General formulations for predicting longevity of submerged superhydrophobic surfaces composed of pores or posts

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.