NANO LETTERS
On The Elastic Modulus of Metallic Nanowires
2008 Vol. 8, No. 11 3613-3618
Matthew T. McDowell,† Austin M. Leach,† and Ken Gall‡,* School of Materials Science and Engineering and George Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332 Received May 28, 2008; Revised Manuscript Received July 24, 2008
ABSTRACT Previous atomistic simulations and experiments have attributed size effects in the elastic modulus of Ag nanowires to surface energy effects inherent to metallic surfaces. However, differences in experimental and computational trends analyzed here imply that other factors are controlling experimentally observed modulus changes. This study utilizes atomistic simulations to determine how strongly nanowire geometry and surface structure influence nanowire elastic modulus. The results demonstrate that although these factors do influence the elastic modulus of Ag nanowires to some extent, they alone are insufficient to explain current experimental trends in nanowire modulus with decreasing dimensional scale. Future work needs to be done to determine whether other factors, such as surface contaminants or oxide layers, contribute to the experimentally observed elastic modulus increase.
Metallic nanowires have remained a subject of intense research efforts on various experimental and theoretical fronts. In order to successfully integrate nanowires into emerging devices, it is imperative to obtain a comprehensive understanding of their size-dependent properties. The mechanical behavior of nanowires is an especially important aspect of their structural performance and functional behavior. In particular, the elastic modulus, or stiffness, of nanowires controls their capacity for deformation under load and their resonance frequency under oscillatory loading often imposed during sensing and actuation. The effects of free surfaces have been shown to impact the elastic modulus of metallic nanowires and other structures with nanometer scale dimensions.1,2 Atoms at or near a free surface have reduced coordination relative to interior lattice atoms. This reduced coordination gives rise to surface energy. Although the effect of surface energy on mechanical properties is usually insignificant on a bulk scale, it has been found that surface energy and associated surface relaxations influence the mechanical behavior of nanostructures.2-4 These effects are realized because the surface area to volume ratio increases dramatically when size decreases to the nanometer range; this causes a proportional increase in the role that surfaces play during mechanical deformation. Atomistic studies have shown that surface stress and surface energy contribute to changing both the elastic modulus and strength of nanowires as their characteristic size decreases to below * Corresponding author. E-mail:
[email protected]. † School of Materials Science and Engineering. ‡ School of Materials Science and Engineering and George W. Woodruff School of Mechanical Engineering. 10.1021/nl801526c CCC: $40.75 Published on Web 10/24/2008
2008 American Chemical Society
about 10 nm.3,5,6 For example, Miller and Shenoy3 developed a theoretical model to predict changes in elastic properties due to surface effects in nanowires over varying length scales, and atomistic simulations were performed to support the model. In addition to the role of surface energy on the elastic modulus of nanowires, a computational study by Liang et al.7 demonstrates that the nonlinear elastic response of the metallic nanowire core also plays a significant role in either causing a decrease or increase in elastic modulus with decreasing nanowire size. In the equilibrium state, tensile surface stress in nanowires causes an axial compressive strain in the nanowire core. As a result of the increased influence of surface stress in smaller nanowires, this compressive strain increases in magnitude with decreasing nanowire size. Due to the inherent nonlinear elastic response of many single crystal bulk metals under compression, the elastic modulus of the nanowire core changes with increasing compressive stress, which is driven by decreasing nanowire size. Other atomistic studies have shown variations in nanowire elastic modulus with size and have explained the results by using the ideas of nonlinear elasticity, surface effects, or a combination of the two. Kulkarni et al.8 demonstrate that the elastic modulus of ZnO nanobelts increases with decreasing lateral dimension, and they attribute this effect to surface stress-induced compressive stresses in the nanowire core. In another study, bending and tensile simulations designed to measure the elastic modulus of Ag nanowires result in modulus values that increase or decrease from the bulk value with decreasing nanowire size depending on axial orientation.9 In all of these computational studies, size-dependent
