A mathematical framework is developed to predict the longevity of a submerged superhydrophobic surface made up of parallel grooves. Time-dependent integrodifferential 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.

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A mathematical framework developed to calculate the shape of the air-water interface and predict the stability of a microfabricated superhydrophobic surface with randomly distributed posts of dissimilar diameters and heights is presented. Using the Young-Laplace equation, a second-order partial differential equation is derived and solved numerically to obtain the shape of the interface, and to predict the critical hydrostatic pressure at which the superhydrophobicity vanishes in a submersed surface. Two examples are given for demonstration of the method's capabilities and accuracy.

To better understand the role of fiber orientation on the stability of superhydrophobic electrospun coatings under hydrostatic pressures, an integro-differential equation is developed from the balance of forces across the air–water interface between the fibers. This equation is solved numerically for a series of superhydrophobic electrospun coatings comprised of random and orthogonal fiber orientations to obtain the exact 3D shape of the air–water interface as a function of hydrostatic pressure. More important, this information is used to predict the pressure at which the coatings start to transition from the Cassie state to the Wenzel state, i.e., the so-called critical transition pressure. Our results indicate that coatings composed of orthogonal fibers can withstand higher elevated hydrostatic pressures than those made up of randomly orientated fibers. Our results also prove that thin superhydrophobic coatings can better resist the elevated pressures. The modeling methodology presented here can be used to design nanofibrous superhydrophobic coatings for underwater applications.

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