Abstract

The range of validity of the Rayleigh–Debye–Gans approximation for the optical cross sections of fractal aggregates (RDG-FA) that are formed by uniform small particles was evaluated in comparison with the integral equation formulation for scattering (IEFS), which accounts for the effects of multiple scattering and self-interaction. Numerical simulations were performed to create aggregates that exhibit mass fractallike characteristics with a wide range of particle and aggregate sizes and morphologies, including x p = 0.01–1.0, |m − 1| = 0.1–2.0, N = 16–256, and D f = 1.0–3.0. The percent differences between both scattering theories were presented as error contour charts in the |m − 1|x p domains for various size aggregates, emphasizing fractal properties representative of diffusion-limited cluster–cluster aggregation. These charts conveniently identified the regions in which the differences were less than 10%, between 10% and 30%, and more than 30% for easy to use general guidelines for suitability of the RDG-FA theory in any scattering applications of interest, such as laser-based particulate diagnostics. Various types of aggregate geometry ranging from straight chains (D f ≈ 1.0) to compact clusters (D f ≈ 3.0) were also considered for generalization of the findings. For the present computational conditions, the RDG-FA theory yielded accurate predictions to within 10% for |m − 1| to approximately 1 or more as long as the primary particles in aggregates were within the Rayleigh scattering limit (x p ≤ 0.3). Additionally, the effect of fractal dimension on the performance of the RDG-FA was generally found to be insignificant. The results suggested that the RDG-FA theory is a reasonable approximation for optics of a wide range of fractal aggregates, considerably extending its domain of applicability.

© 1996 Optical Society of America

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References

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  1. Ü. Ö. Köylü, G. M. Faeth, “Radiative properties of flame-generated soot,” J. Heat Transfer 115, 409–417 (1993).
    [CrossRef]
  2. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983).
  3. R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).
  4. Ü. Ö. Köylü, G. M. Faeth, “Structure of overfire soot in buoyant turbulent diffusion flames at long residence times,” Combust. Flame 89, 140–156 (1992).
    [CrossRef]
  5. Ü. Ö. Köylü, G. M. Faeth, T. L. Farias, M. G. Carvalho, “Fractal and projected structure properties of soot aggregates,” Combust. Flame 100, 621–633 (1995).
    [CrossRef]
  6. R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
    [CrossRef]
  7. R. A. Dobbins, C. M. Megaridis, “Absorption and scattering of light by polydisperse aggregates,” Appl. Opt. 30, 4747–4754 (1991).
    [CrossRef] [PubMed]
  8. Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant turbulent diffusion flames at long residence times,” J. Heat Transfer 116, 152–159 (1994).
    [CrossRef]
  9. Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant laminar diffusion flames,” J. Heat Transfer 116, 971–979 (1994).
    [CrossRef]
  10. C. M. Sorensen, J. Cai, N. Lu, “Light-scattering measurements of monomer size, monomers per aggregate, and fractal dimension for soot aggregates in flames,” Appl. Opt. 31, 6547–6557 (1992).
    [CrossRef] [PubMed]
  11. R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Analysis of light scattering from soot using optical cross sections for aggregates,” in Proceedings of the Twenty-Third Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, Pa., 1990), pp. 1525–1532.
  12. S. B. Singham, C. F. Bohren, “Scattering of unpolarized and polarized light by particle aggregates of different size and fractal dimension,” Langmuir 9, 1431–1435 (1993).
    [CrossRef]
  13. G. W. Mulholland, C. F. Bohren, K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
    [CrossRef]
  14. T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, G. M. Faeth, “Computational evaluation of approximate Rayleigh-Debye-Gans/fractal-aggregate theory for the absorption and scattering properties of soot,” J. Heat Transfer 117, 152–159 (1995).
    [CrossRef]
  15. A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London 366, 111–127 (1979).
    [CrossRef]
  16. D. S. Saxon, “Lectures on the scattering of light,” in The UCLA International Conference on Radiation and Remote Probing of the Atmosphere, J. G. Kuriyan, ed. (U. California Press, Los Angeles, Calif., 1974), pp. 227–308.
  17. J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
    [CrossRef]
  18. W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
    [CrossRef]
  19. T. L. Farias, Ü. Ö. Köylü, M. G. Carvalho, “Effects of polydispersity of primary particle and aggregate sizes on radiative properties of simulated soot,” J. Quant. Spectrosc. Radiat. Transfer 55, 357–371 (1995).
    [CrossRef]
  20. T. L. Farias, “Evaluation of light scattering theories for fractal aggregates,” Ph.D. dissertation (Instituto Superior Técnico, Lisbon, Portugal, 1996), in preparation.
  21. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  22. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  23. Ü. Ö. Köylü, Y. Xing, D. E. Rosner, “Fractal morphology analysis of combustion-generated aggregates using angular light scattering and electron microscope images,” Langmuir 11, 4848–4854 (1995).
    [CrossRef]
  24. M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
    [CrossRef]

