J. Phys. Chem. C 2009, 113, 11995–12001
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ARTICLES Real-Time Monitoring of the Nucleation and Growth of ZnO Nanoparticles Using an Optical Hyper-Rayleigh Scattering Method Doris Segets, Lars Martinez Tomalino, Johannes Gradl, and Wolfgang Peukert* Institute of Particle Technology, Friedrich-Alexander-UniVersity Erlangen-Nuremberg, Cauerstrasse 4, 91058 Erlangen, Germany ReceiVed: February 3, 2009; ReVised Manuscript ReceiVed: April 27, 2009
Incoherent nonlinear hyper-Rayleigh scattering (HRS) has been applied to monitor the different stages in the synthesis of sub-10 nm ZnO nanoparticles. During time-resolved investigation of the ripening process, we measured an increase in the particle hyperpolarizability, βZnO, with increasing particle size. Millisecond time resolution measurements revealed regions of prevalent nucleation, growth, and ripening. Using the sizedependent hyperpolarizabilities, a new general approach is introduced for the quantitative determination of nucleation, growth, and ripening rates via in situ HRS measurements. The derived solubility also allowed the determination of reasonable values of the ZnO surface energy. Introduction ZnO semiconductor quantum dots have attracted considerable attention over the past 10 years due to their unique electronic and optical properties resulting from the quantum size effect. In particular, electronic devices such as solar cells, lightemitting, or laser diodes represent promising applications.1,2 Consequently, much research has been carried out to understand and optimize the slow ripening kinetics of ZnO particles, especially concerning different synthesis routes3 and correlations of optical properties with disperse properties.4 Although quantitative studies of nucleation as well as rapid growth kinetics at early stages of solid formation for nanoscale zinc oxide and other nanocrystallites are rare,5,6 they are essential for the formation of well-defined particle properties.7,8 In the past few years, there have been publications investigating the nucleation and growth (e.g. of gold nanoparticles or CdSe in the millisecond range9-11), but the mechanisms and kinetics of nucleation and their dependence on the thermodynamic driving force (i.e. the chemical potential and the supersaturation, respectively) are largely unknown due to the fast and transient character of these processes. Classical nucleation theory gives only a rough framework. In this work, hyper-Rayleigh scattering (HRS) as a nonlinear optical effect observed in isotropic solutions is shown to be a far superior technique to resolve the underlying kinetics, as compared to most other methods available for monitoring nanoparticle formation in liquids. Also regarding different nonlinear measuring techniques, HRS seems to be the most promising method for the investigated particle system due to its incoherent nature because the particle diameters below 5 nm would not lead to measurable signals; for example, in the case of second harmonic generation (SHG) or sum frequency generation (SFG). For SHG, we found a lower detection limit of 30 nm for silica particles, whereas the smallest particle size * To whom correspondence should be addressed. Phone: +49 9131 8529400. Fax: +49 9131 85 29402. E-mail:
[email protected].
investigated with SFG is 36 nm.12 SFG is used mostly for the characterization of particle layers on a substrate and has just started to become a method for the characterization of colloids. Hyper-Raman scattering intensities usually are smaller than hyper-Rayleigh scattering intensities; therefore, it is assumed to be difficult to obtain enough signal for analysis. An interesting method to obtain concurrent information about both particle size and crystal structure could be polarized parametric light scattering (PLS) in the context of an extension of the HRS method, but this technique is used mainly for the investigation of organic molecules in solution.13 It has the advantage that more components of the hyperpolarizability tensor, βi,j,k, can be obtained by using two laser beams with different frequencies and polarizations. However, detailed studies about the correlation between the hyperpolarizability tensor, βi,j,k, and the crystal structure would be necessary. Regarding the HRS signal, it has its roots in temporal and spatial fluctuations, leading to instantaneous breaks in the system’s averaged centrosymmetry. HRS was first observed in 196514,15 and has become increasingly popular since the early 1990s, when it was established as the method of choice for the determination of molecular hyperpolarizabilities.16 Recently, the application of this method to study nanoparticulate systems has gained in importance due to its high sensitivity toward small changes in size, structure, or shape.17 A variety of different materials have already been studied, as shown by Eisenthal.18 Thereby, gold represents the most investigated material system, with research performed on the effect of the wavelength of the incident laser light,19 the signal increase due to agglomeration events,20,21 or the applicability of HRS in terms of biological labeling and sensing.22-24 Furthermore, the dependency of the nonlinear signals on the particle size has been studied for gold, silver, copper, CdS, CdSe, CdZnSe, and TiO2,25-31 but little focus has been spent on the early stages of particle formation. Zhang et al. performed the first time-resolved HRS measurement on the formation of CdS nanoparticles,27 albeit with relatively poor time resolution and
10.1021/jp9009965 CCC: $40.75 2009 American Chemical Society Published on Web 06/11/2009
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Figure 1. Absorbance spectra measured at different stages of the ZnO nanoparticle ripening process.
