Abstract

An analytical model of the scattering structure factor for an assembly of noninteracting hard disks has recently appeared in the literature [Phys. Rev. A 42, 5978–5989 (1990)]. We employ this model to calculate correlated light scattering by monodispersions and binary mixtures of condensation droplets on a window pane. We find that an area fraction of f ≥ 0.6 is required for producing the near-forward direction scattering suppression and that a moderately wide polydispersion of droplet sizes is capable of producing the experimentally observed bright ring of colored light.

© 1994 Optical Society of America

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References

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  1. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), article 162.
  2. J. J. M. Reesinck, D. A. de Vries, “The diffraction of light by a large number of circular objects,” Physica 7, 603–608 (1940).
    [CrossRef]
  3. J. A. Prins, “Scattering by an assembly of molecules,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 295–303.
  4. R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), Chap. 5.
  5. J. A. Lock, L. Yang, “Mie theory model of the corona,” Appl. Opt. 30, 3408–3414 (1991).
    [CrossRef] [PubMed]
  6. The difference between correlated scattering and other types of scattering is stated especially clearly in the Introduction of C. Smart, R. Jacobsen, M. Kerker, J. P. Kratohvil, E. Matijevic, “Experimental study of mulitple light scattering,” J. Opt. Soc. Am. 55, 947–955 (1965).
    [CrossRef]
  7. T. Alfrey, E. B. Bradford, J. W. Vanderhoff, G. Oster, “Optical properties of uniform particle-size latexes,” J. Opt. Soc. Am. 44, 603–609 (1954).
    [CrossRef]
  8. I. M. Krieger, F. M. O'Neill, “Diffraction of light by arrays of colloidal spheres,” J. Am. Chem. Soc. 90, 3114–3120 (1968).
    [CrossRef]
  9. J. W. Goodwin, R. H. Ottewill, A. Parentich, “Optical examination of structured colloidal dispersions,” J. Phys. Chem. 84, 1580–1586 (1980).
    [CrossRef]
  10. E. D. Freund, R. L. McCally, R. A. Farrell, “Direct summation of fields for light scattering by fibrils with applications to normal corneas,” Appl. Opt. 25, 2739–2746 (1986).
    [CrossRef] [PubMed]
  11. R. W. Hart, R. A. Farrell, “Light scattering in the cornea,” J. Opt. Soc. Am. 59, 766–774 (1969).
    [CrossRef] [PubMed]
  12. W. W. Wood, “Monte Carlo studies of simple liquid models,” in The Physics of Simple Liquids, H. N. V. Temperley, J. S. Rowlinson, G. S. Rushbrooke, eds. (North-Holland, Amsterdam, 1968), Chap. 5.
  13. D. G. Chae, F. H. Ree, T. Ree, “Radial distribution functions and equation of state of the hard-disk fluid,” J. Chem. Phys. 50, 1581–1589 (1969).
    [CrossRef]
  14. W. W. Wood, “NpT—ensemble Monte Carlo calculations for the hard-disk fluid,” J. Chem. Phys. 52, 729–741 (1970).
    [CrossRef]
  15. Y. Uehara, T. Ree, F. H. Ree, “Radial distribution function for hard disks from the BGY2 theory,” J. Chem. Phys. 70, 1876–1883 (1979).
    [CrossRef]
  16. Y. Rosenfeld, “Free-energy model for the inhomogeneous hard-sphere fluid in D dimensions: Structure factors for the hard disk (D = 2) mixtures in simple explicit form,” Phys. Rev. A 42, 5978–5989 (1990).
    [CrossRef] [PubMed]
  17. J.-L. Barrat, H. Xu, J.-P. Hansen, M. Baus, “Freezing of binary hard-disc alloys: I. The equation of state and pair structure of the fluid phase,” J. Phys. C 21, 3165–3176 (1988).
    [CrossRef]
  18. V. Twersky, “Transparency of pair-correlated, random distributions of small scatterers, with applications to the cornea,” J. Opt. Soc. Am. 65, 524–530 (1975), Eq. (22).
    [CrossRef] [PubMed]
  19. General Electric Lamp Specification Bull. LSB#206-0155R1 (General Electric, Cleveland, Ohio, 1980).
  20. D. E. Gray, ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1963), pp. 7.122–7.125.
  21. See also Sky and Telescope, December 1993, p. 52.
  22. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), p. 532.

