Where the linearized Poisson-Boltzmann cell model fails: (I) spurious phase separation in charged colloidal suspensions

TL;DR

This paper analyzes the limitations of the linearized Poisson-Boltzmann cell model, revealing that it predicts spurious phase separation and negative compressibility in charged colloidal suspensions, which are artifacts not present in the nonlinear solution.

Contribution

The study identifies and explains the failure modes of the linearized Poisson-Boltzmann model, highlighting its inaccuracies in predicting phase behavior in colloidal suspensions.

Findings

Linearized model predicts spurious phase separation at high colloidal charges.

Negative compressibility and osmotic pressure differences are artifacts of linearization.

These artifacts are linked to the non-fulfillment of linearization assumptions.

Abstract

We perform a linearization of the Poisson-Boltzmann (PB) density functional for spherical Wigner-Seitz cells that yields Debye-H\"uckel-like equations agreeing asymptotically with the PB results in the weak-coupling (high-temperature) limit. Both the canonical (fixed number of microions) as well as the semi-grand-canonical (in contact with an infinite salt reservoir) cases are considered and discussed in a unified linearized framework. In the canonical case, for sufficiently large colloidal charges 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. In the semi-grand-canonical case it is predicted that the isothermal compressibility and the osmotic-pressure difference between the colloidal suspension and the salt reservoir become negative in the low-temperature, high-surface charge or infinite-dilution (of polyions) limits. As already pointed out in the literature for the latter case, these features are in disagreement with the exact nonlinear PB solution inside a Wigner-Seitz cell and are thus artifacts of the linearization. By using explicitly gauge-invariant forms of the electrostatic potential we show that these artifacts, although thermodynamically consistent with quadratic expansions of the nonlinear functional and osmotic pressure, may be traced back to the non-fulfillment of the underlying assumptions of the linearization.