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Size-dependent elastic moduli and vibrational properties of fivefold twinned copper nanowires
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Nanotechnology Nanotechnology 25 (2014) 315701 (8pp)
doi:10.1088/0957-4484/25/31/315701
Size-dependent elastic moduli and vibrational properties of fivefold twinned copper nanowires Y G Zheng1,2, Y T Zhao1, H F Ye1 and H W Zhang1,2 1
State Key Laboratory of Structure Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, People’s Republic of China E-mail: [email protected] and [email protected] Received 30 December 2013, revised 25 May 2014 Accepted for publication 9 June 2014 Published 17 July 2014 Abstract
Based on atomistic simulations, the elastic moduli and vibration behaviors of fivefold twinned copper nanowires are investigated in this paper. Simulation results show that the elastic (i.e., Young’s and shear) moduli exhibit size dependence due to the surface effect. The effective Young’s modulus is found to decrease slightly whereas the effective shear modulus increases slightly with the increase in the wire radius. Both moduli tend to approach certain values at a larger radius and can be suitably described by core-shell composite structure models. Furthermore, we show by comparing simulation results and continuum predictions that, provided the effective Young’s and shear moduli are used, classic elastic theory can be applied to describe the smallamplitude vibration of fivefold twinned copper nanowires. Moreover, for the transverse vibration, the Timoshenko beam model is more suitable because shear deformation becomes apparent. Keywords: fivefold twinned nanowires, size dependent elastic modulus, vibrational property, classic elastic theory, atomistic simulation (Some figures may appear in colour only in the online journal) transducers, field emission components, and nanosize electrodes [18–25]. Among these potential applications, the highfrequency resonators/actuators are one of the most significant from a mechanical point of view. Thus, it is of the utmost importance to understand the mechanical vibrational properties of FTNs and related nanodevices. Up to now, a great deal of experimental and simulation work has been done to investigate the mechanical vibrational properties of various nanowires, and significant advancements have been achieved. For example, Feng et al have experimentally demonstrated that single-crystal silicon nanowires have resonances as high as several hundred megahertz [24]. Belliard et al have shown by femtosecond transient reflectivity measurements that freestanding copper nanowires can give ultrafast responses to vibration and that these structures can be used to detect the breathing mode up to several tens of gigahertz [25]. Conley et al have experimentally studied the nonlinear dynamics of suspended nanowire resonators and have found that they can suddenly change from a planar
1. Introduction As a special kind of nanowire, fivefold twinned nanowires (FTNs) such as Cu FTNs [1, 2], Ag FTNs [3–5], Pt FTNs [5], Au FTNs [6–8], Fe FTNs [9], and B4C FTNs [10] have been extensively synthesized and have attracted increasing interest in recent years. These nanostructures are composed of multiple twin boundaries (TBs) that have low energy and improved thermal stability [11, 12]. It has been found from experimental and simulation studies that these structures have many enhanced mechanical properties compared with twinfree nanowires, including enhanced elastic modulus, high strength, and improved mechanical stability [13–17]. Due to these superior properties, they are expected to have more potential applications than twin-free nanowires as key building blocks and active components in various nanodevices, such as high-frequency resonators/actuators, acoustic 2
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motion to a whirling motion [26]. Park et al have studied the flexural and longitudinal vibrations of silicon nanocantilevers by using molecular dynamics (MD) simulations and have deduced the size-dependent elastic properties based on continuum theories [27]. Olsson [28] and Yu et al [29] have performed MD simulations to study the transverse vibration behaviors of gold and silicon nanowires, and they have found that the MD simulation results are inconsistent with the predictions from continuum theory for most cases. Zhan et al report a novel beat phenomenon in [110] oriented silver nanowires by MD simulations and have developed a discrete moment of inertia model based on the hard sphere assumption to explain this phenomenon [30, 31]. It is noted that previous studies mainly focus on the mechanical vibrational properties of twin-free nanowires. In addition, although many studies have been conducted to investigate the mechanical properties of FTNs [14–17], these works mainly focus on their plastic deformation behaviors, such as the double-edge effect of TBs on yield stress and ductility in copper FTNs [14], the enhancement of yield stress due to the presence of TBs in iron FTNs [15], and the formation mechanisms of quasi-icosahedral structures with multi-conjoint fivefold deformation twins in various metallic FTNs [16]. So far, experimental or even simulation studies on the mechanical vibrational properties of FTNs have rarely been reported (except in our previous work on the torsional vibration properties of copper FTNs [32]), and such studies may suspend the practical use of FTNs as high-frequency resonators and actuators. Moreover, from a practical point of view, it is quite necessary to assess the validity of classical continuum models or develop new continuum models to describe the vibration of individual FTNs so that the vibration response of nanodevices that are composed of many FTNs can be effectively analyzed by using these simple models rather than time-consuming atomistic methods. In addition, it is well known that theoretical predictions based on continuum models can be strongly influenced by the elastic moduli chosen, and at the same time previous studies have shown that the elastic moduli of materials may undergo drastic change when the characteristic size of nanowires decreases [33–35]. Hence, the elastic moduli of various FTNs and their size dependence should also be well understood. In the present work, molecular static simulations are carried out to evaluate the effective elastic moduli of copper FTNs, and the core-shell composite concept is applied to model the size dependence of these moduli. Furthermore, the longitudinal and transverse vibrational properties of copper FTNs are investigated based on molecular dynamics simulations, and the applicability of classic elastic models based on effective elastic moduli to model FTNs is analyzed and discussed.
Figure 1. (a) The unit cell and (b) the entire structure of a copper
FTN with a circular cross-section. The atoms in light white and green are those in local fcc and hcp lattices, respectively, whereas the disordered atoms in red represent surfaces and dislocation cores, which are identified by common neighbor analysis.
configuration figures). The unit cell is cut from bulk copper ¯ ], and [110]/[112 ¯ ], and with two cutting planes, i.e. [110]/[112 a specified radius. Then the copper FTNs are constructed by copying and rotating the unit cell along the [110] direction. Thus the normal direction of the TBs is perpendicular to the z axis, and the central line of fivefold twins coincides with the z axis. The profile of the nanowire in the xy plane is a circle with the specified radius r. The length along the [110] direction is l. The interatomic interactions among copper atoms are modeled by using the embedded-atom potential [37] parameterized by Mishin et al [38], which has been widely used to study the mechanical properties and deformation mechanisms of nanocrystalline and nanowire copper systems. The simulations are conducted by using the molecular static (MS) method with conjugated gradient algorithm and/or the molecular dynamics (MD) method with velocity-Verlet integrator as implemented in the LAMMPS code [39]. To obtain the effective Young’s modulus and shear modulus, the tensile/ compressive and torsional deformations are simulated by using the MS method. In the tensile/compressive simulations, the periodic boundary condition is imposed in the z direction and thus the copper FTNs are regarded as infinitely long, which can effectively reduce the boundary effect along the axial direction. The tensile/compressive deformations of copper FTNs are achieved via changing first the simulation box size along the z direction to a specified length and then relaxing the stretched/compressed samples. Thereafter, the potential energy and stress of the equilibrated samples are computed and recorded. Here the Virial stress formula is used to calculate the stress. It should be noted that the model used in the tensile/compressive tests can predict the elastic and plastic deformations of copper FTNs that are similar to previous work [14]. In the torsional deformations of copper FTNs, one end of the wires is fixed and the remaining part is rotated. Then, keeping the two ends of the wires fixed, the samples are relaxed to obtain an equilibrium configuration. In previous work, short samples and the data (i.e., the length and potential energy) corresponding to the entire mobile parts have been used to extract the effective shear modulus of FTNs [32]. However, to reduce the influence of fixed boundaries, relatively long samples are used and only the data from the
2. Simulation methodology Figure 1 shows the unit cell and the entire structure of copper FTNs with a circular cross-section (here the atomistic configuration viewer Atomeye [36] is used to draw the 2
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Figure 2. (a) Relative potential energy versus strain and (b) stress versus strain curves of copper FTNs with different radii, in which the solid lines are the fitting curves.
