A

Fig. 11. High strain rate tension of a two-wall tube begins from the outermost layer, nucleating a crack precursor (a), where the atomic size is reduced to make the internal layer visible. Eventually it leads to the formation of monatomic chains (b) (from [77])

iments [53], where the electrostatic force unravels the tube like the sleeve of a sweater. Notably, the breaking strain in such fast-snap simulations is about 30%, and varies with temperature and the strain rate. (For a rope of nanotubes this translates to a more than 150 GPa breaking stress.) This high breaking strain value is consistent with the stability limit (inflection point on the energy curve) of 28% for symmetric low-temperature expansion of graphene sheet [64], and with some evidence of stability of highly stresses graphene shells in irradiated fullerene onions [5].

4.4 Yield Strength and Relaxation Mechanisms in Nanotubes

Fast strain rate (in the range of 100 MHz) simulations correspond to the elongation of the tubule at percents of the speed of sound. In contrast to such "molecular tension test", materials engineering is more concerned with the static or slow tension conditions, when the sample is loaded during significantly longer time. Fracture, of course, is a kinetic process where time is an important parameter. Even a small tension, as any non-hydrostatic stress, makes a nanotube thermodynamically meta-stable and a generation of defects energetically favorable. In order to study a slow strength-determining relaxation process, preceding the fast fracture, one should either perform extensive simulations at exceedingly elevated temperature [9,10], or apply dislocation failure theory [79,81]. It has been shown that in a crystal lattice such as the wall of a CNT, a yield to deformation must begin with a homogeneous nucleation of a slip by the shear stress present. The non-basal edge dislocations emerging in such a slip have a well-defined core, a pentagonheptagon pair, 5/7. Therefore, the prime dipole is equivalent to the Stone-Wales (SW) defect [20] (Fig. 12). The nucleation of this prime dislocation dipole "unlocks" the nanotube for further relaxation: either brittle cleavage or a plastic flow. Remarkably, the latter corresponds to a motion of dislocations along the helical paths (glide "planes") within the nanotube wall. This causes a stepwise (quantized) necking, when the domains of different chi-ral symmetry and, therefore, different electronic structure are formed, thus coupling the mechanical and electrical properties [79,80]. It has further been shown [10,51,62,79,80,81,85] that the energetics of such nucleation explicitly depend on nanotube helicity.

Below, we deduce [79,81], starting with dislocation theory, the atomistics of mechanical relaxation under extreme tension. Locally, the wall of a nano-tube differs little from a single graphene sheet, a two-dimensional crystal of carbon. When a uniaxial tension a (N/m — for the two-dimensional wall it is convenient to use force per unit length of its circumference) is applied it can be represented as a sum of expansion (locally isotropic within the wall) and a shear of a magnitude a/2 (directed at ±45° with respect to tension). Generally, in a macroscopic crystal the shear stress relaxes by a movement of dislocations, the edges of the atomic extra-planes. Burgers vector b quantifies the mismatch in the lattice due to a dislocation [32]. Its glide requires only

Fig. 12. Stone-Wales (SW) dipole embedded in a nanotube hexagonal wall [67]

local atomic rearrangements and presents the easiest way for strain release, provided there is sufficient thermal agitation. In an initially perfect lattice such as the wall of a nanotube, a yield to a great axial tension begins with a homogeneous nucleation of a slip, when a dipole of dislocations (a tiny loop in three-dimensional case) first has to form. The formation and further glide are driven by the reduction of the applied-stress energy, as characterized by the elastic Peach-Koehler force on a dislocation failure. The force component along b is proportional to the shear in this direction and thus depends on the angle between the Burgers vector and the circumference of the tube,

The max \fb\ is attained on two ±45° lines, which mark the directions of a slip in an isotropic material under tension.

