Session R14: Geometry and Mechanics of Folded Filaments, Writhing Ribbons and Braided Bundles

Collective Behavior of Hair, and Ponytail Shape and Dynamics.

I will discuss how we can build a mathematical model of the behaviour of a bundle of hair, comparing the results with experimental studies of the shape and dynamics of human ponytails. We treat the individual fibers as elastic filaments with random intrinsic curvature, in which the balance of bending elasticity, gravity, orientational disorder and inertia is recast as a differential equation for the envelope of the fibre bundle. The static elements of this work were first reported in R.E. Goldstein, P.B. Warren and R.C. Ball, Physical Review Letters 108, 078101 (2012). The compressibility of the bundle enters through an “equation of state” whose empirical form is shown to arise from a Confined Helix Model, in which the constraint of the surrounding hair is on a given fibre is represented as a confining cylinder. Using this model we find the ponytail shape is well fit with only one adjustable parameter, which is the degree to which the confining cylinders over fill space. The dynamics of driven vertical ponytail motion is well reproduced provided we introduce some damping, and we find the level of damping required is consistent with that arising from viscous drag of the lateral motion of the hair fibres through the interstitial air. Most of our match with experiment is achieved by approximating the fibre density of the ponytail to to be uniform across its cross-section, and to vary only length-wise. However we show that detail near the exit from a confining clamp (aka hairband) is only captured by computing the full cross-sectional variation.   

Equilibrium theory for braided elastic filaments

Motivated by supercoiling of DNA and other filamentous structures, we formulate a theory for equilibria of 2-braids, i.e., structures formed by two elastic rods winding around each other in continuous contact and subject to a local interstrand interaction. Unlike in previous work no assumption is made on the shape of the contact curve. Rather, this shape is found as part of the solution. The theory is developed in terms of a moving frame of directors attached to one of the strands with one of the directors pointing to the position of the other strand. The constant-distance constraint is automatically satisfied by the introduction of what we call braid strains. The price we pay is that the potential energy involves arclength derivatives of these strains, thus giving rise to a second-order variational problem. The Euler-Lagrange equations for this problem give balance equations for the overall braid force and moment referred to the moving frame as well as differential equations that can be interpreted as effective constitutive relations encoding the effect that the second strand has on the first as the braid deforms under the action of end loads. Simple analytical cases are discussed first and used as starting solutions in parameter continuation studies to compute classes of both open and closed (linked or knotted) braid solutions.   

Deformation and transport of micro-fibers and helices in viscous flows

Fluid-structure interactions between flexible objects and viscous flows are, to a large extent, governed by the shape of the flexible object. Using microfabrication methods, we obtain complex ``particles'' in fiber and helix form with perfect control not only over the material properties, but also the particle geometry. We then perform an experimental study on the deformation and transport of these particles in microfluidic flows. Fibers are shown to drift laterally in confined flows due to the transport anisotropy of the elongated object. When these fibers interact with lateral walls, complex dynamics are observed, such as fiber oscillation. Fiber flexibility modifies these dynamics. Flexible microhelices are easily stretched by a viscous flow and we characterize the overall shape as a function of the frictional properties. The deformation of these helices is well-described by non-linear finite extensibility. Due to the non-uniform distribution of the pitch of a helix subject to viscous drag, linear and nonlinear behavior is identified along the contour length of a single helix. When a polymer solution is used for the viscous flow, an interesting multiscale problem arises and the typical polymer size needs to be compared not only to the global size of the helix, but also to the dimensions of the ribbon.   

Wrinkles, loops, and topological defects in twisted ribbons

Nature abounds with elastic ribbon like shapes including double-stranded semiflexible polymers, graphene and metal oxide nanoribbons which are examples of elongated elastic structures with a strongly anisotropic cross-section. Due to this specific geometry, it is far from trivial to anticipate if a ribbon should be considered as a flat flexible filament or a narrow thin plate. We thus perform an experiment in which a thin elastic ribbon is loaded using a twisting and traction device coupled with a micro X-ray computed tomography machine allowing a full 3D shape reconstruction. A wealth of morphological behaviors can be observed including wrinkled helicoids, curled and looped configurations, and faceted ribbons. In this talk, I will show that most morphologies can be understood using a far-from-threshold approach and simple scaling arguments. Further, we find that the various shapes can be organized in a phase diagram using the twist, the tension, and the geometry of the ribbon as control parameters. Finally, I will discuss the spontaneous formation of topological defects with negatively-signed Gaussian charge at large twist and small but finite stretch.