## Fast statistical measurement of aspect ratio distribution of gold nanorod ensembles by optical extinction spectroscopy |

Optics Express, Vol. 21, Issue 3, pp. 2987-3000 (2013)

http://dx.doi.org/10.1364/OE.21.002987

Acrobat PDF (1192 KB)

### Abstract

Fast and accurate geometric characterization and metrology of noble metal nanoparticles such as gold nanorod (NR) ensembles is highly demanded in practical production, trade, and application of nanoparticles. Traditional imaging methods such as transmission electron microscopy (TEM) need to measure a sufficiently large number of nanoparticles individually in order to characterize a nanoparticle ensemble statistically, which are time-consuming and costly, though accurate enough. In this work, we present the use of optical extinction spectroscopy (OES) to fast measure the aspect ratio distribution (which is a critical geometric parameter) of gold NR ensembles statistically. By comparing with the TEM results experimentally, it is shown that the mean aspect ratio obtained by the OES method coincides with that of the TEM method well if the other NR structural parameters are reasonably pre-determined, while the OES method is much faster and of more statistical significance. Furthermore, the influences of these NR structural parameters on the measurement results are thoroughly analyzed and the possible measures to improve the accuracy of solving the ill-posed inverse scattering problem are discussed. By using the OES method, it is also possible to determine the mass-volume concentration of NRs, which is helpful for improving the solution of the inverse scattering problem while is unable to be obtained by the TEM method.

© 2013 OSA

## 1. Introduction

3. X. Huang, S. Neretina, and M. A. El-Sayed, “Gold nanorods: From synthesis and properties to biological and biomedical applications,” Adv. Mater. **21**, 4880–4910 (2009). [CrossRef]

*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

4. L. B. Scaffardi, N. Pellegri, O. de Sanctis, and J. O. Tocho, “Sizing gold nanoparticles by optical extinction spectroscopy,” Nanotech. **16**, 158–163 (2005). [CrossRef]

5. W. Haiss, N. T. K. Thanh, J. Aveyard, and D. G. Fernig, “Determination of size and concentration of gold nanoparticles from uv-vis spectra,” Anal. Chem. **79**, 4215–4221 (2007). [CrossRef] [PubMed]

*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

6. N. G. Khlebtsov, “Determination of size and concentration of gold nanoparticles from extinction spectra,” Anal. Chem. **80**, 6620–6625 (2008). [CrossRef] [PubMed]

*σ*> 0.1, the polydispersity should also be taken into account [6

6. N. G. Khlebtsov, “Determination of size and concentration of gold nanoparticles from extinction spectra,” Anal. Chem. **80**, 6620–6625 (2008). [CrossRef] [PubMed]

*et al*. [7

7. 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. S. Link and M. A. El-Sayed, “Simulation of the optical absorption spectra of gold nanorods as a function of their aspect ratio and the effect of the medium dielectric constant,” J. Phys. Chem. B **109**, 10531C10532 (2005). [CrossRef]

11. 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]

12. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**, 1491–1499 (1994). [CrossRef]

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

15. W. Yanpeng and N. Peter, “Finite-difference time-domain modeling of the optical properties of nanoparticles near dielectric substrates,” J. Phys. Chem. C **114**, 7302–7307 (2010). [CrossRef]

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18. M. Karamehmedović, R. Schuh, V. Schmidt, T. Wriedt, C. Matyssek, W. Hergert, A. Stalmashonak, G. Seifert, and O. Stranik, “Comparison of numerical methods in near-field computation for metallic nanoparticles,” Opt. Express **19**, 8939–8953 (2011). [CrossRef]

19. T. Wriedt and U. Comberg, “Comparison of computational scattering methods,” J. Quant. Spectrosc. Radiat. Transfer **60**, 411 – 423 (1998). [CrossRef]

21. M. Karamehmedović, R. Schuh, V. Schmidt, T. Wriedt, C. Matyssek, W. Hergert, A. Stalmashonak, G. Seifert, and O. Stranik, “Comparison of numerical methods in near-field computation for metallic nanoparticles,” Opt. Express **19**, 8939–8953 (2011). [CrossRef]

11. 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]

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

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]

*et al*. [22

22. 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]

23. R. Gans, “*Ü*ber die form ultramikroskopischer goldteilchen,” Annalen der Physik **342**, 881–900 (1912). [CrossRef]

*et al*. [11

11. 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]

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

*et al*. [22

22. 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]

*a priori*information and obtained an excellent agreement between the simulated and measured spectra.

