Models of uniformity for grain boundary distributions

Progress in experimental methods of serial sectioning and orientation determination opens new opportunities to study inter-crystalline boundaries in polycrystalline materials. In particular, macroscopic boundary parameters can now be measured automatically. With sufficiently large data sets, statistical analysis of interfaces between crystals is possible. The most basic and interesting issue is to find out the probability of occurrence of various boundaries in a given material. In order to define a boundary density function, a model of uniformity is needed. A number of such models can be conceived. It is proposed to use those derived from an assumed metric structure of the interface manifold. Some basic metrics on the manifold are explicitly given, and a number of notions and constructs needed for a strict definition of the boundary density function are considered. In particular, the crucial issue of the impact of symmetries is examined. The treatments of homo- and hetero-phase boundaries differ in some respects, and approaches applicable to each of these two cases are described. In order to make the abstract matter of the paper more accessible, a concrete boundary parameterization is used and some examples are given.