Abstract

This paper presents a new approach to characterize nanoparticles using derivatives of scattering profiles of evanescent waves/surface plasmons. We start the procedure using the scattering profiles for an unknown configuration of nanoparticles, either from physical experiments or numerical simulations conducted for different nanoparticles on surfaces. We apply the statistical technique of compound estimation to recover the derivatives of scattering profiles. The L1 discrepancies with the corresponding curves from known configurations are used to identify the most plausible configuration of particles that could yield the “experimental” profiles. We conduct a simulation study to see how often the new procedure correctly recovers the agglomeration level for gold spherical nanoparticles on a thin gold film. The results suggest that first derivatives are much more effective for characterization than undifferentiated profiles and that M33 is the most useful element for distinguishing among configurations. The proposed compound estimation technique is more effective than typical inverse analyses based on look-up tables and can be used effectively in nanoparticle characterization platforms.

© 2007 Optical Society of America

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References

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  1. G. Videen, M. M. Aslan, and M. P. Mengüç, "Characterization of metallic nanoparticles via surface wave scattering: A. Theoretical framework," J. Quant. Spectrosc. Radiat. Transf. 93, 195-206 (2005).
    [CrossRef]
  2. M. M. Aslan, M. P. Mengüç, and G. Videen, "Characterization of metallic nanoparticles via surface wave scattering: B. Physical concept and numerical experiments," J. Quant. Spectrosc. Radiat. Transf. 93, 207-217 (2005).
    [CrossRef]
  3. P. G. Venkata, M. M. Aslan, M. P. Mengüç, and G. Videen, "Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates," ASME J. Heat Transfer 129, 60-70 (2007).
    [CrossRef]
  4. M. P. Mengüç, S. Manickavasagam, and D. A. D'sa, "Determination of radiative properties of pulverized coal particles from experiments," Fuel 73, 613-625 (1994).
    [CrossRef]
  5. B. M. Agarwal and M. P. Mengüç, "Single and multiple scattering of collimated radiation in an axisymmetric system," Int. J. Heat Mass Transfer 34, 633-647 (1991).
    [CrossRef]
  6. C. Saltiel, Q. Chen, S. Manickavasagam, L. S. Schandler, R. W. Siegel, and M. P. Mengüç, "Identification of dispersion behavior of surface-treated nano-scale powders," J. Nanopart. Res. 6, 35-46 (2004).
    [CrossRef]
  7. M. M. Aslan, M. P. Mengüç, S. Manickavasagam, and C. Saltiel, "Size and shape prediction of colloidal metal oxide MgBaFeO particles from light scattering measurements," J. Nanopart. Res. 8, 981-994 (2006).
    [CrossRef]
  8. C. Saltiel, S. Manickavasagam, M. P. Mengüç, and R. Andrews, "Light scattering and dispersion behavior of multi-walled carbon nanotubes," J. Opt. Soc. Am. A 22, 1546-1554 (2005).
    [CrossRef]
  9. M. M. Aslan, C. Crofcheck, D. Tao, and M. P. Mengüç, "Evaluation of micro bubble size and gas hold up in two phase gas-liquid columns via scattered light measurements," J. Quant. Spectrosc. Radiat. Transf. 101, 527-539 (2006).
    [CrossRef]
  10. S. Subramaniam and M. P. Mengüç, "Solution of inverse radiation problem for inhomogeneous and anisotropically scattering medium using a Monte-Carlo technique," Int. J. Heat Mass Transfer 34, 253-266 (1991).
    [CrossRef]
  11. M. P. Mengüç and S. Manickavasagam, "Inverse radiation problem in axisymmetric cylindrical media," AIAA J. Thermophy. Heat Transfer 7, 479-486 (1993).
    [CrossRef]
  12. M. P. Mengüç and P. Dutta, "Scattering tomography and application to sooting diffusion flames," ASME J. Heat Transfer 116, 144-151 (1994).
    [CrossRef]
  13. S. Manickavasagam and M. P. Mengüç, "Scattering matrix elements of fractal-like soot agglomerates," Appl. Opt. 36, 1337-1351 (1997).
    [CrossRef] [PubMed]
  14. M. Francoeur, P. G. Venkata, and M. P. Mengüç, "Sensitivity analysis for characterization of gold nanoparticles and 2D-agglomerates via surface plasmon scattering patterns," J. Quant. Spectrosc. Radiat. Transf. 106, 44-55 (2007).
    [CrossRef]
  15. R. Charnigo and C. Srinivasan, "Local and global analytic curve estimation," submitted and available via http://www.ms.uky.edu/~richc/LACE.pdf.
  16. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  17. D. W. Mackowski, "Calculation of total cross sections of multiple-sphere clusters," J. Opt. Soc. Am. A 11, 2851-2861 (1994).
    [CrossRef]

