Hydrodynamic-Colloidal Interactions of an Oil Droplet and a Membrane Surface

Membranes have been shown to be exceptionally successful in the challenging separation of stable oil/water emulsions but suffer from severe fouling that limits their performance. Understanding the mechanisms leading to oil deposition on the membrane surface, as influenced by hydrodynamics and colloidal surface interactions, is imperative for informing better engineered membrane surfaces and process conditions. Here, we study the interactions between an oil droplet and a membrane surface. Hydrodynamics within the water film, confined between the droplet and the membrane, are captured within the framework of the lubrication approximation, coupled with the van der Waals (vdW) and electrostatic interactions through the droplet shape, which is governed by an augmented Young-Laplace equation. The model is used to calculate possible equilibrium positions, where the droplet is held at a finite distance from the membrane by a balance of the forces present. An equilibrium phase diagram is constructed as a function of various process parameters and is shown in terms of the scaled permeation rate through the membrane. The phase diagram identifies the range of conditions leading to deposition, characterized by a "critical" permeation rate, beyond which no equilibrium exists. When equilibrium positions are permitted, we find that these may be classified as stable/unstable, in the kinetic sense. Further, our results demonstrate the link between the deformation of the droplet and the stability of equilibria. An upward deflection of the droplet surface, owing to a dominant, long-range repulsion, has a stabilizing effect, as it maintains the separation between the droplet and membrane. Conversely, a downward deflection is destabilizing because of the self-amplifying effect of strongly increasing attractive forces with separation distance-as the surfaces are pulled together because of deformation, the attractive force increases, causing further deformation. This is also manifested by a dependence of the bistable region on the deformability of the droplet, which is represented by a capillary number, modified so as to account for the effect of the permeable boundary. As the droplet becomes more easy to deform, the transition from an unconditionally stable region of the phase diagram to a point beyond which there is no equilibrium (interpreted as deposition) becomes abrupt. These results provide valuable physical insights into the mechanisms that govern oil fouling of membrane surfaces.