Abstract

Soot particles can be formed in hydrocarbon flames as a result of an inefficient combustion process. The particles are near-spherical, and at later stages in the soot growth process, they form chainlike sparse aggregates. When applying optical diagnostic methods, this aggregation influences the evaluation of soot properties based on assumptions of isolated particles. In this paper an efficient and accurate method for calculating scattering of light from these structures is presented. The method can handle aggregates with several hundred subparticles with no restrictions on shape, internal structure, or coagulation of the subparticles. The basic idea is that the induced dipole moments of the subparticles are determined from the solution of a quasi-static problem that can be solved with high accuracy by, e.g., the finite element method.

© 2009 Optical Society of America

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References

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  1. A. D'Alessio, A. D. Lorenzo, A. Borghese, F. Beretta, and S. Masi, “Study of the soot nucleation zone of rich methane-oxygen flames,” Proc. Combust. Inst. 16, 695-708 (1977).
  2. R. A. Dobbins, R. J. Santoro, and H. Semerijan, “Interpretation of optical measurements of soot in flames,” Progress in Astronautics and Aeronautics 92, 208-237 (1984).
  3. P.-E. Bengtsson and M. Alden, “Application of a pulsed laser for soot measurements in premixed flames,” Appl. Phys. B: Lasers Opt. 48, 155-164 (1989).
    [CrossRef]
  4. H. Bladh, J. Johnsson, and P.-E. Bengtsson, “Influence of spatial laser energy distribution on evaluated soot particle sizes using two-colour laser-induced incandescence in a flat premixed ethylene/air flame,” submitted to Appl. Phys. B.
  5. T. T. Charalampopoulos, “Morphology and dynamics of agglomerated particulates in combustion systems using light scattering techniques,” Prog. Energy Combust. Sci. 18, 13-45 (1992).
    [CrossRef]
  6. C. M. Sorensen, “Light scattering by fractal aggregate: A review,” Aerosol Sci. Technol. 35, 648-687 (2001).
    [CrossRef]
  7. K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using a LL/LIS technique,” Meas. Sci. Technol. 16, 519-526 (2005).
    [CrossRef]
  8. B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661-3678 (1973).
    [CrossRef]
  9. T. W. A. Doicu and Y. A. Eremin, Light Scattering by Systems of Particles (Springer-Verlag, 2006).
    [CrossRef]
  10. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491-1499 (1994).
    [CrossRef]
  11. T. Weiland, “A discretization method for the solution of Maxwell's equations for six-components fields,” Electronics and Communications AEÜ 31, 116-120 (1977).
  12. J. D. Jackson, Classical Electrodynamics (Wiley, 1975).
  13. T. T. Charalampopoulos and J. D. Felske, “Refractive indices of soot particles deduced from in-situ laser light scattering measurements,” Combust. Flame 68, 283-294 (1987).
    [CrossRef]
  14. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
    [CrossRef]

2005 (1)

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using a LL/LIS technique,” Meas. Sci. Technol. 16, 519-526 (2005).
[CrossRef]

2001 (1)

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

1994 (1)

1992 (1)

T. T. Charalampopoulos, “Morphology and dynamics of agglomerated particulates in combustion systems using light scattering techniques,” Prog. Energy Combust. Sci. 18, 13-45 (1992).
[CrossRef]

1989 (1)

P.-E. Bengtsson and M. Alden, “Application of a pulsed laser for soot measurements in premixed flames,” Appl. Phys. B: Lasers Opt. 48, 155-164 (1989).
[CrossRef]

1987 (1)

T. T. Charalampopoulos and J. D. Felske, “Refractive indices of soot particles deduced from in-situ laser light scattering measurements,” Combust. Flame 68, 283-294 (1987).
[CrossRef]

1984 (1)

R. A. Dobbins, R. J. Santoro, and H. Semerijan, “Interpretation of optical measurements of soot in flames,” Progress in Astronautics and Aeronautics 92, 208-237 (1984).

1977 (2)

A. D'Alessio, A. D. Lorenzo, A. Borghese, F. Beretta, and S. Masi, “Study of the soot nucleation zone of rich methane-oxygen flames,” Proc. Combust. Inst. 16, 695-708 (1977).

T. Weiland, “A discretization method for the solution of Maxwell's equations for six-components fields,” Electronics and Communications AEÜ 31, 116-120 (1977).

