Effects of particle size ratio on single particle motion in colloidal dispersions

The motion of a single Brownian particle of arbitrary size through a dilute colloidal dispersion of neutrally buoyant bath spheres of another characteristic size in a Newtonian solvent is examined in two contexts. First, the particle in question, the probe particle, is subject to a constant applied external force drawing it through the suspension as a simple model for active and nonlinear microrheology. The strength of the applied external force, normalized by the restoring forces of Brownian motion, is the Péclet number, Pe. This dimensionless quantity describes how strongly the probe is upsetting the equilibrium distribution of the bath particles. The mean motion and fluctuations in the probe position are related to interpreted quantities of an effective viscosity of the suspension. These interpreted quantities are calculated to first order in the volume fraction of bath particles and are intimately tied to the spatial distribution, or microstructure, of bath particles relative to the probe. For weak Pe, the disturbance to the equilibrium microstructure is dipolar in nature, with accumulation and depletion regions on the front and rear faces of the probe, respectively. With increasing applied force, the accumulation region compresses to form a thin boundary layer whose thickness scales with the inverse of Pe. The depletion region lengthens to form a trailing wake. The magnitude of the microstructural disturbance is found to grow with increasing bath particle size -- small bath particles in the solvent resemble a continuum with effective microviscosity given by Einstein's viscosity correction for a dilute dispersion of spheres. Large bath particles readily advect toward the minimum approach distance possible between the probe and bath particle, and the probe and bath particle pair rotating as a doublet is the primary mechanism by which the probe particle is able to move past; this is a process that slows the motion of the probe by a factor of the size ratio. The intrinsic microviscosity is found to force thin at low Péclet number due to decreasing contributions from Brownian motion, and force thicken at high Péclet number due to the increasing influence of the configuration-averaged reduction in the probe's hydrodynamic self mobility. Nonmonotonicity at finite sizes is evident in the limiting high-Pe intrinsic microviscosity plateau as a function of bath-to-probe particle size ratio. The intrinsic microviscosity is found to grow with the size ratio for very small probes even at large-but-finite Péclet numbers. However, even a small repulsive interparticle potential, that excludes lubrication interactions, can reduce this intrinsic microviscosity back to an order one quantity. The results of this active microrheology study are compared to previous theoretical studies of falling-ball and towed-ball rheometry and sedimentation and diffusion in polydisperse suspensions, and the singular limit of full hydrodynamic interactions is noted.

Second, the probe particle in question is no longer subject to a constant applied external force. Rather, the particle is considered to be a catalytically-active motor, consuming the bath reactant particles on its reactive face while passively colliding with reactant particles on its inert face. By creating an asymmetric distribution of reactant about its surface, the motor is able to diffusiophoretically propel itself with some mean velocity. The effects of finite size of the solute are examined on the leading order diffusive microstructure of reactant about the motor. Brownian and interparticle contributions to the motor velocity are computed for several interparticle interaction potential lengths and finite reactant-to-motor particle size ratios, with the dimensionless motor velocity increasing with decreasing motor size. A discussion on Brownian rotation frames the context in which these results could be applicable, and future directions are proposed which properly incorporate reactant advection at high motor velocities.

Complex behavior and perception in Drosophila emerges from iterative feedback-regulated reflexes

For a hungry fruit fly, locating and landing on a fermenting fruit where it can feed, find mates, and lay eggs, is an essential and difficult task requiring the integration of both olfactory and visual cues. Understanding how flies accomplish this will help provide a comprehensive ethological context for the expanding knowledge of their neural circuits involved in processing olfaction and vision, as well as inspire novel engineering solutions for control and estimation in computationally limited robotic applications. In this thesis, I use novel high throughput methods to develop a detailed overview of how flies track odor plumes, land, and regulate flight speed. Finally, I provide an example of how these insights can be applied to robotic applications to simplify complicated estimation problems. To localize an odor source, flies exhibit three iterative, reflex-driven behaviors. Upon encountering an attractive plume, flies increase their flight speed and turn upwind using visual cues. After losing the plume, flies begin zigzagging crosswind, again using visual cues to control their heading. After sensing an attractive odor, flies become more attracted to small visual features, which increases their chances of finding the plume source. Their changes in heading are largely controlled by open-loop maneuvers called saccades, which they direct towards and away from visual features. If a fly decides to land on an object, it begins to decelerate so as to maintain a stereotypical ratio of expansion to retinal size. Once they reach a stereotypical distance from the target, flies extend their legs in preparation for touchdown. Although it is unclear what cues they use to trigger this behavior, previous studies have indicated that it is likely under visual control. In Chapter 3, I use a nonlinear control theoretic analysis and robotic testbed to propose a novel and putative mechanism for how a fly might visually estimate distance by actively decelerating according to a visual control law. Throughout these behaviors, a common theme is the visual control of flight speed. Using genetic tools I show that the neuromodulator octopamine plays an important role in regulating flight speed, and propose a neural circuit for how this controller might be implemented in the flies brain. Two general biological and engineering principles are evident across my experiments: (1) complex behaviors, such as foraging, can emerge from the interactions of simple independent sensory-motor modules; (2) flies control their behavior in such a way that simplifies complex estimation problems.

