Abstract

Solar reflectance spectra of pigmented coatings have been obtained from spectroscopic measurements involving integrating sphere attachments. We demonstrate that measured and computed reflectances of an extended four-flux model [Appl. Opt. 37, 2615 (1998)] whose average path-length parameters (APP’s) and forward-scattering ratios (FSR’s) are explicitly evaluated from a multiple-scattering approach at the front or back interface of the particulate coatings display fairly good agreement. The agreement of these properties in a standard four-flux model [Appl. Opt. 23, 3353 (1984)], which neglects the spectral dependence of the APP and FSR, is found in the near infrared. Good agreement between these two four-flux approaches over the solar spectral range is obtained when the mean values of the APP’s and FSR’s are used in the standard model.

© 2001 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  6. M. K. Gunde, J. K. Logar, C. Orel, B. Orel, “Optimum thickness determination to maximise the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
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  13. J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and absorption of light in turbid media,” in Proceedings of the Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 765–784.
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  28. B. R. Palmer, P. Stamatakis, C. F. Bohren, G. C. Salzman, “A multiple-scattering model for opacifying particles in polymer films,” J. Coatings Technol. 61, 41–47 (1989).
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  36. A. R. Dagheidy, “Radiative transfer in a scattering medium with angle-dependent reflective boundaries,” Waves Random Media 7, 579–591 (1997).
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  39. M. I. Mishchenio, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
    [CrossRef]
  40. M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
    [CrossRef] [PubMed]
  41. M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
    [CrossRef]
  42. D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical optics approximation,” Appl. Opt. 36, 4605–4613 (1997).
  43. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
    [CrossRef]
  44. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
    [CrossRef]
  45. D. W. Mackowski, “Electrostatic analysis of radiative absorption by sphere clusters in the Rayleigh limit: application to soot particles,” Appl. Opt. 34, 3535–3545 (1995).
    [CrossRef] [PubMed]
  46. D. W. Mackowski, M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
    [CrossRef]
  47. M. I. Mishchenko, D. W. Mackowski, “Electromagnetic scattering by randomly oriented bispheres: comparisons of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
    [CrossRef]

2000 (1)

W. E. Vargas, P. Greenwood, J. E. Otterstedt, G. A. Niklasson, “Light scattering in pigmented coatings: experiments and theory,” Solar Energy 68, 553–561 (2000).
[CrossRef]

1999 (3)

1998 (3)

C. A. Arancibia, J. C. Ruiz-Suarez, “Spectral selectivity of cermets with large metallic inclusions,” J. Appl. Phys. 83, 5421–5426 (1998).
[CrossRef]

W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
[CrossRef]

E. S. Thiele, R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” J. Am. Ceram. Soc. 81, 469–479 (1998).
[CrossRef]

1997 (9)

W. E. Vargas, G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[CrossRef] [PubMed]

W. E. Vargas, G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

A. R. Dagheidy, “Radiative transfer in a scattering medium with angle-dependent reflective boundaries,” Waves Random Media 7, 579–591 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[CrossRef]

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

W. E. Vargas, G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14, 2243–2252 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Intensity of diffuse radiation in particulate media,” J. Opt. Soc. Am. A 14, 2253–2262 (1997).
[CrossRef]

R. W. Johnson, E. S. Thiele, R. H. French, “Light-scattering efficiency of white pigments: an analysis of model core-shell pigments vs. optimized rutile TiO2,” TAPPI J. 80, 233–239 (1997).

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical optics approximation,” Appl. Opt. 36, 4605–4613 (1997).

1996 (5)

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

D. W. Mackowski, M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[CrossRef]

M. I. Mishchenko, D. W. Mackowski, “Electromagnetic scattering by randomly oriented bispheres: comparisons of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
[CrossRef]

M. K. Gunde, J. K. Logar, C. Orel, 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 (4)

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

M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Application of the Kubelka–Munk theory to thickness-dependent diffuse reflectance of black paints in the mid-IR,” Appl. Spectrosc. 49, 623–629 (1995).
[CrossRef]

T. M. J. Nilsson, G. A. Niklasson, “Radiative cooling during the day: simulations and experiments on pigmented polyethylene cover foils,” Solar Energy Mater Solar Cells 37, 93–118 (1995).
[CrossRef]

