Interface dislocations and grain boundary disconnections using Smith normal bicrystallography

The CSL/DSCL model for interfaces in crystalline materials offers a unified framework to study interface dislocations in phase boundaries and disconnections in grain boundaries. The model relies on the existence of a coincidence relation between the two lattices that meet at an interface. The model's ability to quantitatively predict the thermodynamics and kinetics of interfaces has been demonstrated for a limited set of symmetric tilt grain boundaries (STGBs) in cubic materials and twin boundaries. However, the lack of a general framework of interface defects prevents its applicability to arbitrary rational boundaries. In this paper, we present a mathematical framework based on the Smith normal form (SNF) for integer matrices to study the bicrystallography of rational phase and grain boundaries. One of the main results of the paper is a constructive proof of the invariance of the coincident site lattice (CSL) under discrete relative displacements of the parent lattices (of possibly different kind) by a displacement shift complete lattice (DSCL) vector. In addition, we obtain necessary and sufficient conditions on two lattices, related by not only rotations but also lattice distortions, for the existence of a coincidence relation. We first apply these results to explore coincidence relations in arbitrary phase boundaries, and study interface dislocations. In particular, we demonstrate the framework for a phase boundary formed by a strained hexagonal lattice and a square lattice. As a second application, we show how to enumerate all possible (geometric) disconnection modes in arbitrary rational grain boundaries, including glide and non-glide modes in both STGBs and asymmetric-tilt grain boundaries (ATGBs). The constructive nature of the framework lends itself to an algorithmic implementation based exclusively on integer matrix algebra to construct all interfaces that admit CSLs up to a prescribed size, and determine disconnection modes in grain boundaries. We demonstrate the use of SNF bicrystallography on selected bicrystal misorientation axes/angles in face-centered cubic (fcc), body-centered cubic (bcc), and hexagonal (hex) lattices.