Abstract

The light-scattering properties of pigment suspensions are discussed in terms of an approximate theory, with special emphasis on the role of the size of the scattering particles and their refractive index relative to the surrounding matrix. Experiments are described which confirm that the theory is a useful basis for understanding the behavior of pigment suspensions.

© 1961 Optical Society of America

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References

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  1. R. G. Giovanelli, Optica Acta 2, 153 (1955), Table I.
    [Crossref]
  2. R. G. Giovanelli, private communication.
  3. S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950). Chap. VI.
  4. H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).
  5. A. N. Lowan, Tables of Scattering Functions for Spherical Particles [Natl. Bur. Standards (U.S.), Appl. Math. Ser. 4, Washington, 1948].
  6. C. Chu, G. C. Clark, and A. W. Churchill, Angular Distribution Coefficients for Light-Scattering by Spheres (University of Michigan Press, Ann Arbor, Michigan, 1957).
  7. W. J. Pangonis, W. Heller, and A. W. Jacobson, Tables of Light-Scattering Functions for Spherical Particles (Wayne University Publication, Detroit, Michigan, 1957).
  8. R. O. Gumprecht and C. N. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of Michigan Press, Ann Arbor, Michigan, 1951).
  9. R. B. Penndorf, New Tables of Mie Scattering Functions for Spherical Particles (, Part 6, U. S. Air Force Cambridge Research Center, Bedford, Massachusetts).
  10. G. F. A. Stutz and A. H. Pfund, Ind. Eng. Chem. 19, 51 (1927).
    [Crossref]
  11. D. H. Clewell, J. Opt. Soc. Am. 31, 521 (1941).
    [Crossref]
  12. A. E. Jacobsen, Canadian Chemistry and Process Industries 33, 124 (1949).
  13. W. R. Blevin, J. Oil and Colour Chemists’ Assoc. 41, 850 (1958).
  14. J. W. T. Walsh, D.S.I.R.(G.B.) Illum. Research Comm., Tech. Paper2, 10 (1926).
  15. R. G. Giovanelli, Australian J. Phys. 10, 227 (1957).
    [Crossref]
  16. W. R. Blevin and W. J. Brown, J. Opt. Soc. Am. 51, 129 (1961).
    [Crossref]
  17. W. E. K. Middleton and C. L. Sanders, J. Opt. Soc. Am. 41, 419 (1951).
    [Crossref] [PubMed]

1961 (1)

1958 (1)

W. R. Blevin, J. Oil and Colour Chemists’ Assoc. 41, 850 (1958).

1957 (1)

R. G. Giovanelli, Australian J. Phys. 10, 227 (1957).
[Crossref]

1955 (1)

R. G. Giovanelli, Optica Acta 2, 153 (1955), Table I.
[Crossref]

1951 (1)

1949 (1)

A. E. Jacobsen, Canadian Chemistry and Process Industries 33, 124 (1949).

1941 (1)

1927 (1)

G. F. A. Stutz and A. H. Pfund, Ind. Eng. Chem. 19, 51 (1927).
[Crossref]

Blevin, W. R.

W. R. Blevin and W. J. Brown, J. Opt. Soc. Am. 51, 129 (1961).
[Crossref]

W. R. Blevin, J. Oil and Colour Chemists’ Assoc. 41, 850 (1958).

Brown, W. J.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950). Chap. VI.

Chu, C.

C. Chu, G. C. Clark, and A. W. Churchill, Angular Distribution Coefficients for Light-Scattering by Spheres (University of Michigan Press, Ann Arbor, Michigan, 1957).

Churchill, A. W.

C. Chu, G. C. Clark, and A. W. Churchill, Angular Distribution Coefficients for Light-Scattering by Spheres (University of Michigan Press, Ann Arbor, Michigan, 1957).

Clark, G. C.

C. Chu, G. C. Clark, and A. W. Churchill, Angular Distribution Coefficients for Light-Scattering by Spheres (University of Michigan Press, Ann Arbor, Michigan, 1957).

Clewell, D. H.

Giovanelli, R. G.

