S

Diameter (A)

Fig. 8. Ab initio results for the total strain energy per atom as a function of the tubule diameter, d, for C- (solid circle), BC3- (solid triangle) and BN-(open circle) tubules. The data points are fitted to the classical elastic function 1/d2. The inset shows in a log plot more clearly the 1/d2 dependence of the strain energy for all these tubes. We note that the elasticity picture holds down to sub-nanometer scale. The three calculations for BC3 tubes correspond to the (3, 0), (2, 2) and (4, 0) tubes (adapted from [7,46,47,56])

Fig. 8 that the armchair (n, n) tubes are energetically more stable as compared to other chiralities with the same radius. This difference is, however, very small and decreases as the tube diameter increases. This is expected, since in the limit of large radii the same graphene value is attained, regardless of chirality. It is to some extent surprising that the predictions from elasticity theory are so similar to those of the detailed ab initio calculations. In [1] a complementary explanation based on microscopic arguments is provided. In a very simplified model the energetics of many different fullerene structures depend on a single structural parameter: the planarity 4>n, which is the angle formed by the ^-orbitals of neighbor atoms. Assuming that the change in total energy is mainly due to the change in the nearest neighbor hopping interaction between these orbitals, and that this change is proportional to cos(^n), the d-2 behavior is obtained. By using non-self-consistent first-principles calculations they have obtained a value of D' = 0.085 eVnm2/ atom, similar to the self-consistent value given above.

4.2 Nonlinear Elastic Deformations and Shell Model

Calculations of the elastic properties of carbon nanotubes confirm that they are extremely rigid in the axial direction (high tensile) and more readily dis

tort in the perpendicular direction (radial deformations), due to their high aspect ratio. The detailed studies, stimulated first by experimental reports of visible kinks in the molecules, lead us to conclude that, in spite of their molecular size, nanotubes obey very well the laws of continuum shell theory [2,39,70].

One of the outstanding features of fullerenes is their hollow structure, built of atoms densely packed along a closed surface that defines the overall shape. This also manifests itself in dynamic properties of molecules, which greatly resemble 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, who discovered the elastic instability. A rod subject to longitudinal compression remains straight but shortens by some fraction e, proportional to the force, until a critical value (Euler force) is reached. It then becomes unstable and buckles sideways at e > ecr, while the force almost does not vary. For hollow tubules there is also a possibility of local buckling in addition to buckling as a whole. Therefore, more than one bifurcation can be observed, thus causing an overall nonlinear response of nanotubes to the large deforming forces (note that local mechanics of the constituent shells may well still remain within the elastic domain).

In application to fullerenes, the theory of shells now serves a useful guide [16,25,63,75,76,78], but its relevance for a covalent-bonded system of only a few atoms in diameter was far from being obvious. MD simulations seem better suited for objects that small. Perhaps the first MD-type simulation indicating the macroscopic scaling of the tubular motion emerged in the study of nonlinear resonance [65]. Soon results of detailed MD simulations for a nanotube under axial compression allowed one to introduce concepts of elasticity of shells and to adapt them to nanotubes [75,76]. MD results for other modes of load have also been compared with those suggested by the continuum model and, even more importantly, with experimental evidence [34] (see Fig. 4 in Sect. 3.2).

Figure 9 shows a simulated nanotube exposed to axial compression. The atomic interaction was modeled by the Tersoff-Brenner potential, which reproduces the lattice constants, binding energies, and the elastic constants of graphite and diamond. The end atoms were shifted along the axis by small steps and the whole tube was relaxed by the conjugate-gradient method while keeping the ends constrained. At small strains the total energy (Fig. 9a) grows as E( e) = (t))E" -e2, where E" = 59 eV/atom. The presence of four singularities at higher strains was quite a striking feature, and the patterns (b)-(e) illustrate the corresponding morphological changes. The shading indicates 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 Ci. This evolution of the molecular structure can be described within the framework of continuum elasticity.

Fig. 9. Simulation of a (7, 7) nanotube exposed to axial compression, L = 6nm. The strain energy (a) displays four singularities corresponding to shape changes. At ec = 0.05 the cylinder buckles into the pattern (b), displaying two identical flattenings, "fins", perpendicular to each other. Further increase of e enhances this pattern gradually until at e2 = 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 squashed asymmetric configuration forms (e) (from [75])

Fig. 9. Simulation of a (7, 7) nanotube exposed to axial compression, L = 6nm. The strain energy (a) displays four singularities corresponding to shape changes. At ec = 0.05 the cylinder buckles into the pattern (b), displaying two identical flattenings, "fins", perpendicular to each other. Further increase of e enhances this pattern gradually until at e2 = 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 squashed asymmetric configuration forms (e) (from [75])

