Consolidation of hydrophobic transition criteria by using an approximate energy minimization approach

Recent experimental work has successfully revealed pressure induced transition from Cassie to Wenzel state on rough hydrophobic substrates. Formulas, based on geometric considerations and imposed pressure, have been developed as transition criteria. In the past, transition has also been considered as a process of overcoming the energy barrier between the Cassie and Wenzel states. A unified understanding of the various considerations of transition has not been apparent. To address this issue, in this work, we consolidate the transition criteria with a homogenized energy minimization approach. This approach decouples the problem of minimizing the energy to wet the rough substrate, from the energy of the macroscopic drop. It is seen that the transition from Cassie to Wenzel state, due to depinning of the liquid-air interface, emerges from the approximate energy minimization approach if the pressure-volume energy associated with the impaled liquid in the roughness is included. This transition can be viewed as a process in which the work done by the pressure force is greater than the barrier due to the surface energy associated with wetting the roughness. It is argued that another transition mechanism, due to a sagging liquid-air interface that touches the bottom of the roughness grooves, is not typically relevant if the substrate roughness is designed such that the Cassie state is at lower energy compared to the Wenzel state.