Material properties in codimension > 0: Graphene edge properties

Abstract

When 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 SnO2 nanobelts, 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 codimension c = 1 material; the codimension is c = D-d = 3-2 = 1, where D is the dimensionality of the space in which the material is embedded and d is the dimensionality of the object. By contrast, a flat graphene sheet has c = 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 with c = 0 and c = 1. The edge stress for all edge orientations is compressive with c = 0. Both edge energy and stresses are in reasonable agreement with DFT calculations. The compressive edge stresses in c = 0 lead to edge buckling (out-of-the-plane of the graphene sheet) for all edge orientations (c = 1). The edge buckling in c = 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. ©2010 Materials Research Society.