Date of Award
Doctor of Philosophy (PhD)
asymptotics;bound states;free energy;geometry;soft-matter physics;wkb/semiclassical method
Condensed Matter Physics | Physical Sciences and Mathematics | Physics
This dissertation is concerned with two problems that lie at the interface of soft-matter physics, geometry, and asymptotic analysis, but otherwise have no bearing on one another. In the first problem, I consider the equilibrium thermal fluctuations of deformable mechanical frameworks. These frameworks have served as highly idealized representations of mechanical structures that underlie a plethora of soft, few-body systems at the submicron scale such as colloidal clusters and DNA origami. When the holonomic constraints in a framework cease to be linearly independent, singularities can appear in its configuration space, where it becomes energetically softer. Consequently, the framework's free-energy landscape becomes dominated by the neighborhoods of points corresponding to these singularities. In the second problem, I study the localization of elastic waves in thin elastic structures with spatially varying curvature profiles, using a curved rod and a uniaxially-curved shell as concrete examples. Waves propagating on such structures have multiple components owing to the curvature-mediated coupling of the tangential and normal components of the displacement field. Here, using the semiclassical approximation, I show that these waves form localized, bound states around points where the absolute curvature of the structure has a minimum. Both these problems exemplify the subtle interplay between the mechanical properties of soft materials and their geometry, which further sets the stage for many interesting consequences.
Mannattil, Manu, "Asymptotics, Geometry, and Soft Matter" (2023). Dissertations - ALL. 1792.