Thermodynamic analysis of bridging bubbles and a quantitative comparison with the measured hydrophobic attraction

The shape of a bubble bridging two colloidal spheres is obtained by minimization of the constrained Gibbs free energy. Bubbles bridging hydrophobic surfaces are concave, whereas hydrophilic surfaces give rise to convex bubbles. Owing to an energy barrier, metastable submicroscopic bubbles are prevented from expanding to microscopic size, and vice versa, and hysteresis is found on approach and on separation of the hydrophobic spheres. The force due to the bridging bubble is generally attractive, except at small separations for both pinned bubbles and hydrophilic surfaces. An analytic approximation for the force and bridging bubble size is derived and shown to be much more accurate for colloidal spheres than the classic formula for the capillary adhesion. Dynamic drag on the expanding bubble is shown to reduce the attraction and to give a repulsion at small separations. The dynamic results give a quantitative account of the force measured between hydrophobic surfaces in water with an atomic force microscope.