OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 7 — Apr. 7, 2014
  • pp: 7925–7930
« Show journal navigation

Calculation and experimental validation of spectral properties of microsize grains surrounded by nanoparticles

Haitong Yu, Dong Liu, Yuanyuan Duan, and Xiaodong Wang  »View Author Affiliations


Optics Express, Vol. 22, Issue 7, pp. 7925-7930 (2014)
http://dx.doi.org/10.1364/OE.22.007925


View Full Text Article

Acrobat PDF (947 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Opacified aerogels are particulate thermal insulating materials in which micrometric opacifier mineral grains are surrounded by silica aerogel nanoparticles. A geometric model was developed to characterize the spectral properties of such microsize grains surrounded by much smaller particles. The model represents the material’s microstructure with the spherical opacifier’s spectral properties calculated using the multi-sphere T-matrix (MSTM) algorithm. The results are validated by comparing the measured reflectance of an opacified aerogel slab against the value predicted using the discrete ordinate method (DOM) based on calculated optical properties. The results suggest that the large particles embedded in the nanoparticle matrices show different scattering and absorption properties from the single scattering condition and that the MSTM and DOM algorithms are both useful for calculating the spectral and radiative properties of this particulate system.

© 2014 Optical Society of America

1. Introduction

Opacified aerogels (Fig. 1(a)
Fig. 1 TiO2 opacified aerogels: (a) monolithic slab sample; (b) SEM picture of an opacifier grain surrounded by the aerogel; (c) geometric model used in the MSTM calculations.
) are porous materials with ultra-low thermal conductivities which are composed by the silica aerogel (porosity≥90%) and opacifier grains [1

1. J. Kuhn, T. Gleissner, M. C. Arduinischuster, S. Korder, and J. Fricke, “Integration of mineral powders into SiO2 aerogels,” J. Non-Cryst. Solids 186, 291–295 (1995). [CrossRef]

]. The aerogel consists of packed silica nanoparticles (diameters d = 5-50 nm) [2

2. J. Zhao, Y. Duan, X. Wang, and B. Wang, “Effects of solid-gas coupling and pore and particle microstructures on the effective gaseous thermal conductivity in aerogels,” J. Nanopart. Res. 14(8), 1–15 (2012). [CrossRef] [PubMed]

,3

3. J. Fricke and T. Tillotson, “Aerogels: production, characterization, and applications,” Thin Solid Films 297(1-2), 212–223 (1997). [CrossRef]

]. The opacifiers are mineral grains with large scattering and/or absorption efficiencies in the near-infrared to enhance the aerogel’s extinction coefficients for thermal radiation to achieve low thermal conductivities at high temperatures [1

1. J. Kuhn, T. Gleissner, M. C. Arduinischuster, S. Korder, and J. Fricke, “Integration of mineral powders into SiO2 aerogels,” J. Non-Cryst. Solids 186, 291–295 (1995). [CrossRef]

]. Opacifier grains usually have diameters of 1-5 μm and occupy only a small volume fraction (≈1.5%) in the material to maintain the material’s high porosity (Fig. 1(b)).

Accurate calculations of the opacifier’s optical properties are important for predicting and optimizing the material’s insulation ability. Materials with large spectral extinction coefficients tend to have low radiative flux intensities, while materials with large albedo and strong backward scattering tend to absorb less energy and are resistant to overheating caused by thermal impact. In opacified aerogels, the optical properties of the micrometer-sized opacifier particles are influenced by the surrounding nanoparticles. Scattering interactions exist between the opacifier and the aerogel, but the refraction and scattering of silica is so weak in the near infrared that the aerogel’s influence on the opacifier’s radiative properties have been neglected by most studies [4

4. J. Zhao, Y. Duan, X. Wang, X. Zhang, Y. Han, Y. Gao, Z. Lv, H. Yu, and B. Wang, “Optical and radiative properties of infrared opacifier particles loaded in silica aerogels for high temperature thermal insulation,” Int. J. Therm. Sci. 70, 54–64 (2013). [CrossRef]

