Abstract

Multiangle dynamic light scattering (MDLS) can provide better results for particle size distribution (PSD) determination than single-angle dynamic light scattering. Proper analysis of MDLS data requires data from each measurement angle to be appropriately weighted according to the intensity scattered by the particles at each scattering angle. The angular weighting coefficients may be determined by measuring the angular dependence of the scattered light intensity or estimated in various ways. In either case, any noise on the weighting coefficients will adversely affect the PSD determination. We propose a new iterative recursion method for estimating the weighting coefficients and demonstrate its effectiveness for recovering PSDs from both simulated and real experimental data. The new method gives better PSD results than those found using other weighting estimates.

© 2012 Optical Society of America

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References

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  1. R. Pecora, Dynamic Light Scattering: Applications of Photo 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 (American Chemical Society, 2004).
  3. P. G. Cummins and E. J. Staples, “Particle size distributions determined by a multiangle analysis of photon correlation spectroscopy data,” Langmuir 3, 1109–1113 (1987).
    [CrossRef]
  4. S. E. Bott, “Submicron particle sizing by photo correlation spectroscopy: use of multiple angle detection,” in Particle Size Distribution (American Chemical Society, 1987), pp. 74–88.
  5. R. Finsy, P. D. Groen, and L. Deriemaeker, “Data analysis of multi-angle photo correlation measurements without and with prior knowledge,” Part. Part. Syst. Charact. 9, 237–251 (1992).
    [CrossRef]
  6. C. Wu, K. Unterforsthuber, and D. Lilge, “Determination of particle size distribution by the analysis of intensity-constrained multi-angle photon correlation spectroscopic data,” Part. Part. Syst. Charact. 11, 145–149 (1994).
    [CrossRef]
  7. R. Buttgereit, 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]
  8. G. Bryant and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering,” Langmuir 11, 2480–2485 (1995).
    [CrossRef]
  9. G. Bryant, C. Abeynayake, and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering. 2. Refinements and applications,” Langmuir 12, 6224–6228 (1996).
    [CrossRef]
  10. L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
    [CrossRef]
  11. L. A. Clementi, J. R. Vega, and L. M. Gugliotta, “A Bayesian inversion method for estimating the particle size distribution of latexes from multiangle dynamic light scattering measurements,” Chemometr. Intell. Lab. Syst. 107, 165–173 (2011).
    [CrossRef]
  12. J. R. Vega, L. M. Gugliotta, V. D. Gonzalez, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: a novel data processing for multiangle measurements,” J. Colloid Interface Sci. 261, 74–81 (2003).
    [CrossRef]
  13. W. J. Wiscombe, “Improved Mie scattering algorithm,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef]
  14. P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
    [CrossRef]
  15. 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]
  16. X. J. Zhu, J. Shen, W. Liu, X. M. Sun, and Y. J. 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]
  17. W. Liu, X. M. Sun, and J. Shen, “A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing,” Opt. Laser Technol. 44, 1–5 (2012).
    [CrossRef]
  18. S. Suparno, K. Deurloo, P. Stamatelopolous, R. Srivastva, and J. C. Thomas, “Light scattering with single-model fiber collimators,” Appl. Opt. 33, 7200–7205 (1994).
    [CrossRef]

2012

W. Liu, X. M. Sun, and J. Shen, “A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing,” Opt. Laser Technol. 44, 1–5 (2012).
[CrossRef]

2011

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

2010

2009

L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

2003

J. R. Vega, L. M. Gugliotta, V. D. Gonzalez, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: a novel data processing for multiangle measurements,” J. Colloid Interface Sci. 261, 74–81 (2003).
[CrossRef]

2001

R. Buttgereit, 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]

1996

G. Bryant, C. Abeynayake, and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering. 2. Refinements and applications,” Langmuir 12, 6224–6228 (1996).
[CrossRef]

