Exact solutions for the wrinkle patterns of confined elastic shells

Thin elastic membranes form complex wrinkle patterns when put on substrates of different shapes. Such patterns continue to receive attention across science and engineering. This is due, in part, to the promise of lithography-free micropatterning, but also to the observation that similar patterns arise in biological systems from growth. The challenge is to explain the patterns in any given setup, even when they fail to be robust. Building on the theoretical foundation of [Tobasco, to appear in Arch. Ration. Mech. Anal., arXiv:1906.02153], we derive a complete and simple rule set for wrinkles in the model system of a curved shell on a liquid bath. Our rules apply to shells whose initial Gaussian curvatures are of one sign, such as cutouts of saddles and spheres. They predict the surprising coexistence of orderly wrinkles alongside disordered regions where the response appears stochastic, which we confirm in experiment and simulation. They also unveil the role of the shell's medial axis, a distinguished locus of points that we show is a basic driver in pattern selection. Finally, they explain how the sign of the shell's initial curvature dictates the presence or lack of disorder. Armed with our simple rules, and the methodology underlying them, one can anticipate the creation of designer wrinkle patterns.Comment: Extended text including Supplementary Information. Heavily revised to focus the exposition and incorporate new results; title chang

Crumples as a generic stress-focusing instability in confined sheets

Thin elastic solids are easily deformed into a myriad of three-dimensional shapes, which may contain sharp localized structures as in a crumpled candy wrapper, or have smooth and diffuse features like the undulating edge of a flower. Anticipating and controlling these morphologies is crucial to a variety of applications involving textiles, synthetic skins, and inflatable structures. Here we show that a "wrinkle-to-crumple" transition, previously observed in specific settings, is a ubiquitous response for confined sheets. This unified picture is borne out of a suite of model experiments on polymer films confined to liquid interfaces with spherical, hyperbolic, and cylindrical geometries, which are complemented by experiments on macroscopic membranes inflated with gas. We use measurements across this wide range of geometries, boundary conditions, and lengthscales to quantify several robust morphological features of the crumpled phase, and we build an empirical phase diagram for crumple formation that disentangles the competing effects of curvature and compression. Our results suggest that crumples are a generic microstructure that emerge at large curvatures due to a competition of elastic and substrate energies.Comment: 12 pages, 7 figure

Rheology of grain-loaded liquid/air interfaces : towards the consolidation of an aerated medium

Les mousses liquides sont dans un état métastable. Un moyen de les stabiliser est d’y incorporer des particules solides. Lorsqu’elles sont hydrophobes, ces particules s’attachent aux interfaces liquide-air de la mousse et changent radicalement la rhéologie des films séparant les bulles. L’objectif de cette thèse est d’étudier expérimentalement deux composants élémentaires des mousses chargées en particules, à savoir les films et les bulles. La rhéologie des films particulaires est sondée grâce à des expériences d’éclatement et de compression alors que la résistance des bulles armurées est étudiée en faisant varier leur pression interne. En utilisant l’imagerie rapide, nous montrons d’abord que les particules qui pontent les deux interfaces du film peuvent inhiber l’ouverture d’un trou et que pour les autres configurations, la dynamique de rétraction peut être décrite par un équilibre inertio-capillaire pour des fractions surfaciques en particules <0.6 environ. L’étude de la dynamique de rétraction des films particulaires par Particle Image Velocimetry a permis de les caractériser par une viscosité effective qui diverge à la transition de jamming. De plus, les films particulaires flambent à fraction surfacique en particules élevée indiquant une transition d’un comportement liquide vers un comportement solide. En ce qui concerne les bulles particulaires, nous avons montré l’existence de pressions critiques 10 fois plus grandes que la pression de Laplace avant que les bulles ne se déforment aussi bien en dépression qu’en surpression ; la tension effective dans le plan de la coque granulaire est donc 10 fois plus importante que la tension dans un film liquide. Une fracture, correspondant à l’étirement du film liquide, est par ailleurs observée sur les bulles particulaires en surpression. En adoptant une approche élastique, le critère de Griffith permet de retrouver le bon ordre de grandeur de la pression de fractureLiquid foams are in a metastable state. One way to stabilize them is incorporating solid particles. When hydrophobic, these particles attach to liquid-air interfaces of the particle-laden foam inducing a drastic change in the rheology of the films between bubbles. The aim of this thesis is to study experimentally two elementary components of particle-laden foams, namely soap films and bubbles. The rheology of particulate films is probed through bursting and compression experiments while the resistance of armored bubbles is studied by inner pressure variations. Using high speed photography, we first show that particles bridging both liquid-air interfaces of a liquid film can resist hole opening and that the retraction dynamics for the other configurations can be described by a balance between inertia and capillarity for surface fractions of particles <0.6 approximately. Then studying the retraction dynamics of particulate soap films by Particle Image Velocimetry, we characterized these systems by an effective viscosity that diverges at the jamming transition. Moreover, buckling is observed at high surface fraction of particles indicating a transition from liquid-like to solid-like behavior. Concerning particulate bubbles, we showed the existence of pressure thresholds 10 times greater than Laplace pressure that need to be exceeded in order to observe a deformation in depression and overpressure experiments; the effective tension in the granular shell is thus 10 times greater that the tension in a liquid film. When inflated, a fracture corresponding to the stretching of the liquid film appears on particulate bubbles. Through an elastic approach, Griffith’s criterion gives the right order of magnitude of fracture pressure

