SILICA-BASED NANOPOROUS COMPOSITE MATERIALS
Following from: Spectral radiative properties of some important materials: Experimental data and theoretical models
Various porous materials are widely used as thermal insulation for many engineering problems (Bynum, 2001). The majority of these materials are made of substances characterized by low thermal conductivity. Thus, the resulting thermal conductivity of the insulation decreases. However, the low-conductivity substances are usually semitransparent in the visible and near-infrared spectral ranges. The latter makes it especially important to correctly account for the thermal radiation mode of heat transfer in the porous material, with specific attention to spectral radiative characteristics in the near-infrared. One can formulate the following typical features of scattering and absorption characteristics of semi-transparent porous materials:
- The absorption coefficient is practically independent of the material morphology and can be calculated as
(1) - The characteristics of scattering are insensitive to weak absorption typical for semitransparent materials. The scattering coefficient σ_{λ} and asymmetry factor of scattering μ can usually be predicted by analysis of the material morphology. It is especially important to take into account the contribution of the grains or particle aggregates, pores, and cracks of size comparable with the radiation wavelength.
The above general statements are confirmed by the experience of the authors in experimental study and theoretical modeling of various dispersed materials such as foams, porous ceramics, etc. (see articles Semitransparent media containing bubbles and Low-porosity ceramics for thermal barrier coatings). It should be noted that a relative mutual independence of absorption and scattering is not a specific property of some semitransparent porous materials, but it is a general behavior of disperse systems containing the so-called optically soft particles which satisfy the conditions of the Rayleigh-Gans theory (see the article Rayleigh-Gans scattering).
As engineers continue to work on developing advanced thermal insulations, the composite materials based on nanoporous matrixes are considered as potentially very interesting “superinsulating” materials for applications in aerospace and other important technologies (Reim et al., 2005; Enguehard, 2007; Wiener et al., 2009). The low conductivity of these materials is explained by a very small area of contact between single particles because of highly fragmented solid matter and high porosity (~85%-90%) and also by the low contribution of gas in pores because of the very small size of the pores (less than the free path of gas molecules) equivalent to vacuum conditions. As a result, thermal radiation may proivide an important contribution to heat transfer through this highly porous material. It goes without saying that an understanding of spectral radiative properties of advanced materials based on nanoporous matrixes is of great importance.
The recently reported results of experimental identification of the wide-range spectral properties of nanoporous silica (Lallich et al., 2009) showed an unusual behavior of the transport scattering coefficient σ_{λ}^{tr} = σ_{λ}(1 - μ) both in the visible and near-infrared spectral ranges. Relatively strong scattering in the visible range is explained by formation of aggregates of primary nanoparticles of silica. This statement has been confirmed by a detailed analysis reported by Lallich et al. (2009). However, strange behavior of the scattering coefficient in the near-infrared range, where the local maxima of scattering appear to be at the wavelength of absorption peaks, is still awaiting the physical explanation. Following our recent paper (Dombrovsky et al., 2010), we consider a new physical model of resonance behavior of scattering of nanoporous silica. This model should give an explanation of the correlation between the “resonances” of scattering and the spectral peaks of the material absorption.
The traditional experimental procedure based on spectral measurements of both directional-hemispherical reflectance and transmittance has been employed. Two spectrometers, for the wavelength ranges 0.25 < λ < 2.5 μm and λ > 1.6 μm, were equipped with integrating spheres that collect the radiation propagating through or reflected by the samples. One can find some details of the experiment in paper by Lallich et al. (2009). The identification of the main radiative properties of the samples was based on the radiative transfer equation (RTE) and the transport approximation for scattering function. In the traditional procedure, the numerical solution for the direct radiative transfer problem is obtained using the well-known discrete ordinates method (DOM) (Draine, 2000; Modest, 2003).
The samples of the studied materials were produced by packing silica nanoparticles under the pressure of ~60 bars. Different silica powders from three manufacturers were used to produce the samples (see Table 1 for more details). The silica nanoparticles are hydrophilic: Their surface chemistry results from the presence of silanol groups Si-OH. The concentration of silanol at the surface of the silica particles determines its moisture adsorption capacity. If the ambient gas contains water vapor, it takes only a few minutes to adsorb water in the powder. It is a complicated task to dry the samples and to maintain low water content during the experiments. Therefore, all the results were obtained for hydrated samples at atmospheric conditions.
