Mechanical Definitions Theoretical Predictions for NT Mechanical Properties and Experimental Studies

As noted by Salvetat et al., [92], knowledge of the Young's modulus of a material is the first step toward its use in practical structural applications. For a long, thin rod of arbitrary cross-sectional area Ao, the Young's modulus is defined as the ratio of the axially applied stress (F/Ao) to the fractional axial deformation (8l/lo): E = (Flo)/(8l/Ao). The Young's modulus can also be defined in terms of the bulk modulus K and the shear modulus / of a material: E = 9K//(3K + /). In the simplest terms, the shear modulus is the ratio of the applied stress to the strain of a material in the case where a deformation changes the shape of a body, but not its volume. The opposite case is true for the bulk modulus. It represents the stress/strain ratio for a deformation changing the volume of a body, but not its shape (hydrostatic). Both K and / are defined assuming a linear relationship between the applied stress and strain. The Young's modulus for both SWNTs and MWNTs has been calculated, simulated, and measured in a variety of methods which are described in more detail below. Some groups model nan-otubes as deformed sheets wherein the strain energy due to bending is linearly proportional to a bending modulus k, defined as the second derivative of the surface energy density with respect to curvature. For a tube, k = Ea/(1 — v2), where v is Poisson's ratio and a is the tube wall thickness. Also of interest for the nanomechanics of NTs is the tensile or yield strength (defined as the stress beyond which the strain varies nonlinearly or, alternately, as the maximum stress which may be applied to the material without perturbing its stability [93]).

One of the earliest theoretical studies of the mechanical properties of SWNTs was carried out by Robertson et al. [83]. They investigated the lattice energetic of carbon NTs for all (n, m) nanotubes with radii less than 0.9 nm using two related many-body empirical potentials and local density functional (LDF) theory [94]. The many-body potentials included atomic core-core repulsive terms and attractive terms related to the C valence electrons. It was found that the tensile or compressive strain energy per carbon atom of an SWNT relative to an unstrained tube varied as 1/R2, where R is the tube radius. In addition, it was found that the Young's modulus associated with tube elongation decreased with decreasing radius, that is, smaller nanotubes became softer, depending on the helical (n, m) structure of the nanotube. The former observation agrees well with the expectation of continuum elastic models for the strain energy density of a uniform tube [83, 95]

Ea 24R2

where A is the area of the NT, a is the representative thickness of the NT wall (~0.34 nm), and E is the Young's modulus. Robertson noted that agreement between simulation and continuum elastic theory persisted to radii as low as 0.2 nm. This result was not completely surprising. A tube consisting of a graphene sheet would be expected to maintain a Young's modulus near that of the Cu elastic constant in graphite (corresponding to the elastic modulus parallel to the basal plane), as long as the atomic level deformation induced by the wrapping was low. This is obviously the case for large radii.

In studying the deformation properties of carbon SWNTs, Bernholc and co-workers calculated the Young's modulus and Poisson's ratio via a molecular dynamics framework based on many-body interatomic potentials [96-98]. As with the earlier work of Robertson et al. [83], a high degree of correspondence between the modeled mechanical response and results of classical mechanical theory was found. Their MD simulations predicted a modulus of 5.5 TPa and a Poisson's ratio of 0.19 [96]. The latter is fully 550% larger than the Cu elastic constant (basal plane modulus) of graphite, while the Poisson's ratio is significantly lower than the value of 0.33-0.34 of graphite [50]. At this point, a comment is required regarding the calculation of E for atomi-cally thin tubular shells. From the perspective of continuum elastic energetics, the Young's modulus is also defined as the second derivative of the system energy with respect to strain at equilibrium normalized to system volume. This is the practical definition used for virtually all simulations or analytical models as the energy functional is directly accessible. However, it opens the question of the actual wall thickness of an SWNT. At molecular-length scales, it may not be reasonable to characterize the thickness of an SWNT with the lattice spacing of MWNTs or the c-axis spacing of graphite. Consequently, the use of an elastic modulus such as E, which is, by definition, normalized to a classically defined cross-sectional area, may have little meaning in that a well-defined value for the NT cross-sectional area does not exist. More recently, several groups working on this problem from a simulation or theoretical perspective have stated results only in terms of the product aE to avoid the ambiguity inherent in applying a classical "wall thickness" to a structure of single-atom thickness [99-100]. For the MD simulations previously described, a wall thickness of 0.066 nm was used. When replaced with the value of 0.34 nm (an upper estimate for the wall thickness in MWNTs) and akin to the c-axis spacing in graphite, a value for E of 1.07 TPa is obtained. Suffice it to say that, when compared in terms of aE, the MD work presented here continues to agree with more recent theoretical (and, as will be shown, experimental) results.

