1. Introduction

Metal nanoparticles (NPs), especially noble metal NPs, have important applications nowadays in various fields such as catalysis, medical diagnosis and therapy, biosensing, and drug delivery and release [1]. Reliable, fast, and accurate measurement methods and the related metrology standards are highly demanded for the production, characterization, and commercial use of metal NPs. Since the properties of metal NPs highly depend on their geometric characteristics due to the strong shape- and size-dependent localized surface plasmon resonance (LSPR) of the NPs [2], the dimensional metrology of metal NPs is of high importance.

So far, the most commonly used dimensional metrological methods for metal NPs are microscopic imaging methods such as transmission electron microscopy (TEM), scanning electron microscopy, and scanning probe microscopy. These methods, though with high precision, can only measure individual NPs or a small number of NPs locally. Hence, these methods are slow and costly when they are used to measure large amount of NPs (or the so-called NP ensembles). In addition, when the NPs are prepared for microscopic measurement, the NPs may aggregate strongly after the NP colloid is coated on a substrate and the solvent is evaporated, which is disadvantageous for accurate characterization of the NP geometry. Furthermore, for NPs with non-uniform geometries (i.e., the so-called polydisperse NP ensembles), it is often needed to characterize the distribution function of some geometric parameters of the NPs statistically, which is obviously hard to do with the microscopic methods due to the required large amount of sampling NPs. To achieve this goal, some methods based on scatterometry (i.e., the technique of retrieving the geometrical parameters of NPs from their scattering spectra) have been proposed, such as optical extinction spectroscopy (OES), small-angle x-ray scattering method, and dynamic light scattering (DLS) method. Among these methods, the DLS method is probably the most widely used one because of its versatility of measuring various materials of NPs. However, since the DLS method measures the hydrodynamic size of NPs in a liquid environment by detecting and analyzing the Brownian motion of the NPs, it can only give the equivalent spherical diameter of the measured NPs no matter what practical shape the NPs may have. Therefore, it cannot measure the shape of the NPs. For the characterization of non-spherical NPs, some other scatterometric methods such as the OES method have to be developed.

Gold nanospheres and nanorods (NRs) are typical noble metal NPs that are widely used nowadays in various applications such as biomedical diagnosis and therapy [3]. An ensemble of NPs are referred to as monodisperse if the NPs have the same size and shape, or otherwise as polydisperse. For polydisperse NP ensembles, a probability density function (PDF, which refers to the percentage of the NPs with certain size and shape specifications in the whole NP ensemble) is used to describe the size and shape distribution of the NPs. The standard deviation σ of the PDF therefore characterizes the polydispersity of the sample. In practice, when the standard deviation of the PDF is small enough, the NP ensemble can be regarded as monodisperse. According to previous studies [4, 5], for monodisperse gold nanospheres with σ < 0.1, the mean diameter can be determined accurately by the OES method in a broad range of diameters (3 nm ∼ 100 nm). However, when the NP geometry deviates from an ideal sphere, the shape deviation should be taken into account [6]. On the other hand, for a polydisperse nanosphere ensemble with σ > 0.1, the polydispersity should also be taken into account [6]. Recently, Peña et al. [7] proposed a multivariate optimization algorithm to retrieve the average diameter and the diameter PDF of polydisperse metal nanoshpere ensembles. They showed that the OES method can measure the PDF of a large-amount NP ensemble accurately enough, while the TEM method is more useful for the characterization of a small amount of gold nanospheres directly.

