In part I (Lima et al., Transp Porous Media, 2009), a three-scale model governing the movement of an aqueous saline solution containing four monovalent species (Na + , H + , Cl − , OH − ) in kaolinite clays was derived. Unlike purely macroscopic approaches, the novelty of the formulation relied on the double averaging of the nanoscopic electrochemistry of particle/electrolyte solution interface ruled by the electrical double layer coupled with protonation/deprotonation reactions. The passage from the nano to the micro (pore)-scale gave rise to ion-sorbed concentrations and slip velocity at the solid/fluid interface which are coupled with the microscopic Stokes problem and Nernst-Planck equations governing the hydrodynamics and ion transport in the micropores. Application of a formal homogenization procedure led to macroscopic governing equations with effective electro-chemical parameters, such as retardation coefficients, electro-osmotic permeability, and electric conductivity. In this study, we reconstruct the constitutive laws of the macroscopic coefficients by solving the nano and microscopic closure problems. New generalized isotherms for Na + and H + − OH − sorption are build-up based on a perturbation approach and the limitations 123 46 S. A. de Lima et al. of classical Freundlich isotherm for modeling ion sorption at the solid/fluid interface are discussed. The macroscopic governing equations are discretized by the finite volume method and numerical simulations of a transient electroosmosis experiment for desalination of a clay sample by electrokinetics are presented.
List of Symbolsa fs Surface area [m −1 ] f Characteristic function m, n, p Exponents of the adsorption isotherms k e Isoelectric point p Thermodynamic pressure [Pa] t Time [s] x, y Parallel and orthogonal coordinates [m] u Auxiliary perturbation parameter C ib Ionic bulk concentration with i = Na,H [mol/m 3 ] C H + 0 H + concentration at the solid surface [mol/l] C b Bulk concentration [mol/m 3 ] F Faraday constant [C/mol] H Half distance between the particles [m] K Equilibrium constant [l/mol] K W Ionic product of water [(mol/l) 2 ] L D Debye's length [m] L Sample length [m] M Metallic ions at the surface R ideal gas constant [J/(mol K)] R N , R H Ionic retardation coefficients T Temperature [K] Y Cell domain Y f , Y s Fluid and solid subdomains ∂Y fs Fluid/solid interface x Macroscopic coordinate [m] y Microscopic coordinate [m] A eff First Onsager coefficient [(C·m 2 )/(mol·s)] B eff Second Onsager coefficient [(C·m 2 )/(mol·s)] C eff Effective electric conductivity [C/(m·s)] D eff Net effective diffusivity H + -OH + [m 2 /s] D eff Na Effective sodium diffusivity [m 2 /s] K eff P Hydraulic conductivity [m 2 /(Pa·s)] K eff E Electroosmotic permeability [m/(V·s)] I eff f Effective electric current [C/(m 2 ·s)] J eff Effective ionic flux [mol/(m 2 ·s)] 123 A Three-Scale Model of Ion Transport with Equilibrium Adsorption in 1:1 Clays 47