Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

Abstract

A numerical scheme for the transient solution of a generalized version of the Poisson-Nernst-Planck (PNP) equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The PNP equations represent a set of diffusion equations for charged species, i.e. dissolved ions, present in the pore solution of a rigid porous material in which the surface charge can be assumed neglectable. These equations are coupled to the 'internally' induced electrical field and to the velocity field of the fluid. The Nernst-Planck equations describing the diffusion of the ionic species and Gauss' law in use are, however, coupled in both directions. The governing set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). The simplifications used here mainly concerns ignoring the deformation and stresses in the porous material in which the ionic diffusion occurs. The HMT is a special version of the more 'classical' continuum mixture theories in the sense that it works with averaged equations at macroscale and that it includes the volume fractions of phases in its structure. The background to the PNP equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with Gauss' law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasistatic version of Maxwell's equations making it suitable for analyses of the kind addressed in this work. Within the framework of HTM, constitutive equations have been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.

Use of a multi-species reactive transport model to simulate chloride ingress in mortar exposed to NaCl solution or sea-water

“…A weak form of Eqs. (1) and (2) is needed for discretization of the problem, see e.g., Samson et al [13] and Johannesson [14]. The state variables in the weak formulations are approximated by the general expansion Na where N is the global shape function and a contains the state variables at the nodal points of the domain.…”

“…(3). A modified Newton-Raphson scheme, which ignore the higher order terms in the Taylor expansion, is employed for this study, see e.g., Ottosen and Ristinmaa [15] and Johannesson [14]. The used Newton-Raphson scheme require a truly implicit time integration scheme, h ¼ 1 ð Þ , which reduces Eq.…”

Singular perturbation solutions of steady-state Poisson-Nernst-Planck systems

“…,λ n−1 . Finally, we will determine the integration constants d i in (30) in terms of the roots λ i .…”

“…Thus, the crucial part of this subsection is to find a system of algebraic equations for λ i . First, substituting (30) into (25)…”

Modeling of Electric Double-Layers Including Chemical Reaction Effects

“…The widely known analytical solution of the GC model derives from the Nernst-Planck-Poisson (NPP) system of equations, which mathematically describes the reactive-transport phenomena of chemical species in multi-species electrolytes [7][8][9][10][11]. In the GC model, the Boltzmann distribution and the Poisson-Boltzmann equations are used to describe respectively the ionic concentration and the electric potential in the diffuse layer, and they derive from the NPP system for the steady-state case.…”