Capillary deformations of bendable films

Peeling from a liquid

We establish the existence of a cusp in the curvature of a solid sheet at its contact with a liquid subphase. We study two configurations in floating sheets where the solid-vapor-liquid contact line is a straight line and a circle, respectively. In the former case, a rectangular sheet is lifted at its one edge, whereas in the latter a gas bubble is injected beneath a floating sheet. We show that in both geometries the derivative of the sheet's curvature is discontinuous. We demonstrate that the boundary condition at the contact is identical in these two geometries, even though the shape of the contact line and the stress distribution in the sheet are very different.

Nanoscale Elasto-Capillarity in the Graphene-Water System under Tension: Revisiting the Assumption of a Constant Wetting Angle

Wetting highly compliant surfaces can cause them to deform. Atomically thin materials, such as graphene, can have exceptionally small bending rigidities, leading to elasto-capillary lengths of a few nanometers. Using large-scale molecular dynamics (MD), we have studied the wetting and deformation of graphene due to nanometer-sized water droplets, focusing on the wetting angle near the vesicle transition. Recent continuum theories for wetting of flexible membranes reproduce our MD results qualitatively well. However, we find that when the curvature is large at the triple-phase contact line, the wetting angle increases with decreasing tension. This is in contrast to existing macroscopic theories but can be amended by allowing for a variable wetting angle.

Effect of external tension on the wetting of an elastic sheet

Recent studies of elastocapillary phenomena have triggered interest in a basic variant of the classical Young-Laplace-Dupré (YLD) problem: the capillary interaction between a liquid drop and a thin solid sheet of low bending stiffness. Here we consider a two-dimensional model where the sheet is subjected to an external tensile load and the drop is characterized by a well-defined Young's contact angle θ_{Y}. Using a combination of numerical, variational, and asymptotic techniques, we discuss wetting as a function of the applied tension. We find that, for wettable surfaces with 0<θ_{Y}<π/2, complete wetting is possible below a critical applied tension due to the deformation of the sheet in contrast with rigid substrates requiring θ_{Y}=0. Conversely, for very large applied tensions, the sheet becomes flat and the classical YLD situation of partial wetting is recovered. At intermediate tensions, a vesicle forms in the sheet, which encloses most of the fluid, and we provide an accurate asymptotic description of this wetting state in the limit of small bending stiffness. We show that bending stiffness, however small, affects the entire shape of the vesicle. Rich bifurcation diagrams involving partial wetting and "vesicle" solution are found. For moderately small bending stiffnesses, partial wetting can coexist with both the vesicle solution and complete wetting. Finally, we identify a tension-dependent bendocapillary length, λ_{BC}, and find that the shape of the drop is determined by the ratio A/λ_{BC}^{2}, where A is the area of the drop.

Shapes of a filament on the surface of a bubble

The shape assumed by a slender elastic structure is a function both of the geometry of the space in which it exists and the forces it experiences. We explore, by experiments and theoretical analysis, the morphological phase space of a filament confined to the surface of a spherical bubble. The morphology is controlled by varying bending stiffness and weight of the filament, and its length relative to the bubble radius. When the dominant considerations are the geometry of confinement and elastic energy, the filament lies along a geodesic and when gravitational energy becomes significant, a bifurcation occurs, with a part of the filament occupying a longitude and the rest along a curve approximated by a latitude. Far beyond the transition, when the filament is much longer than the diameter, it coils around the selected latitudinal region. A simple model with filament shape as a composite of two arcs captures the transition well. For better quantitative agreement with the subcritical nature of bifurcation, we study the morphology by numerical energy minimization. Our analysis of the filament’s morphological space spanned by one geometric parameter, and one parameter that compares elastic energy with body forces, may provide guidance for packing slender structures on complex surfaces.

