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1 2 Nanotube diameter (nm) Fig. 19.8. (Continued).

13 different caps, two of which are shown in Fig. 19.9. As another example, the projection mapping of an icosahedral cap for a (10,5) tubule is shown in Fig. 3.9 and 3.10(c). A schematic diagram of the (10,5) tubule is shown in Fig. 19.10, where the chirality of this tubule can be clearly seen.

By adding many rows of hexagons parallel to AB and CD in Fig. 3.9, a properly capped chiral graphene tubule is obtained [19.34], as shown in Fig. 19.10. The projection method illustrated in Fig. 3.10(c) can be extended to generate all possible chiral tubules specified in Fig. 19.2(b). As noted above, many of the chiral tubules can have a multiplicity of caps, each cap joining smoothly to the same vector Ch — «a, + ma2 and hence specifying the same chiral tubule [19.6,35]. We note that for nanotubes with smaller diameters than that of C60, there are no caps containing only pentagons and hexagons that can be fit continuously to such a small carbon nanotube (n, m). For this reason it is expected that the observation of very small diameter (< 7 A) carbon nanotubes is unlikely. For example, the (4,2) chiral vector shown in Fig. 19.2(a) does not have a proper cap and therefore

Fig. 19.9. A projection mapping of two possible caps that join continuously to a (7,5) chiral tubule and satisfy the isolated pentagon rule, (see §3.3).

is not expected to correspond to a physical carbon nanotube. One reason why tubules of larger diameter (> 10 A) are predominantly observed may be related to the larger number of ways that caps can be formed for the larger-diameter tubules; this probability should increase very rapidly with increasing tubule diameter, just as the number of isomers obeying the isolated pentagon rule increases rapidly as the number of carbon atoms in the fullerene increases [19.36].

Most of the tubules that have been discussed in the literature have closed caps. Open-ended tubules have also been reported using high-resolution TEM [19.37] and STM [19.15] techniques. The open ends of tubules appear as "highlighted" edges in the STM pattern due to the dangling bonds at the open ends. A number of recurring cap shapes have been reported in the literature for the closed-end tubules [19.33,38,39]. Some typical examples are shown in Fig. 19.11. Following the discussion of Euler's theorem in §3.1, six pentagons (each corresponding to a disclination of — 7r/3) are needed to form a single cap crowning a carbon tubule. Shown in Fig. 19.11 are a symmetric cap (a), an asymmetric cap (b), and a flat cap (c) to a carbon nanotube. The caps for tubules of larger diameter tend to be more flat than

19. Qo-Related Tubules and Spherules (b)

19. Qo-Related Tubules and Spherules (b)

Fig. 19.10. (a) A projection mapping of the C]40 fullerene. (b) A model of CM0 with icosahedral I symmetry, (c) A schematic diagram of the (10,5) single-wall tubule capped by hemispheres of the icosahedral C140 fullerene at both ends [19.28],

for those with smaller diameters, which are more round. For example, the cap of a tubule of diameter dt = 42 A corresponds to the "hemisphere" of a large fullerene (approximately Qooo)- Some of the common caps, such as in Fig. 19.11(a), have the shape of a cone, which is discussed further in §19.2.4.

The bill-like cap shown in Fig. 19.12 is explained in terms of the placement of a pentagon at "B" (introducing a +77/3 disclination) and a heptagon at C (with a disclination of -7r/3). The narrow tube termination above points "B" and "C" is a typical polyhedral cap with six pentagons. We can understand the relation of the heptagon and pentagon faces by noting that a cap normally is formed by sue pentagons each contributing a disclination of ir/3, yielding a total disclination of 27r, which is the solid angle of a hemisphere. Each heptagon contributes a disclination of —77/3, so that an additional pentagon is needed to offset each heptagon in forming a closed surface.

A rare although important tubule termination is the semitoroidal type shown in Fig. 19.13, which has also been described theoretically [19.40], Figure 19.13 shows a few of the inner cylinders with normal polyhedral

Fig. 19.11. Transmission electron microscopy pictures of carbon nanotubes with three common cap terminations: (a) a symmetric polyhedral cap, (b) an asymmetric polyhedral cap, (c) a symmetrical flat cap [19.37].

terminations and six cylinders with semitoroidal terminations, creating a lip on the tubule cap, which is clarified by the schematic diagram shown in Fig. 19.13(b).

Iijima reports [19.1,9] that the majority of the carbon tubules, which he has observed by electron diffraction in a transmission electron microscope, have screw axes and chirality [i.e., d is neither 0 nor v/6 in Eq. (19.3)]. It was further reported that in multiwall tubules there appears to be no particular correlation between the chirality of adjacent cylindrical planes [19.1,9]. Iijima has also reported an experimental method for measurement of the chiral angle 6 of the carbon nan-otubules based on electron diffraction techniques [19.1,37]. Dravid et al. [19.38], using a similar technique to Iijima, have also found that most of their nanotubes have a chiral structure. However, the TEM studies by Amelinckx and colleagues [19.41,42] produced the opposite result, that nonchiral tubules (6 = 0 or 77-/6) are favored in multiwall nanotubes.