An empirical force-constant model, wherein the atomic interactions near an SWNT equilibrium structure are approximated by the sum of pairwise harmonic potentials, was used by Lu to calculate elastic properties of SWNTs and nanoropes [101]. Values of E = 0.97 TPa were obtained using an SWNT wall thickness of 0.34 nm, and were virtually constant as a function of SWNT helicity and radius. A value for Poisson's ratio of 0.28 was determined that possessed the same insensitivity to (n, m). The Young's moduli of crystalline nanoropes of SWNTs were calculated, and ranged from 0.80 to 0.43 TPa as the number tubes per rope varied from 5 to 15, respectively.

More recently, a broad range of computational and analytical approaches has been applied to the accurate calculation of Young's modulus in CNTs. Yao and Lordi took a finite temperature, molecular dynamics (MD) simulation approach for the calculation of E for SWNTs [102]. Using the universal force field of Rappe and co-workers [17], simulations were carried out on SWNTs 10 nm in length, ranging in diameter from approximately 0.2 nm for a (5, 5) tube to 1.4 nm for a (20, 20) tube. For relatively large (20, 20) tubes, the Young's modulus determined from the simulations was approximately 0.97 TPa. This is quite close to the value of Cu measured in graphite (~1 TPa [104]). However, as the SWNT radius was decreased, the MD simulations predicted a 15% increase in E to 1.12 TPa for the (5, 5) SWNT. The increase was attributed in large part to corresponding increases in torsional strain that develop as the C sp2 bonds are deformed at a smaller tube radius. The variation of strain energy with NT size had been previously studied [105106], but the ability for the MD simulation to potentially predict this behavior more accurately was attributed directly to the inclusion of a torsional term in the bond energy calculation, which turns out to be most sensitive to the variation of E with decreasing radius. Yao and Lordi also simulated thermal vibrations of SWNTs clamped at one end, similar to TEM-based techniques, to experimentally measure E (see below). Their simulations, ranging from temperatures of 300 to 2000 K, yielded temperature-independent vibra-tional mode frequencies that matched well with continuum elastic model predictions.

Similar studies were carried out by Goddard and co-workers using molecular dynamics and molecular mechanics techniques, but for (n, n), (n, 0), and (2n, n) SWNTs [107]. They obtained values for the bending modulus k of 0.96 TPa for (n, n) tubes, 0.91 TPa for (n, 0) tubes, and 0.94 TPa for (2n, n) tubes. They also calculated Young's moduli for SWNT bundles or ropes. Within the bundle, the SWNT axes define a triangular lattice. The lattice spacing varied from 1.68 nm for (n, n) tube bundles to 1.65 nm for (n, 0) and (2n, n) tube bundles. Strain energies for these ropes were calculated, and Young's moduli of 640, 648, and 673 GPa for the (n, n), (n, 0), and (2n, n) ropes, respectively, were obtained. The comparison between the bending moduli calculated in this work and Young's moduli of other techniques is complicated by a dimensional inconsistency. Although the strain energies are defined conventionally, Goddard and co-workers report the bending moduli in units of gigapascals, and do not define the normalization of the wall thickness required for conventional definitions of k [107].

