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.

http://dx.doi.org/10.1098/rsta.1996.0088