SWNTs

20 40

60 80 100

Fig. 8. Measured low-temperature thermal conductivity of SWNTs, compared to a two-band model [19]

20 40

60 80 100

it is frozen out at low temperatures, and begins to contribute near 35 K. The solid line is the sum of the two contributions, and is quite successful in fitting the experimental data below ~100K. The phonon energies, and temperature scale of the observed linear behavior, are higher in the thermal conductivity measurements than in the heat capacity measurements (Sect. 1.4). This may be due to the preferential weighting of higher-velocity modes in the thermal conductivity, although more detailed modeling is needed to resolve this issue. The measured linear slope can be used to calculate the scattering time, or, equivalently, a scattering length. A room-temperature thermal conductivity of 200 W/mK (as is seen in the bulk aligned samples) implies a phonon scattering length of 30 nm, although this value is likely to be higher for single tubes.

We have seen above that the small size of nanotubes causes phonon quantization which can be observed both in the heat capacity and in the thermal conductivity at low temperatures. The restricted geometry of the tubes may also affect the thermal conductivity at high temperature since Umklapp scattering should be suppressed in one dimensional system because of the unavailability of states into which to scatter [16,22]. Extension of these measurements to higher temperatures, as well as additional theoretical modeling, should prove interesting.

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Mechanical Properties of Carbon Nanotubes

Boris I. Yakobson1 and Phaedon Avouris2

1 Center for Nanoscale Science and Technology and Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX, 77251-1892, USA [email protected]

2 IBM T.J. Watson Research Center Yorktown Heights, NY 10598, USA [email protected]

Abstract. This paper presents an overview of the mechanical properties of carbon nanotubes, starting from the linear elastic parameters, nonlinear elastic instabilities and buckling, and the inelastic relaxation, yield strength and fracture mechanisms. A summary of experimental findings is followed by more detailed discussion of theoretical and computational models for the entire range of the deformation amplitudes. Non-covalent forces (supra-molecular interactions) between the nanotubes and with the substrates are also discussed, due to their significance in potential applications.

It is noteworthy that the term resilient was first applied not to nanotubes but to smaller fullerene cages, when Whetten et al. studied the high-energy collisions of C60, C70, and C84 bouncing from a solid wall of H-terminated diamond [6]. They observed no fragmentation or any irreversible atomic rearrangement in the bouncing back cages, which 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, since their elongated shape, with the aspect ratio close to a thousand, makes the mechanical properties especially interesting and important due to potential structural applications.

1 Mechanical Properties and Mesoscopic Duality of Nanotubes

The utility of nanotubes as the strongest or stiffest elements in nanoscale devices or composite materials remains a powerful motivation for the research in this area. While the jury is still out on practical realization of these applications, an additional incentive comes from the fundamental materials physics. There is a certain duality in the nanotubes. On one hand they have molecular size and morphology. At the same time possessing sufficient translational

M. S. Dresselhaus, G. Dresselhaus, Ph. Avouris (Eds.): Carbon Nanotubes, Topics Appl. Phys. 80, 287-327 (2001) © Springer-Verlag Berlin Heidelberg 2001

symmetry to perform as very small (nano-) crystals, with a well defined primitive cell, surface, possibility of transport, etc. Moreover, in many respects they can be studied as well defined engineering structures and many properties can be discussed in traditional terms of moduli, stiffness or compliance, geometric size and shape. The mesoscopic dimensions (a nanometer scale diameter) combined with the regular, almost translation-invariant morphology along their micrometer scale lengths (unlike other polymers, usually coiled), make nanotubes a unique and attractive object of study, including the study of mechanical properties and fracture in particular.

Indeed, fracture of materials is a complex phenomenon whose theory generally requires a multiscale description involving microscopic, mesoscopic and macroscopic modeling. Numerous traditional approaches are based on a macroscopic continuum picture that provides an appropriate model except at the region of actual failure where a detailed atomistic description (involving real chemical bond breaking) is needed. Nanotubes, due to their relative simplicity and atomically precise morphology, offer us the opportunity to address the validity of different macroscopic and microscopic models of fracture and mechanical response. Contrary to crystalline solids where the structure and evolution of ever-present surfaces, grain-boundaries, and dislocations under applied stress determine the plasticity and fracture of the material, nano-tubes possess simpler structure while still showing rich mechanical behavior within elastic or inelastic brittle or ductile domains. This second, theoretical-heuristic value of nanotube research supplements their importance due to anticipated practical applications. A morphological similarity of fullerenes and nanotubes to their macroscopic counterparts, geodesic domes and towers, compels one to test the laws and intuition of macro-mechanics in the scale ten orders of magnitude smaller.

