Abstract

The optical properties of black-pigmented solar absorbing paint were analyzed phenomenologically by use of the Kubelka–Munk theory, including correction for reflection on front and rear surfaces. The effective absorption and scattering coefficients and the efficiency curves for absorption and scattering were calculated for coatings with different pigment-to-volume concentration ratios. The dependence of absorption and scattering efficiency on the pigment-to-volume concentration ratio was analyzed by reference to theoretical data in the literature. It was concluded that, during drying and curing of coatings, spherical primary pigment particles most likely collect in elongated groups oriented perpendicularly to the coating surface. Formation of such groups helps in understanding the independent measurements of solar absorptance.

© 2000 Optical Society of America

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References

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  1. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  2. D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coatings Technol. 67, 53–60 (1995).
  3. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  4. P. Kubelka, “New contributions to the optics of intensely scattering materials. I,” J. Opt. Soc. Am. 38, 448–457 (1984).
    [CrossRef]
  5. H. G. Völz, “Practical pigment testing with the aid of the Kubelka–Munk theory,” Prog. Org. Coat. 13, 135–169 (1985).
    [CrossRef]
  6. J. H. Nobbs, “Kubelka–Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66–75 (1985).
    [CrossRef]
  7. H. G. Völz, “The principle of spectral evaluation in pigment testing. A review of 20 years’ application of a successful method,” Prog. Org. Coat. 15, 99–124 (1987).
    [CrossRef]
  8. F. B. Stieg, “Predicting white hiding power,” J. Coat. Technol. 60, 95–99 (1988).
  9. M. Taira, M. Yamaki, “Studies on optical properties of a visible-light-cured dental composite resin by diffuse reflectance measurements,” J. Mater. Sci. Lett. 14, 198–200 (1995).
    [CrossRef]
  10. H. R. Kang, “Kubelka–Munk modelling of ink jet ink mixing,” J. Imaging Sci. Technol. 17, 76–83 (1991).
  11. M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Flocculation gradient technique in terms of Kubelka–Munk coefficients: quantifying black-pigmented dispersions,” Appl. Spectrosc. 49, 1756–1761 (1995).
    [CrossRef]
  12. M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Optimum thickness determination to maximise the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
    [CrossRef]
  13. Z. Crnjak Orel, M. Klanjšek Gunde, B. Orel, “Application of Kubelka–Munk theory for the determination of the optical properties of solar absorbing paints,” Prog. Org. Coat. 30, 59–66 (1997).
    [CrossRef]
  14. F. B. Stieg, “Pigment/binder geometry” in Pigment Handbook, T. C. Patton, ed. (Wiley, New York, 1973), Vol. III, pp. 203–217.
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  18. S. E. Orchard, “The missing variable: internal surface reflection,” Color Res. Appl. 2, 26–31 (1977).
  19. H. S. Shah, F. W. Billmeyer, “Kubelka–Munk analysis of absorbance in the presence of scattering, including surface-reflection correction to transmittance,” Color Res. Appl. 10, 26–31 (1985).
    [CrossRef]
  20. R. T. Marcus, P. E. Pierce, “An analysis of the first surface correction for the color matching of organic coatings from the viewpoint of radiative transfer theory,” Prog. Org. Coat. 23, 239–264 (1994).
    [CrossRef]
  21. W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
    [CrossRef] [PubMed]
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    [CrossRef]
  23. K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. I. Theory for external aggregation,” J. Opt. Soc. Am. A 11, 3251–3260 (1994).
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  27. P. B. Mitton, “Opacity, hiding power, and tinting strength,” in Pigment Handbook, T. C. Patton, ed. (Wiley, New York, 1973), Vol. III, pp. 289–339.
  28. M. Q. Brewster, C. L. Tien, “Radiative transfer in packed fluidized beds: dependent versus independent scattering,” J. Heat Transfer 104, 573–579 (1982).
    [CrossRef]

1999 (1)

Z. Crnjak Orel, “Preparation of high-temperature-resistant selective paints for solar absorbers,” Sol. Energy Mater. Sol. Cells 57, 290–301 (1999).

1997 (2)

Z. Crnjak Orel, M. Klanjšek Gunde, B. Orel, “Application of Kubelka–Munk theory for the determination of the optical properties of solar absorbing paints,” Prog. Org. Coat. 30, 59–66 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[CrossRef] [PubMed]

1996 (2)

P. Mazeron, S. Muller, “Light scattering by ellipsoids in a physical optics approximation,” Appl. Opt. 35, 3726–3735 (1996).
[CrossRef] [PubMed]

M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Optimum thickness determination to maximise the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
[CrossRef]

1995 (3)

D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coatings Technol. 67, 53–60 (1995).

