FEATURE ARTICLE

# Fullerene Nanotubes: C_{1,000,000} and Beyond

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

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 *v*ertices (here, carbon atoms),
*e*dges (covalent bonds), and *f*aces: *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 5*p* + 6(*f* - *p*), which also
equals the tripled number of vertices (each trivalent carbon is
shared by three adjacent faces).

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 C_{60} in half and inserting 10
more carbon atoms in the breach to get a rugby-ball-like
C_{70}, then adding another belt to make C_{80}, 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,

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.

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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