Letter pubs.acs.org/NanoLett
Crystal Structure Anisotropy Explains Anomalous Elastic Properties of Nanorods Serguei V. Goupalov*,†,‡ †
Department of Physics, Jackson State University, Jackson, Mississippi 39217, United States A.F. Ioffe Physico-Technical Institute,194021 St. Petersburg, Russia
‡
ABSTRACT: It is demonstrated that the frequency of the extensional vibrational mode of a nanorod made of an elastically anisotropic crystalline material deviates widely from the predictions of the theories based on the analysis of the long-wavelength limit. The dispersion relation for the fundamental extensional mode of a gold rod grown in the [100] direction is calculated and found to be in excellent agreement with experimental data obtained from the transient optical absorption measurements on gold nanorods. This explains an anomaly in the elastic properties of nanorods which was previously attributed to a 26% decrease in Young’s modulus for nanorods compared to its bulk value. The developed approach allows one to investigate the role of the crystal structure anisotropy for acoustic phonons in nanorods and nanowires made of any metal or semiconductor material having cubic crystal structure. KEYWORDS: Nanorods, gold nanoparticles, acoustic vibrations, anisotropy
N
surface plasmon resonance, the transient absorption measurements primarily provide information about those vibrational modes which are accompanied by a substantial change in the nanoparticle volume. For cylindrical nanorods, the periods of the fundamental extensional and radial breathing modes have been measured with this technique.5−11 Theoretically, the acoustic vibrations of nanoparticles can be treated using continuum mechanics. The finite length of a nanorod can be taken into account by sampling the dispersion relation describing plane wave propagation in an infinite rod at wave numbers kz = πn/L, where L is the nanorod length and n is an integer. This dispersion relation can be found by two different approaches. The first one12−15 originates from considering the motion of an element of the structure and implies making certain assumptions on the kinematics of deformation. Following ref 15 we can say that the governing theory in this case is based on the ‘strength of materials’ considerations. This approach is valid for long-wavelength vibrations and for rods of arbitrary cross-section and amounts to special consideration of the extensional, flexural, and torsional waves. An alternative approach14,15 explores three-dimensional equations of motion of continuous elastic medium subject to cylindrically symmetric boundary conditions. Its application is limited to rods of cylindrical cross-section but yields universal dispersion equations valid for all eigenmodes and wavelengths.
anomaterials research has evolved tremendously during the last 30 years. Since the initial discovery of size-related properties and establishing solid grounds for understanding correlation between nanoparticle size and the physical properties of materials, a focus of research has shifted toward finding new effects that, while being directly determined by the nanoscale dimensions of the materials, are strongly affected by other parameters. Among these parameters, shape has been identified as a major tool for tailoring the mechanical, optical, and electronic properties of nanomaterials. However, when the nanoparticle shape anisotropy meshes with the crystal structure anisotropy of the constituent material, it can lead to nontrivial physical consequences often requiring an in-depth analysis. The interplay between the shape and crystal structure anisotropies is particularly important for quasi-one-dimensional nanostructures, such as metal and semiconductor nanowires and nanorods. Recent advances in optical spectroscopy made it possible to study transient optical