without any conclusion about kinetic data. Recently, Wallis et al. pointed out the promise of the method to monitor crystal nucleation in solution.32 In this paper, HRS from nanoscale ZnO particles is reported. The hyperpolarizabilities, βZnO, of different sized particles have been determined, and an increase in βZnO with increasing particle size was observed. A simultaneous long-term time-resolved observation of the ripening process by HRS and spectrophotometry leads to a quantitative expression for the size dependence of βZnO. Further time-resolved measurements revealed the distinct steps of the particle formation process and allowed, for the first time, the calculation of nucleation, growth, and ripening rates out of the nonlinear signals, which enable a first estimation of solubility and surface energy of small ZnO nanoparticles. In principle, the approach we present to determine kinetic data quantitatively from the combination of HRS and spectrophotometry is applicable to a variety of systems of materials, especially to the aforementioned semiconductor nanoparticles. Experimental Section Ethanol (99.98%, VWR, Germany) was used for the preparation of all reactant solutions. Before the synthesis, a 0.1 M zinc acetate dihydrate (ACS grade, 98.0-101.0% VWR, Germany) precursor stock solution was prepared and mixed with an equimolar amount of lithium hydroxide (98%, VWR, Germany) at 293 K, resulting in instantaneous particle formation. Absorbance spectra between 250 and 400 nm were measured using a Cary 100 Scan UV-visible spectrophotometer (Varian Deutschland GmbH, Germany). For the HRS measurements, the beam of a short-pulse Ti:Sa laser (Spectra Physics, Tsunami) was focused into a glass cuvette with an optical path length of 10 mm. The wavelength was set to 800 nm, and the mean laser power was 1.7 W at a pulse length of 80 fs. Wavelength separation of the incident light and HRS signal was performed by a blue glass filter of the type BG38 (SchottAG, Germany) and a monochromator. Photon counting was used to detect the nonlinear signal at 400 nm. A detailed specification of the experimental setup can be found in the literature.33,34 Calibration of the system was carried out by determining the hyperpolarizability of crystal violet in methanol using the internal reference method.16,34,35 Results and Discussion Figure 1 shows the absorbance spectra of the investigated ZnO nanoparticles in ethanolic solution at different times during Ostwald ripening. Agglomeration of the particles did not occur under the chosen experimental conditions, as confirmed by the presence of many isolated particles in TEM images (see inset
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Figure 2. Comparison between particle size distributions of ZnO nanoparticles after 890 min of aging determined by image analysis (bars, 221 particles) and TBM (solid line), the inset shows the TEMmicrograph used for the analysis.
