Vibrational Properties of Metal Nanoparticles: Atomistic Simulation

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Vibrational Properties of Metal Nanoparticles: Atomistic Simulation and Comparison with Time-Resolved Investigation Huziel E. Sauceda,† Denis Mongin,‡ Paolo Maioli,‡ Aurélien Crut,‡ Michel Pellarin,‡ Natalia Del Fatti,‡ Fabrice Vallée,‡ and Ignacio L. Garzón*,† †

Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, D. F., México Université Lyon 1, CNRS, LASIM, 43 bd du 11 Novembre 1918, 69622 Villeurbanne cedex, France

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W Web-Enhanced Feature * S Supporting Information *

ABSTRACT: Knowledge of the vibrational spectrum of metal clusters and nanoparticles is of fundamental interest since it is a signature of their morphology, and it can be used to determine their mechanical, thermodynamical, and other physical properties. It is expected that such a vibrational spectrum depends on the material, size, and shape of clusters and nanoparticles. In this work, we report the vibrational spectra and density of states of Au, Pt, and Ag nanoparticles in the size range of 0.5−4 nm (13−2057 atoms), with icosahedral, Marks decahedral, and FCC morphologies. The vibrational spectra were calculated through atomistic simulations (molecular dynamics and a normal-mode analysis) using the many-body Gupta potential. A discussion on the dependence of the vibrational spectrum on the material, size, and shape of the nanoparticle is presented. Linear relations with the nanoparticle diameter were obtained for the periods of two characteristic oscillations: the quasi-breathing and the lowest frequency (acoustic gap) modes. These linear behaviors are consistent with the calculation of the periods corresponding to the breathing and acoustic gap modes of an isotropic, homogeneous metallic nanosphere, performed with continuous elastic theory using bulk properties. Additionally, experimental results on the period corresponding to isotropic volume oscillations of Au nanoparticles measured by time-resolved pump−probe spectroscopy are presented, indicating a linear variation with the mean diameter in the size range of 2−4 nm. These, and similar results previously obtained for Pt nanoparticles with size between 1.3 and 3 nm, are in good agreement with the calculated quasibreathing mode periods of the metal nanoparticles, independently of their morphologies. On the other hand, the calculated period of the mode with the highest (cutoff) frequency displays weak size and shape dependencies up to ∼4 nm, for all nanoparticles under study. In contrast with the behavior of other physicochemical properties, the clear consistency between experiments with atomistic and continuous media approaches resulting from this work indicates the existence of simple relations with size and weak dependence with the material and shape, for vibrational properties of metal nanoparticles.

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INTRODUCTION The main challenge in the study of free metal clusters and nanoparticles is to elucidate and understand the dependence of their physicochemical properties on the material (chemical composition), size, and shape. A great amount of results have been already reported for structural, electronic, and optical properties. Moreover, there is also notoriously an increasing interest in the vibrational properties of metal clusters and nanoparticles, raising the key question of the transition from a solid to a molecular type of vibrational behavior with size reduction. Recent experimental achievements in this direction were the development of multiple photon dissociation (MPD) spectroscopy in the far-infrared (FIR) (also called tera-Hertz region) to measure the vibrational spectra of small metal clusters in the gas phase1 and the use of noncontact timeresolved pump−probe spectroscopy to study the acoustic vibrations of size-controlled metal nanoparticles of 1−4 nm prepared by physical methods.2 Other studies employed © 2012 American Chemical Society

inelastic neutron scattering to investigate the acoustic vibrations of anatase TiO2 nanoparticles,3 Raman scattering to probe the crystallinity of gold nanoparticles through their vibrations,4 and He atom scattering to investigate the surface vibrations of Ar clusters.5 On the theoretical side, although continuous elasticity theory (CET) has been extensively used to describe the vibrational response of nanoparticles,2,6 most of the studies on the size dependence of vibrational spectra in the small size range correspond to atomistic calculations.7−17 They are based on the use of many-body model potentials, although quantum mechanical calculations for small metal clusters had also been reported.18−21 While MPD-FIR spectroscopy assisted by density functional theory (DFT) calculations has allowed the structure determiReceived: September 24, 2012 Revised: November 2, 2012 Published: November 2, 2012 25147

