprehensive review of broad activities inthe field. This presentation is mainly based on the author’s research started at North Carolina State

FIGURE 17.1 Molecular mechanics calculations on the axial and radial deformation of single-wall carbon nano-tubes. (a) Axial deformation resulting from the crossing of two (10,10) nanotubes. (b) Perspective close up of thesame crossing showing that both tubes are deformed near the contact region. (c) Computed radial deformations ofsingle-wall nanotubes adsorbed on graphite. (Adapted from Hertel, T., R.E. Walkup, and P. Avouris, Deformation ofcarbon nanotubes by surface van der Waals forces. Phys. Rev. B, 1998. 58(20): pp. 13870–13873. With permission.)

17

-4

Handbook of Nanoscience, Engineering, and Technology

University and continued at Rice University. Broader or more comprehensive discussion can be foundin other relatively recent reviews by the author6,7 (see also Chapters 18 and 19 in this book.)

17.2 Linear Elastic Properties

Numerous theoretical calculations are dedicated to linear elastic properties, when displacements (strain)are proportional to forces (stress). We recently revisited8 this issue in order to compare, with the samemethod, the elasticity of three different materials: pure carbon (C), boron–nitride (BN), and fluorinatedcarbon (C2F). Due to obvious uncertainty in definition of a nanotube cross-section, the results shouldbe represented by the values of in-plane stiffness, C (J/m2). The values computed with Gaussian-baseddensity functional theory are C = 345 N/m, 271 N/m, and 328 N/m for C, BN, and C2F, respectively.These values in ab initio calculations are almost independent of nanotube diameter and chirality (con-sistent with the isotropic elasticity of a hexagonal, two-dimensional lattice), somewhat in contrast toprevious reports based on tight-binding or classical empirical approximations. Notably, substantial flu-orination causes almost no change in the in-plane stiffness, because the bonding involves mainly p-system while the stiffness is largely due to in-plane s-system. For material property assessments, thevalues of bulk moduli (based on a graphite-type 0.34 nm spacing of layers) yield 1029 GPa, 810 GPa,and 979 GPa — all very high. Knowing the elastic shell parameter C immediately leads to accuratecalculation of a nanotube-beam bending stiffness K (proportional to the cube of diameter, ~d3), asdiscussed later in Sections 17.3 and 17.7. It also allows us to compute vibration frequencies of the tubules,e.g. symmetric breathing mode frequency, f ~ 1/d,8 detectable in Raman spectroscopy.

An unexpected feature discovered in the course of that study is the localized strain induced by theattachment of fluorine. This shifts the energy minimum of the C2F shell lattice from an unstrained sheettoward the highly curved polygonal cylinders (for C2F composition of a near square shape, Figure 17.2).Equilibrium free angle is ~72°.

Theoretical values agree reasonably well with experimental values of Young’s modulus. It was firstestimated9 by measuring free-standing room-temperature vibrations in a transmission electron micro-scope (TEM). The motion of a vibrating cantilever is governed by the known fourth-order wave equation,

ytt = –(YI/rA)yxxxx

where A is the cross-sectional area and r is the density of the rod material. For a clamped rod theboundary conditions are such that the function and its first derivative are zero at the origin, and

FIGURE 17.2 Geometries of the polygonal fluorinated carbon tubes: (a) square F4–(10,10), and (b) pentagonalF5–(10,10). (From Kudin, K.N., G.E. Scuseria, and B.I. Yakobson, C2F, BN and C nanoshell elasticity by ab Initiocomputations. Phys. Rev. B, 2001. 64: p. 235406. With permission.)

a)b)

Nanomechanics

17

-5

the second and third derivatives are zero at the end of the rod. Thermal nanotube vibrations areessentially elastic relaxed phonons in equilibrium with the environment; therefore, the amplitudeof vibration changes stochastically with time. The amplitude of those oscillations was defined bymeans of careful TEM observations of a number of CNTs and yields the values of moduli within arange near 1 Tpa.

Another way to probe the mechanical properties of nanotubes is to use the tip of an AFM (atomic-force microscope) to bend an anchored CNT while simultaneously recording the force exerted by thetube as a function of the displacement from its equilibrium position.10 Obtained values also vary fromsample to sample but generally are close to Y = 1 TPa. Similar values have been obtained with yetanother accurate technique11 based on a resonant electrostatic deflection of a multi-wall carbonnanotube under an external AC field. The detected decrease in stiffness must be related to the emergenceof a different bending mode for the nanotube. In fact, this corresponds to a wave-like distortion–buck-ling of the inner side of the CNT. Nonlinear behavior is discussed in more detail in the next section.Although experimental data on elastic modulus are not very uniform, they correspond to the valuesof in-plane rigidity C = 340 – 440 N/m, to the values Y = 1.0 – 1.3 GPa for multiwall tubules, and toY = 4C/d = (1.36 – 1.76) TPa nm/d for SWNTs of diameter d.

17.3 Nonlinear Elasticity and Shell Model

Almost any molecular structure can sustain very large deformations, compared with the rangecommon in macroscopic mechanics. A less obvious property of CNTs is that the specific featuresof large nonlinear strain can be understood and predicted in terms of continuum theories. One ofthe outstanding features of nanotubes is their hollow structure, built of atoms densely packed alonga closed surface that defines the overall shape. This also manifests itself in dynamic properties ofmolecules, highly resembling the macroscopic objects of continuum elasticity known as shells.Macroscopic shells and rods have long been of interest: the first study dates back to Euler, whodiscovered the elastic instability. A rod subject to longitudinal compression remains straight butshortens by some fraction e, proportional to the force, until a critical value (the Euler force) isreached. It then becomes unstable and buckles sideways at e > ecr, while the force almost does notvary. For hollow tubules there is also a possibility of local buckling in addition to buckling as awhole. Therefore, more than one bifurcation can be observed, thus causing an overall nonlinearresponse to the large deforming forces (note that local mechanics of the constituent shells may wellstill remain within the elastic domain).

