Icosahedral nano-micro capsules with minimal bending energy in regular polyhedra

The lattice model proposed by Lenosky et al. [Nature 355, 333(1992)] have been derived to describe the curvature elastic energy of smoothly curved fullerene surface by Ou-Yang et al. [Phys. Rev. Lett. 78, 4055(1997)]. Recently, we have extended the lattice model to describe the elastic energy of regular polyhedron. This paper will give the detail derivation of the elastic energy of discrete polyhedric structure. It is shown that the icosahedron is the most optimized structure with minimum elastic energy. We also obtain the threshold radius below which the spherical capsule will transform into icosahedron.