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

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    [CrossRef]
  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]
  6. N. G. Khlebtsov, “Determination of size and concentration of gold nanoparticles from extinction spectra,” Anal. Chem.80, 6620–6625 (2008).
    [CrossRef] [PubMed]
  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]
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    [CrossRef]
  10. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002).
  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. C115, 6317–6323 (2011).
    [CrossRef]
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  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.37, 1792–1805 (2008).
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    [CrossRef]
  23. R. Gans, “Über die form ultramikroskopischer goldteilchen,” Annalen der Physik342, 881–900 (1912).
    [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. C112, 12760–12768 (2008).
    [CrossRef]
  25. M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A8, 871–882 (1991).
    [CrossRef]
  26. F. Kuik, J. F. Dehaan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer47, 477–489 (1992).
    [CrossRef]
  27. I. R. Ciric and F. R. Cooray, “Benchmark solutions for electromagnetic scattering by systems of randomly oriented spheroids,” J. Quant. Spectrosc. Radiat. Transfer63, 131–148 (1999).
    [CrossRef]
  28. P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B6, 4370–4379 (1972).
    [CrossRef]
  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]
  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]
  32. P. C. Hansen, “Regularization tools: A matlab package for analysis and solution of discrete ill-posed problems,” NUMER ALGORITHMS6, 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]
  34. P. Gill, W. Murray, and M. Wright, Numerical Linear Algebra and Optimization (Addison Wesley, 1991).
  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]
  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. C111, 11516–11527 (2007).
    [CrossRef]

2012

2011

2010

W. Yanpeng and N. Peter, “Finite-difference time-domain modeling of the optical properties of nanoparticles near dielectric substrates,” J. Phys. Chem. C114, 7302–7307 (2010).
[CrossRef]

N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer111, 1–35 (2010).
[CrossRef]

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

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. A27, 2293–2303 (2010).
[CrossRef]

2009

T. Wriedt, “Light scattering theories and computer codes,” J. Quant. Spectrosc. Radiat. Transfer110, 833 – 843 (2009). Light Scattering: Mie and More Commemorating 100 years of Mie’s 1908 publication.
[CrossRef]

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]

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]

2008

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

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.37, 1792–1805 (2008).
[CrossRef] [PubMed]

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. C112, 12760–12768 (2008).
[CrossRef]

2007

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]

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. C111, 11516–11527 (2007).
[CrossRef]

2006

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]

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]

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

2005

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B72, 195429 (2005).
[CrossRef]

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. B109, 10531C10532 (2005).
[CrossRef]

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]

2003

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]

1999

I. R. Ciric and F. R. Cooray, “Benchmark solutions for electromagnetic scattering by systems of randomly oriented spheroids,” J. Quant. Spectrosc. Radiat. Transfer63, 131–148 (1999).
[CrossRef]

1998

T. Wriedt and U. Comberg, “Comparison of computational scattering methods,” J. Quant. Spectrosc. Radiat. Transfer60, 411 – 423 (1998).
[CrossRef]

1996

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]

1994

P. C. Hansen, “Regularization tools: A matlab package for analysis and solution of discrete ill-posed problems,” NUMER ALGORITHMS6, 1–35 (1994).
[CrossRef]

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A11, 1491–1499 (1994).
[CrossRef]

1992

F. Kuik, J. F. Dehaan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer47, 477–489 (1992).
[CrossRef]

1991

1972

P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

1912

R. Gans, “Über die form ultramikroskopischer goldteilchen,” Annalen der Physik342, 881–900 (1912).
[CrossRef]

Arenas-Alatorre, J.

Aveyard, J.

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]

Bogatyrev, V. A.

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]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Cheang-Wong, J. C.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Ciric, I. R.

I. R. Ciric and F. R. Cooray, “Benchmark solutions for electromagnetic scattering by systems of randomly oriented spheroids,” J. Quant. Spectrosc. Radiat. Transfer63, 131–148 (1999).
[CrossRef]

Comberg, U.

T. Wriedt and U. Comberg, “Comparison of computational scattering methods,” J. Quant. Spectrosc. Radiat. Transfer60, 411 – 423 (1998).
[CrossRef]

Cooray, F. R.

I. R. Ciric and F. R. Cooray, “Benchmark solutions for electromagnetic scattering by systems of randomly oriented spheroids,” J. Quant. Spectrosc. Radiat. Transfer63, 131–148 (1999).
[CrossRef]

Coronado, E. A.

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]

Crespo-Sosa, A.

de Sanctis, O.

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]

Dehaan, J. F.

F. Kuik, J. F. Dehaan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer47, 477–489 (1992).
[CrossRef]

Draine, B. T.

Dykman, L. A.

N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer111, 1–35 (2010).
[CrossRef]

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]

El-Sayed, M. A.

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]

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]

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. B109, 10531C10532 (2005).
[CrossRef]

Eustis, S.

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]

Fernig, D. G.

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]

Flatau, P. J.

Funston, A. M.

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.37, 1792–1805 (2008).
[CrossRef] [PubMed]

Gans, R.