In nanomechanics, the theory of shells was first applied in our early analysis of buckling and sincethen has served as a useful guide.12–15 Its relevance for a covalent-bonded system only a few atoms indiameter was far from being obvious. Molecular dynamics (MD) simulations seem better suited forobjects that small.

Figure 17.3 shows a simulated nanotube exposed to axial compression. The atomic interaction wasmodeled by the Tersoff–Brenner potential, which reproduces the lattice constants and binding energies ofgraphite and diamond. The end atoms were shifted along the axis by small steps, and the whole tube wasrelaxed by a conjugate-gradient method while keeping the ends constrained. At small strains the total energy(Figure 17.3a) grows as E(e) = 1/2 E”·e2. The presence of four singularities at higher strains was quite astriking feature and the patterns (b) – (e) illustrate the corresponding morphological changes. The shadingindicates strain energy per atom, equally spaced from below 0.5 eV (brightest) to above 1.5 eV (darkest).The sequence of singularities in E(e) corresponds to a loss of molecular symmetry from D•h to S4, D2h, C2h,and C1. This evolution of the molecular structure can be put in the framework of continuum elasticity.

The intrinsic symmetry of a graphite sheet is hexagonal, and the elastic properties of two-dimensionalhexagonal structures are isotropic. A curved sheet can also be approximated by a uniform shell with onlytwo elastic parameters — flexural rigidity, D, and its in-plane stiffness, C. The energy of a shell is givenby a surface integral of the quadratic form of local deformation:

17

-6

Handbook of Nanoscience, Engineering, and Technology

(17.1)

where k is the curvature variation, e is the in-plane strain, and x and y are local coordinates. In orderto adapt this formalism to a graphitic tubule, the values of D and C can be identified by comparisonwith the detailed ab initio and semi-empirical studies of nanotube energetics at small strains. Indeed,the second derivative of total energy with respect to axial strain corresponds to the in-plane rigidityC (Section 17.2). Similarly, the strain energy as a function of tube diameter d corresponds to 2D/d2

in Equation (17.1). Using recent ab initio calculations, one obtains C = 56 eV/atom = 340 J/m2 andD = 1.46 eV. The Poisson ratio n = 0.15 was extracted from a reduction of the diameter of a tubestretched in simulations. A similar value is obtained from experimental elastic constants of singlecrystal graphite. One can make a further step toward a more tangible picture of a tube as having wallthickness h and Young modulus Ys. Using the standard relations D = Ysh3/12(1 – n2) and C = Ysh, onefinds Ys = 3.9 TPa and h = 0.089 nm. With these parameters, linear stability analysis allows one toassess the nanotube behavior under strain.

FIGURE 17.3 Simulation of a (7,7) nanotube exposed to axial compression, L = 6 nm. The strain energy (a) displaysfour singularities corresponding to shape changes. At ec = 0.05, the cylinder buckles into the pattern (b), displaying twoidentical flattenings —“fins” perpendicular to each other. Further increase of e enhances this pattern gradually until ate2 = 0.076 the tube switches to a three-fin pattern (c), which still possesses a straight axis. In a buckling sideways at e3

= 0.09, the flattenings serve as hinges, and only a plane of symmetry is preserved (d). At e4 = 0.13, an entirely squashedasymmetric configuration forms (e). (From Yakobson, B.I., C.J. Brabec, and J. Bernholc, Nanomechanics of carbontubes: instabilities beyond the linear response. Phys. Rev. Lett., 1996. 76(14): pp. 2511–2514. With permission.)

E 12-- D kx ky+( )2 2 1 n–( ) kxky kxy

2–( )–[ ]{ÚÚ=

C

1 n2–( )------------------ ex ey+( )2 2 1 n–( ) exey exy

2–( )–[ ] }dS+

Nanomechanics 17-7

To illustrate the efficiency of the shell model, consider briefly the case of imposed axial compression.A trial perturbation of a cylinder has a form of Fourier harmonics, with M azimuthal lobes and Nhalfwaves along the tube (Figure 17.4, inset), i.e., sines and cosines of arguments 2My/d and Npx/L. Ata critical level of the imposed strain, ec(M,N), the energy variation (Equation 17.1) vanishes for thisshape disturbance. The cylinder becomes unstable and lowers its energy by assuming an (M,N) pattern.For tubes of d = 1 nm with the shell parameters identified above, the critical strain is shown in Figure17.4. According to these plots, for a tube with L > 10 nm, the bifurcation is first attained for M = 1, N= 1. The tube preserves its circular cross-section and buckles sideways as a whole; the critical strain isclose to that for a simple rod

ec = 1/2(pd/L)2 (17.2)

or four times less for a tube with hinged (unclamped) ends. For a shorter tube the situation is different.The lowest critical strain occurs for M = 2 (and N ≥ 1, see Figure 17.4), with a few separated flatteningsin directions perpendicular to each other, while the axis remains straight. For such a local buckling, incontrast to (Equation 17.2), the critical strain depends little on length and estimates to ec = d–1=(2/÷3)(1 – n2)–1/2 h·d–1 in the so-called Lorenz limit. For a nanotube one finds

ec = 0.1 nm/d. (17.3)

Specifically, for the 1 nm wide tube of length L = 6 nm, the lowest critical strains occur for M = 2and N = 2 or 3 (Figure 17.4). This is in accord with the two- and three-fin patterns seen in Figure 17.3b,c.Higher singularities cannot be quantified by the linear analysis, but they look like a sideways beambuckling, which at this stage becomes a nonuniform object.

