# Modelling of Ion Transport in Electromembrane Systems: Impacts of Membrane Bulk and Surface Heterogeneity

^{1}

^{2}

^{*}

## Abstract

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## 1. Introduction: Heterogeneity and Multiscale Nature of Ion Transport in Membrane Systems

## 2. Structure of Ion-Exchange Membranes

_{3}H, -COOH, -NH

_{3}OH and others. Protons or ОН

^{−}- ions of these groups can be replaced by cations or anions, respectively, from bathing solutions.

^{®}membranes [10]. Heterogeneous membranes include macroparticles (with the size from 1 to 50 μm) of different polymer materials. For example, the MA-40 anion-exchange membranes (AEMs) (Shchekinoazot, Tula Region, Russia) contain particles of polystyrene crosslinked with divinylbenzene (DVB) anion-exchange resin and polyethylene used as the binder (Figure 1b). Both homogeneous and heterogeneous membranes contain reinforcing cloth (Figure 1a,b). This cloth not only plays an important role as a means to improve the mechanical properties but also causes a certain undulation of the membrane surface, which seems important for initiating electroconvection in overlimiting current regime [11].

^{®}is a perfluorinated sulfonated ion-exchange dense polymer generally considered as homogeneous [28]. Nevertheless, when swelled in water, it contains hydrophilic pores/channels confined within a hydrophobic matrix [5]. These pores/channels provide for the transport of ion and water through a Nafion membrane. The presence of internal solution within a hydrophobic matrix allows one to speak about phase separation [5]. From this point of view, this membrane is multiphase and heterogeneous, however, this heterogeneity appears on a scale of the order of 10 nm.

^{®}, which generally complies with the Gierke model, is made by Kreuer [36]. The clusters, channels, some defects of the structure, spaces between ion-exchange resin particles, the binder and the fabrics of the cloth form a system of pores in an IEMs, whose size varies from a few nm to 1–2 μm [13,37,38,39]. The investigations of the pore size distribution made using the standard contact porosimetry (Divisek et al. [40]; Berezina et al. [37] and Kononenko et al. [38,39]) and differential scanning calorimetry (DSC)-based thermoporosimetry (Bryk et al. [41]; Berezina et al. [42]; Kononenko et al. [39]) methods have shown that the homogeneous IEMs (namely, perfluorinated MF-4SK membranes (Plastpolymer, Russia) [39,41,42]), as well as Nafion-112, Nafion-115, Nafion-117 membranes [40]) do not contain macropores (the pores, whose effective radius size is larger than 100 nm), while heterogeneous membranes have such pores. It was found [37,39,42] that the pore size distribution of Russian heterogeneous MK-40 and MA-40 membranes has two maxima, the first one is at about 10 nm (which is the range of micro- and mesopores) and the second one relates to macropores with size of about 1000 nm. According to [39], the pores of the first type are situated within the ion-exchange resin particles and the pores of the second type are formed by the spaces between different membrane constituents (resin, binder, cloth).

_{3}O+ (or H

_{5}O

_{2}+ or even greater) complex to a neighbouring one rather than moves within its aqueous environment-hydration shell [44]. The contribution of the vehicular transport increases with increasing the degree of water structuring, which takes place in the vicinity of the fixed charged groups. The volume fraction of bulk-like water depends on the membrane water content. Electronic structure calculations [45] show that 2–3 water molecules (n) per sulfonic acid group are needed in perfluorinated membranes (such as Nafion

^{®}) for proton dissociation. The dissociated proton is separated from the sulfonate anion, when 6 water molecules are added in a membrane. According to Kreuer et al. [5], only when n > 14, one can speak of a two-phase system where the bulk-like water is clearly distinguishable in the pore.

_{B}, is the distance at which the electrostatic potential energy between two charges equals the thermal energy scale [50]. According to Bjerrum, if a counter-ion is closer to the fixed charge than λ

_{B}, it is regarded as “associated”: its electrostatic interaction with the fixed ion is so great that it cannot diffuse away from the fixed charge [48]. If the distance between two ions is less than λ

_{B}, the energy of electrostatic interaction exceeds the kinetic energy of thermal motion, which reduces the mobility of ions. Manning [51,52] proposed a model for polyelectrolyte known as the Manning condensation theory [53]. Kamcev et al. [47,48,49] have applied recently this theory to ion-exchangers. According to [49], if the distance between two fixed charges (b) is less than λ

_{B}, counterions condense (or localize) on the polymer chain, which is a result of strong electrostatic interaction between the fixed ions and mobile counterions. When b > λ

_{B}, counterion condensation does not occur. The value of λ

_{B}in water is 0.71 nm, in IEMs with the IEC close to 2 meq. (g dry membrane)

^{−1}] its value is 1.2–1.3 nm [49] (due to lower dielectric permittivity in IEMs). The value of b in the IEMs studied by Kamcev et al., was evaluated between 0.43 and 0.57 nm [49].

## 3. Irreversible Thermodynamics Approach

#### 3.1. The Onsager Phenomenological Equations

_{i}, of any mobile species i (an ion or a molecule) is the same in every point of the membrane material. If nonzero gradients dμ

_{i}/dx are applied, fluxes of different species appear. If the system is close to equilibrium, linear relations between the thermodynamic forces, F

_{j}= −dμ

_{i}/dx and resulting fluxes, J

_{i}, can be written [56,61]:

_{ij}are the phenomenological conductance coefficients. In addition to the material properties, L

_{ij}, can depend on the species concentrations, temperature and pressure but not on fluxes and forces.

_{i}is the same at every point of the plane normal to this direction, regardless of the phase through which the plane passes (Figure 3). dx in Equation (1) can be interpreted as a distance between two planes perpendicular to the transport axis x, the first one corresponding to the value of μ

_{i}on the plane at the distance x, μ

_{i}(x), the second one to μ

_{i}(x + dx) [62,63].

_{i}depends not only on the force applied to species i, F

_{i}but also on all forces applied to other species.

_{ij}coefficients as functions of the “virtual” electroneutral solution concentration. This solution is assumed to be in local equilibrium with a small volume of the membrane. The notion of virtual solution was first introduced by Kedem and Katchalsky, who named it “corresponding solution” [64]. This solution may be physically present in the central zone of meso- and macropores (where the pore radius is higher than the Debye length) (Figure 3); if not (the EDLs at the opposite walls of the pore are overlapped), it may be considered as a hypothetical solution. Since usually local equilibrium at external membrane interfaces takes place, the virtual solution is identical to the external bathing solution at each membrane side.

_{i}in Equation (1) can be present as a function of activity a

_{i}of species i, electrical potential ϕ and pressure p in the virtual solution:

_{i}is the charge of species i; F, R and T are the Faraday constant, the gas constant and the temperature, respectively.

