# Mathematical Modeling of the Effect of Water Splitting on Ion Transfer in the Depleted Diffusion Layer Near an Ion-Exchange Membrane

^{1}

^{2}

^{*}

## Abstract

**:**

^{+}(OH

^{−}) ions generated in WS is presented. The model is based on the Nernst–Planck and Poisson equations; it takes into account deviation from local electroneutrality in the depleted diffusion boundary layer (DBL). The current transported by water ions is given as a parameter. Numerical and semi-analytical solutions are developed. The analytical solution is found by dividing the depleted DBL into three zones: the electroneutral region, the extended space charge region (SCR), and the quasi-equilibrium zone near the membrane surface. There is an excellent agreement between two solutions when calculating the concentration of all four ions, electric field, and potential drop across the depleted DBL. The treatment of experimental partial current–voltage curves shows that under the same current density, the surface space charge density at the anion-exchange membrane is lower than that at the cation-exchange membrane. This explains the negative effect of WS, which partially suppresses EC and reduces salt ion transfer. The restrictions of the analytical solution, namely, the local chemical equilibrium assumption, are discussed.

## 1. Introduction

_{lim}are electroconvection [22,23,24,25,26] and water splitting [27,28,29,30]. Electroconvection is the fluid transfer occurring under the action of an electric force on the space charge in solution. The main mechanism of EC is electroosmotic slip, which takes place when an electric force is applied to the space charge in the depleted solution located at the membrane surface [31]. In the literature, two kinds of electroosmosis in membrane systems are distinguished: the electroosmosis of the first kind, when the space charge exists independently on the applied current, and electroosmosis of the second kind, when an extended SCR is formed by an (overlimiting) current [24,32]. EC occurring in the first case is also called equilibrium electroconvection [33], while that in the second case is called non-equilibrium electroconvection [34,35]. Electroconvection is the main effect enhancing mass transfer in membrane systems in intensive current regimes. The micrometer-scale electroconvective vortices mix the fluid near the membrane surface [36,37,38,39]. Often this effect of EC mixing, which leads to the formation of a flattened concentration profile in the depleted region adjacent to the membrane surface, is interpreted as the reduction in the effective thickness of the solution diffusion layer [22,38,40,41]. Interestingly, the dominant diffusion zone is offset from the membrane surface, so that the diffusion layer is no longer the boundary one.

^{+}and OH

^{–}ions, with the exception of some cases where the pH is adjusted to a targeted value [42], is an undesirable process during electrodialysis. This phenomenon results in a decrease in current efficiency and a change in the pH of the solutions. The latter often causes precipitation of sparingly soluble salts and their deposition on the membrane surface and sometimes within the membrane pores (membrane scaling). In addition, under pH changes, the deposition of organic matter is also possible, the phenomenon known as fouling [43]. However, it was shown in a number of papers [44,45,46] that EC not only enhances the mass transfer rate, but also decreases the water splitting rate, and, hence, reduces scaling and fouling.

^{+}and OH

^{−}ions are generated in proton-transfer reactions between the membrane functional groups and water in a thin, a few nanometers thick, boundary layer of the membrane. The weaker the functional ionogenic groups, the easier the generation of H

^{+}and OH

^{–}ions. The strong electric field in the reaction layer can increase the overall effective water splitting rate constant by several orders [51] via facilitating the favorable water molecules orientation and accelerating the rate of evacuation of the hydrogen and hydroxyl ions from the reaction layer (the second Wien effect) [50]. The amount of the H

^{+}and OH

^{–}ions generated in the depleted boundary solution is essentially less, since the rate constant of water dissociation in free solution (equal to 2 × 10

^{−5}s

^{−1}) is much lower than the effective rate constant in the membrane, k

_{d}

^{*}, where water splitting is facilitated by the proton-transfer reactions. For example, in the case of an anion-exchange membrane with tertiary amino groups, k

_{d}≈ 1 s

^{−1}[28]. However, as it was shown recently by Urtenov et al. [56], the amount of the H

^{+}and OH

^{−}generated in the depleted solution can be significant at high voltages of the order of 10 V over a membrane. As it was found in [56], the water-splitting reaction takes place within overall SCR, which can reach up to 10 µm at high voltages.

