Ni-Al based alloys remain a material of technological importance for high strength and high temperature applications. A design that optimizes desirable properties, such as strength, or creep and oxidation resistance, requires a deep understanding of thermodynamics and kinetics. This thesis uses ab initio methods to model both phase stability and diffusion processes. Starting from Density Functional Theory (DFT), formation energies of different crystal orderings are used to parameterize cluster expansion models. Used in conjunction with Monte Carlo techniques, these models can be used to derive macro scale properties at finite temperature.

We use DFT calculations to explore orderings on FCC and BCC, first for the Ni-Al binary, and then the Ni-Al-Cr ternary. Our calculations not only predict the stability of known phases, but also reveal new families of previously unknown ordered phases. We introduce strain order parameters to systematically analyze orderings on both FCC and BCC lattices. Many of these orderings are predicted to be unstable around x N i = 0.625, where a martensitic phase transformation is known to occur.

DFT calculations serve as training data for cluster expansion effective Hamiltonians, which accurately predict formation energies of arbitrary orderings without costly computations. We use these effective Hamiltonians in Monte Carlo simulations to calculate free energies, with which we construct phase diagrams. The difficulty of parameterizing a cluster expansion grows exponentially with the number of possible components. We introduce a recursive approach to parameterize multi-component alloy Hamiltonians using interaction parameters from simpler subsystems as Bayesian informative priors. We applied this approach to expand the statistical mechanics study of Ni-Al to Ni-Al-Cr, and explore Cr behavior within the L12 ordering.

The same cluster expansion methods can be used to model hop barriers of kinetic processes as a function of the ordering on the crystal. Diffusion barriers for Al hops have a strong dependence on the immediate local composition, while Ni hops are largely independent of their local ordering. We have developed a rigorous Kinetic Monte Carlo model that incorporates these relationships of hop barriers with local ordering, and applied it to the Ni-Al binary. Ni and Al exhibit different diffusion properties in the γ and γ' phases, which we discuss using various metrics of diffusion.

Renewable energy sources are generally abundant but intermittent, with peak production and peak demand occurring at different times. Therefore, storage and controlled distribution of energy is an important component of the shift away from on-demand fuels like petroleum and natural gas. Secondary electrochemical batteries are one convenient method for closing the gap between energy supply and demand. Battery performance and cost are largely restricted by the materials, especially those used for the electrodes. Part of the problem is material instability upon cycling. As the battery is charged (and discharged), the composition of the electrode changes, often resulting in reversible and irreversible phase transitions which result in material degradation. Therefore, battery capacity and lifetime can be limited by thermodynamic instabilities. We employ first-principles methods like density functional theory and Monte Carlo to study phase stability in layered electrode materials. We look at stacking-sequence changes, ion ordering, and atom migration to better understand bulk degradation mechanisms in Li- and Na-ion materials. We use NaxTiS2, NaxTiO2, and LixMO2 (M = Co, Ni, Mn) as model systems to explore phenomena present in a variety of layered transition-metal oxides and sulfides. Our work aids in interpreting experimental observations and in generating design rules for more robust electrode materials.

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.

Modern engineering applications use alloys with up to 10 different elements to meet performance requirements. Designing new alloys in such a vast multicomponent space is difficult. Even with first-principles methods to expedite the quest for new alloys, improvements to materials search algorithms are required to fully sift through the multidimensional chemical space. One possible search strategy is to exploit the natural hierarchy between crystal structures. Many crystal structures are related to a high symmetry parent crystal structure via a group/subgroup symmetry relationship. While many ordered structures are traditionally viewed as distinct crystals, they can be thought of as derivative structures of a parent crystal. The first half of this dissertation describes a framework for alloy design that is based on a parent-derivative crystal structure hierarchy. Chemical trends in this hierarchy are illuminated and new structural transformation pathways between important parent crystals are described.