_{i}) and conjugated fluxes (J

_{i}) in Equations (1) may be different but not arbitrary: when the choice of J

_{i}and F

_{i}is correct, the sum of products J

_{i}F

_{i}gives the dissipation function [56,61]. The transport coefficients’ values depend on the choice of forces and fluxes. Narebska et al. [65] carried out delicate and laborious experiments and found coefficients L

_{ij}of the Onsager equations, Equation (1), for a Nafion

^{®}120/NaCl membrane system, as functions of NaCl solution concentration. For the case of use other forms of transport equations, with other set of the forces and fluxes, the phenomenological coefficients were also reported in Refs. [66,67].

#### 3.2. The Kedem-Katchalsky Equations

_{v}, J

_{i}and i (which are the volume and solute flux densities and the current density, respectively); c

_{s}and c

_{i}are the molar concentration of salt and ion i, respectively, in the virtual solution of the membrane; p and π are the hydrostatic and osmotic pressures and ϕ is the electric potential in the virtual solution, respectively; ν = ν

_{+}+ ν

_{−}is the stoichiometric number; d is the membrane thickness; subscripts “+” and “−” relate to cation and anion, respectively. Equations (4)–(6) can be applied to a CEM and an AEM, in which cations and anions act as counterions, respectively.

_{+}); electrical conductivity (κ); and Staverman reflection coefficient (σ). The meaning of the latter can be understood from considering two limiting cases: σ = 1, when the membrane completely reflects the solute transported with convective flow across the membrane and σ = 0, when the retention is zero. The physical meaning of coefficients P, κ and L

_{p}is the same as that in the linear equations of Fick, Ohm and Darcy, respectively. These linear equations are easily derived from Equations (4)–(6), if only one of three driving force is kept non-zero. t

_{+}is the fraction of electric charge carried by cation under the condition that the gradients of concentration and pressure are zero; β, the electroosmotic permeability, should be determined as the proportionality coefficient between the volume flux and the current, when the gradients of concentration and pressure are zero.

_{+}[68].

_{i}and P with the Onsager coefficients are as follows [67]:

_{±}is the mean (molar) activity coefficient of electrolyte in the virtual solution; ${t}_{+app}$ is the apparent transport number of counter-ion (cation in a CEM). According to the Scatchard equation, ${t}_{+app}$ = ${t}_{+}$ − mM

_{w}t

_{w}, where m is the virtual solution molality (in eq/kg H

_{2}O), M

_{w}is the water molar mass and t

_{w}is the water transport number in the membrane.

^{−1}. Similar expressions for L

_{p}, β and σ are presented in Ref. [67].

_{−+}coefficient and the m

_{s}M

_{w}L

_{−w}term in Equation (10) is small compared to the L

_{−−}diagonal coefficient [67,77]:

^{®}120 and MF-4SK membranes. It was interpreted as a consequence of a relative hydrophobicity of these membranes according to Kedem and Perry [72].

_{max}is the theoretical value of E for an ideal membrane not permeable for co-ions. The activity factor, g, can be found from the literature [75,76,78]. For LiCl, NaCl and NaCl solutions, g equals 1 at zero concentration, it passes through a minimum close to 0.9 (in the range 0.1–0.2 M) and regains the value 1 at 1 M. Thus, if we use the approximation ${t}_{+app}=g=1$, when applying Equation (13) in the case where ${t}_{+app}$ is not too low (>0.9), we find a slightly underestimated (by 10–20%) value of ${t}_{-}$. The true counterion transport number, ${t}_{+}=1-{t}_{-}$, will be find with a reasonable accuracy: it would be overestimated by 1–3%. The accuracy can be improved when using approximation ${t}_{+app}$ ≈ ${t}_{+}=1-{t}_{-}$ In this case Equation (13) is reduced to:

_{int}, is measured in conditions where one of the bathing solutions is of concentration c

_{s}and the other one is water. To find the differential (local) permeability, P, the concentration dependence of P

_{int}should be measured. Then [37,76]:

#### 3.3. The Nernst-Planck Equation

_{i}) or concentration (as in Equation (17)) and the electrical potential (ϕ) in any constituent phase of the membrane material. The use of the virtual solution is often convenient, since in this case a

_{i}(c

_{i}) and ϕ have no jumps at the external membrane boundaries (if only the condition of local equilibrium is satisfied at these boundaries, which is verified for the under-limiting current densities [81,82,83]). More often, Equation (17) is applied to a solution and/or a membrane considered as a homogeneous medium [31,57,84]. In this case, the concentration and electric potential refer to this medium (which can be imagine as a swollen sponge with a charged matrix [31]) and not to a particular phase. To indicate the fact that a quantity refers to the membrane material (or its part) considered as a homogeneous medium, we will use an over-bar: ${\overline{c}}_{i}$, $\overline{\phi}$ and ${\overline{D}}_{i}$. The models, which use the approach where the permeants are assumed to be dissolved in the membrane material and transported there under the action of concentration gradient and/or external electric field, are called the “solution-diffusion” models [85,86]. This term is used in contrast to the term “pore-flow” models, in which the transport of species is considered as occurring within a single membrane pore. Usually, this terminology is used in the context of the pressure-driven processes [86,87], while it is also applied to the electromembrane processes [88,89], where the transport occurs under the action of the electrochemical potential gradient.

_{i}) (the second way of writing), the differential diffusion permeability coefficient, P, is used instead of the diffusion coefficient, $\overline{D}$. In both cases, the transport number has the same meaning: it is the fraction of electric charge transported by ion i in the membrane in conditions where the concentration gradient is zero. It is possible also to write this equation for a solution. In this case, the electrolyte diffusion coefficient in solution is used instead of P.

^{−3}), that is, the ion-exchange capacity (IEC).

## 4. Modelling the Structure-Property Relationships

#### 4.1. “Solution-Diffusion” Models

#### 4.1.1. Teorell-Meyer-Sievers (TMS) Model

_{D}is the Donnan equilibrium coefficient; ${\overline{c}}_{i}^{k}$ and ${c}_{i}^{k}$ are the concentrations of ion i at interface k, from the side of the membrane and solution, respectively [105].

^{−3}) in Equation (26) instead of activities (${a}_{i}={c}_{i}{y}_{i}$ and ${\overline{a}}_{i}={\overline{c}}_{i}{\overline{y}}_{i}$), we arrive at Equation (24) with the Donnan equilibrium coefficient, K

_{D}, expressed through the ratio of mean ionic activity coefficients:

_{B}. The counterions located closer than λ

_{B}to a functional charged group do not have sufficient thermal energy to freely diffuse away from this group. They are considered as “associated” as opposed to “free” ions, which are separated from this group by a distance greater than λ

_{B}[51,107,108].

^{®}CR61 membrane, as 0.73 nm [47].

