A three-scale model for pH-dependent steady flows in 1:1 clays

A new three-scale model to describe the coupling between electro-chemistry and hydrodynamics in non-swelling kaolinite clays in steady-state conditions is proposed. The medium is characterized by three separate nano-micro and macroscopic length scales. At the pore (micro)-scale the portrait of the clay consists of micro-pores saturated by an aqueous solution containing four monovalent ions (Na⁺, H⁺, Cl⁻, OH⁻) and charged solid particles surrounded by thin electrical double layers. The movement of the ions is governed by the Nernst–Planck equations and the influence of the double layers upon the hydrodynamics is modeled by a slip boundary condition in the tangential velocity governed by the Stokes problem. To capture the correct form of the interface condition we invoke the nanoscopic modeling of the thin electrical double layer based on Poisson–Boltzmann problem with varying surface charge density ruled by the protonation/deprotonation reactions which occur at the surface of the particles. The two-scale nano/micro model is homogenized to the macroscale leading to a precise derivation of effective governing equations. The macroscopic model is discretized by the finite volume method and applied to numerically simulate desalination of a clay sample induced by an external electrical field generated by the placement of electrodes. Numerical results indicate strong pH-dependence of the electrokinetics.