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Characterizing Gold Nanorods in Solution Using Depolarized Dynamic Light Scattering Michael Glidden and Martin Muschol* Department of Physics, University of South Florida, Tampa, Florida 33620, United States ABSTRACT: The ability to custom-tailor surface plasmon resonances of metallic nanorods via their geometry has made nanorods a prominent platform for the design of novel nanomaterials with wide-ranging potential applications. Characterization of nanorod geometries within their native solution environments is essential for targeting nanorod properties to the specific application and for improved monitoring of nanorod synthesis. Here we use a custom-designed depolarized dynamic light scattering setup for measuring nanorod translational and rotational diffusion in situ. Using either straight cylinders or prolate ellipsoids as model geometries, we developed an approach to convert diffusion measurements directly into predictions of nanorod lengths and aspect ratios. Neither of these two geometrical models was able to properly reproduce real nanorod diffusion. Yet, these discrepancies mostly precluded reliable predictions of nanorod aspect ratios while derived nanorod lengths were within 10−20% of their values determined separately from transmission electron microscopy.

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INTRODUCTION Many metallic and semiconducting materials develop intriguing optical, electrical, and magnetic properties at the nanoscale, making them prime targets for the development of nanomaterials with novel functionalities. Among the potential applications of nanorods are novel surface coatings,1 including superhydrophobic coatings,2 electrical and magnetic nanowires,3,4 and their use as optical metamaterials for long-distance imaging with near-field resolution.5 Interest in the pronounced surface plasmon resonances of metallic nanoparticles reaches back to classical work by Faraday6 and Mie.7 Selecting the length or aspect ratio of metallic nanorods provides a particularly straightforward method for modulating their prominent optical absorption bands.8 Their near-infrared absorption peaks are particularly favorable for biomedical applications (for a recent review, see ref 9). The observation that anisotropic nanorods are more readily taken up into cells than their spherical counterparts has provided additional impetus for development of biomedical applications.10 The reliable characterization of the physical and optical properties of nanorods, ideally in their natural solution environment, is critical for evaluating their performance in virtually all of the above-mentioned applications. Commonly, the optical absorption spectrum of metallic nanorods itself and, in particular, the peak wavelength of the longitudinal surface plasmon resonance (LSPR), is used to derive aspect ratios for gold nanorods. Yet, the relation between theoretically predicted and experimentally observed absorption spectra for metallic nanorods remains controversial. Counter to long-held assumptions, recent theoretical and experimental efforts indicate that the absorption peak of the surface plasmon resonance is not uniquely determined by the geometrical aspect ratio (length/diameter) of the nanorods.11,12 In addition, the © 2012 American Chemical Society

optical absorption spectrum does not provide a direct measure of the heterogeneity of sizes and shapes often encountered in practice. Electron microscopy does provide an unbiased measure of particle morphologies and size distributions, though it does not yield information about growth kinetics. In addition, it does not indicate whether surfactants, commonly used during synthesis or for maintaining solubility of nanoparticles, adsorb onto nanorod surfaces and whether nanorods remain dispersed or begin to interact and/or aggregate in suspension. Dynamic light scattering (DLS) measurements, in contrast, permit the determination of particles sizes and their distributions in situ. More specifically, the presence of a depolarized component to the light scattering signal allows assessment of both the translational and rotational diffusion of gold nanorods. As a result, depolarized dynamic light scattering can be used not just to obtain the hydrodynamic radius of the equivalent sphere but to determine the actual length and diameter of diffusing nanorods. Hence, depolarized dynamic light scattering can in principle yield both the lengths and aspect ratios of rodlike particles, can provide information about particle polydispersity, can detect potential surface adsorption of any cosolutes, and can indicate whether rods remain soluble. DLS also permits one to monitor the growth kinetics of gold nanorods in situ, thereby providing valuable insights into the growth process itself.13,14 Yet, wider application of depolarized DLS for characterizing nonspherical nanoparticles has been stunted by the intrinsic challenges of measuring the typically weak depolarized scattering signals and by the difficulties of Received: November 30, 2011 Revised: March 6, 2012 Published: March 12, 2012 8128

