### abstract

- Inexpensive highly permselective heterogeneous ion-exchange membranes are prohibitively highly polarizable by a dc current for being used in electrodialysis. According to recent experiments, polarizability of these membranes may be considerably reduced by casting on their surface a thin layer of crosslinked polyelectrolyte, slightly charged with the same sign as the membrane's charge. The present paper is concerned with this effect. Concentration polarization of a permselective heterogeneous ion-exchange membrane by a dc current is determined by geometric factors, such as, the typical size of the ion-permeable ``gates'' at the membrane surface relative to the separation distance between them and the diffusion layer thickness. The main quantitative characteristic of polarizability of a heterogeneous membrane is its voltage/currrent curve with its typical saturation at the limiting current, which is lower than that for a homogeneous membrane. In the present study we modify the previously developed two-dimensional model of ionic transport in a diffusion layer at a heterogeneous ion-exchange membrane by including into consideration a homogeneous ion-exchange layer adjacent to the membrane surface. A numerical solution of the respective boundary value problem shows that, indeed, adding even a very thin and weakly charged layer of this kind increases the value of the limiting current, to that of a homogeneous membrane. What differs, for different values of coating parameters, is the form of the voltage/current curves but not the value of the limiting current, namely: the thinner is the coating and the lower the fixed charge density in it, the ``slower'' is the approach of the limiting current. In order to explain this feature, a simple limiting model of modified membrane is derived from the original two-layer model. In this limiting model, asymptotically valid for a thin coating, solution of the ionic transport equations in it is replaced, via a suitable averaging procedure, by a single nonlinear boundary condition for the membrane/solution interface. Rigorous analysis shows that the aforementioned property of the limiting current is an exact mathematical feature of this limiting model, when the underlying physical phenomenon is the funneling of counterions by the charged layer from the impermeable parts of the membrane towards the ``entrance gates.'' An approximate analytical solution, developed for this model, compares well with the exact numerical one.