Theoretical study on critical radius of nanotube-induced self-scrolling of graphene flake

Based on the van der Waals (vdW) interaction potential and the bending rigidity of graphene, the self-scrolling mechanical behaviors of monolayer graphene flake induced by carbon nanotube (CNT)/black phosphorus nanotube (BPNT) are studied, and the analytical expressions of critical radius of CNT/BPNT are derived when the graphene flakes spontaneously scroll round them. The factors affecting the critical radius of nanotubes are systematically analyzed, which include the axial length ratio between nanotube and graphene, wrapping angle of graphene, temperature, type and layer number of nanotube. It is found that the axial length ratio between nanotube and graphene has a significant effect on the critical radius. The critical radius of nanotubes increases with the decrease of the axial length ratio, and the critical radius of double-walled carbon nanotube (DWCNT) increases faster than that of single-walled carbon nanotube (SWCNT) and BPNT. When the SWCNT and graphene have the same axial length, the critical radius of SWCNT induced completely wrapping of monolayer graphene is about 0.4757 nm, which is in good agreement with the simulation results in the literature. The wrapping angle of graphene can also affect the critical radius of nanotubes, which increases with the increase of wrapping angle. Furthermore, the influence of environmental temperature, the type and layer number of nanotubes on the critical radius of nanotubes is discussed, respectively.