Abstract

The scattering cross sections of optically dense polystyrene-latex suspensions are determined as a function of wavelength, particle size, and particle concentration by using the anisotropic diffusion approximation to the radiative transport equation (or photon diffusion model) in conjunction with reflectance measurements from a thin, parallel-sided suspension cell. Corrections for internal and external surface reflection effects are made. Corresponding calculations are also made of the scattering cross sections based on published values of the Mie-scattering coefficient and phase function for single-particle scattering by making allowance for the fact that the effective refractive index of the medium surrounding a particle varies linearly with the volume concentration of surrounding scatterers. General agreement is found between the experimental and calculated values of the scattering cross sections. In particular a quantitative explanation is obtained for the decrease of scattering efficiency with volume concentration of scatterers.

© 1973 Optical Society of America

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References

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  1. F. B. Stieg, Official Dig. Fed. Paint Varnish 31 (1), 52 (1959).
  2. F. R. Marchetti and W. R. Willets, J. Tech. Assoc. Pulp Paper Ind. 43, 273 (1960).
  3. H. G. Völz, Ber. Bunsenges. 71, 326, (1967).
  4. W. D. Ross, J. Paint Tech. 43, 50 (1971).
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    [Crossref]
  6. W. R. Blevin and W. J. Brown, J. Opt. Soc. Am. 51, 129 (1961).
    [Crossref]
  7. S. W. Churchill, G. C. Clark, and C. M. Sliepcevich, Discuss. Faraday Soc. 30, 192 (1960).
    [Crossref]
  8. I. A. Vasalos, Ph.D. thesis, MIT (1969).
  9. C. Smart, R. Jacobsen, M. Kerker, J. P. Kratohvil, and E. Matijević, J. Opt. Soc. Am. 55, 947 (1965).
    [Crossref]
  10. S. E. Orchard, J. Opt. Soc. Am. 59, 1584 (1969).
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  11. P. S. Mudgett and L. W. Richards, Appl. Opt. 10, 1485 (1971).
    [Crossref] [PubMed]
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    [Crossref]
  13. P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948).
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    [Crossref]
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    [Crossref]
  16. B. J. Brinkworth, Appl. Opt. 11, 1434 (1972).
    [Crossref] [PubMed]
  17. D. B. Judd, J. Res. Nat. Bur. Stand. (U. S.) 29, 329 (1942).
    [Crossref]
  18. P. Anne Hiltner and Irwin M. Krieger, J. Phys. Chem. 73, 2386 (1969).
    [Crossref]
  19. W. Heller, Phys. Rev. 68, 5 (1945).
    [Crossref]
  20. W. M. Irvine, Bull. Astron. Insts. Netherlands 17, 176 (1963).
  21. C. M. Chu, G. C. Clark, and S. W. Churchill, Tables of Angular Distribution Coefficients for Light Scattering by Spheres (Engineering Research Institute, University of Michigan, 1957).
  22. E. J. Moss, Ph.D. thesis, University of Bristol (1970).

1972 (2)

1971 (4)

B. J. Brinkworth, J. Phys. D 4, 1105 (1971).
[Crossref]

P. S. Mudgett and L. W. Richards, Appl. Opt. 10, 1485 (1971).
[Crossref] [PubMed]

L. F. Gate, J. Phys. D 4, 1049 (1971).
[Crossref]

W. D. Ross, J. Paint Tech. 43, 50 (1971).

1969 (2)

P. Anne Hiltner and Irwin M. Krieger, J. Phys. Chem. 73, 2386 (1969).
[Crossref]

S. E. Orchard, J. Opt. Soc. Am. 59, 1584 (1969).
[Crossref]

1967 (1)

H. G. Völz, Ber. Bunsenges. 71, 326, (1967).

1965 (1)

1963 (1)

W. M. Irvine, Bull. Astron. Insts. Netherlands 17, 176 (1963).

1961 (1)

1960 (3)