In the following, Sect. 2 provides a background for the discussion of nanotubes: basic concepts from materials mechanics and definitions of the main properties. We then present briefly the experimental techniques used to measure these properties and the results obtained (Sect. 3). Theoretical models, computational techniques, and results for the elastic constants, presented in Sect. 4, are compared wherever possible with the experimental data. In theoretical discussion we proceed from linear elastic moduli to the nonlinear elastic behavior, buckling instabilities and shell model, to compressive/bending strength, and finally to the yield and failure mechanisms in tensile load. After the linear elasticity, Sect. 4.1, we outline the non-linear buckling instabilities, Sect. 4.2. Going to even further deformations, in Sect. 4.3 we discuss irreversible changes in nanotubes, which are responsible for their inelastic relaxation and failure. Fast molecular tension tests (Sect. 4.3) are followed by the theoretical analysis of relaxation and failure (Sect. 4.4), based on intramolecular dislocation failure concept and combined with the computer simulation evidence. We discuss the mechanical deformation of the nanotubes caused by their attraction to each other (supramolecular interactions) and/or to, the substrates, Sect. 5.1. Closely related issues of manipulation of the tubes position and shape, and their self-organization into ropes and rings, caused by the seemingly weak van der Waals forces, are presented in the Sects. 5.2,5.3. Finally, a brief summary of mechanical properties is included in Sect. 6.

2 Mechanics of the Small: Common Definitions

Nanotubes are often discussed in terms of their materials applications, which makes it tempting to define "materials properties" of a nanotube itself. However, there is an inevitable ambiguity due to lack of translational invariance in the transverse directions of a singular nanotube, which is therefore not a material, but rather a structural member.

A definition of elastic moduli for a solid implies a spatial uniformity of the material, at least in an average, statistical sense. This is required for an accurate definition of any intensive characteristic, and generally fails in the nanometer scale. A single nanotube possesses no translational invariance in the radial direction, since a hollow center and a sequence of coaxial layers are well distinguished, with the interlayer spacing, c, comparable with the nanotube radius, R. It is essentially an engineering structure, and a definition of any material-like characteristics for a nanotube, while heuristically appealing, must always be accompanied with the specific additional assumptions involved (e.g. the definition of a cross-section area). Without it confusion can easily cripple the results or comparisons. The standard starting point for defining the elastic moduli as 1 /V d2E/de2 (where E is total energy as a function of uniform strain e) is not a reliable foothold for molecular structures. For nanotubes, this definition only works for a strain e in the axial direction; any other deformation (e.g. uniform lateral compression) induces non-uniform strain of the constituent layers, which renders the previous expression misleading. Furthermore, for the hollow fullerene nanotubes, the volume V is not well defined. For a given length of a nanotube L, the cross section area A can be chosen in several relatively arbitrary ways, thus making both volume V = LA and consequently the moduli ambiguous. To eliminate this problem, the intrinsic elastic energy of nanotube is better characterized by the energy change not per volume but per area S of the constituent graphitic layer (or layers), C = 1/S d2E/de2. The two-dimensional spatial uniformity of the graphite layer ensures that S = IL, and thus the value of C, is unambiguous. Here l is the total circumferential length of the graphite layers in the cross section of the nanotube. Unlike more common material moduli, C has dimensionality of surface tension, N/m, and can be defined in terms of measurable characteristics of nanotube,

The partial derivative at zero strain in all dimensions except along e yields an analog of the elastic stiffness C\\ in graphite, while a free-boundary (no lateral traction on the nanotube) would correspond to the Young's modulus Y = S-1 (Si i being the elastic compliance). In the latter case, the nanotube Young's modulus can be recovered and used, but the non-unique choice of cross-section A or a thickness h must be kept in mind. For the bending stiffness K correspondingly, one has (k being a beam curvature), where the integration on the right hand side goes over the cross-section length of all the constituent layers, and y is the distance from the neutral surface. Note again, that this allows us to completely avoid the ambiguity of the mono-atomic layer "thickness", and to relate only physically measurable quantities like the nanotube energy E, the elongation e or a curvature k. If one adopts a particular convention for the graphene thickness h (or equivalently, the cross section of nanotube), the usual Young's modulus can be recovered, Y = C/h. For instance, for a bulk graphite h = c = 0.335 nm, C = 342 N/m and Y = 1.02 GPa, respectively. This choice works reasonably well for large diameter multiwall tubes (macro-limit), but can cause significant errors in evaluating the axial and especially bending stiffness for narrow and, in particular, singlewall nanotubes.

