## 3 Irreversible Thermodynamics Approach
### The Nernst-Planck Equation  
The Nernst-Planck equation is a partial case of the transport equations presented above. It is most simple to derive this equation from the Onsager Equation (1) [80]. If one neglects the cross phenomenological coefficients in Equation (1), sets \(p=const\) and assumes that the activity coefficients are constant, one immediately finds [78; 80]:  
\[J_{i}=-D_{i}\left(\frac{dc_{i}}{dx}+z_{i}c_{i}\frac{F}{RT}\frac{d\varphi}{dx}\right) \tag{17}\]  
where \(D_{i}=\frac{L_{i}RT}{c_{i}}\) is the diffusion coefficient of ion \(i\). The electrochemical potential gradient can be expressed through the ion activity (\(a_{i}\)) or concentration (as in Equation (17)) and the electrical potential (\(\phi\)) in any constituent phase of the membrane material. The use of the virtual solution is often convenient, since in this case \(a_{i}\) (\(c_{i}\)) and \(\phi\) have no jumps at the external membrane boundaries (if only the condition of local equilibrium is satisfied at these boundaries, which is verified for the under-limiting current densities [81; 82; 83]). More often, Equation (17) is applied to a solution and/or a membrane considered as a homogeneous medium [31; 57; 84]. In this case, the concentration and electric potential refer to this medium (which can be imagine as a swollen sponge with a charged matrix [31]) and not to a particular phase. To indicate the fact that a quantity refers to the membrane material (or its part) considered as a homogeneous medium, we will use an over-bar: \(\overline{c}_{i}\), \(\overline{\varphi}\) and \(\overline{D}_{i}\). The models, which use the approach where the permeants are assumed to be dissolved in the membrane material and transported there under the action of concentration gradient and/or external electric field, are called the \"solution-diffusion\" models [85; 86]. This term is used in contrast to the term \"pore-flow\" models, in which the transport of species is considered as occurring within a single membrane pore. Usually, this terminology is used in the context of the pressure-driven processes [86; 87], while it is also applied to the electromembrane processes [88; 89], where the transport occurs under the action of the electrochemical potential gradient. The Nernst-Planck equation is used as the governing equation in both types of the models mentioned above [90]. In the \"solution-diffusion\" models, it is applied in the membrane material considered as quasi-homogeneous medium (effective medium approach [91]), where the assumption of local electroneutrality (LEN) is commonly used. In the \"pore-flow\" models, this equation is applied within a single liquid phase, which is the pore solution. Usually, it is coupled there with the Poisson equation, thus taking into account the electric space charge in the EDL at the charged pore wall. It is also often convenient to use the transport equation in the membrane in the reduced Kedem-Katchalsky form:  
\[J_{i}=-\overline{D}\frac{d\overline{c}_{i}}{dx}+\frac{i\overline{t}_{i}}{z_{i} \overline{F}}=-P\frac{dc_{i}}{dx}+\frac{i\overline{t}_{i}}{z_{i}\overline{F}} \tag{18}\]  
Equation (18) is obtained from Equation (5) of the Kedem-Katchalsky set of equations when neglecting the term containing \(dp/dx\). Equation (18) is written in two ways. The first one is applied when the membrane is considered as a quasi-homogeneous gel, in this case the quantities are denoted with an over-bar; the concentrations refer to the number of moles per unit volume of the gel. When the concentrations refer to the virtual electroneutral solution (\(c_{i}\)) (the second way of writing), the differential diffusion permeability coefficient, \(P\), is used instead of the diffusion coefficient, \(\overline{D}\). In both cases, the transport number has the same meaning: it is the fraction of electric charge transported by ion \(i\) in the membrane in conditions where the concentration gradient is zero. It is possible also to write this equation for a solution.