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Current–Voltage Curves for Ion-Exchange Membranes: A Method for Determining the Limiting Current Density
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
205, 365–373 (1998)
CS985649
Current–Voltage Curves for Ion-Exchange Membranes: A Method for Determining the Limiting Current Density V. M. Barraga´n and C. Ruı´z-Bauza´1 Department of Applied Physics I, Faculty of Physics, University Complutense of Madrid, 28040 Madrid, Spain E-mail: [email protected] Received December 31, 1997; accepted May 8, 1998
The curves (V, I) of different ion-exchange membranes have been determined under different experimental conditions. Taking into account the classical theory of concentration polarization, a method has been developed to adjust the (V, I) data, up to a certain value of I, which permits us to obtain easily the value of the limiting current density, IL, in every experimental situation. From these values, the dependence of the limiting current density of the membranes used on the concentration and stirring rate of the solutions has been studied. To check the reliability of the method, the values of IL thus obtained have been compared with those obtained by the usual methods; the similitude between both values is high. © 1998 Academic Press Key Words: current–voltage curves; limiting current density; ion-exchange membrane; concentration polarization.
INTRODUCTION
Concentration polarization is a well-known phenomenon arising at the interface between an ion-exchange membrane and an electrolyte solution when an electric current passes through the system. This phenomenon has been widely studied in the last decades (1–3) with the purpose of establishing the factors that determine it and resolving the problems that concentration polarization causes in the membrane technology, in general, and in the separation processes, in particular. According to the classical theory of concentration polarization (4), the electric current, I, should increase linearly with voltage, V, at low voltages, then increase more slowly, and finally reach a determined limiting value. In practice, such a limiting value is not observed, but the curves (V, I) for the ion-exchange membranes have a characteristic shape on which three regions can be clearly distinguished: (I) a first region of approximately ohmic behavior, transforms as voltage is increased into a second region (II), in which the current varies very slowly with voltage (“plateau” corresponding to the limiting current), followed by a third region (III) of rapid current increase. This implies that the limiting current is a rather 1
To whom correspondence should be addressed.
ill-defined parameter, and it is necessary to use indirect methods, such as Cowan plots or the tangent method, to determine its value. Currents above the limiting one are referred to as “overlimiting.” In spite of that, considerations about concentration polarization always lead to the study of the so-called limiting current and the conditions under which the transport occurs at current densities equal to or greater than this limiting value. So, it is important to determine, in the easier and more accurate way, the limiting current that corresponds, in every experimental condition, to the membrane system studied. In this work, the curves (V, I) for different ion-exchange membranes have been studied under different experimental conditions. A method has been developed to adjust the (V, I) data, which permits us to obtain easily the value of the limiting current density, IL, in every case. The values of IL thus obtained have been compared with those obtained by the usual methods. BASIC EQUATIONS
The system studied consists of an ion-exchange membrane separating two equal 1–1 electrolyte aqueous solutions of bulk concentration c0, which are maintained at the same temperature and hydrostatic pressure. When an electric current passes through the membrane system, from the left compartment to the right one, the current is carried in the solution by both types of ions (positive and negative), whereas in the membrane, assumed to be selective, it is carried mainly, if not exclusively, by the counterions. The difference between the ion mobilities of the counterion, corresponding to the membrane and free solution phases, gives rise to a depletion in the electrolyte concentration on one side of the membrane (c1) and to an enrichment on the opposite side (c2). As a consequence, a concentration gradient (Fig. 1) is established through each one of the thin films that adjoin the membrane at the two sides (polarization layers). In the polarization layers, the electric potential gradient drives positive and negative ions in opposite directions, while
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placed in both sides of the membrane in the corresponding bulk solution is measured. Before the polarization layers are formed, i.e., at the initial moment t 5 0, the value obtained is V 0AB, whereas when the layers have been formed and the stationary state has been reached, the value obtained is V ` AB, ` 0 5 V AB 1 V *AB , V AB
[5]
where V*AB represents the electric potential difference created solely by the formation of the polarization layers. This contribution can be expressed as V *AB 5 V P 1 IDR .
[6]
VP is the polarization potential established through the membrane system and, assuming ideal behavior for the solutions, is given by FIG. 1. Sketch of the concentration profile in the studied system.
