# Mathematical Modeling of the Effect of Pulsed Electric Field on the Specific Permselectivity of Ion-Exchange Membranes

^{*}

## Abstract

**:**

## 1. Introduction

- -
- Reduction of concentration polarization. Relaxation of the concentration profile occurs at the membrane surface during the pause lapse. The concentrations of ion species resume partially or completely to the initial values;
- -
- Intensification of electroconvection at overlimiting current regimes, which in addition to the increase in mass transfer helps to wash out the scale and foulant components from the membrane surface;
- -
- Reduction of water splitting (pH values at which some components can precipitate are not reached).

_{lim}, the specific selectivity is lost [33,34] due to increasing concentration polarization and the transition of the ion transfer control from the membrane to the depleted diffusion layer [35]. The possibility to have a high permselectivity at elevated current densities is of great practical interest. The authors associated the enhancement of permselectivity by PEF with the partial restoration of ion concentration profiles in the depleted boundary solution during the pause. As a consequence, the membrane partially restores control over the kinetics of ion separation. Earlier [17], was studied only the mode where a constant electric current was applied during the pulses. The concentration profiles were calculated, and it was found that during one pulse, there is a gain in the membrane specific permselectivity compared to the conventional continuous current ED mode. However, the averaged in time-specific permselectivity was not quantified. In this paper, a simulation was carried out using a non-stationary mathematical model based on the Nernst–Planck and Poisson equations. For the first time, two PEF modes are compared when current or voltage pulses are applied.

## 2. Experimental Results

^{2}. According to the estimation by the method of Cowan and Brown, the limiting current density was 13.5 mA/cm

^{2}; hence the ratio of the current density to its limiting value at the beginning of the desalination process was i/i

_{lim}≈ 0.6. However, as far as the solution was desalted, the value of i

_{lim}decreased. When the total degree of desalination was 60%, i approached i

_{lim}, and i/i

_{lim}became approximately equal to 1.5. The same current density of 8.0 mA/cm

^{2}was applied during the pulses in PEF mode, while the current was zero during the pauses.

_{k}, are given in CC mode and in PEF mode for the pulse/pause combination, 1 s–1 s (0.5 Hz).

_{k}value, the greater the transport number of ion k (the fraction of the electric charge transported by this ion through the membrane).

^{+}ion is the dominant cation in the sweet whey. Since its mobility in the membrane is relatively high, the demineralization rate for this ion is the highest. The DD by this ion is higher than the total DD, θ

_{K}> 1, and this parameter changes a little with decreasing the total concentration and increasing the i/i

_{lim}ratio. In CC mode, the values of θ

_{Na}and θ

_{Ca}are close at i/i

_{lim}≈ 0.6, but θ

_{Na}is significantly higher than θ

_{Ca}at i/i

_{lim}≈ 1.5. That is, with an increase in the i/i

_{lim}ratio, the selectivity of Ca

^{2+}transfer through the membrane decreases in favor of Na

^{+}. A similar situation occurs when comparing the competitive transport of Na

^{+}and Mg

^{2+}. In the study by Lemay et al., a cation-exchange membrane was used whose properties were close to that of the Neosepta CMX membrane [17]. It is a homogenous membrane (produced by Astom, Tokuyama Soda, Japan) made by the “paste method” [36,37]. Initially, the paste contains styrene monomer with functional groups (which are subsequently grafted with ion-exchange groups), divinylbenzene (45–65%) as a crosslinking agent, a radical polymerization initiator and powdered polyvinyl chloride (45–55%). The paste is deposited on the reinforcing polyvinyl chloride fabric. The copolymerization is carried out before the sulfonation. The membrane consists of two interpenetrating phases: ion exchange material and polyvinyl chloride, PVC, having a particle diameter of 100 nm or less [38]. The structural inhomogeneities in the membrane volume do not exceed 1 micron [39]. An exception is a reinforcing fabric having fibers of 25–30 µm in diameter [40]. The results presented in Table 1 show that in CC mode, this permselectivity decreases both in the cases of Na

^{+}and Ca

^{2+}and Na

^{+}and Mg

^{2+}. However, in PEF mode, the membrane permselectivity for the Ca

^{2+}and Mg

^{2+}cations is significantly greater than that in CC mode if the same current density is applied in CC mode and in PEF mode during the pulses. Note that the average current density applied in PEF mode was two times lower since, during the pauses, the current was zero. It is important that the total duration of the ED process needed to obtain total DD ≈ 70% is nearly the same in both modes [17], whereas in PEF mode, the current flows across the membranes only for a certain fraction of the total processing time.