Figure 1. Experimental and computational normalized elastic modulus data for nanowires with thicknesses ranging from 2 to 200 nm. The experimental data were collected from four different studies, and the computational simulations on 〈110〉 oriented nanowires with square cross sections were performed by the authors. The normalized elastic modulus is calculated by dividing the calculated value by the bulk value. For 〈110〉 axially oriented Ag nanowires (from refs 11 and 12 and in the simulations), the bulk modulus value was 84.1 GPa.22 For the Ag nanowires with unspecified crystallinity (ref 10), a bulk value of 76 GPa was used.23 For the polycrystalline Au nanowires, a bulk value of 78 GPa was used.24
changes in elastic modulus were observed in nanowires with cross-sectional dimensions below about 10 nm. Although well-founded theoretical and computational models predict a strong size scale effect below 10 nm, such results have not been critically compared to experimental studies emerging in the literature. Several experimental studies have been performed to measure the elastic modulus of nanowires. Consistent with atomistic studies, many of these experiments have revealed changes in the elastic modulus of metallic nanowires, relative to the expected bulk value, as dimensional scale is reduced. A study by Cuenot et al.10 shows that the measured elastic modulus values for both Ag and Cu nanowires begins to increase when the diameter of the nanowires falls below about 150 nm. In work by Jing et al.,11 the elastic modulus for Ag nanowires is shown to increase over a similar decreasing size scale. Another experimental study presents the elastic modulus of Ag nanowires with cross-sectional dimensions ranging from 22 to 36 nm, and the average value of the elastic modulus was found to be higher than the bulk value.12 Even experimental studies of ceramic nanowires, including CuO and ZnO, have also demonstrated an increase in elastic modulus with decreasing nanowire diameter.13,14 These experimental works have attributed the variations in nanowire elastic modulus to a multitude of factors. All studies acknowledge that surface effects play a role in determining the elastic modulus of nanowires, although the precise origin of the surface effect is often elusive. Some of the studies ascribe experimentally observed changes in elastic modulus in smaller nanowires to surface stress effects; in other words, the modulus is different for smaller nanowires due to the increased influence of under-coordinated surface 3614
atoms as the surface to volume ratio increases.10,13,14 This explanation is analogous to results on sub 10 nm wires from theoretical and computational studies. In the experimental study by Jing et al. regarding Ag nanowires, the elastic modulus size dependence is proposed to be due to a combination of the effects of surface stress, surface roughness, and an oxide layer.11 An explanation based on roughness and oxide brings to light factors that are typically not considered in computational simulations. In an attempt to better understand size effects on the elastic modulus of nanowires, it is useful to compare experimentally obtained modulus values from the literature with modulus values calculated from computational simulations. Figure 1 is a graph of metallic nanowire elastic modulus data over a range of cross-sectional sizes collected from four experimental studies10-12,15 and from representative molecular statics simulations performed by the authors (simulation procedures are described in a following section). In Figure 1, the normalized elastic modulus is defined as the ratio of the calculated modulus to the modulus for the bulk material, which varies slightly depending on orientation and crystallinity. The computational simulations and three of the experimental studies tested Ag nanowires, while the fourth experimental study tested Au nanowires. Ag and Au nanowires were chosen for comparison because of their structural similarity and the abundance of experimental data documenting their elastic properties. Figure 1 shows a conglomeration of data from different experiments and experimental conditions, such as nanowire orientation and structure. As such, the experimental data in this figure should not be interpreted as showing a continuous trend. However, valuable insight into the current understandNano Lett., Vol. 8, No. 11, 2008
ing of the elastic properties of nanowires (particularly metallic nanowires) can be gained by examining the data in Figure 1. The experimental data for the Au and Ag nanowires show that for nanowires with thicknesses greater than approximately 80-90 nm, the average measured elastic modulus values are similar to the bulk value for all nanowire orientations or structures. For nanowires with thicknesses between 20 and 80-90 nm, the three experimental Ag nanowire studies show that the average elastic modulus values increase with decreasing thickness. For the Au nanowires in this size range, the modulus values remain similar to the bulk value. The Ag nanowire simulation data are similar to the experimental data for Ag nanowires in that the simulations also show a sharp increase in elastic modulus from the bulk value as thickness decreases; these