Nucleation is the first step in the conversion of a metastable phase to a more stable one. It involves the formation of a post-critical nucleus that is thermodynamically fa- vorable to grow and is important in most phase transitions. Our understanding of the thermodynamics and kinetics of nucleation is largely based on classical nucleation theory (CNT). It was formulated on the basic principle developed by Gibbs, who reasoned that the free energy required to form a nucleus involves two competing contributions: (1) a favorable bulk driving force and (2) an unfavorable surface penalty, and the subsequent work by Volmer and Weber, Farkas, Becker and Doring, and Zeldovich that developed rate laws. Almost 100 years later, most studies of nucleation still borrow at least some el- ements of CNT. Although CNT provides a simple and intuitive framework to understand nucleation, it is not applicable in many important situations.

In this thesis, we develop corollary theories that adapt CNT to understand solute precipitate nucleation when different factors affect the driving force or surface energy. We introduce a spatio-temporal survival probability model to provide the first stochastic model of nucleation due to solute enrichment ahead of a crystallizing front. The model predicts the distribution of nucleation times as a function of CNT rate parameters and growth conditions. Finally, we investigate how adsorbing additives can promote nucle- ation. We created a theoretical framework for modeling the thermodynamics and kinetics of solute precipitate nucleation when molecular surfactants can adsorb onto nuclei and reduce the surface energy. We also used lattice simulations to study how properties of adsorbing molecular and oligomeric additives affect nucleation barriers.

Two heterogeneous catalysts, the Phillips ethylene polymerization catalyst and supported rhenium olefin-metathesis catalyst, are computationally investigated.

The mechanism of ethylene polymerization by the Phillips catalyst remains unknown despite numerous hypotheses and studies since its discovery sixty years ago. This work uses density functional and small chromasiloxane cluster models to compare initiation, propagation, and termination pathways for several previously proposed mechanisms and for two newly proposed mechanisms. Where possible, complete catalytic cycles and predicted kinetics, molecular weights, and site abundances are compared to properties of the Phillips catalyst. Prohibitively high activation barriers for propagation rule out chromacycle ring expansion and Green-Rooney mechanisms (alternating alkylidene/chromacycle). A new oxachromacycle ring expansion mechanism has a favorable plausible propagation barrier, but initiation for the oxachromacycle expansion is prohibitively slow. Chain growth is fast on a recently proposed bridging hydroxyl Si(OH)CrIII-alkyl site that is initiated by proton transfer from ethylene. However, the initiation step is extremely slow and uphill, and termination is even faster than propagation, so essentially all sites remain trapped in a dormant state. A new Si(OH)CrII-alkyl site also has a small barrier for Cossee-Arlman type chain growth, but it forms only oligomers because termination by proton transfer back to the alkyl chain is faster than propagation, and initiation is slow. Only the monoalkylchromium(III) site (≡SiO)2Cr-alkyl is viable as an active site for polymerization, however the nature of the initiation mechanism remains unknown.

CH3ReO3 (MTO) interacting with the surface of extensively chlorinated γ-Al2O3 is a highly active catalyst for olefin metathesis. Its activity, selectivity and stability exceed that of any other reported MTO-based catalyst as measured in propene metathesis at 0 °C, including MTO/SiO2-Al2O3 and MTO/ZnCl2/-Al2O3, and are far higher than MTO/-Al2O3. DFT calculations support facile Cl-O ligand exchange between MTO and Cl-Al2O3. The calculations further suggest that chlorination at Re promotes active site formation via tautomerization.

Polymorphs are different crystal structures for the same molecule or compound. Many molecules can crystallize into several different polymorphs, each with different solubility and growth/dissolution kinetics. Polymorphism is especially important in pharmaceuticals

as many drugs are administered as crystalline solids. Relative stability of polymorphs affects solubility which consecutively affects the bioavailability of the solid form. Ab initio calculations typically infer stability from static energy calculations or harmonically

approximate the entropy. Polymorphs of organic molecular crystals may be separated by small free energy differences and hence it is important to account for the entropy exactly. Classical force field based approaches enable complete free energy calculations

with advanced sampling methods. These methods, however, typically use thermodynamic integration and are expensive multi-stage simulations.

We discuss a method to directly yield the polymorph free energy difference in this thesis. The framework combines ideas from the well-established lattice-switch Monte Carlo procedure and relationships that have been previously used in the electron transfer

literature. We demonstrate the applicability of the method by computing Helmholtz free energy differences for the Gaussian core solid using just two unbiased simulations. We generalize the framework to accurately calculate Gibbs free energy differences of arbitrary

triclinic polymorphs. Finally, we extend the method to yield relative stability of organic molecular crystals. We demonstrate the method for benzene polymorphs and all five experimentally reported polymorphs of pharmaceutically relevant carbamazepine.

One of the most awe-inspiring class of materials are crystals. These highly ordered groups of atoms/molecules propel our lives through myriad products we humans rely upon. Right from the salt and sugar we consume to elevate the taste of our foods, the silicon chips that form the brains of every computing device we use, to the life saving drugs that have prevented millions if not billions of deaths, all belong to the humble class of materials — the crystal. Therefore, the engineering of this material is crucial to — improve the manufacturing of products that touch our lives daily — ultimately improving the quality of every human life!

The two physical attributes of a crystalline material that have a major impact on its processability and performance are its shape and size. For e.g., in a crystalline catalyst one wants to engineer the shape of the crystal to maximize the area of its reactive surfaces. In pharmaceutical applications, the crystal size distribution may determine the rate of plasma uptake of a drug when the process is dissolution rate limited. Therefore, engineering the shape and size of the crystals is of immense consequence.

