Mathematical Modeling of the Selective Transport of Singly Charged Ions Through Multilayer Composite Ion-Exchange Membrane during Electrodialysis

The deposition of several alternating anion- and cation-exchange surface layers (layer-by-layer method) is a promising technique for the modification of ion-exchange membranes, which makes it possible to essentially increase their selectivity to singly charged ions. This paper presents a one-dimensional model, which is based on the Nernst–Planck–Poisson equations and describes the competitive transfer of singly and doubly charged ions through a multilayer composite ion-exchange membrane. It has been revealed for the first time that, as in the earlier studied case of a bilayer membrane, the dependence of the specific permselectivity coefficient (P1/2) of a multilayer membrane on the electrical current density passes through a maximum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {P_{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}^{{\max }}} \right).$$\end{document} It has been shown that an increase in the number of nanosized modification bilayers n leads to the growth of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}^{{\max }},$$\end{document} but the flux of a preferably transferred ion decreases in this case. It has been established that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}^{{\max }}$$\end{document} is attained at underlimiting current densities and relatively low potential drop. The simulated dependences \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}^{{\max }}$$\end{document}(n) qualitatively agree with the known literature experimental and theoretical results.


INTRODUCTION
Today, the membrane methods of purification, separation, and concentration are the most environmentally friendly and economically promising methods. One of such methods is electrodialysis (ED), whose efficiency is confirmed on an industrial scale by rather prolonged experience of its application [1][2][3].
Traditional electrodialysis desalination implies the removal of any ions and, thereby, decreases the total mineralization of a solution [1]. However, there also exist such spheres of application, where it is important to remove the ions of certain kind. Thus, when preparing the ground water for irrigation, singly charged Na + and Clions are removed from it to prevent the soil salinization. At the same time, the removal of multicharged hardness cations and sulfate anions is undesirable, because it is necessary for the optimal growth of plants [4]. Another example is dairy industry, where NaCl and KCl are removed from milk whey [5] to leave calcium and organic ions as valuable components for nutrition.
The need for the removal of singly charged ions in the mentioned and many other spheres has caused the creation of a new type of electrodialysis known in the foreign literature as selectodialysis [6,7]. It consists in the use of a special configuration of stacks of ionexchange membranes (IEMs) selective to singly charged ions. This new method essentially improves the separation and extraction of phosphates from waste water [8], increases the efficiency of the purification of industrial processing brines [7], and solves many other problems in the field of ecology and chemical and food industry [9][10][11][12][13][14][15]. Baromembrane processes, such as low-pressure reverse osmosis and nanofiltration also provide the separation of singly charged ions from a mixture with doubly charged ones [16]. From the viewpoint of practice, the distinction between these methods is that, in electrodialysis with membranes selective to the transfer of singly charged ions, they are removed from a solution being desalted and can be concentrated in a concentrate compartment. In nanofiltration, on the contrary, a singly charged ion is removed from the initial solution together with an essential water flow, and a doubly ( ) max 1 2 . P max 1 2 , P max 1 2 P max 1 2 charged ion is retained in the initial solution, which is concentrated in this case. In the processing of solutions with a high concentration, baromembrane methods are limited by the need for very high pressures, which must exceed the osmotic pressure of a solution. In electrodialysis, the limitation associated with high concentrations are much milder. Moreover, the separation factor in the case of electrodialysis may be much higher than in the case of baromembrane methods [6]. Hybrid processes combining selectodialysis with reverse osmosis are very promising [17]. The growing application of membranes, whose permselectivity for singly charged ions is much higher than for multicharged ones has provoked increased interest in the development of such ion-exchange membranes and the understanding of the mechanism of their specific permselectivity [14,[18][19][20][21][22].