Strength and particularly tensile strength of a solid material, similarly to the elastic constants, must ultimately depend on the strength of its interatomic forces/bonds. However, this relationship is far less direct than in the case of linear-elastic characteristics; it is greatly affected by the particular arrangement of atoms in a periodic but imperfect lattice. Even scarce imperfections in this arrangement play a critical role in the material nonlinear response to a large force, that is, plastic yield or brittle failure. Without it, it would be reasonable to think that a piece of material would break at Y/8-Y/15 stress, that is about 10% strain [3]. However, all single-phase solids have much lower aY values, around Y/104, due to the presence of dislocations, stacking-faults , grain boundaries, voids, point defects, etc. The stress induces motion of the pre-existing defects, or a nucleation of the new ones in an almost perfect solid, and makes the deformation irreversible and permanent. The level of strain where this begins to occur at a noticeable rate determines the yield strain eY or yield stress aY . In the case of tension this threshold reflects truly the strength of chemical-bonds, and is expected to be high for C-C based material.

A possible way to strengthen some materials is by introducing extrinsic obstacles that hinder or block the motion of dislocations [32]. There is a limit to the magnitude of strengthening that a material may benefit from, as too many obstacles will freeze (pin) the dislocations and make the solid brittle. A single-phase material with immobile dislocations or no dislocations at all

breaks in a brittle fashion, with little work required. The reason is that it is energetically more favorable for a small crack to grow and propagate. Energy dissipation due to crack propagation represents materials toughness, that is a work required to advance the crack by a unit area, G > 2y (which can be just above the doubled surface energy 7 for a brittle material, but is several orders of magnitude greater for a ductile material like copper). Since the c-edge dislocations in graphite are known to have very low mobility, and are the so called sessile type [36], we must expect that nanotubes per se are brittle, unless the temperature is extremely high. Their expected high strength does not mean significant toughness, and as soon as the yield point is reached, an individual nanotube will fail quickly and with little dissipation of energy. However, in a large microstructured material, the pull-out and relative shear between the tubes and the matrix can dissipate a lot of energy, making the overall material (composite) toughness improved. Although detailed data is not available yet, these differences are important to keep in mind.

Compression strength is another important mechanical parameter, but its nature is completely different from the strength in tension. Usually it does not involve any bond reorganization in the atomic lattice, but is due to the buckling on the surface of a fiber or the outermost layer of nanotube. The standard measurement [37] involves the so called "loop test" where tightening of the loop abruptly changes its aspect ratio from 1.34 (elastic) to higher values when kinks develop on the compressive side of the loop. In nanotube studies, this is often called bending strength, and the tests are performed using an atomic force microscope (AFM) tip [74], but essentially in both cases one deals with the same intrinsic instability of a laminated structure under compression [62].

These concepts, similarly to linear elastic characteristics, should be applied to carbon and composite nanotubes with care. At the current stage of this research, nanotubes are either assumed to be structurally perfect or to contain few defects, which are also defined with atomic precision (the traditional approach of the physical chemists, for whom a molecule is a well-defined unit). A proper averaging of the "molecular" response to external forces, in order to derive meaningful material characteristics, represents a formidable task for theory. Our quantitative understanding of inelastic mechanical behavior of carbon, BN and other inorganic nanotubes is just beginning to emerge, and will be important for the assessment of their engineering potential, as well as a tractable example of the physics of fracture.

3 Experimental Observations

There is a growing body of experimental evidence indicating that carbon nanotubes (both MWNT and SWNT) have indeed extraordinary mechanical properties. However, the technical difficulties involved in the manipulation of these nano-scale structures make the direct determination of their mechanical properties a rather challenging task.

3.1 Measurements of the Young's modulus

Nevertheless, a number of experimental measurements of the Young's modulus of nanotubes have been reported.