Axially compressed tubes of greater length and/or tubes simulated with hinged ends (equivalent to adoubled length) first buckle sideways as a whole at a strain consistent with (Equation 17.2). After that thecompression at the ends results in bending and a local buckling inward. This illustrates the importanceof the “beam-bending” mode, the softest for a long molecule and most likely to attain significant amplitudesdue to either thermal vibrations or environmental forces. In simulations of bending, a torque rather thanforce is applied at the ends and the bending angle q increases step-wise. While a notch in the energy plotcan be mistaken for numerical noise, its derivative dE/dq drops significantly. This unambiguously showsan increase in tube compliance — a signature of a buckling event. In bending, only one side of a tube iscompressed and thus can buckle. Assuming that it buckles when its local strain, e = k·d/2, where k is thelocal curvature, is close to that in axial compression, Equation (17.3), we estimate the critical curvature as:

FIGURE 17.4 The critical strain levels for a continuous, 1 nm wide shell tube as a function of its scaled length L/N. A buckling pattern (M,N) is defined by the number of half-waves 2M and N in y and x directions, respectively;e.g., a (4,4) pattern is shown in the inset. The effective moduli and thickness are fit to graphene. (From Yakobson,B.I., C.J. Brabec, and J. Bernholc, Nanomechanics of carbon tubes: instabilities beyond the linear response. Phys. Rev.Lett., 1996. 76(14): pp. 2511–2514. With permission.)

D C§

17-8 Handbook of Nanoscience, Engineering, and Technology

k c = 0.2 nm/d2. (17.4)

In simulation of torsion, the increase of azimuthal angle f between the tube ends results in abrupt changesof energy and morphology.12,13,16 In continuum model, the analysis based on Equation (17.1) is similarto that outlined above, except that it involves skew harmonics of arguments such as Npx/L±2My/d. Foroverall beam-buckling (M = 1),

fc = 2(1+n)p (17.5)

and for the cylinder-helix flattening (M = 2),

fc = 0.06 nm3/2 L/d5/2. (17.6)

The latter should occur first for L < 140 d5/2 nm, which is true for all tubes we simulated. However,in simulations it occurs later than predicted by Equation (17.6). The ends, kept circular in simulation,which is physically justifiable, by a presence of rigid caps on normally closed ends of a molecule, deterthe through flattening necessary for the helix to form (unlike the local flattening in the case of an axialload).

Experimental evidence provides sufficient support to the notion of high resilience of SWNT. An earlyobservation of noticeable flattening of the walls in a close contact of two multi-walled nanotubes(MWNT) has been attributed to van der Walls forces pressing the cylinders to each other.17 Collapsedforms of the nanotube (nanoribbons), also caused by van der Waals attraction, have been observed inexperiment, and their stability can be explained by the competition between the van der Waals and elasticenergies.18 Any additional torsional strain imposed on a tube in an experimental environment also favorsflattening12,13 and facilitates the collapse. Graphically more striking evidence of resilience is provided bybent structures19 as well as the more detailed observations that actually stimulated our research innanomechanics.20 An accurate measurement with the atomic-force microscope (AFM) tip detects the“failure” of a multi-wall tubule in bending, which essentially represents nonlinear buckling on thecompressive side of the bent tube. The estimated measured local stress is 15–28 GPa, very close to thecalculated value. Buckling and ripple of the outermost layers21,22 in a dynamic resonant bending havebeen directly observed and are responsible for the apparent softening of MWNT of larger diameters.7,23

17.4 Atomic Relaxation and Failure Mechanisms

The important issue of ultimate tensile strength of CNTs is inherently related to the atomic relaxationin the lattice under high strain. This thermally activated process was first predicted to consist of a sequenceof individual bond rotations in the approach based on dislocation theory.21,24,25 Careful computer simu-lations demonstrate feasibility of this mechanism and allow us to quantify important energy aspects.26,27

It has been shown that in a crystal lattice, such as the wall of a CNT, a yield to deformation must beginwith a homogeneous nucleation of a slip by the shear stress present. The nonbasal edge dislocationsemerging in such slip have a well-defined core, a pentagon–heptagon pair (5/7). Therefore, the primedipole is equivalent to the Stone–Wales (SW) defect. The nucleation of this prime dislocation dipoleunlocks the nanotube for further relaxation — either brittle cleavage or a plastic flow. Remarkably, thelatter corresponds to a motion of dislocations along the helical paths (glide planes) within the nanotubewall. This causes a step-wise (quantized) necking, when the domains of different chiral symmetry (andtherefore different electronic structure) are formed, thus coupling the mechanical and electrical proper-ties.21,24,25 It has further been shown21,22,24,27–29 that the energy of such nucleation explicitly depends onCNT helicity (chirality).

Below, starting with dislocation theory, we deduce the atomistics of mechanical relaxation underextreme tension. Locally, the wall of a nanotube differs little from a single graphene sheet, a two-dimensional crystal of carbon. When a uniaxial tension s (N/m — for the two-dimensional wall, it isconvenient to use force per unit length of its circumference) is applied, it can be represented as a sum

Nanomechanics 17-9

of expansion (locally isotropic within the wall) and a shear of a magnitude s/2 (directed at ±45∞ withrespect to tension). Generally, in a macroscopic crystal the shear stress relaxes by a movement of dislo-cations, the edges of the atomic extra-planes. Burger’s vector b quantifies the mismatch in the lattice dueto a dislocation. Its glide requires only local atomic rearrangements and presents the easiest way for strainrelease, with sufficient thermal agitation. In an initially perfect lattice such as the wall of a nanotube, ayield to a great axial tension begins with a homogeneous nucleation of a slip, when a dipole of dislocations(a tiny loop in three-dimensional case) first has to form. The formation and further glide are driven bythe reduction of the applied-stress energy, as characterized by the elastic Peach–Koehler force on adislocation. The force component along b is proportional to the shear in this direction and thus dependson the angle between the Burger’s vector and the circumference of the tube

fb = – 1/2s|b| sin 2q (17.7)

The max |fb| is attained on two ±45∞ lines, which mark the directions of a slip in an isotropic materialunder tension.

