Objects that float at the interface between a liquid and a gas interact because of interfacial deformation and the effect of gravity. We highlight the crucial role of buoyancy in this interaction, which, for small particles, prevails over the capillary suction that is often assumed to be the dominant effect. We emphasize this point using a simple classroom demonstration, and then derive the physical conditions leading to mutual attraction or repulsion. We also quantify the force of interaction in some particular instances and present a simple dynamical model of this interaction. The results obtained from this model are then validated by comparison to experimental results for the mutual attraction of two identical spherical particles. We conclude by looking at some of the applications of the effect that can be found in the natural and manmade worlds.I.

PACS. 62.20.Dc -Elasticity, elastic constants. PACS. 68.03.-g -Gas-liquid and vacuum-liquid interfaces. PACS. 46.32.+x -Static buckling and instability.Abstract. -We study the collective behaviour of a close packed monolayer of non-Brownian particles at a fluid-liquid interface. Such a particle raft forms a two-dimensional elastic solid and can support anisotropic stresses and strains, e.g. it buckles in uniaxial compression and cracks in tension. We characterise this solid in terms of a Young's modulus and Poisson ratio derived from simple theoretical considerations and show the validity of these estimates by using an experimental buckling assay to deduce the Young's modulus.Introduction. -Particle covered liquid interfaces are increasingly being exploited in a wide variety of technological and medical applications [1]. Coating a liquid drop with a hydrophobic powder renders the drop non-wetting with the resulting liquid marbles free to roll on rigid surfaces or even float on water [2], a feature that has also proven useful as an adaptation to life on small scales [3]. In a similar vein, it is hoped that by encapsulating the active ingredients of drugs within a monolayer of colloidal particles (thereby forming a colloidosome [4]) more medicines will soon be administered by inhalation, so improving their efficacy. In the latter case, the particle coated interface exists only in the preliminary stages of manufacture, but in neither case have previous investigations been concerned with understanding the properties of particle monolayers at liquid interfaces. In this article, we show that in fact such particle monolayers behave collectively like 2-dimensional elastic solids and further, we characterize the properties of this two-dimensional solid.A variety of solid-like behaviours are observed for monolayers of hydrophobic particles sprinkled densely onto an air-water interface for a wide range (2.5 µm -6 mm) of particle sizes. For example, such a monolayer buckles under sufficient static compressive loading (see figs. 1 (a) and (b)) demonstrating that it can support an anisotropic stress. This stress state can only be supported by a material with a non-zero shear modulus, which is the signature of a solid. Once the compressive stress is removed, the monolayer returns rapidly (∼ O(0.1)s) to the undeformed state, ironing out the wrinkles formed by the buckling. This elasticity is also reminiscent of a solid and is in stark contrast to what is commonly observed in both dry

The buckling and wrinkling of thin films has recently seen a surge of interest among physicists, biologists, mathematicians, and engineers. This activity has been triggered by the growing interest in developing technologies at ever-decreasing scales and the resulting necessity to control the mechanics of tiny structures, as well as by the realization that morphogenetic processes, such as the tissue-shaping instabilities occurring in animal epithelia or plant leaves, often emerge from mechanical instabilities of cell sheets. Although the most basic buckling instability of uniaxially compressed plates was understood by Euler more than two centuries ago, recent experiments on nanometrically thin (ultrathin) films have shown significant deviations from predictions of standard buckling theory. Motivated by this puzzle, we introduce here a theoretical model that allows for a systematic analysis of wrinkling in sheets far from their instability threshold. We focus on the simplest extension of Euler buckling that exhibits wrinkles of finite length-a sheet under axisymmetric tensile loads. The first study of this geometry, which is attributed to Lamé, allows us to construct a phase diagram that demonstrates the dramatic variation of wrinkling patterns from near-threshold to far-from-threshold conditions. Theoretical arguments and comparison to experiments show that the thinner the sheet is, the smaller is the compressive load above which the far-from-threshold regime emerges. This observation emphasizes the relevance of our analysis for nanomechanics applications.pattern formation | thin-film buckling T hin films are among the ubiquitous examples of flexible structures that buckle under compressive loads. More interestingly, these buckling instabilities usually develop into wrinkled patterns that provide a dramatic display of the applied stress field (1, 2). Wrinkles align perpendicularly to the compression direction, depicting the principal lines of stress and providing a geometric tool for mechanical characterization. Traditional buckling theory is regularly used to understand these patterns in the near-threshold (NT) regime, in which the deformations are small perturbations of the initial flat state. However, it has been known since Wagner (3, 4) that, when the exerted loads are well in excess of those necessary to initiate buckling, the asymptotic state of the plate is very different from the one observed under NT conditions. In this far-from-threshold (FFT) regime, the stress nearly vanishes in the compression direction and wrinkles mark the region where the compressive stress has collapsed.Two complementary approaches have provided some insight into wrinkled sheets under FFT loading conditions. In a 1961 paper (5), Stein and Hedgepeth computed the asymptotic stress field in infinitely thin sheets under compression by assuming a vanishing component of the stress tensor along the compression direction. They further showed how such an asymptotic stress field yields the extent of wrinkles in several basic examples. A s...