The specific permselectivity coefficient P 1/2 characterizing the capability of a membrane for the selective transfer of singly charged ions (ions 1) as compared to doubly charged ions (ions 2) is determined through the ratio of their fluxes J i , partial current densities j i , or their effective transport numbers T i ( is the current fraction transferred by the ion i due to diffusion and electromigration, and j is the current density) [23]: (1) where and are the molar and equivalent concentrations of the ion i in the bulk solution, respectively.
The general principle of the manufacturing of ionexchange membranes selective to a certain kind of ions consists in the formation of a thin active surface layer serving as a barrier for the transfer of ions, which are counterions to the substrate membrane [24]. This barrier creates only a negligible hindrance for singly charged ions, but is a serious one for doubly and especially triply charged ions. Such a barrier can be formed by the deposition of a hydrophobic film [25] and also a thin layer containing fixed ions charged oppositely the substrate membrane [21,[26][27][28][29][30]. The oppositely charged layer creates the effect of the electrostatic exclusion of coions (Donnan exclusion effect [31]), which is more essential for multicharged ions and less essential for singly charged ones.
In the study [21], it has been experimentally and theoretically revealed for the first time that the dependence of P 1/2 on the current density j for the membranes modified with a thin oppositely charged ionexchange layer has the extremum. The transfer process modeling [22] has shown that, at low currents, the limiting stage is transfer through the cation-exchange substrate membrane, which is selective to doubly charged cations. When the current grows, kinetic control is passed to the anion-exchange modification layer, which creates a high barrier for the transport of doubly charged cations. In such a regime, the Na + flux grows with increasing current, whereas the Ca 2+ flux is close to zero, and Р 1/2 increases. However, when the Na + concentration near the membrane surface approaches zero, the flux of these ions and Р 1/2 attain a maximum When the potential drop is further increased, the flux of Ca 2+ ions grows, and kinetic control is passed to the diffusion layer, and Р 1/2 decreases. Specific permselectivity is lost, when the limiting current density j lim is attained.
The producing of membranes with the deposition of alternating anion-and cation-exchange surface layers (layer-by-layer (LbL) method) is promising [14,[32][33][34][35][36][37]. An advantage of this method is very high selectivity attained due to the sequentially alternated Donnan exclusion effect in each layers. This experimental fact was confirmed by the results of direct numerical simulation based on the "E n PE n " mathematical model developed by Femmer et al. [38]. White et al. [37] modified the Nafion 115 commercial cationexchange membrane with several alternating layers of poly-4-styrenesulfonate/protonated polyallylamine to attain selectivity in the electrodialysis separation of ions K + /Mg 2+ > 1000. Ding et al. [34] have demonstrated that Nafion membranes modified with several films of identical modifiers can separate equimolar K + -Mg 2+ or Li + -Mg 2+ mixtures to obtain a 99.5-% output concentration of singly charged ions. In this case, the current efficiency for such separation is nearly 70%.
Abdu et al. [36] have studied the change in the specific selectivity to Na + and Ca 2+ ions for the CMX commercial cation-exchange membrane with multilayer films of hyperbranched sodium polyethyleneimine/poly-4-styrenesulfonate on its surface. They have demonstrated [36] that the specific permselectivity coefficient of the modified membrane grows with an increase in the number of bilayers n, and this effect is most pronounced at n < 3. However, no increase in P 1/2 is already observed at n > 10.
In this paper, a new modification is proposed for the earlier developed model of the competitive ion transport through a bilayer membrane [22]. In contrast to the earlier proposed mathematical descriptions [32,38], the model [22] takes into account the activity coefficients of ions to provide a correct description for the sorption of individual kinds of ions by a substrate membrane. There also is the known thin-pore membrane model [39] taking into account these coefficients; they are used in the boundary conditions at phase interfaces. This model also uses the electroneutrality condition, which disables its application for describing the ion transport through a multilayer membrane, which has layers with a thickness close to the thickness of the electric double layer (EDL). max 1 2 . P The model presented in our paper describes the selective transport of singly charged ions in the system with a multilayer composite IEM. Based on numerical simulation results, we demonstrated for the first time that the dependence of Р 1/2 on the electrical current density passed through a maximum as in the case of bilayer membranes [21,22]. Primary attention is paid in this study to analyzing the dependence of on the number of modifying layers deposited onto the surface of the substrate membrane.