Magnetic alignment of high-aspect ratio microwires into vertical arrays

Fundamental studies of magnetic alignment of highly anisotropic mesostructures can enable the clean-room-free fabrication of flexible, array-based solar and electronic devices, in which preferential orientation of nano- or microwire-type objects is desired. In this study, ensembles of 100 micron long Si microwires with ferromagnetic Ni and Co coatings are oriented vertically in the presence of magnetic fields. The degree of vertical alignment and threshold field strength depend on geometric factors, such as microwire length and ferromagnetic coating thickness, as well as interfacial interactions, which are modulated by varying solvent and substrate surface chemistry. Microwire ensembles with vertical alignment over 97% within 10 degrees of normal, as measured by X-ray diffraction, are achieved over square cm scale areas and set into flexible polymer films. A force balance model has been developed as a predictive tool for magnetic alignment, incorporating magnetic torque and empirically derived surface adhesion parameters. As supported by these calculations, microwires are shown to detach from the surface and align vertically in the presence of magnetic fields on the order of 100 gauss. Microwires aligned in this manner are set into a polydimethylsiloxane film where they retain their vertical alignment after the field has been removed and can subsequently be used as a flexible solar absorber layer. Finally, these microwires arrays can be protected for use in electrochemical cells by the conformal deposition of a graphene layer.

Homo- and Heteronuclear Transition Metal Complexes Supported by Multinucleating Ligands

This dissertation is mainly divided into two sub-parts: organometallic and bioinorganic/materials projects. The approach for the projects involves the use of two different multinucleating ligands to synthesize mono- and multinuclear complexes. Chapter 2 describes the synthesis of a multinucleating tris(phosphinoaryl)benzene ligand used to support mono-nickel and palladium complexes. The isolated mononuclear complexes were observed to undergo intramolecular arene C¬–H to C–P functionalization. The transformation was studied by nuclear magnetic resonance spectroscopy and X-ray crystallography, and represents a rare type of C–H functionalization mechanism, facilitated by the interactions of the group 10 metal with the arene π–system.

Chapter 3 describes the construction of multinickel complexes supported by the same triphosphine ligand from Chapter 2. This chapter shows how the central arene in the ligand’s triarylbenzene framework can interact with dinickel and trinickel moieties in various binding modes. X-ray diffraction studies indicated that all compounds display strong metal–arene interactions. A cofacial triangulo nickel(0) complex supported by this ligand scaffold was also isolated and characterized. This chapter demonstrates the use of an arene as versatile ligand design element for small molecular clusters.

Chapter 4 presents the syntheses of a series of discrete mixed transition metal Mn oxido clusters and their characterization. The synthesis of these oxide clusters displaying two types of transition metals were targeted for systematic metal composition-property studies relevant to mixed transition metal oxides employed in electrocatalysis. A series of heterometallic trimanganese tetraoxido cubanes capped with a redox-active metal [MMn₃O₄] (M = Fe, Co, Ni, Cu) was synthesized starting from a [CaMn₃O₄] precursor and structurally characterized by X-ray crystallography and anomalous diffraction to conclusively determine that M is incorporated at a single position in the cluster. The electrochemical properties of these complexes were studied via cyclic voltammetry. The redox chemistry of the series of complexes was investigated by the addition of a reductant and oxidant. X-ray absorption and electron paramagnetic resonance spectroscopies were also employed to evaluate the product of the oxidation/reduction reaction to determine the site of electron transfer given the presence of two types of redox-active metals. Additional studies on oxygen atom transfer reactivities of [MMn₃O₄] and [MMn₃O₂] series were performed to investigate the effect of the heterometal M in the reaction rates.

Chapter 5 focuses on the use of [CoMn₃O₄] and [NiMn₃O₄] cubane complexes discussed in Chapter 4 as precursors to heterogeneous oxygen evolution reaction (OER) electrocatalysts. These well-defined complexes were dropcasted on electrodes with/without heat treatment, and the OER activities of the resulting films were evaluated. Multiple spectroscopic techniques were performed on the surface of the electrocatalysts to gain insight into the structure-function relationships based on the heterometallic composition. Depending on film preparation, the Co-Mn-oxide was found to change metal composition during catalysis, while the Ni-Mn oxide maintained the NiMn3 ratio. These studies represent the use of discrete heterometallic-oxide clusters as precursors for heterogeneous water oxidation catalysts.