of Figure 2) as well as the absence of a scattering component in the absorbance (here, equivalent to extinction) spectra. The absorption edge is red-shifted with increasing particle size as a consequence of the quantum size effect.4 The strong discretization of the band structure in very small semiconducting particles results in an increasing band gap and, thus, in sizedependent absorption behavior. This can be utilized to calculate the particle size distribution from the absorbance spectra, applying the correlation between characteristic absorbance wavelengths and particle sizes on the basis of the tight binding model (TBM) of Viswanatha.4,36 Figure 2 illustrates the accuracy of this approach. The calculated particle size distribution (solid lines) is in good agreement with the image analysis of TEM micrographs (bars) of particles examined after 890 min of aging. The hyperpolarizability, βZnO, was determined by measuring the quadratic intensity dependence of the signal with time according to16,25
I2ω ) GfIω2(Nsolventβsolvent2 + Nsoluteβsolute2)
(1)
where Iω is the intensity of the incident light, I2ω is the intensity of the HRS signal, Ni is the number density of species i, βi is the corresponding hyperpolarizability, and Gf is a geometrical factor taking into account the instrumental conditions and local field effects. Calibration of the system was carried out by determining the hyperpolarizability of crystal violet in methanol using the internal reference method.16,35 An extinction correction was not necessary because there was negligible absorption or scattering at the fundamental or harmonic frequency, respectively. In the case of particulate systems, the number of molecules per particle, cp, has to be considered leading to an expression including the particle number density, NP, and the particulate hyperpolarizability, βZnO:
(
I2ω ) GfIω2 NEtOHβEtOH2 +
NP β 2 cP ZnO
)
(2)
The value of βEtOH for ethanol has been determined to be 0.32 × 10-30 esu by applying the external reference method.34 Knowing this, the ripening behavior could then be investigated. For an appropriate interpretation of the measured HRS data, the size dependence of the hyperpolarizability, βZnO, has to be known. Therefore, the reactant solutions were mixed, and the intensity dependence of the HRS signal at 293 K was recorded
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Figure 3. Relationship between hyperpolarizability, βZnO, and particle size, x; evolution of size (inset, b) and number concentration (inset, 9) during ripening.
with an automated measuring setup every 12 min. Additionally, the absorbance spectra were monitored simultaneously at the same temperature. Small influences of the laser beam on the sample due to local heating in the focus point cannot be excluded completely, but the effects are believed to be negligible because the pulse duration of 80 fs is short. If this were not the case, it would be expected that the final particles would be larger and would rapidly become unstable due to agglomeration known to occur during ripening of ZnO quantum dots at elevated temperatures. The mean particle size as a function of time was calculated from the absorbance spectra as described above. Since reactant conversion is assumed to be complete, the mass concentration of the particles in the suspension is known and, assuming spherical particles, the mean number concentration of the particles can be determined. The inset of Figure 3 shows the evolution of the particle number concentration, NP(t), calculated from the mean volume weighted particle size x1,3(t). The solid lines represent fits (by power functions) to the data, which were performed to give a mathematical expression for the quantities required for the calculation of βZnO(x, t). This results in a correlation between hyperpolarizability βZnO and particle diameter x (see Figure 3). An increase in particle size leads to an increase in βZnO fitted by a power law (R2 ) 0.95) according to
( nmx )
βZnO(x) ) 2 × 10-28esu
2.8
( nmx )
)a
2.8
(3)
The fact that the exponent approaches the value of 3 is a strong indication that the measured intensities are predominantly volume signals. The deviation from an integer value is believed to be due to the structure and intermolecular interactions of the particles. This agrees with the results of Jacobsohn, who showed that the HRS signal is the sum of a bulk and a surface contribution.29 Knowing the relationship between βZnO and x for the observed particle system, a time-resolved HRS measurement was applied directly to investigate the rapid solid formation process. Figure 4 shows the temporal evolution of the HRS signal for three different time resolutions of 20, 50, and 100 ms, respectively. In region I, a constant solvent signal is observed because the zinc acetate reactant solution gives approximately the same signal intensities as the solvent: I 0 I2ω (t) ) GfIω2NEtOHβEtOH2 ) I2ω
(4)
Figure 4. HRS signals during the early stages of precipitation monitored for three different time resolutions (circles, 100 ms; squares, 50 ms; crosses, 20 ms). Four different regions can be discerned: (I) only one reactant, (II) nucleation, (III) growth, (IV) ripening. Note that the absence of data points (100 ms resolution) between 11 and 13 s was due to drop-out of the laser.