D. W. Mackowski, “Electrostatic analysis of radiative absorption by sphere clusters in the Rayleigh limit: application to soot particles,” Appl. Opt. 34, 3535–3545 (1995).
[CrossRef] [PubMed]

1994 (4)

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
[CrossRef]

M. I. Mishchenio, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

H. Tang, K. Prasad, R. Sanjines, P. E. Schmid, F. Levy, “Electrical and optical properties of TiO2 anatase thin films,” J. Appl. Phys. 75, 2042–2047 (1994).
[CrossRef]

N. P. Ryde, E. Matijevic, “Color effects of uniform colloidal particles of different morphologies packed into films,” Appl. Opt. 33, 7275–7281 (1994).
[CrossRef] [PubMed]

1993 (2)

1991 (2)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
[CrossRef]

1989 (1)

B. R. Palmer, P. Stamatakis, C. F. Bohren, G. C. Salzman, “A multiple-scattering model for opacifying particles in polymer films,” J. Coatings Technol. 61, 41–47 (1989).

1984 (1)

1976 (1)

J. H. Joseph, W. J. Wiscombe, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

1973 (2)

1971 (1)

1970 (1)

E. P. Shettle, J. A. Weinman, “The transfer of solar irradiance through inhomogenous turbid atmospheres evaluated by Eddington’s approximation,” J. Atmos. Sci. 27, 1048–1055 (1970).
[CrossRef]

1940 (1)

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

1931 (1)

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

Arancibia, C. A.

C. A. Arancibia, J. C. Ruiz-Suarez, “Spectral selectivity of cermets with large metallic inclusions,” J. Appl. Phys. 83, 5421–5426 (1998).
[CrossRef]

Arancibia-Bulnes, C. A.

Atkins, J. T.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and absorption of light in turbid media,” in Proceedings of the Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 765–784.

Beasley, J. K.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and absorption of light in turbid media,” in Proceedings of the Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 765–784.

Billmeyer, F. W.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and absorption of light in turbid media,” in Proceedings of the Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 765–784.

Bohren, C. F.

B. R. Palmer, P. Stamatakis, C. F. Bohren, G. C. Salzman, “A multiple-scattering model for opacifying particles in polymer films,” J. Coatings Technol. 61, 41–47 (1989).

Carlson, B. E.

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical optics approximation,” Appl. Opt. 36, 4605–4613 (1997).

Dagheidy, A. R.

A. R. Dagheidy, “Radiative transfer in a scattering medium with angle-dependent reflective boundaries,” Waves Random Media 7, 579–591 (1997).
[CrossRef]

Egan, W. G.

Ferber, J.

H. R. Wilson, J. Ferber, W. Platzer, “Optical properties of thermotropic layers,” in Optical Materials Technology for Energy Efficiency and Solar Energy Conversion XIII, Proc. SPIE2255, 473–484 (1994).
[CrossRef]

French, R. H.

E. S. Thiele, R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” J. Am. Ceram. Soc. 81, 469–479 (1998).
[CrossRef]

R. W. Johnson, E. S. Thiele, R. H. French, “Light-scattering efficiency of white pigments: an analysis of model core-shell pigments vs. optimized rutile TiO2,” TAPPI J. 80, 233–239 (1997).

Gouesbet, G.

Greenwood, P.

W. E. Vargas, P. Greenwood, J. E. Otterstedt, G. A. Niklasson, “Light scattering in pigmented coatings: experiments and theory,” Solar Energy 68, 553–561 (2000).
[CrossRef]

Gunde, M. K.

M. K. Gunde, J. K. Logar, C. Orel, 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. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Application of the Kubelka–Munk theory to thickness-dependent diffuse reflectance of black paints in the mid-IR,” Appl. Spectrosc. 49, 623–629 (1995).
[CrossRef]

Hartel, W.

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

Hilgeman, T.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, Chap. 10.

Johnson, R. W.

R. W. Johnson, E. S. Thiele, R. H. French, “Light-scattering efficiency of white pigments: an analysis of model core-shell pigments vs. optimized rutile TiO2,” TAPPI J. 80, 233–239 (1997).

