Two-dimensional capillary origami

• Full solution set of the two-dimensional capillary origami problem.
• Fluid does not necessarily wet the entire plate.
• Global energy approach provides exact differential equations satisfied by minimizers.
• Bifurcation diagrams highlight three different regimes.
• Conditions for spontaneous encapsulation are identified.

We describe a global approach to the problem of capillary origami that captures all unfolded equilibrium configurations in the two-dimensional setting where the drop is not required to fully wet the flexible plate. We provide bifurcation diagrams showing the level of encapsulation of each equilibrium configuration as a function of the volume of liquid that it contains, as well as plots representing the energy of each equilibrium branch. These diagrams indicate at what volume level the liquid drop ceases to be attached to the endpoints of the plate, which depends on the value of the contact angle. As in the case of pinned contact points, three different parameter regimes are identified, one of which predicts instantaneous encapsulation for small initial volumes of liquid.