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Dilly É, Neukirch S, Derr J, Zanchi D. Traveling Perversion as Constant Torque Actuator. Phys Rev Lett 2023; 131:177201. [PMID: 37955474 DOI: 10.1103/physrevlett.131.177201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2023] [Revised: 08/19/2023] [Accepted: 09/15/2023] [Indexed: 11/14/2023]
Abstract
Mechanical stress and conformation of helical elastic rods clamped at both ends were studied upon unwinding. By axial rotation of one end, the winding number was progressively changed from the natural one (n=n_{0}) to complete chirality inversion (n=-n_{0}) while keeping the total elongation fixed and monitoring the applied torque M and tension T. Along the unwinding process, the system crosses three distinct states: natural helix (+), mixed state (+/-), and inverted helix (-). The mixed state involves two helices with opposite chiralities spatially connected by a perversion (helicity inversion). Upon unwinding, the perversion is "injected" (nucleated) from one side and travels toward the opposite side where it is eventually "absorbed" (annihilated), leaving the system in the (-) state. In the mixed state, the profile of M(n) is almost flat: the system behaves as a constant torque actuator. The three states are quantitatively well described in the framework of a biphasic model which neglects the perversion energy and finite size effects. The latter are taken into account in a numerical simulation based on the Kirchhoff theory of elastic rods. The traveling perversion in helical elastic rods and related topological phenomena are universal, with applications from condensed matter to biological and bioinspired systems, including in particular mechanical engineering and soft robotics.
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Affiliation(s)
- Émilien Dilly
- Laboratoire Matière et Systèmes Complexes, UMR 7057, Université Paris Cité, CNRS, F-75205 Paris Cedex 13, France
| | - Sébastien Neukirch
- Institut Jean Le Rond d'Alembert, CNRS (UMR 7190), Sorbonne Université, 75005 Paris, France
| | - Julien Derr
- Laboratoire Reproduction et Développement des Plantes, École Normale Supérieure de Lyon, CNRS, INRAE, Inria, 69364 Lyon Cedex 07, France
| | - Dražen Zanchi
- Laboratoire Matière et Systèmes Complexes, UMR 7057, Université Paris Cité, CNRS, F-75205 Paris Cedex 13, France
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Neukirch S, Audoly B. A convenient formulation of Sadowsky's model for elastic ribbons. Proc Math Phys Eng Sci 2021. [DOI: 10.1098/rspa.2021.0548] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Elastic ribbons are elastic structures whose length-to-width and width-to-thickness aspect ratios are both large. Sadowsky proposed a one-dimensional model for ribbons featuring a nonlinear constitutive relation for bending and twisting: it brings in both rich behaviours and numerical difficulties. By discarding non-physical solutions to this constitutive relation, we show that it can be inverted; this simplifies the system of differential equations governing the equilibrium of ribbons. Based on the inverted form, we propose a natural regularization of the constitutive law that eases the treatment of singularities often encountered in ribbons. We illustrate the approach with the classical problem of the equilibrium of a Möbius ribbon, and compare our findings with the predictions of the Wunderlich model. Overall, our approach provides a simple method for simulating the statics and the dynamics of elastic ribbons.
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Affiliation(s)
- Sébastien Neukirch
- D’Alembert Institute for Mechanics, CNRS and Sorbonne Université, Paris UMR 7190, France
| | - Basile Audoly
- Laboratoire de Mécanique des Solides, CNRS, Institut Polytechnique de Paris, Palaiseau 91120, France
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Hourlier-Fargette A, Dervaux J, Antkowiak A, Neukirch S. Extraction of Silicone Uncrosslinked Chains at Air-Water-Polydimethylsiloxane Triple Lines. Langmuir 2018; 34:12244-12250. [PMID: 30199255 DOI: 10.1021/acs.langmuir.8b02128] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Silicone elastomers such as polydimethylsiloxane (PDMS) are convenient materials routinely used in laboratories that combine ease of preparation, flexibility, transparency, and gas permeability. However, these elastomers are known to contain a small fraction of uncrosslinked low-molecular-weight oligomers, the effects of which are not completely understood, particularly when used in contact with liquids. Here, we show that triple lines involving air, water, and PDMS elastomers are responsible for the contamination of water-air interfaces by uncrosslinked silicone oligomers through a capillarity-induced extraction mechanism. We investigate both the case of static and moving contact lines and study various geometries ranging from partially immersed PDMS plates to water droplets or air bubbles deposited on PDMS plates, all involving air-water-elastomer triple lines. We demonstrate experimentally that the contamination timescale is strikingly shorter for moving contact lines than in the static case. Eventually, we propose a simple poroelastic model capturing the main features of contamination observed in experiments.