1973 (1)

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661-3678 (1973).
[CrossRef]

Alden, M.

P.-E. Bengtsson and M. Alden, “Application of a pulsed laser for soot measurements in premixed flames,” Appl. Phys. B: Lasers Opt. 48, 155-164 (1989).
[CrossRef]

Barber, P. W.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Bengtsson, P.-E.

P.-E. Bengtsson and M. Alden, “Application of a pulsed laser for soot measurements in premixed flames,” Appl. Phys. B: Lasers Opt. 48, 155-164 (1989).
[CrossRef]

H. Bladh, J. Johnsson, and P.-E. Bengtsson, “Influence of spatial laser energy distribution on evaluated soot particle sizes using two-colour laser-induced incandescence in a flat premixed ethylene/air flame,” submitted to Appl. Phys. B.

Beretta, F.

A. D'Alessio, A. D. Lorenzo, A. Borghese, F. Beretta, and S. Masi, “Study of the soot nucleation zone of rich methane-oxygen flames,” Proc. Combust. Inst. 16, 695-708 (1977).

Bladh, H.

H. Bladh, J. Johnsson, and P.-E. Bengtsson, “Influence of spatial laser energy distribution on evaluated soot particle sizes using two-colour laser-induced incandescence in a flat premixed ethylene/air flame,” submitted to Appl. Phys. B.

Borghese, A.

A. D'Alessio, A. D. Lorenzo, A. Borghese, F. Beretta, and S. Masi, “Study of the soot nucleation zone of rich methane-oxygen flames,” Proc. Combust. Inst. 16, 695-708 (1977).

Charalampopoulos, T. T.

T. T. Charalampopoulos, “Morphology and dynamics of agglomerated particulates in combustion systems using light scattering techniques,” Prog. Energy Combust. Sci. 18, 13-45 (1992).
[CrossRef]

T. T. Charalampopoulos and J. D. Felske, “Refractive indices of soot particles deduced from in-situ laser light scattering measurements,” Combust. Flame 68, 283-294 (1987).
[CrossRef]

Chung, J.

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using a LL/LIS technique,” Meas. Sci. Technol. 16, 519-526 (2005).
[CrossRef]

D'Alessio, A.

A. D'Alessio, A. D. Lorenzo, A. Borghese, F. Beretta, and S. Masi, “Study of the soot nucleation zone of rich methane-oxygen flames,” Proc. Combust. Inst. 16, 695-708 (1977).

Dobbins, R. A.

R. A. Dobbins, R. J. Santoro, and H. Semerijan, “Interpretation of optical measurements of soot in flames,” Progress in Astronautics and Aeronautics 92, 208-237 (1984).

Doicu, T. W. A.

T. W. A. Doicu and Y. A. Eremin, Light Scattering by Systems of Particles (Springer-Verlag, 2006).
[CrossRef]

Draine, B. T.

Eremin, Y. A.

T. W. A. Doicu and Y. A. Eremin, Light Scattering by Systems of Particles (Springer-Verlag, 2006).
[CrossRef]

Felske, J. D.

T. T. Charalampopoulos and J. D. Felske, “Refractive indices of soot particles deduced from in-situ laser light scattering measurements,” Combust. Flame 68, 283-294 (1987).
[CrossRef]

Flatau, P. J.

Han, Y.

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using a LL/LIS technique,” Meas. Sci. Technol. 16, 519-526 (2005).
[CrossRef]

Hill, S. C.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Johnsson, J.

H. Bladh, J. Johnsson, and P.-E. Bengtsson, “Influence of spatial laser energy distribution on evaluated soot particle sizes using two-colour laser-induced incandescence in a flat premixed ethylene/air flame,” submitted to Appl. Phys. B.

Lee, C.

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using a LL/LIS technique,” Meas. Sci. Technol. 16, 519-526 (2005).
[CrossRef]

Lee, K.

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using a LL/LIS technique,” Meas. Sci. Technol. 16, 519-526 (2005).
[CrossRef]

Lee, W.

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using a LL/LIS technique,” Meas. Sci. Technol. 16, 519-526 (2005).
[CrossRef]

Lorenzo, A. D.

A. D'Alessio, A. D. Lorenzo, A. Borghese, F. Beretta, and S. Masi, “Study of the soot nucleation zone of rich methane-oxygen flames,” Proc. Combust. Inst. 16, 695-708 (1977).

