Evolution of bivariate particle size distributions
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I.
INTRODUCTION
MICROSTRUCTURES containing an ensemble of discrete convex particles are frequently observed in nature. For example, precipitation hardenable alloys contain population of the precipitates of one phase in the matrix of another phase. The particulate microstructures evolve during the transformations, such as nucleation and growth, dissolution, coarsening, etc. In order to understand microstructure-properties-processing correlations, it is of interest to relate the kinetic variables, such as nucleation rate and growth rate (associated with the processing), to the evolution of the particle size distribution (which affects the properties) in a rigorous and quantitative manner. DeHoff t~j has given a simple and elegant procedure (called "growth path analysis") for the calculation of individual particle growth rates in an evolving population from a series of experimentally measured particle size distributions during a microstructural transformation. Diwan, t2] Sanders and Gokhale, t3J and Fang eta/. [4] have utilized DeHoff's growth path analysis to calculate the individual particle growth/shrinkage rates during particle coarsening; these calculations permit verification of the particle coarsening theories at their roots. DeHoff's analysis also permits calculation of the particle size distribution function and its evolution from the quantitative mathematical models for the nucleation rate and growth rate; such techniques are useful for the modeling of global microstructural evolution. A relationship between the particle size distribution function, particle shrinkage rate, and initial size distribution has been derived for the precipitate dissolution process I5~ as well. DeHoff's analysis t~l is based on the following assumptions. (1) The geometric state of any given particle in the population can be specified unambiguously by just one size parameter (say, R), and hence, the particle growth paths (R vs t plots) are plane curves and the particle size distribution function is univariate in size (i.e., it is a function of only one size parameter R). (2) Only one growth path can originate from any given nucleation time (say, z). (3) Growth paths do not cross. ARUN M. GOKHALE, Professor, is with the School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245. Manuscript submitted October 29, 1991. METALLURGICAL TRANSACTIONS A
It is the purpose of this article to extend DeHoff's analysis to bivariate particle size distributions. Note that DeHoff's analysis, as well as the present work, invokes a "mean field" description of the microstructural transformation process. Microstructures often contain a population of particles having different sizes and shapes, and in such cases, more than one size parameter is required to uniquely identify any given particle in the population, t6'~4j Such microstructures are characterized by bivariate or multivariate size distribution functions. A bivariate size distribution is a function of two independent size parameters (say, R and L); a r
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