Abstract

A laboratory prototype of a novel experimental apparatus for the analysis of spherical and axisymmetric nonspherical particles in liquid suspensions has been developed. This apparatus determines shape, volume, and refractive index, and this is the main difference of this apparatus from commercially available particle analyzers. Characterization is based on the scattering of a monochromatic laser beam by particles [which can be inorganic, organic, or biological (such as red blood cells and bacteria)] and on the strong relation between the light-scattering pattern and the morphology and the volume, shape, and refractive index of the particles. To keep things relatively simple, first we focus attention on axisymmetrical particles, in which case hydrodynamic alignment can be used to simplify signal gathering and processing. Fast and reliable characterization is achieved by comparison of certain properly selected characteristics of the scattered-light pattern with the corresponding theoretical values, which are readily derived from theoretical data and are stored in a look-up table. The data in this table were generated with a powerful boundary-element method, which can solve the direct scattering problem for virtually arbitrary shapes. A specially developed fast pattern-recognition technique makes possible the on-line characterization of axisymmetric particles. Successful results with red blood cells and bacteria are presented.

© 1998 Optical Society of America

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  1. P. Barber, E. Miller, T. Sarkar, eds., Feature on Scattering by Three-Dimensional Objects,J. Opt. Soc. Am. A11, 1380–1545 (1994).
  2. E. D. Hirleman, C. F. Bohren, eds., Optical Particle Sizing,Appl. Opt.30, 4685–4986 (1991).
  3. N. Chigier, G. Stewart, eds., Feature on Particle Sizing and Spray Analysis,Opt. Eng.23, 554–640 (1984).
  4. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  5. H. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  6. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  7. H. M. Al-Rizzo, J. M. Tranquilla, “Electromagnetic wave scattering by highly elongated and geometrically composite objects of large size parameters: the generalized multipole technique,” Appl. Opt. 34, 3502–3521 (1995).
    [CrossRef] [PubMed]
  8. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  9. J. M. Tranquilla, H. M. Al-Rizzo, “Electromagnetic scattering from dielectric-coated axisymmetric objects using the generalized point-matching technique (GPMT),” IEEE Trans. Antennas Propag. 43, 63–71 (1995).
    [CrossRef]
  10. S. V. Tsinopoulos, S. E. Kattis, D. Polyzos, “Three dimensional boundary element analysis of electromagnetic wave scattering by penetrable bodies,” Computation. Mechan. 21, 306–315 (1998).
    [CrossRef]
  11. V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys. 8, 589–610 (1967).
    [CrossRef]
  12. G. C. Hsiao, R. E. Kleinman, “Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics,” IEEE Trans. Antennas Propag. 45, 316–328 (1997).
    [CrossRef]
  13. K. D. Paulsen, D. R. Lynch, J. W. Strohbehn, “Three dimensional finite, boundary and hybrid solutions of the Maxwell equations for lossy dielectric media,” IEEE Trans. Microwave Theory Tech. 36, 682–693 (1988).
    [CrossRef]
  14. G. D. Manolis, D. E. Beskos, Boundary Element Methods in Elastodynamics (Unwin-Hyman, London, 1988).
  15. G. Dassios, K. Kiriaki, “The low-frequency theory of elastic wave scattering,” Q. Appl. Math. 42, 225–248 (1984).
  16. A. V. Chernyshev, V. I. Prots, A. A. Doroshkin, V. P. Maltsev, “Measurement of scattering properties of individual particles with a scanning flow cytometer,” Appl. Opt. 34, 6301–6305 (1995).
    [CrossRef] [PubMed]
  17. International Mathematical and Statistical Library, IMSL Math/Library User’s Manual, Version 3.0 (Visual Numerics, Inc., Houston, Texas, 1994).
  18. Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues (Springer-Verlag, New York, 1981).
  19. D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
    [CrossRef]
  20. A. Morel, Y.-H. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Marine Res. 48, 145–175 (1990).
    [CrossRef]
  21. H. G. Schlegel, General Microbiology, 6th ed. (Cambridge U. Press, Cambridge, UK, 1986).