^{−}ions generated at a cation-exchange membrane (CEM) and moving into the depleted solution attract the salt cations from the bulk and increase their flux towards the membrane surface. This effect, called the exaltation of the limiting current [57], is described in the case of a 1:1 electrolyte and neutral bulk solution (pH = 7) by a simple relation [29,57]:

_{OH}are the current density and the diffusion coefficient of the water ion (OH

^{–}in the case of the cation-exchange membrane) in the depleted diffusion layer; δ is the thickness of the diffusion layer. When obtaining Equation (1), it is assumed that the partial current density of the Cl

^{−}ions through the membrane is negligible. Note that if ${I}_{w}=0$ (no water splitting occurs), Equation (1) reduces to the well-known Peers equation [29]: ${I}_{+}=\frac{2{D}_{+}{C}_{+}^{0}F}{\delta}.$

_{H}should replace D

_{OH}in Equation (1). The membrane is supposed not permeable for salt co-ions. The first term represents the contribution of the electrodiffusion through the diffusion layer; the second term is the exaltation current, which is obviously zero in the absence of the water splitting.

## 2. Mathematical Description

#### 2.1. Formulation of the Problem. Governing Equations and Additional Conditions

_{i}of salt cations (+) and anions (−), as well as the fluxes of the H

^{+}and OH

^{−}ions in the diffusion layer, are described by the Nernst–Planck equation:

_{i}, z

_{i}, and C

_{i}are the diffusion coefficient, charge, and concentration of ion I, respectively; E is the electric field, the notations F, R, and T refer to the Faraday constant, gas constant, and absolute temperature, respectively. The normal coordinate X takes its zero value in the bulk solution (to the left of the membrane), X = δ on the membrane surface, Figure 1b. The case of 1:1 electrolyte is considered, hence z

_{+}= 1, z

_{–}= −1.

^{+}and OH

^{−}ions are variable due to the generation or recombination of these ions, however, the sum ${J}_{W}={J}_{H}-{J}_{OH}$ remains constant because the number of H

^{+}ions appearing or disappearing at a point X is equal to the corresponding number of OH

^{−}ions.

^{+}and OH

^{−}ions’ generation per unit volume of solution in the diffusion layer, ${R}_{H}$, can be written as follows:

_{r}is the rate constant for the recombination reaction equal to 1.1 × 10

^{11}L·mol

^{−1}·s

^{−1}, ${C}_{w}$ = 55.6 mol·L

^{−1}is the water concentration. The rate constant for the dissociation reaction, ${k}_{d}$, is generally a function of the electric field, E. This dependence is often described by an exponential function [49,50,52]:

^{−8}m·V

^{−1}[52]. The estimations made by Strathmann [51] show that the electric field in the interface layer of the bipolar membrane is in the range 6 × 10

^{8}−9 × 10

^{8}(V·m

^{−1}). Thus, the ${k}_{d}\left(E\right)/{k}_{d}\left(0\right)$ ratio in bipolar membranes can be equal to several tens of times or more. However, in the depleted solution adjacent to a monopolar membrane, this ratio is not much greater than 1, since the electric field at the interface is about two orders of magnitude lower, as we will see in Section 3. The reason is that in the bipolar membrane, the main potential drop occurs in the bipolar junction, while in the solution/monopolar membrane system, the region with the highest resistance is the extended SCR in the depleted solution. In addition, there is a relationship between the values of the electric field in the interface between solution (E

_{m}) and membrane (${\overline{E}}_{m}$), $\overline{\epsilon}{\overline{E}}_{m}=\epsilon {E}_{m}$ (Equation (A9) in Appendix B). According to Simons [48], the relative dielectric permeability in the membrane is essentially smaller than in solution (about 20). Hence, the value of E

_{m}is $\epsilon /\overline{\epsilon}$ times smaller than ${\overline{E}}_{m}$.

