## Retrieval of particle size distribution in the dependent model using the moment method

Optics Express, Vol. 15, Issue 18, pp. 11507-11516 (2007)

http://dx.doi.org/10.1364/OE.15.011507

Acrobat PDF (194 KB)

### Abstract

The problem of determining particle size distribution using the moment method in the spectral extinction technique is studied. The feasibility and reliability of the retrieval of spherical particle size distribution using the moment method are investigated. The single spherical particle extinction efficiency, which is derived theoretically using the Mie’s solution to Maxwell’s equation, is approximated with a higher order polynomial in order to apply the moment method. Simulation and experimental results indicate that a fairly reasonable representation of the particle size distribution can be obtained using the moment method in the dependent model algorithm. The method has advantages of simplicity, rapidity, and suitability for in-line particle size measurement.

© 2007 Optical Society of America

## 1. Introduction

1. F. Ferri, A. Bassini, and E. Paganini, “Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing,” Appl. Opt. **34**, 5829–5839 (1995). [CrossRef] [PubMed]

4. B. N. Khlebtsov, L. A. Kovler, V. A. Bogatyrev, N. G. Khlebtsov, and S. Y. Shchyogolev, “Studies of phosphatidylcholine vesicles by spectroturbidimetric and dynamic light scattering methods,” J. Quant. Spectrosc. Radiat. Transf. **79**–**80**, 825–838 (2003). [CrossRef]

5. A. K. Roy and S. K. Sharma, “A simple analysis of the extinction spectrum of a size distritution of Mie particles,” J. Opt. A: Pure Appl. Opt. **7**, 675–684 (2005). [CrossRef]

6. M. L. Arias and G. L. Frontini, “Particle size distribution retrieval from elastic light scattering measurements by a modified regularization method,” Part. Part. Syst. Charact. **23**, 374–380 (2007). [CrossRef]

7. A. P. Nefedov, O. F. Petrov, and O. S. Vaulina, “Analysis of particle sizes, concentration, and refractive index in measurement of light transmittance in the forward-scattering-angle range,” Appl. Opt. **36**, 1357–1366 (1997). [CrossRef] [PubMed]

## 2. Theory

*I*passes through a suspension of particle system (thickness

_{0}*L*) with an refraction index different from that of the dispersant medium, scattering and absorption lead to an attenuation of the transmitted light. According to the Lambert-Beer law, if the suspension of particle system is polydisperse spherical and the multiple scattering and interaction effects can be neglected, the transmitted light intensity

*I*is defined as follows[10

10. E. Marioth, B. Koenig, H. Krause, and S. Loebbecke, “Fast particle size and droplet size measurements in supercritical CO_{2},” Ind. Eng. Chem. Res. **339**, 4853–4857 (2000). [CrossRef]

*I*is the incident light intensity at wavelength

_{0}(λ)*λ*(i.e. the intensity of transmitted light through the suspending medium-in the absence of suspended particles), the extinction value

*I*(

*λ*)/

*I*(

_{0}*λ*) is obtained by actual measurements,

*Q*(

_{ext}*λ,m,D*) is the Mie extinction efficiency of a single particle which is a complex function of particle diameter

*D*, wavelength

*λ*in the medium and relative refractive index

*m*(the ratio between the particle and medium refractive index),

*N*is the total number of particles, the lower and upper integration limits are denoted by

_{D}*D*

_{min}and

*D*

_{max},

*f*(

*D*) is the volume frequency distribution of particle system with a diameter between

*D*and

*D*+d

*D*, which is the particle size distribution function to be determined[11

11. A. Katz, A. Alimova, M. Xu, P. Gottlieb, E. Rudolph, J. C. Steiner, and R. R. Alfano, “In situ determination of refractive index and size of bacillus spores by light transmission,” Opt. Lett. **30**,589–591 (2005). [CrossRef] [PubMed]

13. W. Liang, Y. Xu, Y. Y. Huang, A. Yariv, J. G. Fleming, and S. W. Yu, “Mie scattering analysis of spherical Bragg “onion” resonators,” Opt. Express **12**, 657–669 (2004). [CrossRef] [PubMed]

*I(λ)/I*. Therefore the retrieval of Eq. (1) is not a trivial task, because the retrieved distribution might be highly unstable. Hence, frequently one has to resort to approximation schemes. To retrieve the particle size distribution from spectral extinction data in the dependent model one need to solve the optimal problem. The optimization method proceeds to fit the spectral extinction data by varying the form of particle size distribution until the best characteristic parameters are achieved.

