SEMITRANSPARENT MEDIA CONTAINING BUBBLES
Following from: Spectral radiative properties of some important materials: Experimental data and theoretical models
In many natural phenomena as well as materials processing and manufacturing situations, the presence of bubbles affects the thermophysical and radiative properties of the two-phase system and hence the transport phenomena. It is well known that radiation scattering by bubbles in the visible and infrared spectral ranges affects the optical properties of semitransparent substances. Several issues are important to keep in mind: the influence of bubbles on light scattering in the ocean (Zhang et al., 1998), the role of vapor bubbles in high-temperature radiative heating of boiling water (Dombrovsky, 2004), and the glass melting process in industrial furnaces, where bubbles are generated by chemical reactions (Fedorov and Pilon, 2002). Similar structures with numerous bubbles or hollow microspheres in a semitransparent host medium are considered as advanced thermal insulation materials (German and Grinchuk, 2002; Papadopoulos, 2005); many aerated foods containing gas bubbles represent the height of the culinary art (Campbell and Mougeot, 1999). Following the recent works by Pilon and Viskanta (2003), Baillis et al. (2004), and Dombrovsky et al. (2005), we focus mainly on applications for the glass industry. At the same time, some methodological and physical results may be interesting in the other previously mentioned fields.
For the disperse systems considered in this section, a combination of a traditional identification procedure and theoretical predictions can be used (see the articles A basis of experimental characterization and identification procedure and Identification procedure). Spherical bubbles in a weakly absorbing medium are ideal objects for applying the Mie theory. The analysis for the most interesting range of parameters presented in the article Radiative properties of gas bubbles is semi-transparent medium showed that scattering properties of polydisperse bubbles do not depend on radiation absorption, whereas the absorption is insensitive to the size distribution of bubbles. These results were used by Dombrovsky et al. (2005) to suggest an improved identification procedure for the directional-hemispherical measurements.
The above-mentioned detailed analysis of the effect of nonabsorbing gas bubbles on absorption and scattering characteristics of a semitransparent absorbing medium yields the following approximate expressions for the absorption coefficient and transport scattering coefficients:
(1) |
where α_{λ}^{0} = 4πκ_{0}/ λ is the absorption coefficient of the matrix. It is important that absorption does not depend on the bubbles size distribution and scattering does not depend on the matrix absorption index. The only parameter related to the bubbles which affects the transport scattering coefficient of the medium is the ratio of the volume fraction of bubbles to their average radius: f_{v}/ a_{32}. To analyze the effects of possible impurities in the material containing bubbles we also consider the case of an absorbing and refracting substance present inside the bubbles. We assume that this substance has a comparably large absorption index κ_{i} >> κ_{0} because it is the only case for which a considerable effect on the absorption coefficient of the medium containing bubbles can be observed. Some results of calculations of relative efficiency factors for the simple case of a homogeneous particle (when a substance with m_{i} = n_{i} - iκ_{i} fills the entire bubble volume) are shown in Figs. 1 and 2. We limit our consideration to cases where n_{i} < n_{0}, κ_{i} << 1, 2κ_{i} x << 1, and plot the ratios of the efficiency factors to the following values:
(2) |
FigureÂ 1. Effect of the absorbing and refracting medium inside a bubble on its efficiency factor of absorption for n_{0} = 1.4: (a) κ_{i} = 10^{-4}, (b) κ_{i} = 10^{-3}; (1) n_{i} = 1.1, (2) n_{i} = 1.2.
FigureÂ 2. Effect of refracting medium inside a bubble on its transport efficiency factor of scattering: (a) n_{0} = 1.4, (b) n_{0} = 1.5; (1) n_{i} = 1, (2) n_{i} = 1.2.
