Abstract

Particle-doped thin films that are translucent and diffusive have applications in cosmetics, coatings, and display technologies, but finding material combinations that produce these effects simultaneously is challenging: formulations tend to be either transparent or opaque. Using a combination of Mie scattering calculations and spectral transmission measurements on monodisperse colloidal suspensions, we demonstrate that the two characteristic optical properties of the films, total transmittance and haze, scale with the effective backscattering and forward scattering cross sections, both of which are properties of single particles. These scalings enable an efficient computational search for combinations of particle sizes, concentrations, and refractive indices that break the trade-off between translucency and diffusion. The optimum particle sizes and concentrations obey power-law dependences on the refractive index difference, a result of the interference condition for resonances in the scattering cross sections. The power laws serve as design equations for formulating particle-doped thin films.

© 2014 Optical Society of America

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References

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  1. P. Yeh and C. Gu, Optics of Liquid Crystal Displays, vol. 67 (John Wiley & Sons, 2010).
  2. R. Emmert, “Quantification of the soft-focus effect,” Cosmetics and Toiletries 111, 57–61 (1996).
  3. S. Chandrasekhar, Radiative Transfer (Courier Dover Publications, 1960).
  4. P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).
  5. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448 (1948).
    [Crossref] [PubMed]
  6. L. Yang and B. Kruse, “Revised Kubelka–Munk theory. I. Theory and application,” J. Opt. Soc. Am. A 21, 1933–1941 (2004).
    [Crossref]
  7. M. Yamada, M. D. Butts, and K. K. Kalla, “Spatial and angular distribution of light incident on coatings using Mie-scattering Monte Carlo simulations,” J. Cosmet. Sci. 56, 193–204 (2005).
    [PubMed]
  8. G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Annalen der Physik (1908).
    [Crossref]
  9. M. Egen and R. Zentel, “Surfactant-free emulsion polymerization of various methacrylates: Towards monodisperse colloids for polymer opals,” Macromolecular Chemistry and Physics 205, 1479–1488 (2004).
    [Crossref]
  10. S. Jiang, E. D. Sudol, V. L. Dimonie, and M. S. El-Aasser, “Seeding as a means of controlling particle size in dispersion polymerization,” Journal of Applied Polymer Science 108, 4096–4107 (2008).
    [Crossref]
  11. S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Optical Materials 29, 1481–1490 (2007).
    [Crossref]
  12. F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
    [Crossref]
  13. J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics Express 19, 8051–8065 (2011).
    [Crossref] [PubMed]
  14. R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discussions 159, 211–234 (2012).
    [Crossref]
  15. E. Hecht, Optics (Addison-Wesley, 2002).
  16. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH Verlag GmbH, 2004).
  17. P. Kaplan, A. Dinsmore, A. Yodh, and D. Pine, “Diffuse-transmission spectroscopy: A structural probe of opaque colloidal mixtures,” Physical Review E 50, 4827 (1994).
    [Crossref]
  18. H. C. Hulst and H. Van De Hulst, Light Scattering by Small Particles (Courier Dover Publications, 1957).
  19. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” The Astrophysical Journal 186, 705–714 (1973).
    [Crossref]
  20. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic Press, 1999).
  21. Z. Gimbutas and L. Greengard, “Fast multi-particle scattering: A hybrid solver for the maxwell equations in microstructured materials,” Journal of Computational Physics 232, 22–32 (2013).
    [Crossref]
  22. A. Small, S. Hong, and D. Pine, “Scattering properties of core–shell particles in plastic matrices,” Journal of Polymer Science Part B: Polymer Physics 43, 3534–3548 (2005).
    [Crossref]

2013 (1)

Z. Gimbutas and L. Greengard, “Fast multi-particle scattering: A hybrid solver for the maxwell equations in microstructured materials,” Journal of Computational Physics 232, 22–32 (2013).
[Crossref]

2012 (1)

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discussions 159, 211–234 (2012).
[Crossref]

2011 (2)

F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
[Crossref]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

2008 (1)

S. Jiang, E. D. Sudol, V. L. Dimonie, and M. S. El-Aasser, “Seeding as a means of controlling particle size in dispersion polymerization,” Journal of Applied Polymer Science 108, 4096–4107 (2008).
[Crossref]

2007 (1)

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Optical Materials 29, 1481–1490 (2007).
[Crossref]

2005 (2)

M. Yamada, M. D. Butts, and K. K. Kalla, “Spatial and angular distribution of light incident on coatings using Mie-scattering Monte Carlo simulations,” J. Cosmet. Sci. 56, 193–204 (2005).
[PubMed]

A. Small, S. Hong, and D. Pine, “Scattering properties of core–shell particles in plastic matrices,” Journal of Polymer Science Part B: Polymer Physics 43, 3534–3548 (2005).
[Crossref]

2004 (2)