1998 (1)

S. V. Tsinopoulos, S. E. Kattis, D. Polyzos, “Three dimensional boundary element analysis of electromagnetic wave scattering by penetrable bodies,” Computation. Mechan. 21, 306–315 (1998).
[CrossRef]

1997 (1)

G. C. Hsiao, R. E. Kleinman, “Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics,” IEEE Trans. Antennas Propag. 45, 316–328 (1997).
[CrossRef]

1995 (3)

1991 (1)

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

1990 (1)

A. Morel, Y.-H. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Marine Res. 48, 145–175 (1990).
[CrossRef]

1988 (1)

K. D. Paulsen, D. R. Lynch, J. W. Strohbehn, “Three dimensional finite, boundary and hybrid solutions of the Maxwell equations for lossy dielectric media,” IEEE Trans. Microwave Theory Tech. 36, 682–693 (1988).
[CrossRef]

1984 (1)

G. Dassios, K. Kiriaki, “The low-frequency theory of elastic wave scattering,” Q. Appl. Math. 42, 225–248 (1984).

1967 (1)

V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys. 8, 589–610 (1967).
[CrossRef]

Ahn, Y.-H.

A. Morel, Y.-H. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Marine Res. 48, 145–175 (1990).
[CrossRef]

Al-Rizzo, H. M.

J. M. Tranquilla, H. M. Al-Rizzo, “Electromagnetic scattering from dielectric-coated axisymmetric objects using the generalized point-matching technique (GPMT),” IEEE Trans. Antennas Propag. 43, 63–71 (1995).
[CrossRef]

H. M. Al-Rizzo, J. M. Tranquilla, “Electromagnetic wave scattering by highly elongated and geometrically composite objects of large size parameters: the generalized multipole technique,” Appl. Opt. 34, 3502–3521 (1995).
[CrossRef] [PubMed]

Barber, P. W.

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

Beskos, D. E.

G. D. Manolis, D. E. Beskos, Boundary Element Methods in Elastodynamics (Unwin-Hyman, London, 1988).

Bohren, C. F.

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

Chernyshev, A. V.

Dassios, G.

G. Dassios, K. Kiriaki, “The low-frequency theory of elastic wave scattering,” Q. Appl. Math. 42, 225–248 (1984).

Doroshkin, A. A.

Fung, Y. C.

Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues (Springer-Verlag, New York, 1981).

Hill, S. C.

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

Hsiao, G. C.

G. C. Hsiao, R. E. Kleinman, “Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics,” IEEE Trans. Antennas Propag. 45, 316–328 (1997).
[CrossRef]

Huffman, D. R.

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

Kattis, S. E.

S. V. Tsinopoulos, S. E. Kattis, D. Polyzos, “Three dimensional boundary element analysis of electromagnetic wave scattering by penetrable bodies,” Computation. Mechan. 21, 306–315 (1998).
[CrossRef]

Kerker, M.

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

Kiefer, D. A.

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

Kiriaki, K.

G. Dassios, K. Kiriaki, “The low-frequency theory of elastic wave scattering,” Q. Appl. Math. 42, 225–248 (1984).

Kleinman, R. E.

G. C. Hsiao, R. E. Kleinman, “Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics,” IEEE Trans. Antennas Propag. 45, 316–328 (1997).
[CrossRef]

Lynch, D. R.

K. D. Paulsen, D. R. Lynch, J. W. Strohbehn, “Three dimensional finite, boundary and hybrid solutions of the Maxwell equations for lossy dielectric media,” IEEE Trans. Microwave Theory Tech. 36, 682–693 (1988).
[CrossRef]

Maltsev, V. P.

Manolis, G. D.

G. D. Manolis, D. E. Beskos, Boundary Element Methods in Elastodynamics (Unwin-Hyman, London, 1988).

Morel, A.

A. Morel, Y.-H. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Marine Res. 48, 145–175 (1990).
[CrossRef]

Paulsen, K. D.

K. D. Paulsen, D. R. Lynch, J. W. Strohbehn, “Three dimensional finite, boundary and hybrid solutions of the Maxwell equations for lossy dielectric media,” IEEE Trans. Microwave Theory Tech. 36, 682–693 (1988).
[CrossRef]

Polyzos, D.

S. V. Tsinopoulos, S. E. Kattis, D. Polyzos, “Three dimensional boundary element analysis of electromagnetic wave scattering by penetrable bodies,” Computation. Mechan. 21, 306–315 (1998).
[CrossRef]

Prots, V. I.

