Surface effects on the elasticity of nanosprings

Nanosprings made of metallic or semiconducting materials hold promise for a wide range of important applications. In this letter, we extend the classical Kirchhoff rod model by incorporating surface effects on the mechanical response of quasi–one-dimensional nanomaterials. The refined model is then employed to derive the elastic constants of a nanospring by accounting for the effect of surface elasticity and residual surface stresses. The results demonstrate that the stiffness of nanosprings may exhibit remarkable dependence on their cross-sections sizes. This study is helpful not only for understanding the size-dependent behavior of nanosprings but also for their applications in micro/nano-electro-mechanical systems. Copyright c © EPLA, 2010 Quasi–one-dimensional nanomaterials with chiral morphologies such as nanosprings, helical nanobelts, and nanotubes are found widely in both synthesized and biological materials. Typical examples of such nanohelices are ZnO and SiOx nanowires with helical shape [1–3], and amorphous boron carbide nanosprings [4,5]. Due to their asymmetric geometry and tiny size, such helical nanomaterials as nanosprings and helical nanobelts exhibit some unusual properties and functions. They hold promise for diverse applications in nanoengineering and biological areas as biosensors, biological force probes, and functional elements in nanoelectronics and nanoelectromechanical systems. In recent years, much effort has been devoted to the synthesis of nanosized helical materials and understanding the physical mechanisms underlying the formation of their asymmetric morphologies [5–9]. As a consequence, nanosprings made of metallic, semiconducting and polymer materials have been fabricated and employed in a wide variety of engineering applications. The identification of the elastic property of such nanosprings is a major concern in their applications. Seto et al. [10,11] used the nanoindentation technique to measure the mechanical stiffness of an array of helical nanosprings. They found that the measured value of a single nanospring had a distinct (a)E-mail: [email protected] difference from that predicted by the classical theory of elasticity. The elastic properties of coiled multiwalled nanotubes, silicon nanosprings and carbon nanocoils have been experimentally characterized using atomic force microscopy [12–14]. Zhang and Zhao [15] showed that the boundary conditions can significantly affect the measured stiffness values. da Fonseca et al. [16,17] investigated the mechanical properties of amorphous nanosprings using the conventional Kirchhoff rod model. Wang et al. [9] addressed the significant effect of anisotropic surface stress on the chiral morphology of some quasi–one-dimensional nanomaterials, and the results were verified by recent experiments [18]. As is well known, surface effects often play a significant role in the mechanical and physical properties of nanomaterials due to their large surface-to-volume ratios. Surface stress and surface energy often lead to size dependences at microand nanoscales. Gurtin and Murdoch developed a surface/interface theory of elasticity [19], in which a surface/interface is modeled as a zero-thickness layer which has different elastic properties and is ideally bonded to the bulk material. The theory has been widely adopted to predict the mechanical behavior of microor nanosized elements, e.g., nanobeams and nanoplates [20,21]. To consider the effect of residual surface stresses on mechanical response, for example, Wang and Feng developed a surface-layer model to calculate the natural