6

6. V. Napp, R. Caps, H. P. Ebert, and J. Fricke, “Optimization of the thermal radiation extinction of silicon carbide in a silica powder matrix,” J. Therm. Anal. Calorim. 56(1), 77–85 (1999). [CrossRef]

] which have mainly characterized the opacified aerogels’ extinction coefficient with the single scattering model. Neither have been other optical properties such as the spectral albedo and the phase functions studied in detail in these studies. Dlugach et al. [7

7. J. M. Dlugach, M. I. Mishchenko, and D. W. Mackowski, “Scattering and absorption properties of polydisperse wavelength-sized particles covered with much smaller grains,” J. Quant. Spectrosc. Radiat. Transf. 113(18), 2351–2355 (2012). [CrossRef]

] calculated the small particles influence on the optical properties of a larger particle, but the sizes, geometries and optical constants were very different from the case of opacified aerogels.

The significant difference in sizes and spatial distributions of aerogel and opacifier particles prohibits characterization by simple dependent scattering models [8

8. C. Tien and B. L. Drolen, “Thermal Radiation in Particulate Media with Dependent and Independent Scattering,” in Annual Review of Numerical Fluid Mechanics and Heat Transfer, T. C. Chawla, ed. (Hemisphere, 1987).

]. Approximating this problem as particle scattering in an effective absorbing media [9

9. M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express 15(20), 13188–13202 (2007). [CrossRef] [PubMed]

,10

10. M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 1. Foldy-Lax equations, order-of-scattering expansion, and coherent field,” Opt. Express 16(3), 2288–2301 (2008). [CrossRef] [PubMed]

] is also not rigorous because it neglects the scattering by aerogel particles. A recent work by Yu et al. [11

11. H. Yu, D. Liu, Y. Duan, and X. Wang, “Theoretical model of radiative transfer in opacified aerogel based on realistic microstructures,” Int. J. Heat Mass Tran. 70, 478–485 (2014). [CrossRef]

] generated a geometric model to represent the material microstructure and used the multi-sphere T-matrix (MSTM) algorithm [12

12. D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T-matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2182–2192 (2011). [CrossRef]

] to characterize the extinction coefficient and the radiative conductivity. The present work analyzes the surrounding nanoparticles’ influence on the spectral properties of a large particle in more detail by characterizing the spectral properties including the extinction coefficient, albedo and phase function. The results of this theoretical model are validated by comparing the experimental measured reflectance with the solution for radiative transfer equation (RTE) in an opacified aerogel slab with the 1-D discrete ordinate method (DOM) [13

13. W. A. Fiveland, “Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media,” J. Heat Transfer-Trans. ASME 109(3), 809–812 (1987). [CrossRef]

] using optical properties calculated by MSTM algorithm.

2. Theoretical model

This work used TiO2 opacified aerogel samples with apparent densities of 300-350 kg·m−3. The TiO2 opacifier (rutile type) with diameters of 1-3 μm had a 20% weight fraction in the material. SEM pictures (Fig. 1(b)) show that the opacifier particles are approximately spherical and are surrounded by aerogel nanoparticles. The opacifier surface is partly exposed in the image because of damage caused by the SEM sample preparation.

After the geometry was generated, the spectral properties of a TiO2 sphere (d = 1 μm) surrounded by silica nanoparticles were calculated using the multi-sphere T-matrix (MSTM) code [12

12. D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T-matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2182–2192 (2011). [CrossRef]

], a numerically accurate method capable of parallel computations for large numbers of spheres with extreme size difference. The MSTM results were compared with the single scattering solutions (Mie theory) [17