Van der Waals interaction affects wrinkle formation in two-dimensional materials

Nonlinear mechanics of solids is an exciting field that encompasses both beautiful mathematics, such as the emergence of instabilities and the formation of complex patterns, as well as multiple applications. Two-dimensional crystals and van der Waals (vdW) heterostructures allow revisiting this field on the atomic level, allowing much finer control over the parameters and offering atomistic interpretation of experimental observations. In this work, we consider the formation of instabilities consisting of radially oriented wrinkles around mono- and few-layer "bubbles" in two-dimensional vdW heterostructures. Interestingly, the shape and wavelength of the wrinkles depend not only on the thickness of the two-dimensional crystal forming the bubble, but also on the atomistic structure of the interface between the bubble and the substrate, which can be controlled by their relative orientation. We argue that the periodic nature of these patterns emanates from an energetic balance between the resistance of the top membrane to bending, which favors large wavelength of wrinkles, and the membrane-substrate vdW attraction, which favors small wrinkle amplitude. Employing the classical "Winkler foundation" model of elasticity theory, we show that the number of radial wrinkles conveys a valuable relationship between the bending rigidity of the top membrane and the strength of the vdW interaction. Armed with this relationship, we use our data to demonstrate a nontrivial dependence of the bending rigidity on the number of layers in the top membrane, which shows two different regimes driven by slippage between the layers, and a high sensitivity of the vdW force to the alignment between the substrate and the membrane.

Wetting and wrapping of a floating droplet by a thin elastic filament

We study the wetting of a thin elastic filament floating on a fluid surface by a droplet of another, immiscible fluid. This quasi-2D experimental system is the lower-dimensional counterpart of the wetting and wrapping of a droplet by an elastic sheet. The simplicity of this system allows us to study the phenomenology of partial wetting and wrapping of the droplet by measuring angles of contact as a function of the elasticity of the filament, the applied tension and the curvature of the droplet. We find that a purely geometric theory gives a good description of the mechanical equilibria in the system. The estimates of applied tension and tension in the filament obey an elastic version of the Young-Laplace-Dupré relation. However, curvatures close to the contact line are not captured by the geometric theory, possibly because of 3D effects at the contact line. We also find that when a highly-bendable filament completely wraps the droplet, there is continuity of curvature at the droplet-filament interface, leading to seamless wrapping as observed in a 3D droplet.

Geometry underlies the mechanical stiffening and softening of an indented floating film

A basic paradigm underlying the Hookean mechanics of amorphous, isotropic solids is that small deformations are proportional to the magnitude of external forces. However, slender bodies may undergo large deformations even under minute forces, leading to nonlinear responses rooted in purely geometric effects. Here we study the indentation of a polymer film on a liquid bath. Our experiments and simulations support a recently-predicted stiffening response [D. Vella and B. Davidovitch, Phys. Rev. E, 2018, 98, 013003], and we show that the system softens at large slopes, in agreement with our theory that addresses small and large deflections. We show how stiffening and softening emanate from nontrivial yet generic features of the stress and displacement fields.

Mesoscale structure of wrinkle patterns and defect-proliferated liquid crystalline phases

Thin solids often develop elastic instabilities and subsequently complex, multiscale deformation patterns. Revealing the organizing principles of this spatial complexity has ramifications for our understanding of morphogenetic processes in plant leaves and animal epithelia and perhaps even the formation of human fingerprints. We elucidate a primary source of this morphological complexity-an incompatibility between an elastically favored "microstructure" of uniformly spaced wrinkles and a "macrostructure" imparted through the wrinkle director and dictated by confinement forces. Our theory is borne out of experiments and simulations of floating sheets subjected to radial stretching. By analyzing patterns of grossly radial wrinkles we find two sharply distinct morphologies: defect-free patterns with a fixed number of wrinkles and nonuniform spacing and patterns of uniformly spaced wrinkles separated by defect-rich buffer zones. We show how these morphological types reflect distinct minima of a Ginzburg-Landau functional-a coarse-grained version of the elastic energy, which penalizes nonuniform wrinkle spacing and amplitude, as well as deviations of the actual director from the axis imposed by confinement. Our results extend the effective description of wrinkle patterns as liquid crystals [H. Aharoni et al, Nat. Commun. 8, 15809 (2017)], and we highlight a fascinating analogy between the geometry-energy interplay that underlies the proliferation of defects in the mechanical equilibrium of confined sheets and in thermodynamic phases of superconductors and chiral liquid crystals.

Quick liquid packaging: Encasing water silhouettes by three-dimensional polymer membranes

One of the most important substances on Earth is water. It is an essential medium for living microorganisms and for many technological and industrial processes. Confining water in an enclosed compartment without manipulating it or by using rigid containers can be very attractive, even more if the container is biocompatible and biodegradable. Here, we propose a water-based bottom-up approach for facile encasing of short-lived water silhouettes by a custom-made adaptive suit. A biocompatible polymer self-assembling with unprecedented degree of freedom over the water surface directly produces a thin membrane. The polymer film could be the external container of a liquid core or a free-standing layer with personalized design. The membranes produced have been characterized in terms of physical properties, morphology and proposed for various applications from nano- to macroscale. The process appears not to harm cells and microorganisms, opening the way to a breakthrough approach for organ-on-chip and lab-in-a-drop experiments.