## 3. Theoretical Part

_{k}, c

_{k}, D

_{k}, z

_{k}, and γ

_{k}are the flux density, concentration, diffusion coefficient, charge number and activity coefficient of ion k (k = Na

^{+}, Ca

^{2+}, ${\mathrm{Cl}}^{-}$), respectively; R is the gas constant; T is the temperature; F is the Faraday constant; φ is the electric potential; ε

_{0}is the vacuum permittivity; ε is the solution relative permittivity; $E=-\frac{\partial \phi}{\partial x}$ is the electric field strength; z

_{m}is the charge number of the membrane; $\overline{Q}$ is the exchange capacity of the membrane; F is the Faraday constant; t is the time.

^{−2}s are considered, while the displacement current occurs in a very short time interval, less than 10

^{−4}s. This process should be accounted for when considering, for example, a high-frequency impedance of membrane systems [41].

^{I}and δ

^{II}be the thicknesses of the depleted and enriched diffusion layers, respectively, and the origin of coordinates is at the solution/membrane boundary (Figure 1). Then, the boundary conditions for the system under study will be:

_{k}(x, t) and φ (x, t) are continuous on the interval (x = −δ

^{I}, x = d + δ

^{II}). The potential is assumed zero at the left edge of the three-layer system (Equation (6)).

_{pulse}and U

_{av}are the voltage applied during the pulse, and the average voltage, respectively; $\alpha $ is the duty cycle.

_{k}, in the membrane. T

_{k}is defined as a fraction of current transported by ion k:

## 4. Results and Discussions

_{k}(or effective transport numbers, T

_{k}), computed in the membrane at a distance of 1 μm from the depleted solution/membrane interface (left-hand interface). The latter shows the rate of ion transport through the left-hand interface. The point located at 1 μm is chosen since, at a shorter distance, the errors in the calculated values of i

_{k}(T

_{k}) are relatively high due to very large concentration and potential gradients.

^{+}and Ca

^{2+}in the membrane are determined from the electrical conductivity of a Neosepta CMX membrane [17]. The activity coefficients of the cations in the membrane are chosen from the following consideration. The activity coefficients in the membrane and solution are linked with the ion-exchange constant, K

_{N}(which is also called the Nikolsky’s constant [43,44]) as follows [45,46]:

_{N}enters the relation of local thermodynamic equilibrium, which holds at the solution/membrane interfaces:

^{2+}in the membrane is about 10 times higher than that of Na

^{+}(as experiment shows in the case of ion-exchangers with sulfonate fixed groups [45]), when their equivalent fractions in the equilibrium bathing solution are the same at I = 0 and when the concentrations of both ions in the solution were 0.02 eq/L, which approximately corresponds to the experimental conditions [17].

_{lim}, of the system was calculated using Equation (12). This equation is applied for systems with ternary electrolyte (e.g., CaCl

_{2}and NaCl), under the assumption that the membrane is impermeable to coion [47]:

^{+}or Ca

^{2+}) and A to coion (Cl

^{−}); δ is the thickness of the depleted diffusion layer (δ = δ

^{I}), which was estimated from the hydrodynamic parameters of the experimental system [17]. Since the model does not take into account all the components of the experimental solution, in particular K

^{+}, which is the dominant cation [17], the calculated limiting current density i

_{lim}= 6.52 mA/cm

^{2}is lower than the experimental one (13.5 mA/cm

^{2}). However, in this paper, we are not aiming for quantitative agreement with the experiment. The goal is to understand whether taking into account electromigration and diffusion is sufficient to obtain a qualitatively correct picture of the effect of PEF on membrane permselectivity for a specific ion.

