The performances of graphene sheet in micro- and nano-electronics and devices are significantly affected by its morphology, which depends on the surface features of the supporting substrate. The substrates with non-developable concave surface are widely used with graphene sheet in applications but rarely studied. Therefore, a theoretical model is established based on the energy analysis to explain the adhesion mechanisms and predict the morphology of the graphene sheet on a non-developable concave surface. Four different morphologies of the graphene sheet are revealed, and the critical conditions are established to predict which morphology the graphene/substrate system belongs to. For the monolayer graphene sheets much larger than the concave of substrate, the final equilibrium morphology is dominated by the half cone angle of the concave. The graphene sheet conforms completely to the SiO2 substrate if the half cone angle is less than 27.5° and spans over the concave if the angel is larger than 27.5°. For graphene sheets smaller than the concave, they fall into the concave and the final morphology depends only on the ratio of graphene radius to concave radius. The monolayer graphene sheet conforms to the concave if the radius ratio is less than 0.51 and wrinkles if the ratio is larger than 0.51. The theoretical results are verified by a series of molecular dynamics simulations on various graphene/substrate systems. This work can provide guidelines to design high quality graphene-coated functional materials and devices, and can offer criterion for graphene-derived nano-electronics and nano-sensors.

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