The graphene wall of the nanotube is not isotropic; its hexagonal symmetry governs the three glide planes — the three lines of closest zigzag atomic packing, oriented at 120° to each other (corresponding to the {101 /} set of planes in three-dimensional graphite). At non-zero shear these directions are prone to slip. The corresponding c-axis edge dislocations involved in such a slip are indeed known in graphite [21,36]. The six possible Burgers vectors l/3o(211 0} have a magnitude b = a = 0.246 nm (lattice constant), and the dislocation core is identified as a 5/7 pentagon-heptagon pair in the honeycomb lattice of hexagons. Therefore, the primary nucleated dipole must have a 5/7/7/5 configuration (a 5/7 attached to an inverted 7/5 core). This configuration is obtained in the perfect lattice (or a nanotube wall) by a 90° rotation of a single C-C bond, well known in fullerene science as a Stone-Wales diatomic interchange [20]. One is led to conclude that the SW

transformation is equivalent to the smallest slip in a hexagonal lattice and must play a key role in the nanotube relaxation under external force.

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

per one displacement, a. Here e is the applied strain, and C = Yh = 342 N/m can be derived from the Young modulus of Y = 1020 GPa and the inter-layer spacing h = 0.335 nm in graphite; one then obtains Ca?/2 = 64.5 eV. Equation (12) allows one to compare different nanotubes (assuming a similar amount of pre-existing dislocations); the more energetically favorable is the glide in a tube, the earlier it must yield to applied strain.

In a pristine nanotube-molecule, the 5/7 dislocations have first to emerge as a dipole, by a prime SW transformation. Topologically, the SW defect is equivalent to either one of the two dipoles, each formed by an ~ a/2 slip. Applying (11) to each of the slips one finds,

The first two terms, the zero-strain formation energy and possible isotropic dilation, do not depend on nanotube symmetry. The symmetry-dependent third term, which can also be derived as a leading term in the Fourier series, describes the essential effect: SW rotation gains more energy in the armchair (0 = 30°) nanotube, making it the weakest, most inclined to SW nucleation of the dislocations, in contrast to the zigzag (0 = 0) where the nucleation is least favorable.

Consider, for example, a (c, c) armchair nanotube as a typical representative (we will also see below that this armchair type can undergo a more general scenario of relaxation.) The initial stress-induced SW rotation creates a geometry that can be viewed as either a dislocation dipole or a tiny crack along the equator. Once "unlocked", the SW defect can ease further relaxation. At this stage, both brittle (dislocation pile-up and crack extension), or plastic (separation of dislocations and their glide away from each other) routes are possible, the former usually at larger stress and the latter at higher temperatures [9,10,79,80,81].

Formally, both routes correspond to a further sequence of SW switches. The 90° rotation of the bonds at the "crack tip" (Fig. 13, left column) will result in a 7/8/7 flaw and then 7/8/8/7 etc. This further strains the bondspartitions between the larger polygons, leading eventually to their breakage, with the formation of greater openings like 7/14/7 etc. If the crack, represented by this sequence, surpasses the critical Griffith size, it cleaves the tubule.

Fig. 13. SW transformations of an equatorially oriented bond into a vertical position creates a nucleus of relaxation (top left corner). It evolves further as either a crack (brittle fracture route, left column) or as a couple of dislocations gliding away along the spiral slip plane (plastic yield, top row). In both cases only SW rotations are required as elementary steps. The stepwise change of the nanotube diameter reflects the change of chirality (bottom right image) causing the corresponding variations of electrical properties [81]

Fig. 13. SW transformations of an equatorially oriented bond into a vertical position creates a nucleus of relaxation (top left corner). It evolves further as either a crack (brittle fracture route, left column) or as a couple of dislocations gliding away along the spiral slip plane (plastic yield, top row). In both cases only SW rotations are required as elementary steps. The stepwise change of the nanotube diameter reflects the change of chirality (bottom right image) causing the corresponding variations of electrical properties [81]

In a more interesting distinct alternative, the SW rotation of another bond (Fig. 13, top row) divides the 5/7 and 7/5, as they become two dislocation cores separated by a single row of hexagons. A next similar SW switch results in a double-row separated pair of the 5/7's, and so on. This corresponds, at very high temperatures, to a plastic flow inside the nanotube-molecule, when the 5/7 and 7/5 twins glide away from each other driven by the elastic forces, thus reducing the total strain energy [cf. (12)]. One remarkable feature of such glide is due to mere cylindrical geometry: the glide "planes" in case of nanotubes are actually spirals, and the slow thermally-activated Brownian walk of the dislocations proceeds along these well-defined trajectories. Similarly, their extra-planes are just the rows of atoms also curved into the helices.

