Elsevier

Powder Technology

Volume 107, Issue 3, 1 February 2000, Pages 243-250
Powder Technology

Number of particles for the determination of size distribution from microscopic images

https://doi.org/10.1016/S0032-5910(99)00192-8Get rights and content

Abstract

The aim of this study was to determine the minimal number of particles required for assessing a reliable particle size distribution from images acquired using microscopy. The proposed methodology used the one-sample Kolmogorov–Smirnov statistic to choose an a priori number of particles and the two-sample Kolmogorov–Smirnov test to validate the sub-sampling procedure inherent to microscopic preparation. The methodology was applied to number-based particle size distributions of starch granules.

Introduction

Determination of particle size distribution is of growing interest in the food industry due to its influence on the processing properties of powders. Microscopy is the only method that provides direct observation of the particles and is therefore considered as a reference method. Image analysis associated with microscopy allows the determination of particle size distribution. In this case, the usual procedure consists in assessing size parameters such as area, length or width for each individual particle analysed. The resulting particle size frequencies may be expressed as number percentage leading to number based distributions, or as percentage of a given measurement, i.e., total area, leading to measurement based distributions. As a result, the comparison of image resulting particle size distribution to other kinds of measurements such as those obtained by sieving or laser light diffraction requires the numerical transformation of the data into volume-based distributions [1]. In the present work, number-based distributions were considered as basically measured by microscopy.

A relevant determination of the size distribution requires the analysis of a sufficient amount of particles. As microscopy is a time-consuming method, the determination of the minimum number of particles to be analysed is a key point of the technique. In practice, an average number of at least 1500 particles and sometimes more [2] is recommended. Barreiros et al. [3], however, noted no difference between results generated from about 1500 particles and those obtained by counting only a few hundred particles. The objective of this work was to propose a method for determining the minimal number of particles required for a relevant estimation of the number based size distribution. Error in the estimation of particle size distribution has been investigated by Paine [4] on the basis of the volume median diameter and the geometrical standard deviation. The method proposed by Paine, however, only applies to log–normal distribution. Another approach exists which considers the uncertainty of the whole distribution function evaluated by the confidence band without reference to any theoretical model. The one-sample Kolmogorov–Smirnov statistic can be used to construct a (1−α) confidence band for an observed cumulative distribution function 5, 6 which depends on the number of particles.

The size distribution is usually assumed to result from a single sample of n particles. The microscopic analysis, however, requires several slides to be observed. The final distribution is estimated by pooling the slide sub-samples. The homogeneity of the sub-samples can be advantageously controlled in order to validate the slide sampling procedure. In the present work, the comparison of slide distributions was performed using pairwise two-sample Kolmogorov–Smirnov tests.

In the proposed methodology, an a priori number of particles is chosen from the one-sample Kolmogorov–Smirnov theory and the two-sample Kolmogorov–Smirnov test is used for experimental procedure validation. Starch granules are chosen for illustration.

Section snippets

Theoretical approach: Kolmogorov–Smirnov statistic

Comparisons of observed with theoretical distributions as well as comparisons of two experimental distributions measured from two independent samples are usually performed by the χ2 test. The Kolmogorov–Smirnov test is another available procedure. The book of Daniel [6] gives a useful discussion and illustration of both tests.

The Kolmogorov–Smirnov test was designed for use with continuous data (here, the particle size), while the χ2 test was designed for frequency histograms. The latter,

Experimental

The proposed methodology was applied to the analysis of Waxy cornstarch (Ref. Clearam CH20, Roquette, Lestrem, France) composed of granules fairly polyhedral in shape with a small dispersion size.

The powder was observed in liquid dispersion to limit aggregation, using water with tensio-active. Images were obtained with an image analysis system consisting of a CCD camera and an image digitizer. The recorded images were tables of 512×512 pixels which corresponded to areas of 154×154 μm. Ten views

Conclusion

In order to determine the number of particles required for assessing number based particle size distribution by image analysis, one has to reach a compromise between operating time and reliability of experimental results. Using the one-sample Kolmogorov–Smirnov statistic, a confidence band was constructed over the whole distribution function. The relationship between the width of the band, the risk and the sample size was used to determine an initial number of particles to get a reliable

References (10)

  • C. Loisel, E. Vigneau, P. Cantoni, M.F. Devaux, PARTEC'98, 7th European Symposium Particle Characterization, II,...
  • M. Wedd, Particle Sizing, in: Pinder, A.C., Godfrey, G. (Eds.), Food Process Monitoring Systems, Blackie Academic and...
  • F.M. Barreiros et al.

    Part. Part. Syst. Charact.

    (1996)
  • A.J. Paine

    Part. Part. Syst. Charact.

    (1993)
  • F.J. Massey

    American Statistical Association Journal

    (1951)
There are more references available in the full text version of this article.

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