Droplets Capped with an Elastic Film Can Be Round, Elliptical, or Nearly Square

We present experiments that show that the partial wetting of droplets capped by taut elastic films is highly tunable. Adjusting the tension allows the contact angle and droplet morphology to be controlled. By exploiting these elastic boundaries, droplets can be made elliptical, with an adjustable aspect ratio, and can even be transformed into a nearly square shape. This system can be used to create tunable liquid lenses and, moreover, presents a unique approach to liquid patterning.

Cracks in Tension-Field Elastic Sheets

We consider the deformation of a thin elastic sheet which is stiff in traction but very soft in compression, as happens in the presence of wrinkling. We use the tension-field material model and explore numerically the response of a thin sheet containing multiple cracks of different geometries, when subjected to applied tension. With a single crack, the stress concentrates along a St. Andrew's cross-shaped pattern, whose branches extend from the crack tips to the corners of the domain; at a (small) distance r from the crack tip, the stress displays the usual r^{-1/2} stress singularity but with an unusual and nonuniversal angular dependence. A strong interaction between multiple cracks is reported and discussed: in particular, for certain configurations of the cracks, the tensile stiffness of a cracked sheet can be zero even though the sheet is made up of a single component.

Microscopic details of a fluid/thin film triple line

In recent years, there has been a considerable interest in the mechanics of soft objects meeting fluid interfaces (elasto-capillary interactions). In this work we experimentally examine the case of a fluid resting on a thin film of rigid material which, in turn, is resting on a fluid substrate. To simplify complexity, we adapt the experiment to a one-dimensional contact geometry and examine the behaviour of polystyrene and polycarbonate films directly with confocal microscopy. We find that the fluid meets the film in a manner consistent with the Young-Dupré equation when the film is thick, but transitions to what appears similar to a Neumann-like balance when the thickness is decreased. However, on closer investigation we find that the true contact angle is always given by the Young construction. The apparent paradox is a result of macroscopically measured angles not being directly related to true microscopic contact angles when curvature is present. We model the effect with an Euler-Bernoulli beam on a Winkler foundation as well as with an equivalent energy-based capillary model. Notably, the models highlight several important lengthscales and the complex interplay of tension, gravity, and bending in the problem.

Arcuate wrinkling on stiff film/compliant substrate induced by thrust force with a controllable micro-probe

Wrinkling patterns are widely observed in nature and can be used in many high-tech applications such as microfluidic channel, self-assembly ordered microstructures and improved adhesives. In order to use the wrinkling patterns for these applications, it is necessary to precisely control the formation and geometry of the wrinkles. In this paper, we investigate the localized wrinkling of a stiff film/compliant substrate system subjected to a thrust force with a controllable micro-probe. A thin Au film is deposited onto a thick PDMS substrate attached to a glass to form the stiff film/compliant substrate system. And a micro-probe is controlled by a piezoelectric microrobotic system to exert a point force onto the stiff film/compliant substrate to demonstrate the evolution of the localized wrinkles. The experiments show that the film will wrinkle into orthoradial morphology spontaneously when it is deformed in the vertical direction, and then it will wrinkle into arcuate morphology with deformation in the horizontal direction. Since the compressive stress and tensile stress of the film are generated simultaneously, the evolution of the arcuate wrinkles is always accompanied by some radial cracks. The morphological characteristic, formation mechanism and dynamic evolution of the arcuate wrinkles are demonstrated and discussed in detail.

Partial wetting of thin solid sheets under tension

We consider the equilibrium of liquid droplets sitting on thin elastic sheets that are subject to a boundary tension and/or are clamped at their edge. We use scaling arguments, together with a detailed analysis based on the Föppl-von-Kármán equations, to show that the presence of the droplet may significantly alter the stress locally if the tension in the dry sheet is weak compared to an intrinsic elasto-capillary tension scale γ2/3(Et)1/3 (with γ the droplet surface tension, t the sheet thickness and E its Young modulus). Our detailed analysis suggests that some recent experiments may lie in just such a "non-perturbative" regime. As a result, measurements of the tension in the sheet at the contact line (inferred from the contact angles of the sheet with the liquid-vapour interface) do not necessarily reflect the true tension within the sheet prior to wetting. We discuss various characteristics of this non-perturbative regime.

