Abstract

This work uses the discrete dipole approximation (DDA) to examine the internal electric field within a simulated carbon soot fractal aggregate in fixed and random orientations. For fixed orientations, deviations of the internal field magnitude up to ±50% from that assumed by the Rayleigh–Debye–Gans Approximation (RDGA) are found. Given the refractive index of the aggregate monomers and conditions for the validity of the approximation, such large deviations are no surprise. Yet despite this deviation, the far-field scattered intensity from such aggregates agrees surprisingly well with that described by the RDGA. Moreover, if the average over an ensemble of many random aggregate-orientations is calculated, both the DDA and RDGA scattered intensities obey the well-known power-law functionality in terms of the scattering wave vector and show a forward-angle intensity-maximum proportional to the square of the number of monomers. The explanation for this lies in the over and under estimations made by the approximation of the internal field, which apparently mostly cancel upon integration to yield the scattered intensity. It is shown that this error cancellation is related to the fractal structure of the aggregate and that the agreement between the DDA and RDGA improves with aggregates of increasing size provided the fractal dimension is less than two. Overall, the analysis suggests that both the special fractal character of the aggregate and its orientational averaging is important to account for the experimentally observed validity of the RDGA despite its poor description of the internal fields.

© 2013 Optical Society of America

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  1. T. C. Bond and R. W. Bergstrom, “Light absorption by carbonaceous particles: an investigative review,” Aerosol Sci. Technol. 40, 27–67 (2006).
    [CrossRef]
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    [CrossRef]
  3. L. Liu and M. I. Mishchenko, “Effects of aggregation on scattering and radiative properties of soot aerosols,” J. Geophys. Res. 110 D11211 (2005).
    [CrossRef]
  4. M. V. Berry and I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
    [CrossRef]
  5. T. L. Farias, Ü. Ö. Köylü, and M. G. Carvalho, “Range of validity of the Rayleigh–Debye–Gans theory for optics of fractal aggregates,” Appl. Opt. 35, 6560–6567 (1996).
    [CrossRef]
  6. G. Wang and C. M. Sorensen, “Experimental test of the Rayleigh–Debye–Gans theory for light scattering by fractal aggregates,” Appl. Opt. 41, 4645–4651 (2002).
    [CrossRef]
  7. H. Y. Chen, M. F. Iskander, and J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke,” J. Mod. Opt. 37, 171–181 (1990).
    [CrossRef]
  8. Y. Zhao and M. Lin, “Assessment of two fractal scattering models for the prediction of the optical characteristics of soot aggregates,” J. Quant. Spectrosc. Radiat. Transfer 110, 315–322 (2009).
    [CrossRef]
  9. R. Dhaubhadel, C. S. Gerving, A. Chakrabarti, and C. M. Sorensen, “Aerosol gelation: synthesis of a novel, lightweight, high specific surface area material,” Aerosol Sci. Technol. 41, 804–810 (2007).
    [CrossRef]
  10. F. G. Pierce, “Aggregation in colloids and aerosols,” Ph.D. dissertation (Kansas State University, Manhattan, Kansas, 2007).
  11. C. M. Sorensen and G. C. Roberts, “The prefactor of fractal aggregates,” J. Colloid Interface Sci. 186, 447–452 (1997).
    [CrossRef]
  12. S. P. Kearney and F. Pierce, “Evidence of soot superaggregates in a turbulent pool fire,” Combust. Flame 159, 3191–3198 (2012).
    [CrossRef]
  13. C. M. Sorensen, W. Kim, D. Fry, D. Shi, and A. Chakrabarti, “Observations of soot superaggregates with a fractal dimension of 2.6 in laminar acetylene/air diffusion flames,” Langmuir 19, 7560–7563 (2003).
    [CrossRef]
  14. R. Dhaubhadel, F. Pierce, A. Chakrabarti, and C. M. Sorensen, “Hybrid superaggregate morphology as a result of aggregation in a cluster-dense aerosol,” Phys. Rev. E 73, 011404 (2006).
    [CrossRef]
  15. A. M. Brasil, T. L. Farias, M. G. Carvalho, and U. O. Koylu, “Numerical characterization of the morphology of aggregated particles,” Aerosol Sci. Technol. 32, 489–508 (2001).
    [CrossRef]
  16. H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, and J. F. Merklin, “In situ optical structure factor measurement of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
    [CrossRef]
  17. C. M. Sorensen, J. Cai, and 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]
  18. K. C. Smyth and C. R. Shaddix, “The elusive history of m=1.57−0.56i for the refractive index of soot,” Combust. Flame 107, 314–320 (1996).
    [CrossRef]
  19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
  20. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).
  21. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
    [CrossRef]
  22. M. J. Berg, “Power-law patterns in electromagnetic scattering: a selected review and recent progress,” J. Quant. Spectrosc. Radiat. Transfer 113, 2292–2309 (2012).
    [CrossRef]
  23. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Reflection symmetry of a sphere’s internal field and its consequences on scattering: a microphysical approach,” J. Opt. Soc. Am. A 25, 98–107 (2008).
    [CrossRef]
  24. M. J. Berg, “Reflection symmetry of a sphere’s internal field and its consequences on scattering: behavior of the Stokes parameters,” in Polarimetric Detection, Characterization and Remote Sensing, M. I. Mishchenko, Y. S. Yatskiv, V. K. Rosenbush, and G. Videen, eds., NATO Science for Peace and Security Series C: Environmental Security (Springer, 2011), pp. 31–48.
  25. N. Lu and C. M. Sorensen, “Depolarized light scattering from fractal soot aggregates,” Phys. Rev. E 50, 3109 (1994).
    [CrossRef]
  26. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  27. N. G. Khlebtsov, “Optics of fractal clusters in the anomalous diffraction approximation,” J. Mod. Opt. 40, 2221–2235 (1993).
    [CrossRef]
  28. N. G. Khlebtsov and A. G. Melnikov, “Structure factor and exponent of scattering by polydisperse fractal colloidal aggregates,” J. Colloid Interface Sci. 163, 145–151 (1994).
    [CrossRef]
  29. M. I. Mishchenko, D. M. Mackowski, and L. D. Travis, “Scattering of light by bi-spheres with touching and separated components,” Appl. Opt. 34, 4589–4599 (1995).
    [CrossRef]
  30. D. M. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
    [CrossRef]