Mathematical Model
A multilayer system composed of a cation-exchange membrane (CEM) substrate, n modifying ionexchange bilayers, and two diffusion layers (DL) adjacent to the surface of modified membrane ( Fig. 1) is considered. The overall thickness of LbL films is denoted as d ML in Fig. 1. Each bilayer is composed of anion-and cation-exchange layers with thicknesses d AEL and d CEL , respectively. The studied system is surrounded by two identical bulk solutions containing two types of cations (Na + and Ca 2+ ) and one anion (Cl -).
The basic assumptions used in the problem formulation are the same as used in the papers [22,40]: -A substrate and modification layers are considered as homogeneous media, in which fixed charged groups are uniformly distributed; -The solvent flux through the membrane is assumed to be negligibly small, so osmosis and electroosmosis phenomena are not taken into account; -The effect of convective transport in solution is taken into account in an implicit form through the dif-max 1 2 P max 1 2 P fusion layer thickness, which is considered to be independent of the applied voltage; -Temperature, pressure, and solution density gradients are not taken into consideration; and -Water splitting and electroconvection are disregarded.

System of Equations
The stationary transport of ions in the studied system is described by the Nernst-Planck-Poisson and material balance equations: where J i , c i , D i , z i , and γ i are the flux density, molar concentration, diffusion coefficient, charge number, and activity coefficient of the ion i, where i = 1 and i = 2 are the Na + and Ca 2+ counterions, respectively, and i = 3 is the Clcoion, R is the universal gas constant, T is the temperature, F is the Faraday constant, ε 0 is the dielectric permittivity of vacuum, ε is the relative dielectric permittivity of a medium, is the electrical field intensity, ϕ is the electrical potential, and and are the charge number and concentration of fixed groups of the membrane. Equations (2)-(4) are equally true for the diffusion layers, substrate membrane, and modification layers. 1. Schematization of the simulated system including a CEM substrate and n modifying layers with thicknesses d and d ML , respectively; depleted and enriched diffusion layers with thicknesses δ I and δ II , respectively, and bulk solution. Direction of electrical current with a density j is shown with an arrow. However, the parameters and depend on the coordinate: and are assumed to be zero in diffusion layers; and in the anion-and cation-exchange modification layers, respectively; and in the membrane substrate. The diffusion coefficients of ions D i are also changed in the same fashion: in the diffusion layers; in the modification layers; in the substrate membrane. The diffusion coefficients were calculated from the experimental data on electroconductivity of the Neosepta CMX commercial homogeneous cation-exchange membrane (Astom, Tokuyama Soda, Japan); since electroconductivity of modification layers can not be measured, it is assumed that is τ times smaller than in solution, and τ = 3 for all the considered ions and cases. This parameter can be interpreted as the tortuosity factor. The activity coefficients of ions in the diffusion and modification layers are assumed to be unit, being in the membrane, where may be other than unit. To provide a smoother change in these parameters at the boundaries of corresponding layers, the weight function "boxcar" (rectangular wave) is used, and the thickness of all the interphase transition regions is chosen to be 0.5 nm, which is close to the thickness of the dense part of the electric double layer [41] (Fig. 2).

Bulk solution
The current density j in the system includes the contributions of all the ions into charge transfer, i.e.,

Boundary Conditions
The thicknesses of the depleted and enriched diffusion layers are δ I and δ II , respectively; all the calculations were performed for δ I = δ II = δ. The coordinate origin is set at the substrate membrane/modification layer interface (Fig. 1). It is assumed that the concentration of components in the bulk solution are known constants; the electrical potential is zero at the left system's boundary and is equal to ϕ 0 at its right boundary (potentiostatic electrical regime), i.e., where d and d ML are the thicknesses of the substrate membrane and modification layers, respectively. In calculations, ϕ 0 was varied from 0 to 3 V with a step of 0.005 V.
The functions γ i c i (x) and ϕ(x) are continuous throughout the entire multilayer system (between and ) including the interfaces solution/modification layers and modification layers/substrate membrane.
The introduction of activity coefficients into Eq. (2) with consideration for their continuous distribution in system's layers makes it possible to describe the selective sorption of individual kinds of ions by the substrate membrane [32,38].
The system of Eqs. (2)-(4) with boundary conditions (6)-(8) represents a boundary-value problem for ordinary differential equations, whose numerical solution was found by the commercially available COMSOL Multiphysics 5.6 software suite.

Model Parameters
All the calculations were carried out with the input parameters listed in Table 1.

