Defect-driven shape instabilities of bundles

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 saddle-like shapes respectively. A recently uncovered mapping between the inter-element 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 nano- or micro-structured 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 inter-filament to intra-filament elastic energies. Along with a host of previously unknown shape equilibria---the filamentous analogs to the conical and saddle-like 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 non-linear dependence on the sign of the disclination charge is shown to have a purely geometric origin, stemming the from the distinct compatibility (or incompatibility) of effectively positive- (or negative-) curvature geometries with lengthwise-constant filament spacing.