FEATURE ARTICLE
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
Crunchy Molecules
The carbon atoms in diamond are tetrahedrally bonded to their four
nearest neighbors, forming the rigid three-dimensional lattice that
gives diamond its unparalleled hardness. By contrast, the
two-dimensional arrangement of atoms in a graphene
(graphite-sheet-like) nanotube wall permits some out-of-plane
flexibility. Combined with the strength of the constituent bonds,
this promises spectacular mechanical properties.
Of course, common graphite doesn't strike one as a very strong
material. Everyone has seen a pencil lead broken by a gentle bending
effort. But as Uzi Landman from the Georgia Institute of Technology
once put it, "small is different." The minuscule diameter
of a nanotube merely leaves no room for the numerous imperfections
and microcrevices that make a pencil lead so brittle. As early
nanotube studies progressed, several groups reported high-resolution
images of greatly distorted tubes with no traces of fracturing.
However, it was not always clear whether the deformations were truly
elastic or partially caused by embedded defects such as pentagonal
or heptagonal rings.
In 1994, at one of the international meetings, Iijima approached
Jerzy Bernholc of North Carolina State University with a few
beautiful transmission-electron-microscope shots of nanotubes, each
showing all the distinct features of a nice elastic bend (Figure
7). Meanwhile, in France, a group led by J. F. Despres reported
similar observations. In experiments, applying controllable forces
to a tiny nanotube is difficult, and so most of the evidence of the
behavior of nanotubes as a material relies on still-life images of
the singular victims of mechanical duress during sample preparation.
If similar shapes could be simulated on a computer, one could learn
about the forces involved and judge whether the deformation is
indeed elastic and reversible.
The forces and stresses in molecules (the title of Richard Feynman's
at-first-unnoticed and controversial undergraduate thesis, which
eventually became the famous Hellman-Feynman theorem) are a subtle
problem in quantum mechanics. Although numerical methods have
progressed dramatically in this field, a first-principles treatment
for a molecule containing thousands of atoms often remains
prohibitively expensive. To get around this obstacle, one can resort
to a recipe for interatomic forces, F, if such a recipe
exists and is well tested, and then simply apply classical mechanics
in the form of Newton's familiar second law of motion, F =
ma (force equals mass times acceleration), to every one of
thousands of atoms, thus computing their motion step by step. This
is what classical molecular dynamics does. One of us (Yakobson),
together with Charles Brabec of North Carolina State University, has
been involved in such modeling.
Our calculations predict the energy cost of deforming a nanotube,
and its elastic parameters agree with those known for graphite or
found by first-principle theoretical methods. We were surprised,
however, to see humps and bumps on the strain-energy curves beyond
what Hooke's law would predict: that each displacement would
generate a proportional restoring (elastic) force.
This indicated that there must be some abrupt changes in the
molecule under mechanical load. Indeed, each singularity in the
stress-strain curve appears to correspond to a sudden shape switch
of an initially perfect cylinder. All generic modes of mechanical
load have been studied this way: bending (Figure 7),
torsion (Figure 8) and axial compression (Figure
10). In the simulations the nanotube is seen to snap from one
shape to the next, emitting acoustic waves along its walls at every
"crunch." These "crunchy molecules" never
actually break, but reversibly accommodate to external stress. It
became clear that, besides the similarity with the patterns seen in
experiments, the observed buckling phenomena resemble the
instabilities well known in macroscopic elasticity of the hollow
objects, thin shells.
The hollow structure is indeed an outstanding feature of fullerene
molecules and of nanotubes in particular. On the other hand, the
interest in the elasticity of macroscopic shells and rods dates back
to the 18th-century work of Leonhard Euler (whose rule for polyhedra
we just discussed above). He discovered the phenomenon of elastic
instability: A rod or column compressed axially remains straight
until a critical force is reached. It then becomes unstable
(undergoes bifurcation, mathematically speaking) and buckles
sideways. The behavior of the hollow tubules is more complex, but
still predictable with continuum-elasticity methods. In its
application to nanotubes, the correspondence between the
elastic-shell model and molecular dynamics is remarkable. The laws
of continuum mechanics are amazingly robust and allow one to treat
even intrinsically discrete objects only a few atoms in diameter.
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