dx.doi.org/10.1021/jp211533d | J. Phys. Chem. C 2012, 116, 8128−8137

The Journal of Physical Chemistry C

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polarized (v-v) or horizontally depolarized (v-h) scattered light. An adjustable iris diaphragm (A2) selects the solid angle of scattered light imaged onto the photon detector, thereby determining the number of coherence areas that are contributing to the signal. Lens L2 (f = 81 mm) focuses the scattered light onto the input of a multimode optical fiber (diameter =100 μm) coupled to a Perkin-Elmer SPCM-AQR13-FC avalanche photodiode (APD). This APD has a typical dark-count rate of 250 cps and a dead-time of 50 ns. The transistor-to-transistor logic (TTL) output pulses from the APD (>2.5 V, 35 ns wide) are fed into a commercial counter/ timer board (USB-6210, National Instruments) that is connected via a USB interface to a personal computer. Data acquisition and processing are performed using custom functions written for Igor Pro data acquisition and processing software (Wavemetrics, Lake Oswego, OR). Photon Counting and Processing. Photon counting and processing utilizes a time-tagged time-resolved (TTTR) approach commonly used for low-intensity fluorescence lifetime measurements.15 To obtain the TTTR photon count history the TTL output from the APD photon detector is used as the gate input for the internal clock of the counter/timer board running either at 20 or 80 MHz. The raw TTTR data are streamed and stored directly to a computer hard disk for subsequent off-line processing. To avoid data overflow, individual measurement segments were limited to a total of 5 × 105 photon arrivals before data were stored in individual files on the computer hard disk. Extended measurement durations were obtained by simply concatenating individual measurement segments. Details regarding TTTR photon counts and efficient computational algorithms for calculating correlation functions from raw TTTR photon count histories have been described by Wahl et al.15 All software routines for controlling data acquisition and for generating correlation functions from the raw TTTR photon counts were written using the programming language provided by the data processing program Igor (Wavemetrics). The algorithm used to generate the autocorrelation function from TTTR labeled photon counts followed the approach outlined by Wahl et al.15 Gold Nanorod Samples and DDLS Measurements. Gold (Au) nanorods with nominal dimensions of 45 nm length ×10 nm diameter (prod. #30-10-850, lot R01141B1) and 135 nm length × 20 nm diameter (prod. #30-HAR-1064, lot #RPB049 and prod. #30-HAR-1000, lot #HAR 95) were purchased from Nanopartz, Inc. (Loveland, OH). Au rod dimensions were chosen so that, in principle, the intrinsic absorption of the Au nanorods was near a minimum at the laser wavelength of λ = 633 nm used for light scattering experiments (see Figure 3). Subsequently, the three different samples are referred to by the respective peaks in their absorption spectra (Figure 3) as Au 823 (45 × 10 nm sample), Au 855 (135 × 20 nm, 30 HAR 1064), and Au 923 (135 × 20 nm, 30-HAR 1000). For DDLS measurements, Au stock solutions were diluted 3− 6-fold into deionized water containing 0.1% of the stabilizing detergent cetyl trimethylammonium bromide (CTAB) as stabilizing agent (Fisher Scientific, Pittsburgh, PA). The solvent was filtered through 0.22 μm PTFE syringe filters before mixing with the Au nanorods. Solutions were centrifuged at 1400 rpm and 30 °C for 15 min to remove any undissolved aggregates or trapped air bubbles. Relative Au rod concentrations for solutions were characterized using sample absorbance deter-

inverting experimental diffusivities into direct predictions of nanorod dimensions. In the following, we report on our efforts to ameliorate both of these limitations. To improve hardware sensitivity, we have built a custom-designed light scattering setup using a timetagged time-resolved (TTTR) photon collection and processing scheme. In addition, we propose a novel approach toward analyzing the experimental translational and rotational diffusivities and comparing them with currently available theoretical predictions for either straight cylinders or prolate ellipsoids. We applied this approach to determine the length and aspect ratios of three commercial samples of gold nanorods. Our novel approach toward data analysis provided two intriguing results. First, independent of the geometrical model used to analyze the data, we can extract the length of our nanorod samples to within 10−20% of their bare dimensions seen in electron micrographs. At the same time, the analysis suggests that the values of nanorod aspect ratios are much more sensitive to the geometrical model used to predict their values and that neither of these simplified geometrical models can properly reproduce them.

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EXPERIMENTAL METHODS Optical Design of the Depolarized Dynamic Light Scattering Setup. Figure 1 gives an overview of the hardware

Figure 1. Optical design of the depolarized dynamic light scattering setup. The beam of a HeNe laser (633 nm) is passed through a GlanThompson prism (GT1) and focused with a lens (L1) onto a thermostated sample cuvette. Scattered light is collected at 90°, with its polarization selected by another Glan-Thompson prism (GT2). The spatial angle of scattered light admitted to the detector is set by the adjustable aperture A2. Lens L2 focuses the scattered light onto an avalanche photodiode detector (APD). The output pulses from the APD are fed into a counter-timer board and streamed to the hard disk of a personal computer.

setup for the custom-built depolarized dynamic light scattering instrument. The optical design closely matches that for standard dynamic light scattering setups using a 90° scattering geometry. The beam from a 35 mW single-mode HeNe laser at λ = 633 nm (JDS Uniphase, model 1145P) passes through a Glan-Thompson prism (GT1), increasing the polarization purity to a nominal ratio of 100 000:1. The polarized beam is then focused onto the sample chamber using a lens (L1) with f = 100 mm, which results in a beam waist of approximately 80 μm diameter at the center of the sample cuvette. Aperture A1 prevents stray light from entering the focusing lens L1. The cuvette holder is a commercial unit with a Peltier temperature controller (TLC-50, Quantum Northwest). Scattered light is collected at 90° relative to the incoming beam. A second GlanThompson prism (GT2) is used to select either vertically 8129



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Figure 2. Bursts in raw photon counts and effects on the correlation function. (A) Photon count history for an Au nanorod solution. Prominent bursts in the count rate most likely due to residual air bubbles are readily apparent. Correlation functions for the photon count in part A calculated either (B) prior to or (C) after removing bursts from the raw photon count.