Table 1. Properties of Powders and Samples under Investigation
Powder name | Wacker HDK-T30 | Cabot Cab-O-Sil EH5 | Degussa Aerosil COK84 |
Chemical composition (% in weight) | ≥99.8% SiO_{2} | ≥99.8% SiO_{2} | 84% SiO_{2} and 16% Al_{2}O_{3} |
Diameter of particles | 9 nm | 7 nm | 13.5 nm |
Characteristic aggregate length | 120 nm | 200–300 nm | – |
Sample thickness | 2 mm | 2 mm | 2 mm |
Sample porosity | 0.87 | 0.86 | 0.86 |
It is known that the hydroxyl in silanol groups is one of the major impurities in silica glass, increasing its optical losses in the near and middle infrared (Davis and Tomozawa, 1996; Zhuravlev, 2000; Plotnichenko et al., 2000; Tomozawa et al., 2001). In addition, the vibrational absorption bands of hydroxyl are observed at the fundamental absorption band at λ = 2.72 μm and at some shorter wavelengths: 2.22, 1.39, and 0.9 μm. Water can exist in silica not only as hydroxyl in the form of silanol groups, but also as molecular water which diffuses in the material and partially reacts with the silica molecules generating various chemical bonds. It is interesting that water species can be detected by near-infrared spectroscopy: Hydroxyl in silica glasses has absorption bands at λ = 2.72 μm and 2.22 μm, while molecular water has absorption bands at λ = 2.94 μm and 1.85 μm. Even the interaction of water with bulk silica (the so-called surface hydroxylation) is a specific complex problem that has been studied in some detail for many years (Zhuravlev, 2000; Peng et al., 2009). As for micro- and nanoporous silica matrixes, there are no similar data in the literature. One can only expect that the hydroxylation and water diffusion in highly porous samples may lead to much stronger effects of the material optical properties than in the case of samples of bulk silica.
It is important to note that there are no isolated primary nanoparticles in the samples. It appears that single particles are collected in relatively stable aggregates. The aggregate structure of the material is illustrated in Fig. 1. This specific morphology of the nanoporous material under investigation should be taken into account in analysis of the experimentally obtained radiative properties. In addition to the parameters of powders, some characteristics of the samples are also presented in Table 1. In this paper, we consider only one representative sample for every powder, but two samples of different thickness were actually examined for HDK and COK powders.
Figure 1. TEM micrograph of the Wacker HDK-T30 powder.
We will not discuss here the mathematical procedure employed to solve the inverse problem for RTE to obtain the material radiative properties from the measurements of reflectance and transmittance for single samples. However, it is important to note that an alternative identification procedure based on the modified two-flux approximation (instead of complete RTE; see the article Hemispherical transmittance and reflectance at normal incidence) was also employed in the present study. It was found that this simplified analytical approach gives exactly the same results as the procedure based on high-order DOM calculations. The latter can be considered as a verification of the mathematics of the inverse problem solution and confirms that the spectral values of both absorption coefficient and transport scattering coefficient presented in Fig. 2 are reliable.
Figure 2. Spectral dependences of the absorption coefficient (a) and the transport scattering coefficient (b) of samples prepared from various powders: (1) HDK-T30, (2) COK84, (3) EH5.