R. G. Giovanelli, Australian J. Phys. 10, 227 (1957).
[Crossref]

R. G. Giovanelli, Optica Acta 2, 153 (1955), Table I.
[Crossref]

R. G. Giovanelli, private communication.

Gumprecht, R. O.

R. O. Gumprecht and C. N. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of Michigan Press, Ann Arbor, Michigan, 1951).

Heller, W.

W. J. Pangonis, W. Heller, and A. W. Jacobson, Tables of Light-Scattering Functions for Spherical Particles (Wayne University Publication, Detroit, Michigan, 1957).

Jacobsen, A. E.

A. E. Jacobsen, Canadian Chemistry and Process Industries 33, 124 (1949).

Jacobson, A. W.

W. J. Pangonis, W. Heller, and A. W. Jacobson, Tables of Light-Scattering Functions for Spherical Particles (Wayne University Publication, Detroit, Michigan, 1957).

Lowan, A. N.

A. N. Lowan, Tables of Scattering Functions for Spherical Particles [Natl. Bur. Standards (U.S.), Appl. Math. Ser. 4, Washington, 1948].

Middleton, W. E. K.

Pangonis, W. J.

W. J. Pangonis, W. Heller, and A. W. Jacobson, Tables of Light-Scattering Functions for Spherical Particles (Wayne University Publication, Detroit, Michigan, 1957).

Penndorf, R. B.

R. B. Penndorf, New Tables of Mie Scattering Functions for Spherical Particles (, Part 6, U. S. Air Force Cambridge Research Center, Bedford, Massachusetts).

Pfund, A. H.

G. F. A. Stutz and A. H. Pfund, Ind. Eng. Chem. 19, 51 (1927).
[Crossref]

Sanders, C. L.

Sliepcevich, C. N.

R. O. Gumprecht and C. N. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of Michigan Press, Ann Arbor, Michigan, 1951).

Stutz, G. F. A.

G. F. A. Stutz and A. H. Pfund, Ind. Eng. Chem. 19, 51 (1927).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

Walsh, J. W. T.

J. W. T. Walsh, D.S.I.R.(G.B.) Illum. Research Comm., Tech. Paper2, 10 (1926).

Australian J. Phys. (1)

R. G. Giovanelli, Australian J. Phys. 10, 227 (1957).
[Crossref]

Canadian Chemistry and Process Industries (1)

A. E. Jacobsen, Canadian Chemistry and Process Industries 33, 124 (1949).

Ind. Eng. Chem. (1)

G. F. A. Stutz and A. H. Pfund, Ind. Eng. Chem. 19, 51 (1927).
[Crossref]

J. Oil and Colour Chemists’ Assoc. (1)

W. R. Blevin, J. Oil and Colour Chemists’ Assoc. 41, 850 (1958).

J. Opt. Soc. Am. (3)

Optica Acta (1)

R. G. Giovanelli, Optica Acta 2, 153 (1955), Table I.
[Crossref]

Other (9)

R. G. Giovanelli, private communication.

S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950). Chap. VI.

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

A. N. Lowan, Tables of Scattering Functions for Spherical Particles [Natl. Bur. Standards (U.S.), Appl. Math. Ser. 4, Washington, 1948].

C. Chu, G. C. Clark, and A. W. Churchill, Angular Distribution Coefficients for Light-Scattering by Spheres (University of Michigan Press, Ann Arbor, Michigan, 1957).

W. J. Pangonis, W. Heller, and A. W. Jacobson, Tables of Light-Scattering Functions for Spherical Particles (Wayne University Publication, Detroit, Michigan, 1957).

R. O. Gumprecht and C. N. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (University of Michigan Press, Ann Arbor, Michigan, 1951).

R. B. Penndorf, New Tables of Mie Scattering Functions for Spherical Particles (, Part 6, U. S. Air Force Cambridge Research Center, Bedford, Massachusetts).

J. W. T. Walsh, D.S.I.R.(G.B.) Illum. Research Comm., Tech. Paper2, 10 (1926).

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Figures (8)

Fig. 1
Fig. 1

The smooth curve shows theoretical values for the total reflectance of a semi-infinite diffuser in which single scattering is isotropic, plotted against the ratio α/σe of the absorption coefficient to the effective scattering coefficient. The points plotted are for diffusers having different angular distributions for singly scattered light. Air matrix; normal illumination.