The wrinkling and delamination of stiff thin films adhered to a polymer substrate have important applications in "flexible electronics." The resulting periodic structures, when used for circuitry, have remarkable mechanical properties because stretching or twisting of the substrate is mostly accommodated through bending of the film, which minimizes fatigue or fracture. To date, applications in this context have used substrate patterning to create an anisotropic substrate-film adhesion energy, thereby producing a controlled array of delamination "blisters." However, even in the absence of such patterning, blisters appear spontaneously, with a characteristic size. Here, we perform well-controlled experiments at macroscopic scales to study what sets the dimensions of these blisters in terms of the material properties and explain our results by using a combination of scaling and analytical methods. Besides pointing to a method for determining the interfacial toughness, our analysis suggests a number of design guidelines for the thin films used in flexible electronic applications. Crucially, we show that, to avoid the possibility that delamination may cause fatigue damage, the thin film thickness must be greater than a critical value, which we determine.adhesion | elasticity | stretchable electronics | blistering | buckling T hin films are adhered to substrates in a range of technological applications either to enhance the mechanical properties of bulk materials or to give the surface novel properties (1). Traditionally, the buckling and delamination of these thin coatings have been viewed as an inconvenience with research focusing on how they might be avoided. More recently, however, both the wrinkling and delamination of such films under compression have been exploited in the development of "flexible electronic" devices (2-7). The goal for these systems is to develop electronic circuits on flexible circuit boards ultimately leading to the manufacture of, among other things, flexible displays and electronic paper (8-10). A major technological challenge limiting the development of such devices is the requirement that the substrate be able to flex without stretching and damaging the wires that make up the circuit. One way to overcome this challenge is to use a polymer substrate that is first stretched and then coated with wires according to the required pattern. On releasing the strain in the substrate, the wires are relatively stiff in compression and so buckle out of the plane to accommodate the imposed deformation. This leads first to a well-studied wrinkling instability (7, 11) and subsequently to the formation of delamination "blisters" (12): localized regions where the film and substrate are no longer bonded. Once formed, these blisters facilitate the flexion of the substrate because the wires can accommodate deformation by bending rather than stretching.Here, we focus on the features of delamination by characterizing the relationship between the blisters' size and the material properties and their evolution. Th...

Elastic capsules, prepared from droplets or bubbles attached to a capillary (as in a pendant drop tensiometer), can be deflated by suction through the capillary. We study this deflation and show that a combined analysis of the shape and wrinkling characteristics enables us to determine the elastic properties in situ. Shape contours are analyzed and fitted using shape equations derived from nonlinear membrane-shell theory to give the elastic modulus, Poisson ratio and stress distribution of the membrane. We include wrinkles, which generically form upon deflation, within the shape analysis. Measuring the wavelength of wrinkles and using the calculated stress distribution gives the bending stiffness of the membrane. We compare this method with previous approaches using the Laplace-Young equation and illustrate the method on two very different capsule materials: polymerized octadecyltrichlorosilane (OTS) capsules and hydrophobin (HFBII) coated bubbles. Our results are in agreement with the available rheological data. For hydrophobin coated bubbles, the method reveals an interesting nonlinear behavior consistent with the hydrophobin molecules having a rigid core surrounded by a softer shell.