Analysis of absorption. The specific process of silica powder preparation makes it very problematic to determine the bulk material optical constants. The simplest way is to use the optical constants of pure silica from the literature (Kitamura et al., 2007). As for the index of refraction, it can be determined because this physical quantity is weakly sensitive to water content and small impurities in the material. In contrast to the index of refraction, the index of absorption is much more sensitive to water content and internal hydroxylation of the highly porous matrix. Small absorbing additives of another nature may also yield a significant contribution to the material absorption, especially in the short-wave range where pure silica is a weakly absorbing substance. It seems to be more realistic to use our experimental data for the absorption coefficient of a nanoporous matrix to estimate an equivalent index of absorption of a conventional bulk material. This procedure is based on the approximate equation (1) and the known relation α_{λ}^{0} = 4πκ/ λ between the absorption coefficient and the index of absorption of a homogeneous medium. Obviously, this approach yields us an upper estimate of the value of κ because the material is not optically soft over the whole spectrum. But the resulting spectral dependences κ(λ) are expected to be sufficiently accurate for further computational analysis. The values of the index of absorption determined for samples made of silica powders HDK-T30 and EH5 are presented in Fig. 3. The values of κ appear to be much greater than the known data for pure silica. One can try to explain so large an absorption by the presence of water molecules in the nanoporous samples by using theoretical estimates based on the Maxwell-Garnett mixing rule (Modest, 2003). It was done using the values of water content of 2% and 8% for the samples made of HDK and EH5 powders, respectively. Figure 3 indicates that this approach is too crude to obtain correct values of the absorption index. Nevertheless, it is clear that molecular water and silanol groups are really responsible at least for high infrared absorption. As for the abnormally high level of short-wave absorption, especially in the visible range, it is not explained so simply by high water content in the samples and may be a subject of a separate study based on detailed chemical analysis. Generally speaking, it seems natural that the role of water is much greater in the case of nanoporous silica matrixes than it has been observed earlier for ordinary silica materials.
Figure 3. Spectral index of absorption of silica determined from the experimental data for absorption coefficient: (1) HDK-T30, (2) EH5 (theoretical estimates based on the Maxwell-Garnett mixing rule are shown by the corresponding thin lines), (3), (4) experimental data for pure silica [(3) Khashan and Nassif (2001), (4) Dombrovsky et al. (2005)].
Analysis of scattering. It has been already shown by Lallich et al. (2009) that relatively high short-wave scattering of radiation by nanoporous silica is a result of dependent scattering of radiation by primary nanoparticles collected in large aggregates. The known discrete-dipole approximation (DDA) (Draine, 2000) was employed to calculate the radiative characteristics of these aggregates. In the present article, we repeat the DDA calculations by using the above determined index of absorption. Of course, the role of the absorption index is very small because the scattering by particle aggregates depends mainly on the index of refraction. It is assumed that the short-wave index of refraction is insensitive to water content and the spectral dependence n(λ) can be calculated by the know three-term Sellmeier equation suggested by Malitson (1965). A comparison of the computational results with experimental data for a sample made of silica powders HDK-T30 is presented in Fig. 4. One can see that theoretical model yields the transport scattering coefficient which is very close to the experimental data in the wavelength range λ < 1.4 μm, but the high scattering at larger wavelengths cannot be treated as a scattering by the nanoparticle aggregates. Note that good agreement of the calculated absorption coefficient with the experimental data confirms the hypothesis (1) of additive absorption.
Figure 4. A comparison of the experimental data (1) and theoretical predictions based on (2) detailed modeling of nanoparticle aggregates and (3) a simplified physical model of thin-wall hollow spheres.
As an alternative of the detailed scattering model described in the paper by Lallich et al. (2009), we suggest a simplified physical model. It is supposed that radiative properties of complex aggregates of particles randomly oriented in space are similar to the properties of hollow microspheres of wall thickness equal to the diameter d of the primary particles. The external radius of these microspheres is determined by the value of characteristic aggregate length:
(2) |
Equation (2) means that long aggregates of the same length are treated as circular chains forming the hollow microsphere. The volume fraction of hollow microspheres f_{v} is calculated by taking into account the particle “porosity” p_{1} = (1 -δ)^{3}, where δ = d/ a is the relative thickness of the particle wall. The resulting relation for the particle volume fraction is as follows:
(3) |
and the equation for transport scattering coefficient is:
(4) |
where Q_{s}^{tr} is the transport efficiency factor of scattering. Obviously, we can consider only the microspheres with p_{1} < p but this condition is satisfied in our problem. One can see in Fig. 4 that the approximate model gives practically the same results as the detailed DDA calculations.