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Kubelka, P.

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

Letoulouzan, J. N.

Levy, F.

H. Tang, K. Prasad, R. Sanjines, P. E. Schmid, F. Levy, “Electrical and optical properties of TiO2 anatase thin films,” J. Appl. Phys. 75, 2042–2047 (1994).
[CrossRef]

Logar, J. K.

M. K. Gunde, J. K. Logar, C. Orel, 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. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Application of the Kubelka–Munk theory to thickness-dependent diffuse reflectance of black paints in the mid-IR,” Appl. Spectrosc. 49, 623–629 (1995).
[CrossRef]

Macke, A.

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical optics approximation,” Appl. Opt. 36, 4605–4613 (1997).

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

D. W. Mackowski, M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[CrossRef]

M. I. Mishchenko, D. W. Mackowski, “Electromagnetic scattering by randomly oriented bispheres: comparisons of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
[CrossRef]

D. W. Mackowski, “Electrostatic analysis of radiative absorption by sphere clusters in the Rayleigh limit: application to soot particles,” Appl. Opt. 34, 3535–3545 (1995).
[CrossRef] [PubMed]

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
[CrossRef]

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

Maheu, B.

Matijevic, E.

Mischenko, M. I.

Mishchenio, M. I.

M. I. Mishchenio, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Mishchenko, M. I.

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical optics approximation,” Appl. Opt. 36, 4605–4613 (1997).

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

D. W. Mackowski, M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[CrossRef]

M. I. Mishchenko, D. W. Mackowski, “Electromagnetic scattering by randomly oriented bispheres: comparisons of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
[CrossRef]

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
[CrossRef]

Mudgett, P. S.

Munk, F.

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

Niklasson, G. A.

W. E. Vargas, P. Greenwood, J. E. Otterstedt, G. A. Niklasson, “Light scattering in pigmented coatings: experiments and theory,” Solar Energy 68, 553–561 (2000).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Intensity of diffuse radiation in particulate media,” J. Opt. Soc. Am. A 14, 2253–2262 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14, 2243–2252 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[CrossRef]

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

W. E. Vargas, G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[CrossRef] [PubMed]

W. E. Vargas, G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

T. M. J. Nilsson, G. A. Niklasson, “Radiative cooling during the day: simulations and experiments on pigmented polyethylene cover foils,” Solar Energy Mater Solar Cells 37, 93–118 (1995).
[CrossRef]

Nilsson, T. M. J.

T. M. J. Nilsson, G. A. Niklasson, “Radiative cooling during the day: simulations and experiments on pigmented polyethylene cover foils,” Solar Energy Mater Solar Cells 37, 93–118 (1995).
[CrossRef]

Orel, B.

M. K. Gunde, J. K. Logar, C. Orel, 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. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Application of the Kubelka–Munk theory to thickness-dependent diffuse reflectance of black paints in the mid-IR,” Appl. Spectrosc. 49, 623–629 (1995).
[CrossRef]

Orel, C.

M. K. Gunde, J. K. Logar, C. Orel, 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]

Orel, Z. C.

Otterstedt, J. E.

W. E. Vargas, P. Greenwood, J. E. Otterstedt, G. A. Niklasson, “Light scattering in pigmented coatings: experiments and theory,” Solar Energy 68, 553–561 (2000).
[CrossRef]

Palmer, B. R.

B. R. Palmer, P. Stamatakis, C. F. Bohren, G. C. Salzman, “A multiple-scattering model for opacifying particles in polymer films,” J. Coatings Technol. 61, 41–47 (1989).

Platzer, W.

H. R. Wilson, J. Ferber, W. Platzer, “Optical properties of thermotropic layers,” in Optical Materials Technology for Energy Efficiency and Solar Energy Conversion XIII, Proc. SPIE2255, 473–484 (1994).
[CrossRef]

Prasad, K.

H. Tang, K. Prasad, R. Sanjines, P. E. Schmid, F. Levy, “Electrical and optical properties of TiO2 anatase thin films,” J. Appl. Phys. 75, 2042–2047 (1994).
[CrossRef]

Reichman, J.

Ribarsky, M. W.

