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Author ORCID Identifier
Open Access Dissertation
Doctor of Philosophy (PhD)
Year Degree Awarded
Month Degree Awarded
Gregory M. Grason
Christian D. Santangelo
Geometry and Topology | Statistical, Nonlinear, and Soft Matter Physics
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'' 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.
Atkinson, Daria, "FILAMENTS, FIBERS, AND FOLIATIONS IN FRUSTRATED SOFT MATERIALS" (2020). Doctoral Dissertations. 1990.
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