_{H/OH}= 0, and

^{+}ions) and the product of water splitting (the OH

^{−}ions in the case of CEM considered here) are set at the left boundary of the considered region:

_{+}and I

_{w}are considered as parameters. Their values can be found experimentally. An example of treatment of the experimental I–V curve, when the partial currents are measured, will be studied below.

^{−}ions was set at X = 1 as and not at X = 0, as in this paper. The value of I

_{OH}at X = 0 corresponds exactly to the experimentally measured current transferred by the products of water splitting. However, the value of I

_{OH}at X = 1 can be different as a number of the H

^{+}and OH

^{−}ions is generated within the depleted diffusion layer.

#### 2.2. Transformation and Integration of the Equations. Relationships between the Fluxes

_{lim}($I<{I}_{lim}=2{D}_{+}F{C}_{+}^{0}/\delta $), L is of the order of the Debye length:

_{+s}) (Figure 1b). If $I\ge {I}_{lim}$, L takes macroscopic values comparable with the DBL thickness δ [68,70]. As the analytical solution shows (Appendix D, Equation (A19)), the concentration profile of the salt counterion is linear in the electroneutral zone; its linear extrapolation gives the intersection point with the X axis at distance ${\delta}^{\prime}$ from the bulk solution ($X=0).$

_{i}, including the co-ion flux (J

_{–}) which is assumed zero, and summing the results yields:

^{−}ions generated in water splitting at the solution/membrane interface move towards the center of the desalination compartment. When ${C}_{OH}^{0}\ge {C}_{H}^{0}$ in the bulk, the ${C}_{OH}>{C}_{H}$ inequality is valid in the whole electroneutral zone of the DBL. The value of ${J}_{OH}$ in this region is comparable with J

_{+}, while ${J}_{H}$ is negligible. Introducing ${J}_{W}={J}_{H}-{J}_{OH}\approx -{J}_{OH}$ in Equation (15) leads to Equation (16):

#### 2.3. Approximate Solution. The Diffusion Layer Structure

#### 2.3.1. Equations in Dimensionless Form

^{+}ions is small as compared to that of the other ions ( ${C}_{H}\ll {C}_{k}$) and is neglected as well as the flux of these ions. The flux of salt co-ions is assumed zero (j

_{–}= 0), j

_{+}> 0, and j

_{OH}< 0 when the current density I is positive. The dimensionless values of the limiting salt counterion flux and limiting current density are equal to j

_{+lim}= 2 and i

_{lim}= 2, respectively. $\tilde{\epsilon}$ is a small parameter, its value ranges from 5 × 10

^{−12}to 5 × 10

^{−5}, the extremum cases being ${C}_{1}^{0}$ = 1 mol·L

^{−1}, δ = 200 μm, and ${C}_{1}^{0}$ = 10

^{−5}mol·L

^{−1}, δ = 20 μm, respectively.

^{+}and OH

^{−}ions.

#### 2.3.2. Thicknesses of Different Zones. Stitching of Solutions

_{1}; the electromigration zone of space charge region of thickness δ

_{2}, where the diffusion contribution to the fluxes is much smaller than the electromigration contribution; and the quasi-equilibrium part of the SCR adjacent to the membrane surface (quasi-equilibrium electric double layer) of thickness δ

_{3}, the values of fluxes in this zone are much smaller than their diffusion and electromigration components

_{+s}. The detailed deduction of the approximate solution of the NPP equations in each zone is presented in the Appendix A, Appendix B, Appendix C and Appendix D. In particular, the following expressions are found for the thicknesses of the mentioned above three zones:

_{+}and j

_{OH}.

