Elastic warping of graphitic carbon sheets: Relative energies of some fullerenes, schwarzites and buckytubes

The energies of hypothetical and the already found 'graphitic' (sp(2)) carbon networks with hyperbolic ('schwarzites'), elliptic ('fullerenes') and parabolic ('buckytubes') curvatures are compared with a simple elastic model of the warping energy of a spontaneously flat graphite sheet, analogous to bending energy models for amphiphilic membranes, invoking both bend and twist contributions. Splay and saddle splay moduli are estimated to be 190kT and -65kT, respectively, an order of magnitude stiffer than typical amphiphilic bilayer membranes. The model accounts adequately for relative self-energies of most curved graphitic networks. Many proposed allotropes of carbon may thus be treated as two dimensional warped layer-structures.