Shape- and Symmetry-Dependent Mechanical Properties of Metallic

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Shape and Symmetry Dependent Mechanical Properties of Metallic Gold and Silver on the Nanoscale Mahmoud A. Mahmoud, Daniel O'Neil, and Mostafa A. El-Sayed Nano Lett., Just Accepted Manuscript • Publication Date (Web): 13 Dec 2013 Downloaded from http://pubs.acs.org on December 20, 2013

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Shape and Symmetry Dependent Mechanical Properties of Metallic Gold and Silver on the Nanoscale Mahmoud A. Mahmoud,† Daniel O'Neil, † Mostafa A. El-Sayed Laser Dynamics Laboratory, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400 *E-mail: [email protected]

Abstract The mechanical properties of anisotropic nanoparticles like gold nanorods (AuNRs) and silver nanorods (AgNRs) are different from those of isotropic shapes such as nanospheres. We probed the coherent lattice oscillations of nanoparticles by following the modulation of the plasmonic band frequency using ultrafast laser spectroscopy. We found that while the frequency of the longitudinal vibration mode of AgNRs is higher than that of AuNRs of similar dimensions, similarly sized gold and silver nanospheres have similar lattice vibration frequencies. Lattice vibrations calculated by finite element modeling showed good agreement with the experimental results for both AgNRs and AuNRs. The accuracy of the calculations was improved by using actual pentagonal shapes rather than cylinders that did not agree well with the experimental results. As the plasmon energy is transferred into lattice vibrations, the temperature of the nanoparticle necessarily increases as a result of this electron-phonon relaxation process. This results in a decrease in the Young’s modulus which was accounted for in the calculations. Calculations showed that the tips of the nanorods are ‘softer’ than the rest of the nanorod. Since the tips comprise a larger portion of the overall rod in the smaller rods, the smaller rods were more affected by the tip effects. Keywords: gold nanorods, silver nanorods, mechanical properties, lattice vibration, pump-probe

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Introduction When plasmonic metallic nanoparticles are irradiated with electromagnetic radiation of resonant frequency, their free electrons oscillate collectively.1-3 In the first few femtoseconds after excitation of the plasmon, the motion is primarily constrained to the electrons rather than the nuclei.4, 5 The oscillating electrons dephase due to electron-surface scattering, electron crystal boundary scattering in polycrystalline nanoparticles, and electron-hole pair generation on a time scale of less than 5 fs.6 If electron-hole pairs are generated in the conduction band (intra-band transition) or in both the d and conduction bands (inter-band-type excitations), the incident photon is absorbed.12,13,6 The excited electrons relax by either electron-electron collision or electron-phonon collisions. Each relaxation pathway has a characteristic lifetime. For instance, the electron−electron scattering and electron diffusion redistribute the energy, leading to a thermalized hot electron distribution within a few hundred fs.7 The excited electrons can also relax by coupling with phonons within 1-5 ps.8-10 Acoustic lattice vibrations occur because the temperature of the electrons is higher than that of the surrounding lattice.11 The resulting lattice vibrations take place at a longer time scale (10-100 ps). Ultrafast femtosecond laser spectroscopy has been used to study both the relaxation processes and the resultant phonons.12 These techniques rely upon changes in the nanoparticle’s oscillating size.13 For instance, the breathing vibrational mode of the nanospheres’ causes rapid changes in radius.12, 14, 15 Some studies have shown that the electron-phonon coupling is changed when the nanoparticle is placed in MgSO49, 16, polymer8, and glass matrices17. Another study showed that the coupling is shape and size independent.18, 19 The lattice vibration modes of nanoparticles are related to the intrinsic mechanical properties of the material and the shape of the nanoparticle. In order to study these features, the


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nanoparticles are excited with a femtosecond laser. The perturbation induced by the laser pulse causes the vibration of the nanoparticle’s lattice as described above.20 These vibrations cause a periodic change in the nanoparticle size and thus a corresponding shift in the LSPR spectrum.20 This time dependent shift is followed by measuring the changes in the intensity of the spectrum at a fixed wavelength at different delay times after the excitation of the LSPR spectrum. The lattice frequencies of isotropic shapes such as gold and silver nanospheres21, 22 and gold and silver nanoprisms23 have been measured by this technique. Gold and silver have comparable mechanical properties, so the lattice vibration frequency of isotropic shapes of the same size had little dependence upon the material. Unlike spheres, many other nanoparticle morphologies are not single crystals, this can also impact the mechanical properties.24 In this letter we study the optomechanical properties of anisotropic nanoparticle shapes experimentally, by ultrafast time-resolved pump-probe spectroscopy, and theoretically, by finite element simulation. We compare our results for silver nanorods with the results of Hu et al.25 for similarly sized gold nanorods. The mechanical properties of the anisotropic shapes are compared with those of isotropic nanoparticles. The effect of the length of gold and silver nanorods on their mechanical properties is discussed. We also study the effect of changing the temperature has on the lattice vibrations.