1995 (4)

T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, G. M. Faeth, “Computational evaluation of approximate Rayleigh-Debye-Gans/fractal-aggregate theory for the absorption and scattering properties of soot,” J. Heat Transfer 117, 152–159 (1995).
[CrossRef]

T. L. Farias, Ü. Ö. Köylü, M. G. Carvalho, “Effects of polydispersity of primary particle and aggregate sizes on radiative properties of simulated soot,” J. Quant. Spectrosc. Radiat. Transfer 55, 357–371 (1995).
[CrossRef]

Ü. Ö. Köylü, Y. Xing, D. E. Rosner, “Fractal morphology analysis of combustion-generated aggregates using angular light scattering and electron microscope images,” Langmuir 11, 4848–4854 (1995).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, T. L. Farias, M. G. Carvalho, “Fractal and projected structure properties of soot aggregates,” Combust. Flame 100, 621–633 (1995).
[CrossRef]

1994 (4)

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant turbulent diffusion flames at long residence times,” J. Heat Transfer 116, 152–159 (1994).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant laminar diffusion flames,” J. Heat Transfer 116, 971–979 (1994).
[CrossRef]

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

G. W. Mulholland, C. F. Bohren, K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

1993 (2)

S. B. Singham, C. F. Bohren, “Scattering of unpolarized and polarized light by particle aggregates of different size and fractal dimension,” Langmuir 9, 1431–1435 (1993).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Radiative properties of flame-generated soot,” J. Heat Transfer 115, 409–417 (1993).
[CrossRef]

1992 (3)

Ü. Ö. Köylü, G. M. Faeth, “Structure of overfire soot in buoyant turbulent diffusion flames at long residence times,” Combust. Flame 89, 140–156 (1992).
[CrossRef]

C. M. Sorensen, J. Cai, N. Lu, “Light-scattering measurements of monomer size, monomers per aggregate, and fractal dimension for soot aggregates in flames,” Appl. Opt. 31, 6547–6557 (1992).
[CrossRef] [PubMed]

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

1991 (1)

1988 (1)

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

1986 (1)

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

1979 (1)

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London 366, 111–127 (1979).
[CrossRef]

Berry, M. V.

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Bohren, C. F.

G. W. Mulholland, C. F. Bohren, K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

S. B. Singham, C. F. Bohren, “Scattering of unpolarized and polarized light by particle aggregates of different size and fractal dimension,” Langmuir 9, 1431–1435 (1993).
[CrossRef]

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

Botet, R.

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

Cai, J.

Carvalho, M. G.

Ü. Ö. Köylü, G. M. Faeth, T. L. Farias, M. G. Carvalho, “Fractal and projected structure properties of soot aggregates,” Combust. Flame 100, 621–633 (1995).
[CrossRef]

T. L. Farias, Ü. Ö. Köylü, M. G. Carvalho, “Effects of polydispersity of primary particle and aggregate sizes on radiative properties of simulated soot,” J. Quant. Spectrosc. Radiat. Transfer 55, 357–371 (1995).
[CrossRef]

T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, G. M. Faeth, “Computational evaluation of approximate Rayleigh-Debye-Gans/fractal-aggregate theory for the absorption and scattering properties of soot,” J. Heat Transfer 117, 152–159 (1995).
[CrossRef]

Charalampopoulos, T. T.