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Affiliation(s)
- Aurélie Hourlier-Fargette
- Sorbonne Université, Centre National de la Recherche Scientifique, UMR 7190, Institut Jean Le Rond d'Alembert , F-75005 Paris , France
- Département de Physique, École Normale Supérieure, CNRS, PSL Research University , F-75005 Paris , France
| | - Julien Dervaux
- Matière et Systèmes Complexes, CNRS UMR 7057, Université Paris Diderot, Sorbonne Paris Cité University , F-75013 Paris , France
| | - Arnaud Antkowiak
- Sorbonne Université, Centre National de la Recherche Scientifique, UMR 7190, Institut Jean Le Rond d'Alembert , F-75005 Paris , France
- Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain , F-93303 Aubervilliers , France
| | - Sébastien Neukirch
- Sorbonne Université, Centre National de la Recherche Scientifique, UMR 7190, Institut Jean Le Rond d'Alembert , F-75005 Paris , France
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Grandgeorge P, Antkowiak A, Neukirch S. Auxiliary soft beam for the amplification of the elasto-capillary coiling: Towards stretchable electronics. Adv Colloid Interface Sci 2018; 255:2-9. [PMID: 28947256 DOI: 10.1016/j.cis.2017.08.011] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2017] [Revised: 08/17/2017] [Accepted: 08/30/2017] [Indexed: 10/18/2022]
Abstract
A flexible fiber carrying a liquid drop may coil inside the drop thereby creating a drop-on-fiber system with an ultra-extensible behavior. During compression, the excess fiber is spooled inside the droplet and capillary forces keep the system taut. During subsequent elongation, the fiber is gradually released and if a large number of spools is uncoiled a high stretchability is achieved. This mechanical behaviour is of interest for stretchable connectors but information, may it be electronic or photonic, usually travels through stiff functional materials. These high Young's moduli, leading to large bending rigidity, prevent in-drop coiling. Here we overcome this limitation by attaching a beam of soft elastomer to the functional fiber, thereby creating a composite system which exhibits in-drop coiling and carries information while being ultra-extensible. We present a simple model to explain the underlying mechanics of the addition of the soft beam and we show how it favors in-drop coiling. We illustrate the method with a two-centimeter long micronic PEDOT:PSS conductive fiber joined to a PVS soft beam, showing that the system conveys electricity throughout a 1900% elongation.
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Grandgeorge P, Krins N, Hourlier-Fargette A, Laberty-Robert C, Neukirch S, Antkowiak A. Capillarity-induced folds fuel extreme shape changes in thin wicked membranes. Science 2018; 360:296-299. [DOI: 10.1126/science.aaq0677] [Citation(s) in RCA: 39] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2017] [Accepted: 02/22/2018] [Indexed: 01/15/2023]
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Elettro H, Vollrath F, Antkowiak A, Neukirch S. Drop-on-coilable-fibre systems exhibit negative stiffness events and transitions in coiling morphology. Soft Matter 2017; 13:5509-5517. [PMID: 28744539 DOI: 10.1039/c7sm00368d] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We investigate the mechanics of elastic fibres carrying liquid droplets. In such systems, buckling may localize inside the drop cavity if the fibre is thin enough. This so-called drop-on-coilable-fibre system exhibits a surprising liquid-like response under compression and a solid-like response under tension. Here we analyze this unconventional behavior in further detail and find theoretical, numerical and experimental evidence of negative stiffness events. We find that the first and main negative stiffness regime owes its existence to the transfer of capillary-stored energy into mechanical curvature energy. The following negative stiffness events are associated with changes in the coiling morphology of the fibre. Eventually coiling becomes tightly locked into an ordered phase where liquid and solid deformations coexist.