middle part of the wires is used for subsequent analyses. In the elastic vibration tests of clamped-free copper FTNs, the atoms are first relaxed at 0 K using the MS method to obtain an initial equilibrium configuration. Then, with one end of the wires fixed, the remaining part is deformed statically and subsequently released suddenly to trigger the vibration of the wires. That is, in the longitudinal vibration simulations, a linearly distributed axial displacement along the z direction (it changes linearly from zero at one side of the remaining part closest to the fixed end of the wires to a prescribed nonzero but small displacement at the other side of the remaining part) is initially applied to the remaining part and then the remaining part is allowed to vibrate freely. The vibration process is simulated by using the MD method. Because an initial axial displacement is used, the main vibration mode should be longitudinal. In the transverse vibration simulations, a linearly distributed transverse displacement along the axial direction is initially applied to the remaining part. During the vibration process, the potential energy of the system is recorded and analyzed to obtain the vibration frequency. The time step used in the MD simulations is 2 fs.
with different radii. It can be seen from the solid curves shown in figure 2(a) that the relative potential energy Erp versus the tensile strain ε can be suitably fitted by Erp = VEeff ε 2 /2, in which V is the volume of the wire and Eeff is a fitting parameter (i.e., the effective Young’s modulus). That is, during the tension of copper FTNs, the potential energy of the system is approximately proportional to the square of the tensile strain, which is consistent with the prediction from classic elastic theory. In addition, the stressstrain curves are presented in figure 2(b). It can be seen that the stress is almost linearly dependent on the strain over a strain range of [−0.01, 0.01], which is similar to the observation of a small tensile strain range in previous work despite distinct nonlinear elastic behaviors at large strains [14]. As a result, it is evident that these curves can be suitably fitted by straight lines, which further confirms the applicability of classical theory to describe the statically small elastic tensile deformation of copper FTNs. The slopes of the fitted lines are thus the effective Young’s moduli for various FTNs. Figure 3 shows the effective Young’s modulus as a function of the wire radius. It can be seen that the effective Young’s moduli obtained from the potential energy-strain curves and the stress-strain curves are substantially in agreement with each other. Young’s modulus shows strong size dependence for the simulated samples with a radius less than 20 nm, and it decreases from about 150 GPa to 145 GPa as the wire radius increases. In addition, Young’s modulus of copper FTNs is higher than that of bulk polycrystalline copper (Emarco ≈ 120 GPa) and is slightly higher than Cao and Wei’s average rough estimation for copper FTNs with a square cross-section of 8 nm × 8 nm [14]. Nevertheless, a more accurate estimation can be made based on the stress-strain curves presented in [14], which shows that the slope of the curves at zero tensile strain is very close to our predictions. The observed tendency is similar to that of silver FTNs as revealed by experiments [35]. However, the size dependence for copper FTNs is much weaker than that for silver FTNs. The size dependence of Young’s modulus for copper FTNs can be ascribed to the surface effect. That is, in smallradius copper FTNs, the surface effect becomes evident due to
3. Results and discussions 3.1. Size-dependent effective elastic moduli of copper FTNs
FTNs have a large surface-to-volume ratio; that is, abundant atoms are located at the surfaces. Thus, the elastic moduli of copper FTNs should be different from those of their counterparts with a large radius. Here the MS simulations are conducted to compute the potential energy or stress of the equilibrated copper FTNs at a specified strain or torsional angle. Thus, the effective elastic moduli (i.e., Young’s modulus and shear modulus) can be obtained by fitting the energystrain (or stress-strain) and energy-torsional angle curves. To evaluate Young’s modulus for copper FTNs, tensile deformation along the axial direction is simulated. The relative potential energies (i.e., the differences between the potential energies of deformed and initial systems) at various strains are plotted as symbols in figure 2(a) for copper FTNs 3
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Similar to Young’s modulus, the effective shear modulus can also be described by a core-shell model as GIP = G0I0p + GsIsp, in which G0 and Gs are the shear moduli of the bulk material and the surface shell, respectively. I0p and Isp are the polar moments of inertia of the core and the shell, respectively. The size-dependent shear modulus can be fitted according to the following equation: ⎡ ⎛G ⎞⎛ r r3 r4 ⎞ ⎤ 3r 2 Geff = G0 ⎢ 1+8 ⎜ s − 1⎟ ⎜ s − s2 + s 3 − s 4 ⎟ ⎥ (2) ⎢⎣ 4r 2r 8r ⎠ ⎥⎦ ⎝ G0 ⎠ ⎝ 2r
The fitted results are shown in figure 4(b), and the fitting parameters are G0 = 26.95 GPa, Gs = 22.5 GPa, and rs = 0.8246 nm. It is observed that the shear modulus of the surface shell is smaller than that of the bulk materials and that ¯ ] the shell thickness is about 7 atom layers along the [112 direction. These results are different from those for Young’s modulus, which may be ascribed to the anisotropic properties of copper FTNs, and need to be further investigated.
Figure 3. Effective Young’s modulus of cylinder copper FTNs as a
function of radius (from stress-strain and energy-strain curves) and the fitted curve according to equation (1).
a large surface-to-volume ratio; thus, Young’s modulus is heavily dependent on the radius. With the increase in the radius, the surface effect can be ignored and the change of Young’s modulus with size becomes unobvious. Young’s modulus can be analyzed in terms of an approximate coreshell composite nanowire model as [33] EeffI = E0I0 + EsIs, in which E0 and Es are the moduli of the bulk material and the surface shell, respectively. I0 and Is are the area moments of inertia of the core and the shell, respectively. Thus, the effective Young’s modulus of copper FTNs can be expressed as
3.2. Longitudinal vibrational properties of copper FTNs
The frequency response of copper FTNs is obtained by performing fast Fourier transformation analysis of the potential energy during vibration [29]. Then the fundamental vibration frequency can be determined as the frequency corresponding to the peak in the frequency response curve. It should be recalled that the relative potential energy of the system is approximately proportional to the square of the tensile strain (i.e., equivalent to the displacement). Thus, the oscillation period of the potential energy is about half that of the longitudinal displacement, and consequently, the actual vibration frequency should be half of that obtained from the frequency response curve of the potential energy. Figure 5(a) shows a portion of the potential energy of a copper FTN with a length of l = 84 nm and a radius of r = 4 nm as a function of time, and figure 5(b) shows the frequency response curve (here the xcoordinate is scaled by 0.5 to convert the vibration frequency of the potential energy to that of the displacement). It can be observed that the vibration modes are mainly the first three longitudinal vibration modes. However, it should be mentioned that other vibration modes have also been excited due to the exciting method used in the present work; i.e., a linearly distributed axial displacement along the z direction is applied to the sample to trigger the vibration (see section 2 for more details). The fundamental longitudinal vibration frequencies as a function of the length and radius of the nanowire are shown in figure 6, and the corresponding values are listed in table 1. It can be seen that the frequency decreases with the increase in length, whereas it is almost independent of the radius. Based on classic elastic theory, the longitudinal vibration of a onedimensional structure can be described by the following governing equation [40]:
⎡ ⎛E ⎞⎛ r 3r 2 r3 r4 ⎞ ⎤ Eeff = E0 ⎢ 1+8 ⎜ s − 1⎟ ⎜ s − s2 + s 3 − s 4 ⎟ ⎥ (1) ⎢⎣ 4r 2r 8r ⎠ ⎥⎦ ⎝ E0 ⎠ ⎝ 2r
in which rs is the thickness of the shell. From figure 3, it can be seen that Young’s modulus of copper FTNs obtained from MS simulations can be suitably fitted to the foregoing equation (here the average values of the simulation results based on the energy-strain and stress-strain curves are used). The corresponding fitting parameters are E0 = 145.3 GPa, ES = 161.2 GPa, and rs = 0.2625 nm, which indicate that Young’s modulus for the surface shell is larger than that for the bulk materials and the shell thickness is about 2 atom ¯ ] direction, i.e., the radial direction. layers along the [112 Similar to the evaluation of the effective Young’s modulus of copper FTNs, the torsional deformation is simulated and the relative potential energies at various torsional angles for different copper FTNs are recorded to estimate the effective shear modulus. Figure 4(a) shows the predicted energy-torsional angle curves and the corresponding fitting curves according to Erp = GeffIpθ2/2L, in which Geff is the fitted effective shear modulus and Ip is the polar moment of inertia of the wire. Figure 4(b) shows effective shear modulus as a function of the wire radius. It can be seen that the effective shear modulus increases slightly with the increase in the wire radius and has almost the same average value as that predicted in our previous work [32] and is much lower than that of bulk polycrystalline copper (Gmacro ≈ 45.32 GPa).