mined at λ = 633 nm with a UV-1 spectrophotometer (ThermoElectron Corp.). DDLS and DLS Measurements of Au Samples. For dynamic light scattering measurements, Au solutions were transferred to clean 10 mm square glass cuvettes and placed in a thermostated sample holder. DDLS measurements were performed using the custom-built 90° scattering setup, which yields a scattering vector q of 1.87 × 107 m−1 or 1/(53.6 nm) for n = 1.33, λ = 633 nm. Polarized (v-v) and depolarized (v-h) TTTR photon counts were accumulated for durations ranging from 90 s (v-v) to 20 min (v-h), using collection apertures of 1 mm (v-v) or 2 mm (v-h), respectively. The corresponding correlation functions were calculated using custom programs written in a C-language hybrid (Igor programming language). In addition, a commercial DLS light scattering unit (Nanosizer S, Malvern Instruments) using a backscattering geometry (θ = 173°) was used to derive distributions of relaxation rates and to collect data at a different scattering wave vector [q = 2.68 × 107 m−1 or 1/(37.9 nm) for n = 1.33, λ = 633 nm]. All measurements were performed at 30 °C to maintain CTAB solubility. Optical Absorbance and TEM Measurements. All optical absorbance measurements were performed using a UV-1 spectrophotometer from Thermo Electron Corp. Au nanorod stock solutions were diluted into 0.1% CTAB/water solvent solutions until absorbance was less than 0.5. All absorbance readings were corrected for any background contributions from the solvent. Beyond 1050 nm, spectral readings were not very reliable. Transmission electron microscopy measurements were performed at the USF Nanotechnology and Research Center using a Tecnai T20 G2 transmission electron microscope run at

200 kV ac. Ten mictoliters of 10−100-fold diluted Au samples were deposited on sample grids, dried with dry nitrogen, and then introduced into the vacuum chamber for imaging.

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RESULTS Custom Depolarized Dynamic Light Scattering Setup Using TTTR Photon Counting. One of the practical challenges faced when using depolarized dynamic light scattering (DDLS) is the notoriously low photon count rates associated with depolarized light scattering signals. Using Gaussian statistics to approximate photon counting noise (S/ N ∝ 1/√N), we can estimate that about N = 400 photons are needed to achieve a modest 5% noise level for a given decay channel. For the comparatively strongly scattering Au nanorods used in our experiments, typical depolarized count rates are around f N = 104 cps (see Figure 2A). For rotational decay rates Γ as high as 100 kHz, the system only accumulates f N/Γ = 0.1 cps in that decay channel. To reach the modest 5% S/N level, signal needs to be accumulated for 400/0.1 s ≈ 1 h. Hence, efficient processing and counting of scattered photons is essential. To address this concern, we present a custom DDLS setup using time-tagged time-resolved photon counting. Figure 1 gives an overview of the optical and hardware arrangements of the apparatus used in our measurements. With additional details provided under Experimental Methods, the optical design closely follows typical setups for 90° light scattering. However, photon counting and processing departs in two significant ways from traditional correlator hardware. First, instead of counting the number of photons arriving within fixed time bins, the temporal sequence of photon arrival times is labeled using the TTTR scheme commonly used for low8130



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intensity fluorescence lifetime measurements.15,16 To obtain the TTTR photon count history the TTL output from an avalanche photodiode is used as the gate input to the internal clock of a counter/timer board (USB-6210, National Instruments) with either 20 or 80 MHz resolution. Hence, photon counts are recorded as a series of delay times between subsequent photon events. This dramatically diminishes the large overhead of empty time bins common for time-binned data at low count rates and, therefore, the acquisition times required to accumulate DDLS correlation functions. As a result, the corresponding demands on bus speeds for data transfer and on space for data storage, as well as the computational time required to calculate the correlation function, are reduced. The second difference in photon count processing is that the raw TTTR data are subjected to preprocessing prior to calculation of the intensity correlation functions. One immediate advantage of acquiring the raw photon count history is the ability to eliminate photon bursts due to large aggregates or air bubbles which otherwise distort the correlation functions derived from weak v-h signals. Again, given the extended acquisition times needed for such measurements, even weak contamination by dust and air bubbles will distort correlation functions during acquisition periods as short as 1 min. This is shown in Figure 2A for a short photon count history from a gold nanorod sample. Even after filtration and centrifugation, the sample displayed significant burst events. The bursts are likely due to air bubbles, the formation of which is facilitated by the presence of the CTAB surfactant required to keep Au nanorods in suspension. As indicated in Figure 2B, these burst events induce long, slow relaxation components and distort the baseline in the corresponding correlation function. We removed these effects by restricting the deviations from the mean count rate in the photon count history to approximately three standard deviations and by concatenating the resulting disjointed segments of the raw TTTR vector of arrival times. For the bimodal decays we typically encountered, the effect of bursts on the fast relaxation component was quite modest (221 vs 226 kHz), but upon removal of spurious counts, the slow relaxation component decreased from 5.4 to 4.1 kHz and the correlation function approached a well-defined baseline (Figure 2C). For weak depolarized light scattering signals, these manipulations of count rate histories are much more efficiently performed using TTTR photon counts. One added advantage of this custom setup is the control over the solid angle used to collect the weak v-h scattering signals. For a 1 mm aperture in our setup, only a single coherence area of scattered light is collected. The resulting A/B ratio of the experimentally observed intensity correlation function G2(τ), i.e. the ratio of [G2(0) − G2(∞)]/G2(∞), is near its theoretical limit of 1. However, the resolution of the various relaxation rates contributing to G2(τ) contained within the weak depolarized scattering signals is typically not limited by the A/B ratio but by several other factors. These include the intrinsic shot noise of the scattering signal, the contributions from photon detector dark counts to the signal, and the presence of spurious counts arising from larger impurities. The first two of these latter noise limitations can be significantly diminished by increasing the total flux of scattered photons regulated by aperture A2. Minimizing spurious bursts from large impurities requires meticulous filtering and cleaning of the sample solutions. As discussed above, acquisition and preprocessing of raw photon counts help to reduce this latter