The first such study [71] correlated the amplitude of the thermal vibrations of the free ends of anchored nanotubes as a function of temperature with the Young's modulus. Regarding a MWNT as a hollow cylinder with a given wall thickness, one can obtain a relation between the amplitude of the tip oscillations (in the limit of small deflections), and the Young's modulus. In fact, considering the nanotube as a cylinder with the high elastic constant c\\ = 1.06 TPa and the corresponding Young's modulus 1.02 TPa of graphite and using the standard beam deflection formula one can calculate the bending of the nanotube under applied external force. In this case, the deflection of a cantilever beam of length L with a force F exerted at its free end is given by S = FL3/(3YI), where I is the moment of inertia. The basic idea behind the technique of measuring free-standing room-temperature vibrations in a TEM, is to consider the limit of small amplitudes in the motion of a vibrating cantilever, governed by the well known fourth-order wave equation, ytt = -(YI/gA)yxxxx, where A is the cross sectional area, and g is the density of the rod material. For a clamped rod the boundary conditions are such that the function and its first derivative are zero at the origin and the second and third derivative are zero at the end of the rod. Thermal nanotube vibrations are essentially elastic relaxed phonons in equilibrium with the environment; therefore the amplitude of vibration changes stochastically with time. This stochastic driven oscillator model is solved in [38] to more accurately analyze the experimental results in terms of the Gaussian vibrational-profile with a standard deviation given by with Do and D; the outer and inner radii, T the temperature and an the standard deviation. An important assumption is that the nanotube is uniform along its length. Therefore, the method works best on the straight, clean nanotubes. Then, by plotting the mean-square vibration amplitude as a function of temperature one can get the value of the Young's modulus.

This technique was first used in [71] to measure the Young's modulus of carbon nanotubes. The amplitude of those oscillations was defined by means of careful TEM observations of a number of nanotubes. The authors obtained an average value of 1.8 TPa for the Young's modulus, though there was significant scatter in the data (from 0.4 to 4.15 TPa for individual tubes). Although this number is subject to large error bars, it is nevertheless indicative of the exceptional axial stiffness of these materials. More recently studies

Fig. 1. Top panel: bright field TEM images of free-standing multi-wall carbon nanotubes showing the blurring of the tips due to thermal vibration, from 300 to 600 K. Detailed measurement of the vibration amplitude is used to estimate the stiffness of the nanotube beam [71]. Bottom panel: micrograph of single-wall nanotube at room temperature, with the inserted simulated image corresponding to the bestsquares fit adjusting the tube length L, diameter d and vibration amplitude (in this example, L = 36.8nm, d = 1.5nm, a = 0.33 nm, and Y = 1.33 ± 0.2TPa) [38]

Fig. 1. Top panel: bright field TEM images of free-standing multi-wall carbon nanotubes showing the blurring of the tips due to thermal vibration, from 300 to 600 K. Detailed measurement of the vibration amplitude is used to estimate the stiffness of the nanotube beam [71]. Bottom panel: micrograph of single-wall nanotube at room temperature, with the inserted simulated image corresponding to the bestsquares fit adjusting the tube length L, diameter d and vibration amplitude (in this example, L = 36.8nm, d = 1.5nm, a = 0.33 nm, and Y = 1.33 ± 0.2TPa) [38]

on SWNT's using the same technique have been reported, Fig. 1 [38]. A larger sample of nanotubes was used, and a somewhat smaller average value was obtained, Y = 1.25 — 0.35/+0.45 TPa, around the expected value for graphite along the basal plane. The technique has also been used in [14] to estimate the Young's modulus for BN nanotubes. The results indicate that these composite tubes are also exceptionally stiff, having a value of Y around 1.22 TPa, very close to the value obtained for carbon nanotubes.

