Buoyant force and sinking conditions of a hydrophobic thin rod floating on water

Owing to the superhydrophobicity of their legs, such creatures as water striders and fisher spiders can stand effortlessly, walk and jump quickly on water. Directed toward understanding their superior repellency ability, we consider hydrophobic thin rods of several representative cross sections pressing a water surface. First, the shape function of the meniscus surrounding a circular rod is solved analytically, and thereby the maximal buoyant force is derived as a function of the Young's contact angle and the rod radius. Then we discuss the critical conditions for a rod to sink into water, including the maximal volume condition and the meniscus-contact condition. Furthermore, we study the sinking conditions and the maximal buoyant forces of hydrophobic long rods with elliptical, triangular, or hexagonal cross-section shapes. The theoretical solutions are quantitatively consistent with existing experimental and numerical results. Finally, the optimized structures of water strider legs are analyzed to elucidate why they can achieve a very big buoyant force on water.