Defect-Driven Shape Instabilities of Bundles

Abstract

Topological defects are crucial to the thermodynamics and structure of condensed matter systems. For instance, when incorporated into crystalline membranes like graphene, disclinations with positive and negative topological charge elastically buckle the material into conical and saddlelike shapes, respectively. A recently uncovered mapping between the interelement spacing in 2D columnar structures and the metric properties of curved surfaces motivates basic questions about the interplay between defects in the cross section of a columnar bundle and its 3D shape. Such questions are critical to the structure of a broad class of filamentous materials, from biological assemblies like protein fibers to nanostructured or microstructured synthetic materials like carbon nanotube bundles. Here, we explore the buckling behavior for elementary disclinations in hexagonal bundles using a combination of continuum elasticity theory and numerical simulations of discrete filaments. We show that shape instabilities are controlled by a single material-dependent parameter that characterizes the ratio of interfilament to intrafilament elastic energies. Along with a host of previously unknown shape equilibria—the filamentous analogs to the conical and saddlelike shapes of defective membranes—we find a profoundly asymmetric response to positive and negative topologically charged defects in the infinite length limit that is without parallel to the membrane analog. The highly nonlinear dependence on the sign of the disclination charge is shown to have a purely geometric origin, stemming from the distinct compatibility (or incompatibility) of effectively positive- (or negative-)curvature geometries with lengthwise-constant filament spacing.