Another way to probe the mechanical properties of nanotubes is to use the tip of an AFM (atomic force microscope) to bend anchored CNT's while simultaneously recording the force exerted by the tube as a function of the displacement from its equilibrium position. This allows one to extract the Young's modulus of the nanotube, and based on such measurements [74] have reported a mean value of 1.28±0.5 TPa with no dependence on tube diameter for MWNT, in agreement with the previous experimental results. Also [60] used a similar idea, which consists of depositing MWNT's or SWNT's bundled in ropes on a polished aluminum ultra-filtration membrane. Many tubes are then found to lie across the holes present in the membrane, with a fraction of their length suspended. Attractive interactions between the nanotubes and the membrane clamp the tubes to the substrate. The tip of an AFM is then used to exert a load on the suspended length of the nanotube, measuring at the same time the nanotube deflection. To minimize the uncertainty of the applied force, they calibrated the spring constant of each AFM tip (usually 0.1 N/m) by measuring its resonant frequency. The slope of the deflection versus force curve gives directly the Young's modulus for a known length and tube radius. In this way, the mean value of the Young's modulus obtained for arc-grown carbon nanotubes was 0.81 ±0.41 TPa. (The same study applied to disordered nanotubes obtained by the catalytic decomposition of acetylene gave values between 10 to 50 GPa. This result is likely due to the higher density of structural defects present in these nanotubes.) In the case of ropes, the analysis allows the separation of the contribution of shear between the constituent SWNT's (evaluated to be close to G =1 GPa) and the tensile modulus, close to 1 TPa for the individual tubes. A similar procedure has also been used [48] with an AFM to record the profile of a MWNT lying across an electrode array. The attractive substrate-nanotube force was approximated by a van der Waals attraction similar to the carbon-graphite interaction but taking into account the different dielectric constant of the SiO2 substrate; the Poisson ratio of 0.16 is taken from ab initio calculations. With these approximations the Young modulus of the MWNT was estimated to be in the order of 1 TPa, in good accordance with the other experimental results.

An alternative method of measuring the elastic bending modulus of nano-tubes as a function of diameter has been presented by Poncharal et al. [52]. The new technique was based on a resonant electrostatic deflection of a multiwall carbon nanotube under an external ac-field. The idea was to apply a time-dependent voltage to the nanotube adjusting the frequency of the source to resonantly excite the vibration of the bending modes of the nano-tube, and to relate the frequencies of these modes directly to the Young modulus of the sample. For small diameter tubes this modulus is about 1 TPa, in good agreement with the other reports. However, this modulus is shown to decrease by one order of magnitude when the nanotube diameter increases (from 8 to 40 nm). This decrease must be related to the emergence of a different bending mode for the nanotube. In fact, this corresponds to a wave-like distortion of the inner side of the bent nanotube. This is clearly shown in Fig. 2. The amplitude of the wave-like distortion increases uniformly from essentially zero for layers close to the nanotube center to about 2-3 nm for the outer layers without any evidence of discontinuity or defects. The non-linear behavior is discussed in more detail in the next section and has been observed in a static rather than dynamic version by many authors in different contexts [19,34,41,58].

Although the experimental data on elastic modulus are not very uniform, overall the results correspond to the values of in-plane rigidity (2) C = 340 — 440 N/m, that is to the values Y =1.0 — 1.3 GPa for multiwall tubules, and to Y = 4C/d = (1.36 — 1.76) TPa nm/d for SWNT's of diameter d.

3.2 Evidence of Nonlinear Mechanics and Resilience of Nanotubes

Large amplitude deformations, beyond the Hookean behavior, reveal nonlinear properties of nanotubes, unusual for other molecules or for the graphite fibers. Both experimental evidence and theory-simulations suggest the ability

Fig. 2. A: bending modulus Y for MWNT as a function of diameter measured by the resonant response of the nanotube to an alternating applied potential (the inset shows the Lorentzian line-shape of the resonance). The dramatic drop in Y value is attributed to the onset of a wave-like distortion for thicker nanotubes. D: highresolution TEM of a bent nanotube with a curvature radius of 400 nm exhibiting a wave-like distortion. B,C: the magnified views of a portion of D [52]

Fig. 2. A: bending modulus Y for MWNT as a function of diameter measured by the resonant response of the nanotube to an alternating applied potential (the inset shows the Lorentzian line-shape of the resonance). The dramatic drop in Y value is attributed to the onset of a wave-like distortion for thicker nanotubes. D: highresolution TEM of a bent nanotube with a curvature radius of 400 nm exhibiting a wave-like distortion. B,C: the magnified views of a portion of D [52]

of nanotubes to significantly change their shape, accommodating to external forces without irreversible atomic rearrangements. They develop kinks or ripples (multiwalled tubes) in compression and bending, flatten into deflated ribbons under torsion, and still can reversibly restore their original shape. This resilience is unexpected for a graphite-like material, although folding of the mono-atomic graphitic sheets has been observed [22]. It must be attributed to the small dimension of the tubules, which leaves no room for the stress-concentrators — micro-cracks or dislocation failure piles (cf. Sect. 4.4), making a macroscopic material prone to failure. A variety of experimental evidence confirms that nanotubes can sustain significant nonlinear elastic deformations. However, observations in the nonlinear domain rarely could directly yield a measurement of the threshold stress or the force magnitudes. The facts are mostly limited to geometrical data obtained with high-resolution imaging.