problem further. All DLS and DDLS light scattering measurements at 90° were performed using the above custom light scattering setup. Characterization of Gold Nanorods Using DDLS. Optical Absorption Spectra of Gold Nanorod Samples. The absorption peaks arising from longitudinal surface plasmon resonance (LSPR) are frequently used to obtain an estimate of the geometric aspect ratio AR = L/d for metallic nanorods, where L and d are the length and diameter of the nanorods, respectively. To minimize absorption effects on the scattering measurements, we selected Au nanorod dimensions likely to produce minima in their absorption spectra near the HeNe wavelength (λ = 633 nm) used for light scattering measurements. We confirmed the absorption spectra of the three specific Au nanorod sample lots used in our experiments (Figure 3). The transverse (TSPR) and longitudinal (LSPR)

Figure 3. Optical absorbance spectra of the three Au nanorod samples used in experiments. The optical absorbance was normalized to their LSPR peak in the near-infrared region. While two of the three samples (Au-855 and Au-953) had nominally similar geometrical specifications, their absorbance spectra were dramatically different from each other and from that of the Au-855 sample and even from the sample spectrum provided by the manufacturer.

surface plasmon resonance peaks for the 45 × 10 nm sample were near 515 and 823 nm, respectively, with an absorbance minimum near 630 nm. This minimum indeed coincided with the wavelength of the HeNe laser. The absorption spectra for the two nominally similar 135 × 20 nm samples were dramatically different from each other, with LSPR maxima at 855 and 953 nm, respectively. It is notable that the spectrum with the 855 nm peak also deviated significantly from the quality control spectrum provided by the manufacturer. We therefore chose to identify these three specific sample lots by their actual LSPR peaks as Au-823, Au-855, and Au-953. The nominal aspect ratios and rod diameters quoted by the manufacturer were AR = 4.3, d = 10 nm (Au-823); AR = 5.8, d = 14 nm (Au-855); and AR = 6.6, d = 20 nm (Au-953), respectively. As detailed further below, independent electron microscopy measurements on the specific batches used in our experiments yielded quite different results for dry nanorod dimensions with AR = 3.5, d = 12 nm (Au-823); AR = 2.7, d = 41 nm (Au-855); and AR = 5.7, d = 8 nm (Au-953), respectively. 8131



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Polarized and Depolarized Intensity Correlation Functions for Au Nanorods. Figure 4 displays typical unnormalized

Figure 5. Lack of concentration dependence of relaxation rates. Exponential relaxation rates averaged over three independent measurements, as a function of the optical absorbance of the samples at 633 nm. Relaxation rates were extracted from fits to the polarized (v-v) and depolarized (v-h) intensity correlation function G2(τ) using either single- or double-exponential fits (see eqs 1 and 2). Neither the fast rotational nor the slow translational decay rates showed any discernible dependence on sample concentration or on whether the values were extracted from v-v or v-h measurements.

Figure 4. Polarized (v-v) and depolarized (v-h) intensity correlation functions, G2,v‑v(τ) and G2,v‑h(τ), for Au nanorod sample Au-823 in 0.1% CTAB/water at 30 °C. Notice the rapid decay of the depolarized correlation function compared to the much slower and bimodal decay of its polarized (v-v) counterpart. Solid lines represent least-squares fits through the data using squared single-exponential (v-h) and double-exponential (v-v) functions with offsets (see eqs 1 and 2).

The experimental relaxation rates are related to the translational and rotational diffusion constants Dtr and Drot via

polarized (v-v) and depolarized (v-h) intensity correlation functions obtained for the 43 × 10 nm Au nanorod solution (Au-823). It is immediately apparent that the depolarized correlation function G2,v‑h(τ) decays significantly faster than the polarized correlation function G2,v‑v(τ). In addition, the decay of the v-h correlation function appears nearly monomodal while the v-v decay contains at least two distinct relaxation rates. The solid lines through the intensity correlation functions in Figure 4 were obtained by directly fitting the raw intensity correlation functions to the square of either a single or doubleexponential function with adjustable offset, i.e. G2,v‐h(τ) = (A exp−(Γtr +Γrot)τ)2 + B

Γtr = Dtr q2

(3)

Γrot = 6Drot

(4)

The magnitude of the wave vector q in eq 3 is given by q = (4πn/λ) sin(θ/2)