_{0}(λ)*f(D)*is described by:

*D*is the particle diameter(in µm),

*u*is the geometric number mean diameter(in µm),σ is the geometric standard deviation.

14. D. L. Wright, S. C. Yu, P. S. Kasibhatla, R. Mcgraw, S. E. Schwartz, V. K. Saxena, and G. K. Yue, “Retrieval of aerosol properties from moments of the particle size distribution for kernel involving the step function:cloud droplet activation,” J. Aerosol Sci. **33**, 319–337 (2002). [CrossRef]

*f(D)*of spherical particles, the

*l*th origin moment M

*is written as:*

_{l}*l*is an arbitrary real number. If

*l*becomes 0, M

*=1. When*

_{0}*l*is 1, M

*is equivalent to the mean value of*

_{1}*f(D)*, and when

*l*is 2, the variance of

*f(D)*is M

_{2}-(M

_{1})

^{2}.

14. D. L. Wright, S. C. Yu, P. S. Kasibhatla, R. Mcgraw, S. E. Schwartz, V. K. Saxena, and G. K. Yue, “Retrieval of aerosol properties from moments of the particle size distribution for kernel involving the step function:cloud droplet activation,” J. Aerosol Sci. **33**, 319–337 (2002). [CrossRef]

15. D. L. Wright, “Retrieval of optical properties of atmospheric aerosols from moments of the particle size distribution,” J. Aerosol Sci. **31**, 1–18 (2000). [CrossRef]

*u*andσ of L-N distribution can be derived by the first-order and second-order moments:

*u*and geometric standard deviation σ can be obtained. With the values of

*u*and σ given by Eqs. (5) and (6), it is possible to construct the L-N size distribution function.

*f(D)*is defined as follows[16

16. J. P. Wang, S. Z. Xie, Y. M. Zhang, and W. Li, “Improved projection to invert forward scattered light for particle sizing,” Appl. Opt. **40**, 3937–3945 (2001). [CrossRef]

*D*is the particle diameter(in µm),

*D*is the is the characteristic diameter of the distribution (in µm), k is narrowness index of the distribution.

*Q*

_{ext}(

*λ,m,D*) can be obtained from the complicated Mie’s solution to Maxell’s equation for spherical particle. However, the mathematical expression of

*Q*

_{ext}(

*λ,m,D*) requires the summation of an infinite series of complicated terms containing spherical Bessel functions, and the integration equation cannot be evaluated as a moment formula in its present from. For this reason,

*Q*

_{ext}(

*λ,m,D*) should be expressed as a function of the power of particle diameter in order to apply the moment method. In this study, the single particle extinction efficiency

*Q*

_{ext}(

*λ,m,D*) which is derived theoretically using Mie’s theory, is approximated with a higher order polynomial. Subsequently, the extinction efficiency can be obtained as follows [8

8. C. H. Jung and Y. P. Kim, “Numerical estimation of the effects of condensation and coagulation on visibility using the moment method,” J. Aerosol Sci. **37**, 143–161 (2006). [CrossRef]

*i*=0…

*P*,

*j*is the jth wavelength, A

_{ij}is the approximated coefficient in regard to P order polynomials which is known beforehand and varied with

*λ*, and the particle size range. The P value also depends on

_{j}, m*λ*, and the particle size range.

