## 1 Introduction: Heterogeneity and Multiscale Nature of Ion Transport in Membrane Systems  
The main aim of this paper is to give a review of the well-established models as well as new approaches for describing the transport phenomena in charged membranes. These materials include ion-exchange (IEM), nanofiltration (NF) and reverse osmosis (RO) membranes, which are widely used in separation and energy production processes.  
The peculiarity of mass transfer phenomenon is membrane systems is the multiscale character of ion and water transport, which is due to the heterogeneity at multiple scales [1]. The charged membranes are self-assembled nanostructured materials constructed from macromolecules [2]. It is remarkable that the structure and, hence, the properties of both biological and artificial membranes show considerable similarities [3; 4]. The main components of the structure are amphiphilic compounds, which have both hydrophilic and hydrophobic constituents. The mutual interaction of the hydrophobic segments and the interaction of polar hydrophilic groups with aqueous medium lead to a self-organized stable structure. In particular, nanoscale hydrophilic channels which are selectively permeable to water and ions are formed within a hydrophobic matrix [5]. The size of transport channels is of a few nanometres, so that the electrical double layers (EDL) formed at the opposing pore walls overlap. Overlapping EDLs leads to an essentially higher concentration of the ions whose sign of the charge is opposite to the that of the fixed ions (the counterions) in comparison with the concentration of the ions having the same sign of the charge as the fixed ions (the co-ions).  
The second scale is due to the diffusion boundary layers (DBLs) at the membrane surface formed under the action of the electric current flowing across the membrane. This scale is of the order of 100 \(\upmu\)m. When an overlimiting current density flows, an intermediate scale determined by the thickness of the extended space charge (of the order of 1 \(\upmu\)m) appears.  
Finally, there is a scale of the order of 10 cm determined by the length of a compartment in an electrodialysis (ED) stack used in industrial process.  
The processes occurring at one scale are coupled with the processes occurring at other scales. For example, selective ion transport in membrane pores causes concentration polarization, that is, formation of concentration gradients in the DBLs of bathing solution.  
In this paper, we consider the transport at all the mentioned above scales and examine the possibilities of multiscale modelling. The latter is rather difficult, as such modelling needs large computer resources. We give the mathematical description of ion transport in the membranes taking into account the structure-properties relationships. Concentration polarization occurring at the scale of DBL and related surface phenomena such as water splitting along with current induced convection are described as well. Their mechanisms are analysed in close relation with the membrane surface properties. Here we focus on the role of the membrane surface heterogeneity, which has been subject to intensive attention in the literature of recent years. Effects associated with desalination of the solution in the longitudinal direction along the length of an ED compartment are also studied.