THERMODYNMICS AND STRUCTURE OF POLY(ETHYLENE OXIDE) IN MIXTURES OF WATER AND ETHANOL

Poly(ethylene oxide) (PEO) is one of the most researched synthetic polymers due to the complex behavior which arises from the interplay of the hydrophilic and hydrophobic sites on the polymer chain. PEO in ethanol forms an opaque gel-like mixture with a partially crystalline structure. Addition of a small amount of water disrupts the gel: 5 wt % PEO in ethanol becomes a transparent solution with the addition of 4 vol % water. The phase behavior of PEO in mixed solvents have been studied using small-angle neutron scattering (SANS). PEO solutions (5 wt % PEO) which contain 4 vol % - 10 vol % (and higher) water behave as an athermal polymer solution and the phase behavior changes from UCST to LCST rapidly as the fraction of water is increased. 2 wt % PEO in water and 10 wt % PEO in ethanol/ water mixtures are examined to assess the role of hydration. The observed phase behavior is consistent with a hydration layer forming upon the addition of water as the system shifts from UCST to LCST behavior. At the molecular level, two or three water molecules can hydrate one PEO monomer (water molecules form a sheath around the PEO macromolecule) which is consistent with the suppression of crystallization and change in the mentioned phase behavior as observed by SANS. The clustering effect of aqueous PEO solution (M.W of PEO = 90,000 g/mol) is monitored as an excess scattering intensity at low-Q. Clustering intensity at Q = 0.004 Å^-1 is used for evaluating the clustering effect. The clustering intensity is proportional to the inverse temperature and levels off when the temperature is less than 50 ˚C. When the temperature is increased over 50 ˚C, the clustering intensity starts decreasing. The clustering of PEO is monitored in ethanol/ water mixtures. The clustering intensity increases as the fraction of water is increased. Based on the solvation intensity behavior, we confirmed that the ethanol/ water mixtures obey a random solvent mixing rule, whereby solvent mixtures are better at solvating the polymer that any of the two solvents. The solution behavior of PEO in ethanol was investigated in the presence of salt (CaCl2) using SANS. Binding of Ca2+ ions to the PEO oxygens transforms the neutral polymer to a weakly charged polyelectrolyte. We observed that the PEO/ethanol solution is better solvated at higher salt concentration due to the electrostatic repulsion of weakly charged monomers. The association of the Ca2+ ions with the PEO oxygen atoms transforms the neutral polymer to a weakly charged polyelectrolyte and gives rise to repulsive interactions between the PEO/Ca2+ complexes. Addition of salt disrupts the gel, which is consistent with better solvation as the salt concentration is increased. Moreover, SANS shows that the phase behavior of PEO/ethanol changes from UCST to LCST as the salt concentration is increased.

Spectral Frame Analysis and Learning through Graph Structure

This dissertation investigates the connection between spectral analysis and frame theory. When considering the spectral properties of a frame, we present a few novel results relating to the spectral decomposition. We first show that scalable frames have the property that the inner product of the scaling coefficients and the eigenvectors must equal the inverse eigenvalues. From this, we prove a similar result when an approximate scaling is obtained. We then focus on the optimization problems inherent to the scalable frames by first showing that there is an equivalence between scaling a frame and optimization problems with a non-restrictive objective function. Various objective functions are considered, and an analysis of the solution type is presented. For linear objectives, we can encourage sparse scalings, and with barrier objective functions, we force dense solutions. We further consider frames in high dimensions, and derive various solution techniques. From here, we restrict ourselves to various frame classes, to add more specificity to the results. Using frames generated from distributions allows for the placement of probabilistic bounds on scalability. For discrete distributions (Bernoulli and Rademacher), we bound the probability of encountering an ONB, and for continuous symmetric distributions (Uniform and Gaussian), we show that symmetry is retained in the transformed domain. We also prove several hyperplane-separation results. With the theory developed, we discuss graph applications of the scalability framework. We make a connection with graph conditioning, and show the in-feasibility of the problem in the general case. After a modification, we show that any complete graph can be conditioned. We then present a modification of standard PCA (robust PCA) developed by Cand\`es, and give some background into Electron Energy-Loss Spectroscopy (EELS). We design a novel scheme for the processing of EELS through robust PCA and least-squares regression, and test this scheme on biological samples. Finally, we take the idea of robust PCA and apply the technique of kernel PCA to perform robust manifold learning. We derive the problem and present an algorithm for its solution. There is also discussion of the differences with RPCA that make theoretical guarantees difficult.