M. Taira, M. Yamaki, “Studies on optical properties of a visible-light-cured dental composite resin by diffuse reflectance measurements,” J. Mater. Sci. Lett. 14, 198–200 (1995).
[CrossRef]

M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Flocculation gradient technique in terms of Kubelka–Munk coefficients: quantifying black-pigmented dispersions,” Appl. Spectrosc. 49, 1756–1761 (1995).
[CrossRef]

1994 (2)

K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. I. Theory for external aggregation,” J. Opt. Soc. Am. A 11, 3251–3260 (1994).
[CrossRef]

R. T. Marcus, P. E. Pierce, “An analysis of the first surface correction for the color matching of organic coatings from the viewpoint of radiative transfer theory,” Prog. Org. Coat. 23, 239–264 (1994).
[CrossRef]

1991 (1)

H. R. Kang, “Kubelka–Munk modelling of ink jet ink mixing,” J. Imaging Sci. Technol. 17, 76–83 (1991).

1988 (1)

F. B. Stieg, “Predicting white hiding power,” J. Coat. Technol. 60, 95–99 (1988).

1987 (1)

H. G. Völz, “The principle of spectral evaluation in pigment testing. A review of 20 years’ application of a successful method,” Prog. Org. Coat. 15, 99–124 (1987).
[CrossRef]

1985 (3)

H. G. Völz, “Practical pigment testing with the aid of the Kubelka–Munk theory,” Prog. Org. Coat. 13, 135–169 (1985).
[CrossRef]

J. H. Nobbs, “Kubelka–Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66–75 (1985).
[CrossRef]

H. S. Shah, F. W. Billmeyer, “Kubelka–Munk analysis of absorbance in the presence of scattering, including surface-reflection correction to transmittance,” Color Res. Appl. 10, 26–31 (1985).
[CrossRef]

1984 (2)

1982 (1)

M. Q. Brewster, C. L. Tien, “Radiative transfer in packed fluidized beds: dependent versus independent scattering,” J. Heat Transfer 104, 573–579 (1982).
[CrossRef]

1979 (1)

1977 (1)

S. E. Orchard, “The missing variable: internal surface reflection,” Color Res. Appl. 2, 26–31 (1977).

1971 (1)

E. D. Campbell, F. W. Billmeyer, “Fresnel reflection coefficients for diffuse and collimated light,” J. Color Appearance 1, 39–41 (1971).

1960 (1)

1942 (1)

1931 (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Asano, S.

Billmeyer, F. W.

H. S. Shah, F. W. Billmeyer, “Kubelka–Munk analysis of absorbance in the presence of scattering, including surface-reflection correction to transmittance,” Color Res. Appl. 10, 26–31 (1985).
[CrossRef]

E. D. Campbell, F. W. Billmeyer, “Fresnel reflection coefficients for diffuse and collimated light,” J. Color Appearance 1, 39–41 (1971).

Bohren, C. F.

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

Brewster, M. Q.

M. Q. Brewster, C. L. Tien, “Radiative transfer in packed fluidized beds: dependent versus independent scattering,” J. Heat Transfer 104, 573–579 (1982).
[CrossRef]

Campbell, E. D.

E. D. Campbell, F. W. Billmeyer, “Fresnel reflection coefficients for diffuse and collimated light,” J. Color Appearance 1, 39–41 (1971).

Crnjak Orel, Z.

Z. Crnjak Orel, “Preparation of high-temperature-resistant selective paints for solar absorbers,” Sol. Energy Mater. Sol. Cells 57, 290–301 (1999).

Z. Crnjak Orel, M. Klanjšek Gunde, B. Orel, “Application of Kubelka–Munk theory for the determination of the optical properties of solar absorbing paints,” Prog. Org. Coat. 30, 59–66 (1997).
[CrossRef]

M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Optimum thickness determination to maximise the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
[CrossRef]

M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Flocculation gradient technique in terms of Kubelka–Munk coefficients: quantifying black-pigmented dispersions,” Appl. Spectrosc. 49, 1756–1761 (1995).
[CrossRef]

Fuller, K. A.

Guesebet, G.

Huffman, D. R.

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

Kang, H. R.

H. R. Kang, “Kubelka–Munk modelling of ink jet ink mixing,” J. Imaging Sci. Technol. 17, 76–83 (1991).

Klanjšek Gunde, M.

Z. Crnjak Orel, M. Klanjšek Gunde, B. Orel, “Application of Kubelka–Munk theory for the determination of the optical properties of solar absorbing paints,” Prog. Org. Coat. 30, 59–66 (1997).
[CrossRef]

M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Optimum thickness determination to maximise the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
[CrossRef]

M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Flocculation gradient technique in terms of Kubelka–Munk coefficients: quantifying black-pigmented dispersions,” Appl. Spectrosc. 49, 1756–1761 (1995).
[CrossRef]

Kottler, F.

