Accurate geometric characterization of gold nanorod ensemble by an inverse extinction/scattering spectroscopic method

Ninghan Xu; Benfeng Bai; Qiaofeng Tan; Guofan Jin

doi:10.1364/OE.21.021639

1. Introduction

Fast and accurate geometric characterization of noble metal nanoparticles such as gold nanorod (NR) ensembles is highly demanded in practical production and trade of nanoparticles, for their important applications in various fields such as catalysis, medical diagnosis and therapy, biosensing, and drug delivery and release [1]. The NR geometry is defined in Fig. 1, which is a cylinder with its end caps in different shapes, depending on the different generation methods of the NRs. Usually, three critical parameters (i.e., aspect ratio, width, and end-cap factor) are defined to characterize the geometry of gold NRs.

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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In our previous work [2], the aspect ratio distribution (ARD) of gold NR ensembles was determined effectively and statistically by an optical extinction spectroscopic (OES) method. It is found that with the OES method, the measured ARD would depend on the nominal mean width and mean end-cap factor of the NRs, although both of them are considered to affect the magnitude and spectral position of the localized surface plasmon resonance (LSPR) of the NRs weakly. Thus we suggested that the measurement accuracy of the ARD can be further improved if the mean width and the mean end-cap factor of the NRs could be pre-determined as a priori information.

To achieve this goal, an intuitive idea is to use a microscopic imaging method such as transmission electron microscopy (TEM) to characterize the width and end-cap shape of individual NRs directly [3, 4]. However, with such imaging methods it is hard to obtain the probability distribution function (PDF) of polydisperse NR ensembles statistically, because a large amount (to the order of 10³) of sampling NRs should be characterized, which is time consuming and expensive. Therefore, in this work we aim to develop a method based on scatterometry to achieve this goal more conveniently.

It is known that the extinction (which is contributed by both absorption and scattering) property of metallic NRs is strongly dependent on the shape and size of the NRs. However, the geometric parameter dependence of the extinction property and that of the scattering property are different, which has been investigated in many previous works [5, 6]. For the extinction spectra of NRs, the aspect ratio is the most sensitive parameter affecting the magnitude and position of the longitudinal mode LSPR (named in this work as L-LSPR) peak. However, for the scattering spectra, the NR width is the most sensitive parameter affecting the magnitude of the L-LSPR peak while the aspect ratio is still the most sensitive parameter affecting the position of the L-LSPR [5]. According to these properties, since the OES method has been developed [2, 7, 8] to retrieve the ARD of the polydisperse gold NR ensembles, a method based on optical scattering spectroscopy (OSS) has the potential to be used as an auxiliary method to determine the mean width of the gold NR ensembles.

Compared with the OES method that can be applied conveniently based on, for example, a UV-VIS spectrophotometer [2], the OSS setup is more complicated. To measure the scattering cross section of the gold nanoparticle ensembles, two alternative approaches have been reported. One is to measure the total scattering energy integrated over the 4π solid angle [9], while the other approach is to measure the angular scattering energy within a small solid angle [10, 11]. It is obvious that the latter approach is easier to implement, where the integrating-sphere is not required. However, for the randomly oriented gold NRs, the simulation of the differential scattering cross section in the second approach is more complicated than the simulation of total scattering cross section in the first approach [12, 13].

In this work, we apply the second scheme of the OSS method. By combing the OES and OSS measurements, we propose a so-called optical extinction/scattering spectroscopic (OESS) method to fast characterize the ARD as well as the mean width of gold NR ensembles simultaneously. The main difference between our approach and the previous works [10, 11, 14, 15] is that we solve the inverse scattering problem by using a fast and reliable optimization procedure so that there is no need to know the a priori information about the geometric values of NRs (such as the nominal width of the NRs) beforehand. When constructing the database of the scattering and extinction spectra of the NRs, the T-matrix method [12] was used to simulate the extinction and differential scattering cross sections of gold NR ensembles rigorously. By using the OESS method to characterize different NR ensemble samples experimentally, it is shown that the measurement results coincide with those of the TEM method quite well, while the OESS method is much faster and more cost effective. The OESS method is also more accurate than the OES method to determine the two critical parameters (aspect ratio and mean width) of NRs.

