Wetting transitions on textured hydrophilic surfaces

We consider the quasi-static energy of a drop on a textured hydrophilic surface, with taking the contact angle hysteresis (CAH) into account. We demonstrate how energy varies as the contact state changes from the Cassie state (in which air is trapped at the drop bottom) to the Wenzel state (in which liquid fills the texture at the drop bottom) assuming that the latter state nucleates from the center of the drop bottom. When the textured substrate is hydrophilic enough to allow spontaneous penetration of liquid film of the texture thickness, the present theory asserts that the drop develops into an experimentally observed state in which a drop looks like an egg fried without flipped over (sunny-side up) with a well-defined radius of the egg yolk. Otherwise, the final contact state of the drop becomes like a Wenzel state, but with the contact circle smaller than the original Wenzel state due to the CAH. We provide simple analytical estimations for the yolk radius of the sunny-side-up state and for the final radius of the contact circle of the pseudo-Wenzel state.