**115**, 6317–6323 (2011). [CrossRef]

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

*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

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

*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

**115**, 6317–6323 (2011). [CrossRef]

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

27. I. R. Ciric and F. R. Cooray, “Benchmark solutions for electromagnetic scattering by systems of randomly oriented spheroids,” J. Quant. Spectrosc. Radiat. Transfer **63**, 131–148 (1999). [CrossRef]

28. P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

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]

3. X. Huang, S. Neretina, and M. A. El-Sayed, “Gold nanorods: From synthesis and properties to biological and biomedical applications,” Adv. Mater. **21**, 4880–4910 (2009). [CrossRef]

6. N. G. Khlebtsov, “Determination of size and concentration of gold nanoparticles from extinction spectra,” Anal. Chem. **80**, 6620–6625 (2008). [CrossRef] [PubMed]

**115**, 6317–6323 (2011). [CrossRef]

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

*γ*of gold NR is increased and can be modified by [4

4. L. B. Scaffardi, N. Pellegri, O. de Sanctis, and J. O. Tocho, “Sizing gold nanoparticles by optical extinction spectroscopy,” Nanotech. **16**, 158–163 (2005). [CrossRef]

*γ*

_{bulk}is the damping constant of bulk gold,

*υ*

_{F}is the electron velocity at the Fermi surface,

*A*

_{s}is the surface electron scattering constant, and

*L*

_{eff}is the effective mean free path for collisions with the boundary. We take the values

*γ*

_{bulk}= 1.64 × 10

^{14}s

^{−1}and

*υ*

_{F}= 1.41 × 10

^{15}nm s

^{−1}from Ref. [4

4. L. B. Scaffardi, N. Pellegri, O. de Sanctis, and J. O. Tocho, “Sizing gold nanoparticles by optical extinction spectroscopy,” Nanotech. **16**, 158–163 (2005). [CrossRef]

*L*

_{eff}= 4

*V/S*was given by Ref. [29

29. E. A. Coronado and G. C. Schatz, “Surface plasmon broadening for arbitrary shape nanoparticles: A geometrical probability approach,” J. Chem. Phys. **119**, 3926–3934 (2003). [CrossRef]

*V*and

*S*are the volume and the surface area of the NR, respectively. The surface electron scattering constant

*A*

_{s}can be considered as a free parameter with the value varying around 1. The value of

*A*

_{s}= 0.3 [30

30. C. Novo, D. Gomez, J. Perez-Juste, Z. Zhang, H. Petrova, M. Reismann, P. Mulvaney, and G. V. Hartland, “Contributions from radiation damping and surface scattering to the linewidth of the longitudinal plasmon band of gold nanorods: a single particle study,” Phys. Chem. Chem. Phys. **8**, 3540–3546 (2006). [CrossRef] [PubMed]

31. N. G. Khlebtsov, V. A. Bogatyrev, L. A. Dykman, and A. G. Melnikov, “Spectral extinction of colloidal gold and its biospecific conjugates,” J. Colloid Interface Sci. **180**, 436 – 445 (1996). [CrossRef]

*λ*is the wavelength of light in nanometers.

### 2.2. Calculation of the absorbance of NR ensembles

22. 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]

*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

*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.

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]

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

**100**, 044324 (2006). [CrossRef]

*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: By this treatment, we just need to discretize Eq. (6) with respect to

*AR*and

*λ*. Then the condition number of the linear system would be much smaller and the optimization process would be faster and more stable.

**A**and

**P**are M × 1 and N × 1 vectors, respectively, and

**C**is a M × N matrix. The vector

**A**contains the measured extinction values at different

*λ*and the matrix

_{m}**C**consists of the calculated extinction cross sections

*C*

_{ext}for NRs with each pair of

*λ*and

_{m}*AR*. The vector

_{n}**P**is the PDF to be solved. Their specific expressions are shown below

*m*and

*n*are integers, and the superscript T means the transpose of the vectors. Here we consider M > N so that Eq. (7) is an overdetermined system with N unknowns.