For gold NRs, the light scattering behavior and LSPR property are more complicated than those of nanospheres. In the past few years, several electromagnetic methods have been used for simulating the LSPRs of metallic NRs, such as the Rayleigh-Gans approximation (RGA) method [8,9], the T-matrix method [10,11], the discrete dipole approximation method [12–14], the finite-difference time-domain method [15], the integral boundary element method [16, 17], and the finite element method [18]. The advantages and disadvantages of these methods in simulating NPs are described detailedly in Refs. [19–21]. Previous research works [11, 14] have shown that the geometric parameters of gold NRs such as the width, the aspect ratio (the length of the NR divided by its width), and the end-cap shape can affect the LSPR of NRs. Specifically, when the aspect ratio of gold NR is lager than about 1.5, there would appear two distinct LSPR peaks in the extinction spectra of the NRs [1]: a transverse-mode LSPR peak (named T-LSPR) and a longitudinal-mode LSPR peak (L-LSPR, which is usually stronger) due to the resonant oscillations of electrons along the short and long axes of the NR, respectively. Since the L-LSPR peak of gold NRs is usually more sensitive to the geometry change than the T-LSPR peak, it can be measured to estimate the mean width and aspect ratio of a monodisperse NR ensemble.

For polydisperse gold NR ensembles, Susie Eustis et al. [22] have used a theoretical fit of the measured L-LSPR to quickly determine the aspect ratio distribution (ARD). When retrieving the structural parameters, they used the RGA method that is only applicable to small spheroidal NPs [23] to simulate the LSPRs of NRs, so that the influences by the other geometric parameters of NRs such as the width and the end-cap shape were ignored. Instead of the RGA method, Boris Khlebtsove et al. [11] applied the T-matrix method (which is a rigorous semi-analytical method) to simulate the LSPR of the gold NPs ensemble based on thousands of TEM images. It shows that the width, the end-cap shape, and the surface electron scattering constant of the NRs may also affect the LSPRs, though weaker than the aspect ratio. Owing to the high calculation efficiency and high accuracy of the T-matrix method, they suggested to apply it to determine the ARD of NRs from extinction experiments by using an optimization procedure. Furthermore, in a previous report of them [24], they applied a modified approach of Susie Eustis et al. [22] to obtain the ARD by fitting the depolarization spectra in addition to the extinction spectra. When retrieving the ARD by optical data, they used the TEM-based data as a priori information and obtained an excellent agreement between the simulated and measured spectra.

These works show that the OES method is more convenient than the TEM method to characterize the size and shape distribution of polydisperse gold NR ensembles statistically. Here, we apply the same strategy by using the T-matrix method to rigorously calculate the extinction spectra of the NR ensembles when solving the inverse scattering problem. However, the main difference between our approach and the previous works [11, 24] is that we use a fast and reliable optimization procedure so that no a priori information about the geometric values (such as the TEM-based width of the NRs) is needed beforehand. We study the use of the OES method to perform fast measurement of the ARD of polydisperse gold NR ensembles statistically, where the constrained nonnegative regularized least-square procedure was applied. The influences by the width, the end-cap shape, and the surface electron scattering constant of the NRs on the ARD measurement are thoroughly analyzed. The measurement results are compared with those obtained by the TEM method, showing the reliability of the OES method. The measures for further improving the solution accuracy of the ill-posed inverse scattering problem are discussed.

2. Theoretical method

The objective of the OES method is to retrieve the geometric parameters of the gold NRs from the measured extinction spectra. It is in essence an inverse scattering problem, which is solved by using the methods below.

2.1. Calculation of the extinction cross section of a single NR

To solve the inverse scattering problem, the premise is that the related directly problem is well solved, i.e., the extinction response of a NR can be precisely modeled and calculated. The NR geometry is defined in Fig. 1, which is a cylinder with its end caps in oblate spheroidal, prolate spheroidal, or spherical shapes, depending on the different generation methods of the NRs. Three parameters are defined to describe the NR geometry, i.e., the width D, the aspect ratio AR = H/D (where H is the length of the NR), and the eccentricity of the end cap e = 2L/D. Clearly, when e = 0, the NR is a perfect cylinder; and when e = 1, the NR is a cylinder with two semi-spherical end caps.

 

Fig. 1 Geometric model of the NR. Several NRs with the same width D and aspect ratio AR but different end-cap factor e are demonstrated.

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The extinction cross section C_ext is defined as the ratio of the radiant power being extinct by a particle to the radiant power incident on the particle in the process of scattering [8]. To rigorously calculate C_ext of a single NR or an ensemble of randomly oriented discrete gold NRs in a monodisperse system, some numerical methods such as the T-matrix method [10] can be used. The T-matrix method, a rigorous semi-analytical method, is used in our simulation because it is much faster for modeling randomly oriented NR ensembles than the other methods [10, 11].