A nanotube with a 5/7 defect in its wall loses axial symmetry and has a bent equilibrium shape; the calculations show [12] the junction angles < 15°. Interestingly then, an exposure of an even achiral nanotube to the axially symmetric tension generates two 5/7 dislocations, and when the tension is removed, the tube "freezes" in an asymmetric configuration, S-shaped or C-shaped, depending on the distance of glide, that is time of exposure. Of course the symmetry is conserved statistically, since many different shapes form under identical conditions.

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

with the corresponding reduction of diameter, d. While the dislocations of the first dipole glide away, a generation of another dipole results, as shown above, in further narrowing and proportional elongation under stress, thus forming a neck. The orientation of a generated dislocation dipole is determined every time by the Burgers vector closest to the lines of maximum shear (±45° cross at the end-point of the current circumference-vector c). The evolution of a (c, c) tube will be: (c, c) ^ (c,c — 1) ^ (c,c — 2) ^ ... (c, 0) ^ [(c — 1,1) or (c, —1)] ^ (c — 1, 0) ^ [(c — 2,1) or (c — 1, —1)] ^ (c — 2,0) ^ [(c — 3,1) or (c — 2, —1)] ^ (c — 3, 0) etc. It abandons the armchair (c, c) type entirely, but then oscillates in the vicinity of to be zigzag (c,0) kind, which appears a peculiar attractor. Correspondingly, the diameter for a (10,10) tube changes stepwise, d = 1.36, 1.29, 1.22, 1.16 nm, etc., the local stress grows in proportion and this quantized necking can be terminated by a cleave at late stages. Interestingly, such plastic flow is accompanied by the change of electronic structure of the emerging domains, governed by the vector (c1, c2). The armchair tubes are metallic, others are semiconducting with the different band gap values. The 5/7 pair separating two domains of different chirality has been discussed as a pure-carbon heterojunction [11,12]. It is argued to cause the current rectification detected in a nanotube nanodevice [15] and can be used to modify, in a controlled way, the electronic structure of the tube. Here we see how this electronic heterogeneity can arise from a mechanical relaxation at high temperature: if the initial tube was armchair-metallic, the plastic dilation transforms it into a semiconducting type irreversibly.

Computer simulations have provided a compelling evidence of the mechanisms discussed above. By carefully tuning the tension in the tubule and gradually elevating its temperature, with extensive periods of MD annealing, the first stages of the mechanical yield of CNT have been observed [9,10]. In simulation of tensile load the novel patterns in plasticity and breakage, just described above, clearly emerge.

Classical MD simulations have been carried out for tubes of various geometries with diameters up to 13 nm. Such simulations, although limited by the physical assumptions used in deriving the interatomic potential, are still invaluable tools in investigating very large systems in the time scales that are characteristic of fracture and plasticity phenomena. Systems containing up to 5000 atoms have been studied for simulation times of the order of nanoseconds. The ability of the classical potential to correctly reproduce the energetics of the nanotube systems has been verified through comparisons with TB and ab initio simulations [9,10].

Beyond a critical value of the tension, an armchair nanotube under axial tension releases its excess strain via spontaneous formation of a SW defect through the rotation of a C-C bond producing two pentagons and two heptagons, 5/7/7/5 (Fig. 14). Further, the calculations [9,10] show the energy of the defect formation, and the activation barrier, to decrease approximately linearly with the applied tension; for (10,10) tube the formation energy can be approximated as Esw(eV) = 2.3 — 40e. The appearance of a SW defect represents the nucleation of a (degenerate) dislocation loop in the planar hexagonal network of the graphite sheet. The configuration 5/7/7/5 of this primary dipole is a 5/7 core attached to an inverted 7/5 core, and each 5/7 defect can indeed further behave as a single edge dislocation in the graphitic plane. Once nucleated, the dislocation loop can split in simulations into two

Fig. 14. Kinetic mechanism of 5/7/7/5 defect formation from an ab-initio quantum mechanical molecular dynamics simulation for the (5, 5) tube at 1800 K [10]. The atoms that take part in the Stone-Wales transformation are in lighter gray. The four snapshots show the various stages of the defect formation, from top to bottom: system in the ideal configurations (t = 0 ps); breaking of the first bond (t = 0.10 ps); breaking of the second bond (t = 0.15 ps); the defect is formed (t = 0.20 ps)