Parameters of the Substrate Membrane and Modification Layers
The thickness d of the substrate membrane and the concentration of its fixed ion groups Q m were taken to be the same as for the Neosepta CMX commercial homogeneous cation-exchange membrane [42]. This membrane contains sulfonate fixed ion groups and is selectively permeable for multicharged cations [43].
The concentration of fixed ion groups in the cation-exchange modification layers Q CEL = 1.1 mol/L was taken the same as for the Nafion 117 commercial cation-exchange membrane [44]. Austing et al. [45] have demonstrated that the thickness of Nafion films after layer-by-layer surface modification may attain approximately several tens of nanometers. In our model, the thickness of each cation-exchange modification layer was taken equal to 50 nm. The thickness of each anion-exchange modification layer was taken equal to 5 nm, being approximately comparable with the layer of Ca 2+ ions adsorbed on the surface of ion-exchange membranes [46]. Such a layer has a charge of opposite sign with respect to the sub-   strate membrane and can potentially increase its specific permselectivity to the transport of singly charged ions. Since the concentration of fixed ion groups in such layer Q AEL can not be sufficiently precisely estimated, it is assumed in our model as a parameter variable within a range of 5-10 mol/L of a pore solution (i.e., the pore space fraction filled with a charged solution is approximately 1/3 and, therefore, the concentration of fixed groups per 1 L of membrane volume is multiplied by 3). The lower limit (5 mol/L) was estimated from the experimental data on the zeta-potential ζ of the CMX substrate membrane, on which Ca 2+ were adsorbed [46]. According to these data, ζ changes from -28.3 to +48 mV after 12 h of membrane operation in a CaCl 2 solution to evidence not only the change of the charge sign, but also an essential increment in the surface charge by absolute value. To provide the mentioned growth of ζ, it is necessary for the charge density of adsorbed Ca 2+ ions to be 3 times higher than the charge density of fixed ions in the CMX substrate membrane, whose exchange capacity is 1.86 mol/L. However, the non-uniform adsorption of Са 2+ ions may lead to the formation of some local areas, where an increment in the positive charge density is much higher. For this reason, the value of 10 mol/L was chosen as the upper limit of varying Q AEL .
As mentioned earlier, the diffusion coefficients of ions in the substrate membrane were calculated from the electroconductivity and diffusion permeability of the Neosepta CMX membrane [42]. In the modification layers, the diffusion coefficients are taken to be 3 times lower than in the solution ( ). This value was chosen on the basis of reasonings that a modification layer is not usually so dense as the substrate membrane, so the diffusion coefficients in it is only slightly lower than in the solution [32,38].

Estimating the Activity Coefficients of Ions
The relation between the activity coefficients of counterions in the solution and membrane follows from the condition of continuity for the activities and electrical potential at the solution/membrane interface to result in the ion-exchange equilibrium equation (which is also called Nikolsky equation [47]): (9) where c i and are the molar concentrations of the ion i in the solution and membrane, respectively, and the lower subscripts 1 and 2 correspond to singly and doubly charged ions, respectively.
The thermodynamic equilibrium constant K 21 can be expressed through the activities of ions [31] as (10) When considering the regularities of ion exchange for the membrane and solution phases, it is reasonable to select the equivalent fractions of ions as concentration units, as the activities of components in the solution are always equal to unit only in this case [31]. Then Eq. (9) takes the form  respectively, and is the ion exchange equilibrium coefficient. For ion exchangers with sulfonate ion groups, > 1 [31]. As pointed out earlier, the activities of ions in the solution and modification layers are taken equal to unit, and the activities of counterions in the substrate membrane were selected such that the equivalent fraction of Са 2+ in the membrane substrate was 20 times higher than for Na + ( ≈ 20) (at equal equivalent concentrations of these ions in an equilibrium solution at a level of 0.02 eq/L) to correspond to the known experimental data for sulfonate cation-exchange materials [48].