An early observation of noticeable flattening of the walls in a close contact of two MWNT has been attributed to van der Walls forces pressing the cylinders to each other [59]. Similarly, a crystal-array [68] of parallel nanotubes will flatten at the lines of contact between them so as to maximize the attractive van der Waals intertube interaction (see Sect. 5.1). Collapsed forms of the nanotube ("nanoribbons"), also caused by van der Waals attraction, have been observed in experiment (Fig. 3d), and their stability can be explained by the competition between the van der Waals and elastic energies (see Sect. 5.1).

Graphically more striking evidence of resilience is provided by bent structures [19,34], Fig. 4. The bending seems fully reversible up to very large bending angles, despite the occurrence of kinks and highly strained tubule regions

Fig. 3. Simulation of torsion and collapse [76]. The strain energy of a 25 nm long (13, 0) tube as a function of torsion angle f (a). At the first bifurcation the cylinder flattens into a straight spiral (b) and then the entire helix buckles sideways, and coils in a forced tertiary structure (c). Collapsed tube (d) as observed in experiment [13]

Fig. 3. Simulation of torsion and collapse [76]. The strain energy of a 25 nm long (13, 0) tube as a function of torsion angle f (a). At the first bifurcation the cylinder flattens into a straight spiral (b) and then the entire helix buckles sideways, and coils in a forced tertiary structure (c). Collapsed tube (d) as observed in experiment [13]

in simulations, which are in excellent morphological agreement with the experimental images [34]. This apparent flexibility stems from the ability of the sp2 network to rehybridize when deformed out of plane, the degree of sp2-sp3 rehybridization being proportional to the local curvature [27]. The accumulated evidence thus suggests that the strength of the carbon-carbon bond does not guarantee resistance to radial, normal to the graphene plane deformations. In fact, the graphitic sheets of the nanotubes, or of a plane graphite [33] though difficult to stretch are easy to bend and to deform.

A measurement with the Atomic Force Microscope (AFM) tip detects the "failure" of a multiwall tubule in bending [74], which essentially represents nonlinear buckling on the compressive side of the bent tube. The measured local stress is 15-28 GPa, very close to the calculated value [62,79]. Buckling and rippling of the outermost layers in a dynamic resonant bending has been directly observed and is responsible for the apparent softening of MWNT of larger diameters. A variety of largely and reversibly distorted (estimated up to 15% of local strain) configurations of the nanotubes has been achieved with AFM tip [23,30]. The ability of nanotubes to "survive the crash" during the impact with the sample/substrate reported in [17] also documents their ability to reversibly undergo large nonlinear deformations.

Fig. 4. HREM images of bent nanotubes under mechanical duress. (a) and (b) single kinks in the middle of SWNT with diameters of 0.8 and 1.2 nm, respectively. (c) and (d) MWNT of about 8nm diameter showing a single and a two-kink complex, respectively [34]

3.3 Attempts of Strength Measurements

Reports on measurements of carbon nanotube strength are scarce, and remain the subject of continuing effort. A nanotube is too small to be pulled apart with standard tension devices, and too strong for tiny "optical tweezers", for example. The proper instruments are still to be built, or experimentalists should wait until longer nanotubes are grown.

A bending strength of the MWNT has been reliably measured with the AFM tip [74], but this kind of failure is due to buckling of graphene layers, not the C-C bond rearrangement. Accordingly, the detected strength, up to 28.5 GPa, is two times lower than 53.4 GPa observed for non-laminated SiC nanorods in the same series of experiments. Another group [23] estimates the maximum sustained tensile strain on the outside surface of a bent tubule as large as 16%, which (with any of the commonly accepted values of the Young's modulus) corresponds to 100-150 GPa stress. On the other hand, some residual deformation that follows such large strain can be an evidence of the beginning of yield and the 5/7-defects nucleation. A detailed study of the failure via buckling and collapse of matrix-embedded carbon nanotube must be mentioned here [41], although again these compressive failure mechanisms are essentially different from the bond-breaking yield processes in tension (as discussed in Sects. 4.3,4.4).

