Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films

https://doi.org/10.1016/j.jmps.2005.02.012Get rights and content

Abstract

Atoms at a free surface experience a different local environment than do atoms in the bulk of a material. As a result, the energy associated with these atoms will, in general, be different from that of the atoms in the bulk. The excess energy associated with surface atoms is called surface free energy. In traditional continuum mechanics, such surface free energy is typically neglected because it is associated with only a few layers of atoms near the surface and the ratio of the volume occupied by the surface atoms and the total volume of material of interest is extremely small. However, for nano-size particles, wires and films, the surface to volume ratio becomes significant, and so does the effect of surface free energy. In this paper, a framework is developed to incorporate the surface free energy into the continuum theory of mechanics. Based on this approach, it is demonstrated that the overall elastic behavior of structural elements (such as particles, wires, films) is size-dependent. Although such size-dependency is negligible for conventional structural elements, it becomes significant when at least one of the dimensions of the element shrinks to nanometers. Numerical examples are given in the paper to illustrate quantitatively the effects of surface free energy on the elastic properties of nano-size particles, wires and films.

Introduction

Nanomaterials in general can be roughly classified into two categories. If the characteristic length of the microstructure, such as the grain size of a polycrystal, is in the nanometer range, it is called a nano-structured material. If at least one of the overall dimensions of a structural element is in the nanometer range, it may be called a nano-sized structural element. This may include nano-particles (Alymov and Shorshorov, 1999, Pei and Hwang, 2003), nano-belts (Pan et al., 2001), nano-wires, nano-films, etc. Clearly, nano-sized structural elements must, by necessity, be made of nano-structured materials.

This paper is primarily concerned with the elastic behavior of nano-sized structural elements such as nano-particles, nano-wires and nano-films. In particular, the size dependency of the overall elastic behavior of such nano-sized structural elements will be investigated.

The elastic behavior of a material is characterized by its elastic modulus, which defines the proportionality between the stress and strain when the material is subjected to external loads. Strictly speaking, modulus is an intensive property defined at each material point when the material is assumed to be a continuum. Therefore, it should be independent of the size of the material sample being considered. However, for inhomogeneous materials such as composites, it is often convenient for engineering design to define the overall (or effective) modulus of the material. Such effective modulus of a composite may depend on the properties of its constituents and the relative volume fraction of each constituent.

The reduced coordination of atoms near a free surface induces a corresponding redistribution of electronic charge, which alters the binding situation (Sander, 2003). As a result, the energy of these atoms will, in general, be different from that of the atoms in the bulk. Thus, the elastic moduli of the surface region may differ from those of the bulk. In this sense, all structural elements (large or small), are not strictly homogeneous. However, the surface region is typically very thin, only a few atomic layers. It is thus perfectly acceptable to neglect the surface region and to use the bulk modulus of a structural element as its overall modulus, when the size of the element is in micrometers or larger. For nano-sized structural elements, however, the surface-to-volume ratio is much higher and the surface region can no longer be neglected when considering the overall elastic behavior of nano-sized structural elements. Consequently, the effective modulus of nano-sized structural elements should be considered, and by definition becomes size-dependent.

To include the surface region in modeling nano-sized structural elements inevitably involves discrete (or atomistic) analysis because the boundary region is only a few atomic layers thick. So, one of the fundamental issues that needs to be addressed in modeling the macroscopic mechanical behavior of nano-sized structural elements is the large difference in length scales. To establish a link between the atomistic structure of surfaces and macroscopic bulk elastic behavior, we propose a two-step approach. First, the surface atomistic structure and interactions should be captured and cast into surface free energy, a thermodynamic quantity of a continuum. Then, this surface free energy will be included in the phenomenological description of strain energy density in modeling the macroscopic behavior of nano-sized structural elements. The surface energy calculation based on molecular dynamics will be presented in a separate paper (Dingreville et al., 2005). In the present paper, we focus on the second step, namely, developing a continuum framework that incorporates the surface free energy into the analysis of macroscopic deformation of nano-sized structural elements. In particular, we study the effects of surface free energy on the effective modulus of nano-particles, nano-wires and nano-films.

