Critical Mechanical Structures -Thermal Fluctuations and Self-Assembly

Abstract

Critical mechanical structures are structures on the verge of mechanical instability. Mechanical instability governs many fascinating phenomena in nature, including jamming, rigidity percolation, glass transitions, and structural phase transitions. Close to mechanical instability, the mechanical response and properties of these critical structures are dominated by the emergence of floppy modes, which are structural deformations that cost very little elastic energy. Although mechanical instability in athermal systems is well understood, how thermal fluctuations interact with floppy modes to modify transitions associated with the point of instability remains largely unexplored. To this end, we study the effect of thermal fluctuations on the phase transitions of various critical mechanical systems.

The first project presented in this dissertation concerns the buckling of rods at finite temperature. Thermal fluctuations can play an important role in the buckling of elastic objects at small scales, such as polymers or nanotubes. We study the finite-temperature buckling transition of an extensible rod by analyzing fluctuation corrections to the elasticity of the rod. We find that, in both two and three dimensions, thermal fluctuations delay the buckling transition, and near the transition, there is a critical regime in which fluctuations are prominent and make a contribution to the effective force that is of the order of the square root of the temperature. We verify our theoretical prediction of the phase diagram with Monte Carlo simulations.

The second project discussed in this dissertation examines how thermal fluctuations change structural transitions in lattices. We present an analytic study of the finite-temperature structural transition for the kagome lattice. Our model exhibits a zero-temperature continuous twisted-untwisted transition as the sign of the next-nearest-neighbor spring constant changes. At finite temperature, we show that the divergent contribution of floppy modes to the vibrational entropy renormalizes this spring constant, resulting in a first-order transition.

Another expanding area of research is the self-assembly of open structures and mechanical metamaterials with novel properties at small scales. In large part to floppy modes, these intriguing properties include negative Poisson's ratio and tunable topological mechanical properties. Employing self-assembly allows for the expedient synthesis of such structures. To this end, we study different techniques currently used in self-assembly. We propose an experimental manifestation of the twisted kagome lattice via self-assembling tri-block Janus particles with offset attractive patches. We also characterize the twisted-untwisted phase transition that these particles can undergo. This may lead to a novel smart material with, for instance, a Poisson's ratio that is tunable between positive and negative values.

The third project discussed in this dissertation explores the self-assembly of open structures using triangular prisms on an air-water interface. We present our theoretical and numerical analysis of how capillary interactions between these prisms, mediated by the deformation of the interface around the prisms, lead to directional binding and the self-assembly of large-scale open structures. We show how particle bowing and contact-line pinning yields a capillary hexapole-like interaction that results in two sets of distinct, highly-directional binding events: tip-to-tip and tip-to-edge-midpoint. We analyze the collapse of these binding events to edge-sharing configurations that impede the formation of ordered, open structures such as the kagome lattice, and we briefly discuss design principles that can be used to stabilize such interactions.