2012

S. P. Kearney and F. Pierce, “Evidence of soot superaggregates in a turbulent pool fire,” Combust. Flame 159, 3191–3198 (2012).
[CrossRef]

M. J. Berg, “Power-law patterns in electromagnetic scattering: a selected review and recent progress,” J. Quant. Spectrosc. Radiat. Transfer 113, 2292–2309 (2012).
[CrossRef]

2009

Y. Zhao and M. Lin, “Assessment of two fractal scattering models for the prediction of the optical characteristics of soot aggregates,” J. Quant. Spectrosc. Radiat. Transfer 110, 315–322 (2009).
[CrossRef]

2008

2007

R. Dhaubhadel, C. S. Gerving, A. Chakrabarti, and C. M. Sorensen, “Aerosol gelation: synthesis of a novel, lightweight, high specific surface area material,” Aerosol Sci. Technol. 41, 804–810 (2007).
[CrossRef]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
[CrossRef]

2006

R. Dhaubhadel, F. Pierce, A. Chakrabarti, and C. M. Sorensen, “Hybrid superaggregate morphology as a result of aggregation in a cluster-dense aerosol,” Phys. Rev. E 73, 011404 (2006).
[CrossRef]

T. C. Bond and R. W. Bergstrom, “Light absorption by carbonaceous particles: an investigative review,” Aerosol Sci. Technol. 40, 27–67 (2006).
[CrossRef]

