Article pubs.acs.org/JPCC
Map of the Structural and Optical Properties of Gold Nanoparticles at Thermal Equilibrium A. L. González,† C. Noguez,*,‡ and A. S. Barnard§ †
Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J48, 72570, Puebla, México, Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, México DF 01000, México § CSIRO Materials Science and Engineering, 343 Royal Parade, Parkville, VIC, 3052, Australia ‡
S Supporting Information *
ABSTRACT: The remarkable relationship among the size, shape, and optical properties of gold nanoparticles is proving to be very useful in a range of highperformance applications. Considerable effort and investment is focused on delivering gold nanoparticles with precise morphologies. However, the reliability of these particles is contingent upon the morphological stability, particularly against variations in the thermodynamic environment, such as changes in temperature. Presented here are results from a combination of computational and theoretical techniques showing how the optical properties of gold nanoparticles respond to changes in the size, shape, or temperature, obtained by sampling the optical spectrum over large configuration space, in accordance with the nanoscale phase diagram. We find that spectrum from morphologies expected at small sizes is robust against temperature fluctuations, unless the concentration is very high. At larger sizes, the color will likely change with temperature, due to the accompanying change in particle shape, and this change will be noticeable when the concentration is low.
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INTRODUCTION Among the variety of noble metal nanoparticles (NPs) produced today, gold is one of the most studied due to a number of interesting physical, chemical, and mechanical properties that show great promise for a range of nanoscale applications.1 Studies have been reported of size-dependent electronic, magnetic, and optical properties as well as chemical compatibility with certain biomolecules.2 Many of the desirable properties are strongly linked with the nanomorphology, including such features as size, geometric shape, and degree of dimensional anisotropy (aspect ratio). The relation between properties (and ultimately applications) and the physical structure is central to the chemistry3−8 and biochemistry9 of nanogold, and relating these characteristics has been the focus of numerous studies.2,10,11 This includes the range of interesting optical properties,12 which are also intrinsically linked to size and shape. Considerable effort is being directed to understanding how engineering the size and shape of nanogold can improve performance.13−17 Typically gold NPs are present as icosahedral and truncated icosahedral structures,18−22 decahedral and truncated decahedral structures,1,19,20,23 truncated octahedra and cuboctahedra (or variants of these shapes),24−27 or as singly or multiply twinned fcc structures.18,28 These morphologies are shown in Figure 1. All of these shapes are dominated by various fraction of {111} and {100} facets, which have different surface atom densities, electronic structure, bonding, chemical reactivities, and thermodynamic properties. Both decahedra and icosahedra are based on fcc units separated by (111) twin planes. The © 2012 American Chemical Society
most effective way of representing all possible structures is through a nanoscale phase diagram,29 which provides a 2D graphical representation of chemical equilibrium. They are commonly used to describe the different phases of a multicomponent system (such as alloys and compounds) but are also useful in describing the phases and motifs (and shapes) of homoelemental systems. This is ideally done using theory and simulation30 because it is possible to control the phase, shape, size, and temperature independently and systematically include regions on a phase diagram for different crystalline polymorphs (or polymotifs) and melts,29 which is extremely challenging (and potentially costly) to do experimentally. The value in this type of structural prediction is that it facilitates the development of robust and functional structure/ property relationships. Using the phase diagram as a basis for a structure/property map can be useful in identifying the right size and conditions required to deliver specific properties, such as optical properties, required for certain applications.31 The optical properties of gold NPs are primarily determined by their surface plasmon resonances (SPRs), which are intrinsically linked to the geometry of individual NPs.32,33 For example, the presence of acute edges and corners is known to enhance the electromagnetic field,34,35 as observed using surface-enhanced Raman spectroscopy36 and plasmon-controlled fluorescence.37 This relationship between the structure and the optical properties has introduced new methods of tuning the optical Received: May 17, 2012 Published: June 22, 2012 14170
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Figure 1. Typical morphologies observed in gold nanoparticles, including the (a) octahedron, (b) truncated octahedron, (c) cuboctahedron, (d) truncated cube, (e) cube, (f) singly twinned truncated octahedron, (g) multiply twinned decahedron, and (h) quasicrystalline icosahedron. The reentrant edges of the twin planes are highlighted in blue.
