We know the Nernst Planck equation is $$ \frac{\partial c}{\partial t} = - \nabla \cdot J \quad | \quad J = -\left[ D \nabla c - u c + \frac{Dze}{k_\mathrm{B} T}c\left(\nabla \phi+\frac{\partial \mathbf A}{\partial t}\right) \right] $$ $$\iff\frac{\partial c}{\partial t} = \nabla \cdot \left[ D \nabla c - u c + \frac{Dze}{k_\mathrm{B} T}c\left(\nabla \phi+\frac{\partial \mathbf A}{\partial t}\right) \right] $$ We know this is basically a continuity equation where a partial of a conserved density quantity is equal to the divergence of its flux. The first term is derived from Ficks Law, the second term is simply a convection term flux=velocity*density and the final term is due to Einstein's Relation stating $$ D = \mu \, k_\text{B} T$$ Where μ is the "mobility", or the ratio of the particle's terminal drift velocity to an applied force, μ = vd/F and we know F=EQ and V=ED so the third term is derived from these equations. I am having trouble with the modified Nernst Planck equation for porous media where we have $$ \nabla (\kappa_{eff}\:\ln C)=\mbox{Current Density}$$ where $$ \kappa_{eff}=\frac{2RT\kappa}{F}\left(t_+-1\right)\left(1+\frac{d \ln f}{d \ln C}\right)=D\left(1+\frac{d \ln f}{d \ln C}\right)$$ Where does this term come from? Logarithm of the activity would imply a chemical potential but how is this related to diffusion?
2 Answers
The Nernst-Planck equation assumes an ideal thermodynamic behavior of ions, i.e., unit activity coefficients. Additionally, it uses self diffusivities, $D$, to describe the response of ions to an applied electric field.
This interpretation is at best valid for dilute electrolytes.
A generalized treatment of ion motion employs Stefan-Maxwell diffusivities, $\mathfrak{D}_{ij}$, and non-ideal energies of the ions, i.e., $f \left(c\right) \neq 1$. This leads to concentrated solution theory (https://onlinelibrary.wiley.com/doi/abs/10.1002/bbpc.19650690712). In this description, the total ionic current density is defined as \begin{equation} i = - \kappa \left( \nabla \phi - \frac{2RT}{F} \left(1-t_+^0\right) \left(1 + \frac{\mathrm{d} \ln f}{\mathrm{d} \ln c}\right) \nabla \ln c \right) \end{equation}. Note that this equation is only valid for an electrolyte with a monovalent cation, monovalent anion, and a solvent. One can derive equivalent expressions for other electrolytes.
Starting from the constitutive relations of the concentration solution theory, one can derive conservation equations in porous electrodes as described in https://iopscience.iop.org/article/10.1149/1.2221597/meta.
I see now that for Chemical Systems of non-ideal solutions/mixtures Ficks First Law is: $$ J_i = - \frac{D c_i}{RT} \frac{\partial \mu_i}{\partial x}$$ This is a restatement of the First Law as $$ J_i = - \frac{D c_i}{RT} \frac{\partial \mu_i}{\partial x}= - \frac{D c_i}{RT} \frac{RT\partial \ln C}{\partial x} \implies d \ln C =\frac{dC}{C}\implies J_i = - D \frac{\partial C}{\partial x}.$$