_{+s}(Appendix A). The sense of ${\delta}_{1}$ and ${\delta}^{\prime}$ is also clear from Figure 1b. Since at $i\ge {i}_{lim}$, c

_{+s}<<1, the values of ${\delta}_{1}$ and ${\delta}^{\prime}$ are very close.

_{+}and j

_{OH}, yields the value of ${c}_{+s}$. When knowing ${c}_{+s}$, it is possible to calculate the potential drops in different zones and in the whole system (see the next section), as well as to find the concentration profiles of all ions present in the system (see Appendix, the last section).

#### 2.3.3. Potential Drops

_{3}. Instead, it is possible to find the sum of interfacial potential drops at the left-hand and right-hand membrane boundaries. At the left-hand interface, we consider the region between the point x = x

_{s}(Figure 1b) and the nearest point inside the membrane where the electroneutral condition is hold (the left-hand limit of the membrane bulk). The Donnan potential drop there is $\Delta {{\phi}^{\prime}}_{D}=-\frac{RT}{{z}_{+}F}\mathrm{ln}\frac{{{\overline{C}}^{\prime}}_{+}}{{C}_{+s}}$, where ${\overline{C}}_{+}^{\prime}$ is the concentration of the salt counterion in the membrane bulk next to the left-hand boundary. At the right-hand boundary of the membrane, no space charge appears outside the double electric layer, and the potential drop is expressed by the Donnan equation: $\Delta {{\phi}^{\u2033}}_{}{}_{D}=-\frac{RT}{{z}_{+}F}\mathrm{ln}\frac{{C}_{+s}^{b}}{{{\overline{C}}^{\u2033}}_{+}}$, where ${\overline{C}}_{+}^{\u2033}$ is the salt counterion concentration in the membrane bulk next to the right-hand boundary, ${C}_{+s}^{b}$ is the salt counterion boundary concentration in the concentration (brine) compartment. Assuming that ${\overline{C}}_{+}^{\prime}={\overline{C}}_{+}^{\u2033}$, the sum of two Donnan potential drops is

## 3. Experimental CVC Treatment

^{−1}passed between the membranes. The potential drop across the cell-pair was measured using two Luggin capillaries whose positions are shown with points 1 and 2 in Figure 1b. The total and partial current densities of the Na

^{+}and H

^{+}ions through the MK-40 membrane and those of the Cl

^{−}and OH

^{−}ions through the MA-40 membrane were found [72] by using the constant pH method, where the pH of the feed solution circulating through the desalination compartment was maintained constant [73]. The partial fluxes/currents were calculated, knowing the rate of decrease in the conductivity of the diluate over time and the rate of addition of a NaOH solution to the diluate, which was necessary to maintain its pH constant.

^{−}ion, which was the counterion for the AEM, was approximately 1.5 times higher than that of Na

^{+}, the counterion for the CEM, one can expect that at the same current density, the potential drop across the AEM will be lower than across the CEM. Following [68], we assumed that 2/3 of the total potential drop referred to the CEM and 1/3, to the AEM. To meet the conditions of the model, Equations (27)–(29), the potential drop in the bulk solution of the desalination compartment was subtracted.

_{+}in the case of CEM, J

_{−}in the case of AEM) and the H

^{+}and OH

^{−}ions in the CEM and AEM, respectively, it is possible to calculate the effective thickness of the diffusion layer, ${\delta}^{\prime}$, by using Equation (1) for each membrane. Then the value of the diffusion layer thicknesses, δ, for the AEM and CEM, are found by fitting the experimental values of the potential drop across the AEM and CEM, respectively.