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nation of small transition metal clusters,22 noncontact timeresolved spectroscopy, using a femtosecond optical pump− probe technique,6,23 has measured a linear size dependence of the isotropic volume oscillation period for Pt nanoparticles in the range 1.3−3 nm.2 The latter results are revealing since they are in agreement with a theoretical calculation using CET and bulk elastic constants, suggesting its validity at the nanoscale and a weak impact of the size reduction on the elastic properties of nanoparticles with a few hundred atoms.2,17 The validity of CET at the nanometer scale had also been theoretically examined by investigating the vibrational properties of Ag and Au spherical nanoparticles in the size range 1.5− 4 nm, through atomistic simulations using an embedded-atom potential.13 The results obtained suggest that the applicability of CET to the nanometer scale depends on the specific vibrational mode considered, giving good agreement with the atomistic simulations for the fundamental breathing mode.13 Nevertheless, a breakdown due to surface effects was found for critical sizes below 2 nm depending on the material.13 Other atomistic simulations, using an embedded-atom potential, on Ag nanoparticles in the size range of 150−1400 atoms had analyzed how their morphology introduces a high degree of complexity into the vibrational spectra.12 In particular, it was shown that for single-crystalline faceted shapes the frequencies of the nearly radial breathing mode and of the acoustic gap are around 10% lower than those predicted by CET.12 Additionally, an atomistic simulation study of the breathing mode frequency for Au−Ag nanoalloys in the size range 1−4 nm showed that CET accounts reasonably well for its size dependence, although finite size corrections that scale as powers of the nanoparticle inverse radius were reported.15 More recently, the laser-induced breathing modes of larger Ag and Au nanospheres in the size range of ∼6 − 46 nm were theoretically studied using a symmetric molecular dynamics method and a long-range empirical potential based on the second-moment approximation of the electronic density of states.16 In this large size regime, the calculated breathing oscillations, which are caused by a symmetric expansion corresponding to the excitation of the A1g modes with the largest in-phase radial displacement of atoms in the nanospheres, were found in good agreement with CET and experimental results.16 On the other hand, the breathing mode frequency had also been studied for small (up to n = 74 atoms) Na, Al, and Cs clusters through first-principles calculations, displaying a n−1/3 dependence, i.e., a linear dependence with the inverse of the cluster size.18,19 Although the linear size dependence of the isotropic volume oscillation period for Pt nanoparticles in the range of 1.3−3 nm, measured by a femtosecond optical pump−probe technique,2 has been well-reproduced by the calculated period of the fundamental radial breathing mode of free (and matrixisolated) elastic isotropic homogeneous nanospheres using continuum mechanics,2 it would be insightful to perform additional atomistic simulations on the acoustic vibration of small metal nanoparticles. One motivation is related with the theoretical prediction on the nonexistence of a mode with a uniform radial expansion and contractionthe breathing mode (BM)in metal clusters and nanoparticles with more than 13 atoms, as was recently reported using a general analytical approach.24 Instead, a possible theoretical interpretation of the experimentally measured radial expansion and contraction the monopole oscillationof metal nanoparticles would require the calculation of the so-called quasi-BM (QBM), defined as the mode with the smallest deviation from the BM.24

An atomistic simulation study of the size and shape dependences of the QBM for metal nanoparticles would provide further insights into the atomic vibrations characterizing their elastic response. In this work, we calculate the vibrational spectra, from a normal-mode frequency analysis (diagonalization of the dynamical matrix), for Au, Ag, and Pt nanoparticles in the size range of 0.5−4 nm (13−2057 atoms), with icosahedral (ICO), Marks decahedral (DEC), and FCC truncatedoctahedral (TOC) morphologies. The objective is to elucidate the dependence of the vibrational spectra on the nanoparticle material, size, and shape and, in particular, to investigate the relation between the calculated QBM and the experimental data for the isotropic volume oscillation (IVO) periods measured in time-resolved pump−probe experiments. This is done using previously reported results for Pt nanoparticles in the size range of 1.3−3 nm2 and new ones reported here for Au nanoparticles in the range 2−4 nm, using the same approach. In both cases clusters were prepared using a physical method avoiding the presence of molecules bound at their surface, which can alter the vibrational response of the metal part, making theoretical− experimental comparison fully relevant. This comparison would be useful not only to gain insights into the acoustic response of metal nanoparticles but also to test the capability of atomistic simulations based on the many-body Gupta potential, containing parameters fitted to bulk properties, to describe their vibrational properties in the size range between 0.5 and 4 nm. Since an overall agreement between calculated and measured periods is obtained, additional reliable predictions on the behavior of other features of the nanoparticle vibrational spectra can be provided.