The second half of this dissertation uses a similar philosophy to explore a different set of materials; it examines the effects of chemical trends on the electrochemical properties of materials used for energy storage. To meet rising energy demands with intermittent renewable energy sources such as solar and wind power, society needs to create high performance secondary batteries for energy storage. Li-ion batteries have quickly risen to be a prime candidate to meet energy storage needs, and new advances are being made to promote sustainable earth-abundant materials, higher charge rates, and larger capacities. Some of these new advances include the investigation of Na-ion and Mg-ion batteries. Spinel intercalation compounds are well-known to facilitate high rate and high voltage Li-ion batteries, but less is known about their Na-ion and Mg-ion counterparts. When exploring new materials, it helps to have a strong fundamental understanding of how changes in chemistry or structure affect electrochemical performance of battery materials. This dissertation uses a combination of electronic structure calculations and statistical mechanics methods to study the thermodynamic and kinetic properties of a wide range of spinel intercalation compounds.

The identification of a crystal structure hierarchy was achieved by analyzing existing crystal structure databases using a crystallographic mapping algorithm to find representative parent crystal structures from which many derivative structures are formed. We found that 73% of binary intermetallic compounds found on the ICSD to be derived from only 20 unique parent crystal structures. There are important crystallographical relationships, such as the Bain path and the Burgers path, that connect parent crystals within our hierarchy. These crystallographic relationships facilitate structural phase transformations which are important for shape-memory alloys, magnetocalorics, or self-assembling block co-polymers. A crystal structure mapping algorithm was used to show the existence of 7 new structural transformation pathways between orderings on simple parent crystal structures to others belonging to the top 20 most common parent crystal structures. We use high-throughput density functional theory (DFT) to probe which elemental combinations are most likely to cause spontaneous structural transformations without an energy barrier. We find multiple chemical combinations that lead to barrierless transformation pathways. These barrierless transformations suggest a nucleation mechanism that does not require large structural fluctuations that are often energetically costly.

In our investigation of electrochemical properties of energy storage materials, we examine the spinel structure which has a general formula AMX2 where A is a guest cation, M is a transition metal and X is a chalcogenide. The electrochemical properties of spinel can be affected by many factors including (i) ionicity of the MX2 framework, (ii) guest cation radius, and (iii) guest cation oxidation state. We conducted a systematic study and determined that guest cation radius and MX2 ionicity play a significant role in guest cation site preference which, in turn, affects the electrochemical properties of spinel. The insights of this study suggest that large cations in an oxide spinel creates a desirable energy landscape for high-rate capable batteries. Using kinetic Monte Carlo simulations and a model cluster expansion Hamiltonian, we identify a topological pitfall that makes spinel prone to highly correlated cation diffusion at intermediate to high guest cation concentrations. This undesirable property is dependent on the strength of the nearest-neighbor repulsion of diffusing cations within spinel. A strong repulsion results in a dependence on large vacancy clusters to mediate diffusion. This dependence indicates that a percolating network of large vacancy clusters is necessary to facilitate long range diffusion. Our results show that the detrimental effects can be mitigated by reducing the strength of the repulsion or increasing the connectivity of the percolating network. Although our focus is on the fully-ordered spinel structure, the conclusions drawn from this study apply to other close-packed anion hosts such as disordered rocksalt electrodes and partially ordered spinels.

Niobium oxides and alloys are of growing interest to both battery engineers and high temperature structural alloy designers in recent years. The niobium oxide Wadsley-Roth crystallographic shear phases are promising next generation Li-ion battery anode materials while a new class of refractory multi-principal metals alloys is of interest for extreme temperature aerospace applications. The phase stability of niobium and oxygen was originally studied during alloy development efforts in the 1950s and 1960s, but has not been recently revisited with modern first principles and statistical mechanics tools. Both niobium and other refractory metals used in these alloys can dissolve significant quantities of oxygen and readily react to form complex and often highly non-protective oxide scales. It is thus of great interest to revisit and evaluate the phase stability and defect behavior of niobium alloys and related oxides.

Throughout this work, a computational approach was applied to determine not only thermodynamic phase stability, but to illuminate the underlying structural and chemical factors which drive this behavior.From first principles and statistical mechanics, a revised phase diagram of the Nb-O system is proposed. This includes a solubility curve determined from a Bayesian cluster expansion approach which allowed the uncertainty of the solubility curve to be directly probed. We also determined a systematic way to classify and enumerate Wadsley-Roth block structures and then identified electrostatic repulsion and lattice relaxations as the key factors determining phase stability. At the other compositional extreme, the multifaceted interactions between interstitial oxygen atoms dissolved in niobium were analyzed.

