A New Metric for the Space of Macroscopic Parameters of Grain Interfaces
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Crystallographic description of grain interfaces in polycrystalline materials is frequently based on macroscopic interface parameters.[1,2] In the recent past, numerous sets of macroscopic grain boundary data have been collected and analyzed; see, e.g., References 3–9 and references therein. These analyses require identifying similar interfaces. For this purpose, a measure of interface similarity is needed. It is convenient to assess the similarity using a formally defined metric or a distance between interfaces. Although this is not always explicitly stated in reports on large interface datasets, the notion of a distance is fundamental for rigorous analysis of the data. The distance is linked to volume in the space of macroscopic parameters, and the volume in turn is necessary for estimating frequencies of occurrence of different boundary types. In particular, the interface distances are essential for determining the
A. MORAWIEC is with the Institute of Metallurgy and Materials Science, Polish Academy of Sciences, Reymonta 25, 30-059 Krako´w, Poland. Contact e-mail: [email protected] Manuscript submitted April 1, 2019. Article published online July 16, 2019 4012—VOLUME 50A, SEPTEMBER 2019
frequencies based on the kernel density estimation.[8–10] They are also crucial for interpolation of functions on the space of macroscopic parameters; see, e.g., Reference 11. A number of distance functions have been proposed in the past.[12–18] This paper introduces a new metric with a simple geometric interpretation. The distance between two interfaces represents a magnitude of crystallite rotations needed for transforming one interface into the other. The macroscopic parameters of an interface between two (centrosymmetric) crystals are given in the (Cartesian) reference frame of one of the crystals. They specify the misorientation of the crystal on the other side of the interface and the outward directed normal to the interface plane. In description of homophase interfaces, the cases with zero misorientation angles (‘‘no-boundary singularity’’) are excluded from the macroscopic domain. With M denoting a 3 3 special orthogonal matrix representing the misorientation and n1 being a unit vector normal in the frame of the first crystal, the pair ðM; n1 Þ identifies macroscopically the interface from the viewpoint of the first grain. From the viewpoint of the second grain, this interface is represented by the pair ðMT ; n2 Þ; where n2 ¼ MT n1 :[12] By definition, a metric space is a set of points with a non-negative distance function d such that dðx; xÞ ¼ 0; dðx; yÞ ¼ 0 implies x ¼ y (identity of indiscernibles), dðx; yÞ ¼ dðy; xÞ (symmetry) and dðx; yÞ þ dðy; zÞ dðx; zÞ (triangle inequality) for every x, y and z.[19] A distance function in the space of macroscopic parameters needs to satisfy some additional symmetry conditions. Generally, a physically unique interface has multiple symmetrically equivalent representations in the space of macroscopic parameters, and the distance between given two points in that space must be equ
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