VP 5 the concentration gradient drives, by diffusion, the ions of both signs in the same direction. For a membrane selective to specific ions (i), and supposing the thicknesses d of the both layers are equal, the current density, I, can be expressed by I5
FD ~c 0 2 c 1 ! FD ~c 2 2 c 0 ! 5 , Dt i d Dt i d
IL 5
FDc 0 . Dt i d
From [1] and [2], it follows that, for a current density I , IL, the concentrations at the membrane/solution interfaces can be expressed as c1 5 c0
S S
I 12 IL
c2 5 c0 1 1
D D
I . IL
[3] [4]
Consider now the experimental situation in which a constant current density, I, is injected through the membrane system and the electric potential difference, VAB, between two fixed points
ES 0
DR 5 L 21
[2]
[7]
where R and T are the gas constant and absolute temperature, respectively. Variable DR is the change in the ohmic resistance as a consequence of the formation of the layers, and it is given by
[1]
where D is the salt diffusion coefficient, Dt i 5 #t i 2 t i is the difference between the counterion transport numbers in the membrane and in the free solution, and F is the Faraday constant. According to the classic theory of polarization, the current density reaches a limiting value, IL, when concentration c1 becomes zero. From [1], IL is given by
RT c2 2Dt iln F c1
2d
D
1 1 dx 1 L 21 2 c ~ x ! c0
E S d1d
d
D
1 1 dx, 2 c ~ x ! c0 [8]
where L is the equivalent conductance and c(x) the local concentration in the layers, given by c ~ x ! 5 c1 2
c0 2 c1 x d
c ~ x ! 5 c2 2
c2 2 c0 ~x 2 d! d
for 2d , x , 0 for d , x , d 1 d .
[9] [10]
From [2]–[4] and [8]–[10], it is obtained that DR 5
S
D
FD 1 c 2 2 . ln 2 LDt i I c1 IL
[11]
Taking into account Eqs. [3] and [4] and Nernst-Einstein’s equation which relates the equivalent conductance L and the diffusion coefficient D of a monovalent electrolyte by means of L 5 L1 1 L2 5
~D 1 1 D 2 ! F 2 2DF 2 5 , RT RT
[12]
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CURRENT–VOLTAGE CURVES
TABLE 1 Characteristic Parameters for All the Studied Membranes Membrane
61 AZL 183
61 CZL 386
204 SXZL 386
103 QZL 386
204 U 386
Selectivitya (%) Electrical resistencea (V cm2) Capacitya (meq/g) Membrane thickness (mm) Fraction of unfilled volume (1022)
98 5.6 3.0 600 48
94 11 2.7 520 32
96 11 2.2 540 45
97 9 2.1 580 41
97 6 2.8 550 22
a
Value provided by the manufacturer.
EXPERIMENTAL
it is obtained that the contribution IDR is expressed as RT 1 1 ~I/I L ! RT IDR 5 ln I. 2 2FDt i 1 2 ~I/I L ! FDt i I L
Materials [13]
If R0 is the ohmic resistance before the polarization layers are formed, we have 0 V AB 5 IR 0
[14]
Obviously, R ` 5 R 0 1 DR. From Eqs. [5], [6], [13], and [14] it is obtained that
V
` AB
S
D
Apparatus
2 RT RT 5 R0 2 I1 @~2Dt i ! FI L F 1 ~2Dt i ! 21#ln
1 1 ~I/I L ! . 1 2 ~I/I L !
[15]
For a membrane system consisting of a highly selective membrane and a dilute KCl solution, it is obtained that #ti > 1 and ti > 1/2, and so, 2Dti > 1. Under these conditions Eq. [15] can be reduced to
S
` V AB 5 R0 2
D
Five commercial membranes have been studied, two cationexchange membranes, the Nepton cation-exchange membrane 61 AZL 183 and the Ionics cation-exchange membrane 61 CZL 386, and three anion-exchange membranes, the Ionics aniontransfer membranes 204 SXZL 386, 103 QZL 386, and 204 U 386. Their characteristics are given in Table 1. The electrolytic solutions employed in the experiments were aqueous solutions of potassium chloride. Pure proanalysis chemicals and pure water (deionized, doubly distilled, doubly filtrated, and degassed) were used.
2 RT 2 RT 1 1 ~I/I L ! I1 ln . FI L F 1 2 ~I/I L !
[16]
Obviously, Eq. [16] is only applicable to current densities such as |I| , |IL|. When I becomes IL, the polarization potential as well as the resistance of the depletion layer tends to infinity in a logarithmic way. Equation [16] permits us the adjustment of the experimental data (V ` AB, I) up to a certain value of I, in different experimental situations. The adjustment parameters will be R0 and IL. For a given membrane system, and according to the definitions previously given for R0 and IL, R0 may depend only on the bulk concentration, but not on the stirring rate of the solutions. The value of IL, on the contrary, may depend on the concentration as well as the stirring.
