Wigner-Seitz model of charged lamellar colloidal dispersions

A concentrated suspension of lamellar colloidal particles (e.g., clay) is modeled by considering a single, uniformly charged, finite platelet confined with co- and counterions to a Wigner-Seitz (WS) cell. The system is treated within Poisson-Boltzmann theory, with appropriate boundary conditions on the surface of the WS cell, supposed to account for the confinement effect of neighboring platelets. Expressions are obtained for the free energy, osmotic, and disjoining pressures and the capacitance in terms of the local electrostatic potential and the co- and counterion density profiles. Explicit solutions of the linearized Poisson-Boltzmann equation are obtained for circular and square platelets placed at the center of a cylindrical or parallelepipedic cell. The resulting free energy is found to go through a minimum as a function of the aspect ratio of the cell, for any given volume (determined by the macroscopic concentration of platelets), platelet surface charge, and salt concentration. The optimum aspect ratio is found to be nearly independent of the two latter physical parameters. The osmotic and disjoining pressures are found to coincide at the free energy minimum, while the total quadrupole moment of the electric double layer formed by the platelet and the surrounding co- and counterions vanishes simultaneously. The osmotic equation of state is calculated for a variety of physical conditions. The limit of vanishing platelet concentration is considered in some detail, and the force acting between two coaxial platelets is calculated in that limit as a function of their separation.