The graphene wall of the nanotube is not isotropic; its hexagonal symmetry governs the three glideplanes — the three lines of closest zigzag atomic packing, oriented at 120∞ to each other (correspondingto the { } set of planes in three-dimensional graphite). At nonzero shear these directions are proneto slip. The corresponding c-axis edge dislocations involved in such slip are indeed known in graphite.The six possible Burger’s vectors 1/3a< > have a magnitude b = a = 0.246 nm (lattice constant),and the dislocation core is identified as a 5/7 pentagon–heptagon pair in the honeycomb lattice ofhexagons. Therefore, the primary nucleated dipole must have a 5/7/7/5 configuration (a 5/7 attached toan inverted 7/5 core). This configuration is obtained in the perfect lattice (or a nanotube wall) by a 90∞rotation of a single C–C bond, well known in fullerene science as a Stone–Wales diatomic interchange.One is led to conclude that the SW transformation is equivalent to the smallest slip in a hexagonal latticeand must play a key role in the nanotube relaxation under external force.

The preferred glide is the closest to the maximum-shear ±45∞ lines and depends on how the graphenestrip is rolled up into a cylinder. This depends on nanotube helicity specified by the chiral indices (c1,c2) or a chiral angle q indicating how far the circumference departs from the leading zigzag motif a1. Themax |fb| is attained for the dislocations with b = ±(0,1) and their glide reduces the strain energy

Eg = –|fba| = – Ca2/2·sin (2q + 60∞)·e (17.8)

per one displacement, a. Here e is the applied strain, and C = Yh = 342 N/m can be derived from the Youngmodulus of Y = 1020 GPa and the interlayer spacing h = 0.335 nm in graphite. One then obtains Ca2/2 =64.5 eV. Equation 17.8 allows one to compare different CNTs (assuming similar amount of preexistingdislocations): the more energetically favorable is the glide in a tube, the earlier it must yield to applied strain.

In a pristine nanotube molecule, the 5/7 dislocations have to first emerge as a dipole by a prime SWtransformation. Topologically, the SW defect is equivalent to either one of the two dipoles, each formedby a ~a/2 slip. Applying Equation (17.8) to each of the slips, one finds

Esw = Eo – A·e – B·sin (2q + 30∞)·e (17.9)

The first two terms, the zero-strain formation energy and possible isotropic dilation, do not depend onchiral symmetry. The symmetry-dependent third term, which can also be derived as a leading term inthe Fourier series, describes the fact that SW rotation gains more energy in an armchair (q = 30∞) CNT,making it thermodynamically the weakest and most inclined to SW nucleation of the dislocations, incontrast to the zigzag (q = 0), where the nucleation is least favorable.

Consider, for example, a (c, c) armchair CNT as a typical representative (we will also see below thatthis armchair type can undergo a more general scenario of relaxation). The initial stress-induced SWrotation creates a geometry that can be viewed as either a dislocation dipole or a tiny crack along theequator. Once “unlocked,” the SW defect can ease further relaxation. At this stage, either brittle (dislo-

101l

2110

17-10 Handbook of Nanoscience, Engineering, and Technology

cation pileup and crack extension) or plastic (separation of dislocations and their glide away from eachother) routes are possible, the former usually at larger stress and the latter at higher temperatures.

Formally, both routes correspond to a further sequence of SW switches. The 90∞ rotation of the bondsat the “crack tip” (Figure 17.5, left column) will result in a 7/8/7 flaw and then 7/8/8/7, etc. This furtherstrains the bonds partitions between the larger polygons, leading eventually to their breakage, with theformation of greater openings such as 7/14/7. If the crack, represented by this sequence, surpasses thecritical Griffith size, it cleaves the tubule.

In a more interesting distinct alternative, the SW rotation of another bond (Figure 17.5, top row)divides the 5/7 and 7/5, as they become two dislocation cores separated by a single row of hexagons. Anext similar SW switch results in a double-row separated pair of the 5/7s, and so on. This corresponds,at very high temperatures, to a plastic flow inside the nanotube molecule, when the 5/7 and 7/5 twinsglide away from each other driven by the elastic forces, thus reducing the total strain energy (Equation17.8). One remarkable feature of such glide is due to mere cylindrical geometry: the glide “planes” inthe case of nanotubes are actually spirals, and the slow thermally activated Brownian walk of the dislo-cations proceeds along these well-defined trajectories. Similarly, their extra-planes are only the rows ofatoms also curved into the helices.

A nanotube with a 5/7 defect in its wall loses axial symmetry and has a bent equilibrium shape; thecalculations show30 the junction angles <15∞. Interestingly, then, an exposure of an even achiral nanotubeto the axially symmetric tension generates two 5/7 dislocations; and when the tension is removed, thetube freezes in an asymmetric configuration, S-shaped or C-shaped, depending on the distance of glide,that is, time of exposure. This seemingly “symmetry-violating” mechanical test is a truly nanoscalephenomenon. Of course, the symmetry is conserved statistically, because many different shapes formunder identical conditions.

When the dislocations sweep a noticeable distance, they leave behind a tube segment changed strictlyfollowing the topological rules of dislocation theory. By considering a planar development of the tubesegment containing a 5/7, for the new chirality vector c¢ one finds

(c1¢,c2¢) = (c1,c2) – (b1,b2) (17.10)

FIGURE 17.5 SW transformations of an equatorially oriented bond into a vertical position create a nucleus of relax-ation (top left corner). It evolves further as either a crack — brittle fracture route (left column) — or as a couple ofdislocations gliding away along the spiral slip plane (plastic yield, top row). In both cases only SW rotations are requiredas elementary steps. The step-wise change of the nanotube diameter reflects the change of chirality (bottom right image),causing the corresponding variations of electrical properties (Adapted from Yakobson, B.I., Mechanical relaxation and“intramolecular plasticity” in carbon nanotubes. Appl. Phys. Lett., 1998. 72(8): pp. 918–920. With permission.)

