Interfaces, like grain boundaries in polycrystals and heterophase interfaces in multiphase, play a central role in determining material properties that are critical for many applications. Particularly, they significantly influence the mechanical and thermodynamic behavior of crystalline materials.

The traditional Frank-Bilby equation (FBE) is widely used to characterize the interface structure and orientation relationship of heterophase interfaces, where the mismatch between two crystals is accommodated by a network of misfit dislocations. In addition to dislocations, steps are another common type of interfacial defect. Disconnections, which possess both dislocation and step character, provide a more comprehensive description of the defect networks in the imperfect interfaces. To extend the applicability of the FBE analysis system to interfaces with steps, we generalize the Frank-Bilby equation (GFBE) for heterophase interfaces in terms of discrete sets of disconnections. The disconnections and reference structures can be determined through a coincidence site lattice (CSL) construction. By solving the non-linear system of equations of the GFBE, we predict potential heterophase interface structures, inclinations, and orientation relationships. Several examples, such as twin grain boundaries, twist-misfit interfaces, and interfaces of α phase precipitates in β phase titanium, illustrate the effectiveness of the GFBE as a powerful tool for predicting the crystallography of heterophase interfaces.

To further incorporate disconnection reactions and identify minimum energy interface structure, we develop a three-dimensional generalized Peierls-Nabarro model for heterophase interfaces, which considers the stacking fault energy for the disregistry that is not confined in the interface plane. The anisotropic elasticity of periodic interfacial defects pattern is analyzed using the Stroh sextic formalism in a Fourier series framework. The parameterized formula is applied to describe three-dimensional stacking fault energy. We investigate the spiral defect pattern in Cu/Ni {111} heterophase interface and the dissociation of <11 ̅00> dislocation into three partial dislocations on the α-alumina {11 ̅00}/<112 ̅0> symmetric tilt grain boundary. Our model is rigorously validated through molecular dynamic simulations and experimental observations.

Starting from the most fundamental parameters of the crystal structures (e.g., lattice parameters) of two crystalline materials, this thesis provides an integrated mesoscale computational framework for predicting heterophase interface structures and properties applicable to any pair of crystalline materials. The crystallographic theory combines anisotropic elasticity and the three-dimensional generalized Peierls-Nabarro model to predict interface thermodynamics.