Modeling the filtration of deformable and permeable colloidal particles: the case of casein micelles
Résumé
When filtrating a colloidal dispersion, colloids accumulate at the membrane surface, thus forming a polarized layer that can ultimately turns into a deposit (= a "gel") in some extreme cases. Modeling such a filtration has always been a challenge. In recent developments, Bacchin et al. propose a model based on the analogy between permeation in a deposit and diffusion in a polarized layer [1]. The permeation flux is then given by the following equation, which is the analog of the classical Darcy's law: J = k(C)/μ . dΠ(C)/dx (eq.(1)), where x is the distance to the membrane, μ is the solvent viscosity, k(C) and Π(C) are the permeability and colloidal osmotic pressure at distance x, respectively. As for now, the model of Bacchin et al. has been used successfully to describe the filtration of impermeable and "hard" spherical objects like latex particles [2]: the idea is to measure the osmotic pressures Π(C) of the latex dispersions through osmotic stress experiments and to estimate the permeability k(C) using the well-known theoretical expressions of Happel. Knowing Π(C) and k(C), eq.(1) is then used to obtain important information like permeation fluxes or concentration profiles in the accumulated layer. The objective of our study is to adapt such a model to the filtration of more complex colloids, namely, deformable and permeable objects like colloidal microgels. Casein micelles, which represent 80% of the proteins in cow milk, are the perfect candidate for that since they are highly permeable and compressible protein aggregates [3]. Additionally, ultra- and microfiltration are widely used for the concentration of milk and there is a serious need for a model that is able to describe these operations. The osmotic pressure of casein micelle dispersions is now perfectly known [3]. So the main difficulty lies in determining k(C) since no theoretical expression is currently available for deformable and permeable objects. Our presentation focus on the different strategies we used to overcome this difficulty and on the significance of the k(C) values we managed to estimate either indirectly from filtration experiments or directly from osmotic stress experiments. We then discuss on the ability for the resulting model to predict filtrations conducted in different modes (dead-end/cross-flow, constant pressure/constant flux).