*p*(

*AR*) has two physical constraints: the non-negativity constraint

_{n}*p*(

*AR*) ≥ 0 and the standard normalization condition ∑

_{n}*(*

_{n}p*AR*) = 1.

_{n}**C**

*is usually ill-conditioned, the inverse problem formulated in terms of Eq. (6) is ill-posed [32*

_{mn}32. P. C. Hansen, “Regularization tools: A matlab package for analysis and solution of discrete ill-posed problems,” NUMER ALGORITHMS **6**, 1–35 (1994). [CrossRef]

33. J. Mroczka and D. Szczuczynski, “Improved regularized solution of the inverse problem in turbidimetric measurements,” Appl. Opt. **49**, 4591–4603 (2010). [CrossRef] [PubMed]

32. P. C. Hansen, “Regularization tools: A matlab package for analysis and solution of discrete ill-posed problems,” NUMER ALGORITHMS **6**, 1–35 (1994). [CrossRef]

35. 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]

**P**

_{RLS}of Eq. (7) is given as [32

32. P. C. Hansen, “Regularization tools: A matlab package for analysis and solution of discrete ill-posed problems,” NUMER ALGORITHMS **6**, 1–35 (1994). [CrossRef]

_{2}is the Euclidean norm,

*γ*is the regularization factor,

**P**

^{*}is an assumed

*a priori*assumed solution (taken as

**P**

^{*}=

**0**here),

**L**is typically either an identity matrix (as we take here) or a discrete approximation of the derivative operator [32

**6**, 1–35 (1994). [CrossRef]

33. J. Mroczka and D. Szczuczynski, “Improved regularized solution of the inverse problem in turbidimetric measurements,” Appl. Opt. **49**, 4591–4603 (2010). [CrossRef] [PubMed]

**Q**= 2(

**C**

^{T}

**C**+

*γ*

^{2}

**L**

^{T}

**L**) is a symmetrical matrix of size N × N, and

**q**= −2

**C**

^{T}

**A**is a N-dimensional column vector. We use the active set method [34] to find the solution of Eq. (13). A mean square error (

*mse*) defined below is used to evaluate the quality of the solution: where

**A**

_{cal}=

**CP**

_{RLS}is the fitted optimal solution of the absorbance. We set the criteria that if

*mse*is smaller than 1 × 10

^{−3}, the solution

**P**

_{RLS}is acceptable.

## 3. Experiment and results

*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 10

^{10}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.

*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.

*D̄*, the mean end-cap factor

*ē*and the ARD

*p*(

*AR*).

*D̄*and dispersancy.

*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: where

*σ*

_{i,AR}are the mean value and standard deviation of the

*i*th Gaussian function

*p*(

*AR*), respectively. The constant

*w*was chosen such that

_{i}*p*(

*AR*) satisfies the standard normalization condition ∑

*(*

_{n}p*AR*) = 1.

_{n}*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.

*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

*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

*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).

*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

**99**, 123504 (2006). [CrossRef]

*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

*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.

*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), where

*R*

^{2}means the coefficient of determination of the linear fit. According to Eq. (16), we know that a change of 10 nm in

*D̄*leads to around 7% change in the retrieved

*σ*of the retrieved

_{AR}*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.

*N*and the mass-volume concentration

_{v}*C*of NRs on the mean width

_{g}*D̄*. Here,

*N*can be obtained by the optimization progress described in Section 2 and the mass-volume concentration

_{v}*C*is derived as

_{g}*C*=

_{g}*ρNv*

**V**

*·*

_{n}**P**

_{RLS}, where

**V**

*is a row vector consisting of the volume of each nanorod of*

_{n}*AR*and

_{n}*ρ*is the density of bulk gold. Since

*N*has an evident dependence on the mean width

_{v}*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*(= 22.45 ± 0.07

_{g}*μ*g/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

*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:

*D̄*, the fitted

*ē*evidently. According to Eq. (17), a change of

*ē*by 0.1 may lead to the change of

*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*only depend on

_{v}*ē*slightly, as shown in Fig. 7. Although

*N*is also dependent on the end-cap factor

_{v}*ē*, the relativity is much smaller, compared with the influence of

*D̄*. Thus it is difficult to use

*N*as

_{v}*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*= 22.39 ± 0.26

_{g}*μ*g/ml), which changes only a little with respect to

*ē*and the value coincides well with

*C*= 22.45 ± 0.07

_{g}*μ*g/ml obtained in subsection 4.1. Therefore, we can conclude that the mass-volume concentration

*C*of the gold NR ensembles can be determined accurately by the OES method, without knowing the other structrual parameters (

_{g}*ē*and

*D̄*) beforehand.

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

*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*and

_{D}*σ*that are defined as the standard deviations of the PDFs of

_{e}*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*.

*p*(

*AR*) obtained by the OES method, by taking different

*σ*and

_{D}*σ*. The

_{e}*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

*σ*of

_{AR}*p*(

*AR*) are around 1% when

*σ*and

_{D}*σ*are significantly increased from 0 to 5.4 and to 0.143m, respectively. This shows that the influences by the polydispersity

_{e}*σ*and

_{D}*σ*are pretty small and can be ignored, compared with the influences by the mean width

_{e}*D̄*and the mean end cap

*ē*.

### 4.4. Influence by the surface electron scattering constant A_{s}

*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

**115**, 6317–6323 (2011). [CrossRef]

30. C. Novo, D. Gomez, J. Perez-Juste, Z. Zhang, H. Petrova, M. Reismann, P. Mulvaney, and G. V. Hartland, “Contributions from radiation damping and surface scattering to the linewidth of the longitudinal plasmon band of gold nanorods: a single particle study,” Phys. Chem. Chem. Phys. **8**, 3540–3546 (2006). [CrossRef] [PubMed]

*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

*σ*decreases by 20%. The decrease of

_{AR}*σ*can be explained by Eq. (1): the increase of

_{AR}*A*

_{s}increases the damping constant

*γ*, which leads to the broadening of the L-LSPR [30

30. C. Novo, D. Gomez, J. Perez-Juste, Z. Zhang, H. Petrova, M. Reismann, P. Mulvaney, and G. V. Hartland, “Contributions from radiation damping and surface scattering to the linewidth of the longitudinal plasmon band of gold nanorods: a single particle study,” Phys. Chem. Chem. Phys. **8**, 3540–3546 (2006). [CrossRef] [PubMed]

*σ*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

_{AR}**8**, 3540–3546 (2006). [CrossRef] [PubMed]

36. B. N. Khlebtsov and N. G. Khlebtsov, “Multipole plasmons in metal nanorods: Scaling properties and dependence on particle size, shape, orientation, and dielectric environment,” J. Phys. Chem. C **111**, 11516–11527 (2007). [CrossRef]

*A*

_{s}are pretty small (compared with the influences by

*D̄*and

*ē*) and can also be ignored.

## 5. Conclusions

*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*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.

_{g}*D̄*and mean end-cap factor

*ē*linearly. A change of 10 nm in

*D̄*may lead to around 7% change in the retrieved

*ē*by 0.1 may lead to the change of

*σ*and

_{D}*σ*, 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

_{e}*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*, which can be achieved by the OES method itself. To get a good guess of the end-cap shape

_{v}*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.

## Acknowledgments

## References and links

1. | N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer |

2. | S. A. Maier, |

3. | X. Huang, S. Neretina, and M. A. El-Sayed, “Gold nanorods: From synthesis and properties to biological and biomedical applications,” Adv. Mater. |

4. | L. B. Scaffardi, N. Pellegri, O. de Sanctis, and J. O. Tocho, “Sizing gold nanoparticles by optical extinction spectroscopy,” Nanotech. |

5. | W. Haiss, N. T. K. Thanh, J. Aveyard, and D. G. Fernig, “Determination of size and concentration of gold nanoparticles from uv-vis spectra,” Anal. Chem. |

6. | N. G. Khlebtsov, “Determination of size and concentration of gold nanoparticles from extinction spectra,” Anal. Chem. |