nanostructures, all in excellent agreement with the corresponding experimental observations.39,45−47 In this Article, we have utilized DDA and the phase diagram of nanogold to model how the optical properties of a colloidal solution may be affected by thermal fluctuations. We have considered a colloidal solution consisting of a dispersed phase defined by the concentration of gold NPs and three different continuous phase: air, water, and toluene with refractive index 1.0, 1.333, and 1.5, respectively. It is well-known that as the refractive index increases, the SPR spectrum suffers a red shift and becomes wider. There is also a change in the intensity of the resonances, but the number of SPRs remains invariant.46 In the present study, our results predict that the characteristic color of a nanogold colloidal solution is sensitive to the refractive index of the continuous phase of the colloid, particle concentration, and morphology at each point on the size/temperature phase diagram.
response of gold NPs by engineering their size and shape,10,38−41 but they may also play an important role in determining our ability to anticipate how these properties may be affected by uncontrolled changes in morphology due to thermal fluctuations. Whereas a morphological or phase transformation may be supremely interesting to study from an academic perspective, it is the consequential change in the properties that is of commercial interest, so a predictive map of the optical properties of nanogold as a function of size and temperature will be invaluable. The development of this type of structure/property relationship using exclusively experimental methods is possible in principle but is extremely complicated (and potentially expensive) owing to the need for exquisite structural control over a wide range of sizes and conditions. Fortunately, SPRs are determined by the NP size, shape, and variations of the dielectric function, and the theory surrounding SPRs is welldeveloped.42 It is known that extinction efficiency of light Qext by an NP is defined as the sum of the absorbed Qabs and scattered Qscat light efficiencies. NPs less than 20 nm in average diameter only absorb energy via intraband transitions (giving rise to SPRs), by interband transitions, via surface dispersion, or by scattering of the free or unbound electrons; then, Qext ≈ Qabs. Additionally, the SPRs’ number, position, intensity, and width are determined by the particle shape. Surface dispersion effects broaden the absorption spectrum but do not change the location of the SPRs. Scattering effects, however, are important only for NPs larger than 30 nm in diameter, where electrons are accelerated due to the electromagnetic field. In general, the absorption spectrum becomes less intense, broader, and red-shifted as the size of the NP is increased.42 A depolarization field term provokes the shift to larger wavelengths, whereas radiation damping causes decreasing intensity and broadening of the spectrum.43 Scattering effects dominate the response of NPs over 100 nm in diameter; that is, Qext ≈ Qscat. This means we will expect to see some variations in the optical properties of gold NPs across the configuration space depicted in the phase diagram of nanogold.29 In the past, simulations applying the discrete dipole approximation (DDA) have been successfully used to model the absorption and scattering of a variety of noble metal
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METHODS
Structural Prediction. Modeling of the equilibrium structure of gold NPs in vacuum as a function of size and temperature has been previously reported elsewhere.29 The modeling was undertaken using a shape-dependent thermodynamic model, parametrized using relativistic density functional theory computer simulations44 that included planar defects and the possible formation of multiply twinned particle such as decahedra and icosahedra. By rapidly and exhaustively sampling the configuration space, the thermodynamically stable shape was determined, and mapped as a function of average diameter (⟨D⟩) and temperature (T). The resultant phase diagram was verified experimentally and is used here as the basis for the optical structure/property map. As part of the information extracted from the phase diagram, we assume that the formation of stable spherical structures (with roughened surfaces) occurs within a size range of 5−30 nm and at temperatures above 700 K, whereas nonspherical shapes such as truncated octahedra (Figure 1b), cuboctahedra (Figure 1c), truncated cubes (Figure 1d), and cubes (Figure 1e) are the most stable structures between 15 and 30 nm and at temperatures below 700 K. Optical Modeling. Predictions of the optical properties of gold NPs at various temperatures were obtained using the 14171