2005

L. Liu and M. I. Mishchenko, “Effects of aggregation on scattering and radiative properties of soot aerosols,” J. Geophys. Res. 110 D11211 (2005).
[CrossRef]

2003

C. M. Sorensen, W. Kim, D. Fry, D. Shi, and A. Chakrabarti, “Observations of soot superaggregates with a fractal dimension of 2.6 in laminar acetylene/air diffusion flames,” Langmuir 19, 7560–7563 (2003).
[CrossRef]

2002

2001

C. M. Sorensen, “Light scattering by fractal aggregates: a review,” Aerosol Sci. Technol. 35, 648–687 (2001).
[CrossRef]

A. M. Brasil, T. L. Farias, M. G. Carvalho, and U. O. Koylu, “Numerical characterization of the morphology of aggregated particles,” Aerosol Sci. Technol. 32, 489–508 (2001).
[CrossRef]

1997

C. M. Sorensen and G. C. Roberts, “The prefactor of fractal aggregates,” J. Colloid Interface Sci. 186, 447–452 (1997).
[CrossRef]

1996

1995

1994

N. G. Khlebtsov and A. G. Melnikov, “Structure factor and exponent of scattering by polydisperse fractal colloidal aggregates,” J. Colloid Interface Sci. 163, 145–151 (1994).
[CrossRef]

N. Lu and C. M. Sorensen, “Depolarized light scattering from fractal soot aggregates,” Phys. Rev. E 50, 3109 (1994).
[CrossRef]

1993

N. G. Khlebtsov, “Optics of fractal clusters in the anomalous diffraction approximation,” J. Mod. Opt. 40, 2221–2235 (1993).
[CrossRef]

1992

1990

H. Y. Chen, M. F. Iskander, and J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

1988

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, and J. F. Merklin, “In situ optical structure factor measurement of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

1986

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

Berg, M. J.

M. J. Berg, “Power-law patterns in electromagnetic scattering: a selected review and recent progress,” J. Quant. Spectrosc. Radiat. Transfer 113, 2292–2309 (2012).
[CrossRef]

M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Reflection symmetry of a sphere’s internal field and its consequences on scattering: a microphysical approach,” J. Opt. Soc. Am. A 25, 98–107 (2008).
[CrossRef]

M. J. Berg, “Reflection symmetry of a sphere’s internal field and its consequences on scattering: behavior of the Stokes parameters,” in Polarimetric Detection, Characterization and Remote Sensing, M. I. Mishchenko, Y. S. Yatskiv, V. K. Rosenbush, and G. Videen, eds., NATO Science for Peace and Security Series C: Environmental Security (Springer, 2011), pp. 31–48.

Bergstrom, R. W.

T. C. Bond and R. W. Bergstrom, “Light absorption by carbonaceous particles: an investigative review,” Aerosol Sci. Technol. 40, 27–67 (2006).
[CrossRef]

Berry, M. V.

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

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Bond, T. C.

T. C. Bond and R. W. Bergstrom, “Light absorption by carbonaceous particles: an investigative review,” Aerosol Sci. Technol. 40, 27–67 (2006).
[CrossRef]

Brasil, A. M.

A. M. Brasil, T. L. Farias, M. G. Carvalho, and U. O. Koylu, “Numerical characterization of the morphology of aggregated particles,” Aerosol Sci. Technol. 32, 489–508 (2001).
[CrossRef]

Cai, J.

Carvalho, M. G.

A. M. Brasil, T. L. Farias, M. G. Carvalho, and U. O. Koylu, “Numerical characterization of the morphology of aggregated particles,” Aerosol Sci. Technol. 32, 489–508 (2001).
[CrossRef]

T. L. Farias, Ü. Ö. Köylü, and M. G. Carvalho, “Range of validity of the Rayleigh–Debye–Gans theory for optics of fractal aggregates,” Appl. Opt. 35, 6560–6567 (1996).
[CrossRef]

Chakrabarti, A.

