## 6 Membranes with Modified Surface
succeeded in increasing the Li\({}^{+}\)/K\({}^{+}\) selectivity of membranes using polyanionite modified with crown ethers with a multilayer composition [327]. However, due to the availability of recent detailed reviews and limited space in this publication, we will consider only some theoretical aspects of this phenomenon. Experiments and mathematical modelling show that the main role in increasing selectivity is played by the first surface bilayer [316; 317; 319]. However, the deposition of additional bilayers continues to increase the specific selectivity up to 10 bilayers [319]. Compromise between the Specific Permselectivity and Permeability. Impact of Concentration Polarization  
For the membranes selective for monovalent ions, as well as for other types of membranes [80; 81; 82; 82], there is a compromise between selectivity and the flux of the target component. As Table 3 shows, the higher the density of the ion flux through the membranes, the less specific permselectivity. In particular, the greater the driving force (voltage/current in the case of IEMs or pressure in the case of nanofiltration (NF) membranes), the less specific permselectivity in most cases. In a number of works [52; 281; 319; 320; 321; 322; 323; 324; 325; 326; 327; 328; 329; 330; 333], the IEM selective permeability significantly decreases with current density increasing and practically disappears when the limiting current density is reached. One of the reasons for the decrease in selective permeability is that the control of mass transfer passes from the membrane to the diffusion layer when approaching the limiting state [281; 329; 330]. Another reason is water splitting (generation of H\({}^{+}\) and OH\({}^{-}\) ions). To increase specific permselectivity, the formation of bi-layer structure on the membrane surface is quite effective [319]. However, the intensity of water splitting increases with an increase of the number of bi-layers because of increasing the number of bipolar junctions [319; 320]. This reduces the transfer of the target ion. In addition, the change in pH in the diffusion layer caused by water splitting can affect the properties of the modifying layer [52; 320]. Thus, an increase in its swelling and a decrease in charge density can significantly reduce the effect of monovalent permselectivity. Quite interesting is the \"odd-even\" effect of the LBL modified membranes. [PERSON] et al. [319] describe the case where a CMX cation-exchange membrane was coated by several bilayers of a hyperbranched poly(ethyleneimine) (PEI) bearing positive fixed charges, then by a poly(4-styrenesulfonate) (PSS) layer bearing negative fixed charges (of the same sign as those of the CMX). The specific permselectivity and water splitting were higher in the cases of PEI-terminated bilayers as compared to their values for the PSS-terminated bilayers. Thus, changing the nature of the terminating layer allows switching on and turning off water splitting at the surface of IEMs [7]. To understand the effect of the value of driving force on the membrane specific permselectivity, it is necessary to take into account the current-induced concentration polarization of IEMs in mixed solutions. Mathematical modeling of ion transfer through IEMs in a mixture of several electrolytes was considered in a number of studies [330, 332, 338, 339, 340]. For the first time, the dependence of the effective transport numbers of competing ions in a membrane system on the current density was described by [PERSON] and [PERSON] [338]. The authors used the Nernst-Planck equations and the electroneutrality condition written for three types of ions in the membrane and in the diffusion layers adjacent to it. Both diffusion and migration contributions to the ion transport were taken into account in the diffusion layers, and only ion migration in the membrane. A similar model was developed in [330], where both migration and diffusion transport in the membrane were accounted for.