In silico tools that can predict the shape and size of a crystal based on inputs such as crystal structure, temperature, supersaturation, etc. are vital to efficiently navigate the process design space. Such tools help achieve the efficiency gains by being a guiding light to experimentalists, thus enabling cheaper and more effective screening. This dissertation lays out the digital design framework to make these predictions starting from a molecule. It focuses on the development of a computational toolkit to predict driving forces for crystallization — a key prediction to enable size predictions — of complex molecules harnessing atomistic simulations.

Over the past few decades there have been tremendous advances in the accuracy of electronic structure calculations, rigorous methods to assess proposed reaction mechanisms, and techniques to create representative models of active sites for homogeneous and ordered heterogeneous catalysts. Advances on all these fronts have greatly improved our understanding of numerous catalysts. However, most of these methods assume that the active sites are both well-defined and uniform (or are in an ensemble of uniform sites). Catalysts with quenched disorder have non-uniform active sites. Most established modeling techniques are inapplicable and many important catalysts in this category remain poorly understood. The major challenges of modeling catalysts with quenched disorder are (i) the nature of the disorder is out-of-equilibrium and unknown; (ii) each active site has a different local environment and activity; (iii) active sites are rare, often less than ~20% of potential sites, depending on the catalyst and preparation method.

In this thesis, we develop new methods to efficiently predict kinetic properties of the distribution of quenched-disordered sites and how they affect different steps of catalyst operation. We combine population balance modeling techniques with machine learning methods to learn how the population of grafted sites evolves in time. We develop a new algorithm to efficiently converge site-averaged kinetics for a given distribution of grafted sites, the Importance Learning algorithm. A combination of Importance sampling and machine Learning are used to efficiently and accurately learn activation parameters of rare sites that dominate site-averaged kinetics. Finally, we develop a protocol to create cluster models of active sites that most accurately preserve quenched degrees of freedom.

Chemical reactions, mass transport in solids, protein folding, and nucleation in first-order phase transitions are examples of processes characterized by multiple, long-lived states. Transitions from one (meta)stable state to another are rare and brief, making them difficult to resolve experimentally. And yet the short transition paths contain valuable structural and dynamic information that governs the lifetimes of the stable states. Transition state theory provides a valuable framework for analyzing rare events, provided that an exact dividing surface with no-recrossing can be found. Direct simulation of rare events processes are complicated by the long waiting times for a transition to spontaneously occur. Simulation methods that introduce a bias decrease the waiting time but also risk altering the mechanism. The reactive flux correlation function provides a two-step recipe for computing rates from simulation using an approximate dividing surface, but may miss important details about the physical reaction mechanism. Transition path sampling (TPS) was developed specifically to sample unbiased dynamical reactive trajectories and in combination with likelihood maximization provides an optimized reaction coordinate model.

We present new, simple TPS methods that reduce the computational expense of simulating rare events over existing methods. We apply these methods to study rare events in condensed phases and analyze the resulting data with likelihood maximization. The mechanism for vacancy migration in a single domain crystal by activated hops is examined. We find that accurately locating the donor and acceptor sites has a dramatic effect on identifying the mechanism. We next investigate the role of water in ion-pair dissociation, uncovering two solvent mechanisms that influence ion-pair transition states. The resulting dividing surface does not eliminate recrossing, so we present a test for the existence of a no-recrossing surface. It is revealed that an exact dividing surface does not exist for ion-pair dissociation. We discuss the ramifications for transition state theory, Grote-Hynes theory and the relationship between them.

Several industrially important catalysts are single metals dispersed on amorphous supports. For example, Cr dispersed on amorphous SiO2 is used to catalyze ethene polymerization and W dispersed on amorphous Al2O3 and SiO2 is used to catalyze olefin metathesis. Despite their extensive use in the industry, these catalysts have largely been intractable to both, experimental and modeling investigations. In particular, they present the following challenges: (i) an unknown quenched disordered structure of the amorphous support, (ii) metal atoms attach to various surface grafting sites with different rates and have different activation and catalytic reaction kinetics, and (iii) only a small fraction of the sites are active. These challenges particularly render ab initio computational tools, routinely applied to study homogeneous and ordered heterogeneous catalysts, inefficient and impractical. The overarching goal of this thesis is developing computational tools to efficiently model the synthesis and reactivity of atomically dispersed catalysts on amorphous supports.

Atomically dispersed amorphous catalysts are synthesized by grafting organometallic complexes onto amorphous supports. We develop a machine learning (ML) parametrized population balance model to predict the evolving population of active sites during catalyst synthesis. We apply the population balance modeling framework to model the grafting of TiCl onto amorphous silica. Equilibrium predictions of the model agree with experimentally determined populations of grafted Ti sites. Additionally, we develop an Importance Learning (IL) algorithm to efficiently calculate the site-averaged activation barrier for amorphous catalysts. IL uses a combination of ML and importance sampling to discover rare and active catalytic sites which dominate kinetics.

Different studies have generated atomistic models of amorphous silica using different simulation protocols. Most studies have claimed that their models are representative of real silicas. Using statistical hypothesis testing, we show that different protocols lead to models with different structural features. We discuss the effects of these structural variations on the grafted catalyst and outline experimental metrics that can be used to validate models in future studies.

Finally, we present a site balance algebra to quantifying amounts of different species generated in grafting experiments. We quantify the amounts of [≡SiOTiCl3] and [([≡SiO)2TiCl2] sites obtained on grafting TiCl4 onto amorphous silica.