Size and Shape Dependence of the Vibrational Spectrum and Low

Article pubs.acs.org/JPCC

Size and Shape Dependence of the Vibrational Spectrum and LowTemperature Specific Heat of Au Nanoparticles Huziel E. Sauceda,† Fernando Salazar,‡ Luis A. Pérez,† and Ignacio L. Garzón*,† †

Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México D. F., México Escuela Superior de Ingeniería Mecánica y Eléctrica − Culhuacán, Instituto Politécnico Nacional, Av. Santa Ana No. 1000, 04430 México D. F., México

‡

W Web-Enhanced Feature * S Supporting Information *

ABSTRACT: The vibrational spectra of metal nanoparticles are a signature of their structures and determine the lowtemperature behavior of their thermal properties. In this work, we report a theoretical study on the size evolution of the vibrational spectrum and density of states (VDOS) of Au nanoparticles in the range of 1−4 nm. Our study focuses on truncated octahedral (FCC), decahedral, and icosahedral nanoparticles. The structural optimization was performed through atomistic simulations using molecular dynamics and the many-body Gupta potential, whereas the vibrational frequency spectrum was obtained within the harmonic approximation through a diagonalization of the dynamical matrix. The calculated frequency spectra are discrete, have a finite acoustic gap (lowest frequency value), and extend up to a maximum frequency in the range of ∼140−185 cm−1, depending on the nanoparticle morphology. The VDOS evolves from a multiple-peak line shape for small sizes to a characteristic profile for the larger nanoparticles that anticipates the well-known VDOS of the bulk Au metal. The frequency spectrum was used to quantify the specific heat at low temperatures for the Au nanoparticles, displaying small variations with size and shape. Further analysis of these results indicates that the acoustic gap is responsible of a slight reduction in the specific heat with respect to bulk in the temperature range, 0 < T < Tr, (Tr ≈ 5 K for Au nanoparticles with size ∼1.4 nm). Also, the well-known increment in the specific heat of metal nanoparticles with respect to the bulk value, caused by the enhancement of the VDOS at low frequencies, is recovered for Tr < T < Ts (Ts ≈ 35−45 K). Moreover, it is also found that for T > Ts the calculated specific heat of all Au nanoparticles under study is again smaller than the bulk value. This oscillating behavior in the specific heat of Au nanoparticles is related to the differences in their VDOS line shape with respect to the one of the bulk phase. The usefulness of the equivalent (temperature-dependent) Debye temperature of Au nanoparticles to describe the temperature behavior of their specific heat is also discussed.

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INTRODUCTION Vibrational thermodynamics of materials is a well-established research field in continuous progress due to the existence of accurate and efficient methods to calculate phonons and of increasing advances in experimental techniques like calorimetry, inelastic neutron scattering, and inelastic X-ray scattering.1 A fundamental question is to what extent the thermodynamics of nanomaterials differs from that of bulk materials and what characterizes its size and shape dependence.1−3 In this respect, theoretical and experimental studies have shown that indeed thermal properties like the lattice specific heat,4−12 vibrational entropy,1,3,12−14 atomic mean displacement and Debye temperature,15−19 thermal expansion,20,21 melting temperature,22,23 and lattice thermal conductivity24,25 are modified at the nanoscale. Because the low-temperature behavior of these thermal properties depends on the atomic vibrations, systematic studies on the vibrational spectrum and vibrational density of states (VDOS) as a function of size and shape of nanomaterials are required. For example, several investigations on this topic have already shown, both theoretically and experimentally, that the VDOS of FCC, BCC, and icosahedral © 2013 American Chemical Society

(ICO) metal nanoparticles in the size range of 1−6 nm displays an enhancement at low frequencies and the appearance of a high-frequency tail as compared with the bulk VDOS.3,14−17,26,27 Moreover, it has been found that the VDOS enhancement at low frequencies, which can be associated with the lower average atomic coordination displayed by systems with a finite surface to volume ratio, is responsible for the higher values of the heat capacity measured at low temperatures in metal nanoparticles in comparison with its bulk value.4−12 Also, it has been obtained that the vibrational properties of metal nanoclusters show a strong shape dependence, particularly in the high-frequency region, where the lattice contraction together with the internal pressure explain the enhanced high-energy tail in the VDOS of FCC nanoparticles, resembling to a certain extent a bulk behavior, but not in the case of ICO nanoparticles.27 Received: September 7, 2013 Revised: October 28, 2013 Published: November 1, 2013 25160