results are consistent with other published computational studies.3,7 However, the thickness range over which the data from the simulations increase from the bulk value is much narrower than the experimental thickness range. In addition, the elastic modulus in the simulations increases at significantly smaller thickness values compared to the experiments. This difference is highlighted by comparing the elastic modulus values for the smallest experimentally tested nanowires to the modulus value for the largest computationally tested nanowire, which have similar thicknesses (∼20 nm). The smallest experimentally tested nanowires have modulus values that are nearly twice the bulk value, while the largest simulated nanowire has an elastic modulus that is much closer to the bulk value. Attempting to understand the differences in the calculated elastic modulus values of Ag nanowires in simulations and experiments will provide insight into the factors driving the elastic response of metallic nanowires. As previously stated, most computational and experimental works have explained the modulus size effect by using the idea of increasingly predominant “surface energy contributions” in smaller nanowires. The noted disparities between the computational and compiled experimental data in Figure 1 indicate that factors other than inherent metallic surface energy driven by low atomic coordination may be influencing the measured elastic modulus in experiments. Simulations allow for precise manipulation of perfectly crystalline nanowires to measure elastic properties; because of this, there are relatively few variables besides surface energy that could cause a size effect on elastic properties. Nanowires used in experiments are inherently more complex due to factors such as surface contaminants and crystal defects.10,11 The geometry and surface structure of actual Ag nanowires could influence their elastic properties, but many simulations have not taken experimentally observed nanowire structures into account. Most nanowire simulations have tested single-crystal nanowires with rectangular cross sections; this nanowire model is significantly simplified compared to actual nanowires.3,5,7,16 To investigate whether the cross-sectional geometry of nanowires influences the elastic modulus, 〈110〉 axially oriented Ag nanowires with different geometries were created for simulations. Three different nanowire types are shown in Figure 2. Both the rhombic and pentagonal Nano Lett., Vol. 8, No. 11, 2008
Figure 2. 〈110〉 axially oriented Ag nanowires tested in the simulations to determine the elastic modulus. (a) shows a nanowire with a square cross section laterally bound by {100} and {110} surfaces, (b) shows a nanowire with a rhombic cross section laterally bound by {111} surfaces, and (c) shows a nanowire with a pentagonal cross section laterally bound by {100} surfaces. The “pentagonal” nanowire in panel c has a penta-twinned internal structure. The nanowires are colored according to the centrosymmetry parameter, where red indicates a high degree of undercoordination and blue indicates full coordination with respect to a fcc lattice.
nanowires (Figure 2b,c) were chosen because of their similarities to experimentally observed nanowires. Metallic nanowires with rhombic geometry have been observed to result from top-down methods for nanowire production, such as electron irradiation.17 Stable Ag nanowires with pentagonal geometry have been chemically grown, which represents a bottom-up method.18 In addition, penta-twinned Ag nanowires similar to the pentagonal nanowire shown in Figure 2c were used in the experiments by Wu et al. to obtain data included in Figure 1.12 The square nanowire geometry shown in Figure 2a was included in the simulations for comparison; it is simply a section of an Ag face centered cubic (fcc) crystal lattice. To test the effects of surface structure on elastic properties, nanowires with surface steps were created by removing surface atoms from a rectangular nanowire similar to the nanowire shown in Figure 2a. Rectangular sections of atoms were removed from each of the four lateral surfaces to create nanowires with varying degrees of surface roughness. These nanowires were designed to replicate the imperfect surface structure of actual nanowires, which can have steps and ledges. In addition to the geometrical similarities of the nanowires in Figure 1 to certain experimentally observed nanowires, 3615
Table 1. Elastic Modulus Values for Nanowires with an Increasing Quantity of Surface Atoms Removed to Create Surface Steps.
Figure 3. Elastic modulus data from molecular statics simulations of 〈110〉 axially oriented nanowires with square, rhombic, and pentagonal cross-sectional geometry. “Nanowire thickness” is the shortest lateral dimension of each cross section. The data values from the square nanowires are also shown in Figure 1 for comparison with experiments. To obtain the normalized elastic modulus, the measured modulus value was divided by the bulk value for 〈110〉 Ag, 84.1 GPa.