Atomic resolution STM measurements (see §17.4) done under highresolution conditions have been used to measure the chiral angle of the cylindrical shell on the surface of carbon nanotubes [19.12,43]. Both chiral and nonchiral tubules have been observed. For the case of a singlewall tube, a zigzag nanotube with an outer diameter of 10 A has been

Fig. 19.12. (a) Transmission electron micrograph showing a bill-like termination of a nanotube consisting of three parts, a tube, a cone, and a smaller tube. Transitions in the shape of the tube are caused by a pentagon at "B" and a heptagon at "C." (b) Schematic illustration of the transition region showing the pentagon and heptagon responsible for the curvatures at "B" and "C" [19.37].

Fig. 19.12. (a) Transmission electron micrograph showing a bill-like termination of a nanotube consisting of three parts, a tube, a cone, and a smaller tube. Transitions in the shape of the tube are caused by a pentagon at "B" and a heptagon at "C." (b) Schematic illustration of the transition region showing the pentagon and heptagon responsible for the curvatures at "B" and "C" [19.37].

imaged directly [19.12], Using a moiré pattern technique which gives the interference pattern between the stacking of adjacent layers, it was shown that the tubule cylinder and the adjacent inner cylinder were both helical with chiral angles of 6 — 5° and 6 = -4°, respectively, yielding a relative chiral angle of 9° as shown in Fig. 19.14(c) [19.12], Atomic resolution STM observations are very important in demonstrating networks of perfect honeycomb structures (see Fig. 19.14) and showing that a multilayer nanotube may consist of cylinders with different chiral angles [19.4,12],

Several theoretical arguments favoring chiral nanotubes have been given [19.4,6,46,47], based on the many more combinations of caps that

Fig. 19.13. (a) Transmission electron micrograph of the semitoroidal termination of a tube which consists of six graphene shells, (b) Schematic illustration of a semitoroidal termination of a tubule which is caused by six pentagon-heptagon pairs in a hexagonal network [19.37].

can be formed for chiral rather than nonchiral nanotubes, especially at larger diameters d„ and the more favorable growth opportunities that are possible for chiral nanotubes (see §19.3). Although most workers have discussed carbon nanotubes with circular cross sections normal to the tubule axis, some reports show evidence for faceted polygonal cross sections, including cross sections with fivefold symmetry [19.48], Polygonal cross sections have been known for vapor-grown carbon fibers, upon heat treatment to ~3000°C [19.17,49], For the case of carbon fibers with high structural order, the faceting introduces interlayer longrange order which results in a lower free energy, despite the strong curvatures at the polygon corners. It would seem that polygonal cross sections are more common in large-diameter tubules, where most of the carbon atoms would be on planar surfaces of a faceted nanotube. Nanotubes with polygonal cross sections could be of interest as possible hosts for intercalated guest species between planar regions on adjacent tubules.

Because of the special atomic arrangement of the carbon atoms in a C60-based tubule, substitutional impurities are inhibited by the small size of the carbon atoms. Furthermore, the screw axis dislocation, the most common defect found in bulk graphite, should be inhibited by the monolayer structure of the C60 tubule. Thus, the special geometry of C60 and of the Qo-related carbon nanotubes should make these structures strong and stiff along the tubule axis and relatively incompressible to hydrostatic stress, when compared to other materials [19.50] (see §19.8).

19.2.4. Carbon Nanocones

Carbon nanocones are found on the caps of nanotubes (see §19.2.3) and also as free-standing structures generated in a carbon arc. These cones have been studied predominantly by high-resolution TEM [19.39,51] and by high-resolution STM [19.52]. The observed cones normally have opening angles of ~19° and can be quite long (~240 A) [19.15], Cones are formed from hexagons of the honeycomb lattice by adding fewer pentagons than the six needed by Euler's theorem to form a cylinder, which is the basic constituent of a nanotube. The addition or removal of a pentagon thus corresponds to a change in solid angle of 47t/12 or 77-/3. Thus the addition of 12 pentagons corresponds to a solid angle of 4tr and the formation of a closed cage. The solid angle fl subtended by the cone is given by where np is the number of pentagons in the cone, 28p is the cone angle, and 77 — 29p is the opening angle of the cone. The cone angle 20p is then given by where 0p is the angle of the cone with respect to the z axis, which is taken as the symmetry axis of the cone. Values for the cone angles 28p and for the opening angles tt - 26p in degrees are given in Table 19.1, and each of the entries for 7r - 26p (np = 1,2,3,4,5) is shown in Fig. 19.15. The limiting cases for the cones in Table 19.1 are 7r -26p = 180° corresponding to a flat plane and 7r — 20p = 0° corresponding to np = 6 and the formation of a cylindrical carbon nanotube rather than a cone. An example of a cone structure, as observed under high-resolution TEM, is shown in Fig. 19.16(a), and the schematic representation of this cone in the context of Eq. (19.4) is shown in Fig. 19.16(b).

Fig. 19.14. (a) Atomic resolution STM image of a carbon nanotube 35 Â in diameter. In addition to the atomic honeycomb structure, a zigzag superpattern along the tube axis can be seen, (b) A ball-and-stick structural model of a C^-based carbon tubule, (c) Structural model of a giant superpattern produced by two adjacent misoriented graphene sheets. The carbon atoms in the first layer are shaded and the second layer atoms are open. Between the two dashed lines are highlighted those first layer white atoms that do not overlap with second layer atoms. Because of their higher local density of states at the Fermi level, these atoms appear particularly bright in STM images [19.44,45]. This moiré pattern results in a zigzag superpattern along the tube axis within the two white dashed lines as indicated [19.12].

Cone angles in degrees for nanocones with various numbers of pentagons.

np

20,

W- 20,

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