Dispersion of charged tracers in charged porous media

We report a lattice-Boltzmann scheme to compute the dispersion of charged tracers in charged porous media under the combined effect of advection, diffusion and electro-migration. To this end, we extend the moment propagation approach, introduced to study the dispersion of neutral tracers (Lowe C. and Frenkel D., Phys. Rev. Lett., 77 (1996) 4552), to include the effect of electrostatic forces. This method allows us to compute the velocity autocorrelation function of the charged tracers with high accuracy. The algorithm is validated studying the dispersion coefficient in the case of electro-osmotic flow in a slit without added salt. We find excellent agreement between the numerical and analytical results. This method also provides the full time dependence of the diffusion coefficient, including for charged tracers. We illustrate on the slit case how D(t), which is measured by NMR to probe the geometry of porous media, reflects how the porosity explored by tracers depends on their charge. Copyright c © EPLA, 2008 The transport of charged particles in charged porous media, under the combined effect of advection, diffusion and migration, is encountered in many situations of technological interest, such as the separation of species by electro-osmotic flow in capillaries, microfluidic devices and ion-exchange membranes. Contamination of soils by toxic or radioactive waste, which often consist of charged species, are examples of transport of charged tracers in porous media with considerable environmental relevance. Similarly, a better understanding of the electro-kinetic removal of such contaminants is of considerable practical importance. Even in complex biological systems such as the living cell, electro-kinetic transport (often in the nonlinear regime) plays a relevant role. A saturated porous media consists by definition of a solid phase and a fluid phase. At mean-field level and neglecting the finite size of the particles, the dynamics of charged species in the fluid phase can be described in terms of the species density ρ(r, t), which follows a convection-diffusion equation ∂tρ(r, t)+∇·J(r, t) = 0, (1a) J= ρu−D∇ρ+ ρβD[qE−∇V ], (1b) (a)E-mail: [email protected] where D is the particle’s molecular diffusion coefficient, q its charge, and β = 1/kBT . In addition to the fluxes due to advection (ρu) and diffusion (−D∇ρ), two forcing terms contribute to the charged species dynamics. The first includes the effect of particle interactions with the rest of the system, as the gradient of the mean-field potential V . This potential is in fact the excess part of the tracer chemical potential. It is not necessarily limited to the electrostatic term qψ where ψ satisfies the Poisson equation ∆ψ= ρel/ 0 r, with ρel the charge density, 0 and r the void and fluid relative permittivities, respectively. The second contribution accounts for the effect of an external electric field, which does not necessarily derive from a potential, e.g. for infinite systems. Although the dispersion of tracers could in principle be investigated by studying the spreading of a tracer pulse, the evaluation of the dispersion coefficient requires solving eq. (1), e.g. with a finite element method, for a large number of initial conditions, which rapidly becomes computationally intractable for complex porous media. An alternative is to evaluate it for each tracer charge q from the tracer velocity auto-correlation function (VACF)