Schlegel, H. G.

H. G. Schlegel, General Microbiology, 6th ed. (Cambridge U. Press, Cambridge, UK, 1986).

Stramski, D.

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

Strohbehn, J. W.

K. D. Paulsen, D. R. Lynch, J. W. Strohbehn, “Three dimensional finite, boundary and hybrid solutions of the Maxwell equations for lossy dielectric media,” IEEE Trans. Microwave Theory Tech. 36, 682–693 (1988).
[CrossRef]

Tranquilla, J. M.

J. M. Tranquilla, H. M. Al-Rizzo, “Electromagnetic scattering from dielectric-coated axisymmetric objects using the generalized point-matching technique (GPMT),” IEEE Trans. Antennas Propag. 43, 63–71 (1995).
[CrossRef]

H. M. Al-Rizzo, J. M. Tranquilla, “Electromagnetic wave scattering by highly elongated and geometrically composite objects of large size parameters: the generalized multipole technique,” Appl. Opt. 34, 3502–3521 (1995).
[CrossRef] [PubMed]

Tsinopoulos, S. V.

S. V. Tsinopoulos, S. E. Kattis, D. Polyzos, “Three dimensional boundary element analysis of electromagnetic wave scattering by penetrable bodies,” Computation. Mechan. 21, 306–315 (1998).
[CrossRef]

Twersky, V.

V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys. 8, 589–610 (1967).
[CrossRef]

van de Hulst, H.

H. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Appl. Opt. (2)

Computation. Mechan. (1)

S. V. Tsinopoulos, S. E. Kattis, D. Polyzos, “Three dimensional boundary element analysis of electromagnetic wave scattering by penetrable bodies,” Computation. Mechan. 21, 306–315 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

G. C. Hsiao, R. E. Kleinman, “Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics,” IEEE Trans. Antennas Propag. 45, 316–328 (1997).
[CrossRef]

J. M. Tranquilla, H. M. Al-Rizzo, “Electromagnetic scattering from dielectric-coated axisymmetric objects using the generalized point-matching technique (GPMT),” IEEE Trans. Antennas Propag. 43, 63–71 (1995).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. D. Paulsen, D. R. Lynch, J. W. Strohbehn, “Three dimensional finite, boundary and hybrid solutions of the Maxwell equations for lossy dielectric media,” IEEE Trans. Microwave Theory Tech. 36, 682–693 (1988).
[CrossRef]

J. Marine Res. (1)

A. Morel, Y.-H. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Marine Res. 48, 145–175 (1990).
[CrossRef]

J. Math. Phys. (1)

V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys. 8, 589–610 (1967).
[CrossRef]

Prog. Oceanogr. (1)

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

Q. Appl. Math. (1)

G. Dassios, K. Kiriaki, “The low-frequency theory of elastic wave scattering,” Q. Appl. Math. 42, 225–248 (1984).

Other (11)

G. D. Manolis, D. E. Beskos, Boundary Element Methods in Elastodynamics (Unwin-Hyman, London, 1988).

H. G. Schlegel, General Microbiology, 6th ed. (Cambridge U. Press, Cambridge, UK, 1986).

International Mathematical and Statistical Library, IMSL Math/Library User’s Manual, Version 3.0 (Visual Numerics, Inc., Houston, Texas, 1994).

Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues (Springer-Verlag, New York, 1981).

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

P. Barber, E. Miller, T. Sarkar, eds., Feature on Scattering by Three-Dimensional Objects,J. Opt. Soc. Am. A11, 1380–1545 (1994).

E. D. Hirleman, C. F. Bohren, eds., Optical Particle Sizing,Appl. Opt.30, 4685–4986 (1991).

N. Chigier, G. Stewart, eds., Feature on Particle Sizing and Spray Analysis,Opt. Eng.23, 554–640 (1984).

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

H. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of the apparatus for particle analysis.

Fig. 2
Fig. 2

Drawing of the flow chamber with RBC’s flowing through it and of the scattering pattern. Snapshots of (a) a stagnant isotonic sodium chloride solution containing red blood cells and (b) the same solution flowing at a velocity of 0.1 mm/s.

Fig. 3
Fig. 3

(a) Light-scattering pattern of spherical latex particles, D = 4.99 μm, suspended in water and (b) comparison of experimental results and theoretical predictions.

