Abstract

An important step in analyzing data from dynamic light scattering is estimating the relaxation time spectrum from the correlation time function. This estimation is frequently done by regularization methods. To obtain good results with this step, the statistical errors of the correlation time function must be taken into account [J. Phys. A 6, 1897 (1973)]. So far error models assuming independent statistical errors have been used in the estimation. We show that results for the relaxation time spectrum are better if correlation between statistical errors is taken into account. There are two possible ways to obtain the error sizes and their correlations. On the one hand, they can be calculated from the correlation time function by use of a model derived by Schätzel. On the other hand, they can be computed directly from the time series of the scattered light. Simulations demonstrate that the best results are obtained with the latter method. This method requires, however, storing the time series of the scattered light during the experiment. Therefore a modified experimental setup is needed. Nevertheless the simulations also show improvement in the resulting relaxation time spectra if the error model of Schätzel is used. This improvement is confirmed when a lattice with a bimodal sphere size distribution is applied to experimental data.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Schätzel, E. O. Schulz-Du Bois, “Improvements of photon correlation techniques,” Infrared Phys. 32, 409–416 (1991).
    [CrossRef]
  2. K. Schätzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
    [CrossRef]
  3. K. Schätzel, “New concepts in correlator design,” Inst. Phys. Conf. B 77, 175–184 (1985).
  4. K. Schätzel, R. Peters, “Noise on multiple-τ photon correlation data,” Photon Corr. Spectrosc. Multicomponent Syst. 1430, 109–115 (1991).
    [CrossRef]
  5. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  6. D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
    [CrossRef]
  7. I. D. Morrison, E. F. Grabowski, C. A. Herb, “Improved analysis for particle size determination by quasi-elastic light scattering,” Langmuir 4, 496–501 (1985).
    [CrossRef]
  8. R. Finsey, P. de Groen, L. Deriemaker, M. Van Laethem, “Singular value analysis and reconstruction of photon correlation data equidistant in time,” J. Chem. Phys. 91, 7374–7383 (1989).
    [CrossRef]
  9. J. Honerkamp, D. Maier, J. Weese, “A nonlinear regularization method for the analysis of photon correlation spectroscopic data,” J. Chem. Phys. 98, 865–872 (1993).
    [CrossRef]
  10. 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]
  11. S. W. Provencher, “contin: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations,” Comput. Phys. Commun. 27, 229–242 (1982).
    [CrossRef]
  12. V. A. Morozov, Methods for Solving Incorrectly Posed Problems (Springer-Verlag, Berlin, 1984).
    [CrossRef]
  13. K. Schätzel, R. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
    [CrossRef]
  14. K. Schätzel, “Noise on photon correlation data: I. Autocorrelation functions,” Quantum Opt. 2, 287–305 (1990).
    [CrossRef]
  15. C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, London, 1984).
  16. J. Weese, “A regularization method for nonlinear ill-posed problems,” Comput. Phys. Commun. 77, 429–440 (1993).
    [CrossRef]
  17. J. Honerkamp, J. Weese, “Tikhonov’s regularization method for ill-posed problems,” Continuum Mech. Thermodyn. 2, 17–30 (1990).
    [CrossRef]
  18. J. Honerkamp, Stochastic Dynamical Systems (VCH, New York, 1993).
  19. A. Einstein, “Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes,” Ann. Phys. 13, 1275–1298 (1910).
    [CrossRef]
  20. E. Jakeman, “Theory of optical spectroscopy by digital autocorrelation of photon-counting fluctuations,” J. Phys. Gen. Phys. 3, 201–215 (1970).
    [CrossRef]
  21. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  22. Keithley Instruments GmbH, Germering, Germany.