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

E = \jj ' r> K< + ««)* ~ 2(! " ")(«*«« " </)] (5) C

+ (1 - 2(1 - v){ex£y - 4y)}}dS, where k is the curvature variation, e is the in-plane strain, and x and y are local coordinates). In order to adapt this formalism to a graphitic tubule, the values of D and C are identified by comparison with the detailed ab initio and semi-empirical studies of nanotube energetics at small strains [1,54]. Indeed, the second derivative of total energy with respect to axial strain corresponds to the in-plane rigidity C (cf. Sect. 3.1). Similarly, the strain energy as a function of tube diameter d corresponds to 2D/d? in (5). Using the data of [54], one obtains C = 59 eV/atom = 360 J/m2, and D = 0.88 eV. The Poisson ratio v = 0.19 was extracted from a reduction of the diameter of a tube stretched in simulations. A similar value is obtained from experimental elastic constants of single crystal graphite [36]. One can make a further step towards a more tangible picture of a tube as having wall thickness h and Young's modulus Ys. Using the standard relations D = Yh3/12(1 — v2) and C = Ysh, one finds Ys = 5.5 TPa and h = 0.067 nm. With these parameters, linear stability analysis [39,70] allows one to assess the nanotube behavior under strain.

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 N half-waves along the tube (Fig. 10, inset), i.e. sines and cosines of arguments 2My/d and Nnx/L. At a critical level of the imposed strain, ec(M, N), the energy variation (4.1) vanishes for this shape 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 Fig. 10. 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

Fig. 10. The critical strain levels for a continuous, 1nm 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 [75])

Fig. 10. The critical strain levels for a continuous, 1nm 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 [75])

buckles sideways as a whole; the critical strain is close to that for a simple rod,

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 Fig. 10), with a few separated flattenings in directions perpendicular to each other, while the axis remains straight. For such a local buckling, in contrast to (6), the critical strain depends little on length and estimates to ec = 4y/D/C dr1 = (2/v/3)(l - v2)-1!2 hdr1 in the so-called Lorenz limit. For a nanotube one finds, ec = 0.077 nm/d. (7)

Specifically, for the 1 nm wide tube of length L = 6 nm, the lowest critical strains occur for the M = 2 and N = 2 or 3 (Fig. 10), and are close to the value obtained in MD simulations, (Fig. 9a). This is in accord with the two- and three-fin patterns seen in Figs.9b,c. Higher singularities cannot be quantified by the linear analysis, but they look like a sideways beam buckling, which at this stage becomes a non-uniform object.

Axially compressed tubes of greater length and/or tubes simulated with hinged ends (equivalent to a doubled length) first buckle sideways as a whole at a strain consistent with (6). After that the compression at the ends results in bending and a local buckling inward. This illustrates the importance of the "beam-bending" mode, the softest for a long molecule and most likely to attain significant amplitudes due to either thermal vibrations or environmental forces. In simulations of bending, a torque rather than force is applied at the ends and the bending angle 0 increases stepwise. While a notch in the energy plot can be mistaken for numerical noise, its derivative dE/d0 drops significantly, which unambiguously shows an increase in tube compliance — a signature of a buckling event. In bending, only one side of a tube is compressed and thus can buckle. Assuming that it buckles when its local strain, e = K ■ (d/2), where K is the local curvature, is close to that in axial compression, (7), we estimate the critical curvature as

This is in excellent agreement (within 4%) with extensive simulations of single wall tubes of various diameters, helicities and lengths [34]. Due to the end effects, the average curvature is less than the local one and the simulated segment buckles somewhat earlier than at 0c = KcL, which is accurate for longer tubes.

In simulations of torsion, the increase of azimuthal angle $ between the tube ends results in energy and morphology changes shown in Fig. 3. In the continuum model, the analysis based on (5) is similar to that outlined above, except that it involves skew harmonics of arguments like Nnx/L ± 2My/d. For overall beam-buckling (M =1),

The latter should occur first for L < 136 d5/2 nm, which is true for all tubes we simulated. However, in simulations it occurs later than predicted by (10). The ends, kept circular in simulation, which is physically justifiable, by a presence of rigid caps on normally closed ends of a molecule, deter the through flattening necessary for the helix to form (unlike the local flattening in the case of an axial load).

In the above discussion, the specific values of the parameters C and D (or Y and h) are chosen to achieve the best correspondence between the elastic-shell and the MD simulation within the same study, performed with the Tersoff-Brenner potential. Independent studies of nanotube dynamics under compression generally agree very well with the above description, although they reveal reasonable deviations in the parameter values [16,25]. More accurate and realistic values can be derived from the TB or the ab initio calculations [1,7,57] of the elastic shell, and can be summarized in the somewhat "softer but thicker" shell [76]. Based on a most recent study [28] one obtains effective shell parameters C = 415 J/m2 and D = 1.6 eV = 2.6 x 10~19 J, that is correspondingly Ys = 4.6 TPa and h = 0.09 nm, cf. Sect. 4.1.