∂ ∂x
⎡ ∂u ( x , t ) ⎤ ∂ 2u ( x , t ) ⎥⎦ = ρA ⎢⎣ EA ∂x ∂t 2
(3)
in which E is Young’s modulus, A is the area of the cross4
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Figure 4. (a) Relative potential energy versus torsional angle curves of copper FTNs with different radii, in which the solid lines are the fitting curves; (b) effective shear modulus of FTNs as a function of radius and the fitted curve according to equation (2).
Figure 5. (a) Potential energy of a copper FTN with a length of l = 84 nm and a circular cross-section of r = 3 nm as a function of time and (b)
fast Fourier transformation of the potential energy.
Figure 6. Fundamental tensile vibration frequency as a function of (a) the wire length for FTNs with a radius of r ≈ 3 nm and (b) the wire
radius for copper FTNs with a length of l ≈ 96 nm. The results predicted by classic theory for different material properties are also shown.
5
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or the Timoshenko beam model. The corresponding governing equations are [41] Euler-Bernoulli
Table 1. Fundamental longitudinal vibration frequencies of copper FTNs.
Theoretical frequency (GHz)
No.
Length (∼nm)
Radius (∼nm)
MD Frequency (GHz)
1 2 3 4 5 6 7
48 60 72 84 96 96 96
3 3 3 3 3 4 6
21.36 17.17 14.11 12.21 10.68 10.68 10.68
∂ 2 ⎡ ∂ 2w (x , t ) ⎤ ∂ 2w (x , t ) =0 EI ⎥ + ρA 2 ⎢ 2 ∂x ⎣ ∂x ∂t 2 ⎦
Eeff
Emacro
Timoshenko
21.35 17.08 14.23 12.20 10.67 10.64 10.60
19.09 15.27 12.73 10.91 9.55 9.55 9.55
⎧ ∂ 2w (x , t ) ⎪ ρA − k′GA ∂t 2 ⎪ ⎪ ⎡ ∂ 2w (x , t ) ∂α ( x , t ) ⎤ ⎪ ×⎢ − ⎥ = 0 2 ⎪ ∂x ⎦ ∂x ⎣ ⎨ ⎪ ∂ 2α (x , t ) − k′GA ⎪ ρI ∂t 2 ⎪ ⎤ ∂ 2α ( x , t ) ⎡ ∂w ( x , t ) ⎪ α × − = 0 x t ( , ) ⎥⎦ − ⎢ ⎪ ⎣ ∂x ∂x 2 ⎩
section, ρ is the mass density, and u(x,t) is the axial displacement at point x and time t. The corresponding vibration angular frequency for a clamped-free one-dimensional structure/beam can be written as ωi =
(2i − 1) π 2L
E (i = 1,2, …) ρ
(5)
(6)
4
in which I = π4r is the area moment of inertia of the crosssection about the neutral axis, G is the shear modulus, k′ is the shear factor, and w(x, t) and α(x, t) are the deflection angle and rotation angle of the cross-section at point x and time t, respectively. The corresponding vibration frequency for a clamped-free beam can be written as [41] Euler-Bernoulli
(4)
in which L is the length of the beam. From the frequency response curve of the potential energy versus time curve of the aforementioned FTN, i.e., the curve shown in figure 5(b), it can be observed that the first three longitudinal frequencies are about 12.21 GHz, 36.62 GHz, and 60.94 GHz, respectively. These frequencies possess the relationships f2 ≈ 3f1 and f3 ≈ 5f1, which are consistent with equation (4). In addition, the theoretical prediction indicates that the fundamental longitudinal vibration frequency is inversely proportional to the length and independent of the wire radius, which is also valid for copper FTNs. To make a quantitative comparison, figure 6 and table 1 also display the predictions from classic elastic theory, with Emarco and Eeff for copper FTNs with different lengths and radii. We can see that the predictions using the effective Young’s modulus Eeff agree quite well with the MD calculations, whereas those calculated based on Emarco diverge significantly from the MD results. From the foregoing analysis, it can be concluded that, provided an effective Young’s modulus that considers the discrete nature and surface effect of nanowires is used rather than Young’s modulus of bulk polycrystalline copper materials, classic elastic theory can be applied to modeling the longitudinal vibration of copper FTNs.
ωi =
k i2 L2
⎞ EI ⎛ 2i − 1 ⎜ k = 1.875,4.694, π for j ⩾ 3⎟ (7) i ⎝ ⎠ 2 ρA
Timoshenko ωi =
1 L
E ρ
ai2 − bi2 (i = 1,2, …) 1 + γ2
(8)
in which γ 2 = k E′ G and ai and bi can be determined by solving the following nonlinear equations: ⎧ a 2 + γ 2b 2 a 2 γ 2 + b 2 ⎪ − s2 = 0 ⎪ a 2 − b2 1 + γ 2 ⎪ ⎪ a 2 − b2 sin a sinh b ⎨ ⎪ a 4 + a 4γ 4 + 4γ 2a 2b2 + b4γ 4 + b4 ⎪ − ab ⎪ a 2 + γ 2b2 a 2γ 2 + b2 ⎪ ⎩ × cos a cosh b − 2ab = 0
(
(
(
)( )(
)
)
) (
(9)
)
(
)(
)
It should be noted that the preceding equations hold for a one-dimensional structure with a small slenderness ratio s = L A /I , which is the case considered in the present work. Figure 7 and table 2 present the predictions from the classic Euler-Bernoulli beam theory with Emarco and Eeff and the classic Timoshenko beam theory with Emarco, Eeff, Gmarco, and Geff. We can see that the predictions using the effective Young’s modulus and shear modulus are in reasonable agreement with the MD calculations. However, the EulerBernoulli beam theory is mainly applicable to copper FTNs with a large slenderness ratio. The Timoshenko beam theory is valid for all copper FTNs simulated in the present work,
3.3. Transverse vibrational properties of copper FTNs
The fundamental transverse vibration frequencies as a function of length and radius of copper FTNs are shown in figure 7, and the corresponding values are listed in table 2. It can be seen that the frequency decreases quadratically with the increase in length for copper FTNs, whereas it increases almost linearly with the increase in radius. From classic elastic theory, the transverse vibration of a one-dimensional structure can be described by the Euler-Bernoulli beam model 6
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Figure 7. Fundamental transverse vibration frequency as a function of (a) the wire length for copper FTNs with a radius of r ≈ 2 nm and (b)
the wire radius for copper FTNs with a length of l ≈ 40 nm. The results predicted by classic theory for different material properties are also shown. Table 2. Fundamental transverse vibration frequencies of copper FTNs.
Theoretical frequency (GHz) Euler Bernoulli
Timoshenko
No.
Length (∼nm)
Radius (∼nm)
MD frequency (GHz)
Eeff
Emacro Gmacro
Eeff
Emacro Gmacro
1 2 3 4 5 6
24 32 40 48 40 40
2 2 2 2 3 4
3.82 2.15 1.37 0.95 2.03 2.63
3.94 2.21 1.42 0.99 2.08 2.81
3.52 1.97 1.26 0.88 1.86 2.52
3.82 2.17 1.40 0.98 2.03 2.69
3.46 1.96 1.26 0.88 1.84 2.46
longitudinal and transverse vibration of copper FTNs, it has been found that the vibration frequencies predicted by using classic elastic theories with the obtained effective elastic moduli are consistent with atomistic simulation results, which validates the applicability of the continuum theories at the nanoscale and suggests that the constructed core-shell composite structure models can be effectively used further for the mechanical analysis of copper FTNs based on continuum theories. Furthermore, we found that the Timoshenko beam model is more appropriate to describe the transverse vibration. These findings should be helpful for the construction of a hierarchical multiscale coupling method of atomistic and continuum theories that can be used to effectively model large systems comprising many FTNs and is of practical importance for the design of FTN-based vibration nanodevices.
taking into account reasonable consideration of shear deformation.
