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Reliable access to clean water is a major and growing societal challenge. Selective membrane technologies are expected to play a critical role in sustaining the water economy due to their energy efficient filtration of wastewater. High performing water filtration membranes require both high water permeability and high size-selectivity to ensure that purified water is produced in a cost-effective manner. Existing ultrafiltration membranes contain continuous and interconnected pores that allow for the rapid transport of water, satisfying the requirement of high water permeability. However, they typically exhibit broad pore size distributions that limit their size-selectivity and prohibit their application in highly precise separations. Block polymers represent a potentially powerful alternative class of materials for improved size-selectivity due to their self-assembly into well-defined domains of uniform size at the nanoscale. Removing one of the blocks generates the uniform pores required for precise separations. By coating a thin block polymer selective layer onto a commercially available ultrafiltration membrane, it may be possible to simultaneously obtain both high water permeability and high size-selectivity in a single membrane. However, commercialization of block polymer membranes has been impeded by technological challenges associated with producing continuous pores in an industrially scalable fabrication process from the typically observed ordered block polymer morphologies. Rather than targeting these ordered morphologies, this thesis aims to utilize the disordered state of block polymers to produce higher performing and potentially more scalable membranes. By kinetically trapping disordered state composition fluctuations, a disordered and co-continuous morphology can be obtained and subsequently converted into uniform and continuous pores without the need for challenging processing techniques. Chapter 1 introduces key concepts in block polymer self-assembly, including the order-disorder transition and composition fluctuations. Chapter 2 provides a summary of the technological requirements of an ideal water filtration membrane and discusses various strategies to integrate block polymers into these systems. Chapter 3 contains an overview of the various synthetic, processing, and characterization techniques employed throughout the thesis. Chapter 4 describes proof-of-concept results demonstrating that thermal cross-linking can be used to kinetically trap disordered state composition fluctuations. Chapter 5 details a strategy that introduces temporal control by using thermally stable photocuring strategies to arrest the disordered state. Chapter 6 describes a fundamental investigation into the temperature dependent morphological evolution of block polymers in the disordered state. Chapter 7 examines the use of large amplitude oscillatory shear to precisely control the domain structure of disordered block polymers. Chapter 8 integrates all these findings into the development of a novel co-casting technique to fabricate composite membranes with both high water permeability and high size-selectivity in a potentially scalable manner.

This dissertation discusses the application of dynamical systems in three different fields. We study the cell cycle dynamics in a cell-cell coupling system, heteroclinic binding networks modeling sequential working memory and the tensor approximation problems using the gradient flow.We consider the cell cycle coupling model for budding yeast (Saccharomyces cerevisiae), where the rates of progression for cells in a certain phase (responsive region R) of the cell division cycle are influenced by cells in another phase (signaling region S). We first study a model to mimic the situation where there is a delay between cells in S producing the chemical agents and cells in R experiencing the feedback from the agents by including a gap between the two regions. We analyze the dynamics of this system for the k-periodic solutions and compare the dynamics with the system without a gap.Second, we study a model where two biologically motivated sources of noise are introduced into the cell cycle system and compare them with the model that has Gaussian white noise perturbations. In simulations, we explore how an ordered two-cluster periodic state of cells becomes disordered by the increase of noise and a uniform distribution emerges for large noise. As a corollary to the results, we can estimate the coupling strength of the cell cycle in yeast autonomous oscillation experiments where clustering is observed.We also study high-dimensional dynamical systems with the existence of a heteroclinic network in the phase space under certain assumptions. This kind of dynamical systems can be used to describe the phenomenon of winnerless competition. We study the heteroclinic network that is considered as a heteroclinic chain of heteroclinic cycles: Starting from each saddle point that is not in the last heteroclinic cycle, there are two heteroclinic orbits, one stays in its heteroclinic cycle and the other one goes to the next heteroclinic cycle; there is only one heteroclinic orbit starting from each saddle point in the last heteroclinic cycle. We prove under certain technical assumptions that for each collection of successive heteroclinic orbits inside this network, there is an open set of initial points such that any trajectory starting from each of them follows the prescribed collection of heteroclinic orbits and stays in a small neighborhood of it. We show also that the symbolic complexity function of the system restricted to this small neighborhood is a polynomial of degree L-1, where L is the number of heteroclinic cycles in the network.Approximations of a high-dimensional tensor by sums of separable tensors can be used to avoid the curse of dimensionality in numerical computations. We study the approximation of a rank-2 symmetric tensor using a rank-1 tensor according to the gradient flow of the regularized error function. It had been previously known that at least one local minimum is attained in the symmetric set. We prove that the non-symmetric equilibria to the gradient flow are either saddle points or the maximum points. This implies that outside of the symmetric set there do not exist local minimum points. Furthermore, for these saddle points, we study the possibility of the so-called swamp behavior, where the trajectory stays in a neighborhood of the saddle point for a relatively long time before it leaves the neighborhood.