absorption in metal nanoparticles following coherent excitation of vibrational resonances by ultrafast laser pulses.1−11 In such experiments the ultrafast excitation of metal nanoparticles causes an energy flow out of the electron subsystem and into the crystal lattice within a few picoseconds. This time scale is faster than periods of fundamental vibrational eigenmodes of nanoparticles. Thus, heating and consequent expansion of the crystal lattice lead to excitation of the vibrational resonances. The excited modes give rise to modulations in the transient absorption signal. From these modulations one can measure the periods of the vibrational modes.1−11 As the modulations result from the periodic change in the volume of nanoparticles shifting the spectral position of the localized © 2014 American Chemical Society
Received: January 2, 2014 Revised: February 12, 2014 Published: February 24, 2014 1590
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A similar approach is usually employed to find vibrational modes of spherical nanoparticles.14,16−18 For the fundamental extensional mode of an infinite rod, the first approach yields the dispersion relation of12−15 ω=
E kz ρ
while for the two transverse solutions the eigenvectors can be chosen as follows ⎡ kz −iφ ⎤ e JM − 1(Qr )⎥ ⎢ ⎢ 2 ⎥ ei(kzz + Mφ) ⎢ ⎥ (v1) iQJM (Qr ) uM (r , φ , z) = ⎢ ⎥ Q 2 + kz2 ⎢ ⎥ kz iφ ⎢ e JM + 1(Qr ) ⎥ ⎣ 2 ⎦
(1)
where E is Young’s modulus along the rod and ρ is the mass density. For a rod made of elastically isotropic material, the second approach yields the same result in the long-wavelength limit.14,15 However, reduced symmetry of crystalline materials leads to appreciable anisotropy in their elastic properties.13 In particular, the values of Young’s modulus of bulk gold for the different crystallographic directions are E[100] = 42 GPa, E[110] = 81 GPa, and E[111] = 115 GPa.7 In ref 7 the transient optical absorption was used to measure the vibrational period of the fundamental extensional mode for single-crystalline gold nanorods grown along the [100] crystallographic direction. Different nanorod samples with the mean nanorod diameters in the range from 81 to 183 Å and the mean nanorod aspect ratios from 1.7 to 4.6 were studied. Provided eq 1 is valid, these measurements enable one to extract the value of Young’s modulus. This value was found to be approximately 26% smaller than the bulk value of E[100].7,8 Similar results were obtained for gold nanorods grown along the [110] direction.6−8 It was speculated that elastic properties of nanostructured materials may be different from those of the bulk ones.6,7 In this Letter we show that, when crystal structure anisotropy is large enough, eq 1 becomes invalid. We find the dispersion relation for the extensional vibration of a gold nanorod grown in the [100] direction taking the anisotropy into account. This dispersion relation turns out to be in excellent agreement with the experimental results.7,8 We start with a brief outline of the approach exploiting equations of motion of continuous elastic medium for isotropic media. These equations can be cast in the form13,18 ̂ + (ct2 − c l2)(Ĵ ∇)2 ]u = −ω 2 u Λ̂ 0u ≡ [c l2 IΔ
⎡ e−iφJ
u(Mv2)(r , φ , z) =
i(kzz + Mφ) ⎢
2
M−1
(5)
Here M is the projection of the total angular momentum onto the z axis, JM(x) is the Bessel function of order M, and the wave numbers q and Q satisfy the dispersion relations ω = cl (q2 + k2z)1/2 and ω = ct (Q2 + k2z)1/2, respectively. The dispersion relations for vibrations of an infinite cylindrical rod of radius R are obtained (v1) (v2) by imposing upon the linear combination au(l) M + buM + cuM of these three solutions the boundary condition requiring that the traction forces vanish on the rod surface. In cylindrical coordinates, the components of the traction force are as follows ⎧ ⎡ ⎪ FM , r = ρei(kzz + Mφ)⎨a⎢(c l2 − 2ct2) q2 + kz2 JM (qR ) ⎪ ⎢⎣ ⎩ ⎤ d 2JM (qR ) ⎥ 2ct2kz 2ct2 b − − dR2 ⎥⎦ Q Q 2 + kz2 q2 + kz2 ×