We developed our own T-matrix numerical codes based on the algorithms reported in [10], which were calibrated with the benchmark results reported in Ref. [25–27]. In our calculation, the real and imaginary parts of the complex dielectric function of bulk gold were taken, which were calculated from the experimental values given by Johnson and Christy [28]. According to previous works [1,3–6,11,24], this dielectric function should be corrected of gold NRs. Due to the mean free path limitation for electrons, the damping constant γ of gold NR is increased and can be modified by [4]

2.2. Calculation of the absorbance of NR ensembles

In the OES measurement, the measurand is the absorbance A of the sample, by which to retrieve the structural parameters of the NRs. Therefore, we should get the relation between A and the extinction cross section C_ext of a single NR. For a monodisperse ensemble of NRs, the transmittance (defined as the ratio of the transmitted intensity over the incident light intensity I/I_inc), the absorbance A, and the total extinction A_ext of the sample medium with length l are related by [8]

Obviously, Eq. (4) is valid only for a monodisperse system. If the NR ensemble is polydisperse, the measured absorbance spectrum is the superposition of the absorbance spectra of the composing NRs with different sizes and shapes [22]. Therefore, for a polydisperse NR ensemble, the total absorbance A is calculated by integrating the contribution from the composing NRs according to the PDF p(D,AR,e) as [10]:

2.3. Solution of the inverse scattering problem

In the inverse scattering problem of determining the NR geometry by scatterometry, the three structural parameters of NR as well as the PDF are unknowns. Therefore, the inverse problem is generally ill-posed so that the solution is not unique or a small perturbation of the measurand may result in a large variation of the retrieved parameters. Furthermore, Eq. (5) usually cannot be solved analytically and should be discretized for numerical solution. If we discretize it with three variables AR, D, and e and use, for example, an optimization process to search for the solution without any a priori information of the PDF, the size of the matrix in calculation would be very large so that the condition number of the linear system is too large to give an accurate and stable solution. To avoid this problem, a model of the PDF p(D,AR,e) of the NRs may be adopted, either according to preliminary experimental statistics (by, for example, TEM or dark-field microscopy) or by reasonable assumption based on the production method of the NPs. Then an optimization process can be launched to search for the solution. Therefore, the accuracy and stability of the solution are dependent on the adopted PDF model.

On the other hand, according to our study and many previous works [1, 14, 22], the aspect ratio AR is the primary parameter affecting the extinction of the NR ensemble. Hence, to a first approximation, we may fix the width D and the end-cap factor e as their mean values D̄ and ē, so that these two variables can be separated from the integral equation:

The discretization of Eq. (6) results in the following system of linear algebraic equations:

Since the discretized matrix C_mn is usually ill-conditioned, the inverse problem formulated in terms of Eq. (6) is ill-posed [32]. In order to find a unique and accurate solution of the inverse problem, one of the commonly used numerical techniques is Tikhonov regularization [33]. Here we just briefly summarize the process of Tikhonov regularization in our problem and more details can be found in Refs. [32–35].

The regularized least-squared solution P_RLS of Eq. (7) is given as [32]:

3. Experiment and results

We applied the OES method to measure the ARD p(AR) of gold NR ensemble samples and compared the results with those directly obtained by the TEM method. 30 samples of gold NR ensembles were measured and analyzed, in which each sample contains approximately 10¹⁰ NRs per millilitre. Here, without loss of generality, the results of three samples with different D, AR and e are demonstrated. The three samples designated as NR-40-700, NR-20-700, and NR-10-750 were obtained from NanoSeedz Ltd., which have the nominal width D of 40 nm, 20 nm, and 10 nm and the expected L-LSPR wavelengthes of 700 nm, 700 nm, and 750 nm, respectively.