Fig. 2
Fig. 2

The theoretical scattering efficiency, [(1− μ ¯)K(x,n)]/nx, of a monodisperse suspension of nonabsorbing spherical particles, plotted against the particle size parameter nx for several values of n, the refractive index of the particles relative to the matrix.

Fig. 3
Fig. 3

Theoretical data for a monodisperse suspension of nonabsorbing spherical particles. The effective scattering coefficient per unit cross-sectional area of particle, σe/(Nπr2), is plotted against the ratio n of the particle refractive index to the matrix refractive index, for several values of the particle size parameter x. Each curve relates to a constant particle diameter provided the matrix refractive index is constant.

Fig. 4
Fig. 4

Theoretical data for a monodisperse suspension of nonabsorbing spherical particles. The effective scattering coefficient per unit cross-sectional area of particle, σe/(Nπr2), is plotted against the ratio n of the particle refractive index to the matrix refractive index, for several values of the particle size parameter nx. Each curve relates to a constant particle diameter provided the particle refractive index is constant, in which case an increase in the value of n denotes a decrease in the matrix refractive index.

Fig. 5
Fig. 5

Theoretical values for the total reflectances of five semi-infinite diffusers, plotted against the ratio n of the scattering particle refractive index to the matrix refractive index. The light is incident normally; the diffuser matrix and the medium outside the diffuser have the same refractive index; the scattering particles are monodisperse, nonabsorbing spheres of size parameter nx equal to 3.

Fig. 6
Fig. 6

The diffuse reflectances of semi-infinite diffusers, plotted against the ratio it of the scattering particle refractive index to the matrix refractive index. The light is incident normally, with air outside the diffuser. The full-line curves show theoretical data for diffusers in which the scattering particles are monodisperse, nonabsorbing spheres of refractive index np equal to 2.00 and of size parameter nx equal to 6. The broken-line curves show experimental data from Fig. 7 for diffusers in which the scattering is due to polydisperse zinc oxide particles (np=2.01, nx≅6).

Fig. 7
Fig. 7

Experimental data on three opaque suspensions of zinc oxide and carbon, having matrices of air, water, and glycerine respectively. The diffuse reflectance of each suspension, for almost normally incident light of wavelength 500 mμ, is plotted against the concentration of carbon relative to that of zinc oxide.

Fig. 8
Fig. 8

Experimental data on three opaque suspensions of barium sulfate and carbon, having matrices of air, water, and glycerine, respectively. The diffuse reflectance of each suspension, for almost normally incident light of wavelength 700 mμ, is plotted against the concentration of carbon relative to that of barium sulfate.

Tables (2)

Tables Icon

Table I Ratio of the effective scattering coefficient σe, for use in isotropic theory, to the true scattering coefficient σ, for a monodisperse suspension of nonabsorbing spheres.

Tables Icon

Table II Theoretical data for a normally illuminated semi-infinite diffuser, whose surface is optically smooth and whose matrix is of refractive index nm. The total reflectance Rm of the diffuser when the outside medium is also of index nm, is compared with the total reflectance R1 when the outside medium is air.

Equations (9)

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σ e = ( 1 - μ ¯ ) σ ,
= - 1 + 1 I ( μ ) μ d μ - 1 + 1 I ( μ ) d μ .
σ = N π r 2 K ( x , n ) ,
σ e = N π r 2 ( 1 - μ ¯ ) K ( x , n ) .
σ e V = 3 π n p 2 λ 0 ( 1 - μ ¯ ) K ( x , n ) n x .
2 r λ 0 / n p .
K ( n , x ) = 8 3 ( n 2 - 1 n 2 + 2 ) 2 x 4 .
n m = n p / ( 1 + 3 ) 1 2 n p / 1.65.
R 1 = r + R m τ n m - 2 ( 1 - r ) 1 - R d ( 1 - τ n m - 2 ) ,