The electronic structures, many-body interactions and Fermi surface topologies of Au/Mo(112) were investigated in detail and were found to play important roles in the newly discovered order-disorder structural phase transition of the system. First, the high-resolution angle-resolved photoemission spectroscopy was utilized to characterize the electronic band structure of Mo(112) in far greater details than before. This elucidated the existence of several surface-derived states and their dispersion relations in high precisions near the Fermi level, as well as the symmetries of the bulk and surface electronic states, which are in good quantitative agreement with the ab-initio calculations. Such thorough understanding of the electronic states on Mo(112) made it possible to investigate the more complex electronic structure and many-body interactions in the Au overlayers formed on the Mo(112) surface and their interface. Upon the Au adsorption on Mo(112) substrate, the Au overlayer states are seen to hybridize with those of Mo substrate, which resulted in the formation of the several surface resonance bands, exhibiting high electronic localization near the surface and interface of the combined system. Furthermore, the electron-phonon coupling, involving these surface resonance states, is found to cause strong effective mass enhancement of the electrons near the Fermi level, which can contribute significantly to the surface lattice instability. In particular, for the (4x1) Au overlayer on Mo(112), the noticeable temperature-dependent changes in the Fermi surface contours were observed near the room temperature and were seen to act in favor of the stronger nesting condition and phonon-induced lattice distortions. The combination of the identified strong electron-phonon coupling and the critical Fermi surface topology near the room temperature likely relates to the overlayer lattice instability on the Au/Mo(112) system. In accord with the above general expectation, the order-disorder structural phase transitions were identified on Au/Mo(112) above the room temperature, which is characterized by the abrupt changes in the effective surface Debye temperature, indicative of significant softening of phonons on Au/Mo(112) across the transition. The sequence of these studies likely evidences that the strong electron-phonon coupling and the temperature-dependent Fermi surface topology are indispensable in driving the order-disorder transitions on Au/Mo(112). Advisor: Peter A. Dowben

We investigate the dynamics of a three-state stochastic lattice gas, consisting of holes and two oppositely "charged" species of particles, under the influence of an "electric" field, at zero total charge. Interacting only through an excluded volume constraint, particles can hop to nearest neighbor empty sites, but particle-particle exchange between oppositely charged particles is also allowed on a separate time scale. Controlled by this relative time scale, particle density and drive, the system orders into a charge-segregated state. Using a combination of Monte Carlo simulations and continuum field theory techniques, we study the order of these transitions and map out the steady state phase diagram of the system. On a single sheet of transitions, a line of multicritical points is found, separating the first order and continuous transitions. Furthermore, we study the steady-state structure factors in the disordered phase where homogeneous configurations are stable against small harmonic perturbations. The average structure factors show a discontinuity singularity at the origin which in real space predicts an intricate crossover between power laws of different kinds. We also seek for generic statistical properties of these quantities. The probability distributions of the structure factors are universal asymmetric exponential distributions. This research was supported in part by grants from the National Science Foundation through the Division of Materials Research.