Experimental Silver nanorods with similar diameters but different lengths were prepared by ethylene glycol (EG) reduction of silver nitrate in the presence of polyvinyl pyrrolidone at 175 oC as reported in our earlier publication.26 Different lengths of AgNRs were prepared from the same batch; the length of the rods was increased by increasing the reduction time of Ag ions by EG.


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The elongation of the rods was stopped by quenching in ice. The AgNRs were cleaned from EG and by-products by water dilution and precipitation by centrifugation, and then dispersion in water. Figure 1 shows the high resolution TEM of a single AgNR, the rod has pentatwinned shape. Gold nanorods (AuNRs) of two different lengths were prepared and cleaned as reported earlier.27, 28 Zeiss Ultra 60 was used to image the AgNRs (see figure S1), while TEM was used to image the AuNRs (see figure S2). A Tecnai F30 was used for high-resolution transmission electron microscopy (HR-TEM) imaging.

Figure 1 High-resolution TEM of a single silver nanorod. The rod has pentatwinned structure with a truncated tip.

The vibrational dynamics were determined for the AgNRs and AuNRs. A Coherent Libra-HE regenerative-amplified Ti:sapphire laser system produced the pump and probe pulses for transient absorption measurements. This pulse compression system produces 4 mJ, sub50 ps pulses of 800 nm-centered light with a repetition rate of 1 kHz. Approximately 25% of this beam was frequency-doubled by a BBO crystal and then filtered to yield a 400 nm pump beam. Every other pump beam pulse was blocked by an optical chopper. Less than 5% of the 800 nm 4 ACS Paragon Plus Environment

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fundamental was directed onto a delay stage and then to a sapphire crystal to produce a supercontinuum from 450 to 820 nm. This probe beam spectrum was band-pass filtered as needed. The two beams were focused and overlapped on the sample. The 1/e2 radii of the pump and probe beams were 850 ± 50 µm and 275 ± 25 µm, respectively. The transmitted probe beam was focused into a fiber optic cable and measured by a Si array detector. ∆A was calculated as a function of delay time by comparing the transmitted probe beam when the pump beam was and was not blocked by the chopper. Solution measurements were performed in 2 mm optical crown glass cuvettes. Solutions were diluted to peak optical densities of < 1 AU. Pump beam fluences were kept below20 µJ/cm2 to minimize sample damage. Sample dynamics were studied for a single wavelength between 15 and 40 nm below the sample’s extinction maximum. The ∆A data was empirically modeled as two exponential decays and an exponentially decaying sinusoid:29 ∆ cos2 ⁄ ⁄ ⁄ Bn, τn, f, ϕ, and c are the pre-exponential constants, time constants, frequency, phase, and offset, respectively. Fits were performed using custom Python code utilizing the Scipy and Numpy libraries.30 By probing all samples on the same side of their extinction maxima, we minimized phase and frequency differences.29


