  Spontaneous polygonization for a multi-walled carbon nanotubes (MWCNTs) have
been observed for about two decades. In present manuscript, this phenomenon is
understood by the competition between cohesion energy (with lattice mismatching
effect) and curvature elastic energy of tubes inside the MWCNTs. Based on a
continuum elastic model and perturbation analysis, the crucial conditions for
spontaneous polygonization of MWCNT is expressed by fundamental parameters of
MWCNTs, which are in reasonable agreement with all relevant experiments and
computer simulations reported in literatures. Present approach can be used in
further studies of crossectional geometry-dependent properties of MWCNTs.
