CaRelated Tubules and Spherules

In addition to ball-like fullerenes, it is possible to synthesize tubular fullerenes and nested concentric fullerenes (see §19.2 and §19.10). The field of carbon tubule research was greatly stimulated by the initial report of the existence of carbon tubules or nanotubes [19.1] and the subsequent report of conditions for the synthesis of large quantities of nanotubes [19.2,3], Various experiments carried out thus far [transmission electron spectroscopy (TEM), scanning tunneling microscopy (STM), resistivity, Raman scattering, and susceptibility] are consistent with identifying the carbon nanotubes with cylindrical graphene sheets of sp2-bonded carbon atoms. In this chapter we review the present state of knowledge of carbon nanotubes (both monolayer and multilayer) and nested concentric fullerenes.

Formally, carbon nanotubes and fullerenes have a number of common features and also many differences. In reviewing the theoretical literature, the focus is on single-wall tubules, cylindrical in shape, either infinite in length or with caps at each end, such that the two caps can be joined to form a fullerene. Formally, the cylindrical portions of the tubules consist of a single graphene sheet, rolled to form the cylinder. The various types of cylindrical shells that can be formed are reviewed in §19.1, which is followed in §19.2 and §19.2.2 by a review of experimental observations in multiwall and single-wall tubules, respectively. Following a discussion of the synthesis (§19.2.5), possible growth mechanisms (§19.3) are considered. Next, the symmetry properties of carbon tubules are summarized (§19.4), followed by the remarkable electronic structure as predicted (§19.5) and observed (§19.6). The phonon dispersion relations are reviewed in §19.7, including the infrared and Raman spectroscopy in §19.7.3, and elastic properties in §19.8. The opening and filling of carbon nanotubes are discussed in §19.9 and nested carbon spherules in §19.10. The possibility of superconductivity in C60-related tubules is considered in §19.11.

19.1. Relation between Tubules and Fullerenes

In this section we consider first two simple examples of single-wall carbon nanotubes based on the C60 fullerene. The concept of a single-wall nano-tube is then generalized to specify the idealized structure of single-wall nanotubes in general.

In analogy to a C60 molecule, we can specify a single-wall C60-derived tubule by bisecting a C60 molecule at the equator and joining the two resulting hemispheres with a cylindrical tube one monolayer thick and with the same diameter as C60. If the C60 molecule is bisected normal to a fivefold axis, the "armchair" tubule shown in Fig. 19.1(a) is formed, and if the C60 molecule is bisected normal to a threefold axis, the "zigzag" tubule in Fig. 19.1(b) is formed [19.4], Armchair and zigzag carbon nanotubules of larger diameter, and having correspondingly larger caps, are defined below.

In addition to the armchair and zigzag tubules, a large number of chi-ral carbon nanotubes can be formed with a screw axis along the axis of the tubule and with a variety of "hemispherical"-like caps. These general carbon nanotubules can be specified mathematically in terms of the tubule diameter d, and chiral angle 0, which are shown in Fig. 19.2(a), where the chiral vector Ck

is shown, as well as the basic translation vector T for the tubule, which is discussed below. In Fig. 19.2(a), the vector Ch connects two crystallo-graphically equivalent sites O and A on a two-dimensional (2D) graphene sheet where a carbon atom is located at each vertex of the honeycomb structure [19.4]. The construction in Fig. 19.2(a) shows the chiral angle 0 of the nanotube with respect to the zigzag direction (0 = 0) and the unit vectors a, and a2 of the hexagonal honeycomb lattice. The armchair tubule [Fig. 19.1(a)] corresponds to 0 = 30° on this construction. An ensemble of possible chiral vectors can be specified by Eq. (19.1) in terms of pairs of integers (n, m) and this ensemble is shown in Fig. 19.2(b) [19.6]. Each pair of integers (n, m) defines a different way of rolling the graphene sheet to form a carbon nanotube. We now show how the construction in Fig. 19.2(a) specifies the geometry of the carbon nanotube.

