Poisson-Boltzmann Theory of Charged Colloids: Limits of the Cell Model for Salty Suspensions

Thermodynamic properties of charge-stabilised colloidal suspensions are commonly modeled by implementing the mean-field Poisson-Boltzmann (PB) theory within a cell model. This approach models a bulk system by a single macroion, together with counterions and salt ions, confined to a symmetrically shaped, electroneutral cell. While easing solution of the nonlinear PB equation, the cell model neglects microion-induced correlations between macroions, precluding modeling of macroion ordering phenomena. An alternative approach, avoiding artificial constraints of cell geometry, maps a macroion-microion mixture onto a one-component model of pseudo-macroions governed by effective interactions. In practice, effective-interaction models are usually based on linear screening approximations, which can accurately describe nonlinear screening only by incorporating an effective (renormalized) macroion charge. Combining charge renormalization and linearized PB theories, in both the cell model and an effective-interaction (cell-free) model, we compute osmotic pressures of highly charged colloids and monovalent microions over a range of concentrations. By comparing predictions with primitive model simulation data for salt-free suspensions, and with predictions of nonlinear PB theory for salty suspensions, we chart the limits of both the cell model and linear-screening approximations in modeling bulk thermodynamic properties. Up to moderately strong electrostatic couplings, the cell model proves accurate in predicting osmotic pressures of deionized suspensions. With increasing salt concentration, however, the relative contribution of macroion interactions grows, leading predictions of the cell and effective-interaction models to deviate. No evidence is found for a liquid-vapour phase instability driven by monovalent microions. These results may guide applications of PB theory to soft materials.