Pressurized elastic capsules arise at scales ranging from the 10 m diameter pressure vessels used to store propane at oil refineries to the microscopic polymeric capsules that may be used in drug delivery. Nature also makes extensive use of pressurized elastic capsules: plant cells, bacteria and fungi have stiff walls, which are subject to an internal turgor pressure. Here, we present theoretical, numerical and experimental investigations of the indentation of a linearly elastic shell subject to a constant internal pressure. We show that, unlike unpressurized shells, the relationship between force and displacement demonstrates two linear regimes. We determine analytical expressions for the effective stiffness in each of these regimes in terms of the material properties of the shell and the pressure difference. As a consequence, a single indentation experiment over a range of displacements may be used as a simple assay to determine both the internal pressure and elastic properties of capsules. Our results are relevant for determining the internal pressure in bacterial, fungal or plant cells. As an illustration of this, we apply our results to recent measurements of the stiffness of baker's yeast and infer from these experiments that the internal osmotic pressure of yeast cells may be regulated in response to changes in the osmotic pressure of the external medium.

Wrinkle patterns in compressed thin sheets are ubiquitous in nature and technology, from the furrows on our foreheads to crinkly plant leaves, from ripples on plastic-wrapped objects to the protein film on milk. The current understanding of an elementary descriptor of wrinkles-their wavelength-is restricted to deformations that are parallel, spatially uniform, and nearly planar. However, most naturally occurring wrinkles do not satisfy these stipulations. Here we present a scheme that quantitatively explains the wrinkle wavelength beyond such idealized situations. We propose a local law that incorporates both mechanical and geometrical effects on the spatial variation of wrinkle wavelength. Our experiments on thin polymer films provide strong evidence for its validity. Understanding how wavelength depends on the properties of the sheet and the underlying liquid or elastic subphase is crucial for applications where wrinkles are used to sculpt surface topography, to measure properties of the sheet, or to infer forces applied to a film.elastic sheets | wrinkles | curved topography W rinkles emerge in response to confinement, allowing a thin sheet to avoid the high energy cost associated with compressing a fraction e Δ of its length ( Fig. 1) (1-7). The wavelength, λ, of wrinkles reflects a balance between two competing effects: the bending resistance, which favors large wavelengths, and a restoring force that favors small amplitudes of deviation from the flat, unwrinkled state. Two such restoring forces are those due to the stiffness of a solid foundation or the hydrostatic pressure of a liquid subphase (Fig. 1A). Cerda and Mahadevan (1) realized that a tension in the sheet can give rise to a qualitatively similar effect ( Fig. 1B) and thereby proposed a universal law that applies in situations where the wrinkled sheet is nearly planar and subjected to uniaxial loading:Here the bending modulus B = Et 3 =½12ð1 − Λ 2 Þ (with E the Young's modulus, t the sheet's thickness, and Λ the Poisson ratio), whereas out-of-plane deformation is resisted by an effective stiffness, K eff , which can originate from a fluid or elastic substrate, an applied tension, or both. Eq. 1 is appealing in its simplicity, but it applies only for patterns that are effectively one-dimensional. In particular, it does not apply when the stress varies spatially or when there is significant curvature along the wrinkles. Here, we study two experimental settings in which these limitations are crucial: (i) indentation of a thin polymer sheet floating on a liquid, which leads to a horn-shaped surface with negative Gaussian curvature, and (ii) a circular sheet attached to a curved liquid meniscus with positive Gaussian curvature. In both cases, wrinkle patterns live on a curved surface, show spatially varying wavelengths, and are limited in spatial extent. The extent of finite wrinkle patterns in a variety of such 2D situations has recently been addressed (6,(8)(9)(10)(11) and was found to depend largely on external forces and boundary conditions. Howeve...

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