## 5 Concentration Polarization in Electrodialysis (ED)  
### Current-Induced Concentration Gradients  
The permselectivity of IEMs towards the ions bearing the charge of a certain sign (counterions) is the main functional property [208]. Namely this property conditions the applications of this kind of membranes in ED, where the feed electrolyte solution is simultaneously demineralized and concentrated in different ED compartments (Figure 8). When a direct current of density \(i\) flows across a membrane, concentration polarization occurs: concentration gradients arise in the boundary solution layers adjacent to the membrane. Immediately after the current is turned on, there are no diffusion fluxes, while the counterion migration flux through the membrane is higher than their flux from the solution bulk to the membrane surface. As a result, the concentration of the counterions (as well as that of co-ions) decreases at one membrane side and increases at the other. The changes of concentration near the membrane continue until the diffusion flux, which grows with time, completely compensates for the difference in migration fluxes in solution and membrane. Figure 7: Schematic depiction of the membrane structure fragment containing pores with charged walls. The polymer matrix is presented by the hatched area. Reproduced with permission from [4], [PERSON], 2012. At any time, the flux density of species \(i\) coming from the solution and entering the membrane/solution interface is equal to the flux density leaving the interface into the membrane bulk. This condition of flux continuity is written as:  
\[\left(J_{i}\right)_{s}=\left(-D\frac{\partial c_{i}}{\partial x}+\frac{it_{i}}{ z_{i}F}\right)_{s}=\frac{iT_{i}}{z_{i}F} \tag{37}\]  
where \(\left(J_{i}\right)_{s}\) is the flux density of ion \(i\) through the interface; the middle part of the Equation (37) refers to the solution: \(D\) is the electrolyte diffusion coefficient in solution, \(C_{i}\) and \(t_{i}\) are the concentration and transport number of ion \(i\) in solution; the right-hand part refers to the membrane: \(T_{i}\) is called integral [31; 209] or effective [54; 210] transport number of this ion in the membrane. \(T_{i}\) is defined as the current fraction carried by ion \(i\) through the interface or the membrane in steady state where no restrictions on the driving forces are imposed:  
\[T_{i}=\frac{z_{i}F(J_{i})_{s}}{i} \tag{38}\]  
\(T_{i}\) can differ from the (electromigration) transport number in the membrane, \(t_{i}^{mb}\) (\(t_{i}\) in Equation (5) written for a membrane), since according to Equation (5), the (electromigration) transport number must be determined under the condition that the concentration gradient and the convective flow across the membrane are zero. Nevertheless, if the external solution concentration is not too high, \(T_{i}\) is quite close to \(t_{i}^{mb}\). Normally, commercial membranes are highly permselective to counterions [76]: the values of \(T_{1}\) and \(t_{i}^{mb}\) approach 1 when the solution is diluted. According to classical electrochemistry, the occurrence of concentration gradients near the surface of a membrane (or an electrode) results from the limitation of the current density \(i\). With increasing \(i\), the electrolyte concentration at the interface, \(c_{s}\), decreases (Figure 8). When \(c_{s}\) approaches zero (becomes much smaller than the bulk concentration, \(c^{0}\)), the current density reaches its limiting value  
Figure 8: Scheme of an ED cell with an AEM and a CEM; DC and CC are the desalting and concentrating compartments, respectively. The points 1 and 1\({}^{\prime}\) show where the tips of Luggin’s capillaries are placed for measuring PD. Redrawn from [4]. (\(i_{\rm lim}\)).