EIS simulations for ion-selective site-based membranes by a numerical solution of the coupled Nernst–Planck–Poisson equations

Theory of the formation of the electric double layer at the ion exchange membrane-solution interface

This work aims to extend the study of the formation of the electric double layer at the interface defined by a solution and an ion-exchange membrane on the basis of the Nernst-Planck and Poisson equations, including different values of the counter-ion diffusion coefficient and the dielectric constant in the solution and membrane phases. The network simulation method is used to obtain the time evolution of the electric potential, the displacement electric vector, the electric charge density and the ionic concentrations at the interface between a binary electrolyte solution and a cation-exchange membrane with total co-ion exclusion. The numerical results for the temporal evolution of the interfacial electric potential and the surface electric charge are compared with analytical solutions derived in the limit of the shortest times by considering the Poisson equation for a simple cationic diffusion process. The steady-state results are justified from the Gouy-Chapman theory for the diffuse double layer in the limits of similar and high bathing ionic concentrations with respect to the fixed-charge concentration inside the membrane. Interesting new physical insights arise from the interpretation of the process of the formation of the electric double layer at the ion exchange membrane-solution interface on the basis of a membrane model with total co-ion exclusion.

Parallel Adaptive Finite Element Algorithms for Solving the Coupled Electro-diffusion Equations

rithms for solving the 3D electro-diffusion equations such as the Poisson-Nernst-Planck equations and the size-modified Poisson-Nernst-Planck equations in simulations of biomolecular systems in ionic liquid. A set of transformation methods based on the generalized Slotboom variables is used to solve the coupled equations. Calculations of the diffusion-reaction rate coefficients, electrostatic potential and ion concentrations for various systems verify the method’s validity and stability. The iterations between the Poisson equation and the Nernst- Planck equations in the primitive method and in the transformation method are compared to illustrate how the new method accelerates the convergence of the solution. To speed up the convergence, we introduce the DIIS (direct inversion of the iterative subspace) method including Simple Mixing and Anderson Mixing as under-relaxation techniques, the effectiveness of which on acceleration is shown by numerical tests. It is worth noting that the primitive method fails to solve the size-modified Poisson-Nernst-Planck equations for real protein systems but the transformation method succeeds in the simulations of the ACh-AChE reaction system and the DNA fragment. To improve the accuracy of the solution, we introduce high order elements and mesh adaptation based on an a posteriori error estimator. Numerical results indicate that our mesh adaptation process leads to quasi-optimal convergence. We implement our algorithms using the parallel adaptive finite element package PHG [53] and high parallel efficiency is obtained.

Computer Simulation of Ion-Selective Membrane Electrodes and Related Systems by Finite-Element Procedures

A simple but powerful numerical simulation for analyzing the electrochemical behavior of ion-selective membranes and liquid junctions is presented. The computer modeling makes use of a finite-element procedure in the space and time domains, which can be easily processed (e. g., with MS Excel software) without the need for complex mathematical evaluations. It leads to convincing results on the dynamic evolution of concentration profiles, potentials, and fluxes in the studied systems. The treatment accounts for influences of convection, flow, or stirring in the sample solution that act on the boundary diffusion layer and it is even capable of including the effects of an electrolyte flow through the whole system. To minimize the number of arbitrary parameters, interfacial reactions are assumed to be near local equilibrium, and space-charge influences are considered via phase-boundary potential differences. The applicability of the computer simulation is demonstrated for different ion-selective membranes as well as for liquid junctions. The numerical results are in excellent agreement with experimental data.