Engineering materials properties in codimension > 0
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When thin nanomaterials spontaneously deform into nonflat geometries (e.g., nanorods into nanohelices, thin sheets into ruffled forms), their properties may change by orders of magnitude. We discuss this phenomenon in terms of a formal mathematical concept: codimension c 5 D d, the difference between the dimensionality of space D, and that of the object d. We use several independent examples such as the edge stress of graphene nanoribbons, the elastic moduli of nanowires, and the thermal expansion of a modified bead-chain model to demonstrate how this framework can be used to generically understand some nanomaterial properties and how these properties can be engineered by using mechanical constraints to manipulate the codimension of the corresponding structure.
I. INTRODUCTION
Low-dimensional nanomaterials such as carbon nanotubes (CNTs),1,2 nanowires,3,4 and graphene5 have properties that are very different from bulk materials. Although these properties are usually attributed to their high surface area to volume ratio or quantum confinement effects, these materials are distinct from the bulk in another important way. Low-dimensional nanostructures can be flexible and may not be straight/flat at equilibrium. These shapes arise because of a compromise between bending energy and other forces, for example, surface stress, internal stress, electrostatic interaction, or adsorption. There are myriad examples of such nanostructures, including ZnO and SnO2 nanobelts,6,7 graphene nanoribbons (GNRs),8 silica/carbonate helicoids,9 SiGe nanodrills,10 and SiGe/Si/Cr nanocoils.11 Because thermodynamic properties are defined relative to an equilibrium state, the properties of a nanostructure will depend on the shape of that structure. Although these shapes are not unique to the nanometer scale, they are more common in nanostructures because of the strong influence of surfaces and low mechanical rigidity. This observation leads us to classify nanostructures based on the number of dimensions into which they can spontaneously deform c. Formally, c is the codimension of the nanostructure: the difference between the dimensionality of the space in which the material is embedded D (i.e., in which it can deform), and the intrinsic dimensionality of the object d.12,13 The codimension is a measure of the spatial degrees of freedom of the nanostructure. For example, a graphene sheet and a nanowire embedded in a three-dimensional space has codimension c 5 3 2 5 1 and c 5 3 1 5 2, respectively. If a nanowire, in three dimensions, is constrained to remain a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2011.306 J. Mater. Res., Vol. 27, No. 3, Feb 14, 2012
straight (i.e., it is prevented from deforming into the other two dimensions), it has codimension c 5 1 1 5 0. The properties of a nanostructure can be distinct from the properties of the corresponding bulk material. The thermodynamic properties of a nanostructure are defined relative to nanoscale variables. For example, for a nanowire, we may
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