Hydrophilicity and Hydrophobicity at the Nanoscale

Hydrophobicity and hydrophilicity are incompatible. On a hydrophobic substrate, a macroscopic droplet always exhibits a morphology with contact angle higher than 90∘, never lower than 90∘. In this paper, we theoretically demonstrate the possibility that a nanoscale droplet can exhibit a contact angle lower than 90∘ on the same hydrophobic substrate. To demonstrate this, we analyze the morphology and contact angle of a sessile droplet on smooth flat substrates, taking into account disjoining pressure. By constraining the two-dimensional cylindrical droplet and minimizing the free-energy functional, we derive a formula to determine the droplet’s morphology and the boundary between hydrophilic and hydrophobic contact angle for finite-sized droplets. Using this formulation, we reconsider the formula for the macroscopic contact angle, known as the Derjaguin-Frumkin formula. By utilizing a simple disjoing pressure model, we find that the calculated contact angle at the nanoscale is always smaller than the macroscopic contact angle determined by the Derjaguin-Frumkin formula. Consequently, the wettability (hydropilicity/hydrophobicity) differs at the nanoscale compared to the macroscale. We further discuss the implication of our results on the size-dependent contact angle and line tension at the nanoscale.