PLASTICGLIDE

(10,9)

BRITTLE FRACTURE(10,10)

Nanomechanics 17-11

with the corresponding reduction of diameter, d. While the dislocations of the first dipole glideaway, a generation of another dipole results in further narrowing and proportional elongation understress, thus forming a neck as shown above. The orientation of a generated dislocation dipole isdetermined every time by the Burger’s vector closest to the lines of maximum shear (±45∞ cross atthe endpoint of the current circumference-vector c). The evolution of a (c,c) tube will be: (c,c) Æ(c,c–1) Æ (c,c–2) Æ … (c,0) Æ (c–1,1) or (c,–1) Æ (c–1,0) Æ (c–2,1) or (c–1,–1) Æ (c–2,0) Æ[(c–3,1) or (c–2,–1)] Æ (c–3,0), etc. It abandons the armchair (c,c) type entirely, but then oscillatesin the vicinity of the zigzag kind (c,0), which appears to be a peculiar attractor. Correspondingly,the diameter for a (10,10) tube changes step-wise, d = 1.36, 1.29, 1.22, 1.16 nm, etc.; the local stressgrows in proportion, and this quantized necking can be terminated by a cleave at late stages.Interestingly, such plastic flow is accompanied by the change of electronic structure of the emergingdomains, governed by the vector (c1,c2). The armchair tubes are metallic, and others are semicon-ducting with the different band gap values. The 5/7 pair separating two domains of different chiralityhas been discussed as a pure carbon heterojunction, is argued to cause the current rectificationdetected in a nanotube nanodevice,31 and can be used to modify, in a controlled way, the electronicstructure of the tube. Here we see how this electronic heterogeneity can arise from a mechanicalrelaxation at high temperature: if the initial tube were armchair-metallic, the plastic dilation wouldtransform it into a semiconducting type irreversibly.24,25,32

While the above analysis is based on the atomic picture (structure and interactions), recentdevelopments33 offer an approach where the fracture nucleation can be described rather elegantlywithin nonlinear continuum mechanics (based on classical interatomic forces for carbon). Whetherthis approach can describe change in chirality, temperature dependence, or temporal aspects of relax-ation should yet be explored.

The dislocation theory allows one to expand the approach to other materials, and we have recentlyapplied it to boron nitride (BN) nanotubes.34 While the binding energy in BN is lower than in CNT, theformation of a 5/7/7/5 defect can be more costly due to Coulomb repulsion between emerging BB andNN pairs (Figure 17.6a). (Bonding in BN is partially ionic with a strong preference to chemical neighboralternation in the lattice.) Another dislocation pair 4/8/8/4 that preserves the alternation must be con-sidered (Figure 17.6b). It turns out that the quantitative results are sensitive to the level of theory accuracy.Tight-binding approximation35 underestimates the repulsion by almost 3 eV.34 Ab initio DFT calculationsshow that 5/7/7/5 is metastable lowest energy defect in BN, and its formation energy 5.5 eV is higherthan 3.0 eV in carbon,34 thus suggesting higher thermodynamic stability of BN under tensile load.Relaxation under compression is different as it involves skin-type buckling also investigated recently.36

FIGURE 17.6 The geometries of (5,5) BN tubule with (a) 5/7/7/5 defect emerging at high tension and temperature,and (b) 4/8/8/4 dislocation dipole. (From Bettinger, H., T. Dumitrica, G.E. Scuseria, and B.I. Yakobson, Mechanicallyinduced defects and strength of BN nanotubes. Phys. Rev. B, 2002. 65 (Rapid Comm.): p. 041406. With permission.)

17-12 Handbook of Nanoscience, Engineering, and Technology

17.5 Kinetic Theory of Strength

Computer simulations have provided compelling evidence of the mechanisms discussed above. By carefullytuning a quasi-static tension in the tubule and gradually elevating its temperature with extensive periods ofMD annealing, the first stages of the mechanical yield of CNT have been observed (Figure 17.7).26,27 Insimulation of tensile load, the novel patterns in plasticity and breakage, as described above, clearly emerge.At very high strain rate the details of primary defects cannot be seen; and they only begin to emerge at ahigher strain level, giving the impression of exceedingly high breaking strain.16

Fracture, of course, is a kinetic process where time is an important parameter. Even a small tension,as any nonhydrostatic stress, makes material thermodynamically meta-stable and a generation of defectsenergetically favorable. Thus the important issue of strength remains beyond the defect formation energyand its reduction with the applied tension. Recently we developed kinetic theory in application to CNT.29,37

In this approach we evaluate conditions (strain e, temperature T) when the probability P of defectformation becomes significant within laboratory test time Dt:

P = n Dt NB/3 Sm exp [–Em(e, c)/kbT] ~ 1. (17.11)

Here n = kbT/h is the usual attempted frequency, and NB is the number of bonds in the sample. Activationbarrier Em(e,c) must be computed as a function of strain and chirality c of the tubule, and then thesolution of this equation with respect to e gives the breaking strain values. This approach involvedsubstantial computational work in finding the saddle points and energies (Figure 17.8a) for a variety ofconditions and for several transition-state modes (index m in the summation above). Obtained yieldstrain near 17% (Figure 17.8b)29,37 is in reasonable agreement with the series of experimental reports.

FIGURE 17.7 T = 3000 K, strain 3%, plastic flow behavior (about 2.5 ns). The shaded area indicates the migrationpath of the 5/7 edge dislocation. (Adapted from Nardelli, M.B., B.I. Yakobson, and J. Bernholc, Brittle and ductilebehavior in carbon nanotubes. Phys. Rev. Lett., 1998. 81(21): pp. 4656–4659. With permission.)

Nanomechanics 17-13

We currently are implementing a similar approach with an ab initio level of saddle point barrierscalculations.38 Preliminary data show higher 8–9 eV barriers, but their reduction with tension is also faster.