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METHODOLOGIES Theoretical. The vibrational spectra of each nanoparticle were calculated through a normal-mode analysis through a direct diagonalization of the dynamical matrix. The atomic interactions in the metal nanoparticle were described by a many-body Gupta potential using parameters fitted to bulk properties given by Cleri and Rosato.25 Nanoparticles with ICO, DEC, and TOC morphologies containing 13−2057 atoms were structurally optimized, without symmetry constraints, by a simulated thermal quenching technique26,27 using constant-energy molecular dynamics before the frequency normal modes analysis was performed. The vibrational density of states (VDOS) was constructed using a Gaussian broadening (with a width of 2 cm−1) of the 3n-6 eigenvalues obtained from the diagonalization of the dynamical matrix. Quantification of the vibrational periods for the QBM was done for all nanoparticle sizes and shapes by locating, within the whole vibrational spectrum, the mode with the smallest deviation from the BM using the procedure described in ref 24. This procedure counts the number of antiphase oscillating atoms and calculates the deviation from radiality and uniformity, by projecting the atomic equilibrium vector positions into each eigenvector obtained from the diagonalization of the dynamical matrix19,24 (see Supported Information (SI) for additional details of this calculation). It will be assumed that the calculated QBM period would correspond to the IVO, which is induced by a fast laser pulse as is the case in the time-resolved pump− probe spectroscopy measurements.2,16 25148

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Figure 1. Morphology of the metal nanoparticles under study for a fixed geometry: truncated octahedra (TOC, left column), Marks decahedra (DEC, middle column), and Mackay icosahedra (ICO, right column). Size (number of atoms) and diameter (maximum interatomic distance in nanometers).

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EXPERIMENTAL SECTION

nanoparticles is sufficiently low (less than 0.5%) to assume they are isolated and noninteracting. Importantly, the cluster surface is free from any bound molecules or ligands whose presence can strongly modify their optical and vibrational responses.30 Further details of the experimental setup are similar to those used for the synthesis of Pt nanoparticles.2 The femtosecond optical pump−probe technique was utilized for the measurements of the nanoparticle acoustic vibrations. In this approach, a first pump pulse excites the conduction electrons of the nanoparticle. A second timedelayed probe pulse follows the change of the sample optical transmission ΔT/T, yielding information on the electron energy relaxation process and on the mechanical movement of the particles.6,23,31 The experimental setup is based on an amplified Ti:sapphire oscillator. This delivers pulses of about

Nanoparticle samples are synthesized by a physical route using the low energy cluster beam deposition (LECBD) technique.28 Gold clusters are produced in a conventional laser vaporization source from bulk metallic targets. Adjusting the cluster production and selection conditions, Au nanoparticles with mean diameter of 2, 3, and 4 nm were synthesized. Size selection is performed on singly charged cationic clusters with an electrostatic deviator.29 The residual size dispersion is about 10% of the average value (fwhm) which can be accurately controlled with a time-of-flight spectrometer. Nanostructured samples consist of clusters embedded in a silica film (a few hundreds nanometers thick) codeposed by vaporization with an electron gun over a silica slide. The volume density of metallic 25149

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100 fs, with a wavelength at 800 nm at a repetition rate of 250 kHz. The output train is split into two parts. The first part is frequency-doubled to 400 nm in a 200 μm thick BBO crystal and subsequently used as the pump pulses. The other part is used to pump an optical parametric amplifier, which generates pulses in the visible range (450−700 nm), used as probe pulses. The time delay between the pump and probe pulses is varied using a mechanical delay stage. High sensitivity detection of the pump-induced changes of the probe pulse transmission is achieved by mechanical chopping of the pump beam at 40 kHz and lock-in detection of the probe beam transmission change.