1991 (1)

1990 (1)

Y. Rosenfeld, “Free-energy model for the inhomogeneous hard-sphere fluid in D dimensions: Structure factors for the hard disk (D = 2) mixtures in simple explicit form,” Phys. Rev. A 42, 5978–5989 (1990).
[CrossRef] [PubMed]

1988 (1)

J.-L. Barrat, H. Xu, J.-P. Hansen, M. Baus, “Freezing of binary hard-disc alloys: I. The equation of state and pair structure of the fluid phase,” J. Phys. C 21, 3165–3176 (1988).
[CrossRef]

1986 (1)

1980 (1)

J. W. Goodwin, R. H. Ottewill, A. Parentich, “Optical examination of structured colloidal dispersions,” J. Phys. Chem. 84, 1580–1586 (1980).
[CrossRef]

1979 (1)

Y. Uehara, T. Ree, F. H. Ree, “Radial distribution function for hard disks from the BGY2 theory,” J. Chem. Phys. 70, 1876–1883 (1979).
[CrossRef]

1975 (1)

1970 (1)

W. W. Wood, “NpT—ensemble Monte Carlo calculations for the hard-disk fluid,” J. Chem. Phys. 52, 729–741 (1970).
[CrossRef]

1969 (2)

R. W. Hart, R. A. Farrell, “Light scattering in the cornea,” J. Opt. Soc. Am. 59, 766–774 (1969).
[CrossRef] [PubMed]

D. G. Chae, F. H. Ree, T. Ree, “Radial distribution functions and equation of state of the hard-disk fluid,” J. Chem. Phys. 50, 1581–1589 (1969).
[CrossRef]

1968 (1)

I. M. Krieger, F. M. O'Neill, “Diffraction of light by arrays of colloidal spheres,” J. Am. Chem. Soc. 90, 3114–3120 (1968).
[CrossRef]

1965 (1)

1954 (1)

1940 (1)

J. J. M. Reesinck, D. A. de Vries, “The diffraction of light by a large number of circular objects,” Physica 7, 603–608 (1940).
[CrossRef]

Alfrey, T.

Barrat, J.-L.

J.-L. Barrat, H. Xu, J.-P. Hansen, M. Baus, “Freezing of binary hard-disc alloys: I. The equation of state and pair structure of the fluid phase,” J. Phys. C 21, 3165–3176 (1988).
[CrossRef]

Baus, M.

J.-L. Barrat, H. Xu, J.-P. Hansen, M. Baus, “Freezing of binary hard-disc alloys: I. The equation of state and pair structure of the fluid phase,” J. Phys. C 21, 3165–3176 (1988).
[CrossRef]

Bradford, E. B.

Chae, D. G.

D. G. Chae, F. H. Ree, T. Ree, “Radial distribution functions and equation of state of the hard-disk fluid,” J. Chem. Phys. 50, 1581–1589 (1969).
[CrossRef]

de Vries, D. A.

J. J. M. Reesinck, D. A. de Vries, “The diffraction of light by a large number of circular objects,” Physica 7, 603–608 (1940).
[CrossRef]

Farrell, R. A.

Freund, E. D.

Goodwin, J. W.

J. W. Goodwin, R. H. Ottewill, A. Parentich, “Optical examination of structured colloidal dispersions,” J. Phys. Chem. 84, 1580–1586 (1980).
[CrossRef]

Hansen, J.-P.

J.-L. Barrat, H. Xu, J.-P. Hansen, M. Baus, “Freezing of binary hard-disc alloys: I. The equation of state and pair structure of the fluid phase,” J. Phys. C 21, 3165–3176 (1988).
[CrossRef]

Hart, R. W.

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), p. 532.

Jacobsen, R.

Kerker, M.

Kratohvil, J. P.

Krieger, I. M.

I. M. Krieger, F. M. O'Neill, “Diffraction of light by arrays of colloidal spheres,” J. Am. Chem. Soc. 90, 3114–3120 (1968).
[CrossRef]

Lock, J. A.

Matijevic, E.

McCally, R. L.

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), article 162.

O'Neill, F. M.

I. M. Krieger, F. M. O'Neill, “Diffraction of light by arrays of colloidal spheres,” J. Am. Chem. Soc. 90, 3114–3120 (1968).
[CrossRef]

Oster, G.

Ottewill, R. H.

J. W. Goodwin, R. H. Ottewill, A. Parentich, “Optical examination of structured colloidal dispersions,” J. Phys. Chem. 84, 1580–1586 (1980).
[CrossRef]

Parentich, A.

J. W. Goodwin, R. H. Ottewill, A. Parentich, “Optical examination of structured colloidal dispersions,” J. Phys. Chem. 84, 1580–1586 (1980).
[CrossRef]

Prins, J. A.

J. A. Prins, “Scattering by an assembly of molecules,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 295–303.

Ree, F. H.