Evolution of local wrinkles near defects on stiff film/compliant substrate

Disordered wrinkles are widely observed in stiff film deposited onto a thermally expanded polymer when compressive stress exceeds the critical wrinkling stress of the film. Highly ordered wrinkles can be fabricated by introducing regularly arranged patterns on the polymer before deposition. However, the study on the morphological evolution of localized wrinkling patterns near defects on the stiff film/compliant substrate is neglected. In this paper, we show two morphological transitions of the local wrinkles induced by defects on an Au film/PDMS substrate. The observation shows that the straight wrinkles form perpendicularly to the line defects and the radial wrinkles form near spot-like defects. We observe that the extended radial wrinkles tend to split and evolve into branching patterns, this limits the deviation of the local wrinkle wavelength from the equilibrium wrinkle wavelength and causes the wrinkle wavelength to be always maintained in a narrow interval. Because the herringbone patterns have the minimum energy state, the straight and radial wrinkles evolve into herringbone wrinkles spontaneously. The morphological characteristic and evolution mechanism of the local wrinkles are described in detail. The observation may provide some clues to the formation and evolution of some localized wrinkling patterns in nature and multilayer materials.

Surface Stress and Surface Tension in Polymeric Networks

Understanding of how surface properties could change upon deformation is of paramount importance for controlling adhesion, friction, and lubrication of soft polymeric materials (i.e., networks and gels). Here, we use a combination of the theoretical calculations and coarse-grained molecular dynamics simulations to study surface stress dependence on deformation in films made of soft and rigid polymeric networks. Simulations have shown that films of polymeric networks could demonstrate surface properties of both polymer melts and elastic solids depending on their deformation. In particular, at small film deformations the film surface stress ϒ is equal to the surface tension obtained at zero film strains, γ₀, and surface properties of networks are similar to those of polymer melts. The surface stress begins to show a strain dependence when the film deformation ratio λ approaches its maximum possible value λ_max corresponding to fully stretched network strands without bond deformations. In the entire film deformation range the normalized surface stress ϒ(λ)/γ₀ is a universal function of the ratio λ/λ_max. Analysis of the simulation data at large film deformations points out that the significant increase in the surface stress can be ascribed to the onset of the bond deformation. In this deformation regime network films behave as elastic solids.

Indentation of a floating elastic sheet: geometry versus applied tension

The localized loading of an elastic sheet floating on a liquid bath occurs at scales from a frog sitting on a lily pad to a volcano supported by the Earth's tectonic plates. The load is supported by a combination of the stresses within the sheet (which may include applied tensions from, for example, surface tension) and the hydrostatic pressure in the liquid. At the same time, the sheet deforms, and may wrinkle, because of the load. We study this problem in terms of the (relatively weak) applied tension and the indentation depth. For small indentation depths, we find that the force-indentation curve is linear with a stiffness that we characterize in terms of the applied tension and bending stiffness of the sheet. At larger indentations, the force-indentation curve becomes nonlinear and the sheet is subject to a wrinkling instability. We study this wrinkling instability close to the buckling threshold and calculate both the number of wrinkles at onset and the indentation depth at onset, comparing our theoretical results with experiments. Finally, we contrast our results with those previously reported for very thin, highly bendable membranes.

Liquid droplets on a free-standing glassy membrane: Deformation through the glass transition

In this study, micro-droplets are placed on thin, glassy, free-standing films where the Laplace pressure of the droplet deforms the free-standing film, creating a bulge. The film's tension is modulated by changing temperature continuously from well below the glass transition into the melt state of the film. The contact angle of the liquid droplet with the planar film as well as the angle of the bulge with the film are measured and found to be consistent with the contact angles predicted by a force balance at the contact line.

Regimes of wrinkling in pressurized elastic shells

We consider the point indentation of a pressurized elastic shell. It has previously been shown that such a shell is subject to a wrinkling instability as the indentation depth is quasi-statically increased. Here we present detailed analysis of this wrinkling instability using a combination of analytical techniques and finite-element simulations. In particular, we study how the number of wrinkles observed at the onset of instability grows with increasing pressurization. We also study how, for fixed pressurization, the number of wrinkles changes both spatially and with increasing indentation depth beyond onset. This 'Far from threshold' analysis exploits the largeness of the wrinkle wavenumber that is observed at high pressurization and leads to quantitative differences with the standard 'Near threshold' stability analysis.This article is part of the themed issue 'Patterning through instabilities in complex media: theory and applications.'