The latter approximation of Q_{s}^{tr} is not as exact for n_{i} > 1 as for n_{i} = 1 but it can be used as a first-order approximation. The small matrix absorption coefficient has no effect on the applicability of the approximate equations (2). In addition, the value of Q_{s}^{tr} is insensitive to the weak absorption of the particle substance. The approximate expression of Q_{a}^{i} in general form, applicable for arbitrary values of κ_{i}x, can be written as follows (Dombrovsky et al., 2003):
(3) |
In the cases where 2κ_{i}x << 1, the resulting approximate relations for the absorption and transport scattering coefficients of a weakly absorbing media with bubbles can be expressed as
(4) |
where f_{v} is the total volume fraction of bubbles, f_{v}^{i} is the volume fraction of bubbles filled by the absorbing substance, and α_{λ}^{i} = 4πκ_{i}/ λ is the absorption coefficient of the filling substance. One can see that the absorption coefficient does not depend on the size distribution of the bubbles but it is important to know the volume fractions of bubbles filled by an absorbing substance. In contrast with the absorption coefficient, the transport scattering coefficient is directly proportional to the total concentration of bubbles and inversely proportional to their average radius. Note that the scattering function does not depend on the volume fraction f_{v}. The details of the bubble size distribution affect neither the absorption nor the scattering characteristics of the medium.
The above analysis shows that the size distribution of bubbles is not important but we have to know the total volume fraction of bubbles f_{v}, the average radius a_{32}, as well as the volume fraction f_{v}^{i} of possible bubbles filled by an absorbing substance. Note that one can use the volume-averaged values of f_{v}, f_{v}^{i}, and a_{32} without accounting for their spatial variation because of the small optical thickness of the samples.
The total volume fraction f_{v} can be evaluated directly by measuring the sample density but one can also use indirect evaluation of the same value by calculating the surface concentration f_{s} of bubbles. It is clear that the concentration of bubbles on the sample surfaces (they look like defects of the surface) is proportional to f_{v}. For a large number of bubbles, the average surface concentration of bubbles f_{s} for an arbitrary cut of the sample does not depend on the possible order in spatial distribution of the bubbles. For this reason, we can consider the simplest cubic structure. For samples with a uniform volume distribution of polydisperse bubbles, one can write
(5) |
where d is the distance between neighboring bubbles (step of the cubic structure). While “cutting” the sample by planes, which are parallel to one of the sides of the cubic structure, the probability that a bubble is located on the sample surface is p = 2a_{10}/ d and the value of the average surface concentration of bubbles is determined by
(6) |
Comparison of Eqs. (5) and (6) gives the following relation between the volume fraction and the average surface concentration of bubbles:
(7) |
We consider two samples of different thickness: z_{0} = 5 and 10 mm. The size distribution of bubbles for the thin sample was determined by analyzing high-resolution digital photographs. The resulting normalized distribution function based on measurements of N = 212 bubbles is shown in Fig. 3. Simple calculations give the following values of the average radii of bubbles: a_{21} = 0.56 mm and a_{32} = 0.64 mm. Note that previous measurements reported by Baillis et al. (2004) gave slightly greater values (a_{21} = 0.70 mm, a_{32} = 0.75 mm) mainly because the significant contribution of small bubbles was ignored. In the case of thin samples, image analysis also enables the direct determination of the volume fraction:
(8) |
where S is the sample surface area. The results of the different methods are as follows: f_{v} = (4.6 ± 1.1)% from density measurements, f_{v} ≈ 3.2% from measurements of surface defects, and f_{v} = (4.3± 0.2)% from photographs. The simplest method, using surface defects, underestimates the volume fraction of bubbles because of the disappearance of small defects after polishing the sample surfaces. Both of the other methods give approximately the same results. In the analysis of experimental data for infrared properties of fused quartz samples containing bubbles, we will use the approximate value of a_{32} = 0.64 mm and the range of volume fraction f_{v} = 3.5% - 5%.
FigureÂ 3. Normalized size distribution of bubbles in the fused quartz sample.