M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Reflection symmetry of a sphere’s internal field and its consequences on scattering: a microphysical approach,” J. Opt. Soc. Am. A 25, 98–107 (2008).
[CrossRef]

R. Dhaubhadel, C. S. Gerving, A. Chakrabarti, and C. M. Sorensen, “Aerosol gelation: synthesis of a novel, lightweight, high specific surface area material,” Aerosol Sci. Technol. 41, 804–810 (2007).
[CrossRef]

R. Dhaubhadel, F. Pierce, A. Chakrabarti, and C. M. Sorensen, “Hybrid superaggregate morphology as a result of aggregation in a cluster-dense aerosol,” Phys. Rev. E 73, 011404 (2006).
[CrossRef]

C. M. Sorensen, W. Kim, D. Fry, D. Shi, and A. Chakrabarti, “Observations of soot superaggregates with a fractal dimension of 2.6 in laminar acetylene/air diffusion flames,” Langmuir 19, 7560–7563 (2003).
[CrossRef]

Chen, H. Y.

H. Y. Chen, M. F. Iskander, and J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

Dhaubhadel, R.

R. Dhaubhadel, C. S. Gerving, A. Chakrabarti, and C. M. Sorensen, “Aerosol gelation: synthesis of a novel, lightweight, high specific surface area material,” Aerosol Sci. Technol. 41, 804–810 (2007).
[CrossRef]

R. Dhaubhadel, F. Pierce, A. Chakrabarti, and C. M. Sorensen, “Hybrid superaggregate morphology as a result of aggregation in a cluster-dense aerosol,” Phys. Rev. E 73, 011404 (2006).
[CrossRef]

Farias, T. L.

A. M. Brasil, T. L. Farias, M. G. Carvalho, and U. O. Koylu, “Numerical characterization of the morphology of aggregated particles,” Aerosol Sci. Technol. 32, 489–508 (2001).
[CrossRef]

T. L. Farias, Ü. Ö. Köylü, and M. G. Carvalho, “Range of validity of the Rayleigh–Debye–Gans theory for optics of fractal aggregates,” Appl. Opt. 35, 6560–6567 (1996).
[CrossRef]

Fry, D.

C. M. Sorensen, W. Kim, D. Fry, D. Shi, and A. Chakrabarti, “Observations of soot superaggregates with a fractal dimension of 2.6 in laminar acetylene/air diffusion flames,” Langmuir 19, 7560–7563 (2003).
[CrossRef]

Gerving, C. S.

R. Dhaubhadel, C. S. Gerving, A. Chakrabarti, and C. M. Sorensen, “Aerosol gelation: synthesis of a novel, lightweight, high specific surface area material,” Aerosol Sci. Technol. 41, 804–810 (2007).
[CrossRef]

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Iskander, M. F.

H. Y. Chen, M. F. Iskander, and J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

Kearney, S. P.

S. P. Kearney and F. Pierce, “Evidence of soot superaggregates in a turbulent pool fire,” Combust. Flame 159, 3191–3198 (2012).
[CrossRef]

Kerker, M.

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

Khlebtsov, N. G.

N. G. Khlebtsov and A. G. Melnikov, “Structure factor and exponent of scattering by polydisperse fractal colloidal aggregates,” J. Colloid Interface Sci. 163, 145–151 (1994).
[CrossRef]

N. G. Khlebtsov, “Optics of fractal clusters in the anomalous diffraction approximation,” J. Mod. Opt. 40, 2221–2235 (1993).
[CrossRef]

Kim, W.

C. M. Sorensen, W. Kim, D. Fry, D. Shi, and A. Chakrabarti, “Observations of soot superaggregates with a fractal dimension of 2.6 in laminar acetylene/air diffusion flames,” Langmuir 19, 7560–7563 (2003).
[CrossRef]

Koylu, U. O.