L. Yang and B. Kruse, “Revised Kubelka–Munk theory. I. Theory and application,” J. Opt. Soc. Am. A 21, 1933–1941 (2004).
[Crossref]

M. Egen and R. Zentel, “Surfactant-free emulsion polymerization of various methacrylates: Towards monodisperse colloids for polymer opals,” Macromolecular Chemistry and Physics 205, 1479–1488 (2004).
[Crossref]

1996 (1)

R. Emmert, “Quantification of the soft-focus effect,” Cosmetics and Toiletries 111, 57–61 (1996).

1994 (1)

P. Kaplan, A. Dinsmore, A. Yodh, and D. Pine, “Diffuse-transmission spectroscopy: A structural probe of opaque colloidal mixtures,” Physical Review E 50, 4827 (1994).
[Crossref]

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” The Astrophysical Journal 186, 705–714 (1973).
[Crossref]

1948 (1)

1931 (1)

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

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH Verlag GmbH, 2004).

Butts, M. D.

M. Yamada, M. D. Butts, and K. K. Kalla, “Spatial and angular distribution of light incident on coatings using Mie-scattering Monte Carlo simulations,” J. Cosmet. Sci. 56, 193–204 (2005).
[PubMed]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Courier Dover Publications, 1960).

Cheong, F. C.

F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
[Crossref]

Dimiduk, T. G.

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discussions 159, 211–234 (2012).
[Crossref]

Dimonie, V. L.

S. Jiang, E. D. Sudol, V. L. Dimonie, and M. S. El-Aasser, “Seeding as a means of controlling particle size in dispersion polymerization,” Journal of Applied Polymer Science 108, 4096–4107 (2008).
[Crossref]

Dinsmore, A.

P. Kaplan, A. Dinsmore, A. Yodh, and D. Pine, “Diffuse-transmission spectroscopy: A structural probe of opaque colloidal mixtures,” Physical Review E 50, 4827 (1994).
[Crossref]

Egen, M.

M. Egen and R. Zentel, “Surfactant-free emulsion polymerization of various methacrylates: Towards monodisperse colloids for polymer opals,” Macromolecular Chemistry and Physics 205, 1479–1488 (2004).
[Crossref]

El-Aasser, M. S.

S. Jiang, E. D. Sudol, V. L. Dimonie, and M. S. El-Aasser, “Seeding as a means of controlling particle size in dispersion polymerization,” Journal of Applied Polymer Science 108, 4096–4107 (2008).
[Crossref]

Emmert, R.

R. Emmert, “Quantification of the soft-focus effect,” Cosmetics and Toiletries 111, 57–61 (1996).

Fung, J.

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discussions 159, 211–234 (2012).
[Crossref]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

Gimbutas, Z.

Z. Gimbutas and L. Greengard, “Fast multi-particle scattering: A hybrid solver for the maxwell equations in microstructured materials,” Journal of Computational Physics 232, 22–32 (2013).
[Crossref]

Greengard, L.

Z. Gimbutas and L. Greengard, “Fast multi-particle scattering: A hybrid solver for the maxwell equations in microstructured materials,” Journal of Computational Physics 232, 22–32 (2013).
[Crossref]

Grier, D. G.

F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
[Crossref]

Gu, C.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays, vol. 67 (John Wiley & Sons, 2010).

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002).

Hong, S.

A. Small, S. Hong, and D. Pine, “Scattering properties of core–shell particles in plastic matrices,” Journal of Polymer Science Part B: Polymer Physics 43, 3534–3548 (2005).
[Crossref]

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic Press, 1999).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH Verlag GmbH, 2004).

Hulst, H. C.

H. C. Hulst and H. Van De Hulst, Light Scattering by Small Particles (Courier Dover Publications, 1957).

Ivanov, C. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Optical Materials 29, 1481–1490 (2007).
[Crossref]

Jiang, S.

S. Jiang, E. D. Sudol, V. L. Dimonie, and M. S. El-Aasser, “Seeding as a means of controlling particle size in dispersion polymerization,” Journal of Applied Polymer Science 108, 4096–4107 (2008).
[Crossref]

Kalla, K. K.

M. Yamada, M. D. Butts, and K. K. Kalla, “Spatial and angular distribution of light incident on coatings using Mie-scattering Monte Carlo simulations,” J. Cosmet. Sci. 56, 193–204 (2005).
[PubMed]

Kaplan, P.

P. Kaplan, A. Dinsmore, A. Yodh, and D. Pine, “Diffuse-transmission spectroscopy: A structural probe of opaque colloidal mixtures,” Physical Review E 50, 4827 (1994).
[Crossref]

Kasarova, S. N.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Optical Materials 29, 1481–1490 (2007).
[Crossref]

Kaz, D. M.

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

Kruse, B.

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448 (1948).
[Crossref] [PubMed]

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

Manoharan, V. N.