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

Dobbins, R. A.

R. A. Dobbins, C. M. Megaridis, “Absorption and scattering of light by polydisperse aggregates,” Appl. Opt. 30, 4747–4754 (1991).
[CrossRef] [PubMed]

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Analysis of light scattering from soot using optical cross sections for aggregates,” in Proceedings of the Twenty-Third Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, Pa., 1990), pp. 1525–1532.

Faeth, G. M.

T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, G. M. Faeth, “Computational evaluation of approximate Rayleigh-Debye-Gans/fractal-aggregate theory for the absorption and scattering properties of soot,” J. Heat Transfer 117, 152–159 (1995).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, T. L. Farias, M. G. Carvalho, “Fractal and projected structure properties of soot aggregates,” Combust. Flame 100, 621–633 (1995).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant turbulent diffusion flames at long residence times,” J. Heat Transfer 116, 152–159 (1994).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant laminar diffusion flames,” J. Heat Transfer 116, 971–979 (1994).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Radiative properties of flame-generated soot,” J. Heat Transfer 115, 409–417 (1993).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Structure of overfire soot in buoyant turbulent diffusion flames at long residence times,” Combust. Flame 89, 140–156 (1992).
[CrossRef]

Farias, T. L.

Ü. Ö. Köylü, G. M. Faeth, T. L. Farias, M. G. Carvalho, “Fractal and projected structure properties of soot aggregates,” Combust. Flame 100, 621–633 (1995).
[CrossRef]

T. L. Farias, Ü. Ö. Köylü, M. G. Carvalho, “Effects of polydispersity of primary particle and aggregate sizes on radiative properties of simulated soot,” J. Quant. Spectrosc. Radiat. Transfer 55, 357–371 (1995).
[CrossRef]

T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, G. M. Faeth, “Computational evaluation of approximate Rayleigh-Debye-Gans/fractal-aggregate theory for the absorption and scattering properties of soot,” J. Heat Transfer 117, 152–159 (1995).
[CrossRef]

T. L. Farias, “Evaluation of light scattering theories for fractal aggregates,” Ph.D. dissertation (Instituto Superior Técnico, Lisbon, Portugal, 1996), in preparation.

Fuller, K. A.

G. W. Mulholland, C. F. Bohren, K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

Huffman, D. R.

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

Jones, A. R.

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London 366, 111–127 (1979).
[CrossRef]

Jullien, R.

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

Kerker, M.

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

Köylü, Ü. Ö.

Ü. Ö. Köylü, Y. Xing, D. E. Rosner, “Fractal morphology analysis of combustion-generated aggregates using angular light scattering and electron microscope images,” Langmuir 11, 4848–4854 (1995).
[CrossRef]

T. L. Farias, Ü. Ö. Köylü, M. G. Carvalho, “Effects of polydispersity of primary particle and aggregate sizes on radiative properties of simulated soot,” J. Quant. Spectrosc. Radiat. Transfer 55, 357–371 (1995).
[CrossRef]

T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, G. M. Faeth, “Computational evaluation of approximate Rayleigh-Debye-Gans/fractal-aggregate theory for the absorption and scattering properties of soot,” J. Heat Transfer 117, 152–159 (1995).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, T. L. Farias, M. G. Carvalho, “Fractal and projected structure properties of soot aggregates,” Combust. Flame 100, 621–633 (1995).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant turbulent diffusion flames at long residence times,” J. Heat Transfer 116, 152–159 (1994).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant laminar diffusion flames,” J. Heat Transfer 116, 971–979 (1994).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Radiative properties of flame-generated soot,” J. Heat Transfer 115, 409–417 (1993).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Structure of overfire soot in buoyant turbulent diffusion flames at long residence times,” Combust. Flame 89, 140–156 (1992).
[CrossRef]

Ku, J. C.

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

Lou, W.

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

Lu, N.

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983).

