Stereological analysis of spherical particles: Experimental assessment and comparison to laser diffraction
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I. INTRODUCTION
THE measurement of two-dimensional (2-D) sections of particles or precipitates produces size distributions with many cross sections that are actually sections of larger particles. If the “three-dimensional” (3-D) particle-size distribution is to be estimated, a stereological unfolding procedure such as the Schwartz–Saltykov (SS) technique[1,2,3] or the method developed by Harayama[4] and applied by Basak and Sengupta[5] (HBS) can be applied to the 2-D distribution. The SS technique is a classical stereology procedure, a basic requirement of which is that the particles are spherical (circular cross sections). Even with this constraint, the stereological methods may be useful for analysis of polymer powders, atomized metal powders, spherical pores in metals or ceramics,[6] and graphite nodules in cast iron,[5,7,8] depending on the degree to which the features deviate from spherical. Stereological procedures for particles with nonspherical shapes have also been developed, often with the aid of computer simulation. Examples include techniques for ellipsoidal particles[9,10] and cubo-spherical particles produced by liquidphase sintering.[11,12] Many methods are available for particle-size analysis, such as laser diffraction, sedimentation, photon-correlation spectroscopy, and sieving techniques.[13–17] Of these, laser diffraction (LD) is the most common and widely accepted method. Most of these techniques rely on an effect of particle size, e.g., light scattering, rather than the direct measurement of the particles themselves. Microscopy/image analysis (IA) is a direct method of measuring particle size, and it is generally regarded as highly accurate.[18] Some comparisons of microscopy, laser diffraction, sieve techniques, etc., have
shown good agreement in particle- or pore-size measurement.[14,15,16] In other studies, however, experimental limitations with IA gave mixed results for particle- and pore-size analysis compared with other methods.[13,17] While traditional microscopic analysis involves samples prepared on slides and analyzed by light microscopy or scanning electron microscopy (SEM), several researchers have analyzed cross sectioned precipitates or particles prepared in a packed bed using stereological techniques.[19–23] To date, a direct comparison of IA/stereology of sectioned particles with LD of loose powders has not been performed. The objectives of this work were to assess the viability of IA and the SS or HBS techniques as quantitative particle-analysis methods and to compare them to LD particle-size analysis. With spherical particles, there is no shape-related ambiguity in the LD results or the stereology process and it, thus, provides for a good comparison. The experimental approach allows for assessment of some difficulties in IA such as noise in the images due to sample-preparation artifacts, statistical noise in the data, etc. Also, unlike model distributions which are often approximated as log-normal, experimental particlesize distributions may contain irregularities and deviatio
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