Charge transport through a spatially periodic porous medium : electrokinetic and convective dispersion phenomena

Abstract

A general theory is outlined for the transport of charge and the convective dispersion of charged species through a spatially periodic porous medium under the influence of a homogeneous, Darcy-scale electric field Ē , as well as a homogeneous applied pressure-gradient field ∇ ¯ p ¯ . The particulate surfaces of the porous medium are characterized as possessing a non-uniform surface charge, with thin, Helmholtz double layers bordering the charged surfaces in the interstitial fluid phase. The theory uses a straightforward application of macrotransport theory, as well as standard methods of analysis of transport phenomena in spatially periodic systems, to derive, first, general expressions for the following four Darcy-scale, electromechanical-transduction property dyadics: (i) the effective electrical conductivity σ ¯ ; (ii) the hydraulic permeability K ¯ ; (iii) the ‘streaming potential’ coupling dyadic K ¯ P C ; and (iv) the ‘electroosmotic’ coupling dyadic K ¯ E C . General formulas for these gross-scale, phenomenological coefficients are provided in terms of four spatially periodic, microscale dyadic fields ( ∇ g , ∇ h , V , V E ). Unit-cell, boundary-value problems are derived for determining these latter dyadics as functions of the microscale geometrical and physicochemical nature of the porous material. In addition, formulas for computing the mean velocity U ¯ ∗ and dispersivity D ¯ ∗ of a charged, convecting and diffusing Brownian particle (or cluster of particles) are presented. Two explicit examples are offerred to illustrate the implementation of the theory. In the first example, a charged, pointsize, Brownian particle is imagined as convecting and diffusing within a porous medium composed of parallel, charged, rectilinear plates between which a Newtonian fluid flows and an electric field is applied. In the second example, leading-order expressions are derived for the electrokinetic transductive properties ( U ¯ ∗ ) of a highly porous two-dimensional array of charged circular cylinders though which a Newtonian fluid flows. These leading-order results are found to be in agreement with results appearing in the literature.