Y. Uehara, T. Ree, F. H. Ree, “Radial distribution function for hard disks from the BGY2 theory,” J. Chem. Phys. 70, 1876–1883 (1979).
[CrossRef]

D. G. Chae, F. H. Ree, T. Ree, “Radial distribution functions and equation of state of the hard-disk fluid,” J. Chem. Phys. 50, 1581–1589 (1969).
[CrossRef]

Ree, T.

Y. Uehara, T. Ree, F. H. Ree, “Radial distribution function for hard disks from the BGY2 theory,” J. Chem. Phys. 70, 1876–1883 (1979).
[CrossRef]

D. G. Chae, F. H. Ree, T. Ree, “Radial distribution functions and equation of state of the hard-disk fluid,” J. Chem. Phys. 50, 1581–1589 (1969).
[CrossRef]

Reesinck, J. J. M.

J. J. M. Reesinck, D. A. de Vries, “The diffraction of light by a large number of circular objects,” Physica 7, 603–608 (1940).
[CrossRef]

Rosenfeld, Y.

Y. Rosenfeld, “Free-energy model for the inhomogeneous hard-sphere fluid in D dimensions: Structure factors for the hard disk (D = 2) mixtures in simple explicit form,” Phys. Rev. A 42, 5978–5989 (1990).
[CrossRef] [PubMed]

Smart, C.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), Chap. 5.

Twersky, V.

Uehara, Y.

Y. Uehara, T. Ree, F. H. Ree, “Radial distribution function for hard disks from the BGY2 theory,” J. Chem. Phys. 70, 1876–1883 (1979).
[CrossRef]

Vanderhoff, J. W.

Wood, W. W.

W. W. Wood, “NpT—ensemble Monte Carlo calculations for the hard-disk fluid,” J. Chem. Phys. 52, 729–741 (1970).
[CrossRef]

W. W. Wood, “Monte Carlo studies of simple liquid models,” in The Physics of Simple Liquids, H. N. V. Temperley, J. S. Rowlinson, G. S. Rushbrooke, eds. (North-Holland, Amsterdam, 1968), Chap. 5.

Xu, H.

J.-L. Barrat, H. Xu, J.-P. Hansen, M. Baus, “Freezing of binary hard-disc alloys: I. The equation of state and pair structure of the fluid phase,” J. Phys. C 21, 3165–3176 (1988).
[CrossRef]

Yang, L.

Appl. Opt. (2)

J. Am. Chem. Soc. (1)

I. M. Krieger, F. M. O'Neill, “Diffraction of light by arrays of colloidal spheres,” J. Am. Chem. Soc. 90, 3114–3120 (1968).
[CrossRef]

J. Chem. Phys. (3)

D. G. Chae, F. H. Ree, T. Ree, “Radial distribution functions and equation of state of the hard-disk fluid,” J. Chem. Phys. 50, 1581–1589 (1969).
[CrossRef]

W. W. Wood, “NpT—ensemble Monte Carlo calculations for the hard-disk fluid,” J. Chem. Phys. 52, 729–741 (1970).
[CrossRef]

Y. Uehara, T. Ree, F. H. Ree, “Radial distribution function for hard disks from the BGY2 theory,” J. Chem. Phys. 70, 1876–1883 (1979).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Phys. C (1)

J.-L. Barrat, H. Xu, J.-P. Hansen, M. Baus, “Freezing of binary hard-disc alloys: I. The equation of state and pair structure of the fluid phase,” J. Phys. C 21, 3165–3176 (1988).
[CrossRef]

J. Phys. Chem. (1)

J. W. Goodwin, R. H. Ottewill, A. Parentich, “Optical examination of structured colloidal dispersions,” J. Phys. Chem. 84, 1580–1586 (1980).
[CrossRef]

Phys. Rev. A (1)

Y. Rosenfeld, “Free-energy model for the inhomogeneous hard-sphere fluid in D dimensions: Structure factors for the hard disk (D = 2) mixtures in simple explicit form,” Phys. Rev. A 42, 5978–5989 (1990).
[CrossRef] [PubMed]

Physica (1)

J. J. M. Reesinck, D. A. de Vries, “The diffraction of light by a large number of circular objects,” Physica 7, 603–608 (1940).
[CrossRef]

Other (8)

J. A. Prins, “Scattering by an assembly of molecules,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 295–303.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), Chap. 5.

W. W. Wood, “Monte Carlo studies of simple liquid models,” in The Physics of Simple Liquids, H. N. V. Temperley, J. S. Rowlinson, G. S. Rushbrooke, eds. (North-Holland, Amsterdam, 1968), Chap. 5.