A. M. Brasil, T. L. Farias, M. G. Carvalho, and U. O. Koylu, “Numerical characterization of the morphology of aggregated particles,” Aerosol Sci. Technol. 32, 489–508 (2001).
[CrossRef]

Köylü, Ü. Ö.

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Lin, M.

Y. Zhao and M. Lin, “Assessment of two fractal scattering models for the prediction of the optical characteristics of soot aggregates,” J. Quant. Spectrosc. Radiat. Transfer 110, 315–322 (2009).
[CrossRef]

Liu, L.

L. Liu and M. I. Mishchenko, “Effects of aggregation on scattering and radiative properties of soot aerosols,” J. Geophys. Res. 110 D11211 (2005).
[CrossRef]

Lu, N.

Mackowski, D. M.

Melnikov, A. G.

N. G. Khlebtsov and A. G. Melnikov, “Structure factor and exponent of scattering by polydisperse fractal colloidal aggregates,” J. Colloid Interface Sci. 163, 145–151 (1994).
[CrossRef]

Merklin, J. F.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, and J. F. Merklin, “In situ optical structure factor measurement of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

Mishchenko, M. I.

L. Liu and M. I. Mishchenko, “Effects of aggregation on scattering and radiative properties of soot aerosols,” J. Geophys. Res. 110 D11211 (2005).
[CrossRef]

D. M. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[CrossRef]

M. I. Mishchenko, D. M. Mackowski, and L. D. Travis, “Scattering of light by bi-spheres with touching and separated components,” Appl. Opt. 34, 4589–4599 (1995).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Olivier, B. J.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, and J. F. Merklin, “In situ optical structure factor measurement of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

Penner, J. E.

H. Y. Chen, M. F. Iskander, and J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

Percival, I. C.

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

Pierce, F.

S. P. Kearney and F. Pierce, “Evidence of soot superaggregates in a turbulent pool fire,” Combust. Flame 159, 3191–3198 (2012).
[CrossRef]

R. Dhaubhadel, F. Pierce, A. Chakrabarti, and C. M. Sorensen, “Hybrid superaggregate morphology as a result of aggregation in a cluster-dense aerosol,” Phys. Rev. E 73, 011404 (2006).
[CrossRef]

Pierce, F. G.

F. G. Pierce, “Aggregation in colloids and aerosols,” Ph.D. dissertation (Kansas State University, Manhattan, Kansas, 2007).

Ramer, E. R.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, and J. F. Merklin, “In situ optical structure factor measurement of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

Roberts, G. C.

C. M. Sorensen and G. C. Roberts, “The prefactor of fractal aggregates,” J. Colloid Interface Sci. 186, 447–452 (1997).
[CrossRef]

Shaddix, C. R.

K. C. Smyth and C. R. Shaddix, “The elusive history of m=1.57−0.56i for the refractive index of soot,” Combust. Flame 107, 314–320 (1996).
[CrossRef]

Shi, D.

C. M. Sorensen, W. Kim, D. Fry, D. Shi, and A. Chakrabarti, “Observations of soot superaggregates with a fractal dimension of 2.6 in laminar acetylene/air diffusion flames,” Langmuir 19, 7560–7563 (2003).
[CrossRef]

Smyth, K. C.

K. C. Smyth and C. R. Shaddix, “The elusive history of m=1.57−0.56i for the refractive index of soot,” Combust. Flame 107, 314–320 (1996).
[CrossRef]

Sorensen, C. M.