Liquid Droplets Act as "Compass Needles" for the Stresses in a Deformable Membrane

We examine the shape of droplets atop deformable thin elastomeric films prepared with an anisotropic tension. As the droplets generate a deformation in the taut film through capillary forces, they assume a shape that is elongated along the high tension direction. By measuring the contact line profile, the tension in the membrane can be completely determined. Minimal theoretical arguments lead to predictions for the droplet shape and membrane deformation that are in excellent agreement with the data. On the whole, the results demonstrate that droplets can be used as probes to map out the stress field in a membrane.

Indentation metrology of clamped, ultra-thin elastic sheets

We study the indentation of ultrathin elastic sheets clamped to the edge of a circular hole. This classical setup has received considerable attention lately, being used by various experimental groups as a probe to measure the surface properties and stretching modulus of thin solid films. Despite the apparent simplicity of this method, the geometric nonlinearity inherent in the mechanical response of thin solid objects renders the analysis of the resulting data a nontrivial task. Importantly, the essence of this difficulty is in the geometric coupling between in-plane stress and out-of-plane deformations, and hence is present in the behaviour of Hookean solids even when the slope of the deformed membrane remains small. Here we take a systematic approach to address this problem, using the membrane limit of the Föppl-von-Kármán equations. This approach highlights some of the dangers in the use of approximate formulae in the metrology of solid films, which can introduce large errors; we suggest how such errors may be avoided in performing experiments and analyzing the resulting data.

Geometry-Driven Folding of a Floating Annular Sheet

Predicting the large-amplitude deformations of thin elastic sheets is difficult due to the complications of self contact, geometric nonlinearities, and a multitude of low-lying energy states. We study a simple two-dimensional setting where an annular polymer sheet floating on an air-water interface is subjected to different tensions on the inner and outer rims. The sheet folds and wrinkles into many distinct morphologies that break axisymmetry. These states can be understood within a recent geometric approach for determining the gross shape of extremely bendable yet inextensible sheets by extremizing an appropriate area functional. Our analysis explains the remarkable feature that the observed buckling transitions between wrinkled and folded shapes are insensitive to the bending rigidity of the sheet.

Interaction of Droplets Separated by an Elastic Film

The Laplace pressure of a droplet placed on one side of an elastic thin film can cause significant deformation in the form of a bulge on its opposite side. Here, we show that this deformation can be detected by other droplets suspended on the opposite side of the film, leading to interaction between droplets separated by the solid (but deformable) film. The interaction is repulsive when the drops have a large overlap and attractive when they have a small overlap. Thus, if two identical droplets are placed right on top of each other (one on either side of the thin film), they tend to repel each other, eventually reaching an equilibrium configuration where there is a small overlap. This observation can be explained by analyzing the energy landscape of the droplets interacting via an elastically deformed film. We further demonstrate this idea by designing a pattern comprising a big central drop with satellite droplets. This phenomenon can lead to techniques for directed motion of droplets confined to one side of a thin elastic membrane by manipulations on the other side.

Pattern transitions in a compressible floating elastic sheet

Compressible thin layers floating on a liquid surface develop wrinkled and folded patterns under lateral pressure.

Increased Hierarchical Wrinklons on Stiff Metal Thin Film on a Liquid Meniscus

Wrinklons-the hierarchical merging of wrinkles-are observed on several surfaces including thin films, curtains, graphene sheets, and skin. Wrinklons are a consequence of the interplay between bending, stretching, and gravitational energies and generally exhibit 1 to 2 hierarchical transitions (λn+1 = 2λn). Here we show that parallel and self-similar wrinklons on ultrathin cobalt/chromium film atop a contracting silicone oil meniscus can produce up to 5 hierarchical wrinklon transitions near the fluid-solid boundary. Further, these wrinklons do not follow the standard von-Kármán wrinklon scaling near the edge, attributed to the added surface energy (L/λ ∝ (A/t)(0.31)). A model developed via scale analysis shows (a) the relationship between wavelength and length of the wrinkles and (b) a linear relation between the amplitude and the length of wrinkles at all observed hierarchic levels (L ∝ A), fitted well with previous literature results. This work provides a mechanism for thin-film metal wrinkling on liquids and shows that surface stretching effects can allow increased hierarchical levels in wrinklons.

