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 minimiz ation of the constrained Gibbs free energy. Bubbles bridging hydrophobic su rfaces are concave, whereas hydrophilic surfaces give rise to convex bubble s. Owing to an energy barrier, metastable submicroscopic bubbles are preven ted from expanding to microscopic size, and vice versa, and hysteresis is f ound on approach and on separation of the hydrophobic spheres. The force du e to the bridging bubble is generally attractive, except at small separatio ns for both pinned bubbles and hydrophilic surfaces. An analytic approximat ion 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 capill ary adhesion. Dynamic drag on the expanding bubble is shown to reduce the a ttraction and to give a repulsion at small separations. The dynamic results give a quantitative account of the force measured between hydrophobic surf aces in water with an atomic force microscope.