#### 4.1.2. Multiphase Models

_{i}, which characterizes a membrane layer of thickness dx as a function of coefficients ${L}_{i}^{j}$ (characterizing an individual phase j), as well as the structural and geometric parameters, describing the shape and mutual position of the phases. This formulation addresses the effective-medium approach [91,126] intensively developed in relation to a great variety of systems and transfer phenomena: electrical conduction [127], diffusion [128], heat transfer [129], optics and other [130,131]. This approach was firstly applied by Maxwell [132], who, as early as in 1873, has theoretically found the conductivity of a system containing a conductive continuum medium and dispersed spheres, the conductivity of which differed from that of the medium. Later, Rayleigh, Lichtenecker, Bruggeman, Landau and Lifshitz, Mackie and Meares [133] and other [134,135] contributed to the development of effective-medium approach. The hypothesis called “principle of generalized conductivity” [135] is often applied for generalization of experimental results. According to this hypothesis, the function relating L

_{i}to ${L}_{i}^{j}$ does not depend on both the nature of applied force and the nature of the transported species.

_{i}and ${L}_{i}^{j}$ instead of particular transport coefficients. Along with the membrane electrical conductivity [54,138,139,140,141,142,143], this approach also allows one to find electrolyte diffusion permeability [54,114], permselectivity (ion transport numbers) [54,144] and some other properties [74,145,146,147]. In the case of electrolyte sorption, conductivity, diffusion permeability and transport numbers, it is possible to quantitatively describe all these four characteristics as functions of bathing solution concentration starting from a single set of structural and kinetic parameters [147].

_{i}, is expressed through ${L}_{i}^{j}$ in the following form [63,135]:

_{1}and f

_{2}are the volume fractions of the corresponding phases: f

_{1}= V

_{g}/V

_{m}, f

_{2}= V

_{s}/V

_{m}, f

_{1}+ f

_{2}= 1 (where V

_{g}, V

_{s}and V

_{m}are the volumes of gel, inter-gel solution and the membrane, respectively, see Figure 3); α is the structural parameter depending on the position of the phases with respect to the axis of transport: when the phases are parallel to this axis, α = 1; when they are in serial disposition, α = −1; in other cases −1 < α < 1.

^{−1}or higher. Co-ions are therefore strongly excluded, so that their concentration in the gel phase is small compared to $\overline{Q}$, at least in relatively dilute solutions. When assuming ${\overline{c}}_{-}$ << $\overline{Q}$ and ${\overline{c}}_{+}\approx \overline{Q}$, Equation (24) can be simplified. In the case of a symmetrical electrolyte (${z}_{+}=-{z}_{-}=z$) one gets [46,148]:

_{A}refers to the co-ion concentration in the inter-gel solution (assumed to be the same as the external solution).

_{s}, involving the co-ion concentration in the membrane, ${c}_{A}^{\ast}$, can be found using Equation (31) [63]:

_{D}are the IEC of the gel phase and the Donnan coefficient, respectively. Despite of small value of f

_{2}(typically less than 0.1 in homogeneous membranes and close to 0.2 in heterogeneous ones), the sorption of electrolyte by the inter-gel spaces is dominant, especially in diluted solutions, because of the co-ion exclusion from the gel involving EDL in micro- and mesopores [46,148,149] is small. For conventional IEMs, Equation (31) is verified for external concentrations up to 1−2 M. Note that an equation similar to Equation (32) was proposed by Geise et al. [150], where the second term on the right-hand side of this equation was attributed to an “uncharged material,” which can sorb electrolyte stronger than the charged gel.

_{s}with increasing the external concentration. At low external electrolyte concentration K

_{s}is close to f

_{2}, that is, it value is in the vicinity of 0.1 (or lower) for the homogeneous membranes and 0.2 for the heterogeneous ones [46,148,149].

_{D}and ${Q}^{g}$ (thermodynamics coefficients); two structural, f

_{1}and α; and two kinetic ones, the diffusion coefficients of counterion, ${D}_{1}^{g}$ and co-ion, ${D}_{A}^{g}$, in the gel phase. The diffusion coefficients in the inter-gel solution, ${D}_{i}^{s}$, are assumed the same as in free solution. When these parameters are known, L

_{i}coefficients can be calculated. Then the transport membrane characteristics κ, t

_{i}and P, can be found by using Equations (8)–(10) giving the links between the Onsager and Kedem-Katchalsky conductance coefficients. Equations (8)–(10) can be applied not only in the case, where the membrane is equilibrated with a bathing solution but also when there is a concentration gradient across the membrane. Then Equations (8)–(10) are applied at any coordinate in the membrane for a local concentration с of the inter-gel solution. In this way, it is possible to incorporate the microheterogeneous model into a boundary-value problem for modelling ion and solvent transport in membrane systems [46,151].

^{g}, only slightly depends on electrolyte concentration (since the co-ion uptake by the gel is quite low—when the external concentration is not high, usually <1 M), the lgκ − lgκ

^{s}correlation, according to Equation (33), should be linear with f

_{2}as the coefficient. Numerical calculations indicate that near the “isoconductance point” (where $\kappa ={\kappa}^{g}={\kappa}^{s}$), $\kappa $ is almost independent on α [63]. In the range 0.1${c}_{iso}$ < c < 10${c}_{iso}$, the lgκ − lgκ

^{s}dependence can be approximated by a straight line with slope f

_{2}, if $\left|\alpha \right|$ ≤ 0.2. For most IEMs, α lays in the range 0.1−0.3 [37,54], hence, Equation (33) can be confidently applied, when the concentration is sufficiently close to the isoconductance concentration ${c}_{iso}$. Figure 4 shows Lgκ − Lgс dependencies for different IEMs taken from different papers. Since ${\kappa}^{s}$ is approximately proportional to the concentration of bathing solution, c, the slope of these curves should be close to f

_{2}. Taking into account that some approximations have been made, we will call the value of f

_{2}found as the slope of Lgκ − Lgс linear regression the “apparent volume fraction of the inter-gel solution, f

_{2app}”.

_{2app}and the conductivity of the gel phase, ${\kappa}^{g}$. To a first approximation (justified when the contribution of co-ions in the conductivity of the gel can be neglected), the value of ${\kappa}^{g}$ may be considered as a constant equal to the conductivity of the membrane at the isoconductance point.

_{2}found from conductivity measurements when applying Equation (33) and when using other methods, namely electrolyte sorption, differential scanning calorimetry (DSC) and standard contact porosimetry [37,38,156]. Relatively close values of f

_{2}were reported. For example, for a heterogeneous MK-40 membrane, the authors found f

_{2}equal to 0.17 ± 0.02 from conductivity measurements [41], 0.10 ± 0.02 [148] and 0.26 ± 0.02 [41] from sorption, 0.21 from DSC [41] and 0.23 ± 0.05 from porosimetry [41]. These results show that the microheterogeneous model adequately describes the membrane properties when operating with reasonable structure and kinetic parameters. The ease of applying Equation (33) for processing experimental conductivity data, the possibility to determine the parameters having a clear sense, the ability to describe a set of transport properties, involving the diffusion permeability and ion transport numbers, makes this model suitable for IEMs characterization [8,37,138,142,143,144,157,158,159,160,161].