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Results and Discussion Coherent lattice vibrations of gold and silver nanorods of similar diameter and different length It is important to understand the mechanical properties of nanostructures before using them in applications. Although the properties of bulk gold and silver are well known, it is not completely clear whether the optomechanical properties of silver and gold nanoparticles depend on their shape, size, and thermal properties. In order to study the optomechanical properties of silver and gold on the nanoscale, isotropic shapes such as gold and silver nanorods, of similar diameter and different length, were studied as an example. A time-resolved femtosecond pump-probe technique was used to measure the optomechanical properties of AgNRs and AuNRs. Coherent lattice vibrations were first induced with a 405 nm femtosecond laser pulse and their frequencies were probed by a second pulse, with a wavelength in resonance with the LSPR of the nanoparticles, at variable time delay. Since the LSPR peak changes along with the lattice vibrations, following the change in absorption near the LSPR is akin to measuring the phonon frequency. The LSPR spectrum of AgNRs of similar diameter and different lengths are shown in Figure S3. Figure 2 A shows the transient absorption of silver nanorods of ~21 nm diameter and 39.6±3.2 and 86.0±9.7 nm length pumped at 405 nm and probed at wavelengths of 529 and 787 nm, respectively. The vibration lattice frequency is calculated from the transient absorption spectrum using Eq. 1. The Fouriertransform of the modulating transient spectra showed one primary frequency corresponding to the longitudinal mode of vibration.25 This longitudinal mode is expected to vary with length (not diameter) but all rods examined had similar diameters to control as many variables as possible. These same measurements were conducted on AgNRs of different lengths and on two sizes of AuNRs. To make a full comparison between gold and silver we compare our data to that 6 ACS Paragon Plus Environment

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obtained for AuNRs by Hartland’s group.25 We obtained similar values for our two AuNRs as were found by Hu et al.25 Figure 2B shows the relationship between the lattice vibration frequency and the length of AuNRs and AgNRs of similar diameter and different length. The lattice vibrational frequency of AgNRs is higher than that for AuNRs of similar length. For both metals, the frequency increases with decreasing length, but silver’s frequency is more sensitive to length changes than that of the gold. This behavior differs from gold and silver nanospheres, which have the same phonon frequency for a given size.21, 22 The reason for this difference in behavior between shapes is partly due to geometric reasons as is discussed below. Theoretical calculations in the next section will help to explain the unusual optomechanical properties of the anisotropic gold and silver shapes.

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Figure 2. A) Transient absorption dynamics for aqueous solutions of silver nanorods of similar diameter ~ 21 nm and different length 39.6±3.2 (blue) and 86.0±9.7 (red) nm, excited by 405 nm femtosecond laser and probed at 529 and 787 nm. The solid lines represent fits of each data set to an exponentially decaying sine function. B) The relationship between length and extensional frequency is approximately inverse. Silver rods (in black) have higher frequencies than gold rods and the difference decreases as length increases. Theoretical calculation of the lattice frequency of pentagonal gold and silver nanorods The equation for the natural frequencies of cylinders has been analytically determined. In particular, the first longitudinal (extensional) mode is given by the following equation which 7 ACS Paragon Plus Environment

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relates frequency (f) to length (L), Young’s modulus (E) and density (ρ). This equation has been used to model gold nanorods by assuming that nanorods are cylindrical.25

2 Similarly, the analytical equation for the completely symmetric “breathing” oscillation of spheres has been solved.

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Where f is the frequency of the breathing modes, R is the radius, cl is the longitudinal speed of sound, ct is the transverse speed of sound, and χ are the eigenvalues for which there is one for each order mode. These are found by numerically solving equation 4. Typically, bulk values for polycrystalline metals are used: density equals 19.3 and 10.5 g/cm3 for gold and silver, respectively, and Young’s modulus is 79 and 83 GPa for these metals. As shown in Figure 3C, the analytical equation for spheres fits the experimental data extremely well. Unfortunately, the analytical equation for rods above does not exactly fit the experimental results (see figure S4). Interestingly, we see a greater deviation from this equation at shorter lengths. The main reason for the poor fit is that gold and silver nanorods are pentagonal (as shown by HR-TEM (Figure 1)) and cannot be accurately modeled as cylinders. Additionally, the cylinder model is evaluated assuming that the nanorods are isotropic. While this would be true if they were composed of randomly oriented crystal grains, the actual rods consist of five aligned crystalline domains. Thus, it is better to treat the rods as if they were anisotropic single crystals. Petrova et al.24 have used the cylinder model to show that the data for gold nanorods is better fit if you assume that the gold is aniostropic. Crut et al.31 used finite-