The cell employed in these measurements consisted of two glass chambers of ;100 cm3 each. Both chambers were separated by a methacrylate membrane holder, where the membrane was positioned. Each chamber was provided with four orifices communicating to the exterior. Two of them were used as solution inlet and outlet by means of a peristaltic pump. Two Ag/AgCl electrodes were introduced in the other two orifices. One of them had a large active surface in order to inject the current, and the other was used to measure the electric potential difference values. An HP 6186C DC current source was used to maintain a constant electric current, and a Keithley 195 system DMM multimeter was used to measure the voltage. All the experiments were carried out under isothermal conditions (30°C). The temperature requirements were achieved by introducing the complete unit in a large ambient thermostat. The stirring of the solutions was assessed by circulating them by means of a peristaltic pump. RESULTS AND DISCUSSION
The curves (V, I) for the studied ion-exchange membranes have been studied under different experimental conditions. Figure 2 shows the curves (V, I) obtained, in the absence of stirring, for the cation-exchange membrane 61 CZL 386 at different KCl concentrations. Figures 3–7 show the curves (V,
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FIG. 2. Curves (V, I) for membrane 61 CZL 386, in the absence of stirring, with different KCl concentrations. (E) 5 3 1023 mol/liter; (F) 1022 mol/liter; (1) 2 3 1022 mol/liter; (h) 5 3 1022 mol/liter. The lines correspond to Eq. [16].
FIG. 4. Curves (V, I) for membrane 61 AZL 183, at different stirring rates, with aqueous 1022 mol/liter KCl. (F) 0 ml/min; (E) 102 ml/min; (■) 195 ml/min; (h) 305 ml/min. The lines correspond to Eq. [16].
I) obtained for the different studied membranes at various stirring rates and with 1022 mol/liter concentration of KCl. As can be observed in each figure, the obtained curves (V, I) show the three usual characteristic regions defined in the introduction for this kind of membranes. In addition, a visual inspection of these figures permits us to establish that the
second region, corresponding to the “plateau,” occurs at greater current densities and becomes less appreciable when the concentration of the solution and the stirring rate increase. This behavior is similar for cationic and anionic membranes. From the data (V, I) obtained in the first region, the value of
FIG. 3. Curves (V, I) for membrane 61 CZL 386, at different stirring rates, with aqueous 1022 mol/liter KCl. (E) 0 ml/min; (F) 128 ml/min; (h) 208 ml/min. The lines correspond to Eq. [16].
FIG. 5. Curves (V, I) for membrane 204 SXZL 386, at different stirring rates, with aqueous 1022 mol/liter KCl. (F) 0 ml/min; (E) 129 ml/min; (!) 201 ml/min. The lines correspond to Eq. [16].
CURRENT–VOLTAGE CURVES
FIG. 6. Curves (V, I) for membrane 103 QZL 386, at different stirring rates, with aqueous 1022 mol/liter KCl. (ƒ) 0 ml/min; (Œ) 207 ml/min; () 236 ml/min; () 349 ml/min. The lines correspond to Eq. [16].
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FIG. 8. Values of IL obtained, in the absence of stirring, for the two studied cation-exchange membranes at different KCl concentrations. From the adjustment to Eq. [16]: (F) membrane 61 AZL 183; (E) membrane 61 CZL 386. From the usual methods: (3) membrane 61 AZL 183; (1) membrane 61 CZL 386.
the limiting current density can be obtained under every studied experimental situation by fitting these data to Eq. [16] by means of a minimization x2 method. The adjustment parameters are IL and R0, as was previously indicated. Some of the curves obtained are shown in Figs. 2–7 together with the corresponding experimental values (V, I).
The values of IL obtained from the fits for the two cationexchange membranes, in the absence of stirring and at the different studied concentrations, are shown as a function of log c0 in Fig. 8. Together with these values of the limiting current density, those obtained from the usual methods under the same
FIG. 7. Curves (V, I) for membrane 204 U 386, at different stirring rates, with aqueous 1022 mol/liter KCl. ({) 0 ml/min; (}) 96 ml/min; () 152ml/ min; () 373 ml/min. The lines correspond to Eq. [16].
FIG. 9. Values of R0 as a function of log c0 obtained, in the absence of stirring, for the two studied cation-exchange membranes. (F) membrane 61 AZL 183; (E) membrane 61 CZL 386.