Rhéologie d'interface liquide/air chargées de grains : vers la consolidation d'un milieu aéré

Liquid foams are in a metastable state. One way to stabilize them is incorporating solid particles. When hydrophobic, these particles attach to liquid-air interfaces of the particle-laden foam inducing a drastic change in the rheology of the films between bubbles. The aim of this thesis is to study experimentally two elementary components of particle-laden foams, namely soap films and bubbles. The rheology of particulate films is probed through bursting and compression experiments while the resistance of armored bubbles is studied by inner pressure variations. Using high speed photography, we first show that particles bridging both liquid-air interfaces of a liquid film can resist hole opening and that the retraction dynamics for the other configurations can be described by a balance between inertia and capillarity for surface fractions of particles <0.6 approximately. Then studying the retraction dynamics of particulate soap films by Particle Image Velocimetry, we characterized these systems by an effective viscosity that diverges at the jamming transition. Moreover, buckling is observed at high surface fraction of particles indicating a transition from liquid-like to solid-like behavior. Concerning particulate bubbles, we showed the existence of pressure thresholds 10 times greater than Laplace pressure that need to be exceeded in order to observe a deformation in depression and overpressure experiments; the effective tension in the granular shell is thus 10 times greater that the tension in a liquid film. When inflated, a fracture corresponding to the stretching of the liquid film appears on particulate bubbles. Through an elastic approach, Griffith’s criterion gives the right order of magnitude of fracture pressuresLes mousses liquides sont dans un état métastable. Un moyen de les stabiliser est d’y incorporer des particules solides. Lorsqu’elles sont hydrophobes, ces particules s’attachent aux interfaces liquide-air de la mousse et changent radicalement la rhéologie des films séparant les bulles. L’objectif de cette thèse est d’étudier expérimentalement deux composants élémentaires des mousses chargées en particules, à savoir les films et les bulles. La rhéologie des films particulaires est sondée grâce à des expériences d’éclatement et de compression alors que la résistance des bulles armurées est étudiée en faisant varier leur pression interne. En utilisant l’imagerie rapide, nous montrons d’abord que les particules qui pontent les deux interfaces du film peuvent inhiber l’ouverture d’un trou et que pour les autres configurations, la dynamique de rétraction peut être décrite par un équilibre inertio-capillaire pour des fractions surfaciques en particules <0.6 environ. L’étude de la dynamique de rétraction des films particulaires par Particle Image Velocimetry a permis de les caractériser par une viscosité effective qui diverge à la transition de jamming. De plus, les films particulaires flambent à fraction surfacique en particules élevée indiquant une transition d’un comportement liquide vers un comportement solide. En ce qui concerne les bulles particulaires, nous avons montré l’existence de pressions critiques 10 fois plus grandes que la pression de Laplace avant que les bulles ne se déforment aussi bien en dépression qu’en surpression ; la tension effective dans le plan de la coque granulaire est donc 10 fois plus importante que la tension dans un film liquide. Une fracture, correspondant à l’étirement du film liquide, est par ailleurs observée sur les bulles particulaires en surpression. En adoptant une approche élastique, le critère de Griffith permet de retrouver le bon ordre de grandeur de la pression de fractur

Viscosity of particle laden films

We perform retraction experiments on soap films where large particles bridge the two interfaces. Local velocities are measured by PIV during the unstationnary regime. The velocity variation in time and space can be described by a continuous fluid model from which effective viscosity (shear and dilatational) of particulate films is measured. The 2D effective viscosity of particulate films η2D increases with particle surface fraction ϕ: at low ϕ, it tends to the interfacial dilatational viscosity of the liquid/air interfaces and it diverges at the critical particle surface fraction ϕc ≃ 0.84. Experimental data agree with classical viscosity laws of hard spheres suspensions adapted to the 2D geometry, assuming viscous dissipation resulting from the squeeze of the liquid/air interfaces between the particles. Finally, we show that the observed viscous dissipation in particulate films has to be considered to describe the edge velocity during a retraction experiment at large particle coverage