It is important that the above approximate model is physically sound and there are no free/fitting parameters in the model. Moreover, one can use only a very simple analytical equation for the efficiency factor of Rayleigh scattering for hollow spherical particles (see the article Rayleigh scattering):
(5) |
where m = n-iκ is the complex index of refraction. It goes without saying that this approach is much simpler than DDA calculations for complex aggregates of nanoparticles. It means that this approximation can be recommended for engineering estimates of intense scattering typical of pressed nanoporous silica in the range of ~0.3 < λ < 1.4 μm. Unfortunately, both models of radiation scattering by aggregates of nanoparticles do not explain the observed behavior of scattering at wavelengths >~2 μm.
We were trying to find some micron-size morphological objects (single cavities, hollow particles, or cracks) which might be responsible for high scattering in the near-infrared. Thin cracks at the side surface of some samples were actually observed. That is why we have analyzed the radiation scattering (reflection) by polydisperse parallel cracks at normal incidence. Obviously, the material including some cracks oriented parallel to the sample surface is anisotropic. Nevertheless, having in mind a predominant role of radiation propagating in the normal direction to the sample surface, we consider only the scattering characteristics in this direction. The relations for the normal reflection coefficient of a thin plane-parallel gap inside a homogeneous absorbing and refracting medium are well known and can be found elsewhere (Modest, 2003). One should remember the relation between the reflectance of a single crack and transport efficiency factor of scattering (Dombrovsky, 2004): Q_{s}^{tr} = 2R. The reflectance of polydisperse cracks of various thicknesses Δ can be calculated as follows:
(6) |
where F(Δ) is the normalized distribution of the gap thickness. The choice of the lower limit of integration is explained by the fact that every long crack of variable thickness can be considered as a polydisperse system of relatively short cracks and the minimum thickness is very small. It is also clear that the physical result should not depend on the details of the gap size distribution. Assuming the simplest size distribution of the gaps,
(7) |
where Θ is the Heaviside function, one can write:
(8) |
In the calculations, we used the subtractive Kramers-Krönig analysis (Ahrenkiel, 1971) to determine the spectral dependence of the index of refraction n(λ) of silica in the nanoporous silica matrix under investigation. This analysis is based on the equation for the difference between n(λ) and the reliable measured value of the refractive index at a certain wavelength λ_{1}. The calculations were based on Eq. (5) from the article Spectral radiative properties of diesel fuel droplets with λ_{min} = 0.28 μm and λ_{max} = 7.1 μm. It was assumed n(λ_{1}) = 1.488 at λ_{1} = 0.3 μm [from the dispersion relation by Malitson (1965) for pure silica]. The results of the calculations are shown in Fig. 5(a). One can see that strong absorption bands of hydroxylated silica lead to nonmonotonic spectral variation of the refractive index and the local maxima of the index of refraction appear at the wavelength corresponding to the absorption peaks. At the same time, the amplitude of the oscillations of n(λ) is very small and one cannot expect a considerable effect from these spectral variations of index of refraction on scattering characteristics of some particles.
Figure 5. Spectral dependences of (a) the index of refraction of pure silica and (b) the reflection coefficient of polydisperse cracks in nanoporous silica at normal incidence: (1) Δ_{max} = 1 μm, (2) Δ_{max} = 5 μm, (3) Δ_{max} = 10μm.
The calculations of reflectance of polydisperse cracks within nanoporous silica matrix [see Fig. 5(b)] indicate smooth spectral dependences without strong peaks observed in the experimental scattering curves. There is practically no effect of small spectral variations of the index of refraction [Fig. 5(a)]. The latter is an additional confirmation of the above formulated statement that a weak uniform absorption cannot yield a considerable contribution to scattering in disperse/porous media. Of course, the cracks of volume fraction f_{v,cr} give a contribution to the transport scattering coefficient in the near-infrared according to the obvious equation
(9) |
but the spectral dependence of σ_{λ,cr}^{tr} has no strong maxima in the absorption bands. The expected maximum value of the ratio of f_{v,cr}/ Δ ~ 10^{4} can be obtained from the minimum experimental values of σ_{λ}^{tr} = 43 m^{-1} at λ ≈ 1.7 μm and σ_{λ}^{tr} = 53 m^{-1} at λ ≈ 4.0μm for the sample made of HDK powder. In the case of Δ ~ 1 μm, we find a realistic value of f_{v,cr}^{max} ≈ 1%.