these 〈110〉 axially oriented nanowires were chosen because of the drastic observed elastic modulus increase with decreasing size compared to nanowires with other orientations.7,9 Metallic nanowires with different crystallographic orientations have also been studied, and elastic modulus size effects have been observed. For instance, simulations and experiments of 〈100〉 axially oriented Au and Ag nanowires have shown that the elastic modulus decreases slightly with decreasing size.7,9,19 In this work, we focus only on nanowires with crystallographic orientations that are matched between experiments and simulations. The embedded atom method (EAM) was used to perform the molecular statics simulations in this study.20 In this framework, the total energy of an atom is a summation of a simple pair potential term and an embedding energy term. The embedding energy represents the energy required to place one atom into the spherically averaged background electron density of surrounding atoms. The EAM is useful for determining properties of materials on the nanometer scale because of its relatively accurate depiction of surface electron density effects due to imperfect coordination of surface atoms. However, when compared to ab initio calculations, the EAM slightly underestimates the surface stress. In the simulations to calculate the elastic modulus, the nanowires were created by placing atoms in positions corresponding to a bulk FCC crystal lattice with no periodic boundary conditions. The nanowires were then relaxed by using the conjugate gradient energy minimization method. After reaching equilibrium configurations, the nanowires underwent quasi-static loading along the axial direction. The loading method consisted of consecutive increments in which the nanowire was alternately strained and relaxed. To strain the nanowire along the axial direction, a few layers of atoms at one end of the nanowire were fixed, and a linearly varying displacement profile was imposed along the axial direction of the remainder of the nanowire. For each increment, the strain was zero at the fixed end and 0.2% at the opposite 3616
end. After displacement, a few layers of atoms at the displaced end were fixed and the nanowire was statically relaxed with the conjugate gradient method. At the end of each increment, the total potential energy of the strained nanowire was calculated. This process was repeated for ten increments. After loading, the elastic modulus was calculated by utilizing an energy-based continuum-mechanical method developed by Diao et al.5 Using this method, the elastic modulus can be calculated by equating the change in potential energy ∆U in the nanowire during each increment to the work done by the externally applied load ∆U )
∫
∆L
0
F d(∆L) )
∫ Aσ dε ) ∫ Vσ dε ε
ε
0
0
(1)
where F is the applied load, A is the cross-sectional area of the nanowire, L is the nanowire length, σ is the average axial stress, ε is the axial strain, and V is the volume of the nanowire. Expanding σ and V as functions of ε allows for eq 1 to be written as ∆U 1 1 ) E ε2 + βε3 V0 2 3
(2)
where E is the elastic modulus at ε ) 0, V0 is the initial volume, and β is a constant. E was calculated for each nanowire by fitting the strain energy versus applied strain data to eq 2. Using these methods, the elastic modulus was calculated for square, rhombic, and pentagonal nanowires (the geometries shown in Figure 2, panels a, b, and c, respectively) over a range of sizes. The elastic modulus data are shown in Figure 3; the data from the square nanowires are also shown in Figure 1. Figure 3 shows that the elastic modulus values for nanowires with similar thicknesses are comparable for all considered cross-sectional geometries. These results indicate that nanowire cross section does not significantly affect the elastic modulus for ξ axially oriented Ag nanowires and that geometry cannot be a driving force for the difference in simulation and experimental data brought to light in Figure 1. The elastic modulus was also calculated for nanowires with surface steps. The calculated elastic modulus values of stepped nanowires with initial lateral dimensions of 3.9 nm by 4.4 nm are shown in Table 1 along with an image of a Nano Lett., Vol. 8, No. 11, 2008
stepped nanowire. Each stepped nanowire exhibits an elastic modulus that is within 1.5% of the modulus of the initial nanowire without surface steps. As shown in the last row of Table 1, the elastic modulus is highest when all the surface atoms are removed. This is in accordance with the trend in Figure 3; a smaller nanowire should have a higher modulus value. Also, the elastic modulus values of the nanowires with stepped surfaces lie between the modulus values of the smaller and larger nanowires without stepped surfaces. From this data, it is clear that surface steps do not significantly affect the elastic modulus of Ag nanowires. Other bulk defects such as vacancies have also been considered, and the impact on measured elastic modulus is also negligible relative to the difference in computational and experimental results in Figure 1. It should be noted that the continuum-based method used to calculate the elastic modulus is less accurate for stepped nanowires than for nanowires with uniform cross sections along their axes. This is because the analysis assumes that the applied stress during tensile deformation is uniformly distributed over the cross section along the entire length of the nanowire. The volume term used in eq 1 for the stepped nanowires was calculated by simply subtracting the volume of the removed atoms from the initial volume of the perfectly crystalline nanowire. Even though this method is expected to add some amount of error to the calculation, it is not expected to substantially change the results. The results of these simulations have significant ramifications on our understanding of the difference between experimentally and computationally measured elastic modulus values for metallic nanowires, as discussed in the introduction and shown in Figure 1. The simulations in this study show that the elastic modulus is similar for nanowires with the same size but different geometry or surface structure. Therefore, the dissimilarity in elastic modulus observed between simulations and experiments cannot be due to the differences in cross-sectional geometry or surface structure of experimentally and computationally tested nanowires; nanowire geometry and metallic surface structure or roughness can be nearly eliminated as contributing factors to the experimentally observed increase in elastic modulus as size decreases. Although eliminating these variables is important, the mechanism causing the increase in elastic modulus in experiments has yet to be identified. Even though surface energy plays a critical role at very small nanowire sizes (