2007 (2)

P. G. Venkata, M. M. Aslan, M. P. Mengüç, and G. Videen, "Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates," ASME J. Heat Transfer 129, 60-70 (2007).
[CrossRef]

M. Francoeur, P. G. Venkata, and M. P. Mengüç, "Sensitivity analysis for characterization of gold nanoparticles and 2D-agglomerates via surface plasmon scattering patterns," J. Quant. Spectrosc. Radiat. Transf. 106, 44-55 (2007).
[CrossRef]

2006 (2)

M. M. Aslan, M. P. Mengüç, S. Manickavasagam, and C. Saltiel, "Size and shape prediction of colloidal metal oxide MgBaFeO particles from light scattering measurements," J. Nanopart. Res. 8, 981-994 (2006).
[CrossRef]

M. M. Aslan, C. Crofcheck, D. Tao, and M. P. Mengüç, "Evaluation of micro bubble size and gas hold up in two phase gas-liquid columns via scattered light measurements," J. Quant. Spectrosc. Radiat. Transf. 101, 527-539 (2006).
[CrossRef]

2005 (3)

G. Videen, M. M. Aslan, and M. P. Mengüç, "Characterization of metallic nanoparticles via surface wave scattering: A. Theoretical framework," J. Quant. Spectrosc. Radiat. Transf. 93, 195-206 (2005).
[CrossRef]

M. M. Aslan, M. P. Mengüç, and G. Videen, "Characterization of metallic nanoparticles via surface wave scattering: B. Physical concept and numerical experiments," J. Quant. Spectrosc. Radiat. Transf. 93, 207-217 (2005).
[CrossRef]

C. Saltiel, S. Manickavasagam, M. P. Mengüç, and R. Andrews, "Light scattering and dispersion behavior of multi-walled carbon nanotubes," J. Opt. Soc. Am. A 22, 1546-1554 (2005).
[CrossRef]

2004 (1)

C. Saltiel, Q. Chen, S. Manickavasagam, L. S. Schandler, R. W. Siegel, and M. P. Mengüç, "Identification of dispersion behavior of surface-treated nano-scale powders," J. Nanopart. Res. 6, 35-46 (2004).
[CrossRef]

1997 (1)

1994 (3)

M. P. Mengüç and P. Dutta, "Scattering tomography and application to sooting diffusion flames," ASME J. Heat Transfer 116, 144-151 (1994).
[CrossRef]

D. W. Mackowski, "Calculation of total cross sections of multiple-sphere clusters," J. Opt. Soc. Am. A 11, 2851-2861 (1994).
[CrossRef]

M. P. Mengüç, S. Manickavasagam, and D. A. D'sa, "Determination of radiative properties of pulverized coal particles from experiments," Fuel 73, 613-625 (1994).
[CrossRef]

1993 (1)

M. P. Mengüç and S. Manickavasagam, "Inverse radiation problem in axisymmetric cylindrical media," AIAA J. Thermophy. Heat Transfer 7, 479-486 (1993).
[CrossRef]

1991 (2)

S. Subramaniam and M. P. Mengüç, "Solution of inverse radiation problem for inhomogeneous and anisotropically scattering medium using a Monte-Carlo technique," Int. J. Heat Mass Transfer 34, 253-266 (1991).
[CrossRef]

B. M. Agarwal and M. P. Mengüç, "Single and multiple scattering of collimated radiation in an axisymmetric system," Int. J. Heat Mass Transfer 34, 633-647 (1991).
[CrossRef]

Agarwal, B. M.