M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Reflection symmetry of a sphere’s internal field and its consequences on scattering: a microphysical approach,” J. Opt. Soc. Am. A 25, 98–107 (2008).
[CrossRef]

R. Dhaubhadel, C. S. Gerving, A. Chakrabarti, and C. M. Sorensen, “Aerosol gelation: synthesis of a novel, lightweight, high specific surface area material,” Aerosol Sci. Technol. 41, 804–810 (2007).
[CrossRef]

R. Dhaubhadel, F. Pierce, A. Chakrabarti, and C. M. Sorensen, “Hybrid superaggregate morphology as a result of aggregation in a cluster-dense aerosol,” Phys. Rev. E 73, 011404 (2006).
[CrossRef]

C. M. Sorensen, W. Kim, D. Fry, D. Shi, and A. Chakrabarti, “Observations of soot superaggregates with a fractal dimension of 2.6 in laminar acetylene/air diffusion flames,” Langmuir 19, 7560–7563 (2003).
[CrossRef]

G. Wang and C. M. Sorensen, “Experimental test of the Rayleigh–Debye–Gans theory for light scattering by fractal aggregates,” Appl. Opt. 41, 4645–4651 (2002).
[CrossRef]

C. M. Sorensen, “Light scattering by fractal aggregates: a review,” Aerosol Sci. Technol. 35, 648–687 (2001).
[CrossRef]

C. M. Sorensen and G. C. Roberts, “The prefactor of fractal aggregates,” J. Colloid Interface Sci. 186, 447–452 (1997).
[CrossRef]

N. Lu and C. M. Sorensen, “Depolarized light scattering from fractal soot aggregates,” Phys. Rev. E 50, 3109 (1994).
[CrossRef]

C. M. Sorensen, J. Cai, and 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]

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, and J. F. Merklin, “In situ optical structure factor measurement of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, D. M. Mackowski, and L. D. Travis, “Scattering of light by bi-spheres with touching and separated components,” Appl. Opt. 34, 4589–4599 (1995).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Wang, G.

Yurkin, M. A.

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
[CrossRef]

Zhang, H. X.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, and J. F. Merklin, “In situ optical structure factor measurement of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

Zhao, Y.

Y. Zhao and M. Lin, “Assessment of two fractal scattering models for the prediction of the optical characteristics of soot aggregates,” J. Quant. Spectrosc. Radiat. Transfer 110, 315–322 (2009).
[CrossRef]

Aerosol Sci. Technol.

T. C. Bond and R. W. Bergstrom, “Light absorption by carbonaceous particles: an investigative review,” Aerosol Sci. Technol. 40, 27–67 (2006).
[CrossRef]

C. M. Sorensen, “Light scattering by fractal aggregates: a review,” Aerosol Sci. Technol. 35, 648–687 (2001).
[CrossRef]

R. Dhaubhadel, C. S. Gerving, A. Chakrabarti, and C. M. Sorensen, “Aerosol gelation: synthesis of a novel, lightweight, high specific surface area material,” Aerosol Sci. Technol. 41, 804–810 (2007).
[CrossRef]

A. M. Brasil, T. L. Farias, M. G. Carvalho, and U. O. Koylu, “Numerical characterization of the morphology of aggregated particles,” Aerosol Sci. Technol. 32, 489–508 (2001).
[CrossRef]

Appl. Opt.

Combust. Flame

K. C. Smyth and C. R. Shaddix, “The elusive history of m=1.57−0.56i for the refractive index of soot,” Combust. Flame 107, 314–320 (1996).
[CrossRef]

S. P. Kearney and F. Pierce, “Evidence of soot superaggregates in a turbulent pool fire,” Combust. Flame 159, 3191–3198 (2012).
[CrossRef]

J. Colloid Interface Sci.

C. M. Sorensen and G. C. Roberts, “The prefactor of fractal aggregates,” J. Colloid Interface Sci. 186, 447–452 (1997).
[CrossRef]

N. G. Khlebtsov and A. G. Melnikov, “Structure factor and exponent of scattering by polydisperse fractal colloidal aggregates,” J. Colloid Interface Sci. 163, 145–151 (1994).
[CrossRef]

J. Geophys. Res.

L. Liu and M. I. Mishchenko, “Effects of aggregation on scattering and radiative properties of soot aerosols,” J. Geophys. Res. 110 D11211 (2005).
[CrossRef]