#### 4.1. CC Mode

_{av}, which is linked with the voltage during the pulse, U

_{pulse}, as U

_{av}= αU

_{pulse}.

^{2+}in the membrane (due to great value of K

_{N}, Table 2) leads to a higher flux of this ion compared to the Na

^{+}flux in conditions where the equivalent concentrations of both ions in the solution bulk are the same; thus, the membrane shows permselectivity for specific ions [48], the divalent Ca

^{2+}cation in the considered case. However, with increasing current density, the permselectivity decreases, and it is nearly completely lost at I = i

_{lim}[33,34,47]. This loss is due to a particular developing concentration polarization of the membrane. Since Ca

^{2+}is the preferentially transferred ion over Na

^{+}, the applied electric current causes a faster decrease in the concentration of Ca

^{2+}at the depleted solution/membrane interface than the decrease in the concentration of Na

^{+}. Moreover, the computation shows (and experimental data [49,50] confirm that) that at low voltages (e.g., at U

_{av}= 111 mV in Figure 2a), the concentration of Na

^{+}at the depleted interface can be even higher than its bulk value. Thus, under an applied current, the equilibrium at this interface shifts in favor of Na

^{+}, whose relative concentration in the depleted solution and in the membrane increases. As a result, the ratio between the fluxes of Ca

^{2+}and Na

^{+}(and between the transport numbers) changes in favor of Na

^{+}: preferential transfer of Ca

^{2+}at low current densities (low voltages) is lost, the transfer rate of Na

^{+}at i = i

_{lim}becomes even greater than that of Ca

^{2+}(Figure 3). At the limiting current density, the fluxes of the competing ions are controlled by the depleted diffusion layer and do not depend on the membrane properties Equation (9); the role of the membrane in terms of selective transfer is reduced to being a barrier for co-ions.

#### 4.2. PEF Mode (Current Pulses)

_{CC}= i

_{pulse}= 0.6i

_{lim}, in CC mode, the T

_{Ca}:T

_{Na}ratio is 0.52:0.48, while in PEF mode this ratio is essentially higher: 0.65:0.34; the parameters of the PEF mode are as follows: the duty cycle α = 0.5 and frequency f = 0.5 Hz. This theoretical result is in qualitative agreement with the experiment [17] described above (Section 2). The PEF effect is due to partial restoration of concentrations during a pause lapse. The restoration occurs near the surface; a little increase in concentration produces an important decrease in resistance. As a consequence, after a pause, the profiles are closer to those, which occur at a lower CC current density, which explains a higher permselectivity. An important feature is that the Na

^{+}partial current density is negative during the pause lapse, while the Ca

^{2+}partial current density remains positive (Figure 5a). This explains the increase in permselectivity attained in the PEF mode. However, when calculating the values of the transport numbers in the CC and PEF modes under the condition that the average over the period current density i

_{av}in PEF mode is equal to the constant current density in CC mode, no difference was found between the transport numbers in CC and PEF modes. When we have taken i

_{av}= i

_{CC}= 0.3i

_{lim}(in this case the current density during the pulse is i

_{pulse}= 0.6i

_{lim}), α = 0.5 and f = 0.5 Hz, we find that the T

_{Ca}:T

_{Na}ratio is the same in both modes and equal to 0.65:0.34. The same T

_{Ca}:T

_{Na}ratio (0.65:0.34) is found at a higher frequency f = 10 Hz (α = 0.5), when i

_{pulse}= 0.6i

_{lim}. The calculations show that the T

_{Ca}:T

_{Na}ratio depends only on the i

_{av}value and does not depend on the frequency and duty cycle; it does not matter if PEF or CC mode is used.

#### 4.3. PEF Mode (Voltage Pulses)

^{+}and Ca

^{2+}ions at a frequency of 0.5 Hz are shown in Figure 7.