1993 (2)

J. Honerkamp, D. Maier, J. Weese, “A nonlinear regularization method for the analysis of photon correlation spectroscopic data,” J. Chem. Phys. 98, 865–872 (1993).
[CrossRef]

J. Weese, “A regularization method for nonlinear ill-posed problems,” Comput. Phys. Commun. 77, 429–440 (1993).
[CrossRef]

1991 (2)

K. Schätzel, E. O. Schulz-Du Bois, “Improvements of photon correlation techniques,” Infrared Phys. 32, 409–416 (1991).
[CrossRef]

K. Schätzel, R. Peters, “Noise on multiple-τ photon correlation data,” Photon Corr. Spectrosc. Multicomponent Syst. 1430, 109–115 (1991).
[CrossRef]

1990 (2)

J. Honerkamp, J. Weese, “Tikhonov’s regularization method for ill-posed problems,” Continuum Mech. Thermodyn. 2, 17–30 (1990).
[CrossRef]

K. Schätzel, “Noise on photon correlation data: I. Autocorrelation functions,” Quantum Opt. 2, 287–305 (1990).
[CrossRef]

1989 (1)

R. Finsey, P. de Groen, L. Deriemaker, M. Van Laethem, “Singular value analysis and reconstruction of photon correlation data equidistant in time,” J. Chem. Phys. 91, 7374–7383 (1989).
[CrossRef]

1988 (1)

K. Schätzel, R. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
[CrossRef]

1987 (1)

K. Schätzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
[CrossRef]

1985 (2)

K. Schätzel, “New concepts in correlator design,” Inst. Phys. Conf. B 77, 175–184 (1985).

I. D. Morrison, E. F. Grabowski, C. A. Herb, “Improved analysis for particle size determination by quasi-elastic light scattering,” Langmuir 4, 496–501 (1985).
[CrossRef]

1982 (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]

S. W. Provencher, “contin: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations,” Comput. Phys. Commun. 27, 229–242 (1982).
[CrossRef]

1972 (1)

D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

1970 (1)

E. Jakeman, “Theory of optical spectroscopy by digital autocorrelation of photon-counting fluctuations,” J. Phys. Gen. Phys. 3, 201–215 (1970).
[CrossRef]

1910 (1)

A. Einstein, “Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes,” Ann. Phys. 13, 1275–1298 (1910).
[CrossRef]

Berne, B. J.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

de Groen, P.

R. Finsey, P. de Groen, L. Deriemaker, M. Van Laethem, “Singular value analysis and reconstruction of photon correlation data equidistant in time,” J. Chem. Phys. 91, 7374–7383 (1989).
[CrossRef]

Deriemaker, L.

R. Finsey, P. de Groen, L. Deriemaker, M. Van Laethem, “Singular value analysis and reconstruction of photon correlation data equidistant in time,” J. Chem. Phys. 91, 7374–7383 (1989).
[CrossRef]

Drewel, R.

K. Schätzel, R. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
[CrossRef]

Einstein, A.

A. Einstein, “Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes,” Ann. Phys. 13, 1275–1298 (1910).
[CrossRef]

Finsey, R.

R. Finsey, P. de Groen, L. Deriemaker, M. Van Laethem, “Singular value analysis and reconstruction of photon correlation data equidistant in time,” J. Chem. Phys. 91, 7374–7383 (1989).
[CrossRef]

Grabowski, E. F.

I. D. Morrison, E. F. Grabowski, C. A. Herb, “Improved analysis for particle size determination by quasi-elastic light scattering,” Langmuir 4, 496–501 (1985).
[CrossRef]

Groetsch, C. W.

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, London, 1984).

Herb, C. A.

I. D. Morrison, E. F. Grabowski, C. A. Herb, “Improved analysis for particle size determination by quasi-elastic light scattering,” Langmuir 4, 496–501 (1985).
[CrossRef]

Honerkamp, J.

J. Honerkamp, D. Maier, J. Weese, “A nonlinear regularization method for the analysis of photon correlation spectroscopic data,” J. Chem. Phys. 98, 865–872 (1993).
[CrossRef]

J. Honerkamp, J. Weese, “Tikhonov’s regularization method for ill-posed problems,” Continuum Mech. Thermodyn. 2, 17–30 (1990).
[CrossRef]

J. Honerkamp, Stochastic Dynamical Systems (VCH, New York, 1993).

Jakeman, E.

