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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

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.

Figure 7. Graphite is a very brittle materialClick to Enlarge Image

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.

Figure 8. In torsionClick to Enlarge Image

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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