The two maxima observed in the bubble size distribution shown in Fig. 3 can be interpreted as a result of strong compression of bubbles of average size before solidification of the quartz melt. The surface tension of the melt leads to a significant increase in gas pressure inside the collapsing bubbles (Fedorov and Pilon, 2002; Pilon and Viskanta, 2003). Thus, one cannot exclude the possibility that the smallest bubbles contain a condensed substance which absorbs radiation in the near infrared. We will consider this hypothesis by analyzing the experimental results for the spectral absorption coefficient of fused quartz samples containing bubbles.
The samples of fused quartz containing bubbles were illuminated by a normally incident collimated beam. The experimental setup consists of two main parts: a BIO-RAD FTS 60A FTIR spectrometer and a gold-coated integrating sphere CSTM RSA-DI-40D which collects hemispherically the radiation crossing, or reflected by, the sample onto a detector placed on the wall of the sphere. The incident beam was not parallel but convergent with an angle of 2.25°. The diameter of the sample area subjected to a normally incident beam was ~28 mm and ~16 mm for transmittance and reflectance measurements, respectively. Fused quartz samples containing bubbles were prepared with special attention to the quality of their surfaces as described by Baillis et al. (2004). Transmittance and reflectance spectra have been acquired several times for different positions and orientations of the samples. Because of “noisy” transmittance and reflectance spectra, the data have been smoothed by using the BOXCAR procedure available in the WIN-IR PRO software. The smoothing parameters were chosen to eliminate the numerical noise but not to affect the physical behavior of the spectra. The results of directional-hemispherical measurements in the spectral range 2 < λ < 4 μm are presented in Fig. 4. Note that only the average values are shown. The standard absolute deviation is ~3%-8% for transmittance and 8%-15% for reflectance. The analysis of the experimental results is based mainly on the more reliable transmittance measurements whereas the reflectance data are used only for evaluating the volume fraction of bubbles.
FigureÂ 4. Directional-hemispherical transmittance and reflectance for two samples of fused quartz containing bubbles: (1) z_{0} = 5 mm, (2) 10 mm.
Similar measurements were reported by Dombrovsky et al. (2005) for determining the index of absorption κ_{0} of fused quartz samples without bubbles cut from the same piece as that used for the samples containing bubbles. The absorption index was calculated from the transmittance data. The three-term dispersion relation suggested by Malitson (1965) for the index of refraction of fused quartz was used instead of the measurements of reflectance, which are too noisy. The standard absolute deviation of transmittance from the average values was <5%. Note that the applicability of Malitson’s dispersion relation has been confirmed in several more recent papers (Tan, 1998; Tan and Arndt, 2001). The values of the absorption index of fused quartz determined by Dombrovsky et al. (2005) are in good agreement with published data (see article Near-infrared properties of quartz fibers for more details).
Because of a small volume fraction of randomly placed bubbles, the radiation transfer theory can be used for calculating the reflection and transmission of infrared radiation in glass containing bubbles. The problem statement for radiation transfer in a plane-parallel slab of an absorbing, refracting, and anisotropically scattering medium at normal illumination by randomly polarized radiation was considered in detail in the article Hemispherical transmittance and reflectance at normal incidence. One can also find there the general expressions for directional-hemispherical transmittance and reflectance and a description of some computational models. Following our paper (Dombrovsky et al., 2005), we consider two alternative models for calculating the diffuse component of the radiation intensity: (1) the numerical solution which uses the discrete ordinates method (DOM) and (2) the analytical solution based on the modified two-flux approximation (MDP_{0}). In the first case, two approximations of scattering function are considered: the transport approximation and the Henyey-Greenstein approximation. It was shown that the error of the transport approximation increases with optical thickness but it is insignificant in the range of weak extinction which is the most important for the problem under consideration. Good agreement between calculations using the Henyey-Greenstein and the transport approximation showed that separate determination of the scattering coefficient and of the asymmetry factor using hemispherical measurements is practically impossible. Fortunately, one does not need these data in the usual radiation heat transfer calculations. It was also shown that the modified two-flux approximation can be used in analyzing the experimental data for the directional-hemispherical transmittance and reflectance of fused quartz samples containing bubbles.
