Stereological Characterization

Stereology refers to the study of the three-dimensional structure of materials from two-dimensional sections or projections. This discipline is primarily concerned with the geometrical evaluation of microstructural features, based primarily on geometrical probabilities. However, statistics, topology, and projection geometry also contribute to the overall quantitative characterization (Ref 20, 31). Quantitative characterization of the shape of microstructural features (e.g., pore space, precipitates, phase regions, and grains) and of particles on a substrate or embedded in a resin by means of data derived from planar images (sections or projections) is the prime consideration of stereology.

Stereology usually does not apply to the geometric nature of single particles, but rather to the geometric characteristics of a large number of particles or to the geometric characteristics of the "average particle" in a powder mass. Detailed information on the fundamentals, instrumentation, and applications of quantitative stereology can be found in Ref 18, 19,

Numerous shape parameters have been proposed, but only a few are practical. A useful stereological shape parameter (factor) must fulfill several requirements:

• Shape sensitivity: The value of a particular parameter must vary systematically with changes in shape--it should be sensitive to specific aspects such as elongation, bulkiness, and symmetry.

• Independence of other geometric properties: Size, size distribution, volume fraction, or other non-shape-related geometric characteristics should not influence the value for a given shape parameter.

• Accessibility: Quantities from which a stereological shape parameter is calculated must be available by simple measurements performed on planar images.

Other less significant requirements are that shape parameters must be dimensionless and independent of rotation or translation of the objects. Furthermore, it should be easy to visualize the significance of a shape parameter in terms of interpretable microstructural shape changes. The range for typical microstructural changes should be wide compared to statistical fluctuations and measuring errors.

From the parameters listed in standard textbooks (Ref 22, 32) and from parameters proposed in literature (Ref 34, 35, 36, 37), none conforms to these requirements completely. In the following sections, some of the more useful stereological shape parameters, their limitations, and their practical alternatives (fingerprinting by means of simple two-dimensional parameters) are discussed.

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