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Superhydrophobic materials are often inspired by nature, whereas metamaterials are engineered to have properties not usually found in naturally occurring materials. In both cases, the key that unlocks their unique properties is structure. Here, we show that a negative Poisson's ratio (auxetic) mechanical metamaterial is capable of transforming into a unique type of superhydrophobic material. When stretched its surface has the counterintuitive property that it also expands in the orthogonal lateral direction. We model the change in the solid surface fraction as strain is applied and show it decreases as the space between solid elements of the auxetic lattice expands. This results in a unique dependence of the superhydrophobicity on strain. We construct experimental models illustrating the relationship between different states of strain and superhydrophobicity as the lattice structure transitions from an auxetic to a conventional (positive Poisson's ratio) one. The principles we have discovered offer a new approach to designing superhydrophobic materials for self-cleaning surfaces, droplet transportation, droplet encapsulation and oil-water separation., Comment: 10 pages, 4 figures, 13 page supplementary information

We theoretically investigate the apparent contact angle of droplets on liquid infused surfaces as a function of the relative size of the wetting ridge and the deposited droplet. We provide an intuitive geometrical interpretation whereby the variation in the apparent contact angle is due to the rotation of the Neumann triangle. We also derive linear and quadratic corrections to the apparent contact angle as power series expansion in terms of pressure differences between the lubricant, droplet and gas phases. These expressions are much simpler and more compact compared to those previously derived by Semprebon et al. [Soft Matter, 2017, 13, 101-110].

We demonstrate spontaneous bidirectional motion of droplets on liquid infused surfaces in the presence of a topographical gradient, in which the droplets can move either toward the denser or the sparser solid fraction area. Our analytical theory explains the origin of this bidirectional motion. Furthermore, using both lattice Boltzmann simulations and experiments, we show that the key factor determining the direction of motion is the wettability difference of the droplet on the solid surface and on the lubricant film. The bidirectional motion is shown for various combinations of droplets and lubricants, as well as for different forms of topographical gradients.

The transport of small quantities of liquid on a solid surface is inhibited by the resistance to motion caused by the contact between the liquid and the solid. To overcome such resistance, motion can be externally driven through gradients in electric fields, but these all inconveniently involve the input of external energy. Alternatively, gradients in physical shape and wettability - the conical shape of cactus spines to create self-propelled motion. However, such self-propelled motion to date has limited success in overcoming the inherent resistance to motion of the liquid contact with the solid. Here we propose a simple solution in the form of shaped-liquid surface, where solid topographic structures at one length scale provides the base for a smaller length-scale liquid conformal layer. This dual-length scale render possible slippery surfaces with superhydrophobic properties. Combined to an heterogeneous topography, it provides a gradient in liquid-on-liquid wettability with minimal resistance to motion and long range directional self-propelled droplet transport. Moreover, the liquid-liquid contact enables impacting droplets to be captured and transported, even when the substrate is inverted. These design principles are highly beneficial for droplet transport in microfluidics, self-cleaning surfaces, fog harvesting and in heat transfer., Comment: 11 pages, 4 figures

We demonstrate the continuous translational invariance of the energy of a capillary surface in contact with reconfigurable solid boundaries. We present a theoretical approach to find the energy-invariant equilibria of spherical capillary surfaces in contact with solid boundaries of arbitrary shape and examine the implications of dynamic frictional forces upon a reconfiguration of the boundaries. Experimentally, we realise our ideas by manipulating the position of a droplet in a wedge geometry using lubricant-impregnated solid surfaces, which eliminate the contact-angle hysteresis and provide a test bed for quantifying dissipative losses out of equilibrium. Our experiments show that dissipative energy losses for an otherwise energy-invariant reconfiguration are relatively small, provided that the actuation timescale is longer than the typical relaxation timescale of the capillary surface. We discuss the wider applicability of our ideas as a pathway for liquid manipulation at no potential energy cost in low-pinning, low-friction situations., Comment: 12 pages, 4 figures. To appear in Physical Review Letters

We present a theoretical study of the statics and dynamics of a partially wetting liquid droplet, of equilibrium contact angle $\theta_{\rm e}$, confined in a solid wedge geometry of opening angle $\beta$. We focus on a mostly non-wetting regime, given by the condition $\theta_{\rm e} - \beta > 90^\circ$, where the droplet forms a liquid barrel -- a closed shape of positive mean curvature. Using a quasi-equilibrium assumption for the shape of the liquid-gas interface, we compute the surface energy landscapes experienced by the liquid upon translations along the symmetry plane of the wedge. Close to equilibrium, our model is in good agreement with numerical calculations of the surface energy minimisation subject to a constrained position of the centre of mass of the liquid. Beyond the statics, we put forward a Lagrangian description for the droplet dynamics. We focus on the the over-damped limit, where the driving capillary force is balanced by the frictional forces arising from the bulk hydrodynamics, the corner flow near the contact lines and the contact line friction. Our results provide a theoretical framework to describe the motion of partially wetting liquids in confinement, and can be used to gain further understanding on the relative importance of dissipative processes that span from microscopic to macroscopic length scales., Comment: 33 pages, 6 figures

We theoretically investigate the apparent contact angle and contact angle hysteresis of a droplet placed on a liquid infused surface. We show that the apparent contact angle is not uniquely defined by material parameters, but also has a strong dependence on the relative size between the droplet and its surrounding wetting ridge formed by the infusing liquid. We derive a closed form expression for the contact angle in the limit of vanishing wetting ridge, and compute the correction for small but finite ridge, which corresponds to an effective line tension term. We also predict contact angle hysteresis on liquid infused surfaces generated by the pinning of the contact lines by the surface corrugations. Our analytical expressions for both the apparent contact angle and contact angle hysteresis can be interpreted as `weighted sums' between the contact angles of the infusing liquid relative to the droplet and surrounding gas phases, where the weighting coefficients are given by ratios of the fluid surface tensions.