After 10 s, an equimolar amount of lithium hydroxide solution (which also has HRS signal intensities similar to the pure solvent) was added to the sample cuvette by a computercontrolled dosing unit. Due to the low solubility of zinc oxide in ethanol, the local supersaturation at this stage is extremely high; thus, nucleation is expected to dominate in comparison to growth processes. Region III is related to particle growth in the supersaturated solution, leading to a signal increase with increasing particle diameter. As the supersaturation is continuously reduced, particle growth slows down, and on reaching a sufficiently low supersaturation level, the smallest particles are able to dissolve due to their high surface energy. This so-called Ostwald ripening occurs in region IV, and the overall signal decreases due to a reduction in particle number. The transition between the dashed regions of nucleation and growth in Figure 4 is gradual and cannot be determined accurately without applying a second measurement method, such as spectrophotometry, to obtain absorbance spectra simultaneously. Nevertheless, we have demonstrated the capability of HRS to study precipitation processes, including nucleation, growth, and ripening, due to its high time resolution and sensitivity toward particle size and concentration. To quantify the kinetics of the particle formation process shown in Figure 4, we assume spherical particles of a mean particle diameter x(t) and a particle number concentration NP(t) (see eq 2). The time-dependent signal is obtained by combining eqs 2-4 and by introducing the molar mass, M, and the density, F:
I2ω(t) ) Gf Iω2 NP(t)
a2M 0 x(t)2.6 + I2ω π FN 6 A
(5)
Equation 5 can be differentiated to determine the nucleation rate, B; the growth rate, G; and the ripening rate, R, with time. Here, the nucleation rate is defined as the derivative of the number of particles with respect to time; the growth and ripening rates are the derivative of the particle diameter with respect to time. The difference between the latter two parameters is that growth takes place at global supersaturations, S > 1 leading to a net mass transfer between liquid and solid, whereas ripening is an equilibrium process with a constant amount of ZnO in the solid state. In our analysis, we first determine the ripening rate, assuming the incorporation of all available molecular ZnO into particles during region IV (cZnO ) c0). Thus, the particle number
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density can be related to the particle size (see the Appendix), and the derivative of the particle size with respect to time (i.e., the ripening rate) is obtained by differentiating the nonlinear signals (see eq 6). In addition, the growth and nucleation rates are determined, provided the following assumptions, which allow the two processes to be differentiated, are made: during growth (region III) the particle number density, NP, is set constant and equal to the particle number density at the beginning of ripening (region IV). Furthermore, the particle diameter, x, during nucleation (region II) is assumed to be constant, corresponding to the particle diameter at the beginning of growth. This approach leads to the following expressions: Figure 5. Growth (circles) and ripening rates (stars) during the first 30 s of the precipitation process determined by eqs 6 and 7. The inset shows the concentration decrease in dissolved ZnO or the ZnO precursor, respectively.
dx )R dt
( )
0 -5/2 c0Gf MaIω d((I2ω(t) - I2ω ) ) NA π dt F 6 0 -5/2 d((I2ω(t) - I2ω ) ) ) cR dt 5
)
(6)
dx )G dt III-IV 0 -75/26 - I2ω × ) (I2ω )
( )
5
0 5/13 c0Gf MaIω d((I2ω(t) - I2ω ) ) NA π dt F 6 0 5/13 d((I2ω(t) - I2ω ) ) ) cG dt
B) )
dNP dt
( π6 MF )
14
×
II-III 0 - I2ω (I2ω )(Iωa)15 III-IV 0 - I2ω (I2ω )NA 15
(
) cB
(7)
Gf 0 d(I2ω(t) - I2ω ) dt
)
c0
-13