Fig. 14. Kinetic mechanism of 5/7/7/5 defect formation from an ab-initio quantum mechanical molecular dynamics simulation for the (5, 5) tube at 1800 K [10]. The atoms that take part in the Stone-Wales transformation are in lighter gray. The four snapshots show the various stages of the defect formation, from top to bottom: system in the ideal configurations (t = 0 ps); breaking of the first bond (t = 0.10 ps); breaking of the second bond (t = 0.15 ps); the defect is formed (t = 0.20 ps)

dislocation cores, 5/7/7/5 ^ 5/7+ 7/5, which are then seen to glide through successive SW bond rotations. This corresponds to a plastic flow of dislocations and gives rise to possible ductile behavior. The thermally activated migration of the cores proceeds along the well-defined trajectories (Fig. 15) and leaves behind a tube segment changed according to the rules of dislocation theory, (14). The tube thus abandons the armchair symmetry (c,c) and undergoes a visible reduction of the diameter, a first step of the possible quantized necking in "intramolecular plasticity" [79,80,81].

The study, based on the extensive use of classical, tight-binding and ab initio MD simulations [10], shows that the different orientations of the carbon bonds with respect to the strain axis (in tubes of different symmetry) lead to different scenarios. Ductile or brittle behaviors can be observed in nano-tubes of different indices under the same external conditions. Furthermore, the behavior of nanotubes under large tensile strain strongly depends on their symmetry and diameter. Several modes of behavior are identified, and a map of their ductile-vs-brittle behavior has been proposed. While graphite is brittle, carbon nanotubes can exhibit plastic or brittle behavior under deformation, depending on the external conditions and tube symmetry. In the case of a zig-zag nanotube (longitudinal tension), the formation of the SW defect is strongly dependent on curvature, i.e., on the diameter of the tube and gives rise to a wide variety of behaviors in the brittle-vs-ductile map of stress response of carbon nanotubes [10]. In particular, the formation energy of the off-axis 5/7/7/5 defect (obtained via the rotation of the C-C bond oriented 120° to the tube axis) shows a crossover with respect to the diameter.

Fig. 15. Evolution of a (10,10) nanotube at T = 3000 K, strain 3 % within about 2.5 ns time. An emerging Stone-Wales defect splits into two 5/7 cores which migrate away from each other, each step of this motion being a single-bond rotation. The shaded area indicates the migration path of the 5/7 edge dislocation failure [9] and the resulting nanotube segment is reduced to the (10,9) in accord with (14) [80,81]

Fig. 15. Evolution of a (10,10) nanotube at T = 3000 K, strain 3 % within about 2.5 ns time. An emerging Stone-Wales defect splits into two 5/7 cores which migrate away from each other, each step of this motion being a single-bond rotation. The shaded area indicates the migration path of the 5/7 edge dislocation failure [9] and the resulting nanotube segment is reduced to the (10,9) in accord with (14) [80,81]

It is negative for (c, 0) tubes with c < 14 (d < 1.1 nm). The effect is clearly due to the variation in curvature, which in the small-diameter tubes makes the process energetically advantageous. Therefore, above a critical value of the curvature a plastic behavior is possible and the tubes can be ductile.

Overall, after the nucleation of a first 5/7/7/5 defect in the hexagonal network either brittle cleavage or plastic flow are possible, depending on tube symmetry, applied tension and temperature. Under high strain and low temperature conditions, all tubes are brittle. If, on the contrary, external conditions favor plastic flow, such as low strain and high temperature, tubes of diameter less than approximately 1.1 nm show a completely ductile behavior, while larger tubes are moderately or completely brittle depending on their symmetry.

5 Supramolecular Interactions

Most of the theoretical discussions of the structure and properties of carbon nanotubes involve free unsupported nanotubes. However, in almost all experimental situations the nanotubes are supported on a solid substrate with which they interact. Similarly, nanotubes in close proximity to each other will interact and tend to associate and form larger aggregates [69,82].

