Mie Theory

    Mie theory provides rigorous solutions for light scattering by an isotropic sphere embedded in a homogeneous medium. 10-13 Extensions of Mie theory include solutions for core/shell spheres14 and gradient-index spheres. 15 Although these theories are restricted to the case of a perfect sphere, the results have provided insight into the scattering and absorption properties for a wide variety of pigment systems, including non-spherical pigments.16 In most paper applications, where TiO2 concentrations are relatively low ( < 15% by volume), theoretical calculations predict the relative effects of particle size, particle composition, composition of the surrounding medium and wavelength of light. These trends correlate well with experimental data.17 In applications with TiO2 concentrations greater than ~15% by volume, near-field optical interactions between neighboring particles become significant and can dramatically impact macroscopic optical properties. The optical theories applied in the present study describe the light scattering properties of an isolated spherical particle and therefore cannot be applied to systems in which the particles are crowded together and near-field interactions between particles are significant.

    The concepts of geometrical optics (refraction by lenses and reflection by mirrors) that are familiar in the macroscopic world do not adequately describe the interactions of particles with light when the particle size is comparable to the wavelength of the light. Rigorous optical theories such as Mie theory address the full complexity of vector electromagnetic quantities interacting with a particle.. The mathematics of Mie theory is straightforward but tedious, requiring the computation of a potentially large number of series expansions. Digital computers are ideally suited to this task. In the present study, the computer codes BHMIE an BHCOAT provided by Bohren and Huffman have been used to compute the scattering results presented, after slight modification to incorporate computation of the asymmetry parameter.13

    In general, when the light wavelength is similar to the particle diameter, light interacts with the particle over a cross-sectional area larger than the geometric cross section of the particle. The Mie calculation output provides this scattering cross section, Csca. Often this parameter is divided by the geometric cross-sectional area to give a dimensionless scattering efficiency parameter, Qsca.

Qsca = Csca/p r2

    However, in pigment applications, the formulation properties and costs depend on particle volume rather than cross-sectional area. Therefore , a more meaningful efficiency parameter is the scattering coefficient per micron, SCPM, defined as the scattering cross section divided by particle volume.

SCPM = Csca/(4p r3/3) = 3Qsca/4r

    Since the intensity of scattered light varies with the scattering angle, the asymmetry parameter must be considered 5 to give the scattering coefficient, S(mm-1), which is the preferred parameter for correlations with experimental data for systems in which multiple scattering is predominant. The relationship is 

S(mm-1) = 0.75 . (1 - cos q) . SCPM

where the asymmetry parameter, cos q, is the average cosine of the scattering angle, weighted by the intensity of the scattered light as a function of angle.

References

1. Houle, L.L., Boudreau, J., and Shukla, G., TAPPI 1990 Coating Conference Proceedings, TAPPI Press, Atlanta, p.259. 

2. Kwoka, R.A., TAPPI 1990 Dyes, Fillers and Pigments Short Course, TAPPI Press, Atlanta, p.21. 

3. Kwoka, R.A., TAPPI 1993 Coating Conference Proceedings, TAPPI Press, Atlanta, p.133. 

4. Kwoka, R.A. and Logan, T.W., TAPPI Journal, 77(10):136(1994). 

5. Ross, W.D., J. Paint Technology, 43(563):51 (December 1971). 

6. Ross, W.D., I&EC Product R&D, 13(1):45 (March 1974). 

7. Hsu, W.P., Yu, R. and Matijevic, E., J. Colloid & Interface Sci., 156:56(1993). 

8. Matijevic, E., Hsu, P., and Kuehnle, W., US patent 5,248,556 (1993). 

9. Otterstedt, J., Otterstedt, O., Sterte, P., and Greenwood, P., International patent application WO 94/21733 (1994).

10. Mie, G., Ann.Physik, [4] 25, 377(1908).

11. van de Hulst, H.C., "Light Scattering by Small Particles," Chapters 9 and 10, Wiley, New York, 1957.

12. Kerker, M., "The Scattering of Light and Other Electromagnetic Radiation," Chapters 3 and 4, Academic Press, New York, 1969.

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

Aden, A.L. and Kerker, M., "Scattering of Electromagnetic Waves from Two Concentric Spheres," J. Appl. Phys., 22, 1242 (1951).

15. Kai, L. and d’Alessio, A., "Extinction Efficiency of Gradient-Index Microspheres," Particle and Particle Systems Characterization, 12, 119-122 (1995).

16. Braun, J.H., "Introduction to Pigments," Federation of Societies for Coatings Technology, Blue Bell, PA, 1993, p.16.

17. Fields, D.P., Buchacek, R.J., and Dickinson, J.G., JOCCA, 1993(2):87(1993).

Figure 6. Finite element model and near field scattering results for a 197-nm diameter, n=2.74 sphere in n=1.514 resin; the light wavelength is 560 nm. On the left are three dimensional views of the model and scattered light intensity, while on the right are cross sectional views of the model and scattered light intensity.

Figure 2. Scattering coefficient S versus sphere diameter for an optically isotropic sphere from Mie theory, using the average index approximation.

Figure 3. Angle-weighted scattering coefficient s versus sphere diameter for an optically isotropic sphere from Mie theory, using the average index approximation.

Comment: (c) 2010 Roger H. French , frenchrh@lrsm.upenn.edu
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