In the OES measurement, a UV-VIS spectrophotometer (PekinElmer LAMBDA 950) was used to measure the extinction spectra of the samples. For each sample, the measurement was repeated six times in one hour and the average value is used. The measurement range of wavelength λ was 400 nm – 1000 nm and the step was taken as 1 nm. The most time-consuming process of the OES method is to prepare the extinction spectra database [corresponding to the matrix C in Eq. (7)] of the gold NRs with different values of the width, the aspect ratio and the end-cap factor D, AR, and e. Fortunately, the database just need to be calculated once. By using a dual-core 2.13GHz Intel Xeon CPU with 80Gb RAM, it takes about 12 seconds to calculate a single extinction spectrum of NRs in the wavelength range of the 400 nm – 1000 nm, with a 1 nm spectral resolution, and a relative calculation accuracy of better than 1%. Based on the measurement data, the optimization process described above was implemented to retrieve the ARD p(AR) of the samples, where AR was discretized in the range of 1 to 5, with a step of 0.1. The inverse algorithm was run on a 3.00GHz Intel Core2 Duo CPU with 4Gb RAM and the average time consumption is about 0.25 seconds for a single measured spectra.

In the TEM experiment, a transmission electron microscope (Hitachi H-7650B) was used to get the images of the NRs. For each of the three samples, ten TEM images were taken. Therefore, we analyzed altogether 788, 896, and 804 NRs in samples NR-400-700, NR-20-700, and NR-10-750, respectively to get the mean width D̄, the mean end-cap factor ē and the ARD p(AR).

The TEM images of the three samples as well as their extinction spectra measured by the OES method are shown in Fig. 2. In Figs. 2(a)–(c), it is clearly seen that the gold NR ensembles are polydisperse. In Fig. 2(d), the measured L-LSPR extinction peaks of the three samples are very close to their nominal values. NR-40-700 (red line) and NR-20-700 (black line) have the same resonance wavelength but different linewidths, i.e., the full width at half maximum (FWHM) of the resonance peak, because of their different D̄ and dispersancy.

 

Fig. 2 TEM images of the three gold NR ensemble samples: (a) NR-40-700, (b) NR-20-700, (c) NR-10-750. (d) Experimentally measured extinction spectra (dots) of the samples as well as the corresponding numerically reproduced extinction spectra (lines) according to the retrieved ARD functions p(AR) based on the OES results.

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The AR distributions p(AR) of the samples were retrieved from the OES results, with the procedure presented in Section 2. In Fig. 3, the red columns and curves are the retrieved results obtained by the OES method while the black ones are the results obtained by TEM. We use a sum of Gaussian functions to fit the discrete results:

 

Fig. 3 Comparison of the measured AR distribution functions of three gold NR ensemble samples obtained by the OES method (red) and those obtained by the TEM method (black). In each subfigure, both the discrete AR distribution and a Gaussian fit of it are given. The values in parentheses ( $\overline{AR}$, σ_AR) give the mean AR and the standard deviation of the PDF obtained by the two methods.

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Furthermore, with these retrieved AR distribution functions p(AR), the extinction spectra of the three samples were numerically reproduced by Eq. (6), as shown in Fig. 2(d), which coincide with the measured extinction spectra quite well. Therefore, both Fig. 3 and Fig. 2(d) show that the retrieved results by the OES method are reliable in our characterization.

By comparing the OES results with those obtained by the TEM method in detail, it is seen that the mean AR values derived by the two methods coincide with each other well, with their relative difference as 0.70%, 0.26% and 1.10% for samples NR-40-700, NR-20-700, and NR-10-750, respectively. The relative difference here is calculated by $\left({\overline{AR}}_{\text{OES}}-{\overline{AR}}_{\text{TEM}}\right)/{\overline{AR}}_{\text{OES}}$. It is worth noting that in Figs. 3(b) and 3(c), the OES results show significant ARD between 1 and 1.5, while the TEM results have few NPs in this range. The main reason is that in the counting process of the TEM method, we ignored most byproducts (such as spheres, cubes etc.) in the samples, as shown in the TEM images of Figs. 2(b) and 2(c). However, these byproducts also contribute to the extinction spectra and thus can be detected by the OES method. The standard deviation of the OES results (σ_OES,AR = 0.144, 0.221, and 0.306) are smaller than those of the TEM results (σ_TEM,AR = 0.267, 0.345, and 0.364), with their relative difference as 46.1%, 35.9%, and 15.9% for the three samples. The possible reason is that many factors such as the deviation between the real shape of the gold NRs and our calculation shape model and the correction method of the dielectric function could influence the extinction spectra and thus also influence the retrieved ARD. Therefore, we proceed to discuss the influences by these parameters.