#### 4.2. Modelling of Properties of Ion-Exchange (IEMs) Containing Nanoparticles

_{2}, ZrO

_{2}, TiO

_{2}and other nanoparticles showed better water retention and proton conductivity, especially at low water vapor pressure, in a rather large temperature interval up to 140 °C [162,163,164,165,166,167], which sometimes was accompanied with increasing IEC [168].

_{2}or ZrO

_{2}, where it has a negative charge), the EDL around the particle is considered to be complementary to the EDL occurring at the charged pore walls (Figure 5).

_{i}and then the membrane conductivity, electrolyte diffusion permeability, transport numbers.

_{1}), this parameter should be higher in the membranes doped with nanoparticles, since even when the EDL thickness is low, the increase in t

_{1}would be conditioned by the fact that a nonconductive nanoparticle replaces the electroneutral solution present in the inter-gel spaces. Therefore, contribution of the non-selective ion transport in this solution will be reduced in favour of highly selective transport in the charged gel phase.

_{p}, is an important parameter. With increasing f

_{p}, the membrane conductivity initially increases (because the total EDL volume increases), then it passes through a maximum and declines. The latter is due to the saturation of the pore space available for nanoparticles. When f

_{p}becomes large enough, the nanoparticles block ion conductivity in the pores, which they occupy. Then ion transport occurs only across the microporous medium (the “gel phase”), which is assumed not available for nanoparticles. An example of such a dependence is given in Figure 6a. It involves experimental data for a Nafion

^{®}membrane containing nanoparticles of caesium acid salt of phosphotungstic heteropolyacid [170] and the results of calculation according to the model described above [174]. Note that a similar dependence was found for a perfluorinated sulfocation-exchange MF-4SK membrane (analogue of Nafion

^{®}): the proton conductivity curve as a function of additive content has a maximum at 2.6 vol%, when it is doped with SiO

_{2}and 1.5 vol%, when it is doped with PAni [175]. Parizian et al. [176] studied an experimental IEM modified with ZnO nanoparticles. Similarly as Safronova et al. [170] and Novikova et al. [175], they found that the dependence of water content, membrane potential, transport number and selectivity of this membrane initially increased with increasing the content of added ZnO nanoparticles, then these characteristics reach their maximum and declined when the nanoparticles’ content was rather large.

#### 4.3. “Pore-flow” Models

_{ij}coefficients for a Nafion membrane as functions of the membrane nanostructure parameters (which are the pore radius, pore wall charge density and the tortuosity factor). Their comparison with the experimental coefficients determined by Narebska et al. [65] has shown a rather good agreement. Thus, it becomes possible to bridge the gap between two different approaches, that is, the microscopic model description and the irreversible thermodynamics.

## 5. Concentration Polarization in Electrodialysis (ED)

#### 5.1. Current-Induced Concentration Gradients

_{i}and t

_{i}are the concentration and transport number of ion i in solution; the right-hand part refers to the membrane: T

_{i}is called integral [31,209] or effective [54,210] transport number of this ion in the membrane. T

_{i}is defined as the current fraction carried by ion i through the interface or the membrane in steady state where no restrictions on the driving forces are imposed:

_{i}can differ from the (electromigration) transport number in the membrane, ${t}_{i}^{mb}$ (${t}_{i}$ in Equation (5) written for a membrane), since according to Equation (5), the (electromigration) transport number must be determined under the condition that the concentration gradient and the convective flow across the membrane are zero. Nevertheless, if the external solution concentration is not too high, T

_{i}is quite close to ${t}_{i}^{mb}$. Normally, commercial membranes are highly permselective to counterions [76]: the values of T

_{1}and ${t}_{i}^{mb}$ approach 1 when the solution is diluted.

_{s}, decreases (Figure 8). When c

_{s}approaches zero (becomes much smaller than the bulk concentration, c

^{0}), the current density reaches its limiting value (i

_{lim}). The expression for i

_{lim}can be obtained from Equations (37), (38), when writing the concentration gradient of the counterion (1) at the membrane surface as:

_{1s}= 0 yields:

^{0}= ${z}_{+}{c}_{+}^{0}=-{z}_{-}{c}_{-}^{0}$ is the bulk electrolyte concentration in eq L

^{−1}. Equation (40) was first obtained by Peers in 1956 [211].

_{lim}), the PD over a membrane surrounded by two DBLs tends to infinity. However, in real membrane or electrode systems, the limiting current density can be exceeded in several times [212,213,214,215]. The causes of the overlimiting current are vividly discussed in literature [212,214,216,217,218,219]. The main effects are current-induced convection (electro- and gravitational convection) and generation of additional charge carriers (e.g., water splitting at the depleted solution/membrane interface, which produces the H

^{+}and OH

^{−}ions). While current-induced convection enhances the mass transfer across the membrane, water splitting in most cases is undesirable effect causing pH changes and the loss in current efficiency.

^{−}ions generated in water splitting at the depleted membrane interface may deprotonate these functional groups, that is, discharge them. This produces: (1) a decrease in the surface charge density, which reduces electroconvection developing as electroosmosis of both the 1st and 2nd kind [235,236] and (2) a decrease in IEC, which results in increasing co-ion back diffusion through the membrane. The latter leads to a decrease in T

_{1}, hence, in an increase of the limiting current density according to Equation (40). Note that electroosmosis of the 2nd kind is named also induced-charge electroosmosis [237,238], since this effect is due to the extended space charge region [239], which emerges next to the equilibrium EDL at overlimiting currents.

#### 5.2. 2D Modelling of Electrodialysis with Homogeneous Membranes

#### 5.2.1. Governing Equations and General Boundary Conditions

^{+}(OH

^{−}) transport across the membrane. For this, experimental [76,265] or theoretical [266] evaluation of ${T}_{1}^{c}$ and ${T}_{2}^{a}$ should be carried out.

_{el})/n, where U is the voltage measured between the polarizing electrodes of the ED cell, Δϕ

_{el}is the PD on the electrodes and n is the number of cell pairs. It is assumed that any plane passing through the centre of an CC is equipotential and the current density and ionic fluxes along this plane are zero.

_{a}, R

_{c}and R

_{cc}are the resistances of the AEM, CEM and the concentration compartment, respectively (all are assumed known), $\Delta {\phi}_{D}$ is the sum of (Donnan) PDs over the solution/membrane interfaces for AEM and CEM, $E=-\frac{\partial \phi}{\partial x}$ is the electric field; $\underset{0}{\overset{h}{\int}}E}dx$ is the PD in the DC. The diffusion transport in the membranes and the CC is not taken into account, the $\overrightarrow{i}$ and $\overrightarrow{E}$ vectors are assumed directed normally to the membrane surface. According to Equation (27),

^{−1}).

#### 5.2.2. Concentration Distribution and Diffusion Layer Thickness

_{lim}or ${\delta}_{N}$ found numerically from a convection-diffusion problem where zero concentration at the wall is used as the boundary condition. In this case, the limiting current density is in excellent agreement with Equations (60), while the values of the local and average ${\delta}_{N}$ are slightly (about 4%) higher than those presented by Equations (61) due to the effect of the omitted second term. Namely, instead of 0.68 and 1.02, the numerical solution gives 0.71 and 1.06, for the local and the average ${\delta}_{N}$, respectively [269].

