Dehydration-induced corrugated folding in Rhapis excelsa plant leaves.

Plant leaves, whose remarkable ability for morphogenesis results in a wide range of petal and leaf shapes in response to environmental cues, have inspired scientific studies as well as the development of engineering structures and devices. Although some typical shape changes in plants and the driving force for such shape evolution have been extensively studied, there remain many poorly understood mechanisms, characteristics, and principles associated with the vast array of shape formation of plant leaves in nature. Here, we present a comprehensive study that combines experiment, theory, and numerical simulations of one such topic-the mechanics and mechanisms of corrugated leaf folding induced by differential shrinking in Rhapis excelsa. Through systematic measurements of the dehydration process in sectioned leaves, we identify a linear correlation between change in the leaf-folding angle and water loss. Building on experimental findings, we develop a generalized model that provides a scaling relationship for water loss in sectioned leaves. Furthermore, our study reveals that corrugated folding induced by dehydration in R. excelsa leaves is achieved by the deformation of a structural architecture-the "hinge" cells. Utilizing such connections among structure, morphology, environmental stimuli, and mechanics, we fabricate several biomimetic machines, including a humidity sensor and morphing devices capable of folding in response to dehydration. The mechanisms of corrugated folding in R. excelsa identified in this work provide a general understanding of the interactions between plant leaves and water. The actuation mechanisms identified in this study also provide insights into the rational design of soft machines.

Delamination from an adhesive sphere: Curvature-induced dewetting versus buckling.

Everyday experience confirms the tendency of adhesive films to detach from spheroidal regions of rigid substrates-what is a petty frustration when placing a sticky band aid onto a knee is a more serious matter in the coating and painting industries. Irrespective of their resistance to bending, a key driver of such phenomena is Gauss' Theorema Egregium, which implies that naturally flat sheets cannot conform to doubly curved surfaces without developing a strain whose magnitude grows sharply with the curved area. Previous attempts to characterize the onset of curvature-induced delamination, and the complex patterns it gives rise to, assumed a dewetting-like mechanism in which the propensity of two materials to form contact through interfacial energy is modified by an elastic energy penalty. We show that this approach may characterize moderately bendable sheets but fails qualitatively to describe the curvature-induced delamination of ultrathin films, whose mechanics is governed by their propensity to buckle and delaminate partially, under minute levels of compression. Combining mechanical and geometrical considerations, we introduce a minimal model for curvature-induced delamination accounting for the two buckling motifs that underlie partial delamination: shallow "rucks" and localized "folds". We predict nontrivial scaling rules for the onset of curvature-induced delamination and various features of the emerging patterns, which compare well with experiments. Beyond gaining control on the use of ultrathin adhesives in cutting-edge technologies such as stretchable electronics, our analysis is a significant step toward quantifying the multiscale morphology that emerges upon imposing geometrical and mechanical constraints on highly bendable solid objects.

Insect-scale jumping robots enabled by a dynamic buckling cascade.

Millions of years of evolution have allowed animals to develop unusual locomotion capabilities. A striking example is the legless-jumping of click beetles and trap-jaw ants, which jump more than 10 times their body length. Their delicate musculoskeletal system amplifies their muscles' power. It is challenging to engineer insect-scale jumpers that use onboard actuators for both elastic energy storage and power amplification. Typical jumpers require a combination of at least two actuator mechanisms for elastic energy storage and jump triggering, leading to complex designs having many parts. Here, we report the new concept of dynamic buckling cascading, in which a single unidirectional actuation stroke drives an elastic beam through a sequence of energy-storing buckling modes automatically followed by spontaneous impulsive snapping at a critical triggering threshold. Integrating this cascade in a robot enables jumping with unidirectional muscles and power amplification (JUMPA). These JUMPA systems use a single lightweight mechanism for energy storage and release with a mass of 1.6 g and 2 cm length and jump up to 0.9 m, 40 times their body length. They jump repeatedly by reengaging the latch and using coiled artificial muscles to restore elastic energy. The robots reach their performance limits guided by theoretical analysis of snap-through and momentum exchange during ground collision. These jumpers reach the energy densities typical of the best macroscale jumping robots, while also matching the rapid escape times of jumping insects, thus demonstrating the path toward future applications including proximity sensing, inspection, and search and rescue.