1995

G. Bryant and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering,” Langmuir 11, 2480–2485 (1995).
[CrossRef]

1994

C. Wu, K. Unterforsthuber, and D. Lilge, “Determination of particle size distribution by the analysis of intensity-constrained multi-angle photon correlation spectroscopic data,” Part. Part. Syst. Charact. 11, 145–149 (1994).
[CrossRef]

S. Suparno, K. Deurloo, P. Stamatelopolous, R. Srivastva, and J. C. Thomas, “Light scattering with single-model fiber collimators,” Appl. Opt. 33, 7200–7205 (1994).
[CrossRef]

1993

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

1992

R. Finsy, P. D. Groen, and L. Deriemaeker, “Data analysis of multi-angle photo correlation measurements without and with prior knowledge,” Part. Part. Syst. Charact. 9, 237–251 (1992).
[CrossRef]

1987

P. G. Cummins and E. J. Staples, “Particle size distributions determined by a multiangle analysis of photon correlation spectroscopy data,” Langmuir 3, 1109–1113 (1987).
[CrossRef]

1980

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]

Abeynayake, C.

G. Bryant, C. Abeynayake, and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering. 2. Refinements and applications,” Langmuir 12, 6224–6228 (1996).
[CrossRef]

Bott, S. E.

S. E. Bott, “Submicron particle sizing by photo correlation spectroscopy: use of multiple angle detection,” in Particle Size Distribution (American Chemical Society, 1987), pp. 74–88.

Bryant, G.

G. Bryant, C. Abeynayake, and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering. 2. Refinements and applications,” Langmuir 12, 6224–6228 (1996).
[CrossRef]

G. Bryant and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering,” Langmuir 11, 2480–2485 (1995).
[CrossRef]

Buttgereit, R.

R. Buttgereit, 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]

Clementi, L. A.

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

L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

Crassous, J.

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

Cummins, P. G.

P. G. Cummins and E. J. Staples, “Particle size distributions determined by a multiangle analysis of photon correlation spectroscopy data,” Langmuir 3, 1109–1113 (1987).
[CrossRef]

Deriemaeker, L.

R. Finsy, P. D. Groen, and L. Deriemaeker, “Data analysis of multi-angle photo correlation measurements without and with prior knowledge,” Part. Part. Syst. Charact. 9, 237–251 (1992).
[CrossRef]

Deurloo, K.

Finsy, R.

R. Finsy, P. D. Groen, and L. Deriemaeker, “Data analysis of multi-angle photo correlation measurements without and with prior knowledge,” Part. Part. Syst. Charact. 9, 237–251 (1992).
[CrossRef]

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]

Gonzalez, V. D.

J. R. Vega, L. M. Gugliotta, V. D. Gonzalez, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: a novel data processing for multiangle measurements,” J. Colloid Interface Sci. 261, 74–81 (2003).
[CrossRef]

Groen, P. D.

R. Finsy, P. D. Groen, and L. Deriemaeker, “Data analysis of multi-angle photo correlation measurements without and with prior knowledge,” Part. Part. Syst. Charact. 9, 237–251 (1992).
[CrossRef]

Gugliotta, L. M.

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

L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

J. R. Vega, L. M. Gugliotta, V. D. Gonzalez, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: a novel data processing for multiangle measurements,” J. Colloid Interface Sci. 261, 74–81 (2003).
[CrossRef]

Hansen, P. C.

P. C. Hansen and D. P. O’Leary, “The use of the 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.

R. Buttgereit, 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]

Lilge, D.

C. Wu, K. Unterforsthuber, and D. Lilge, “Determination of particle size distribution by the analysis of intensity-constrained multi-angle photon correlation spectroscopic data,” Part. Part. Syst. Charact. 11, 145–149 (1994).
[CrossRef]

Liu, W.

W. Liu, X. M. Sun, and J. Shen, “A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing,” Opt. Laser Technol. 44, 1–5 (2012).
[CrossRef]

X. J. Zhu, J. Shen, W. Liu, X. M. Sun, and Y. J. 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]

Meira, G. R.