R. Gans, “Über die form ultramikroskopischer goldteilchen,” Annalen der Physik342, 881–900 (1912).
[CrossRef]

Garcia de Abajo, F. J.

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.37, 1792–1805 (2008).
[CrossRef] [PubMed]

Gill, P.

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

Gomez, D.

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]

Haiss, W.

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]

Hansen, P. C.

P. C. Hansen, “Regularization tools: A matlab package for analysis and solution of discrete ill-posed problems,” NUMER ALGORITHMS6, 1–35 (1994).
[CrossRef]

Hartland, G. V.

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]

Hergert, W.

Hohenester, U.

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B72, 195429 (2005).
[CrossRef]

Hovenier, J. W.

F. Kuik, J. F. Dehaan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer47, 477–489 (1992).
[CrossRef]

Huang, X.

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]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Karamehmedovic, M.

Kellermann, G.

Khanadeev, V.

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. C115, 6317–6323 (2011).
[CrossRef]

Khanadeev, V. A.

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. C112, 12760–12768 (2008).
[CrossRef]

Khlebtsov, B.

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. C115, 6317–6323 (2011).
[CrossRef]

Khlebtsov, B. N.

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. C112, 12760–12768 (2008).
[CrossRef]

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. C111, 11516–11527 (2007).
[CrossRef]

Khlebtsov, N.

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. C115, 6317–6323 (2011).
[CrossRef]

Khlebtsov, N. G.

N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer111, 1–35 (2010).
[CrossRef]

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. C112, 12760–12768 (2008).
[CrossRef]

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

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. C111, 11516–11527 (2007).
[CrossRef]

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]

Krenn, J.

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B72, 195429 (2005).
[CrossRef]

Kuik, F.

F. Kuik, J. F. Dehaan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer47, 477–489 (1992).
[CrossRef]

Lacis, A. A.

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

Link, S.

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. B109, 10531C10532 (2005).
[CrossRef]

Liz-Marzan, L. M.

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.37, 1792–1805 (2008).
[CrossRef] [PubMed]

Loke, V. L. Y.

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Matyssek, C.

Melnikov, A. G.

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]

Mengüç, M. P.

Mishchenko, M. I.

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A8, 871–882 (1991).
[CrossRef]

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

Mroczka, J.

Mulvaney, P.

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.37, 1792–1805 (2008).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
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.

Fig. 2
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.

Fig. 3
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 ( A R ¯, σAR) give the mean AR and the standard deviation of the PDF obtained by the two methods.

Fig. 4
Fig. 4

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

Fig. 5
Fig. 5

(a) Dependence of the retrieved mean aspect ratio A R ¯ and the standard deviation σ on the assumed mean width for sample NR-20-700 with assumed ē = 0.6. (b)Dependence of the number of NRs per unit volume Nv and the mass-volume concentration Cg of NRs on the assumed mean width .

Fig. 6
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.

Fig. 7
Fig. 7

(a) Dependence of the retrieved mean aspect ratio A R ¯ and the standard deviation σ on the assumed mean end-cap eccentricity ē for sample NR-20-700 with assumed = 20 nm. (b) Dependence of the number of NRs per unit volume Nv and the mass-volume concentration Cg of NRs on the assumed mean end-cap eccentricity ē.

Fig. 8
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.

Fig. 9
Fig. 9

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

Equations (17)

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γ = γ bulk + A s υ F L eff ,
n s = 1.32334 + 3479 λ 2 5.111 × 10 7 λ 4 ,
I I inc = 10 A = e A ext l ,
A = l ln 10 A ext = l N v ln 10 C ext .
A ( λ , D , A R , e ) = l N v ln 10 D min D max A R min A R max e min e max p ( D , A R , e ) C ext ( λ , D , A R , e ) d D d A R d e .
A ( λ , D ¯ , A R , e ¯ ) = l N v ln 10 A R min A R max p ( D ¯ , A R , e ¯ ) C ext ( λ , D ¯ , A R , e ¯ , λ ) d A R .
A = CP ,
A = [ A ( λ 1 ) A ( λ 2 ) A ( λ m ) A ( λ M ) ] T , m = 1 , 2 , , M ,
P = Δ A R [ p ( A R 1 ) p ( A R 2 ) p ( A R n ) p ( A R N ) ] T , n = 1 , 2 , , N ,
C m n = l N v ln 10 [ C ext ( λ m , D ¯ , A R n , e ¯ ) ] ,
Δ A R = A R max A R min N ,
P RLS = min { A CP 2 2 + γ 2 L ( P P * ) 2 2 } ,
P RLS = min { 1 2 P T QP + q T P } ,
m s e = 1 M m [ A ( λ m ) A cal ( λ m ) A ( λ m ) ] 2 , m = 1 , 2 , , M ,
p ( A R ) = i w i σ i , AR 2 π exp [ ( A R A R ¯ i ) 2 2 σ i , AR 2 ] ,
A R ¯ = 3.01 0.0176 D ¯ ( R 2 = 0.9971 )
A R ¯ = 2.404 + 0.43 e ¯ ( R 2 = 0.9941 ) .

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