Bulk Equation

The previous system of equations, Eqns. 5 and 6, apply only to the electrolyte within the pores. For porous materials, one must be explicit whether the flux equation is for the pore space or for the pore space and solid structure combined (bulk). In this study, a bulk formulation is used. For the electro-diffusive flux in Eqn. 5, it has been shown previously that the diffusion coefficient D_i that appears in Eqn. 5 is the bulk microstructural diffusion coefficient ( = D_i), and is independent of the pore solution composition [7]; interactions within the pore solution are accounted for in the remaining terms in the equation. For a porous material with a formation factor F, the bulk microstructural diffusion coefficient appearing in Eqn. 5 is proportional to the self-diffusion coefficient D^s [7]:

$$D^{\mu}_i = \frac{D^s_i}{F}$$ (7)

The self-diffusion coefficient of an ionic species is its diffusion coefficient in water in the limit of infinite dilution [30]. Equation 7 reflects the fact that the bulk microstructural diffusion coefficient $D^{\mu}_i$ depends on the species i. It is important to realize that $D^{\mu}_i$ is not the apparent Fickian diffusion coefficient D^a_i. Rather, it simply characterizes the diffusion of an ionic species in the absence of electrostatic interactions.

Use of the formation factor originated in geological research and has been defined as the ratio of the conductivity of a pore solution $\sigma _p$_p to the bulk conductivity $\sigma _b$_b of a insulating porous material filled with that pore solution [31]:

$$F= \frac{\sigma_p}{\sigma_b}$$ (8)

In principle, the formation factor is a constant, at constant porosity and tortuosity (absence of chemical reaction with solid microstructure), for a particular porous material, and is independent of ionic species [7]. Since the self-diffusion coefficient of an ionic species is also a constant (at constant temperature), the flux equation for the N species has been reduced from a system of N equations with N apparent diffusion coefficients, as shown in Eqn. 5, to a system of N equations with only a single parameter, the formation factor.

In practice, determining the formation factor must be performed with some care. In porous systems containing sufficient surface charge, the ratio of the pore solution conductivity to the bulk conductivity will depend on the pore solution concentration [32,33]. This concentration dependence is due to a surface conduction contribution to the measured conductivity [34,35]. In principle, this effect can be isolated, yielding the true microstructural formation factor that is required in the equations used here.