Discussion on a mechanical equilibrium condition of a sessile drop on a smooth solid surface

Young's equation describes an interfacial equilibrium condition of a liquid droplet on a smooth solid surface. This relation is derived by Thomas Young in 1805. It has been discussed until today after his work. In general, Young's equation is discussed from the viewpoint of thermodynamics and derived by minimizing the total free energy of the system with intensive parameters in the total free energy kept constant, i.e., the variation in the total free energy is zero. In the derivation, the virtual work variations in the horizontal and vertical directions of the droplet on the smooth solid are considered independently. However, the virtual work variation at the droplet surface depends on the variation of the horizontal and vertical directions, which are related to an incline of the droplet surface. This point has been overlooked in past studies. In this study, by considering this directional dependency, we derive the modified Young's equation based on the thermodynamics. Finally, we evaluate the modified Young's equation by comparing the analytical solution of the relationship between a contact angle and the contact line radii of the droplet with some experimental data. Moreover, we investigated the line tension itself.