Masi, S.

A. D'Alessio, A. D. Lorenzo, A. Borghese, F. Beretta, and S. Masi, “Study of the soot nucleation zone of rich methane-oxygen flames,” Proc. Combust. Inst. 16, 695-708 (1977).

Peterson, B.

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661-3678 (1973).
[CrossRef]

Santoro, R. J.

R. A. Dobbins, R. J. Santoro, and H. Semerijan, “Interpretation of optical measurements of soot in flames,” Progress in Astronautics and Aeronautics 92, 208-237 (1984).

Semerijan, H.

R. A. Dobbins, R. J. Santoro, and H. Semerijan, “Interpretation of optical measurements of soot in flames,” Progress in Astronautics and Aeronautics 92, 208-237 (1984).

Sorensen, C. M.

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

Ström, S.

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661-3678 (1973).
[CrossRef]

Weiland, T.

T. Weiland, “A discretization method for the solution of Maxwell's equations for six-components fields,” Electronics and Communications AEÜ 31, 116-120 (1977).

Aerosol Sci. Technol. (1)

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

Appl. Phys. B: Lasers Opt. (1)

P.-E. Bengtsson and M. Alden, “Application of a pulsed laser for soot measurements in premixed flames,” Appl. Phys. B: Lasers Opt. 48, 155-164 (1989).
[CrossRef]

Combust. Flame (1)

T. T. Charalampopoulos and J. D. Felske, “Refractive indices of soot particles deduced from in-situ laser light scattering measurements,” Combust. Flame 68, 283-294 (1987).
[CrossRef]

Electronics and Communications AEÜ (1)

T. Weiland, “A discretization method for the solution of Maxwell's equations for six-components fields,” Electronics and Communications AEÜ 31, 116-120 (1977).

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using a LL/LIS technique,” Meas. Sci. Technol. 16, 519-526 (2005).
[CrossRef]

Phys. Rev. D (1)

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661-3678 (1973).
[CrossRef]

Proc. Combust. Inst. (1)

A. D'Alessio, A. D. Lorenzo, A. Borghese, F. Beretta, and S. Masi, “Study of the soot nucleation zone of rich methane-oxygen flames,” Proc. Combust. Inst. 16, 695-708 (1977).

Prog. Energy Combust. Sci. (1)

T. T. Charalampopoulos, “Morphology and dynamics of agglomerated particulates in combustion systems using light scattering techniques,” Prog. Energy Combust. Sci. 18, 13-45 (1992).
[CrossRef]

Progress in Astronautics and Aeronautics (1)

R. A. Dobbins, R. J. Santoro, and H. Semerijan, “Interpretation of optical measurements of soot in flames,” Progress in Astronautics and Aeronautics 92, 208-237 (1984).

Other (4)

H. Bladh, J. Johnsson, and P.-E. Bengtsson, “Influence of spatial laser energy distribution on evaluated soot particle sizes using two-colour laser-induced incandescence in a flat premixed ethylene/air flame,” submitted to Appl. Phys. B.

T. W. A. Doicu and Y. A. Eremin, Light Scattering by Systems of Particles (Springer-Verlag, 2006).
[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Pictures analyzed using transmission electron microscopy (TEM). Left: A large number of aggregates, each consisting of a number of primary soot particles. Right: One aggregate consisting of roughly spherical primary particles. The black cube in the corner has a side that corresponds to 50 nm , thus the primary particle sizes are in the range 20 40 nm .

Fig. 2
Fig. 2

Model of a soot particle with subparticles as spheres.

Fig. 3
Fig. 3

Nine spherical subparticles with radius 15 nm . The polarizability matrices and intensities for the index of refraction n = 1.61 are given in Tables 1, 2. The polarizability matrices and intensities for the index of refraction n = 1.61 + i 0.6 are given in Table 5 and Fig. 5.

Fig. 4
Fig. 4

Nine spherical subparticles with corresponding polarizability matrices given in Table 3. The spheres are identical with radius 15 nm and index of refraction 1.61.

Fig. 5
Fig. 5

Averaged normalized intensity by Eq. (34) for the aggregate in Fig. 3 with n = 1.61 + i 0.6 . The normalized length is the total length of the aggregate divided by the wavelength λ = 532 nm .