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Affiliation(s)
- Hervé Elettro
- Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190 Institut Jean Le Rond d'Alembert, F-75005 Paris, France
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Hourlier-Fargette A, Antkowiak A, Chateauminois A, Neukirch S. Role of uncrosslinked chains in droplets dynamics on silicone elastomers. Soft Matter 2017; 13:3484-3491. [PMID: 28440371 DOI: 10.1039/c7sm00447h] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We report an unexpected behavior in wetting dynamics on soft silicone substrates: the dynamics of aqueous droplets deposited on vertical plates of such elastomers exhibits two successive speed regimes. This macroscopic observation is found to be closely related to microscopic phenomena occurring at the scale of the polymer network: we show that uncrosslinked chains found in most widely used commercial silicone elastomers are responsible for this surprising behavior. A direct visualization of the uncrosslinked oligomers collected by water droplets is performed, evidencing that a capillarity-induced phase separation occurs: uncrosslinked oligomers are extracted from the silicone elastomer network by the water-glycerol mixture droplet. The sharp speed change is shown to coincide with an abrupt transition in surface tension of the droplets, when a critical surface concentration in uncrosslinked oligomer chains is reached. We infer that a droplet shifts to a second regime with a faster speed when it is completely covered with a homogeneous oil film.
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Affiliation(s)
- Aurélie Hourlier-Fargette
- Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France. and Département de Physique, École Normale Supérieure, CNRS, PSL Research University, F-75005 Paris, France
| | - Arnaud Antkowiak
- Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France. and Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, F-93303 Aubervilliers, France
| | - Antoine Chateauminois
- ESPCI & CNRS, UMR 7615, Laboratoire de Sciences et Ingénierie de la Matière Molle, F-75005 Paris, France
| | - Sébastien Neukirch
- Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France.
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Bosi F, Misseroni D, Dal Corso F, Neukirch S, Bigoni D. Publisher's Note: Asymptotic self-restabilization of a continuous elastic structure [Phys. Rev. E 94, 063005 (2016)]. Phys Rev E 2017; 95:019905. [PMID: 28208431 DOI: 10.1103/physreve.95.019905] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2017] [Indexed: 06/06/2023]
Abstract
This corrects the article DOI: 10.1103/PhysRevE.94.063005.
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Bosi F, Misseroni D, Dal Corso F, Neukirch S, Bigoni D. Asymptotic self-restabilization of a continuous elastic structure. Phys Rev E 2016; 94:063005. [PMID: 28085333 DOI: 10.1103/physreve.94.063005] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2016] [Indexed: 11/07/2022]
Abstract
A challenge in soft robotics and soft actuation is the determination of an elastic system that spontaneously recovers its trivial path during postcritical deformation after a bifurcation. The interest in this behavior is that a displacement component spontaneously cycles around a null value, thus producing a cyclic soft mechanism. An example of such a system is theoretically proven through the solution of the elastica and a stability analysis based on dynamic perturbations. It is shown that the asymptotic self-restabilization is driven by the development of a configurational force, of similar nature to the Peach-Koehler interaction between dislocations in crystals, which is derived from the principle of least action. A proof-of-concept prototype of the discovered elastic system is designed, realized, and tested, showing that this innovative behavior can be obtained in a real mechanical apparatus.
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Affiliation(s)
- F Bosi
- DICAM-University of Trento, via Mesiano 77, I-38123 Trento, Italy
| | - D Misseroni
- DICAM-University of Trento, via Mesiano 77, I-38123 Trento, Italy
| | - F Dal Corso
- DICAM-University of Trento, via Mesiano 77, I-38123 Trento, Italy
| | - S Neukirch
- Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France
| | - D Bigoni
- DICAM-University of Trento, via Mesiano 77, I-38123 Trento, Italy
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Elettro H, Neukirch S, Antkowiak A, Vollrath F. Adhesion of dry and wet electrostatic capture silk of uloborid spider. Naturwissenschaften 2015; 102:41. [DOI: 10.1007/s00114-015-1291-6] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/24/2015] [Revised: 06/22/2015] [Accepted: 06/24/2015] [Indexed: 11/25/2022]
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Fargette A, Neukirch S, Antkowiak A. Elastocapillary snapping: capillarity induces snap-through instabilities in small elastic beams. Phys Rev Lett 2014; 112:137802. [PMID: 24745456 DOI: 10.1103/physrevlett.112.137802] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2013] [Indexed: 06/03/2023]
Abstract
We report on the capillarity-induced snapping of elastic beams. We show that a millimeter-sized water drop gently deposited on a thin buckled polymer strip may trigger an elastocapillary snap-through instability. We investigate experimentally and theoretically the statics and dynamics of this phenomenon and we further demonstrate that snapping can act against gravity, or be induced by soap bubbles on centimeter-sized thin metal strips. We argue that this phenomenon is suitable to miniaturization and design a condensation-induced spin-off version of the experiment involving a hydrophilic strip placed in a steam flow.