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

]. The optical constants of rutile and silica are from the HOC by Palik [18

18. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

]. Figure 2(a)
Fig. 2 Calculated TiO2 opacifiers optical properties: (a) extinction efficiency, Qext, absorption efficiency, Qabs, and scattering albedo, ωλ, and (b) phase functions, Φλ, at 2 μm, 4 μm and 8 μm.
shows that in the 2-8 μm range, the aerogel particles only weakly influence the opacifier’s extinction efficiency but increase its absorption efficiency by an order of magnitude and decrease its albedo. Thus the aerogel nanoparticles boost the absorption of the opacifier grains which results in stronger total absorption and weaker scattering of thermal radiation. The phase functions Φλ given by the MSTM code and by Mie theory for several wavelengths are almost identical (Fig. 2(b)), possibly because the aerogel particles are nearly isotropically distributed and do not alter the opacifier’s scattered energy distribution.

3. Reflectance prediction and experimental validation

The reflectance of opacified aerogels is more observable than the transmittance and is insensitive to the sample thickness for an optically thick slab. However, various optical properties such as the extinction, albedo and phase functions cannot be determined from only the measured reflectance. Therefore, instead of deriving the spectral properties from measurements, the calculated spectral parameters were used here in the RTE to predict the sample reflectance for comparison with measurements. The discrete ordinate method (DOM) [13

13. W. A. Fiveland, “Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media,” J. Heat Transfer-Trans. ASME 109(3), 809–812 (1987). [CrossRef]

] was used to solve the RTE. Compared to approximations used, the DOM characterizes the scattering phase function’s influence in finer detail.

The RTE for a one dimensional participating media without thermal emission is written as:
μdIλ(z,μ)βλdz=Iλ(z,μ)+ωλ211Φλ(μ,μ')I(z,μ')dμ'
(1)
where z is the coordinate along the sample thickness and μ = cosθ indicates the scattering direction. The input spectral parameters are the extinction coefficient, βλ, the scattering albedo, ωλ, and the phase function, Φλ. The directional, spectral radiative intensity along z-axis, Iλ, is determined from the RTE and the boundary conditions at the sample surfaces:
Iλ(z=0,μ)={I0(|μμ0|<ε)0(μ>0,μμ0),Iλ(z=h,μ)=0forμ0
(2)
where μ0>0 is determined by the incidence direction in the experiment, ε denotes the small angle region where the incidence intensity I0 exists and h is the sample thickness. The hemispherical reflectance, Rλ, is defined as the ratio of the total reflected energy to the incident energy at z = 0:

Rλ=10Iλ(z=0,μ)μdμ/01Iλ(z=0,μ)μdμ
(3)

To solve the RTE, the DOM replaces the angular integral term in Eq. (1) by quadrature [13

13. W. A. Fiveland, “Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media,” J. Heat Transfer-Trans. ASME 109(3), 809–812 (1987). [CrossRef]

]:
11μΦ(μ)dμk=1NwkμkΦ(μk)
(4)
where N is the number of quadrature points in μ∈[-1,1] and the discrete angular ordinates, μk, and their weights, wk, are determined by Fiveland (SN approximation) [13

13. W. A. Fiveland, “Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media,” J. Heat Transfer-Trans. ASME 109(3), 809–812 (1987). [CrossRef]

] as

{wk=2/N(k=1,2,...,N)01μidμ=k=1N/2wkμi(i=0,1,...,N/2)
(5)

The spectral hemispherical reflectance of the opacified aerogel was measured with a Bruker FTIR spectrometer and a gold integration sphere with near normal incident light. Measured wavelength region was 2.0 to 16.7 μm with the short wavelength results (2-8 μm) as the main concern to be compared with the DOM results. The actual opacifiers were not monodisperse particles with the diameters of the TiO2 opacifier in the sample being 1-3 μm (claimed by the manufacturer). Suppose the diameter d obeys the log-normal distribution [6

6. V. Napp, R. Caps, H. P. Ebert, and J. Fricke, “Optimization of the thermal radiation extinction of silicon carbide in a silica powder matrix,” J. Therm. Anal. Calorim. 56(1), 77–85 (1999). [CrossRef]