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element analysis to investigate the vibrational acoustic modes of nanospheres and nanorods in both frequency and time domain experiments; this simulation is based on modifying the different acoustic vibration modes (fundamental and quadrupolar) leading to activation of a quadrupolar-like mode in time-domain. Activation of the quadrupolar mode was found to increase the quality of the theoretical fit of the experimental results. There is little hope for finding an exact expression for the natural frequencies of pentagonal nanorods and so we turned to finite element modeling (FEM). The FEM package COMSOL was used to calculate the eigen frequencies of AuNRs and AgNRs which were treated as pentagonal prisms with pentagonal tips. Instead of utilizing the bulk mechanical constants for the polycrystalline metals, the three elastic constants which completely characterize a cubic solid were used. For gold c11 = 193 GPa, c12 = 164 GPa, c44 = 42 GPa. For gold c11 = 124 GPa, c12 = 93 GPa, c44 = 46 GPa. It was seen that the geometry of the tips of the rods had a significant impact on calculated frequencies. High resolution TEM has shown that AuNRs have truncated tips.32 Rods with this geometry were found to be in better agreement with the experimental data than rods with sharp points. Figure 3A shows the relationship between the experimental and modeled extensional frequencies of AuNRs. Similarly, the theoretical and experimental values of lattice vibrations of AgNRs are shown in figure 3B. Although the theoretical values are close to the experimental results, there is still some discrepancy. The reason of the deviation between the experimental and theoretical values could be due to the following: 1) All of these models (both FEM and analytical equations) have used room temperature values for Young’s modulus. However the oscillation of the nanoparticles is due to energy transfer to the lattice, which, of course, causes a temperature increase. The value of the Young’s modulus is temperature dependent: it decreases as the temperature increases.33 The temperature effect on the Young’s


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modulus will be discussed in the next section. 2) Both the analytical equation (Figure 3C) and FEM simulations for spheres accorded very well with the experimental values published earlier.21, 22 The deviation of the experimental from the theoretical values in the case of AuNRs and AgNRs could be due to their anisotropic structures. 40

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Figure 3. Relationship between the frequencies (black = experimental, red = modeled) of the nanorods and their length for A) gold nanorods, B) silver nanorods. There is good agreement between the modeled results and experimental data. C) The relationship between the experimental (red and blue dots for silver and gold, respectively) and theoretical (red and blue lines for silver and gold, respectively) lattice vibration frequency values for gold and silver nanospheres and the particle radius. Unlike the nanorods, there is no deviation from the values predicted by the analytical equation.

Effect of temperature and geometry on the lattice vibration of gold and silver nanorods As mentioned earlier, the value of the Young’s modulus is not constant, but depends on the temperature of the materials.33 Figure S5 shows the relationship between the value of the Young’s modulus of gold and silver at different temperatures. The relationship is approximately an inverse linear relationship. The individual elastic constants (c11, c12, c44) have similar temperature dependences.

Figure 4A shows the relationship between the variations in the

extensional frequency of AuNRs (of different length) calculated at different temperatures by FEM. The value of the lattice vibration frequency is found to decrease as the temperature is increased; the short rods are more affected by the change of the temperature compared with the long ones. This could describe the possible reason of the mismatching of the experimental and


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theoretical values of shorter nanorods; however the calculation in Figure 3A was carried out at 300 K. Nanoparticles with isotropic shapes are characterized by the presence of a center of symmetry. When their lattice vibrates, the nanoparticle expands and contract around their center of symmetry. The shape deformation during the nanorods oscillation was calculated by FEM. Figure 4B shows the contracted long AuNR or AuNR while Figure 3C shows the elongated nanorod. When the long rod expands longitudinally, the tips expand more than the middle part of the rods. The red indicates maximum displacement while the blue indicates little displacement. The short nanorod oscillates in the same manner as the long ones, but the tips are relatively more affected than in case of the long nanorod. This is due to the fact that the tapered cap constitutes a larger portion of the small nanorods than the long ones. The thinner tips are relatively more important in the small rods and so the frequency is “softer” than what is expected. For this reason the long and short rods have different mechanical behavior. The temperature rise in the nanorod lattice is determined by the number of photons absorbed per unit volume.24 The number of absorbed photons depends on the absorption crosssections of the nanorods. Since the gold and silver nanorods were pumped by different transitions, the length dependences of their absorption cross sections were different. The AuNRs were pumped by exciting their interband transition while transverse plasmon modes of the AgNRs were excited. Discrete dipole approximation (DDA) calculations of AgNRs show that the transverse mode’s cross section has very little dependence on the length of the nanoparticle. Calculations done on rods of similar diameter (21 nm) and different length showed a comparable absorption peak intensity at ~ 400 nm (see figure S6). Since all of the silver rods we studied had approximately the same width they all absorbed similar numbers of photons. However, since the