dx.doi.org/10.1021/jp408976f | J. Phys. Chem. C 2013, 117, 25160−25168

The Journal of Physical Chemistry C

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Ts, (Ts ≈ 35−45 K), the known increment in the specific heat of metal nanoparticles with respect to the bulk value is recovered, which is due to the enhancement at low frequencies displayed by the VDOS of finite size systems. However, it is also found that for T > Ts the calculated specific heat of all Au nanoparticles under study is again smaller than the bulk value. This oscillating behavior, due to the differences between the VDOS of finite systems with respect to the bulk one, is in contrast with the larger values obtained for the specific heat of supported Fe nanoparticles as compared with the bulk Fe value.12 In the present theoretical study, the equivalent (temperaturedependent) Debye temperature of Au nanoparticles is also calculated to describe the specific heat in the whole temperature range under study. Although the concept of the Debye temperature for finite nanoparticles has been questioned,29,30 it would be insightful to examine the size dependence of their equivalent Debye temperatures.

In other studies, the validity of the bulk Debye model for the VDOS of finite size nanoparticles has been investigated, indicating a Debye behavior [g(ω) ≈ ωn with n ≈ 2] for small (∼2 nm) supported Fe nanoparticles but a non-Debye behavior (n ≈ 1.4) for larger ones (∼3−6 nm).12 Other investigation on Ag nanograins (with 150−1400 atoms) pointed out that only those having an irregular shape with grain boundaries and surface defects exhibit Debye (ω2) bulk behavior in their VDOS.28 Debye temperatures had also been extracted from EXAFS measurements of the mean square bond length fluctuations of supported Pt and Fe nanoparticles, although a definitive trend in their values with respect to the bulk Debye temperature has not yet been established.12,18,19 This might be due to the intrinsic limitations of the Debye model (and Debye temperature) when applied to nanoscale systems, which have been pointed out recently.29,30 Although the previously mentioned trends have provided initial insights into the relation between the VDOS and thermal properties like the specific heat and the Debye temperature of nanoparticles, further investigations are still necessary to explore in more detail their behavior at very low temperatures as well as their size and shape dependence. This is also motivated by a recent study indicating that instead of displaying an enhancement the specific heat of a 139-atom charged sodium nanocluster decays much faster with temperature than the bulk one in the very low temperature regime (T < 5 K).30 It was found that this distinct behavior is due to the discreteness of the cluster frequency spectrum and to the finite value of its acoustic gap (lowest frequency value).30 Because it has been already shown that the acoustic gap values decrease linearly with the inverse of the nanoparticle size,31 it would be interesting to investigate the size dependence of the specific heat and its difference with the bulk behavior at very low temperatures. In this work, we present theoretical results on the vibrational frequency spectrum of FCC, ICO, and decahedral (DEC) Au nanoparticles in the size range of ∼1−4 nm (55−2057 atoms). These results are useful to further analyze the size evolution toward the bulk limit as well as the shape dependence of the vibrational properties of Au nanoparticles. However, the main objective of this study is to evaluate the low-temperature dependence of the specific heat of FCC, ICO, and DEC Au nanoparticles as a function of size and to investigate its behavior with respect to the bulk value. This information would be useful not only in the design of nanodevices for heat transport or energy conversion working at low temperatures25 but also to gain additional insights into the relation between the VDOS and the range of low temperatures at which the specific heat of Au nanoparticles is smaller or larger than that of the bulk phase. These theoretical results on the low-temperature dependence of the specific heat of metal nanoparticles are also timely because at the present it is already possible to correlate them with experimental measurements of the caloric behavior of metal nanoclusters down to 6 K.32 In fact, our results show that there is a temperature range, 0 < T < Tr, (Tr ≈ 5 K for the Au nanoparticle with size ∼1.4 nm), that decreases with the nanoparticle size, where there is a slight reduction in the specific heat of Au nanoparticles with respect to bulk value. This decrease is due to the nonzero value of the acoustic gap characterizing the discrete frequency spectrum of finite size nanoparticles.30,31 It is also found that the difference in the specific heats diminishes with the nanoparticle size because the acoustic gap vanishes in the bulk limit. For Tr < T
100 cm−1. These differences may be associated with the larger strain and twining existing in