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TABLE 2 Values of IL and R0 Obtained for the Cation-Exchange Membrane 61 AZL 183 v (ml/min)
R0 (KV cm2)
IL (mA/cm2)
ITL (mA/cm2)
TABLE 3 Values of IL and R0 Obtained for the Cation-Exchange Membrane 61 CZL 386 v (ml/min)
1023 mol/liter 0 168 192 288
5.46 5.27 5.30
— 6 0.06 6 0.05 6 0.06
R0 (KV cm2)
IL (mA/cm2)
ITL (mA/cm2)
5 3 1023 mol/liter — 0.50 6 0.14 0.65 6 0.01 0.77 6 0.21
0.26 0.53 0.59 0.68
0 159 190
2.57 6 0.05 2.74 6 0.01 2.66 6 0.04
0.51 6 0.04 0.76 6 0.01 0.85 6 0.12
0.55 0.75 0.86
1022 mol/liter 23
5 3 10 0 105 200 286
1.37 1.35 1.58 1.36
mol/liter
6 0.01 6 0.03 6 0.01 6 0.02
0.99 6 0.02 1.26 6 0.5 1.47 6 0.06 1.68 6 0.27
1.07 1.26 1.39 1.60
1022 mol/liter 0 102 195 305
0.894 0.885 0.882 0.888
6 0.006 6 0.015 6 0.006 6 0.003
1.45 6 0.02 2.04 6 0.32 2.61 6 0.05 3.27 6 0.05
1.64 2.10 2.60 3.40
5 3 1022 mol/liter 0 100 196 312
0.20 0.204 0.203 0.200
6 0.01 6 0.003 6 0.001 6 0.003
6.53 6 0.02 9.75 6 0.32 15.57 6 0.05 17.85 6 0.05
6.62 10.0 14.3 20.0
1021 mol/liter 0 99 198 206 305
0.085 6 0.001 0.0954 6 0.0008 0.1004 6 0.0001 0.0975 6 0.0005 0.0969 6 0.0002
11.5 6 20.1 6 28.1 6 29.7 6 30.5 6
0.5 0.5 0.3 0.9 0.3
12.0 18.0 24.2 25.2 31.5
experimental conditions are shown; observe that the two sets of values are very similar. For membrane 61 AZL 183, the slope obtained for the curve which represents log IL against log c0 is 1.2, a result which is in agreement with the behavior IL } c5/4 0 predicted by the hydrodynamics theory (5, 6). In Fig. 8, a change of slope can also be observed occurring at concentrations higher than 5 3 1022 mol/liter, obtaining in this case 0.8, a result that can be explained by taking into account the electrogravitational effects. Thus, in this range, the hypothesis of the hydrodynamics theory is not applicable, in agreement with the results obtained by other authors. The values of R0 obtained in the absence of stirring are shown as a function of concentration logarithm in Fig. 9 for the former membranes. As can be expected, the value of R0 decreases when the concentration of the external solution increases. Tables 2 and 3 show the values of IL and R0 obtained from the fit to Eq. [16] for the cation-exchange membranes 61 AZL
0 128 208
1.47 6 0.01 1.40 6 0.01 1.35 6 0.01
1.90 6 0.26 2.12 6 0.02 2.65 6 0.17
1.90 2.10 2.30
183 and 61 CZL 386 at different stirring rates for every concentration studied. In Table 4 the values of IL and R0 obtained for the three anion-exchange membranes studied are shown, with a KCl 1022 mol/liter solution and at various stirring rates. In these tables, the value of the limiting current density obtained from the usual methods is also shown. As can be observed, a great similitude exists between the values obtained from both methods. It can be observed that in all the studied cases the obtained limiting current density increases, for a given concentration, when the stirring rate increases; that is a natural requirement since it is expected that the stirring decreases the thickness of the layers and, so, favors the transport process. On the contrary,
TABLE 4 Values of IL and R0 Obtained for the Anion-Exchange Membranes with a 1022 mol/liter Concentration of KCl v (ml/min)
R0 (KV cm2)
IL (mA/cm2)
ITL (mA/cm2)
204 SXZL 386 0 129 201
1.241 6 0.006 1.241 6 0.005 1.186 6 0.003
0.95 6 0.02 2.16 6 0.07 2.94 6 0.02
0.9 2.2 2.9
103 QZL 386 0 207 236 345
1.141 6 0.003 1.118 6 0.005 1.091 6 0.003 1.174 6 0.003
1.17 6 0.02 3.06 6 0.14 3.50 6 0.11 4.58 6 0.04
1.2 2.9 3.6 4.2
1.16 6 0.04 1.50 6 0.01 1.71 6 0.01 2.59 6 0.01
1.2 1.5 1.6 2.5
204 U 386 0 102 195 305
1.074 6 0.006 1.148 6 0.004 1.136 6 0.002 1.096 6 0.002
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CURRENT–VOLTAGE CURVES
d ~v ! 5
d0 , av 1 1
[17]
where d0 is the thickness of the layer in the absence of stirring. Expression [17] has been previously used to justify the dependence of the electroosmotic permeability of this kind of membrane on the rate of stirring, obtaining satisfactory results (7), and to consider the phenomena of temperature polarization in experiments on thermoosmosis and membrane distillation (8). Taking into account Eqs. [2] and [17], it is obtained that I L 5 I L0 1 b v ,
[18]
I L0 5 ~FDc 0 !/~Dt id 0 !