B. M. Agarwal and M. P. Mengüç, "Single and multiple scattering of collimated radiation in an axisymmetric system," Int. J. Heat Mass Transfer 34, 633-647 (1991).
[CrossRef]

Andrews, R.

Aslan, M. M.

P. G. Venkata, M. M. Aslan, M. P. Mengüç, and G. Videen, "Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates," ASME J. Heat Transfer 129, 60-70 (2007).
[CrossRef]

M. M. Aslan, C. Crofcheck, D. Tao, and M. P. Mengüç, "Evaluation of micro bubble size and gas hold up in two phase gas-liquid columns via scattered light measurements," J. Quant. Spectrosc. Radiat. Transf. 101, 527-539 (2006).
[CrossRef]

M. M. Aslan, M. P. Mengüç, S. Manickavasagam, and C. Saltiel, "Size and shape prediction of colloidal metal oxide MgBaFeO particles from light scattering measurements," J. Nanopart. Res. 8, 981-994 (2006).
[CrossRef]

M. M. Aslan, M. P. Mengüç, and G. Videen, "Characterization of metallic nanoparticles via surface wave scattering: B. Physical concept and numerical experiments," J. Quant. Spectrosc. Radiat. Transf. 93, 207-217 (2005).
[CrossRef]

G. Videen, M. M. Aslan, and M. P. Mengüç, "Characterization of metallic nanoparticles via surface wave scattering: A. Theoretical framework," J. Quant. Spectrosc. Radiat. Transf. 93, 195-206 (2005).
[CrossRef]

Bohren, C. F.

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

Charnigo, R.

R. Charnigo and C. Srinivasan, "Local and global analytic curve estimation," submitted and available via http://www.ms.uky.edu/~richc/LACE.pdf.

Chen, Q.

C. Saltiel, Q. Chen, S. Manickavasagam, L. S. Schandler, R. W. Siegel, and M. P. Mengüç, "Identification of dispersion behavior of surface-treated nano-scale powders," J. Nanopart. Res. 6, 35-46 (2004).
[CrossRef]

Crofcheck, C.

M. M. Aslan, C. Crofcheck, D. Tao, and M. P. Mengüç, "Evaluation of micro bubble size and gas hold up in two phase gas-liquid columns via scattered light measurements," J. Quant. Spectrosc. Radiat. Transf. 101, 527-539 (2006).
[CrossRef]

D'sa, D. A.

M. P. Mengüç, S. Manickavasagam, and D. A. D'sa, "Determination of radiative properties of pulverized coal particles from experiments," Fuel 73, 613-625 (1994).
[CrossRef]

Dutta, P.

M. P. Mengüç and P. Dutta, "Scattering tomography and application to sooting diffusion flames," ASME J. Heat Transfer 116, 144-151 (1994).
[CrossRef]

Francoeur, M.

M. Francoeur, P. G. Venkata, and M. P. Mengüç, "Sensitivity analysis for characterization of gold nanoparticles and 2D-agglomerates via surface plasmon scattering patterns," J. Quant. Spectrosc. Radiat. Transf. 106, 44-55 (2007).
[CrossRef]

Huffman, D. R.

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

Mackowski, D. W.

Manickavasagam, S.

