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In recent years, composite materials containing hollow glass or ceramic microspheres have attracted considerable attention. These materials have very good heat-insulation characteristics, which are largely defined by special features of absorption and scattering of thermal radiation by thin-walled hollow particles (German and Grinchuk, 2002; Dombrovsky, 2005). The paint coatings containing hollow glass microspheres have already found applications for reducing heat loss from the walls of buildings owing to a decrease in thermal radiation at night. Nevertheless, choosing the best disperse composition of microspheres for use in building paints and other polymer coatings remains a problem which requires theoretical and experimental investigation.

Theoretical prediction of wide-range spectral radiative properties of the coatings containing polydisperse hollow microspheres based on the traditional radiation transfer model in combination with the Mie theory for single particles, as was suggested by Dombrovsky (2005), may not be adequate at a high volume fraction of microspheres because of specific scattering effects in the volume and at the coating interface. It is known that the so-called dependent scattering effects can be observed in densely packed disperse systems (Ivanov et al., 1988; Kumar and Tien, 1990; Singh and Kaviany, 1992).

Dombrovsky et al. (2007) have presented probably the first experimental study of the main radiative properties of polymer films with microspheres in the spectral range from 2.6 to 18 μm. The measurements of directional-hemispherical reflectance and transmittance of coatings at various concentrations of hollow glass microspheres were performed by use of Fourier transform infrared (FTIR) spectrometry. The new experimental data were compared with theoretical predictions based on the hypothesis of the negligible role of dependent scattering effects. Two series of hollow glass microspheres produced by the 3M Company were used to prepare the samples of composite material. The normalized size distributions F(a) for microspheres K20 and S38 are shown in Fig. 1. These distributions were obtained from volume distributions provided by 3M. To estimate an average wall thickness of microspheres one can use the following equations assuming a definite relation between the wall thickness δ and particle radius a:

(1)

where ρg is the density of glass, ρm is the apparent density of polydisperse microspheres, and the parameters aij are defined as usual:

(2)

Figure 1. Normalized size distributions of glass microspheres.

It is known that a32=50 μm, ρm=0.202 g/cm3 for K20; a32=40.7 μm, ρm=0.387g/cm3 for S38 microspheres. Substituting these values and ρg=2.54g/cm3 for soda lime borosilicate glass in Eq. (1), one can find the average wall thickness δ:

(3)

We had no data for the dependence of δ(a). For this reason, our analysis was based on the results for the following two assumptions: δ = δ=const or δ = δa/a32. The samples of composite material were made by mixing of glass microspheres in polymer ACRONAL 290D, which is made from an emulsion of styrene-acrylic from BASF Corporation. The mixture was spread out over a flat surface and it was then dried. The resulting parameters of thin square samples of area 16 cm2 are given in Table 1. When the polymer layer dries out, the distance between the microspheres decreases differently in different directions. This results in the anisotropy of the volume concentration of microspheres and in the respective anisotropy of the radiative characteristics of the medium (Dombrovsky, 2005). The estimates have showed that this effect is not significant and we assumed the composite material to be homogeneous and isotropic.

Table 1. Main parameters of the samples

Type of microspheres Sample number Thickness d, μm Thickness deviation Δd, μm Volume fraction of microspheres fv, % Volume fraction uncertainty Δfv, %
K20 1 141.9 3.2 6.6 3.4
2 138.9 8.0 23.3 7.4
3 140.2 5.6 37.4 4.4
4 156.6 4.8 50.1 2.9
5 174.9 3.5 65.4 1.5
S38 1 142.6 5.7 9.6 6.0
2 144.7 2.0 23.4 1.8
3 149.4 4.9 38.2 3.9
4 152.2 5.9 52.8 4.1
5 165.6 4.4 66.4 2.4

Because of the specific manufacturing process, one side of each sample was relatively smooth while the other side appeared to be rougher. Two typical microphotographs of the rough-side view of the samples are presented in Fig. 2. The majority of particles are ideal spheres but some of the large particles are damaged (at least, near the sample surface). One can see also several irregular small fragments of the glass microspheres. The roughness profiles of the sample surfaces h(s) were measured by using a wide-field confocal microscope, MICROSME-2, with the interval Δy = 5 μm at a distance y0 = 10 mm. The average height of the roughness h(y) was determined as follows:

(4)

Figure 2. The fragments of the rough surface of the polymer samples containing microspheres of series K20. Left image: sample 1; right image: sample 5 (see Table 2).