^{+}, Cl

^{−}, H

^{+}, and OH

^{−}ions were taken as 1.33 × 10

^{−5}, 2.05 × 10

^{−5}, 9.31 × 10

^{−5}, and 5.27 × 10

^{−5}(in cm

^{2}·s

^{−1}); bulk solution conductivity κ

_{sol}= 0.025 S·m

^{−1}; the ohmic potential drop in the membranes was neglected, taking into account their high conductivity, about 0.4 S·m

^{−1}, compared to that of the solution. The limiting current density through the AEM and CEM was determined from the experimental CVC (by the point of intersection of the tangents drawn to the initial region and to the inclined plateau of the CVC); it was equal approximately to 0.72 mA·cm

^{−2}and 1.1 mA·cm

^{−2}, respectively. According to the Peers equation, which is a particular case of Equation (1) when I

_{w}= 0, these values of the limiting current density refer to a DBL with a thickness of 72 µm.

_{2}, was larger near the CEM. Moreover, the space charge surface density localized in the depleted solution near the CEM was significantly lower than that near the AEM (Figure 5).

_{w}). Increasing I

_{w}leads to a higher exaltation effect, when the salt counterions are attracted from the solution bulk to the membrane surface by the H

^{+}/OH

^{−}ions generated in WS. One of the possible mechanisms for decreasing ${\delta}^{\prime}$ is electroconvection, which is the most effective in diluted solutions where the EDL thickness is relatively large [75]. The electric current acts on the space charge in the depleted solution near the interface and makes the SCR in motion producing EC, which destroys the DBL from the inside [22,38,40,41]. The more the space charge and the SCR thickness, the more the effect of this motion on the transport enhancement and less ${\delta}^{\prime}$. The space charge and the SCR thickness increase with increasing voltage, as a growing electric field more effectively removes the co-ions from the depleted solution and draws the counterions into this near-surface region.

_{+(−)}/D

_{w}is of the order of 10

^{−1}. For example, when I

_{w}= I

_{+(−)}(hence, 50% of the charge is transferred by the H

^{+}and OH

^{−}ions), the increase in I

_{+(−)}due to the exaltation effect is only about 10%, according to Equation (1). Moreover, as it was found in the literature [29,30], and shown by the experimental data presented in this paper, WS essentially reduces EC.

^{+}ions in the extended SCR near the AEM in comparison to the concentration of the OH

^{−}ions in the SCR near the CEM (compare Figure 3d,f). Since the charge of the H

^{+}ions is opposite to that of the space charge at the AEM (which is determined by the salt anions), the presence of the H

^{+}ions reduces the space charge density. The lower space charge density results in a less intensive EC. Note that this explanation was previously proposed by Mishchuk based on a theoretical consideration [74]. The use of a mathematical model applied to quantitative processing of experimental data provides more arguments to this explanation.

_{w}

^{s}, can be estimated according to [48] and Equation (5) as

_{d}is the water dissociation constant in the solution SCR, C

_{w}is the water concentration, δ

_{SCR}= δ

_{2}+ δ

_{3}is the SCR thickness.

^{+}and OH

^{−}ions in the depleted diffusion layer by water dissociation. The electric field in this SCR is not so strong (of the order of 10

^{6}V·m

^{−1}, Figure 6), as in the interface layer of a bipolar membrane, where it is of the order of 10

^{8}V m

^{−1}[76]. For this reason, the Wien effect described by Equation (6) is not significant in the depleted diffusion layer at a monopolar membrane. Hence, it is possible to set k

_{d}= 2 × 10

^{−5}s

^{−1}in the SCR. The comparison of the calculations using Equation (30) with the numerical solution shows that Equation (30) gives a slightly overestimated value for I

_{w}

^{s}. For example, at I = 9.5 mA·cm

^{−2}, the δ

_{SCR}value was 2.26 µm. Setting C

_{w}= 55.6 mol·L

^{−1}, we found the increase of I

_{w}within the depleted solution I

_{w}

^{s}≈ 0.030 mA·cm

^{−2}. The numerical solution gives 0.021 mA·cm

^{−2}(Figure 7b). The difference is due to the fact that the product C

_{H}C

_{OH}in the SCR is not much less than K

_{w}(as it is assumed when obtaining Equation (30)), but is comparable with K

_{w}(Figure 8). The value of I

_{w}

^{s}is much lower than the experimental value of I

_{w}= 4.1 mA·cm

^{−2}. This means that nearly all the H

^{+}and OH

^{−}ions were generated in the interfacial layer of the membrane, where the increase in I

_{w}was equal to 4.05 mA·cm

^{−2}(Figure 7a).