5.1 Nanotube—Substrate and Nanotube—Nanotube Interactions: Binding and Distortions

These nanotube-substrate interactions can be physical or chemical. So far, however, only physical interactions have been explored. The large polariz-ability of carbon nanotubes (see article by S. Louie in this volume) implies that these physical interactions (primarily van der Waals forces) are significant. One very important consequence of the strong adhesive forces with which carbon nanotubes bind to a substrate is the deformation of the atomic structure of the nanotube itself. An experimental demonstration of this effect is given in Fig. 16, which shows non-contact AFM images of two pairs of overlapping multi-wall nanotubes deposited on an inert H-passivated silicon surface. The nanotubes are clearly distorted in the overlap regions with the upper nanotubes bending around the lower ones [30,31]. These distortions arise from the tendency of the upper CNTs to increase their area of contact with the substrate so as to increase their adhesion energy. Counteracting this tendency is the rise in strain energy produced from the increased curvature of the upper tubes and the distortion of the lower tube. The total energy of the system can be expressed as an integral of the strain energy U(k) and the adhesion energy V(z) over the entire tube profile: E = f {U(k) + V[z(x)]dx}. Here, k is the local tube curvature and V[z(x)] the nanotube-substrate interaction potential at a distance z above the surface. Using the experimental value of Young's modulus for MWNTs [71,74] and by fitting to the experimentally observed nanotube profile, one can estimate the binding energy from the observed distortion. For example, for a 100 A diameter MWNT a binding energy of about 0.8 eV/A is obtained. Therefore, van der Waals binding energies, which for individual atoms or molecules are weak (typically 0.1 eV), can be quite strong for mesoscopic systems such as the CNTs. High binding energies imply that strong forces are exerted by nanotubes on underlying surface features such as steps, defects, or other nanotubes. For example, the force leading to the compression of the lower tubes in Fig. 16a is estimated to be as high as 30 nN. The effect of these forces can be observed as a reduced inter-tube electrical resistance in crossed tube configurations similar to those shown in Fig. 16 [24].

The axial distortions of CNTs observed in AFM images are also found in molecular dynamics and molecular mechanics simulations. Molecular mechanics represents a simple alternative to the Born-Oppenheimer approximation-based electronic structure calculations. In this case, nuclear motion is studied assuming a fixed electron distribution associated with each atom. The molecular system is described in terms of a collection of spheres representing the atoms, which are connected with springs to their neighbors. The motion of the atoms is described classically using appropriate potential energy functions. The advantage of the approach is that very large systems (many thousands of atoms) can be easily simulated. Figure 17a,b show the results of such simulations involving two single-walled (10,10) CNTs crossing each other over a graphite slab [31]. In addition to their axial distortion, the two nanotubes develop a distorted, non-circular cross-section in the overlap region. Further results on the radial distortions of single-walled nanotubes due to van der Waals interactions with a graphite surface are shown in Fig. 17c. The adhesion forces tend to flatten the bottom of the tubes so as to increase the area of contact. At the same time, there is an increase in the curvature of the tube and therefore a rise in strain energy ES. The resulting overall shape is again dictated by the optimization of these two opposing trends. Small diameter

Fig. 16. AFM non-contact mode images of two overlapping multi-wall nanotubes. The upper tubes are seen to wrap around the lower ones which are slightly compressed. The size of image (a) is 330 nm x 330 nm and that of (b) is 500 nm x 500 nm [4]

Fig. 16. AFM non-contact mode images of two overlapping multi-wall nanotubes. The upper tubes are seen to wrap around the lower ones which are slightly compressed. The size of image (a) is 330 nm x 330 nm and that of (b) is 500 nm x 500 nm [4]

Fig. 17. Molecular mechanics calculations on the axial and radial deformation of single-wall carbon nanotubes. (a) Axial deformation resulting from the crossing of two (10,10) nanotubes. (b) Perspective close up of the same crossed tubes showing that both tubes are deformed near the contact region. (c) Computed radial deformations of single-wall nanotubes adsorbed on graphite [4]

tubes that already have a small radius of curvature RC resist further distortion (Es « R-2), while large tubes flatten out and increase considerably their binding energy [by 115% in the case of the (40,40) tube]. In the case of MWNTs, we find that as the number of carbon shells increases, the overall gain in adhesion energy due to distortion decreases as a result of the rapidly increasing strain energy [31].

