Abstract

Particle-size distribution and the concentration of polystyrene particles suspended in water were accurately recovered from the inversion of spectral extinction data measured with a commercial spectrophotometer. The instrument was modified by placing a spatial filter in the collection optics to prevent low-angle scattered light from affecting the measurement of transmitted power. The data were inverted by use of a nonlinear iterative algorithm. When the extinction coefficient is measured in the λ range of 0.3–1.1 µm, the particle distributions can be retrieved over a diameter range of 0.6–2.8 µm for a wide interval of sample concentrations. The average diameters are recovered with a precision of better than ±1% and with accuracies consistent with the uncertainties by which the nominal diameters are known. The relative standard deviations of distributions corresponding to monodisperse samples are ±5–10%, whereas the accuracy on the measured concentrations is ∼5%.

© 1997 Optical Society of America

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References

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  1. K. Leschonki, ed., Preprints of the Fifth European Symposium on Particle Characterization (Numberg Messe, Numberg, Germany, 1992).
  2. E. D. Hirleman, ed., Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State University Printing Services, Tempe, Ariz., 1990).
  3. M. Maeda, S. Nakae, M. Ikegami, eds., Proceedings of the Third International Congress on Optical Particle Sizing, 23–26 August 1993, Yokohama, Japan.
  4. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977).
  5. F. Ferri, A. Bassini, E. Paganini, “Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing,” Appl. Opt. 34, 5829–5839 (1995).
    [CrossRef] [PubMed]
  6. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 9, p. 127.
  7. M. T. Chahine, “Determination of the temperature profile in an atmosphere from its outgoing radiance,” J. Opt. Soc. Am. 58, 1634–1637 (1968).
    [CrossRef]
  8. M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
    [CrossRef]
  9. W. H. Richardson, “Bayesian based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  10. L. B. Lucy, “An iterative technique for the rectification of the observed distributions,” Astron. J. 79, 745–754 (1974).
    [CrossRef]
  11. R. H. Partridge, “Vacuum-ultraviolet absorption spectrum of polystyrene,” J. Chem. Phys. 47, 4223–4227 (1967).
    [CrossRef]
  12. I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985); Astron. J. 79, 745–754 (1974).
    [CrossRef]
  13. R. H. Boundy, R.F. Boyer, eds., Styrene, Its Polymers, Copolymers and Derivatives (Reinhold, New York, 1952), p. 524.

1995 (1)

1985 (1)

I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985); Astron. J. 79, 745–754 (1974).
[CrossRef]

1974 (1)

L. B. Lucy, “An iterative technique for the rectification of the observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

1972 (1)

1970 (1)

M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

1968 (1)

1967 (1)

R. H. Partridge, “Vacuum-ultraviolet absorption spectrum of polystyrene,” J. Chem. Phys. 47, 4223–4227 (1967).
[CrossRef]

Bassini, A.

Chahine, M. T.

M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

M. T. Chahine, “Determination of the temperature profile in an atmosphere from its outgoing radiance,” J. Opt. Soc. Am. 58, 1634–1637 (1968).
[CrossRef]

Ferri, F.

Grigull, U.

I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985); Astron. J. 79, 745–754 (1974).
[CrossRef]

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of the observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Paganini, E.

Partridge, R. H.

R. H. Partridge, “Vacuum-ultraviolet absorption spectrum of polystyrene,” J. Chem. Phys. 47, 4223–4227 (1967).
[CrossRef]

Richardson, W. H.

Straub, J.

I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985); Astron. J. 79, 745–754 (1974).
[CrossRef]

Thormahlen, I.

I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985); Astron. J. 79, 745–754 (1974).
[CrossRef]

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 9, p. 127.

Appl. Opt. (1)

Astron. J. (1)

L. B. Lucy, “An iterative technique for the rectification of the observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

J. Atmos. Sci. (1)

M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

J. Chem. Phys. (1)

R. H. Partridge, “Vacuum-ultraviolet absorption spectrum of polystyrene,” J. Chem. Phys. 47, 4223–4227 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. Chem. Ref. Data (1)

I. Thormahlen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985); Astron. J. 79, 745–754 (1974).
[CrossRef]

Other (6)

R. H. Boundy, R.F. Boyer, eds., Styrene, Its Polymers, Copolymers and Derivatives (Reinhold, New York, 1952), p. 524.

K. Leschonki, ed., Preprints of the Fifth European Symposium on Particle Characterization (Numberg Messe, Numberg, Germany, 1992).