M. M. Aslan, M. P. Mengüç, S. Manickavasagam, and C. Saltiel, "Size and shape prediction of colloidal metal oxide MgBaFeO particles from light scattering measurements," J. Nanopart. Res. 8, 981-994 (2006).
[CrossRef]

C. Saltiel, S. Manickavasagam, M. P. Mengüç, and R. Andrews, "Light scattering and dispersion behavior of multi-walled carbon nanotubes," J. Opt. Soc. Am. A 22, 1546-1554 (2005).
[CrossRef]

C. Saltiel, Q. Chen, S. Manickavasagam, L. S. Schandler, R. W. Siegel, and M. P. Mengüç, "Identification of dispersion behavior of surface-treated nano-scale powders," J. Nanopart. Res. 6, 35-46 (2004).
[CrossRef]

S. Manickavasagam and M. P. Mengüç, "Scattering matrix elements of fractal-like soot agglomerates," Appl. Opt. 36, 1337-1351 (1997).
[CrossRef] [PubMed]

M. P. Mengüç, S. Manickavasagam, and D. A. D'sa, "Determination of radiative properties of pulverized coal particles from experiments," Fuel 73, 613-625 (1994).
[CrossRef]

M. P. Mengüç and S. Manickavasagam, "Inverse radiation problem in axisymmetric cylindrical media," AIAA J. Thermophy. Heat Transfer 7, 479-486 (1993).
[CrossRef]

Mengüç, M. P.

M. Francoeur, P. G. Venkata, and M. P. Mengüç, "Sensitivity analysis for characterization of gold nanoparticles and 2D-agglomerates via surface plasmon scattering patterns," J. Quant. Spectrosc. Radiat. Transf. 106, 44-55 (2007).
[CrossRef]

P. G. Venkata, M. M. Aslan, M. P. Mengüç, and G. Videen, "Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates," ASME J. Heat Transfer 129, 60-70 (2007).
[CrossRef]

M. M. Aslan, C. Crofcheck, D. Tao, and M. P. Mengüç, "Evaluation of micro bubble size and gas hold up in two phase gas-liquid columns via scattered light measurements," J. Quant. Spectrosc. Radiat. Transf. 101, 527-539 (2006).
[CrossRef]

M. M. Aslan, M. P. Mengüç, S. Manickavasagam, and C. Saltiel, "Size and shape prediction of colloidal metal oxide MgBaFeO particles from light scattering measurements," J. Nanopart. Res. 8, 981-994 (2006).
[CrossRef]

C. Saltiel, S. Manickavasagam, M. P. Mengüç, and R. Andrews, "Light scattering and dispersion behavior of multi-walled carbon nanotubes," J. Opt. Soc. Am. A 22, 1546-1554 (2005).
[CrossRef]

M. M. Aslan, M. P. Mengüç, and G. Videen, "Characterization of metallic nanoparticles via surface wave scattering: B. Physical concept and numerical experiments," J. Quant. Spectrosc. Radiat. Transf. 93, 207-217 (2005).
[CrossRef]

G. Videen, M. M. Aslan, and M. P. Mengüç, "Characterization of metallic nanoparticles via surface wave scattering: A. Theoretical framework," J. Quant. Spectrosc. Radiat. Transf. 93, 195-206 (2005).
[CrossRef]

C. Saltiel, Q. Chen, S. Manickavasagam, L. S. Schandler, R. W. Siegel, and M. P. Mengüç, "Identification of dispersion behavior of surface-treated nano-scale powders," J. Nanopart. Res. 6, 35-46 (2004).
[CrossRef]

S. Manickavasagam and M. P. Mengüç, "Scattering matrix elements of fractal-like soot agglomerates," Appl. Opt. 36, 1337-1351 (1997).
[CrossRef] [PubMed]

M. P. Mengüç and P. Dutta, "Scattering tomography and application to sooting diffusion flames," ASME J. Heat Transfer 116, 144-151 (1994).
[CrossRef]

M. P. Mengüç, S. Manickavasagam, and D. A. D'sa, "Determination of radiative properties of pulverized coal particles from experiments," Fuel 73, 613-625 (1994).
[CrossRef]

M. P. Mengüç and S. Manickavasagam, "Inverse radiation problem in axisymmetric cylindrical media," AIAA J. Thermophy. Heat Transfer 7, 479-486 (1993).
[CrossRef]

S. Subramaniam and M. P. Mengüç, "Solution of inverse radiation problem for inhomogeneous and anisotropically scattering medium using a Monte-Carlo technique," Int. J. Heat Mass Transfer 34, 253-266 (1991).
[CrossRef]

B. M. Agarwal and M. P. Mengüç, "Single and multiple scattering of collimated radiation in an axisymmetric system," Int. J. Heat Mass Transfer 34, 633-647 (1991).
[CrossRef]

Saltiel, C.