M. W. Ribarsky, “Titanium dioxide (TiO2) (rutile),” in Handbook of Optical Constants, E. D. Palik, ed. (Academic, New York, 1985), pp. 795–804.
[CrossRef]

Rich, D. C.

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

Richards, L. W.

Roos, A.

A. Roos, “Use of an integrating sphere in solar energy research,” Solar Energy Mater. Solar Cells 30, 77–94 (1993).
[CrossRef]

Ruiz-Suarez, J. C.

C. A. Arancibia, J. C. Ruiz-Suarez, “Spectral selectivity of cermets with large metallic inclusions,” J. Appl. Phys. 83, 5421–5426 (1998).
[CrossRef]

Ruiz-Suárez, J. C.

Ryde, N. P.

Salzman, G. C.

B. R. Palmer, P. Stamatakis, C. F. Bohren, G. C. Salzman, “A multiple-scattering model for opacifying particles in polymer films,” J. Coatings Technol. 61, 41–47 (1989).

Sanjines, R.

H. Tang, K. Prasad, R. Sanjines, P. E. Schmid, F. Levy, “Electrical and optical properties of TiO2 anatase thin films,” J. Appl. Phys. 75, 2042–2047 (1994).
[CrossRef]

Schmid, P. E.

H. Tang, K. Prasad, R. Sanjines, P. E. Schmid, F. Levy, “Electrical and optical properties of TiO2 anatase thin films,” J. Appl. Phys. 75, 2042–2047 (1994).
[CrossRef]

Shettle, E. P.

E. P. Shettle, J. A. Weinman, “The transfer of solar irradiance through inhomogenous turbid atmospheres evaluated by Eddington’s approximation,” J. Atmos. Sci. 27, 1048–1055 (1970).
[CrossRef]

Stamatakis, P.

B. R. Palmer, P. Stamatakis, C. F. Bohren, G. C. Salzman, “A multiple-scattering model for opacifying particles in polymer films,” J. Coatings Technol. 61, 41–47 (1989).

Tang, H.

H. Tang, K. Prasad, R. Sanjines, P. E. Schmid, F. Levy, “Electrical and optical properties of TiO2 anatase thin films,” J. Appl. Phys. 75, 2042–2047 (1994).
[CrossRef]

Thiele, E. S.

E. S. Thiele, R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” J. Am. Ceram. Soc. 81, 469–479 (1998).
[CrossRef]

R. W. Johnson, E. S. Thiele, R. H. French, “Light-scattering efficiency of white pigments: an analysis of model core-shell pigments vs. optimized rutile TiO2,” TAPPI J. 80, 233–239 (1997).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenio, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Vargas, W. E.

W. E. Vargas, P. Greenwood, J. E. Otterstedt, G. A. Niklasson, “Light scattering in pigmented coatings: experiments and theory,” Solar Energy 68, 553–561 (2000).
[CrossRef]

W. E. Vargas, “Diffuse radiation intensity propagating through a particulate slab,” J. Opt. Soc. Am. A 16, 1362–1372 (1999).
[CrossRef]

W. E. Vargas, “Two-flux radiative transfer model under nonisotropic propagating diffuse radiation,” Appl. Opt. 38, 1077–1085 (1999).
[CrossRef]

W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14, 2243–2252 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Intensity of diffuse radiation in particulate media,” J. Opt. Soc. Am. A 14, 2253–2262 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[CrossRef] [PubMed]

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

W. E. Vargas, “Light scattering and absorption in pigmented coatings,” Ph.D. dissertation (Uppsala University, Uppsala, Sweden, 1997).

Weinman, J. A.

E. P. Shettle, J. A. Weinman, “The transfer of solar irradiance through inhomogenous turbid atmospheres evaluated by Eddington’s approximation,” J. Atmos. Sci. 27, 1048–1055 (1970).
[CrossRef]

Wielaard, D. J.

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical optics approximation,” Appl. Opt. 36, 4605–4613 (1997).

Wilson, H. R.

H. R. Wilson, J. Ferber, W. Platzer, “Optical properties of thermotropic layers,” in Optical Materials Technology for Energy Efficiency and Solar Energy Conversion XIII, Proc. SPIE2255, 473–484 (1994).
[CrossRef]

Wiscombe, W. J.

