Abstract

We study theoretically the light-scattering efficiency of paper coatings made of starch acetate pigments. For the light-scattering code we use a discrete dipole approximation method. The coating layer is assumed to consists of roughly equal-sized spherical pigments packed either at a packing density of 50% (large cylindrical slabs) or at 37% or 57% (large spheres). Because the scanning electron microscope images of starch acetate samples show either a particulate or a porous structure, we model the coatings in two complementary ways. The material can be either inside the constituent spheres (particulate case) or outside of those (cheeselike, porous medium). For the packing of our spheres we use either a simulated annealing or a dropping code. We can estimate, among other things, that the ideal sphere diameter is in the range 0.250.4  μm.

© 2006 Optical Society of America

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References

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  1. European Fibre Flow Chart 2003, Special recycling 2003 statistics, Confederation of European Paper Industries, available http://www.cepi.org.
  2. H. Mälkki and R. Lehtilä, "Organic pigment and a method for its preparation," U.S. patent 6,582,509, B2, 24 June2003.
  3. H. Bengs and J. Grande, "Method for the production of spherical microparticles consisting totally or partly of at least one water insoluble polyglucan containing branches and microparticles produced according to said method," U.S. patent 6,562,459, B1, 13 May2003.
  4. F. Giezen, R. Jongboom, H. Feil, K. Gotlieb, and A. Boersma, "Biopolymers nanoparticles," U.S. patent 6,677,386, 13 January 2004.
  5. S. Bloembergen, F. Kappen, and M. Van Leeuwen, "Method for production of corrugated board and product obtained thereof," U.S. patent application 20040241382, 2 December 2004.
  6. B. T. Draine and P. J. Flatau, "The discrete dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994).
    [CrossRef]
  7. C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, New York, 1999).
  8. A. Penttilä and K. Lumme, "Numerical solution to the first-order light scattering by rough and particular surfaces," in Proceedings of the Eigth Conference on Electromagnetic and Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, F.Moreno, J.J.López-Moreno, O.Muñoz, and A.Molina, eds. (University of Granada, Spain, 2005), pp. 265-268.
    [PubMed]
  9. L. Yang and B. Kruse, "Revised Kubelka-Munk theory. I. Theory and application," J. Opt. Soc. Am. A. 21, 1933-1941 (2004).
    [CrossRef]
  10. L. Yang, B. Kruse, and S. J. Miklavcic, "Revised Kubelka-Munk theory. II. Unified framework for homogeneous and inhomogeneous optical media," J. Opt. Soc. Am. A 21, 1942-1952 (2004).
    [CrossRef]
  11. J. W. Hovenier, K. Lumme, M. I. Mishchenko, N. V. Voshchinnikov, D. W. Mackowski, and J. Rahola, "Computations of scattering matrices or four types of non-spherical particles using diverse methods," J. Quant. Spectrosc. Radiat. Transfer 55, 695-705 (1996).
    [CrossRef]
  12. M. J. Collinge and B. T. Draine, "Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry," J. Opt. Soc. Am. A 21, 2023-2028 (2004).
    [CrossRef]
  13. B. T. Draine and J. Goodman, "Beyond Clasius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
    [CrossRef]
  14. P. Karvinen, H. Mikkonen, and R. Silvennoinen, "Determination of the refractive index of starch acetate: Biodegradable pigment of paper," in Technical Digest of the Sixth Japan- Finland Joint Symposium on Optics in Engineering (OIE05) (Hokkai-Gakuen University, 2005), pp. 5-6.
  15. B. T. Draine and P. J. Flatau, "User guide for the discrete dipole approximation code DDSCAT (Version 6.1) (2004)," available at http://arxiv.org/abs/astro-ph/0409262.
  16. J. Rahola, "Efficient solution of dense systems of linear equations in electromagnetic scattering calculations," Ph.D. dissertation (Helsinki University of Technology, 1996).

2004 (3)

2003 (2)

H. Mälkki and R. Lehtilä, "Organic pigment and a method for its preparation," U.S. patent 6,582,509, B2, 24 June2003.

H. Bengs and J. Grande, "Method for the production of spherical microparticles consisting totally or partly of at least one water insoluble polyglucan containing branches and microparticles produced according to said method," U.S. patent 6,562,459, B1, 13 May2003.