Rubio and co-workers have carried out predictive calculations for E of SWNTs using tight-binding (TB) and pseudopotential-based density function theory methodologies [99]. To avoid the wall thickness problem, the Young's modulus, alternately defined in elastic systems as the volume-normalized second derivative of the system energy with respect to strain (at equilibrium), is instead normalized to the SWNT surface area. Their TB model followed that of Porezag et al. [108], containing band-energy terms and a repulsive pair potential. As with similar techniques used for band calculations of conventional materials [109], the successful application of this technique depends upon a generalized eigenvalue equation for a nonorthogonal basis set. As with previous continuum elastic and calcula-tional approaches, an inverse quadratic dependence of strain energy on tube diameter was found [83]. Assuming a carbon SWNT wall thickness of 0.34 nm, their corresponding results for E ranged from 1.26 TPa for a (20, 0) carbon SWNT to 1.22 TPa for a (6, 6) SWNT. In addition, the authors calculated the Poisson's ratio, defined here as

The Poisson's ratio varied from 0.27 for a (20, 0) tube to 0.25 for a (6, 6) tube. Considering the sidewall thickness used for the determination of E, these results agreed reasonably well with other approaches, and confirmed that, in certain respects, the elastic behavior of carbon SWNTs should mirror that of in-plane graphite. An important aspect of this work is the prediction that E will decrease rapidly as the tube radius decreases. This tends to agree with earlier comments made by Robertson et al. [83], but contradicts the intuitively appealing result of other groups that E should increase with smaller radius tubes due to the increased energy associated with the rolling strain [102]. Despite the significant theoretical and modeling efforts ongoing for nanotube structures, this issue seems not to be resolved.

Mechanical strain energies and Young's modulus for straight and bent SWNTs were also calculated by Xin et al.

[110]. In contrast to prior investigations that used continuum elastic theory, molecular dynamics simulations, or empirical many-body potentials, a band-theory approach was taken here [111-112]. It was noted from a straightforward calculation that the strain energy of straight SWNTs originates from the curvature-induced electronic energy change. Consequently, determination of the electronic energy of all of the occupied bands was carried out to provide predictions for SWNT mechanical properties. The band-structure calculation utilized a nearest neighbor tight-binding approach

[111]. Such approaches have been widely used to calculate graphene and SWNT electrical properties. Strain energies were calculated for a tube of cross-sectional radius R subjected to a bending deformation (bending radius = p). The R dependence of the bending energy Eb related to the strain energy Es of a straight tube determined from the tight-binding model agreed well with continuum elastic theory predictions [113]

From the tight-binding calculation carried out by Xin and co-workers, A = 0.7 eV/atom. This is a further confirmation that continuum elastic theory provides a valid framework within which to predict nanotube mechanical behavior.

In this same work, however, a Young's modulus of approximately 5 TPa is predicted. This magnitude exceeds other estimates by a factor of 5. However, it should be noted that the SWNT wall thickness calculated by Xin et al. is approximately 0.07 nm, and differs significantly from the conventional value of a = 0.34 nm used for the calculation of E by most other authors (noted previously). In terms of the product aE, the results from this tight-binding model match well with other simulations.

Theoretical calculations of SWNT mechanical properties have also been carried out using local density approximation cluster models. As applied by Zhou et al., a linear combination of atomic orbitals is used to solve the Schroedinger equation directly, applying appropriate boundary conditions to remedy the problem of dangling bonds present at the edges of the cluster [93]. Such an approach makes fewer initial restrictions on the form of the electron wave function as do tight-binding approaches. A 156 atom NT was used for purposes of calculation [114-115]. A value of Young's modulus of 0.76 TPa was calculated, somewhat lower than previously discussed values (assuming a wall thickness of 0.34 nm). A value of Poisson's ratio of 0.32 (slightly lower than that of graphite, ~0.33-0.34) was also determined from these calculations. It should be noted that, unlike molecular dynamics approaches, the density-of-states methodology used here also provides specific information regarding carbon-carbon bonding states. It was found that, upon rolling the graphene sheet into an (n, n) conformation, the overlap integral of a bonds in the graphene plane decreased by approximately 7%. In contrast, the overlap integral of w bonds increased by about 2.4%, and the total overlap integral decreased by approximately 4%. The decrease of the overall overlap integral leads to a lower net bonding energy compared to graphite. In addition, the reduced bond overlap will broaden the effective interatomic electrostatic potential well, and reduce the Young's modulus compared with that of basal plane graphite (~1 TPa). It is interesting to note that, in this work, the curvature effect tends to weaken the a bonding, but strengthens the w bonding, which may have implications with respect to the electronic functionality of NTs.