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RESULTS AND DISCUSSION Structure and Energetics. In the first stage of the present study, the relaxed structures of Au, Ag, and Pt nanoparticles with TOC, ICO, and DEC morphologies, in the size range of 13−2057 atoms, were obtained through the simulated thermal quenching,26,27 using constant-energy molecular dynamics simulation and the many-body Gupta potential with parameters fitted to bulk properties by Cleri and Rosato.25 Figure 1 displays the relaxed geometries, size (number of atoms), and diameter (maximum interatomic distance) for 26 nanoparticles belonging to the three structural families. While the number of atoms of each nanoparticle within the ICO family corresponds to closed shell Mackay icosahedra, the specific size for the nanoparticles within the TOC and DEC families was chosen to construct nearly isotropic (spherical) closed shell FCC and Marks decahedral structures, respectively. As expected, for a given size and shape the nanoparticle diameter increases from Pt to Au, to Ag, in accordance with the order of the known values for the corresponding bulk lattice constant. On the other hand, for a similar number of atoms in the nanoparticle, the diameter of the Marks decahedron (n = 318) is slightly larger than that of the icosahedral (n = 309) and the truncatedoctahedral (n = 314) structures. The energetic stability of metal nanoparticles with TOC, ICO, and DEC morphologies has been extensively studied using a variety of semiempirical potentials.32−34 Here we report additional results on the size dependence of the nanoparticle binding energy from the present calculations. Figure 2 shows the quantity Δ = (Etot − nEbulk)/n2/3, where Etot is the nanoparticle total energy after relaxation of the structure and Ebulk is the cohesive energy per atom in the bulk, as a function of size n. The quantity Δ, being an excess energy roughly divided by the number of surface atoms, has been introduced to compare the energetics of clusters with different sizes n.32,33 Our results indicate that for Au and Pt nanoparticles DEC morphologies are preferred along the whole size range investigated (except n = 49), whereas for Ag clusters, ICO shapes are the most stable ones up to n = 309. The strong oscillations shown by the Δ quantity for TOC structures suggest that nearly isotropic FCC geometries would not yet be favored for n < 2000 atoms. These results are in overall agreement with those obtained with similar semiempirical potentials,32−34 indicating that for silver nanoparticles the ICO shape is the most favored at small sizes. For the three metal nanoparticles, the TOC geometries are more stable for larger sizes, and the DEC morphologies could be favored in the intermediate region. Nevertheless, the crossover sizes between the different motifs depend strongly on the type of metal: in Ag the icosahedral interval is rather large, followed by a very wide decahedral window, but on the contrary, in Au and Pt the icosahedral interval is much smaller.33 It is worthwhile to

Figure 2. Nanoparticle excess energy roughly divided by the number of surface atoms as a function of size n. Etot is the total energy of the nanoparticle after relaxation of the structure, and Ebulk is the cohesive energy per atom in the bulk. Color code: red (TOC); yellow (DEC); blue (ICO).

mention that the higher stability of the DEC morphology in the intermediate size region has been recently confirmed through an electron microscopy study for Au923.35 Also, it is convenient to point out that although the above discussion refers to the metal nanoparticle energetics calculated at T = 0 K, free energy calculations within the harmonic approximation indicate that no changes between the relative stability of the different structural motifs are expected up to room temperature. The SI shows free energy calculations for Au nanoparticles based on the harmonic approximation, using a procedure previously applied to small clusters.36,37 Since the variations in the binding energy per atom are only of a few meV/atom along the size range investigated, all morphologies were utilized for the vibrational properties study. Vibrational Properties. To show the effect of material, size, and shape on the vibrational properties of metal nanoparticles, the normal modes spectrum and the VDOS were calculated for all geometries within the three structural families mentioned above. Figures 3a−3c display the size evolution of vibrational spectra and VDOS for Au, Pt, and Ag nanoparticles with DEC, TOC, and ICO morphologies, respectively. These spectra spread over the far-infrared region with maximum frequencies lying in a broad range of 160−250 cm−1, depending on the metal and morphology. As expected, the different range of values of the vibrational frequencies for the Au, Pt, and Ag nanoparticles depends on the parameters of the Gupta potential, which were fitted to the cohesive energy, 25150

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Figure 3. Vibrational spectrum (blue bars) and VDOS (red line) of (a) Marks decahedral (DEC) Au nanoparticles, (b) truncated octahedral (TOC) Pt nanoparticles, and (c) Mackay icosahedra (ICO) Ag nanoparticles. Yellow bars correspond to the acoustic gap (lowest), quasi-breathing, and cutoff (highest) frequencies.