Fig. 6
Fig. 6

Left figure depicts the aggregate with six arms of non-overlapping spheres used for the graphs in Figs. 5, 7, 8. The right figure depicts the corresponding aggregate with overlapping spheres considered in Fig. 9. The volume of each subparticle is the same in the left and right aggregates. The spheres overlap such that the dashed lines in the right figure are perpendicular.

Fig. 7
Fig. 7

Averaged normalized intensity by Eq. (34) for an aggregate with six straight arms of non-overlapping spherical subparticles (solid line). The dashed line is the corresponding intensity using Rayleigh scattering with no interaction between subparticles. The normalized length is the length of the aggregate along a pair of arms divided by the wavelength λ = 532 nm .

Fig. 8
Fig. 8

Averaged normalized absorbed power by Eq. (35) for an aggregate with six straight arms of non-overlapping spherical subparticles. The normalized length is the length of the aggregate along a pair of arms divided by the wavelength λ = 532 nm .

Fig. 9
Fig. 9

Averaged normalized intensity by Eq. (34) for an aggregate with six straight arms of overlapping spherical subparticles. The normalized length is the length of the aggregate along a pair of arms divided by the wavelength λ = 532 nm .

Fig. 10
Fig. 10

Intensity I ( θ ) from a plane wave scattered from a prolate spheroid with half-axes 75 nm and 15 nm and index of refraction n = 1.61 + i 0.60 . The intensity is normalized with the intensity of a single sphere I = r 2 5.38 10 20 . Curves A, C, and E are generated by the T-matrix method and B, D, and F by the present method. The incident plane wave propagates in the positive z direction with polarization in the x direction. The angle θ is the polar angle measured from the z axis in the y z plane for curves A, B, E, and F and in the x z plane for curves C and D. The spheroid has its major axis along the z axis for curves A and B and along the x axis for curves C, D, E, and F.

Fig. 11
Fig. 11

Particle in Fig. 4 formed by concatenation of three parts. In a FEM calculation on each of the three parts, the polarizability matrices for the filled spheres differs by less than 2% from the corresponding values in Table 4.

Tables (5)

Tables Icon

Table 1 Polarizability Matrices for the Soot Particle Aggregate in Fig. 3 a

Tables Icon

Table 2 Normalized Intensities by Eq. (15) for the Soot Particle in Fig. 3 for a Wavelength of 532 nm

Tables Icon

Table 3 Polarizability Matrices for the Configuration in Fig. 4 a

Tables Icon

Table 4 Normalized Intensities by Eq (15) for the Soot Particle in Fig. 4 and Table 3

Tables Icon

Table 5 Polarizability Matrices for the Soot Particle in Fig. 3 a

Equations (53)