Actual tensile load can be applied to the nanotube immersed in matrix materials, provided the adhesion is sufficiently good. Such experiments, with stress-induced fragmentation of carbon nanotube in a polymer matrix has been reported, and an estimated strength of the tubes is 45 GPa, based on a simple isostrain model of the carbon nanotube-matrix. It has also to be remembered that the authors [72] interpret the contrast bands in HRTEM images as the locations of failure, although the imaging of the carbon nano-tube through the polymer film limits the resolution in these experiments.

While a singular single-wall nanotube is an extremely difficult object for mechanical tests due to its small molecular dimensions, the measurement of the "true" strength of SWNTs in a rope-bundle arrangement is further complicated by the weakness of inter-tubular lateral adhesion. External load is likely to be applied to the outermost tubules in the bundle, and its transfer and distribution across the rope cross-section obscures the interpretation of the data. Low shear moduli in the ropes (1 GPa) indeed has been reported [60].

Recently, a suspended SWNT bundle-rope was exposed to a sideways pull by the AFM tip [73]. It was reported to sustain reversibly many cycles of elastic elongation up to 6%. If this elongation is actually transferred directly to the individual constituent tubules, the corresponding tensile strength then is above 45 GPa. This number is in agreement with that for multiwalled tubes mentioned above [72], although the details of strain distribution can not be revealed in this experiment.

Fig. 5. A: SEM image of two oppositely aligned AFM tips holding a MWCNT which is attached at both ends on the AFM silicon tip surface by electron beam deposition of carbonaceous material. The lower AFM tip in the image is on a soft cantilever whose deflection is used to determine the applied force on the MWCNT. B-D: Large magnification SEM image of the indicated region in (A) and the weld of the MWCNT on the top AFM tip [84]

Fig. 5. A: SEM image of two oppositely aligned AFM tips holding a MWCNT which is attached at both ends on the AFM silicon tip surface by electron beam deposition of carbonaceous material. The lower AFM tip in the image is on a soft cantilever whose deflection is used to determine the applied force on the MWCNT. B-D: Large magnification SEM image of the indicated region in (A) and the weld of the MWCNT on the top AFM tip [84]

A direct tensile, rather than sideways, pull of a multiwall tube or a rope has a clear advantage due to simpler load distribution, and an important step in this direction has been recently reported [84]. In this work tensile-load experiments (Fig. 5) are performed for MWNTs reporting tensile strengths in the range of 11 to 63 GPa with no apparent dependence on the outer shell diameter. The nanotube broke in the outermost layer ("sword in sheath" failure) and the analysis of the stress-strain curves (Fig. 6) indicates a Young's modulus for this layer between 270 and 950 GPa. Moreover, the measured strain at failure can be as high as 12% change in length. These high breaking strain values also agree with the evidence of stability of highly stressed graphene shells in irradiated fullerene onions [5].

In spite of significant progress in experiments on the strength of nano-tubes that have yielded important results, a direct and reliable measurement remains an important challenge for nanotechnology and materials physics.

Fig. 6. A: A schematic explaining the principle of the tensile-loading experiment. B: Plot of stress versus strain curves for individual MWCNTs [84]

4 Theoretical and Computational Models

4.1 Theoretical Results on Elastic Constants of Nanotubes

An early theoretical report based on an empirical Keating force model for a finite, capped (5,5) tube [49] could be used to estimate a Young's modulus about 5 TPa (five times stiffer than iridium). This seemingly high value is likely due to the small length and cross-section of the chosen tube (only 400 atoms and diameter d = 0.7 nm). In a study of structural instabilities of SWNT at large deformations (see Sect. 4.2) the Young's modulus that had to be assigned to the wall was 5 TPa, in order to fit the results of molecular dynamics simulations to the continuum elasticity theory [75,76]. From the point of view of elasticity theory, the definition of the Young's modulus involves the specification of the value of the thickness h of the tube wall. In this sense, the large value of Y obtained in [75,76] is consistent with a value of h = 0.07 nm for the thickness of the graphene plane. It is smaller than the value used in other work [28,42,54] that simply took the value of the graphite interlayer spacing of h = 0.34 nm. All these results agree in the values of inherent stiffness of the graphene layer Yh = C, (2), which is close to the value for graphite, C = Yh = 342 N/m. Further, the effective moduli of a material uniformly distributed within the entire single wall nanotube cross section will be Yt = 4C/d or Yb = 8C/d, that is different for axial tension or bending, thus emphasizing the arbitrariness of a "uniform material" substitution.

