Abstract

Static light scattering is widely used for sizing of particles with radii in the range of 50 nm up to several micrometers. These experiments usually require very low particle concentrations (<10−4) for prevention of multiple scattering. As a consequence, nonabsorbing samples that are suited for light-scattering investigations must be transparent so that the transmittance of the incident light is typically above 95%. Investigations of less translucent samples require corrective terms for the beginning of multiple scattering to retrieve the particle-size distribution successfully. We applied a computationally convenient first-order approximation for the multiple-scattering problem that has Hartel’s approach in its first steps. When incorporated into our inversion technique, this approximation functions well for samples with transmittances above 30%. We present examples of applications to experimental data.

© 1995 Optical Society of America

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References

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  1. L. Lorenz, Oevres Scientifiques de L. Lorenz, Revues et Annotes par H. Valentiner (Librairie Lehman et Stage, Copenhague, 1898).
  2. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloider Metalllösungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  3. H. C. van de Hulst, Light Scattering of Small Particles (Wiley, New York, 1957).
  4. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  6. H. Schnablegger, O. Glatter, “Optical sizing of small colloidal particles: an optimized regularization technique,” Appl. Opt. 30, 4889–4896 (1991).
    [CrossRef] [PubMed]
  7. H. Schnablegger, O. Glatter, “Simultaneous determination of size distribution and refractive index of colloidal particles from static light scattering experiments,” J. Colloid Interface Sci. 158, 228–242 (1993).
    [CrossRef]
  8. O. Glatter, “A new method for the evaluation of small-angle scattering data,” J. Appl. Crystallogr. 10, 415–421 (1977).
    [CrossRef]
  9. W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten, besonders Trübgläser,” Licht 10, 141–143, 165, 190–191, 214–215, 232–234 (1940).
  10. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  11. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).
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  14. H. H. Theissing, “Macrodistribution of light scattered by dispersions of spherical dielectric particles,” J. Opt. Soc. Am. 40, 232–243 (1950).
    [CrossRef]
  15. H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
    [CrossRef]
  16. T. N. E. Greville, Theory and Application of Spline Functions (Academic, New York, 1969).
  17. J. Schelten, F. Hossfeld, “Application of spline functions to the correction of resolution errors in small-angle scattering,” J. Appl. Crystallogr. 4, 210–223 (1971).
    [CrossRef]
  18. C. de Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978).
    [CrossRef]
  19. Ph. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).
  20. O. Glatter, J. Sieberer, H. Schnablegger, “A comparative study on different scattering techniques and data evaluation methods for sizing of colloidal systems using light scattering,” Part. Part. Syst. Charact. 8, 274–281 (1991).
    [CrossRef]
  21. C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  22. S. W. Provencher, “A constrained regularization method for inverting data represented by linear algebraic or integral equation,” Comput. Phys. Commun. 27, 213–227 (1982).
    [CrossRef]

1993 (1)

H. Schnablegger, O. Glatter, “Simultaneous determination of size distribution and refractive index of colloidal particles from static light scattering experiments,” J. Colloid Interface Sci. 158, 228–242 (1993).
[CrossRef]

1991 (2)

H. Schnablegger, O. Glatter, “Optical sizing of small colloidal particles: an optimized regularization technique,” Appl. Opt. 30, 4889–4896 (1991).
[CrossRef] [PubMed]

O. Glatter, J. Sieberer, H. Schnablegger, “A comparative study on different scattering techniques and data evaluation methods for sizing of colloidal systems using light scattering,” Part. Part. Syst. Charact. 8, 274–281 (1991).
[CrossRef]

1983 (1)

H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
[CrossRef]

1982 (1)

S. W. Provencher, “A constrained regularization method for inverting data represented by linear algebraic or integral equation,” Comput. Phys. Commun. 27, 213–227 (1982).
[CrossRef]

1977 (1)

O. Glatter, “A new method for the evaluation of small-angle scattering data,” J. Appl. Crystallogr. 10, 415–421 (1977).
[CrossRef]

1971 (1)

J. Schelten, F. Hossfeld, “Application of spline functions to the correction of resolution errors in small-angle scattering,” J. Appl. Crystallogr. 4, 210–223 (1971).
[CrossRef]

1957 (1)

1955 (1)

1950 (1)

1940 (1)

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten, besonders Trübgläser,” Licht 10, 141–143, 165, 190–191, 214–215, 232–234 (1940).

