A survey on properties of Nernst–Planck–Poisson system. Application to ionic transport in porous media

Abstract

The aim of this paper is to study the properties of Nernst-Planck-Poisson system describing the evolution of ionic concentrations in porous media. In particular, the well-posedness character of the system is proved and some qualitative properties of the global solution are established (energy and entropy laws, stationary states, Boltzmann distribution, influence of an external electrical field). We prove also that a superficial accumulation of ions occurs when an important external electrical field is applied.Nernst-Planck-Poisson system, cementitious materials, wellposedness, energy and entropy laws 1 Corresponding author. Email address : olivier.millet@univ-lr.fr (O. Millet). 2 The condition of non-negativity, linked to the well-posedness character of the system, is also crucial in the numerical solution of NPP system. 1 analysis of the macroscopic system is due to H. Gajewski and K. Gröger [20] in 1986. Specifically, the crucial step of their proofs consists in finding appropriate a priori estimates in a analytical relevant framework (Dubinskii's compactness results and Schauder's fixed point theorem, Banach and Hilbert spaces, Sobolev functions, in spirit of J.-L. Lions variational methods [30]): boundedness properties by the Moser's iteration technique ([24], p.21-24), estimates by means of a physically motived Liapunov function. A nonlinear version is obtained in 1997 by Dıaz, J.I., Galiano, G. and Jungel, A. [15][16] for a quasilinear degenerate system arising in semiconductors theory; a degenerate parabolic system consisting in two equations for densities of charged particles and in the Poisson equation for an electric field is considered. The proof are based on energy methods. In this vein, M. Schmuck provided a particularly interesting extension by investigating the Navier-Stokes-Nernst-Planck-Poisson system ([41] and his doctoral dissertation [43]). However, these purely mathematical papers need to be adapted to the physical framework and to the context of electrochemistry in order to respect consistency of formulas with the physical units and the dimensionnal analysis without loss of mathematical rigor. Due to their technological relevance and the industrial stakes, these systems received and are currently receiving considerable emphasis on the side of numerical analysis. A. Prohl and M. Schmuck [36][37] propose different finite element convergent discretizations in order to reproduce such sui generis behaviors as the energy and entropy principles by means of suitable fixed point algorithms and stopping criteria. Further finite element convergent discretizations are presented by these authors in the broader framework of the Navier-Stokes-Nernst-Planck-Poisson system [36][37]. Generally, it is convenient to assume that porous media have a periodic microstructure. Consequently, many papers are concerned with the Nernst-Planck-Poisson model posed in periodic porous media since the continuity equations of the system are written at the microscopic level whereas only macroscopic physical quantities are of ...