d 2JM (QR ) dR2
−c
⎫ 2Mct2 d ⎛ JM (QR ) ⎞⎪ ⎟⎬ ⎜ ⎪ Q dR ⎝ R ⎠⎭
(6)
⎡ ⎛ J (qR ) dJ (qR ) ⎞ 2M ⎟ ⎜ M FM , φ = iρct2ei(kzz + Mφ)⎢a − M ⎢ q2 + k 2 R ⎝ R dR ⎠ ⎣ z
(2)
Here u is the vector of displacement understood as a threecomponent column, cl and ct are, respectively, the longitudinal and transverse sound velocities, I ̂ is the unit matrix, Δ is the scalar Laplace operator, Jα̂ (α = x,y,z) are the matrices of the angular momentum J = 1, and ω is the vibration frequency. The eigenvectors of the matrix operator Λ̂0 describe displacements in the longitudinal and transverse sound waves propagating in an infinite isotropic elastic medium. We are interested in the eigenvectors having cylindrical symmetry and regular on the z axis. They can be easily constructed if we use the cyclic vector components rather than the Cartesian ones and notice that the eigenvectors should satisfy one of the conditions curl u = 0 (for longitudinal sound) or div u = 0 (for transverse sound). The eigenvector for the longitudinal solution is ⎡ q −iφ ⎤ ⎢ 2 e JM − 1(qr )⎥ ⎢ ⎥ ei(kzz + Mφ) ⎢ ⎥ − ik J ( qr ) u(l) ( r , φ , z ) = zM M ⎥ q2 + kz2 ⎢ q ⎢ ⎥ iφ ⎢⎣ 2 e JM + 1(qr ) ⎥⎦
e
(Qr ) ⎤ ⎥ ⎢ ⎥ 0 ⎢ iφ ⎥ ⎢⎣−e JM + 1(Qr )⎥⎦
(4)
+b
⎛ J (QR ) dJ (QR ) ⎞ ⎟ ⎜ M − M dR ⎠ Q 2 + kz2 QR ⎝ R 2kzM
⎛ 2 d 2J (QR ) ⎞⎤ − cQ ⎜⎜ 2 M 2 + JM (QR )⎟⎟⎥ dR ⎝Q ⎠⎥⎦
(7)
⎡ dJM (qR ) 2kz Q 2 − kz2 FM , z = iρct2ei(kzz + Mφ)⎢− a +b ⎢ dR q2 + kz2 Q Q 2 + kz2 ⎣ ×
dJM (QR ) dR
− cMkz
⎤ JM (QR ) ⎥ QR ⎥⎦
(8)
Here the constants a, b, and c have the dimension of length and FM is the traction force, or force per unit area, acting on the area element of the cylindrical surface facing the radial direction. For M = 0 eqs 6 and 8 have no contribution of the solution (eq 5), while eq 7 has no contributions of the solutions (eqs 3 and 4). Therefore, the solutions are partially decoupled. The purely transverse displacement wave which is only contributed by the solution (eq 5) describes torsional vibrations, and the extensional
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in agreement with eq 1. Further substituting ω2 = ω20 + Δ(ω2) into eq 9 and keeping terms linear in Δ(ω2) and up to quadratic in QR, qR, we get 2Δ(ω2) = −ω20σ2(kzR)2, or (cf. refs 12, 14, and 15)
mode is among the remaining solutions. While the extensional mode in the long-wavelength limit has a linear dispersion, the other remaining solutions do not. In the limit of kz → 0 they further decouple (cf. eqs 6 and 8). The solution which in this limit becomes purely longitudinal describes the radial breathing mode. The remaining purely transverse solution is not accompanied by a change in the rod volume, and therefore, its chances to be observed in the transient absorption measurements are very low. When the values of kz are not restricted, we obtain the following dispersion equation
⎛ σ 2(kzR )2 ⎞ ⎟ ω = ω0⎜1 − 4 ⎝ ⎠
where
⎡⎛ dJ (qR ) ⎤ 2 c2 ⎞ ⎢⎜1 − 2 t2 ⎟ω 2J (qR ) + 2ct2q 1 ⎥(Q − kz2)J (QR ) 1 ⎢⎣⎝ dR ⎥⎦ cl ⎠ 0 +
4ct2kz2qJ1(qR )
dJ1(QR ) dR
=0
σ=
c l2 ct2
−1 (10)
3c l2 − 4ct2 c l2 − ct2
ctkz ≡
E kz ρ
2(c l2 − ct2)
(13)
λ − λ1 λ ̂ + 3 Λ̂ = 1 IΔ ρ ρ
the value of q becomes imaginary. Therefore, in this domain, one has to substitute q → iκ, JM(qR) → iMIM(κR) in eq 9, where IM(x) is the modified Bessel function of order M. [Note that, in order to find dispersion relation for the flexural mode with |M| = 1 and superlinear (quadratic) dispersion, one has to make such substitutions for both q and Q.] In Figure 1 are the dispersion curves for the three lowest vibrational modes described by eq 9, including the fundamental extensional and radial breathing modes, for gold cylindrical rods. The values of sound velocities used for this calculation are cl = 3.24 × 105 cm/s and ct = 1.2 × 105 cm/s. Using the value of the mass density ρ = 19.7 g/cm3, this translates to the values of Young’s modulus E ≈ 80.6 GPa and Poisson’s ratio σ ≈ 0.42. In the long-wavelength limit the dispersion relation for the extensional mode can be found by expanding the Bessel functions in eq 9 into series up to the terms linear in QR, qR: ω0 =