4. Conclusions In summary, we have used classical molecular static and dynamic simulations to investigate the elastic and vibrational properties of copper FTNs. Simulation results reveal size dependencies of Young’s and shear moduli of copper FTNs; that is, Young’s modulus decreases with the increase in radius, while the shear modulus increases with the increase in radius. Empirical core-shell composite structure models have been constructed, and they can describe the observed size dependencies fairly well because these models give reasonable consideration to the surface effect of FTNs. These models show that there is a transition from surface elastic moduli to bulk elastic moduli and that the transition size has been found to be about 10 ∼ 20 nm. However, it should be noted that, owing to the anisotropy of copper FTNs, the surface thicknesses of the core-shell models predicted from the size-dependent Young’s modulus and shear modulus are different from each other. Regarding the small-amplitude
Acknowledgments The support of the National Natural Science Foundation of China (Nos. 11272003, 11232003, and 11302037), the 111 Project (No. B08014), the National Basic Research Program of China (Nos. 2010CB832704 and 2011CB013401), the Program for New Century Excellent Talents in University 7
Nanotechnology 25 (2014) 315701
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(NCET-13-0088), the PhD Programs Foundation of the Ministry of Education of China (20130041110050), and the Fundamental Research Funds for the Central Universities is gratefully acknowledged.
[21] [22]
References [23]
[1] Lisiecki I, Filankembo A, Sack-Kongehl H, Weiss K, Pileni M P and Urban J 2000 Structural investigations of copper nanorods by high-resolution TEM Phys. Rev. B 61 4968–74 [2] Kim C, Gu W H, Briceno M, Robertson I M, Choi H and Kim K 2008 Copper nanowires with a five-twinned structure grown by chemical vapor deposition Adv. Mater. 20 1859–63 [3] Chen H Y, Gao Y, Zhang H R, Liu L B, Yu H C, Tian H F, Xie S S and Li J Q 2004 Transmission-electron-microscopy study on fivefold twinned silver nanorods J. Phys. Chem. B 108 12038–43 [4] Wiley B, Sun Y G, Mayers B and Xia Y N 2005 Shapecontrolled synthesis of metal nanostructures: the case of silver Chem. Eur. J. 11 454–63 [5] Chen J Y, Wiley B and Xia Y N 2007 One-dimensional nanostructures of metals: large-scale synthesis and some potential applications Langmuir 23 4120–9 [6] Hu M, Hillyard P and Hartland G V 2004 Determination of the elastic constants of gold nanorods produced by seed mediated growth Nano Lett. 4 2493–7 [7] Hartland G V 2004 Measurements of the material properties of metal nanoparticles by time-resolved spectroscopy Phys. Chem. Chem. Phys. 6 5263–74 [8] Huang X, Neretina S and El-Sayed M A 2009 Gold nanorods: from synthesis and properties to biological and biomedical applications Adv. Mater. 21 4880–910 [9] Ling T, Yu H M, Liu X H, Shen Z Y and Zhu J 2008 Five-fold twinned nanorods of fcc fe: synthesis and characterization Cryst. Growth Des. 8 4340–2 [10] Fu X, Jiang J, Liu C and Yuan J 2009 Fivefold twinned boron carbide nanowires Nanotechn. 20 365707 [11] Anderoglu O, Misra A, Wang H and Zhang X 2008 Thermal stability of sputtered Cu films with nanoscale growth twins J. Appl. Phys. 103 094322 [12] Kulkarni Y, Asaro R J and Farkas D 2009 Are nanotwinned structures in fcc metals optimal for strength, ductility and grain stability? Scr. Mater. 60 532–5 [13] Yoo J H, Oh S I and Jeong M S 2010 The enhanced elastic modulus of nanowires associated with multitwins J. Appl. Phys. 107 094316 [14] Cao A and Wei Y 2006 Atomistic simulations of the mechanical behavior of fivefold twinned nanowires Phys. Rev. B 74 214108 [15] Wu J Y, Nagao S, He J Y and Zhang Z L 2011 Role of fivefold twin boundary on the enhanced mechanical properties of fcc fe nanowires Nano Lett. 11 5264–73 [16] Jiang S, Shen Y G, Zheng Y G and Chen Z 2013 Formation of quasi-icosahedral structures with multi-conjoint fivefold deformation twins in fivefold twinned metallic nanowires Appl. Phys. Lett. 103 041909 [17] Wen Y H, Huang R, Zhu Z Z and Wang Q 2012 Mechanical properties of platinum nanowires: an atomistic investigation on single-crystalline and twinned structures Comput. Mater. Sci. 55 205–10 [18] Lieber C M 2003 Nanoscale science and technology: building a big future from small things MRS Bull. 28 486–91 [19] Craighead H G 2000 Nanoelectromechanical systems Science 290 1532–5 [20] Husain A, Hone J, Postma H W C, Huang X M H, Drake T, Barbic M, Scherer A and Roukes M L 2003 Nanowire-based
[24] [25]
[26] [27]
[28] [29]
[30] [31] [32]
[33] [34]
[35]
[36] [37] [38]
[39] [40] [41]
8
very-high-frequency electromechanical resonator Appl. Phys. Lett. 83 1240–2 Chang P C, Fan Z Y, Wang D W, Tseng W Y, Chiou W A, Hong J and Lu J G 2004 ZnO nanowires synthesized by vapor trapping CVD method Chem. Mater. 16 5133–7 Li Y, Tan B and Wu Y 2006 Freestanding mesoporous quasisingle-crystalline Co3O4 nanowire arrays J. Am. Chem. Soc. 128 14258–9 Qu L T, Shi G Q, Wu X F and Fan B 2004 Facile route to silver nanotubes Adv. Mater. 16 1200–3 Feng X L, He R R, Yang P D and Roukes M L 2007 Very high frequency silicon nanowire electromechanical resonators Nano Lett. 7 1953–9 Belliard L, Cornelius T W, Perrin B, Kacemi N, Becerra L, Thomas O, Toimil-Molares M E and Cassinelli M 2013 Vibrational response of free standing single copper nanowire through transient reflectivity microscopy J. Appl. Phys. 114 193509 Conley W G, Raman A, Krousgrill C M and Mohammadi S 2008 Nonlinear and nonplanar dynamics of suspended nanotube and nanowire resonators Nano Lett. 8 1590–5 Park S H, Kim J S, Park J H, Lee J S, Choi Y K and Kwon O M 2005 Molecular dynamics study on sizedependent elastic properties of silicon nanocantilevers Thin Solid Films 492 285–9 Olsson P A T 2010 Transverse resonant properties of strained gold nanowires J. Appl. Phy. 108 034318 Yu H, Zhang W W, Lei S Y, Lu L B, Sun C and Huang Q A 2012 Study on vibration behavior of doubly clamped silicon nanowires by molecular dynamics J. Nanomater. 2012 342329 Zhan H F and Gu Y T 2012 A fundamental numerical and theoretical study for the vibrational properties of nanowires J. Appl. Phys. 111 124303 Zhan H F, Gu Y T and Park H S 2012 Beat phenomena in metal nanowires, and their implications for resonance-based elastic property measurements Nanoscale 4 6779 Zheng Y G, Zhao Y T, Ye H F, Zhang H W and Fu Y F 2014 Atomistic simulation of torsional vibration and plastic deformation of five-fold twinned copper nanowires Int. J. Comput. Methods 11 1344010 Chen C Q, Shi Y, Zhang Y S, Zhu J and Yan Y J 2006 Size dependence of Young’s modulus in ZnO nanowires Phys. Rev. Lett. 96 075505 Gavan K B, Westra H J R, van der Drift E W J M, Venstra W J and van der Zant H S J 2009 Size-dependent effective Young’s modulus of silicon nitride cantilevers Appl. Phys. Lett. 94 233108 Zhu Y, Qin Q, Xu F, Fan F, Ding Y, Zhang T, Wiley B J and Wang Z L 2012 Size effects on elasticity, yielding, and fracture of silver nanowires: In situ experiments Phys. Rev. B 85 045443 Li J 2003 AtomEye: an efficient atomistic configuration viewer Modelling Simul. Mater. Sci. Eng. 11 173–7 Daw M S and Baskes M I 1984 Embedded-atom method— derivation and application to impurities, surfaces, and other defects in metals Phys. Rev. B 29 6443–53 Mishin Y, Mehl M J, Papaconstantopoulos D A, Voter A F and Kress J D 2001 Structural stability and lattice defects in copper: ab initio, tight-binding, and embedded-atom calculations Phys. Rev. B 63 224106 Plimpton S 1995 Fast parallel algorithms for short-range molecular-dynamics J. Comput. Phys. 117 1–19 Beards C F 1996 Structural Vibration: Analysis and Damping (Oxford: Butterworth-Heinemann) Han S M, Benaroya H and Wei T 1999 Dynamics of transversely vibrating beams using four engineering theories J. Sound Vib. 225 935–88
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Size-dependent elastic moduli and vibrational properties of fivefold twinned copper nanowires
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Nanotechnology Nanotechnology 25 (2014) 315701 (8pp)
doi:10.1088/0957-4484/25/31/315701
Size-dependent elastic moduli and vibrational properties of fivefold twinned copper nanowires Y G Zheng1,2, Y T Zhao1, H F Ye1 and H W Zhang1,2 1
State Key Laboratory of Structure Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, People’s Republic of China E-mail: [email protected] and [email protected] Received 30 December 2013, revised 25 May 2014 Accepted for publication 9 June 2014 Published 17 July 2014 Abstract