As was discussed above, the scattering characteristics of a weakly absorbing medium containing bubbles do not depend on the absorption characteristics of the continuous or dispersed phases. This is very important because small impurities in either phase can affect the absorption coefficient in the semitransparency region of the spectrum and one cannot be sure of the theoretical predictions based on the absorption properties of the medium components. For this reason, it is suggested that only the predictions for the scattering characteristics but not for the absorption are used in the identification procedure.
Let us assume that the scattering characteristics of the heterogeneous medium can be determined from the approximate relations suggested above. Three values of the volume fraction of bubbles are considered: f_{v} = 3.5%, 4%, and 4.5% and a fixed value of a_{32} = 0.64 μm. It is also assumed that f_{v}^{i} = 0. The choice of f_{v} is based on the estimates that give the best-curve fit for the measured reflectance spectra. The averaged experimental values of the directional-hemispherical transmittance at each wavelength are considered to be exact. The spectral dependency of T_{d-h}(λ) is used for determining the transport albedo and the absorption coefficient of the heterogeneous medium. It should be noted that this procedure does not require solving the ill-posed inverse problem as in the case of several unknown optical parameters. All the calculations are performed using the analytical solution of the MDP_{0} approximation.
Comparison between the calculated reflectance and the experimental measurements shown in Fig. 5 enables the evaluation of the bubble volume fraction: It is of the order of 4% for both samples. Note that it is difficult to curve fit R_{d-h}(λ) by choosing the value of f_{v}; however, one should keep in mind that experimental error associated to the reflectance measurements is much greater than that for the transmittance. Therefore, the reflectance measurements were used only to evaluate the volume fraction of bubbles f_{v}. It is important that this value is in good agreement with independent estimations based on density measurements and on image analysis of high-resolution photographs of the thin sample.
FigureÂ 5. Directional-hemispherical reflectance for two samples of fused quartz containing bubbles: (a) z_{0} = 5 mm, (b) 10 mm; (1) measurement, (2) calculation for f_{v} = 3.5%, (3) for f_{v} = 4.5%.
The values of the transport albedo and of the absorption coefficient determined by the identification procedure for f_{v}=4% are presented in Figs. 6 and 7. In the latter, the theoretical value of the absorption coefficient predicted by Eq. (1) is also shown. One can see that the difference between experimental and theoretical values of the absorption coefficient Δα_{λ} is relatively small, except around the narrow absorption peak near λ = 2.7 μm. This result can provide an estimate of the maximum value of the integral parameters in the event impurities are present in the medium containing bubbles. Assuming the presence of an absorbing medium inside the bubbles, one can obtain from Eq. (4) the following approximate relation:
(9) |
FigureÂ 6. Transport albedo of fused quartz containing bubbles (f_{v} = 4%): (1) z_{0} = 5 mm, (2) 10 mm.
FigureÂ 7. Absorption coefficient of fused quartz containing bubbles (f_{v} = 4%): (a) z_{0} = 5 mm, (b) 10 mm; (1) experiment, (2) theoretical prediction.
It should be noted that even the presence of molecular water impurities in the fused quartz should be accounted for in the absorption spectrum of glass (Plotnichenko et al., 2000; Tomozawa et al., 2001; Kournyts’kyi et al., 2005). To obtain some quantitative estimates, consider water as a model condensed substance inside some part of small bubbles. Using the data by Hale and Querry (1973) (n_{i} = 1.3, κ_{i} = 10^{-3} at λ = 2μm and n_{i} = 1.2, κ_{i} = 0.02 at λ = 2.7μm) and assuming f_{v}^{i} = 0.4% we find Δα_{λ} ~ 1 m^{-1} at λ = 2 μm and Δα_{λ} ~ 30 m^{-1} at λ = 2.7 μm. These evaluations correlate rather well with the level of differences between experimental data for fused quartz containing bubbles and the theoretical predictions based on the assumption f_{v}^{i} = 0. This suggests that the experimental data for the absorption coefficient of fused quartz samples containing bubbles does not exclude the presence of some radiation-absorbing impurities entrapped in the bubbles.
