Stereology of dense polycrystalline materials - from interface density and mean curvature integral density to Rayleigh distributions of grain sizes

Based on planar sections, the microstructure of dense polycrystalline materials can be described by two global metric microstructural descriptors, the interface density and the mean curvature integral density, which is closely related to the edge line density (= volumetric density of the triple junction line length) [1]. The former is related to the mean chord length of the grains or crystallites, while the latter is related to the so-called Jeffries grain size [2,3]. Both these direct grain size measures can be transformed into indirect grain size measures, so-called grain size numbers (linear and planar), and moreover, the Jeffries size concept can be generalized to two-phase microstructures, e.g. porous media [4]. In this work the aforementioned microstructural descriptors have been determined for transparent YAG ceramics intended for laser applications. Absolute errors, both expected and observed, have been calculated using the standard error and the normalized deviation (t-variate) according to Student's t- distribution. Interface densities are in the range 60.6-281.9 mm?1, mean curvature integral densities in the range 5081-104639 mm?2, mean chord lengths 7.2-33.2 µm, Jeffries grain sizes 7.8-35.5 µm, linear grain size numbers 6.54-10.98 and planar grain size numbers 6.69-11.06. The ratio of mean chord length and Jeffries grain size, which is characteristic of the microstructure, is determined to be 0.927 ± 0.034. For selected samples the size distribution of grain sections is measured via area-equivalent circle diameters (Heywood diameters) and, considering the isometric grain shape as an approximation to spherical shape, the 2D grain section size distribution is transformed via the Saltykov transformation [1] to obtain the true 3D grain size distribution in terms of equivalent sphere diameters. It is found that in this case the Saltykov transformation has almost no effect, i.e. the grain size distribution, which corresponds to the final microstructure after sintering, is very close to a Rayleigh distribution. This is proved by fitting the measured grain section size distribution with a two-parameter Weibull function. It can be conjectured that it is a very general law of nature that due to Ostwald ripening (growth of large grains at the cost of small grains) the grain size distribution in dense single-phase polycrystalline materials with mature microstructures tends to approach a Rayleigh distribution. References: [1] Russ J. C., Dehoff R. T.: Practical Stereology. Second edition. Kluwer Academic / Plenum Press 2000. [2] ASTM E-112-96: Standard test methods for determining average grain size. American Soceity for Testing of Materials, West Conshohoken 1996 (reapproved 2004). [3] Uhlí?ová T., Hosta?a J., Pabst W.: Characterization of the microstructure of YAG ceramics via stereology-based image analysis, Ceram. Silik. 58 (3), 173-183 (2014). [4] Uhlí?ová T., Gregorová E., Pabst W., Ne?ina V.: Preparation of cellular alumina ceramics via biological foaming with yeast and its microstructural characterization via stereological relations, J. Eur. Ceram. Soc. 35, 187-196 (2015).