frequency modes originate from vibrations involving the atoms localized at the center of the ICO configurations. To investigate additional effects of the nanoparticle structure on the VDOS line shapes, Figures 4a and 4b display the frequency spectra and VDOS for DEC, TOC, and ICO morphologies with a similar size for Pt (∼300 atoms) and Au (∼900 atoms), respectively. In both figures, the close similarity between the line shapes corresponding to DEC and TOC morphologies and the distinct behavior of the ICO one can be further appreciated. For example, the depression and broadening of the high frequency peak in the Pt309 and Au923 ICO nanoparticles is evident, as well as the appearance of high frequency localized modes around 240 and 155 cm−1, respectively. On the other hand, the overall similarity between the VDOS of the Au928 TOC and Au906 DEC nanoparticles is remarkable, only displaying small differences around the maximum frequency region (140−150 cm−1). Further insights into the size and shape dependence of the vibrational properties of metal nanoparticles can be obtained from the calculated periods of three characteristic modes: the QBM and those with the lowest (acoustic gap) and highest (cutoff) frequencies. As was mentioned in the Methodology section, the QBM was searched within the whole vibrational spectrum by locating the eigenvector, which produced the maximum projection (scalar product) with the vector position of each nanoparticle in its equilibrium configuration.19,24 The SI contains more details on the location of the QBM frequency. Figure 5 displays a noteworthy result of the present study: a linear variation with size (nanoparticle diameter) of the periods corresponding to the QBM and acoustic gap for Au, Pt, and Ag nanoparticles, independently of their morphology. This result is significant since it not only points out the existence of a linear relation between those vibrational periods and the nanoparticle diameter in the size range of 0.5−4 nm but also coincides with a similar linear dependence, obtained for the periods of the breathing and acoustic gap modes of a metallic isotropic homogeneous sphere, using the equations from CET.38,39 A closer examination of Figure 5 indicates that there is a slight

lattice constant, and elastic constants of each bulk metal (see Table 2 of ref 25). On the other hand, Figures 3a−3c also show the characteristic degeneracy of the vibrational modes corresponding to the different point symmetry of DEC, TOC, and ICO configurations. In particular, the existence of several 5-fold degeneracy modes for the ICO nanoparticles can be noticed, including the one with the lowest frequency (acoustic gap), as well as its splitting into 2-fold and 3-fold degeneracy modes for DEC and TOC morphologies. The VDOS variation with size for DEC Au nanoparticles shown in Figure 3a illustrates not only the evolution of a multiple-peaks structure to a nearly continuous line shape due to the increase in the number of normal modes, but also the formation of a two-band shape with an intermediate minimum configuring the whole spectrum that corresponds to the nanoscale signatures of the transverse and longitudinal phonons, characteristic of bulk metals.12 Moreover, for the larger decahedra a quasi-linear behavior in the low frequency region of the VDOS is also notorious, which had also been obtained for FCC and BCC metal nanocrystals.10 The size evolution of the line shapes for the VDOS of Pt nanoparticles with TOC morphologies, displayed in Figure 3b, also shows the formation of a double band spectra, analogous to that seen for the Au DEC nanoparticles. This result indicates that a similar behavior would be expected for the vibrational properties of Pt and Au nanoparticles with DEC and TOC morphologies, respectively, up to sizes ∼3 nm. In contrast, Figure 3c, displaying the size dependence of the VDOS for Ag nanoparticles with ICO morphologies, shows the formation of a line shape qualitatively different from those obtained for DEC and TOC structures. For example, the appearance of a broad asymmetric bell shape in the central part of the spectrum and a smear of the high frequency peak are noticeable. This difference may be associated to the larger strain and twining existing in the ICO morphologies.12 Moreover, the VDOS of ICO nanoparticles also presents a frequency tail beyond the highest frequency in DEC and TOC morphologies. These high 25151

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Figure 4. Vibrational spectrum (blue bars) and VDOS (red line) of (a) Pt nanoparticles and (b) Au nanoparticles of similar size with truncated octahedra (TOC), Marks decahedra (DEC), and Mackay icosahedra (ICO) shapes. Yellow bars correspond to the acoustic gap (lowest), quasi-breathing, and cutoff (highest) frequencies. Figure 5. Periods of different vibrational modes as a function of diameter for Au, Pt, and Ag nanoparticles (top, middle, and bottom panels, respectively) with TOC (red), DEC (yellow), and ICO (blue) shapes: acoustic gap (triangles), quasi-breathing mode (squares), and cutoff frequency (diamonds). The linear behavior of the period of the acoustic gap and breathing modes for a homogeneous isotropic sphere predicted by the continuous elastic theory, using the slopes reported in Table 1, is displayed with continuous (C1 values) and dashed (C2 values) lines.

