## 4 Modelling the Structure-Property Relationships
### \"Solution-Diffusion\" Models
#### 4.1.1 Teorell-Meyer-Sievers (TMS) Model
The counterions located closer than \(\lambda_{\text{B}}\) to a functional charged group do not have sufficient thermal energy to freely diffuse away from this group. They are considered as \"associated\" as opposed to \"free\" ions, which are separated from this group by a distance greater than \(\lambda_{\text{B}}\)[51; 107; 108]. The critical linear charge density characterizing the onset of counterion condensation in polyelectrolyte solutions was found experimentally by a number of researchers confirming [PERSON]'s counterion condensation theory. A comprehensive review of these publications is given by [PERSON] [109]. While the model is rather simple and uses no adjustable parameters, there is a good agreement between theoretical and experimental values of activity coefficients for many polyelectrolytesystems [51; 109; 110]. The projected length of one monomer unit was evaluated as 0.25 nm [111]; the average distance between fixed charges for a Nepton(r) CR61 membrane, as 0.73 nm [47]. The TMS models gives an adequate qualitative description of such fundamental membrane properties as electric conductivity, transport number and membrane potential [112]. When two kinds of counterions, 1 and 2, are present in the membrane, the Nernst-Planck equation together with Equations (24) and (27) allow description of counterion competitive transfer. For more information on the TMS model, see Refs. [31; 55; 57; 93; 113]. Similar assumptions are accepted in some recent models of ion sorption and transfer in charged membranes regarded as homogeneous medium [114; 115; 116; 117; 118; 119]. [PERSON] et al. [114; 115; 116; 117] use the so-called fine-pore model, which allows description of some special cases, for example, when an organic IEM contains inorganic nanoparticles. [PERSON] and [PERSON] [118; 119] use the same assumption as the TMS model but apply the extended Nernst-Planck equation, which contains the convective term. That is important when considering the application of IEMs in chlor-alkali electrolysis. There are models considering the membrane as a single phase (charged gel), in which the concentration of fixed charges and/or diffusion coefficients continuously change along the normal coordinate [120; 121]. In this case, the TMS theory can be applied locally. Generally, these models show that a heterogeneity in the fixed charge distribution leads to an increase of permselectivity in comparison with a membrane wearing homogeneously distributed fixed charges of the same average concentration. The explanation is that the permselectivity is controlled by the layer having the highest local concentration of fixed charge, while the layers with low fixed charge concentrations have a minor impact on the global membrane behaviour. For example, the use of conical and double-conical nanopores in NF membranes allows improving the performance of these membranes (the salt rejection) [122]. Two different phenomena are at the basis of the filtration rectification properties of conical nanopores: the co-ion exclusion at the pore mouth and the sign and the magnitude of the (pressure-induced) electric field arising through the nanopores. Hourglass-shape nanopores show improved separation performance compared with cylindrical and conical nanopores having identical average diameter, length and surface charge density. Notably, they exhibit a higher salt rejection with a smaller driving force than the other pore geometries. That makes them attractive candidates for the design of advanced NF membranes.