(5)

where n is the refractive index of the solution, λ the wavelength of the incident light, and θ is the scattering angle between the incident and scattered light directions. The value for the wave vector in the DDLS set up was q = 1.87 × 107 m−1 or 1/(53.6 nm). Usually, measurements at multiple angles are required to separate the (q-dependent) translational and (q-independent) rotational relaxation components. This dependence is typically rather weak over the range of experimentally accessible qvalues, making the separation a nontrivial and time-consuming endeavor. Separate v-v and v-h measurements, instead, readily identify the translational and mixed translational/rotational relaxation components (Figure 4). The translational and rotational relaxation rates, Γtr and Γrot, obtained from singleor double exponential fits to the raw intensity correlation functions from polarized (v-v) and depolarized (v-h) DDLS measurements are summarized in Table 1. In addition, we used a commercial dynamic light scattering unit without polarizers that essentially collects v-v data. Due to the backscattering geometry of the DLS setup (θ = 173°) the corresponding value for the wave vector was q = 2.64 × 107 m−1 or 1/(37.9 nm). These latter measurements yielded bimodal distributions of relaxation rates with two well-defined peaks (Figure 6). Comparing relaxation rates to those obtained with DDLS, we assigned the faster of the two decay rates to Γmix = Γtr + Γrot and the slower one to Γtr. Table 1 provides a summary of the peak values for the experimental decay rates measured for the three different Au samples. Consistent with the q2-dependence of Γtr (eq 3), the translational relaxation rates at q = 173° are about twice as fast as those at 90°. As expected from eq 4, the

(1)

and G2,v‐v (τ) = (A1 exp−Γtrτ +A2 exp−(Γtr +Γrot)τ)2 + B (2)

In the fits, the amplitudes A and B and the relaxation rates Γtr and Γrot are adjustable parameters. To determine whether residual effects of Au nanorod absorption or the presence of interparticle interactions among the charged nanorods might affect relaxation rates at our sample concentrations, we repeated DDLS measurements for a series of progressive sample dilutions. As shown in Figure 5, the translational and rotational relaxation rates Γtr and Γrot for the shorter nanorods had no discernible dependence on Au nanorod concentration. Hence, within the concentration range we measured, the effects of Au absorption or particle−particle interactions on nanorod diffusion are negligible. Decay rates for the longer nanorod samples did show a very weak but systematic increase of their decay rates with concentration, consistent with either weak hydrodynamic or electrostatic repulsion (data not shown). To derive the corresponding relaxation rates in the absence of interactions, the experimental rates were linearly extrapolated to their infinite dilution limits (C = 0) prior to analysis. 8132



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CTAB in theoretical calculations was suggested in order to match rotational diffusion data, but noticeable deviations from translational diffusion rates remained.18 A more rational approach was suggested by de la Torre et al.19,20 using the ratio of the translational and the cube-root of the rotational diffusion coefficients to obtain an estimate for a function f(AR) that is only dependent on AR. That function could be numerically calculated and inverted to determine AR directly. This approach was used to determine the dimensions of short DNA oligonucleotides.21 While the specific ratio proposed by de la Torre is mathematically exact, the resulting dependence of f(AR) on AR is rather weak, varying by less than a factor of 2 for aspect ratios AR ranging between 2 and 20 (see Figure 6 in ref 19). Here we propose an analysis that also combines the two experimentally measured relaxation rates, Γtr and Γmix, into a single ratio. As shown below, this ratio can be used to extract the overall length L for the nanorods which, in turn, allows extraction of the corresponding aspect ratio AR. We find that the ratio (Γmix − Γtr)/Γtr of the experimentally observed relaxation rates serves this purpose. The starting point for the analysis is the expressions for translational and rotational diffusion coefficients, Dtr and Drot, using two different geometrical models of elongated nanoparticles: straight cylinders and prolate ellipsoids. In the following, L is the length (major diameter) of the cylindrical (ellipsoidal) nanoparticle, d is the (minor) diameter, and AR = L/d is the cylinder’s or ellipsoid’s aspect ratio. Predictions for straight cylinders use extrapolating polynomials derived by de la Torre et al. from simulation results with the bead−shell model. These extrapolations are valid for aspect ratio AR between 2 and 2020,22 and, more recently, for AR values down to 0.1.19 Expressions for prolate ellipsoids date back to closed-form expressions obtained by Perrin.23 The theoretical expressions for the translational diffusion constants are of the form

Table 1. DDLS and DLS Measurements of Au Nanorods Relaxation Rates sample

Γtr (kHz)a

Γrot (kHz)a

Γrot (kHz)b

Γtr (kHz)c

Au823 Au855 Au953

3.27 ± 0.13

145.7 ± 4.2

138 ± 8.1

6.44

2.07 ± 0.35

23.8 ± 1.5

4.09 ± 0.14

212.2 ± 0.8

17 ± 0.65 189 ± 6.3

4.96 8.68

Γmix (kHz)c 163 37.7 215

From DDLS measurements at θ = 90°, with v-v polarization and double-exponential fit. bSame as footnote a but using v-h polarization. c From DLS measurements at θ = 173°; relaxation rates are from the peak of the log-normal size distribution. a

Figure 6. Distribution of relaxation rates for each of the three Au nanorod samples using a commercial DLS unit and built-in Laplace inversion algorithms. Size distributions obtained as output of the inversion algorithms were reconverted into distributions of relaxation rates using eq 14. Translational (Γtr = Dtrq2) or mixed (Γmix = 6Drot + Dtrq2) relaxation peaks were assigned on the basis of v-v vs v-h relaxation rates identified with the DDLS setup (see Table 1). Peak areas represent the relative scattering intensity associated with a given relaxation mode.