Elastic membranes in confinement

An elastic membrane stretched between two walls takes a shape defined by its length and the volume of fluid it encloses. Many biological structures, such as cells, mitochondria and coiled DNA, have fine internal structure in which a membrane (or elastic member) is geometrically 'confined' by another object. Here, the two-dimensional shape of an elastic membrane in a 'confining' box is studied by introducing a repulsive confinement pressure that prevents the membrane from intersecting the wall. The stage is set by contrasting confined and unconfined solutions. Continuation methods are then used to compute response diagrams, from which we identify the particular membrane mechanics that generate mitochondria-like shapes. Large confinement pressures yield complex response diagrams with secondary bifurcations and multiple turning points where modal identities may change. Regions in parameter space where such behaviour occurs are then mapped.

Surface tension measurement from the indentation of clamped thin films

We developed an indentation technique to measure the surface tension of relatively stiff solids. In the proposed method, a suspended thin solid film is indented by a rigid sphere and its deflection is measured by optical interferometry. The film deflection is jointly resisted by surface tension, elasticity and residual stress. Using a version of nonlinear von Karman plate theory that includes surface tension, we are able to separate the contribution of elasticity to the total tension in the film. Surface tension is determined by extrapolating the sum of surface tension and residual stress to zero film thickness. We measured the surface tension of polydimethylsiloxane (PDMS) using this technique and obtained a value of 19.5 ± 3.6 mN m(-1), consistent with the surface energy of PDMS reported in the literature.

Capillary-induced deformations of a thin elastic sheet

We develop a three-dimensional model for capillary origami systems in which a rectangular plate has finite thickness, is allowed to stretch and undergoes small deflections. This latter constraint limits our description of the encapsulation process to its initial folding phase. We first simplify the resulting system of equations to two dimensions by assuming that the plate has infinite aspect ratio, which allows us to compare our approach to known two-dimensional capillary origami models for inextensible plates. Moreover, as this two-dimensional model is exactly solvable, we give an expression for its solution in terms of its parameters. We then turn to the full three-dimensional model in the limit of small drop volume and provide numerical simulations showing how the plate and the drop deform due to the effect of capillary forces.

Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets

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.

Optimal wrapping of liquid droplets with ultrathin sheets

Elastic sheets offer a path to encapsulating a droplet of one fluid in another that is different from that of traditional molecular or particulate surfactants. In wrappings of fluids by sheets of moderate thickness with petals designed to curl into closed shapes, capillarity balances bending forces. Here, we show that, by using much thinner sheets, the constraints of this balance can be lifted to access a regime of high sheet bendability that brings three major advantages: ultrathin sheets automatically achieve optimally efficient shapes that maximize the enclosed volume of liquid for a fixed area of sheet; interfacial energies and mechanical properties of the sheet are irrelevant within this regime, thus allowing for further functionality; and complete coverage of the fluid can be achieved without special sheet designs. We propose and validate a general geometric model that captures the entire range of this new class of wrapped and partially wrapped shapes.

Wrinkles and folds in a fluid-supported sheet of finite size

A laterally confined thin elastic sheet lying on a liquid substrate displays regular undulations, called wrinkles, characterized by a spatially extended energy distribution and a well-defined wavelength λ. As the confinement increases, the deformation energy is progressively localized into a single narrow fold. An exact solution for the deformation of an infinite sheet was previously found, indicating that wrinkles in an infinite sheet are unstable against localization for arbitrarily small confinement. We present an extension of the theory to sheets of finite length L, accounting for the experimentally observed wrinkle-to-fold transition. We derive an exact solution for the periodic deformation in the wrinkled state, and an approximate solution for the localized, folded state. We find that a second-order transition between these two states occurs at a critical confinement Δ(F)=λ(2)/L.

Curvature-induced symmetry breaking determines elastic surface patterns

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.

Elastocapillary deformations on partially-wetting substrates: rival contact-line models

A partially-wetting liquid can deform the underlying elastic substrate upon which it rests. This situation requires the development of theoretical models to describe the wetting forces imparted by the drop onto the solid substrate, particularly those at the contact-line. We construct a general solution using a displacement potential function for the elastic deformations within a finite elastic substrate associated with these wetting forces, and compare the results for several different contact-line models. Our work incorporates internal contributions to the surface stress from both liquid/solid Σls and Σsg solid/gas solid surface tensions (surface stress), which results in a non-standard boundary-value problem that we solve using a dual integral equation. We compare our results to relevant experiments and conclude that the generalization of solid surface tension Σls ≠ Σsg is an essential feature in any model of partial-wetting. The comparisons also allow us to systematically eliminate some proposed contact-line models.