The cylinder connecting the two hemispherical caps of Fig. 19.1 is formed by superimposing the two ends OA of the vector Ch. The cylinder joint is made by joining the line AB' to the parallel line OB in Fig. 19.2(a),

Fig. 19.1. By rolling a graphene sheet (a single layer from a 3D graphite crystal) into a cylinder and capping each end of the cylinder with half of a fullerene molecule, a "fullerene-derived tubule," one atomic layer in thickness, is formed. Shown here is a schematic theoretical model for a single-wall carbon tubule with the tubule axis normal to: (a) the 8 = 30° direction (an "armchair" tubule), (b) the 0 = 0° direction (a "zigzag" tubule), and (c) a general direction 0 < 8 < 30° (see Fig. 19.2) (a "chiral" tubule). The actual tubules shown in the figure correspond to (n, m) values of: (a) (5, 5), (b) (9, 0), and (c) (10, 5) [19.5].

Fig. 19.1. By rolling a graphene sheet (a single layer from a 3D graphite crystal) into a cylinder and capping each end of the cylinder with half of a fullerene molecule, a "fullerene-derived tubule," one atomic layer in thickness, is formed. Shown here is a schematic theoretical model for a single-wall carbon tubule with the tubule axis normal to: (a) the 8 = 30° direction (an "armchair" tubule), (b) the 0 = 0° direction (a "zigzag" tubule), and (c) a general direction 0 < 8 < 30° (see Fig. 19.2) (a "chiral" tubule). The actual tubules shown in the figure correspond to (n, m) values of: (a) (5, 5), (b) (9, 0), and (c) (10, 5) [19.5].

where lines OB and AB' are perpendicular to the vector Ch at each end [19.4]. The chiral tubule thus generated has no distortion of bond angles other than distortions caused by the cylindrical curvature of the tubule. Differences in chiral angle 8 and in the tubule diameter d, give rise to differences in the properties of the various carbon nonotubes. In the (n, m) notation for specifying the chiral vector Ch in Eq. (19.1), the vectors (n, 0) denote zigzag tubules and the vectors (n, n) denote armchair tubules, and the larger the value of n, the larger the tubule diameter. Both the (n, 0) and (n, n) nanotubes have especially high symmetry, as discussed in §19.4.3, and exhibit a mirror symmetry plane normal to the tubule axis. All other vectors (n, m) correspond to chiral tubules [19.6]. Since both right- and left-handed

® :metal • :semiconductor armchair

Fig. 19.2. (a) The chiral vector OA or Ck = na, + ma, is defined on the honeycomb lattice of carbon atoms by unit vectors a, and a2 and the chiral angle 0 with respect to the zigzag axis. Along the zigzag axis, 6 = 0°. Also shown are the lattice vector QB= T of the ID tubule unit cell and the rotation angle *// and the translation r which constitute the basic symmetry operation R = (i//|r) for the carbon nanotube. The diagram is constructed for (n, m) = (4,2). (b) Possible vectors specified by the pairs of integers {n, m) for general carbon tubules, including zigzag, armchair, and chiral tubules. Below each pair of integers (n, m) is listed the number of distinct caps that can be joined continuously to the carbon tubule denoted by (n, m) [19.4], as discussed in §19.2.3. The encircled dots denote metallic tubules while the small dots are for semiconducting tubules.

chirality is possible for ehiral tubules, it is expected that chiral tubules are optically active to either right or left circularly polarized light propagating along the tubule axis. In terms of the integers (n, m), the tubule diameter d, is given by d, = Ch/tr = \/3ac_c(m2 + mn + /j2)1/2/tt (19.2)

where is the nearest-neighbor C-C distance (1.421 A in graphite), Ch is the length of the chiral vector Ch, and the chiral angle 6 is given by

For example, a zigzag tubule (0 = 0°) specified by (9,0) has a theoretical tubule diameter of d, = 9x/3ac_c/7r = 7.15 A, while an armchair tubule specified by (5,5) has d, — \5ac_c/v = 6.88 A, both derived from hemispherical caps for the C60 molecule and assuming an average ac_c = 1.44 A appropriate for C60. If the graphite value of ac c = 1.421 A is used, slightly smaller values for d, are obtained. Substitution of (n,m) = (5,5) into Eq. (19.3) yields 0 = 30° while substitution of (n,m) = (9,0) and (0,9) yields 0 = 0° and 60°, respectively. The tubules (0,9) and (9,0) are equivalent, because of the sixfold symmetry of the graphene layer. Because of the point group symmetry of the honeycomb lattice, several different integers (n, m) will give rise to equivalent nanotubes. To define each nanotube once and once only, we restrict ourselves to consideration of nanotubes arising from the 30° wedge of the 2D Bravais lattice shown in Fig. 19.2(b). Because of the small diameter of a carbon nanotube (~10 A) and the large length-to-diameter ratio (> 103), carbon nanotubes provide an important system for studying one-dimensional physics, both theoretically and experimentally.