Megaridis, C. M.

Mountain, R. D.

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

Mulholland, G. W.

G. W. Mulholland, C. F. Bohren, K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

Percival, I. C.

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Rosner, D. E.

Ü. Ö. Köylü, Y. Xing, D. E. Rosner, “Fractal morphology analysis of combustion-generated aggregates using angular light scattering and electron microscope images,” Langmuir 11, 4848–4854 (1995).
[CrossRef]

Santoro, R. J.

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Analysis of light scattering from soot using optical cross sections for aggregates,” in Proceedings of the Twenty-Third Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, Pa., 1990), pp. 1525–1532.

Saxon, D. S.

D. S. Saxon, “Lectures on the scattering of light,” in The UCLA International Conference on Radiation and Remote Probing of the Atmosphere, J. G. Kuriyan, ed. (U. California Press, Los Angeles, Calif., 1974), pp. 227–308.

Semerjian, H. G.

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Analysis of light scattering from soot using optical cross sections for aggregates,” in Proceedings of the Twenty-Third Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, Pa., 1990), pp. 1525–1532.

Shim, K.-H.

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

Singham, S. B.

S. B. Singham, C. F. Bohren, “Scattering of unpolarized and polarized light by particle aggregates of different size and fractal dimension,” Langmuir 9, 1431–1435 (1993).
[CrossRef]

Sorensen, C. M.

Xing, Y.

Ü. Ö. Köylü, Y. Xing, D. E. Rosner, “Fractal morphology analysis of combustion-generated aggregates using angular light scattering and electron microscope images,” Langmuir 11, 4848–4854 (1995).
[CrossRef]

Appl. Opt. (2)

Combust. Flame (2)

Ü. Ö. Köylü, G. M. Faeth, “Structure of overfire soot in buoyant turbulent diffusion flames at long residence times,” Combust. Flame 89, 140–156 (1992).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, T. L. Farias, M. G. Carvalho, “Fractal and projected structure properties of soot aggregates,” Combust. Flame 100, 621–633 (1995).
[CrossRef]

J. Heat Transfer (4)

Ü. Ö. Köylü, G. M. Faeth, “Radiative properties of flame-generated soot,” J. Heat Transfer 115, 409–417 (1993).
[CrossRef]

T. L. Farias, M. G. Carvalho, Ü. Ö. Köylü, G. M. Faeth, “Computational evaluation of approximate Rayleigh-Debye-Gans/fractal-aggregate theory for the absorption and scattering properties of soot,” J. Heat Transfer 117, 152–159 (1995).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant turbulent diffusion flames at long residence times,” J. Heat Transfer 116, 152–159 (1994).
[CrossRef]

Ü. Ö. Köylü, G. M. Faeth, “Optical properties of soot in buoyant laminar diffusion flames,” J. Heat Transfer 116, 971–979 (1994).
[CrossRef]

J. Phys. D (1)

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

T. L. Farias, Ü. Ö. Köylü, M. G. Carvalho, “Effects of polydispersity of primary particle and aggregate sizes on radiative properties of simulated soot,” J. Quant. Spectrosc. Radiat. Transfer 55, 357–371 (1995).
[CrossRef]

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

Langmuir (4)

Ü. Ö. Köylü, Y. Xing, D. E. Rosner, “Fractal morphology analysis of combustion-generated aggregates using angular light scattering and electron microscope images,” Langmuir 11, 4848–4854 (1995).
[CrossRef]

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

S. B. Singham, C. F. Bohren, “Scattering of unpolarized and polarized light by particle aggregates of different size and fractal dimension,” Langmuir 9, 1431–1435 (1993).
[CrossRef]

G. W. Mulholland, C. F. Bohren, K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10, 2533–2546 (1994).
[CrossRef]

Opt. Acta (1)

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Proc. R. Soc. London (1)

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London 366, 111–127 (1979).
[CrossRef]

Other (7)

D. S. Saxon, “Lectures on the scattering of light,” in The UCLA International Conference on Radiation and Remote Probing of the Atmosphere, J. G. Kuriyan, ed. (U. California Press, Los Angeles, Calif., 1974), pp. 227–308.

