General Variational Formulation of Axisymmetric Capillary Bridges: Modeling Contact Angle Hysteresis and Capillary Forces

This work presents a general framework for deriving the Young–Laplace equation and the Young's equations for an axisymmetric capillary bridge between two parallel plates by minimizing the system's total energy. These Young's equations naturally emerge as boundary conditions associated with the Young–Laplace equation. An additional term accounting for the contact line energy is incorporated into the total energy to model the contact angle hysteresis observed experimentally (advancing and receding contact angles). Minimization of this augmented total energy leads to generalized Young's equations. The proposed approach also provides a consistent and general definition of the corresponding capillary forces.