[19]
where
b 5 I L0 a . FIG. 10. Values of IL as a function of stirring rate for membrane 61 AZL 183 at different KCl concentrations. (1) 1023 mol/liter; (E) 5 3 1023 mol/liter; (F) 1022 mol/liter; (Œ) 5 3 1022 mol/liter; (ƒ) 1021 mol/liter.
the value of R0 for every concentration is, as could be expected, practically independent of the stirring rate. The values of IL obtained for the cation-exchange membrane 61 AZL 183 have been shown versus stirring rate for the different concentrations used, Fig. 10. The observation of the experimental data of this figure induces one to think of a possible linear relation between the limiting current density and the stirring rate. For this reason, the data have been fitted to straight lines, obtaining in all cases a satisfactory result. The straight lines found are shown in Fig. 10 together with the values of IL. The finding of this linear behavior can be justified taking into account that, for a determined membrane separating a solution of given nature and concentration, the value of the limiting current density is only a function, according to Eq. [2], of the thickness of the polarization layers. These thicknesses only depend on the stirring rate, so that, the stirring process causes a decrease of the layer thickness. This last fact can be taking into account in a quantitative way by supposing the following relation between the thickness of the layers for a given stirring rate, d(v), and the stirring rate, v,
[20]
Equation [18] expresses a linear relation between the limiting current density and the stirring rate, just as it is obtained in the experiments. The fit of the values of IL against stirring rate to Eq. [18] permits us to determine the parameters IL0 and b for each situation. From Eqs. [19] and [20] the values of a and the thickness of the polarization layer in the absence of stirring, d0, can be estimated. The results obtained for all the used membranes with a 1022 mol/liter KCl concentration are shown in Table 5. Table 6 shows the results obtained with the cationexchange membrane 61 AZL 183 and different KCl concentrations. As can be observed in Tables 5 and 6, the thickness of the polarization layers in the absence of stirring is of the same order of magnitude as those obtained by other authors [9]. Likewise, as can be observed in Table 4, this thickness is the greater the greater the concentration of the external solution in contact with the membrane is. It can be also deduced, from the results shown in Tables 5 and 6, that the values of a and b depend, for the same concentration, on the membrane used, and for a given membrane, depend on the value of the concentration, with a trend to increase with the increase of concentration. This result is in agreement with those obtained in [7], where the value of a was estimated from the study of the influence of the polarization layers on the electroosmotic transport for the two cation-exchange membranes used in this work.
TABLE 5 Results Obtained for All the Studied Membranes with a 1022 mol/liter Concentration of KCl Membrane
61 AZL 183
61 CZL 386
204 SXZL 386
103 QZL 386
204 U 386
IL0 (mA/cm2) a (1023 min/ml) b (1023 min A/cm5) d0 (mm)
1.443 6 0.005 4.15 6 0.03 5.99 6 0.02 287
1.84 6 0.18 1.9 6 0.9 3.4 6 1.3 217
0.95 6 0.03 10.4 6 0.5 9.9 6 0.2 435
1.17 6 0.01 8.4 6 0.2 9.78 6 0.08 353
1.12 6 0.01 3.5 6 0.1 3.93 6 0.05 369
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TABLE 6 Results Obtained for the Cation-Exchange Membrane 61 AZL 183 at Different Concentrations of KCl c0 (mol/liter)
1023
5 3 1023
1022
5 3 1021
IL0 (mA/cm2) a (1023 min/ml) b (1023 min A/cm5) d0 (mm)
0.26 6 0.04 7 62 1.8 6 0.3 162
0.990 6 0.005 2.43 6 0.02 2.40 6 0.01 210
1.443 6 0.005 4.15 6 0.03 5.99 6 0.02 287
6.53 6 0.11 5.6 6 0.2 3.63 6 0.05 313
21
10
11.1 6 0.6 8.1 6 0.7 8.9 6 0.3 365
The limiting current, IL, is associated with the absence of a transport mechanism in the limiting diffusion layer that makes it possible for the electric current to increase over IL. However, as was said in the introduction, the limiting current is not so limiting in practice, and there continues to exist an additional transport of charge which causes electric currents over the “limiting” value. These are the so-called overlimiting currents, the origin of which is still not well understood. The existence of overlimiting currents has been appointed to very different causes in the literature, such as water splitting (10, 11), electroosmotic transport (12), loss of selectivity of the membrane (13), interactions between the membrane matrix and its electroactive part (14), the variation of the thickness of the diffusion-limiting layer with current (15), deviations of the hypotheses of local electroneutrality and local equilibrium (1, 16). In relation to the overlimiting current, it is interesting to emphasize the differences existing between the behaviors of the cationic and the anionic membranes, differences which are usually associated with the different chemical groups fixed in the membrane matrix. In anion-exchange membranes, the main mechanism of transport seems to be the water splitting, favored by
