## 3 Irreversible Thermodynamics Approach
### The Onsager Phenomenological Equations  
The strength but also the weakness of the description of ion and water transport across the membranes in the framework of irreversible thermodynamics is that information about the membrane structure and transport mechanism is not used. This approach gives general relationships between fluxes and driving forces, which are valid for any type of membrane. The specificity of a given membrane is taken into account through a set of phenomenological coefficients, which cannot be found within this approach. Usually, these coefficients are determined in specially designed experiments, which is sufficient for engineering purposes. However, a more theoretical way is possible, when the transport equations of irreversible thermodynamics are completed with other equations/boundary conditions, taking into account the details of membrane structure. At the equilibrium state, the electrochemical potential, \(\mu_{i}\), of any mobile species \(i\) (an ion or a molecule) is the same in every point of the membrane material. If nonzero gradients \(d\mu_{i}/dx\) are applied, fluxes of different species appear. If the system is close to equilibrium, linear relations between the thermodynamic forces, \(F_{j}=-d\mu_{i}/dx\) and resulting fluxes, \(J_{i}\), can be written [56; 61]:  
\[J_{i}=-\sum_{j}L_{ij}d\mu_{j}/dx \tag{1}\]  
where \(L_{ij}\) are the phenomenological conductance coefficients. In addition to the material properties, \(L_{ij}\), can depend on the species concentrations, temperature and pressure but not on fluxes and forces. It is assumed that no net transfer occurs in the direction parallel to the membrane surface, and, therefore, the value of \(\mu_{i}\) is the same at every point of the plane normal to this direction, regardless of the phase through which the plane passes (Figure 3). \(dx\) in Equation (1) can be interpreted as a distance between two planes perpendicular to the transport axis \(x\), the first one corresponding to the value of \(\mu_{i}\) on the plane at the distance \(x\), \(\mu_{i}(x)\), the second one to \(\mu_{i}(x+dx)\)[62; 63]. Note, that Equation (1) shows that \(J_{i}\) depends not only on the force applied to species \(i\), \(F_{i}\) but also on all forces applied to other species. It is suitable to present the \(L_{ij}\) coefficients as functions of the \"virtual\" electroneutral solution concentration. This solution is assumed to be in local equilibrium with a small volume of the membrane. The notion of virtual solution was first introduced by [PERSON] and [PERSON], who named it \"corresponding solution\" [64]. This solution may be physically present in the central zone of meso- and macroproes (where the pore radius is higher than the Debye length) (Figure 3); if not (the EDLs at the opposite walls of the pore are overlapped), it may be considered as a hypothetical solution. Since usually local equilibrium at external membrane interfaces takes place, the virtual solution is identical to the external bathing solution at each membrane side. The electrochemical potential \(\mu_{i}\) in Equation (1) can be present as a function of activity \(a_{i}\) of species \(i\), electrical potential \(\phi\) and pressure \(p\) in the virtual solution:  
\[\mu_{i}=\mu_{i}^{0}+RT\ln a_{i}+z_{i}F\phi+\overline{V}_{i}p \tag{2}\]  
where \(\overline{V}_{i}\) is the partial molar volume and \(z_{i}\) is the charge of species \(i\); \(F\), \(R\) and \(T\) are the Faraday constant, the gas constant and the temperature, respectively. The choice of thermodynamic forces (\(F_{i}\)) and conjugated fluxes (\(J_{i}\)) in Equations (1) may be different but not arbitrary: when the choice of \(J_{i}\) and \(F_{i}\) is correct, the sum of products \(J_{i}F_{i}\) gives the dissipation function [56; 61]. The transport coefficients' values depend on the choice of forces and fluxes. [PERSON] et al.