_{w}

^{s}with the current density transferred by salt counterions. In the considered case, the solution (a 0.002 M NaCl) if rather dilute, but the salt counterion current density, I

_{+}, was relatively high, 5.4 mA cm

^{−2}. It is due to good hydrodynamic conditions in the experimental cell and intensive EC, which provide a very thin diffusion layer, of the thickness of about 14 µm. However, if the conditions for salt ion transfer are not so favorable, and the feed solution concentration is lower, the WS in the depleted solution may be significant. For example, in the case of a 5 × 10

^{−4}M NaCl and a diffusion layer thickness δ = 100 µm, the limiting current density I

_{lim}found as 2 D

_{+}F${C}_{+}^{0}$/δ (the Peers equation) was approximately equal to 0.13 mA·cm

^{−2}. Then I

_{w}

^{s}= 0.02 mA·cm

^{−2}becomes a significant contribution to the total current, which can noticeably change the pH of the diluate/concentrate solutions.

_{D}and decreasing δ. L

_{D}increases with decreasing the solution concentration; hence, it can be predicted that electroconvection enhances with diluting the feed solution. The situation is more complicated concerning the value of δ. This value can be decreased when increasing the mean velocity of the feed solution forced flow in the desalination compartment. However, a greater forced flow rate could hinder the development of electroconvection. As for the rate of water splitting, it should increase when the feed solution concentration decreases, and decrease with increasing electroconvection. The latter was indirectly shown experimentally [46]. Note that neither the effect of concentration nor the influence of flow velocity has been studied systematically in the literature. Nevertheless, this topic seems important and interesting both for the theory and practice of electrodialysis.

## 4. Conclusions

^{+}(OH

^{−}) ions in the depleted diffusion layer near an ion-exchange membrane offers a relatively easy way to obtain important information about the distribution of ions and electric field in the vicinity of the membrane. The knowledge of the local values of ion concentrations (e.g., of the OH

^{−}and doubly charges cations) allows one to predict whether the precipitation of the sparingly soluble compounds will occur. The model also makes it possible to calculate the potential drop across an ion-exchange membrane on the conditions that the partial current densities of the salt and water ions are specified. The treatment of experimental partial current–voltage curves shows that under the same current density, the surface space charge density in the solution at the anion-exchange membrane was lower than that at the cation-exchange membrane. This should be due to the fact that the WS rate at the AEM was significantly higher than that at the CEM. A higher WS rate results in a higher concentration of the H

^{+}ions, which are co-ions, in the extended SCR at the AEM. This reduces the space charge density and partially suppresses EC. Thus, water splitting not only reduces the current efficiency and increases the risk of precipitation, but also reduces the mass transfer through suppressing electroconvection.

_{H}C

_{OH}product remains comparable with the K

_{w}value in the solution space charge region where water splitting takes place.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Thickness of Different Zones of the Depleted Diffusion Layer

^{+}ions are neglected. By integrating Equation (A1) from x = 0 to x = x

_{s}(x

_{s}is the point where $d{c}_{+}/dx=0$, Figure 1b) while expressing (${c}_{+}-{c}_{-}-{c}_{OH}$) from Equation (18) and presenting ede/dx as (½)de

^{2}/dx, we obtain:

_{s}, index “0” corresponds to x = 0. At point x

_{s}as follows from Equation (17) and condition $d{c}_{+}/dx=0$, the electric field is

_{s}is

_{s}is small compared to that in the bulk (x = 0); however, the term $\tilde{\epsilon}\left({e}_{s}^{2}-{e}_{0}^{2}\right)/2$ in this equation cannot be neglected. Nevertheless, electric field ${e}_{s}$ at x = x

_{s}is great compared to the ${e}_{0}$ value, which allows obtaining the following expression from Equation (A2):

_{s}. The expressions for ${\tilde{\delta}}_{1}$ and ${\tilde{\delta}}_{2}$ in dimension form are presented by Equations (21) and (22), respectively.