]:
p(d)d(lnd)=12πσexp[(lndlnd0)22σ2]d(lnd)
(7)
where p(d) is the probability density function, d0 is the median diameter and σ is the standard deviation. To enable 95% of the particle population to be 1-3 μm, d0 = 1.73 μm and σ = 0.27. The size-averaged optical properties were calculated by numerical integration:
X¯λ(d0,σ)=0X¯λ(d)p(d)d(lnd)i=1NX¯λ(d)p(d)Δlnd
(8)
where Xλ stands for a monodisperse spectral property (βλ, ωλ, or Φλ) and X¯(d0,σ) is the corresponding sized averaged property. In this case 12 integration points di were selected from 1 μm to 3 μm with a constant log-interval Δlnd = di+1-di. The original geometric model was linearly enlarged for each opacifier diameter di to calculate the monodisperse spectral properties using MSTM and then averaged over the sizes. The spectral parameters of the opacified aerogel were determined from the individual optical properties of the opacifier and the aerogel and their volume fractions in the material, where the aerogel spectral parameters were calculated as described previously [11

11. H. Yu, D. Liu, Y. Duan, and X. Wang, “Theoretical model of radiative transfer in opacified aerogel based on realistic microstructures,” Int. J. Heat Mass Tran. 70, 478–485 (2014). [CrossRef]

]. The resulting optical properties were then used in the DOM to predict the reflectance, where an S12 approximation was used for the anisotropic scattering and the incident boundary conditions were μ0 = μ1 and ε→0 to approximate normal incidence.

As shown in Fig. 3
Fig. 3 DOM results from Mie and T-matrix calculations compared with the measured reflectance.
, the DOM results based on the MSTM calculation are in better agreement with the measurements than based on Mie theory. The single scattering properties from Mie theory resulted in reflectance predictions higher than the measurement. The comparison verifies the spectral properties shown in Fig. 2 which shows that the surrounding nanoparticles result in stronger absorption and lower albedo of the opacifier. In case that the errors in Mie results was caused by the fixed opacifier sizes, the calculations were further done with a range of median diameters, d0, and standard deviations, σ. Figure 4(a)
Fig. 4 Size parameters’ influence on the Mie results: (a) varied σ with given d0; (b) varied d0 with given σ.
and 4(b) show that even adjusting the input sizes does not enable the Mie results to agree with the measurements at all wavelengths with the predicted reflectance with showing agreement at short wavelengths (such as d0 = 2.5 μm in Fig. 4(b)) but with large errors at long wavelength, and vice versa.

4. Summary

In this work, the spectral properties of a micrometer-scale opacifier particle surrounded by silica nanoparticles were calculated using the MSTM code based on a computer generated microstructure. This model was validated by comparing the DOM predictions with measured opacified aerogel reflectance. The results show that even the small effect of refraction and scattering by silica nanoparticles affected the optical properties of the opacifier grains inside the aerogel. The absorption efficiency and scattering albedo show observable differences from single scattering properties, while the phase functions are almost unchanged.

The agreement between the theory and experiment also shows the ability of this method to predict the optical and radiative properties. The T-matrix code is useful for systems with large numbers of particles even with extremely different sizes. The discrete ordinate method can be used to predict the radiative properties with known optical parameters and shows satisfactory agreement with measurements.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant No. 51236004 and 51321002).

References and links

1.

J. Kuhn, T. Gleissner, M. C. Arduinischuster, S. Korder, and J. Fricke, “Integration of mineral powders into SiO2 aerogels,” J. Non-Cryst. Solids 186, 291–295 (1995). [CrossRef]

2.

J. Zhao, Y. Duan, X. Wang, and B. Wang, “Effects of solid-gas coupling and pore and particle microstructures on the effective gaseous thermal conductivity in aerogels,” J. Nanopart. Res. 14(8), 1–15 (2012). [CrossRef] [PubMed]

3.