c l2 − 2ct2
is the Poisson ratio. The result of eq 12 is sometimes referred to as the Rayleigh correction.15 This approximate solution is shown in Figure 1 by the dashed line. One can see that it gives a very good approximation of the exact solution even for relatively short wavelengths. Meanwhile, the departure of the dispersion curve for the extensional mode from the linear one is substantial even for the long wavelengths. For cubic crystals the operator Λ̂0 in eq 2 is replaced by13,20
(9)
This equation (or its equivalent) is sometimes referred to as the Pochhammer frequency equation,15 first published in 1876.19 One can consider the left-hand side of eq 9 as a function of QR and look for its zeros. However, when 0 < Q < kz
λ + λ3 −2 2 ρ
∑ Ĵα2 ∇α2 α
∑ {Jα̂ Jβ̂ }∇α ∇β β>α
(14)
where the axes x, y, and z are chosen along the directions [100], [010], and [001], respectively, {Jα̂ Jβ̂ } = (Jα̂ Jβ̂ + Jβ̂ Jα̂ )/2, and λ1, λ2, and λ3 are the three independent components of the elastic stiffness tensor:13 λ1 = λxxxx, λ2 = λxxyy, λ3 = λxyxy. To see how this form of the operator Λ̂ affects the solutions of eq 2 let us study its action on the vector v = ei(Mφ + ikzz)[A e−iφJM − 1(kr )e+1 + BJM (kr )e0 + C eiφJM + 1(kr )e−1]
where e±1 = ∓(ex ± i ey)/√2, e0 = ez are the cyclic orts, and A, B, and C are understood as the vector’s coordinates in the basis thus defined. We find Λ̂ = Λ̂1 + Λ̂2 with
(11)
⎡ λ k 2 + λ (k 2 + 2k 2) i(λ + λ )k k (λ + λ )k 2 ⎤⎡ A ⎤ 3 z 2 3 z 1 2 ⎢ 1 ⎥⎢ ⎥ 2 2 4 ⎢ ⎥⎢ ⎥ ⎡ A⎤ ⎢ ⎥⎢ ⎥ i(λ + λ3)kzk i(λ + λ3)kzk ⎢ ⎥ ⎥⎢ B ⎥ Λ̂1⎢ B ⎥ = −ρ−1⎢− 2 λ1kz2 + λ3k 2 − 2 ⎢ ⎥⎢ ⎥ 2 2 ⎣C ⎦ ⎢ ⎥⎢ ⎥ ⎢ (λ1 + λ 2)k 2 i(λ 2 + λ3)kzk λ1k 2 + λ3(k 2 + 2kz2) ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ C ⎦ 4 2 2
(15)
model the frequency of the extensional mode is much lower than the frequency of any mode with |M| = 4, this operator can be safely neglected. As a last resort, it can be taken into account in the second order of the perturbation theory. Note, however, that this operator is proportional to the parameter λ2 − λ1 + 2 λ3 which serves as the measure of the anisotropy for cubic crystals.13 If this parameter is zero, then the limit of an isotropic material is regained. More precisely, the limit of the isotropic medium is obtained as follows: λ1 → ρc2l , λ2 → ρ(c2l − 2c2t ), λ3 → ρc2t .
λ − λ1 + 2λ3 2 i(Mφ + ikzz) i3φ Λ̂ 2v = 2 ke (C e JM + 3(kr )e+1 4ρ + A e−i3φJM − 3(kr )e−1)
(12)
(16)
The operator Λ̂2 leads to admixture of the solutions with |M| = 4 to the solutions with M = 0. However, since in the isotropic 1592
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A numerical procedure to solve eq 20 can be implemented as follows. One changes the variable QR and finds ω from eq 18. This value is fed into eq 17 which, after substituting eq 19, becomes a quadratic equation on q2. When choosing one of the two roots of this equation we are guided by the limiting case of the isotropic model, which makes the choice unambiguous. If the root is negative, then one replaces q with iκ. Once q (or κ) is determined, one can evaluate the left-hand side of eq 20 and, eventually, find its zeros. In Figure 2 by solid red line is shown the dispersion curve resulting from the numerical solution of eq 20 for gold rods.
Figure 1. Dispersion curves for the three lowest vibrational modes described by eq 9 of the isotropic model for gold cylindrical rods. The sound velocities are cl = 3.24 × 105 cm/s and ct = 1.2 × 105 cm/s. By dashed line is shown the long-wavelength asymptotic solution of eq 12.
