Energetics of nanoscale graphitic tubules

Using both empirical potentials and first-principles total-energy methods, we have examined the energetics and elastic properties of all possible graphitic tubules with radii less than 9 Å. We find that the strain energy per carbon atom relative to an unstrained graphite sheet varies as 1/${\mathit{R}}^2$ (where R is the tubule radius) and is insensitive to other aspects of the lattice structure, indicating that relationships derivable from continuum elastic theory persist well into the small-radius limit. We also predict that this strain energy is much smaller than that in highly symmetric fullerene clusters with similar radii, suggesting a possible thermodynamic preference for tubular structures rather than cage structures. The empirical potentials further predict that the elastic constants along the tubule axis generally soften with decreasing tubule radius, although with a distinct dependence on helical conformation.