Flattened and Wrinkled Encapsulated Droplets: Shape Morphing Induced by Gravity and Evaporation.

We report surprising morphological changes of suspension droplets (containing class II hydrophobin protein HFBI from Trichoderma reesei in water) as they evaporate with a contact line pinned on a rigid solid substrate. Both pendant and sessile droplets display the formation of an encapsulating elastic film as the bulk concentration of solute reaches a critical value during evaporation, but the morphology of the droplet varies significantly: for sessile droplets, the elastic film ultimately crumples in a nearly flattened area close to the apex while in pendant droplets, circumferential wrinkling occurs close to the contact line. These different morphologies are understood through a gravito-elastocapillary model that predicts the droplet morphology and the onset of shape changes, as well as showing that the influence of the direction of gravity remains crucial even for very small droplets (where the effect of gravity can normally be neglected). The results pave the way to control droplet shape in several engineering and biomedical applications.

Wrinkling composite sheets.

We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation. The buckling modes are controlled by the bending stiffness of the sheet, the density of the substrate, and the size and the spatially dependent elastic coefficients of the sheet. We solve the beam equation describing the mechanical equilibrium of a sheet when its bending stiffness varies parallel to the direction of confinement. The case of a homogeneous bending stiffness exhibits a degeneracy of wrinkled states for certain lengths of the confined sheet; we explain this degeneracy using an asymptotic analysis valid for long sheets, and show that it corresponds to the switching of the sheet between symmetric and antisymmetric buckling modes. This degeneracy disappears for spatially dependent elastic coefficients. Medium length sheets buckle similarly to their homogeneous counterparts, whereas the wrinkled states in large length sheets concentrate the bending energy towards the soft regions of the sheet.

Effect of Macroscopic Surface Heterogeneities on an Advancing Contact Line.

The shape of a liquid-air interface advancing on a heterogeneous surface was studied experimentally, together with the force induced by the pinning of the contact line to surface defects. Different surfaces were considered with circular defects introduced as arrays of cocoa butter patches or small circular holes. These heterogeneous surfaces were submerged in aqueous ethanol solutions while measuring the additional force arising from the deformation of the advancing contact line and characterizing the interface shape and its pinning on the defects. Initially, the submersion force is linear with submerged depth, suggesting a constant defect-induced stiffness. This regime ends when the contact line depins from the defects. A simple scaling is proposed to describe the depinning force and the depinning energy. As the defect separation increases, the interface stiffness is found to increase too, with a weak dependency on the defect radius. This interaction between defects cannot be captured by simple scaling but can be well predicted by a theory considering the interface deformation in the presence of a periodic arrays of holes. Creating a four-phase contact line by including solid defects (cocoa butter) reduced pinning forces. The radius of the defect had a nonlinear effect on the depinning depth. The four-phase contact line resulted in depinning before the defects were fully submerged. These experimental results and the associated theory help to understand quantitatively the extent to which surface heterogeneities can slow down wetting. This in turn paves the way to tailoring the design of heterogeneous surfaces toward desired wetting performances.

Compression of a pressurized spherical shell by a spherical or flat probe.

Measuring the mechanical properties of cells and tissues often involves indentation with a sphere or compression between two plates. Different theoretical approaches have been developed to retrieve material parameters (e.g., elastic modulus) or state variables (e.g., pressure) from such experiments. Here, we extend previous theoretical work on indentation of a spherical pressurized shell by a point force to cover indentation by a spherical probe or a plate. We provide formulae that enable the modulus or pressure to be deduced from experimental results with realistic contact geometries, giving different results that are applicable depending on pressure level. We expect our results to be broadly useful when investigating biomechanics or mechanobiology of cells and tissues.

Dynamics of wrinkling in ultrathin elastic sheets.

The wrinkling of thin elastic objects provides a means of generating regular patterning at small scales in applications ranging from photovoltaics to microfluidic devices. Static wrinkle patterns are known to be governed by an energetic balance between the object's bending stiffness and an effective substrate stiffness, which may originate from a true substrate stiffness or from tension and curvature along the wrinkles. Here, we investigate dynamic wrinkling induced by the impact of a solid sphere onto an ultrathin polymer sheet floating on water. The vertical deflection of the sheet's center induced by impact draws material radially inward, resulting in an azimuthal compression that is relieved by the wrinkling of the entire sheet. We show that this wrinkling is truly dynamic, exhibiting features that are qualitatively different to those seen in quasistatic wrinkling experiments. Moreover, we show that the wrinkles coarsen dynamically because of the inhibiting effect of the fluid inertia. This dynamic coarsening can be understood heuristically as the result of a dynamic stiffness, which dominates the static stiffnesses reported thus far, and allows control of wrinkle wavelength.