4. Discussion: influences by the other structural parameters

In the AR retrieval process of the OES method, the mean width values D̄_OES that we adopted are 46.0 nm, 20.0 nm, and 22.0 nm for samples NR-40-700, NR-20-700,and NR-10-750, respectively, but not the nominal values, so as to obtain the best-fit results. By TEM imaging, the measured mean width and the standard deviation (D̄_TEM ± σ_TEM,D) of the three NR ensemble samples are (47.0±4.1 nm), (19.5±3.0 nm), and (18.7±1.9 nm), which are also different from the nominal values but close to our OES results. This, on the other hand, shows that our OES measurement results are reliable. For the end-cap eccentricity, the best-fit values of ē_OES that we used in the OES method are 0.9, 0.6, and 0.3 for the three samples, while the corresponding TEM measurement results ē_TEM are 0.8, 0.6, and 0.4, respectively. The two sets of end-cap eccentricity values also coincide with each other relatively well (where the small difference may be owing to the in-sufficient sampling in the TEM method).

However, from another point of view, these calculations show that the selection of the width D̄ and the end-cap factor ē is important in the retrieval process, although D̄ and ē are considered to affect the LSPR response of the gold NR ensembles weakly at the beginning [14]. In the following, we detailedly analyze the influences of D̄ and ē on the retrieval results. We analyze the value of mean width D̄ ranged from 5 nm to 50nm with a step 5 nm and the mean end-cap factor ē ranged from 0 to 1 with a step of 0.1. Without loss of generality, we choose sample NR-20-700 in the following analysis.

4.1. Influence by the selection of mean width D̄

Here, the NRs have a fixed end-cap eccentricity ē = 0.6 and the mean width D̄ is varied. In the retrieval calculation, ten different values of D̄ were selected to solve the inverse problem. The obtained mean square error mse values are summarized in Fig. 4(b). It shows that four of them (for 10 ≤ D̄ ≤ 25 nm) are acceptable with mse ≤ 1 × 10⁻³ while the others (for D̄ ≥ 30 nm or D̄ ≤ 5 nm) are unacceptable.

 

Fig. 4 (a) Comparison of the retrieved ARD p(AR) obtained by the OES method using different assumed mean width D̄ and the p(AR) directly measured by the TEM method. (b) Dependence of the mean square error mse on the assumed mean width D̄.

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Figure 4(a) compares the retrieved aspect ratio distribution p(AR) obtained by the OES method using five different assumed mean width D̄ with the p(AR) directly measured by the TEM method. It is seen that when D̄ increases from 10 nm to 30 nm, the retrieved p(AR) is left shifted and the FWHM decreases. By linear fitting, we find that the shift bears a linear relation with respect to D̄, as shown in Fig. 5(a),

 

Fig. 5 (a) Dependence of the retrieved mean aspect ratio $\overline{AR}$ and the standard deviation σ on the assumed mean width D̄ for sample NR-20-700 with assumed ē = 0.6. (b)Dependence of the number of NRs per unit volume N_v and the mass-volume concentration C_g of NRs on the assumed mean width D̄.

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In addition, the standard deviation σ_AR of the retrieved p(AR) also decreases with the increase of D̄, as shown in Fig. 5(a). These are owing to the ill-posedness of the inverse problem. Therefore, if we want to get accurate solution of p(AR) without knowing the value of D̄, some other a priori information about the gold NR ensembles should be determined beforehand.

