Abstract

Dynamic light scattering (DLS) from colloidal particles often contains noise, which makes inversion of the correlation function to obtain the particle size distribution (PSD) unreliable. In this work, poor-quality correlation function data with baseline error were analyzed using constrained regularization techniques. The effect of baseline error was investigated, and two strategies were proposed to compensate for baseline error. One strategy is based on edge proportion detection of spurious peaks at large size in the PSD, and the other is based on the solution norm. Results from simulated and experimental data demonstrate the effectiveness of our proposed strategies. The L-curve rules for standard Tikhonov and for constrained regularization, the generalized cross-validation (GCV) rule, and the robust GCV rule were investigated for determination of the regularization parameter. A comparison of these rules was done using both simulated and experimental data. It is shown that correction of baseline error with baseline compensation as well as a reasonable regularization parameter choice improves the accuracy of PSD recovery in poor-quality DLS data analysis.

© 2012 Optical Society of America

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  1. R. Pecora, Dynamic Light Scattering: Application of Photon Correlation Spectroscopy (Plenum, 1985).
  2. F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (ACS, 2004).
  3. J. C. Thomas, “Photon correlation spectroscopy: technique and instrumentation,” Proc. SPIE 1430, 2–18 (1991).
    [CrossRef]
  4. Y. Li, V. Lubchenko, and P. G. Vekilov, “The use of dynamic light scattering and Brownian microscopy to characterize protein aggregation,” Rev. Sci. Instrum. 82, 053106(2011).
    [CrossRef]
  5. 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]
  6. D. A. Ross and H. S. Dhadwal, “Regularized inversion of the Laplace transform: accuracy of analytical and discrete inversion,” Part. Part. Syst. Charact. 8, 282–286 (1991).
    [CrossRef]
  7. H. Schnablegger and O. Glatter, “Optical sizing of small colloidal particles: an optimized regularization technique,” Appl. Opt. 30, 4889–4896 (1991).
    [CrossRef]
  8. S. W. Provencher and P. Stepanek, “Global analysis of dynamic light scattering autocorrelation functions,” Part. Part. Syst. Charact. 13, 291–294 (1996).
    [CrossRef]
  9. M. J. Fernandes, N. C. Santos, and M. Castanho, “Continuous particle size distribution analysis with dynamic light scattering: MAXAMPER: a regularization method using the maximum amplitude for the average error and the Lagrange’s multipliers method,” J. Biochem. Biophys. Methods 36, 101–117 (1998).
    [CrossRef]
  10. V. M. Gun’ko, A. V. Klyueva, Y. N. Levchuk, and R. Leboda, “Photon correlation spectroscopy investigations of proteins,” Adv. Colloid Interface Sci. 105, 201–328 (2003).
    [CrossRef]
  11. X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
    [CrossRef]
  12. X. Zhu, J. Shen, W. Liu, X. Sun, and Y. Wang, “Nonnegative least-squares truncated singular value decomposition to particle size distribution inversion from dynamic light scattering data,” Appl. Opt. 49, 6591–6596 (2010).
    [CrossRef]
  13. A. E. Smart, R. V. Edwards, and W. V. Meyer, “Quantitative simulation of errors in correlation analysis,” Appl. Opt. 40, 4064–4078 (2001).
    [CrossRef]
  14. H. Ruf, “Effects of normalization errors on size distributions obtained from dynamic light scattering data,” Biophys. J. 56, 67–78 (1989).
    [CrossRef]
  15. B. B. Weiner and W. W. Tscharnuter, Uses and Abuses of Photon Correlation Spectroscopy in Particle Sizing (ACS, 1987).
  16. H. Ruf, B. Gould, and W. Haase, “The effect of nonrandom errors on the results from regularized inversions of dynamic light scattering data,” Langmuir 16, 471–480(2000).
    [CrossRef]
  17. H. Ruf, “Treatment of contributions of dust to dynamic light scattering data,” Langmuir 18, 3804–3814 (2002).
    [CrossRef]
  18. D. A. Lowther, R. D. Throne, L. G. Olson, and J. R. Windle, “A comparison of two methods for choosing the regularization parameter for the inverse problem of electrocardiography,” Biomed. Sci. Instrum. 38, 257–261 (2002).
  19. C. G. Farquharson and D. W. Oldenburg, “A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems,” Geophys. J. Int. 156, 411–425 (2004).
    [CrossRef]
  20. H. G. Choi, A. N. Thite, and D. J. Thompson, “Comparison of methods for parameter selection in Tikhonov regularization with application to inverse force determination,” J. Sound Vib. 304, 894–917 (2007).
    [CrossRef]
  21. F. Bauer and M. A. Lukas, “Comparing parameter choice methods for regularization of ill-posed problems,” Math. Comput. Simul. 81, 1795–1841 (2011).
    [CrossRef]
  22. K. Schatzel, M. Drewel, and S. Stimac, “Photon correlation measurements at large lag times: improving statistical accuracy,” J. Mod. Opt. 35, 711–718 (1988).
    [CrossRef]
  23. P. Stepanek, Z. Tuzar, P. Kadlec, and J. Kriz, “A dynamic light scattering study of fast relaxations in polymer solutions,” Macromolecules 40, 2165–2171 (2007).
    [CrossRef]
  24. K. Sumitomo, K. Mayumi, and H. Yokoyama, “Dynamic light scattering measurement of sieving polymer solutions for protein separation on SDS CE,” Electrophoresis 30, 3607–3612 (2009).
    [CrossRef]
  25. T. Roths and J. Honerkamp, “Simultaneous regularization method for the determination of radius distributions from experimental multiangle correlation functions,” Phys. Rev. E 64, 041404 (2001).
    [CrossRef]
  26. P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
    [CrossRef]
  27. G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
    [CrossRef]
  28. M. A. Lukas, “Robust generalized cross-validation for choosing the regularization parameter,” Inverse Probl. 22, 1883–1902 (2006).
    [CrossRef]
  29. P. C. Hansen, “Regularization Tools Version 4.0 for Matlab 7.3,” Numer. Algorithms 46, 189–194 (2007).
    [CrossRef]
  30. A. B. Yu and N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
    [CrossRef]
  31. Y. Sun and J. Walker, “Maximum likelihood data inversion for photon correlation spectroscopy,” Meas. Sci. Technol. 19, 115302 (2008).
    [CrossRef]
  32. L. A. Clementi, J. R. Vega, L. M. Gugliotta, and H. Orlande, “A Bayesian inversion method for estimating the particle size distribution of latexes from multiangle dynamic light scattering measurements,” Chemom. Intell. Lab. Syst. 107, 165–173 (2011).
    [CrossRef]

