Hydrophilicity and Hydrophobicity at the Nanoscale: A Theoretical Study on Two-Dimensional Cylindrical Droplets with Disjoining Pressure Effects

Hydrophobicity and hydrophilicity are incompatible in the sense that a single substrate cannot exhibit both characteristics simultaneously. On a hydrophobic substrate, for example, a macroscopic droplet always exhibits a morphology with a 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 of Lennard–Jones type. 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 angles 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 disjoining 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 (hydrophilicity/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.