1999 (1)

C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, New York, 1999).

1996 (1)

J. W. Hovenier, K. Lumme, M. I. Mishchenko, N. V. Voshchinnikov, D. W. Mackowski, and J. Rahola, "Computations of scattering matrices or four types of non-spherical particles using diverse methods," J. Quant. Spectrosc. Radiat. Transfer 55, 695-705 (1996).
[CrossRef]

1994 (1)

1993 (1)

B. T. Draine and J. Goodman, "Beyond Clasius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

Bengs, H.

H. Bengs and J. Grande, "Method for the production of spherical microparticles consisting totally or partly of at least one water insoluble polyglucan containing branches and microparticles produced according to said method," U.S. patent 6,562,459, B1, 13 May2003.

Bloembergen, S.

S. Bloembergen, F. Kappen, and M. Van Leeuwen, "Method for production of corrugated board and product obtained thereof," U.S. patent application 20040241382, 2 December 2004.

Boersma, A.

F. Giezen, R. Jongboom, H. Feil, K. Gotlieb, and A. Boersma, "Biopolymers nanoparticles," U.S. patent 6,677,386, 13 January 2004.

Casella, G.

C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, New York, 1999).

Collinge, M. J.

Draine, B. T.

M. J. Collinge and B. T. Draine, "Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry," J. Opt. Soc. Am. A 21, 2023-2028 (2004).
[CrossRef]

B. T. Draine and P. J. Flatau, "The discrete dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994).
[CrossRef]

B. T. Draine and J. Goodman, "Beyond Clasius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

B. T. Draine and P. J. Flatau, "User guide for the discrete dipole approximation code DDSCAT (Version 6.1) (2004)," available at http://arxiv.org/abs/astro-ph/0409262.

Feil, H.

F. Giezen, R. Jongboom, H. Feil, K. Gotlieb, and A. Boersma, "Biopolymers nanoparticles," U.S. patent 6,677,386, 13 January 2004.

Flatau, P. J.

B. T. Draine and P. J. Flatau, "The discrete dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994).
[CrossRef]

B. T. Draine and P. J. Flatau, "User guide for the discrete dipole approximation code DDSCAT (Version 6.1) (2004)," available at http://arxiv.org/abs/astro-ph/0409262.

Giezen, F.

F. Giezen, R. Jongboom, H. Feil, K. Gotlieb, and A. Boersma, "Biopolymers nanoparticles," U.S. patent 6,677,386, 13 January 2004.

Goodman, J.

B. T. Draine and J. Goodman, "Beyond Clasius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

Gotlieb, K.

F. Giezen, R. Jongboom, H. Feil, K. Gotlieb, and A. Boersma, "Biopolymers nanoparticles," U.S. patent 6,677,386, 13 January 2004.

Grande, J.

H. Bengs and J. Grande, "Method for the production of spherical microparticles consisting totally or partly of at least one water insoluble polyglucan containing branches and microparticles produced according to said method," U.S. patent 6,562,459, B1, 13 May2003.

Hovenier, J. W.

J. W. Hovenier, K. Lumme, M. I. Mishchenko, N. V. Voshchinnikov, D. W. Mackowski, and J. Rahola, "Computations of scattering matrices or four types of non-spherical particles using diverse methods," J. Quant. Spectrosc. Radiat. Transfer 55, 695-705 (1996).
[CrossRef]

Jongboom, R.

F. Giezen, R. Jongboom, H. Feil, K. Gotlieb, and A. Boersma, "Biopolymers nanoparticles," U.S. patent 6,677,386, 13 January 2004.

Kappen, F.

S. Bloembergen, F. Kappen, and M. Van Leeuwen, "Method for production of corrugated board and product obtained thereof," U.S. patent application 20040241382, 2 December 2004.

Karvinen, P.

P. Karvinen, H. Mikkonen, and R. Silvennoinen, "Determination of the refractive index of starch acetate: Biodegradable pigment of paper," in Technical Digest of the Sixth Japan- Finland Joint Symposium on Optics in Engineering (OIE05) (Hokkai-Gakuen University, 2005), pp. 5-6.

