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The free-energy barrier of filling a spherical cavity having an inner wall ofvarious wettabiities is studied. The morphology and free energy of alens-shaped droplet are determined from the minimum of the free energy. Theeffect of line tension on the free energy is also studied. Then, theequilibrium contact angle of the droplet is determined from the generalizedYoung's equation. By increasing the droplet volume within the spherical cavity,the droplet morphology changes from spherical with an equilibrium contact angleof $180^{\circ}$ to a lens with a convex meniscus, where the morphologicalcomplete drying transition occurs. By further increasing the droplet volume,the meniscus changes from convex to concave. Then, the lens-shaped droplet withconcave meniscus spreads over the whole inner wall resulting in an equilibriumcontact angle of $0^{\circ}$ to leave a spherical bubble, where themorphological complete wetting transition occurs. Finally, the whole cavity isfilled with liquid. The free energy shows a barrier from complete drying tocomplete wetting as a function of droplet volume, which corresponds to theenergy barrier between the Cassie and Wenzel state of the superhydrophobicsurface with spherical cavities. The free-energy maximum occurs when themeniscus of the droplet becomes flat and it is given by an analytic formula.The effect of line tension is expressed by the scaled line tension, and thiseffect is largest at the free-energy maximum. The positive line tensionincreases the free-energy maximum, which thus, increases the stability of theCassie superhydrophobic state, whereas the negative line tension destabilizesthe superhydrophobic state.

Free-energy barrier of filling a spherical cavity in the presence of  line tension: Implication to the energy barrier between the Cassie and Wenzel  state on a superhydrophobic surface with spherical cavities