Systematic Modeling of Incommensurate Grain Boundaries in Magnesium Oxide via Farey Hierarchy

Periodic atomistic simulations of incommensurate grain boundaries (GBs) require an appropriate rational approximation of lattice matching across the interface. In a previous study, the {100} || {110}[001] GB in MgO was successfully modeled using a 5:7 rational approximation of the interplanar spacing ratio, providing important insight into the atomic structure of this incommensurate interface. Building upon this work, the present study examines the same GB within a framework based on rational approximation and the Farey hierarchy. The irrational spacing ratio d(110)/d(100) = √2 /2 is systematically approximated using continued-fraction expansion, yielding a sequence of rational approximants. Periodic GB models corresponding to selected low-order approximants are constructed and optimized using first-principles calculations. The resulting structures reveal that higher-order approximants are composed of structural units already present in lower-order models, reflecting the hierarchical structure of the Farey diagram. The calculated GB energies do not exhibit a monotonic dependence on the global misfit parameter. The relatively small energy differences among several approximants indicate that interfacial stability is not determined solely by lattice mismatch. Instead, the arrangement and relative abundance of local structural units play an important role in determining interfacial energetics. Atomic-resolution ABF-STEM observations further confirm the presence of similar structural units at the GB. These results suggest that rational approximation in modeling incommensurate GBs serves not only to reduce lattice mismatch but also to reveal the structural hierarchy underlying the interface. The present approach provides a systematic framework for interpreting periodic approximations of incommensurate GBs and may be applicable to other interfaces with irrational lattice relationships.