The AFM results and the molecular mechanics calculations indicate that carbon nanotubes in general tend to adjust their structure to follow the surface morphology of the substrate. One can define a critical radius of surface curvature Rcrt above which the nanotube can follow the surface structure or roughness. Given that the strain energy varies more strongly with tube diameter (<x d4) than the adhesion energy (<x d), the critical radius is a function of the NT diameter. For example, Rcrt is about (12d)-1 for a CNT with a d = 1.3 nm, while it is about (50d)-1 for a CNT with d = 10 nm.

5.2 Manipulation of the Position and Shape of Carbon Nanotubes

A key difference between the mechanical properties of CNTs and carbon fibers is the extraordinary flexibility and resistance to fracture of the former. Furthermore, the strong adhesion of the CNTs to their substrate can stabilize highly strained configurations. Deformed, bent and buckled nanotubes were clearly observed early in TEM images [34]. One can also mechanically manipulate and deform the CNTs using an AFM tip and then study the properties

of the deformed structures using the same instrument [23,30]. For this purpose one uses the AFM in the so-called contact mode with normal forces of the order of 10-50 nN [30]. It was found that most MWNTs can sustain multiple bendings and unbending without any observable permanent damage. Bending of MWNTs induces buckling, observed in the form of raised points along the CNT, due to the collapsing of shells. When the bending curvature is small a series of regularly spaced buckles appear on the inside wall of the nanotube [23]. This phenomenon is analogous to axial bifurcations predicted by a continuum mechanics treatment of the bending of tubes [39].

In studies of electrical or other properties of individual CNTs it is highly desirable to be able to manipulate them and place them in particular positions of the experimental setup, such as on metal electrodes in conductance studies, or in order to build prototype electronic devices structures. Again the AFM can be used for this purpose. The shear stress of CNTs on most surfaces is high, so that not only can one control the position of the nanotubes at even elevated temperatures, but also their shape.

In Fig. 18, a MWNT is manipulated in a series of steps to fabricate a simple device [4]. While highly distorted CNT configurations were formed during the manipulation process, no obvious damage was induced in the CNT. The same conclusion was reached by molecular dynamics modeling of the bending of CNTs [34]. The ability to prepare locally highly strained configurations stabilized by the interaction with the substrate, and the well known dependence of chemical reactivity on bond strain suggest that manipulation may be used to produce strained sites and make them susceptible to local chemistry. Furthermore, bending or twisting CNTs changes their electrical properties [35,55] and, in principle, this can be used to modify the electrical behavior of CNTs through mechanical deformation.

5.3 Self-Organization of Carbon Nanotubes: Nanotube Ropes, Rings, and Ribbons

Van der Waals forces play an important role not only in the interaction of the nanotubes with the substrate but also in their mutual interaction [68]. The different shells of a MWNT interact primarily by van der Waals forces; single-walled tubes form ropes for the same reason [69]. In these ropes the nanotubes form a regular triangular lattice. Calculations have shown that the binding forces in a rope are substantial. For example, the binding energy of 1.4 nm diameter SWNTs is estimated to be about 0.48eV/nm, and rises to 1.8eV/nm for 3 nm diameter tubes [68]. The same study showed that the nanotubes may be flattened at the contact areas to increase adhesion [68]. Aggregation of single-walled tubes in ropes is also expected to affect their electronic structure. When a rope is formed from metallic (10,10) nanotubes a pseudogap of the order of 0.1 eV is predicted to open up in the density of states due to the breaking of mirror symmetry in the rope [18].

Fig. 18. AFM manipulation of a single multi-wall carbon nanotube such that electrical transport through it can be studied. Initially, the nano-tube is located on the insulating (SiO2) part of the sample. In a stepwise fashion (not all steps are shown) it is dragged up the 80 A high metal thin film wire and finally is stretched across the oxide barrier [4]

Fig. 18. AFM manipulation of a single multi-wall carbon nanotube such that electrical transport through it can be studied. Initially, the nano-tube is located on the insulating (SiO2) part of the sample. In a stepwise fashion (not all steps are shown) it is dragged up the 80 A high metal thin film wire and finally is stretched across the oxide barrier [4]

A different manifestation of van der Waals interactions involves the self-interaction between two segments of the same single-wall CNT to produce a closed ring (loop) [44,45]. Nanotube rings were first observed in trace amounts in the products of laser ablation of graphite and were assigned a toroidal structure [40]. More recently, rings of SWNTs were synthesized with large yields (up to 50%) from straight nanotube segments, Fig. 19. These rings were shown to be coils not tori [45].