It is important that the above-reported research of absorption and scattering of near-infrared radiation by fused quartz containing bubbles is based on a combination of experimental measurements and theoretical analysis using the Mie theory. The Mie calculations over a wide range of parameters enable the formulation of approximate relations for the main radiative characteristics of semitransparent media containing large polydisperse bubbles, including those filled with an absorbing and refracting substance. It is shown that radiation scattering by bubbles is independent of the weak absorption by the matrix and the potential absorbing substance inside the bubbles. The use of theoretically predicted scattering characteristics of the heterogeneous medium makes it possible to avoid large errors in the identification procedure caused by noisy experimental data for the small values of reflectance.
Application of the suggested identification procedure for studying the near-infrared properties of fused quartz samples containing bubbles provides new data on spectral single-scattering albedo and for the absorption coefficient of the heterogeneous medium in the spectral range from 2 to 4 μm. The volume fraction of bubbles obtained from directional-hemispherical reflectance of fused quartz samples are in good agreement with both density measurements and image analysis of high-resolution photographs of the thin sample. Comparison between experimentally determined and predicted values of the absorption coefficient of fused quartz containing bubbles enables the evaluation of the integral characteristic of possible impurities which are treated as an absorbing substance entrapped inside the bubbles. To our mind, the suggested procedure combining experimental measurements and theoretical analysis of infrared radiative properties of semitransparent substances containing bubbles can be used for both controlling the optical purity of the medium and estimating the volume fraction of polydisperse bubbles in various applications.
REFERENCES
Baillis, D., Pilon, L., Randrianalisoa, H., Gomez, R., and Viskanta, R., Measurements of radiation characteristics of fused quartz containing bubbles, J. Opt. Soc. Am. A, vol. 21, no. 1, pp. 149-159, 2004.
Campbell, G. M. and Mougeot, E., Creation and characterisation of aerated food products, Trends Food Sci. Technol., vol. 10, no. 9, pp. 283-296, 1999.
Dombrovsky, L. A., The propagation of infrared radiation in a semitransparent liquid containing gas bubbles, High Temp., vol. 42, no. 1, pp. 133-139, 2004.
Dombrovsky, L. A., Sazhin, S. S., Mikhalovsky, S. V., Wood, R., and Heikal, M. R., Spectral properties of diesel fuel droplets, Fuel, vol. 82, no. 1, pp. 15-22, 2003.
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.
Fedorov, A. G. and Pilon, L., Glass foam: Formation, transport properties, and heat, mass, and radiation transfer, J. Non-Cryst. Solids, vol. 311, no. 2, pp. 154-173, 2002.
German, M. L. and Grinchuk, P. S., Mathematical model for calculating the heat-protection properties of the composite coating “ceramic microspheres--binder,” J. Eng. Phys. Thermophys., vol. 75, no. 6, pp. 1301-1313, 2002.
Hale, G. M. and Querry, M. P., Optical constants of water in the 200 nm to 200 μm wavelength region, Appl. Opt., vol. 12, no. 3, pp. 555-563, 1973.
Kournyts’kyi, T., Melnik, R. V. N., and Gachkevich, A., Thermal behavior of absorbing and scattering glass media containing molecular water impurity, Int. J. Therm. Sci., vol. 44, no. 2, 107-114, 2005.
Malitson, I. H., Interspecimen comparison of the refractive index of fused silica, J. Opt. Soc. Am., vol. 55, no. 10, pp. 1205-1209, 1965.
Papadopoulos, A. M., State of the art in thermal insulation materials and aims for future developments, Energy and Buildings, vol. 37, no. 1, pp. 77-86, 2005.
Pilon, L. and Viskanta, R., Radiation characteristics of glass containing bubbles, J. Am. Ceramic Soc., vol. 86, no. 8, pp. 1313-1320, 2003.
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.