Fewer theoretical and simulation efforts have been carried out with respect to the elastic moduli of MWNTs. Using the empirical force constant model referenced earlier, Lu predicted an increase of E with the number of walls from a value of 0.97 TPa for a single (5, 5) SWNT to a value of 1.11 TPa for an MWNT with seven nested tubes about a (5, 5) core [101]. A Poisson's ratio value of 0.27 was insensitive to the number of MWNT walls. For these studies, as most others, a 0.34 nm spacing is assumed. The critical aspect of MWNT modeling is proper inclusion of the electrostatic energies associated with the adjacent graphitic walls. No dramatic effects have been predicted with respect to linear response to elastic strain. However, several groups point to significant effects of wall-wall coupling in nonlinear (inelastic) deformation, and they will be discussed next.

Experimental determination of the Young's modulus for SWNTs and MWNTs has utilized a wide array of approaches. The groundbreaking work was carried out on MWNTs by Treacy et al., resulting from an observation that the free ends of cantilevered SWNTs appeared fuzzy in a TEM [116]. Subsequent thermal studies revealed that poor TEM focus resulted from thermal vibrations. Using classical mechanical beam theory, the vibration frequency spectrum wn of a uniform clamped tube [116, 95]

was used to estimate the E experimentally, assuming a ther-modynamic equipartition concerning the SWNT vibrational mode energies. Above, Rt and Ro represent the inner and outer SWNT tube radii, respectively, p is the in-plane mass density, L is the tube length, and pn is a set of numerical coefficients. Fitting the observed thermal dependence of the vibration amplitude to the above expression, Emeasured varied from 0.4 to 3.7 TPa for wall numbers ranging from approximately 5 to >25. Although the uncertainties in the data were quite large and dominated by experimental errors, there was an increase in E attributed to thinner tubes. These studies were extended and refined by Krishnan et al. with respect to SWNTs [117]. Using the same basic experimental techniques and stochastically driven oscillator theory for the SWNTs, a refined error analysis was presented, and an average Young's modulus of 1.3-0.4/+0.6 TPa was determined. Similar to the work on MWNTs, measured values of E for SWNTs ranged from ~0.3 to 2.7 TPa. The origin of experimental errors stems from the accurate determination of the thermal vibration amplitude to the accurate calculation of the free length of individual NTs. Regardless, these studies represent a breakthrough for nanomechanical measurement. Note that the clamped beam models used to calculate E from experimental data require similar assumptions regarding the NT wall thickness as did the theoretical efforts [95]. All experimental work summarized here typically used a value of 0.34 nm SWNT wall thickness (or MWNT wall spacing). Hence, their elasticity data should be compared with correspondingly normalized values of E from theoretical calculations.

More direct measurements of SWNT mechanical response were undertaken by Salvetat et al. that exploited the surface adsorption of SWNT ropes to nanoporous templates to form suspended SWNT rope "beams" clamped at both ends by surface van der Waal forces [118]. The ropes were modeled as a continuous cylindrical object characterized by an elastic modulus Er (Young's modulus renormalized by bending) and a shear modulus The elastic modulus decreased from a maximum of 1.2 TPa (for a 3 nm diameter rope) to values as low as 0.07 TPa for a 20 nm diameter SWNT rope. The measured shear modulus acted in many respects similarly, ranging from 6.5 to 0.7 GPa for ropes of diameter 4.5 and 20 nm, respectively. Although the error values for these elastic constant measurements are reported at ±50%, the estimation of Er for the thinnest ropes is consistent with that of individual NTs. In this work, the shearing was assumed to occur between SWNTs only, that is, shearing of individual SWNTs was ignored. An inherent difficulty in work of this sort is the possibility of defects or breaks of single SWNTs in the ropes. Such effects are cited as a possible source for error, and are primarily responsible for the low value of / .