J. H. Joseph, W. J. Wiscombe, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
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Appl. Opt. (14)

N. P. Ryde, E. Matijevic, “Color effects of uniform colloidal particles of different morphologies packed into films,” Appl. Opt. 33, 7275–7281 (1994).
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J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 1. Theory,” Appl. Opt. 12, 1811–1815 (1973).
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W. G. Egan, T. Hilgeman, J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 2. Experiment,” Appl. Opt. 12, 1816–1823 (1973).
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P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[CrossRef] [PubMed]

W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
[CrossRef]

W. E. Vargas, “Two-flux radiative transfer model under nonisotropic propagating diffuse radiation,” Appl. Opt. 38, 1077–1085 (1999).
[CrossRef]

B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 (1984).
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C. A. Arancibia-Bulnes, J. C. Ruiz-Suárez, “Average path-length parameter of diffuse light in scattering media,” Appl. Opt. 38, 1877–1883 (1999).
[CrossRef]

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

W. E. Vargas, G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[CrossRef] [PubMed]

M. I. Mischenko, “Light scattering by size-shape distributions of randomly oriented axially symmetric particles of a size comparable to a wavelength,” Appl. Opt. 32, 4652–4666 (1993).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical optics approximation,” Appl. Opt. 36, 4605–4613 (1997).

D. W. Mackowski, “Electrostatic analysis of radiative absorption by sphere clusters in the Rayleigh limit: application to soot particles,” Appl. Opt. 34, 3535–3545 (1995).
[CrossRef] [PubMed]

Appl. Spectrosc. (1)

J. Am. Ceram. Soc. (1)

E. S. Thiele, R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” J. Am. Ceram. Soc. 81, 469–479 (1998).
[CrossRef]

J. Appl. Phys. (2)

H. Tang, K. Prasad, R. Sanjines, P. E. Schmid, F. Levy, “Electrical and optical properties of TiO2 anatase thin films,” J. Appl. Phys. 75, 2042–2047 (1994).
[CrossRef]

C. A. Arancibia, J. C. Ruiz-Suarez, “Spectral selectivity of cermets with large metallic inclusions,” J. Appl. Phys. 83, 5421–5426 (1998).
[CrossRef]

J. Atmos. Sci. (2)

E. P. Shettle, J. A. Weinman, “The transfer of solar irradiance through inhomogenous turbid atmospheres evaluated by Eddington’s approximation,” J. Atmos. Sci. 27, 1048–1055 (1970).
[CrossRef]

J. H. Joseph, W. J. Wiscombe, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

J. Coat. Technol. (1)

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

J. Coatings Technol. (1)

B. R. Palmer, P. Stamatakis, C. F. Bohren, G. C. Salzman, “A multiple-scattering model for opacifying particles in polymer films,” J. Coatings Technol. 61, 41–47 (1989).

J. Opt. Soc. Am. A (6)

J. Phys. Condens. Matter (2)

W. E. Vargas, G. A. Niklasson, “Forward-scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

M. I. Mishchenko, D. W. Mackowski, “Electromagnetic scattering by randomly oriented bispheres: comparisons of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

Licht (1)

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

Opt. Commun. (1)

M. I. Mishchenio, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

Solar Energy (1)

W. E. Vargas, P. Greenwood, J. E. Otterstedt, G. A. Niklasson, “Light scattering in pigmented coatings: experiments and theory,” Solar Energy 68, 553–561 (2000).
[CrossRef]

Solar Energy Mater Solar Cells (1)

T. M. J. Nilsson, G. A. Niklasson, “Radiative cooling during the day: simulations and experiments on pigmented polyethylene cover foils,” Solar Energy Mater Solar Cells 37, 93–118 (1995).
[CrossRef]

Solar Energy Mater. Solar Cells (1)

A. Roos, “Use of an integrating sphere in solar energy research,” Solar Energy Mater. Solar Cells 30, 77–94 (1993).
[CrossRef]

TAPPI J. (1)

R. W. Johnson, E. S. Thiele, R. H. French, “Light-scattering efficiency of white pigments: an analysis of model core-shell pigments vs. optimized rutile TiO2,” TAPPI J. 80, 233–239 (1997).

