The effect of nonuniform surface elasticity on buckling of ZnO nanowires

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Abstract

This paper investigates the effect of surface elasticity on the Euler buckling of ZnO nanowires under axial compressive loads, in particular, on the effective elastic modulus, the critical compressive load, and the critical stress. Two modes of boundary conditions are studied in this paper: the fixed–fixed nanowire mode (mode 1) and the fixed–pinned nanowire mode (mode 2). Depending on a group of fitting surface parameters, analytical solutions for the effective elastic modulus, the critical buckling load and the critical stress with the effect of surface elasticity are presented. It is found that the surface elasticity plays a significant role in the buckling of ZnO nanowires because of the large surface-to-volume ratio. This study is expected to be helpful for further research of nanosized elements and nanobeam-based devices.

Graphical abstract

Surface elasticity has a significant impact on the critical stresses for given buckling ZnO nanowires under different boundary condition modes.

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Highlights

► We investigate the effect of surface elasticity on the Euler buckling of ZnO nanowires under axial compressive loads. ► Modified core–shell model that introduces an exponential surface elasticity into the surface region is presented. ► Analytical results indicate that the surface elasticity affects the buckling behaviors of the nanowires significantly.

Introduction

Nanowires or nanobeams, as the basic elements of nanoelectromechanical systems (NEMS), have attracted much interest from researchers. Consequently, investigating the exact characterization of the mechanical properties of nanowires is becoming a critical issue. Presently, there are many reports on the mechanical properties of ZnO nanowires [1], [2], [3], [4], [5]. For the application of nanowires in NEMS, axial buckling of the nanowires is a major topic of particular focus. Recent experimental reports [5], [6], [7] have discussed the buckling of such ZnO nanowires under uniaxial compression, and proposed a new method to observe the critical load, elastic modulus, and stress or strain of the nanowires based on the Euler or Johnson buckling theory. As is widely known, when the size of a material is reduced to the nanoscale, owing to the large surface-to-volume ratio, surface effects will play important roles in the nanostructures. Many experimental measurements [8], [9], [10], [11], [12], [13], [14] have shown that the elastic property of nanostructures is markedly size-dependent, especially their lateral dimensions, which are down to several hundreds of nanometers or less. Among numerous models of surface effects, the theory of surface elasticity proposed by Gurtin et al. [15] has been widely employed and developed to account for the surface effects on nanoscaled elements. Since then, other theoreticians have further analyzed the surface effects, including the surface tension and surface elasticity affecting the buckling or bending of nanowires [16], [17], [18], [19], [20], [21], [22]. Wang and Feng [16], [17], [18] analyzed the impacts of surface elasticity and residual surface tension on the vibration frequency and buckling of nanowires based on the Young–Laplace equation and on an assumption that the surface layer has zero thickness (i.e. the core–surface model). He and Lilley [19], [20] studied the influences of surface stress and surface elasticity on the static and dynamic bending of nanowires by incorporating the generalized Young–Laplace equation into the Euler–Bernoulli beam theory. Recently, Li et al. [21] utilized an energy model to investigate the effects of surface elasticity and residual surface tension on the post-buckling of nanowires under uniaxial compression. Park [22] studied how surface stresses impacted on the critical buckling strains of silicon nanowires based upon the recently-developed surface Cauchy–Born model. Pradhan and Murmu. [23] investigated the buckling behavior of single-layered graphene sheet which is embedded in an elastic medium based on nonlocal elasticity theory. All of the above results indicate that surface effects play an important role in the bending or buckling of nanostructures. In summary, most of the above theories have a common basis, that is, the surface layer of the nanostructures, which ideally adheres to the bulk material, is regarded as being of zero thickness or approaching zero thickness. This assumption of a zero thickness surface layer is expected to lead to an unrealistic physical image. This is of particular note in the case of nanomaterials, where such an assumption is considered to be invalid because of the small size of the nanomaterials, for instance, the cross section of a nanowire or nanobeam is typically of a few tens of nanometers or less in size. At the nanoscale, the atoms within the thin layer near the surface of a nanomaterial exhibit a different mechanical behavior from those of atoms in the bulk material and will thus start to play a prominent role. As such, the surface shell thickness may greatly affect the mechanical properties of nanomaterials and should not be ignored. Therefore, it is necessary to introduce a certain surface thickness into such models to account for the effect of surface elasticity. Theoretically, the core–shell model [11], [12], [13], [14], [24] is often employed to quantify the elasticity of bulk and surface regions; however, it is probably more suitable for composite materials because of the step-like elastic modulus.

In the present paper, we adopt a core–shell structure to divide the cross section of the nanowire into two parts: the bulk region and the surface region. However, the elasticity in the surface region of the nanowires is not uniform but gradually increases or decreases exponentially. The trend of nonlinear behavior of surface elasticity has been suggested in previous research [25], which implies the elasticity in the non-bulk layer can be expressed by an exponential decay function in Si nanoplates. However, the same research did not investigate how this behavior in the non-bulk region affects the elastic modulus of nanostructures. Based on the assumption of exponential surface elasticity, we derived analytical solutions for the effective elastic modulus, the critical load and the critical stress when the nanowire is buckled under an external load. Using ZnO nanowires as an example, two different boundary conditions (fixed–fixed and fixed–pinned) are selected to show how the surface elasticity impacts on the effective elastic modulus, critical loads and stresses. The conclusions we derived indicate that the surface elasticity has an important and distinct impact on the buckling of the nanowires with decreasing nanowire cross-section and slenderness ratio.

Section snippets

Theory basis

It is well known that the atoms within a very thin layer near the surface of nanomaterials experience a different circumstance from those of the bulk atoms; consequently, the mechanical properties in the surface region are distinct from those in the bulk region. This observation is usually attributed to the effects of surface energy, which leads to the formation of surface stresses. Generally, the surface stresses include two portions: “strain-independent” surface stress, i.e., residual surface

Numerical results and discussion

We know that the stiffening elastic property for ZnO nanowires with decreasing feature size is verified by many experiments [11], [12], [13], [14]. This observation means that the sign of the inhomogeneous degree parameter α0 is positive, which physically describes the degree of inward compaction of atoms around the surface of the nanowires. First, we will examine the normalized critical load of buckling as a function of the nanowire diameter D, and the surface thickness rs, taking the

Summary

In summary, we have investigated the effect of surface elasticity on the buckling of ZnO nanowires under an axial compressive load based on the buckling theory for an Euler beam. Unlike the previously reported results, the surface elastic modulus in this paper is not a uniform quantity but a variable quantity that varies exponentially in the surface region (shell) of the nanowires. On this assumption, the analytical results indicate that the surface elasticity affects the buckling behaviors of

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 11072104, and the Program for Innovative Research Team of the Inner Mongolia University under Grant no. 10013-12110605. H.Y. Yao also gratefully acknowledges the “211 project” Innovative Talents Training Program of the Inner Mongolia University, China.

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