2011

F. Bauer and M. A. Lukas, “Comparing parameter choice methods for regularization of ill-posed problems,” Math. Comput. Simul. 81, 1795–1841 (2011).
[CrossRef]

L. A. Clementi, J. R. Vega, L. M. Gugliotta, and H. Orlande, “A Bayesian inversion method for estimating the particle size distribution of latexes from multiangle dynamic light scattering measurements,” Chemom. Intell. Lab. Syst. 107, 165–173 (2011).
[CrossRef]

Y. Li, V. Lubchenko, and P. G. Vekilov, “The use of dynamic light scattering and Brownian microscopy to characterize protein aggregation,” Rev. Sci. Instrum. 82, 053106(2011).
[CrossRef]

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
[CrossRef]

2010

2009

K. Sumitomo, K. Mayumi, and H. Yokoyama, “Dynamic light scattering measurement of sieving polymer solutions for protein separation on SDS CE,” Electrophoresis 30, 3607–3612 (2009).
[CrossRef]

2008

Y. Sun and J. Walker, “Maximum likelihood data inversion for photon correlation spectroscopy,” Meas. Sci. Technol. 19, 115302 (2008).
[CrossRef]

2007

P. Stepanek, Z. Tuzar, P. Kadlec, and J. Kriz, “A dynamic light scattering study of fast relaxations in polymer solutions,” Macromolecules 40, 2165–2171 (2007).
[CrossRef]

P. C. Hansen, “Regularization Tools Version 4.0 for Matlab 7.3,” Numer. Algorithms 46, 189–194 (2007).
[CrossRef]

H. G. Choi, A. N. Thite, and D. J. Thompson, “Comparison of methods for parameter selection in Tikhonov regularization with application to inverse force determination,” J. Sound Vib. 304, 894–917 (2007).
[CrossRef]

2006

M. A. Lukas, “Robust generalized cross-validation for choosing the regularization parameter,” Inverse Probl. 22, 1883–1902 (2006).
[CrossRef]