2. Methods

It is in essence an inverse scattering problem by using the OESS method to retrieve the geometric parameters of the gold NRs from the measured extinction and differential scattering spectra. The objective is to retrieve the PDF p(D, AR, e) with respect to the three structural quantities: the width D, the aspect ratio AR and the end-cap factor e of the NRs as well as determining the estimates of the parameters D, AR and e. The measurement method and the numerical retrieving algorithm applied in the OESS method are presented below.

2.1. Measurement method and setup

We apply a commercial UV-VIS spectrophotometer (PekinElmer LAMBDA 950) to measure the extinction spectra of the sample, as shown in Fig. 2(a). The setup is the same as that used in the OES method presented in our previous work [2]. The measurand of this optical setup is the absorbance A(λ) of the sample medium with a length l, which can be expressed as

A (λ) = - log [\frac{I_{e} (λ) I_{r 0} (λ)}{I_{r 1} (λ) I_{m 0} (λ)}] = \frac{l N_{v}}{ln 10} 〈 C_{ext} (λ) 〉,

where λ is the wavelength of light, N_v is the number of NRs per unit volume, 〈C_ext〉 is the average extinction cross section of the sample, and I(λ) with different subscripts represent the intensities of different light beams shown in Fig. 2(a).

Fig. 2 Optical setups used in the OESS method for measuring (a) the optical extinction spectra and (b) the angular light scattering spectra at an angle of 90° of the gold NRs.

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To perform the OSS measurement, we modify the spectrophotometer setup as shown in Fig. 2(b), which can measure the angular scattering spectra at different angles [10]. Here we measure the scattering at an angle of 90°. Two reflective mirrors are added in the measurement beam to change the incident angle of the beam onto the sample. The same sample is inserted in the reference beam with a neutral density (ND) filter to attenuate the beam. The measurand is the transmittance T₀(λ) and can be expressed by

T_{0} (λ) = \frac{I_{s} (λ)}{I_{r 2} (λ)} \frac{I_{r 0} (λ)}{I_{m 0} (λ)},

where I(λ) with different subscripts represent the intensities of different light beams shown in Fig. 2(b). I_r2(λ) and I_s(λ) in Eq. (2) can be expressed as

\begin{array}{l} I_{r 2} (λ) = I_{r 0} (λ) T_{nd} (λ) T_{ext} (λ), \\ I_{s} (λ) = I_{m 0} (λ) R_{1} (λ) R_{2} (λ) T_{lens} (λ) T_{ext} (λ) α_{sca}^{90 °} (λ), \end{array}

where T_nd(λ), T_ext(λ), and T_lens(λ) denote the transmittances of the ND filter, the sample, and the lens, respectively, R₁(λ) and R₂(λ) are the reflectances of the mirrors, and

α_{sca}^{90 °} (λ)

is the angular scattering efficiency.

When measuring $α_{sca}^{90 °} (λ)$ , the same sample was inserted in the reference beam to correct the extinction of the gold NR sample in the measurement beam, as shown in Fig. 2(b). This is because that the scattering spectra measured without correction would differ radically from the real spectra if the extinction of the measurement volume is not taken into account [10]. Besides, here we use a lens to enhance the measurement beam and use a ND filter to attenuate the reference beam. The reason is that the value of $α_{sca}^{90 °} (λ)$ is quite small (due to the weak scattering of NRs at 90°) compared with the value of I_r2(λ) so that the signal-to-noise ratio of the detector is poor when measuring the transmittance T₀(λ) directly. By using the lens and the ND filter, the measurement values of I_s(λ) and I_r2(λ) can be adjusted to the same order of magnitude so as to get the best detection response.

In the OSS measurement, the angular scattering efficiency $α_{sca}^{90 °} (λ)$ is in proportion to the solid angle Ω_90° centered around the direction of scattering, the number of NRs per unit volume N_v, and the average differential scattering cross section 〈dS(λ)〉, i.e., [12]

α_{sca}^{90 °} (λ) \propto N_{v} Ω_{90 °} 〈 d S (λ) 〉 .