^{2+}gradually decreases, while that of Na

^{+}increases. Qualitatively, this behavior is similar to that occurring in PEF mode, when constant current pulses are applied. Similar to above, the reason for the Ca

^{2+}current diminution is a more rapid decrease in its concentration compared to that of Na

^{+}in the depleted diffusion layer and membrane. During a pause lapse, the partial current density of Ca

^{2+}increases, while that of Na

^{+}decreases so that i

_{Na}becomes negative. It should be noted that the total current density during pulse lapse is significantly higher than that at CC mode at the same average value of the voltage, U

_{av}.

^{+}current density during a pulse lapse becomes greater than the current density of Ca

^{2+}at a higher value of the average voltage (Figure 8c) than in the case of 0.5 Hz (Figure 7b).

_{av}, in PEF mode is higher than that in CC mode at a given average voltage, U

_{av}; moreover, i

_{av}increases with increasing frequency that agrees with the results of Sistat et al. [16]. This increase is explained by the fact that only the first moments (a few hundredths of a second) are profitable for obtaining a high current density (Figure 6b). With increasing duration of the pulse, the current density decreases rapidly due to increasing concentration polarization. However, when comparing the T

_{Ca}:T

_{Na}ratios obtained in PEF mode and CC mode, it is found that this ratio is independent of how the PEF mode is applied, whether current or voltage pulses are applied, and what are the frequency and duty cycle. This ratio is a function of only one quantity, the average current density.

^{2+}is improved in PEF mode. However, there is controversy about the effect of the frequency. The simulation shows that the membrane specific permselectivity in PEF mode does not depend on the frequency; only the average current density is important. However, the experiment indicates that under conditions that i

_{av}is the same, the permselectivity for Ca

^{2+}in the case of f = 0.5 Hz is greater than in the case of f = 5 Hz. Evidently, the mechanism of the effects observed in [17] is more complicated than the developed model suggests. It is possible that an important role in these phenomena is played by electroconvection. As mentioned in the introduction, at overlimiting average currents, electroconvection can increase the mass transfer rate by about 33% [25,26]. Electroconvective micro-vortices could essentially improve mixing the depleted solution near the membrane surface. The enhancement of convective mass transfer from the bulk to the membrane surface could shift the Ca

^{2+}:Na

^{+}concentration ratio in favor of Ca

^{2+}. However, modeling competitive ion transport with taking into account electroconvection is not an easy task; it can be the subject of another publication.

## 5. Conclusions

_{1}and T

_{2}) in PEF mode are equal to their values in CC mode. Moreover, the values of T

_{1}and T

_{2}do not depend on whether current or voltage is applied during the pulses; they are also independent of the frequency and duty cycle used in PEF mode. The last results do not agree with the experiment of Lemay et al. [17]. It was found in [17], that the membrane permselectivity for Ca

^{2+}in the case of f = 0.5 Hz is greater than in the case of f = 5 Hz. Therefore, taking into account only electromigration and diffusion as the mechanisms of ion transport seems insufficient. It is possible that this discrepancy is due to electroconvection. Electroconvective micro-vortices could deliver a fresh solution from the solution bulk to the depleted membrane interface, which can maintain relatively high Ca

^{2+}:Na

^{+}concentration and flux ratios at this interface.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

