Anomalous Flexural Elasticities of Graphene Membranes Unveiled by Manipulating Topology

Mechanical behavior of atomically thin membranes is governed by bending rigidity and the Gaussian modulus. However, owing to methodological drawbacks, these two parameters have not been investigated sufficiently. We employed atomic force microscopy to demonstrate that the bending rigidity can be extracted from a quadratic relationship of adhesion energy with monolayer curvatures of rolled and unrolled graphene. The tip-induced topological defects revealed the Gaussian modulus; to the best of our knowledge, this is the first study on these parameters. Our study may hold great significance because existing investigations have been performed only on flat graphene. The configurational (strain) energy was evaluated via changes in the surface geometry, with subatomic resolution, by three-dimensional analyses of attractive interatomic forces. The mechanical parameters, evaluated at the hollow sites of the honeycomb lattice, were consistent with the isotropic elastic attributes. The remarkably large negative Gaussian modulus, observed when a single carbon atom was located at the center of the tip-induced bump, revealed attractive interactions between the topological defects and geometric potentials of the Gaussian curvature. Our approach will aid in developing two-dimensional materials and understanding cell biology.