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloider Metalllösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Bevington, Ph. R.

Ph. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Chu, C. M.

Churchill, S. W.

Clark, G. C.

de Boor, C.

C. de Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978).
[CrossRef]

Glatter, O.

H. Schnablegger, O. Glatter, “Simultaneous determination of size distribution and refractive index of colloidal particles from static light scattering experiments,” J. Colloid Interface Sci. 158, 228–242 (1993).
[CrossRef]

H. Schnablegger, O. Glatter, “Optical sizing of small colloidal particles: an optimized regularization technique,” Appl. Opt. 30, 4889–4896 (1991).
[CrossRef] [PubMed]

O. Glatter, J. Sieberer, H. Schnablegger, “A comparative study on different scattering techniques and data evaluation methods for sizing of colloidal systems using light scattering,” Part. Part. Syst. Charact. 8, 274–281 (1991).
[CrossRef]

O. Glatter, “A new method for the evaluation of small-angle scattering data,” J. Appl. Crystallogr. 10, 415–421 (1977).
[CrossRef]

Greville, T. N. E.

T. N. E. Greville, Theory and Application of Spline Functions (Academic, New York, 1969).

Hanson, R. J.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Hartel, W.

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten, besonders Trübgläser,” Licht 10, 141–143, 165, 190–191, 214–215, 232–234 (1940).

Hecht, H. G.

H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
[CrossRef]

Hossfeld, F.

J. Schelten, F. Hossfeld, “Application of spline functions to the correction of resolution errors in small-angle scattering,” J. Appl. Crystallogr. 4, 210–223 (1971).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Lawson, C. L.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Lorenz, L.

L. Lorenz, Oevres Scientifiques de L. Lorenz, Revues et Annotes par H. Valentiner (Librairie Lehman et Stage, Copenhague, 1898).

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloider Metalllösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Provencher, S. W.

S. W. Provencher, “A constrained regularization method for inverting data represented by linear algebraic or integral equation,” Comput. Phys. Commun. 27, 213–227 (1982).
[CrossRef]

Schelten, J.

J. Schelten, F. Hossfeld, “Application of spline functions to the correction of resolution errors in small-angle scattering,” J. Appl. Crystallogr. 4, 210–223 (1971).
[CrossRef]

Schnablegger, H.

H. Schnablegger, O. Glatter, “Simultaneous determination of size distribution and refractive index of colloidal particles from static light scattering experiments,” J. Colloid Interface Sci. 158, 228–242 (1993).
[CrossRef]

H. Schnablegger, O. Glatter, “Optical sizing of small colloidal particles: an optimized regularization technique,” Appl. Opt. 30, 4889–4896 (1991).
[CrossRef] [PubMed]

O. Glatter, J. Sieberer, H. Schnablegger, “A comparative study on different scattering techniques and data evaluation methods for sizing of colloidal systems using light scattering,” Part. Part. Syst. Charact. 8, 274–281 (1991).
[CrossRef]

Sieberer, J.

O. Glatter, J. Sieberer, H. Schnablegger, “A comparative study on different scattering techniques and data evaluation methods for sizing of colloidal systems using light scattering,” Part. Part. Syst. Charact. 8, 274–281 (1991).
[CrossRef]

Theissing, H. H.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

H. C. van de Hulst, Light Scattering of Small Particles (Wiley, New York, 1957).

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloider Metalllösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Appl. Opt. (1)

Comput. Phys. Commun. (1)

S. W. Provencher, “A constrained regularization method for inverting data represented by linear algebraic or integral equation,” Comput. Phys. Commun. 27, 213–227 (1982).
[CrossRef]

J. Appl. Crystallogr. (2)

J. Schelten, F. Hossfeld, “Application of spline functions to the correction of resolution errors in small-angle scattering,” J. Appl. Crystallogr. 4, 210–223 (1971).
[CrossRef]

