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.

Superhydrophobicity can arise from the ability of a submerged rough hydrophobic surface to trap air in its surface pores, and thereby reduce the contact area between the water and the frictional solid walls. A submerged surface can only remain superhydrophobic (SHP) as long as it retains the air in its pores. SHP surfaces have a short underwater life, and their longevity depends strongly on the hydrostatic pressure at which they operate. In this work, a comprehensive mathematical framework is developed to predict the mechanical stability and the longevity of submerged SHP surfaces with arbitrary pore or post geometries. We start by deriving an integro-partial differential equation for the 3-D shape of the air-water interface, and use this information to predict the rate of dissolution of the entrapped air into the ambient water under different hydrostatic pressures. For the special case of circular pores, the above integro-partial differential equation is reduced to easy-to-solve ordinary differential equations. In addition, approximate nonlinear algebraic solutions are also obtained for surfaces with circular pores or posts. The effects of geometrical parameters and hydrostatic conditions on surface stability and longevity are discussed in detail. Moreover, a simple equivalent pore diameter method is developed for SHP surfaces composed of posts with ordered or random configuration--an otherwise complicated task requiring the solution of an integro-partial differential equation.

Superhydrophobic (SHP) surfaces are known for their drag-reducing attributes thanks to their ability to trap air in their surface pores and thereby reduce the contact between water and the frictional solid area. SHP surfaces are prone to failure under elevated pressures or because of air-layer dissolution into the surrounding water. Slippery liquid-infused porous surfaces (SLIPS) or liquid-infused surfaces (LIS) in which the trapped air is replaced with a lubricant have been proposed in the literature as a way of eliminating the air dissolution problem as well as improving the surface stability under pressure. While an LIS surface has been shown to reduce drag for flow of water-glycerol mixture (ref 18), no significant drag reduction has yet been reported for the flow of water (a lower viscosity fluid) over LIS. In this concern, we have designed a new surface in which a layer of air is trapped underneath the infused lubricant to reduce the frictional forces preventing the LIS to provide drag reduction for water or any fluid with a viscosity less than that of the lubricant. Drag reduction performance of such surfaces, referred to here as liquid-infused surfaces with trapped air (LISTA), is predicted by solving the biharmonic equation for the water-oil-air three-phase system in transverse grooves with enhanced meniscus stability thanks to double-reentry designs. For the arbitrary dimensions considered in our proof-of-concept study, LISTA designs showed 20-37% advantage over their LIS counterparts.

While the air–water interface over superhydrophobic surfaces decorated with hierarchical micro- or nanosized geometrical features have shown improved stability under elevated pressures, their underwater longevity—-the time that it takes for the surface to transition to the Wenzel state—-has not been studied. The current work is devised to study the effects of such hierarchical features on the longevity of superhydrophobic surfaces. For the sake of simplicity, our study is limited to superhydrophobic surfaces composed of parallel grooves with side fins. The effects of fins on the critical pressure—-the pressure at which the surface starts transitioning to the Wenzel state—-and longevity are predicted using a mathematical approach based on the balance of forces across the air–water interface. Our results quantitatively demonstrate that the addition of hierarchical fins significantly improves the mechanical stability of the air–water interface, due to the high advancing contact angles that can be achieved when an interface comes in contact with the fins sharp corners. For longevity on the contrary, the hierarchical fins were only effective at hydrostatic pressures below the critical pressure of the original smooth-walled groove. Our results indicate that increasing the length of the fins decreases the critical pressure of a submerged superhydrophobic groove but increases its longevity. Increasing the thickness of the fins can improve both the critical pressure and longevity of a submerged groove. The mathematical framework presented in this paper can be used to custom-design superhydrophobic surfaces for different applications.

No abstract