Dtr = (kBT /3πηL)F(AR)

(6)

where F(AR) is a model-dependent function of the aspect ratio given by the following equation for straight cylinders FSC(AR) = ln(AR) + 0.312 + 0.565/AR − 0.1/AR2

rotational relaxation rates (Γrot = Γmix − Γtr) are essentially independent of the scattering angle. Unified Data Analysis of Translational and Rotational Au Nanorod Diffusion. The main goal of the subsequent data analysis is to relate these experimental relaxation rates to the corresponding geometrical parameters (length L and aspect ratio AR) of the nanorods in their solution environment via comparison to theoretical predictions for translational and rotational diffusion of either straight cylinders or prolate ellipsoids. Previous reports on translational and rotational diffusion of elongated metallic nanoparticles could not fully resolve discrepancies between experimentally measured and theoretically predicted diffusion rates.17,18 One of the challenges of the data analysis is to find the best match of two experimental parameters (Γtr and Γmix) to the predictions for the rotational and translational diffusion which, in turn, depend nonlinearly on two underlying geometrical parameters (e.g., L and AR). One approach has been to use the adjustable geometrical parameters (L and AR) in order to match one set of data (e.g., rotational relaxation) and see whether this provided reasonable agreement for the other data set (e.g., translation). For example, the inclusion of a detergent layer of

(7a)

2 FSC(AR) = 3 AR2 (f /f0 )−1 3

(7b)

with f /f0 = 1.009 + 0.01395 ln(AR) + 0.0788 ln(AR)2 + 0.006040 ln(AR)3

and by the following for prolate ellipsoids FPE(AR) = AR × S(AR)

(8)

with S(AR) = ln[AR +

AR2 − 1 ]/ AR2 − 1

The corresponding expressions for rotational diffusion are Drot = (3kBT /πηL3)G(AR) 8133

(9)



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have very similar AR dependencies, their ratio H(AR) is nearly constant for either cylinders or ellipsoids (see Figure 7). Interestingly, the value of H(AR) is also rather insensitive to the specific theoretical model used to analyze the data (here, straight cylinders or prolate ellipsoids). As a result, the ratio of decay rates Γrot/Γtr is strongly dependent on the overall length L of the nanoparticle but only varies weakly with the aspect ratio AR. Both factors represent significant advantages for obtaining robust estimates for L. The nanoparticle length L can therefore be expressed as

where G(AR) is given by the following equation for a straight cylinder GSC(AR) = ln(AR) − 0.662 + 0.917/AR − 0.05/AR2 (10a)

GSC(AR) =

2 2 AR (τ/τ0)−1 9

(10b)

with the above ratio of rotational relaxation times τ/τ0 given by τ/τ0 = 1.18 + 1.116(ln(AR) + 0.2877)2 − 0.9729

L = q−1 54(Γrot/Γtr)−1H(AR)

(ln(AR) + 0.2877)3 + 0.4954(ln(AR) + 0.2877)4

Using eq 6, the estimates obtained for L can then be used to determine F(AR)

and the following equation for prolate ellipsoids GPE(AR) =

1 2 AR [((2AR2 − 1)/AR)S(AR) − 1] 2 2

/(AR − 1)

(12)

F(AR) = (3πη /kBT )(ΓtrL /q2) (11)

(13)

Equations 12 and 13 represent our basic approach toward converting the experimental relaxation rates into values for nanoparticle length L and aspect ratio AR. The results obtained for F(AR) can be used in two ways. First, the specific values for F(AR) derived from eq 13 need to fall within the range predicted from the theoretical model. This indicates whether the underlying geometrical model (straight cylinder or ellipsoid) can possibly match the actual nanorod geometry. In addition, by tabulating F(AR) vs AR, this relationship can then be inverted to obtain actual predictions for the aspect ratio AR. Au Nanorod Length and Aspect Ratios determined from DDLS and DLS. Using the relaxation rates in Table 1, we determined the corresponding ratios Γrot/Γtr or (Γmix − Γtr)/Γtr listed in Table 2. Estimates for the corresponding Au nanorod length L from eq 12 were derived using the Γrot/Γtr ratio from the v-v measurements obtained from DDLS measurements. Results using either the v-h measurements or the DLS measurements taken with a commercial unit yielded comparable results. Equation 12 was evaluated for each of the three Au nanorod samples and for either straight cylinders (LSC) or prolate ellipsoids (LPE). Values for H(AR) were taken for either AR = 2.5 or 4, which we consider the limits of reasonable estimates for the nanorods’ AR given their original specifications. The range of values for the overall length L represents those obtained for H(2.5) and H(4), respectively. Using eq 13, we then determined the corresponding range of values for F(AR) for either straight cylinders or prolate ellipsoids. The results of these calculations are summarized in Table 2, as well. While the estimates for the overall nanorod length are different between the two models, they are both reasonable (see also TEM analysis below). However, the values for F(AR) derived from these models fall well outside the range of theoretical values obtained for either straight cylinders or prolate ellipsoids. The same result was obtained with each of the three Au nanorod samples we investigated. Characterizing Sample Polydispersity. One potential reason for the observed mismatch between the theoretical predictions and experimental results could be the intrinsic polydispersity of the sizes and shapes of the nanorod samples. Using transmission electron microscopy we determined the actual dimensions and shapes for each of the three Au nanorod samples in these experiments (Figure 8). The dimensions for the Au-823 samples were reasonably uniform and close to their specifications but did contain some more compact aggregates with AR ratios closer to 1 (Figure 8A). The higher aspect nanorods (Au-855) were noticeably more polydisperse both in

