Influence of Size, Composition, and Chemical Order on the Vibrational

ARTICLE pubs.acs.org/JPCC

Influence of Size, Composition, and Chemical Order on the Vibrational Properties of Gold
Silver Nanoalloys F. Calvo* LASIM, Universit
e de Lyon and CNRS UMR 5579, 43 Bd du 11 Novembre 1918, F69622 Villeurbanne Cedex, France ABSTRACT: The vibrations of a metallic nanoparticle are strongly affected by its size and shape. In the present work, the respective roles of size, chemical order, and composition have been theoretically examined in the case of gold
silver nanoalloys by means of atomistic simulations. Whereas the vibrational density of states exhibits some qualitative differences between alloyed and segregated (core
shell) particles, the breathing frequency varies smoothly but nonlinearly with composition in all cases considered. Elasticity theory accounts reasonably well for the size dependencies, with finite size corrections scaling as powers of the inverse radius. These deviations are found to vary with composition through a simple quadratic expansion.

1. INTRODUCTION Since the early era of metallurgy, mixing two or more elements has constituted a major way of modifying and improving the physical and chemical properties of materials. On the nanoscale, where size and shape are naturally important factors, composition and chemical order provide additional control parameters that could be even more consequential. The possibilities offered by these so-called nanoalloys have already been exploited in various fields ranging from magnetism to catalysis and life sciences.1,2 The optical response of bimetallic nanoparticles is a typical example of a property tunable by changing composition and chemical order.3
6 The vibrational response, obviously, has no reason not to show any sensitivity. Vibrational properties of nanoparticles are important, at a fundamental level, for characterization purposes and, in a more applied context, because they largely rule energy conversion and heat transport. Several methods are available to measure vibrational properties of nanoparticles, including infrared spectroscopy,7 pump
probe techniques,8 inelastic neutron scattering,9 and Raman scattering.10,11 Unfortunately, studies on bimetallic particles are relatively scarce and, at most, semiquantitative. Low-frequency Raman scattering experiments on Ni
Ag nanoparticles10,12 have confirmed the preference for a core
shell structure suggested by optical measurements. In the case of glass-embedded Ag
Au particles, the composition had to be estimated from the position of the surface plasmon resonance peak.13 From the theoretical perspective, the usual approach based on elasticity theory has been rather successful in predicting the acoustic modes of homogeneous nanoparticles with diameter exceeding a few nanometers,14 but its application to bimetallic particles seems more problematic.10 Atomistic descriptions based on force fields have been especially used in the nonscalable, small sizes regime15
17 r 2011 American Chemical Society

and also, for larger clusters, to determine vibrational densities of states.18
20 The atomistic approach has been shown to confirm generally the overall validity of the continuum models and suggested that the deviations are ascribable to surface effects.21 To the best of our knowledge, no such studies at the atomistic level have yet been attempted for bimetallic particles. Our goal in this Article is to elucidate the combined influences of composition and chemical order on the vibrational properties of Ag
Au nanoparticles, systematically exploring various concentrations in the size range of 1
4 nm. Gold and silver have been two of the most studied metals on the nanoscale because of their great versatility and the possibility to control their size, shape, and chemical ordering by various synthesis techniques. In particular, chemical reduction6,22
24 and laser-assisted methods5,25,26 have allowed a broad variety of segregation patterns to be explored, ranging from fully segregated (core
shell with either metal at the core) to fully alloyed (solid solution). To extract meaningful information about size and composition effects and, more importantly, to determine quantitative trends, we performed most calculations on sequences of nanoparticles with a fixed geometrical pattern, namely, multishell icosahedra, but we also considered cuboctahedral particles for comparison. The theoretical methods are described in the next section, the results are presented and discussed in Section 3, and some concluding remarks are given in Section 4.