The value of h for a pure polymer film appears to be equal to 3.5 μm for the rough side and 0.7 μm for the smooth side. The other values obtained for the samples containing glass microspheres are given in Table 2. One can see that the height of roughness increases with the volume fraction and with the average size of the microspheres. To analyze the effect of the surface roughness on reflectance and transmittance, all the measurements were performed for two orientations of the samples with respect to the collimated incident radiation.

Table 2. Average height of sample roughness

Type of microspheres Sample number h, μm
Smooth surface Rough surface
K20 1 - 5.0
2 - 7.9
3 - 9.2
4 - 14.8
5 - 18.9
S38 1 0.7 3.5
2 0.9 5.4
3 1.1 6.4
4 1.5 10.2
5 2.0 14.0

The samples of polymer film containing glass microspheres were illuminated by a collimated beam at an angle 10° incident to the normal direction. The experimental setup consisted 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. Not only the total reflectance and transmittance were measured but also their diffuse components, which do not include the specularly reflected radiation and the radiation transmitted in the direction of the incident collimated beam. In contrast to similar measurements for fused quartz containing gas bubbles (see article Semitransparent media containing bubbles) the spectra of directional-hemispherical reflectance and transmittance were not very noisy. The standard absolute deviation was <5% for both transmittance and reflectance in the entire spectral range.

Spectral optical constants of the polymer were obtained from the measurements of reflectance and transmittance for a pure polymer film of 20-μm thickness. The well-known relations were used for calculating the index of refraction n0 and the index of absorption κ0 of the polymer. The data obtained from the measurements at various angles of incidence were averaged to find more accurate results, shown in Fig. 3 The glass microspheres are made from soda lime borosilicate glass (75% SiO2, 4% Na2O, 15% CaO, and 6% B2O3). To the best of our knowledge, there are no data for optical constants of this glass in the literature. The wide-range infrared optical constants of soda lime silica glasses were investigated by Rubin (1985). The tabulated results for a typical clear window glass (73% SiO2, 15% Na2O, 10% CaO, and 2% Al2O3) are plotted in Fig. 4. We also performed the measurements for low-expansion borosilicate glass supplied by the Verre Labo Mula Company: 81% SiO2, 4% Na2O, 13% B2O3, and 2% Al2O3. The optical constants of this glass in the wavelength range 3 < λ < 5 μm were determined from the reflectance and transmittance measurements for the sample of thickness 0.7 mm by use of the same experimental technique as that used for polymer. The optical constants in the range 7 < λ < 18 μm were obtained by use of the least-squares optimization technique and experimental data for specular reflectance in the angular range from 13° to 80°. The wide-range optical constants of borosilicate glass are also presented in Fig. 4. The spectral behavior of optical constants for two types of glass is similar but one can see considerable quantitative differences both in the range of semitransparency and in the long-wave region. The effect of the optical constant’s uncertainty on radiative properties of microspheres and composite material will be analyzed below.

Figure 3. Optical constants of polymer

Figure 4. Optical constants of glasses: (1) soda lime silica glass (Rubin, 1985), (2) borosilicate glass (Dombrovsky et al., 2007).