Dynamic Buckling of an Elastic Ring in a Soap Film.

Dynamic buckling may occur when a load is rapidly applied to, or removed from, an elastic object at rest. In contrast to its static counterpart, dynamic buckling offers a wide range of accessible patterns depending on the parameters of the system and the dynamics of the load. To study these effects, we consider experimentally the dynamics of an elastic ring in a soap film when part of the film is suddenly removed. The resulting change in tension applied to the ring creates a range of interesting patterns that cannot be easily accessed in static experiments. Depending on the aspect ratio of the ring's cross section, high-mode buckling patterns are found in the plane of the remaining soap film or out of the plane. Paradoxically, while inertia is required to observe these nontrivial modes, the selected pattern does not depend on inertia itself. The evolution of this pattern beyond the initial instability is studied experimentally and explained through theoretical arguments linking dynamics to pattern selection and mode growth. We also explore the influence of dynamic loading and show numerically that, by imposing a rate of loading that competes with the growth rate of instability, the observed pattern can be selected and controlled.

Tapered elasticæ as a route for axisymmetric morphing structures.

Transforming flat two-dimensional (2D) sheets into three-dimensional (3D) structures by combining carefully made cuts with applied edge-loads has emerged as an exciting manufacturing paradigm in a range of applications from mechanical metamaterials to flexible electronics. In Kirigami, patterns of cuts are introduced that allow solid faces to rotate about each other, deforming in three dimensions whilst remaining planar. In other scenarios, however, the solid elements bend in one direction. In this paper, we model such bending deformations using the formulation of an elastic strip whose thickness and width are tapered (the 'tapered elastica'). We show how this framework can be exploited to design the tapering patterns required to create planar sheets that morph into desired axisymmetric 3D shapes under a combination of horizontal and vertical edge-loads. We exhibit this technique by recreating miniature structures with positive, negative, and variable apparent Gaussian curvatures. With sheets of constant thickness, the resulting morphed shapes may leave gaps between the deformed elements. However, by tapering the thickness of the sheet too, these gaps can be closed, creating tessellated three-dimensional structures. Our theoretical approaches are verified by both numerical simulations and physical experiments.

Cloaking by coating: how effectively does a thin, stiff coating hide a soft substrate?

From human tissue to fruits, many soft materials are coated by a thin layer of a stiffer material. While the primary role of such a coating is often to protect the softer material, the thin, stiff coating also has an important effect on the mechanical behaviour of the composite material, making it appear significantly stiffer than the underlying material. We study this cloaking effect of a coating for the particular case of indentation tests, which measure the 'firmness' of the composite solid: we use a combination of theory and experiment to characterize the firmness quantitatively. We find that the indenter size plays a key role in determining the effectiveness of cloaking: small indenters feel a mixture of the material properties of the coating and of the substrate, while large indenters sense largely the unadulterated substrate.

Fluctuation spectra and force generation in nonequilibrium systems.

Many biological systems are appropriately viewed as passive inclusions immersed in an active bath: from proteins on active membranes to microscopic swimmers confined by boundaries. The nonequilibrium forces exerted by the active bath on the inclusions or boundaries often regulate function, and such forces may also be exploited in artificial active materials. Nonetheless, the general phenomenology of these active forces remains elusive. We show that the fluctuation spectrum of the active medium, the partitioning of energy as a function of wavenumber, controls the phenomenology of force generation. We find that, for a narrow, unimodal spectrum, the force exerted by a nonequilibrium system on two embedded walls depends on the width and the position of the peak in the fluctuation spectrum, and oscillates between repulsion and attraction as a function of wall separation. We examine two apparently disparate examples: the Maritime Casimir effect and recent simulations of active Brownian particles. A key implication of our work is that important nonequilibrium interactions are encoded within the fluctuation spectrum. In this sense, the noise becomes the signal.

Wettability-Independent Droplet Transport by Bendotaxis.