M. M. Aslan, M. P. Mengüç, S. Manickavasagam, and C. Saltiel, "Size and shape prediction of colloidal metal oxide MgBaFeO particles from light scattering measurements," J. Nanopart. Res. 8, 981-994 (2006).
[CrossRef]

C. Saltiel, S. Manickavasagam, M. P. Mengüç, and R. Andrews, "Light scattering and dispersion behavior of multi-walled carbon nanotubes," J. Opt. Soc. Am. A 22, 1546-1554 (2005).
[CrossRef]

C. Saltiel, Q. Chen, S. Manickavasagam, L. S. Schandler, R. W. Siegel, and M. P. Mengüç, "Identification of dispersion behavior of surface-treated nano-scale powders," J. Nanopart. Res. 6, 35-46 (2004).
[CrossRef]

Schandler, L. S.

C. Saltiel, Q. Chen, S. Manickavasagam, L. S. Schandler, R. W. Siegel, and M. P. Mengüç, "Identification of dispersion behavior of surface-treated nano-scale powders," J. Nanopart. Res. 6, 35-46 (2004).
[CrossRef]

Siegel, R. W.

C. Saltiel, Q. Chen, S. Manickavasagam, L. S. Schandler, R. W. Siegel, and M. P. Mengüç, "Identification of dispersion behavior of surface-treated nano-scale powders," J. Nanopart. Res. 6, 35-46 (2004).
[CrossRef]

Srinivasan, C.

R. Charnigo and C. Srinivasan, "Local and global analytic curve estimation," submitted and available via http://www.ms.uky.edu/~richc/LACE.pdf.

Subramaniam, S.

S. Subramaniam and M. P. Mengüç, "Solution of inverse radiation problem for inhomogeneous and anisotropically scattering medium using a Monte-Carlo technique," Int. J. Heat Mass Transfer 34, 253-266 (1991).
[CrossRef]

Tao, D.

M. M. Aslan, C. Crofcheck, D. Tao, and M. P. Mengüç, "Evaluation of micro bubble size and gas hold up in two phase gas-liquid columns via scattered light measurements," J. Quant. Spectrosc. Radiat. Transf. 101, 527-539 (2006).
[CrossRef]

Venkata, P. G.

P. G. Venkata, M. M. Aslan, M. P. Mengüç, and G. Videen, "Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates," ASME J. Heat Transfer 129, 60-70 (2007).
[CrossRef]

M. Francoeur, P. G. Venkata, and M. P. Mengüç, "Sensitivity analysis for characterization of gold nanoparticles and 2D-agglomerates via surface plasmon scattering patterns," J. Quant. Spectrosc. Radiat. Transf. 106, 44-55 (2007).
[CrossRef]

Videen, G.

P. G. Venkata, M. M. Aslan, M. P. Mengüç, and G. Videen, "Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates," ASME J. Heat Transfer 129, 60-70 (2007).
[CrossRef]

G. Videen, M. M. Aslan, and M. P. Mengüç, "Characterization of metallic nanoparticles via surface wave scattering: A. Theoretical framework," J. Quant. Spectrosc. Radiat. Transf. 93, 195-206 (2005).
[CrossRef]

M. M. Aslan, M. P. Mengüç, and G. Videen, "Characterization of metallic nanoparticles via surface wave scattering: B. Physical concept and numerical experiments," J. Quant. Spectrosc. Radiat. Transf. 93, 207-217 (2005).
[CrossRef]

AIAA J. Thermophy. Heat Transfer (1)

M. P. Mengüç and S. Manickavasagam, "Inverse radiation problem in axisymmetric cylindrical media," AIAA J. Thermophy. Heat Transfer 7, 479-486 (1993).
[CrossRef]

Appl. Opt. (1)

ASME J. Heat Transfer (2)