2. METHODS The present modeling starts with a consideration of nanoparticle sizes, which for the sake of computer efficiency had to be limited to a few thousands atoms or a radius of ∼4 nm. Our main Received: June 16, 2011 Revised: July 23, 2011 Published: August 10, 2011 17730

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The Journal of Physical Chemistry C interest lies in quantifying size effects, and for this purpose, it is more efficient to deal with nanoparticles that belong to a same structural family, thus lying in the scalable regime. At small sizes (below 103
104 atoms depending on the material), most metal clusters exhibit icosahedral shapes before undergoing a transition toward bulk-like structures.27 Previous studies of Ag
Au clusters28,29 support icosahedra as the most stable structure in the small size range. The face-centered cubic geometry is expected to be recovered mostly as truncated octahedra, which are the optimal Wulff construction for the fcc structure. Cuboctahedral nanoparticles, although not as stable as truncated octahedra, share the same magic numbers as icosahedra and were chosen for comparison. The atomic interactions were modeled using the many-body potential derived in the second moment approximation of the electronic density of states in tight-binding theory, with parameters taken from Rossi and coworkers.30 This potential predicts a surface enrichment in silver at very low temperatures, which is consistent with the lower surface energy of this metal relative to gold. However, at finite temperature, the clusters undergo a smooth order to disorder transition.29 Although the alloyed state may be thermodynamically the most stable even on the nanoscale, the experimental feasibility6 of both AgshellAucore and AgcoreAushell types led us to consider those metastable segregated structures as well. To avoid any neglect of low-frequency modes, we used the potential without any truncation. The vibrational modes of an n-atom nanoparticle were obtained for a set of locally stable structures by diagonalizing the dynamical matrix Kij = [∂2V/∂ri∂rj]/(mimj)1/2, where the ri and mi values are the 3n Cartesian coordinates and atomic masses, respectively, V(R) being the potential energy at configuration R = {ri}. Most of the sampling problem is then related to finding a set of relevant isomers corresponding to a prescribed number of atoms, composition, and type of chemical ordering. These socalled homotops31 were located by a basin-hopping procedure32 combining Monte Carlo moves affecting only chemical identities and followed by a gradient minimization. Exchange-only basinhopping, which has already been used in the past to locate the most stable homotops of binary clusters,33,34 is a straightforward approach in the case of alloyed nanoparticles because all atoms on the lattice are given a chance to swap identities. However, homotops also exist in the case of segregated core
shell nanoparticles, except at accidental concentrations that exactly fill magic numbers. Sampling the homotops of core
shell structures was achieved by allowing only the atoms of the intermediate mixed shell separating the pure core and outer layers to be part of the Monte Carlo process. Note that even in this situation all atoms were subsequently relaxed to their local equilibrium position. The nanoparticles consisted of 3 to 12 icosahedral or cuboctahedral shells, or equivalently 147 to 3871 atoms. For each size, the relative Ag
Au composition was varied by steps of 10%, and the number of homotops sampled was taken as 1000 up to 6 shells and 500 in larger particles. Only the 100 energetically lowest homotops were kept for further vibrational analysis. The density of vibrational states was obtained by integration over all normal-mode frequencies and the set of homotops, and a discretization into bins of 20 cm
1 was used. Although the number of homotops sampled represents only a tiny fraction of the available set, the vibrational density of states and especially the breathing frequency turned out to converge surprisingly fast with the number of homotops in the set.

ARTICLE

Figure 1. Densities of vibrational states of icosahedral, 923-atom Ag
Au nanoalloys, discretized into 20 cm
1 bins, for 50 (left panel) and 20% (right) silver compositions and averaged over chemical disorder. The lowermost, central, and uppermost curves correspond to AgshellAucore, alloy and AgcoreAushell particles, respectively, with typical results from an individual homotop represented with dashed lines. Vertical dotted lines locate the breathing frequencies.

Only a selection of cuboctahedral nanoparticles were investigated for comparison with the icosahedral case, and we systematically studied the pure nanoparticles in the entire size range together with the six-shell nanoalloys in the entire composition range.