_{lim}and ${\delta}_{N}$ are inversely proportional to each other, Equation (40), we can arrive at the meaning of the average value of Nernst’s DBL thickness: ${\delta}_{N\text{}av}^{-1}=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int}}{\delta}_{N\text{}loc}^{-1}\left(y\right)dy}$. On the other hand, ${\delta}_{N\text{}av}$ can be also understood as the effective DBL thickness, ${\delta}_{N\text{}ef}$, in a channel where ${\delta}_{N}$ is a function of the distance from the entrance: the substitution of ${\delta}_{N\text{}av}$ in the Peers equation, Equation (40), gives the correct value of ${i}_{\mathrm{lim}\text{}av}$.

_{av}is the average value of Sh in a channel of length L, ν is the kinematic viscosity.

^{−2}cm/s to 2 cm/s. The intermembrane distance (h = 5−6 mm) and the channel length (20 mm) give the dimensionless length Y in the interval 10

^{−5}–10

^{−2}, where the Lévêque approximations (Equations (60)) are well satisfied. The possibility of knowing the theoretical limiting current density and the DBL thickness is very important, as it allows evaluation the coupled effects of concentration polarization. Indeed, the theoretical values of i

_{lim}and δ are determined for the case of an “ideal” IEM, which does not generate current-induced convection and does not split water molecules. Hence, the comparison of the experimental and theoretical i

_{lim}can give some ideas, for example about the intensity of electroconvection [217]. The role of the Lévêque equations for the slot cells with laminar flow is similar to the Levich equation obtained for the rotating disk electrode [271] or rotating membrane disc [225,272,273].

#### 5.3. ED with Heterogeneous Membranes

^{®}AMX and CMX membranes have PVC particles with a diameter of about 100 nm. The presence of non-conductive areas leads to the accumulation of the electric current lines on the conductive regions and to the curvature of these lines. On one hand, this “funnel effect” [274] increases membrane concentration polarization and water splitting rate, while reducing the limiting current density [274]. On the other hand, tangential electric field enhances electroconvection and mass transfer [232,236,275,276]. From what has been said before, the electrical heterogeneity of the membrane surface is subject to optimization. Too low surface fraction of membrane conductive area (γ) will lead to very high concentration polarization, which could be remediated by electroconvection. However, one can imagine that a certain heterogeneity of the surface could effectively enhance electroconvection, while only slightly increase concentration polarization. Simulations made by Davidson et al. [233] and Zabolotsky et al. [272] predict that the rate of electroconvection and, as a consequence, the rate of mass transfer pass through a maximum with increasing γ at a fixed voltage. According to [233], the maximum is close to γ = 0.5; according to [272], it is close to γ = 0.8. The both models are based on the same governing equations (the Nernst-Planck-Poisson-Navier-Stokes equations). Perhaps, the difference is due to the fact that Davidson et al. [233] do not take into account the forced flow between the membranes, while Zabolotsky et al. [272] do.

#### Effect of Electrical Surface Heterogeneity on the Limiting Current Density and Diffusion Layer Thickness

_{0}flows between the membrane, the flow being laminar and there is Poiseuille velocity distribution, Equation (46). A PD $\Delta \phi $ is applied between two planes passing through the centres of the concentration compartments adjacent to the central DC.

_{1m}at the CEM and c

_{2m}at the AEM, their values are considered as parameters). This condition may be called the Dirichlet-Rubinstein boundary condition [214]. It is applied also by Pham et al. [277] and Kwak et al. [230]. Hence, for the conductive surface regions of the CEM and the whole surface of the AEM, we have:

_{a}and N

_{c}show how many times ${c}_{2m}$ and ${c}_{1m}$ are higher than the inlet electrolyte concentration (c

_{0}), respectively. Subscript “cond” refers to a conductive surface region.

_{0}. Under these assumptions, the PD between the plane passing through the centre of the CC and x = 0, $\Delta {\phi}_{c}$ and the PD between the plane passing through the centre of the CC and x = h, $\Delta {\phi}_{a}$, read as follows [214]:

^{−1}; the length of the conductive surface regions is fixed: a = 10 μm.

_{tot}, is often used. δ

_{tot}can be defined [268], as the distance from the surface to the point in solution where c = 0.99c

_{0}. The results of the computations of ${\delta}_{N}^{\prime}$ and δ

_{tot}for the cross section at a distance of 550 μm from the ED channel entrance are shown in Table 2.

## 6. Effect of Surface Heterogeneity on the Transient Behaviour of IEMs

#### 6.1. Modelling of Chronopotentiograms for Homogeneous Membranes. Effect of Finite-Length Diffusion Layer

_{lim}), the electrolyte diffusion reaches such a rate, that the difference between the counterion migration fluxes in the membrane and solution gets compensated. The system reaches a steady state whereas the main mechanism of current transfer remains electrodiffusion. However, if the current density is higher than i

_{lim}, at a certain time called “transition time” there is a change in current transfer mechanism across the membrane. At i > i

_{lim}, the electrolyte diffusion is not sufficient to stop the decrease of concentration at the membrane interface. When the electrolyte concentration at this interface reaches a rather small critical value [216,277], current-induced mechanisms of overlimiting current conductance such as coupled convection (gravitational convection and electroconvection), water splitting [24,220,279] and others [213,234] occur additionally to the electrodiffusion. Current-induced convection enhances the delivery of electrolyte to the membrane surface [217,230,232]. Since the rate of electrolyte delivery via current-induced convection increases as the current density increases, the system reaches a steady-state where its resistance is higher than that at i < i

_{lim}but remains finite.

_{lim}, the ion concentration at the interface never reaches zero and PD is not tends to infinity [239,280].

_{k}was first made apparently by Lerche and Wolf [285] and later by Krol et al. [282].

_{Don}) at the membrane/depleted solution interface. When an overlimiting current density is applied and the time approaches the transition time, current-induced convection (namely, electroconvection) arises near the depleted interface and causes a decrease of δ. The δ(Δϕ

_{Don}) function is assumed a linear one containing two adjustable parameters: a threshold value of Δϕ

_{Don}, which relates to the onset of electroconvection and a value of Δϕ

_{Don}related to the steady state.

_{lim}in Figure 14), the concentration profile develops so rapidly that the DBL thickness does not reach the values, where the contribution of forced convection is significant; therefore, at i > 1.5 i

_{lim}, the model of Mareev et al. [269] and the Sand theory give the same results and both agree with the experiment. The same conclusion concerning the agreement between the Sand theory and a model taking into account the finite length of δ was obtained by van Soestbergen et al. [290] in the case of electrode systems.

