Finite Element Simulation of Modified Poisson-Nernst-Planck/Navier-Stokes Model for Compressible Electrolytes under Mechanical Equilibrium

TL;DR

The paper develops a finite element method for a thermodynamically consistent modified Poisson–Nernst–Planck/Navier–Stokes model tailored to compressible electrolytes under mechanical equilibrium. By replacing the classical diffusion flux with the implicit Dreyer–Guhlke–Müller flux, the authors obtain a nonlinear, coupled framework that admits a reduced modified Poisson–Boltzmann formulation at equilibrium and a dimensionless, dimensionally robust set of equations. The method, implemented in FEniCS, handles both compressible and incompressible regimes via a bulk modulus $K$ and demonstrates convergence, boundary-layer behavior, and meaningful differences from classical NP, including cross-diffusion and saturation effects in concentrated electrolytes. Numerical experiments across 1D–3D domains, including annular geometries and temperature variations, reveal how compressibility and temperature influence space-charge layers and double-layer capacitance, with results aligning with theoretical predictions in the incompressible limit. The work provides a practical, scalable tool for simulating complex electrokinetic systems and offers insights for battery design and energy storage in systems where density variations are important.

Abstract

This work presents a finite element method for a modified Poisson-Nernst-Planck/Navier-Stokes (PNP/NS) model under the mechanical equilibrium, developed for compressible electrolytes. The modification is based on the new model proposed by Dreyer, Guhlke and Muller [39], where the diffusion flux in the classical PNP system is replaced with an implicitly involved new diffusion flux, leading to fractional nonlinearity. He and Sun [42] previously developed a numerical approach for another type of modification, where the Poisson equation in the PNP system was substituted with a fourth-order elliptic equation. Another key contribution of this work is the reduction of the equilibrium system to a modified Poisson-Boltzmann system. The proposed numerical scheme is capable of handling both compressible and incompressible regimes by employing a bulk modulus parameter, which governs the fluid's compressibility and enables seamless transition between these regimes. To emphasize practical relevance, we discuss the implications of compressible electrolytes in the context of double-layer capacitance behavior. We also conduct numerical simulations over various domains to demonstrate its applicability under various operating conditions, including temperature fluctuations and variations in the bulk modulus. The numerical results validate the accuracy and robustness of our computational scheme and demonstrate that the observed limiting behavior for the incompressible regime aligns with the theoretical trends anticipated by Dreyer et al. [39].

Figures (13)

• Figure 1: Schematic of a simple electrochemical cell with zinc and copper electrodes in NaCl electrolyte solution.
• Figure 2: Log-Log plot showing second order of convergence in $L^{2}$ norm in example \ref{['Sec:6.1']}.
• Figure 3: Boundary conditions on the electric potential $\varphi$ for the unit square domain in Example \ref{['Sec:6.2']}. All other boundary conditions for each variable are the same as those in Section \ref{['Bdry']}.
• Figure 4: Numerical solutions of (a) cation $y_c$, (b) anion $y_a$, (c) electric potential $\varphi$, and (d) total number density $n$ for different applied potentials on the unit interval in Example \ref{['Sec:6.2']}. The parameter settings are detailed in Table \ref{['Compressible']}, and the boundary conditions are described in Section \ref{['Bdry']}.
• Figure 5: Numerical solutions of (a) cation $y_c$, (b) anion $y_a$, (c) electric potential $\varphi$, and (d) total number density $n$ with applied potential $\pm 1$ unit on the 2D square domain in Example \ref{['Sec:6.2']}. For parameter values, see Table \ref{['Compressible']}; for boundary conditions, see Section \ref{['Bdry']} and Fig. \ref{['Fig:3']}.