Institute of Metals Division - The Estimation of Particle-Size Distributions from Simple Counting Measurements Made on Random Plane Sections

A new approach to the measurement of the size distribution of particulate structures imbedded in opaque bodies is developed. The method is based upon the assumption of a general two-parameter distribution curve, and the estimation of these two parameters from three well-known counting measurements made upon metallogvaphic sections. The total number of particles in the system is also a product of the analysis. Application of the method with the assumption of the logarithmiconorma1 disfribution function is developed in detail. A complete description of the geometry of a particulate structure requires the determination of the size distribution of particles. The problem of estimating the size distribution of particles of a particular phase in a multiphase system, or the grain-size distribution in a homogeneous system, has been approached by many investigators.'-'0 All of the solutions proposed by these investigators share the following characteristics. a) A minimum of several hundred particle sections in the microstructure must be examined, and some characteristic dimension recorded, such as area,3'5 intercept length,6"9 or diameter.1'2'4'10 b) The resulting data must undergo a series of fairly complex calculations to generate the empirical histogram which describes the size distribution in three dimensions. c) In order to manipulate the result, the histogram must be fitted to some assumed mathematical formula, using the attendant statistical procedures. d) With two notable exceptions,8,10 the particles must be spherical. The measurements and calculations involved are in general very tedious and time-consuming, with the logical result that much effort has been expended in the development of equipment which will perform these tasks automatically and semiautomati-cally.9,11-15 considerable success has been achieved in this area, but the equipment is in general expensive, and usually places stringent requirements On the quality of microstructure which can be reliably analyzed. In laboratories which do not possess such equip- ment, the analysis of geometry is usually limited to measuring total properties of the microstructures, such as volume fraction and interfacial area, because the measurements required are much simpler to make. The determination of size distributions rarely is undertaken. The development which is presented in this paper permits the estimation of size distributions of three-dimensional structures from the same simple counting measurements which are required to determine the surface area and volume fraction of the particulate structure, and one other counting measurement. The basic approach that is used is to assume a general mathematical form for the distribution, which is described by two parameters (as the normal distribution is described by assigning values to the mean and the standard deviation), and then estimate these two parameters from the results of the counting measurements. The introduction of a third counting measurement also permits the evaluation of the number of particles per unit volume, Nv.