The electrical double-layer free energy is computed numerically for a model colloidal suspension of charged, hexagonally-packed monodisperse cylinders in a symmetric electrolyte solution using the Poisson-Boltzmann equations. Comparison is made to the energies estimated from sums of pair interactions and from two versions of a radially-symmetric cell model: one in which the outer boundary of the cylindrical cell is chosen to make the particle volume fraction Φ in the approximate model agree with that of the full lattice and the other in which the position of the outer boundary is chosen as half the nearest neighbor separation in the lattice. Calculations are performed for ranges of particle volume fractions from infinite dilution to 98% of closest packing, ionic strengths (in terms of the ratio of Debye length to particle radius λ/a) from 0.1 to 10.0, and dimensionless surface potentials and charge densities from 0.1 to 10.0.
The pair approximation is valid only when typical interparticle spacings are greater than the Debye length. The first implementation of the cell model proves quite accurate, while the second is only qualitatively correct. The magnitude of particle surface charge density or potential has little effect on these conclusions.