Material Properties in Codimension > 0: Graphene Edge Properties

Abstract

AbstractWhen materials are very thin in one or more dimensions, their equilibrium shapes are often curved/bent. Such shapes commonly represent a compromise between elastic strain energy and other thermodynamic forces (e.g. related to surface stresses, electrostatic interactions, or adsorption). Examples include ZnO and SnO2nanobelts, silica/carbonate helicoids, and graphene sheets and nanoribbons. Here, we demonstrate that when the equilibrium shape of a nanomaterial is not flat/straight, important fundamental material properties may be orders of magnitude different from their bulk counterparts. We focus here primarily on the graphene edges. Graphene in three dimensions is a codimensionc= 1 material; the codimension isc = D – d= 3 – 2 = 1, whereDis the dimensionality of the space in which the material is embedded anddis the dimensionality of the object. By contrast, a flat graphene sheet hasc= 2 – 2 = 0. We use the REBO-II interatomic potential to calculate the edge orientation dependence of the edge energy and edge stresses of graphene withc= 0 andc= 1. The edge stress for all edge orientations is compressive withc= 0. Both edge energy and stresses are in reasonable agreement with DFT calculations. The compressive edge stresses inc= 0 lead to edge buckling (out-of-the-plane of the graphene sheet) for all edge orientations (c= 1). The edge buckling inc= 1 reduces all edge energies and dramatically reduces all edge stresses to near zero (more than an order of magnitude drop). We also report the effect of codimension on the free energy and entropy of a graphene sheet and the elastic properties of ZnO nanohelices.