Abstract

The Mie model is widely used to analyze light scattering from particulate aerosols. The Diesel particle scatterometer, for example, determines the size and optical properties of Diesel exhaust particles that are characterized by the measurement of three angle-dependent elements of the Mueller scattering matrix. These elements are then fitted by Mie calculations with a Levenburg–Marquardt optimization program. This approach has achieved good fits for most experimental data. However, in many cases, the predicted complex index of refraction was smaller than that for solid carbon. To understand this result and explain the experimental data, we present an assessment of the Mie model by use of a light-scattering model based on the coupled-dipole approximation. The results indicate that the Mie calculation can be used to determine the largest dimension of irregularly shaped particles at sizes characteristic of Diesel soot and, for particles of known refractive index, tables can be constructed to determine the average porosity of the particles from the predicted index of refraction.

© 2004 Optical Society of America

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References

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  1. C. A. Pope, D. Dockery, J. Schwartz, “Review of epidemiological evidence of health effects of particulate air pollution,” Inhalation Toxicol. 7, 4–18 (1995).
    [CrossRef]
  2. D. S. Shprentz, Breath-Taking: Premature Mortality Due to Particulate Air Pollution in 239 American Cities (National Resources Defense Council, Publications Department, New York, 1996).
  3. Environmental Protection Agency, “National air quality and emissions trends report,” EPA-454/R-95-011 (Environmental Protection Agency, Springfield, Va., 1995).
  4. W. H. Lipkea, J. H. Johnson, C. T. Vuk, “The physical and chemical character of diesel particulate emissions—Measurement techniques and fundamental considerations,” Paper 780108 (Society of Automotive Engineers, International, Warrendale, Pa., 1979).
  5. J. B. Heywood, Internal Combustion Engine Fundamentals (McGraw-Hill, New York, 1988).
  6. A. J. Hunt, M. S. Quinby-Hunt, I. G. Shepherd, “Polarized light scattering for Diesel exhaust particulate characterization,” in Proceedings of the Diesel Engine Emissions Reduction Workshop, J. Fairbanks, ed., DOE/EE-0191 (Department of Energy, Washington, D.C., 1999).
  7. A. J. Hunt, D. R. Huffman, “A new polarization-modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
    [CrossRef]
  8. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  9. R. J. Perry, A. J. Hunt, D. R. Huffman, “Experimental determination of Mueller scattering matrices for nonspherical particles,” Appl. Opt. 17, 2700–2710 (1978).
    [CrossRef] [PubMed]
  10. M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Polarized light scattering in the marine environment,” in Light Scattering by Non-Spherical Particles: Theory, Measurements, and Geophysical Applications, M. L. Mischenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 19.
    [CrossRef]
  11. M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Predicting polarization properties of marine aerosols,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 735–746 (1994).
    [CrossRef]
  12. M. S. Quinby-Hunt, L. L. Erskine, A. J. Hunt, “Polarized light scattering by aerosols in the marine atmospheric boundary layer,” Appl. Opt. 36, 5168–5184 (1997).
    [CrossRef] [PubMed]
  13. W. H. Press, B. R. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Fortran version) (Cambridge U. Press, Cambridge, UK, 1989).
  14. D. C. Sigla, G. W. Smith, Particulate Carbon (Plenum, New York, 1981).
    [CrossRef]
  15. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  16. P. G. Hull, M. S. Quinby-Hunt, D. B. Shapiro, “Coupled dipole approximation: predicting scattering by nonspherical marine organisms,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. SPIE1537, 21–30 (1991).
    [CrossRef]
  17. S. B. Singham, G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled-dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
    [CrossRef]

1997 (1)

1995 (1)

C. A. Pope, D. Dockery, J. Schwartz, “Review of epidemiological evidence of health effects of particulate air pollution,” Inhalation Toxicol. 7, 4–18 (1995).
[CrossRef]

1986 (1)

S. B. Singham, G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled-dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

1978 (1)

1973 (2)

A. J. Hunt, D. R. Huffman, “A new polarization-modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Bohren, C. F.

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

Dockery, D.