Possibly, the underlying mechanisms may be more complex. According to Spanhel and Anderson,3,37 nucleation and growth (and ripening) may follow a cluster-cluster aggregation process. At first, Zn4O(Ac)6 tetrahedrons form, which then aggregate through a series of magic clusters. For the time, a quantitative model for this detailed growth chemistry does not exist. Therefore, we will rely upon classical theory to show the strength of our method, which will be used in the future to develop better and more detailed models of the early stages of particle formation. Thus, knowing the kinetic data, it is possible to calculate the absolute concentration decrease in ZnO molecules, cZnO,liquid, that are not incorporated into particles with time inside the suspension, as shown in the inset of Figure 5. To validate our approach, we derived the surface energy, γZnO, from the measured data to prove if the calculation gives reasonable values. Assuming diffusion-controlled growth, it is possible to determine the growth (xjgrowth) ) 2.2 × 10-7 M from the growth rates. solubility cZnO We are aware that this value depends on the particle diameter, but this will be neglected for a first approximation. Because flat ) 4.9 × 10-13 M, we know the equilibrium concentration, cZnO from long-term ripening experiments,36 the surface energy during the growth phase can be calculated via Kelvin’s equation:
0 d(I2ω(t) - I2ω ) dt
γ)
(8)
The detailed derivation of these expressions is found in the Appendix. Our approach is based on reasonable physical assumptions and does not assume any nucleation mechanism. The equations are applicable for cases in which a thermodynamic nucleation barrier does exist (classical nucleation theory) and for cases when this barrier does not exist. In the latter, the formation of stable nuclei is controlled by reaction kinetics. The constants determined for the different regions are cB ) 2.16 × 1021 m-3, cG ) 1.77 × 10-11 m, and cR ) 1.76 m. The nucleation rate, B, is calculated as a constant mean value of 3.35 × 1025 m-3 s-1 for the first few seconds. The results of growth and ripening rates are shown in Figure 5. Following nucleation, the particle diameter grows at a rate of 1.35 × 10-10 m s-1, leading to a reduction in the supersaturation. Consequently, the mean growth rate decreases until supersaturation is reduced sufficiently low, whereupon Ostwald ripening commences.
growth cZnO jx · kBT · ν · ln flat 4Vm cZnO
(9)
where γ is the surface energy, x is the particle diameter, kB is Boltzmann’s constant, T is the temperature, Vm is the molar volume,38 and ν is the dissociation number of ZnO. The surface energy was calculated to be 0.47 J m-2, which compares reasonably to literature values38,39 and shows the great potential of our approach to determine fundamental, interfacial properties. Conclusion In conclusion, HRS of ZnO nanoparticles was measured, and the size dependency of the hyperpolarizability, βZnO, in ethanol was obtained quantitatively by measuring a slowly ripening ZnO suspension. An increasing power law was determined, indicating a dominating volume contribution to the overall HRS signal. Furthermore, it was shown that HRS provides a spectroscopic tool to resolve fast precipitation processes, including nucleation and growth. Different stages of particle evolution could be monitored by time-resolved HRS down to the range of milliseconds, and a preliminary approach for the quantitative
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determination of the nucleation, growth and ripening rates has been performed. Out of this data, it is possible to determine realistic size-dependent surface energies that are experimentally otherwise inaccessible. Our work demonstrates the great potential of HRS to study transient and highly dynamic nanoparticulate systems with relatively small experimental effort compared to small angle scattering techniques. In this technique, data are evaluated according to classical homogeneous nucleation, growth, and Ostwald ripening models. Our evaluation satisfies mass balances, is based on the observed absorbance spectra and the calculated PSDs, and allows us to extract quantitative kinetic and thermodynamic data so far inaccessible. Appendix The overall intensity of the nonlinear signal I2ω(t) is given by
I2ω(t) ) Gf Iω2 NP(t)
βZnO2(t) 0 + I2ω cP(t)
Ripening Rate. For the derivation of the ripening rate, it is assumed that all the provided Zn2+ is converted to ZnO and that all ZnO molecules are incorporated into particles:
cZnO ) c0 ) const,
conversion ) complete
The ripening rate is formulated as the change in particle diameter with time:
dx dt
R)
The number density of particles during ripening is expressed by the particle diameter, assuming a constant concentration of ZnO in the solid phase (during ripening, the molecules of the dissolving particles are incorporated into larger crystals; thus, the solid concentration is kept constant).
