Where the linearized Poisson-Boltzmann cell model fails: (II) the planar case as a prototype study

TL;DR

This paper analyzes the limitations of linearized Poisson-Boltzmann models for charged planes, showing they fail in certain regimes and produce unphysical results, despite being accurate in weak-coupling limits.

Contribution

It provides an exact nonlinear solution for charged planes and critically assesses the validity of linearized models, highlighting their failures in specific regimes.

Findings

Linearized models agree with nonlinear results in weak-coupling limits.

Artifacts such as negative osmotic pressure arise in linearized models outside these limits.

Explicit gauge-invariant analysis explains the origin of linearization artifacts.

Abstract

The classical problem of two uniformly charged infinite planes in electrochemical equilibrium with an infinite monovalent salt reservoir is solved exactly at the mean-field nonlinear Poisson-Boltzmann (PB) level, including an explicit expression of the associated nonlinear electrostatic contribution to the semi-grand-canonical potential. A linearization of the nonlinear functional is presented that leads to Debye-H\"uckel-like equations agreeing asymptotically with the nonlinear PB results in the weak-coupling (high-temperature) and counterionic ideal-gas limits. This linearization scheme yields artifacts in the low-temperature, large-separation or high-surface charge limits. In particular, the osmotic-pressure difference between the interplane region and the salt reservoir becomes negative in the above limits, in disagreement with the exact (at mean-field level) nonlinear PB solution. 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 -- can be traced back to the non-fulfillment of the underlying assumptions of the linearization. Explicit comparison between the analytical expressions of the exact nonlinear solution and the corresponding linearized equations allows us to show that the linearized results are asymptotically exact in the weak-coupling and counterionic ideal-gas limits, but always fail otherwise, predicting negative osmotic-pressure differences.