Viscosity of particulate soap films : approaching the jamming of 2D capillary suspensions

We compute the effective viscosity of particulate soap films thanks to local velocity fields obtained by Particle Image Velocimetry (PIV) during film retraction experiments. We identify the jamming of these 2D capillary suspensions at a critical particle surface fraction (0.84) where effective viscosity diverges. Pair correlation function and number of neighbors in contact or close to contact reveal the cohesive nature of this 2D capillary granular media. The experimental 2D dynamic viscosities can be predicted by a model considering viscous dissipation at the liquid interfaces induced by the motion of individual particles.We compute the effective viscosity of particulate soap films thanks to local velocity fields obtained by Particle Image Velocimetry (PIV) during film retraction experiments. We identify the jamming of these 2D capillary suspensions at a critical particle surface fraction ([similar, equals]0.84) where effective viscosity diverges. Pair correlation function and number of neighbors in contact or close to contact reveal the cohesive nature of this 2D capillary granular media. The experimental 2D dynamic viscosities can be predicted by a model considering viscous dissipation at the liquid interfaces induced by the motion of individual particles

Gas Marbles : Much Stronger than Liquid Marbles

Enwrapping liquid droplets with hydrophobic particles allows the manufacture of so-called " liquid marbles " [Aussillous and Quéré Nature (London) 411, 924 (2001); Mahadevan 411, 895 (2001)]. The recent intensive research devoted to liquid marbles is justified by their very unusual physical and chemical properties and by their potential for various applications, from microreactors to water storage, including water pollution sensors [Bormashenko Curr. Opin. Colloid Interface Sci. 16, 266 (2011)]. Here we demonstrate that this concept can be successfully applied for encapsulating and protecting small gas pockets within an air environment. Similarly to their liquid counterparts, those new soft-matter objects, that we call " gas marbles, " can sustain external forces. We show that gas marbles are surprisingly tenfold stronger than liquid marbles and, more importantly, they can sustain both positive and negative pressure differences. This magnified strength is shown to originate from the strong cohesive nature of the shell. Those interesting properties could be exploited for imprisoning valuable or polluted gases or for designing new aerated materials

Bursting of particulate films

International audienceWhen covered with solid particles forming a 2D granular media, free liquid interfaces acquire interesting properties. As the particles concentration increases, a transition from liquid like behavior to solid like behavior is observed [1]. Modifying free liquid interfaces properties is interesting in systems such as foams that undergo aging mechanisms or flow. During all this destabilizing mechanisms, liquid films, which are two interfaces trapping liquid, are subject to high deformation. Former studies have shown the great ability of solid particles to stabilize foams [2]. The aim of our work is to understand the stabilizing mechanisms at the film scale. To do so, we perform two experiments on single films laden with hydrophobic solid particles that can either penetrate the film or stick to the two interfaces depending on particle density and diameter, and the liquid film thickness. On one hand, we study the opening dynamic of a hole in a hexagonal particles laden film, and observe that the opening can expand homothetically or instead a branching fracture can occur depending on the microstructure of the film. On a second hand, we focus on the retraction in a rectangular laden film limited by a mobile stick tracked by surface tension. For high particle coverage rate and large aspect ratio initial state, the sick can be stopped: the film is jammed at its final state. The particle laden film microstructure influences retraction dynamic, as well, while folds are created in almost all the experiments. Moreover, we study the motions of the particles in the dynamic phase using the PIV technique to characterize the rheology of such 2D granular media, in aim to correlate initial state and final state of our experiments. [1] Cicuta, P., Stancik, E. J., & Fuller, G. G. (2003). Shearing or compressing a soft glass in 2D: time-concentration superposition. Physical review letters, 90(23), 236101.[2] Stocco, A., Rio, E., Binks, B. P., & Langevin, D. (2011). Aqueous foams stabilized solely by particles. Soft Matter, 7(4), 1260-1267

Opening and retraction of particulate soap films

International audienceWe study for the first time the bursting dynamics of thin liquid films laden with hy-drophobic micronic particles either with free or constrained edges. We highlight that the particles can arrange in bilayer or monolayer configurations and explore a range of particles coverage from zero to random close packing. When the particles bridge the two interfaces (monolayer configuration) of free-edge films, the hole opens intermittently. For the other cases, we observe constant retraction velocities, modeled by balancing liquid and particles inertia against surface tension as in Taylor-Culick theory. But, this approach is only valid up to a critical value of particles coverage due to the interplay between the interfaces and the friction between particles