J. Mod. Opt.

H. Y. Chen, M. F. Iskander, and J. E. Penner, “Light scattering and absorption by fractal agglomerates and coagulations of smoke,” J. Mod. Opt. 37, 171–181 (1990).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Open-flame acetylene soot aggregate. (b) DLCA-simulated aggregate. The simulated aggregate contains Nm=99 monomers with a fractal dimension of Df=1.8. Also shown in (b) is the DDA lattice residing within the monomers discussed in Section 4. In this case, the number of lattice sites is Ndda=3323 corresponding to approximately 34 sites per monomer. The aggregate’s radius of gyration Rg is shown by the large gray sphere positioned on the center of mass.

Fig. 2.
Fig. 2.

Percent deviation of the monomer internal field from the RDGA. Shown here is the DLCA-simulated aggregate in Fig. 1(b) with each monomer color-coded according to Eq. (5).

Fig. 3.
Fig. 3.

Comparison between the DDA and RDGA scattered intensity. Both intensities are calculated using Eq. (6), except that the RDGA of the internal field is replaced by the incident field. Thus this calculation of the RDGA scattered intensity is essentially the square of the Fourier transform of the aggregate’s spatial structure (see [22]). Plot (a) presents the comparison for an aggregate in a single orientation and averaged over Nori=500 orientations. Plot (b) shows the DDA-calculated scattered intensity for a variety of aggregate orientations to demonstrate the effect that this averaging has to yield agreement with the power-law functionality of Eq. (2).

Fig. 4.
Fig. 4.

Survey of the internal field and scattered wave’s polarization state as a function of aggregate orientations. Row (a) shows the internal field directions given by Eq. (4), and row (b) shows histogram plots of the number of DDA lattice sites Ns with internal field deviation δi [Eq. (3)] in 5% bins. Row (c) shows the polarization state of the far-field scattered wave across the unit sphere as calculated by the DDA via the Stokes parameters.

Fig. 5.
Fig. 5.

Comparison of the DDA and RDGA scattered intensity and internal field as a function of increasing aggregate size. The parent aggregate is shown in IV and is the same aggregate used in Figs. 14. Smaller child aggregates are generated from this parent by truncating the parent with a sphere of radius Rt. The middle plots show the percent deviation δI [Eq. (8)] of DDA and RDGA scattered intensity for aggregates in a single orientation and an ensemble of 500 orientations. The bottom plots show the number of lattice sites Ns with internal field magnitude deviations δi [Eq. (3)] in 5% bins.

Fig. 6.
Fig. 6.

Relative error between DDA scattered intensity curves using successively greater numbers of dipoles per monomer Ndda/Nm. The upper plot shows the convergence error as described by Eq. (9) where the number of dipoles per monomer is indicated on each curve. The aggregate is the parent aggregate of Fig. 5. The lower plot shows the scattered intensity curve for the largest, i.e., most accurate, value for Ndda/Nm. Here, one can see the correlation between the minima in the scattered intensity and the spikes in the convergence error above.

Equations (9)

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Nm=ko(RgRm)Df,
I(qRg){Nm2qRg<1Nm2(qRg)Df1qRgqRmNm2(qRg)4qRm<qRg.
δi=1Norin=1Nori|Enint(ri)||Eninc(ri)||Eninc(ri)|.
ei=1Norin=1NoriRe{Enint(ri)}|Re{Enint(ri)}|.
Δi=1VmVm|Eint(ri)||Einc(ri)||Einc(ri)|dV,
E1sca(r^)=k24πεo(Ir^r^)·i=1Nddapiexp(ikr^·ri),
ρeff=2k|m1|Rm3DfRgDf2.
δI(qRt)=Idda(qRt)Irdga(qRt)Irdga(qRt).
Idda(8)(qRt)Idda(4)(qRt)Idda(8)(qRt)×100,

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