E. Jakeman, “Theory of optical spectroscopy by digital autocorrelation of photon-counting fluctuations,” J. Phys. Gen. Phys. 3, 201–215 (1970).
[CrossRef]

Kerker, M.

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

Koppel, D. E.

D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

Maier, D.

J. Honerkamp, D. Maier, J. Weese, “A nonlinear regularization method for the analysis of photon correlation spectroscopic data,” J. Chem. Phys. 98, 865–872 (1993).
[CrossRef]

Morozov, V. A.

V. A. Morozov, Methods for Solving Incorrectly Posed Problems (Springer-Verlag, Berlin, 1984).
[CrossRef]

Morrison, I. D.

I. D. Morrison, E. F. Grabowski, C. A. Herb, “Improved analysis for particle size determination by quasi-elastic light scattering,” Langmuir 4, 496–501 (1985).
[CrossRef]

Pecora, R.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Peters, R.

K. Schätzel, R. Peters, “Noise on multiple-τ photon correlation data,” Photon Corr. Spectrosc. Multicomponent Syst. 1430, 109–115 (1991).
[CrossRef]

Provencher, S. W.

S. W. Provencher, “contin: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations,” Comput. Phys. Commun. 27, 229–242 (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]

Schätzel, K.

K. Schätzel, R. Peters, “Noise on multiple-τ photon correlation data,” Photon Corr. Spectrosc. Multicomponent Syst. 1430, 109–115 (1991).
[CrossRef]

K. Schätzel, E. O. Schulz-Du Bois, “Improvements of photon correlation techniques,” Infrared Phys. 32, 409–416 (1991).
[CrossRef]

K. Schätzel, “Noise on photon correlation data: I. Autocorrelation functions,” Quantum Opt. 2, 287–305 (1990).
[CrossRef]

K. Schätzel, R. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
[CrossRef]

K. Schätzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
[CrossRef]

K. Schätzel, “New concepts in correlator design,” Inst. Phys. Conf. B 77, 175–184 (1985).

Schulz-Du Bois, E. O.

K. Schätzel, E. O. Schulz-Du Bois, “Improvements of photon correlation techniques,” Infrared Phys. 32, 409–416 (1991).
[CrossRef]

Stimac, S.

K. Schätzel, R. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
[CrossRef]

Van Laethem, M.

R. Finsey, P. de Groen, L. Deriemaker, M. Van Laethem, “Singular value analysis and reconstruction of photon correlation data equidistant in time,” J. Chem. Phys. 91, 7374–7383 (1989).
[CrossRef]

Weese, J.

J. Honerkamp, D. Maier, J. Weese, “A nonlinear regularization method for the analysis of photon correlation spectroscopic data,” J. Chem. Phys. 98, 865–872 (1993).
[CrossRef]

J. Weese, “A regularization method for nonlinear ill-posed problems,” Comput. Phys. Commun. 77, 429–440 (1993).
[CrossRef]

J. Honerkamp, J. Weese, “Tikhonov’s regularization method for ill-posed problems,” Continuum Mech. Thermodyn. 2, 17–30 (1990).
[CrossRef]

Ann. Phys. (1)

A. Einstein, “Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes,” Ann. Phys. 13, 1275–1298 (1910).
[CrossRef]

Appl. Phys. B (1)

K. Schätzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
[CrossRef]

Comput. Phys. Commun. (3)

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]

S. W. Provencher, “contin: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations,” Comput. Phys. Commun. 27, 229–242 (1982).
[CrossRef]

J. Weese, “A regularization method for nonlinear ill-posed problems,” Comput. Phys. Commun. 77, 429–440 (1993).
[CrossRef]

Continuum Mech. Thermodyn. (1)

