Thickness of graphene and single-wall carbon nanotubes

Young's modulus and the thickness of single wall carbon nanotubes (CNTs) obtained from prior atomistic studies are largely scattered. In this paper we establish an analytic approach to bypass atomistic simulations and determine the tension and bending rigidities of graphene and CNTs directly from the interatomic potential. The thickness and elastic properties of graphene and CNTs can also be obtained from the interatomic potential. But the thickness, and therefore elastic moduli, also depend on type of loading (e.g., uniaxial tension, uniaxial stretching, equibiaxial stretching), as well as the nanotube radius R and chirality when R<1 nm. This explains why the thickness obtained from prior atomistic simulations is scattered. This analytic approach is particularly useful in the study of multiwall CNTs since their stress state may be complex even under simple loading (e.g., uniaxial tension) due to the van der Waals interactions between nanotube walls. The present analysis also provides an explanation of Yakobson's paradox that the very high Young's modulus reported from the atomistic simulations together with the shell model may be due to the not-well-defined CNT thickness.