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Affiliation(s)
- Aurélie Fargette
- Département de Physique, École Normale Supérieure, 24 rue Lhomond, 75005 Paris, France and CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France and UPMC Université Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France
| | - Sébastien Neukirch
- CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France and UPMC Université Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France
| | - Arnaud Antkowiak
- CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France and UPMC Université Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France
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Neukirch S, Antkowiak A, Marigo JJ. Soft beams: when capillarity induces axial compression. Phys Rev E Stat Nonlin Soft Matter Phys 2014; 89:012401. [PMID: 24580231 DOI: 10.1103/physreve.89.012401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/11/2013] [Indexed: 06/03/2023]
Abstract
We study the interaction of an elastic beam with a liquid drop in the case where bending and extensional effects are both present. We use a variational approach to derive equilibrium equations and constitutive relation for the beam. This relation is shown to include a term due to surface energy in addition to the classical Young's modulus term, leading to a modification of Hooke's law. At the triple point where solid, liquid, and vapor phases meet, we find that the external force applied on the beam is parallel to the liquid-vapor interface. Moreover, in the case where solid-vapor and solid-liquid interface energies do not depend on the extension state of the beam, we show that the extension in the beam is continuous at the triple point and that the wetting angle satisfies the classical Young-Dupré relation.
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Affiliation(s)
- S Neukirch
- CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France and UPMC Université Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France
| | - A Antkowiak
- CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France and UPMC Université Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France
| | - J-J Marigo
- CNRS, Ecole Polytechnique, UMR 7649, Laboratoire de Mécanique des Solides, F-91128 Palaiseau Cedex, France
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Marko JF, Neukirch S. Global force-torque phase diagram for the DNA double helix: structural transitions, triple points, and collapsed plectonemes. Phys Rev E Stat Nonlin Soft Matter Phys 2013; 88:062722. [PMID: 24483501 PMCID: PMC3936674 DOI: 10.1103/physreve.88.062722] [Citation(s) in RCA: 38] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/03/2013] [Indexed: 06/01/2023]
Abstract
We present a free energy model for structural transitions of the DNA double helix driven by tensile and torsional stress. Our model is coarse grained and is based on semiflexible polymer descriptions of B-DNA, underwound L-DNA, and highly overwound P-DNA. The statistical-mechanical model of plectonemic supercoiling previously developed for B-DNA is applied to semiflexible polymer models of P- and L-DNA to obtain a model of DNA structural transitions in quantitative accord with experiment. We identify two distinct plectonemic states, one "inflated" by electrostatic repulsion and thermal fluctuations and the other "collapsed," with the two double helices inside the supercoils driven to close contact. We find that supercoiled B and L are stable only in the inflated form, while supercoiled P is always collapsed. We also predict the behavior and experimental signatures of highly underwound "Q"-DNA, the left-handed analog of P-DNA; as for P, supercoiled Q is always collapsed. Overstretched "S"-DNA and strand-separated "stress-melted" DNA are also included in our model, allowing prediction of a global phase diagram for forces up to 1000 pN and torques between ±60 pN nm, or, in terms of linking number density, from σ=-5 to +3.
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Affiliation(s)
- John F Marko
- Department of Physics and Astronomy and Department of Molecular Biosciences, Northwestern University, Evanston, Illinois 60208, USA
| | - Sébastien Neukirch
- CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France and UPMC Université Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France
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Abstract
We study the interaction of a liquid drop with an elastic beam in the case where bending effects dominate. We use a variational approach to derive equilibrium equations for the system in the presence of gravity and in the presence or absence of contact line pinning. We show that the derived equilibrium equations for the beam subsystem reveal the external forces applied on the beam by the liquid and vapour phases. Among these, the force applied at the triple line (the curve where the three phases meet) is found to lie along the liquid–vapour interface.