Many of the experimentally observed carbon tubules are multilayered, consisting of capped concentric cylinders separated by ~ 3.5 A. In a formal sense, each of the constituent cylinders can be specified by the chiral vector Ch in terms of the indices (n, m) of Eq. (19.1), or equivalently by the tubule diameter dt and chiral angle 0. Because of the different numbers of carbon atoms around the various concentric tubules, it is not possible to achieve the ABAB... interlayer stacking of graphite in carbon nanotubes. Thus, an in-terlayer spacing closer to that of turbostratic graphite (3.44 A) is expected, subject to the quantized nature of the (n, m) integers, which determine Ch. We illustrate this quantum constraint by considering the nesting of the (9,0) zigzag tubule (which has d, = 7.15 A) within an adjacent zigzag tubule of larger diameter. From the turbostratic constraint, we know that the adjacent nested tubule must have a diameter in excess of [7.15+2(3.44)] A= 14.03 A. From Eq. (19.2), the zigzag tubule (18,0) has a diameter 14.29 A which meets the turbostratic constraint, while the (17,0) tubule has a diameter of only 13.50 A, which is too small to accommodate a concentric (9,0) tubule. In fact, Eq. (19.2) shows that the (18,0) tubule has the smallest diameter subject to the turbostratic constraint (d, > 13.93 A), even when all possible (n,m) indices are considered, thus leading to a minimum interlayer separation distance of 3.57 A. In the next two sections, we review experimental observations on multiwall (§19.2) and single-wall (§19.4.1) carbon nanotubes. Further discussion of Fig. 19.2(a) is given in §19.4.1 and §19.4.2 in terms of the symmetry of carbon nanotubes.

19.2. Experimental Observation of Carbon Nanotubes

Most of the experimental observations have been on multiwall carbon nanotubes and bundles of nanotubes, which are discussed initially. We then proceed to review the more recent work on single-wall tubules, which relate more closely to the theoretical calculations. Since much attention has been given to the structure of tubule caps, this topic is also reviewed. Finally, methods for synthesis of carbon nanotubes are discussed.

19.2.1. Observation of Multiwall Carbon Nanotubes

The earliest observations of carbon tubules with very small (nanometer) diameters [19.1,7,8] were based on high-resolution transmission electron microscopy (TEM) measurements on material produced in a carbon arc. This work provided strong evidence for yam-long tubules, with cross sections showing several coaxial tubes and a hollow core. In Fig. 19.3, the first published observations of carbon nanotubes are shown [19.1]. Here we see only multilayer carbon nanotubes, but one tubule has only two coaxial carbon cylinders [Fig. 19.3(b)], and another has an inner diameter of only 23 A [Fig. 19.3(c)] [19.1]. Typically, the outer diameter of carbon nanotubes prepared by a carbon arc process ranges between 20 and 200 A and the inner diameter ranges between 10 and 30 A [19.9], Typical lengths of the arc-grown tubules are ~1 /im, giving rise to an aspect ratio (length-to-diameter ratio) of 102 to 103. Because of their small diameter, involving only a small number of carbon atoms, and because of their large aspect ratio, carbon nanotubes are classified as ID carbon systems. Most of the theoretical work on carbon nanotubes emphasizes their ID properties. In the multiwall carbon nanotubes, the measured (by high-resolution TEM) interlayer distance is 3.4 A [19.1], in good agreement with the value of 3.39 A for the average equilibrium interlayer separation, obtained from self-consistent electronic structure calculations [19.10,11].

Scanning tunneling microscopy (STM) [19.3,12-15] and atomic force microscopy (AFM) [19.15,16] have also provided powerful local probes of

Fig. 19.3. The observation of N coaxial carbon tubules with various inner diameters d, and outer diameters d„ reported by Iijima using TEM: (a) N = 5, da = 67 A, (b) N = 2, da = 55 A, and (c) N = 7, d, = 23 A, d„ = 65 A. The sketch (d) indicates how the interference patterns for the parallel planes labeled H are used to determine the chiral angle 6, which in turn is found from the orientation of the tubule axis relative to the nearest zigzag axis defined in Fig. 19.2(a). The interference patterns that are labeled V determine the interplanar distances [19.1].

