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Dependent light scattering in white paint films: clarification and application of the theoretical concepts

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Abstract

Among the numerous publications analyzing the causes and consequences of titanium dioxide crowding on the optical properties of white paint films, one notes some inconsistencies. First, a significant number of studies are inclined to describe “dependent” and “multiple” scattering of light as distinct phenomena. Second, the transition from independent to dependent light-scattering is often associated with an ill-defined “threshold” concentration. The aim of this study is to clarify the intricate connections between these two scattering regimes and in particular to show that for white paint films loaded with rutile titanium dioxide pigments, “dependent” light scattering is merely a particular manifestation of multiple scattering processes. We also clarify that the transition from independent to dependent scattering is a continuous process that cannot be formally related to a specific threshold in the pigment volume concentration. Finally, we propose a simple method based on the dependent scattering amplitude to assist paint formulators facing the task of improving the hiding power of a white paint either by increasing the quantity of pigments or by improving their spatial state of dispersion.

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References

  1. Van de Hulst, RC, Light Scattering by Small Particles. Dover Publications, New York, 1957

    Google Scholar 

  2. Orchard, SE, “A New Look at Pigment Optics.” J. Oil Colour Chem. Assoc., 54 44–60 (1968)

    Google Scholar 

  3. Fitzwater, S, Hook, JW, III, “Dependent Scattering Theory: A New Approach to Predicting Scattering in Paints.” J. Coat. Technol., 57 (721) 39–47 (1985)

    CAS  Google Scholar 

  4. Drolen, B, Tien, CL, “Independent and Dependent Scattering in Packed-Sphere Systems.” J. Thermophys. Heat Transf., 1 (1) 63–68 (1987)

    Article  CAS  Google Scholar 

  5. Tinsley, SG, Bowman, A, Phil, D, “Rutile Type Titanium Pigments.” J. Oil Colour Chem. Assoc., 32 (348) 233–270 (1949)

    CAS  Google Scholar 

  6. Ishimaru, A, Wave Propagation and Scattering in Random Media, Vols. 1 & 2. Academic Press, 1978

  7. Chandrasekhar, S, Radiative Transfer. Oxford University Press, Oxford, 1950

    Google Scholar 

  8. Tsang, L, Kong, JA, Ding, K-H, Scattering of Electromagnetic Waves: Theories and Application, Vol. 1. John Wiley and Sons, 2000

  9. Mackowski, DW, Mishchenko, MI, “Calculation of the T Matrix and the Scattering Matrix for Ensembles of Spheres.” J. Opt. Soc. Am. A, 13 (11) 2266–2278 (1996)

    Article  Google Scholar 

  10. Tsang, L, Kong, J, Shin, R, Theory of Microwave Remote Sensing. Wiley Series in Remote Sensing. John Wiley & Sons, 1985

  11. Kubelka, P, Munk, F, “Ein Beitrag zur Optik der Farbanstriche.” Zeit. f. tech. Physik, 12 593 (1931)

    Google Scholar 

  12. Saunderson, JL, “Calculation of the Color of Pigmented Plastics.” J. Opt. Soc. Am., 32 (4) 727–736 (1942)

    Article  Google Scholar 

  13. Mudgett, PS, Richards, LW, “Multiple Scattering Calculations for Technology.” Appl. Opt., 10 1485–1501 (1971)

    Article  CAS  Google Scholar 

  14. Vargas, WE, Niklasson, GA, “Optical Properties of Nano-Structured Dye-Sensitized Solar Cells.” Sol. Energy Mater. Sol. Cells, 69 147–163 (2001)

    Article  CAS  Google Scholar 

  15. Bruls, WAG, van der Leun, JC, “Forward Scattering Properties of Human Epidermal Layers.” Photochem. Photobiol., 40 231–242 (1984)

    Article  CAS  Google Scholar 

  16. Magnain, C, Elias, M, Frigerio, J-M, “Skin Color Modeling Using the Radiative Transfer Equation Solved by the Auxiliary Function Method.” J. Opt. Soc. Am. A, 24 (8) 2196–2205 (2007)

    Article  Google Scholar 

  17. Mishchenko, MI, Travias, LD, Lacis, AA, Multiple Scattering of Light by Particles, Radiative Transfer abd Coherent Backscatttering. Cambridge University Press, Cambridge, 2006

    Google Scholar 

  18. Mishchenko, MI, “Gustave Mie and the Fundamental Concept of Electromagnetic Scattering by Particles: A Perspective.” JQSRT, 110 1210–1222 (2009)

    CAS  Google Scholar 

  19. Braun, JH, White Pigments. Federation of Societies for Coatings Technology, Blue Bell, PA, 1995

    Google Scholar 

  20. ASTM D2805-88, Standard Test Method for Hiding Power of Paints by Reflectometry. Book of ASTM Standards

  21. Hottel, HC, Sarofim, AF, Dalzell, WH, Vasalos, IA, “Optical Properties of Coatings, Effect of pigment Concentration.” AIAA J., 9 (10) 1895–1898 (1970)

    Article  Google Scholar 

  22. Auger, JC, Barrera, RG, Stout, B, “Scattering Efficiencies of Aggregates of Spherical Particles.” J. Quant. Spectrosc. Radiat. Transf., 79–80 521–531 (2003)