In contrast to the beam-bending approach utilized above, Ruoff and co-workers conducted a series of impressive experiments wherein SWNT ropes were literally welded to a cantilevered probe tip inside an SEM, and carried out direct tensile testing driven by a piezoelectric deflector [119]. In these experiments, the base of the nanorope undergoing testing was entangled within a nanorope mat produced via the standard arc-discharge method. By knowledge of the force constant of the cantilevered probe tip, the Young's modulus of multiple nanoropes (assumed to be comprised of (10, 10) SWNTs) was directly measured, and varied from 0.32 to 1.47 TPa, with a mean of 1.0 TPa. These data are in excellent agreement with other experimental measurements and a larger body of theoretical work.

Since the increased size of multiwalled carbon nanotubes and corresponding MWNT ropes increases experimental accessibility, a wide range of mechanical measurements have been performed on these systems. Pan et al. pulled individual MWNT ropes as long as 2 mm to carry out direct tensile strength measurements using conventional test kits [120]. The linear portion of the stress-strain curve thus generated was used to directly measure E. The average value thus determined was 0.45 ± 0.23 TPa. Akita and co-workers used a slightly more elegant approach by attaching an MWNT segment directly to a micromachined cantilever while using a second cantilever applied to the tube to carry out compressive strain testing. The MWNT was considered as a classic beam, and the measured buckling force was assumed described by Euler beam theory (-fbuckling ~ t2EI/2L2; L = length, I = MWNT inertial moment) to extract a Young's modulus of 0.46 TPa (no errors were quoted). Using an experimental approach similar to Treacy and co-workers to extract elastic moduli from nanotubes, Gao considered MWNTs grown pyrolytically [121]. Similar thermal vibrations were observed via TEM from individual MWNTs, and the conventional analogy between the MWNT and a vibrating uniform beam clamped at one end was used to infer the modulus from the thermally stimulated vibration amplitudes [95]. In this work, the authors erroneously label the modulus used in such a model as a bending modulus. For the classical clamped beam problem, the modulus is defined as Young's modulus [4, 95]. Their results suggest an average value of E for their tubes of 0.03 TPa. This value is dramatically lower than prior experiments. The authors note that the expected high density of structural defects in pyrolyti-cally grown MWNTs is the most likely cause. For MWNTs possessing so-called volume defects (areas of the tube with reduced radii), the effective value of E was as low as 0.002 TPa. It is clear that structural defects in NT structures can have dramatic effects on the mechanical properties.

The previous discussions have centered on nanotubes made from carbon. However, there is no physical principle limiting the composition of such structures to carbon. Vaccarini et al. have nicely summarized theoretical and some experimental results for nanotube compositions ranging from the standard carbon to BN, BN3, BCN2, GaSe, WS2, and MoSe. Hernandez and co-workers carried out theoretical investigations of BN, BC3, BCN2, and C3N4 SWNTs as well [88]. From a physical point of view, all compounds exhibiting layered structures in the bulk phase are likely to form nanotubes, with the caveat that the "layered" structure possesses dramatically anisotropic bonding as is found in graphite. For example, BN carbon nanotubes possess the same honeycomb sheet structure of carbon NTs; however, B and N atoms populate alternating sites. The same (n, n) notation can be used to describe the tube structure, although the "unit hexagon" is composed of equal B and N atoms. Utilizing TB models (see above), the BN SWNT Young's modulus varied from 0.75 to 0.92 TPa for tube radii varying from 0.5 to 2.5 nm, respectively. The increase of modulus with increasing tube radius mirrored theoretical results for corresponding carbon SWNTs using the same computational approach. The same authors note that the Poisson's ratio was similar for all NT compositions studied [88]. And once again, the results for BN nanotubes compared well with simple continuum elastic approaches.