Previous work most relevant to this paper is the study on surface and interface stress effects in thin films. It has been found (Cammarata and Sieradzki, 1989, Cammarata and Sieradzki, 1994, Kosevich and Kosevich, 1989, Banerjea and Smith, 1987, Nix and Gao, 1998) that the surface free energy could increase the apparent in-plane bi-axial modulus of a Cu (1 0 0) free standing film of 2 nm thick by about 15–25%. Some experimental work (Catlin and Walter, 1960) seems to indicate that the modulus enhancement could be as much as 50%, although it has been pointed out by later studies (Itozaki, 1982, Baker et al., 1993) that such a large enhancement might be due to experimental errors. When the thickness reduces to below 5 nm, modulus enhancement/reduction of 20% was also predicted (Streitz et al., 1994a, Streitz et al., 1994b; Cammarata and Sieradzki, 1989) and confirmed experimentally for several multilayered metal films such as Cu–Nb (Fartash et al., 1991). More recently, Miller and Shenoy (2000) developed a simple model to incorporate surface stress in determining the size-dependent elastic modulus of plates and rods. Using molecular dynamic simulations, Zhou and Huang (2004) have shown that, depending upon the crystallographic orientations, the effective elastic modulus of a thin free-standing film can either increase or decrease as the film thickness decreases. The effect of surface stress on the growth of thin films has been investigated by several researchers (e.g., Cammarata, 1997, Nix and Clemens, 1999, Cammarata et al., 2000).

Another relevant area of research is the examination of elastic properties of grain boundaries. A number of publications have suggested that the elastic moduli in the grain boundary domain may differ significantly from those of the bulk. Wolf and co-workers (Wolf et al., 1989, Wolf and Lutsko, 1989, Kluge et al., 1990, Wolf and Kluge, 1990), who studied superlattices of (0 0 1) twist boundaries, as well as Adams et al. (1989), who examined the Σ=5(001) twist boundary in a thin film of copper, have found an increase of the Young's modulus perpendicular to the boundary plane and a substantial decrease of the shear modulus in the boundary plane in the atomic layers adjacent to the boundary. Bassani and co-workers (Alber et al., 1992, Bassani et al., 1992, Vitek et al., 1994, Marinopoulos et al., 1998) defined the local elastic modulus tensor and determined the values of the local elastic modulus tensor near grain boundaries in several face center cubic metals using molecular dynamic simulations. They, too, found that the local elastic moduli are significantly different for atoms near the grain boundaries.

Since grain boundaries have distinct elastic properties, the effective modulus of polycrystalline materials should also be dependent on the grain size because the interface-(grain boundary) to-volume ratio is inversely proportional to the grain size. Unlike the thin film case, however, existing literature has shown mixed results on the dependency of modulus on grain sizes. Some have reported reduction of elastic modulus by as much as 30% (Suryanarayana, 1995, Gleiter, 1989, Korn et al., 1988) for nano-structured materials. Others (e.g., Nieman et al., 1991, Krstic et al., 1993, Fougere et al., 1995) argued that such reduction is purely due to porosities. However, careful molecular dynamic simulations of copper polycrystal (Schiøtz et al., 1998) have shown that the Young's modulus is indeed reduced by over 25% when the grain size is reduced to 5 nm, even when the polycrystal is fully dense. Similar reduction is seen in simulations where the nanocrystalline metal is grown from a molten phase (Phillpot et al., 1995).

It should be mentioned in passing that an elegant mathematical theory incorporating surface stress and interfacial energy into the continuum mechanics formulation was proposed in the 1970s by Gurtin and his co-workers (e.g., Gurtin and Murdoch, 1975, Gurtin and Murdoch, 1978, Murdoch, 1976, Gurtin et al., 1998). Based on this theory, Sharma and Ganti (2003) have developed the size-dependent Eshelby's tensor for embedded nano-inclusions incorporating interfacial energy. The size-dependent effective modulus of an elastic matrix containing spherical nano-cavities at dilute concentration was obtained by Yang (2004).

