Abstract

Optical particle sizing in the range of 10 nm up to several micrometers by means of quasi-elastic and elastic light scattering requires sophisticated data inversion techniques. We have developed an optimized regularization technique that can be used for the inversion of such light-scattering data. The technique has been successfully tested for a large number of simulated and measured data. It is easy to handle. Typical problems that arise in practical applications are discussed.

© 1991 Optical Society of America

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References

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  1. B. E. Dahneke, Measurement of Suspended Particles by Quasi-Elastic Light Scattering (Wiley, New York, 1983).
  2. S. W. Provencher, “A constrained regularization method for inverting data represented by linear algebraic or integral equations,” Comput. Phys. Commun. 27, 213–227 (1982).
    [CrossRef]
  3. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
    [CrossRef]
  4. S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
    [CrossRef]
  5. O. Glatter, H. 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. Charac. (to be published).
  6. C. de Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978).
    [CrossRef]
  7. T. N. E. Greville, Theory and Application of Spline Functions (Academic, New York, 1969).
  8. 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]
  9. H. Greschonig, O. Glatter, “Determination of equivalence points of sigmoidal potentiometric titration curves,” Microchem. Acta 2, 389–399 (1986).
  10. O. Glatter, M. Hofer, “Interpretation of elastic light scattering data. III. Determination of size distributions of polydisperse systems,” J. Colloid Interface Sci. 122, 496–506 (1988).
    [CrossRef]
  11. P. W. Barber, S. C. Hill, Computational Light Scattering (World Scientific, Singapore, 1990).
  12. E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” in Optical Particle SizingTheory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).
  13. O. Glatter, “Data evaluation in small angle scattering: Calculation of the radial electron density distribution by means of indirect Fourier transformation,” Acta Phys. Austriaca 47, 83–102 (1977).
  14. O. Glatter, “A new method for the evaluation of small-angle scattering data,” J. Appl. Crystallogr. 10, 415–421 (1977).
    [CrossRef]
  15. O. Glatter, “Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method,” J. Appl. Crystallogr. 13, 7–11 (1980).
    [CrossRef]
  16. O. Glatter, “Data treatment,” in Small Angle X-Ray Scattering (Academic, New York, 1982), Chap. 4.
  17. C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  18. J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
    [CrossRef]
  19. M. Bertero, P. Boccacci, C. De Mol, E. R. Pike, “Extraction of polydispersity information in photon correlation spectroscopy,” in Optical Particle Sizing, Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).
  20. K. Müller, O. Glatter, “Practical aspects to the use of indirect Fourier transformation methods,” Makromol. Chem. 183, 465–479 (1982).
    [CrossRef]
  21. J. L. Shi, J. H. Gao, Z. X. Lin, “Formation of monosized spherical aluminum hydroxide particles by urea method,” Solid State Ionics 32/33, 537–543 (1989).
    [CrossRef]
  22. R. C. Weast, Handbook of Chemistry and Physics (Chemical Rubber, Cleveland, 1982).

1989

J. L. Shi, J. H. Gao, Z. X. Lin, “Formation of monosized spherical aluminum hydroxide particles by urea method,” Solid State Ionics 32/33, 537–543 (1989).
[CrossRef]

1988

O. Glatter, M. Hofer, “Interpretation of elastic light scattering data. III. Determination of size distributions of polydisperse systems,” J. Colloid Interface Sci. 122, 496–506 (1988).
[CrossRef]

1986

H. Greschonig, O. Glatter, “Determination of equivalence points of sigmoidal potentiometric titration curves,” Microchem. Acta 2, 389–399 (1986).

1982

K. Müller, O. Glatter, “Practical aspects to the use of indirect Fourier transformation methods,” Makromol. Chem. 183, 465–479 (1982).
[CrossRef]

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

1980

O. Glatter, “Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method,” J. Appl. Crystallogr. 13, 7–11 (1980).
[CrossRef]

1978

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

1977

O. Glatter, “Data evaluation in small angle scattering: Calculation of the radial electron density distribution by means of indirect Fourier transformation,” Acta Phys. Austriaca 47, 83–102 (1977).

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

1971

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]

1963

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

1962

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

Barber, P. W.

P. W. Barber, S. C. Hill, Computational Light Scattering (World Scientific, Singapore, 1990).

Bertero, M.

M. Bertero, P. Boccacci, C. De Mol, E. R. Pike, “Extraction of polydispersity information in photon correlation spectroscopy,” in Optical Particle Sizing, Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

Boccacci, P.

M. Bertero, P. Boccacci, C. De Mol, E. R. Pike, “Extraction of polydispersity information in photon correlation spectroscopy,” in Optical Particle Sizing, Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

Dahneke, B. E.