Based on atomistic simulations, the elastic moduli and vibration behaviors of fivefold twinned copper nanowires are investigated in this paper. Simulation results show that the elastic (i.e., Young’s and shear) moduli exhibit size dependence due to the surface effect. The effective Young’s modulus is found to decrease slightly whereas the effective shear modulus increases slightly with the increase in the wire radius. Both moduli tend to approach certain values at a larger radius and can be suitably described by core-shell composite structure models. Furthermore, we show by comparing simulation results and continuum predictions that, provided the effective Young’s and shear moduli are used, classic elastic theory can be applied to describe the smallamplitude vibration of fivefold twinned copper nanowires. Moreover, for the transverse vibration, the Timoshenko beam model is more suitable because shear deformation becomes apparent. Keywords: fivefold twinned nanowires, size dependent elastic modulus, vibrational property, classic elastic theory, atomistic simulation (Some figures may appear in colour only in the online journal) transducers, field emission components, and nanosize electrodes [18–25]. Among these potential applications, the highfrequency resonators/actuators are one of the most significant from a mechanical point of view. Thus, it is of the utmost importance to understand the mechanical vibrational properties of FTNs and related nanodevices. Up to now, a great deal of experimental and simulation work has been done to investigate the mechanical vibrational properties of various nanowires, and significant advancements have been achieved. For example, Feng et al have experimentally demonstrated that single-crystal silicon nanowires have resonances as high as several hundred megahertz [24]. Belliard et al have shown by femtosecond transient reflectivity measurements that freestanding copper nanowires can give ultrafast responses to vibration and that these structures can be used to detect the breathing mode up to several tens of gigahertz [25]. Conley et al have experimentally studied the nonlinear dynamics of suspended nanowire resonators and have found that they can suddenly change from a planar
1. Introduction As a special kind of nanowire, fivefold twinned nanowires (FTNs) such as Cu FTNs [1, 2], Ag FTNs [3–5], Pt FTNs [5], Au FTNs [6–8], Fe FTNs [9], and B4C FTNs [10] have been extensively synthesized and have attracted increasing interest in recent years. These nanostructures are composed of multiple twin boundaries (TBs) that have low energy and improved thermal stability [11, 12]. It has been found from experimental and simulation studies that these structures have many enhanced mechanical properties compared with twinfree nanowires, including enhanced elastic modulus, high strength, and improved mechanical stability [13–17]. Due to these superior properties, they are expected to have more potential applications than twin-free nanowires as key building blocks and active components in various nanodevices, such as high-frequency resonators/actuators, acoustic 2
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motion to a whirling motion [26]. Park et al have studied the flexural and longitudinal vibrations of silicon nanocantilevers by using molecular dynamics (MD) simulations and have deduced the size-dependent elastic properties based on continuum theories [27]. Olsson [28] and Yu et al [29] have performed MD simulations to study the transverse vibration behaviors of gold and silicon nanowires, and they have found that the MD simulation results are inconsistent with the predictions from continuum theory for most cases. Zhan et al report a novel beat phenomenon in [110] oriented silver nanowires by MD simulations and have developed a discrete moment of inertia model based on the hard sphere assumption to explain this phenomenon [30, 31]. It is noted that previous studies mainly focus on the mechanical vibrational properties of twin-free nanowires. In addition, although many studies have been conducted to investigate the mechanical properties of FTNs [14–17], these works mainly focus on their plastic deformation behaviors, such as the double-edge effect of TBs on yield stress and ductility in copper FTNs [14], the enhancement of yield stress due to the presence of TBs in iron FTNs [15], and the formation mechanisms of quasi-icosahedral structures with multi-conjoint fivefold deformation twins in various metallic FTNs [16]. So far, experimental or even simulation studies on the mechanical vibrational properties of FTNs have rarely been reported (except in our previous work on the torsional vibration properties of copper FTNs [32]), and such studies may suspend the practical use of FTNs as high-frequency resonators and actuators. Moreover, from a practical point of view, it is quite necessary to assess the validity of classical continuum models or develop new continuum models to describe the vibration of individual FTNs so that the vibration response of nanodevices that are composed of many FTNs can be effectively analyzed by using these simple models rather than time-consuming atomistic methods. In addition, it is well known that theoretical predictions based on continuum models can be strongly influenced by the elastic moduli chosen, and at the same time previous studies have shown that the elastic moduli of materials may undergo drastic change when the characteristic size of nanowires decreases [33–35]. Hence, the elastic moduli of various FTNs and their size dependence should also be well understood. In the present work, molecular static simulations are carried out to evaluate the effective elastic moduli of copper FTNs, and the core-shell composite concept is applied to model the size dependence of these moduli. Furthermore, the longitudinal and transverse vibrational properties of copper FTNs are investigated based on molecular dynamics simulations, and the applicability of classic elastic models based on effective elastic moduli to model FTNs is analyzed and discussed.
Figure 1. (a) The unit cell and (b) the entire structure of a copper
FTN with a circular cross-section. The atoms in light white and green are those in local fcc and hcp lattices, respectively, whereas the disordered atoms in red represent surfaces and dislocation cores, which are identified by common neighbor analysis.
configuration figures). The unit cell is cut from bulk copper ¯ ], and [110]/[112 ¯ ], and with two cutting planes, i.e. [110]/[112 a specified radius. Then the copper FTNs are constructed by copying and rotating the unit cell along the [110] direction. Thus the normal direction of the TBs is perpendicular to the z axis, and the central line of fivefold twins coincides with the z axis. The profile of the nanowire in the xy plane is a circle with the specified radius r. The length along the [110] direction is l. The interatomic interactions among copper atoms are modeled by using the embedded-atom potential [37] parameterized by Mishin et al [38], which has been widely used to study the mechanical properties and deformation mechanisms of nanocrystalline and nanowire copper systems. The simulations are conducted by using the molecular static (MS) method with conjugated gradient algorithm and/or the molecular dynamics (MD) method with velocity-Verlet integrator as implemented in the LAMMPS code [39]. To obtain the effective Young’s modulus and shear modulus, the tensile/ compressive and torsional deformations are simulated by using the MS method. In the tensile/compressive simulations, the periodic boundary condition is imposed in the z direction and thus the copper FTNs are regarded as infinitely long, which can effectively reduce the boundary effect along the axial direction. The tensile/compressive deformations of copper FTNs are achieved via changing first the simulation box size along the z direction to a specified length and then relaxing the stretched/compressed samples. Thereafter, the potential energy and stress of the equilibrated samples are computed and recorded. Here the Virial stress formula is used to calculate the stress. It should be noted that the model used in the tensile/compressive tests can predict the elastic and plastic deformations of copper FTNs that are similar to previous work [14]. In the torsional deformations of copper FTNs, one end of the wires is fixed and the remaining part is rotated. Then, keeping the two ends of the wires fixed, the samples are relaxed to obtain an equilibrium configuration. In previous work, short samples and the data (i.e., the length and potential energy) corresponding to the entire mobile parts have been used to extract the effective shear modulus of FTNs [32]. However, to reduce the influence of fixed boundaries, relatively long samples are used and only the data from the
2. Simulation methodology Figure 1 shows the unit cell and the entire structure of copper FTNs with a circular cross-section (here the atomistic configuration viewer Atomeye [36] is used to draw the 2
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Figure 2. (a) Relative potential energy versus strain and (b) stress versus strain curves of copper FTNs with different radii, in which the solid lines are the fitting curves.