## Appendix B. Space Charge and Relation between the Strengths of Electric Field in the Membrane and Solution

_{m}= E(δ).

## Appendix C. Potential Drops in Different Zones

_{k}and finding $\frac{{j}_{+}}{{d}_{+}}-\frac{{j}_{-}}{{d}_{-}}-\frac{{j}_{OH}}{{d}_{OH}}$ yield:

#### Appendix C.1. Electroneutral Zone ($0\le x\le {\tilde{\delta}}_{1}$)

#### Appendix C.2. Electromigration Space Charge Region (${\tilde{\delta}}_{1}\le x\le {x}_{s}$)

## Appendix D. Concentration Profiles

#### Appendix D.1. Electroneutral Zone ($0\le x\le {\tilde{\delta}}_{1}$)

_{H}<< 1. The profile of counter-ion concentration is obtained from Equation (A2) by applying the electroneutrality condition (${c}_{+}={c}_{-}+{c}_{OH}$):

#### Appendix D.2. Electromigration Space Charge Region (${\tilde{\delta}}_{1}\le x\le {x}_{s}$)

_{+}and c

_{OH}in this region, following from neglecting the diffusion terms in the Nernst–Planck equation:

^{−}ions in the depleted diffusion layer, respectively. These transport numbers are determined from the experimental partial current densities: ${T}_{+}={I}_{+}/I$, ${T}_{OH}={I}_{OH}/I$. Note that the transport number of H

^{+}ions in the CEM is equal to the transport number of OH

^{−}ions in the depleted solution.

^{+}ions in this region are negligible.

#### Appendix D.3. Equilibrium Electric Double Layer (${x}_{s}\le x\le 1$)

_{+}, in Equation (17), is much smaller than two of its contributions, the diffusion and migration ones. Then we obtain:

^{−}are co-ions, their concentration is very low, and the concentration of the H

^{+}ions is low because they are generated in the membrane. In these conditions, the Poisson equation (Equation (18)) can be written as

_{+}, we find:

_{m}= e(1) is the value of the electric field at the membrane surface.

_{+}(x) through the derivative of the electric field, Equation (A26), which is found using Equation (A29), we find an equation for the calculation of ${c}_{+}\left(x\right)$ in the range ${x}_{s}\le x\le 1$:

_{m}can be evaluated using the first integral of the Nernst–Planck–Poisson equations for a binary electrolyte [64,77], which in the used designations can be written as:

_{−}<< c

_{+}=c

_{+m}. As the dimensionless limiting current density, I, is close to its limiting value 2, the (I−2) term is small, and we find

_{s}, where c

_{+}= c

_{+s}, it is possible to express the thickness of the equilibrium part of the EDL, δ

_{3}, as

_{+m}>> c

_{+s}, the second term in the right-hand side of Equation (A33) can be neglected, and we arrive at Equation (23).

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**Figure 1.**An electrodialysis cell with anion-exchange membranes (AEM) and cation-exchange membranes (CEM) forming desalination (DC) and concentration (CC) compartments (

**a**); concentration profiles of a salt counterion in the diffusion boundary layer (DBL) at a CEM schematically shown at different current densities (

**b**). “1”, “2”, and “3” indicate the points between which the potential drop is determined. δ′ is the effective DBL thickness; δ

_{1}, δ

_{2}, and δ

_{3}are the thicknesses of the electroneutral zone, extended space charge region (SCR), and the equilibrium part of the electric double layer, respectively, of the DBL. Redrawn from [68].

