Where the linearized Poisson–Boltzmann cell model fails: Spurious phase separation in charged colloidal suspensions

Abstract

The Poisson–Boltzmann (PB) spherical Wigner–Seitz cell model—introduced to theoretically describe suspensions of spherical charged colloidal particles—is investigated at the nonlinear and linearized levels. The linearization of the mean-field PB functional yields linearized Debye–Hückel-type equations agreeing asymptotically with the nonlinear PB results in the weak-coupling (high-temperature) limit. Both the canonical (fixed number of microions) as well as the semigrand-canonical (in contact with an infinite salt reservoir) cases are considered and discussed in a unified linearized framework. In disagreement with the exact nonlinear PB solution inside a Wigner–Seitz cell, the linearized theory predicts the occurrence of a thermodynamical instability with an associated phase separation of the homogeneous suspension into dilute (gas) and dense (liquid) phases, being thus a spurious result of the linearization. We show that these artifacts, although thermodynamically consistent with quadratic expansions of the nonlinear functional and osmotic pressure, may be traced back to the nonfulfillment of the underlying assumptions of the linearization. This raises questions about the reliability of the prediction of gas/liquid-like phase separation in deionized aqueous suspensions of charged colloids mediated by monovalent counterions obtained by linearized theories.