O. Glatter, “A new method for the evaluation of small-angle scattering data,” J. Appl. Crystallogr. 10, 415–421 (1977).
[CrossRef]

J. Colloid Interface Sci. (1)

H. Schnablegger, O. Glatter, “Simultaneous determination of size distribution and refractive index of colloidal particles from static light scattering experiments,” J. Colloid Interface Sci. 158, 228–242 (1993).
[CrossRef]

J. Opt. Soc. Am. (3)

Licht (1)

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten, besonders Trübgläser,” Licht 10, 141–143, 165, 190–191, 214–215, 232–234 (1940).

Opt. Acta (1)

H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
[CrossRef]

Part. Part. Syst. Charact. (1)

O. Glatter, J. Sieberer, H. Schnablegger, “A comparative study on different scattering techniques and data evaluation methods for sizing of colloidal systems using light scattering,” Part. Part. Syst. Charact. 8, 274–281 (1991).
[CrossRef]

Other (10)

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

L. Lorenz, Oevres Scientifiques de L. Lorenz, Revues et Annotes par H. Valentiner (Librairie Lehman et Stage, Copenhague, 1898).

T. N. E. Greville, Theory and Application of Spline Functions (Academic, New York, 1969).

C. de Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978).
[CrossRef]

Ph. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

H. C. van de Hulst, Light Scattering of Small Particles (Wiley, New York, 1957).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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Figures (8)

Fig. 1
Fig. 1

Scattering geometry of a cylindrical sample cell with radius d. A narrow primary beam enters the sample at point A and leaves at point D. At point B′ the effective intensity of the multiply scattered light is represented by an equivalent virtual source beam characterized by the parameters z 1, z 2, and α. The experimental intensity is detected along the +x axis pointing to the detector.

Fig. 2
Fig. 2

Scattering geometry for the description of the reflected-light contributions. In the center of the cylindrical glass cell the laser light is scattered in all directions. The intensity arriving at the detector consists of the directly scattered intensity with scattering angle θ and the reflected intensity from the glass wall opposite to the direction of detection with scattering angle θ r = 180° − θ.

Fig. 3
Fig. 3

Mean deviation surface as a function of the transmittance T and the stability parameter λ [see Eq. (30)]. Each curve represents the achieved mean deviation with a set of T values as input parameters and with λ kept constant. The top curve corresponds to the highest λ value.

Fig. 4
Fig. 4

Experimental scattering function of toluene between 7.5° and 150° as a check of the alignment quality of the scattering device. The scattering function should be a constant or at least a set of uncorrelated data points if the light-scattering device is perfectly aligned.

Fig. 5
Fig. 5

Differential scattering curves of increasingly concentrated latex suspensions (symbols) and the fitting functions (curves). The concentrations of both (a) the monodisperse lattices and (b) the 2:1 mixture of two monodisperse lattices were 2.0 × 10−4 (crosses), 5.0 × 10−3 (pluses), 2.0 × 10−2 (triangles), and 5.0 × 10−2 (diamonds) mg/mL sulfate-polystyrene particles.

Fig. 6
Fig. 6

Volume distribution functions D v (R) of both (a) the monodisperse lattices and (b) the 2:1 mixture of two monodisperse lattices determined from the experimental data (Fig. 5) with the assumption of single scattering. With increasing concentration [5.0 × 10−3 (pluses) and 2.0 × 10−2 (triangles) mg/mL], artificial peaks appear because of an increasing degree of multiple scattering.

Fig. 7
Fig. 7

Same as in Fig. 6 but calculated with the multiple-scattering correction presented in this paper. In comparison with the results in Fig. 6, the effects of multiple scattering can be suppressed down to a limiting transmittance value of approximately 30% (triangles).

Fig. 8
Fig. 8

Reconstructed single-scattering curves (symbols) of both (a) the monodisperse lattices and (b) the 2:1 mixture of two monodisperse lattices obtained from the experimental scattering curves in Fig. 5 in comparison with the scattering curve (curves) of the lowest-concentrated sample (the reflected-light contributions are already subtracted). The concentrations were 5.0 × 10−3 mg/mL (pluses) and 2.0 × 10−2 mg/mL (triangles).