Consider first the total and diffuse directional-hemispherical characteristics for the case of orientation of the samples toward the incident radiation by a relatively smooth surface (the case of smooth front surface). The data for samples of number 1, 3, and 5 are presented in Fig. 5 (intermediate curves for samples 2 and 4 are not shown in the graph). One can see significant increase in directional-hemispherical reflectance due to scattering of radiation by microspheres. This increase in reflectance is approximately proportional to the volume fraction of microspheres:

(5)

where R0 is the specular reflectance of the polymer sample without microspheres. Such behavior of diffuse reflectance with particle concentration is typical for independently scattering particles in a weakly absorbing medium. It is interesting that the value of r1 is not sensitive to the type of microspheres. The peak of r1(λ) at the wavelength 4.5 μm is a result of intense scattering of the radiation by hollow glass microspheres. The microspheres decrease the directional-hemispherical transmittance of the polymer samples. This effect depends on the type of microspheres (at least for fv ≥ 30%) and cannot be described by the linear function of the volume fraction fv as was done for the reflectance. It is important that the specular component of transmitted radiation is not small: Approximately half of the transmitted radiation is concentrated near the forward direction.

Figure 5. Total (a), (b) and diffuse (c), (d) reflectance and transmittance of polymer samples containing glass microspheres: 1, 3, 5 are the sample numbers. The samples are oriented to the incident radiation by their smooth surfaces.

The experimental data for the case of rougher front surface of the samples are presented in Fig. 6. In this case, there is no specular reflection from the front surface. At the same time, the radiation propagating through the front surface is mainly concentrated near the forward direction and the reflection from smoother back (shadow) surface provides the specular component in the total reflectance of the sample. The last effect is considerable only for a small volume fraction of microspheres. Note that reflectance in the range 9 < λ < 13 μm is sensitive to the type of microspheres. This effect was not observed when the samples are oriented to the incident radiation by their smoother surface. It is interesting that the short-wave peak of total reflectance at λ = 4.5 μm is insensitive to both the sample orientation and the type of microspheres. The same statement is true for the corresponding peak of total transmittance. In contrast to the reflection, one can observe considerable specular transmission of the radiation in the case of rough front surface and smooth back surface of the sample. This result is clear if we remember that the rougher front surface does not prevent concentration of the propagated radiation near the forward direction. As for diffuse transmittance, it is practically the same for various orientations of the sample.

Figure 6. Total (a), (b) and diffuse (c,) (d) reflectance and transmittance of polymer samples containing microspheres: 1, 3, 5 are the sample numbers. The samples are oriented to the incident radiation by their rough surfaces.

Following our paper (Dombrovsky et al., 2007), we consider a theoretical model based on a modified two-flux approximation for the radiation transfer and the Mie theory for scattering of radiation by single hollow glass particles embedded in the polymer matrix. This approach can be used when dependent scattering effects are not significant. The latter assumption should be verified by comparison of theoretical predictions with experimental data.

The modified two-flux approach suggested for the normally incident collimated radiation (see article Hemispherical transmittance and reflectance at normal incidence) was applied to the radiation transfer equation for the model transport scattering function. We have used the derived analytical solution to calculate the directional-hemispherical reflectance and transmittance of a homogeneous medium layer with perfectly smooth surfaces. It is known that the modified two-flux approximation usually gives rather accurate results for the directional-hemispherical characteristics. For the values of reflectance and transmittance typical for semitransparency ranges of the polymer film, the error of this approach is <3%.

Three spectral characteristics of an absorbing and anisotropically scattering medium are included as coefficients in the modified two-flux approximation: the index of refraction of the host medium n0, the absorption coefficient of the composite material α, and the transport scattering coefficient of polydisperse particles σtr. The absorption and transport scattering coefficients can be determined as follows:

(6)

where α0 = 4πκ0/ λ is the absorption coefficient of the host medium (polymer matrix), and Qa, Qstr are the efficiency factor of absorption and the transport efficiency factor of scattering for single spherical particles, respectively. The efficiency factors Qa, Qstr can be calculated by the Mie theory generalized to the case of a refracting and absorbing medium as described in the article Radiative properties of gas bubbles in semitransparent medium. In the monodisperse approximation, which is often used in engineering calculations, Eqs. (6) are reduced to the following (see article Radiative properties of polydisperse systems of independent particles):

(7)

It is important that the monodisperse approximation gives the exact results for polydisperse systems in the case of constant efficiency factors as well as in the case when Qa, Qstr are directly proportional to the particle radius. In the latter case, the corresponding coefficients do not depend on particle size distribution.

