Applications of Bicrystallography: Revealing Generic Similarities in Coincidence Site Lattice Boundaries of all Holohedral Cubic Materials and Facilitating the Design of 3D Printed Models of such Grain Boundaries

Abstract

The application of bicrystallography [1-3] makes structural units [4][5][6] and dislocations [7] superfluous as descriptors of grain boundaries. All there is around an ideal (un-relaxed) coincidence site lattice (CSL) grain boundary are predicable atomic positions with certain black-white symmetries. Free energy minimization driven relaxations of such a hypothetical grain boundary structure may lead to the breaking of some or all of the black-white symmetries, just as monochrome space group symmetries may be broken locally in the two real single crystals that make up the bicrystal. Any relaxed (real) atomic position will, however, be very close to the predicted un-relaxed (ideal) atomic position as long as that position remains occupied. Grain boundary structures with nine degrees of freedom as predicted by bicrystallography in three dimensions (3D) are, therefore, ideal as atomic level starting structures for free energy minimization calculations! Bicrystallography in two dimensions (2D) allows for both visualizations of edge-on projections of CSL tilt boundaries [3] and the extension of the atomic column indexing procedure of ref.[8] to all experimental 2D images of such boundaries. In order to illustrate the predictive power of bicrystallography in 2D briefly, we reproduce below in Figure 1 migration related segments of translation averaged Z-STEM images of a Σ 13a (510) tilt boundary in SrTiO3 in [001] projection from ref. [3] (with permission from the publisher).A range of CSL [001] tilt boundaries in CeO2 were imaged with an aberration corrected Z-contrast Scanning Transmission Electron Microscope (Z-STEM) and striking visual similarities of their (projected) structural units to those of (analogous orientation relationship) grain boundaries in face-center cubic metals, yttriastabilized zirconia, and SrTiO3 (Fig. 1) were recently noted in ref. [4]. There is, however, some arbitrariness in the choosing of structural units [7], despite their emergence in connection with free energy minimization calculation [5] and their identification as cores of grain boundary dislocations [6]. In essence, they are arrangements of atoms from both sides of a grain boundary with geometries that can be chosen "simply for convenience" [7]. The predictive power of structural units is, accordingly, limited to series of grain boundaries in the same material when only one of the five macroscopic degrees of freedom is varied [7]. For predictions of the structures of tilt boundaries on the basis of structural units, the parameter that varies in a series is most often the tilt angle, e.g. see ref. [4].Our bicrystallography analysis, on the other hand, shows that these structural similarities are simply byproducts of the edge-on projection of the black-white layer groups of the bicrystals in these materials [3]. In short, whenever Bärnighausen trees [9] reveal structural relations (i.e. similarities) between the space groups of different materials, there will (per bicrystallography) also be predicable structural similarities in their ana...