Kožar Logar, J.

M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Optimum thickness determination to maximise the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
[CrossRef]

M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Flocculation gradient technique in terms of Kubelka–Munk coefficients: quantifying black-pigmented dispersions,” Appl. Spectrosc. 49, 1756–1761 (1995).
[CrossRef]

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely scattering materials. I,” J. Opt. Soc. Am. 38, 448–457 (1984).
[CrossRef]

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Letoulouzan, J. N.

Maheu, B.

Marcus, R. T.

R. T. Marcus, P. E. Pierce, “An analysis of the first surface correction for the color matching of organic coatings from the viewpoint of radiative transfer theory,” Prog. Org. Coat. 23, 239–264 (1994).
[CrossRef]

Mazeron, P.

Mitton, P. B.

P. B. Mitton, “Opacity, hiding power, and tinting strength,” in Pigment Handbook, T. C. Patton, ed. (Wiley, New York, 1973), Vol. III, pp. 289–339.

Muller, S.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Niklasson, G. A.

Nobbs, J. H.

J. H. Nobbs, “Kubelka–Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66–75 (1985).
[CrossRef]

Orchard, S. E.

S. E. Orchard, “The missing variable: internal surface reflection,” Color Res. Appl. 2, 26–31 (1977).

Orel, B.

Z. Crnjak Orel, M. Klanjšek Gunde, B. Orel, “Application of Kubelka–Munk theory for the determination of the optical properties of solar absorbing paints,” Prog. Org. Coat. 30, 59–66 (1997).
[CrossRef]

M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Optimum thickness determination to maximise the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
[CrossRef]

M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Flocculation gradient technique in terms of Kubelka–Munk coefficients: quantifying black-pigmented dispersions,” Appl. Spectrosc. 49, 1756–1761 (1995).
[CrossRef]

Pierce, P. E.

R. T. Marcus, P. E. Pierce, “An analysis of the first surface correction for the color matching of organic coatings from the viewpoint of radiative transfer theory,” Prog. Org. Coat. 23, 239–264 (1994).
[CrossRef]

Rich, D. C.

D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coatings Technol. 67, 53–60 (1995).

Saunderson, J. L.

Shah, H. S.

H. S. Shah, F. W. Billmeyer, “Kubelka–Munk analysis of absorbance in the presence of scattering, including surface-reflection correction to transmittance,” Color Res. Appl. 10, 26–31 (1985).
[CrossRef]

Stieg, F. B.

F. B. Stieg, “Predicting white hiding power,” J. Coat. Technol. 60, 95–99 (1988).

F. B. Stieg, “Pigment/binder geometry” in Pigment Handbook, T. C. Patton, ed. (Wiley, New York, 1973), Vol. III, pp. 203–217.

Taira, M.

M. Taira, M. Yamaki, “Studies on optical properties of a visible-light-cured dental composite resin by diffuse reflectance measurements,” J. Mater. Sci. Lett. 14, 198–200 (1995).
[CrossRef]

Tien, C. L.

M. Q. Brewster, C. L. Tien, “Radiative transfer in packed fluidized beds: dependent versus independent scattering,” J. Heat Transfer 104, 573–579 (1982).
[CrossRef]

Vargas, W. E.

Völz, H. G.

H. G. Völz, “The principle of spectral evaluation in pigment testing. A review of 20 years’ application of a successful method,” Prog. Org. Coat. 15, 99–124 (1987).
[CrossRef]

H. G. Völz, “Practical pigment testing with the aid of the Kubelka–Munk theory,” Prog. Org. Coat. 13, 135–169 (1985).
[CrossRef]

Yamaki, M.

M. Taira, M. Yamaki, “Studies on optical properties of a visible-light-cured dental composite resin by diffuse reflectance measurements,” J. Mater. Sci. Lett. 14, 198–200 (1995).
[CrossRef]

Appl. Opt. (4)

Appl. Spectrosc. (1)

Color Res. Appl. (2)

S. E. Orchard, “The missing variable: internal surface reflection,” Color Res. Appl. 2, 26–31 (1977).

H. S. Shah, F. W. Billmeyer, “Kubelka–Munk analysis of absorbance in the presence of scattering, including surface-reflection correction to transmittance,” Color Res. Appl. 10, 26–31 (1985).
[CrossRef]

J. Coat. Technol. (1)

F. B. Stieg, “Predicting white hiding power,” J. Coat. Technol. 60, 95–99 (1988).

J. Coatings Technol. (1)

D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coatings Technol. 67, 53–60 (1995).