the electric fields developed in the membrane–solution interface and catalyzed by the membrane fixed groups. In cation-exchange membranes, this splitting is scarcely found and another mechanism appears, of a seemingly convective nature. To analyze this difference in the overlimiting behavior, the overpotential V*AB has been shown against the reduced current, Ir 5 I/IL, for a cation-exchange membrane, Fig. 11, and for an anion-exchange membrane, Fig. 12, with a 1022 mol/liter concentration of KCl and the different studied stirring rates. In both cases, it can be observed that the overpotential is practically zero up to values of the current close to the limiting current density, Ir > 1. At this value occurs a sudden step in V*AB and afterwards, there is an increase in V*AB at higher values of Ir . Although this behavior is similar for both kinds of membranes, the observed step is higher for the cation-exchange membrane than for the anionexchange membrane, which may indicate that the transport mechanism starts before for the latter membrane. On the other hand, relating to the influence of the stirring rate, it is observed that an increase in the stirring rate seems to increase the overpotential in the cation-exchange membrane,
FIG. 11. Overpotential V*AB as a function of the reduced current density Ir for the cation-exchange membrane 61 CZL 386 with aqueous 1022 mol/liter KCl. (F) 0 ml/min; (E) 128 ml/min; (1) 208 ml/min.
FIG. 12. Overpotential V*AB as a function of the reduced current density Ir for the anion-exchange membrane 103 QZL 386 with aqueous 1022 mol/liter KCl. (F) 0 ml/min; (E) 207 ml/min; (1) 349 ml/min.
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CURRENT–VOLTAGE CURVES
whereas the opposite behavior seems to occur in the anionexchange membrane. SUMMARY
1. The current–voltage curves obtained show the three usual characteristic regions: the first region of approximately ohmic behavior, the second region corresponding to the “plateau,” followed by the third region of rapid current increase. 2. A new method has been developed for determining the limiting current density, IL, from the adjustment of the data corresponding to the first region of the current–voltage curves. 3. The values obtained for IL under the different experimental conditions considered are close to those obtained from the usual methods, ITL, under the same conditions, increasing when the solution concentration increases. 4. The values of the limiting current density depend linearly on the stirring rate of the solutions. The fit of these values to straight lines allows for a determination of— among other parameters—the thickness of the polarization layers in the absence of stirring.
REFERENCES 1. Rubistein, I., and Shtilman, L., J. Chem. Soc. Faraday Trans. 2 75, 231 (1979). 2. Aguilella, V. M., Mafe´, S., Manzanares, J. A., and Pellicer, J., J. Membrane Sci. 61, 177 (1991). 3. Tanaka, Y., J. Membrane Sci. 57, 217 (1991). 4. Helfferich, F., “Ion-Exchange.” McGraw-Hill, New York, 1962. 5. Ibl, N., Electrochim. Acta 1, 117 (1959). 6. Ibl, N., Electrochim. Acta 1, 3 (1959). 7. Barraga´n, V. M., Ruı´z-Bauza´, C., and Mengual, J. I., J. Colloid Interface Sci. 168, 458 (1994). 8. Ortı´z-Za´rate, J. M., Garcı´a-Lo´pez, F., and Mengual, J. I., J. Membrane. Sci. 56, 181 (1991). 9. Inenaga, K., and Yoshida, N., J. Membrane Sci. 6, 271 (1980). 10. Simons, R., Electrochim. Acta 29, 151 (1984). 11. Tanaka, Y., and Seno, M., J. Chem. Eng. Jpn. 22, 352 (1989). 12. Frilette, V. J., J. Phys. Chem. 61, 168 (1957). 13. Partridge, S. M., and Peers, A. M., J. Appl. Chem. 8, 49 (1958). 14. Forgacs, C., in “Proceedings, First International Symposium on Water Desalination,” Vol. 3, p. 155. Washington DC, 1967. 15. Lerche, D., and Wolf, H., Bioelectrochem. Bioenergetics 2, 293 (1975). 16. Bassignaga, I. C., and Reiss, H., J. Phys. Chem. 87, 136 (1983).