where S(AR) was defined in eq 8. The two separate equations for F(AR) and G(AR) (eqs 7a,7b and 10a, 10b) cover different ranges of AR, with AR = 2−20 and 0.1−20, respectively. We found that the two extrapolation polynomials agreed well in the overlap region of AR = 2−5, but that eq 10b developed an unphysical undulation for AR > 5. This is probably due to the paucity of data points for large AR used to derive the extrapolation polynomials for f/f 0 and τ/τ0. In our analysis we therefore switched from eqs (7b/10b) to (7a/10a) for AR > 5. Plots for F(AR) and G(AR) for both models are shown in Figure 7.

Figure 7. Dependence of translational and rotational diffusivities on the aspect ratio for either cylindrical rods or prolate ellipsoids. Plot of the dimensionless functions F(AR) and G(AR) vs the aspect ratio AR = L/d from theoretical predictions for straight cylinders and prolate ellipsoids given in eqs (7, 8, 10, and 11). In addition, the dimensionless ratios H(AR) ≡ G(AR)/F(AR) for straight cylinders and prolate ellipsoids are shown.

We propose to combine the experimentally derived values for Γrot (or Γmix) and Γtr into the simple ratio Γrot/Γtr = (Γmix − Γtr)/ Γtr. The relaxation rates are related to the corresponding diffusion constants via Γtr = Dtrq2 and Γmix = 6Drot + Dtrq2 (see eqs 3−5), where q is the scattering vector given in eq 5. Hence Γrot/Γtr = (Γmix − Γtr)/ Γtr = (6/q2)(Drot/Dtr) = 54H(AR)/ (Lq)2 which follows from eqs 6 and 9 and by defining H(AR) = G(AR)/F(AR). Importantly, since both G(AR) and F(AR) 8134



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Table 2. Theoretical Predictions of Rod Length L and Translational Modulation Factor F(AR) sample

Γrot / Γtra

Γrot / Γtrb

(Γmix − Γtr )/Γtrc

LSC (nm)d

FSCe

LPE (nm)d

FPEe

Au-823 Au-855 Au-953

44.6 11.5 51.9

42.2 8.2 46.2

24.3 6.6 23.7

39−43 77−85 36−41

0.65−0.72 0.82−0.90 0.76−0.87

52−53 100−103 47−48

0.85−0.87 1.07−1.10 1.05−1.08

a Γrot(v-v)/Γtr(v-v) (DDLS). bΓrot(v-v)/Γtr(v-h) (DDLS). cFrom DLS. dUsing Γrot(v-v)/Γtr(v-v) from the first column in eq 12 and HT(AR) for AR = 2.5 and 4. Results using data from the second or third column are within 2−3 nm. eUsing eq 13 and HT(AR) at AR = 2.5 and 4.

We proposed a new unified approach for directly deriving particle dimensions from translational and rotational relaxation rates, using different models for the particle geometry. In particular, we considered theoretical predictions for straight cylinders and prolate ellipsoids. Specifically, eq 12 provides a direct way to extract the overall length (long dimension) of elongated nanoparticles from the ratio of rotational and translational relaxation rates. Table 3 indicates that, with the exception of the highly polydisperse sample Au-853, the overall lengths of the nanorods predicted from either the straight cylinders or prolate ellipsoids model agreed fairly closely with each other and with the results of direct TEM measurements. In contrast, the values for F(AR) derived from eq 13 fell outside the range of meaningful values for either of the two geometrical models. Using the derived values for L to determine G(AR) instead of F(AR) yielded values for G(AR) that again were too small to match with physically plausible AR ratios. The TEM measurements and the analysis of sample polydispersity indicate that this discrepancy, with the possible exception of the Au-853 sample lot, is unlikely to be due to heterogeneities in Au rod dimensions or geometry. It is notable that we also tried the approach taken by Rodriguez-Fernandez et al.18 of directly matching the rotational diffusion data to model predictions for cylinders. As they observed, this required the assumption of an adsorbed CTAB layer in the theoretical calculation to obtain a reasonable match. There are several reasons why we argue that this assumption is not consistent with our data. First, as these authors have observed, we could not find values of L and AR that would provide simultaneously good fits to both translational and rotational diffusion constants. More importantly, our revised analysis indicates that the actual length predicted by the straight cylinder geometry is close to the bare nanorod dimensions, in direct contradiction to the idea of a rigid CTAB layer. The inability of either the straight cylinder or the prolate ellipsoid model to provide reasonable estimates for the aspect ratios of the nanorods is particularly intriguing. The TEM images suggest some possible explanations for this discrepancy. The actual geometry of gold nanorods shows protruding, faceted end-caps and clearly faceted sides. Hence, actual nanorod geometries are distinct from either prolate ellipsoids or straight cylinders with flat end-caps. Sensitivity to such boundary conditions has been shown to explain discrepancies in theoretical predictions for cylinder diffusion by Broersma vs those by de la Torre, particularly for rods with modest aspect ratios.20 Yet, the robustness of the overall lengths derived from either ellipsoidal or cylindrical geometries implies that the ratios of translational and rotational diffusion rates for either geometry are dominated by overall nanorod length. This is consistent with eq 12, which suggests that H(AR) is nearly constant and does not vary dramatically between different models for elongated nanoparticles. In contrast, reasonable estimates of the aspect ratio apparently require that the model geometry closely matches the actual morphologies of the

