Trapped liquid drop in a microchannel: Multiple stable states

A liquid drop trapped in a microchannel, in which both contact angle (wettability) and opening angle (geometry) can vary with position, is investigated based on the minimization of free energy. The calculus of variation yields the Young-Laplace equation and its further integration leads to the general force balance. The equilibrium position of the trapped drop is determined by the balance between the area-mean capillary force and the area-mean hydrostatic pressure difference. Trapped liquid drops in truncated cones and hyperboloids are studied to elucidate our theory. As the volume of the drop trapped in the hydrophilic cones is increased, four regimes separated by three critical volumes are identified. The drop is either trapped at the narrow end or away from the cone top. The solution at the cone top satisfies the force balance by adjusting the upper contact angle, which is experimentally observed and verified by Surface Evolver (SE) simulations. Multiple stable states can exist in a particular regime. The hyperboloid tube in which the opening angle varies with position is also considered. As the gravitational strength is increased in hydrophilic hyperboloid, four regimes separated by three critical gravitational strengths are identified. The drop is either trapped near the neck or below the neck. Unlike hydrophilic cones, the drop stays near the neck of the hyperboloid due to varying opening angles. Multiple stable states are also observed. For both cone and hyperboloid, hydrophobic cases are studied as well and all theoretical solutions of the force balance agree well with SE simulation outcomes.