One should remember that the physical approach discussed above is applicable only for a weakly absorbing host medium. The optical thickness of the medium at distances comparable with the particle size must be small: τ0 = α0a << 1. In the opposite case, one cannot consider far-field characteristics of single particles and the traditional radiation transfer theory is inapplicable (Mishchenko et al., 2004). It is a limiting case of an opaque medium when the volume radiative properties do not determine the reflection of radiation from the composite material.

Let us consider the most interesting spectral ranges where the polymer is semitransparent and considerable values of both reflectance and transmittance are observed in the experiments: λ = 4.5 μm and λ = 11.5 μm. The first of these ranges is characterized by the strongest peak of reflectance and it is a good chance to verify the theoretical model in the near infrared. The peak of reflectance at λ = 11.5 μm is not so strong, but this spectral range is important for the problem of night cooling of buildings because of the transparency of clear sky in the range 8.5 < λ < 13.5 μm (Berdahl and Fromberg, 1982; Skartveit et al., 1996; Berger and Bathiebo, 2003).

The results of the calculations are presented in Figs. 7 and 8. Two different assumptions were used for the microsphere wall thickness: The wall thickness was assumed constant or directly proportional to the radius of the microsphere. The calculations were performed for two variants of glass optical constants corresponding to soda lime silica glass and low-expansion borosilicate glass. The scattering is characterized by the transport efficiency factor Qstr (Fig. 7) and the absorption is characterized by the absorption parameter α1 = -Qa/ / (Fig. 8). The physical sense of the last parameter is clear from the expression for the absorption coefficient of the composite material (see article Radiative properties of gas bubbles in semitransparent medium):

(8)

One can see in Fig. 7 that variation of Qstr with particle radius is not strong in the range of a > 20 μm both for constant and variable δ, and the value of Qstr is not sensitive to the type of microspheres. The difference between two variants of glass optical constants is small at the wavelength λ = 4.5 μm and significant at λ = 11.5 μm. In the first spectral range, one can use the value Qstr = 0.45 for microspheres of radius a32, whereas at a wavelength of 11.5 μm the following average values should be used: Qstr = 0.25 for soda lime silica glass and Qstr = 0.1 for borosilicate glass.

Figure 7. Transport efficiency factor of scattering for single microspheres at λ = 4.5 μm (a), (b) and λ = 11.5 μm (c), (d) in the cases of δ = δ (a), (c) and δ = δa/ a32 (b), (d): (1) δ = 1.3 μm (K20), (2) δ = 2.1 μm (S38); (I) soda lime silica glass, (II) borosilicate glass.

In contrast to the case of large bubbles in a weakly absorbing medium, the absorption parameter α1 for hollow glass microspheres is not equal to unity even in the range λ = 4.5 μm [Fig. 8(a)] and α1 < 0 at the wavelength 11.5 μm [Fig. 8(b)]. The negative values of α1 show that absorption coefficient increases when glass microspheres are present in the polymer. In our calculations, we used the average value α1 = 0.9 at λ = 4.5 μm and the following values at λ = 11.5 μm: α1 = 0 (K20), α1 = -0.4 (S38) for soda lime silica glass and α1 = -0.3 (K20), α1 = -0.6 (S38) for borosilicate glass. The results of the calculations of total directional-hemispherical reflectance and transmittance by use of the modified two-flux approximation are presented in Figs. 9 and 10. The calculations were performed for the average values of sample thickness d and volume fraction of microspheres fv from Table 1.

Figure 8. Absorption parameter for single microspheres at the wavelength λ = 4.5 μm (a) and λ = 11.5 μm (b): (1) δ = 1.3μm (K20), (2) δ = 2.1μm (S38); (I) soda lime silica glass, (II) borosilicate glass.