2004

C. G. Farquharson and D. W. Oldenburg, “A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems,” Geophys. J. Int. 156, 411–425 (2004).
[CrossRef]

2003

V. M. Gun’ko, A. V. Klyueva, Y. N. Levchuk, and R. Leboda, “Photon correlation spectroscopy investigations of proteins,” Adv. Colloid Interface Sci. 105, 201–328 (2003).
[CrossRef]

2002

H. Ruf, “Treatment of contributions of dust to dynamic light scattering data,” Langmuir 18, 3804–3814 (2002).
[CrossRef]

D. A. Lowther, R. D. Throne, L. G. Olson, and J. R. Windle, “A comparison of two methods for choosing the regularization parameter for the inverse problem of electrocardiography,” Biomed. Sci. Instrum. 38, 257–261 (2002).

2001

T. Roths and J. Honerkamp, “Simultaneous regularization method for the determination of radius distributions from experimental multiangle correlation functions,” Phys. Rev. E 64, 041404 (2001).
[CrossRef]

A. E. Smart, R. V. Edwards, and W. V. Meyer, “Quantitative simulation of errors in correlation analysis,” Appl. Opt. 40, 4064–4078 (2001).
[CrossRef]

2000

H. Ruf, B. Gould, and W. Haase, “The effect of nonrandom errors on the results from regularized inversions of dynamic light scattering data,” Langmuir 16, 471–480(2000).
[CrossRef]

1998

M. J. Fernandes, N. C. Santos, and M. Castanho, “Continuous particle size distribution analysis with dynamic light scattering: MAXAMPER: a regularization method using the maximum amplitude for the average error and the Lagrange’s multipliers method,” J. Biochem. Biophys. Methods 36, 101–117 (1998).
[CrossRef]

1996

S. W. Provencher and P. Stepanek, “Global analysis of dynamic light scattering autocorrelation functions,” Part. Part. Syst. Charact. 13, 291–294 (1996).
[CrossRef]

1993

P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

1991

D. A. Ross and H. S. Dhadwal, “Regularized inversion of the Laplace transform: accuracy of analytical and discrete inversion,” Part. Part. Syst. Charact. 8, 282–286 (1991).
[CrossRef]

J. C. Thomas, “Photon correlation spectroscopy: technique and instrumentation,” Proc. SPIE 1430, 2–18 (1991).
[CrossRef]

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

1990

A. B. Yu and N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
[CrossRef]

1989

H. Ruf, “Effects of normalization errors on size distributions obtained from dynamic light scattering data,” Biophys. J. 56, 67–78 (1989).
[CrossRef]

1988

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

1982

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]

1979

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Bauer, F.

F. Bauer and M. A. Lukas, “Comparing parameter choice methods for regularization of ill-posed problems,” Math. Comput. Simul. 81, 1795–1841 (2011).
[CrossRef]

Castanho, M.

M. J. Fernandes, N. C. Santos, and M. Castanho, “Continuous particle size distribution analysis with dynamic light scattering: MAXAMPER: a regularization method using the maximum amplitude for the average error and the Lagrange’s multipliers method,” J. Biochem. Biophys. Methods 36, 101–117 (1998).
[CrossRef]

Choi, H. G.

H. G. Choi, A. N. Thite, and D. J. Thompson, “Comparison of methods for parameter selection in Tikhonov regularization with application to inverse force determination,” J. Sound Vib. 304, 894–917 (2007).
[CrossRef]

Clementi, L. A.

L. A. Clementi, J. R. Vega, L. M. Gugliotta, and H. Orlande, “A Bayesian inversion method for estimating the particle size distribution of latexes from multiangle dynamic light scattering measurements,” Chemom. Intell. Lab. Syst. 107, 165–173 (2011).
[CrossRef]

Crassous, J.

F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (ACS, 2004).

Dhadwal, H. S.

D. A. Ross and H. S. Dhadwal, “Regularized inversion of the Laplace transform: accuracy of analytical and discrete inversion,” Part. Part. Syst. Charact. 8, 282–286 (1991).
[CrossRef]

Drewel, M.

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

Edwards, R. V.

Farquharson, C. G.

C. G. Farquharson and D. W. Oldenburg, “A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems,” Geophys. J. Int. 156, 411–425 (2004).
[CrossRef]

Fernandes, M. J.