J. Color Appearance (1)

E. D. Campbell, F. W. Billmeyer, “Fresnel reflection coefficients for diffuse and collimated light,” J. Color Appearance 1, 39–41 (1971).

J. Heat Transfer (1)

M. Q. Brewster, C. L. Tien, “Radiative transfer in packed fluidized beds: dependent versus independent scattering,” J. Heat Transfer 104, 573–579 (1982).
[CrossRef]

J. Imaging Sci. Technol. (1)

H. R. Kang, “Kubelka–Munk modelling of ink jet ink mixing,” J. Imaging Sci. Technol. 17, 76–83 (1991).

J. Mater. Sci. Lett. (1)

M. Taira, M. Yamaki, “Studies on optical properties of a visible-light-cured dental composite resin by diffuse reflectance measurements,” J. Mater. Sci. Lett. 14, 198–200 (1995).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Prog. Org. Coat. (4)

H. G. Völz, “The principle of spectral evaluation in pigment testing. A review of 20 years’ application of a successful method,” Prog. Org. Coat. 15, 99–124 (1987).
[CrossRef]

H. G. Völz, “Practical pigment testing with the aid of the Kubelka–Munk theory,” Prog. Org. Coat. 13, 135–169 (1985).
[CrossRef]

Z. Crnjak Orel, M. Klanjšek Gunde, B. Orel, “Application of Kubelka–Munk theory for the determination of the optical properties of solar absorbing paints,” Prog. Org. Coat. 30, 59–66 (1997).
[CrossRef]

R. T. Marcus, P. E. Pierce, “An analysis of the first surface correction for the color matching of organic coatings from the viewpoint of radiative transfer theory,” Prog. Org. Coat. 23, 239–264 (1994).
[CrossRef]

Rev. Prog. Color. Relat. Top. (1)

J. H. Nobbs, “Kubelka–Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66–75 (1985).
[CrossRef]

Sol. Energy Mater. Sol. Cells (1)

Z. Crnjak Orel, “Preparation of high-temperature-resistant selective paints for solar absorbers,” Sol. Energy Mater. Sol. Cells 57, 290–301 (1999).

Thin Solid Films (1)

M. Klanjšek Gunde, Z. Crnjak Orel, J. Kožar Logar, B. Orel, “Optimum thickness determination to maximise the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
[CrossRef]

Z. Tech. Phys. (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Other (3)

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

F. B. Stieg, “Pigment/binder geometry” in Pigment Handbook, T. C. Patton, ed. (Wiley, New York, 1973), Vol. III, pp. 203–217.

P. B. Mitton, “Opacity, hiding power, and tinting strength,” in Pigment Handbook, T. C. Patton, ed. (Wiley, New York, 1973), Vol. III, pp. 289–339.

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Figures (6)

Fig. 1
Fig. 1

Scanning-electron micrograph of the pigment used in powder form.

Fig. 2
Fig. 2

Reflectance spectra of paint coatings of several thicknesses at a pigment-to-volume concentration ratio of 0.2. Layer thicknesses are 1.1, 1.2, 1.6, 2.0, 2.1, 3.2, 5.1, and 7.2 µm from top to bottom. The reflectance of the bare substrate (R g ) is shown also. All spectra were measured with diffuse illumination and an 8° viewing angle in the specular-included mode.

Fig. 3
Fig. 3

Spectral dependence of KM (a) absorption and (b) scattering coefficients calculated for coatings with several pigment loadings f. The 4% uncertainty in layer thickness was allowed.

Fig. 4
Fig. 4

Efficiencies of (a) absorption Q abs and (b) scattering Q sca for coatings with several concentrations of pigment particles f.

Fig. 5
Fig. 5

Regions in the (f, x) plane where dependent or independent scattering prevails (according to the criterion from the literature27). Our measurements are performed for values in straight lines (shown dashed).

Fig. 6
Fig. 6

Pigment particle distribution within dry coatings when the pigment-to-volume concentration ratio (f) increases. Two-dimensional sketches of side and top views correspond to values of f of (a) 0.18 and (b) 0.28, respectively.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

RKM=1-RgA-B cothBSdA+B cothBSd-Rg,
R=k1-D+1-k11-k2Rth1-Rthk2,
Qext=Cext/G, Qabs=Cabs/G, Qsca=Csca/G, Qext=Qabs+Qsca,
K=3f4πa3 Cabs,  S=3f4πa3 Csca.
Qext=4a3fK+S,  Qabs=4a3f K,  Qsca=4a3f S.
x=ka=2πaλ=2πNaλ0,
cλ0=x(0.905-f3πf3<0.30,

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