#### 6.2. Impact of Surface Heterogeneity on the Transition Time of IEM

#### The Use of Electric Current Stream Function. Comparison with the Sand and Choi and Moon Theories

_{1}within a nonconductive ring of external radius R

_{2}. The DBL thickness, δ, is assumed identical from both membrane sides. Using cylindrical symmetry, the system under study may be presented in two coordinates r and z.

_{av}, by the local current density, ${i}_{local}={i}_{av}/\gamma $, across the conductive areas:

^{®}perfluorinated resin solution of thickness d

_{2}about 3 μm, the Nafion

^{®}solution filled also the pores. Thus, a membrane with a smooth homogeneous permselective surface on one side and a heterogeneous surface on the other was obtained. The circular conductive areas filled with the Nafion

^{®}material occupy about ≈7.7% of the surface, 92.3% of the surface was non-conductive.

_{1}= 12.71 μm, is below the experimental points; that for the upper bound, R

_{1}= 13.15 μm, is above. The transition times calculated using the Sand equation, Equation (76), are essentially higher than the experimental ones and that calculated using the Choi-Moon equation are much lower (Figure 16a). The calculated values of τ using the 3D model [303] but under condition (80) are lower too. In fact, Equations (76) and (81) give two extreme asymptotes of the model [303]. As we mention in the previous section, the Sand Equation (76) holds, when the surface is homogeneous and the current density i > (1.5 ÷ 2)${i}_{av\text{}\mathrm{lim}}$ [269] (Figure 16b,c). The Sand equation holds also when γ < 1 and the ratio R

_{1}/δ is sufficiently small (Figure 16c). In this case, the concentration distribution far from the heterogeneous surface is the same as in the case of homogeneous surface. The Choi-Moon theory [291] assumes the current lines normal to the conductive surface, hence, the funnel effect [274] is not taken into account and tangential transport is considered negligible. This condition is satisfied, when R

_{1}/δ and i are both sufficiently great (Figure 16b,c).

#### 6.3. Phenomenon of Two Transition Times of Chronopotentiograms for Membranes with Heterogeneous Surface

_{1}, is attained, when the electrolyte concentration at the conductive regions of the surface becomes sufficiently low to give rise current-induced convection developed locally at the boundaries between the conductive and non-conductive regions. The second transition time, τ

_{2}, is reached when a critically small concentration is attained at the whole membrane surface, including the conductive and non-conductive regions [314]. As the 3D model [303] predicts, when there is no current-induced convection, the electrolyte concentration remains rather elevated at the non-conductive regions, while it reaches very small values at the conductive regions (Figure 17). According to this model, only one transition time τ

_{1}< ${\tau}_{Sand}$ can occur in a heterogeneous membrane system if the ions in the depleted DBL are transferred by electrodiffusion solely. The decrease of concentration near the non-conductive regions may be caused by the current-induced convection, apparently electroconvection occurring as electroosmosis of the first kind [314]. Electroconvection enhances the exchange between the solution domains, which are located near the conductive and non-conductive surface regions. The onset of this enhancement (occurring at $t={\tau}_{1}<{\tau}_{Sand}$) reduces the rate of the PD growth, which gives the first inflection point on the chronopotentiogram. The appearance of larger electroconvective vortices, when the concentration reaches a critically small value on the entire membrane surface ($t={\tau}_{2}\approx {\tau}_{Sand}$), gives the second inflection point. τ

_{2}is close to ${\tau}_{Sand}$, since the latter event occurs also at homogeneous membrane surface. However, electroconvection occurring when the near-surface concentration is small everywhere can enhance mass transfer and slows down the rate of concentration decrease resulting in increasing τ

_{2}higher than ${\tau}_{Sand}$. This case is shown in Figure 18 for the M1 and M2 membranes.

_{2}are essentially higher than the results of calculations for these membranes using the 3D model [303], which takes into account the electrical heterogeneity of the surface but not account for electroconvection. Note that the increase in the values of τ

_{2}with decreasing the $i/{i}_{\mathrm{lim}}$ ratio is due to the effect of the finite length of DBL described in Section 6.1. The behaviour of the MA-41 membrane is similar: at $i/{i}_{\mathrm{lim}}\le $ 1.25, there are two transition times. However, the $\tau /{\tau}_{Sand}$ vs. $i/{i}_{\mathrm{lim}}$ curve passes lower than the similar curves for the M2 membrane, which is apparently due to a low fraction of the conductive surface area.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## List of Symbols

Abbreviations | |

AEM | anion-exchange membrane |

CEM | cation-exchange membrane |

DBL | diffusion boundary layer |

DC | desalting compartment |

DVB | divinylbenzene |

ED | electrodialysis |

EDL | electrical double layer |

IEM | ion-exchange membrane |

IEC | ion-exchange capacity |

LEN | local electroneutrality (assumption) |

NF | nanofiltration |

PD | potential difference |

PVC | polyvinylchloride |

RO | reverse osmosis |

TMS | (model of) Teorell-Meyer-Sievers |

Symbols | |

${a}_{i}$, ${a}_{\pm}$ | ion and electrolyte molar activity, respectively |

${a}_{\pm}^{I}$, ${a}_{\pm}^{II}$ | electrolyte activities in the left- and right-hand side bathing solutions |

b | the distance between two neighbouring fixed ions |

c_{i} | molar concentration of ion i |

c_{s} | molar concentration of salt |

c | electrolyte concentration in the virtual solution |

${c}^{0}$, ${c}_{1}^{0}$ | concentration of electrolyte, counterion in the solution bulk, respectively |

d | membrane thickness |

D | diffusion coefficient of electrolyte |

D_{i} | diffusion coefficient of ion i |

E | potential difference |

E | electric field strength |

E_{max} | theoretical value of E for an ideal membrane not permeable for co-ions |

f_{i} | volume fraction of the gel (1) and intergel solution (2) phase |

${f}_{EDLp}$ | volume fraction of the nanoparticle’s EDL |

F | Faraday constant |

F_{j} | thermodynamic force of j kind |

i | current density |

i_{lim} | limiting current density |

g | activity factor |

h | intermembrane distance |

j_{i} | flux density of ion “i” |

J_{i} | flux density of ion “i” |

J_{v} | volume flux density |

K | equilibrium constant |

K_{D} | Donnan equilibrium coefficient |

K_{S} | electrolyte partition coefficient |

L_{D} | thickness of the diffuse part of the EDL |

L_{p} | hydraulic permeability coefficient |

L_{ij} | phenomenological conductance coefficient |

m | virtual solution molality |

M_{w} | water molar mass |

n | number of molecules per fixed group |

p | hydrostatic pressure |

P | diffusion permeability |

P_{int} | integral (or global) permeability |

Q | ion-exchange capacity |

r | pore radius |

R | universal gas constant |

R_{a}, R_{c}, R_{cc} | resistances of the AEM, CEM and the concentration compartment |