J. Honerkamp, J. Weese, “Tikhonov’s regularization method for ill-posed problems,” Continuum Mech. Thermodyn. 2, 17–30 (1990).
[CrossRef]

Infrared Phys. (1)

K. Schätzel, E. O. Schulz-Du Bois, “Improvements of photon correlation techniques,” Infrared Phys. 32, 409–416 (1991).
[CrossRef]

Inst. Phys. Conf. B (1)

K. Schätzel, “New concepts in correlator design,” Inst. Phys. Conf. B 77, 175–184 (1985).

J. Chem. Phys. (3)

D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

R. Finsey, P. de Groen, L. Deriemaker, M. Van Laethem, “Singular value analysis and reconstruction of photon correlation data equidistant in time,” J. Chem. Phys. 91, 7374–7383 (1989).
[CrossRef]

J. Honerkamp, D. Maier, J. Weese, “A nonlinear regularization method for the analysis of photon correlation spectroscopic data,” J. Chem. Phys. 98, 865–872 (1993).
[CrossRef]

J. Mod. Opt. (1)

K. Schätzel, R. Drewel, S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
[CrossRef]

J. Phys. Gen. Phys. (1)

E. Jakeman, “Theory of optical spectroscopy by digital autocorrelation of photon-counting fluctuations,” J. Phys. Gen. Phys. 3, 201–215 (1970).
[CrossRef]

Langmuir (1)

I. D. Morrison, E. F. Grabowski, C. A. Herb, “Improved analysis for particle size determination by quasi-elastic light scattering,” Langmuir 4, 496–501 (1985).
[CrossRef]

Photon Corr. Spectrosc. Multicomponent Syst. (1)

K. Schätzel, R. Peters, “Noise on multiple-τ photon correlation data,” Photon Corr. Spectrosc. Multicomponent Syst. 1430, 109–115 (1991).
[CrossRef]

Quantum Opt. (1)

K. Schätzel, “Noise on photon correlation data: I. Autocorrelation functions,” Quantum Opt. 2, 287–305 (1990).
[CrossRef]

Other (6)

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, London, 1984).

V. A. Morozov, Methods for Solving Incorrectly Posed Problems (Springer-Verlag, Berlin, 1984).
[CrossRef]

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

Keithley Instruments GmbH, Germering, Germany.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

J. Honerkamp, Stochastic Dynamical Systems (VCH, New York, 1993).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Experimental setup of the DLS experiment.

Fig. 2
Fig. 2

Given discrete values h i of the relaxation time spectrum used to simulate data.

Fig. 3
Fig. 3

Given relaxation time spectrum h(ln α) (solid curve) and the estimated spectrum (error bars) for the time series with M = 5 × 105: (a) conventional method; (b), (c) spectra estimated from the new method from the same time series. In (b) the covariance matrix is estimated with the model of Schätzel, and in (c) it is directly estimated from the time series of the scattered photons. The dashed line marks the limit of the accessible range.

Fig. 4
Fig. 4

Given relaxation time spectrum h(ln α) (solid curve) and the estimated spectrum (error bars) for the time series with M = 106: (a) conventional method; (b), (c) spectra estimated with the new method from the same time series. In (b) the covariance matrix is estimated with the model of Schätzel, and in (c) it is directly estimated from the time series of the scattered photons. The dashed line marks the limit of the accessible range.

Fig. 5
Fig. 5

Given relaxation time spectrum h(ln α) (solid curve) and the estimated spectrum (error bars) for the time series with M = 5 × 106: (a) conventional method; (b), (c) spectra estimated with the new method from the same time series. In (b) the covariance matrix is estimated with the model of Schätzel, and in (c) it is directly estimated from the time series of the scattered photons. The dashed line marks the limit of the accessible range.

Fig. 6
Fig. 6

Approximate sphere size distribution p(R) for the experimental data from Dow Chemical Company.

Fig. 7
Fig. 7

DLS data for θ = 90° of the bimodal lattice.