J. Fricke and T. Tillotson, “Aerogels: production, characterization, and applications,” Thin Solid Films 297(1-2), 212–223 (1997). [CrossRef]

4.

J. Zhao, Y. Duan, X. Wang, X. Zhang, Y. Han, Y. Gao, Z. Lv, H. Yu, and B. Wang, “Optical and radiative properties of infrared opacifier particles loaded in silica aerogels for high temperature thermal insulation,” Int. J. Therm. Sci. 70, 54–64 (2013). [CrossRef]

5.

X. Wang, D. Sun, Y. Duan, and Z. Hu, “Radiative characteristics of opacifier-loaded silica aerogel composites,” J. Non-Cryst. Solids 375, 31–39 (2013). [CrossRef]

6.

V. Napp, R. Caps, H. P. Ebert, and J. Fricke, “Optimization of the thermal radiation extinction of silicon carbide in a silica powder matrix,” J. Therm. Anal. Calorim. 56(1), 77–85 (1999). [CrossRef]

7.

J. M. Dlugach, M. I. Mishchenko, and D. W. Mackowski, “Scattering and absorption properties of polydisperse wavelength-sized particles covered with much smaller grains,” J. Quant. Spectrosc. Radiat. Transf. 113(18), 2351–2355 (2012). [CrossRef]

8.

C. Tien and B. L. Drolen, “Thermal Radiation in Particulate Media with Dependent and Independent Scattering,” in Annual Review of Numerical Fluid Mechanics and Heat Transfer, T. C. Chawla, ed. (Hemisphere, 1987).

9.

M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express 15(20), 13188–13202 (2007). [CrossRef] [PubMed]

10.

M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 1. Foldy-Lax equations, order-of-scattering expansion, and coherent field,” Opt. Express 16(3), 2288–2301 (2008). [CrossRef] [PubMed]

11.

H. Yu, D. Liu, Y. Duan, and X. Wang, “Theoretical model of radiative transfer in opacified aerogel based on realistic microstructures,” Int. J. Heat Mass Tran. 70, 478–485 (2014). [CrossRef]

12.

D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T-matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2182–2192 (2011). [CrossRef]

13.

W. A. Fiveland, “Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media,” J. Heat Transfer-Trans. ASME 109(3), 809–812 (1987). [CrossRef]

14.

T. A. Witten Jr and L. M. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47(19), 1400–1403 (1981). [CrossRef]

15.

A. Emmerling and J. Fricke, “Scaling properties and structure of aerogels,” J. Sol-Gel Sci. Techn. 8, 781–788 (1997).

16.

M. I. Mishchenko, L. Liu, and G. Videen, “Conditions of applicability of the single-scattering approximation,” Opt. Express 15(12), 7522–7527 (2007). [CrossRef] [PubMed]

17.

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

18.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

19.

S. Lallich, F. Enguehard, and D. Baillis, “Experimental determination and modeling of the radiative properties of silica nanoporous matrices,” J. Heat Transfer-Trans. ASME 131(8), 082701 (2009). [CrossRef]

20.

A. Tamanai, H. Mutschke, J. Blum, and R. Neuhäuser, “Experimental infrared spectroscopic measurement of light extinction for agglomerate dust grains,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 373–381 (2006). [CrossRef]

OCIS Codes
(030.5620) Coherence and statistical optics : Radiative transfer
(160.6060) Materials : Solgel
(290.4210) Scattering : Multiple scattering

ToC Category:
Spectroscopy

History
Original Manuscript: February 12, 2014
Manuscript Accepted: March 18, 2014
Published: March 27, 2014

Virtual Issues
Vol. 9, Iss. 6 Virtual Journal for Biomedical Optics

Citation
Haitong Yu, Dong Liu, Yuanyuan Duan, and Xiaodong Wang, "Calculation and experimental validation of spectral properties of microsize grains surrounded by nanoparticles," Opt. Express 22, 7925-7930 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-7925


