CHARGE-TRANSPORT THROUGH A SPATIALLY PERIODIC POROUS-MEDIUM - ELECTROKINETIC AND CONVECTIVE DISPERSION PHENOMENA

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 (E) over bar, as well as a homogeneous applied pressure-gradient field <(del)over bar>(p) over bar. 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 <(sigma)over bar>; (ii) the hydraulic permeability (K) over bar; (iii) the 'streaming potential' coupling dyadic (K) over bar(P)(c) and (iv) the 'electroosmotic' coupling dyadic (K) over bar(E)(c). General formulas for these gross-scale, phenomenological coefficients are provided in terms of four spatially periodic, microscale dyadic fields (del g, del 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) over bar* and dispersivity (D) over bar* 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 (<(sigma)over bar>, (K) over bar, (K) over bar(P)(c), (K) over bar(E)(c)) 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.