(10) NP(t) )
Thereby is Gf a geometrical factor, Iω the intensity of the incident light, NP the number density of the particles, βZnO the first order hyporpolarizability of the ZnO particles, cP the number of ZnO 0 the nonlinear signal from molecules per ZnO particle, and I2ω the solvent. We assume that the particle size can be representated by the diameter, x, of an equivalent sphere of similar properties as the ZnO particles. From long-term ripening experiments, the dependency between hyperpolarizability, βZnO, and particle size x is known:
c0M
(14)
π F x(t)3 6
Inserting eq 14 into 13, the resulting eq 15 depends only on the particle diameter, x: 0 I2ω(t) - I2ω ) GfIω2
c0M
a2M x(t)2.6 π π F x(t)3 FNA 6 6
(15)
Solving eq 15 for x yields
βZnO(t) ) ax(t)2.8
Thereby is a a constant (2 × 10-28 esu) and x the particle diameter. The number of molecules in a particle can be expressed by the concentration of ZnO molecules in the solid phase, cZnO, and the particle diameter, x:
π NA FZnO x(t)3 Nmolecule cZnONA 6 cP(t) ) ) ) Nparticle cZnOM M π F x(t)3 6
(12)
Thereby is Nmolecule the number of ZnO molecules inside the sample volume, Nparticle the number of ZnO particles inside the sample volume, cZnO the molar concentration of ZnO, NA the Avogadro number, M the molar mass, and F the density of ZnO. Inserting eqs 11 and 12 into eq 10, an expression for the nonlinear signal is derived that contains only the particle number density and the particle diameter as variables:
a2 (x(t)2.8)2 M π NA F x(t)3 6 2 aM ) GfIω2 NP(t) x(t)2.6 π FN 6 A
(
(11)
c0M2a2GfIω2 x) 2 0 (I2ω(t) - I2ω ) π6 F NA
( )
)
5/2
(16)
Building the derivative,
dx )R) dt
( ) c0Gf MaIω NA π F 6
5
0 -5/2 d((I2ω(t) - I2ω ) ) dt
(17)
Growth Rate. For the derivation of the growth rate, it is assumed that the number density of particles during the growth phase (III) is constant and equal to the number density at the beginning of the ripening phase (IV):
NPgrowth ) const ) NPIII-IV The growth rate is formulated as the change in particle diameter of the equivalent spherical particle with time:
0 I2ω(t) - I2ω ) GfIω2 NP(t)
G)
dx dt
(13) The signal at the transition between growth and ripening is expressed according to eq 18,
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Segets et al.
a2(xIII-IV)2.6 M π FN 6 A
(18)
Nucleation Rate. The critical particle diameter during nucleation (II) is constant and equal to the particle diameter at the beginning of the growth phase (III): xnuc ) xII-III ) const. The nucleation rate is formulated as the change in the number density of particles with time:
which directly leads to
NPIII-IV
B)
π - ) FNA ( 6 ) GfIω2a2(xIII-IV)2.6 M III-IV I2ω
0 I2ω
(19) Expressing xII-III with eq 25,
Inserting eq 19 into eq 13 yields
0 I2ω(t) - I2ω
III-IV 0 -75/26 xII-III ) (I2ω - I2ω )