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Affiliation(s)
- Sébastien Neukirch
- CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, 75005 Paris, France
- UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, 75005 Paris, France
| | - Arnaud Antkowiak
- CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, 75005 Paris, France
- UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, 75005 Paris, France
| | - Jean-Jacques Marigo
- CNRS, Ecole Polytechnique, UMR 7649, Lab. Méca. Solides, 91128 Palaiseau Cedex, France
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Abstract
We study the deformation of an elastic strip by a liquid drop. At small enough scales, capillarity is the dominant fluid effect and surface tension forces may be sufficient to fold the beam, resulting in the wrapping of the drop by the beam. However, wrapping of the drop can be inhibited by the weight of the beam, which creates an energy barrier. The barrier can be overcome by input of kinetic energy in the form of impact of the drop. We introduce a semi-analytical model to study equilibria and their stability in three drop-beam systems: evaporation of a drop wetting and bending an elastic beam; impact of a drop on an elastic beam; lifting of a heavy elastic beam by a drop and we show the model reproduces experimental data. In relevant cases, we use the concept of suddenly applied load to discuss dynamic instabilities.
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Affiliation(s)
- Marco Rivetti
- UPMC Univ Paris 06, Institut Jean Le Rond d'Alembert, 75005 Paris, France
- CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, 75005 Paris, France
| | - Sébastien Neukirch
- UPMC Univ Paris 06, Institut Jean Le Rond d'Alembert, 75005 Paris, France
- CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, 75005 Paris, France
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Marko JF, Neukirch S. Competition between curls and plectonemes near the buckling transition of stretched supercoiled DNA. Phys Rev E Stat Nonlin Soft Matter Phys 2012; 85:011908. [PMID: 22400592 PMCID: PMC3763912 DOI: 10.1103/physreve.85.011908] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2011] [Indexed: 05/28/2023]
Abstract
Recent single-molecule experiments have observed that formation of a plectonemically supercoiled region in a stretched, twisted DNA proceeds via abrupt formation of a small plectonemic "bubble." A detailed mesoscopic model is presented for the formation of plectonemic domains, including their positional entropy, and the influence of small chiral loops or "curls" along the extended DNA. Curls begin to appear just before plectoneme formation, and are more numerous at low salt concentrations (<20 mM univalent ions) and at low forces (<0.5 pN). However, plectonemic domains quickly become far more stable slightly beyond the transition to supercoiling at moderate forces and physiological salt conditions. At the supercoiling transition, for shorter DNAs (2 kb) only one supercoiled domain appears, but for longer DNAs at lower forces (<0.5 pN) positional entropy favors formation of more than one plectonemic domain; a similar effect occurs for low salt. Although they are not the prevalent mode of supercoiling, curls are a natural transition state for binding of DNA-loop-trapping enzymes; we show how addition of loop-trapping enzymes can modify the supercoiling transition. The behavior of DNA torque is also discussed, including the effect of the measurement apparatus torque stiffness, which can play a role in determining how large the torque "overshoot" is at the buckling transition.
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Affiliation(s)
- John F Marko
- Department of Physics and Astronomy and Department of Molecular Biosciences, Northwestern University, Evanston, Illinois 60208, USA
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Neukirch S, Marko JF. Analytical description of extension, torque, and supercoiling radius of a stretched twisted DNA. Phys Rev Lett 2011; 106:138104. [PMID: 21517425 PMCID: PMC3120040 DOI: 10.1103/physrevlett.106.138104] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2010] [Indexed: 05/30/2023]
Abstract
We study the mixture of extended and supercoiled DNA that occurs in a twisted DNA molecule under tension. Closed-form asymptotic solutions for the supercoiling radius, extension, and torque of the molecule are obtained in the high-force limit where electrostatic and elastic effects dominate. We demonstrate that experimental data obey the extension and torque scaling laws apparent in our formulas, in the regime where thermal fluctuation effects are quenched by applied force.