E. D. Hirleman, ed., Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State University Printing Services, Tempe, Ariz., 1990).

M. Maeda, S. Nakae, M. Ikegami, eds., Proceedings of the Third International Congress on Optical Particle Sizing, 23–26 August 1993, Yokohama, Japan.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 9, p. 127.

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Figures (11)

Fig. 1
Fig. 1

Modified optical layout of the spectrophotometer. In the measuring arm a spatial filter was added to keep low-angle scattered light from falling onto the detector.

Fig. 2
Fig. 2

Behavior of the measured extinction coefficient α(λ) versus the vacuum wavelength λ for suspensions of polystyrene spheres with diameters of (a) 1.06 µm and (b) 2.75 µm. The different symbols refer to measurements taken by placing different pinhole sizes in the measuring arm of the spectrophotometer (see text and Fig. 1). The solid lines are guides to the eye, passing through the data corresponding to the smallest pinhole size. The sample concentrations are 3.81 and 4.21 × 107 cm-3.

Fig. 3
Fig. 3

Retrieved particle distributions obtained by inverting two of the data shown in Fig. 2(a). The solid line shows the distribution when a 0.5-mm pinhole is used, whereas the dotted line shows the distribution when no pinhole is used.

Fig. 4
Fig. 4

Behavior of the parameters characterizing the retrieved distributions for 1.06-µm particles with a number concentration of 3.81 × 107 cm-3 as a function of the pinhole size. The inversion was carried out by inverting the same data shown in Fig. 2(a).

Fig. 5
Fig. 5

(a) Retrieved distributions and (b) reconstructed signals obtained by inverting the data corresponding to 1.06-µm particles taken with a 1-mm pinhole size. The solid lines show the results when the inversion was carried out by taking into account the dependence of the refractive indices of both the solvent and polystyrene as a function of λ. In this case the retrieved distribution is fairly accurate (see Fig. 4), and the match between the experimental data (circles) and the reconstructed signals is excellent. Vice versa the dotted lines show the results when the refractive indices are kept constant (n solv = 1.33, n polyst = 1.584). In this second case the quality of the signal reconstruction is fairly poor, whereas the retrieved distribution is fairly different and far from that expected.

Fig. 6
Fig. 6

(a) Retrieved distributions and (b) reconstructed signals obtained by inverting the data corresponding to a sample made of a mixture of particles of two different diameters, 1.06 and 1.78 µm, and concentrations equal to 1.50 and 1.53 × 107 cm-3, respectively. The average diameters corresponding to the two peaks are recovered with accuracies of ∼5% and ∼4%. The rms deviations between the experimental data and the retrieved signals are ∼1%.

Fig. 7
Fig. 7

Comparison between the experimental results and the performances of our inversion algorithm as estimated by computer simulations and described in Ref. 5. The nominal distribution corresponding to particles with a 1.06-µm diameter and a relative standard deviation 2.2% (dotted line), the retrieved distribution obtained by inverting the simulated data generated according to the nominal distribution (dashed curve), and the retrieved distribution obtained by inverting the experimental data (solid line) is shown.

Fig. 8
Fig. 8

Deviations of the recovered average diameters with respect to their nominal values as a function of the sample concentration for eight different particle diameters. All the data were taken with a 1-mm pinhole and 10-mm cell.

Fig. 9
Fig. 9

Relative standard deviations of the retrieved distributions as a function of the sample concentration for the same samples of Fig. 8.

Fig. 10
Fig. 10

Deviations of the retrieved signals with respect to the experimental data as a function of the sample concentration for the same samples in Fig. 8.

Fig. 11
Fig. 11

Errors on the recovered volume fraction concentration as a function of the sample concentration for the same samples as in Fig. 8.

Equations (3)

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PT=Po exp-αλL,
αλ= πr2Qextr, λ, mNrdr,
Njp+1=Njpi=1q Wijαmeasλiαcalcpλi  j=1, 2, , q,

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