3. RESULTS AND DISCUSSION We have represented in Figure 1 the vibrational state densities for (AgxAu1
x)n nanoparticles at silver compositions of x = 0.5 and 0.2 and for six icosahedral shells (n = 923). These densities generally display a broad asymmetric bell shape centered near 60 cm
1 depending on size and composition, extending in the infrared range up to ∼200 wavenumbers. Those distributions are in semiquantitative agreement with the results previously obtained by various authors,18,20 differences being mostly due to the different lattice symmetries, but they also exhibit significant variations with the type of chemical ordering. Overall, the densities are smoothest in the alloyed case, which we interpret as reflecting the chemically most homogeneous phase once homotops are averaged out. At this stage, it is important to notice that the deviations in the individual densities around the statistical average are rather minor in all cases considered. This illustrates the rather fast convergence of the vibrational properties with only a modest number of homotops. In the segregated particles, the presence of an interface leads to rougher densities, especially for AgcoreAushell nanoparticles. These systems are energetically the most strained, hence also likely those in which the vibrational modes of both metals do not naturally mix with each other. The vibrational densities of states of the corresponding cuboctahedral nanoparticles are displayed in Figure 2. The detailed shape of the vibrational states density thus strongly depends on the nanoparticle shape, and for cuboctahedra, they resemble much more the two-peak shapes obtained by Kara and 17731

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Figure 4. Breathing frequency of cuboctahedral, 923-atom Ag
Au nanoalloys as a function of Ag concentration, and for different types of chemical ordering. The error bars indicate the standard deviation due to averaging over homotops.

Figure 2. Densities of vibrational states of cuboctahedral, 923atom Ag
Au nanoalloys, discretized into 20 cm
1 bins, for 50 (left panel) and 20% (right) silver compositions and averaged over chemical disorder. The lowermost, central, and uppermost curves correspond to AgshellAucore, alloy, and AushellAgcore particles, respectively, with typical results from an individual homotop represented with dashed lines. Vertical dotted lines locate the breathing frequencies.

mode does not necessarily lie among the normal modes. Alternatively, it is possible to extract the frequency of homogeneous radial motion by a simple Taylor expansion of the classical Hamiltonian H of the system. This is justified by the recent molecular dynamics investigation by Ng and Chang,37 who inferred the breathing mode from the power spectrum of the velocity autocorrelation function after such a purely radial expansion. Scaling all coordinates by a factor 1 + ε(t) with |ε| , 1 leads to H ¼

∑i 2 mi ri2ε_2 þ V ðR 0 Þ 1

þ ε

∑i


∂V

ri
∂ri


R0

ε2 þ 2

∑i, j


∂2 V

ri rj
∂ri ∂rj


ð1Þ

R0

where R0 is the equilibrium geometry and with the sums running over all 3n coordinates. Because all gradients vanish at equilibrium, H can be written up to an unimportant additive constant as H = 1/2Iε.2 + 1/2Wε2, with I¼

∑i mi ri2

W¼

Figure 3. Breathing frequency of icosahedral Ag
Au nanoalloys as a function of Ag concentration and for different types of chemical ordering. The results are shown for 561-atom and 923-atom clusters as dashed and solid lines, respectively, and the error bars indicate the standard deviation due to averaging over homotops.

Rahman18 and by Meyer and Entel20 for one-component crystalline nanomaterials. However, even in this case alloying produces much smoother densities than either of the segregated patterns, thus highlighting the role of the interface. Unfortunately, the entire vibrational density of states is difficult to determine experimentally, whereas specific modes can be measured by dedicated techniques. In particular, the breathing frequency, although not IR active, can be accessed by Raman scattering and ultrafast pump
probe methods. Its theoretical evaluation is convenient with elasticity approaches, assuming spherical35 or anisotropic21,36 geometries, but it does not seem so clear in atomistic approaches, essentially because the breathing

∑i;j

ri rj

ð2Þ
∂2 V

∂ri rj
R0

and the oscillation frequency reads pffiffiffiffiffiffiffiffiffi ωbreathing ¼ W=I

ð3Þ

ð4Þ

This procedure for calculating the breathing frequency is computationally inexpensive because it does not involve diagonalizing the dynamical matrix or solving equations of motion. The average breathing frequency of the six-shell particles discussed in Figure 1 falls in the 30
40 cm
1 range, with a marked dependence on composition and a smaller, albeit visible, dependence on chemical order. Both dependencies are better seen by varying the composition, still fixing the number of shells to six. The average breathing frequency of such nanoparticles is represented in Figure 3 for nanoalloys and the two core
shell segregated particles. Those variations are very smooth and monotonic but with a clear nonlinear character even in the alloyed case. The residual mixing effect remains within 10% but is clearly significant over most of the composition range. In both kinds of segregated nanoparticles, the nonlinear behavior is 17732