Equations (37)

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- 1 μ ( s · ) Φ ( r , s ) = Φ ( r , s ) - 1 4 π p ( s , s ) Φ ( r , s ) d ω s ,
- d Φ k ( z , θ , ϕ ) μ d z = Φ k ( z , θ , ϕ ) - 1 4 π × θ = 0 π ϕ = 0 2 π I 1 ( θ , ϕ , θ , ϕ ) Φ k - 1 ( z , θ , ϕ ) × sin θ d θ d ϕ ,
Φ k ( z , θ , ϕ ) = Q k ( z ) I k ( θ , ϕ ) .
- d Q k ( z ) μ d z = Q k ( z ) - Q k - 1 ( z ) ,
I k ( θ , ϕ ) = 1 4 π θ = 0 π ϕ = 0 2 π I 1 ( θ , ϕ , θ , ϕ ) I k - 1 ( θ , ϕ ) × sin θ d θ d ϕ ,
I 1 ( θ ) = n = 0 A n P n ( cos θ ) .
I k ( θ ) = 1 4 π n = 0 ( 2 n + 1 ) B n k P n ( cos θ ) .
Q k ( z ) = ( μ z ) k k ! exp ( - μ z ) ,
Φ ( z , θ ) = k = 1 Q k ( z ) I k ( θ ) = 1 4 π n = 0 ( 2 n + 1 ) { exp [ - μ z ( 1 - B n ) ] - exp ( - μ z ) } P n ( cos θ ) ,
k = 1 ( μ z B n ) k k ! = exp ( μ z B n ) - 1.
I 1 ( θ ) = R = 0 D ( R ) w ( R ) I 1 ( θ , R , m ) d R ,
A n = R = 0 D ( R ) w ( R ) A n ( R ) d R ,
D ( R ) = i = 1 q a i φ i ( R ) ,
A n = i = 1 q a i γ n , i ,
γ n , i = R = 0 w ( R ) φ i ( R ) A n ( R ) d R .
i s ( θ ) = i 0 exp ( - μ d ) Φ ( d , θ ) / sin θ + i 0 ( - d 0 Φ ( z 1 , α ) × Φ ( z 2 , θ - α ) exp { - μ [ ( d - x ) - z 2 ] } d x + 0 d Φ ( z 1 , α ) Φ ( d - x , θ - α ) d x ) / 2 d ,
z 1 = ( d 2 + x 2 + 2 d x cos θ ) 1 / 2 ,
z 2 = ( d 2 - x 2 ) / z 1 ,
cos ( α ) = ( d + x cos θ ) / z 1 ,
cos ( θ - α ) = ( x + d cos θ ) / z 1 .
i s ( θ ) = i s ( θ ) + c r R p exp ( - 2 μ d ) i s ( θ r ) ,
R p = ( n s - n g n s + n g ) 2 ,
y ( θ ) = f ( θ , c ) + ( θ ) ,
y ( θ ) = f ( θ , c 0 ) + i = 1 N f ( θ , c ) / c i c = c 0 ( c i - c 0 i ) + ( θ ) ,
y Δ = A c Δ + ,
y Δ = y ( θ ) - f ( θ , c 0 ) ,
c Δ = c - c 0
A j i = f j ( c ) / c i c = c 0 ,
= A c Δ - y Δ = min ,
( A λ B ) c Δ - ( y Δ λ h ) = min
B = [ - 1 1 0 0 0 - 1 1 0 0 0 - 1 1 ] ,             h = ( 0 0 0 ) ,
G ( c Δ + c 0 ) k ,
G c Δ - G c 0 ,
y ( θ ) = 0 D ( R ) w ( R ) I 1 ( θ , R , m ) d R + ( θ ) ,
D ( R ) = i = 1 q a i φ i ( R ) ,
ψ i ( θ , m ) = 0 φ i ( R ) w ( R ) I 1 ( θ , R , m ) d R .
y ( θ ) = i = 1 q a i ψ i ( θ , m ) + ( θ ) .

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