205, 365–373 (1998)
CS985649
Current–Voltage Curves for Ion-Exchange Membranes: A Method for Determining the Limiting Current Density V. M. Barraga´n and C. Ruı´z-Bauza´1 Department of Applied Physics I, Faculty of Physics, University Complutense of Madrid, 28040 Madrid, Spain E-mail: [email protected] Received December 31, 1997; accepted May 8, 1998
The curves (V, I) of different ion-exchange membranes have been determined under different experimental conditions. Taking into account the classical theory of concentration polarization, a method has been developed to adjust the (V, I) data, up to a certain value of I, which permits us to obtain easily the value of the limiting current density, IL, in every experimental situation. From these values, the dependence of the limiting current density of the membranes used on the concentration and stirring rate of the solutions has been studied. To check the reliability of the method, the values of IL thus obtained have been compared with those obtained by the usual methods; the similitude between both values is high. © 1998 Academic Press Key Words: current–voltage curves; limiting current density; ion-exchange membrane; concentration polarization.
INTRODUCTION
Concentration polarization is a well-known phenomenon arising at the interface between an ion-exchange membrane and an electrolyte solution when an electric current passes through the system. This phenomenon has been widely studied in the last decades (1–3) with the purpose of establishing the factors that determine it and resolving the problems that concentration polarization causes in the membrane technology, in general, and in the separation processes, in particular. According to the classical theory of concentration polarization (4), the electric current, I, should increase linearly with voltage, V, at low voltages, then increase more slowly, and finally reach a determined limiting value. In practice, such a limiting value is not observed, but the curves (V, I) for the ion-exchange membranes have a characteristic shape on which three regions can be clearly distinguished: (I) a first region of approximately ohmic behavior, transforms as voltage is increased into a second region (II), in which the current varies very slowly with voltage (“plateau” corresponding to the limiting current), followed by a third region (III) of rapid current increase. This implies that the limiting current is a rather 1
To whom correspondence should be addressed.
ill-defined parameter, and it is necessary to use indirect methods, such as Cowan plots or the tangent method, to determine its value. Currents above the limiting one are referred to as “overlimiting.” In spite of that, considerations about concentration polarization always lead to the study of the so-called limiting current and the conditions under which the transport occurs at current densities equal to or greater than this limiting value. So, it is important to determine, in the easier and more accurate way, the limiting current that corresponds, in every experimental condition, to the membrane system studied. In this work, the curves (V, I) for different ion-exchange membranes have been studied under different experimental conditions. A method has been developed to adjust the (V, I) data, which permits us to obtain easily the value of the limiting current density, IL, in every case. The values of IL thus obtained have been compared with those obtained by the usual methods. BASIC EQUATIONS
The system studied consists of an ion-exchange membrane separating two equal 1–1 electrolyte aqueous solutions of bulk concentration c0, which are maintained at the same temperature and hydrostatic pressure. When an electric current passes through the membrane system, from the left compartment to the right one, the current is carried in the solution by both types of ions (positive and negative), whereas in the membrane, assumed to be selective, it is carried mainly, if not exclusively, by the counterions. The difference between the ion mobilities of the counterion, corresponding to the membrane and free solution phases, gives rise to a depletion in the electrolyte concentration on one side of the membrane (c1) and to an enrichment on the opposite side (c2). As a consequence, a concentration gradient (Fig. 1) is established through each one of the thin films that adjoin the membrane at the two sides (polarization layers). In the polarization layers, the electric potential gradient drives positive and negative ions in opposite directions, while
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´ N AND RUI´Z-BAUZA ´ BARRAGA
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placed in both sides of the membrane in the corresponding bulk solution is measured. Before the polarization layers are formed, i.e., at the initial moment t 5 0, the value obtained is V 0AB, whereas when the layers have been formed and the stationary state has been reached, the value obtained is V ` AB, ` 0 5 V AB 1 V *AB , V AB
[5]
where V*AB represents the electric potential difference created solely by the formation of the polarization layers. This contribution can be expressed as V *AB 5 V P 1 IDR .
[6]
VP is the polarization potential established through the membrane system and, assuming ideal behavior for the solutions, is given by FIG. 1. Sketch of the concentration profile in the studied system.