S. W. Churchill, G. C. Clark, and C. M. Sliepcevich, Discuss. Faraday Soc. 30, 192 (1960).
[Crossref]

R. H. Harding, B. Golding, and R. A. Morgen, J. Opt. Soc. Am. 50, 446 (1960).
[Crossref]

F. R. Marchetti and W. R. Willets, J. Tech. Assoc. Pulp Paper Ind. 43, 273 (1960).

1959 (1)

F. B. Stieg, Official Dig. Fed. Paint Varnish 31 (1), 52 (1959).

1948 (1)

1945 (1)

W. Heller, Phys. Rev. 68, 5 (1945).
[Crossref]

1942 (1)

D. B. Judd, J. Res. Nat. Bur. Stand. (U. S.) 29, 329 (1942).
[Crossref]

Blevin, W. R.

Brinkworth, B. J.

Brown, W. J.

Chu, C. M.

C. M. Chu, G. C. Clark, and S. W. Churchill, Tables of Angular Distribution Coefficients for Light Scattering by Spheres (Engineering Research Institute, University of Michigan, 1957).

Churchill, S. W.

S. W. Churchill, G. C. Clark, and C. M. Sliepcevich, Discuss. Faraday Soc. 30, 192 (1960).
[Crossref]

C. M. Chu, G. C. Clark, and S. W. Churchill, Tables of Angular Distribution Coefficients for Light Scattering by Spheres (Engineering Research Institute, University of Michigan, 1957).

Clark, G. C.

S. W. Churchill, G. C. Clark, and C. M. Sliepcevich, Discuss. Faraday Soc. 30, 192 (1960).
[Crossref]

C. M. Chu, G. C. Clark, and S. W. Churchill, Tables of Angular Distribution Coefficients for Light Scattering by Spheres (Engineering Research Institute, University of Michigan, 1957).

Gate, L. F.

L. F. Gate, J. Phys. D 5, 837 (1972).
[Crossref]

L. F. Gate, J. Phys. D 4, 1049 (1971).
[Crossref]

Golding, B.

Harding, R. H.

Heller, W.

W. Heller, Phys. Rev. 68, 5 (1945).
[Crossref]

Hiltner, P. Anne

P. Anne Hiltner and Irwin M. Krieger, J. Phys. Chem. 73, 2386 (1969).
[Crossref]

Irvine, W. M.

W. M. Irvine, Bull. Astron. Insts. Netherlands 17, 176 (1963).

Jacobsen, R.

Judd, D. B.

D. B. Judd, J. Res. Nat. Bur. Stand. (U. S.) 29, 329 (1942).
[Crossref]

Kerker, M.

Kratohvil, J. P.

Krieger, Irwin M.

P. Anne Hiltner and Irwin M. Krieger, J. Phys. Chem. 73, 2386 (1969).
[Crossref]

Kubelka, P.

Marchetti, F. R.

F. R. Marchetti and W. R. Willets, J. Tech. Assoc. Pulp Paper Ind. 43, 273 (1960).

Matijevic, E.

Morgen, R. A.

Moss, E. J.

E. J. Moss, Ph.D. thesis, University of Bristol (1970).

Mudgett, P. S.

Orchard, S. E.

Richards, L. W.

Ross, W. D.

W. D. Ross, J. Paint Tech. 43, 50 (1971).

Sliepcevich, C. M.

S. W. Churchill, G. C. Clark, and C. M. Sliepcevich, Discuss. Faraday Soc. 30, 192 (1960).
[Crossref]

Smart, C.

Stieg, F. B.

F. B. Stieg, Official Dig. Fed. Paint Varnish 31 (1), 52 (1959).

Vasalos, I. A.

I. A. Vasalos, Ph.D. thesis, MIT (1969).

Völz, H. G.

H. G. Völz, Ber. Bunsenges. 71, 326, (1967).

Willets, W. R.