Thin Solid Films (1)

M. K. Gunde, J. K. Logar, C. Orel, 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]

Waves Random Media (1)

A. R. Dagheidy, “Radiative transfer in a scattering medium with angle-dependent reflective boundaries,” Waves Random Media 7, 579–591 (1997).
[CrossRef]

Z. Tech. Phys. (1)

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

Other (5)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, Chap. 10.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and absorption of light in turbid media,” in Proceedings of the Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 765–784.

M. W. Ribarsky, “Titanium dioxide (TiO2) (rutile),” in Handbook of Optical Constants, E. D. Palik, ed. (Academic, New York, 1985), pp. 795–804.
[CrossRef]

W. E. Vargas, “Light scattering and absorption in pigmented coatings,” Ph.D. dissertation (Uppsala University, Uppsala, Sweden, 1997).

H. R. Wilson, J. Ferber, W. Platzer, “Optical properties of thermotropic layers,” in Optical Materials Technology for Energy Efficiency and Solar Energy Conversion XIII, Proc. SPIE2255, 473–484 (1994).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Spectral dependence of (a) the total reflectance of a pigmented coating containing commercial rutile titania particles, where R EXP denotes experimental values and R MLG and R EFF correspond to values calculated from the MLG and EFF models, respectively; the (b) average path-length parameters and forward-scattering ratios (evaluated at τ = τ′) that correspond to diffuse radiation intensities propagating in the forward and backward directions; and (c) effective and intrinsic scattering coefficients per unit length of the particulate coating (S and α, respectively) in units of inverse micrometers. The mean diameter of the pigments is 0.30 µm, the pigment weight (volume) fraction is 0.05 (0.013), and the coating thickness is 67 µm.

Fig. 2
Fig. 2

Orientational average for the scattering efficiency (a) of polydispersions of spheroidal TiO2 particles hosted in a polymer matrix, as a function of the aspect ratio (as shown) and (b) of a cluster of seven TiO2 particles surrounded by a polymer matrix. The refractive indices of the particles and the matrix were as 2.75 + i0.0 and 1.35, respectively. The free-space wavelength of the incident radiation was set to 0.55 µm.

Fig. 3
Fig. 3

(a) Calculated total reflectance spectra of the coating in Fig. 1 evaluated from the EFF model with average path-length parameters and forward-scattering ratios calculated at three different optical depths, as indicated. Optical depth dependence of forward and backward average path-length parameters at free space wavelengths of (b) 0.61 and (c) 2.23 µm. Solid (dashed) curves, correspond to forward (backward) average path-length parameters.

Fig. 4
Fig. 4

Diffuse intensity patterns at optical depths close to (a) the illuminated and (b) the back interfaces of the pigmented coating treated in Figs. 13. The values of the relative optical depths, τ/τ′, are indicated. The free-space wavelength of the incoming radiation is 0.61 µm.

Fig. 5
Fig. 5

Measured and computed reflectance spectra of a polymeric film containing rutile titania pigments at a particle weight (volume) concentration of 0.50 (0.19). The thickness of the film is 60 µm. R EXP, experimental values; R MLG and R EFF, values calculated from the MLG and EFF models, respectively.

Equations (8)

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

Iτ, μ=n=0 cn+τPnμ,  0<μ1,
Jτ, μ=n=0 cn(-)τPnμ,  with -1μ<0,
ξ+=01 Iτ, μdμ01 μIτ, μdμ, σ+=01dμ 01 Iτ, μpμ, μdμ01dμ -11 Iτ, μpμ, μdμ,
ξ(-)=--10 Jτ, μdμ-10 μJτ, μdμ, σ(-)=-10dμ -10 Jτ, μpμ, μdμ-10dμ -11 Jτ, μpμ, μdμ,
dIcdz=α+βIc,
dJcdz=-α+βJc,
dIddz=ξ+β+1-σ+αId-ξ(-)1-σ(-)αJd-σcαIc-1-σcαJc,
dJddz=-ξ(-)β+1-σ(-)αJd+ξ+1-σ+αId+1-σcαIc+σcαJc,

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