Theoretical model of viscous friction inside steadily sheared foams and concentrated emulsions

In a recent Letter [N. D. Denkov, Phys. Rev. Lett. 100, 138301 (2008)] we calculated theoretically the macroscopic viscous stress of steadily sheared foam or emulsion from the energy dissipated inside the transient planar films, formed between neighboring bubbles or drops in the shear flow. The model predicts that the viscous stress in these systems should be proportional to Ca 1/2, where Ca is a capillary number and n=1/2 is the power-law index. In the current paper we explain our model in detail and develop it further in several aspects: First, we extend the model to account for the effects of viscous friction in the curved meniscus regions, surrounding the planar films, on the dynamics of film formation and on the total viscous stress. Second, we consider the effects of surface forces (electrostatic, van der Waals, etc.) acting between the surfaces of the neighboring bubbles or drops and show that these forces could be important in emulsions, due to the relatively small thickness of emulsion films (often comparable to the range of action of surface forces). In contrast, the surface forces are usually negligible in steadily sheared foams, because the dynamic foam films are thicker than the extent of surface forces, except for foams containing micrometer-sized bubbles and/or at very low shear rates. Third, additional consideration is made for bubbles or drops exhibiting high surface viscosity, for which we demonstrate an additional contribution to the macroscopic viscous stress, created by the surface dissipation of energy. The new upgraded model predicts that the energy dissipation at the bubble or drop surface leads to power-law index n1/2 , which explains the rich variety of foam or emulsion behaviors observed upon steady shear. Various comparisons are made between model predictions and experimental results for both foams and emulsions, and very good agreement is found.