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Fullerene Nanotubes: C1,000,000 and Beyond

Some unusual new molecules—long, hollow fibers with tantalizing electronic and mechanical properties—have joined diamonds and graphite in the carbon family

Boris Yakobson, Richard Smalley

Nanotube Shape and Structure

Nanotubes are giant linear fullerenes. A fullerene, by definition, is a closed, convex cage molecule containing only hexagonal and pentagonal faces. (This definition intentionally leaves out possible heptagons, which are responsible for the concave parts and are treated as defects.) Like any simple polyhedron, a fullerene cage or a nanotube satisfies Euler's theorem (earlier proved by Descartes) relating the number of vertices (here, carbon atoms), edges (covalent bonds), and faces: v - e + f = 2. If the number of pentagons is p, and the other (f - p) faces are all hexagonal, then the doubled number of edges (each edge belongs to two faces) is 5p + 6(f - p), which also equals the tripled number of vertices (each trivalent carbon is shared by three adjacent faces).

Figure 4. NanotubesClick to Enlarge Image

A simple accounting then yields p = 12, and therefore a nice, defectless nanotube must have exactly 12 pentagons, the same dozen as in the buckyball! The strict rules of topology impose this family trait on all fullerenes. An even more obvious trait the nanotubes inherit from another ancestor, graphite, is a hexagonal pattern on their walls. Figure 4 illustrates this by showing two possible ways of constructing a nanotube from a precursor form of carbon.

One can start by cutting C60 in half and inserting 10 more carbon atoms in the breach to get a rugby-ball-like C70, then adding another belt to make C80, and repeating this process indefinitely to create a buckytube of unlimited length. Or a nanoscopic tailor might start with a nice big piece of a one-atom-thick sheet of graphite, cut a long strip out of it and roll it up into a cylinder with no stitches left. The tailor has a decision to make, though: whether to choose the strip width parallel to the dense zigzag row of bonds, perpendicular to it, or at some angle θ. The first choice results in a so-called zigzag nanotube, the second an armchair tubule; the third arrangement is chiral and results from turning the sheet at an angle somewhere between the angles of the zigzag (0 degrees) and the armchair (30 degrees). The oriented width is specified by a rollup vector (n,m), which records the number of steps along the a and b directions. Its integer components uniquely define the tubule diameter d and its corkscrew symmetry, called helicity, or chiral angle θ. As follows from elementary geometry and an assumption that the C-C bond has its normal length of 0.14 nanometers,

Click to Enlarge Image

For example, the integers (9,0) or (5,5) correspond respectively to zigzag (θ = 0 degrees) or armchair (θ = 30 degrees) buckytubes that are roughly 0.7 nanometers in diameter. Adding the terminating caps completes the job. Any such seamless, coherent arrangement of atoms has a certain helicity, and recognition of this, based on observed electron-diffraction patterns, was perhaps even more important in Iijima's discovery than the fact that the nanotubes are thin cylinders.

It is the chemical genius of carbon—the ability to satisfy its four valence electrons by bonding with three neighbors—that makes the above structures possible in graphite and its fullerene relatives. Each of three electrons is assigned to a partner; the fourth is shared by everybody, being delocalized all over the network. These shared electrons, called π electrons, make fullerenes aromatic and allow some nanotubes to conduct electricity.

Although sp 2-carbon (the term for the bonding arrangement just described) is happiest in a flat hexagonal tiling, rolling the sheet up expends relatively little elastic energy, which is generously returned when all the dangling unhappy bonds at the edges are eliminated in a seamless cylinder. Furthermore, carbon doesn't really mind a few pentagons or heptagons (seven edges). Geometrically these insertions allow for a local Gaussian curvature, positive and caplike in the case of a pentagon or negative and saddlelike where a heptagon is involved (Figure 5). With these two elements, all kinds of equilibrium shapes and plumbing become possible in carbon construction, giving rise in the literature to numerous fantasies.

Figure 5. Cap- or saddle-like curvatureClick to Enlarge Image Figure 6. Catalytic metal (nickel-cobalt) clusterClick to Enlarge Image

Both the morphology of nanotubes and their helical structure are inherently related to the mechanisms by which they grow. The formation of these nanoscopic stalagmites in most cases requires an open end where carbon atoms arriving from the gas phase can coherently land. The growth of nested multiwall nanotubes can be stabilized by the strained "lip-lip" bonding between the coaxial edges, highly fluctuating and therefore accessible for new atoms. In general, the open end can be maintained either by a high electric field, by the entropy opposing the orderly cap termination or by the presence of a metal catalyst. The last mechanism, in the form of a tiny metal (nickel-cobalt) cluster scooting around the nanotube edge, is apparently responsible for the high-yield growth of (10,10) tubes recently reported by one of the authors' (Smalley's) groups. The alloy cluster anneals all unfavorable structures into hexagons, which in turn welcome the newcomers and promote the continuous growth of a straight cylindrical tower (Figure 6).

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