Predicting longevity of submerged superhydrophobic surfaces with parallel grooves

A mathematical framework is developed to predict the longevity of a submerged superhydrophobic surface made up of parallel grooves. Time-dependent integro-differential equations predicting the instantaneous behavior of the air–water interface are derived by applying the balance of forces across the air–water interface, while accounting for the dissolution of the air in water over time. The calculations start by producing a differential equation for the initial steady-state shape and equilibrium position of the air–water interface at t = 0. Analytical and/or numerical solutions are then developed to solve the time-dependent equations and to compute the volume of the trapped air in the grooves over time until a Wenzel state is reached as the interface touches the groove's bottom. For demonstration, a superhydrophobic surface made of parallel grooves is considered, and the influence of the groove's dimensions on the longevity of the surface under different hydrostatic pressures is studied. It was found that for grooves with higher width-to-depth ratios, the critical pressure (pressure at which departure from the Cassie state starts) is higher due to stronger resistance to deflection of the air–water interface from the air trapped in such grooves. However, grooves with higher width-to-depth ratios reach the Wenzel state faster because of their greater air–water interface areas.