The Wigner - Seitz model for concentrated clay suspensions

The model of a single uniformly charged finite platelet confined with its counter-ions and added salt to a Wigner - Seitz cell is treated within linearized Poisson - Boltzmann (or Debye - Hückel) theory. We consider circular (disc-like) and square platelets placed at the centre of a cylindrical or parallelepipedic cell of volume fixed by the macroscopic clay concentration. For a given volume the free energy F is minimized with respect to the aspect ratio of the cell. We find that the quadrupole moment Q of the total charge distribution always vanishes at the free-energy minimum, and that for discs, Q and F are practically identical for the two cell geometries at any given volume and salt concentration. Finally we propose a hybrid Poisson - Boltzmann/Debye - Hückel formulation which allows non-linearities to be approximately accounted for.