The VDOS displays a behavior that evolves from a multiplepeak line shape for small sizes to a characteristic profile for the larger nanoparticles that anticipates the well-known VDOS of a FCC bulk metal. In fact, the VDOS of the largest FCC nanoparticle under study (Au1926) shows an overall line shape resembling that one of bulk Au shown at the bottom panel of Figure 1. Despite this similarity with the bulk metal, the VDOS of Au1926 still displays significant differences with respect to the bulk one. These include a nonzero value for the acoustic gap, an enhancement at low frequencies, and additional high frequency peaks, which are all a direct consequence of the nanoparticle finite size. For example, the high-frequency peak at ω ≈ 136 cm−1 in the VDOS of Au1926 (see Figure 1) has no bulk analogue, and it corresponds to oscillations of the less coordinated edge atoms, whereas the peak around ω ≈ 125 cm−1 is populated by localized vibrations of atoms forming the surface square [100] faces. An animation file displaying these high-frequency modes (labeled 1 and 199 for the frequencies 136 and 125 cm−1, respectively) is available in the HTML version of this paper. Other normal modes of interest are those with frequencies where the VDOS displays features like those that in the bulk limit correspond to the van Hove singularities as well as the mode corresponding to the acoustic gap frequency. In this regard, the VDOS maximum of Au1926 (see Figure 1) at ω ≈ 110 cm−1 (mode labeled 821 in the animation file) corresponds to transversal atomic oscillations with respect to symmetry axes, whereas at the VDOS minimum (mode labeled 1461 in the animation file) at ω ≈ 100 cm−1 the atomic vibrations are mainly localized along the directions connecting opposite [100] 25163



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the ICO morphologies. The VDOS of ICO nanoparticles also presents a frequency tail beyond the highest frequency obtained for FCC and DEC nanoparticles.31 This peculiarity in the vibrational properties of ICO morphologies can be better appreciated in Figure 3, where a comparison between the

Figure 4. Size dependence of the (a) acoustic gap (lowest frequency) and (b) maximum frequency for Au nanoparticles with FCC (red), DEC (orange), and ICO (blue) morphologies. Figure 3. Vibrational frequency spectra (blue bars) and VDOS (red line) of Au nanoparticles with FCC truncated-octahedral (TOC), decahedral (DEC), and icosahedral (ICO) morphologies. The VDOSs were scaled to obtain the same value after integration over the whole frequency range.

calculated for the bulk Au metal using the VDOS displayed at the bottom panel of Figure 1. The result is shown through the dotted lines in the six panels of Figure 5. It can be appreciated that the specific heat of the bulk Au metal displays similar temperature dependence as that of the Au nanoparticles, independently of their size and shape over a range of temperatures up to 150 K (right panels). Nevertheless, small deviations from the bulk behavior can be seen in the left panels of Figure 5, displaying in more detail the specific heat in the low-temperature regime. In fact, it is found that the specific heat of the bulk Au metal is lower than those of the Au nanoparticles in the temperature range ∼5−35 K. This behavior is a direct consequence of the enhancement in the nanoparticle VDOS with respect to the bulk one, which is due to their finite size. This result confirms a similar trend that had been obtained both theoretically and experimentally on different metal nanoparticles.4−12 Figure 6 shows the difference between the specific heat of the FCC Au nanoparticles and that of the bulk Au metal as a function of temperature, indicating that their specific heat is smaller than the bulk value at very low temperatures ( 35 K. At very low temperatures (see the inset in Figure 6), the specific heat of Au nanoparticles decays according to (hωAG/kBT)2 exp(−hωAG/ kBT), where ωAG is the acoustic gap (lowest vibrational frequency).41 For the values of ωAG shown in Figure 4, this decrease is faster than the Debye T3 law characterizing the specific heat of the bulk metal at very low temperatures. Another important result obtained from the present study is the smaller value of the specific heat of Au nanoparticles with respect to the bulk value for T > 35 K. (See Figure 6.) Although, Figure 6 shows this result for FCC nanoparticles, a similar result was obtained for DEC and ICO morphologies, indicating that this behavior is independent of the nanoparticle shape. Figure 6 also shows that the difference between the smaller value of the specific heat of the Au nanoparticles and the bulk one decreases as the nanoparticle size increases. These results are in contrast with the trend extracted from 57Fe nuclear resonant inelastic X-ray scattering (NRIXS) measurements for supported Fe nanoparticles in the size range of 2−6