7. | 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. |

8. | C. F. Bohren and D. R. Huffman, |

9. | S. Link and M. A. El-Sayed, “Simulation of the optical absorption spectra of gold nanorods as a function of their aspect ratio and the effect of the medium dielectric constant,” J. Phys. Chem. B |

10. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

11. | 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 |

12. | B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

13. | V. L. Y. Loke and M. P. Mengüç, “Surface waves and atomic force microscope probe-particle near-field coupling: discrete dipole approximation with surface interaction,” J. Opt. Soc. Am. A |

14. | S. W. Prescott and P. Mulvaney, “Gold nanorod extinction spectra,” J. Appl. Phys. |

15. | W. Yanpeng and N. Peter, “Finite-difference time-domain modeling of the optical properties of nanoparticles near dielectric substrates,” J. Phys. Chem. C |

16. | U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B |

17. | V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. Garcia de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. |

18. | M. Karamehmedović, R. Schuh, V. Schmidt, T. Wriedt, C. Matyssek, W. Hergert, A. Stalmashonak, G. Seifert, and O. Stranik, “Comparison of numerical methods in near-field computation for metallic nanoparticles,” Opt. Express |

19. | T. Wriedt and U. Comberg, “Comparison of computational scattering methods,” J. Quant. Spectrosc. Radiat. Transfer |

20. | T. Wriedt, “Light scattering theories and computer codes,” J. Quant. Spectrosc. Radiat. Transfer |

21. | M. Karamehmedović, R. Schuh, V. Schmidt, T. Wriedt, C. Matyssek, W. Hergert, A. Stalmashonak, G. Seifert, and O. Stranik, “Comparison of numerical methods in near-field computation for metallic nanoparticles,” Opt. Express |

22. | 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. |

23. | R. Gans, “ |

24. | B. N. Khlebtsov, V. A. Khanadeev, and N. G. Khlebtsov, “Observation of extra-high depolarized light scattering spectra from gold nanorods,” The J. Phys. Chem. C |

25. | M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A |

26. | F. Kuik, J. F. Dehaan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer |

27. | I. R. Ciric and F. R. Cooray, “Benchmark solutions for electromagnetic scattering by systems of randomly oriented spheroids,” J. Quant. Spectrosc. Radiat. Transfer |

28. | P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B |

29. | E. A. Coronado and G. C. Schatz, “Surface plasmon broadening for arbitrary shape nanoparticles: A geometrical probability approach,” J. Chem. Phys. |

30. | C. Novo, D. Gomez, J. Perez-Juste, Z. Zhang, H. Petrova, M. Reismann, P. Mulvaney, and G. V. Hartland, “Contributions from radiation damping and surface scattering to the linewidth of the longitudinal plasmon band of gold nanorods: a single particle study,” Phys. Chem. Chem. Phys. |

31. | N. G. Khlebtsov, V. A. Bogatyrev, L. A. Dykman, and A. G. Melnikov, “Spectral extinction of colloidal gold and its biospecific conjugates,” J. Colloid Interface Sci. |

32. | P. C. Hansen, “Regularization tools: A matlab package for analysis and solution of discrete ill-posed problems,” NUMER ALGORITHMS |

33. | J. Mroczka and D. Szczuczynski, “Improved regularized solution of the inverse problem in turbidimetric measurements,” Appl. Opt. |

34. | P. Gill, W. Murray, and M. Wright, |

35. | J. Mroczka and D. Szczuczynski, “Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements,” Appl. Opt. |

36. | B. N. Khlebtsov and N. G. Khlebtsov, “Multipole plasmons in metal nanorods: Scaling properties and dependence on particle size, shape, orientation, and dielectric environment,” J. Phys. Chem. C |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(290.3200) Scattering : Inverse scattering

(290.5850) Scattering : Scattering, particles

(160.4236) Materials : Nanomaterials

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: November 27, 2012

Revised Manuscript: January 18, 2013

Manuscript Accepted: January 21, 2013

Published: January 31, 2013

**Virtual Issues**

Vol. 8, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Ninghan Xu, Benfeng Bai, Qiaofeng Tan, and Guofan Jin, "Fast statistical measurement of aspect ratio distribution of gold nanorod ensembles by optical extinction spectroscopy," Opt. Express **21**, 2987-3000 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2987

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### References

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