effect of the nanoparticle shape on the acoustic gap periods, giving a small dispersion in their linear behaviors with size for Pt and a more pronounced one for Ag nanoparticles. Nevertheless, since a qualitative agreement has been obtained in the linear dependences with size between the QBM and acoustic gap periods of the metal nanoparticles and those obtained for the BM and acoustic gap of a metallic isotropic homogeneous sphere, it would be insightful to perform a quantitative comparison between the vibrations described by both the atomistic and continuous medium approaches. To this end, CET linear relations for the BM and acoustic gap periods as a function of the nanosphere diameter, given in refs 38 and 39, respectively, can be utilized to calculate the corresponding slopes using the transverse and longitudinal sound velocities for the three bulk metals (see the SI for details of these calculations). Calculated slope values are reported in Table 1 using two sets of bulk elastic constant values,40,41 together with the slopes fitted to the data shown in Figure 5 for the Au, Pt, and Ag nanoparticles. A comparison of the CQBM, C1BM, and C2BM values reported in Table 1 indicates that there is a good quantitative agreement between the linear behavior of the QBM of the three metal nanoparticles and the BM of a metallic isotropic homogeneous isotropic sphere. This result would point out that the BM mode of a homogeneous isotropic sphere is correctly modeling the QBM of faceted DEC, TOC, and ICO metal nanoparticles. On the other hand, the comparison of the linear relations for the periods of the acoustic gap shows that the corresponding slopes

(CAG‑nano values) calculated for the metal nanoparticles are slightly larger than those obtained for isotropic, homogeneous metallic spheres from CET (C1AG‑bulk and C2AG‑bulk values). These larger differences can be explained by considering that, according to the present calculations, the acoustic gap of the Au, Pt, and Ag nanoparticles corresponds to bending modes, whereas in isotropic homogeneous metallic spheres the lowest frequency mode had been found to be a torsional one.12,39 Moreover, torsional modes are also present in metal nanoparticles, but in this study, they appear with higher frequencies. An animation file for characteristic acoustic gap modes is available in the HTML version of the paper. Another noticeable result from this investigation is the weak dependence on size and shape of the period corresponding to the highest frequency (cutoff) mode, which can be appreciated from the values shown in the bottom data (diamonds) on each panel of Figure 5. Table 1 displays their average and standard deviation, indicating 10% dispersion for Au and Pt and 12% for Ag nanoparticles due to their sizes and shapes. Moreover, the 25152

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Table 1. Slopes of the Fitted Linear Variation with Nanoparticle Size for the Periods of the QBM (CQBM) and Acoustic Gap (CAG‑nano) and Average and Standard Deviation Values of the Periods Corresponding to the Highest (Cutoff) Frequency Mode (τcutoff), Obtained from the Data Shown in Figure 5a Au Pt Ag

CQBM (ps/nm)

CAG‑nano (ps/nm)

τcutoff (ps)

C1BM (ps/nm)

C1AG‑bulk (ps/nm)

C2BM (ps/nm)

C2AG‑bulk (ps/nm)

τDebyeb (ps)

0.323 0.253 0.287

1.207 0.890 0.858

0.216 ± 0.021 0.161 ± 0.016 0.175 ± 0.021

0.322 0.263 0.304

1.045 0.729 0.741

0.290 0.291 0.301

1.060 0.745 0.740

0.291 0.200 0.213

a

C1BM, C1AG‑bulk and C2BM, C2AG‑bulk were calculated using two different sets of bulk elastic constants40,41 and the linear relations38,39 for the BM and acoustic gap for an isotropic sphere using the equations from continuous elastic theory (see the SI for details of these calculations). τDebye is the period calculated from the bulk Debye frequency. bτDebye was calculated using the bulk values from ref 42.