C. A. Pope, D. Dockery, J. Schwartz, “Review of epidemiological evidence of health effects of particulate air pollution,” Inhalation Toxicol. 7, 4–18 (1995).
[CrossRef]

Erskine, L. L.

Flannery, B. R.

W. H. Press, B. R. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Fortran version) (Cambridge U. Press, Cambridge, UK, 1989).

Heywood, J. B.

J. B. Heywood, Internal Combustion Engine Fundamentals (McGraw-Hill, New York, 1988).

Huffman, D. R.

R. J. Perry, A. J. Hunt, D. R. Huffman, “Experimental determination of Mueller scattering matrices for nonspherical particles,” Appl. Opt. 17, 2700–2710 (1978).
[CrossRef] [PubMed]

A. J. Hunt, D. R. Huffman, “A new polarization-modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

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

Hull, P. G.

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Predicting polarization properties of marine aerosols,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 735–746 (1994).
[CrossRef]

P. G. Hull, M. S. Quinby-Hunt, D. B. Shapiro, “Coupled dipole approximation: predicting scattering by nonspherical marine organisms,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. SPIE1537, 21–30 (1991).
[CrossRef]

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Polarized light scattering in the marine environment,” in Light Scattering by Non-Spherical Particles: Theory, Measurements, and Geophysical Applications, M. L. Mischenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 19.
[CrossRef]

Hunt, A. J.

M. S. Quinby-Hunt, L. L. Erskine, A. J. Hunt, “Polarized light scattering by aerosols in the marine atmospheric boundary layer,” Appl. Opt. 36, 5168–5184 (1997).
[CrossRef] [PubMed]

R. J. Perry, A. J. Hunt, D. R. Huffman, “Experimental determination of Mueller scattering matrices for nonspherical particles,” Appl. Opt. 17, 2700–2710 (1978).
[CrossRef] [PubMed]

A. J. Hunt, D. R. Huffman, “A new polarization-modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Polarized light scattering in the marine environment,” in Light Scattering by Non-Spherical Particles: Theory, Measurements, and Geophysical Applications, M. L. Mischenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 19.
[CrossRef]

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Predicting polarization properties of marine aerosols,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 735–746 (1994).
[CrossRef]

A. J. Hunt, M. S. Quinby-Hunt, I. G. Shepherd, “Polarized light scattering for Diesel exhaust particulate characterization,” in Proceedings of the Diesel Engine Emissions Reduction Workshop, J. Fairbanks, ed., DOE/EE-0191 (Department of Energy, Washington, D.C., 1999).

Johnson, J. H.

W. H. Lipkea, J. H. Johnson, C. T. Vuk, “The physical and chemical character of diesel particulate emissions—Measurement techniques and fundamental considerations,” Paper 780108 (Society of Automotive Engineers, International, Warrendale, Pa., 1979).

Lipkea, W. H.

W. H. Lipkea, J. H. Johnson, C. T. Vuk, “The physical and chemical character of diesel particulate emissions—Measurement techniques and fundamental considerations,” Paper 780108 (Society of Automotive Engineers, International, Warrendale, Pa., 1979).

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Perry, R. J.

Pope, C. A.

C. A. Pope, D. Dockery, J. Schwartz, “Review of epidemiological evidence of health effects of particulate air pollution,” Inhalation Toxicol. 7, 4–18 (1995).
[CrossRef]

Press, W. H.

W. H. Press, B. R. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Fortran version) (Cambridge U. Press, Cambridge, UK, 1989).

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Quinby-Hunt, M. S.