Limiting Current Density
Since a rather low electrolyte solution concentration (0.02 eq/L) is considered in this study, the transfer of coions through the membrane may be neglected. The limiting current density for the system with an unmodified membrane (n = 0) was calculated by Eq. (12) derived for the case with a membrane impermeable for coions [49] (12)

RESULTS AND DISCUSSION
In INTRODUCTION it has been pointed out that the dependence P 1/2 (j) has the extremum for the membranes modified with a thin oppositely charged ionexchange layer. Numerical simulation results show that an identical dependence is also observed for several layers with alternating charge signs of fixed ion groups (Fig. 3).
Such a character of dependence for multilayer membranes is explained by the same reasons as in the case of bilayer membranes: kinetic control over the fluxes of singly and doubly charged ions is passed from the substrate membrane to the modification layer and further to the depleted diffusion layer [22].
It can be seen from Fig. 3 that an increase in the number of modification bilayers n leads to an increase in This is due to the fact that the growth of n produces a more appreciable relative decrease in the partial current density of Na + ions as compared to Ca 2+  4a): a decrease in the current of Na + on transition from n = 0.5 (one anion-exchange modifying layer) to n = 5.5 is almost 7% (Fig. 4a), whereas the current of Ca 2+ decreases by 68%. According to Eq. (1), such a change in the currents leads to the growth of A reason for a decrease in the current density attained for Na + ions (the prime means that this value corresponds to the point ) at a maximum point of the curve is that the Na + concentration drop in this layer grows with an increase in the overall thickness of the modification layer. This increase is caused by that the curve attains a maximum point, when the concentration of Na + ions at the substrate membrane surface becomes close to zero (Fig. 5). In this case, the concentration of Na + ions at the surface of the outer modification layer grows (Fig. 4b). Correspondingly, decreases (Fig. 4a). As for Ca 2+ ions, an increase in the overall thickness of the modification layer also produces a decrease in their current, which becomes small at a maximum point of as soon as at n = 0.5. A decrease in with increasing n is small in absolute value, but a relative decrease in this parameter, as pointed out above, is very essential.
It should be pointed out that the growth of , which is accompanied by a decrease in the current density of a preferably transferred ion, agrees with the principle of trade-off between the perm selectivity and permeability of a membrane [15,18]. This principle can be briefly formulated as follows [18]: "highly permeable membranes have no high selectivity and vice versa." The effect of the number of bilayers on and the potential drop in the membrane system at different Q AEL is illustrated in Fig. 6. Figure 6a indicates that, at an odd number of layers (n = 0.5, 1.5, …, 5.5), i.e., when the layer adjacent to the depleted diffusion layer is anion-exchange one, the specific selectivity of the modified membrane essentially grows. Notably, a maximum increment in is observed when passing from n = 0.5 to n = 1.5. At the same time, the selectivity slightly decreases when passing from an odd number of layers to the closest even number (when the depleted diffusion layer is adjacent to a cationexchange layer). Such dependences qualitatively agree with the known experimental [36] and theoretical [38] data.
It can also be seen from Fig. 6a that an increase in the concentration of fixed ion groups in the anionexchange modifying layers leads to the appreciable growth of It should be pointed out that such high selectivities correspond to rather low potential drop (Fig. 6b). This is due to the fact that the modification layers create not very high additional resistance in the system in virtue of their small (nanometer) thicknesses. The transition from odd n to its even value leads to increment of the layer, where the charge of fixed groups has the same sign as in the substrate membrane. For this reason, the potential drop gains an insufficient increment after this transition (Fig. 6b).

CONCLUSIONS
Based on the developed 1D-model and the Nernst-Planck-Poisson equations, it has been shown that the dependence of the specific permselectivity of a multilayer composite ion-exchange membrane on the electrical current density passes through a maximum as in the case of membranes modified with a single thin oppositely charged layer.
Modeling results qualitatively confirm the trend of growth revealed in the specific selectivity of a membrane with an increase in the number of modification bilayers in the earlier known theoretical and experimental studies. It has been shown that an increase in the number of bilayers is accompanied by a small increment in the potential drop at a maximum point  , if the thickness of a modification layer with an opposite charge (with respect to the substrate) is nearly 10 nm. The growth of with increasing n is accompanied by a decrease in the current density of a preferably transferred ion, being in agreement with the principle of trade-off between the perm selectivity and permeability of a membrane.

CONFLICT OF INTEREST
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