  A concentrated suspension of lamellar colloidal particles (e. g. clay) is
modelled 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
neighbouring 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 (LPB) equation are obtained for
circular and square platelets placed at the centre 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.
