Electrochemical energy storage technology must continue to improve in order to meet increasing demand across sectors, while balancing performance with cost and resource constraints. For large-scale stationary applications such as grid storage of renewable energy, Na- and K-ion batteries have received attention as potential alternatives to Li-ion batteries mainly due to the far greater abundance of those elements. Research efforts for these "beyond Li-ion" technologies include developing robust electrode materials that can undergo many cycles without degrading. One promising class of candidate materials are layered oxide intercalation compounds, which have been widely adopted in positive electrodes for commercial Li-ion batteries. However, when intercalated with the larger Na+ and K+ ions, these materials often exhibit additional structural phase transitions and ion-vacancy orderings that are not encountered with Li+. These effects have important implications for the voltage profile, degradation mechanisms, and rate capability.

In this dissertation, we study thermodynamic and kinetic properties of layered oxides for Na- and K-ion battery electrodes using first-principles techniques. Density functional theory calculations provide energies and relaxed geometries of ordered configurations at varying Na/K concentration (corresponding to different states of charge), as well as barriers for ion migration. Statistical mechanics methods, namely cluster expansion effective Hamiltonians and Monte Carlo simulations, are employed to efficiently model finite-temperature behavior and predict ground state configurations.

In the NaxCoO2, NaxCrO2, and KxCrO2 systems (0 ≤ x ≤ 1), we examine phase stability among layered structures that host Na/K in octahedral or prismatic coordination, as well as ion-vacancy orderings within them. We establish a comprehensive description of ordering in these systems, in which most of the stable ordered phases belong to families that are each based on a particular motif. At intermediate x, we identify orderings with Na/K in prismatic coordination that accommodate variations in composition as antiphase boundaries. We demonstrate how the composition can be changed essentially continuously by adjusting the average spacing between boundaries, leading to "Devil's staircase" behavior that agrees well with experimental observations. We predict a similar family of orderings at high x in KxCrO2 that host both prismatically and octahedrally coordinated K within the same intercalation layer, which we find to be a plausible explanation of the experimentally reported structural evolution during cycling. In NaxCrO2, we also confirm a preference for Cr migration to the intercalation layers at low x, which is a key degradation mechanism observed in this material.

Using NaxCoO2 as a model system, we explore Na diffusion within the orderings identified near x = 1/2. While Na mobility is found to be highly restricted, we uncover a mechanism that enables the collective motion of antiphase boundaries through the intercalation layers, the limiting migration barriers for which are relatively low. We simulate the macroscopic diffusion behavior arising from this mechanism using a kinetic Monte Carlo model. Our simulations show that antiphase boundary migration, though quite distinct from textbook atomistic diffusion mechanisms, follows normal Fickian diffusion in one dimension, but with strong composition dependence of the diffusion coefficient. These results lay important groundwork for understanding the effects of ordering and engineering improved battery materials that might take advantage of unconventional diffusion mechanisms in ordered phases.