Abbreviations | |

CC | Continuous current |

DBL | Diffusion boundary layer |

DD | Degree of desalination |

ED | Electrodialysis |

PEF | Pulsed electric field |

PVC | Polyvinyl chloride |

Symbols | |

c_{k} | Concentration of ion k |

c_{k0} | Concentration of ion k in the bulk solution |

d | Membrane thickness |

D_{k} | Diffusion coefficient of ion k in the solution |

${\overline{D}}_{k}$ | Diffusion coefficient of ion k in the membrane |

E | Electric field strength |

f | Frequency |

F | Faraday constant |

i | Current density |

i_{k} | Partial current density of ion k |

i_{lim} | Limiting current density |

J_{k} | Flux density of ion k |

K_{N} | Nikolsky’s constant |

$\overline{Q}$ | Exchange capacity of the membrane |

R | Universal gas constant |

t | Time |

T | Absolute temperature |

T_{k} | Effective transport number of ion k in membrane |

U_{av} | Average voltage |

U_{pulse} | Voltage, applied during the pulse |

z_{k} | Charge number of ion k |

z_{m} | Charge of fixed ions in the membrane |

Greek symbols | |

$\alpha $ | Duty cycle |

${\gamma}_{k}$ | Activity coefficient of ion k in the solution |

${\overline{\gamma}}_{k}$ | Activity coefficient of ion k in the membrane |

${\delta}^{I}$,${\delta}^{II}$ | Thickness of the depleted and enriches diffusion boundary layers, respectively |

$\epsilon $ | Solution relative permittivity |

${\epsilon}_{0}$ | Vacuum permittivity |

${\theta}_{k}$ | Ratio of the degree of desalination by ion k to the total degree of desalination |

$\phi $ | Electric potential |

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**Figure 1.**Schematic representation of the simulated system, including an ion-exchange membrane of thickness d (3) and two DBLs (2 and 4) of the thickness δ

^{I}and δ

^{II}, respectively. Zones (1) and (5) are the bulk solution. The typical concentration profiles are given in the case of CC mode: for ion k in the diffusion layers (k = Na

^{+}, Ca

^{2+}or Cl

^{−}), and for all three ions in the membrane.

**Figure 2.**Concentration profiles of ions in CC mode at: U

_{av}= 111 mV (i = 0.3 i

_{lim}) (

**a**); U

_{av}= 476 mV (i = i

_{lim}) (

**b**). The concentrations are normalized by c

_{0}; c

_{0}= ${c}_{Cl}^{0}$ in the diffusion layers (${c}_{Cl}^{0}$ is the concentration of Cl

^{−}ions in the bulk solution), and c

_{0}= Q in the membrane.

**Figure 3.**Dependence of the transport numbers of Ca

^{2+}and Na

^{+}ions in the membrane on the time–average voltage.

**Figure 4.**Time dependences of the current density (

**a**) and voltage (

**b**) for the pulsed electric field (PEF) mode, when current pulses are applied. i

_{av}is the time–average current density; U

_{av}is the average voltage. The results of calculations at α = 0.5, f = 0.5 Hz and i

_{CC}= i

_{pulse}= 0.6i

_{lim}.

**Figure 5.**Dependences of partial current densities of Na

^{+}and Ca

^{2+}ions on time at the frequencies of PEF of 0.5 Hz (

**a**) and 10 Hz (

**b**) at i

_{av}= 0.3 i

_{lim}.

**Figure 6.**Time dependences of the voltage (

**a**) and current density (

**b**) for PEF mode, when voltage pulses are applied. The results of calculations at α = 0.5, f = 0.5 Hz and U

_{av}= 111 mV (i = 0.3 i

_{lim}in CC mode).

**Figure 7.**Time dependences of normalized partial current densities of Na

^{+}and Ca

^{2+}ions in PEF mode at a frequency of 0.5 Hz and different U

_{av}: U

_{av}= 111 mV (i = 0.3 i

_{lim}in CC mode) (

**a**); U

_{av}= 150 mV (i = 0.4 i

_{lim}in CC mode) (

**b**) and U

_{av}= 238 mV (i = 0.65 i

_{lim}in CC mode) (

**c**). T

_{PEF}is the period PEF.

**Figure 8.**Dependences of normalized partial current densities of Na

^{+}and Ca

^{2+}ions on time in PEF mode at a frequency of 10 Hz and different U

_{av}: U

_{av}= 111 mV (

**a**); U

_{av}= 150 mV (

**b**) and U

_{av}= 238 mV (

**c**).

**Table 1.**The total degree of desalination (DD) of sweet whey and the ratio of the DD by individual ions to the total DD, θ

_{k}.