middle part of the wires is used for subsequent analyses. In the elastic vibration tests of clamped-free copper FTNs, the atoms are first relaxed at 0 K using the MS method to obtain an initial equilibrium configuration. Then, with one end of the wires fixed, the remaining part is deformed statically and subsequently released suddenly to trigger the vibration of the wires. That is, in the longitudinal vibration simulations, a linearly distributed axial displacement along the z direction (it changes linearly from zero at one side of the remaining part closest to the fixed end of the wires to a prescribed nonzero but small displacement at the other side of the remaining part) is initially applied to the remaining part and then the remaining part is allowed to vibrate freely. The vibration process is simulated by using the MD method. Because an initial axial displacement is used, the main vibration mode should be longitudinal. In the transverse vibration simulations, a linearly distributed transverse displacement along the axial direction is initially applied to the remaining part. During the vibration process, the potential energy of the system is recorded and analyzed to obtain the vibration frequency. The time step used in the MD simulations is 2 fs.
with different radii. It can be seen from the solid curves shown in figure 2(a) that the relative potential energy Erp versus the tensile strain ε can be suitably fitted by Erp = VEeff ε 2 /2, in which V is the volume of the wire and Eeff is a fitting parameter (i.e., the effective Young’s modulus). That is, during the tension of copper FTNs, the potential energy of the system is approximately proportional to the square of the tensile strain, which is consistent with the prediction from classic elastic theory. In addition, the stressstrain curves are presented in figure 2(b). It can be seen that the stress is almost linearly dependent on the strain over a strain range of [−0.01, 0.01], which is similar to the observation of a small tensile strain range in previous work despite distinct nonlinear elastic behaviors at large strains [14]. As a result, it is evident that these curves can be suitably fitted by straight lines, which further confirms the applicability of classical theory to describe the statically small elastic tensile deformation of copper FTNs. The slopes of the fitted lines are thus the effective Young’s moduli for various FTNs. Figure 3 shows the effective Young’s modulus as a function of the wire radius. It can be seen that the effective Young’s moduli obtained from the potential energy-strain curves and the stress-strain curves are substantially in agreement with each other. Young’s modulus shows strong size dependence for the simulated samples with a radius less than 20 nm, and it decreases from about 150 GPa to 145 GPa as the wire radius increases. In addition, Young’s modulus of copper FTNs is higher than that of bulk polycrystalline copper (Emarco ≈ 120 GPa) and is slightly higher than Cao and Wei’s average rough estimation for copper FTNs with a square cross-section of 8 nm × 8 nm [14]. Nevertheless, a more accurate estimation can be made based on the stress-strain curves presented in [14], which shows that the slope of the curves at zero tensile strain is very close to our predictions. The observed tendency is similar to that of silver FTNs as revealed by experiments [35]. However, the size dependence for copper FTNs is much weaker than that for silver FTNs. The size dependence of Young’s modulus for copper FTNs can be ascribed to the surface effect. That is, in smallradius copper FTNs, the surface effect becomes evident due to
3. Results and discussions 3.1. Size-dependent effective elastic moduli of copper FTNs
FTNs have a large surface-to-volume ratio; that is, abundant atoms are located at the surfaces. Thus, the elastic moduli of copper FTNs should be different from those of their counterparts with a large radius. Here the MS simulations are conducted to compute the potential energy or stress of the equilibrated copper FTNs at a specified strain or torsional angle. Thus, the effective elastic moduli (i.e., Young’s modulus and shear modulus) can be obtained by fitting the energystrain (or stress-strain) and energy-torsional angle curves. To evaluate Young’s modulus for copper FTNs, tensile deformation along the axial direction is simulated. The relative potential energies (i.e., the differences between the potential energies of deformed and initial systems) at various strains are plotted as symbols in figure 2(a) for copper FTNs 3
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Similar to Young’s modulus, the effective shear modulus can also be described by a core-shell model as GIP = G0I0p + GsIsp, in which G0 and Gs are the shear moduli of the bulk material and the surface shell, respectively. I0p and Isp are the polar moments of inertia of the core and the shell, respectively. The size-dependent shear modulus can be fitted according to the following equation: ⎡ ⎛G ⎞⎛ r r3 r4 ⎞ ⎤ 3r 2 Geff = G0 ⎢ 1+8 ⎜ s − 1⎟ ⎜ s − s2 + s 3 − s 4 ⎟ ⎥ (2) ⎢⎣ 4r 2r 8r ⎠ ⎥⎦ ⎝ G0 ⎠ ⎝ 2r
The fitted results are shown in figure 4(b), and the fitting parameters are G0 = 26.95 GPa, Gs = 22.5 GPa, and rs = 0.8246 nm. It is observed that the shear modulus of the surface shell is smaller than that of the bulk materials and that ¯ ] the shell thickness is about 7 atom layers along the [112 direction. These results are different from those for Young’s modulus, which may be ascribed to the anisotropic properties of copper FTNs, and need to be further investigated.
Figure 3. Effective Young’s modulus of cylinder copper FTNs as a
function of radius (from stress-strain and energy-strain curves) and the fitted curve according to equation (1).
a large surface-to-volume ratio; thus, Young’s modulus is heavily dependent on the radius. With the increase in the radius, the surface effect can be ignored and the change of Young’s modulus with size becomes unobvious. Young’s modulus can be analyzed in terms of an approximate coreshell composite nanowire model as [33] EeffI = E0I0 + EsIs, in which E0 and Es are the moduli of the bulk material and the surface shell, respectively. I0 and Is are the area moments of inertia of the core and the shell, respectively. Thus, the effective Young’s modulus of copper FTNs can be expressed as
3.2. Longitudinal vibrational properties of copper FTNs
The frequency response of copper FTNs is obtained by performing fast Fourier transformation analysis of the potential energy during vibration [29]. Then the fundamental vibration frequency can be determined as the frequency corresponding to the peak in the frequency response curve. It should be recalled that the relative potential energy of the system is approximately proportional to the square of the tensile strain (i.e., equivalent to the displacement). Thus, the oscillation period of the potential energy is about half that of the longitudinal displacement, and consequently, the actual vibration frequency should be half of that obtained from the frequency response curve of the potential energy. Figure 5(a) shows a portion of the potential energy of a copper FTN with a length of l = 84 nm and a radius of r = 4 nm as a function of time, and figure 5(b) shows the frequency response curve (here the xcoordinate is scaled by 0.5 to convert the vibration frequency of the potential energy to that of the displacement). It can be observed that the vibration modes are mainly the first three longitudinal vibration modes. However, it should be mentioned that other vibration modes have also been excited due to the exciting method used in the present work; i.e., a linearly distributed axial displacement along the z direction is applied to the sample to trigger the vibration (see section 2 for more details). The fundamental longitudinal vibration frequencies as a function of the length and radius of the nanowire are shown in figure 6, and the corresponding values are listed in table 1. It can be seen that the frequency decreases with the increase in length, whereas it is almost independent of the radius. Based on classic elastic theory, the longitudinal vibration of a onedimensional structure can be described by the following governing equation [40]:
⎡ ⎛E ⎞⎛ r 3r 2 r3 r4 ⎞ ⎤ Eeff = E0 ⎢ 1+8 ⎜ s − 1⎟ ⎜ s − s2 + s 3 − s 4 ⎟ ⎥ (1) ⎢⎣ 4r 2r 8r ⎠ ⎥⎦ ⎝ E0 ⎠ ⎝ 2r
in which rs is the thickness of the shell. From figure 3, it can be seen that Young’s modulus of copper FTNs obtained from MS simulations can be suitably fitted to the foregoing equation (here the average values of the simulation results based on the energy-strain and stress-strain curves are used). The corresponding fitting parameters are E0 = 145.3 GPa, ES = 161.2 GPa, and rs = 0.2625 nm, which indicate that Young’s modulus for the surface shell is larger than that for the bulk materials and the shell thickness is about 2 atom ¯ ] direction, i.e., the radial direction. layers along the [112 Similar to the evaluation of the effective Young’s modulus of copper FTNs, the torsional deformation is simulated and the relative potential energies at various torsional angles for different copper FTNs are recorded to estimate the effective shear modulus. Figure 4(a) shows the predicted energy-torsional angle curves and the corresponding fitting curves according to Erp = GeffIpθ2/2L, in which Geff is the fitted effective shear modulus and Ip is the polar moment of inertia of the wire. Figure 4(b) shows effective shear modulus as a function of the wire radius. It can be seen that the effective shear modulus increases slightly with the increase in the wire radius and has almost the same average value as that predicted in our previous work [32] and is much lower than that of bulk polycrystalline copper (Gmacro ≈ 45.32 GPa).