M. J. Fernandes, N. C. Santos, and M. Castanho, “Continuous particle size distribution analysis with dynamic light scattering: MAXAMPER: a regularization method using the maximum amplitude for the average error and the Lagrange’s multipliers method,” J. Biochem. Biophys. Methods 36, 101–117 (1998).
[CrossRef]

Glatter, O.

Golub, G. H.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Gould, B.

H. Ruf, B. Gould, and W. Haase, “The effect of nonrandom errors on the results from regularized inversions of dynamic light scattering data,” Langmuir 16, 471–480(2000).
[CrossRef]

Guan, J.

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
[CrossRef]

Gugliotta, L. M.

L. A. Clementi, J. R. Vega, L. M. Gugliotta, and H. Orlande, “A Bayesian inversion method for estimating the particle size distribution of latexes from multiangle dynamic light scattering measurements,” Chemom. Intell. Lab. Syst. 107, 165–173 (2011).
[CrossRef]

Gun’ko, V. M.

V. M. Gun’ko, A. V. Klyueva, Y. N. Levchuk, and R. Leboda, “Photon correlation spectroscopy investigations of proteins,” Adv. Colloid Interface Sci. 105, 201–328 (2003).
[CrossRef]

Haase, W.

H. Ruf, B. Gould, and W. Haase, “The effect of nonrandom errors on the results from regularized inversions of dynamic light scattering data,” Langmuir 16, 471–480(2000).
[CrossRef]

Hansen, P. C.

P. C. Hansen, “Regularization Tools Version 4.0 for Matlab 7.3,” Numer. Algorithms 46, 189–194 (2007).
[CrossRef]

P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

Heath, M.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Honerkamp, J.

T. Roths and J. Honerkamp, “Simultaneous regularization method for the determination of radius distributions from experimental multiangle correlation functions,” Phys. Rev. E 64, 041404 (2001).
[CrossRef]

Kadlec, P.

P. Stepanek, Z. Tuzar, P. Kadlec, and J. Kriz, “A dynamic light scattering study of fast relaxations in polymer solutions,” Macromolecules 40, 2165–2171 (2007).
[CrossRef]

Klyueva, A. V.

V. M. Gun’ko, A. V. Klyueva, Y. N. Levchuk, and R. Leboda, “Photon correlation spectroscopy investigations of proteins,” Adv. Colloid Interface Sci. 105, 201–328 (2003).
[CrossRef]

Kriz, J.

P. Stepanek, Z. Tuzar, P. Kadlec, and J. Kriz, “A dynamic light scattering study of fast relaxations in polymer solutions,” Macromolecules 40, 2165–2171 (2007).
[CrossRef]

Leboda, R.

V. M. Gun’ko, A. V. Klyueva, Y. N. Levchuk, and R. Leboda, “Photon correlation spectroscopy investigations of proteins,” Adv. Colloid Interface Sci. 105, 201–328 (2003).
[CrossRef]

Levchuk, Y. N.

V. M. Gun’ko, A. V. Klyueva, Y. N. Levchuk, and R. Leboda, “Photon correlation spectroscopy investigations of proteins,” Adv. Colloid Interface Sci. 105, 201–328 (2003).
[CrossRef]

Li, Y.

Y. Li, V. Lubchenko, and P. G. Vekilov, “The use of dynamic light scattering and Brownian microscopy to characterize protein aggregation,” Rev. Sci. Instrum. 82, 053106(2011).
[CrossRef]

Liu, W.

Lowther, D. A.

D. A. Lowther, R. D. Throne, L. G. Olson, and J. R. Windle, “A comparison of two methods for choosing the regularization parameter for the inverse problem of electrocardiography,” Biomed. Sci. Instrum. 38, 257–261 (2002).

Lubchenko, V.

Y. Li, V. Lubchenko, and P. G. Vekilov, “The use of dynamic light scattering and Brownian microscopy to characterize protein aggregation,” Rev. Sci. Instrum. 82, 053106(2011).
[CrossRef]

Lukas, M. A.

F. Bauer and M. A. Lukas, “Comparing parameter choice methods for regularization of ill-posed problems,” Math. Comput. Simul. 81, 1795–1841 (2011).
[CrossRef]

M. A. Lukas, “Robust generalized cross-validation for choosing the regularization parameter,” Inverse Probl. 22, 1883–1902 (2006).
[CrossRef]

Mayumi, K.