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E i ( r ) = E 0 e i k k ̂ i r e ̂ ,
q = k ( k ̂ i r ̂ ) .
E s ( r ) = k 2 e i k r 4 π ɛ 0 r r ̂ × ( r ̂ × V e i q r ( ɛ c ( r ) ɛ 0 ) E ( r ) d v ) = k 2 e i k r 4 π ɛ 0 r r ̂ × ( r ̂ × V e i q r P ( r ) d v ) ,
ɛ c ( r ) = ɛ 0 ɛ r ( r ) + i σ ( r ) ω
p n = V n ( ɛ c ( r ) ɛ 0 ) E ( r ) d v ,
E s ( r ) = k 2 e i k r 4 π ɛ 0 r n = 1 N e i q r n ( r ̂ × ( r ̂ × p n ) ) .
γ n = ( γ n x x γ n x y γ n x z γ n y x γ n y y γ n y z γ n z x γ n z y γ n z z ) .
( p n x p n y p n z ) = ɛ 0 γ n ( E x ext E y ext E z ext ) .
× E ( r ) = 0 ,
× H ( r ) = i ω ɛ c ( r ) E ( r ) ,
E ( r ) E ext when r ,
{ ( ɛ c ( r ) V ( r ) ) = 0 V ( r ) E ext when r } .
γ n x x = p n x ɛ 0 ,
γ n y x = p n y ɛ 0 ,
γ n z x = p n z ɛ 0 .
γ n x y = p n x ɛ 0 ,
γ n y y = p n y ɛ 0 ,
γ n z y = p n z ɛ 0 .
γ n x z = p n x ɛ 0 ,
γ n y z = p n y ɛ 0 ,
γ n z z = p n z ɛ 0 .
I ( r ) = E s ( r ) 2 E 0 2 = 1 E 0 2 n = 1 N E n s ( r ) m = 1 N E m s * ( r ) = k 4 ( 4 π E 0 ɛ 0 r ) 2 n = 1 N m = 1 N e i q ( r n r m ) ( r ̂ × ( r ̂ × p n ) ) ( r ̂ × ( r ̂ × p m * ) ) ,
I co = 1 E 0 2 e ̂ E s 2 ,
I cross = 1 E 0 2 k ̂ i E s 2 .
k i = k k ̂ i = k ( sin θ cos φ , sin θ sin φ , cos θ ) ,
e ̂ = ( e x , e y , e z ) = ξ ̂ cos ψ + η ̂ sin ψ .
η ̂ = k ̂ i × z ̂ k ̂ i × z ̂ ,
ξ ̂ = η ̂ × k ̂ i .
I ( r ) = 1 E 0 2 8 π 2 λ 3 0 λ 0 λ 0 λ 0 2 π 0 π 0 2 π E s ( r ) 2 sin θ d φ d θ d ψ d x 0 d y 0 d z 0 ,
p n = ɛ 0 E 0 γ n ( e x e y e z ) ,
I ( r ) = 1 E 0 2 8 π 2 0 2 π 0 π 0 2 π E s ( r ) 2 sin θ d φ d θ d ψ .
ξ ̂ = x ̂ cos θ cos φ y ̂ cos θ sin φ + z ̂ sin θ ,
η ̂ = x ̂ sin φ y ̂ cos φ .
e x = cos θ cos φ cos ψ + sin φ sin ψ ,
e y = cos θ sin φ cos ψ cos φ sin ψ ,
e z = sin θ cos ψ ,
r ̂ = x ̂ ( sin φ cos ψ + cos θ cos φ sin ψ ) + y ̂ ( cos φ cos ψ cos θ sin φ sin ψ ) + z ̂ sin θ sin ψ .
I = 1 E 0 2 8 π 2 0 2 π 0 π 0 2 π E s 2 sin θ d φ d θ d ψ = k 4 ( 4 π E 0 ɛ 0 r ) 2 8 π 2 0 2 π 0 π 0 2 π ( n = 1 N ( p n 2 r ̂ p n 2 ) + 2 Re { n = 1 N 1 m = n + 1 N e i q ( r n r m ) ( p n p m * ( r ̂ p n ) ( r ̂ p m * ) ) } ) sin θ d φ d θ d ψ .
I co = 1 E 0 2 8 π 2 0 2 π 0 π 0 2 π e ̂ E s 2 sin θ d φ d θ d ψ ,
I cross = 1 E 0 2 8 π 2 0 2 π 0 π 0 2 π k ̂ i E s 2 sin θ d φ d θ d ψ .
W abs n = 1 2 V n J E * d V = 1 2 V n σ ( r ) E 2 d V ,
e Re { i ω μ 0 ɛ c δ } = e 1 .
W abs n = 1 2 V n J E * d V p n 2 σ 2 V n ɛ c ɛ 0 2 ,
W abs = 1 8 π 0 2 π 0 π 0 2 π n = 1 N W abs n sin θ d φ d θ d ψ 1 16 π 2 σ ɛ c ɛ 0 2 0 2 π 0 π 0 2 π n = 1 N p n 2 V n sin θ d φ d θ d ψ .
γ n = ( γ n x x 0 0 0 γ n y y 0 0 0 γ n z z ) .
γ = 4 π a 3 ɛ c ɛ 0 ɛ c + 2 ɛ 0 I ,
r 1 = ( 30 , 0 , 90 ) nm , r 2 = ( 30 , 0 , 60 ) nm ,
r 3 = ( 30 , 0 , 30 ) nm ,
r 4 = ( 30 , 0 , 0 ) nm , r 5 = ( 0 , 0 , 0 ) nm , r 6 = ( 30 , 0 , 0 ) nm ,
r 7 = ( 30 , 30 , 0 ) nm , r 8 = ( 30 , 60 , 0 ) nm ,
r 9 = ( 30 , 90 , 0 ) nm .
normalized averaged intensity = I N I sphere ,
normalized absorbed power = W abs N W abs , sphere ,

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