A geometry-based model for spreading drops applied to drops on a silicon wafer and a swellable polymer brush film

We investigate the dynamics of spreading in a regime where the shape of the drop is close to a spherical cap. The latter simplification is applicable in the late (viscous) stage of spreading for highly viscous drops with a diameter below the capillary length. Moreover, it applies to the spreading of a drop on a swellable polymer brush, where the complex interaction with the substrate leads to a very slow spreading dynamics. The spherical cap geometry allows to derive a closed ordinary differential equation (ODE) for the spreading if the capillary number is a function of the contact angle as it is the case for empirical contact angle models. The latter approach has been introduced by de Gennes (Reviews of Modern Physics, 1985) for small contact angles. In the present work, we generalize the method to arbitrary contact angles. The method is applied to experimental data of spreading water-glycerol drops on a silicon wafer and spreading water drops on a PNIPAm coated silicon wafer. It is found that the ODE-model is able to describe the spreading kinetics in the case of partial wetting. Moreover, the model can predict the spreading dynamics of spherical cap-shaped droplets if the relationship between the contact angle and the capillary number is universal.