Electron beam O

the topographical structure. STM techniques have also been used to study the electronic structure through measurements of the electronic density of states and AFM has been used to study the elastic properties of carbon nanotubes.

Carbon nanotubes grown by vapor growth methods have also been reported. In Fig. 19.4(a) we see a thin carbon tubule (b) nucleated on a con-

Fig. 19.4. Two kinds of vapor-grown carbon fibers (VGCF) observed in the as-grown sample: (a) a thick hollow fiber formed by a catalytic metal particle, (b) an exposed nanotube (f) and a pyrolytically-coated segment of the nanotube (44) [19.17].

ventional vapor-grown carbon fiber (a) during the growth process [19.7,8]. The region denoted by the double arrows shows a pyrolytic carbon deposit on the as-grown nanotube. In contrast, the carbon tubules of Fig. 19.3(a), (b), and (c) were nucleated on the surface of the negative carbon electrode in a carbon arc discharge apparatus [19.1]. It is believed that the properties of carbon nanotubes grown by the arc discharge method are similar to those grown from the vapor phase [19.17] (see §19.2.5).

Although very small diameter carbon filaments, such as shown in Fig. 19.5, were observed many years earlier [19.18,19], no detailed systematic studies of such very thin filaments were reported in the 1970s and 1980s. A direct stimulus to a more systematic study of carbon filaments of very small diameters came from the discovery of fullerenes by Kroto, Smalley, and co-workers [19.22] and subsequent developments resulting in the synthesis of gram quantities of fullerenes by Kratschmer, Huffman, and co-workers [19.23]. These developments heralded the entry of many scientists into the field, together with many ideas for new carbon materials, and bringing new importance to carbon systems of nanometer dimensions. Iijima's important contribution [19.1] included an appreciation for the importance of his nanotube work in relation to ongoing fullerene

Fig. 19.5. An early high-resolution TEM micrograph showing carbon nanotubes with diameters less than 10 nm [19.18-21].

studies. Quite independently, Russian workers also reported the discovery of carbon tubules and bundles, but generally having much smaller aspect ratios, and hence they called their tubules "barrelenes" [19.24,25]. These barrelenes have similarities to cylindrical fullerenes reported by Wang and Buseck [19.26] with length-to-diameter ratios or 10 or less.

Soon after the report of carbon nanotube synthesis by the arc discharge method [19.1], a second report was published giving conditions for the synthesis of copious amounts of fullerene tubules [19.27,28] using the carbon arc discharge method. The availability of large quantities of carbon nano-tubules greatly stimulated experimental activity in this field [19.27]. One interesting and useful characteristic of the growth of the carbon nanotubules is the tendency for large numbers of nanotubules to grow parallel to each other, forming a bundle of nanotubules [19.27], perhaps 50 nm in diameter, and reminiscent of the winding of carbon commercial fibers in continuous tows.

It should also be mentioned that the fullerene tubules of Fig. 19.3 differ in a fundamental way from the scroll-like graphite whiskers, synthesized by Bacon many years ago in a dc carbon arc discharge [19.29], under current and voltage conditions similar to those used for the growth of carbon nanotubules [19.27], but operating at much higher gas pressures [19.29], The main difference is that graphite whiskers have been reported to form a scroll-like tube, while the nanotubes are coaxially stacked cylinders (see Fig. 19.3), as has been verified by high-resolution STM observations [19.12], which always show the same number of layers on the left and right hand sides of a carbon nanotube image, as seen in Figs. 19.3(a), (b), and (c). As mentioned above, highly elongated fullerenes with shapes between the traditional icosahedral fullerenes and the carbon nanotubules have been observed by high-resolution transmission electron microscopy [19.26]. It has also been shown that the carbon nanotubes are unstable under high-intensity electron beam irradiation and can thus be transformed into concentric spherical shells (sometimes called onions) [19.30], as discussed in §19.10.

