Poisson–Boltzmann theory of charged colloids: limits of the cell model for saltysuspensions

Abstract

Thermodynamic properties of charge-stabilized colloidal suspensions and polyelectrolytesolutions are commonly modelled by implementing the mean-field Poisson–Boltzmann(PB) theory within a cell model. This approach models a bulk system by a singlemacroion, together with counterions and salt ions, confined to a symmetricallyshaped, electroneutral cell. While easing numerical solution of the nonlinear PBequation, the cell model neglects microion-induced interactions and correlationsbetween macroions, precluding modelling of macroion ordering phenomena. Analternative approach, which avoids the artificial constraints of cell geometry, exploitsthe mapping of a macroion–microion mixture onto a one-component model ofpseudo-macroions governed by effective interparticle interactions. In practice,effective-interaction models are usually based on linear-screening approximations,which can accurately describe strong nonlinear screening only by incorporatingan effective (renormalized) macroion charge. Combining charge renormalizationand linearized PB theories, in both the cell model and an effective-interaction(cell-free) model, we compute osmotic pressures of highly charged colloids andmonovalent microions, in Donnan equilibrium with a salt reservoir, over a range ofconcentrations. By comparing predictions with primitive model simulation data for salt-freesuspensions, and with predictions from nonlinear PB theory for salty suspensions, wechart the limits of both the cell model and linear-screening approximations inmodelling bulk thermodynamic properties. Up to moderately strong electrostaticcouplings, the cell model proves accurate for predicting osmotic pressures of deionized(counterion-dominated) suspensions. With increasing salt concentration, however, therelative contribution of macroion interactions to the osmotic pressure grows, leadingpredictions from the cell and effective-interaction models to deviate. No evidence isfound for a liquid–vapour phase instability driven by monovalent microions. Theseresults may guide applications of PB theory to colloidal suspensions and other softmaterials.