Tan, C. Z., Determination of refractive index of silica glass for infrared wavelength by IR spectroscopy, J. Non-Cryst. Solids, vol. 223, no. 1-2, pp. 158-163, 1998.
Tan, C. Z. and Arndt, J., Refractive index, optical dispersion, and group velocity of infrared waves in silica glass, J. Phys. Chem. Solids, vol. 62, no. 6, pp. 1087-1092, 2001.
Tomozawa, M., Kim, D.-L., and Lou, V., Preparation of high purity, low water content fused silica glass, J. Non-Cryst. Solids, vol. 296, no. 1-2, pp. 102-106, 2001.
Zhang, X., Lewis, M., and Johnson, B., Influence of bubbles on scattering of light in the ocean, Appl. Opt., vol. 37, no. 27, pp. 6525-6536, 1998.
References
- Baillis, D., Pilon, L., Randrianalisoa, H., Gomez, R., and Viskanta, R., Measurements of radiation characteristics of fused quartz containing bubbles, J. Opt. Soc. Am. A, vol. 21, no. 1, pp. 149-159, 2004.
- Campbell, G. M. and Mougeot, E., Creation and characterisation of aerated food products, Trends Food Sci. Technol., vol. 10, no. 9, pp. 283-296, 1999.
- Dombrovsky, L. A., The propagation of infrared radiation in a semitransparent liquid containing gas bubbles, High Temp., vol. 42, no. 1, pp. 133-139, 2004.
- Dombrovsky, L. A., Sazhin, S. S., Mikhalovsky, S. V., Wood, R., and Heikal, M. R., Spectral properties of diesel fuel droplets, Fuel, vol. 82, no. 1, pp. 15-22, 2003.
- 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.
- Fedorov, A. G. and Pilon, L., Glass foam: Formation, transport properties, and heat, mass, and radiation transfer, J. Non-Cryst. Solids, vol. 311, no. 2, pp. 154-173, 2002.
- German, M. L. and Grinchuk, P. S., Mathematical model for calculating the heat-protection properties of the composite coating “ceramic microspheres--binder,” J. Eng. Phys. Thermophys., vol. 75, no. 6, pp. 1301-1313, 2002.
- Hale, G. M. and Querry, M. P., Optical constants of water in the 200 nm to 200 μm wavelength region, Appl. Opt., vol. 12, no. 3, pp. 555-563, 1973.
- Kournyts’kyi, T., Melnik, R. V. N., and Gachkevich, A., Thermal behavior of absorbing and scattering glass media containing molecular water impurity, Int. J. Therm. Sci., vol. 44, no. 2, 107-114, 2005.
- Malitson, I. H., Interspecimen comparison of the refractive index of fused silica, J. Opt. Soc. Am., vol. 55, no. 10, pp. 1205-1209, 1965.
- Papadopoulos, A. M., State of the art in thermal insulation materials and aims for future developments, Energy and Buildings, vol. 37, no. 1, pp. 77-86, 2005.
- Pilon, L. and Viskanta, R., Radiation characteristics of glass containing bubbles, J. Am. Ceramic Soc., vol. 86, no. 8, pp. 1313-1320, 2003.
- 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.
- Tan, C. Z., Determination of refractive index of silica glass for infrared wavelength by IR spectroscopy, J. Non-Cryst. Solids, vol. 223, no. 1-2, pp. 158-163, 1998.
- Tan, C. Z. and Arndt, J., Refractive index, optical dispersion, and group velocity of infrared waves in silica glass, J. Phys. Chem. Solids, vol. 62, no. 6, pp. 1087-1092, 2001.
- Tomozawa, M., Kim, D.-L., and Lou, V., Preparation of high purity, low water content fused silica glass, J. Non-Cryst. Solids, vol. 296, no. 1-2, pp. 102-106, 2001.
- Zhang, X., Lewis, M., and Johnson, B., Influence of bubbles on scattering of light in the ocean, Appl. Opt., vol. 37, no. 27, pp. 6525-6536, 1998.