Capillary origami: superhydrophobic ribbon surfaces and liquid marbles

In the wetting of a solid by a liquid it is often assumed that the substrate is rigid. However, for an elastic substrate the rigidity depends on the cube of its thickness and so reduces rapidly as the substrate becomes thinner as it approaches becoming a thin sheet. In such circumstances, it has been shown that the capillary forces caused by a contacting droplet of a liquid can shape the solid rather than the solid shaping the liquid. A substrate can be bent and folded as a (pinned) droplet evaporates or even instantaneously and spontaneously wrapped on contact with a droplet. When this effect is used to create three dimensional shapes from initially flat sheets, the effect is called capillary origami or droplet wrapping. In this work, we consider how the conditions for the spontaneous, capillary induced, folding of a thin ribbon substrate might be altered by a rigid surface structure that, for a rigid substrate, would be expected to create Cassie-Baxter and Wenzel effects. For smooth thin substrates, droplet wrapping can occur for all liquids, including those for which the Young's law contact angle (defined by the interfacial tensions) is greater than 90 and which would therefore normally be considered relatively hydrophobic. However, consideration of the balance between bending and interfacial energies suggests that the tendency for droplet wrapping can be suppressed for some liquids by providing the flexible solid surface with a rigid topographic structure. In general, it is known that when a liquid interacts with such a structure it can either fully penetrate the structure (the Wenzel case) or it can bridge between the asperities of the structure (the Cassie-Baxter case). In this report, we show theoretically that droplet wrapping should occur with both types of solid-liquid contact. We also derive a condition for the transition between the Cassie-Baxter and Wenzel type droplet wrapping and relate it to the same transition condition known to apply to superhydrophobic surfaces. The results are given for both droplets being wrapped by thin ribbons and for solid grains encapsulating droplets to form liquid marbles.