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Figure 5. Breathing frequency of icosahedral Ag
Au nanoalloys at fixed Ag concentration of 0.5 as a function of n
1/3 µ 1/D, where D is the inverse cluster diameter, and for different types of chemical ordering. The dashed straight line shows the perfect 1/D behavior. The individual nanoparticle sizes are also indicated.

more apparent as a stronger slope when the impurities remain at the surface and a gentler slope once the outer layer is chemically uniform. This suggests that surface composition contributes more to the breathing frequency than the core or the interface. The vibrational densities for the cuboctahedral particles at the same size and composition, shown in Figure 2, indicate that there is some additional dependence of ωbreathing on morphology. However, the same trend is found as the nanoparticles change from AgshellAucore to the alloy and finally to AgcoreAushell, with two successive increases in ωbreathing of 1 to 1.5 cm
1. For a given composition and size, the breathing frequency of the cuboctahedral particle decreases by 0.4 and 0.6 cm
1 in the x = 0.2 and 0.5 cases, respectively. The behavior in the entire composition range is generally similar and is depicted in Figure 4. On the basis of these results, chemical ordering has a stronger influence on the breathing frequency than structure itself. However, size effects are very important for the breathing frequency, and a comparison between the results found for the five- and six-shell nanoparticles is shown in Figure 3. The curves for the 561-atom systems appear mostly shifted and scaled with respect to the larger nanoparticles, the magnitude of the variations with composition increasing now to five wavenumbers. The higher breathing frequency of the smaller nanoparticle is expected based on elasticity arguments, with Lamb’s theory predicting exactly a 1/D scaling for spherical particles with diameter D.35 Fixing the composition and the type of chemical ordering, the validity of Lamb’s theory can be quantitatively examined by looking into a broader size range 147 e n e 3871. The average breathing frequency of the (Ag0.5Au0.5)n nanoparticles is plotted against n
1/3 in Figure 5 for alloyed and core
shell segregated systems. The n
1/3 parameter was preferred here over the inverse diameter D because it is much less ambiguous and does not involve any density consideration that may vary depending on chemical ordering. As can be seen from this Figure, the breathing frequency is relatively well-described by a linear dependence on n
1/3, deviations becoming increasingly important for smaller nanoparticles. This is in keeping with previous studies on

Figure 6. Leading (ω) and correcting (R and γ) terms in the expansion of the breathing frequency versus nanoparticle size, as a function of silver concentration and for different types of chemical ordering. The solid lines are best fit quadratic expansions with fixed end points.

different systems21,38 and intuitively confirms that the continuum approach improves with increasing nanoparticle size. It has been conjectured21 that the deviations to Lamb’s theory can be ascribed to surface effects. Other properties of nanoscale systems, such as the melting temperature or the binding energy, also belong to this class.27 A suitable way to model those effects consists of expanding the property in question in powers of n
1/3 that successively account for the surface/volume, edge/volume, and vertices/volume ratios. In the present work, the breathing frequency of (AgxAu1
x)n was therefore described as ωbreathing ðn, xÞ ¼ ωðxÞn
1=3  ½1 þ RðxÞn
1=3 þ γðxÞn
2=3 

ð5Þ

where ω(x), R(x), and γ(x) are expansion coefficients that, besides composition, depend only on the type of chemical ordering. The quality of this fitting template in reproducing the calculated data can be gauged from visual inspection of Figure 5. The variations of the three parameters ω, R, and γ with composition are shown in Figure 6 for alloyed and segregated icosahedral particles. Consistently with the breathing frequency at fixed size, the variations of these parameters exhibit a pronounced nonlinear character, which is well-described by simple quadratic functions. It is thus possible to write the leading term ω(x) as a function of pure elements contributions ωAg and ωAu, together with a mixing term ωmix as ωðxÞ ¼ xωAg þ ð1
xÞωAu þ xð1
xÞωmix

ð6Þ

and similar expressions can be used for the higher-order terms R(x) and γ(x). Fixing the coefficients for the pure elements, only the mixed coefficients have to be numerically determined, and the resulting values are lumped together in 17733

