Mechanics of nanoscale wrinkling of graphene on a non-developable surface

As a two-dimensional crystal, the graphene sheet is often used with substrate materials because the freestanding graphene tends to corrugate and can hardly display its extraordinary properties in devices. However, the substrate is rarely perfectly-flat, but has microscopic roughness. Whether or not the graphene sheet can conform fully to a substrate with nano- and micro-roughness is essential for the performance of the graphene-based devices. In this paper, a theoretical model is developed to predict the morphology of a monolayer graphene sheet attaching to the substrate with microscopic non-developable roughness by an energy-based analysis. The final graphene morphology is revealed to result from the competition between two energy terms: the adhesion energy between graphene and substrate, and the strain energy stored in the graphene due to the deformations. Thus, by accounting for these two parts of energy, the critical condition to predict the morphology conversion from full conformation to wrinkling is established, which agrees well with the results of molecular dynamics simulations. This study has significant meanings for design and fabrication of high quality nanostructured coatings on substrates with complex surfaces, and can offer a guide for designing new functional graphene electronical devices such as nano-sensors and nano-switches as well.