Flux Equation

The electro-diffusion equation for an electrolyte, referred to elsewhere as the extended Nernst-Planck equation [19,20,24], is a transport equation that characterizes flux due to a gradient in the total chemical potential that is composed of the internal chemical potential and the external chemical potential due to an electrical potential [25]. Within the electrolyte, the flux j_i of the i-th species is proportional to the gradient in the internal chemical potential µ _i (molar basis) and the gradient in the diffusive electrical potential $\psi_D$_D (electro-chemical potential) [26]:

$${\bf j}_i = \frac{-D_i}{RT} c_i \nabla \left(\mu_i + z_i {\cal F}\psi_D\right)$$ (2)

The quantity D_i is the diffusion coefficient within the electrolyte, z_i is the ionic valence, ${\cal F}$ is the Faraday constant, c_i is the ionic amount-of-substance concentration, R is the gas constant, and T is the absolute temperature. The diffusion potential $\psi_D$_D arises due to variations in the self-diffusion coefficients of the various ionic species. The flux relationship in Eqn. 2 has implicitly incorporated the Einstein relation between the diffusion coefficient D_i and the conventional ionic mobility u_i[26]:

$$D_i = RT\,u_i$$ (3)

Although this relationship is not exact for increasing ionic strengths, it is sufficiently accurate to capture the salient behavior of experimental systems, as will be demonstrated subsequently.

The flux in Eqn. 2 can be expressed in experimentally measurable variables by converting from chemical potential µ_i to chemical activity a_i, which can then be related to concentration through the mathematical construction of the activity coefficient $\gamma_i$_i [27]:

$$a_i = \gamma_i c_i$$ (4)

Making these substitutions leads to the electro-diffusive flux equation that is similar in form to those that have appeared elsewhere [4,7,19,20]:

$${\bf j}_i = -D_i \left(1+\frac{\partial \ln\gamma_i}{\parti... ...t) \nabla c_i - \frac{{\cal F}}{RT} z_i c_i D_i \nabla\psi_D$$ (5)

The activity coefficients can be calculated using the Pitzer equations [28] and the method can be incorporated into a computer code such as was done for the PHRQPITZ program [29]. The above flux equation, in conjunction with the diffusion potential $\psi_D$_D, is sufficient to describe transport in nonreactive porous systems with negligible binding.

For this study, the diffusion potential was chosen in such a manner as to ensure that the electrolytic solution remains neutral; there is no excess charge. This is accomplished by enforcing zero total current I_T everywhere (nil current condition) [7]:

$${\bf I}_T = {\cal F}\sum_i z_i {\bf j}_i = 0$$ (6)

If the system is initially neutral ( $\sum z_i c_i = 0$ z_ic_i = 0), the nil current condition will ensure electro-neutrality [8,24]. This approach is sufficient for diffusing species. An alternative approach is to solve the Nernst-Planck-Poisson system of equations [17,19,20].