M. P. Mengüç and P. Dutta, "Scattering tomography and application to sooting diffusion flames," ASME J. Heat Transfer 116, 144-151 (1994).
[CrossRef]

P. G. Venkata, M. M. Aslan, M. P. Mengüç, and G. Videen, "Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates," ASME J. Heat Transfer 129, 60-70 (2007).
[CrossRef]

Fuel (1)

M. P. Mengüç, S. Manickavasagam, and D. A. D'sa, "Determination of radiative properties of pulverized coal particles from experiments," Fuel 73, 613-625 (1994).
[CrossRef]

Int. J. Heat Mass Transfer (2)

B. M. Agarwal and M. P. Mengüç, "Single and multiple scattering of collimated radiation in an axisymmetric system," Int. J. Heat Mass Transfer 34, 633-647 (1991).
[CrossRef]

S. Subramaniam and M. P. Mengüç, "Solution of inverse radiation problem for inhomogeneous and anisotropically scattering medium using a Monte-Carlo technique," Int. J. Heat Mass Transfer 34, 253-266 (1991).
[CrossRef]

J. Nanopart. Res. (2)

C. Saltiel, Q. Chen, S. Manickavasagam, L. S. Schandler, R. W. Siegel, and M. P. Mengüç, "Identification of dispersion behavior of surface-treated nano-scale powders," J. Nanopart. Res. 6, 35-46 (2004).
[CrossRef]

M. M. Aslan, M. P. Mengüç, S. Manickavasagam, and C. Saltiel, "Size and shape prediction of colloidal metal oxide MgBaFeO particles from light scattering measurements," J. Nanopart. Res. 8, 981-994 (2006).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transf. (4)

G. Videen, M. M. Aslan, and M. P. Mengüç, "Characterization of metallic nanoparticles via surface wave scattering: A. Theoretical framework," J. Quant. Spectrosc. Radiat. Transf. 93, 195-206 (2005).
[CrossRef]

M. M. Aslan, M. P. Mengüç, and G. Videen, "Characterization of metallic nanoparticles via surface wave scattering: B. Physical concept and numerical experiments," J. Quant. Spectrosc. Radiat. Transf. 93, 207-217 (2005).
[CrossRef]

M. M. Aslan, C. Crofcheck, D. Tao, and M. P. Mengüç, "Evaluation of micro bubble size and gas hold up in two phase gas-liquid columns via scattered light measurements," J. Quant. Spectrosc. Radiat. Transf. 101, 527-539 (2006).
[CrossRef]

M. Francoeur, P. G. Venkata, and M. P. Mengüç, "Sensitivity analysis for characterization of gold nanoparticles and 2D-agglomerates via surface plasmon scattering patterns," J. Quant. Spectrosc. Radiat. Transf. 106, 44-55 (2007).
[CrossRef]

Other (2)

R. Charnigo and C. Srinivasan, "Local and global analytic curve estimation," submitted and available via http://www.ms.uky.edu/~richc/LACE.pdf.

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

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Figures (10)

Fig. 1
Fig. 1

(a) Schematic of the scattering from a group of spherical nanoparticles of diameters d m located at a distance h above a metallic thin film of thickness t. (b) Different patterns of agglomeration.

Fig. 2
Fig. 2

Reference curves (estimated scattering profiles) M 11 , p { 0 } ( θ ) , M 12 , p { 0 } ( θ ) , M 33 , p { 0 } ( θ ) , M 34 , p { 0 } ( θ ) for agglomeration levels p = 0 , 25 , 75 , 100 .

Fig. 3
Fig. 3

Reference curves (estimated first derivatives of scattering profiles) M 11 , p { 1 } ( θ ) , M 12 , p { 1 } ( θ ) , M 33 , p { 1 } ( θ ) , M 34 , p { 1 } ( θ ) for agglomeration levels p = 0 , 25 , 75 , 100 .