average values are remarkably close (in the range of 0.16−0.22 ps) for the three metal nanoparticles, not reflecting the larger variation in the corresponding values obtained from the inverse of the bulk Debye frequency, also shown in Table 1. A possible explanation for this behavior is that in ICO nanoparticles the highest frequency mode corresponds to oscillations of the internal 13-atom icosahedral core, whereas in TOC and DEC nanoparticles collective vibrations of low-coordinated vertex atoms together with their nearest neighbors characterize this cutoff frequency mode. In both cases, the vibrational periods of these localized modes are expected to have a small size and shape dependence. An animation file for characteristic cutoff frequency modes is also available in the HTML version of this paper. Comparison between Theory and Experiment. Until now it has been difficult to measure the whole vibrational spectrum or VDOS of metal nanoparticles. Nevertheless, some characteristic modes have been already experimentally investigated.6,17,23 They include the ones detected for Au and Ag nanoparticles (with diameter ∼5 nm) using low-frequency Raman scattering4 and the IVO of Pt nanoparticles (in the size range 1.3−3 nm) probed through noncontact time-resolved spectroscopy.2 In the latter case, a linear relation between the period of the IVO and the mean diameter of Pt nanoparticles was found.2 That investigation has been extended in the present study to measure the corresponding periods of Au nanoparticles synthesized with mean diameters in the range of 2−4 nm. The elastic response of gold nanoparticles was investigated using a femtosecond optical pump−probe technique. The probe transmission change ΔT/T measured in such an experiment is illustrated in Figure 6(a) in the case of 4.0 nm diameter Au particles embedded in silica. The fast ΔT/T rise and fall on a few picoseconds reflects heating of the nanoparticle electrons by the pump pulse and their subsequent cooling by the electron−lattice interaction inside each nanoparticle.43 The resulting ultrafast heating of the lattice launches the dilation of the particles, which subsequently undergo radial contractions and expansions around their new equilibrium size, corresponding to the fundamental quasi-breathing mode of the particles, i.e., their isotropic volume oscillations.44 This mechanical movement induces a periodic modulation of the metal dielectric function, which shows up by oscillations of the differential probe transmission ΔT/T. This oscillating part is more clearly seen after subtraction of the short time-delay response reflecting electron heating and cooling (see Figure 6(b)). The period T0 of the oscillations is determined by fitting the oscillating part of the signal in the time domain with a damped sinusoidal function (see Figure 6(b)), leading to a breathing mode period T0 ≈ (1.35 ± 0.06) ps. The same

Figure 6. (a) Time-resolved change of the sample probe transmission ΔT/T as a function of the pump−probe delay measured in a sample of silica-embedded 4 nm gold nanoparticles. The red dashed line is a fit of the exponential decay associated with electron−lattice thermalization. (b) Oscillating part of the ΔT/T signal together with a fit by a damped sinusoidal function (red dashed line). Pump pulses are tuned to 400 nm with an average power of 285 μW. Probe pulse wavelength is 460 nm.

measurements were performed in other samples with 2 and 3 nm diameter Au nanoparticles. Although the time-resolved pump−probe spectroscopy only allowed the measuring of IVO periods for three Au nanoparticle sizes, a clear linear relation between these quantities was observed, in qualitative agreement with the measurements for Pt nanoparticles.2 Figure 7 (top panel) shows the experimental data obtained for the IVO periods together with the calculated values of the QBM for Au nanoparticles with DEC, TOC, and ICO morphologies. A similar comparison using the data reported in ref 2 is presented in Figure 7 (bottom panel) for Pt nanoparticles. The linear dependence of the period for a metallic isotropic homogeneous nanosphere predicted by CET is also displayed in Figure 7, using two sets of bulk elastic constants40,41 (dotted and dashed lines, see SI for details) and the data from Table 1. 25153

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mimic the corresponding atomic interactions. Indeed, after reviewing the procedure reported in ref 25 for the fitting of the Gupta potential parameters to the elastic constants of bulk metals, it was found that for Au the fitted parameters very well reproduced its mechanical properties, whereas for Pt fitting discrepancies were obtained (see, for example, Table 2 of ref 25). Therefore, a similar capability of the Gupta potential to model the atomic interactions in Au than in Pt nanoparticles would not be expected.