VDOS of FCC, DEC, and ICO Au nanoparticles with size ∼4 nm is displayed. The visual analysis of the high-frequency modes at ω > 160 cm−1 of the ICO Au2057 indicates that they originate from vibrations involving the core atoms localized at the center of the ICO morphology (mode labeled 1 in the animation file). Nevertheless, the 10 modes obtained at ω ≈ 154 cm−1 are mainly due to vibrations of the vertex atoms of the icosahedral structure (mode labeled 13 in the animation file). An animation file displaying these high-frequency modes is available in the HTML version of this paper. Further insights into the size and shape dependence of the vibrational properties of metal nanoparticles can be obtained from the calculated values of the acoustic gap and the maximum (cutoff) frequency. Figure 4 shows this dependence for FCC, DEC, and ICO Au nanoparticles in the size range ∼1−4 nm, indicating a shape-independent monotonic decrease with size (∼1/diameter) of the acoustic gap. In contrast, the maximum frequency values show a small variation with size but strong shape dependence with ICO morphologies displaying larger values than those found for FCC and DEC nanoparticles.

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THERMAL PROPERTIES The calculated frequency spectrum can be utilized to evaluate the specific heat of the Au nanoparticles using eq 1. The left panels of Figure 5 display the low-temperature dependence of this quantity divided by the number of vibrational modes (3N6), whereas the right panels show its temperature dependence up to 150 K for different sizes and shapes. It is noteworthy that the calculated specific heat displays only small variations both with size, in the range of 100−2000 atoms (1−4 nm), and with shape (FCC, DEC, ICO), despite the VDOS line shapes showing stronger size and shape dependences. To investigate how different the specific heat of Au nanoparticles from the bulk behavior is, this quantity was 25164



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Figure 5. Low-temperature dependence (left panels) and temperature dependence up to 150 K (right panels) of the specific heat for Au nanoparticles with different sizes and shapes.

Figure 7. Difference between the integrated values of the VDOS’s of FCC Au nanoparticles and integrated values of the VDOS of bulk Au as a function of frequency.

Figure 6. Temperature dependence of the difference in the FCC Au nanoparticles specific heat with respect to the bulk value.

(see bottom panel of Figure 1), all of the vibrational modes contributed to the specific heat of the bulk metal, whereas for the nanoparticles there are higher frequency vibrational modes that have not yet been accumulated in the specific heat. Then, the differences between bulk and nanoscale specific heat behavior are just reflections of the different cumulative densities of states. The reliability of the previously mentioned results depends on the capability of the Gupta many-body potential to describe the atomic interactions in the Au nanoparticles as well as on the validity of the harmonic approximation along the range of temperatures in which the specific heat was calculated. With respect to the reliability of the Gupta many-body potential, it was already mentioned that this potential was recently tested in a combined theoretical−experimental study on the periods of the quasi-breathing vibrational motions of Au nanoparticles in the size range of 2−4 nm.31 In that study, it was shown that the

nm, indicating that their excess vibrational specific heat with respect to bulk BCC Fe is positive up to 700 K.12 To gain insight into our theoretical results, we analyzed the difference between the integrated values of the VDOSs of FCC Au nanoparticles and those corresponding to the integrated values of the VDOS of the bulk Au as a function of frequency (Figure 7). It can be noticed that up to ω ≈ 80 cm−1, a larger number of vibrational modes are being accumulated for nanoparticles than for bulk, whereas for ω > 80 cm−1, there is a larger accumulation of vibrational modes in the bulk metal than in nanoparticles, changing the difference in the integrated values of the corresponding VDOSs to negative values. This result is mainly due to the existence of the characteristic optical peak in the VDOS of FCC metals where the maximum is reached, whereas in the nanoparticles case this peak is still growing as their size increases. Moreover, for ω ≈ 115 cm−1 25165



