Edge Width Induced Warping Of Graphene Sheets

Graphene, as one of the most promising new material, exhibits unique and excellent performance in optics, electricity, thermology, and mechanics. It has broad applications in the fields of nanoelectronic devices, highly sensitive sensors, transparent conductive films, functional composite materials, energy storage, catalysis and so on. Studies show that, in the case of no thermal effect (at absolute zero temperature), due to the presence of edge stress, the independent boundary of graphene films still exhibits instability, which then influences the atomic structure of the boundary and the properties of grapheme. The present two-dimensional analysis methods are that:employing elastic plate model to the bulk graphene and beam model to the edges which have no geometry dimensions only area concept incorporated in its edge stress and elastic modulus, thus to determine the scale relations when the edges of graphene film are warped.However, these methods do not fully characterize the real condition of atomic structure of the graphene film’s edges. In the actual boundary of graphene films the atomic structure is different from interior. It implies that the actual edges of graphene thin film should have a certain geometrical width. Therefore the warped morphology of edges should be taken serious considerations in the study. The main work is as follows:(1) This paper starts with theoretical analysis. First of all, the graphene’s boundaries are simplified into ideal beam model, which has no geometry but area through establishing a two-dimensional thin plate model for the interior. A function describing the warping morphology is introduced, thus to investigate the relationships among the wavelength of warped geometry, the attenuation depth, and the warping amplitude when the graphene film of limited size reaches a stable situation.(2) A real two-dimensional plate model is established for graphene edges which can have definite geometrical width involved, in addition to the interior based on the theory of linear elasticity. The similar procedure is followed then to analyze the effect of edge width on warping morphology and relationships among the wavelength, the attenuation depth and the warping amplitude for armchair and zigzag edges, respectively.(3) As the edge width increases, the material of edges approaches that of interior, we can deal with the elastic modulus of graphene’s edge as a related function of the interior elastic modulus, then similar analyses are carried out. This treatment can enrich the definition scope of elastic modulus of graphene’s edge. At the same time the impact of real border width on the warping morphology is elaborated by considering the high-order strain components along the thickness direction of three-dimensional graphene film.Studies show that when the structure of graphene is stable, the warp wavelength of armchair edges is significantly longer than zigzag edges. And the same manifestation is observed in the warp amplitude. Border width shows an approximately linear relation to the total strain energy of structure, to the attenuation depth and to the warp amplitude. However, the warp amplitudechanges not obvious when border width changesin the case of armchair edges.