Substituting Eq. (3) and Eq. (4) into Eq. (2), we can obtain

T_{0} (λ) \propto S_{90} (λ),

where S₉₀(λ) = N_v〈dS(λ)〉. Eq. (5) connects the angular scattering spectra S₉₀(λ) with the measurand T₀(λ). However, it should be noted that the absolute value of S₉₀(λ) cannot be measured directly but should be calibrated. A simple way is to measure a standard sample whose angular scattering cross section is known beforehand. In this work, we perform calibration by using the standard polystyrene microspheres (PS) from Thermo Fisher Scientific (China) Ltd., whose diameter is 102 nm and whose average differential scattering cross section 〈dS(λ)〉 can be calculated from the average scattering cross section 〈C_sca(λ)〉 by [12, 13]:

〈 d S^{ps} (λ) 〉 = \frac{a_{1}^{ps} (λ, 90 °) 〈 C_{sca}^{ps} (λ) 〉}{4 π},

where the superscript ps stands for the PS sample. In Eq. (6), a₁(λ, 90°) represents the element at the first row and the first column of the Mueller matrix (or phase matrix), which can be calculated conveniently by the T-matrix method [12]. The detailed calculation steps can be found in, for example, Ref. [13].

For the PS sample, the imaginary part of the refractive index is so small that the average absorption cross section can almost be ignored, which means $〈 C_{abs}^{ps} (λ) 〉 = 0$ . Therefore, the extinction of the PS sample is contributed only by scattering. Taking into account Eq. (1), we can easily obtain

〈 C_{sca}^{ps} (λ) 〉 = 〈 C_{ext}^{ps} (λ) 〉 - 〈 C_{abs}^{ps} (λ) 〉 = 〈 C_{ext}^{ps} (λ) 〉 = \frac{ln 10}{l N_{v}^{ps}} A^{ps} (λ) .

Combining Eqs. (5)–(7), we can obtain the absolute value of the angular scattering spectra of the gold NR ensemble as

S_{90}^{g} (λ) = N_{v}^{g} 〈 d S^{g} (λ) 〉 = \frac{ln 10}{l} \frac{T_{0}^{g} (λ)}{T_{0}^{ps} (λ)} \frac{A^{ps} (λ) a_{1}^{ps} (λ)}{4 π},

where the superscripts g and ps stand for the gold NRs and the PS sample, respectively.

2.2. Inverse-problem solution algorithm

With the measured extinction spectra and angular scattering spectra, the estimates of the geometric parameters as well as the PDF p(D, AR, e) of the gold NR ensemble can be retrieved. To solve this inverse scattering problem, we perform a similar retrieving procedure as that presented in our previous work [2]. The main steps are still based on the constrained non-negative regularized least-square procedure. But the integral discrete method and the objective function of the optimization are modified as follows.

For a polydisperse NR ensemble, the total absorbance A is calculated by integrating the contribution of all the composing NRs according to the PDF p(D, AR, e) [2]:

\begin{array}{l} A^{g} (λ) & = \frac{l N_{v}^{g}}{ln 10} 〈 C_{ext}^{g} (λ) 〉 \\ = \frac{l N_{v}^{g}}{ln 10} \int_{D_{min}}^{D_{max}} \int_{{AR}_{min}}^{{AR}_{max}} \int_{e_{min}}^{e_{max}} p (D, A R, e) C_{ext}^{g} (λ, D, A R, e) d D d A R d e . \end{array}

Similarly, the total angular scattering cross section S₉₀(λ) of a polydisperse NR ensemble can be calculated by

S_{90}^{g} (λ) = N_{v}^{g} \int_{D_{min}}^{D_{max}} \int_{{AR}_{min}}^{{AR}_{max}} \int_{e_{min}}^{e_{max}} p (D, A R, e) d S^{g} (λ, D, A R, e) d D d A R d e .

In general, the integrals in Eq. (9) and Eq. (10) cannot be solved analytically and should be discretized for numerical solution. If they are discretized with respect to all the three variables AR, D, and e, the condition number of the reduced linear system is usually too large to produce an accurate and stable solution. Thus in many previous works [2, 4, 7], the width D and the end-cap factor e were usually fixed as their mean values (as we have done in our previous work [2]) or even were ignored, for the reason that the aspect ratio AR is the primary parameter affecting the extinction of the NR ensemble [6]. However, as we have pointed out, the estimate of D and e is significant to the retrieving result of the PDF of AR.