M. S. Quinby-Hunt, L. L. Erskine, A. J. Hunt, “Polarized light scattering by aerosols in the marine atmospheric boundary layer,” Appl. Opt. 36, 5168–5184 (1997).
[CrossRef] [PubMed]

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Predicting polarization properties of marine aerosols,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 735–746 (1994).
[CrossRef]

A. J. Hunt, M. S. Quinby-Hunt, I. G. Shepherd, “Polarized light scattering for Diesel exhaust particulate characterization,” in Proceedings of the Diesel Engine Emissions Reduction Workshop, J. Fairbanks, ed., DOE/EE-0191 (Department of Energy, Washington, D.C., 1999).

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Polarized light scattering in the marine environment,” in Light Scattering by Non-Spherical Particles: Theory, Measurements, and Geophysical Applications, M. L. Mischenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 19.
[CrossRef]

P. G. Hull, M. S. Quinby-Hunt, D. B. Shapiro, “Coupled dipole approximation: predicting scattering by nonspherical marine organisms,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. SPIE1537, 21–30 (1991).
[CrossRef]

Salzman, G. C.

S. B. Singham, G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled-dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

Schwartz, J.

C. A. Pope, D. Dockery, J. Schwartz, “Review of epidemiological evidence of health effects of particulate air pollution,” Inhalation Toxicol. 7, 4–18 (1995).
[CrossRef]

Shapiro, D. B.

P. G. Hull, M. S. Quinby-Hunt, D. B. Shapiro, “Coupled dipole approximation: predicting scattering by nonspherical marine organisms,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. SPIE1537, 21–30 (1991).
[CrossRef]

Shepherd, I. G.

A. J. Hunt, M. S. Quinby-Hunt, I. G. Shepherd, “Polarized light scattering for Diesel exhaust particulate characterization,” in Proceedings of the Diesel Engine Emissions Reduction Workshop, J. Fairbanks, ed., DOE/EE-0191 (Department of Energy, Washington, D.C., 1999).

Shprentz, D. S.

D. S. Shprentz, Breath-Taking: Premature Mortality Due to Particulate Air Pollution in 239 American Cities (National Resources Defense Council, Publications Department, New York, 1996).

Sigla, D. C.

D. C. Sigla, G. W. Smith, Particulate Carbon (Plenum, New York, 1981).
[CrossRef]

Singham, S. B.

S. B. Singham, G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled-dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

Smith, G. W.

D. C. Sigla, G. W. Smith, Particulate Carbon (Plenum, New York, 1981).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. R. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Fortran version) (Cambridge U. Press, Cambridge, UK, 1989).

Vetterling, W. T.

W. H. Press, B. R. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Fortran version) (Cambridge U. Press, Cambridge, UK, 1989).

Vuk, C. T.

W. H. Lipkea, J. H. Johnson, C. T. Vuk, “The physical and chemical character of diesel particulate emissions—Measurement techniques and fundamental considerations,” Paper 780108 (Society of Automotive Engineers, International, Warrendale, Pa., 1979).

Appl. Opt. (2)

Astrophys. J. (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Inhalation Toxicol. (1)

C. A. Pope, D. Dockery, J. Schwartz, “Review of epidemiological evidence of health effects of particulate air pollution,” Inhalation Toxicol. 7, 4–18 (1995).
[CrossRef]

J. Chem. Phys. (1)

S. B. Singham, G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled-dipole approximation,” J. Chem. Phys. 84, 2658–2667 (1986).
[CrossRef]

Rev. Sci. Instrum. (1)

A. J. Hunt, D. R. Huffman, “A new polarization-modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

Other (11)

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

W. H. Press, B. R. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Fortran version) (Cambridge U. Press, Cambridge, UK, 1989).

D. C. Sigla, G. W. Smith, Particulate Carbon (Plenum, New York, 1981).
[CrossRef]

D. S. Shprentz, Breath-Taking: Premature Mortality Due to Particulate Air Pollution in 239 American Cities (National Resources Defense Council, Publications Department, New York, 1996).

Environmental Protection Agency, “National air quality and emissions trends report,” EPA-454/R-95-011 (Environmental Protection Agency, Springfield, Va., 1995).