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Table 1. Parameters of the Quadratic Expansion for the Breathing Frequency As a Function of Size n and Composition x for Icosahedral (ico) Clustersa ω

R

γ

Pure Elements Ag(ico)

506.1


0.197

0.504

Au(ico)

305.6


0.251

0.370

Ag(cubo)

502.3


0.221

0.411

Au(cubo)

302.9


0.282

0.276

Mixed Elements (ico) AgshellAucore Alloy AgcoreAushell

39.6


0.629

1.360


66.6

0.354


0.619


163.0

1.234


1.530

a

Contributions from the pure and mixed elements are given also for cuboctahedral (cubo) clusters, with ω in wavenumbers and both R and γ dimensionless.

Table 1. The parameters obtained for the cuboctahedral nanoparticles are very similar to those for icosahedra, the bulk frequencies being lower by about four wavenumbers. This also matches the results reported above for the 923-atom nanoalloys. It is noteworthy that the mixing coefficients show different signs between the AgshellAucore segregation and the two other chemical orderings, which is consistent with the different curvatures displayed by ωbreathing(x) in Figure 3. The alloy properties also fall in between the values for the two segregated forms. Both features support the idea that the breathing frequency is more sensitive to the surface atoms than to the core. The present results emphasize that the vibrational frequencies of Ag
Au nanoparticles do not obey linear relations with composition and that the presence of an interface between the pure metals contributes to the nonlinearities. For the pure elements, comparison can be attempted for the bulk frequencies ωAg and ωAu based on Lamb’s theory, which predicts that ωLamb = Svl /2πrWS, where vl is the longitudinal sound velocity, rWS is the Wigner
Seitz radius, and S is the eigenvalue of Lamb’s secular equation for the corresponding monopolar mode. Taking for these quantities the values from the literature,39 one finds ωLamb = 319 cm
1 for gold and 340 cm
1 for silver. The agreement is thus good for gold but rather poor for silver, which suggests that the current many-body potential could be partially improved for vibrational properties. Such improvements could include additional parameters aimed at reproducing bulk dispersion curves; however, this would imply a similar effort for the alloy. Alternatively, the potential could account for some residual charge transfer arising from the electronegativity difference between the two metals, an approach already used for modeling atomic clusters40,41 and recently applied to gold
silver nanoalloys by Cerbelaud and coworkers.42 However, it is unclear to which direction and to which extent such another potential will affect the structural, size, and composition dependencies reported in the present work. This clearly calls for further theoretical studies and of course experimental validation to discriminate the various possible models.

4. CONCLUSIONS In summary, the vibrational properties of gold
silver nanoparticles have been theoretically shown to depend on size, composition, and chemical ordering in nontrivial ways. To some

extent, the dependence on size is accounted for by elasticity theory, which predicts a linear scaling with inverse radius. Effects beyond the continuum approach can be quantitatively modeled by additional terms involving the surface/volume ratio, and we found evidence that putting the two metals into contact yields a distinct nonlinear behavior as composition is varied, especially in core
shell structures. The effects of segregation were found to be generally significant and larger in magnitude than structural effects alone at a given size. The breathing frequency thus increases with the number of silver atoms in the outer layers, a feature that could be exploited in the future to tailor nanoparticles with desired acoustic properties. The most obvious perspectives, besides seeking experimental validation, involve improving the quantitative details by using a more accurate potential and look at different morphologies. Although the present study was restricted mostly to icosahedral nanoparticles, the findings should also be relevant in other morphologies and for nonmagic sizes. It would be worth extending these calculations for other bimetallic systems of interest in catalysis, such as Pt- or Pd-containing materials. Highly mismatched metals like Ag
Ni or Ag
Cu could also exhibit contrasted behaviors. Lastly, the influence of the embedding medium on the vibrational properties could be addressed,43 although the fully atomistic approach may become prohibitive relative to the still applicable continuum models. Intermediate methods, in which the environment is treated implicitly,44 may offer appealing alternatives in this respect.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The author wishes to thank Prof. M. Broyer and Dr. A. Tanguy for very useful discussions as well as generous computational resources from the regional P^ole Scientifique de Mod
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