General Electric Lamp Specification Bull. LSB#206-0155R1 (General Electric, Cleveland, Ohio, 1980).

D. E. Gray, ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1963), pp. 7.122–7.125.

See also Sky and Telescope, December 1993, p. 52.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), p. 532.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), article 162.

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Figures (8)

Fig. 1
Fig. 1

Geometry for correlated scattering. The ith scatterer is located at ri, the incident wave vector is k0, and the scattered wave vector is ks. The scattering angle is θ.

Fig. 2
Fig. 2

Pair-correlation function for noninteracting hard disks with area fractions (a) f = 0.05, (b) f = 0.2, (c) f = 0.5 employing the free-energy model of Section 2. The nearest-neighbor peak is denoted by N and the next-nearest-neighbor peak is denoted by NN.

Fig. 3
Fig. 3

Scattered intensity as a function of scattering angle from Eqs. (9), (10), (20), and (21) for a = 7.5 μm, k = 0.6328 μm, and for a number of area fractions in the range 0.001 ≤ f ≤ 0.7. In this model, the long-range order–disorder phase transition occurs at f = 0.754.

Fig. 4
Fig. 4

Scattered intensity as a function of scattering angle from Eqs. (9), (10), (20), and (21) for a = 7.5 μm, λ = 0.6328 μm, and f = 0.7. The first scattering peak occurs at sin θ ≈ λ/2a, and the second scattering peak occurs at sin θ ≈ 1.84λ/2a.

Fig. 5
Fig. 5

Scattered intensity as a function of scattering angle from Eqs. (9), (10), (20), and (21) for a = 7.5 μm, λ = 0.4358 μm (B), λ = 0.5461 μm (G), λ = 0.5791 μm (Y), and λ = 0.6198 μm (R), and for f = 0.7.

Fig. 6
Fig. 6

(a) Scattered intensity as a function of scattering angle for a1 = 6.75 μm, f1 = 0.7 (curve 1); a2 = 7.5 μm, f2 = 0.7 (curve 2); and the binary mixture a1 = 6.75 μm, a2 = 7.5 μm, f1 = 0.3133, f2 = 0.3867, and σ1 = σ2 (dashed curve). All these curves are for λ = 0.6328 μm. The size ratio is a2/a1 = 1.11. (b) Scattered intensity as a function of scattering angle for a1 = 6.0 μm, f1 = 0.7 (curve 1); a2 = 7.5 μm, f2 = 0.7 (curve 2); and the binary mixture a1 = 6.0 μm, a2 = 7.5 μm, f1 = 0.2732, f2 = 0.4268, and σ1 = σ2 (dashed curve). All these curves are for λ = 0 0.6328 μm. The size ratio is a2/a1 = 1.25.

Fig. 7
Fig. 7

(a) Scattered intensity as a function of scattering angle for the binary mixture of Fig. 6(a) and λ = 0.4358 μm (B), λ = 0.5461 μm (G), λ = 0.5891 μm (Y), and λ = 0.6198 μm (R). (b) Scattered intensity as a function of scattering angle for the binary mixture of Fig. 6(b) and the wavelengths of Fig. 7(a).

Fig. 8
Fig. 8

Scattered intensity as a function of scattering angle for a1 = 7.5 μm, f1 = 0.7 (curve 1); a2 = 15.0 μm, f2 = 0.7 (curve) 2); and the binary mixture a1 = 7.5 μm, a2 = 15.0 μm, f1 = 0.275, f2 = 0.425, and σ2 = 2.57 σ1 (dashed curve). All the curves are for λ = 0.6328 μm. The size ratio is a2/a1 = 2.0.

Tables (1)

Tables Icon

Table 1 Average Droplet Radius from Expression (30) based on the Blue Edge (0.4358 μm) and the Red Edge (0.6198 μm) of the Bright Scattering Ring for Eight Photographs of Correlated Scattering Produced by Condensation Droplets on a Camera Lensa

Equations (32)