Appendix A describes the ongoing effort to synthesize a series of heteromultimetallic [MMn₃X] clusters (X = O, S, F). Complexes such as [ZnMn₃O], [CoMn₃O], [Mn₃S], and [Mn₄F] have been synthesized and structurally characterized. An amino-bis-oxime ligand (PRABO) has been installed on the [ZnMn₃O] cluster. Upon the addition of O₂, the desymmetrized [ZnMn₃O] cluster only underwent an outer-sphere, one-electron oxidation. Efforts to build and manipulate other heterometallic [MMn₃X] clusters are still ongoing, targeting O₂ binding and reduction. Appendix B summarizes the multiple synthetic approaches to build a [Co₄O₄]-cubane complex relevant to heterogeneous OER electrocatalysis. Starting with the tricobalt cluster [LCo₃(O₂CR)₃] and treatment various strong oxidants that can serve as oxygen atom source in the presence Co²⁺ salt only yielded tricobalt mono–oxo complexes. Appendix C presents the efforts to model the H-cluster framework of [FeFe]-hydrogenase by incorporating a synthetic diiron complex onto a protein-supported or a synthetic ligand-supported [Fe₄S₄]-cluster. The mutant ferredoxin with a [Fe₄S₄]-cluster and triscarbene ligand have been characterized by multiple spectroscopic techniques. The reconstruction of an H-cluster mimic has not yet been achieved, due to the difficulty of obtaining crystallographic evidence and the ambiguity of the EPR results.

Eigenvalue problems associated with Korn's inequalities in the theory of elasticity

Interest in the possible applications of a priori inequalities in linear elasticity theory motivated the present investigation. Korn's inequality under various side conditions is considered, with emphasis on the Korn's constant. In the "second case" of Korn's inequality, a variational approach leads to an eigenvalue problem; it is shown that, for simply-connected two-dimensional regions, the problem of determining the spectrum of this eigenvalue problem is equivalent to finding the values of Poisson's ratio for which the displacement boundary-value problem of linear homogeneous isotropic elastostatics has a non-unique solution.

Previous work on the uniqueness and non-uniqueness issue for the latter problem is examined and the results applied to the spectrum of the Korn eigenvalue problem. In this way, further information on the Korn constant for general regions is obtained.

A generalization of the "main case" of Korn's inequality is introduced and the associated eigenvalue problem is a gain related to the displacement boundary-value problem of linear elastostatics in two dimensions.

Microscopic Origin of Macroscopic Strength in Granular Media: A Numerical and Analytical Approach

Constitutive modeling in granular materials has historically been based on macroscopic experimental observations that, while being usually effective at predicting the bulk behavior of these type of materials, suffer important limitations when it comes to understanding the physics behind grain-to-grain interactions that induce the material to macroscopically behave in a given way when subjected to certain boundary conditions.

The advent of the discrete element method (DEM) in the late 1970s helped scientists and engineers to gain a deeper insight into some of the most fundamental mechanisms furnishing the grain scale. However, one of the most critical limitations of classical DEM schemes has been their inability to account for complex grain morphologies. Instead, simplified geometries such as discs, spheres, and polyhedra have typically been used. Fortunately, in the last fifteen years, there has been an increasing development of new computational as well as experimental techniques, such as non-uniform rational basis splines (NURBS) and 3D X-ray Computed Tomography (3DXRCT), which are contributing to create new tools that enable the inclusion of complex grain morphologies into DEM schemes.

Yet, as the scientific community is still developing these new tools, there is still a gap in thoroughly understanding the physical relations connecting grain and continuum scales as well as in the development of discrete techniques that can predict the emergent behavior of granular materials without resorting to phenomenology, but rather can directly unravel the micro-mechanical origin of macroscopic behavior.

In order to contribute towards closing the aforementioned gap, we have developed a micro-mechanical analysis of macroscopic peak strength, critical state, and residual strength in two-dimensional non-cohesive granular media, where typical continuum constitutive quantities such as frictional strength and dilation angle are explicitly related to their corresponding grain-scale counterparts (e.g., inter-particle contact forces, fabric, particle displacements, and velocities), providing an across-the-scale basis for better understanding and modeling granular media.

In the same way, we utilize a new DEM scheme (LS-DEM) that takes advantage of a mathematical technique called level set (LS) to enable the inclusion of real grain shapes into a classical discrete element method. After calibrating LS-DEM with respect to real experimental results, we exploit part of its potential to study the dependency of critical state (CS) parameters such as the critical state line (CSL) slope, CSL intercept, and CS friction angle on the grain's morphology, i.e., sphericity, roundness, and regularity.

Finally, we introduce a first computational algorithm to ``clone'' the grain morphologies of a sample of real digital grains. This cloning algorithm allows us to generate an arbitrary number of cloned grains that satisfy the same morphological features (e.g., roundness and aspect ratio) displayed by their real parents and can be included into a DEM simulation of a given mechanical phenomenon. In turn, this will help with the development of discrete techniques that can directly predict the engineering scale behavior of granular media without resorting to phenomenology.