Here, by taking into account the differential scattering spectra of the NR ensemble, the width D is possible to be determined [5] so that we can fix only the end-cap factor e and discretize Eq. (9) and Eq. (10) with respect to AR, D, and λ. Similar with the OES method [2], the adoption of different end-cap factor e in the OESS method can also influence the retrieved results. Even by the OESS measurement, the mean end cap e cannot be determined accurately and should be assumed or be measured by some other methods beforehand.

The discretization results in the following system of linear algebraic equations:

A = C P, S = S_{d} P,

where A and S are M × 1 vectors, C and S_d are M × N matrices, P is a N × 1 vector, and M and N are integer numbers. The vectors A and S contain the values of the measured extinction A(λ) and angular scattering cross section per unit volume S₉₀(λ) at different λ, respectively. The values of extinction cross section of gold NRs

C_{ext}^{g} (λ, D, A R, e)

are stored in the matrix C for various wavelengths λ (rows), various aspect ratios AR (columns) and various widths D (columns) for fixed end-cap shape e. By the same way, the values of differential scattering cross section dS^g(λ, D, AR, e) of gold NRs at the scattering angle 90° are stored in the matrix S_d. The vector P is the PDF to be solved and it has two physical constraints: the non-negativity constraint P ≥ 0 and the standard normalization condition UP = 1 where

U = [\underset{1 \times N}{\underset{︸}{1, 1, \dots, 1}}]

.

Usually the condition number of the linear system Eq. (11) have the order ∼10, thus the problem is an ill-posed inverse problem. Here the constrained non-negative least-square procedure [16] is applied to solve it. The least-squared solution P_RLS can be expressed as [17]:

\begin{array}{l} P_{RLS} & = min_{P} {{‖ A - C P ‖}_{2}^{2} + ω_{S} {‖ S - S_{d} P ‖}_{2}^{2}} \\ = min_{P} {{(A - C P)}^{T} (A - C P) + ω_{S} {(S - S_{d} P)}^{T} (S - S_{d} P)} \\ = min_{P} {P^{T} (C^{T} C + ω_{S} {S_{d}}^{T} S_{d}) P - A^{T} C P - {(A^{T} C P)}^{T} - ω_{S} S^{T} S_{d} P - ω_{S} {(S^{T} S_{d} P)}^{T}}, \end{array}

where ||·||₂ is the Euclidean norm, the superscript T means the transpose of the vectors, and the non-negative weight coefficient ω_S means the weight between the OES and OSS data. The value ω_S = max (A)/max (S) is used here to balance the weight of the OES data and the OSS data, or the amplitude of the OSS data would be much smaller than the amplitude of the OES data and we would not obtain a good optimization result. Note that in Eq. (12), A^TCP and S^TS_dP result in scalars so that their transposes are themselves. Consequently, A^TCP = (A^TCP)^T, S^TS_dP = (S^TS_dP)^T, and Eq. (12) can be written in another equivalent form

\begin{array}{l} P_{RLS} & = min_{P} {P^{T} (C^{T} C + ω_{S} S_{d}^{T} S_{d}) P - 2 (A^{T} C + ω_{S} S^{T} S_{d}) P} \\ = min_{P} {\frac{1}{2} P^{T} Q P + q^{T} P}, \end{array}

where Q = 2(C^TC + ω_SS_d^TS_d) is a symmetric matrix of size N × N, and q = −2(C^TA + ω_SS_d^TS) is a N-dimensional column vector. We use the active set method [16] to find the solution of Eq. (13). Then the retrieved ARD and the mean width D_m can be calculated easily by reshaping the vector P_RLS to a two dimensional matrix which consist of the PDF p(AR, D) and averaging the row and the column of the PDF matrix, respectively.