In Fig. 5 (b), we calculated the dependence of the number of NRs per unit volume N_v and the mass-volume concentration C_g of NRs on the mean width D̄. Here, N_v can be obtained by the optimization progress described in Section 2 and the mass-volume concentration C_g is derived as C_g = ρNvV_n · P_RLS, where V_n is a row vector consisting of the volume of each nanorod of AR_n and ρ is the density of bulk gold. Since N_v has an evident dependence on the mean width D̄, as shown in Fig. 5(b), it can be used as a priori information for the retrieval process. In contrast, the value of C_g (= 22.45 ± 0.07 μg/ml) changes only a little with respect to the change of D̄ when D̄ ≤ 30 nm. Thus it is not suitable to act as a priori information for the determination of the mean width.

4.2. Influence by the mean end-cap eccentricity ē

To study the influence by the mean end-cap factor ē, we fix the mean width value D̄ = 20 nm and vary ē. In the calculation, 11 different values of ē were adopted to solve the inverse problem and the obtained mse values are summarized in Fig. 6(b). It is seen that eight of them (for 0.3 ≤ ē ≤ 1) are acceptable and the others are unacceptable. Figure 6(a) shows the comparison of the retrieved ARD p(AR) obtained by the OES method using eight assumed mean end-cap eccentricities ē and the measured p(AR) obtained by the TEM method. With the increase of ē from 0.3 to 1, the retrieved p(AR) has a right shift while the FWHM only changes a little. The shift also bears a linear relation with respect to ē, following the fitted equation:

 

Fig. 6 (a) Comparison of the retrieved ARD p(AR) obtained by the OES method using different assumed mean end-cap eccentricity ē and the measured p(AR) obtained by the TEM method. (b) Dependence of the mse on the assumed ē.

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Similarly as the influence by D̄, the fitted $\overline{AR}$ is affected by the change of ē evidently. According to Eq. (17), a change of ē by 0.1 may lead to the change of $\overline{AR}$ by 1.6%. It means that the retrieved PDF is dependent not only on the value of mean width D̄, but also on the value of ē. However, different from the influence by D̄, in this case the standard deviation σ and the number of NRs per unit volume N_v only depend on ē slightly, as shown in Fig. 7. Although N_v is also dependent on the end-cap factor ē, the relativity is much smaller, compared with the influence of D̄. Thus it is difficult to use N_v as a priori information to determine the mean end-cap factor ē. Meanwhile the value of the mass-volume concentration can also be obtained (as C_g = 22.39 ± 0.26 μg/ml), which changes only a little with respect to ē and the value coincides well with C_g = 22.45 ± 0.07 μg/ml obtained in subsection 4.1. Therefore, we can conclude that the mass-volume concentration C_g of the gold NR ensembles can be determined accurately by the OES method, without knowing the other structrual parameters (ē and D̄) beforehand.

 

Fig. 7 (a) Dependence of the retrieved mean aspect ratio $\overline{AR}$ and the standard deviation σ on the assumed mean end-cap eccentricity ē for sample NR-20-700 with assumed D̄ = 20 nm. (b) Dependence of the number of NRs per unit volume N_v and the mass-volume concentration C_g of NRs on the assumed mean end-cap eccentricity ē.

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On the other hand, if we want to get accurate solution of $\overline{AR}$ without knowing the value of ē, we need to find another sensitive measurand related to the mean end-cap factor ē. For this aim, we may carry out some auxiliary measurements (such as scattering cross section measurement, and the polarization-dependent or incident-angle-dependent scattering measurements of individual NRs or well-aligned NR array) to facilitate the determination of ē. Or, alternatively, a reliable value of the mean end cap should be obtained beforehand, for example, by TEM imaging of a few sampling NRs.

4.3. Influence by the polydispersity of the width D and end-cap eccentricity e

So far, we have been always considering the fixed values of D̄ and ē. In this subsection, we analyze the influence by the polydispersity of the width D and end-cap eccentricity e, i.e, σ_D and σ_e that are defined as the standard deviations of the PDFs of D and e, respectively. The retrieved p(AR) was obtained by integrating the retrieval results with respect to each discrete pair of D and e.