K. Sumitomo, K. Mayumi, and H. Yokoyama, “Dynamic light scattering measurement of sieving polymer solutions for protein separation on SDS CE,” Electrophoresis 30, 3607–3612 (2009).
[CrossRef]

Meyer, W. V.

O’Leary, D. P.

P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

Oldenburg, D. W.

C. G. Farquharson and D. W. Oldenburg, “A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems,” Geophys. J. Int. 156, 411–425 (2004).
[CrossRef]

Olson, L. G.

D. A. Lowther, R. D. Throne, L. G. Olson, and J. R. Windle, “A comparison of two methods for choosing the regularization parameter for the inverse problem of electrocardiography,” Biomed. Sci. Instrum. 38, 257–261 (2002).

Orlande, H.

L. A. Clementi, J. R. Vega, L. M. Gugliotta, and H. Orlande, “A Bayesian inversion method for estimating the particle size distribution of latexes from multiangle dynamic light scattering measurements,” Chemom. Intell. Lab. Syst. 107, 165–173 (2011).
[CrossRef]

Pecora, R.

R. Pecora, Dynamic Light Scattering: Application of Photon Correlation Spectroscopy (Plenum, 1985).

Provencher, S. W.

S. W. Provencher and P. Stepanek, “Global analysis of dynamic light scattering autocorrelation functions,” Part. Part. Syst. Charact. 13, 291–294 (1996).
[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]

Ross, D. A.

D. A. Ross and H. S. Dhadwal, “Regularized inversion of the Laplace transform: accuracy of analytical and discrete inversion,” Part. Part. Syst. Charact. 8, 282–286 (1991).
[CrossRef]

Roths, T.

T. Roths and J. Honerkamp, “Simultaneous regularization method for the determination of radius distributions from experimental multiangle correlation functions,” Phys. Rev. E 64, 041404 (2001).
[CrossRef]

Ruf, H.

H. Ruf, “Treatment of contributions of dust to dynamic light scattering data,” Langmuir 18, 3804–3814 (2002).
[CrossRef]

H. Ruf, B. Gould, and W. Haase, “The effect of nonrandom errors on the results from regularized inversions of dynamic light scattering data,” Langmuir 16, 471–480(2000).
[CrossRef]

H. Ruf, “Effects of normalization errors on size distributions obtained from dynamic light scattering data,” Biophys. J. 56, 67–78 (1989).
[CrossRef]

Santos, N. C.

M. J. Fernandes, N. C. Santos, and M. Castanho, “Continuous particle size distribution analysis with dynamic light scattering: MAXAMPER: a regularization method using the maximum amplitude for the average error and the Lagrange’s multipliers method,” J. Biochem. Biophys. Methods 36, 101–117 (1998).
[CrossRef]

Schatzel, K.

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

Scheffold, F.

F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (ACS, 2004).

Schnablegger, H.

Schurtenberger, P.

F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (ACS, 2004).

Shalkevich, A.

F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (ACS, 2004).

Shen, J.

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
[CrossRef]

X. Zhu, J. Shen, W. Liu, X. Sun, and Y. Wang, “Nonnegative least-squares truncated singular value decomposition to particle size distribution inversion from dynamic light scattering data,” Appl. Opt. 49, 6591–6596 (2010).
[CrossRef]

Smart, A. E.

Standish, N.

A. B. Yu and N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
[CrossRef]

Stepanek, P.

P. Stepanek, Z. Tuzar, P. Kadlec, and J. Kriz, “A dynamic light scattering study of fast relaxations in polymer solutions,” Macromolecules 40, 2165–2171 (2007).
[CrossRef]

S. W. Provencher and P. Stepanek, “Global analysis of dynamic light scattering autocorrelation functions,” Part. Part. Syst. Charact. 13, 291–294 (1996).
[CrossRef]

Stimac, S.

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

Sumitomo, K.

K. Sumitomo, K. Mayumi, and H. Yokoyama, “Dynamic light scattering measurement of sieving polymer solutions for protein separation on SDS CE,” Electrophoresis 30, 3607–3612 (2009).
[CrossRef]

Sun, X.

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
[CrossRef]

X. Zhu, J. Shen, W. Liu, X. Sun, and Y. Wang, “Nonnegative least-squares truncated singular value decomposition to particle size distribution inversion from dynamic light scattering data,” Appl. Opt. 49, 6591–6596 (2010).
[CrossRef]

Sun, Y.