_{j}, m*I(λ)/I*can be developed with the general form:

_{0}(λ)*m*=1.235. The curve of the extinction efficiency is oscillatory as a function of the particle diameter, and the shape of the extinction efficiency curve is different in different particle size range. So the fitted polynomial is also different in different particle size range. For the case of (a) where the particles are in the range from 0.1~1 um in diameter, the third-order polynomial is used to fit the extinction efficiency. For the case of (b) where the particles are in the range from 1~5 um in diameter, the tenth-order polynomial is used. For the case of (c) and (d), the twentieth-order polynomial is used to fit the extinction efficiency. As shown in Fig. 1, both results show good agreement without a great loss of accuracy in different particle size range. Here, the overall particle size measurement range is limited from 0.1~10 um in diameter, which is the optimal measurement rang in spectral extinction technique. Thus, we can conclude that the approximated polynomial can be alternatively applied to the retrieval of the particle size distribution instead of the Mie solution for spherical particle.

## 3. Computer simulations and experiments

*N*

_{D}is set

*N*

_{D}=1, because

*N*

_{D}is just an arbitrary scaling factor for the integral properties of the particle size distribution,

*L*=10 mm. The second step is that these data is processed and the particle size distribution is retrieved via the optimization algorithm. In order to examine the effect of noise on the retrieval of particle size distribution, 2% random noise is also added to the intensity signals by multiplying each extinction value by (1+0.02*

*R*(0,1)), representing a 2% noise level, where

*R*(0,1) represents a normally distributed random number with mean zero and standard deviation one. The genetic algorithm is superior to these simplex optimization techniques in the search of global optimal values of multi-object function instead of local ones. It is a heuristic combinatorial search technique that comes from the concept of natural genetics and the Darwinian theory of survival of the fittest, and it obtains its solution in a way after such genetic operations as selection, crossover, and mutation. The genetic algorithm has advantages of simplicity, globality, parallelism although it is a little time consuming. In our investigation, the initial population is generated randomly from a uniform distribution where each variable is chosen within the bounds. At the end of each current generation the elitist strategy is applied and the worst individual is replaced by the best one of the former generation to ensure the globe convergence.

*m*=1.235. The order of the fitted polynomial is determined according to the different particle size range. Since the fitted polynomial of the extinction efficiency is still an oscillatory function. The oscillatory nature of the extinction efficiency determines that it needs more than two equations to solve the two parameters

*D*and

*k*. Here four incident wavelengths(0.4 um, 0.55 um, 0.65 um, 0.8 um) are selected in the visible spectrum for the monomodal distributions. It is possible to see that the retrieved size parameter

*D*provides almost perfect agreement with the given one, but the distribution parameter

*k*is slightly sensitive to the noise. From the practical point of view the results can be considered satisfactory in all the size range, especially for the case of putting 2% random noise in the extinction values. It is important to note that the accuracy of the inversion results is increased by using a higher order polynomial as the fitted extinction efficiency. However, further increasing the order of the polynomial leads to the fitted extinction efficiency become more and more oscillatory and thus the quality of the results cannot be improved.

*m*=1.235. The good features of retrieved results mentioned in Table 2 are still presented for L-N distribution function.

*f(D)*is defined as follows:

*D*̄

_{1},

*D*̄

_{2}are the characteristic diameters of the distribution(in µm),

*k*

_{1},

*k*

_{2}are the narrowness indices of the distribution,

*n*is the weight coefficient between the two peaks.

*m*=1.235. There are five parameters (

*D*̄

_{1},

*k*

_{1},

*D*̄,

*k*

_{2},

*n*) to be retrieved and six incident wavelengths(0.4 um, 0.5 um, 0.6 um, 0.7 um, 0.75 um, 0.8 um) are selected for the bimodal distributions. In order to present the results clearly, the retrieved size distributions are also depicted in Fig. 5. The same conclusion can be drawn that it is very feasible to retrieve the particle size distribution using the moment method for the bimodal size distributions. The bimodal R-R distributions inversion results with different random noise for moment and numerical integration method are listed in Table 6.