3. Experimental results and discussions

3.1. Comparison of the OESS and TEM measurements

We applied the OESS method presented above to measure the ARD function p(AR) and the mean width D_m of gold NR ensemble samples. The results are compared with those directly obtained by the TEM method (which is considered as a benchmark). In our comparison experiment, altogether 20 NR samples were measured and analyzed, each of which contains approximately 10¹⁰ NRs per millilitre. Without loss of generality, the results of four samples with different D, AR and e are demonstrated here. The four samples, designated as NR-10, NR-20, NR-30, and NR-40, were obtained from the National Center for Nanoscience and Technology, Beijing, China, whose nominal mean width D_m are 10 nm, 20 nm, 30nm, and 40 nm, respectively.

In the TEM experiment, a transmission electron microscope (Hitachi H-7650B) was used. In order to characterize the samples statistically, about 20 TEM images were taken for each sample and altogether 1522, 1222, 876, and 933 NRs in samples NR-10, NR-20, NR-30, and NR-40 were analyzed, respectively. In the OESS experiment, the measurement of each sample was repeated six times in one hour and the average value of the results was adopted. The measurement range of wavelength λ was 400 nm – 850 nm, with a step of 1 nm. The extinction spectra database and the angular scattering spectra database of the gold NRs [corresponding to the matrices C and S_d in Eq. (11)] were calculated with the T-matrix method. It takes about 30 seconds to calculate a single scattering spectrum of NRs in the wavelength range of 400 nm – 850 nm, with a 1 nm spectral resolution, by using a dual-core 2.13GHz Intel Xeon CPU with 80Gb RAM. After that, the optimization procedure described above was implemented to retrieve the ARD function p(AR) and the mean width D_m of the samples, where AR was discretized in the range of 1 to 5, with a step of 0.1. The inverse-problem solution algorithm was run on a 3.00GHz Intel Core2 Duo CPU with 4Gb RAM and the time consumption is about 5 seconds for each sample.

Table 1 shows the comparison values of the geometric parameters of the four samples measured by the two methods, as well as their relative differences. For the mean end-cap factor e_m, six values with the range 0–1 and the step 0.2 were assumed beforehand. The values of e_m shown in the table 1 were adopted for the reason that the retrieved ARDs by the OESS method coincide with the measured ARDs by the TEM method best.

Table 1. Measurement results of the four gold NR ensemble samples

View Table

It can be seen that the mean width D_m and the mean aspect ratio AR_m derived by the two methods coincide with each other well, with their relative difference smaller than 6% and 2%, respectively. These show that our OESS measurement results are reliable. For the standard deviation σ_AR (which represents the polydispersity of the ARD), the two sets of values also co-incide with each other relatively well, where the small difference may be owing to the deviation between the real shape of the gold NRs and the geometric model adopted in our calculation.

Besides the geometric parameters, the mass-volume concentration C_g of the NRs can also be determined by the OESS method. It can be derived as C_g = ρNvV · P_RLS, where V is a row vector consisting of the volume of each nanorod geometry in the sample and ρ is the density of bulk gold. The values of C_g measured by the OESS method for the four samples NR-10, NR-20, NR-30, NR-40 are 14.54±0.26 μg/ml, 15.78±0.17 μg/ml, 18.03±0.14 μg/ml, and 15.34 ± 0.24 μg/ml, respectively. Since these concentration values cannot be obtained by the TEM method, we cannot make a comparison of the two methods here.

The measurement results of the four samples are shown in detail in Fig. 3 and Fig. 6. From the four TEM images, it is clearly seen that the gold NR ensembles are polydisperse. In the four extinction spectra, the L-LSPR peaks are at different wavelengths, while the transverse mode LSPR (named by T-LSPR here) peaks are nearly at the same wavelength. However, in the four 90° scattering spectra, the L-LSPR peaks are slightly different from those in the extinction spectra and the T-LSPR peaks are much weaker than their counterparts in the extinction spectra. Furthermore, for different samples with different widths, the maximum intensities of the extinction spectra are in a small range (∼0.3 – 0.5) and are close to each other, but the maximum intensities of the scattering spectra are in a relative bigger range (∼0.006 – 0.05) and deviate from each other significantly. These different features of the extinction spectra and the scattering spectra show the potential for retrieving both the ARD function p(AR) and the mean width D_m by the OESS method, as we mentioned before.