It remains to study the effect of Λ̂1. Let us solve the eigenvalue problem for the matrix (eq 15). One of its eigenvectors has the form ⎡ A⎤ ⎡1⎤ 1 ⎢ ⎥ ⎢ ⎥ ⎢ B⎥ = 2 ⎢ 0 ⎥ ⎣−1⎦ ⎣C ⎦
i.e., the corresponding dimensionless displacement coincides with eq 5 (with Q = k). The corresponding eigenvalue is ω2 =
Figure 2. The experimental data of ref 7 (symbols) for gold nanorods of radius R and length L. The product of the vibration frequency for the fundamental extensional mode and the nanorod radius is shown versus the product of the radius and kz = π/L. By solid red line is shown the theoretical dispersion curve obtained by a numerical solution of eq 20. By dashed blue line is shown the prediction of eq 21.
(λ1 − λ 2 + 2λ3)k 2 + 4λ3kz2 4ρ
As in the case of the isotropic medium, for M = 0 this solution turns out to be decoupled from the two other solutions. For these we replace k by q and Q, respectively, in order to obtain the corresponding limits in the isotropic case. We get ω2 =
3λ1 + λ 2 + 6λ3 2 λ + λ3 2 G(q) q + 1 kz + 8ρ 2ρ 8ρ
(17)
ω2 =
3λ1 + λ 2 + 6λ3 2 λ + λ 3 2 G (Q ) Q + 1 kz − 8ρ 2ρ 8ρ
(18)
We have used the following values of the elastic stiffness constants:21 λ1 = 186 GPa, λ2 = 157 GPa, λ3 = 42 GPa, and the value of the mass density ρ = 19.7 g/cm3. Also shown in Figure 2 is experimental data of ref 7 for the frequencies of the fundamental extensional mode of gold nanorods as functions of kzR = πR/L, where L is the nanorod length. One can see that the experimental data are in excellent agreement with the theoretical result. By blue dashed line in Figure 2 is shown the prediction of eq 1 with E = E[100]. Let us study the long-wavelength limit of eq 20. Substituting G(q) and G(Q) from eqs 17 and 18, respectively, and expanding the Bessel functions up to terms linear in qR, QR, we obtain
where G(q) = [(3λ1 + λ 2 − 2λ3)2 q 4 − 8(3λ12 − 5λ1λ3 + λ1λ 2 − 17λ 2λ3 − 6λ32 − 8λ 22)q2kz2 + 16(λ1 − λ3)2 kz4]1/2
(19)
{λ3(λ1 + λ 2)q2 + [λ12 − 2λ 22 + λ 2(λ1 − 2λ3)]kz2
Note that in the isotropic limit G(q) = − (q + k2z ). Now one has to impose the boundary conditions at r = R on a linear combination of the eigenvectors corresponding to the eigenvalues (eqs 17 and 18). A straightforward derivation leads to the following analog of eq 9 4ρ(c2l
c2t )
2
− (λ1 + λ 2)ρω 2 }[λ1kz2 − λ 2Q 2 − ρω 2 ] − {λ3(λ1 + λ 2)Q 2 + [λ12 − 2λ 22 + λ 2(λ1 − 2λ3)]kz2 − (λ1 + λ 2)ρω 2 }[λ1kz2 − λ 2q2 − ρω 2 ] = 0
{[D(q) − G(q)][(λ 2 + λ1)J0 (qR ) + (λ 2 − λ1)J2 (qR )]
One can see that the left-hand side of this equation is proportional to q2 − Q2. Dividing the above equation by this term we obtain
− 16λ 2(λ 2 + λ3)kz2J0 (qR )}J1(QR )q × [D(Q ) + G(Q ) − 8(λ 2 + λ3)Q 2] − {[D(Q ) + G(Q )][(λ 2 + λ1)J0 (QR ) + (λ 2 − λ1)J2 (QR )]
ω=
− 16λ 2(λ 2 + λ3kz2J0 (QR )}J1(qR )Q [D(q) − G(q) − 8(λ 2 + λ3)q2] = 0
(20)
(λ1 + 2λ 2)(λ1 − λ 2) kz ≡ ρ(λ1 + λ 2)
E[100] ρ
kz
(21)
Paradoxically, we have obtained the result that we just claimed to be wrong. At first glance, it seems to invalidate our
where D(q) = q2(2λ3 − λ2 − 3 λ1) + 4k2z (λ1 − λ3). 1593
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findings about the dispersion of the extensional mode. However, an expansion of the left-hand side of eq 20 for small QR implies that it is a smooth function of QR. In reality, there are points where its first derivative is discontinuous. These points occur whenever the values of q switch from real to imaginary ones (or vice versa) and also when the discriminant of the quadratic equation on q2 is zero. This behavior is illustrated in Figure 3 where we have plotted the left-hand side
Figure 4. Comparison of the dispersion curves for the three lowest vibrational modes described by eq 20 for parameters of the isotropic (blue lines) and anisotropic (red lines) models of a gold cylindrical rod.
