We study the emergence of helical structures subjected to a stretching force, demonstrating that the force transforms disk-shaped colloidal membranes into twisted chiral ribbons of predetermined handedness. Using an experimental technique that enforces torque-free boundary conditions we simultaneously measure the force-extension curve and quantify the shape of emergent ribbons. An effective theory that accounts for the membrane bending energy and uses geometric properties of the edge to model the internal liquid crystalline degrees of freedom explains both the measured force-extension curve and shape of the twisted ribbons in response to an applied force.
Twisted ribbons subjected to a tension exhibit a remarkably rich morphology, from smooth and wrinkled helicoids, to cylindrical or faceted patterns. These shapes are intimately related to the instability of the natural, helicoidal symmetry of the system, which generates both longitudinal and transverse stresses, thereby leading to buckling of the ribbon. In this paper, we focus on the tessellation patterns made of triangular facets. Our experimental observations are described within an "asymptotic isometry" approach that brings together geometry and elasticity. The geometry consists of paramet…
We address the mechanics of an elastic ribbon subjected to twist and tensile load. Motivated by the classical work of Green and a recent experiment that discovered a plethora of morphological instabilities, we introduce a comprehensive theoretical framework through which we construct a 4D phase diagram of this basic system, spanned by the exerted twist and tension, as well as the thickness and length of the ribbon. Different types of instabilities appear in various "corners" of this 4D parameter space, and are addressed through distinct types of asymptotic methods. Our theory employs three ins…
We present an analytical study of bent twisted ribbons. We first describe the elastic response of a ribbon within a purely mechanical framework. We then study the role of thermal fluctuations in modifying its elastic response. We predict the moment angle relation of bent and twisted ribbons. Such a study is expected to shed light on the role of twist in the "J factor" for DNA looping and on bending elasticity of twisted graphene ribbons.
In this paper we derive the general equilibrium equations of a polymer chain with a noncircular cross section by the variation of the free energy functional. From the equilibrium equation of the elastic ribbon we derive analytically the equilibrium conformations both of the helical ribbons and the twisted ribbons. We find that the pitch angle of the helical ribbons depends on the ratio of the torsional rigidity to the bending one. For the twisted ribbons, the rotation rate depends on the spontaneous torsion, which is determined by the elastic properties of the polymers. Our results for helical…
A rich zoology of shapes emerges from a simple stretched and twisted elastic ribbon. Despite a lot of interest, all these shape are not understood, in particular the shape that prevails at large tension and twist and that emerges from a transverse instability of the helicoid. Here, we propose a simple description for this cylindrical shape. By comparing its energy to the energy of other configurations, we are able to determine its location on the phase diagram. The theoretical predictions are in good agreement with our experimental results.
A theoretical analysis of the effect of force and torque on spontaneously twisted, fluctuating elastic ribbons is presented. We find that when a filament with a straight center line and a spontaneously twisted noncircular cross section is subjected to a sufficiently strong extensional force, its average elongation exhibits an asymmetric response to large over and undertwist. We construct the stability diagram that describes the buckling transition of such ribbons under the opposing action of force and torque and show that all the predicted behaviors can be understood in terms of continuous tra…
Smectic-A monolayers self-assembled from aqueous solutions of chiral fd viruses and a polymer depletant can assume a variety of shapes such as flat disks and twisted ribbons. A theoretical model based on the de Gennes model for the smectic A phase, the Helfrich model of membrane elasticity and a simple edge energy has been previously used to calculate the disk-ribbon phase diagram. In this paper we apply this model to the nucleation process of ribbons. First, we study the "rippled disks" that have been observed as precursors of ribbons. Using a model shape proposed by Meyer which includes ripp…
We consider a theoretical model for the chiral smectic A twisted ribbons observed in assemblies of fd viruses condensed by depletion forces. The depletion interaction is modeled by an edge energy assumed to be proportional to the depletant polymer in solution. Our model is based on the Helfrich energy for surface bending and the de Gennes model of chiral smectic A liquid crystals with twist penetration at the edge. We consider two variants of this model, one with the conventional Helfrich Gaussian curvature term, and a second with saddle-splay energy. A mean field analysis of both models yield…
Ribbons are topological objects of biological and technological importance. Here, we study the folding of thick ribbons with hydrophobic surfaces in a bad solvent in regimes in which either the ribbon's thickness or the solvent molecule size is not vanishingly small compared to the ribbon's width. Extensive Monte Carlo simulations show that ribbons of various lengths and with a small stiffness adopt several distinct configurations as the ground state that include rolled (Archimedean spiral), curled, twisted and globule conformations. Analytic and numerical calculations based on the considerati…
A novel model, devised to describe spontaneous chirality synchronization in complex liquids and liquid crystals, is proposed and studied. Segments of ribbon-like molecular columns with left- or right-handed 180degree twist lie on the bonds of a honeycomb lattice so that three ribbons meet in a vertex of the hexagonal honeycomb. The energy of each vertex is a minimum if the three ribbons have the same chirality, -E, and is +E otherwise, and the ground state is homochiral, i.e. all ribbons have the same hand. The energy levels for two vertices linked by a single ribbon are either -2E, 0 and +2 E…
Chirality frustrates and shapes the assembly of flexible filaments in rope-like, twisted bundles and fibers by introducing gradients of both filament shape (i.e. curvature) and packing throughout the structure. Previous models of chiral filament bundle formation have shown that this frustration gives rise to several distinct morphological responses, including self-limiting bundle widths, anisotropic domain (tape-like) formation and topological defects in the lateral inter-filament order. In this paper, we employ a combination of continuum elasticity theory and discrete filament bundle simulati…
Helical ribbons arise in many biological and engineered systems, often driven by anisotropic surface stress, residual strain, and geometric or elastic mismatch between layers of a laminated composite. A full mathematical analysis is developed to analytically predict the equilibrium deformed helical shape of an initially flat, straight ribbon, with prescribed magnitudes and orientations of the principal curvatures when subjected to arbitrary surface stress and/or internal residual strain distribution. The helix angle, radius, axis and chirality of the deformed helical ribbons are predicted with…
Ribbons are a class of slender structures whose length, width, and thickness are widely separated from each other. This scale separation gives a ribbon unusual mechanical properties in athermal macroscopic settings, e.g. it can bend without twisting, but cannot twist without bending. Given the ubiquity of ribbon-like biopolymers in biology and chemistry, here we study the statistical mechanics of microscopic inextensible, fluctuating ribbons loaded by forces and torques. We show that these ribbons exhibit a range of topologically and geometrically complex morphologies exemplified by three phas…
We present an experimental study of the quasi-elastic Raman scattering (QES) of plane-wave and twisted light by liquid crystals. Depending on their temperature, these crystals can exhibit isotropic, nematic and chiral nematic phases. The question is addressed of how the phase of a crystal and the state of incident light can affect the quasi-elastic energy spectra of the scattered radiation, whose shape is usually described by the combination of Lorentzian and Gaussian components. Special attention is paid to the \textit{chiral phase}, for which the Raman QES spectrum is dominated by a Lorentzi…