Total DD | ED Mode | K^{+} | Na^{+} | Ca^{2+} | Mg^{2+} |
---|---|---|---|---|---|

30% i/i _{lim} ≈ 0.6 | CC mode | 1.3 | 0.57 | 0.67 | 0.57 |

PEF mode, 1 s: 1 s | 1.3 | 0.17 | 1.0 | 0.88 | |

60% i/i _{lim} ≈ 1.5 | CC mode | 1.2 | 0.90 | 0.67 | 0.58 |

PEF mode, 1 s: 1 s | 1.2 | 0.61 | 0.87 | 0.83 |

Name | Value | Description |
---|---|---|

c_{0} | 0.02 mol/L | Bulk solution concentration |

$\overline{Q}$ | 1.64 mol/L | Exchange capacity of the membrane |

D_{Na} | 1.33 × 10^{−9} m^{2}/s | Ion diffusion coefficients in the solution |

D_{Ca} | 7.96 × 10^{−10} m^{2}/s | |

D_{Cl} | 2.04 × 10^{−9} m^{2}/s | |

${\overline{D}}_{Na}$ | 1.07 × 10^{−10} m^{2}/s | Ion diffusion coefficients in the membrane |

${\overline{D}}_{Ca}$ | 4.93 × 10^{−12} m^{2}/s | |

${\overline{D}}_{Cl}$ | 10^{−11} m^{2}/s | |

${\overline{\gamma}}_{Na}$ | 1 | Ion activity coefficients in the membrane |

${\overline{\gamma}}_{Ca}$ | 0.05 | |

${\overline{\gamma}}_{Cl}$ | 1 | |

δ^{I} | 150 μm | Depleted diffusion layer thickness |

δ^{II} | 150 μm | Enriched diffusion layer thickness |

d | 180 μm | Membrane thickness |

K_{N} | 4.5 | Nikolsky’s constant |

**Table 3.**Results of calculation of the average current density and transport numbers in the membrane in CC and PEF modes at different average voltages, U

_{av}, and different frequencies in Hz: f = 0 (CC mode), 0.5 and 10; α = 0.5.

U_{av}, mV | i_{av}/i_{lim} | T_{Ca}:T_{Na} | T_{Cl} | ||||||
---|---|---|---|---|---|---|---|---|---|

f = 0 | f = 0.5 | f = 10 | f = 0 | f = 0.5 | f = 10 | f = 0 | f = 0.5 | f = 10 | |

111 | 0.3 | 0.303 | 0.307 | 0.646:0.345 | 0.629:0.363 | 0.639:0.352 | 0.0092 | 0.0085 | 0.0085 |

150 | 0.4 | 0.428 | 0.437 | 0.593:0.398 | 0.559:0.432 | 0.569:0.422 | 0.0093 | 0.0086 | 0.0086 |

238 | 0.65 | 0.678 | 0.720 | 0.501:0.490 | 0.485:0.506 | 0.482:0.508 | 0.0098 | 0.0092 | 0.0094 |

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**MDPI and ACS Style**

Gorobchenko, A.; Mareev, S.; Nikonenko, V. Mathematical Modeling of the Effect of Pulsed Electric Field on the Specific Permselectivity of Ion-Exchange Membranes. *Membranes* **2021**, *11*, 115.
https://doi.org/10.3390/membranes11020115

**AMA Style**

Gorobchenko A, Mareev S, Nikonenko V. Mathematical Modeling of the Effect of Pulsed Electric Field on the Specific Permselectivity of Ion-Exchange Membranes. *Membranes*. 2021; 11(2):115.
https://doi.org/10.3390/membranes11020115

**Chicago/Turabian Style**

Gorobchenko, Andrey, Semyon Mareev, and Victor Nikonenko. 2021. "Mathematical Modeling of the Effect of Pulsed Electric Field on the Specific Permselectivity of Ion-Exchange Membranes" *Membranes* 11, no. 2: 115.
https://doi.org/10.3390/membranes11020115