**Figure 2.**Current–voltage curves for a desalination compartment formed of a MA-40 anion-exchange membrane and a MK-40 cation-exchange membrane. The potential drop across both membranes (

**a**) refers to points 1 and 2 in Figure 1a; that across the MK-40 membrane (

**b**), to points 3 and 2; that across the MA-40 membrane (

**c**), to points 1 and 3. In the case of individual membranes, the potential drop in the bulk solution was subtracted. The total and partial currents of the Cl

^{−}and OH

^{−}ions through the MA-40 membrane and Na

^{+}and H

^{+}ions through the MK-40 membrane are shown. The symbols show the experimental data taken from [72]; the lines are a guide to the eye.

**Figure 3.**Concentration profiles of Na

^{+}, Cl

^{−}, H

^{+}, and OH

^{−}ions in the depleted DBL near the AEM (

**a**–

**d**) and CEM (

**e**,

**f**) shown at different scales, at current densities I = 1.8 mA cm

^{−2}(

**a**,

**b**) and I = 9.5 mA cm

^{−2}(

**c**–

**f**). Solid and dashed lines were calculated numerically and analytically, respectively. X = 0 relates to the bulk solution, x = 1, to the membrane surface.

**Figure 4.**The effective thicknesses of the diffusion layer, δ′, and the thickness of the extended SCR, δ

_{2}, as functions of the applied current density, for the AEM and CEM.

**Figure 5.**The extended space charge surface density as a function of the applied current density (

**a**) and the I/I

_{lim}ratio (

**b**), for the AEM and CEM. Numerical calculation.

**Figure 6.**The electric field in the SCR of the depleted solution at the CEM (

**a**) and AEM (

**b**) as a function of the distance at different current densities, the values of which in mA·cm

^{−2}are shown near the curves; x = 0 relates to the bulk solution, x = 1, to the membrane surface. The solid lines are calculated numerically, the dashed lines, using the analytical solution, Equation (A31).

**Figure 7.**The partial current density of the OH

^{−}ions (

**a**) and the increment, I

_{w}−I

_{w0}, of the current densities of the H

^{+}and OH

^{−}ions, I

_{w}, over their values at x = 0, I

_{w0}, as functions of the distance, x, in the depleted diffusion layer at the CEM (

**b**) at different current densities; x = 0 relates to the bulk solution, x = 1, to the membrane surface. The values of the current density in mA cm

^{−2}are shown near the curves.

**Figure 8.**The C

_{H}C

_{OH}/K

_{w}ratio as a function of coordinate x for the case of CEM and I = 9.5 mA·cm

^{−2}. Numerical computation.

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**MDPI and ACS Style**

Nikonenko, V.; Urtenov, M.; Mareev, S.; Pourcelly, G.
Mathematical Modeling of the Effect of Water Splitting on Ion Transfer in the Depleted Diffusion Layer Near an Ion-Exchange Membrane. *Membranes* **2020**, *10*, 22.
https://doi.org/10.3390/membranes10020022

**AMA Style**

Nikonenko V, Urtenov M, Mareev S, Pourcelly G.
Mathematical Modeling of the Effect of Water Splitting on Ion Transfer in the Depleted Diffusion Layer Near an Ion-Exchange Membrane. *Membranes*. 2020; 10(2):22.
https://doi.org/10.3390/membranes10020022

**Chicago/Turabian Style**

Nikonenko, Victor, Mahamet Urtenov, Semyon Mareev, and Gérald Pourcelly.
2020. "Mathematical Modeling of the Effect of Water Splitting on Ion Transfer in the Depleted Diffusion Layer Near an Ion-Exchange Membrane" *Membranes* 10, no. 2: 22.
https://doi.org/10.3390/membranes10020022