∂ ∂x
⎡ ∂u ( x , t ) ⎤ ∂ 2u ( x , t ) ⎥⎦ = ρA ⎢⎣ EA ∂x ∂t 2
(3)
in which E is Young’s modulus, A is the area of the cross4
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Figure 4. (a) Relative potential energy versus torsional angle curves of copper FTNs with different radii, in which the solid lines are the fitting curves; (b) effective shear modulus of FTNs as a function of radius and the fitted curve according to equation (2).
Figure 5. (a) Potential energy of a copper FTN with a length of l = 84 nm and a circular cross-section of r = 3 nm as a function of time and (b)
fast Fourier transformation of the potential energy.
Figure 6. Fundamental tensile vibration frequency as a function of (a) the wire length for FTNs with a radius of r ≈ 3 nm and (b) the wire
radius for copper FTNs with a length of l ≈ 96 nm. The results predicted by classic theory for different material properties are also shown.
5
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or the Timoshenko beam model. The corresponding governing equations are [41] Euler-Bernoulli
Table 1. Fundamental longitudinal vibration frequencies of copper FTNs.
Theoretical frequency (GHz)
No.
Length (∼nm)
Radius (∼nm)
MD Frequency (GHz)
1 2 3 4 5 6 7
48 60 72 84 96 96 96
3 3 3 3 3 4 6
21.36 17.17 14.11 12.21 10.68 10.68 10.68
∂ 2 ⎡ ∂ 2w (x , t ) ⎤ ∂ 2w (x , t ) =0 EI ⎥ + ρA 2 ⎢ 2 ∂x ⎣ ∂x ∂t 2 ⎦
Eeff
Emacro
Timoshenko
21.35 17.08 14.23 12.20 10.67 10.64 10.60
19.09 15.27 12.73 10.91 9.55 9.55 9.55
⎧ ∂ 2w (x , t ) ⎪ ρA − k′GA ∂t 2 ⎪ ⎪ ⎡ ∂ 2w (x , t ) ∂α ( x , t ) ⎤ ⎪ ×⎢ − ⎥ = 0 2 ⎪ ∂x ⎦ ∂x ⎣ ⎨ ⎪ ∂ 2α (x , t ) − k′GA ⎪ ρI ∂t 2 ⎪ ⎤ ∂ 2α ( x , t ) ⎡ ∂w ( x , t ) ⎪ α × − = 0 x t ( , ) ⎥⎦ − ⎢ ⎪ ⎣ ∂x ∂x 2 ⎩
section, ρ is the mass density, and u(x,t) is the axial displacement at point x and time t. The corresponding vibration angular frequency for a clamped-free one-dimensional structure/beam can be written as ωi =
(2i − 1) π 2L
E (i = 1,2, …) ρ
(5)
(6)
4
in which I = π4r is the area moment of inertia of the crosssection about the neutral axis, G is the shear modulus, k′ is the shear factor, and w(x, t) and α(x, t) are the deflection angle and rotation angle of the cross-section at point x and time t, respectively. The corresponding vibration frequency for a clamped-free beam can be written as [41] Euler-Bernoulli
(4)
in which L is the length of the beam. From the frequency response curve of the potential energy versus time curve of the aforementioned FTN, i.e., the curve shown in figure 5(b), it can be observed that the first three longitudinal frequencies are about 12.21 GHz, 36.62 GHz, and 60.94 GHz, respectively. These frequencies possess the relationships f2 ≈ 3f1 and f3 ≈ 5f1, which are consistent with equation (4). In addition, the theoretical prediction indicates that the fundamental longitudinal vibration frequency is inversely proportional to the length and independent of the wire radius, which is also valid for copper FTNs. To make a quantitative comparison, figure 6 and table 1 also display the predictions from classic elastic theory, with Emarco and Eeff for copper FTNs with different lengths and radii. We can see that the predictions using the effective Young’s modulus Eeff agree quite well with the MD calculations, whereas those calculated based on Emarco diverge significantly from the MD results. From the foregoing analysis, it can be concluded that, provided an effective Young’s modulus that considers the discrete nature and surface effect of nanowires is used rather than Young’s modulus of bulk polycrystalline copper materials, classic elastic theory can be applied to modeling the longitudinal vibration of copper FTNs.
ωi =
k i2 L2
⎞ EI ⎛ 2i − 1 ⎜ k = 1.875,4.694, π for j ⩾ 3⎟ (7) i ⎝ ⎠ 2 ρA
Timoshenko ωi =
1 L
E ρ
ai2 − bi2 (i = 1,2, …) 1 + γ2
(8)
in which γ 2 = k E′ G and ai and bi can be determined by solving the following nonlinear equations: ⎧ a 2 + γ 2b 2 a 2 γ 2 + b 2 ⎪ − s2 = 0 ⎪ a 2 − b2 1 + γ 2 ⎪ ⎪ a 2 − b2 sin a sinh b ⎨ ⎪ a 4 + a 4γ 4 + 4γ 2a 2b2 + b4γ 4 + b4 ⎪ − ab ⎪ a 2 + γ 2b2 a 2γ 2 + b2 ⎪ ⎩ × cos a cosh b − 2ab = 0
(
(
(
)( )(
)
)
) (
(9)
)
(
)(
)
It should be noted that the preceding equations hold for a one-dimensional structure with a small slenderness ratio s = L A /I , which is the case considered in the present work. Figure 7 and table 2 present the predictions from the classic Euler-Bernoulli beam theory with Emarco and Eeff and the classic Timoshenko beam theory with Emarco, Eeff, Gmarco, and Geff. We can see that the predictions using the effective Young’s modulus and shear modulus are in reasonable agreement with the MD calculations. However, the EulerBernoulli beam theory is mainly applicable to copper FTNs with a large slenderness ratio. The Timoshenko beam theory is valid for all copper FTNs simulated in the present work,
3.3. Transverse vibrational properties of copper FTNs
The fundamental transverse vibration frequencies as a function of length and radius of copper FTNs are shown in figure 7, and the corresponding values are listed in table 2. It can be seen that the frequency decreases quadratically with the increase in length for copper FTNs, whereas it increases almost linearly with the increase in radius. From classic elastic theory, the transverse vibration of a one-dimensional structure can be described by the Euler-Bernoulli beam model 6
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Figure 7. Fundamental transverse vibration frequency as a function of (a) the wire length for copper FTNs with a radius of r ≈ 2 nm and (b)
the wire radius for copper FTNs with a length of l ≈ 40 nm. The results predicted by classic theory for different material properties are also shown. Table 2. Fundamental transverse vibration frequencies of copper FTNs.
Theoretical frequency (GHz) Euler Bernoulli
Timoshenko
No.
Length (∼nm)
Radius (∼nm)
MD frequency (GHz)
Eeff
Emacro Gmacro
Eeff
Emacro Gmacro
1 2 3 4 5 6
24 32 40 48 40 40
2 2 2 2 3 4
3.82 2.15 1.37 0.95 2.03 2.63
3.94 2.21 1.42 0.99 2.08 2.81
3.52 1.97 1.26 0.88 1.86 2.52
3.82 2.17 1.40 0.98 2.03 2.69
3.46 1.96 1.26 0.88 1.84 2.46
longitudinal and transverse vibration of copper FTNs, it has been found that the vibration frequencies predicted by using classic elastic theories with the obtained effective elastic moduli are consistent with atomistic simulation results, which validates the applicability of the continuum theories at the nanoscale and suggests that the constructed core-shell composite structure models can be effectively used further for the mechanical analysis of copper FTNs based on continuum theories. Furthermore, we found that the Timoshenko beam model is more appropriate to describe the transverse vibration. These findings should be helpful for the construction of a hierarchical multiscale coupling method of atomistic and continuum theories that can be used to effectively model large systems comprising many FTNs and is of practical importance for the design of FTN-based vibration nanodevices.
taking into account reasonable consideration of shear deformation.