Let us now focus on the observed physical effect: a correlation between the scattering and absorption peaks (note that both absorption and scattering peaks are really strong: It does not seem to be evident in the logarithmic scale). One can remember similar behavior of scattering in the case of nonrefracting but absorbing particles. This effect is described by the Mie theory for single spherical particles. Particularly, it was discussed in detail in the book by Dombrovsky (1996) and called there “the scattering by absorption” (see also article Near-infrared properties of quartz fibers). The physical explanation of the scattering by absorption is quite clear: The local absorption leads to deformation of the wave front. It means that the electromagnetic wave near this local region does not propagate in the original direction and there is some scattering.
As applied to our problem, one can imagine that there is an absorbing substance (like water or something else) which is not uniformly distributed in a nonabsorbing (or weakly absorbing) matrix but it is concentrated in some local regions. These local regions (quasiparticles) may have practically the same index of refraction as the index of refraction of the ambient host medium, but the index of absorption of the quasiparticles is greater than the index of absorption of the host medium. It is obvious that our quasiparticles will scatter the radiation and this scattering is directly proportional to the absorption of the quasiparticle material.
The scattering by absorbing quasiparticles can be estimated using the known modification of the Mie theory for the case of a refracting but nonabsorbing ambient medium. It is sufficient to use the relative complex index of refraction m = 1 -iκ_{*}/ n, where κ_{*} >> κ is the index of absorption of the quasiparticle, n is the ambient medium index of refraction. The calculations should be done at the modified value of the diffraction parameter x = nx, where x = 2πa_{*}/ λ, and a_{*} is the quasiparticle radius. For an equivalent optically soft refracting medium, which represents highly porous material, we have n ≈ 1 and there is no need to account for the refraction of the ambient medium. The results of calculations using the Mie theory are presented in Fig. 6. One can see that the value of Q_{s}^{tr} increases very fast with the diffraction parameter and reaches approximately constant value at x ~ 1. Moreover, the dependence of Q_{s}^{tr} on κ_{*} is almost linear and we can use the following approximate relation in further estimates:
(10) |
Figure 6. Transport efficiency factor of scattering for nonrefracting particles as a function of diffraction parameter [(a), (1) κ_{*} = 0.01, (2) κ_{*} = 0.1, (3) κ_{*} = 1)] and index of refraction [(b), (1) - x = 2, (2) x = 3, (3) x = 5)].
We assume that absorption is localized mainly in quasiparticles and Eq. (1) is correct. It enables us to obtain the volume fraction of quasiparticles:
(11) |
Having substituted Eqs. (10) and (11) in the known formula for the transport scattering coefficient of a monodisperse system (see the article Radiative properties of polydisperse systems of independent particles) we obtain
(12) |
One can see that the transport coefficient of scattering is directly proportional to the absorption coefficient and the important ratio S_{λ} = σ_{λ}^{tr}/ α_{λ} = ω_{λ}/ (1 - ω_{λ}) (ω_{λ} is the albedo) does not depend on the material porosity and optical constants. The above relations yield S_{λ} ≈ 0.13 at x = 1 (a_{*} = 1 μm and λ ≈ 6.3 μm). In terms of order of magnitude, this result agrees well with the experimental data. For the sample made of HDK powder we have (see Fig. 2) an experimental value of S_{λ} ≈ 0.22 at the absorption peak of λ = 5.34 μm and S_{λ} ≈ 0.19 at the peak of λ = 6.14 μm. Of course, the above comparison is not quite correct because one should consider not the total values of the absorption and scattering coefficients but only their parts related with hydroxylation. In addition, our assumption of the spherical shape of quasiparticles is not justified and one can consider some elongated quasiparticles oriented mainly along the sample surfaces. In other words, we obtained a physical estimate only. But it is important that our hypothesis of scattering by absorption gives a qualitative explanation of the experimental results.