The moduli C for a SWNT can be extracted from the second derivative of the ab initio strain energy with respect to the axial strain, d?E/de2. Recent calculations [61] show an average value of 56 eV, and a very small variation between tubes with different radii and chirality, always within the limit of accuracy of the calculation. We therefore can conclude that the effect of curvature and chirality on the elastic properties of the graphene shell is small. Also, the results clearly show that there are no appreciable differences between this elastic constant as obtained for nanotubes and for a single graphene sheet. The ab initio results are also in good agreement with those obtained in [54] using Tersoff-Brenner potentials, around 59eV/atom, with very little dependence on radius and/or chirality.

Tight-binding calculations of the stiffness of SWNTs also demonstrate that the Young modulus depends little on the tube diameter and chirality [28], in agreement with the first principles calculations mentioned above. It is predicted that carbon nanotubes have the highest modulus of all the different types of composite tubes considered: BN, BC3, BC2N, C3N4, CN [29]. Those results for the C and BN nanotubes are reproduced in the left panel of Fig. 7. The Young's modulus approaches the graphite limit for diameters of the order of 1.2 nm. The computed value of C for the wider carbon nanotubes is 430 N/m; that corresponds to 1.26 TPa Young's modulus (with h = 0.34 nm), in rather good agreement with the value of 1.28 TPa reported for multi-wall nanotubes [74]. Although this result is for MWNT, the similarity between SWNT is not surprising as the intra-wall C-C bonds mainly determine the moduli. From these results one can estimate the Young's modulus for two relevant geometries: (i) multiwall tubes, with the normal area calculated using the interlayer spacing h approximately equal to the one of graphite, and (ii) nanorope or bundle configuration of SWNTs, where the tubes form a hexagonal closed packed lattice, with a lattice constant of (d + 0.34 nm). The results for these two cases are presented in the right panel of Fig. 7. The MWNT

■•— C <11, n) —o— C (n,0) —■— BN(n.n) —Q— BN (n,0)

ftfl OS Ifl 1,5 18

Tube Diameter (nm)

MWNT

1

D graphite---.

o o %

o

■ i

Fig. 7. Left panel: Young modulus for armchair and zig-zag C- and BN- nanotubes. The values are given in the proper units of TPa • nm for SWNTs (left axis), and converted to TPa (right axis) by taking a value for the graphene thickness of 0.34 nm. The experimental values for carbon nanotubes are shown on the right-hand-side: (a) 1.28TPa [74]; (b) 1.25TPa [38]; (c) 1 TPa for MWNT [48]. Right panel: Young's modulus versus tube diameter in different arrangements. Open symbols correspond to the multi-wall geometry (10 layer tube), and solid symbols for the SWNT crystalline-rope configuration. In the MWNT geometry the value of the Young's modulus does not depend on the specific number of layers (adapted from [61])

geometry gives a value that is very close to the graphitic one. The rope geometry shows a decrease of the Young's modulus with the increasing tube diameter, simply proportional to the decreasing mass-density. The computed values for MWNT and SWNT ropes are within the range of the reported experimental data, (Sect. 3.1).

Values of the Poisson ratio vary in different model computations within the range 0.15-0.28, around the value 0.19 for graphite. Since these values always enter the energy of the tube in combination with unity (5), the deviations from 0.19 are not, overall, very significant. More important is the value of another modulus, associated with the tube curvature rather than in-plane stretching. Fig. 8 shows the elastic energy of carbon and the newer composite BN and BC3 SWNT. The energy is smaller for the composite than for the carbon tubules. This fact can be related to a small value of the elastic constants in the composite tubes as compared to graphite. From the results of Fig. 8 we clearly see that the strain energy of C, BN and BC3 nanotubes follows the D'/d2 law expected from linear elasticity theory, cf. (5). This dependence is satisfied quite accurately, even for tubes as narrow as (4,4). For carbon armchair tubes the constant in the strain energy equation has a value of D' = 0.08 eVnm2/atom (and up to 0.09 for other chiral tubes) [61]. Previous calculations using Tersoff and Tersoff-Brenner potentials [54] predict the same dependence and give a value of D' ~ 0.06 eVnm2/ atom and D' ~ 0.046 eVnm2/ atom. The latter corresponds to the value D = 0.85 eV in the energy per area as in (5), since the area per atom is 0.0262nm2. We note in

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