The continuous modeling of charge-stabilized colloidal suspensions in shear flows

Abstract

Flows of concentrated colloidal suspensions may exhibit a rich set of behaviors due to both hydrodynamic and colloidal interactions between the particles. Colloidal flows are generally modeled with an effective Navier–Stokes equation and a mass balance for the solid phase involving a diffusion coefficient given by the generalized Stokes–Einstein relation. This picture corresponds to a near equilibrium regime in which entropic and colloidal forces dominate over hydrodynamic interactions, the latter being totally ignored. On the other hand, suspension flows far from equilibrium require the modeling of significant hydrodynamic stresses responsible in particular for shear-induced migration, a phenomenon known to occur in some industrial processes involving colloids, such as ultrafiltration. The choice of the proper model ingredients requires a knowledge of the domains in parameter space in which colloidal or hydrodynamic effects are dominant. In this article, such a phase diagram is established for a channel flow of charge-stabilized colloids with a version of the suspension balance model including both colloidal and hydrodynamic effects at the continuous level. It is shown that the classical Peclet number is not sufficient to characterize the flow regime. The phase boundary between the colloidal and hydrodynamic regimes exhibits an original shape explained by the dependence of electrostatic interactions with the colloidal surface charge, and in particular by the phenomenon of ionic condensation. We also show that the phase diagram can be predicted based on the knowledge of a rescaled Peclet number comparing the hydrodynamic stress scale to the bulk modulus of the suspension. The criterion determined here provides important guidelines for the efficient modeling of colloidal flows.