Fig. 8
Fig. 8

Estimated relaxation time spectrum h(ln α) for the light-scattering data of Fig. 7: (a), conventional method; (b) new method. The covariance matrix is estimated from the Schätzel model.

Fig. 9
Fig. 9

Estimated sphere size distribution p(R) obtained from the light-scattering data of Fig. 7: (a) conventional method; (b) new method. The straight line shows the approximate distribution given by Dow Chemical Company.

Fig. 10
Fig. 10

Proposal for a new experimental setup that can store the measured time series.

Tables (1)

Tables Icon

Table 1 Determined Mean Values According to Eq. (21) for the Old and the New Methods

Equations (24)

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

q=|q|=4πn/λ sinθ/2,
α=1/Dq2,
gτ=ntstntst+τn2-1.
gτ=β- exp-τ/αhln αdln α2,
gˆτk=1/M-1j=1M-kntsti-nˆntsti+τk-nˆnˆ2
nˆ=1/M j=1M ntstj.
gˆτk=j=k+1M ntstjntstj-τkM j=1M ntstjj=k+1M ntstj-1,
gˆτk=j=k+1M ntstjntstj-τkMj=1M ntstj2-j=1+kM ntstjj=1M ntstj.
covgˆτk, gˆτl=M-1i=-NN gτigτi+k-l+gτi+k+l+2βˆgτkgτl1/2i=-NNgτi1/2×gτi+k-l1/2+gτi+k+l1/2+2βˆ i=-NNgτigτi+kgτi+lgτi+k+l1/2+4gτkgτli=-NN gτi-4βˆ1/2gτl×gτk1/2i=-NNgτigτi+k1/2-4βˆ1/2gτkgτl1/2i=-NNgτigτi+l1/2+2nˆ-1gτk-l+gτk+l-2gτkgτl+2βˆ1/2gτkgτlgτk-l1/2+2βˆ1/2gτkgτlgτk+l1/2+nˆ-2δkl1+gτk.
βˆ=linτ0 gˆτ.
covgˆτk, gˆτl=M-2nˆ-4×i,j=1Mntstintsti+τkntstjntstj+τl-sˆτksˆτl-2sˆτlnˆi,j=1M×ntstintsti+τkntstj-nˆsˆτk-2sˆτknˆi,j=1Mntstintsti+τlntstj-nˆsˆτl+4sˆτksˆτlnˆ2×i,j=1Mntstintstj-nˆ2
sˆτk=1/M i=1M ntstintsti+τk.
covgˆτk, gˆτl=M-1nˆ-4i=-NN×1jmaxj=1jmax ntstj+N+1ntstj+N+1+τknts×ti+j+N+1ntsti+j+N+1+τl-sˆτksˆτl-2sˆτlnˆi=-NN1jmaxj=1jmax ntstj+N+1nts×tj+N+1+τkntsti+j+N+1-nˆsˆτk-2sˆτknˆi=-NN1jmaxj=1jmax ntstj+N+1nts×tj+N+1+τlntsti+j+N+1-nˆsˆτl+4sˆτksˆτlnˆ2i=-NN1jmaxj=1jmax ntstj+N+1nts×ti+j+N+1-nˆ2.
Vλ=k=1Pgˆτk-βˆ-dln αexp-τk/αhln α2σk2+λ -dln αhln α2
Vλ=k,l=1Pgˆτk-βˆ-dln αexp-τk/αhln α2×cov-1gˆτk, gˆτl×gˆτl-βˆ-dln αexp-τl/αhln α2+λ -dln αhln α2
r˙it=2αi q ηt.
hi=k=12 akexp-lnαi-αk2/2σk22π σk.
et=i=1100hi expiqrit.
pntst|It=exp-γITsγItsnn!.
R=8πkTn2 sin2θ/23λ2η α=const α.
Ri=constpeaki αhln αdln α.
R1total=58.7±2 nm,  R2total=100.6±4 nm.
R1given=63.5±3.15 nm,  R2given=96.5±3 nm.
pi=hi/Ri6.

Metrics