Y. Sun and J. Walker, “Maximum likelihood data inversion for photon correlation spectroscopy,” Meas. Sci. Technol. 19, 115302 (2008).
[CrossRef]

Thite, A. N.

H. G. Choi, A. N. Thite, and D. J. Thompson, “Comparison of methods for parameter selection in Tikhonov regularization with application to inverse force determination,” J. Sound Vib. 304, 894–917 (2007).
[CrossRef]

Thomas, J. C.

J. C. Thomas, “Photon correlation spectroscopy: technique and instrumentation,” Proc. SPIE 1430, 2–18 (1991).
[CrossRef]

Thompson, D. J.

H. G. Choi, A. N. Thite, and D. J. Thompson, “Comparison of methods for parameter selection in Tikhonov regularization with application to inverse force determination,” J. Sound Vib. 304, 894–917 (2007).
[CrossRef]

Throne, R. D.

D. A. Lowther, R. D. Throne, L. G. Olson, and J. R. Windle, “A comparison of two methods for choosing the regularization parameter for the inverse problem of electrocardiography,” Biomed. Sci. Instrum. 38, 257–261 (2002).

Tscharnuter, W. W.

B. B. Weiner and W. W. Tscharnuter, Uses and Abuses of Photon Correlation Spectroscopy in Particle Sizing (ACS, 1987).

Tuzar, Z.

P. Stepanek, Z. Tuzar, P. Kadlec, and J. Kriz, “A dynamic light scattering study of fast relaxations in polymer solutions,” Macromolecules 40, 2165–2171 (2007).
[CrossRef]

Vavrin, R.

F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (ACS, 2004).

Vega, J. R.

L. A. Clementi, J. R. Vega, L. M. Gugliotta, and H. Orlande, “A Bayesian inversion method for estimating the particle size distribution of latexes from multiangle dynamic light scattering measurements,” Chemom. Intell. Lab. Syst. 107, 165–173 (2011).
[CrossRef]

Vekilov, P. G.

Y. Li, V. Lubchenko, and P. G. Vekilov, “The use of dynamic light scattering and Brownian microscopy to characterize protein aggregation,” Rev. Sci. Instrum. 82, 053106(2011).
[CrossRef]

Wahba, G.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Walker, J.

Y. Sun and J. Walker, “Maximum likelihood data inversion for photon correlation spectroscopy,” Meas. Sci. Technol. 19, 115302 (2008).
[CrossRef]

Wang, X.

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Figures (13)

Fig. 1.
Fig. 1.

L-curve rule to choose the regularization parameter.

Fig. 2.
Fig. 2.

Recovered PSDs for a 10 to 1000 nm unimodal PSD. Correlation function data corrupted with uncorrelated noise.

Fig. 3.
Fig. 3.

Recovered PSDs for a 10 to 1000 nm unimodal PSD. Correlation function data corrupted with correlated noise.

Fig. 4.
Fig. 4.

Recovered PSDs for a 10–1000 nm unimodal PSD showing the effect of correlation function data baseline correction. Correlation function baseline error is 4×103.

Fig. 5.
Fig. 5.

Recovered PSDs for a 10–1000 nm unimodal PSD showing the effect of correlation function data baseline correction. Correlation function baseline error is +4×103.

Fig. 6.
Fig. 6.

Edge proportion versus the correlation function baseline compensation for a 10–1000 nm unimodal PSD with (a) a positive baseline error of 3.6×104 and (b) a negative baseline error of 3.6×104.

Fig. 7.
Fig. 7.

Recovered PSDs with (filled circles) and without (open circles) baseline compensation for 10–1000 nm unimodal PSD correlation function data with (a) a positive baseline error of 3.6×104 and (b) a negative baseline error of 3.6×104.

Fig. 8.
Fig. 8.

PSD norm versus baseline compensation for the 10–1000 nm unimodal PSD correlation function with a positive baseline error of 3.6×104.

Fig. 9.
Fig. 9.

Two strategies for compensating the baseline of correlation function data at 120° scattering angle. (a) Edge proportion strategy and (b) PSD norm strategy.

Fig. 10.
Fig. 10.

PSDs recovered from correlation function data at 120° scattering angle that have been baseline error corrected using the edge proportion strategy (open circles) and PSD norm strategy (filled circles).

Fig. 11.
Fig. 11.