19.2.2. Observation of Single-Wall Carbon Nanotubes

We now describe a few of the interesting characteristics of single-wall carbon nanotubes. The single-wall nanotubes, just like the multiwall nanotubes, tend to form bundles of nearly parallel tubules (Fig. 19.6), although single, isolated tubules are also found [19.31]. Soot commonly deposits on the surface of these tubules and also on the inside of tubules [19.31]. Deposits of amorphous carbon on vapor-grown carbon fibers are well known (see Figs. 19.4 and 19.5), and such deposits are used to thicken vapor-grown fibers, after appropriate heat treatment procedures (see §2.5). Furthermore, single-wall tubules are remarkably flexible, and bend into curved arcs with radii of curvature as small as 20 nm, as shown in Fig. 19.7. This flexibility suggests excellent mechanical properties, consistent with the high tensile strength and bulk modulus of commercial and research-grade vapor-grown carbon fibers (see §2.5). The longest reported nanotube length is 700 nm for a 0.9 nm diameter tubule, yielding a large length-to-diameter ratio (aspect ratio) of —800, although not as large as the aspect ratio (~104) found in the larger (>100 nm diameter) vapor-grown carbon fibers [19.32]. The single-wall nanotubes, just like the multiwall nanotubes and the conventional vapor-grown carbon fibers, have hollow cores along the axis of the tubule.

The diameter distribution of single-wall carbon nanotubes is of great interest for both theoretical and experimental reasons. Theoretical studies have shown that the physical properties of carbon nanotubes are strongly a

Fig. 19.6. (a) Transmission electron micrographs of the Co-catalyzed soot for a concentration of 4% Co. The threads in the figure are apparently individual nanotubules or bundles of nanotubes such as shown in (b). Most of the as-prepared nanotubes are covered with carbon soot [19.31],

dependent on tubule diameter. These predictions remain to be well tested experimentally. Because of the difficulty of making physical measurements on individual single-wall nanotubes, a number of exploratory studies have been made on bundles of tubules. Early results on the diameter distribution of Fe-catalyzed single-wall nanotubes [Fig. 19.8(a)] show a diameter range between 7 A and 16 A, with the largest peak in the distribution at 10.5A, and with a smaller peak at 8.5A [19.33]. The smallest reported diameter for the single-wall carbon nanotubes is 7 A [19.33], the same as the smallest diameter expected theoretically (7.0 A for a tubule based on C60) [19.4], and this is further discussed in §19.2.3. Qualitatively similar results, but differing in detail were obtained for the Co-catalyzed b

Fig. 19.6. (Continued).

nanotubes, with a peak in the distribution at 13A, as shown in Fig. 19.8(b) [19.31]. These experimental results indicate that the diameter distribution of the single-wall nanotube involves predominantly small diameter tubules. The single-wall nanotubes have a much narrower diameter distribution which is dependent on the catalyst and preparation conditions, so that explicit measurements of the diameter distribution will be needed for samples used for properties measurements on tubule bundles.

The second characteristic parameter of great importance for properties measurements on single-wall carbon nanotubes is the chiral angle 6. Two experimental problems contribute to the difficulty in determining the chiral angle 6. Because of the small number of carbon atoms available for carrying out a diffraction experiment on a single-wall nanotube, electron diffraction is the favored technique. Furthermore, the low atomic number for carbon (Z = 6) results in a very low electron scattering cross section. Nevertheless, it has been demonstrated [19.33] that electron diffraction measurements with a suitable TEM instrument and using micro-diffraction techniques can provide detailed structural information on an individual single-wall nanotube, including measurement of the tubule diameter d, and the chiral an-

Fig. 19.7. High-resolution TEM micrographs showing nanotubes with a 20 nm radius of curvature [19.31],

gle 0. Based on their observations, Iijima and co-workers claim that most single-wall nanotubes show chirality [19.33] as was previously also claimed for the multiwall nanotubes. Atomic resolution STM techniques also provide sensitive techniques for the measurement of the chiral angle 0 [19.15].

While the ability to measure the diameter d, and chiral angle 0 of individual single-wall tubules has been demonstrated, it remains a major challenge to determine d, and 0 for specific tubules used for an actual property measurement, such as electrical conductivity, magnetoresistance or Raman scattering. It should be emphasized that such properties measurements on individual carbon nanotubes are already very difficult, and adding to these difficulties are further challenges of characterizing each tubule regarding its dt and 0 values.

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