T. L. Farias, “Evaluation of light scattering theories for fractal aggregates,” Ph.D. dissertation (Instituto Superior Técnico, Lisbon, Portugal, 1996), in preparation.

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

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

R. A. Dobbins, R. J. Santoro, H. G. Semerjian, “Analysis of light scattering from soot using optical cross sections for aggregates,” in Proceedings of the Twenty-Third Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, Pa., 1990), pp. 1525–1532.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983).

R. Jullien, R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987).

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Figures (8)

Fig. 1
Fig. 1

Percent deviation contours for the accuracy domains of the RDG-FA approximation for predicting the absorption cross sections of various size DLCC aggregates (D f ≈ 1.8). In region I, the RDG-FA and IEFS theories agree to within 10%; the differences are between 10% and 30% in region II whereas they are greater than 30% in region III.

Fig. 2
Fig. 2

Contour charts for the accuracy domains of the RDG-FA theory for predicting the total scattering cross sections of various size DLCC aggregates. Definitions of the regions are similar to Fig. 1.

Fig. 3
Fig. 3

Contour charts for the accuracy domains of the RDG-FA theory for predicting the vertically polarized angular scattering cross sections at θ = 0° of various size DLCC aggregates. Definitions of the regions are similar to Fig. 1.

Fig. 4
Fig. 4

Same as Fig. 3 except for θ = 45°.

Fig. 5
Fig. 5

Same as Fig. 3 except for θ = 90°.

Fig. 6
Fig. 6

Effect of fractal dimension D f on the accuracy of the RDG-FA theory for predicting the various optical cross sections of fractallike aggregates with |m − 1| = 0.75, x p = 0.3, and N = 64. The simulated aggregate is a straight chain for D f = 1.0 and a compact cube for D f = 3.0.

Fig. 7
Fig. 7

Error contour charts for the accuracy domains of the RDG-FA theory for predicting the absorption cross sections of various aggregate morphologies with 1 ≤ D f ≤ 3. Definitions of the regions are similar to Fig. 1.

Fig. 8
Fig. 8

Error contour charts for the accuracy domains of the RDG-FA theory for predicting the total scattering cross sections of various aggregate morphologies with 1 ≤ D f ≤ 3. Definitions of the regions are similar to Fig. 1.

Equations (11)

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E j = ( 3 m 2 + 2 ) E inc , j + i ( m 2 - 1 m 2 + 2 ) × x p 2 j 1 ( x p ) k = 1 , j N T ¯ j k E k + s j E j ; j = 1 , 2 , , N ,
C abs = 4 π k 2 x p 2 j 1 ( x p ) Im ( m 2 - 1 ) j = 1 N E j 2 ,
C sca = 4 π 3 k 2 x p 4 j 1 2 ( x p ) m 2 - 1 2 j = 1 N k = 1 N E j Re ( T ¯ j k ) E k * ,
C p p ( θ , ϕ ) = 1 k 2 x p 4 j 1 2 ( x p ) m 2 - 1 2 | j = 1 N exp ( - i k r j cos β j ) ( E j , θ θ ^ + E j , ϕ ϕ ^ ) p p | 2 ,
C ext = 4 π k 2 x p 2 j 1 ( x p ) Im [ ( m 2 - 1 ) j = 1 N E inc * · E j ] .
E j = ( 3 m 2 + 2 ) E inc , j .
C abs = N 4 π x p 3 k 2 Im ( m 2 - 1 m 2 + 2 ) = N C abs p ,
C sca = N 2 8 π x p 6 3 k 2 | m 2 - 1 m 2 + 2 | 2 g ( x a ) = N 2 C sca p g ( x a ) ,
C v v = N 2 x p 6 k 2 | m 2 - 1 m 2 + 2 | 2 f ( w a ) = N 2 C v v p f ( w a ) ,
f ( w a ) = { exp ( - w a 2 3 ) ,             Guinier regime w a - D f ,             power - law regime ,
N = k f ( R g d p ) D f .

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