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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

Superstiff Shells

A synergism of atomistic model and macroscopic structural mechanics was achieved with the proper choice of parameters of the continuum shell: a Young's modulus of elasticity (Y) equal to 5 terapascals (a terapascal is a trillion pascals, a unit of pressure), and an effective thickness (h) of 0.07 nanometers. The small thickness simply reflects the fact that flexing is much easier than stretching for a single graphite sheet. The large modulus is in fact consistent with the standard value for graphite, if one takes into account the normal spacing (c) of 0.34 nanometers between the sheets in a stack: Y(h/c) = 1 terapascal.

The shell model has the benefits of any reductionist approach: Instead of dealing with innumerable interatomic forces, one has a smooth piece of uniform material. The insight helps one to handle larger systems, multiwall tubes or onions, sets of cylinders or spheres nested like a Russian doll. For example, this allowed us to calculate a particular hydrostatic compressibility or bending stiffness of a nanotube containing an arbitrary number of walls. The compressibility appears to depend on nanotube diameter and is a mixture of very rigid in-plane behavior and a relatively gentle coupling between the layers (owing to the weak intermolecular van der Waals forces).

Bending stiffness appears to be very high for the thinnest single- or double-walled nanotubes and can surpass the commonly expected level by a factor of four, although it converges to normal values for the thicker nanotubes containing many walls.

Figure 9. Images of free-standing nanotube whiskersClick to Enlarge Image

The above can serve as a partial explanation of recent measurements of an exceptional Young's modulus. The elegant approach of the scientists from NEC has enabled the amplitude of the thermal vibrations of a tiny nanotube whisker to be visualized (Figure 9) and measured. The equipartition theorem of statistical mechanics prescribes that the energy of any degree of freedom is determined by the temperature. The vibration amplitude, then, allows one to assess the stiffness of the cantilever and the effective Young's modulus of the nanotube material. In spite of some consonance with the shell model (which agrees in turn with the common graphite data!), the extracted high values, up to 4 terapascals, cannot be easily explained. There must be more fundamental causes on the chemical-bond level, a matter that requires further study. The same technique was recently used by a group at the University of California at Berkeley, who also report a high Young's modulus of 1.2 terapascals for a nanotube made of boron nitride.

Figure 10. Carbon nanotubesClick to Enlarge Image

The ability of a nanotube to sustain axial force to some level, but then to buckle sideways, suits it well for use as a nanoprobe in a scanning microscope, which studies the response of a sample to carefully controlled disturbance. In the work of the Rice group, a nanotube has been employed as a smart tool whose gentle touch does not damage the sample and allows the probe itself to survive the crash if this happens. At the same time, the tool's slenderness allows it to image sharp topographic details.

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