Molecular dynamics and ab initio total energy methods were applied to BN nanotube modeling by Srivastava et al. [122]. A value E = 1.2 TPa was determined. Contrary to carbon nanotubes, however, the bond strain inherent in confinement of B-N sheets in tube form tends to expel the N atoms and subduct B atoms, providing an atomic-scale corrugation at the tube surface. Hernandez and co-workers obtained lower values for BN SWNT Young's moduli generated from TB models [99]. 0.84 TPa < E < 0.91 TPa were determined for tube geometries ranging from (6, 6) to (20, 0). No significant dependence on size or chirality was found. Typically, other composition SWNTs displayed Young's moduli in similar ranges, with the exception of C3N4 tubes where E dropped to ~0.6 TPa. In all cases, the highest values of E from simulations corresponded to carbon tubes. From an experimental point of view, measurements of BN nanotube elastic moduli were carried out in precise analogy to those for carbon tubes. Using the thermal vibration amplitude methodology, Chopra and Zettl measured the BN MWNT Young's modulus to be 1.22 ± 0.24 TPa [123]. This compares well with the theoretical result of Srivastava et al. [122]; however, it exceeds estimates from other simulations [93]. Still, the body of experimental and theoretical work on noncarbon nanotubes makes clear the ubiquity of extremely high-modulus nanotube structures provided a high degree of structural perfection exists.

From the viewpoint of elastic properties, an impressive degree of agreement between experimental and theoretical studies of carbon (and other) SWNTs and MWNTs is evident. The question arises as to the promise of nanotechnology. Do the high values of E constitute an emergent nanomechanical property? Although the point may be argued, it is clear that easily synthesized arrays of these nanostructures with near-perfect elastic properties can be produced. The fact that the measurements confirm that these materials possess superior properties compared to larger (microscale) carbon fibers justifies the classification of NTs as nanostructured materials with emergent properties.

The remainder of this section will provide a brief overview of investigations of inelastic properties of nanotubes, namely, tensile strength (defined experimentally as the maximum force sustainable during a stress-strain test) and strain-induced defect generation. Utilizing the tensile strain test kit described earlier, Pan et al. measured the tensile strength of MWNT ropes to be 1.33 GPa [120]. Early theoretical studies [124] suggested that the tensile strength of carbon NTs should be similar to carbon whiskers (~20 GPa). Further refinement of the experimental studies by the same group, including a Weibull analysis of a collection of MWNT ropes, yielded an average tensile strength value of 3.6 GPa [125]. More recently, Yu and co-workers undertook highresolution tensile strength studies of carbon SWNT ropes, and measured breaking strengths that ranged from 13 to 52 GPa [126]. The experimental mean value was ~30 GPa. The wide variance of the breaking strength values are attributed to various experimental and SWNT rope geometric factors and, most importantly, the lack of information regarding the continuity of individual SWNTs within the rope. However, their results clearly establish that a 5% strain is reproducibly achievable in carbon NT systems.

This value of this so-called breaking strain has been verified by a number of experimental studies as representative of a strain necessary for crystalline defect generation in the graphene or graphitic sheets of SWNTs or MWNTs, respectively. Before reviewing these results, it is important to appreciate the C-C bond orientation as a function of NT helicity. For the so-called armchair (n, n) NT geometry, C-C bonds are either normal to the axis of the tube or rotated by angles of ±30° to that axis. Zigzag (n, 0) tubes possess C-C bonds either parallel to the tube axis or rotated by angle of ±60° to that axis. The normally oriented bonds of the (n, n) tubes present the greatest moment to axial strain, compressive or tensile. Consequently, inelastic deformation would be expected to vary significantly on the tube helicity. Although the currently available experimental data present only anecdotal corroboration of this observation, theoretical and simulation-based studies confirm, in detail, this effect.

Experimental studies have also demonstrated the incredible flexibility of NTs against nonrecoverable deformation. Falvo et al. reported AFM studies of MWNTs wherein tubes were subjected to large buckling strains by the AFM tip. Kink angles varied from 90 to ~180° [127]. Of particular interest were two phenomena. First, topographic rippling was observed along the tube axis, as is expected from continuum shell theory [128, 129]. Away from the kinks, the rippling was somewhat recoverable; however, topographic variations at the kink positions appeared permanent. In no case was tube breaking observed, a testament to the flexibility and strength of NTs, even in a highly deformed state. In fact, topographical imaging studies have been carried out on completely collapsed MWNTs [126] or nanotube ribbons that possessed remarkable flexibility. It should be noted that an isolated ribbon or collapsed NT is energetically less stable than its tubular analog [107]. The aforementioned experimental analysis explicitly calculated the stabilizing effect of the attractive NT/substrate interaction to demonstrate its crucial role in observing such deformed NT structures.