Previously performed high strain rate simulations have shown temperature dependence of breakingstrain,13,16 consistent with the kinetic theory.29 In a constant strain rate arrangement (when the ends ofthe sample are pulled from the equilibrium with certain rate), the rate equation is slightly modified toits integral form. However, the main contribution comes from the vicinity of the upper limit:

P = n NB/3 Ú Sm exp{–Em[e(t), c]/kbT}dt ~ 1 (17.12)

Simple analysis reveals certain invariant T ¥ log(nDt) of the time of failure and temperature (providedthe constant strain). Detailed simulations could shed additional light on this aspect.39

17.6 Coalescence of Nanotubes as a Reversed Failure

Understanding the details of failure mechanism has led us recently40 to investigate an opposite process,a coalescence of nanoscale clusters analogous to macroscopic sintering or welding. Fusion of smallercomponents into a larger whole is a ubiquitous process in condensed matter. In molecular scale itcorresponds to chemical synthesis, where exact rearrangement of atoms can be recognized. Coalescenceor sintering of macroscopic parts is usually driven by well-defined thermodynamic forces (frequently,surface energy reduction), but the atomic evolution path is redundant and its exact identification isirrelevant. Exploring a possibility of the two particles merging with atomic precision becomes compellingin nanometer scale, where one aspires to “arrange the atoms one by one.” Are the initial and final statesconnected by a feasible path of atomic movements, or separated by insurmountable barriers? Direct MDinvestigation is usually hampered by energy landscape traps and, beyond very few atomic steps, needsto be augmented with additional analysis.

An example of very small particles is offered by fullerene cages and CNTs. Fusion of fullerenes hasbeen previously reported, and the lateral merging (diameter doubling) of CNT has been observed andsimulated.41,42 In contrast, head-to-head coalescence of CNT segments remains unexplored and of par-ticular theoretical and practical interest.

1. Is it permitted by rigorous topology rules to eliminate all the pentagons always present in the CNTends and thus dissolve the caps completely?

FIGURE 17.8 Activation barrier values (here, computed within classical multibody potential, a) serve as inputto the rate Equation (17.11) and the calculation of the yield strain as a function of time (here, from 1 ms to 1year, b), temperature (here, 300 K) and chiral symmetry (c). (From Samsonidze, G.G., G.G. Samsonidze, and B.I.Yakobson, Kinetic theory of symmetry-dependent strength in carbon nanotubes. Phys. Rev. Lett., 2002. 88: p.065501. With permission.)

17-14 Handbook of Nanoscience, Engineering, and Technology

2. Can this occur through a series of well-defined elementary steps, and what is the overall energychange if the system evolves through the intermediate disordered states to the final purely hexag-onal lattice of continuous tubule?

If feasible, such “welding” can lead to increase of connectivity in CNT arrays in bundles/ropes and crystalsand thus significantly improve the mechanical, thermal, and electrical properties of material. In addition,determining the atomistic steps of small-diameter tubes coalescence (with the end-caps identical to half-buckyball) can shed light on the underlying mechanism of condensed phase conversion or CNT synthesisfrom C60 components.

We have reported40 for the first time atomically precise routes for complete coalescence of genericfullerene cages: cap-to-cap CNT and C60 merging together to form a defectless final structure. The entireprocess is reduced to a sequence of Stone–Wales bond switches and, therefore, is likely the lowest energypath of transformation. Several other examples of merging follow immediately as special cases: coales-cence of buckyballs in peapod, joining of the two (5,5) tubes as in Figure 17.9, welding the (10,10) to(10,10) following Figure 17.10, etc. The approach remains valid for arbitrary tubes with the importantconstraint of unavoidable grain boundary for the tubes of different chirality. The junction of (n,m) and(n¢,m¢) must contain 5/7 dislocations or their equivalent of (n-n¢,m-m¢) total Burger’s vector.24 Theproposed mechanism40 has important implications for nanotube material (crystals, ropes) processingand property enhancement, engineering of nanoscale junctions of various types, and possible growthmechanisms with the C60 and other nanoparticles as feedstock. In the context of nanomechanics, aninteresting feature of the late stages of coalescence is the annealing and annihilation of 5/7 pairs in aprocess exactly reverse to the formation and glide of these dislocation cores in the course of yield andfailure under tension.

FIGURE 17.9 Two-dimensional geodesic projection (left) and the actual three-dimensional structures (right) showthe transformations from a pair of separate (5,5) tubes (a) to a single defect-free nanotube. Primary polymerizationlinks form as two other bonds break (b, dashed lines). The p/2-rotations of the links (the bonds subject to such SWflip are dotted) and the SW flips of the four other bonds in (c) produce a (5,0) neck (d). It widens by means of anotherten SW rotations, forming a perfect single (5,5) tubule (not shown). (From Zhao, Y., B.I. Yakobson, R.E. Smalley,Dynamic topology of fullerene coalescence. Phys. Rev. Lett., 88:185501, 2002. With permission.)

Nanomechanics 17-15

17.7 Persistence Length, Coils, and Random FuzzBalls of CNTS

Van der Waals forces play an important role not only in the interaction of the nanotubes with thesubstrate but also in their mutual interaction. The different shells of an MWNT interact primarily byvan der Waals forces; single-wall tubes form ropes for the same reason. A different manifestation ofvan der Waals interactions involves the self-interaction between two segments of the same single-wallCNT to produce a closed ring (loop).43 SWNT rings were first observed in trace amounts in theproducts of laser ablation of graphite and assigned a toroidal structure. More recently, rings of SWNTswere synthesized with large yields (~50%) from straight nanotube segments. These rings were shownto be loops, not tori.43 The synthesis involves the irradiation of raw SWNTs in a sulfuric acid–hydrogenperoxide solution by ultrasound. This treatment both etches the CNTs, shortening their length toabout 3–4 mm, and induces ring formation.

The formation of coils by CNTs is particularly intriguing. While coils of biomolecules and polymersare well-known structures, they are stabilized by a number of interactions that include hydrogen bondsand ionic interactions. On the other hand, the formation of nanotube coils is surprising given the highflexural rigidity (K = Young’s modulus times areal moment of inertia) of CNTs and the fact that CNTcoils can only be stabilized by van der Waals forces. However, estimates based on continuum mechanicsshow that, in fact, it is easy to compensate for the strain energy induced by the coiling process throughthe strong adhesion between tube segments in the coil. Details of this analysis can be found in the originalreports43 or in our recent review.7 Here we will outline briefly a different and more common situationwhere the competition of elastic energy and the intertubular linkage is important. Following our recent

FIGURE 17.10 Two-dimensional projections (left) and the computed three-dimensional intermediate structures(right) in the coalescence of the two (10,10) nanotubes: separate caps (a) in a sequence similar to Figure 17.9 developa (5,5) junction (b), which then shortens (c) and widens into a (10,5) neck (d). Glide of the shaded 5/7 dislocationscompletes the annealing into a perfect (10,10) CNT (not shown). Due to the 5-fold symmetry, only two cells aredisplayed. (From Zhao, Y., B.I. Yakobson, R.E. Smalley, Dynamic topology of fullerene coalescence. Phys. Rev. Lett.,88:185501, 2002. With permission.)