The systematic underestimation of the reflectance in the calculations at the wavelength 4.5 μm [Fig. 9(a)] can be partially explained by a small additional reflection of radiation from the rough surface of the samples. Additional calculations showed that another possible reason of low predicted reflectance is the total internal reflection taken into account in the modified two-flux approximation. One can expect that the effect of total internal refection decreases with the volume fraction of microspheres. The experimental data for directional-hemispherical reflectance do not show any specific behavior at high volume fraction of the microspheres. It means that dependent scattering has no significant effect on the reflectance. Note that this result was not evident. One can remember the study of microsphere ceramics by Dombrovsky (2004). At the same wavelength, the calculations of total directional-hemispherical transmittance yield the results which are closer to the measurements of samples with a smooth front surface [Fig. 9(b)]. The difference between the results for microspheres of series K20 and S38 is approximately the same in the calculations and in the experiment. The role of the optical properties of glass is negligible in this range. The theoretical model gives a rather good qualitative description of the peaks of reflectance and transmittance at λ = 4.5 μm and their dependences on volume fraction and the average size of microspheres.

Figure 9. Total direction-hemispherical reflectance (a) and transmittance (b) at the wavelength λ = 4.5 μm. Comparison of calculations with experimental data for different orientations of the samples (smooth or rough front surface).

At the wavelength 11.5 μm, the discrepancy between calculated and measured reflectance is significant [Fig. 10(a)]. Comparison of the experimental results for different orientations of the samples shows that relatively large measured values of reflectance are explained by the roughness of the front surface of the sample. The effect of roughness on reflectance is greater than the difference between two types of microspheres, but the roughness is not taken into account in the approximate radiation transfer model. The low transmittance at λ = 11.5 μm [Fig. 10(b)] is explained by the absorption of radiation in the polymer and the additional absorption in glass of the microspheres. These effects are well described by the theoretical model which shows that >90% of the radiation is absorbed in the samples with a high volume fraction of microspheres. Both measurements and calculations show that transmittance is considerably less in the case of thick-wall particles.

Figure 10. Total direction-hemispherical reflectance (a) and transmittance (b) at the wavelength λ = 11.5 μm. Comparison of calculations [(1) soda lime silica glass, (2) borosilicate glass] with experimental data for different orientations of the samples (smooth or rough front surface).

It should be noted that the theoretical model cannot give good predictions of reflectance at the wavelength 11.5 μm because of too-high absorption of the polymer. The above mentioned condition τ0 << 1 is not satisfied for particles of average radius a32: τ0 = 0.65 for K20 and τ0 = 0.53 for S38 microspheres.

The experimental results showed a small reflectance from the optically thick layer of the composite material in the spectral range 8.5 < λ < 13.5 μm, which is important for radiative heat losses from buildings. It means that a decrease in integral hemispherical emissivity εh of the polymer coating due to the presence of hollow glass microspheres is not expected to be significant. The integral emissivity of optically thick polymer samples in the wavelength range from 8 to 14 μm was also measured directly by pyrometer CYCLOPS COMPAC 3S for several directions from θ = 0° to 60°. The samples were placed in a heated metal frame whose temperature was maintained at 373 ± 2 K. The measured values of ε(θ) with uncertainty ±0.02 are given in Table 3. The high values of emissivity are in good agreement with the low reflectance of the samples in the range 8 < λ < 14 μm (see Fig. 6).

Table 3. Directional integral emissivity of the polymer samples containing microspheres

Type of microspheres Sample number ε(θ)
θ = 0° 20° 40° 50° 60°
K20 1 0.96 0.96 0.94 0.92 0.86
5 0.89 0.89 0.88 0.87 0.77
S38 1 0.97 0.97 0.96 0.93 0.87
5 0.92 0.92 0.92 0.90 0.82

REFERENCES

Berdahl, P. and Fromberg, R., The thermal radiance of clear skies, Solar Energy, vol. 29, no. 4, pp. 299-314, 1982.