Kruse, B.

Lehtilä, R.

H. Mälkki and R. Lehtilä, "Organic pigment and a method for its preparation," U.S. patent 6,582,509, B2, 24 June2003.

Lumme, K.

J. W. Hovenier, K. Lumme, M. I. Mishchenko, N. V. Voshchinnikov, D. W. Mackowski, and J. Rahola, "Computations of scattering matrices or four types of non-spherical particles using diverse methods," J. Quant. Spectrosc. Radiat. Transfer 55, 695-705 (1996).
[CrossRef]

A. Penttilä and K. Lumme, "Numerical solution to the first-order light scattering by rough and particular surfaces," in Proceedings of the Eigth Conference on Electromagnetic and Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, F.Moreno, J.J.López-Moreno, O.Muñoz, and A.Molina, eds. (University of Granada, Spain, 2005), pp. 265-268.
[PubMed]

Mackowski, D. W.

J. W. Hovenier, K. Lumme, M. I. Mishchenko, N. V. Voshchinnikov, D. W. Mackowski, and J. Rahola, "Computations of scattering matrices or four types of non-spherical particles using diverse methods," J. Quant. Spectrosc. Radiat. Transfer 55, 695-705 (1996).
[CrossRef]

Mälkki, H.

H. Mälkki and R. Lehtilä, "Organic pigment and a method for its preparation," U.S. patent 6,582,509, B2, 24 June2003.

Mikkonen, H.

P. Karvinen, H. Mikkonen, and R. Silvennoinen, "Determination of the refractive index of starch acetate: Biodegradable pigment of paper," in Technical Digest of the Sixth Japan- Finland Joint Symposium on Optics in Engineering (OIE05) (Hokkai-Gakuen University, 2005), pp. 5-6.

Miklavcic, S. J.

Mishchenko, M. I.

J. W. Hovenier, K. Lumme, M. I. Mishchenko, N. V. Voshchinnikov, D. W. Mackowski, and J. Rahola, "Computations of scattering matrices or four types of non-spherical particles using diverse methods," J. Quant. Spectrosc. Radiat. Transfer 55, 695-705 (1996).
[CrossRef]

Penttilä, A.

A. Penttilä and K. Lumme, "Numerical solution to the first-order light scattering by rough and particular surfaces," in Proceedings of the Eigth Conference on Electromagnetic and Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, F.Moreno, J.J.López-Moreno, O.Muñoz, and A.Molina, eds. (University of Granada, Spain, 2005), pp. 265-268.
[PubMed]

Rahola, J.

J. W. Hovenier, K. Lumme, M. I. Mishchenko, N. V. Voshchinnikov, D. W. Mackowski, and J. Rahola, "Computations of scattering matrices or four types of non-spherical particles using diverse methods," J. Quant. Spectrosc. Radiat. Transfer 55, 695-705 (1996).
[CrossRef]

J. Rahola, "Efficient solution of dense systems of linear equations in electromagnetic scattering calculations," Ph.D. dissertation (Helsinki University of Technology, 1996).

Robert, C. P.

C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, New York, 1999).

Silvennoinen, R.

P. Karvinen, H. Mikkonen, and R. Silvennoinen, "Determination of the refractive index of starch acetate: Biodegradable pigment of paper," in Technical Digest of the Sixth Japan- Finland Joint Symposium on Optics in Engineering (OIE05) (Hokkai-Gakuen University, 2005), pp. 5-6.

Van Leeuwen, M.

S. Bloembergen, F. Kappen, and M. Van Leeuwen, "Method for production of corrugated board and product obtained thereof," U.S. patent application 20040241382, 2 December 2004.

Voshchinnikov, N. V.

J. W. Hovenier, K. Lumme, M. I. Mishchenko, N. V. Voshchinnikov, D. W. Mackowski, and J. Rahola, "Computations of scattering matrices or four types of non-spherical particles using diverse methods," J. Quant. Spectrosc. Radiat. Transfer 55, 695-705 (1996).
[CrossRef]

Yang, L.