Fig. 4
Fig. 4

Typical diagram of the dependence of the normalized scattered-light intensity on the scattering angle for a prolate spheroidal particle (D = 3.2 μm, d = 1.6 μm, m = 1.13). The main parameters used in the inversion procedure are defined here (see text).

Fig. 5
Fig. 5

Particle diameter distributions obtained by the commercial particle analyzer (Mastersizer) and the apparatus described here for a suspension containing latex spherical particles with 〈D〉 = 5.08 μm.

Fig. 6
Fig. 6

Particle-diameter distributions and cumulative fractional number concentrations obtained by the commercial particle analyzer (Mastersizer) and the apparatus described here for a suspension containing latex spherical particles with diameters D = 2 and D = 5 μm.

Fig. 7
Fig. 7

Analysis of a suspension containing a 1:1 mixture of latex particles and glass beads, D = 10 μm, suspended in water: (a) scattering pattern, (b) fractional number concentrations calculated by the inversion algorithm, (c) comparison of experimental results with theoretical predictions.

Fig. 8
Fig. 8

(a) Light-scattering pattern of RBC’s suspended in an isotonic saline solution and (b) comparison of experimental results and theoretical predictions.

Fig. 9
Fig. 9

(a) Light-scattering pattern of bacteria (Bacilus subtilis) in aqueous solution and (b) comparison of experimental results and theoretical predictions.

Equations (20)

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× × Ψ ext x - k ext 2 Ψ ext x = 0 , · Ψ ext x = 0 , k ext 2 = ω 2 μ ext ε ext ,     x 3 - V ; × × Ψ int x - k int 2 Ψ int x = 0 , · Ψ int x = 0 ,
k int 2 = ω 2 μ int ε int - i   σ ω = ω 2 μ int ε int * ,     x V ,
n ˆ · Ψ int x = δ n ˆ · Ψ ext x , n ˆ × Ψ int x = n ˆ × Ψ ext x , n ˆ × × Ψ int x = ζ n ˆ × × Ψ ext x ,     x S ,
Ψ ext x = U inc x + U sc x .
Ĩ - C ˜ x · Ψ ext x + - S q ˜ ext y ,   x · Ψ ext y d S y = - S g ˜ ext y ,   x · T ext y d S y + U inc x ,
C ˜ x · Ψ int x + - S q ˜ int y ,   x · Ψ int y d S y = - S g ˜ int y ,   x · T int y d S y ,
C ˜ x = Ĩ x V 0 ˜ x 3 - V 1 / 2 Ĩ x S ,
T x = n ˆ × × Ψ x ,
g ˜ y ,   x = - G y ,   x Ĩ ,
q ˜ y ,   x = y G y ,   x n ˆ - n ˆ y G y ,   x + n ˆ · y G y ,   x Ĩ ,
G r = e - ikr 4 π r ,     r = | y - x | .
Ψ int x = Q ˜ · Ψ ext x , T int x = ζ T ext x ,     x S ,
Q ˜ x = Ĩ - n ˆ n ˆ + δ n ˆ n ˆ .
H ˜ ext G ˜ ext H ˜ int G ˜ int Ψ ext T ext = U inc 0 ,
y - x | y - x | = - x ˆ + O 1 | x | , | y - x | = | x | - x ˆ · y + O 1 | x | ,     | x | ,
U sc R = - exp - ik ext R 4 π R g θ R ˆ ,   k ˆ ;   d ˆ θ ˆ + g φ R ˆ ,   k ˆ ;   d ˆ φ ˆ ,     R ,
    g θ R ˆ ,   k ˆ ;   d ˆ = - S ik ext θ ˆ R ˆ - R ˆ θ ˆ : n ˆ Ψ ext y + θ ˆ · T ext y × exp ik ext R ˆ · y d S y ,
g φ R ˆ ,   k ˆ ;   d ˆ = - S ik ext φ ˆ R ˆ - R ˆ φ ˆ : n ˆ Ψ ext y + φ ˆ · T ext y × exp ik ext R ˆ · y d S y , a b : c d = a · d b · c .
σ D = | g θ R ˆ ,   k ˆ ;   d ˆ | 2 + | g φ R ˆ ,   k ˆ ;   d ˆ | 2 k ext 2 ,
F x = B - A x ,

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