Statistical extreme value modeling of particle size distributions: experimental grain size distribution type estimation and parameterization of sintered zirconia

The grain size distribution of a sintered zirconia is investigated in detail. Microstructural evidence in combination with X-ray diffraction (XRD) data reveal a bimodal microstructure consisting of a polycrystalline matrix of retained untransformed tetragonal zirconia, containing clusters of smaller transformed monoclinic zirconia grains. This necessitates a comparison between the size measurements for the untransformed zirconia and the full set of measurements made on the sintered zirconia microstructure. A statistically consistent, graphical representation of grain size density distributions is shown. The experimental frequency density distributions are compared with the Log Normal and the Gumbel model. Traditional statistical tools are used to estimate the probability of observing the experimental data set from the expected model functions. Parameter estimations for the model functions are made according to a quantile approach, an R-2 fit, a chi (2) fit and a non-linear simplex optimization routine. The bimodality of the experimentally obtained grain size distribution influences the likelihood probabilities of the model functions in accordance with their respective boundary conditions. The Gumbel function is shown to provide the best correspondence to the experimental data, with a slightly higher probability for a unimodal distribution as compared with the bimodal dataset. The Log Normal function is less likely than the Gumbel function in both instances, but has a slightly higher probability for the bimodal data set. It is shown that the quantile estimation is fast and straightforward to implement and allows a full evaluation of the expectation ranges on the estimated parameters. (C) 2000 Elsevier Science Inc. All rights reserved.