The formation of coils by CNTs is particularly intriguing. While coils of biomolecules and polymers are well known structures, they are stabilized by a number of interactions that include hydrogen bonds and ionic interactions [8]. On the other hand, the formation of nanotube coils is surprising, given the high flexural rigidity of CNTs and the fact that CNT coils can only be stabilized by van der Waals forces. However, estimates based on continuum mechanics show that in fact it is easy to compensate for the strain energy induced by the coiling process through the strong adhesion between tube segments in the coil. Figure 20 shows how a given length of nanotube l should be divided between the perimeter of the coil, 2nR, that defines the strain energy and the interaction length, l; = l — 2nR, that contributes to

Fig. 19. Scanning electron microscope images of rings of single-wall nanotubes dispersed on hydrogen-passivated silicon substrates [45]
Fig. 20. Thermodynamic stability limits for rings formed by coiling single wall nanotubes with radii of 0.7 nm (plain line), 1.5 nm (dashed line), and 4.0 nm (dotted line) calculated using a continuum elastic model [45]

the adhesion (see the schematic in the inset) so that a stable structure is formed [45]. From this figure it is clear that the critical radius RC for forming rings is small, especially for small radius CNTs such as the (10,10) tube (r = 0.7nm).

The coiling process is kinetically controlled. The reason is easy to understand; to form a coil the two ends of the tube have to come first very close to each other before any stabilization (adhesion) begins to take place. This bending involves a large amount of strain energy ES <x R-2, and the activation energy for coiling will be of the order of this strain energy (i.e. several eV). Similar arguments hold if, instead of a single SWNT, one starts with a SWNT rope. Experimentally, the coiling process is driven by exposure to ultrasound [44]. Ultrasonic irradiation can provide the energy for thermal activation [66], however, it is unrealistic to assume that the huge energy needed is supplied in the form of heat energy. It is far more likely that mechanical processes associated with cavitation, i.e. the formation and collapse of small bubbles in the aqueous solvent medium that are generated by the ultrasonic waves, are responsible for tube bending [66]. The nanotubes may act as nucle-ation centers for bubble formation so that a hydrophobic nanotube trapped at the bubble-liquid interface is mechanically bent when the bubble collapses. Once formed, a nanotube "proto-ring" can grow thicker by the attachment of other segments of SWNTs or ropes. The synthesis of nanotube rings opens the door for the fabrication of more complex nanotube-based structures relying on a combination of mechanical manipulation and self-adhesion forces.

Finally, we note that opposite sections of the carbon atom shell of a nanotube also attract each other by van der Waals forces, and under certain conditions this attraction energy (Evdw) may lead to the collapse of the nanotube to a ribbon-like structure. Indeed, such structures are often observed in TEM [13] and AFM images [43] of nanotubes (primarily multi-wall tubes). The elastic curvature energy per unit length of a tube is proportional to 1/R (R, radii of the tubes). However, for a fully collapsed single-wall tubule, the energy contains the higher curvature energy due to the edges, independent of the initial radius, and a negative (attractive) van der Waals contribution, £vdw ~ 0.03 — 0.04eV/ atom, that is proportional to R per unit length. Collapse occurs when the latter term prevails above a certain critical tube radii Rc that increases with increasing number N of shells of the nanotube. For example: Rc(N =1) ~ 8dvdw and Rc(N = 8) ~ 19dvdw [13]. The thickness of the collapsed strip-ribbon is obviously (2N — 1)dvdw. Any torsional strain imposed on a tube by the experimental environment favors flattening [55,75,76] and facilitates the collapse. The twisting and collapse of a nanotube brings important changes to its electrical properties. For example, a metallic armchair nanotube opens up a gap and becomes a semiconductor as shown in Fig. 21.

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