III-IV 0 π (I2ω - I2ω ) FNA 2 6 aM 2 ) GfIω x(t)2.6 2 2 III-IV 2.6 π GfIω a M(x ) FN 6 A (20)
I2ω(t) -
0 I2ω
)
III-IV 0 - I2ω (I2ω ) x(t)2.6
(21)
(xIII-IV)2.6
)
x
(
c0Gf NA
)
III-IV 0 π - I2ω ) F √(I2ω 6
0 I2ω(t) - I2ω ) GfIω2 NP(t)
(
III-IV 0 -75/26 - I2ω (I2ω )
NP(t) ) π F 14 6M II-III 0 (I2ω - I2ω )(Iωa)15
III-IV 0 III-IV 0 5/2 - I2ω - I2ω (I2ω ) x(t)2.6 NA5/2 π6 F (I2ω ) 5
( ( )
)
2.6
0 (I2ω(t) - I2ω )
x(t) )
(
III-IV I2ω
-
(
-
0 -75/26 I2ω
)
)
( )( 6.5
( ) c0Gf MaIω NA π F 6
MaIω13 π F 6
B)
)
13
(24)
5
(I2ω(t) -
0 5/13 I2ω
)
(25)
Building the derivative,
dx III-IV 0 -75/26 - I2ω × ) G ) (I2ω ) dt
( ) c0Gf MaIω NA π F 6
5
a2M × π FNA 6 c0Gf MaIω NA π F 6
( )
)
5
2.6
II-III 0 5/13 - I2ω (I2ω )
(28)
(
III-IV 0 - I2ω (I2ω )NA
Gf
)
15
c0-13 ×
(29)
Building the derivative,
Solving eq 23 for x yields
0 7.5 I2ω
(27)
(23)
(c05/2Gf5/2M5a5Iω5)2.6
III-IV I2ω
0 5/13 I2ω )
Solving eq 28 for NP yields
( )
c0Gf NA
II-III (I2ω
(22)
0 I2ω(t) - I2ω )
0 I2ω(t) - I2ω
)
5
5
MaIω
Inserting eq 22 into eq 21 yields
x(t)13/5 )
(
c0Gf MaIω NA π F 6
Inserting eq 27 into eq 13 yields
The unknown xIII-IV is expressed via the measured nonlinear III-IV by eq 16. signal, I2ω
III-IV
dNP dt
0 5/13 d((I2ω(t) - I2ω ) ) dt
(26)
)
dNP dt
( π6 MF )
14
II-III 0 (I2ω - I2ω )(Iωa)15
0 ) d(I2ω(t) - I2ω dt
(
III-IV 0 (I2ω - I2ω )NA Gf
)
15
c0-13 ×
(30)
Acknowledgment. The authors thank the German Research Council (DFG) for their financial support (Projects PE 427/ 11-2 and PE 427/18-2) and for the support within the framework of its Excellence Initiative for the Cluster of Excellence “Engineering of Advanced Materials” (www.eam.uni-erlangen.de) at the University of Erlangen-Nuremberg. We thank Prof. R. N. Klupp Taylor for HRTEM measurements and assistance with preparing the manuscript. References and Notes (1) Cole, J. J.; Wang, X.; Knuesel, R. J.; Jacobs, H. O. Nano Lett. 2008, 8 (5), 1477.
Hyper-Rayleigh Scattering of ZnO Nanoparticles (2) Klingshirn, C. Phys. Status Solidi B 2007, 244 (9), 3027. (3) Spanhel, L. J. Sol-Gel Sci. Technol. 2006, 39, 7. (4) Viswanatha, R.; Sapra, S.; Satpati, B.; Satyam, P. V.; Dev, B. N.; Sarma, D. D. J. Mater. Chem. 2004, 14 (4), 661. (5) Davey, R. J.; Allen, K.; Blagden, N.; Cross, W. I.; Liebermann, H. F.; Quayle, M. J.; Righini, S.; Seton, L.; Tiddy, G. J. T. CrystEngComm 2002, 4 (47), 257. (6) Talanquer, V.; Oxtoby, D. W. J. Chem. Phys. 1998, 109 (1), 223. (7) Yin, Y.; Alivisatos, A. P. Nature 2005, 437 (7059), 664. (8) Davey, R. J.; Blagden, N.; Righini, S.; Alison, H.; Quayle, M. J.; Fuller, S. Cryst. Growth Des. 2001, 1 (1), 59. (9) Abe´cassis, B.; Testard, F.; Spalla, O.; Barboux, P. Nano Lett. 2007, 7 (6), 1723. (10) Qu, L.; Yu, W. W.; Peng, X. Nano Lett. 2004, 4 (3), 465. (11) Tuinenga, C.; Jasinski, J.; Iwamoto, T.; Chikan, V. ACS Nano 2008, 2 (7), 1411. (12) Roke, S.; Berg, O.; Buitenhuis, J.; von Blaaderen, A.; Bonn, M. Proc. Natl. Acad. Sci. U.S.A. 2006, 103 (36), 13311. (13) Verbiest, T.; Kauranen, M.; Persoons, A. J. Chem. Phys. 1994, 101 (2), 1745. (14) Terhune, R. W.; Maker, P. D.; Savage, C. M. Phys. ReV. Lett. 1965, 14 (17), 681. (15) Bersohn, R.; Pao, Y.