4. Conclusions In summary, we have used classical molecular static and dynamic simulations to investigate the elastic and vibrational properties of copper FTNs. Simulation results reveal size dependencies of Young’s and shear moduli of copper FTNs; that is, Young’s modulus decreases with the increase in radius, while the shear modulus increases with the increase in radius. Empirical core-shell composite structure models have been constructed, and they can describe the observed size dependencies fairly well because these models give reasonable consideration to the surface effect of FTNs. These models show that there is a transition from surface elastic moduli to bulk elastic moduli and that the transition size has been found to be about 10 ∼ 20 nm. However, it should be noted that, owing to the anisotropy of copper FTNs, the surface thicknesses of the core-shell models predicted from the size-dependent Young’s modulus and shear modulus are different from each other. Regarding the small-amplitude
Acknowledgments The support of the National Natural Science Foundation of China (Nos. 11272003, 11232003, and 11302037), the 111 Project (No. B08014), the National Basic Research Program of China (Nos. 2010CB832704 and 2011CB013401), the Program for New Century Excellent Talents in University 7
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(NCET-13-0088), the PhD Programs Foundation of the Ministry of Education of China (20130041110050), and the Fundamental Research Funds for the Central Universities is gratefully acknowledged.
[21] [22]
References [23]
[1] Lisiecki I, Filankembo A, Sack-Kongehl H, Weiss K, Pileni M P and Urban J 2000 Structural investigations of copper nanorods by high-resolution TEM Phys. Rev. B 61 4968–74 [2] Kim C, Gu W H, Briceno M, Robertson I M, Choi H and Kim K 2008 Copper nanowires with a five-twinned structure grown by chemical vapor deposition Adv. Mater. 20 1859–63 [3] Chen H Y, Gao Y, Zhang H R, Liu L B, Yu H C, Tian H F, Xie S S and Li J Q 2004 Transmission-electron-microscopy study on fivefold twinned silver nanorods J. Phys. Chem. B 108 12038–43 [4] Wiley B, Sun Y G, Mayers B and Xia Y N 2005 Shapecontrolled synthesis of metal nanostructures: the case of silver Chem. Eur. J. 11 454–63 [5] Chen J Y, Wiley B and Xia Y N 2007 One-dimensional nanostructures of metals: large-scale synthesis and some potential applications Langmuir 23 4120–9 [6] Hu M, Hillyard P and Hartland G V 2004 Determination of the elastic constants of gold nanorods produced by seed mediated growth Nano Lett. 4 2493–7 [7] Hartland G V 2004 Measurements of the material properties of metal nanoparticles by time-resolved spectroscopy Phys. Chem. Chem. Phys. 6 5263–74 [8] Huang X, Neretina S and El-Sayed M A 2009 Gold nanorods: from synthesis and properties to biological and biomedical applications Adv. Mater. 21 4880–910 [9] Ling T, Yu H M, Liu X H, Shen Z Y and Zhu J 2008 Five-fold twinned nanorods of fcc fe: synthesis and characterization Cryst. Growth Des. 8 4340–2 [10] Fu X, Jiang J, Liu C and Yuan J 2009 Fivefold twinned boron carbide nanowires Nanotechn. 20 365707 [11] Anderoglu O, Misra A, Wang H and Zhang X 2008 Thermal stability of sputtered Cu films with nanoscale growth twins J. Appl. Phys. 103 094322 [12] Kulkarni Y, Asaro R J and Farkas D 2009 Are nanotwinned structures in fcc metals optimal for strength, ductility and grain stability? Scr. Mater. 60 532–5 [13] Yoo J H, Oh S I and Jeong M S 2010 The enhanced elastic modulus of nanowires associated with multitwins J. Appl. Phys. 107 094316 [14] Cao A and Wei Y 2006 Atomistic simulations of the mechanical behavior of fivefold twinned nanowires Phys. Rev. B 74 214108 [15] Wu J Y, Nagao S, He J Y and Zhang Z L 2011 Role of fivefold twin boundary on the enhanced mechanical properties of fcc fe nanowires Nano Lett. 11 5264–73 [16] Jiang S, Shen Y G, Zheng Y G and Chen Z 2013 Formation of quasi-icosahedral structures with multi-conjoint fivefold deformation twins in fivefold twinned metallic nanowires Appl. Phys. Lett. 103 041909 [17] Wen Y H, Huang R, Zhu Z Z and Wang Q 2012 Mechanical properties of platinum nanowires: an atomistic investigation on single-crystalline and twinned structures Comput. Mater. Sci. 55 205–10 [18] Lieber C M 2003 Nanoscale science and technology: building a big future from small things MRS Bull. 28 486–91 [19] Craighead H G 2000 Nanoelectromechanical systems Science 290 1532–5 [20] Husain A, Hone J, Postma H W C, Huang X M H, Drake T, Barbic M, Scherer A and Roukes M L 2003 Nanowire-based
[24] [25]
[26] [27]
[28] [29]
[30] [31] [32]
[33] [34]
[35]
[36] [37] [38]
[39] [40] [41]
8
very-high-frequency electromechanical resonator Appl. Phys. Lett. 83 1240–2 Chang P C, Fan Z Y, Wang D W, Tseng W Y, Chiou W A, Hong J and Lu J G 2004 ZnO nanowires synthesized by vapor trapping CVD method Chem. Mater. 16 5133–7 Li Y, Tan B and Wu Y 2006 Freestanding mesoporous quasisingle-crystalline Co3O4 nanowire arrays J. Am. Chem. Soc. 128 14258–9 Qu L T, Shi G Q, Wu X F and Fan B 2004 Facile route to silver nanotubes Adv. Mater. 16 1200–3 Feng X L, He R R, Yang P D and Roukes M L 2007 Very high frequency silicon nanowire electromechanical resonators Nano Lett. 7 1953–9 Belliard L, Cornelius T W, Perrin B, Kacemi N, Becerra L, Thomas O, Toimil-Molares M E and Cassinelli M 2013 Vibrational response of free standing single copper nanowire through transient reflectivity microscopy J. Appl. Phys. 114 193509 Conley W G, Raman A, Krousgrill C M and Mohammadi S 2008 Nonlinear and nonplanar dynamics of suspended nanotube and nanowire resonators Nano Lett. 8 1590–5 Park S H, Kim J S, Park J H, Lee J S, Choi Y K and Kwon O M 2005 Molecular dynamics study on sizedependent elastic properties of silicon nanocantilevers Thin Solid Films 492 285–9 Olsson P A T 2010 Transverse resonant properties of strained gold nanowires J. Appl. Phy. 108 034318 Yu H, Zhang W W, Lei S Y, Lu L B, Sun C and Huang Q A 2012 Study on vibration behavior of doubly clamped silicon nanowires by molecular dynamics J. Nanomater. 2012 342329 Zhan H F and Gu Y T 2012 A fundamental numerical and theoretical study for the vibrational properties of nanowires J. Appl. Phys. 111 124303 Zhan H F, Gu Y T and Park H S 2012 Beat phenomena in metal nanowires, and their implications for resonance-based elastic property measurements Nanoscale 4 6779 Zheng Y G, Zhao Y T, Ye H F, Zhang H W and Fu Y F 2014 Atomistic simulation of torsional vibration and plastic deformation of five-fold twinned copper nanowires Int. J. Comput. Methods 11 1344010 Chen C Q, Shi Y, Zhang Y S, Zhu J and Yan Y J 2006 Size dependence of Young’s modulus in ZnO nanowires Phys. Rev. Lett. 96 075505 Gavan K B, Westra H J R, van der Drift E W J M, Venstra W J and van der Zant H S J 2009 Size-dependent effective Young’s modulus of silicon nitride cantilevers Appl. Phys. Lett. 94 233108 Zhu Y, Qin Q, Xu F, Fan F, Ding Y, Zhang T, Wiley B J and Wang Z L 2012 Size effects on elasticity, yielding, and fracture of silver nanowires: In situ experiments Phys. Rev. B 85 045443 Li J 2003 AtomEye: an efficient atomistic configuration viewer Modelling Simul. Mater. Sci. Eng. 11 173–7 Daw M S and Baskes M I 1984 Embedded-atom method— derivation and application to impurities, surfaces, and other defects in metals Phys. Rev. B 29 6443–53 Mishin Y, Mehl M J, Papaconstantopoulos D A, Voter A F and Kress J D 2001 Structural stability and lattice defects in copper: ab initio, tight-binding, and embedded-atom calculations Phys. Rev. B 63 224106 Plimpton S 1995 Fast parallel algorithms for short-range molecular-dynamics J. Comput. Phys. 117 1–19 Beards C F 1996 Structural Vibration: Analysis and Damping (Oxford: Butterworth-Heinemann) Han S M, Benaroya H and Wei T 1999 Dynamics of transversely vibrating beams using four engineering theories J. Sound Vib. 225 935–88