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d sin θ = λ ,
E ( θ ) = i = 1 N E a ( θ ) exp [ i ( k 0 k s ) · r i ]
Δ ( r j r i ) = g ( r j r i ) σ ( r j ) .
f = π a 2 σ
I ( θ ) = E * ( θ ) · E ( θ ) = I a ( θ ) × { N + i = 1 N j = 1 j i N exp [ i K · ( r j r i ) ] } ,
K k 0 k s
| K | = K = 2 k sin ( θ / 2 ) .
I ( θ ) = I a ( θ ) { N + N σ d 2 r [ g ( r ) 1 ] exp ( i K · r ) + σ 2 | d 2 r exp ( i K · r ) | 2 } ,
I ( θ ) = N I a ( θ ) { 1 + 2 π σ 0 r d r [ g ( r ) 1 ] × J 0 ( 2 kr sin θ ) } = N I a ( θ ) S ( θ ) ,
I a ( θ ) = π 2 a 4 λ 2 R 2 I 0 [ 2 J 1 ( ka sin θ ) ka sin θ ] 2 ,
sin θ 1.22 λ 2 a ,
f 1 = π a 1 2 σ 1 ,
f 2 = π a 2 2 σ 2 .
f = f 1 + f 2 .
f 1 = f 1 + ( a 2 a 1 ) 2 σ 2 σ 1 , f 2 = ( a 2 a 1 ) 2 σ 2 σ 1 f 1 + ( a 2 a 1 ) 2 σ 2 σ 1 .
Δ 11 ( r j r i ) = g 11 ( r j r i ) σ 1 ( r j ) , Δ 21 ( r j r i ) = g 21 ( r j r i ) σ 2 ( r j ) , Δ 12 ( r j r i ) = g 12 ( r j r i ) σ 1 ( r j ) , Δ 22 ( r j r i ) = g 22 ( r j r i ) σ 2 ( r j ) .
g 21 ( r ) = g 12 ( r ) .
I ( θ ) = N 1 I 1 ( θ ) S 11 ( θ ) + N 2 I 2 ( θ ) S 22 ( θ ) + ( N 1 N 2 ) 1 / 2 × [ E 1 * ( θ ) · E 2 ( θ ) + E 2 * ( θ ) · E 1 ( θ ) ] S 12 ( θ ) ,
S ij ( θ ) = δ ij + 2 π ( σ i σ j ) 1 / 2 0 r d r [ g ij ( r ) 1 ] × J 0 ( 2 kr sin θ ) .
S ( θ ) = ( C ( θ ) + 1 ) 1 ,
C ( θ ) = 4 f ( 1 f ) [ 2 J 1 ( 2 ka sin θ ) 2 ka sin θ ] + 4 f 2 ( 1 f ) 2 J 0 ( ka sin θ ) [ 2 J 1 ( ka sin θ ) ka sin θ ] + [ f 2 ( 1 f ) 2 + 2 f 3 ( 1 f ) 3 ] [ 2 J 1 ( ka sin θ ) ka sin θ ] 2 .
S ( 0 ) = ( 1 f ) 3 ( 1 + f )
[ S ij ( θ ) ] = [ C 11 + 1 C 12 C 21 C 22 + 1 ] 1 ,
C ii = χ 2 a i 2 f i [ 2 J 1 ( k a i sin θ ) k a i sin θ ] 2 + 4 χ 1 a i f i J 0 ( k a i sin θ ) × [ 2 J 1 ( k a i sin θ ) k a i sin θ ] + 4 χ 0 f i [ 2 J 1 ( 2 k a i sin θ ) 2 k a i sin θ ] , C 12 = C 21 = χ 2 a 1 a 2 ( f 1 f 2 ) 1 / 2 [ 2 J 1 ( k a 1 sin θ ) k a 1 sin θ ] × [ 2 J 1 ( k a 2 sin θ ) k a 2 sin θ ] + 2 χ 1 a 2 ( f 1 f 2 ) 1 / 2 J 0 ( k a 1 sin θ ) × [ 2 J 1 ( k a 2 sin θ ) k a 2 sin θ ] + 2 χ 1 a 1 ( f 1 f 2 ) 1 / 2 J 0 ( k a 2 sin θ ) × [ 2 J 1 ( k a 1 sin θ ) k a 1 sin θ ] + χ 0 ( f 1 f 2 ) 1 / 2 ( a 1 + a 2 ) 2 a 1 a 2 × [ 2 J 1 ( k a 1 sin θ + k a 2 sin θ ) k ( a 1 + a 2 ) sin θ ] ,
χ 0 = ( 1 f ) 1 , χ 1 = ( f 1 a 1 + f 2 a 2 ) ( 1 f ) 2 , χ 2 = ( f 1 a 1 2 + f 2 a 2 2 ) ( 1 f ) 2 + 2 ( f 1 a 1 + f 2 a 2 ) 2 ( 1 f ) 3 .
I ( θ ) N I a ( θ ) .
sin θ λ 2 a .
sin θ 1.84 λ 2 a .
a ave = ( a 1 4 σ 1 ) a 1 + ( a 2 4 σ 2 ) a 2 a 1 4 σ 1 + a 2 4 σ 2 ,
sin θ λ 2 a ave .
sin θ 1 λ 2 a 1 , sin θ 2 λ 2 a 2 .
W 0.32 λ Δ θ = 50 μ m .

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