It is clear at the moment that there are two major effects which contribute to the radiation scattering in nanoporous silica in the visible and near-infrared: (1) the scattering by submicron aggregates of primary nanoparticles in the range of λ < 1.4 μm and (2) the scattering by micron-size local regions (quasiparticles) of relatively high absorption (mainly by silanol groups), which is the main mode in the infrared range of λ > 2.5 μm. To the best of our knowledge, it is the first case when the effect of scattering by absorption appears to be important for the near-infrared properties of a semitransparent thermal insulation. Note that the materials of this section have been recently reported in more detail by Dombrovsky et al. (2010).
REFERENCES
Ahrenkiel, R. K., Modified Kramers-Krönig analysis of optical spectra, J. Opt. Soc. Am., vol. 61, no. 12, pp. 1651-1655, 1971.
Bynum, R. T., Jr., Insulation Handbook, New York: McGraw-Hill, 2001.
Davis K. M. and Tomozawa, M., An infrared spectroscopic study of water-related species in silica glasses, J. Non-Cryst. Solids, vol. 201, no. 3, pp. 177-198, 1996.
Dombrovsky, L. A. Radiation Heat Transfer in Disperse Systems, New York: Begell House, 1996.
Dombrovsky, L. A., Approximate models of radiation scattering in hollow-microsphere ceramics, High Temp., vol. 42, no. 5, pp. 776-784, 2004.
Dombrovsky, L., Randrianalisoa, J., Baillis, D., and Pilon, L., Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles, Appl. Opt., vol. 44, no. 33, pp. 7021-7031, 2005.
Dombrovsky, L., Lallich, S., Enguehard, F., and Baillis, D.. An effect of “scattering by absorption” observed in near-infrared properties of nanoporous silica, J. Appl. Phys., vol. 107, no. 8, p. 083106, 2010.
Draine, B. T., The discrete dipole approximation for light scattering by irregular targets, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, edited by M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Chap. 5, San Diego: Academic Press, 2000.
Enguehard, F., Multiscale modeling of radiation heat transfer through nanoporous superinsulating materials, Int. J. Thermophys., vol. 28, no. 5, pp. 1693-1717, 2007.
Khashan, M. A. and Nassif, A. Y., Dispersion of the optical constants of quartz and polymethyl methacrylate glasses in a wide spectral range: 0.2-3 μm, Optics Commun., vol. 188, no. 1-4, pp. 129-139, 2001.
Kitamura, R., Pilon, L., and Jonasz, M., Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperatures, Appl. Opt., vol. 46, no. 33, pp. 8118-8133, 2007.
Lallich, S., Enguehard, F., and Baillis, D., Experimental determination and modeling of the radiative properties of silica nanoporous matrixes, ASME J. Heat Transfer, vol. 131, no. 8: p. 082701, 2009.
Malitson, I. H., Interspecimen comparison of the refractive index of fused silica, J. Opt. Soc. Am., vol. 55, no. 10, pp. 1205-1209, 1965.
Modest, M. F., Radiative Heat Transfer, 2nd ed., New York: Academic Press, 2003.
Peng, L., Qisui, W., Xi, L., and Chaocan, Z., Investigation of the states of water and OH groups on the surface of silica, Colloids Surf. A: Physicochem. Eng. Aspects, vol. 334, no. 1-3, pp. 112-115, 2009.
Plotnichenko, V. G., Sokolov, V. O., and Dianov, E. M., Hydroxyl groups in high-purity silica glass, J. Non-Cryst. Solids, vol. 261, no. 1-3, pp. 186-194, 2000.
Reim, M., Körner, W., Manara, J., Korder, S., Arduini-Schuster, M., Ebert, H.-P., and Fricke, J., Silica aerogel granulate material for thermal insulation and daylighting, Solar Energy, vol. 79, no. 2, pp. 131-139, 2005.
Tomozawa, M., Kim, D.-L., Agarwal, A., and Davis, K.M., Water diffusion and surface structural relaxation of silica glasses, J. Non-Cryst. Solids, vol. 288, no. 1-3, pp. 73-80, 2001.
Wiener, M., Reichenauer, G., Braxmeier, S., Hemberger, F., and Ebert, H.-P., Carbon aerogel-based high-temperature thermal insulation, Int. J. Thermophys., vol. 30, no. 4, pp. 1372-1385, 2009.
