Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences

FILAMENTS, FIBERS, AND FOLIATIONS IN FRUSTRATED SOFT MATERIALS

Assemblies of one-dimensional filaments appear in a wide range of physical systems: from biopolymer bundles, columnar liquid crystals, and superconductor vortex arrays; to familiar macroscopic materials, like ropes, cables, and textiles. Interactions between the constituent filaments in such systems are most sensitive to the distance of closest approach between the central curves which approximate their configuration, subjecting these distinct assemblies to common geometric constraints. Dual to strong dependence of inter-filament interactions on changes in the distance of closest approach is their relative insensitivity to reptations, translations along the filament backbone. In this dissertation, after briefly reviewing the mechanics and geometry of frustrated elastic materials relevant for the discussion of fiber geometry and elasticity in Chapter 1, we examine in detail the geometry associated with constant spacing between continuous filament fields, and the associated couplings between stretching of lengths between filaments, symmetries of multi-filament energies, and the shapes adopted by filament bundles. In Chapter 2, we consider two distinct notions of constant spacing in multi-filament packings in three Euclidean dimensions, E3: equidistance, where the distance of closest approach is constant along the length of filament pairs; and isometry, where the distances of closest approach between all neighboring filaments are constant and equal. We show that, although any smooth curve in E3 permits one dimensional families of collinear equidistant curves belonging to a ruled surface, there are only two families of tangent fields with mutually equidistant integral curves in E3. The relative shapes and configurations of curves in these families are highly constrained: they must be either (isometric) developable domains, which can bend, but not twist; or (non-isometric) constant-pitch helical bundles, which can twist, but not bend. Thus, filament textures that are simultaneously bent and twisted, such as twisted toroids of condensed DNA plasmids or wire ropes, are doubly frustrated: twist frustrates constant neighbor spacing in the cross-section, while non-equidistance requires additional longitudinal variations of spacing along the filaments. To illustrate the consequences of the failure of equidistance, we compare spacing in three almost equidistant\u27\u27 ansatzes for twisted toroidal bundles and use our formulation of equidistance to construct upper bounds on the growth of longitudinal variations of spacing with bundle thickness. In Chapter 3, we show that because the elastic response of non-equidistant filament bundles is frustrated, it cannot adequately be described by linearized, two-dimensional strains. To describe non-equidistant configurations, we derive a geometrically nonlinear, coordinate invariant, gauge-like theory for the elasticity of filamentous materials. For small strains, we derive the Euler-Lagrange equations for general, non-equidistant filament bundles, and show that, while force balance is qualitatively similar to that for 2D crystals, there are corrections which account for the non-integrability of twisted filament fields. Because of these corrections, force balance along the filament tangents couples to the convective flow tensor, which measures local deviations from equidistance. Within this framework, we discuss the impact of filament texture on bundle elasticity, and extend the analysis of helical filament bundles to the large twist limit. In Chapter 4, we finally turn our attention to longitudinally frustrated, non-equidistant bundles. Taking twisted toroidal filament bundles, which can be found in condensates of nucleic acids under confinement (e.g., inside a viral capsid), as a geometric prototype for the more general class of non-equidistant filament bundles, we derive the linearized force-balance equations in the limit of small central-filament curvature. While we make substantial progress towards a qualitative understanding of the behavior of non-equidistant filaments, the general solution to the Euler-Lagrange equations remains out of reach due to the presence of singularities at the outer boundary that emerge as a result of our perturbation scheme. We conclude by discussing the progress made in this dissertation in understanding the physics of frustrated fibers, and speculating about the ramifications for more general soft-elastic materials

Clay flocculation effect on microbial community composition in water and sediment

Clay-based flocculation techniques have been developed to mitigate harmful algal blooms; however, the potential ecological impacts on the microbial community are poorly understood. In this study, chemical measurements were combined with 16S rRNA sequencing to characterize the microbial community response to different flocculation techniques, including controls, clay flocculation, clay flocculation with zeolite, and clay flocculation with O2 added zeolite capping. Sediment bacterial biomass measured by PLFA were not significantly altered by the various flocculation techniques used. However, 16S rRNA sequencing revealed differences in water microbial community structure between treatments with and without zeolite capping. The differences were related to significant reductions of total nitrogen (TN), total phosphorus (TP) and ammonia (NH4+) concentration and increase of nitrate (NO3-) concentration in zeolite and O2 loaded zeolite capping. The relative abundance of ammonia oxidizing bacteria increased four-fold in zeolite capping microcosms, suggesting zeolite promoted absorbed ammonia removal in the benthic zone. Zeolite-capping promoted bacteria nitrogen cycling activities at the water-sediment interface. Potential pathogens that are usually adapted to eutrophic water bodies were reduced after clay flocculation. This study demonstrated clay flocculation did not decrease bacterial populations overall and may reduce regulatory indicators and pathogenic contaminants in water. Zeolite capping may also help prevent nutrients from being released back into the water thus preventing additional algal blooms