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Clauvelin N, Audoly B, Neukirch S. Elasticity and electrostatics of plectonemic DNA. Biophys J 2009; 96:3716-23. [PMID: 19413977 PMCID: PMC2711414 DOI: 10.1016/j.bpj.2009.02.032] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2008] [Revised: 01/30/2009] [Accepted: 02/03/2009] [Indexed: 10/20/2022] Open
Abstract
We present a self-contained theory for the mechanical response of DNA in single molecule experiments. Our model is based on a one-dimensional continuum description of the DNA molecule and accounts both for its elasticity and for DNA-DNA electrostatic interactions. We consider the classical loading geometry used in experiments where one end of the molecule is attached to a substrate and the other one is pulled by a tensile force and twisted by a given number of turns. We focus on configurations relevant to the limit of a large number of turns, which are made up of two phases, one with linear DNA and the other one with superhelical DNA. The model takes into account thermal fluctuations in the linear phase and electrostatic interactions in the superhelical phase. The values of the torsional stress, of the supercoiling radius and angle, and key features of the experimental extension-rotation curves, namely the slope of the linear region and thermal buckling threshold, are predicted. They are found in good agreement with experimental data.
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Affiliation(s)
| | | | - S. Neukirch
- Université Pierre et Marie Curie, University Paris, UMR 7190, Institut Jean Le Rond d'Alembert, Paris, France; and Centre National de la Recherche Scientifique, UMR 7190, Institut Jean Le Rond d'Alembert, Paris, France
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Neukirch S, Starostin EL. Writhe formulas and antipodal points in plectonemic DNA configurations. Phys Rev E Stat Nonlin Soft Matter Phys 2008; 78:041912. [PMID: 18999460 DOI: 10.1103/physreve.78.041912] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2008] [Indexed: 05/27/2023]
Abstract
The linking and writhing numbers are key quantities when characterizing the structure of a piece of supercoiled DNA. Defined as double integrals over the shape of the double helix, these numbers are not always straightforward to compute, though a simplified formula was established in a theorem by Fuller [Proc. Natl. Acad. Sci. U.S.A. 75, 3557 (1978)]. We examine the range of applicability of this widely used simplified formula, and show that it cannot be employed for plectonemic DNA. We show that inapplicability is due to a hypothesis of Fuller theorem that is not met. The hypothesis seems to have been overlooked in many works.
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Affiliation(s)
- Sébastien Neukirch
- Institut Jean le Rond d'Alembert, CNRS and UPMC Univ. Paris 6, 4 Place Jussieu (case 162), 75005 Paris, France
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Affiliation(s)
- N. Clauvelin
- Institut Jean le Rond d’Alembert, UPMC Univ. Paris 06 & CNRS, 4, place Jussieu, Paris, France
| | - B. Audoly
- Institut Jean le Rond d’Alembert, UPMC Univ. Paris 06 & CNRS, 4, place Jussieu, Paris, France
| | - S. Neukirch
- Institut Jean le Rond d’Alembert, UPMC Univ. Paris 06 & CNRS, 4, place Jussieu, Paris, France
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Abstract
Coiled coils are important protein-protein interaction motifs with high specificity that are used to assemble macromolecular complexes. Their simple geometric organization, consisting of alpha helices wrapped around each other, confers remarkable mechanical properties. A geometrical and mechanical continuous model taking into account sequence effects and based on the superhelical winding of the constituent helices is introduced, and a continuous family of solutions in which the oligomerization interactions are satisfied is derived. From these solutions, geometric and structural properties, such as the chirality and pitch of the coiled coil and the location of residues, are obtained. The theoretical predictions are compared to x-ray data from the leucine zipper motif.
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Affiliation(s)
- Sébastien Neukirch
- Institut Jean le Rond d'Alembert, Centre National de la Recherche Scientifique and Université Pierre et Marie Curie, Paris, France
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Abstract
We study the mechanical response of elastic rods bent into open knots, focusing on the case of trefoil and cinquefoil topologies. The limit of a weak applied tensile force is studied both analytically and experimentally: the Kirchhoff equations with self-contact are solved by means of matched asymptotic expansions; predictions on both the geometrical and mechanical properties of the elastic equilibrium are compared to experiments. The extension of the theory to tight knots is discussed.
