# On the Accuracy of Describing Polyelectrolyte Systems Based on Cross-Linked Networks in Terms of Linear Differential Equations

## TL;DR

This paper investigates when linear equations can accurately describe complex polyelectrolyte systems, especially those far from equilibrium.

## Contribution

The study derives a generalized linear differential equation valid for non-equilibrium polyelectrolyte systems, avoiding the Boltzmann distribution approximation.

## Key findings

- Linear models are valid under specific conditions even for non-equilibrium systems.
- A generalized linear differential equation was developed for polyelectrolyte systems with steady electric currents.

## Abstract

Theoretical models of polyelectrolyte systems with cross-linked polymer networks are often simplified to linear differential equations by means of the linearized Poisson–Boltzmann approximation, whose validity is traditionally limited to cases where the electrostatic potentials are small. However, the limits of applicability of the linear theory remain debatable in many cases. Moreover, the Poisson–Boltzmann equation is, in principle, not applicable to the description of non-equilibrium systems, particularly those through which an electric current flows. In the present work, a direct comparison is carried out between the exact solution and the approximate solution (i.e., the solution obtained within the framework of the linearization procedure) of the equations describing the contact region between a cross-linked polyelectrolyte network and a low-molecular-mass salt solution. This makes it possible to determine the conditions under which the linear model is applicable, including for the analysis of promising systems in the field of organic electronics. The conclusions obtained in this work are based on basic electrostatics equations and transport equations of low-molecular-mass ions. The proposed approach also makes it possible to obtain a generalized linear differential equation that is not subject to a Boltzmann distribution approximation and is valid for polyelectrolyte systems rather far from thermodynamic equilibrium and even carrying steady electric currents.