Fig. 3 Measurement results of the sample NR-10: (a) the absorbance A measured by the OES method and the 90° scattering intensity S₉₀ (cm⁻¹) measured by the OSS method; (b) the ARD function retrieved by the OESS method; (c) the ARD function measured by the TEM method. In (a), the original measurement data of the extinction spectra A-Exp. (circle dots) and the scattering spectra S90-Exp. (square dots, multiplied by 60), as well as the corresponding numerically reproduced extinction spectra A-Fit (solid line) and scattering spectra S90-Fit (dashed line, multiplied by 60) according to the retrieved NR parameters are given. The inset in (a) shows the TEM image of the sample. In (b) and (c), both the discrete ARD and a Gaussian fit of it are given; the values in parentheses give the mean AR and the standard deviation of the ARD.

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Fig. 4 The same as Fig. 3, but for sample NR-20.

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Fig. 5 The same as Fig. 3, but for sample NR-30.

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Fig. 6 The same as Fig. 3, but for sample NR-40.

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By comparing the OESS results with those obtained by the TEM method in Fig. 3 and Fig. 6, it is seen that the ARD function p(AR) derived by the two methods in general coincide with each other well. With the retrieved ARD function p(AR) and the mean width D_m, the extinction spectra and 90° angular scattering spectra of the four samples were also numerically reproduced by Eq. (9) and Eq. (10), as shown in Fig. 3(a) – Fig. 6(a), which also coincide with the measured extinction spectra and the angular scattering spectra quite well. Thus Table 1 and Fig. 3 and Fig. 6 have shown that the retrieved results by the OESS method are reliable in the characterization of the gold NRs ensembles. However, there are still some small differences between the measurement results derived by the OESS method and the TEM method. In the following, we discuss in detail these differences and their possible causes.

3.2. Discussions

As seen in Fig. 3(b) and (c) as well as Fig. 4(b) and (c), for the samples NR-10 and NR-20, the ARD function obtained by the OESS method has a significant distribution in the range of 1–2, while the TEM results have quite few NRs in this range. The main reason is that in the image processing of the TEM method, one may (as we did) ignore most byproducts (such as spheres, cubes etc.) in the samples, as shown in the inset TEM images in Fig. 3(a) and Fig. 4(a). However, these byproducts also contribute to the extinction and angular scattering spectra and thus can be detected by the OESS method, as indicated by the dashed circles in Fig. 3 and Fig. 4. This, from another point of view, shows an advantage of the OESS method: by measuring the scattering and extinction of a large amount of gold NRs, the global features of the ensemble can be captured statistically without losing the contribution of any composing particles (even the byproducts).

For the sample NR-40 in Fig. 6, the difference between the measured and reproduced extinction spectra of the sample is more significant (especially around the T-LSPR peak). The main reason is that the mean aspect ratio of this sample (which is around 1.6) is much smaller than those of the other three samples so that the L-LSPR peak is very close to the T-LSPR peak. In this case, the disturbance by the byproducts (whose LSPR mainly contributes to the T-LSPR of the extinction spectra) is more significant and is more difficult to be distinguished by the OESS method, as indicated in Figs. 6(b) and (c). Therefore, we can conclude that by using the OESS method, the detection of the byproducts would be more sensitive if the LSPR of the byproducts is not too close to the L-LSPR of the gold NR ensemble.

For the sample NR-30 in Fig. 5, there are three LSPR peaks in the extinction spectra. Obviously, the left LSPR peak (∼520 nm) is mainly contributed by the T-LSPR of the gold NRs (although the LSPR of the byproducts, such as the gold spheres of diameter around 30 nm, would also contribute to this T-LSPR peak). The right LSPR peak (∼720nm) is the L-LSPR of the gold NRs. These two characteristic LSPR peaks can also be seen in the extinction spectra of the other three samples, as shown in Fig. 3(a), Fig. 4(a), and Fig. 6(a). The LSPR peak in the middle (∼610nm), however, is likely to be caused by the LSPR of the cubic byproducts with width 60 nm and aspect ratio around 1.2, as indicated by the dashed circles in Fig. 5(a). In addition to these byproducts, some longer gold NRs with aspect ratio between 1.5 and 2 are also detected by the OESS method (which do not actually exist), as indicated in Fig. 5(b). The main reason is that in our extinction and scattering spectra database, only gold NRs but no rectangular cuboid nanoparticles were calculated. Consequently, the LSPR in the middle was actually fitted by the contribution of longer gold NRs. To confirm this inference, we have calculated the extinction spectra of two gold nanoparticle ensembles: an ensemble of rectangular cuboid gold nanoparticles with D = 60 nm, AR = 1.2, and an ensemble of gold NRs with D = 30 nm, AR = 1.8, e = 0.8, as shown in Fig. 7(a). Indeed, we can see that the LSPR peak around 610 nm is reproduced in both spectra, where the peak of the cubic nanoparticles is even stronger.