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theoretical solution by Mie48 and the numerical model based on the DDA derived by Purcell and Pennypacker.49 The former provides the exact solution for spherical particles of different sizes, whereas the latter gives an excellent approximation for nonspherical shapes.50 Both theories allow us to include the refractive index of the matrix where the particle is. DDA is a very useful model because it allows us to take into account shape, size, and environment effects; its foundations are contained elsewhere.51,52 In this particular case, we calculate the optical properties of nonspherical shapes using the computational code named DDSCAT,53 which is a numerical implementation of DDA. To compute the optical properties, we discretized the NP with a particular shape into a cubic array of point dipoles that reproduces the morphology.54 We then consider a dielectric function of the material, in this case the bulk dielectric function of Au,55 which is properly modified to take into account surface dispersion effects according to the NP size.39 First, the extinction Qext, absorption Qabs, and scattering Qsca efficiencies are calculated for a gold NP with given morphology and size according to the phase diagram; the refractive index of the continuous phase is also considered. We then proceed to obtain the color observed in transmittance for a colloidal solution containing such NPs. For this, we use the Kubelka− Munk theory56 to calculate the transmittance, ; , of a colloidal solution with a specific concentration c of NPs of a particular size and shape dispersed in a medium as56 ;=
b a sinh(bSL) + b cosh(bSL)
Figure 2. Extinction efficiency spectra of single gold NPs in air as a function of wavelength for different morphologies as the temperature increases. (a) Spectra of an NP of 10 nm and different temperatures. (b) Spectra of an NP of 25 nm and different temperatures.
initially at ∼0 K the NP is an icosahedron and suffers a morphological transition at T ≈ 70 K, where Au atoms undergo a spacial rearrangement forming a truncated decahedron. This morphological transformation is shown for larger sized NPs in ref 29. The decahedral structure is retained over a wide range of temperatures until T is close to 800 K, where again the atoms restructured to form either a crystalline or amorphous structure with roughened (amorphous) surfaces and spherical symmetry. The sphere is preserved for temperatures below 1200 K, and at higher temperatures the NP melts entirely. The morphological evolution at this size is reflected in small changes in the extinction efficiency of the gold NP, where the intensity and position of the main SPR change slightly. For instance, the intensity is decreased by ∼10% as the temperature increases, whereas the main resonance is blue-shifted by ∼10 nm. For larger NPs, in the range of 20−30 nm of diameter, the structural evolution as the temperature increases produces a larger quantity of different morphologies that go from a truncated octahedra to cuboctahedra to truncated cubes to cubes and finally to spherical NPs. Figure 2b shows the evolution of the extinction efficiency spectra of a gold NP of 25 nm as it heats from 0 to 1300 K. Between 0 and 560 K the NP is a truncated octahedron, where the degree of truncation depends on the temperature. For instance, lower degrees of the truncation are found at low value of T, whereas the maximum truncation is obtained for temperatures close to 560 K, where cuboctahedral NPs are obtained. Above this temperature, at ∼630 K the NP changes its morphology to a truncated cube. Now the truncation degree decreases when temperature increases, such that when T is between 700 and 770 K the minimum truncation is reached and the NP is closer to a perfect cube. Again, for temperatures above 800 K, the atoms are restructured, forming an amorphous structure with spherical symmetry, which is preserved for temperatures below 1200 K before melting occurs. In Figure 2b, we observe that the position of the main SPRs shifts to the red by ∼10 nm, when the NP heats from 0 to ∼700 K; then, they are blue-shifted by ∼15 nm when a spherical shape is reached at temperatures above 800 K. The variations on the position of the SPRs are quite small; however, the intensity suffers notable changes. The maximum intensity is obtained when cubical NPs are found
(1)
where L is the thickness of the sample; a and b are defined as a= b=
S+K S a2 − 1
(2)
and S and K are the Kubelka−Munk absorption and scattering coefficients.56 S and K are related to the optical efficiencies and the concentration c of NPs as K = 2cQ absπa 2 S = 3Q scaπa 2 /4c
(3)
Using the transmittance of the colloidal suspension, we obtain the color matching functions: r, g, b (red, green, blue).56 Finally, on the basis of these results, we predict the color of a solution with a given concentration c at different temperatures that would be observed by the human eye, where a color map of the phase diagram relating morphology and optical properties is obtained. The results are discussed below.