    Article  Google Scholar 

  23. Cartigny, JD, Yamada, Y, Tien, CL, “Radiative Transfer with Dependent Scattering by Particles: Part 1. Theoretical Investigation.” Trans. ASME, 108 608–613 (1986)

    Article  CAS  Google Scholar 

  24. Holthoff, H, Borkovec, M, Schurtenberger, P, “Determination of Light-Scattering Form Factors of Latex Particle Dimers with Simultaneous Static and Dynamic Light Scattering in an Aggregating Suspension.” Phys. Rev. E, 56 (6) 6945–6953 (1997)

    Article  CAS  Google Scholar 

  25. Blum, L, Stell, G, “Scattering of Polydisperse Fluids.” J. Chem. Phys., 71 (1) 42–46 (1979)

    Article  CAS  Google Scholar 

  26. Auger, JC, Stout, B, “A Recursive Centered T-Matrix Algorithm to Solve the Multiple Scattering Equation : Numerical Validation.” J. Quant. Spectrosc. Radiat. Transf., 79–80 533–547 (2003)

    Article  Google Scholar 

  27. Auger, J-C, Martinez, VA, Stout B, “Theoretical Study of the Scattering Efficiency of Rutile Titanium Pigments as Function of Their Spatial Dispersion.” JCT Research., 6 (1) 89–97 (2009)

    CAS  Google Scholar 

  28. Dick, VP, Ivanov, AP, “Extinction of Light in Dispersive Media with High Particle Concentrations: Applicability Limits of the Interference Approximation.” J. Opt. Soc. Am. A, 16 (5) 1034–1039 (1999)

    Article  Google Scholar 

  29. Göbel, G, Kuhn, J, Fricke, J, “Dependent Scattering Effects in Latex-Sphere Suspensions and Scattering Powders.” Waves Random Media, 5 413–426 (1995)

    Article  Google Scholar 

  30. Auger, J-C, Stout, B, Lafait, J, “Dependent Light Scattering in Dense Heterogeneous Media.” Physica B, 279 21–24 (2000)

    Article  CAS  Google Scholar 

  31. Vargas, WE, “Optical Properties of Pigmented Coatings Taking into Account Particle Interactions.” J. Quant. Spectrosc. Radiat. Transf., 78 (2) 187–195 (2003)

    Article  CAS  Google Scholar 

  32. Tsang, L, Kong, JA, Shin, R, Theory of Microwave Remote Sensing. Wiley-Interscience, New York, 1985

    Google Scholar 

  33. Liu, L, Gong, R, Huang, D, Nie, Y, Liu, C, “Calculation of Emittance of a Coating Layer with the Kubelka-Munk Theory and the Mie-Scattering Model.” J. Opt. Soc. Am. A, 22 (11) 2424–2429 (2005)

    Article  Google Scholar 

  34. Richter, SM, Sevick-Muraca, EM, “Characterization of Concentrated Colloidal Suspensions Using Time-Dependent Photon Migration Measurements.” Colloid Surf. A: Physicochem. Eng. Asp., 172 163–173 (2000)

    Article  CAS  Google Scholar 

  35. McNeil, LE, Hanuska, AR, French, RH, “Orientation Dependence in Near-Field Scattering from TiO2 Particles.” Appl. Opt., 40 3726–3736 (2001)

    Article  CAS  Google Scholar 

  36. Auger, J-C, Stout, B, “Local Field Intensity in Aggregates Illuminated by Diffuse Light: T Matrix Approach.” Appl. Opt., 47 (16) 2897–2905 (2008)

    Article  Google Scholar 

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Acknowledgments

The authors would like to thank Fernando Zaldo and Centro de Investigacion en Polimeros of Grupo Comex who provided the experimental set of data used in this study.

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Authors and Affiliations

  1. Kyoralis Research and Consulting, 5 rue du Loing, 75014, Paris, France

    J.-C. Auger

  2. Institut Fresnel, Aix-Marseille Université, D.U. de St Jérôme UMR 6133, 13397, Marseille Cedex 20, France

    Brian Stout

Authors
  1. J.-C. Auger
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  2. Brian Stout
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Corresponding author

Correspondence to J.-C. Auger.

Appendix

Appendix

Expression of the Pearson correlation coefficient

$$ \varepsilon = \frac{{\sum\nolimits_{k = 1}^{k = K} {\left( {I_{\text{DEP}}^{(k)} - \bar{I}_{\text{DEP}} } \right)\left( {I_{\text{FFD}}^{(k)} - \bar{I}_{\text{FFD}} } \right)} }}{{\sqrt {\sum\nolimits_{k = 1}^{k = K} {\left( {I_{\text{DEP}}^{(k)} - \bar{I}_{\text{DEP}} } \right)^{2} } \sqrt {\sum\nolimits_{k = 1}^{k = K} {\left( {I_{\text{FFD}}^{(k)} - \bar{I}_{\text{FFD}} } \right)^{2} } } } }} $$
(1.1)

For a determined system defined by the set of parameters (n r, ϕ), calculations were performed on K = 1000 different random configurations, which were proven to be sufficient for configuration average statistics.

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Auger, JC., Stout, B. Dependent light scattering in white paint films: clarification and application of the theoretical concepts. J Coat Technol Res 9, 287–295 (2012). https://doi.org/10.1007/s11998-011-9371-9

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