Fig. 7 Comparison of the extinction cross sections of different gold nanoparticles: (a) rectangular cuboid gold nanoparticles and gold NRs, (b) gold NRs with different end caps.

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For the sample NR-30, one may note that the mean end-cap factor e_m obtained by the OESS method (e_m = 0.8) deviates from the value obtained by the TEM method (e_m = 0.4) significantly, as shown in Table 1. The possible reason is that the end cap shape of the NRs in the sample NR-30 is sharper, as shown in the inset of Fig. 5(a), which deviates significantly from our cylindrical NR model with semi-spheroidal end cap. To corroborate this, we have calculated three NR ensembles with different end-cap shapes, as shown in Fig. 7(b). It is seen evidently that for the same value of end-cap factor e_m = 0.4, the L-LSPR peak of gold NRs with a cone-like end cap is blue shifted compared with that of gold NRs with a semi-spheroidal end cap. As a consequence, the extinction spectrum of the NRs with cone-like end cap of factor 0.4 matches the spectrum of the NRs with semi-spheroidal end cap of factor 0.8 better. For this reason, the fitted value of the end-cap facotr that we obtained by the OESS measurement is 0.8, but not 0.4. These imply that in the practical characterization of gold NR ensembles, the database of extinction and scattering spectra should be expanded, by calculating not only NRs but also nanoparticles of other shapes (such as spheres, cubes, and NRs with special end-cap shapes). By taking these measures, the measurement accuracy of the OESS method can be further improved.

4. Conclusion

We have proposed an OESS method by combining the OES and OSS measurements to fast characterize the critical geometric parameters AR and D, as well as the probability density functions p(AR) of gold NR ensembles statistically. To perform the OESS method, the angular scattering spectra at the angle of 90° are measured in addition to the extinction spectra measured by the UV-VIS spectrometer. To solve the inverse scattering problem, the extinction cross section and the differential scattering cross section of polydisperse NR ensembles are calculated rigorously by the T-matrix method. Then the critical parameters are retrieved by an optimization process with data fitting to the measured spectra.

By characterizing different samples of gold NR ensembles experimentally with the OESS method, it is shown that the measured ARD and mean width D_m coincide well with those obtained by the TEM method. By using the OESS method, it is also possible to determine the mass-volume concentration of NRs, which is unable to be measured by the TEM method. The comparison results indicate that the OESS method is fast, accurate, and cost effective to measure the critical geometric parameters of the gold NR ensemble. Further improvement is to enlarge the database of the extinction and scattering spectra of gold nanoparticles of different shapes (such as rectangular cuboid nanoparticles and NRs with special end-cap shapes) by considering the byproducts of practical samples, by which the retrieving accuracy can be further improved. The proposed OESS method has the potential to be developed for characterizing not only gold NR ensembles but also other non-spherical metal nanoparticles such as silver NRs.

Acknowledgments

We would like to thank Prof. Xiaochun Wu in the National Center for Nanoscience and Technology, China for preparing the gold nanorod samples. This work was supported by the Ministry of Science and Technology of China (Project No. 2011BAK15B03) and the Natural Science Foundation of China (Project No. 61161130005).