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longer rods have a larger volume, they will reach a lower temperature than the short rods. Conversely, the absorption cross section of the interband transition for gold will vary proportionally with volume and will have little dependence on aspect ratio and shape.24 Therefore, at a certain pump laser power, the lattice temperature AuNRs of different lengths will be similar, while longer AgNRs will be cooler than shorter silver rods. This explains why when the length of the rods is increased the value of the lattice vibration frequency decreases with higher rate in case of silver than in the case of gold. Because the lattice frequency in case of AuNRs is affected by the length which is inversely proportional to the lattice frequency, while for AgNRs in addition to the rod length effect, the lattice temperature is variable and so affects the lattice frequency.

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Figure 4 A) The relationship between the extensional frequency of silver nanorods calculated at different temperatures compared with that calculated at room temperature (300K). The frequencies of the lattice vibrations of short rods are affected by temperature change more than the long rods. B) A visualization of the rod’s extensional mode. The color indicates the extent of deformation with red indicating much deformation from the equilibrium position and blue showing little deformation. The tips tend to have a greater amount of extension (even taking into account their greater distance from the center) than the body of the rod. In the little rod (C) we see that because the tips comprise a greater percentage of the total rod length, they have a greater effect on frequency.


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Conclusion The lattice vibration frequencies of silver and gold nanospheres of similar sizes are very similar. The analytical solution for a vibrating sphere predicts the experimental results very accurately. Nanorods, however, cannot be accurately modeled by the analytical solution for a vibrating cylinder. Additionally, a large difference between gold and silver nanorods of similar sizes is observed. A large reason for this discrepancy is that nanorods are not cylinders. In order to correct for this, finite element model simulations were carried out to calculate the extensional frequency of gold and silver nanorods. The modeling results of the frequency agree well with the experiment, but not completely. This mismatch results from the fact that the Young’s modulus is temperature dependent, while previous models assume room temperature values for elastic constants. For this reason we have carried out the calculations at different temperatures. The values of the frequency of the lattice vibration of silver and gold nanorods decrease as the length of the rod increases but the decrease in case of silver is larger than that for gold. This is because the gold rods are optically excited in a way that heats rods of all lengths uniformly whereas shorter silver rods reach a higher temperature than long silver rods. Notes †These two authors (M.A.M. and D.O.) contributed equally to this work. Acknowledgment This work was supported by the Office of Basic Energy Sciences of the US Department of Energy under Contract No. DE-FG02-97-ER 14799. Supporting Information Figure S1 SEM of silver nanorods of different lengths and comparable diameters, Figure S2 SEM of gold nanorods of two different aspect ratios. Figure S3 is the LSPR spectrum of silver


