Moiré patterns and carbon nanotube sorting

Moiré patterns (MPs), arising from the superposition of two lattices with close periods, are tightly related to the physicochemical properties of bilayer nanostructures. Here, we develop the theory of complex MPs emerging in twisted bilayer graphene and planar nets of double-walled nanotubes at significant relative twist and/or deformation of layers. The proposed theory clarifies the physicochemical regularities arising at sorting of single-walled carbon nanotubes (SWCNTs) by organic molecules, which self-assemble in regular coatings on both the tubes and planar graphene. We introduce and consider an outer tubular virtual lattice that is a parent structure for the deposited coating and due to this fact, its existence is crucial for the coating formation. As we show, such outer lattices exist only for successfully sorted SWCNTs and the superposition between the outer lattice and SWCNT forms a specific long-period MP. We explain known experimental results of SWCNT sorting by molecules of flavin group, poly(9,9-dioctylfluorene-2,7-diyl), and poly [(m-phenylenevinylene)-alt-(p-phenylenevinylene)]. Also, our approach points out other organic molecules and polymers suitable for effective carbon nanotube sorting.