A new computational approach for evaluating bending rigidity of graphene sheets incorporating disclinations

Two-dimensional (2D) materials exhibit remarkable flexibility and can be transformed into various shapes. Graphene sheets (GSs), in particular, can form conical or saddle-like shapes through the introduction of lattice defects known as disclinations, represented by 5- and 7-membered rings, respectively. These rotational-type lattice defects possess relatively large spontaneous curvature and significantly affect the bending rigidity of the GS. Despite increasing interests in exploiting such deformations for material design, evaluating the bending rigidity of GSs with lattice defects remains challenging owing to the complexity introduced by curvature and defect configurations. In this study, we propose a novel computational method that integrates the Helfrich theory of membranes with molecular-dynamics simulations to analyze the effect of curvature and defect patterns on the bending rigidity of GSs. This hybrid approach enables the direct evaluation of bending rigidity from atomic and geometric structures, eliminating the need for experimental bending tests. Using this method, we reveal, for the first time, contrasting trends in bending rigidity between GSs with monopole and dipole disclinations. In the presence of disclination monopoles, the bending rigidity remains independent of the specific structural pattern. Conversely, disclination dipoles, comprising both conical and saddle-shaped surfaces, induce local shape distortions that lead to localized variations in bending rigidity. These findings provide important guidelines for the design of 2D materials with specific bending rigidities, supporting the development of new materials.