-H.; Frisch, H. L. J. Chem. Phys. 1966, 45 (9), 3184. (16) Clays, K.; Persoons, A. Phys. ReV. Lett. 1991, 66 (23), 2980. (17) Johnson, R. C.; Hupp, J. T. Spectrum 2000, 13 (2), 1. (18) Eisenthal, K. B. Chem. ReV. 2006, 106 (4), 1462. (19) Russier-Antoine, I.; Jonin, C.; Nappa, J.; Benichou, E.; Brevet, P. J. Chem. Phys. 2004, 120, 10748. (20) Rouillat, M.-H.; Russier-Antoine, I.; Benichou, E.; Brevet, P. F., Anal. Sci. 2001, 17, Supplement, i235. (21) Galetto, P.; Brevet, P.-F.; Girault, H. H.; Antoine, R.; Broyer, M. J. Phys. Chem. B 1999, 103 (41), 8706. (22) Russier-Antoine, I.; Huang, J.; Benichou, E.; Bachelier, G.; Jonin, C.; Brevet, P.-F. Chem. Phys. Lett. 2008, 450 (4-6), 345.
J. Phys. Chem. C, Vol. 113, No. 28, 2009 12001 (23) Dannenberger, O.; Buck, M.; Grunze, M. J. Phys. Chem. 1999, 103 (12), 2202. (24) Clark, H. A.; Campagnola, P. J.; Wuskell, J. P.; Lewis, A.; Loew, L. M. J. Am. Chem. Soc. 2000, 122, 10234. (25) Johnson, R. C.; Li, J.; Hupp, J. T.; Schatz, G. C. Chem. Phys. Lett. 2002, 356 (5-6), 534. (26) Chandra, M.; Indi, S. S.; Das, P. K. J. Phys. Chem. C 2007, 111 (28), 10652. (27) Zhang, Y.; Ma, M.; Wang, X.; Fu, D.; Zhang, H.; Gu, N.; Liu, J.; Lu, Z.; Xu, L.; Chen, K. Inorg. Chem. Commun. 2003, 6 (4), 427. (28) Santos, B. S.; Pereira, G. A. L.; Petrov, D. V.; de Mello Donega, C. Opt. Commun. 2000, 178 (1), 187. (29) Jacobsohn, M.; Banin, U. J. Phys. Chem. B 2000, 104 (1), 1. (30) Petrov, D. V.; Santos, B. S.; Pereira, G. A. L.; de Mello Donega, C. J. Phys. Chem. B 2002, 106 (21), 5325. (31) Sahyun, M. R. V. Spectrochim. Acta, Part A 2002, 58 (14), 3149. (32) D’Arrigo, M. C.; Cruickshank, F. R.; Pugh, D.; Sherwood, J. N.; Wallis, J. D.; Mackenzie, C.; Hayward, D. Phys. Chem. Chem. Phys. 2006, 8 (32), 3761. (33) Schneider, L.; Peukert, W. Part. Part. Syst. Charact. 2006, 23 (5), 351. (34) Tomalino, L. M.; Voronov, A.; Kohut, A.; Peukert, W. J. Phys. Chem. B 2008, 112 (20), 6338. (35) Clays, K.; Persoons, A. ReV. Sci. Instrum. 1994, 65 (7), 2190. (36) Segets, D.; Gradl, J.; Klupp Taylor, R.; Vassilev, V.; Peukert, W., ACS Nano. [online early access]. DOI: 10.1021/nn900293d. (37) Spanhel, L.; Anderson, M. A. J. Am. Chem. Soc. 1991, 113 (8), 2826. (38) Hu, Z.; Oskam, G.; Searson, P. C. J. Colloid Interface Sci. 2003, 263 (2), 454. (39) Zangwill, A. Physics at Surfaces; Cambridge University Press: Cambridge, 1988.
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