R_{1} | radius of conductive area |

Re | Reynolds number |

Sh | Sherwood number |

Sc | Schmidt number |

t | time |

t_{+app} | apparent transport number of counter-ion |

t_{i} | transport number of ion “i” in solution or membrane |

t_{w} | water transport number |

T | temperature |

T_{i} | effective transport number of ion “i” in the membrane |

V | velocity |

$\overline{V}$ | average flow velocity |

$\overrightarrow{V}$ | flow velocity |

${\overline{V}}_{i}$ | partial molar volume of species |

${\overline{V}}_{s}$ | partial molar volume of an electrolyte |

${\overline{V}}_{w}$ | partial molar volume of water |

x | normal to membrane coordinate |

y | longitudinal coordinate |

${y}_{\pm}$ | mean molar activity coefficient |

Y | dimensionless distances from the channel entrance |

z_{i} | charge number of ion i |

Greek Symbols | |

α | structural parameter |

β | electro-osmotic permeability |

β | structural parameter (similar to) |

δ | Nernst’s diffusion layer thickness |

$\Delta {\phi}_{D}$ | (Donnan) electric potential difference between two phases |

γ | surface fraction of membrane conductive area |

η | electric current scream function |

ε, ε_{0} | relative dielectric permittivity, vacuum dielectric permittivity |

ξ | parameter related to the linear charge density of the polyelectrolyte chain |

κ | electric conductivity, S m^{−1} |

λ | Debye length |

λ_{B} | Bjerrum length |

μ_{l} | electrochemical potential |

ν | stoichiometric parameter |

ν | kinematic viscosity |

π | osmotic pressure |

σ | Staverman’s reflection coefficient |

τ | transition time |

ϕ | electric potential |

Indices | |

$\overline{a}$ | over-bar means that the parameter refers to the membrane (gel) medium |

a | anion-exchange membrane |

A | co-ion |

c | cation-exchange membrane |

cc | concentration compartment |

D | Donnan |

g | gel phase |

i | ionic species |

iso | isoconductance point |

k | left-hand (I) and right-hand (II) interface |

Lev | Leveque |

lim | limiting |

limloc | limiting local |

limav | limiting average |

m, mb | membrane |

ms | membrane and bulk solution |

N loc | Nernst local |

N av | Nernst average |

p | particle |

pin | particle intergel |

s | interstitial solution |

sin | solution intergel |

w | water |

1 | counterion |

+ | cations |

− | anions |

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**Figure 1.**SEM images of surface and cross-sections of IEMs; (

**a**) a homogeneous Neosepta

^{®}AMX (Astom, Toukyouto, Japan, 2016) AEM, a styrene-DVB copolymer [12], (

**b**) a heterogeneous MA-40 (Shchekinoazot, Russia, 2016) AEM; 1: particles of ion-exchange resin; 2: polyethylene; 3: reinforcing cloth. Reproduced with permission from [4], Wiley, 2012.

**Figure 2.**Distribution of counterion (1) and co-ion (2) within a pore near a charged wall with fixed functional groups (3) shown as distribution of points (

**a**) or with concentration profiles (

**b**). Reproduced with permission from [4], Wiley, 2012.

**Figure 3.**Schematic representation of the IEM structure with its main elements: fixed ions (shown as circles with “−”), EDL formed at the internal interfaces and electroneutral solution in the centre zone of inter-gel spaces. The gel phase includes the polymer matrix bearing the fixed ions and the EDL. Adapted with permission from [46], Elsevier, 2008.

**Figure 4.**Conductivity of IEMs under AC (κ) as a function of bathing solution concentration, c, in Lg-Lg coordinates for different membranes; 1 – CMX in KCl [141], 2 – MK-40 in KCl [141], 3 – Nafion 125 in NaCl [153], 4 – MF-4SK-101 in NaCl [153], 5 – MA-41 in KCl [141], 6 – AMX in KCl [141], 7 – AMV in KCl [144], 8 – AMX in NaCl [154], 9 – AMX in KCl [144], 10 – CMS in NaCl [146], 11 – LNA in NaCl [154].

**Figure 5.**Schematic representation of a fragment of a mesoporous IEM containing a charged nanoparticle surrounded by an EDL. Adapted with permission from [174], Elsevier, 2016.

**Figure 6.**Proton conductivity in water κ (

**a**) and diffusion permeability P (from 0.1 M HCl solution to water) (

**b**) of a Nafion

^{®}membrane doped with caesium acid salt of phosphotungstic heteropolyacid, as functions of nanoparticle volume fraction, f

_{p}. Diamond icons show experimental results [170]; the curves are calculated by applying the model [174] for different particle diameters shown in nm near the curves. The best fitting is obtained, when assuming that the nanoparticle size is 16 nm. Reproduced with permission from [174], Elsevier, 2016.

**Figure 7.**Schematic depiction of the membrane structure fragment containing pores with charged walls. The polymer matrix is presented by the hatched area. Reproduced with permission from [4], Wiley, 2012.

**Figure 8.**Scheme of an ED cell with an AEM and a CEM; DC and CC are the desalting and concentrating compartments, respectively. The points 1 and 1′ show where the tips of Luggin’s capillaries are placed for measuring PD. Redrawn from [4].

**Figure 9.**Concentration profile in the DC of an ED cell computed (at = 0.2 V) using the basic version of convection-diffusion model [247,248,249] at different dimensionless distances from the channel entrance $Y=yD/\left(\overline{V}{h}^{2}\right)$: 1 – 0, 2 – 0.001, 3 – 0.02, 4 – 0.2. AEM and CEM refer to an AEM and a CEM, respectively.

**Figure 11.**Schematic distribution of the electric potential. The variations within quasi-equilibrium interfacial EDLs are shown as vertical lines. ${c}_{2s}$ and ${c}_{1s}$ relate to the positions of counterion minimum concentration near the AEM and the CEM, respectively; ${c}_{1m}$ and ${c}_{2m}$ show the position of the points where the counterion concentration profile passes from the solution to the membrane phase.

**Figure 12.**The positions of the planes normal to the membrane surface for which the concentration profiles are computed. Cross-section 1 is at a distance of 550 μm from the DC entrance.

**Figure 13.**Concentration profiles near a heterogeneous (1–4) and a homogeneous (5) CEM. The fraction of conductive area for the heterogeneous membrane is 0.1 (

**a**), 0.2 (

**b**) and 0.5 (

**c**).

**Figure 14.**Dependence of theoretical (lines) and experimental (points) transition time on the normalized current density for a Neosepta

^{®}CMX membrane in a 0.02 M NaCl, τ

_{Sand}refers to the transition time calculated using the Sand equation, Equation (76); τ

_{Model}refers to the finite-length time-dependent diffusion layer model [269]; the DBL thickness and the limiting current density calculated using the Lévêque Equations (60) and (61) are δ

_{Lev}= 260 μm, i

_{lim}= 2.0 mA/cm

^{2}, respectively. Reproduced with permission from [269], Elsevier, 2016.