17-16 Handbook of Nanoscience, Engineering, and Technology

work,44 we will discuss the connection between the nanomechanics of CNTs and their random curlingin a suspension or a raw synthesized material of buckypaper.

SWNTs are often compared with polymer chains as having very high rigidity and, therefore, largepersistence length. In order to quantify this argument, we note that a defectless CNT has almost no staticflexing, although 5/7 defects, for example, introduce a small kink-angle of 5–15° and could cause somestatic curvature; but their energies are high and concentration is usually negligible. Only dynamic elasticflexibility should be considered. If u(s) is a unit direction vector tangent to the CNT at contour lengthpoint s and the bending stiffness is K, then statistical probability of certain shape u(s) is

P[u(s)] = exp[–1/2(K/kbT) Ú(∂u/∂s)2ds] = exp[–1/2 L Ú(∂u/∂s)2ds] (17.13)

Here persistence length is L = (K/kbT). For a (10,10) SWNT of radius R = 0.7 nm and the effective wallthickness h = 0.09 nm (see Sections 17.2 and 17.3), the bending stiffness is very close to K = pCR3 (C =345 N/m is the in-plane stiffness, based on ab initio calculations). Persistence length at room temperature,therefore, is L1[(10,10), 293 K] ~ 0.1 mm, in the macroscopic range much greater than for most polymermolecules. The result can be generalized for a single SWNT of radius R as

L1 = (30K/T)(R/0.7nm)3 mm (17.14)

or N times more for a bundle of N tubes (assuming additive stiffness for the case of weak lateral cohesionof the constituent SWNTs). For example, for a smallest close-packed bundle of seven (one surroundedby six neighbors), this yields L7 = 1 mm. Such an incoherent bundle and a solid-coherent bundle withperfect lateral adhesion provide the lower and upper limits for the persistence length, NL1 < LN < N2L1.Remarkably, these calculations show that the true thermodynamic persistence length of small CNTbundles or even an individual SWNT is in the macroscopic range from a fraction of a millimeter andgreater. This means that highly curved structures often observed in buckypaper mats (Figure 17.11) areattributed not to thermodynamic fluctuations but rather to residual mechanical forces preventing thesecoils from unfolding. Elastic energy of a typical micron-size (r ~ 1 mm) curl-arc is much greater than

FIGURE 17.11 Raw-produced SWNTs often form ropes/bundles, entangled and bent into a rubbery structure called“buckypaper.” The length scale of bends is much smaller than the persistence length for the constituent filaments.Shown here is material produced by HiPco (high pressure CO) synthesis method. (Adapted from O’Connell, M.J.et al., Chem. Phys. Lett., 342, 265, 2001. With permission.)

Nanomechanics 17-17

thermal, Ucurl ~ kbT(L/r)2 >> kbT. At the same time, a force required to maintain this shape is Fcurl ~ K/r2 = N pN, several piconewtons, where N is the number of SWNTs in each bundle. This is much lessthan a force per single chemical bond (~ 1 nN), and therefore any occasional lateral bonding betweenthe tubules can be sufficient to prevent them from disentanglement.

Acknowledgments

The author appreciates the support from NASA Ames, Air Force Research Laboratory, Office of NavalResearch DURINT grant, and the Nanoscale Science and Engineering Initiative of the National ScienceFoundation, award number EEC-0118007.

References

1. Yakobson, B.I., Morphology and rate of fracture in chemical decomposition of solids. Phys. Rev.Lett., 1991. 67(12): pp. 1590–1593.

2. Yakobson, B.I., Stress-promoted interface diffusion as a precursor of fracture. J. Chem. Phys., 1993.99(9): pp. 6923–6934.

3. Iijima, S., Helical microtubules of graphitic carbon. Nature, 1991. 354: pp. 56–58.4. Phillips, R.B., Crystals, Defects and Microstructures. Cambridge: Cambridge University Press, 2001,

p. 780.5. Hertel, T., R.E. Walkup, and P. Avouris, Deformation of carbon nanotubes by surface van der Waals

forces. Phys. Rev. B, 1998. 58(20): pp. 13870–13873.6. Yakobson, B.I. and R.E. Smalley, Fullerene nanotubes: C1,000,000 and beyond. Am. Sci., 1997. 85(4):

pp. 324–337.7. Yakobson, B.I. and P. Avouris, Mechanical properties of carbon nanotubes, in Carbon Nanotubes,

M.S. Dresselhaus, G. Dresselhaus, and P. Avouris (Eds.), Springer: Berlin, 2001, pp. 287–327.8. Kudin, K.N., G.E. Scuseria, and B.I. Yakobson, C2F, BN and C nanoshell elasticity by ab Initio

computations. Phys. Rev. B, 2001. 64: p. 235406.9. Treacy, M.M.J., T.W. Ebbesen, and J.M. Gibson, Exceptionally high Young’s modulus observed for

individual carbon nanotubes. Nature, 1996. 381: pp. 678–680.10. Wong, E.W., P.E. Sheehan, and C.M. Lieber, Nanobeam mechanics: elasticity, strength and tough-

ness of nanorods and nanotubes. Science, 1997. 277: pp. 1971–1975.11. Poncharal, P., Z.L. Wang, D. Ugarte, and W.A. Heer, Electrostatic deflections and electromechanical

resonances of carbon nanotubes. Science, 1999. 283: pp. 1513–1516.12. Yakobson, B.I., C.J. Brabec, and J. Bernholc, Nanomechanics of carbon tubes: instabilities beyond