J. R. Vega, L. M. Gugliotta, V. D. Gonzalez, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: a novel data processing for multiangle measurements,” J. Colloid Interface Sci. 261, 74–81 (2003).
[CrossRef]

O’Leary, D. P.

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

Pecora, R.

R. Pecora, Dynamic Light Scattering: Applications of Photo Correlation Spectroscopy (Plenum, 1985).

Roths, T.

R. Buttgereit, 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]

Scheffold, F.

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

Schurtenberger, P.

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

Shalkevich, A.

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

Shen, J.

W. Liu, X. M. Sun, and J. Shen, “A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing,” Opt. Laser Technol. 44, 1–5 (2012).
[CrossRef]

X. J. Zhu, J. Shen, W. Liu, X. M. Sun, and Y. J. 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]

Srivastva, R.

Stamatelopolous, P.

Staples, E. J.

P. G. Cummins and E. J. Staples, “Particle size distributions determined by a multiangle analysis of photon correlation spectroscopy data,” Langmuir 3, 1109–1113 (1987).
[CrossRef]

Stegmayer, G. S.

L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

Sun, X. M.

W. Liu, X. M. Sun, and J. Shen, “A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing,” Opt. Laser Technol. 44, 1–5 (2012).
[CrossRef]

X. J. Zhu, J. Shen, W. Liu, X. M. Sun, and Y. J. 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]

Suparno, S.

Thomas, J. C.

G. Bryant, C. Abeynayake, and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering. 2. Refinements and applications,” Langmuir 12, 6224–6228 (1996).
[CrossRef]

G. Bryant and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering,” Langmuir 11, 2480–2485 (1995).
[CrossRef]

S. Suparno, K. Deurloo, P. Stamatelopolous, R. Srivastva, and J. C. Thomas, “Light scattering with single-model fiber collimators,” Appl. Opt. 33, 7200–7205 (1994).
[CrossRef]

Unterforsthuber, K.

C. Wu, K. Unterforsthuber, and D. Lilge, “Determination of particle size distribution by the analysis of intensity-constrained multi-angle photon correlation spectroscopic data,” Part. Part. Syst. Charact. 11, 145–149 (1994).
[CrossRef]

Vavrin, R.

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

Vega, J. R.

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

J. R. Vega, L. M. Gugliotta, V. D. Gonzalez, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: a novel data processing for multiangle measurements,” J. Colloid Interface Sci. 261, 74–81 (2003).
[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]

Wang, Y. J.

Wiscombe, W. J.

Wu, C.

C. Wu, K. Unterforsthuber, and D. Lilge, “Determination of particle size distribution by the analysis of intensity-constrained multi-angle photon correlation spectroscopic data,” Part. Part. Syst. Charact. 11, 145–149 (1994).
[CrossRef]

Zhu, X. J.

Appl. Opt.

Chemometr. Intell. Lab. Syst.

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

J. Colloid Interface Sci.

J. R. Vega, L. M. Gugliotta, V. D. Gonzalez, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: a novel data processing for multiangle measurements,” J. Colloid Interface Sci. 261, 74–81 (2003).
[CrossRef]

Langmuir

P. G. Cummins and E. J. Staples, “Particle size distributions determined by a multiangle analysis of photon correlation spectroscopy data,” Langmuir 3, 1109–1113 (1987).
[CrossRef]

G. Bryant and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering,” Langmuir 11, 2480–2485 (1995).
[CrossRef]

G. Bryant, C. Abeynayake, and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering. 2. Refinements and applications,” Langmuir 12, 6224–6228 (1996).
[CrossRef]

Opt. Laser Technol.

W. Liu, X. M. Sun, and J. Shen, “A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing,” Opt. Laser Technol. 44, 1–5 (2012).
[CrossRef]

Part. Part. Syst. Charact.

