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

Optics Express, Vol. 22, Issue 7, pp. 7925-7930 (2014)

http://dx.doi.org/10.1364/OE.22.007925

Acrobat PDF (947 KB)

### 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

1. J. Kuhn, T. Gleissner, M. C. Arduinischuster, S. Korder, and J. Fricke, “Integration of mineral powders into SiO_{2} aerogels,” J. Non-Cryst. Solids **186**, 291–295 (1995). [CrossRef]

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

1. J. Kuhn, T. Gleissner, M. C. Arduinischuster, S. Korder, and J. Fricke, “Integration of mineral powders into SiO_{2} aerogels,” J. Non-Cryst. Solids **186**, 291–295 (1995). [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]

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]

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]

## 2. Theoretical model

_{2}opacified aerogel samples with apparent densities of 300-350 kg·m

^{−3}. The TiO

_{2}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.

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]

_{2}particle was used as the original seed particle and then, according to the DLA algorithm, silica nanoparticles were cast into the volume region and attached to the previous particle surfaces [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]

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]

_{2}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]

*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.*

_{λ}## 3. Reflectance prediction and experimental validation

1. J. Kuhn, T. Gleissner, M. C. Arduinischuster, S. Korder, and J. Fricke, “Integration of mineral powders into SiO_{2} aerogels,” J. Non-Cryst. Solids **186**, 291–295 (1995). [CrossRef]

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]

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]

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]

*z*is the coordinate along the sample thickness and

*μ*= cos

*θ*indicates the scattering direction. The input spectral parameters are the extinction coefficient,

*β*, the scattering albedo,

_{λ}*ω*, and the phase function, Φ

_{λ}*. The directional, spectral radiative intensity along*

_{λ}*z*-axis,

*I*, is determined from the RTE and the boundary conditions at the sample surfaces:where

_{λ}*μ*

_{0}>0 is determined by the incidence direction in the experiment,

*ε*denotes the small angle region where the incidence intensity

*I*

_{0}exists and

*h*is the sample thickness. The hemispherical reflectance,

*R*, is defined as the ratio of the total reflected energy to the incident energy at

_{λ}*z*= 0:

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]

*N*is the number of quadrature points in

*μ*∈[-1,1] and the discrete angular ordinates,

*μ*, and their weights,

_{k}*w*, are determined by Fiveland (S

_{k}*approximation) [13*

_{N}**109**(3), 809–812 (1987). [CrossRef]

*N*equations. The equation for the

*k*th directiondescribes the radiative intensity in direction

*μ*. The complete RTE can then be solved numerically by replacing the spatial differential d

_{k}*I*/d

_{λ}*z*with a finite difference. This reflectance prediction procedure is independent of any previous optical measurements with only optical constants from the literature and sample gradients parameters needed.

_{2}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*) is the probability density function,

*d*

_{0}is the median diameter and

*σ*is the standard deviation. To enable 95% of the particle population to be 1-3 μm,

*d*

_{0}= 1.73 μm and

*σ*= 0.27. The size-averaged optical properties were calculated by numerical integration:where

*X*stands for a monodisperse spectral property (

_{λ}*β*,

_{λ}*ω*, or Φ

_{λ}*) and*

_{λ}*d*were selected from 1 μm to 3 μm with a constant log-interval Δ

_{i}_{ln}

*=*

_{d}*d*

_{i}_{+1}

*-d*The original geometric model was linearly enlarged for each opacifier diameter

_{i}.*d*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

_{i}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]

*S*

_{12}approximation was used for the anisotropic scattering and the incident boundary conditions were

*μ*

_{0}=

*μ*

_{1}and

*ε*→0 to approximate normal incidence.

*d*

_{0}, and standard deviations,

*σ*. Figure 4(a) 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

*d*

_{0}= 2.5 μm in Fig. 4(b)) but with large errors at long wavelength, and vice versa.

## 4. Summary

## Acknowledgments

## References and links

1. | J. Kuhn, T. Gleissner, M. C. Arduinischuster, S. Korder, and J. Fricke, “Integration of mineral powders into SiO |

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. |

3. | J. Fricke and T. Tillotson, “Aerogels: production, characterization, and applications,” Thin Solid Films |

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. |

5. | X. Wang, D. Sun, Y. Duan, and Z. Hu, “Radiative characteristics of opacifier-loaded silica aerogel composites,” J. Non-Cryst. Solids |

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. |

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. |

8. | C. Tien and B. L. Drolen, “Thermal Radiation in Particulate Media with Dependent and Independent Scattering,” in |

9. | M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express |

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 |

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. |

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. |

13. | W. A. Fiveland, “Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media,” J. Heat Transfer-Trans. ASME |

14. | T. A. Witten Jr and L. M. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. |

15. | A. Emmerling and J. Fricke, “Scaling properties and structure of aerogels,” J. Sol-Gel Sci. Techn. |

16. | M. I. Mishchenko, L. Liu, and G. Videen, “Conditions of applicability of the single-scattering approximation,” Opt. Express |

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

18. | E. D. Palik, ed., |

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 |

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. |

**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

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### References

- 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]
- 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]
- J. Fricke, T. Tillotson, “Aerogels: production, characterization, and applications,” Thin Solid Films 297(1-2), 212–223 (1997). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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).
- M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express 15(20), 13188–13202 (2007). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- T. A. Witten, L. M. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47(19), 1400–1403 (1981). [CrossRef]
- A. Emmerling, J. Fricke, “Scaling properties and structure of aerogels,” J. Sol-Gel Sci. Techn. 8, 781–788 (1997).
- M. I. Mishchenko, L. Liu, G. Videen, “Conditions of applicability of the single-scattering approximation,” Opt. Express 15(12), 7522–7527 (2007). [CrossRef] [PubMed]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
- E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).
- 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]
- 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]

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