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nanorods of similar diameter and different lengths. Figure S4 shows that the values of the lattice vibration of gold and silver nanorods calculated by the analytical equation do not exactly fit the experimental values. The relationship between the value of the Young’s modulus of gold and silver at different temperatures is in Figure S5. The absorption spectrum of silver nanorods of different length and 21 nm diameter calculated by DDA technique is in Figure S6. This information is available free of charge via the Internet at http://pubs.acs.org/. References 1. Malinsky, M. D.; Kelly, K. L.; Schatz, G. C.; Van Duyne, R. P. J. Phys. Chem. B 2001, 105, 2343-2350. 2. Kreibig, U.; Vollmer, M., Optical Properties of Metal Clusters. (Springer Series in Materials Science 25). 1995; p 532. 3. Link, S.; El-Sayed, M. A. Int. Rev. Phys. Chem. 2000, 19, 409-453. 4. Jain, P. K.; Lee, K. S.; El-Sayed, I. H.; El-Sayed, M. A. J. Phys. Chem. B 2006, 110, 7238-7248. 5. Huang, W. Y.; Qian, W.; El-Sayed, M. A.; Ding, Y.; Wang, Z. L. J. Phys. Chem. C 2007, 111, 10751-10757. 6. Hu, M.; Novo, C.; Funston, A.; Wang, H. N.; Staleva, H.; Zou, S. L.; Mulvaney, P.; Xia, Y. N.; Hartland, G. V. J. Mater. Chem. 2008, 18, 1949-1960. 7. Perner, M.; Gresillon, S.; März, J.; von Plessen, G.; Feldmann, J.; Porstendorfer, J.; Berg, K. J.; Berg, G. Physical Review Letters 2000, 85, 792-795. 8. Mohamed, M. B.; Ahmadi, T. S.; Link, S.; Braun, M.; El-Sayed, M. A. Chem. Phys. Lett. 2001, 343, 55-63. 9. Link, S.; Hathcock, D. J.; Nikoobakht, B.; El-Sayed, M. A. Adv. Mater. 2003, 15, 393396. 10. El-Sayed, M. A. Acc. Chem. Res. 2004, 37, 326-333. 11. Groeneveld, R. H. M.; Sprik, R.; Lagendijk, A. Phys.Rev.B 1992, 45, 5079-5082. 12. Ahmadi, T. S.; Logunov, S. L.; El-Sayed, M. A. The J. Phys. Chem. 1996, 100, 80538056. 13. El-Sayed, M. A. Acc. Chem. Res. 2001, 34, 257-264. 14. Zhang, J. Z. Acc. Chem. Res. 1997, 30, 423-429. 15. Hartland, G. V. Chem. Rev. 2011, 111, 3858-3887. 16. Link, S.; Furube, A.; Mohamed, M. B.; Asahi, T.; Masuhara, H.; El-Sayed, M. A. J. Phys. Chem. B 2002, 106, 945-955. 17. Halte, V.; Bigot, J. Y.; Palpant, B.; Broyer, M.; Prevel, B.; Perez, A. Appl. Phys. Lett. 1999, 75, 3799-3801. 18. Link, S.; Burda, C.; Mohamed, M. B.; Nikoobakht, B.; El-Sayed, M. A. Phys. Rev. B 2000, 61, 6086-6090. 19. Hodak, J. H.; Martini, I.; Hartland, G. V. J. Phys. Chem. B 1998, 102, 6958-6967. 20. Heilweil, E. J.; Hochstrasser, R. M. J. Chem. Phys. 1985, 82, 4762-4770. 14 ACS Paragon Plus Environment

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21. Del Fatti, N.; Voisin, C.; Chevy, F.; Vallee, F.; Flytzanis, C. J. Chem. Phys. 1999, 110, 11484-11487. 22. Hodak, J. H.; Henglein, A.; Hartland, G. V. J Chem. Phys. 1999, 111, 8613-8621. 23. Huang, W. Y.; Qian, W.; El-Sayed, M. A. J. Phys. Chem. B 2005, 109, 18881-18888. 24. Petrova, H.; Perez-Juste, J.; Zhang, Z. Y.; Zhang, J.; Kosel, T.; Hartland, G. V. J. Mater. Chem. 2006, 16, 3957-3963. 25. Hu, M.; Wang, X.; Hartland, G. V.; Mulvaney, P.; Juste, J. P.; Sader, J. E. J. Am. Chem.Soc. 2003, 125, 14925-14933. 26. Mahmoud, M. A.; El-Sayed, M. A. J. Phys. Chem. Lett. 2013, 4, 1541–1545. 27. Jana, N. R.; Gearheart, L.; Murphy, C. J. J. Phys. Chem. B 2001, 105, 4065-4067. 28. Nikoobakht, B.; El-Sayed, M. A. Chem. Mater.2003, 15, 1957-1962. 29. Hartland, G. V.; Hu, M.; Wilson, O.; Mulvaney, P.; Sader, J. E. J. Phys. Chem. B 2002, 106, 743-747. 30. Oliphant, T. E. Computing in Science & Engineering 2007, 9, 10-20. 31. Crut, A.; Maioli, P.; Fatti, N. D.; Vallee, F. Phys.Chem. Chem. Phys. 2009, 11, 58825888. 32. Carbó-Argibay, E.; Rodríguez-González, B.; Gómez-Graña, S.; Guerrero-Martínez, A.; Pastoriza-Santos, I.; Pérez-Juste, J.; Liz-Marzán. L.M. Angew. Chem. Int. Ed. 2010, 122, 95879590. 33. Neighbours, J. R.; Alers, G. A. Phys.Rev. 1958, 111, 707-712.


Nano Letters

TOC

AgNRs AuNRs

Lattice vibration frequency (GHz)

35

r lve Si

30

ds ro no na

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

25 20

Go ld n

15 10 5

20

40

ano

rod s 60

80

100

Nanorod Length (nm)

120


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Shape- and Symmetry-Dependent Mechanical Properties of Metallic

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