Astrophys. J. (1)

B. T. Draine and J. Goodman, "Beyond Clasius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

L. Yang and B. Kruse, "Revised Kubelka-Munk theory. I. Theory and application," J. Opt. Soc. Am. A. 21, 1933-1941 (2004).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

J. W. Hovenier, K. Lumme, M. I. Mishchenko, N. V. Voshchinnikov, D. W. Mackowski, and J. Rahola, "Computations of scattering matrices or four types of non-spherical particles using diverse methods," J. Quant. Spectrosc. Radiat. Transfer 55, 695-705 (1996).
[CrossRef]

Other (10)

C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, New York, 1999).

A. Penttilä and K. Lumme, "Numerical solution to the first-order light scattering by rough and particular surfaces," in Proceedings of the Eigth Conference on Electromagnetic and Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, F.Moreno, J.J.López-Moreno, O.Muñoz, and A.Molina, eds. (University of Granada, Spain, 2005), pp. 265-268.
[PubMed]

European Fibre Flow Chart 2003, Special recycling 2003 statistics, Confederation of European Paper Industries, available http://www.cepi.org.

H. Mälkki and R. Lehtilä, "Organic pigment and a method for its preparation," U.S. patent 6,582,509, B2, 24 June2003.

H. Bengs and J. Grande, "Method for the production of spherical microparticles consisting totally or partly of at least one water insoluble polyglucan containing branches and microparticles produced according to said method," U.S. patent 6,562,459, B1, 13 May2003.

F. Giezen, R. Jongboom, H. Feil, K. Gotlieb, and A. Boersma, "Biopolymers nanoparticles," U.S. patent 6,677,386, 13 January 2004.

S. Bloembergen, F. Kappen, and M. Van Leeuwen, "Method for production of corrugated board and product obtained thereof," U.S. patent application 20040241382, 2 December 2004.

P. Karvinen, H. Mikkonen, and R. Silvennoinen, "Determination of the refractive index of starch acetate: Biodegradable pigment of paper," in Technical Digest of the Sixth Japan- Finland Joint Symposium on Optics in Engineering (OIE05) (Hokkai-Gakuen University, 2005), pp. 5-6.

B. T. Draine and P. J. Flatau, "User guide for the discrete dipole approximation code DDSCAT (Version 6.1) (2004)," available at http://arxiv.org/abs/astro-ph/0409262.

J. Rahola, "Efficient solution of dense systems of linear equations in electromagnetic scattering calculations," Ph.D. dissertation (Helsinki University of Technology, 1996).

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Figures (6)

Fig. 1
Fig. 1

SEM images of the two types of starch acetate products: (a) a particulate layer of AP2 pigments, (b) a continuous starch acetate foam with pores.

Fig. 2
Fig. 2

Example of particulate and porous media inside a cylindrical vessel. The vessel in this illustration is larger than the one used in our simulations; this one has ∼1400 pigments or pores. The packing densities in both the cases are ∼50%.

Fig. 3
Fig. 3

Albedos for particulate and porous media in cylindrical (left) and in spherical (right) vessels.

Fig. 4
Fig. 4

Light scattering from particulate and porous media in a cylindrical vessel and with m = 1.47 + i0.001.

Fig. 5
Fig. 5

Light scattering from particulate and porous media in a cylindrical vessel and with m = 1.6 + i0.001.

Fig. 6
Fig. 6

Light scattering for particulate and porous media in a spherical vessel and with two packing densities.

Equations (11)

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R = 4   μm ,
H = 1.24   μm   for   cylinders,
R = 1   μm   for   spheres ,
x c = ( | m | d ) / λ < 0.1 ( our case for cylinder slab ) ,
x c = ( | m | d ) / λ < 0.053 ( our case for sphere vessel ) ,
x c o = ( | m | d ) / λ < 1 / ( 2 π ) ( Draine's condition for the cross sections 15 ) ,
x c = ( | m | d ) / λ < 1 / ( 4 π ) ( Draine's condition for the matrix elements ) .
k sca = C sca / V ,
g = ½ θ d π F 11 ( θ ) sin ( θ ) cos ( θ ) d θ ,
R = ½ π / 2 π F 11 ( θ ) sin ( θ ) ,
T = ½ θ d π / 2 F 11 ( θ ) sin ( θ ) d θ ,

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