*D*=1.98 um and

*D*=3.17 um are produced by Beijing Institute of Nuclear Engineering. The particles are dispersed in water, whose relative refractive indices are

*m*=1.235. The incident wavelengths are selected as

*λ*

_{1}=0.8 um,

*λ*

_{2}=0.6728 um,

*λ*

_{3}=0.5695 um and

*λ*

_{4}=0.4098 um. In this experiment, the extinction intensity ratios at any two wavelengths are used as the input data, and the R-R distribution function is assumed, then the data are processed using the genetic algorithm in combination with the moment relation. The results are presented in Table 7. For a monodisperse or polydisperse particle system, the Sauter mean diameter

*D*

_{32}[defined in Eq. (12)] is the optimal expression of mean diameter for the particle system. As a comparison, we also use the data to retrieve the particle size distribution with numerical integration method. It is possible to see that the retrieved mean diameter

*D*

_{32}is nearly closed to the labeled one. The difference between the retrieved result and the labeled diameter is less than 10%, which satisfies the demand of the standard monodisperse spherical polystyrene particles.

## 4. Conclusion

## Acknowledgments

## References and links

1. | F. Ferri, A. Bassini, and E. Paganini, “Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing,” Appl. Opt. |

2. | A. V. Kharchenko and D. Gresillon, “Nonparticle laser velocimetry and permanent velocity measurement by enhanced light scattering,” Meas. Sci. Technol. |

3. | F. Pedocchi and M. H. Garcia, “Noise-resolution trade-off in projection algorithms for laser diffraction particle sizing,” Appl. Opt. |

4. | B. N. Khlebtsov, L. A. Kovler, V. A. Bogatyrev, N. G. Khlebtsov, and S. Y. Shchyogolev, “Studies of phosphatidylcholine vesicles by spectroturbidimetric and dynamic light scattering methods,” J. Quant. Spectrosc. Radiat. Transf. |

5. | A. K. Roy and S. K. Sharma, “A simple analysis of the extinction spectrum of a size distritution of Mie particles,” J. Opt. A: Pure Appl. Opt. |

6. | M. L. Arias and G. L. Frontini, “Particle size distribution retrieval from elastic light scattering measurements by a modified regularization method,” Part. Part. Syst. Charact. |

7. | A. P. Nefedov, O. F. Petrov, and O. S. Vaulina, “Analysis of particle sizes, concentration, and refractive index in measurement of light transmittance in the forward-scattering-angle range,” Appl. Opt. |

8. | C. H. Jung and Y. P. Kim, “Numerical estimation of the effects of condensation and coagulation on visibility using the moment method,” J. Aerosol Sci. |

9. | S. H. Park, R. Xiang, and K. W. Lee, “Brownian coagulation of fractal agglomerates:analytical solution using the log-normal size distribution assumption,” J. Colloid Interface Sci. |

10. | E. Marioth, B. Koenig, H. Krause, and S. Loebbecke, “Fast particle size and droplet size measurements in supercritical CO |

11. | A. Katz, A. Alimova, M. Xu, P. Gottlieb, E. Rudolph, J. C. Steiner, and R. R. Alfano, “In situ determination of refractive index and size of bacillus spores by light transmission,” Opt. Lett. |

12. | D. Rosskamp, F. Truffer, S. Bolay, and M. Geiser, “Forward scattering measurement device with a high angular resolution,” Opt. Express |

13. | W. Liang, Y. Xu, Y. Y. Huang, A. Yariv, J. G. Fleming, and S. W. Yu, “Mie scattering analysis of spherical Bragg “onion” resonators,” Opt. Express |

14. | D. L. Wright, S. C. Yu, P. S. Kasibhatla, R. Mcgraw, S. E. Schwartz, V. K. Saxena, and G. K. Yue, “Retrieval of aerosol properties from moments of the particle size distribution for kernel involving the step function:cloud droplet activation,” J. Aerosol Sci. |

15. | D. L. Wright, “Retrieval of optical properties of atmospheric aerosols from moments of the particle size distribution,” J. Aerosol Sci. |

16. | J. P. Wang, S. Z. Xie, Y. M. Zhang, and W. Li, “Improved projection to invert forward scattered light for particle sizing,” Appl. Opt. |

**OCIS Codes**

(290.2200) Scattering : Extinction

(290.5820) Scattering : Scattering measurements

(290.5850) Scattering : Scattering, particles

(300.6360) Spectroscopy : Spectroscopy, laser

**ToC Category:**

Scattering

**History**

Original Manuscript: July 3, 2007

Revised Manuscript: August 14, 2007

Manuscript Accepted: August 22, 2007

Published: August 24, 2007

**Virtual Issues**

Vol. 2, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Xiaogang Sun, Hong Tang, and Jingmin Dai, "Retrieval of particle size distribution in the dependent model using the moment method," Opt. Express **15**, 11507-11516 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-18-11507