L. M. Gugliotta, G. S. Stegmayer, and L. A. Clementi, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

R. Finsy, P. D. Groen, and L. Deriemaeker, “Data analysis of multi-angle photo correlation measurements without and with prior knowledge,” Part. Part. Syst. Charact. 9, 237–251 (1992).
[CrossRef]

C. Wu, K. Unterforsthuber, and D. Lilge, “Determination of particle size distribution by the analysis of intensity-constrained multi-angle photon correlation spectroscopic data,” Part. Part. Syst. Charact. 11, 145–149 (1994).
[CrossRef]

Phys. Rev. E

R. Buttgereit, 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]

SIAM J. Sci. Comput.

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

Technometrics

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]

Other

S. E. Bott, “Submicron particle sizing by photo correlation spectroscopy: use of multiple angle detection,” in Particle Size Distribution (American Chemical Society, 1987), pp. 74–88.

R. Pecora, Dynamic Light Scattering: Applications of Photo Correlation Spectroscopy (Plenum, 1985).

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

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Figures (6)

Fig. 1.
Fig. 1.

PSD estimates for a polydisperse 200 nm PSD obtained through the regularization method using different weighting ratios.

Fig. 2.
Fig. 2.

PSD estimates for a polydisperse 600 nm PSD obtained through the regularization method using different weighting ratios.

Fig. 3.
Fig. 3.

PSD estimates for a 500 and 800 nm bimodal PSD obtained through the regularization method using different weighting ratios.

Fig. 4.
Fig. 4.

The angular dependence of light scattering intensity for range of particle sizes.

Fig. 5.
Fig. 5.

PSD estimates for the 300 and 503 nm diameter bimodal mixture obtained from regularization method using different weighting ratios.

Fig. 6.
Fig. 6.

PSD estimates for the 300 and 503 nm diameter bimodal mixtures obtained from (a) the iterative procedure proposed by Bryant et al. and (b) the regularization method using real static intensity data as the weighting coefficients.

Tables (6)

Tables Icon

Table 1. Weighting Ratio Values and Their Relative Errors for a Polydisperse 200 nm PSDa

Tables Icon

Table 2. Weighting Ratio Values and Their Relative Errors for a Polydisperse 600 nm PSDa

Tables Icon

Table 3. Weighting Ratio Values and Their Relative Errors for a 500 and 800 nm Bimodal PSDa

Tables Icon

Table 4. Parameters for the True and Estimated PSDs, the Relative Errors, and the Values of the Performance Index (V) for the Three Size Distributions and the Relative Heights of the Two Peaks for the Bimodal Samplea

Tables Icon

Table 5. True Weighting Ratios, Weighting Ratios Estimated through the Alternative Recursive Least-Squares Method and the Iterative Recursion Method, and the Relative Errorsa

Tables Icon

Table 6. Parameters for the True and Estimated PSD, the Relative Errors, and the Number Ratio of the Bimodal Distributiona

Equations (10)

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

Gθr(2)(τj)=G,θr(2)(1+β|gθr(1)(τj)|)2,r=1,2,,Randj=1,2,,M.
gθr(1)(τj)=kθri=1Nexp[Γ0(θr)τj/Di]CI,θr(Di)f(Di),
Γ0(θr)=16πKT3η[nm(λ)λ]sin2(θr2).
gθr(1)=kθrFθrf,r=1,2,,R.
kθr*=kθrkθ1=(G,θ1(2)G,θr(2))1/2=Iθ1Iθr.
G2=[G1kθ2*Fθ2],
f(Di)=0.75NPDiσ12πexp[[ln(Di/Dg,1)]22σ12]+0.25NPDiσ22πexp[[ln(Di/Dg,2)]22σ22].
G,θr(2)=c[i=1NCI,θr(Di)f(Di)]2,r=1,2,,R.
V=(i=1N[f(Di)f(Di)^]2i=1N[f(Di)]2)1/2,
kθr*=Iθ1Iθr=5×CI,θ1(300nm)+1×CI,θ1(503nm)5×CI,θr(300nm)+1×CI,θr(503nm).

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