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SUMMARY The vibrational spectra and VDOS of Au, Pt, and Ag nanoparticles with DEC, ICO, and TOC morphologies have been calculated in the size range of 0.5−4 nm, through atomistic simulations using a many-body Gupta potential with parameters fitted to bulk metal properties. The dependence of the vibrational spectrum on the material, size, and shape of the nanoparticle was discussed. Our analysis focused on the behavior of the periods of three characteristic vibrational modes: QBM, acoustic gap, and cutoff frequency. For the first two modes the period follows a linear variation with the nanoparticle diameter, almost independently of its morphology, while for the cutoff frequency mode the period showed a weak size and shape dependence. Moreover, the linear behavior of the QBM and acoustic gap periods coincides with the linear dependence of the BM and acoustic gap periods of a metallic isotropic homogeneous nanosphere predicted by CET using bulk elastic constant values. This study also reported the experimental measurements of the IVO period as a function of the mean diameter for Au nanoparticles, synthesized in the size range of 2−4 nm, obtained through time-resolved pump probe spectroscopy. Experimental data show a linear variation of the IVO period, in agreement with a previous result obtained for Pt nanoparticles2 with the same spectroscopic technique. The comparison of theoretical results from the atomistic and continuous medium approaches with experiments indicated a good agreement for Au and Pt nanoparticles. The slight discrepancy found for Pt is attributed to a lower capability of its Gupta potential parameters to describe their atomic interactions in bulk and nanoparticles. In the present atomistic theoretical study, based on the harmonic approximation, the lifetime of the vibrational modes that is responsible for the decay of oscillations shown in Figure 6 was not discussed. Nevertheless, it is well-known that the decay rate increases linearly with size for nanospheres when damping is dominated by energy transfer to the environment. This result was obtained from the continuous mechanics model for embedded nanospheres and confirmed by experiments.6,45 A study of the effect of the nanoparticle shape on this decay mechanism would be very interesting, but it requires extensive additional calculations out of the scope of this paper (note that environment is not included in our atomistic calculation). On the other hand, a recent molecular dynamics atomistic study indicated that polycrystallinity affects the intrinsic decay rate (i.e., due to vibration−vibration coupling) of a nanoparticle.46 Moreover, experiments performed on single nanoparticles always show significant particle to particle variation in damping times,6,47 suggesting either fluctuations of their environment or additional effects inherent to the particles. Additional atomistic studies are required to investigate the correlation of vibrational lifetime measurements on a single nanoparticle to its exact

Figure 7. Quasi-breathing mode periods for Au (top panel) and Pt (bottom panel) nanoparticles with TOC (red); DEC (yellow); and ICO (blue) shapes. Experimental values are in black open triangles. Dotted and dashed lines correspond to the linear variations obtained from continuous elastic theory using the slopes C1BM and C2BM, respectively, reported in Table 1.

Figure 7 (top panel) shows an excellent agreement between the calculated QBM and the measured IVO periods for Au nanoparticles in the size range up to 4 nm, independently of their shape. Moreover, as was mentioned above, the calculated periods of the QBM are also in very good agreement with the linear variation with diameter (dotted and dashed lines in Figure 7 (top panel)), predicted for the BM of a homogeneous, isotropic Au nanosphere. This compatibility of results not only further confirms the validity of the CET at the nanoscale, as was found for Pt nanoparticles,2 but also indicates that the QBM, which was extracted as a characteristic oscillation of the whole vibrational spectrum through a normal-mode analysis, is correctly describing the period of the radial motion (IVO) of Au nanoparticles measured in time-resolved experiments. The comparison of the calculated QBM periods for Pt nanoparticles with experimental data2 and CET values, shown in Figure 7 (bottom panel), also indicates a good agreement in the linear variation with the nanoparticle diameter, although smaller QBM periods were calculated from the present atomistic simulations compared to the corresponding experimental values. The different level of agreement of the present atomistic calculations with the experimental data for Au and for Pt nanoparticles might be related with a distinct capability of the parameters of the many-body Gupta potential to correctly 25154

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morphology that would be available, for example, using highresolution transmission electron microscopy. In sum, the main trend emerging from the present study shows that in contrast with other physicochemical properties of metal nanoparticles like melting temperatures48 or their energeticis on catalyst sintering,49 vibrational properties like the periods of characteristic oscillations display a simple (linear) variation with size and weak material and shape dependence.

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ASSOCIATED CONTENT

S Supporting Information *

Free energy calculations for different sizes and shapes of Au nanoparticles are described. Details on the procedure to locate the quasi-breathing mode and on the calculation of the periods of the breathing and acoustic gap modes from continuous elasticity theory are provided. The complete author list of reference 28 is included. This material is available free of charge via the Internet at http://pubs.acs.org. W Web-Enhanced Features *

An animation file for characteristic vibrational modes is available in the HTML version of the paper.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: garzon@fisica.unam.mx Phone: +52-55-56225147. Fax: +52-55-56161535. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge support from Conacyt-México under Project 177981. Calculations were done using resources from the Supercomputing Center DGTIC-UNAM. We thank Dr. Alfredo Tlahuice for providing us the initial coordinates of some nanoparticles and useful discussions regarding vibrational modes visualization at the early stages of this work. HES acknowledges support from Programa de Doctores Jóvenes de la Universidad Autónoma de Sinaloa.

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REFERENCES

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