Osmotic pressure of charged colloidal suspensions: a unified approach to linearized Poisson-Boltzmann theory

We study theoretically the equation of state of a fluid suspension of charged objects (e.g., colloids, polyelectrolytes, clay platelets, etc.) dialyzed against an electrolyte solution using the cell model and linear Poisson-Boltzmann (PB) theory. From the volume derivative of the grand potential functional of linear theory we obtain two expressions for the osmotic pressure in terms of the potential or ion profiles, neither of which coincides with the expression known from nonlinear PB theory, namely, the density of microions at the cell boundary. We show that the range of validity of linearization depends strongly on the linearization point and prove that expansion about the self-consistently determined average potential is optimal in several respects. For instance, screening inside the suspension is automatically described by the actual ionic strength, resulting in the correct asymptotics at high colloid concentration. Together with the analytical solution of the linear PB equation for cell models of arbitrary dimension and electrolyte composition, explicit and very general formulas for the osmotic pressure ensue. A comparison with nonlinear PB theory is provided. Our analysis also shows that whether or not linear theory predicts a phase separation depends crucially on the precise definition of the pressure, showing that depending on the choice, an artificial phase separation in systems as important as DNA in physiological salt solution may result.