Zhuravlev, L. T., The surface chemistry of amorphous silica. Zhuravlev Model, Colloids Surf. A: Physicochem. Eng. Aspects, vol. 173, no. 1, pp. 1-38, 2000.
References
- Ahrenkiel, R. K., Modified Kramers-Krönig analysis of optical spectra, J. Opt. Soc. Am., vol. 61, no. 12, pp. 1651-1655, 1971.
- Bynum, R. T., Jr., Insulation Handbook, New York: McGraw-Hill, 2001.
- Davis K. M. and Tomozawa, M., An infrared spectroscopic study of water-related species in silica glasses, J. Non-Cryst. Solids, vol. 201, no. 3, pp. 177-198, 1996.
- Dombrovsky, L. A. Radiation Heat Transfer in Disperse Systems, New York: Begell House, 1996.
- Dombrovsky, L. A., Approximate models of radiation scattering in hollow-microsphere ceramics, High Temp., vol. 42, no. 5, pp. 776-784, 2004.
- Dombrovsky, L., Randrianalisoa, J., Baillis, D., and Pilon, L., Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles, Appl. Opt., vol. 44, no. 33, pp. 7021-7031, 2005.
- Dombrovsky, L., Lallich, S., Enguehard, F., and Baillis, D.. An effect of “scattering by absorption” observed in near-infrared properties of nanoporous silica, J. Appl. Phys., vol. 107, no. 8, p. 083106, 2010.
- Draine, B. T., The discrete dipole approximation for light scattering by irregular targets, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, edited by M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Chap. 5, San Diego: Academic Press, 2000.
- Enguehard, F., Multiscale modeling of radiation heat transfer through nanoporous superinsulating materials, Int. J. Thermophys., vol. 28, no. 5, pp. 1693-1717, 2007.
- Khashan, M. A. and Nassif, A. Y., Dispersion of the optical constants of quartz and polymethyl methacrylate glasses in a wide spectral range: 0.2-3 μm, Optics Commun., vol. 188, no. 1-4, pp. 129-139, 2001.
- Kitamura, R., Pilon, L., and Jonasz, M., Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperatures, Appl. Opt., vol. 46, no. 33, pp. 8118-8133, 2007.
- Lallich, S., Enguehard, F., and Baillis, D., Experimental determination and modeling of the radiative properties of silica nanoporous matrixes, ASME J. Heat Transfer, vol. 131, no. 8: p. 082701, 2009.
- Malitson, I. H., Interspecimen comparison of the refractive index of fused silica, J. Opt. Soc. Am., vol. 55, no. 10, pp. 1205-1209, 1965.
- Modest, M. F., Radiative Heat Transfer, 2nd ed., New York: Academic Press, 2003.
- Peng, L., Qisui, W., Xi, L., and Chaocan, Z., Investigation of the states of water and OH groups on the surface of silica, Colloids Surf. A: Physicochem. Eng. Aspects, vol. 334, no. 1-3, pp. 112-115, 2009.
- Plotnichenko, V. G., Sokolov, V. O., and Dianov, E. M., Hydroxyl groups in high-purity silica glass, J. Non-Cryst. Solids, vol. 261, no. 1-3, pp. 186-194, 2000.
- Reim, M., Körner, W., Manara, J., Korder, S., Arduini-Schuster, M., Ebert, H.-P., and Fricke, J., Silica aerogel granulate material for thermal insulation and daylighting, Solar Energy, vol. 79, no. 2, pp. 131-139, 2005.
- Tomozawa, M., Kim, D.-L., Agarwal, A., and Davis, K.M., Water diffusion and surface structural relaxation of silica glasses, J. Non-Cryst. Solids, vol. 288, no. 1-3, pp. 73-80, 2001.
- Wiener, M., Reichenauer, G., Braxmeier, S., Hemberger, F., and Ebert, H.-P., Carbon aerogel-based high-temperature thermal insulation, Int. J. Thermophys., vol. 30, no. 4, pp. 1372-1385, 2009.
- Zhuravlev, L. T., The surface chemistry of amorphous silica. Zhuravlev Model, Colloids Surf. A: Physicochem. Eng. Aspects, vol. 173, no. 1, pp. 1-38, 2000.