F. R. Marchetti and W. R. Willets, J. Tech. Assoc. Pulp Paper Ind. 43, 273 (1960).

Appl. Opt. (2)

Ber. Bunsenges. (1)

H. G. Völz, Ber. Bunsenges. 71, 326, (1967).

Bull. Astron. Insts. Netherlands (1)

W. M. Irvine, Bull. Astron. Insts. Netherlands 17, 176 (1963).

Discuss. Faraday Soc. (1)

S. W. Churchill, G. C. Clark, and C. M. Sliepcevich, Discuss. Faraday Soc. 30, 192 (1960).
[Crossref]

J. Opt. Soc. Am. (5)

J. Paint Tech. (1)

W. D. Ross, J. Paint Tech. 43, 50 (1971).

J. Phys. Chem. (1)

P. Anne Hiltner and Irwin M. Krieger, J. Phys. Chem. 73, 2386 (1969).
[Crossref]

J. Phys. D (3)

L. F. Gate, J. Phys. D 4, 1049 (1971).
[Crossref]

L. F. Gate, J. Phys. D 5, 837 (1972).
[Crossref]

B. J. Brinkworth, J. Phys. D 4, 1105 (1971).
[Crossref]

J. Res. Nat. Bur. Stand. (U. S.) (1)

D. B. Judd, J. Res. Nat. Bur. Stand. (U. S.) 29, 329 (1942).
[Crossref]

J. Tech. Assoc. Pulp Paper Ind. (1)

F. R. Marchetti and W. R. Willets, J. Tech. Assoc. Pulp Paper Ind. 43, 273 (1960).

Official Dig. Fed. Paint Varnish (1)

F. B. Stieg, Official Dig. Fed. Paint Varnish 31 (1), 52 (1959).

Phys. Rev. (1)

W. Heller, Phys. Rev. 68, 5 (1945).
[Crossref]

Other (3)

I. A. Vasalos, Ph.D. thesis, MIT (1969).

C. M. Chu, G. C. Clark, and S. W. Churchill, Tables of Angular Distribution Coefficients for Light Scattering by Spheres (Engineering Research Institute, University of Michigan, 1957).

E. J. Moss, Ph.D. thesis, University of Bristol (1970).

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Figures (8)

F. 1
F. 1

Forward-scattering factor (1 − 〈cosθav) as a function of α and m.

F. 2
F. 2

Mie-scattering coefficient as a function of α and m.

F. 3
F. 3

Scattering cross sections per unit latex volume concentration (Σg/V) as a function of latex volume concentration (r = 0.11 μm, λ = 681 nm). ○—Experimental determinations, full line—calculated.

F. 4
F. 4

Scattering cross sections of latex suspensions as a function of latex concentration (r = 0.11 μm, λ = 681 nm). ○— Experimental determinations, full line—calculated.

F. 5
F. 5

Scattering cross sections of latex suspensions (r = 0.11 μm) as a function of wavelength at two concentrations. ○—Experimental determinations, full line—calculated.

F. 6
F. 6

Scattering cross sections of latex suspensions as a function of wavelength for r=0.156 μm (V = 0.095). ○—Experimental determinations, full line—calculated.

F. 7
F. 7

Scattering cross sections of latex suspensions as a function of wavelength for r = 0.241 μm (V = 0.095).○—Experimental determinations, full line—calculated.

F. 8
F. 8

Scattering cross sections of latex suspensions as a function of wavelength for r=0.395 μm (V = 0.095). ○—Experimental determinations, full line—calculated.

Tables (1)

Tables Icon

Table I Latex scattering cross sections (r = 0.11 μm)

Equations (10)

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R 0 = R 0 + T 0 2 R B 1 R 0 R B .
R 0 R B = ( R 0 + 1 R B ) R 0 + ( T 0 2 R 0 2 ) .
( T 0 2 R 0 2 ) = 1 R 0 ( 1 + R 2 R ) ,
R 0 R B = ( R 0 + 1 R B ) R 0 R 0 ( 1 + R 2 R ) + 1 .
Σ s = 2 3 a ( 1 + R 1 R ) coth 1 ( 1 R 0 1 + R 0 ) ( 1 + R 1 R ) 0.71 a ,
R M = R + ( 1 R ) ( 1 r ) R 0 1 R 0 r .
Σ s = N π r 2 · K ( 1 cos θ av ) .
Σ s V = 3 4 r K ( 1 cos θ av ) .
n m = 1.33 + V ( n s 1.33 ) .
m = n s 1.33 + V ( n s 1.33 ) .