the linear response. Phys. Rev. Lett., 1996. 76(14): pp. 2511–2514.13. Yakobson, B.I., C.J. Brabec, and J. Bernholc, Structural mechanics of carbon nanotubes: from

continuum elasticity to atomistic fracture. J. Comp. Aided Mater. Des., 1996. 3: pp. 173–182.14. Garg, A., J. Han, and S.B. Sinnott, Interactions of carbon-nanotubule proximal probe tips with

diamond and graphite. Phys. Rev. Lett., 1998. 81(11): pp. 2260–2263.15. Srivastava, D., M. Menon, and K. Cho, Nanoplasticity of single-wall carbon nanotubes under

uniaxial compression. Phys. Rev. Lett., 1999. 83(15): pp. 2973–2976.16. Yakobson, B.I., M.P. Campbell, C.J. Brabec, and J. Bernholc, High strain rate fracture and C-chain

unraveling in carbon nanotubes. Comp. Mater. Sci., 1997. 8: pp. 341–348.17. Ruoff, R.S., et al., Radial deformation of carbon nanotubes by van der Waals forces. Nature, 1993.

364: pp. 514–516.18. Chopra, N.G. et al., Fully collapsed carbon nanotubes. Nature, 1995. 377: pp. 135–138.19. Despres, J.F., E. Daguerre, and K. Lafdi, Flexibility of graphene layers in carbon nanotubes. Carbon,

1995. 33(1): pp. 87–92.20. Iijima, S., C.J. Brabec, A. Maiti, and J. Bernholc, Structural flexibility of carbon nanotubes. J. Chem.

Phys., 1996. 104(5): pp. 2089–2092.

17-18 Handbook of Nanoscience, Engineering, and Technology

21. Yakobson, B.I., Dynamic Topology and Yield Strength of Carbon Nanotubes, in Fullerenes, Electro-chemical Society, Pennington, NJ, 1997.

22. Smalley, R.E. and B.I. Yakobson, The future of the fullerenes. Solid State Commn., 1998. 107(11):pp. 597–606.

23. Liu, J.Z., Q. Zheng, and Q. Jiang, Effect of a rippling mode on resonances of carbon nanotubes.Phy. Rev. Lett., 2001. 86: pp. 4843–4846.

24. Yakobson, B.I., Mechanical relaxation and “intramolecular plasticity” in carbon nanotubes. Appl.Phys. Lett., 1998. 72(8): pp. 918–920.

25. Yakobson, B.I., Physical property modification of nanotubes. U. S. Patent 6,280,677 B1, 2001.26. Nardelli, M.B., B.I. Yakobson, and J. Bernholc, Mechanism of strain release in carbon nanotubes.

Phys. Rev. B, 1998. 57: p. R4277.27. Nardelli, M.B., B.I. Yakobson, and J. Bernholc, Brittle and ductile behavior in carbon nanotubes.

Phys. Rev. Lett., 1998. 81(21): pp. 4656–4659.28. Yakobson, B.I., G. Samsonidze, and G.G. Samsonidze, Atomistic theory of mechanical relaxation

in fullerene nanotubes. Carbon, 2000. 38: p. 1675.29. Samsonidze, G.G., G.G. Samsonidze, and B.I. Yakobson, Kinetic theory of symmetry-dependent

strength in carbon nanotubes. Phys. Rev. Lett., 2002. 88: p. 065501.30. Chico, L., et al., Pure carbon nanoscale devices: nanotube heterojunctions. Phys. Rev. Lett., 1996.

76(6): pp. 971–974.31. Collins, P.G., et al., Nanotube nanodevice. Science, 1997. 278: pp. 100–103.32. Tekleab, D., D.L. Carroll, G.G. Samsonidze, and B.I. Yakobson, Strain-induced electronic property

heterogeneity of a carbon nanotube. Phys. Rev. B, 2001. 64: p. 035419.33. Zhang, P., Y. Huang, H. Gao, and K.C. Hwang, Fracture nucleation in SWNT under tension: a

continuum analysis incorporating interatomic potential. ASME Trans. J. Appl. Mechanics, 2002: inpress.

34. Bettinger, H., T. Dumitrica, G.E. Scuseria, and B.I. Yakobson, Mechanically induced defects andstrength of BN nanotubes. Phys. Rev. B, 2002. 65 (Rapid Comm.): p. 041406.

35. Zhang, P. and V.H. Crespi, Plastic deformations of boron-nitride nanotubes: an unexpected weak-ness. Phys. Rev. B, 2000. 62: p. 11050.

36. Srivastava, D., M. Menon, and K.J. Cho, Anisotropic nanomechanics of boron nitride nanotubes:nanostructured “skin” effect. Phys. Rev. B, 2001. 63: p. 195413.

37. Samsonidze, G.G., G.G. Samsonidze, and B.I. Yakobson, Energetics of Stone-Wales defects indeformations of monoatomic hexagonal layers. Comp. Mater. Sci., 23:62, 2002.

38. Dumitrica, T., H. Bettinger, and B.I. Yakobson, Stone-Wales barriers and kinetic theory of strengthfor nanotubes. In progress.

39. Wei, C., K.J. Cho, and D. Srivastava, private communication: xxx.lanl.gov/abs/cond-mat/0202513.40. Zhao, Y., B.I. Yakobson, and R.E. Smalley, Dynamic topology of fullerene coalescence. Phys. Rev.

Lett., 88: 185501, 2002.41. Nikolaev, P., et al., Diameter doubling of single-wall nanotubes. Chem. Phys. Lett., 1997. 266: p. 422.42. Terrones, M., et al., Coalescence of single-walled carbon nanotubes. Science, 2000. 288: pp.

1226–1229.43. Martel, R., H.R. Shea, and P. Avouris, Rings of single-wall carbon nanotubes. Nature, 1999. 398:

p. 582.44. Yakobson, B.I. and L.S. Couchman. Mechanical properties of carbon nanotubes as random coils:

from persistence length to the equation of state. In progress.