their physical dimensions and their morphologies. For example, the longest nanorod in Figure 8B measured about 310 × 27 nm, compared to the average of 135 × 20 nm quoted by the manufacturer. Finally, the last batch of Au-953 was reasonably homogeneous, but significantly shorter and thinner (L = 40−50 nm, d = 5−7 nm) than indicated by the manufacturer’s spec sheet. In order to assess sample polydispersity from the distribution of decay rates of the light scattering signal we utilized the builtin Laplace inversion algorithms of the commercial DLS unit. Laplace inversion converts the field correlation function into distributions of decay rates instead of the two discrete decay rates used for the analysis of the DDLS intensity correlation functions. The Laplace inversion algorithm returns a distribution of hydrodynamic diameters for spherical particles diffusing at the same rate as the measured relaxation rates. To convert these diameters D back into the underlying distribution of relaxation rates, one simply combines eq 4 with the standard Stokes−Einstein relation [eq 6 with D equal to L and F(AR) = 1], to obtain Γ = (kBT /3πη)q2 /D

(14)

As mentioned above, the two peaks in the distributions of relaxation rates arise from the (slower) translational (Γtr) and (faster) mixed modes (Γmix = Γtr + Γrot) (see Figure 6). The more heterogeneous sample Au-853 displays a significantly broader translational peak, particularly when accounting for the logarithmic scale of the relaxation rate axis. However, the maximal relaxation rates for Γtr and Γmix for both peaks closely match those returned by the simplified double-exponential fit to the v-v correlation functions obtained with the DDLS setup. Hence, the large variety of both shapes and sizes present in the Au-855 sample also appears in the broad distribution of relaxation rates seen with DLS. However, both the Au-823 and Au-953 samples were sufficiently homogeneous to permit a meaningful comparison between the theoretical models and the experimental results. Table 3 summarizes the dimensions of the Au nanorods as provided by the manufacturer, direct results of TEM measurements, and the lengths L of the rods derived from DDLS measurements and our analysis, assuming either straight cylinders or prolate ellipsoids as geometrical models.

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DISCUSSION AND CONCLUSIONS We used depolarized and polarized dynamic light scattering to determine the translational and rotational relaxation rates for three commercial samples of faceted Au nanorods. Polarized and depolarized measurements provided comparable decay rates for the rotational diffusion coefficient. However, the use of depolarized light scattering allowed immediate identification of the rotational relaxation component. The results from our custom DDLS setup agreed quantitatively with relaxation rates obtained with a commercial DLS unit in backscattering geometry. 8135



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Table 3. Au Nanorod Dimensions: TEM vs DDLS L × d (nm)

AR

sample

nominala

TEM

LSPRb

TEMc

L (nm)d DDLS

Au-823 Au-855 Au-953

43 × 10 132 × 20 81 × 14

42 × 12 111 × 41 46 × 8

4.3 6.6 5.8

3.5 2.7 5.7

41/53 81/102 39/48

a

Specifications for sample lot provided by the manufacturer. bEstimate of manufacturer based on LSPR peak value. cMeasurement from TEM images of actual sample lots. dAverage length taken from Table 2 for straight cylinder/prolate ellipsoid.

estimate can then be used to evaluate whether the underlying geometrical model can possibly yield reasonable AR values. Given the critical importance of the aspect ratio to many of the optical absorption parameters and the ease with which DLS can provide robust translational and rotational diffusivities, we hope that future simulations with the bead−shell model will clarify how sensitive diffusivities indeed are to the details of nanorod geometries.

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AUTHOR INFORMATION

Corresponding Author

*Tel: (813) 974-2564. Fax: (813) 974-5813. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We are grateful for helpful discussion and suggestions by our colleague Dr. Vinay Gupta in the Department of Chemical Engineering. We also would like to acknowledge the support of Dr. Yusuf Emirov at the USF Nanotechnology Research & Education Center for his TEM imaging of the Au nanorods.

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REFERENCES

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Figure 8. Transmission electron microscopy images of (A) Au-823, (B) Au-855, and (C) Au-953 gold nanorod samples. Scale bars represent 10, 50, and 20 nm, respectively.

particles. This suggests that the approach for analyzing translational and rotational diffusion data introduced here has yet another advantage: instead of trying to determine values for AR that are highly sensitive to the specific model geometry, our approach first estimates the overall particle length L. This 8136


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Ortega, A.; de la Torre, J. G. J. Chem. Phys. 2003, 119, 9914. Tirado, M. M.; Martinez, C. L.; Torre, J. G. d. l. J. Chem. Phys. 81, 2047. Eimer, W.; Pecora, R. J. Chem. Phys. 1991, 94, 2324. de la Torre, J. G.; Bloomfield, V. A. Q. Rev. Biophys. 1981, 14, Perrin, F. J. Phys. Radium 1936, 7, 1.

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