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### References

- F. Ferri, A. Bassini, and E. Paganini, "Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing," Appl. Opt. 34, 5829-5839 (1995). [CrossRef] [PubMed]
- A. V. Kharchenko, and D. Gresillon, "Nonparticle laser velocimetry and permanent velocity measurement by enhanced light scattering," Meas. Sci. Technol. 14, 228-233 (2003). [CrossRef]
- F. Pedocchi, and M. H. Garcia, "Noise-resolution trade-off in projection algorithms for laser diffraction particle sizing," Appl. Opt. 45, 3620-3628 (2006). [CrossRef] [PubMed]
- B. N. Khlebtsov, L. A. Kovler, V. A. Bogatyrev, N. G. Khlebtsov, and S. Y. Shchyogolev, "Studies of phosphatidylcholine vesicles by spectroturbidimetric and dynamic light scattering methods," J. Quant. Spectrosc. Radiat. Transf. 79-80, 825-838 (2003). [CrossRef]
- A. K. Roy and S. K. Sharma, "A simple analysis of the extinction spectrum of a size distritution of Mie particles," J. Opt. A: Pure Appl. Opt. 7, 675-684 (2005). [CrossRef]
- M. L. Arias, and G. L. Frontini, "Particle size distribution retrieval from elastic light scattering measurements by a modified regularization method," Part. Part. Syst. Charact. 23, 374-380 (2007). [CrossRef]
- A. P. Nefedov, O. F. Petrov, and O. S. Vaulina, "Analysis of particle sizes, concentration, and refractive index in measurement of light transmittance in the forward-scattering-angle range," Appl. Opt. 36, 1357-1366 (1997). [CrossRef] [PubMed]
- C. H. Jung and Y. P. Kim, "Numerical estimation of the effects of condensation and coagulation on visibility using the moment method," J. Aerosol Sci. 37, 143-161 (2006). [CrossRef]
- S. H. Park, R. Xiang, and K. W. Lee, "Brownian coagulation of fractal agglomerates:analytical solution using the log-normal size distribution assumption," J. Colloid Interface Sci. 231, 129-135 (2000). [CrossRef] [PubMed]
- E. Marioth, B. Koenig, H. Krause, and S. Loebbecke, "Fast particle size and droplet size measurements in supercritical CO2," Ind. Eng. Chem. Res. 339, 4853-4857 (2000). [CrossRef]
- A. Katz, A. Alimova, M. Xu, P. Gottlieb, E. Rudolph, J. C. Steiner, and R. R. Alfano, "In situ determination of refractive index and size of bacillus spores by light transmission," Opt. Lett. 30, 589-591 (2005). [CrossRef] [PubMed]
- D. Rosskamp, F. Truffer, S. Bolay, and M. Geiser, "Forward scattering measurement device with a high angular resolution," Opt. Express 15, 2683-2690 (2007). [CrossRef] [PubMed]
- W. Liang, Y. Xu, Y. Y. Huang, A. Yariv, J. G. Fleming, and S. W. Yu, "Mie scattering analysis of spherical Bragg "onion" resonators," Opt. Express 12, 657-669 (2004). [CrossRef] [PubMed]
- D. L. Wright, S. C. Yu, P. S. Kasibhatla, R. Mcgraw, S. E. Schwartz, V. K. Saxena, and G. K. Yue, "Retrieval of aerosol properties from moments of the particle size distribution for kernel involving the step function:cloud droplet activation," J. Aerosol Sci. 33, 319-337 (2002). [CrossRef]
- D. L. Wright, "Retrieval of optical properties of atmospheric aerosols from moments of the particle size distribution," J. Aerosol Sci. 31, 1-18 (2000). [CrossRef]
- J. P. Wang, S. Z. Xie, Y. M. Zhang, and W. Li, "Improved projection to invert forward scattered light for particle sizing," Appl. Opt. 40, 3937-3945 (2001). [CrossRef]

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