Liquid-bridge breakup in contact-drop dispensing: Liquid-bridge stability with a free contact line.

The static stability of weightless liquid bridges with a free contact line with respect to axisymmetric and nonaxisymmetric perturbations is studied. Constant-volume and constant-pressure stability regions are constructed in slenderness versus cylindrical volume diagrams for fixed contact angles. Bifurcations along the stability-region boundaries are characterized by the structure of axisymmetric bridge branches and families of equilibria. A wave-number definition is presented based on the pieces-of-sphere states at branch terminal points to classify equilibrium branches and identify branch connections. Compared with liquid bridges pinned at two equal disks, the free contact line breaks the equatorial and reflective symmetries, affecting the lower boundary of the constant-volume stability region where axisymmetric perturbations are critical. Stability is lost at transcritical bifurcations and turning points along this boundary. Our results furnish the maximum-slenderness stability limit for drop deposition on real surfaces when the contact angle approaches the receding contact angle.