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DISCUSSION OF RESULTS According to the phase diagram,29 we know that gold NPs smaller than ∼10 nm in diameter have a morphological evolution from an icosahedron to a truncated decahedron and to a sphere when they heat from 0 to 1200 K. The temperature, at which the morphological transitions from one shape to another occur, depends on the NP size. Assuming that the particle suffers only a morphological evolution, maintaining its volume when the temperature increases, we show in Figure 2a the gradual development of the extinction efficiency of a gold NP of 10 nm in diameter as it heats. For this particular size, 14172
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the dilute regime, where the interactions between particles are negligible. At each concentration, we model the absorbance, as the optical response of one suspended particle multiplied the concentration39 (the absorbance is additive in the dilute regime). The calculation of the color was done as described in the previous section, where we have taken intervals of sizes of 5 nm, three different degrees of truncation for octahedral and one for cubic NPs, because of the considerable time and resources involved in these types of computations. However, the calculated colors for 6 different sizes and 7 different shapes, with a total of 35 optical spectra, are enough to provide trends of the actual color of the corresponding colloidal suspensions as a function of size, temperature, and concentration. The r, g, and b values obtained for these spectra are shown in the Supporting Information. The observed color maps of the solution at three different concentrations, 5 × 1011, 5 × 1012, and 2 × 1013 particles/mL, are illustrated in Figure 3a−c, respectively. In general, we observe that these colloidal solutions are characterized by pink, salmon, and reddish colors. The reddish color in aqueous dispersions of spherical-like Au NPs has been reported by ́ Rodriguez-Gonzá lez and colleagues. 57 For the smallest concentration, we found that particles with sizes around 5 nm absorb very small quantities of electromagnetic radiation so that they become almost transparent, independently of the temperature, because almost the same color is obtained for icosahedra, decahedra, and spherical NPs. This can be inferred by the extinction spectra shown in Figure 2a, where very small variations can be found. However, these slight variations in intensity and wavelength become important for larger concentrations. As the particle’s concentration increases, a hue saturation occurs, resulting in darker colors and thus a larger contrast between different sizes and shapes. In this case, it is clear in Figure 3 that even for small particles one can easily discern different colors at different temperatures. These optical structure/property maps also demonstrate that it is possible to distinguish between different shapes. As observed in Figure 2b, cubic NPs have larger intensities in the extinction efficiency, consistently resulting in a darker color, independent of the size and concentration. There is a slight shift in color as we move to the truncated cubes, but we do not find large color differences between cuboctahedral and truncated octahedral NPs. Furthermore, if we increase the concentration and NP size, then the colloidal suspension becomes very dark, which makes it almost impossible to distinguish between different shapes or identify morphological changes that take place during heating. If one wishes to use optical spectra to identify the shapes of large gold NPS, then one must keep the concentration low. In contrast, regardless of the concentration, the color exhibited by icosahedral and spherical colloids is always similar because the difference in the spectra is only in the intensity, as shown in Figure 2a. This means that when the concentration is low it would be very difficult to distinguish between spherical and icosahedral particles, except that the latter are found at low temperature, whereas spherical particles are obtained at high temperatures. If one wishes to use optical spectra to identify the shapes of small gold NPS, then one must keep the concentration high. As we previously mentioned, the optical properties of an NP also depend on the surrounding medium. As the refractive index of the medium increases, the SPR spectrum suffers a red shift and becomes wider and the intensity changes. This behavior is clearly observed in Figure 4, where the extinction
between 700 and 770 K, but above this temperature the NP experiences a transition from a cube to a sphere and then the intensity of the SPR is drastically reduced by almost 50%. This will produce strong effects in the color observed in transmission by a colloidal solution, as we later show. Phase diagrams relating structural and optical properties of gold colloidal suspensions with different NP concentrations as a function of NP sizes and temperature are shown in Figure 3.