Nanotube kinking was observed via TEM studies soon after their discovery [84, 130]. Continuum shell analysis was typically invoked to describe these deformations, although no formal justification for the applicability of those theories was provided, with the exception of the 1/R2 dependence of the "elastic" nanotube strain energy compared to that of a single graphene sheet. Groundbreaking work regarding a theoretical basis for nanotube deformation was carried out by Bernolc and co-workers through MD simulations and a continuum shell model [96, 130]. The authors considered inelastic deformation formation in carbon SWNTs under compression, bending, and torsion. With respect to bending strain, simulations for a 1.2 nm diameter tube predicted kink formation at a bend angle of ~30°. The critical curvature Cc associated with tube kinking is described by an empirical relation fit to the simulation results

9.89

where d is the tube diameter and h characterizes the helicity ranging from 0 [zigzag or (n, 0)] to n/6 [armchair or (n, n)]. The formation of double kinks was observed and, interestingly, simulations implied that the bending deformations were completely recoverable up to bending angles of 110°. Deformation of SWNTs under compressive strain was also predicted by their simulations. For strain e > 5%, a characteristic double-fin flattening occurred. At higher strains, a three-fin flattening occurred before broad buckling deformations set in. MD simulations of excessive torsional stress resulted in the flattening and twisting of the nanotube into a ribbon-like structure [91]. Utilizing a continuum shell model, the authors considered inelastic deformation perturbations in terms of the stability of axial and azimuthal ripples. A stability analysis against such rippling yielded critical strains ec = (0.077 nm)d-1, which agree well with the MD results noted previously.

MD simulations were further used to investigate strain release in SWNTs through defect generation. Bernholc and co-workers [131] investigated the static and dynamical properties of such tubes under uniaxial tension. At high temperatures (~1800 K), (5, 5) tubes under a 10% axial strain spontaneously generated crystal defects via C-C bond rotation. This defect, referred to as a Stone-Wales transformation, rotates the C-C bond normal to the tube axis, generating a double pentagon-heptagon defect (so-called 5-7-7-5 defect). This is pictured in Figure 17. Using classical potential calculations, the formation energy of 5-7-7-5 defect formation was calculated, and shown to be stable for various (n, n) SWNTs at strains between 5 and 6%. If it is assumed that such defect generation is a precursor to mechanical yield, these calculations support the aforementioned experimental studies. From the onset of the 5-7-7-5 defects, simulations predict unbinding of the defect pairs, and subsequent strain-induced motion along appropriate armchair glide planes. Higher order defect generation (e.g., 5-7-58-5) were also predicted prior to tube "breaking" which, in simulations, proceeded by unraveling of carbon strips from the edge of the tube until final separation occurred. The motion of defects thus implies a certain degree of ductility for SWNTs.

The propensity of (n, n) SWNTs to generate C-C bond rotation defects has important implications regarding the relative ductile behavior of nanotubes as a function of helic-ity [132]. In an extension of the aforementioned simulation work, Nardelli examined defect generation in (n, 0) and (n, m) tubes. As expected, the orientation of C-C bonds in (n, 0) tubes dramatically altered the stability of C-C bond rotation defects under applied strains [132]. In fact, the lack of defect generation and subsequent defect motion in these NTs dramatically reduced their ductile properties. Consequently, a theoretical "ductility phase diagram" was developed, and is adapted in Figure 18 to describe the (n, m) domains where brittle versus ductile behavior is expected. Experimentally, confirmation of this phase diagram is challenging due to the uncertainty of temperature effects and

Figure 17. Schematic of a Stone-Wales transformation occuring in an (n, n) nanotube under axial tensile stress. For sufficient stress, the defects can separate and alter the local chirality of the tube.

