Elastic bending modulus of monolayer graphene

An analytic formula is derived for the elastic bending modulus of monolayer graphene based on an empirical potential for solid-state carbon atoms. Two physical origins are identified for the non-vanishing bending stiffness of the atomically thin graphene sheet, one due to the bond-angle effect and the other resulting from the bond-order term associated with the dihedral angles. The analytical prediction compares closely with ab initio energy calculations. Pure bending of graphene monolayers into cylindrical tubes is simulated by a molecular mechanics approach, showing slight nonlinearity and anisotropy in the tangent bending modulus as the bending curvature increases. An intrinsic coupling between bending and in-plane strain is noted for graphene monolayers rolled into carbon nanotubes. (Some figures in this article are in colour only in the electronic version) Theunique two-dimensional (2D) lattice structure andphysical properties of graphene have attracted tremendous interest recently. In particular, rippling of suspended graphene monolayers has been observed, with mesoscopic amplitude and wavelength [1]. Imaging of monolayer graphene sheets on silicon dioxide has also shown structural corrugation [2, 3]. Theoretical studies [4–6] have suggested that bending stiffness of the monolayer graphene is critical in attaining the structural stability and morphology for both suspended and supported graphene sheets, which in turn could have important impacts on their electronic properties. Furthermore, single-walled carbon nanotubes are essentially graphene monolayers subjected to cylindrical bending along the particular chiral directions. It has been reported that the bending curvature (inverse of the tube radius) and bending orientation (tube chirality) have dramatic effects on both the mechanical and electronic properties of carbon nanotubes [7, 8]. An accurate account of the bending modulus of graphene is thus important for understanding the mechanics of carbon nanotubes. While the in-planemechanical properties (elasticmodulus and strength) of monolayer graphene have been deduced from experiments [9, 10], direct measurement of bending stiffness of monolayer graphene has not been reported in the literature. The often cited experimental value of 1.2 eV (∼0.192 nN nm) was derived from the phonon spectrum of graphite [11]. Theoretically, bending modulus of monolayer graphene has been predicted based on empirical potentials [7, 12–14] and ab initio calculations [15, 16]. The fact that the atomically thin graphene monolayer has a finite bending modulus is in contrast to classical theories for plates and shells [17]. For example, the bending modulus of an elastic thin plate scales with the cube of its thickness, namely,