Fig. 4
Fig. 4

Reference curves (estimated second derivatives of scattering profiles) M 11 , p { 2 } ( θ ) , M 12 , p { 2 } ( θ ) , M 33 , p { 2 } ( θ ) , M 34 , p { 2 } ( θ ) for agglomeration levels p = 0 , 25 , 75 , 100 .

Fig. 5
Fig. 5

Reference curves (estimated third derivatives of scattering profiles) M 11 , p { 3 } ( θ ) , M 12 , p { 3 } ( θ ) , M 33 , p { 3 } ( θ ) , M 34 , p { 3 } ( θ ) for agglomeration levels p = 0 , 25 , 75 , 100 .

Fig. 6
Fig. 6

Reference curves (estimated fourth derivatives of scattering profiles) M 11 , p { 4 } ( θ ) , M 12 , p { 4 } ( θ ) , M 33 , p { 4 } ( θ ) , M 34 , p { 4 } ( θ ) for agglomeration levels p = 0 , 25 , 75 , 100 .

Fig. 7
Fig. 7

Reference curves (estimated fifth derivatives of scattering profiles) M 11 , p { 5 } ( θ ) , M 12 , p { 5 } ( θ ) , M 33 , p { 5 } ( θ ) , M 34 , p { 5 } ( θ ) for agglomeration levels p = 0 , 25 , 75 , 100 .

Fig. 8
Fig. 8

Data obtained under nonideal experimental conditions: M * 33 , 0 ( 1 ) through M * 33 , 0 ( 179 ) and M * 33 , 100 ( 1 ) through M * 33 , 100 ( 179 ) are shown for data sets 1 and 81, with and without noise.

Fig. 9
Fig. 9

Data obtained under nonideal experimental conditions: M * 34 , 0 ( 1 ) through M * 34 , 0 ( 179 ) and M * 34 , 100 ( 1 ) through M * 34 , 100 ( 179 ) are shown for data sets 1 and 81, with and without noise.

Fig. 10
Fig. 10

Solution of the inverse problem: M * 11 , 0 { 0 } ( θ ) for data set 81 is plotted along with the reference curves M 11 , 0 { 0 } ( θ ) , M 11 , 25 { 0 } ( θ ) , M 11 , 75 { 0 } ( θ ) , and M 11 , 100 { 0 } ( θ ) .

Tables (6)

Tables Icon

Table 1 Nonideal Conditions under which Data Sets Were Generated

Tables Icon

Table 2 Correct Classification Rates Using M 11 Alone

Tables Icon

Table 3 Correct Classification Rates Using M 12 Alone

Tables Icon

Table 4 Correct Classification Rates Using M 33 Alone

Tables Icon

Table 5 Correct Classification Rates Using M 34 Alone

Tables Icon

Table 6 Correct Classification Rates Using M 11 , M 12 , M 33 , and M 34

Equations (9)

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( I Q U V ) sca = 1 k 2 r 2 [ S 11 S 12 S 13 S 14 S 21 S 22 S 23 S 24 S 31 S 32 S 33 S 34 S 41 S 42 S 43 S 44 ] ( I Q U V ) inc ,
M obs ( θ ) = M true ( θ ) + ϵ .
Δ 11 { 0 } ( q , p ) = 0 180 M * 11 , q ( θ ) M 11 , p ( θ ) d θ { sup θ , p M 11 , p ( θ ) inf θ , p M 11 , p ( θ ) } ,
Δ 11 { d } ( q , p ) = 0 180 M * 11 , q { d } ( θ ) M 11 , p { d } ( θ ) d θ { sup θ , p M 11 , p { d } ( θ ) inf θ , p M 11 , p { d } ( θ ) }
q 11 { d } = arg min p Δ 11 { d } ( q , p ) ,
q 12 { d } = arg min p Δ 12 { d } ( q , p ) ,
q 33 { d } = arg min p Δ 33 { d } ( q , p ) ,
q 34 { d } = arg min p Δ 34 { d } ( q , p ) ,
q { d } = arg min p { Δ 11 { d } ( q , p ) + Δ 12 { d } ( q , p ) + Δ 33 { d } ( q , p ) + Δ 34 { d } ( q , p ) } .

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