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. J. Kuhn, T. Gleissner, M. C. Arduinischuster, S. Korder, J. Fricke, “Integration of mineral powders into SiO2 aerogels,” J. Non-Cryst. Solids 186, 291–295 (1995). [CrossRef]
  2. J. Zhao, Y. Duan, X. Wang, B. Wang, “Effects of solid-gas coupling and pore and particle microstructures on the effective gaseous thermal conductivity in aerogels,” J. Nanopart. Res. 14(8), 1–15 (2012). [CrossRef] [PubMed]
  3. J. Fricke, T. Tillotson, “Aerogels: production, characterization, and applications,” Thin Solid Films 297(1-2), 212–223 (1997). [CrossRef]
  4. J. Zhao, Y. Duan, X. Wang, X. Zhang, Y. Han, Y. Gao, Z. Lv, H. Yu, B. Wang, “Optical and radiative properties of infrared opacifier particles loaded in silica aerogels for high temperature thermal insulation,” Int. J. Therm. Sci. 70, 54–64 (2013). [CrossRef]
  5. X. Wang, D. Sun, Y. Duan, Z. Hu, “Radiative characteristics of opacifier-loaded silica aerogel composites,” J. Non-Cryst. Solids 375, 31–39 (2013). [CrossRef]
  6. V. Napp, R. Caps, H. P. Ebert, J. Fricke, “Optimization of the thermal radiation extinction of silicon carbide in a silica powder matrix,” J. Therm. Anal. Calorim. 56(1), 77–85 (1999). [CrossRef]
  7. J. M. Dlugach, M. I. Mishchenko, D. W. Mackowski, “Scattering and absorption properties of polydisperse wavelength-sized particles covered with much smaller grains,” J. Quant. Spectrosc. Radiat. Transf. 113(18), 2351–2355 (2012). [CrossRef]
  8. C. Tien and B. L. Drolen, “Thermal Radiation in Particulate Media with Dependent and Independent Scattering,” in Annual Review of Numerical Fluid Mechanics and Heat Transfer, T. C. Chawla, ed. (Hemisphere, 1987).
  9. M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express 15(20), 13188–13202 (2007). [CrossRef] [PubMed]
  10. M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 1. Foldy-Lax equations, order-of-scattering expansion, and coherent field,” Opt. Express 16(3), 2288–2301 (2008). [CrossRef] [PubMed]
  11. H. Yu, D. Liu, Y. Duan, X. Wang, “Theoretical model of radiative transfer in opacified aerogel based on realistic microstructures,” Int. J. Heat Mass Tran. 70, 478–485 (2014). [CrossRef]
  12. D. W. Mackowski, M. I. Mishchenko, “A multiple sphere T-matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2182–2192 (2011). [CrossRef]
  13. W. A. Fiveland, “Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media,” J. Heat Transfer-Trans. ASME 109(3), 809–812 (1987). [CrossRef]
  14. T. A. Witten, L. M. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47(19), 1400–1403 (1981). [CrossRef]
  15. A. Emmerling, J. Fricke, “Scaling properties and structure of aerogels,” J. Sol-Gel Sci. Techn. 8, 781–788 (1997).
  16. M. I. Mishchenko, L. Liu, G. Videen, “Conditions of applicability of the single-scattering approximation,” Opt. Express 15(12), 7522–7527 (2007). [CrossRef] [PubMed]
  17. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  18. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).
  19. S. Lallich, F. Enguehard, D. Baillis, “Experimental determination and modeling of the radiative properties of silica nanoporous matrices,” J. Heat Transfer-Trans. ASME 131(8), 082701 (2009). [CrossRef]
  20. A. Tamanai, H. Mutschke, J. Blum, R. Neuhäuser, “Experimental infrared spectroscopic measurement of light extinction for agglomerate dust grains,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 373–381 (2006). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited