Tendril Perversion in Intrinsically Curved Rods

We investigate the deployment of a thin elastic rod onto a rigid substrate and study the resulting coiling patterns. In our approach, we combine precision model experiments, scaling analyses, and computer simulations toward developing predictive understanding of the coiling process. Both cases of deposition onto static and moving substrates are considered. We construct phase diagrams for the possible coiling patterns and characterize them as a function of the geometric and material properties of the rod, as well as the height and relative speeds of deployment. The modes selected and their characteristic length scales are found to arise from a complex interplay between gravitational, bending, and twisting energies of the rod, coupled to the geometric nonlinearities intrinsic to the large deformations. We give particular emphasis to the first sinusoidal mode of instability, which we find to be consistent with a Hopf bifurcation, and analyze the meandering wavelength and amplitude. Throughout, we systematically vary natural curvature of the rod as a control parameter, which has a qualitative and quantitative effect on the pattern formation, above a critical value that we determine. The universality conferred by the prominent role of geometry in the deformation modes of the rod suggests using the gained understanding as design guidelines, in the original applications that motivated the study.thin rods | elasticity T he laying of the first transatlantic telegraph cable (1) opened the path for fast long-distance communication. Nowadays, submarine fiber-optic cables, a crucial backbone of the international communications (e.g., the Internet) infrastructure, are typically installed from a cable-laying vessel that, as it sails, pays out the cable from a spool downward onto the seabed. The portion of suspended cable between the vessel and the contact point with the seabed takes the form of a catenary (2). Similar procedures can also be used to deploy pipelines (3), an historical example of which is the then highly classified Operation PLUTO (Pipe-Lines Under the Ocean) (4), which provided fuel supplies across the English Channel at the end of World War II. One of the major challenges in the laying process of these cables and pipelines is the accurate control between the translation speed of the ship, v b , and the pay-out rate of the cable, v. A mismatch between the two may lead to mechanical failure due to excessive tension (if v b > v) or buckling (if v b < v), which for the case of communication cables can cause the formation of loops and tangles, resulting in undesirable signal attenuation (5, 6). At the microscale, deployment of nanowires onto a substrate has been used to print stretchable electronic components (7), and both periodic serpentines and coils have been fabricated by the flow-directed deposition of carbon nanotubes onto a patterned substrate (8).The common thread between these engineering systems is the geometry of deployment of the filamentary structure with a kinematic mismatch between the deposition ra...

Section: Static Coilingmentioning

confidence: 99%

Coiling of elastic rods on rigid substrates

Jawed¹,

Joo

et al. 2014

“…As the forces on the filament increase, the first loop of the helix above the motor is stretched and bends around toward a right-handed helix. In elastic filaments, such transitions in helicity under stress are known as "perversions" (39). Here, however, the formation of the perversion is not completed.…”

Section: Msmentioning

confidence: 99%

Bacteria exploit a polymorphic instability of the flagellar filament to escape from traps

Kühn

Schmidt

Eckhardt

et al. 2017

126

136

Many bacterial species swim by rotating single polar helical flagella. Depending on the direction of rotation, they can swim forward or backward and change directions to move along chemical gradients but also to navigate their obstructed natural environment in soils, sediments, or mucus. When they get stuck, they naturally try to back out, but they can also resort to a radically different flagellar mode, which we discovered here. Using high-speed microscopy, we monitored the swimming behavior of the monopolarly flagellated species Shewanella putrefaciens with fluorescently labeled flagellar filaments at an agarose-glass interface. We show that, when a cell gets stuck, the polar flagellar filament executes a polymorphic change into a spiral-like form that wraps around the cell body in a spiral-like fashion and enables the cell to escape by a screw-like backward motion. Microscopy and modeling suggest that this propagation mode is triggered by an instability of the flagellum under reversal of the rotation and the applied torque. The switch is reversible and bacteria that have escaped the trap can return to their normal swimming mode by another reversal of motor direction. The screw-type flagellar arrangement enables a unique mode of propagation and, given the large number of polarly flagellated bacteria, we expect it to be a common and widespread escape or motility mode in complex and structured environments.Shewanella | flagella | motility | structured environment

“…The analytic form of these one-parameter curves ((), ()) is given in ref. 43 The vertical portion of the boundary corresponds to helices with large local curvature ϭ 1 folding on themselves, a regime for which the validity of the rod model is questionable. In contrast, the curved segment of the boundary corresponds to helices resting on themselves, i.e., where contact is from remote parts of the helical tube; the rod model is entirely consistent right up to this boundary.…”

Section: Self-contactmentioning

confidence: 99%

Helices

Chouaieb

Goriely

Maddocks

2006

145

132

Helices are among the simplest shapes that are observed in the filamentary and molecular structures of nature. The local mechanical properties of such structures are often modeled by a uniform elastic potential energy dependent on bending and twist, which is what we term a rod model. Our first result is to complete the semi-inverse classification, initiated by Kirchhoff, of all infinite, helical equilibria of inextensible, unshearable uniform rods with elastic energies that are a general quadratic function of the flexures and twist. Specifically, we demonstrate that all uniform helical equilibria can be found by means of an explicit planar construction in terms of the intersections of certain circles and hyperbolas. Second, we demonstrate that the same helical centerlines persist as equilibria in the presence of realistic distributed forces modeling nonlocal interactions as those that arise, for example, for charged linear molecules and for filaments of finite thickness exhibiting self-contact. Third, in the absence of any external loading, we demonstrate how to construct explicitly two helical equilibria, precisely one of each handedness, that are the only local energy minimizers subject to a nonconvex constraint of self-avoidance.biomolecules ͉ differential geometry ͉ elasticity ͉ filaments ͉ rods S cientists have long held a fascination, sometimes bordering on mystical obsession, for helical structures in nature (1, 2). Helices arise in nanosprings, carbon nanotubes, ␣-helices, DNA double and collagen triple helices, lipid bilayers, bacterial flagella in Salmonella and Escherichia coli, aerial hyphae in actynomycetes, bacterial shape in spirochetes, horns, tendrils, vines, screws, springs, and helical staircases (3-13). Helical structures can be understood from a discrete point of view as a regular, periodic stacking of rigid blocks, such as the bases in DNA strands (14), the tail sheaths of bacteriophages (15), the packing of flagellin subunits (16), or simply the stairs in spiral staircases (1). However, it also can be beneficial to adopt a continuous description in which a filamentary structure is represented by a central space curve along with a frame that captures the orientation of the material cross sections at each point along the curve. Although this description neglects some fine features of deformations in the cross section, it is nevertheless suitable to describe the large-scale geometrical and physical properties of long, thin structures. If the deformations are small enough with respect to the appropriate characteristic length scales of the problem, the physical attributes of the filament such as the stresses acting across cross sections can be averaged and represented as a resultant force and moment acting on the centerline. We will refer to such a description of a filamentary structure as a rod model (see, for instance, refs 5, 7, 8, and 10-13 for examples of such rod theories).In continuum mechanics, a semi-inverse problem is generally taken to mean the study of a special class of solutions w...