Frenkel's method and the spreading of small spherical droplets

Frenkel's method is used to obtain a relation between the edge velocity and the dynamic contact angle for the spreading of small spherical droplets of viscous and non-volatile fluids. The change in surface free energy is equated to the viscous dissipation caused by Poiseuille flow, first in a wedge and then in a cone inscribed within the spherical cap and this introduces a cut-off integral. An equation relating the edge velocity to the dynamic contact angle for both complete and partial wetting is obtained. In the small angle limit this formula corresponds to a generalized Tanner's law and, in the case of the cone model, also predicts the variation of the edge velocity over a wide angular range. A form for the angular dependence in the cut-off integral is suggested and estimates of a related vertical cut-off length are obtained from existing data. Good agreement of the edge velocity angular dependence with literature values covering the range up to 165 degrees for complete spreading is found. In addition, for high-energy surfaces a closed form for the time dependence of the contact angle can be obtained.