Berger, X. and Bathiebo, J., Directional spectral emissivities of clear skies, Renewable Energy, vol. 28, no. 12, pp. 1925-1933, 2003.

Dombrovsky, L. A., Approximate models of radiation scattering in hollow-microsphere ceramics, High Temp., vol. 42, no. 5, pp. 776-784, 2004.

Dombrovsky, L. A., Modeling of thermal radiation of a polymer coating containing hollow microspheres, High Temp., vol. 43, no. 2, pp. 247-258, 2005.

Dombrovsky, L., Randrianalisoa, J., and Baillis, D., Infrared radiative properties of polymer coatings containing hollow microspheres, Int. J. Heat Mass Transfer, vol. 50, no. 7-8, pp. 1516-1527, 2007.

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.

Ivanov, A. P., Loiko, V. A., and Dick, V. P., Propagation of Light in Densely Packed Dispersive Media, Minsk: Nauka i Technika, 1988 (in Russian).

Kumar, S. and Tien, C. L., Dependent absorption and extinction of radiation by small particles, ASME J. Heat Transfer, vol. 112, no. 1, pp. 178-185, 1990.

Mishchenko, M. I., Hovenier, J. W., and Mackowski, D. W., Single scattering by a small volume element, J. Opt. Soc. Am. A, vol. 21, no. 1, pp. 71-87, 2004.

Rubin, M., Optical properties of soda lime silica glasses, Solar Energy Mater., vol. 12, no. 4, pp. 275-288, 1985.

Singh, B. P. and Kaviany, M., Modeling radiative heat transfer in packed beds, Int. J. Heat Mass Transfer, vol. 35, no. 6, pp. 1397-1405, 1992.

Skartveit, A., Olseth, J. A., Czeplak, G., and Rommel, M., On the estimation of atmospheric radiation from surface meteorological data, Solar Energy, vol. 56, no. 4, pp. 349-359, 1996.

References

  1. Berdahl, P. and Fromberg, R., The thermal radiance of clear skies, Solar Energy, vol. 29, no. 4, pp. 299-314, 1982.
  2. Berger, X. and Bathiebo, J., Directional spectral emissivities of clear skies, Renewable Energy, vol. 28, no. 12, pp. 1925-1933, 2003.
  3. Dombrovsky, L. A., Approximate models of radiation scattering in hollow-microsphere ceramics, High Temp., vol. 42, no. 5, pp. 776-784, 2004.
  4. Dombrovsky, L. A., Modeling of thermal radiation of a polymer coating containing hollow microspheres, High Temp., vol. 43, no. 2, pp. 247-258, 2005.
  5. Dombrovsky, L., Randrianalisoa, J., and Baillis, D., Infrared radiative properties of polymer coatings containing hollow microspheres, Int. J. Heat Mass Transfer, vol. 50, no. 7-8, pp. 1516-1527, 2007.
  6. 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.
  7. Ivanov, A. P., Loiko, V. A., and Dick, V. P., Propagation of Light in Densely Packed Dispersive Media, Minsk: Nauka i Technika, 1988 (in Russian).
  8. Kumar, S. and Tien, C. L., Dependent absorption and extinction of radiation by small particles, ASME J. Heat Transfer, vol. 112, no. 1, pp. 178-185, 1990.
  9. Mishchenko, M. I., Hovenier, J. W., and Mackowski, D. W., Single scattering by a small volume element, J. Opt. Soc. Am. A, vol. 21, no. 1, pp. 71-87, 2004.
  10. Rubin, M., Optical properties of soda lime silica glasses, Solar Energy Mater., vol. 12, no. 4, pp. 275-288, 1985.
  11. Singh, B. P. and Kaviany, M., Modeling radiative heat transfer in packed beds, Int. J. Heat Mass Transfer, vol. 35, no. 6, pp. 1397-1405, 1992.
  12. Skartveit, A., Olseth, J. A., Czeplak, G., and Rommel, M., On the estimation of atmospheric radiation from surface meteorological data, Solar Energy, vol. 56, no. 4, pp. 349-359, 1996.
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