Section snippets

Surface free energy and surface stress

There are different ways in which the properties of the surface can be defined and introduced. For example, if one considers an “interface” separating two otherwise homogeneous phases, the interfacial property may be defined either in terms of an inter-phase, or by introducing the concept of a dividing surface. In the first approach, the system is considered to be one in which there are three phases present—the two bulk phases and an inter-phase; the boundaries of the inter-phase are somewhat

Effective modulus of a particle

Conventionally, the elastic modulus of a material is an intensive property. It is defined as a point-wise quantity that relates the stresses and strains at each point in the material. When a material is not homogeneous, such as a composite material, its elastic modulus may vary from point to point. In this case, the concept of effective modulus can be introduced. For example, effective modulus is used to characterize the overall stiffness of a fiber reinforced composite, where the fiber and

Numerical examples and discussion

In this section, several numerical examples for the effective modulus and effective Poisson's ratio of copper spherical particles, wires of square cross-section and films are presented. For the films and wires, it is assumed that they are made of copper single crystals and that their crystallographic directions coincide with the surfaces of the films and wires as shown in Figs. 4 and 5. The cubic (second-order) elastic constants of the copper single crystals are C11=167.38GPa, C12=124.11GPa.

Summary

In this paper, a framework is developed to incorporate the surface free energy into the continuum theory of mechanics. Analytical expressions were derived for the effective elastic modulus tensor of nano-sized structural elements that account for the effects of surface free energy. Explicit expressions of the effective elasticity tensors were obtained for thin films, wires and spherical particles. The solutions derived here show that the overall elastic properties of structural elements (such

Acknowledgements

The work is supported in part by the NSF Packaging Research Center at Georgia Tech. JQ was also supported in part by NSFC through Grant 10228204 and by Harbin Institute of Technology.

References (54)

  • D. Sander

    Surface stress: implications and measurements

    Curr. Opin. Solid State Mater. Sci.

    (2003)
  • V. Vitek et al.

    Relationship between modeling of the atomic structure of grain boundaries and studies of mechanical properties

    J. Phys. Chem. Solids

    (1994)
  • G.J. Ackland et al.

    Semi-empirical calculation of solid surface tensions in B.C.C. transition metals

    Philos. Mag. A

    (1986)
  • J.B. Adams et al.

    Elastic properties of grain boundaries in copper and their relationship to bulk elastic constants

    Phys. Rev. B

    (1989)
  • I. Alber et al.

    Grain boundaries as heterogeneous systems: atomic and continuum elastic properties

    Philos. Trans. R. Soc. London A

    (1992)
  • R. Aris

    Vectors, Tensors, and Basic Equations of Fluid Mechanics

    (1962)
  • Baker, S.P., Small, M.K., Vlassak, B.J., Daniels, B.J., Nix, W.D., 1993. The search for the supermodulus effect. In:...
  • A. Banerjea et al.

    Continuum elasticity analysis of the enhanced modulus effect in metal–alloy superlattice films

    Phys. Rev. B

    (1987)
  • J.M. Blakely

    Introduction to the Properties of Crystal Surfaces

    (1973)
  • M.J. Buerger

    Elementary Crystallography

    (1963)
  • R.C. Cammarata et al.

    Effects of surface stress on the elastic moduli of thin films and superlattices

    Phys. Rev. Lett.

    (1989)
  • R.C. Cammarata et al.

    Surface and interface stresses

    Annu. Rev. Mater. Sci.

    (1994)
  • R.C. Cammarata et al.

    Surface stress model for intrinsic stresses in thin films

    J. Mater. Res.

    (2000)
  • Capolungo, L., Cherkaoui, M., Qu, J., 2005, A self consistent model for the inelastic deformation of nanocrystalline...
  • A. Catlin et al.

    Mechanical properties of thin single-crystal gold films

    J. Appl. Phys.

    (1960)
  • J. Diao et al.

    Surface-stress-induced transformation in metal nanowires

    Nat. Mater.

    (2003)
  • J. Diao et al.

    Atomistic simulation of the structure and elastic properties of gold nanowires

    J. Mech. Phys. Solids

    (2004)
  • Cited by (0)

    View full text