Notably, closed-shell hybridization between oxygen-oxygen pairs drives strong repulsion at short distances which leads to the suppression of high energy pair clusters which persists to high temperatures.Similar closed-shell hybridization is also observed in some repulsive oxygen-solute interactions in dilute alloys, though frequently electrostatic interactions are dominant in this scenario. Finally, through first principles calculations we found that the topology of generalized stacking fault energy curves in BCC Nb are drastically altered when interstitial oxygen atoms are contained within a glide plane. These findings suggest that mechanical deformation mechanisms may be impacted by interstitial oxygen in unexpected ways.

High-throughput density functional theory calculations can provide a method for predicting material properties in alkali-ion electrode and electrolyte chemistries. Through these calculations, paired with cluster expansion models and uncertainty quantification methods, we have tools to understand the fundamental thermodynamic origins of crystallographic deformation and alkali-ion phase stabilities in these materials. In this analysis, we use these methods to understand phase stability and lithiation mechanisms in a family of high power density electrode chemistries called the Wadsley-Roth crystallographic shear phases. Due to the complexity and large cell sizes of Wadsley-Roth phase materials, this combination of methods is useful for tapping the origin of the high charge rates and power densities in these unique structures. The study begins with an examination of stability of compounds in the Ti-Nb-O ternary where we examine the effect of electrostatics and distortions on phase stability of the Wadsley-Roth phases. We then examine the effect of chemical strain in the lithium site-filling mechanism in the highly reversible Wadsley-Roth material, PNb9O25. We extend our analysis to the high-power density, commercialized Wadsley-Roth phase, TiNb2O7, where we examine the effect of metal-metal electronic interactions and pair distances on the complex lithium site filling mechanism present in this compound. We then do an in depth analysis of the effect of lithiation on octahedral distortions to acquire a complete understanding of structural changes upon lithiation. These analyses are paired with experimental results to exhibit the validity of these calculations.

Density functional theory (DFT) is undergoing a shift from a descriptive to a predictive tool in the field of solid state physics, with undertakings like the Materials Project, OQMD, and AFLOW leading the way in utilizing high-throughput data to predict and seek novel materials properties. However, methods to rigorously evaluate the \textit{validity} and \textit{accuracy} of these studies is lacking in both the availability and utilization of techniques. The natural disconnect between simulated and experimental length-scales and temperatures, combined with this lack of validation, raises serious questions when simulation and experiment disagree. In this thesis, we analyze several transition metal systems where simulations and experiments present unusual disagreements, and develop a new formalism for comparing high-temperature measurements to \textit{ab-initio} calculations. Our work aims to broaden the understanding not only of the specific systems discussed, but of how presently available \textit{ab-initio} methods perform for transition metal alloys across \textit{all} systems.

Recent high-throughput ab-initio studies of transition metal binaries have suggested a great number of undiscovered stable phases present in well-studied systems. Co-Pt alloys, especially, have a long experimental history demonstrating three stable mixed phases: L1$_0$ CoPt and L1$_2$ Co$_3$Pt and CoPt$_3$, but density functional theory suggests a set of yet-unobserved long-period $\beta_2$-like superstructures at Pt-rich compositions. We analyze the Co-Pt system in-depth, calculating the energy of over 1,400 structures to thoroughly explore the series of unusual superstructures suggested by DFT\@. Simulated diffraction patterns, analysis of magnetic behavior, and investigation of the density-of-states emphasize the stark differences between measured behaviors and \textit{ab-initio} predictions. By moving up the Jacob's Ladder of functionals, we show that we only replace one set of discrepancies for another, and even the introduction of vibrational degrees of freedom fails to solve the massive differences in predicted phase stability. By fitting the \textit{ab-initio} results to a cluster expansion Hamiltonian and performing Monte Carlo calculations, we show that the resulting high-temperature phase diagram is wholly incompatible with experimental results.

Like Co-Pt, Heusler compounds have unique magnetic properties, resulting in interest for their potential applications as spintronic materials. The pseudo-binary (Mn,Fe)Ru$_2$Sn, formed as a solid solution of the full Heuslers (Mn, Fe)Ru$_2$Sn, has been recently shown to exhibit exchange-hardening implicative of two \textit{magnetic} phases, despite the presence of only one \textit{chemical} phase. Using \textit{ab-initio} calculations we show that the magnetic behavior of this alloy arises from a competition between AFM-favoring Sn-mediated superexchange and FM-favoring RKKY exchange mediated by spin-polarized conduction electrons. Changes in valency upon replacement of Mn with Fe shifts the balance from superexchange-dominated interactions to RKKY-dominated interactions. Using our electronic structure calculations, we parameterize a mixed-basis chemical-and-magnetic cluster expansion, and use Monte Carlo simulations to demonstrate a ferromagnetic (FM) to antiferromagnetic (AFM) behavior dependent on composition with the experimental study. By examining the low-temperature ensemble averages of magnetic and chemical correlations, we identify the mechanism behind magnetic hardening in the solid solution.