W. H. Lipkea, J. H. Johnson, C. T. Vuk, “The physical and chemical character of diesel particulate emissions—Measurement techniques and fundamental considerations,” Paper 780108 (Society of Automotive Engineers, International, Warrendale, Pa., 1979).

J. B. Heywood, Internal Combustion Engine Fundamentals (McGraw-Hill, New York, 1988).

A. J. Hunt, M. S. Quinby-Hunt, I. G. Shepherd, “Polarized light scattering for Diesel exhaust particulate characterization,” in Proceedings of the Diesel Engine Emissions Reduction Workshop, J. Fairbanks, ed., DOE/EE-0191 (Department of Energy, Washington, D.C., 1999).

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Polarized light scattering in the marine environment,” in Light Scattering by Non-Spherical Particles: Theory, Measurements, and Geophysical Applications, M. L. Mischenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 19.
[CrossRef]

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Predicting polarization properties of marine aerosols,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 735–746 (1994).
[CrossRef]

P. G. Hull, M. S. Quinby-Hunt, D. B. Shapiro, “Coupled dipole approximation: predicting scattering by nonspherical marine organisms,” in Underwater Imaging, Photography, and Visibility, R. W. Spinrad, ed., Proc. SPIE1537, 21–30 (1991).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Schematic of the DPS. The baseplate is approximately 2 ft × 2 ft.

Fig. 2
Fig. 2

Mueller matrix elements for scattering from a single particle and an orientation average of the same particle. A Mie calculation (dashed line) is also shown for comparison.

Fig. 3
Fig. 3

Comparison of three matrix elements for a sphere with a size parameter of 1.76 by use of the CD calculation and a Mie calculation. The Mie calculation is the solid line without markers, and the CD calculation is marked by open circles.

Fig. 4
Fig. 4

Sketch of the arrangement of the dipoles on a lattice of 81 positions as dipoles are removed by random selection. The circular line represents the radius of a sphere used in the Mie calculation that best fits the scattering matrix elements.

Fig. 5
Fig. 5

Results of a series of CD calculations of the scattering matrix elements for a sphere with a size parameter of 1.76. Dipoles were removed from the lattice by random selection and the matrix elements recalculated. A density of 100% is a sphere with a completely filled lattice.

Fig. 6
Fig. 6

Index of refraction and absorption of a sphere. The figure shows the decrease in the index of refraction and absorption determined by the L-M fit to a Mie sphere as the CD sphere becomes more porous. The effective radius determined by the L-M fit is approximately that of the CD sphere, and the radius predicted by the model does not change appreciably as the sphere becomes more porous.

Fig. 7
Fig. 7

Results of a series of CD calculations of the scattering matrix elements for an ellipsoid with a ratio of major to minor axis of 2.0. The size parameter of a sphere of equivalent volume is 1.89. Dipoles are removed from the lattice by random selection and the matrix elements recalculated. A density of 100% is an ellipsoid with a completely filled lattice.

Fig. 8
Fig. 8

Index of refraction and absorption of an ellipsoid with a ratio of the major to minor axis of 2.0. The figure shows the decrease in the index of refraction and absorption determined by the L-M fit to a sphere as the ellipsoid becomes more porous. The effective radius determined by the L-M fit is approximately that of a sphere made up of the same number of dipoles. The radius predicted by the model does not change appreciably as the ellipsoid becomes more porous.

Fig. 9
Fig. 9

(a) Matrix element S 22 for ellipsoids of various aspect ratios. The size parameter has been fixed at 2.0, and all ellipsoids have a 100% filled lattice. (b) S 22 for ellipsoids of different fill densities. The size parameter has been fixed at 2.0 and the aspect ratio at 4.0. (c) S 22 for ellipsoids with varying size parameters. All ellipsoids have a 100% filled lattice, and the aspect ratio is fixed at 4.0.

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