B. E. Dahneke, Measurement of Suspended Particles by Quasi-Elastic Light Scattering (Wiley, New York, 1983).

de Boor, C.

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

De Mol, C.

M. Bertero, P. Boccacci, C. De Mol, E. R. Pike, “Extraction of polydispersity information in photon correlation spectroscopy,” in Optical Particle Sizing, Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

Gao, J. H.

J. L. Shi, J. H. Gao, Z. X. Lin, “Formation of monosized spherical aluminum hydroxide particles by urea method,” Solid State Ionics 32/33, 537–543 (1989).
[CrossRef]

Glatter, O.

O. Glatter, M. Hofer, “Interpretation of elastic light scattering data. III. Determination of size distributions of polydisperse systems,” J. Colloid Interface Sci. 122, 496–506 (1988).
[CrossRef]

H. Greschonig, O. Glatter, “Determination of equivalence points of sigmoidal potentiometric titration curves,” Microchem. Acta 2, 389–399 (1986).

K. Müller, O. Glatter, “Practical aspects to the use of indirect Fourier transformation methods,” Makromol. Chem. 183, 465–479 (1982).
[CrossRef]

O. Glatter, “Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method,” J. Appl. Crystallogr. 13, 7–11 (1980).
[CrossRef]

O. Glatter, “Data evaluation in small angle scattering: Calculation of the radial electron density distribution by means of indirect Fourier transformation,” Acta Phys. Austriaca 47, 83–102 (1977).

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

O. Glatter, “Data treatment,” in Small Angle X-Ray Scattering (Academic, New York, 1982), Chap. 4.

O. Glatter, H. 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. Charac. (to be published).

Greschonig, H.

H. Greschonig, O. Glatter, “Determination of equivalence points of sigmoidal potentiometric titration curves,” Microchem. Acta 2, 389–399 (1986).

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).

Hill, S. C.

P. W. Barber, S. C. Hill, Computational Light Scattering (World Scientific, Singapore, 1990).

Hirleman, E. D.

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” in Optical Particle SizingTheory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

Hofer, M.

O. Glatter, M. Hofer, “Interpretation of elastic light scattering data. III. Determination of size distributions of polydisperse systems,” J. Colloid Interface Sci. 122, 496–506 (1988).
[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]

Lawson, C. L.

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

Lin, Z. X.

J. L. Shi, J. H. Gao, Z. X. Lin, “Formation of monosized spherical aluminum hydroxide particles by urea method,” Solid State Ionics 32/33, 537–543 (1989).
[CrossRef]

McWhirter, J. G.

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Müller, K.

K. Müller, O. Glatter, “Practical aspects to the use of indirect Fourier transformation methods,” Makromol. Chem. 183, 465–479 (1982).
[CrossRef]

Phillips, D. L.

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

Pike, E. R.

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

M. Bertero, P. Boccacci, C. De Mol, E. R. Pike, “Extraction of polydispersity information in photon correlation spectroscopy,” in Optical Particle Sizing, Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

Provencher, S. W.

S. W. Provencher, “A constrained regularization method for inverting data represented by linear algebraic or integral equations,” 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.

O. Glatter, H. 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. Charac. (to be published).

Shi, J. L.

J. L. Shi, J. H. Gao, Z. X. Lin, “Formation of monosized spherical aluminum hydroxide particles by urea method,” Solid State Ionics 32/33, 537–543 (1989).
[CrossRef]

Sieberer, H.

O. Glatter, H. 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. Charac. (to be published).

Twomey, S.

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

Weast, R. C.

R. C. Weast, Handbook of Chemistry and Physics (Chemical Rubber, Cleveland, 1982).

Acta Phys. Austriaca

O. Glatter, “Data evaluation in small angle scattering: Calculation of the radial electron density distribution by means of indirect Fourier transformation,” Acta Phys. Austriaca 47, 83–102 (1977).

Comput. Phys. Commun.

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

J. Appl. Crystallogr.

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]

O. Glatter, “Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method,” J. Appl. Crystallogr. 13, 7–11 (1980).
[CrossRef]

J. Assoc. Comput. Mach.

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

J. Colloid Interface Sci.

O. Glatter, M. Hofer, “Interpretation of elastic light scattering data. III. Determination of size distributions of polydisperse systems,” J. Colloid Interface Sci. 122, 496–506 (1988).
[CrossRef]

J. Phys. A

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Makromol. Chem.

K. Müller, O. Glatter, “Practical aspects to the use of indirect Fourier transformation methods,” Makromol. Chem. 183, 465–479 (1982).
[CrossRef]

Microchem. Acta

H. Greschonig, O. Glatter, “Determination of equivalence points of sigmoidal potentiometric titration curves,” Microchem. Acta 2, 389–399 (1986).

