Dynamical equations for the contact line of an evaporating or condensing sessile drop

The equations that govern, away from equilibrium and accounting for dissipation, the evolution of the contact line of an evaporating or condensing sessile drop on a rigid, planar substrate are derived. Aside from the normal and tangential components of the standard (Newtonian) force balance, these include a configurational force balance. At equilibrium, the normal component of the standard force balance reduces to the modified Young's equation first mentioned by Gibbs. The remaining balances are purely dissipative and hence are vacuous in equilibrium. A complete description of contact-line dynamics generally involves all three equations. The theory is embedded in a thermodynamic framework that ensures consistency of all constitutive relations with the second law. In the linearly dissipative case, these involve six contact-line viscosities. When viscous coupling is neglected, only three viscosities remain. One is associated with stretching of the fluid along the contact line. The remaining two are related to dissipation that accompanies mass transfer between liquid and vapour phases during evaporation or condensation.