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parameters of the Gupta many-body potential describing the interaction between Au atoms are good enough to calculate the QBM vibrational frequencies of Au nanoparticles, which are in very good agreement with time-resolved pump−probe spectroscopy data.31 It is known that the Gupta potential parameters for bulk Au cannot precisely reproduce both the C44 elastic constant and the maximum (cutoff) frequency of the VDOS, which, respectively, correspond to properties at the center and at the boundary of the Brillouin zone.33,44 This discrepancy, originated from the intrinsic difficulty of fitting with four Gupta parameters the structural, binding, and elastic properties of bulk Au, is also reflected in the comparison between the theoretical and experimental values of the specific heat of bulk Au, shown in Figure S2 of the SI. Notice that at low temperatures, and also in the range of 120−150 K there is good agreement between calculated and experimental values, but at intermediate temperatures there exists a larger difference between these quantities. For a quantitative comparison of these values, we calculated the weighted profile factor Rw of the theoretical bulk Au specific heat with respect to the experimental one (see the SI for details of this calculation), obtaining a value of 13%, which is an estimation of the overall difference between the compared data. Because to our knowledge there are not yet experimental results available for the specific heat of Au nanoparticles with size in the range of 2−4 nm, it is not possible to make a comparison as in the bulk case. Therefore, at the moment, our theoretical results, shown in Figures 5 and 6 indicating that differences of around +15− 20% at T ≈20 K and of ∼ −1.5% at T ≈ 75 K exist between the specific heat of Au nanoparticles and bulk, might be considered as useful trends waiting for a quantitative experimental confirmation. In fact, the recent experimental results on the caloric behavior of the Na139+ cluster, down to 6 K, suggest that such type of studies would be possible in the near future. Once the low-temperature dependence of the specific heat was obtained, the appropriateness of the bulk Debye model to describe these results can be tested. For this purpose, the theoretical results shown in Figure 5 were fitted to the specific heat expression provided by the bulk Debye model, scaled to the number of degrees of freedom of the nanoparticle:30,32 ⎛ T ⎞3 C D(T , θD) = 3(3N − 6)kB⎜ ⎟ ⎝ θD ⎠

∫0

θD/ T

Figure 8. Equivalent Debye temperature as a function of temperature for FCC Au nanoparticles. The dashed line corresponds to the Au bulk metal obtained from the VDOS shown in the bottom panel of Figure 1

when the temperature increases. Moreover, it turns out that at low temperatures ( Ts the calculated specific heat of all Au nanoparticles under study is again smaller than the bulk value. This oscillating behavior in the specific heat of Au nanoparticles is due to the differences between the VDOS of finite systems and the bulk one. The usefulness of the equivalent (temperature-dependent) Debye temperature of Au nanoparticles to describe the specific heat along the whole temperature range under study was discussed. It is expected that the above-mentioned theoretical results motivate further experimental investigations on the low-temperature behavior of the specific heat of metal nanoparticles, as has been recently studied for size-selected sodium clusters down to 6 K.32 Moreover, our theoretical results would also be valuable in the sense that they provide additional new information toward a reparameterization of the Gupta potential, which should include both bulk and nanoparticles properties in the fittingparameters procedure. Work in such direction might be of interest for future research in the computational nanoscience field.

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ASSOCIATED CONTENT

* Supporting Information S

Comparison of the specific heat for the icosahedral Au55 cluster calculated with the Gupta potential and DFT methodology and the comparison with experimental results of the specific heat for bulk Au calculated with the Gupta potential. The complete list of authors of refs 12 and 19 is presented. This material is available free of charge via the Internet at http://pubs.acs.org. W Web-Enhanced Features *

Animation files displaying high-frequency modes are available in the HTML version of this 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 and DGAPA-UNAM project IN102511. Calculations were done using resources from the Supercomputing Center DGTIC-UNAM. We thank Mr. Jesús Pelayo and Dr. Israel Valencia for performing DFT calculations on the vibrational properties of the Au55 cluster. H.E.S. acknowledges support from Programa de Doctores Jóvenes de la Universidad Autónoma de Sinaloa. F.S. acknowledges support from Secretariá de Ciencia Tecnologiá e Innovación del Distrito Federal under Projects ICyTDF/PICSO12-085 and ICyTDF/ 325/2011 and from SIP-Instituto Politécnico Nacional under the Multidisciplinary Project 2012-1439.

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REFERENCES

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