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discussions 159, 211–234 (2012).
[Crossref]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

Martin, K. E.

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

McGorty, R.

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

Meng, G.

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discussions 159, 211–234 (2012).
[Crossref]

Mie, G.

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Annalen der Physik (1908).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic Press, 1999).

Munk, F.

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

Nikolov, I. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Optical Materials 29, 1481–1490 (2007).
[Crossref]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” The Astrophysical Journal 186, 705–714 (1973).
[Crossref]

Perry, R. W.

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discussions 159, 211–234 (2012).
[Crossref]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

Pine, D.

A. Small, S. Hong, and D. Pine, “Scattering properties of core–shell particles in plastic matrices,” Journal of Polymer Science Part B: Polymer Physics 43, 3534–3548 (2005).
[Crossref]

P. Kaplan, A. Dinsmore, A. Yodh, and D. Pine, “Diffuse-transmission spectroscopy: A structural probe of opaque colloidal mixtures,” Physical Review E 50, 4827 (1994).
[Crossref]

Pine, D. J.

F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
[Crossref]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” The Astrophysical Journal 186, 705–714 (1973).
[Crossref]

Small, A.

A. Small, S. Hong, and D. Pine, “Scattering properties of core–shell particles in plastic matrices,” Journal of Polymer Science Part B: Polymer Physics 43, 3534–3548 (2005).
[Crossref]

Sudol, E. D.

S. Jiang, E. D. Sudol, V. L. Dimonie, and M. S. El-Aasser, “Seeding as a means of controlling particle size in dispersion polymerization,” Journal of Applied Polymer Science 108, 4096–4107 (2008).
[Crossref]

Sultanova, N. G.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Optical Materials 29, 1481–1490 (2007).
[Crossref]

Travis, L. D.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic Press, 1999).

Van De Hulst, H.

H. C. Hulst and H. Van De Hulst, Light Scattering by Small Particles (Courier Dover Publications, 1957).

Xiao, K.

F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
[Crossref]

Yamada, M.

M. Yamada, M. D. Butts, and K. K. Kalla, “Spatial and angular distribution of light incident on coatings using Mie-scattering Monte Carlo simulations,” J. Cosmet. Sci. 56, 193–204 (2005).
[PubMed]

Yang, L.

Yeh, P.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays, vol. 67 (John Wiley & Sons, 2010).

Yodh, A.

P. Kaplan, A. Dinsmore, A. Yodh, and D. Pine, “Diffuse-transmission spectroscopy: A structural probe of opaque colloidal mixtures,” Physical Review E 50, 4827 (1994).
[Crossref]

Zentel, R.

M. Egen and R. Zentel, “Surfactant-free emulsion polymerization of various methacrylates: Towards monodisperse colloids for polymer opals,” Macromolecular Chemistry and Physics 205, 1479–1488 (2004).
[Crossref]

Cosmetics and Toiletries (1)

R. Emmert, “Quantification of the soft-focus effect,” Cosmetics and Toiletries 111, 57–61 (1996).

Faraday Discussions (1)

R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, and V. N. Manoharan, “Real-space studies of the structure and dynamics of self-assembled colloidal clusters,” Faraday Discussions 159, 211–234 (2012).
[Crossref]

J. Cosmet. Sci. (1)

M. Yamada, M. D. Butts, and K. K. Kalla, “Spatial and angular distribution of light incident on coatings using Mie-scattering Monte Carlo simulations,” J. Cosmet. Sci. 56, 193–204 (2005).
[PubMed]

J. Opt. Soc. Am. (1)

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

Journal of Applied Polymer Science (1)

S. Jiang, E. D. Sudol, V. L. Dimonie, and M. S. El-Aasser, “Seeding as a means of controlling particle size in dispersion polymerization,” Journal of Applied Polymer Science 108, 4096–4107 (2008).
[Crossref]

Journal of Computational Physics (1)

Z. Gimbutas and L. Greengard, “Fast multi-particle scattering: A hybrid solver for the maxwell equations in microstructured materials,” Journal of Computational Physics 232, 22–32 (2013).
[Crossref]

Journal of Polymer Science Part B: Polymer Physics (1)

A. Small, S. Hong, and D. Pine, “Scattering properties of core–shell particles in plastic matrices,” Journal of Polymer Science Part B: Polymer Physics 43, 3534–3548 (2005).
[Crossref]

Macromolecular Chemistry and Physics (1)

M. Egen and R. Zentel, “Surfactant-free emulsion polymerization of various methacrylates: Towards monodisperse colloids for polymer opals,” Macromolecular Chemistry and Physics 205, 1479–1488 (2004).
[Crossref]

Optical Materials (1)

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Optical Materials 29, 1481–1490 (2007).
[Crossref]

Optics Express (1)