Figure 17. Schematic of a Stone-Wales transformation occuring in an (n, n) nanotube under axial tensile stress. For sufficient stress, the defects can separate and alter the local chirality of the tube.

Figure 18. Ductility phase diagram for carbon nanotubes adapted from [132]. Brittle behavior is expected as zigzag geometries are approached due to the unfavorable energetics of defect generation through C-C bond rotation.

activation energies for the bond defects. However, it provides an important roadmap for investigation.

In an attempt to quantify the appearance of such defects in SWNTs under applied strain, Zhao et al. investigated the relevant activation energies using large-scale quantum calculations [133]. The generation of 5-7-7-5 defects at large applied strains was investigated for both (n, n) and (n, 0) tubes—specifically, the activation barriers for defect generation computed by constrained relaxation along the simplest kinetic pathway leading to defect formation. The resulting activation energies were prohibitively large, and ranged from 8.6 eV for a (5, 5) SWNT to 9.9 eV for a (9, 0) SWNT at zero strain. Even at strains as large as 10% (far beyond the breaking strain observed in laboratory experiments), the 5-7-7-5 activation energies exceeded 4 eV, implying a van-ishingly small defect generation density at or near room temperature. Consequently, it was concluded that such bond rotation in strained NTs (near room temperature) is unlikely due to the energetics of the problem. The authors note that prior agreement between estimates of the 5% strain required for 5-7-7-5 defect stability and the experimentally observed breaking strain is most likely fortuitous, and note that, for some tubes, the breaking strain exceeds 5%. A more likely scenario describing the mechanical failure of actual SWNTs and MWNTs is the nucleation of yield or failure at frozen-in defects resulting from the growth process. As the field of nanomechanics of nanotubes progresses, it will be critical to further address the roles of defects and defect generation with respect to the inelastic processes.

The validity of continuum elastic theory to treat NTs was specifically considered by Govindjee and Sackman [100]. Their primary result rested on conventional assumptions made by a number of experimental and theoretical works concerning the calculation of the moment of inertia for a tube. Through a rigorous analysis of the error associated with incorrect geometrical assumptions regarding INT, it was shown that errors in E exceeding 200% were possible. The authors suggested a geometrical correction to avoid such errors. With the exception of the geometrical subtleties in correctly calculating inertial moments, the physical validity of linear (elastic) response to describe small deformations in NTs was not challenged. Harik also investigated the applicability of classical "bending beam mechanics" to NTs based on a length scale analysis [134]. Basically, NTs were separated into two classes. For small diameter/length ratios and

Figure 18. Ductility phase diagram for carbon nanotubes adapted from [132]. Brittle behavior is expected as zigzag geometries are approached due to the unfavorable energetics of defect generation through C-C bond rotation.

for NT radii large compared to the hexagonal atomic lattice spacing, it was appropriate to apply beam-like shell theory. For long NTs with radii on the order of the atomic lattice spacing, a solid beam approach was warranted. A scaling analysis of buckling was presented to determine the dependence of critical buckling strain with the NT aspect ratio as a function of the NT diameter.

Due to the challenges in the scanning force microscopy of high aspect-ratio structures, very little direct nanomo-mechanical mechanical imaging has been carried out on nanotubes. Very recently, Ajayan and Geer have undertaken UFM imaging of chemical-vapor-deposited MWNTs [84] to investigate the axial variation of MWNT rigidity associated with volume defects (axial variation of tube radius). Initial results of this work are shown in Figure 19. The MWNT has a nominal diameter of 63 nm, determined by AFM height measurements. The apparent 200 nm width results from tip convolution. The axial variation of the UFM response is axi-ally symmetric over the central uniform region of the tube. The UFM contrast lines parallel to the tube axis are due to variations in the tip-tube contact area during the scan. Near the edges of the scan, however, there are significant variations of the UFM response associated with the radius variations of the volume defects. These variations, in contrast, are associated with modulation of the mechanical response due to the variations in tube radius. It is clear, as noted by Gao et al. [121], that such volume defects could significantly alter the mechanical response (effective Young's modulus) of the tube. Investigations into quantitative modulus extraction from such measurements are continuing.

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