References and links

1. N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer 111, 1–35 (2010). [CrossRef]

2. N. Xu, B. Bai, Q. Tan, and G. Jin, “Fast statistical measurement of aspect ratio distribution of gold nanorod ensembles by optical extinction spectroscopy,” Opt. Express 21, 2987–3000 (2013). [CrossRef] [PubMed]

3. B. Khlebtsov, V. Khanadeev, T. Pylaev, and N. Khlebtsov, “A new t-matrix solvable model for nanorods: Tem-based ensemble simulations supported by experiments,” J. Phys. Chem. C 115, 6317–6323 (2011). [CrossRef]

4. B. N. Khlebtsov, V. A. Khanadeev, and N. G. Khlebtsov, “Observation of extra-high depolarized light scattering spectra from gold nanorods,” The Journal of Physical Chemistry C 112, 12760–12768 (2008). [CrossRef]

5. K.-S. Lee and M. A. El-Sayed, “Dependence of the enhanced optical scattering efficiency relative to that of absorption for gold metal nanorods on aspect ratio, size, end-cap shape, and medium refractive index,” The Journal of Physical Chemistry B 109, 20331–20338 (2005). [CrossRef]

6. S. W. Prescott and P. Mulvaney, “Gold nanorod extinction spectra,” J. Appl. Phys. 99, 123504 (2006). [CrossRef]

7. S. Eustis and M. A. El-Sayed, “Determination of the aspect ratio statistical distribution of gold nanorods in solution from a theoretical fit of the observed inhomogeneously broadened longitudinal plasmon resonance absorption spectrum,” J. Appl. Phys. 100, 044324 (2006). [CrossRef]

8. O. Peña, L. Rodríguez-Fernández, V. Rodríguez-Iglesias, G. Kellermann, A. Crespo-Sosa, J. C. Cheang-Wong, H. G. Silva-Pereyra, J. Arenas-Alatorre, and A. Oliver, “Determination of the size distribution of metallic nanoparticles by optical extinction spectroscopy,” Appl. Opt. 48, 566–572 (2009). [CrossRef] [PubMed]

9. D. D. Evanoff and G. Chumanov, “Size-controlled synthesis of nanoparticles. 2. measurement of extinction, scattering, and absorption cross sections,” The Journal of Physical Chemistry B 108, 13957–13962 (2004). [CrossRef]

10. V. A. Bogatyrev, L. A. Dykman, K. B. N., and N. G. Khlebtsov, “Measurement of mean size and evaluation of polydispersity of gold nanoparticles from spectra of optical absorption and scattering,” Optics and Spectroscopy 96, 128–135 (2004). [CrossRef]

11. G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C 114, 2853–2860 (2010). [CrossRef]

12. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

13. M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991). [CrossRef]

14. A. V. Alekseeva, V. A. Bogatyrev, L. A. Dykman, B. N. Khlebtsov, L. A. Trachuk, A. G. Melnikov, and N. G. Khlebtsov, “Preparation and optical scattering characterization of gold nanorods and their application to a dot-immunogold assay,” Appl. Opt. 44, 6285–6295 (2005). [CrossRef] [PubMed]

15. B. Khlebtsov, V. Khanadeev, and B. N. Khlebtsov, “Tunable depolarized light scattering from gold and gold/silver nanorods,” Physical Chemistry Chemical Physics 12, 3210 (2010). [CrossRef] [PubMed]

16. P. Gill, W. Murray, and M. Wright, Numerical Linear Algebra and Optimization (Addison Wesley, 1991).

17. J. Mroczka and D. Szczuczynski, “Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements,” Appl. Opt. 51, 1715–1723 (2012). [CrossRef] [PubMed]

Sample no.	D_m (nm)			AR_m			σ_AR		e_m

	TEM	OESS	RD*	TEM	OESS	RD	TEM	OESS	TEM	OESS
NR-10	12.5	13.2	5.6%	3.112	3.105	0.2%	0.56	0.54	0.7	0.6
NR-20	17.1	16.8	1.6%	2.536	2.519	0.7%	0.32	0.24	0.8	0.6
NR-30	31.6	30.4	4.1%	2.659	2.610	1.9%	0.35	0.51	0.4	0.8
NR-40	40.9	38.5	5.8%	1.641	1.619	1.3%	0.13	0.16	0.7	0.6

Accurate geometric characterization of gold nanorod ensemble by an inverse extinction/scattering spectroscopic method

Abstract

1. Introduction

2. Methods

2.1. Measurement method and setup

2.2. Inverse-problem solution algorithm

3. Experimental results and discussions

3.1. Comparison of the OESS and TEM measurements

3.2. Discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (13)

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