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Optics Express 19, 8051–8065 (2011).
[Crossref] [PubMed]

Physical Review E (1)

P. Kaplan, A. Dinsmore, A. Yodh, and D. Pine, “Diffuse-transmission spectroscopy: A structural probe of opaque colloidal mixtures,” Physical Review E 50, 4827 (1994).
[Crossref]

Soft Matter (1)

F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
[Crossref]

The Astrophysical Journal (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” The Astrophysical Journal 186, 705–714 (1973).
[Crossref]

Z. Tech. Phys. (Leipzig) (1)

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

Other (7)

P. Yeh and C. Gu, Optics of Liquid Crystal Displays, vol. 67 (John Wiley & Sons, 2010).

S. Chandrasekhar, Radiative Transfer (Courier Dover Publications, 1960).

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Annalen der Physik (1908).
[Crossref]

H. C. Hulst and H. Van De Hulst, Light Scattering by Small Particles (Courier Dover Publications, 1957).

E. Hecht, Optics (Addison-Wesley, 2002).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH Verlag GmbH, 2004).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic Press, 1999).

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Figures (5)

Fig. 1
Fig. 1 Schematic of the experimental apparatus used to make spectral transmission measurements. (a) Total transmittance measurements are made with the sample at port-hole position P1, the light trap at P5 (not shown), and the reference chamber at P2 (not shown). (b) Diffuse transmittance measurements are made with the sample at P1, the light trap at P3, and the reference chamber at P2 (not shown). The reference spectrum is taken with the reference chamber at P1, the sample at P2, and the light trap at P5.
Fig. 2
Fig. 2 Total transmittance (a) and haze (b) of colloidal films are monotonic functions of the particle volume fraction for differently sized polystyrene (open symbols) and PMMA (closed symbols) microspheres in 10 mM NaCl/ Millipore water (particle diameters indicated in (a)). The inset in (b) shows haze as a function of particle diameter D for polystyrene suspensions at a volume fraction of 0.005. All data collapse when plotted against two dimensionless groups, σ b eff ρ L and σ f eff ρ L, and are captured quantitatively (curves) by Eqs. (3) and (5), respectively (c–d). Measurements are performed on samples with a thickness of 40 micrometers. The refractive index of polystyrene is taken from measurements of bulk samples [11]; the refractive index of PMMA is taken from direct measurements of single particles using digital holographic microscopy [12]. The effective backscattering cross section σ b eff and effective forward scattering cross section σ f eff are evaluated using the Mie module of the open-source software package HoloPy (http://manoharan.seas.harvard.edu/holopy) [13, 14]. The computed cross sections represent averages over the visible spectrum.
Fig. 3
Fig. 3 Schematic of the sample chamber (not to scale). Totally transmitted light is attenuated by one of two processes: (1) scattering of light by more than 90° from the direction of incident illumination, or (2) total internal reflection of light scattered at angles exceeding the critical angle θc.
Fig. 4
Fig. 4 Regions of parameter space corresponding to simultaneous translucency and diffusion (characterized by the geometric mean of T and H) are small. (a) shows H × T for suspensions of polystyrene in water; (b) shows H × T for suspensions of poly(methyl methacrylate) in water. Plots are computed using Eqs. (3) and (5) and Mie scattering calculations of σ b eff and σ f eff. From the plots, we define an optimum particle size D* and volume fraction φ* (orange symbols) that maximize the geometric mean of T and H. Calculations are for films of thickness L = 40 μm.
Fig. 5
Fig. 5 The optimum particle size D* (a) and volume fraction φ* (b) of different particle-medium combinations (symbols) follow power law scalings (lines) with the absolute refractive index difference Δn and sample thickness L. The Δn−1 scaling of the optimum diameter results from the interference condition that gives rise to scattering resonances (Eq. (7)). The power-law relationships are fit well by Eqs. (6) and (8) and can be used to engineer the bulk optical properties of suspensions by controlling the single particle attributes (np, D, φ).

Equations (13)

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

H 100 % = T diffuse T .
d I = σ b eff ρ I d x
σ b eff = 2 π θ c π d σ sca d Ω sin θ d θ
A [ I ( x + Δ x ) I ( x ) ] = σ b eff A Δ x ρ I ( x ) .
lim Δ x 0 I ( x + Δ x ) I ( x ) Δ x = d I ( x ) d x = σ b eff ρ I ( x ) .
T 100 % = e σ b eff ρ L ,
d I = σ sca ρ I d x ,
H 100 % = 1 e σ sca ρ L T .
H 100 % = 1 e σ f eff ρ L ,
σ f eff = 2 π 0 θ c d σ sca d Ω sin θ d θ
D * = 0.32 μ m Δ n ,
x ( m 1 ) = ( 2 p + 1 ) π / 2 ,
φ * = 0.11 μ m Δ n 3 / 2 L .

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