Symmetry-breaking transitions associated with the buckling and folding of curved multilayered surfaces-which are common to a wide range of systems and processes such as embryogenesis, tissue differentiation and structure formation in heterogeneous thin films or on planetary surfaces-have been characterized experimentally. Yet owing to the nonlinearity of the underlying stretching and bending forces, the transitions cannot be reliably predicted by current theoretical models. Here, we report a generalized Swift-Hohenberg theory that describes wrinkling morphology and pattern selection in curved elastic bilayer materials. By testing the theory against experiments on spherically shaped surfaces, we find quantitative agreement with analytical predictions for the critical curves separating labyrinth, hybrid and hexagonal phases. Furthermore, a comparison to earlier experiments suggests that the theory is universally applicable to macroscopic and microscopic systems. Our approach builds on general differential-geometry principles and can thus be extended to arbitrarily shaped surfaces.

Low viscosity (<100 cSt) silicon oil droplets are placed on a high viscosity (1000 cSt) oil bath that vibrates vertically. The viscosity difference ensures that the droplet is more deformed than the bath interface. Droplets bounce periodically on the bath when the acceleration of its sinusoidal motion is larger than a threshold value. The threshold is minimum for a particular frequency of excitation: droplet and bath motions are in resonance. The bouncing droplet has been modeled by considering the deformation of the droplet and the lubrication force exerted by the air layer between the droplet and the bath. Threshold values are predicted and found to be in good agreement with our measurements.

We present the results of an experimental investigation on the crystallography of the dimpled patterns obtained through wrinkling of a curved elastic system. Our macroscopic samples comprise a thin hemispherical shell bound to an equally curved compliant substrate. Under compression, a crystalline pattern of dimples selforganizes on the surface of the shell. Stresses are relaxed by both out-of-surface buckling and the emergence of defects in the quasihexagonal pattern. Three-dimensional scanning is used to digitize the topography. Regarding the dimples as point-like packing units produces spherical Voronoi tessellations with cells that are polydisperse and distorted, away from their regular shapes. We analyze the structure of crystalline defects, as a function of system size. Disclinations are observed and, above a threshold value, dislocations proliferate rapidly with system size. Our samples exhibit striking similarities with other curved crystals of charged particles and colloids. Differences are also found and attributed to the far-fromequilibrium nature of our patterns due to the random and initially frozen material imperfections which act as nucleation points, the presence of a physical boundary which represents an additional source of stress, and the inability of dimples to rearrange during crystallization. Even if we do not have access to the exact form of the interdimple interaction, our experiments suggest a broader generality of previous results of curved crystallography and their robustness on the details of the interaction potential. Furthermore, our findings open the door to future studies on curved crystals far from equilibrium. mechanical instabilities | curved surfaces | pattern formation | packing | defects T he classic design of a soccer ball, with its 20 hexagonal (white) patches interspersed with 12 (black) pentagons, the buckminsterfullerene C 60 (1), virus capsules (2), colloidosomes (3), and geodesic architectural domes (4) are all examples of crystalline packings on spherical surfaces. In contrast with crystals on flat surfaces, these structures cannot be constructed from a tiling of hexagons alone. Instead, disclinations--nonhexagonal elements such as the 12 pentagons on a soccer ball--are required by topology (5, 6), which constrains how the crystal order must comply with the geometry of the underlying surface. For example, seeding a hexagonal crystal with a pentagon (fivefold disclination) disrupts the perfect hexagonal symmetry and introduces a localized stress concentrator, which can be relaxed through out-ofplane deformation with positive Gaussian curvature (7,8). Likewise, a heptagon (sevenfold disclination) induces a disturbance with negative Gaussian curvature.An example of a physical realization of curved crystals is found in experiments on colloidal emulsions, where equally charged particles self-organize at the curved interface of two immiscible liquids (3, 9-11). These experiments build upon a wealth of previous theoretical and numerical investigations, as reviewed by Bowick...

The crater formation due to the impact of a water droplet onto a granular bed has been experimentally investigated. Three parameters were tuned: the impact velocity, the size of the droplet, and the size of the grains. The aim is to determine the influence of the kinetic energy on the droplet pattern. The shape of the crater depends on the kinetic energy at the moment the droplet starts to impact the bed. The spreading and recession of the liquid during the impact were carefully analyzed from the dynamical point of view, using image analysis of high-speed video recordings. The different observed regimes are characterized by the balance between the impregnation time of the water by the granular bed by the water and the capillary time responsible for the recession of the drop.