**Figure 15.**Distribution of round conductive areas over the membrane surface in hexagonal geometry (

**a**). The DBL of thickness δ is assumed independent of lateral position (

**b**). Conversion of the 3D system (

**b**) into a 2D simulation problem (

**c**). Reproduced with permission from [303], American Chemical Society, 2016.

**Figure 16.**Transition time normalized by ${\tau}_{Sand}$ as a function of (

**a**) $i/{i}_{av\text{}\mathrm{lim}}$: experimental results (circles) are compared with the simulation using the 3D model [303] with integral current condition (77) (curve 1) or condition [274] (80) (curve 2) and the Choi-Moon theory [291], Equation (81), (curve 3), R

_{2}= 46.53 μm; (

**b**) $i/{i}_{av\text{}\mathrm{lim}}$ at different fraction of conductive area $\gamma =0.1;0.3;0.5;0.7;0.9;1.0$; R

_{1}= 12.71 μm; (

**c**) R

_{1}/δ at different γ (the same as in b) at fixed $i=2{i}_{av\text{}\mathrm{lim}}$; (

**b**,

**c**) are computed using the 3D model [303] with condition (77). The dashed straight lines show the transition time calculated by the Choi-Moon Equation (81). In all cases δ = 260 μm, ${i}_{av\text{}\mathrm{lim}}$ = 19.74 A/m

^{2}. Reproduced with permission from [303], American Chemical Society, 2016.

**Figure 17.**Distribution of electrolyte concentration in the DBL at a heterogeneous membrane surface, calculated using the 3D model [303] at i = 1.25i

_{lim}in moments of time 0.2, 0.8, 1.4 … 5.6 s. The parameters used in simulation relate to a membrane with randomly distributed conductive areas with the radius of 12.95 μm, the fraction of the conductive surface area γ = 0.077, the DBL thickness δ = 260 μm. The centre of the conductive area refers to r = 0, the boundary between the conductive and non-conductive regions is shown with the dashed line. Reproduced with permission from [314], Elsevier, 2018.

**Figure 18.**Dependence of the $\tau /{\tau}_{Sand}$ ratio on the $i/{i}_{\mathrm{lim}}$ ratio for the M1 (the conductive surface area fraction γ = 0.077, red circles) and M2 (γ = 0.318, triangles) membranes (both are lab-made on the basis of track-etched membranes) (

**a**) and for a anion-exchange heterogeneous MA-41 membrane (γ = 0.2) (

**b**). The horizontal dashed line shows the ${\tau}_{Sand}$ value. The solid curves present the results of calculation of τ using the 3D model [303]: curve 1 relates to a hypothetical homogeneous membrane, curves 2 and 3, to the M1 and M2 heterogeneous membranes, respectively. Reproduced with permission from [314], Elsevier, 2018.

**Table 1.**Properties of some homogeneous and heterogeneous ion-exchange membranes (IEMs) in NaCl or KCl solutions. Conductivity was measured under alternative current.

Membrane | Type, Manufacturer | IEC, meq/g | Water Content, g/g _{dry mbr} or % When in g/g_{wet mbr} | Conductivity of Gel Phase, κ^{g}, mS cm^{−1} | Apparent Volume Fraction of Inter-Gel Solution, f_{2 app} |
---|---|---|---|---|---|

AMX, anion-exchange | Homogeneous, Neosepta Astom (Japan) | 1.30 ± 0.05 [144] | 0.10–0.14 [144] | 3.72 * (KCl) [141] 2.19 ± 0.02 (KCl) [144] 3.2 (NaCl) [154] | 0.11 * (KCl) [141] 0.099 (KCl) [144] 0.06 (NaCl) [154] |

CMX, cation-exchange | Idem | 1.65 [153] | 0.275 [153] | 8.79 * (KCl) [141] 6.72 (NaCl) [153] | 0.10 * (KCl) [141] 0.06 (NaCl) [153] |

CMS, cation-exchange | Idem | 2.2 [146] | 22–30% [146] | 2.7 (NaCl) [146] | 0.13 (NaCl) [146] |

AMV anion-exchange | Homogeneous, Selemion Asahi Glass (Japan) | 1.85 ± 0.04 [144] | 0.14–0.18 [144] | 3.80 ± 0.01 (KCl) [144] 3.50 (NaCl) [153] | 0.06 (KCl) [144] 0.07 in NaCl [153] |

Nafion 125, cation-exchange, perfluorinated | Homogeneous, DuPont™ | 0.81 [153] | 0.129 [153] | 6.70 (NaCl) [153] | 0.07 (NaCl) [153] |

MF-4SK, cation-exchange, perfluorinated | Homogeneous, Plastpolymer (Russia) | 0.86 [153] | 0.156 [153] | 6.40 (NaCl) [153] | 0.05 (NaCl) [153] |

MA-41, anion-exchange | Heterogeneous Shchekinoazot (Russia) | 1.25 [155] | 0.29 [155] | 3.99 * (KCl) [141] | 0.24 * (KCl) [141] |

MK-40, cation-exchange | Idem | 2.53 [153] | 0.503 [153] | 5.64 * (KCl) [141] 5.27 [153] | 0.25 * (KCl) [141] 0.18 [153] |

LNA in NaCl | Heterogeneous, Lin’an (China) | 1.63 [154] | 49–55% [154] | 2.2 (NaCl) [154] | 0.28 (NaCl) [154] |

**Table 2.**Thickness of the DBL determined numerically for cross section 1 (Figure 12) passing across the centre of the conductive region at a distance of 550 μm from the ED channel entrance at different values of the fraction of conductive surface area, γ.

γ | ${\mathit{\delta}}_{\mathit{N}}^{\prime},\text{}\mathsf{\mu}\mathbf{m}$ | δ_{tot}, μm |
---|---|---|

0.1 | 54 | 128 |

0.2 | 72 | 134 |

0.5 | 86 | 136 |

1 | 85 | 139 |

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**MDPI and ACS Style**

Nikonenko, V.; Nebavsky, A.; Mareev, S.; Kovalenko, A.; Urtenov, M.; Pourcelly, G.
Modelling of Ion Transport in Electromembrane Systems: Impacts of Membrane Bulk and Surface Heterogeneity. *Appl. Sci.* **2019**, *9*, 25.
https://doi.org/10.3390/app9010025

**AMA Style**

Nikonenko V, Nebavsky A, Mareev S, Kovalenko A, Urtenov M, Pourcelly G.
Modelling of Ion Transport in Electromembrane Systems: Impacts of Membrane Bulk and Surface Heterogeneity. *Applied Sciences*. 2019; 9(1):25.
https://doi.org/10.3390/app9010025

**Chicago/Turabian Style**

Nikonenko, Victor, Andrey Nebavsky, Semyon Mareev, Anna Kovalenko, Mahamet Urtenov, and Gerald Pourcelly.
2019. "Modelling of Ion Transport in Electromembrane Systems: Impacts of Membrane Bulk and Surface Heterogeneity" *Applied Sciences* 9, no. 1: 25.
https://doi.org/10.3390/app9010025