Figure 3. Quantitative color phase maps relating structural and optical properties of Au nanoparticles in air with different concentrations of (a) 5 × 1011, (b) 5 × 1012, and (c) 2 × 1013 particles/mL.
The dispersed phase of the colloidal suspension is defined by the concentration, c, of gold NPs, where it is assumed that all of the NPs have the same specific size and shape (are monodispersed) but are randomly oriented. It is also assumed that the NPs are dispersed in air. For illustrative purposes, we have chosen three different concentrations that correspond to 14173
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ASSOCIATED CONTENT
S Supporting Information *
CIE L*a*b* tristimulus values associated with a colloidal solution of Au NPs with concentrations 5 × 1011, 5 × 1012, and 2 × 1013 particles/mL. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: cecilia@fisica.unam.mx. Notes
The authors declare no competing financial interest.
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Figure 4. Extinction efficiency spectra of a single sphere of diameter 20 nm and embedded in air, water, and toluene. Colored circles represent the color of a colloidal solution when the particle concentration is 5 × 1011 particles/mL.
ACKNOWLEDGMENTS This project has been supported by the Australian Research Council (ARC) under grant number DP0986752, by the Secretary of Public Education of México under PROMEP project BUAP-PTC-265, and DGAPA-UNAM IN104212. Computational resources for morphology modelling have been supplied by the National Computing Infrastructure (NCI) national facility under MAS Grant p00, while DDA calculations were carried out using computational resources at UNAM and BUAP. We thank Guillermo P. Ortiz for his helpful comments about the color theory.
spectra of a single spherical NP in air, water, and toluene are shown. We also illustrate the changes in color for the particular case of a colloidal solution with a concentration of 5 × 1011. The color tends to be darker as the refractive index of the continuous phase becomes larger.
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CONCLUSIONS As we can see, on the basis of a combination of computational and theoretical techniques, the optical properties may respond to changes in the size, shape, temperature, and surrounding medium. By sampling the optical spectrum over large configuration space, in accordance with the equilibrium phase diagram of nanogold, we find that although the morphology of the NPs is sensitive to increases in temperature, the hue of the colloid does not change discernibly, unless the concentration is just right. This means that in general the optical properties of gold NPs may be considered to be robust against temperature fluctuations, provided the concentration is carefully monitored. At small size, the concentration should be low, as the response to environmental changes is only significant when the concentration is high, and at larger sizes, the concentration should be high, as the response to environmental changes is only significant when the concentration is low. Consideration of this additional degree of freedom will enable the design of more reliable devices and more robust technologies based on remarkable properties of nanogold. Whereas a structure/property map can be very instructive, the morphology of metal NPs can often deviate from the thermodynamically ideal configurations and can be specifically engineered via templating or growth kinetics.1 In the former case, we have shown (see Figure 2) how this approach may be used to predict the response of specific morphologies (such as those that may be produced via templating). In the latter case, this approach could also be used to generate a similar structure/ property map based on populations statistics58 or by combining the optical simulations with kinetic nanomorphology models to predict the size and shape as a function of size and temperature.59 Now that the structure/property mapping technique has been established, this is an ideal topic for future work. In the future, we also plan to use these techniques and approach established here to investigate the optical response to changes in other important morphological features, such as variations in dimensional anisotropy and the cross sectional shape of gold nanorods.
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