Figure 8 shows the retrieved ARD p(AR) obtained by the OES method, by taking different σ_D and σ_e. The mse is always smaller than 1 × 10⁻³ so that the retrieved results are acceptable. It is seen that the changes of the mean aspect ratio $\overline{AR}$ and standard deviations σ_AR of p(AR) are around 1% when σ_D and σ_e are significantly increased from 0 to 5.4 and to 0.143m, respectively. This shows that the influences by the polydispersity σ_D and σ_e are pretty small and can be ignored, compared with the influences by the mean width D̄ and the mean end cap ē.

 

Fig. 8 Comparison of the retrieved p(AR) obtained by the OES method by assuming (a) different polydispersities of the width D and a fixed e = 0.6, and (b) different polydispersities of the end-cap eccentricity e and a fixed D = 20 nm.

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4.4. Influence by the surface electron scattering constant A_s

To study the influence by the surface electron scattering constant A_s, we fix D̄ = 20 nm and ē = 0.6. In the calculation, six different values of A_s were used to solve the inverse problem and the obtained mse values are summarized in Fig. 9(b). It is seen that when A_s increases from 0.3 to 1.3, the values for mse also increases. The range of the acceptable values (0.3 ≤ A_s ≤ 0.6) are consistent well with the measurement values determine by Ref. [11, 30].

 

Fig. 9 (a) Comparison of the retrieved ARD p(AR) by the OES method using different surface electron scattering constant A_s. (b) Dependence of the mse on the A_s.

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Figure 9(a) shows the retrieved ARD p(AR) by the OES method, by taking different A_s. It is seen that when A_s is significantly increased from 0.3 to 1.3, the mean aspect ratio $\overline{AR}$ only changes negligibly (smaller than 1%) while the standard deviation σ_AR decreases by 20%. The decrease of σ_AR can be explained by Eq. (1): the increase of A_s increases the damping constant γ, which leads to the broadening of the L-LSPR [30]. Thus the collection of these spectra would be broadened and the σ_AR of the retrieved ARD would decrease. It is worth noting that only the gold NRs narrower than ∼20 nm may have broader resonance due to the surface scattering [30, 36]. However, most of our calculations were performed for NRs with width and lengths exceeding 20 nm, so that the influences by A_s are pretty small (compared with the influences by D̄ and ē) and can also be ignored.

5. Conclusions

We have studied the use of the OES method to measure the ARD of polydisperse gold NR ensembles statistically. To solve the inverse scattering problem, the extinction of the polydisperse NR ensemble is modeled rigorously by the T-matrix method and the AR parameter retrieval is performed by an optimization process with data fitting to the measured extinction spectra. We have shown that, for different NR samples that we have prepared, the retrieved PDF results coincide well with those obtained by the TEM method. The comparison results indicate that the OES method is fast, cost effective, and accurate enough if the mean width D̄ and end-cap shape ē of the NR ensembles are reasonably assumed or pre-determined. Furthermore, the C_g of NRs can also be measured by the OES method, which is useful for improving the solution of the inverse problem while cannot be obtained by the imaging methods.

Detailed analyses of the influences of NR parameters on the retrieval results have shown that the measured mean aspect ratio $\overline{AR}$ depends on the assumed mean width D̄ and mean end-cap factor ē linearly. A change of 10 nm in D̄ may lead to around 7% change in the retrieved $\overline{AR}$ and a change of ē by 0.1 may lead to the change of $\overline{AR}$ by 1.6%. The influences by the polydispersity σ_D and σ_e, however, are pretty small and can be ignored. For gold NRs with the width larger than ∼20 nm, the influence by the surface electron scattering constant A_s is also very small and can be ignored. Based on the analyses, we suggest that the measurement accuracy can be further improved if some a priori information of the NRs can be obtained beforehand. A good guess of the mean width D̄ can be obtained by measuring the number of NRs per unit volume N_v, which can be achieved by the OES method itself. To get a good guess of the end-cap shape ē, some auxiliary measurements (such as scattering cross section measurements, and polarization- or incident-angle-dependent scattering measurements of NRs) could be taken, which are the tasks of our further work.