## 2 Structure of Ion-Exchange Membranes
According to [PERSON] et al. [5], only when n > 14, one can speak of a two-phase system where the bulk-like water is clearly distinguishable in the pore. Schematically, the structure of a functional part of an IEM is shown in Figure 3. It is a fragment including hydrophobic domains of the matrix, micropores (where the EDLs are overlapped) and mesopores (where the EDLs are not overlapped but occupy an essential portion of a pore). This fragment is typical for a homogeneous IEM. Heterogeneous IEMs contain also macropores (spaces between ion-exchange resin particles and reinforcing binder and tissue). Figure 2: Distribution of counterion (1) and co-ion (2) within a pore near a charged wall with fixed functional groups (3) shown as distribution of points (**a**) or with concentration profiles (**b**). Reproduced with permission from [4], [PERSON], 2012. To pass from one fixed ion to a neighbouring fixed ion, a counterion has to overcome a potential barrier. The height of this barrier depends on the energy of electrostatic interactions between the hydrated counterion and the fixed ion, on the chemical interaction between this pair of ions and on the energy required to move the polymer chains to form a sufficiently large ion transport channel in the matrix. The average distance between two neighbouring fixed ions, \(b\), depends on the exchange capacity and membrane morphology. For common membrane materials, \(b\) is in the range 0.5-1.0 nm [3]. This distance for the fixed ions in a hydrated cluster of perfluorosulfonic membranes is about 0.8 nm [3, 5]. [PERSON] et al. [47, 48, 49] propose to use the Bjerrum length as the distance from the fixed ions, which determines the boundary dividing the electrolyte solution in membrane pores into two parts whose properties are different. The Bjerrum length, \(\lambda_{\rm B}\), is the distance at which the electrostatic potential energy between two charges equals the thermal energy scale [50]. According to Bjerrum, if a counter-ion is closer to the fixed charge than \(\lambda_{\rm B}\), it is regarded as \"associated\": its electrostatic interaction with the fixed ion is so great that it cannot diffuse away from the fixed charge [48]. If the distance between two ions is less than \(\lambda_{\rm B}\), the energy of electrostatic interaction exceeds the kinetic energy of thermal motion, which reduces the mobility of ions. [PERSON] [51, 52] proposed a model for polyelectrolyte known as the Manning condensation theory [53]. [PERSON] et al. [47, 48, 49] have applied recently this theory to ion-exchangers. According to [49], if the distance between two fixed charges (\(b\)) is less than \(\lambda_{\rm B}\), counterions condense (or localize) on the polymer chain, which is a result of strong electrostatic interaction between the fixed ions and mobile counterions. When \(b>\lambda_{\rm B}\), counterion condensation does not occur. The value of \(\lambda_{\rm B}\) in water is 0.71 nm, in IEMs with the IEC close to 2 meq (g dry membrane)\({}^{-1}\)] its value is 1.2-1.3 nm [49] (due to lower dielectric permittivity in IEMs). The value of \(b\) in the IEMs studied by [PERSON] et al., was evaluated between 0.43 and 0.57 nm [49].