The equilibrium shape of liquid drops on elastic substrates is determined by minimising elastic and capillary free energies, focusing on thick incompressible substrates. The problem is governed by three length scales: the size of the drop R, the molecular size a, and the ratio of surface tension to elastic modulus γ/E. We show that the contact angles undergo two transitions upon changing the substrates from rigid to soft. The microscopic wetting angles deviate from Young's law when γ/Ea 1, while the apparent macroscopic angle only changes in the very soft limit γ/ER 1. The elastic deformations are worked out in the simplifying case where the solid surface energy is assumed constant. The total free energy turns out lower on softer substrates, consistent with recent experiments.
The coalescence of viscous drops on a substrate is studied experimentally and theoretically. We consider cases where the drops can have different contact angles, leading to a very asymmetric coalescence process. Side view experiments reveal that the "bridge" connecting the drops evolves with self-similar dynamics, providing a new perspective on the coalescence of sessile drops. We show that the universal shape of the bridge is accurately described by similarity solutions of the one-dimensional lubrication equation. Our theory predicts a bridge that grows linearly in time and stresses the strong dependence on the contact angles. Without any adjustable parameters, we find quantitative agreement with all experimental observations. PACS numbers: 47 55.D-DropsThe coalescence or breakup of liquid drops is a fundamental process relevant for the formation of raindrops or sprays, inkjet printing, or stability of foams and emulsions [1][2][3]. The initial stages of coalescence of two spherical drops has been characterized in great detail [4][5][6][7][8][9][10]. After contact, a small liquid bridge connects the two drops and the bridge grows rapidly with time. Depending on the viscosity of the liquid, the radius of the bridge grows as r ∼ t (high viscosity) [5][6][7][8], or r ∼ t 1/2 (low viscosity, inertia dominated) [7][8][9], with a crossover depending on fluid properties and drop size [10].In many cases, however, the coalescing drops are not freely suspended but are in contact with a substrate. Much less is known about the coalescence of such sessile drops. When looking from a top view (perpendicular to the substrate), the coalescence of drops on a substrate looks very similar to the case for spherical drops [3]; yet the bridge dynamics is fundamentally different. Measurements for very viscous drops give a growth r ∼ t 1/2 [11,12], and even smaller exponents have been suggested [13]. The challenge lies in the complications introduced by the presence of the substrate. First, the geometry of the drop is no longer a sphere with an axisymmetric bridge, but a spherical cap with a contact angle θ. As a consequence, a top view of the coalescence process is very different from a side view. Second, the wall slows down the liquid transport towards the bridge [11] and gives rise to the motion of a contact line [14]. At present, it is not clear whether or not this contact line motion affects the initial stages of coalescence, and different predictions for the θ dependence have been reported [11][12][13].In this Letter we resolve the coalescence of viscous drops on a substrate by performing side view experiments, imaging parallel to the substrate (Fig. 1). Our central finding is that the initial stages evolve by a selfsimilar shape of the bridge, with a linear growth of the bridge height h 0 ∼ t. The influence of the contact angle is studied in detail by considering drops with identical or different contact angles, resulting into symmetric or asymmetric coalescence [ Fig. 1(bc)]. Theoretically, we show that all experiments can be d...
Solid particles floating at a liquid interface exhibit a long-ranged attraction mediated by surface tension. In the absence of bulk elasticity, this is the dominant lateral interaction of mechanical origin. Here, we show that an analogous long-range interaction occurs between adjacent droplets on solid substrates, which crucially relies on a combination of capillarity and bulk elasticity. We experimentally observe the interaction between droplets on soft gels and provide a theoretical framework that quantitatively predicts the interaction force between the droplets. Remarkably, we find that, although on thick substrates the interaction is purely attractive and leads to dropdrop coalescence, for relatively thin substrates a short-range repulsion occurs, which prevents the two drops from coming into direct contact. This versatile interaction is the liquid-on-solid analog of the "Cheerios effect." The effect will strongly influence the condensation and coarsening of drops on soft polymer films, and has potential implications for colloidal assembly and mechanobiology.elastocapillarity | wetting | soft matter | mechanosensing | droplets T he long-ranged interaction between particles trapped at a fluid interface is exploited for the fabrication of microstructured materials via self-assembly and self-patterning (1-5) and occurs widely in the natural environment when living organisms or fine particles float on the surface of water (6, 7). In a certain class of capillary interactions, the particles deform the interface because of their shape or chemical heterogeneity (8)(9)(10). In this case, the change in interfacial area upon particle-particle approach causes an attractive capillary interaction between the particles. In the so-called Cheerios effect, the interaction between floating objects is mainly due to the change in gravitational potential energy associated to the weight of the particles, which deform the interface while being supported by surface tension (11), and the same principle applies when the interface is elastic (12-14). The name "Cheerios effect" is reminiscent of breakfast cereals floating on milk and sticking to each other or to the walls of the breakfast bowl.Here, we consider a situation opposite to that of the Cheerios effect, liquid drops deposited on a solid. The solid is sufficiently soft to be deformed by the surface tension of the drops, resulting in a lateral interaction. Recent studies have provided a detailed view of statics of single-drop wetting on deformable surfaces (15-19). The length scale over which the substrate is deformed is set by the ratio of the droplet surface tension γ and the substrate shear modulus G. The deformation can be seen as an elastocapillary meniscus, or "wetting ridge," around the drop (Fig. 1 A and B). Interestingly, the contact angles at the edge of the drop are governed by Neumann's law, just as for oil drops floating on water. In contrast to the statics of soft wetting, its dynamics has only been explored recently. New effects such as stick-slip motion induced...
We uncover how nonlinearities dramatically alter the buckling of elastic beams. First, we show experimentally that sufficiently wide ordinary elastic beams and specifically designed metabeams-beams made from a mechanical metamaterial-exhibit discontinuous buckling, an unstable form of buckling where the postbuckling stiffness is negative. Then we use simulations to uncover the crucial role of nonlinearities, and show that beams made from increasingly nonlinear materials exhibit an increasingly negative postbuckling slope. Finally, we demonstrate that for sufficiently strong nonlinearity, we can observe discontinuous buckling for metabeams as slender as 1% numerically and 5% experimentally.
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