Solid State Ionics 32/33

J. L. Shi, J. H. Gao, Z. X. Lin, “Formation of monosized spherical aluminum hydroxide particles by urea method,” Solid State Ionics 32/33, 537–543 (1989).
[CrossRef]

Other

R. C. Weast, Handbook of Chemistry and Physics (Chemical Rubber, Cleveland, 1982).

M. Bertero, P. Boccacci, C. De Mol, E. R. Pike, “Extraction of polydispersity information in photon correlation spectroscopy,” in Optical Particle Sizing, Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

O. Glatter, “Data treatment,” in Small Angle X-Ray Scattering (Academic, New York, 1982), Chap. 4.

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

P. W. Barber, S. C. Hill, Computational Light Scattering (World Scientific, Singapore, 1990).

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” in Optical Particle SizingTheory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

O. Glatter, H. 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. Charac. (to be published).

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

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

B. E. Dahneke, Measurement of Suspended Particles by Quasi-Elastic Light Scattering (Wiley, New York, 1983).

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Figures (8)

Fig. 1
Fig. 1

Stability plot for finding the correct Lagrange multiplier. The correct Lagrange multiplier λ can be found from the point of inflection of the logarithmic norm Fc (circles) of the first derivative of the solution. The first derivative Fc has a minimum, and the second derivative Fc has a zero at this point. The occurrence of this point of inflection must be in the range where the mean deviation (triangles) is still at a minimum value.

Fig. 2
Fig. 2

Inverse Laplace transforms of correlation functions and their dependence on the Lagrange multiplier. The region of stable solutions is rather small. The correct solution is marked as a bold solid line.

Fig. 3
Fig. 3

Volume and intensity distribution: Even a large volume fraction of small particles (a) can make a small contribution to the signal (b). The solid curves are the given functions, and the pluses symbolize their reconstruction. In many cases it is important to distinguish small particles in the presence of large ones. The fit and the residuals for the volume distribution of simulated data are shown in (c) and (d), respectively.

Fig. 4
Fig. 4

Variation of the base line. The minimum of the combined condition (the sum of the least-squares plus squared norm Nc times the Lagrange multiplier) shows the best apparent base line to be subtracted from g2(t). Taking this base line minimizes the systematic error between data and fit.

Fig. 5
Fig. 5

Inverse Laplace transformation of the data of low quality. The volume distribution function of precipitating calcium oxalate was calculated with (a) the apparent base line (pluses, Rmax = 104 nm) and (b) the measured far point (solid curve, Rmax = 105 nm; pluses, Rmax = 104 nm). The existence of an unavoidable amount of systematic error is not clear from the fit (c) but is more pronounced in the residuals (d).

Fig. 6
Fig. 6

Effect of the constraint of positivity on the stability plot. An inverse Laplace transformation was performed to reconstruct a theoretical bimodal distribution with (a) 22 and (b) 35 splines. The triangles are the mean deviations, and the circles are the logarithmic norm Fc. A small number of splines deactivate the constraint of the minimum first derivative of the solution. The result is one plateau where the activity of both constraints coincides. A point of inflection is found for only 35 splines.

Fig. 7
Fig. 7

Regularization technique applied to the inversion of elastic light-scattering data in the Lorenz–Mie regime. The solid curve is the given function, and the pluses are its reconstruction from simulated data.

Fig. 8
Fig. 8

Inverse Lorenz–Mie transformation of experimental data: (a) the volume distribution function of a suspension of aluminum oxide hydrate and (b) the fit (solid curve) to the measured data (pluses).

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

g 2 ( t ) = B + C g 1 ( t ) 2 ,
g 1 ( t ) = Γ min Γ max G ( Γ ) exp ( Γ t ) d Γ .
Γ = D h 2 ,
h = ( 4 π n / λ 0 ) sin ( ϴ / 2 ) ,
D = k T / 6 π η R H ,
g 1 ( t ) = τ min τ max D ( τ ) W ( τ ) exp ( t / τ ) / τ 2 d τ .
I ( h ) = R min R max D ( R ) W ( R ) Φ ( h , R , m ) d R ,
Φ ( h , R ) = [ 3 sin ( h R ) ( h R ) cos ( h R ) ( h R ) 3 ] 2 .
I ( h ) = R min R max D ( R ) J 1 2 ( 2 π n R ϴ / λ 0 ) ( R / ϴ ) 2 d R .
D ( R ) = i = 1 n c i φ i ( R ) .
ψ i ( x ) = A ( x , R ) φ i ( R ) d R ,
f ( x ) = i = 1 n c i ψ i ( x ) ,
N c = [ i = 1 n 1 ( c i + 1 c i ) 2 ] 1 / 2 ,

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