and thus exhibited extensional frequencies close to these of free rods. Fitting of the measured extensional frequencies for this portion of nanorods with eq 1 yielded the value of Young’s modulus close to the bulk one which allowed Zijlstra et al. to question experimental results of ref 7. However, one cannot completely exclude the influence of the substrate on the results of single particle measurements on immobilized nanorods. Another experiment was done by the same group on gold nanorods in water,10 but nanorod sizes were not well-known for that study. To study acoustic vibrations of rods grown in the [110] direction, one can rewrite the equations of motion in a rotated coordinate frame with the z axis parallel to the [110] direction. The resulting operator, replacing the operator Λ̂0 in eq 2, can be decomposed into two parts, Λ̂1 and Λ̂2, the same way we did it for rods grown in the [100] direction. However, for the [110] rods, Λ̂1 turns out to be almost isotropic, while Λ̂2 couples the extensional mode not only to remote modes with |M| = 4 but also to the modes with |M| = 2. As the lowest mode with |M| = 2 has linear dispersion for infinite cylindrical rods, this coupling can lead to substantial alteration in the period of the low-frequency acoustic mode measured in the transient absorption experiments on nanorods. To conclude, we have demonstrated that the frequency of the extensional vibrational mode of a nanorod made of an elastically anisotropic crystalline material deviates widely from the predictions of the theories based on the analysis of the longwavelength limit. We have calculated a dispersion relation for the fundamental extensional mode of a gold rod grown in the [100] direction and found it to be in an excellent agreement with experimental data obtained from the transient absorption measurements on gold nanorods.7,8 Thus, we explained the anomaly in the elastic properties of nanorods which had been attributed to a 26% decrease in Young’s modulus for nanorods compared to its bulk value.7 The developed approach allows one to investigate the role of the crystal structure anisotropy for acoustic phonons in nanorods and nanowires made of any metal or semiconductor material having cubic crystal structure.
Figure 3. The left-hand side of eq 20 (up to a scale factor) as a function of QR for the parameters of isotropic (blue line) and anisotropic (red line) models of a gold rod and the value of kzR = 0.6.
of eq 20 as a function of QR for the parameters of both the isotropic (blue line) and the anisotropic (red line) models of a gold rod and for the value of kzR = 0.6. For the isotropic model the value of q is imaginary for all values of QR shown in Figure 3 (cf. eq 10), the blue curve is smooth, and its first zero is given by the long-wavelength approximation with a high precision. On the contrary, the behavior of the red curve suddenly changes close to the point where we would anticipate its zero, provided the red curve were smooth. The value of q is imaginary to the right of the point of the derivative discontinuity and real to the left of this point. As a result, the zero of the red curve is shifted toward lower values of QR. In Figure 4 we compare the dispersion curves resulting from the isotropic and the anisotropic models of a gold rod grown in the [100] direction. Interestingly, the dispersion for the extensional mode in the anisotropic model is linear even for relatively short wavelengths, while in the isotropic model it demonstrates a substantial departure from the linear behavior. The frequency of the radial breathing mode in the anisotropic model almost coincides with that of the isotropic model at kz = 0, while for shorter wavelengths the curve of the anisotropic model demonstrates a more pronounced dispersion. It is worth to note that in ref 7 the transient optical absorption measurements were performed on ensembles of nanorods. More recently the acoustic vibration measurements of single gold nanorods grown in the [100] direction and immobilized on a substrate were carried out by Zijlstra et al.9 Fitting of the results for the measured frequencies of the fundamental extensional mode with eq 1 yielded, for most nanorods, the values of Young’s modulus about 60 GPa which is 50% higher than the bulk [100] value. The difference was attributed to the effect of the substrate coupled to the nanorods via van der Waals interactions. It was further proposed9 that some of the nanorods did not rigidly adhere to the substrate
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. 1594
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Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work was supported by the Research Corporation for Science Advancement (award no. 20081) and the National Science Foundation.
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REFERENCES
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