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Affiliation(s)
- B Audoly
- Institut Jean le Rond d'Alembert, UMR 7190, CNRS, Paris, France
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Abstract
Twining plants achieve vertical growth by revolving around supports of different sizes on which they exert a pressure. This observation raises many intriguing questions that are addressed within the framework of elastic filamentary structures by modeling the stem close to the apex as a growing elastic rod. The analysis shows that vertical growth is achieved thanks to discrete contact points and regions with continuous contact, that the contact pressure creates tension in the stem as observed experimentally, and that there is a maximal radius of the pole around which a twiner can climb.
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Affiliation(s)
- Alain Goriely
- Department of Mathematics and Program in Applied Mathematics, University of Arizona, Tucson, Arizona 85721, USA.
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Abstract
When thin brittle rods such as dry spaghetti pasta are bent beyond their limit curvature, they often break into more than two pieces, typically three or four. With the aim of understanding these multiple breakings, we study the dynamics of a bent rod that is suddenly released at one end. We find that the sudden relaxation of the curvature at this end leads to a burst of flexural waves, whose dynamics are described by a self-similar solution with no adjustable parameters. These flexural waves locally increase the curvature in the rod, and we argue that this counterintuitive mechanism is responsible for the fragmentation of brittle rods under bending. A simple experiment supporting the claim is presented.
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Affiliation(s)
- Basile Audoly
- Laboratoire de Modélisation en Mécanique, CNRS/Université Paris VI, 4 place Jussieu, Paris, France
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Abstract
We use an elastic rod model with contact to study the extension versus rotation diagrams of single supercoiled DNA molecules. We reproduce quantitatively the supercoiling response of overtwisted DNA and, using experimental data, we obtain an estimate of the effective supercoiling radius and of the twist rigidity of B-DNA. We find that the twist rigidity of DNA seems to vary widely with the nature and concentration of the salt buffer in which it is immersed.
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Affiliation(s)
- Sébastien Neukirch
- Institute for Mathematics B, Swiss Federal Institute of Technology, Lausanne, Switzerland
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Affiliation(s)
- J. M. T Thompson
- Centre for Nonlinear Dynamics, Civil Engineering Building, University College London, Gower Street, London WC1E 6BT, UK
| | - G. H. M van der Heijden
- Centre for Nonlinear Dynamics, Civil Engineering Building, University College London, Gower Street, London WC1E 6BT, UK
| | - S. Neukirch
- Centre for Nonlinear Dynamics, Civil Engineering Building, University College London, Gower Street, London WC1E 6BT, UK
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Neukirch S. Integrals of motion and semipermeable surfaces to bound the amplitude of a plasma instability. Phys Rev E Stat Nonlin Soft Matter Phys 2001; 63:036202. [PMID: 11308736 DOI: 10.1103/physreve.63.036202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2000] [Indexed: 05/23/2023]
Abstract
We study a dissipative dynamical system that models a parametric instability in a plasma. This instability is due to the interaction of a whistler with the ion acoustic wave and a plasma oscillation near the lower hybrid resonance. The amplitude of these three oscillations obey a three-dimensional system of ordinary differential equations which exhibits chaos for certain parameter values. By using certain "integrability informations" we have on the system, we get geometrical bounds for its chaotic attractor, leading to an upper bound for its Lyapunov dimension. On the other hand, we also obtain ranges of values of the system's parameters for which there is no chaotic motion.
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Affiliation(s)
- S Neukirch
- Centre for Nonlinear Dynamics and its Applications, University College London, Gower Street, London WC1E 6BT, United Kingdom.
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Neukirch S, Giacomini H. Shape of attractors for three-dimensional dissipative dynamical systems. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 2000; 61:5098-107. [PMID: 11031552 DOI: 10.1103/physreve.61.5098] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/22/1999] [Indexed: 11/07/2022]
Abstract
We introduce a method to bound attractors of dissipative dynamical systems in phase and parameter spaces. The method is based on the determination of families of transversal surfaces (surfaces crossed by the flow in only one direction). This technique yields very restrictive geometric bounds in phase space for the attractors. It also gives ranges of parameters of the system for which no chaotic behavior is possible. We illustrate our method on different three-dimensional dissipative systems.
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Affiliation(s)
- S Neukirch
- CNRS Laboratoire de Mathematiques et Physique Theorique, Faculte des Sciences et Techniques, Universite Francois Rabelais, Tours, France.
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