Closed-Form Expressions for Contact Angle Hysteresis: Capillary Bridges between Parallel Platens

Abstract

A closed form expression capable of predicting the evolution of the shape of liquid capillary bridges and the resultant force between parallel platens is derived. Such a scenario occurs within many micro-mechanical structures and devices, for example, in micro-squeeze flow rheometers used to ascertain the rheological properties of pico- to nano-litre volumes of complex fluids, which is an important task for the analysis of biological liquids and during the combinatorial polymer synthesis of healthcare and personal products. These liquid bridges exhibit capillary forces that can perturb the desired rheological forces, and perhaps more significantly, determine the geometry of the experiment. The liquid bridge has a curved profile characterised by a contact angle at the three-phase interface, as compared to the simple cylindrical geometry assumed during the rheological analysis. During rheometry, the geometry of the bridge will change in a complex nonlinear fashion, an issue compounded by the contact angle undergoing hysteresis. Owing to the small volumes involved, ascertaining the bridge geometry visually during experiment is very difficult. Similarly, the governing equations for the bridge geometry are highly nonlinear, precluding an exact analytical solution, hence requiring a substantial numerical solution. Here, an expression for the bridge geometry and capillary forces based on the toroidal approximation has been developed that allows the solution to be determined several orders of magnitude faster using simpler techniques than numerical or experimental methods. This expression has been applied to squeeze-flow rheometry to show how the theory proposed here is consistent with the assumptions used within rheometry. The validity of the theory has been shown through comparison with the exact numerical solution of the governing equations. The numerical solution for the shape of liquid bridges between parallel platens is provided here for the first time and is based on existing work of liquid bridges between spheres.