Bending rigidity and Gaussian bending stiffness of single-layered graphene

Bending rigidity and Gaussian bending stiffness are the two key parameters that govern the rippling of suspended graphene-an unavoidable phenomenon of two-dimensional materials when subject to a thermal or mechanical field. A reliable determination about these two parameters is of significance for both the design and the manipulation of graphene morphology for engineering applications. By combining the density functional theory calculations of energies of fullerenes and single wall carbon nanotubes with the configurational energy of membranes determined by Helfrich Hamiltonian, we have designed a theoretical approach to accurately determine the bending rigidity and Gaussian bending stiffness of single-layered graphene. The bending rigidity and Gaussian bending stiffness of single-layered graphene are 1.44 eV (2.31 x 10(-19) N m) and -1.52 eV (2.43 x 10(-19) N m), respectively. The bending rigidity is close to the experimental result. Interestingly, the bending stiffness of graphene is close to that of lipid bilayers of cells about 1-2 eV, which might mechanically justify biological applications of graphene