Homogenization of the Poisson-Nernst-Planck Equations for Ion Transport in Charged Porous Media

Effective Poisson-Nernst-Planck (PNP) equations are derived for macroscopic ion transport in charged porous media under periodic fluid flow by an asymptotic multi-scale expansion with drift. The microscopic setting is a two-component periodic composite consisting of a dilute electrolyte continuum (described by standard PNP equations) and a continuous dielectric matrix, which is impermeable to the ions and carries a given surface charge. Four new features arise in the upscaled equations: (i) the effective ionic diffusivities and mobilities become tensors, related to the microstructure; (ii) the effective permittivity is also a tensor, depending on the electrolyte/matrix permittivity ratio and the ratio of the Debye screening length to the macroscopic length of the porous medium; (iii) the microscopic fluidic convection is replaced by a diffusion-dispersion correction in the effective diffusion tensor; and (iv) the surface charge per volume appears as a continuous "background charge density", as in classical membrane models. The coefficient tensors in the upscaled PNP equations can be calculated from periodic reference cell problems. For an insulating solid matrix, all gradients are corrected by the same tensor, and the Einstein relation holds at the macroscopic scale, which is not generally the case for a polarizable matrix, unless the permittivity and electric field are suitably defined. In the limit of thin double layers, Poisson's equation is replaced by macroscopic electroneutrality (balancing ionic and surface charges). The general form of the macroscopic PNP equations may also hold for concentrated solution theories, based on the local-density and mean-field approximations. These results have broad applicability to ion transport in porous electrodes, separators, membranes, ion-exchange resins, soils, porous rocks, and biological tissues

New porous medium Poisson-Nernst-Planck equations for strongly oscillating electric potentials

We consider the Poisson-Nernst-Planck system which is well-accepted for describing dilute electrolytes as well as transport of charged species in homogeneous environments. Here, we study these equations in porous media whose electric permittivities show a contrast compared to the electric permittivity of the electrolyte phase. Our main result is the derivation of convenient low-dimensional equations, that is, of effective macroscopic porous media Poisson-Nernst-Planck equations, which reliably describe ionic transport. The contrast in the electric permittivities between liquid and solid phase and the heterogeneity of the porous medium induce strongly oscillating electric potentials (fields). In order to account for this special physical scenario, we introduce a modified asymptotic multiple-scale expansion which takes advantage of the nonlinearly coupled structure of the ionic transport equations. This allows for a systematic upscaling resulting in a new effective porous medium formulation which shows a new transport term on the macroscale. Solvability of all arising equations is rigorously verified. This emergence of a new transport term indicates promising physical insights into the influence of the microscale material properties on the macroscale. Hence, systematic upscaling strategies provide a source and a prospective tool to capitalize intrinsic scale effects for scientific, engineering, and industrial applications

Role of non-ideality for the ion transport in porous media: derivation of the macroscopic equations using upscaling

This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. We first consider equilibrium solutions in the absence of external forces. In such a case, the velocity and diffusive fluxes vanish and the equilibrium electrostatic potential is the solution of a variant of Poisson-Boltzmann equation coupled with algebraic equations. Contrary to the ideal case, this nonlinear equation has no monotone structure. However, based on invariant region estimates for Poisson-Boltzmann equation and for small characteristic value of the solute packing fraction, we prove existence of at least one solution. To our knowledge this existence result is new at this level of generality. When the motion is governed by a small static electric field and a small hydrodynamic force, we generalize O'Brien's argument to deduce a linearized model. Our second main result is the rigorous homogenization of these linearized equations and the proof that the effective tensor satisfies Onsager properties, namely is symmetric positive definite. We eventually make numerical comparisons with the ideal case. Our numerical results show that the MSA model confirms qualitatively the conclusions obtained using the ideal model but there are quantitative differences arising that can be important at high charge or high concentrations.Comment: 46 page

Rigorous Homogenization of a Stokes-Nernst-Planck-Poisson Problem for various Boundary Conditions

We perform the periodic homogenization (i.e. \eps\to 0) of the non-stationary Stokes-Nernst-Planck-Poisson system using two-scale convergence, where \eps is a suitable scale parameter. The objective is to investigate the influence of \textsl{different boundary conditions and variable choices of scalings in \eps} of the microscopic system of partial differential equations on the structure of the (upscaled) limit model equations. Due to the specific nonlinear coupling of the underlying equations, special attention has to be paid when passing to the limit in the electrostatic drift term. As a direct result of the homogenization procedure, various classes of upscaled model equations are obtained.Comment: 29 pages, 1 figur

Numerical homogenization of electrokinetic equations in porous media using lattice-Boltzmann simulations

International audienceWe report the calculation of all the transfer coefficients which couple the solvent and ionic fluxes through a charged pore under the effect of pressure, electrostatic potential, and concentration gradients. We use a combination of analytical calculations at the Poisson-Nernst-Planck and Navier-Stokes levels of description and mesoscopic lattice simulations based on kinetic theory. In the absence of added salt, i.e., when the only ions present in the fluid are the counterions compensating the charge of the surface, exact analytical expressions for the fluxes in cylindrical pores allow us to validate a new lattice-Boltzmann electrokinetics (LBE) scheme which accounts for the osmotic contribution to the transport of all species. The influence of simulation parameters on the numerical accuracy is thoroughly investigated. In the presence of an added salt, we assess the range of validity of approximate expressions of the fluxes computed from the linearized Poisson-Boltzmann equation by a systematic comparison with LBE simulations

Ion transport through deformable porous media: derivation of the macroscopic equations using upscaling

We study the upscaling or homogenization of the transport of a multicomponentelectrolyte in a dilute Newtonian solvent through a deformable porous medium.The pore scale interaction between the flow and the structure deformation is taken into account.After a careful adimensionalization process, we first consider so-called equilibrium solutions,in the absence of external forces, for which the velocity and diffusive fluxes vanish andthe electrostatic potential is the solution of a Poisson-Boltzmann equation.When the motion is governed by a small static electric field and small hydrodynamic and elastic forces,we use O'Brien's argument to deduce a linearized model. Then we perform the homogenizationof these linearized equations for a suitable choice of time scale. It turns out thatthe deformation of the porous medium is weakly coupled to the electrokinetics systemin the sense that it does not influence electrokinetics although the latter one yieldsan osmotic pressure term in the mechanical equations. As a byproduct we find that theeffective tensor satisfies Onsager properties, namely is symmetric positive definite