Charge renormalization, osmostic pressure, and bulk modulus of colloidal crystals: Theory

The interactions between charged colloidal particles with sufficient strength to cause crystallization are shown to be describable in terms of the usual Debye-Huckel approximation, but with a renormalized charge. The effective charge in general is smaller than the actual charge. We calculate the relationship between the actual charge and the renormalized charge by solving the Boltzmann-Poisson equation numerically in a spherical Wigner-Seitz cell. We then relate the numerical solutions and the effective charge to the osmotic pressure and the bulk modulus of the crystal. Our calculations also reveal that the renormalization of the added electrolyte concentration is negligible, so that the effective charge computations are useful even in the presence of salts.