Two strategies for compensating the baseline of correlation function data at 100° scattering angle. (a) Edge proportion strategy and (b) PSD norm strategy.

Fig. 12.
Fig. 12.

PSDs recovered from correlation function data at 100° scattering angle that have been baseline error corrected using the edge proportion strategy (open circles) and PSD norm strategy (filled circles).

Fig. 13.
Fig. 13.

PSDs recovered from original correlation function data at 120° (open circles) and 100° (filled circles) scattering angles.

Tables (8)

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Table 1. Baseline Compensation for Baseline Error Corrupted Correlation Function Data for a 10–1000 nm Unimodal PSD

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Table 2. Baseline Compensation for Baseline Error and Random Noise Corrupted Correlation Function Data for a 100–800 nm Unimodal PSD

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Table 3. Regularization Parameters Determined Using the L-Curve Method for Standard Regularization, for the Constrained Regularization, and for the GCV and the Robust GCV Rules for the 10–1000 nm and 100–800 nm Unimodal, 10–1000 nm Bimodal, and 100 and 500 nm Bimodal PSD Correlation Function Data Corrupted by Uncorrelated Noise at Low Noise Level

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Table 4. Regularization Parameters Determined Using the L-Curve Method for Standard Regularization, for the Constrained Regularization, and for the GCV and the Robust GCV Rules for the 10–1000 nm and 100–800 nm Unimodal, 10–1000 nm Bimodal, and 100 and 500 nm Bimodal PSD Correlation Function Data Corrupted by Uncorrelated Noise at High Noise Level

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Table 5. Regularization Parameters Determined Using the L-Curve Method for Standard Regularization, for the Constrained Regularization, and for the GCV and the Robust GCV Rules for the 10–1000 nm and 100–800 nm Unimodal, 10–1000 nm Bimodal, and 100 and 500 nm Bimodal PSD Correlation Function Data Corrupted by Correlated Noise at High SNR

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Table 6. Regularization Parameters Determined Using the L-Curve Method for Standard Regularization, for the Constrained Regularization and for the GCV and the Robust GCV Rules for the 10–1000 nm and 100–800 nm Unimodal, 10–1000 nm Bimodal, and 100 and 500 nm Bimodal PSD Correlation Function Data Corrupted by Correlate Noise at Low SNR

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Table 7. Diameter and Intensity Ratio for PSDs Obtained from Autocorrelation Function Data for Scattering Angles at 120° and 100° after Baseline Compensation

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Table 8. Regularization Parameters Determined Using the L-Curve for Standard Regularization, for the Constrained Regularization, and for the GCV and Robust GCV Rules for the 198nm/502nm Bimodal Experimental PSD Before and After Baseline Compensation

Equations (22)

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

g(1)(τ)=0G(Γ)exp(Γτ)dΓ,
g(1)(τ)=0f(D)exp(kτ/D)dD,
k=kBT3πη[4πnλosin(θ2)]2.
g(1)(τj)=i=1Nexp(kτj/Di)ai,
G(2)(τ)=n(t)n(t+τ).
g(2)(τ)=G(2)(τ)n(t)2.
g(2)(τ)=1+β|g(1)(τ)|2.
g(2)(τ)=(1ΔBB^)G(2)(τ)n(t)2+EB^.
g(1)(τ)=[(1ΔBB^)G(2)(τ)n(t)2+EB^1]1/2=[(1ΔBB^)β|gt(1)(τ)|2(ΔBB^+EB^)]1/2,
g=Ax,
Mα(x,g)=Axg2+αΩ(x).
Mα(x,g)=Axg2+αΩ(x)s.t.1x0
Mα(x,g)=12Axg2+α2Dx2s.t.1x0.
Mα(x,g)=12||[AαD]x[g0]||2subject toxS1{x1x0},
k(α)=ρθρθ(ρ2+θ2)32,
k(α)=d2ρ/d2θ(1+(dρ/dθ)2)32.
V(α)=n(IA(α))g2(Tr(IA(α)))2.
V(α)=n(IA(α))g2(Tr(IA(α)))2(γ+(1γ)n1Tr(A(α)2)),
f(D)=D2π(DmaxDmin)(t(1t))1exp(0.5(μ+σln(t/(1t)))2),
t=(DDmin)/(DmaxDmin).
y(τ)=g(2)(τ)+δε,
y(τ)=g(2)(τ)+δg(2)(τ)ε/δ.

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