Simulations of nanotubes under mechanical duress lead to shapes very different in appearance. At the same time there are robust traits in common: a deformation, proportional to the force within Hooke's law, eventually leads to a collapse of the cylinder and an abrupt change in pattern, or a sequence of such events. The presence of a snap-through buckling of nanotubes allows for a possibility of "shape memory", when in an unloading cycle the switch between patterns occurs at a somewhat lower level of strain. A small hysteresis observed in simulations is practically eliminated by thermal motion at any finite temperature. However, this hysteresis is greatly enhanced by the presence of van der Waals attraction which causes the tube walls to "stick"-flatten together after the collapse, Fig. 3d [13]. The simulations at even a low temperature (e.g. 50 K) shows strongly enhanced thermal vibrations in the vicinity of every pattern switch, while before and after the transition only barely noticeable ripples are seen. Physically, this indicates softening of the system, when one of the eigenvalues tends to zero in the vicinity of the bifurcation.

While several reports focus on a nonlinear dynamics of an open-end SWNT, when the terminal ring atoms are displaced gradually in simulation, a more realistic interaction of a cap-closed SWNT with the (diamond or graphite) substrates has been studied recently [25]. An inward cap collapse

^c = 2(1 + V )n and for the cylinder-helix flattening (M = 2), = 0.055 nm3/2 L/d5/2 .

and/or sideways sliding of the nanotube tip along the substrate are observed, in addition to the buckling of the tubule itself. Furthermore, an interaction of a small (four SWNT) bundle and a double-wall tubule with the substrates has been also reported [26].

An atomistic modeling of multi-layer tubes remains expensive. It makes extrapolation of the continuum model tempting, but involves an interlayer van der Waals interaction. The flexural rigidity scales as ~ h3 in case of a coherent, and as ~ h for an incoherent stack of layers sliding with respect to each other when the tube is deformed; this affects the mechanical properties and still has to be investigated.

Direct simulations of the tubules under hydrostatic pressure have not been reported to the best of our knowledge. In this scale anisotropic lateral forces in a molecular crystal packing are more plausible than a uniform pressure. An ability of a shell-tubule to bifurcate in a flattened form makes it an example of a two-level system, which manifests in the phase-transition behavior of SWNT crystal, as was first described in [68] and is now indicated by several experimental reports. While the faceting in the triangular crystal packing results in a partial wall flattening, a singular tubule under hydrostatic pressure can collapse completely. One can resort to continuum elasticity and estimate a pressure leading to an inward buckling as pc = 2Y(h/d)3, that is thousands of atmospheres for a nanometer tube. However, it drops fast with the diameter and is assisted by a flattening effects of twisting or bending and by van der Waals attraction between the opposite walls [13]. Such collapse cannot occur simultaneously throughout the significant SWNT length, but rather propagates at a certain speed depending on the ambient over-pressure u oc y/(p — pc)- This pressure dependence [76] is similar to the observations on macroscopic objects like underwater pipelines [50].

4.3 Atomistics of High Strain-Rate Failure

The simulations of compression, torsion, and tension described above (Sect. 4.2) do not show any bond breaking or atoms switching their positions, in spite of the very large local strain in the nanotubes. This supports the study of axial tension, where no shape transformations occur up to an extreme dilation. How strong in tension is a carbon nanotube? Since the tensile load does not lead to any shell-type instabilities, it is transferred more directly to the chemical bond network. The inherent strength of the carbon-carbon bond indicates that the tensile strength of carbon nanotubes might exceed that of other known fibers. Experimental measurements remain complex (Sect. 3.3) due to the small size of the grown single tubes. In the meantime, some tests are being done in computer modeling, especially well suited to the fast strain rate [75,76,77,78]. Indeed, a simulation of an object with thousand atoms even using a classical potential interaction between atoms is usually limited to picoseconds up to nanoseconds of real physical time. This is sufficiently long by molecular standards, as is orders of magnitude greater than the periods of intramolecular vibrations or intermolecular collision times. However, it is still much less than a normal test-time for a material, or an engineering structure. Therefore a standard MD simulation addresses a "molecular strength" of the CNT, leaving the true mechanisms of material behavior to the more subtle considerations (Sect. 4.4).

In MD simulation, the high-strain-rate test proceeds in a very peculiar manner. Fast stretching simply elongates the hexagons in the tube wall, until at the critical point an atomic disorder suddenly nucleates: one or a few C-C bonds break almost simultaneously, and the resulting "hole" in a tube wall becomes a crack precursor (see Fig. 11a). The fracture propagates very quickly along the circumference of the tube. A further stage of fracture displays an interesting feature, the formation of two or more distinct chains of atoms, ... = C = C = C = ..., spanning the two tube fragments, Fig. 11b. The vigorous motion (substantially above the thermal level) results in frequent collisions between the chains; they coalesce, and soon only one such chain survives. A further increase of the distance between the tube ends does not break this chain, which elongates by increasing the number of carbon atoms that pop out from both sides into the necklace. This scenario is similar to the monatomic chain unraveling suggested in field-emission exper-

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