Capillary pressure, osmotic pressure and bubble contact areas in foams

Abstract

The capillary pressure of foams and emulsions is the difference between the average pressure in the dispersed phase and the pressure in the continuous phase. The pressure difference between individual bubbles or drops and the continuous phase is due to interfacial tension, and governs the thickness of films that separate neighbouring particles. Princen and Derjaguin presented an analytic relation, validated for ordered monodisperse foams with face-centered cubic (fcc) structure, that links the capillary pressure to osmotic pressure; they conjectured that it also held for disordered polydisperse foams that are encountered more frequently in nature and applications. Their conjecture is widely accepted. We derive their relation from first principles, and use known empirical expressions for the osmotic pressure to obtain analytic predictions for the capillary pressure and the average bubble contact area in a foam over the full range of liquid fractions. These results are validated using Surface Evolver simulations and previous experimental data. They also apply to emulsions.