Our multiple successes in utilizing cluster expansions, both to deeply analyze failures and successfully describe complex chemical-magnetic interactions, motivates an experiments-driven approach to lattice Hamiltonians. For alloys, cluster expansion Hamiltonians reduce the complex, many-body electron problem of density functional theory to a series of simple site-wise basis functions (e.g., products of site occupancy variables) on an atomic scale. The resulting energy polynomial is computationally inexpensive, and hence suitable for the (tens of) thousands of calculations of large systems required by stochastic methods. We present a new method to run the statistical mechanics problem ``in reverse'', using high-temperature observations and thermodynamic connections to construct an effective Hamiltonian and thereby predict the 0 Kelvin energy spectrum and associated ground states. By re-examining the cluster expansion coefficients as thermodynamic state variables and utilizing entropy-maximization approaches, we develop an algorithm to select clusters and determine cluster interactions using only a few, high-temperature experiments on disordered phases. We demonstrate that our approach can recover not only the stable ground states at 0 Kelvin, but also the full phase behavior for three realistic two-dimensional and three-dimensional alloy test-cases.

Titanium is a lightweight and attractive metal for use in aerospace, chemical processing, and biomedical industries due to its high strength-to-weight ratio, good oxidation and corrosion resistance, as well as biocompatibility. One important problem with the use of titanium is the embrittlement that accompanies its incorporation of interstitials, most notably oxygen and nitrogen. Since titanium readily dissolves both nitrogen and oxygen, understanding these reactions is fundamental to develop strategies to minimize contamination or to take advantage of these reactions to form nitrides, oxides, or oxynitrides that may exhibit interesting properties for energy storage, biomedical, coatings, or other applications. This research is motivated by the need for improving the fundamental understanding of titanium interactions with nitrogen and oxygen and the nature of its developed phases.Reaction studies were performed at 800°C (to remain in the hcp α-Ti phase) using argon-based environments with three distinct low partial pressures of nitrogen and oxygen. These experiments were performed in two furnace configurations, one of which allowed monitoring of scale evolution in situ using Raman spectroscopy. Characterization of the developed scales relied on scanning and transmission electron microscopy, energy dispersive spectroscopy, focused ion beam, X-ray diffraction, electron diffraction, and Raman spectroscopy. Across environments used, the microstructures show the formation of layered structures with varying porosity throughout. When fast-cooling specimens in the gettered Ar environment (the lowest partial pressure used), the scale consists of mainly nitride phases. Slow-cooling enables the formation of a N-rich hcp-based phase, an oxynitride (TiNxOy) phase, Ti2O3 in a corundum structure, Magnéli phases, and rutile. Ultra-high purity Ar studies show an additional range of oxygen orderings in the substrate when slow-cooling the specimen after the desired dwell time. Analysis of the specimens exposed to gettered and ultra-high purity Ar also shows the formation of twinned fcc TiN or oxynitride structures and an orientation relationship between the fcc phase and the underlying hcp Ti substrate. Partial dislocations are proposed to play a role in the hcp Ti to fcc TiN/TiNxOy transformation and potentially in the formation of voids at the metal-scale interface. Reacting titanium to a 1%O2-Ar environment hinders the formation of nitride or oxynitride phases and develops multiple layers of rutile. In situ experiments proved rutile formation occurs at the exposure temperature rather than during cooling and shows coloration changes in the surface throughout the reaction. Blisters also appeared in between the outermost rutile scales and developed at temperature rather than from CTE mismatch between the oxide and metal during cooling. This dissertation has contributed to the understanding of titanium reactions with nitrogen and oxygen at high temperatures and low partial pressures. The experimental approaches used can be applied towards studying further complex systems either in Ti-based alloys or in other systems such as complex concentrated alloys (CCAs).