We demonstrate "bendotaxis," a novel mechanism for droplet self-transport at small scales. A combination of bending and capillarity in a thin channel causes a pressure gradient that, in turn, results in the spontaneous movement of a liquid droplet. Surprisingly, the direction of this motion is always the same, regardless of the wettability of the channel. We use a combination of experiments at a macroscopic scale and a simple mathematical model to study this motion, focusing in particular on the timescale associated with the motion. We suggest that bendotaxis may be a useful means of transporting droplets in technological applications, e.g., in developing self-cleaning surfaces, and discuss the implications of our results for such applications.

Dynamics of droplets on cones: self-propulsion due to curvature gradients.

We study the dynamics of droplets driven by a gradient of curvature, as may be achieved by placing a drop on the surface of a cone. The curvature gradient induces a pressure gradient within the drop, which in turn leads to spontaneous propulsion of the droplet. To investigate the resulting driving force we perform a series of experiments in which we track a droplet's displacement, s, from the apex of a cone whose surface is treated to exhibit near-zero pinning effects. We find an s ∼ t1/4 scaling at sufficiently late times t. To shed light upon these dynamics, we perform an asymptotic calculation of the equilibrium shape of a droplet on a weakly curved cylinder, deriving the curvature-induced force responsible for its propulsion. By balancing this driving force with viscous dissipation, we recover a differential equation for the droplet displacement, whose predictions are found to be in good agreement with our experimental results.

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.

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.

Passive Control of Viscous Flow via Elastic Snap-Through.

We demonstrate the passive control of viscous flow in a channel by using an elastic arch embedded in the flow. Depending on the fluid flux, the arch may "snap" between two states-constricting and unconstricting-that differ in hydraulic conductivity by up to an order of magnitude. We use a combination of experiments at a macroscopic scale and theory to study the constricting and unconstricting states, and determine the critical flux required to transition between them. We show that such a device may be precisely tuned for use in a range of applications, and, in particular, has potential as a passive microfluidic fuse to prevent excessive fluxes in rigid-walled channels.

Floating and Sinking of a Pair of Spheres at a Liquid-Fluid Interface.

Spheres floating at liquid-fluid interfaces cause interfacial deformations such that their weight is balanced by the resultant forces of surface tension and hydrostatic pressure while also satisfying a contact angle condition. Determining the meniscus shape around several floating spheres is a complicated problem because the vertical locations of the spheres and the horizontal projections of the three-phase contact lines are not known a priori. Here, a new computational algorithm is developed to simultaneously satisfy the nonlinear Laplace-Young equation for the meniscus shape, the vertical force balance, and the geometric properties of the spheres. We implement this algorithm to find the shape of the interface around a pair of floating spheres and the horizontal force required to keep them at a fixed center-center separation. Our numerical simulations show that the ability of a pair of spheres to float (rather than sink) depends on their separation. Similar to previous work on horizontal cylinders, sinking may be induced at close range for small spheres that float when isolated. However, we also discover a new and unexpected behavior: at intermediate inter-particle distances, spheres that would sink in isolation can float as a pair. This effect is more pronounced for spheres of radius comparable to the capillary length, suggesting that this effect is a result of hydrostatic pressure, rather than surface tension. An approximate solution confirms these phenomena and shows that the mechanism is indeed the increased supporting force provided by the hydrostatic pressure. Finally, the horizontal force of capillary attraction between the spheres is calculated using the results of the numerical simulations. These results show that the classic linear superposition approximation (due to Nicolson) can lose its validity for relatively heavy particles at close range.

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.

Using evaporation to control capillary instabilities in micro-systems.

The instabilities of fluid interfaces represent both a limitation and an opportunity for the fabrication of small-scale devices. Just as non-uniform capillary pressures can destroy micro-electrical mechanical systems (MEMS), so they can guide the assembly of novel solid and fluid structures. In many such applications the interface appears during an evaporation process and is therefore only present temporarily. It is commonly assumed that this evaporation simply guides the interface through a sequence of equilibrium configurations, and that the rate of evaporation only sets the timescale of this sequence. Here, we use Lattice-Boltzmann simulations and a theoretical analysis to show that, in fact, the rate of evaporation can be a factor in determining the onset and form of dynamical capillary instabilities. Our results shed light on the role of evaporation in previous experiments, and open the possibility of exploiting diffusive mass transfer to directly control capillary flows in MEMS applications.