VP 5 the concentration gradient drives, by diffusion, the ions of both signs in the same direction. For a membrane selective to specific ions (i), and supposing the thicknesses d of the both layers are equal, the current density, I, can be expressed by I5
FD ~c 0 2 c 1 ! FD ~c 2 2 c 0 ! 5 , Dt i d Dt i d
IL 5
FDc 0 . Dt i d
From [1] and [2], it follows that, for a current density I , IL, the concentrations at the membrane/solution interfaces can be expressed as c1 5 c0
S S
I 12 IL
c2 5 c0 1 1
D D
I . IL
[3] [4]
Consider now the experimental situation in which a constant current density, I, is injected through the membrane system and the electric potential difference, VAB, between two fixed points
ES 0
DR 5 L 21
[2]
[7]
where R and T are the gas constant and absolute temperature, respectively. Variable DR is the change in the ohmic resistance as a consequence of the formation of the layers, and it is given by
[1]
where D is the salt diffusion coefficient, Dt i 5 #t i 2 t i is the difference between the counterion transport numbers in the membrane and in the free solution, and F is the Faraday constant. According to the classic theory of polarization, the current density reaches a limiting value, IL, when concentration c1 becomes zero. From [1], IL is given by
RT c2 2Dt iln F c1
2d
D
1 1 dx 1 L 21 2 c ~ x ! c0
E S d1d
d
D
1 1 dx, 2 c ~ x ! c0 [8]
where L is the equivalent conductance and c(x) the local concentration in the layers, given by c ~ x ! 5 c1 2
c0 2 c1 x d
c ~ x ! 5 c2 2
c2 2 c0 ~x 2 d! d
for 2d , x , 0 for d , x , d 1 d .
[9] [10]
From [2]–[4] and [8]–[10], it is obtained that DR 5
S
D
FD 1 c 2 2 . ln 2 LDt i I c1 IL
[11]
Taking into account Eqs. [3] and [4] and Nernst-Einstein’s equation which relates the equivalent conductance L and the diffusion coefficient D of a monovalent electrolyte by means of L 5 L1 1 L2 5
~D 1 1 D 2 ! F 2 2DF 2 5 , RT RT
[12]
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CURRENT–VOLTAGE CURVES
TABLE 1 Characteristic Parameters for All the Studied Membranes Membrane
61 AZL 183
61 CZL 386
204 SXZL 386
103 QZL 386
204 U 386
Selectivitya (%) Electrical resistencea (V cm2) Capacitya (meq/g) Membrane thickness (mm) Fraction of unfilled volume (1022)
98 5.6 3.0 600 48
94 11 2.7 520 32
96 11 2.2 540 45
97 9 2.1 580 41
97 6 2.8 550 22
a
Value provided by the manufacturer.
EXPERIMENTAL
it is obtained that the contribution IDR is expressed as RT 1 1 ~I/I L ! RT IDR 5 ln I. 2 2FDt i 1 2 ~I/I L ! FDt i I L
Materials [13]
If R0 is the ohmic resistance before the polarization layers are formed, we have 0 V AB 5 IR 0
[14]
Obviously, R ` 5 R 0 1 DR. From Eqs. [5], [6], [13], and [14] it is obtained that
V
` AB
S
D
Apparatus
2 RT RT 5 R0 2 I1 @~2Dt i ! FI L F 1 ~2Dt i ! 21#ln
1 1 ~I/I L ! . 1 2 ~I/I L !
[15]
For a membrane system consisting of a highly selective membrane and a dilute KCl solution, it is obtained that #ti > 1 and ti > 1/2, and so, 2Dti > 1. Under these conditions Eq. [15] can be reduced to
S
` V AB 5 R0 2
D
Five commercial membranes have been studied, two cationexchange membranes, the Nepton cation-exchange membrane 61 AZL 183 and the Ionics cation-exchange membrane 61 CZL 386, and three anion-exchange membranes, the Ionics aniontransfer membranes 204 SXZL 386, 103 QZL 386, and 204 U 386. Their characteristics are given in Table 1. The electrolytic solutions employed in the experiments were aqueous solutions of potassium chloride. Pure proanalysis chemicals and pure water (deionized, doubly distilled, doubly filtrated, and degassed) were used.
2 RT 2 RT 1 1 ~I/I L ! I1 ln . FI L F 1 2 ~I/I L !
[16]
Obviously, Eq. [16] is only applicable to current densities such as |I| , |IL|. When I becomes IL, the polarization potential as well as the resistance of the depletion layer tends to infinity in a logarithmic way. Equation [16] permits us the adjustment of the experimental data (V ` AB, I) up to a certain value of I, in different experimental situations. The adjustment parameters will be R0 and IL. For a given membrane system, and according to the definitions previously given for R0 and IL, R0 may depend only on the bulk concentration, but not on the stirring rate of the solutions. The value of IL, on the contrary, may depend on the concentration as well as the stirring.
The cell employed in these measurements consisted of two glass chambers of ;100 cm3 each. Both chambers were separated by a methacrylate membrane holder, where the membrane was positioned. Each chamber was provided with four orifices communicating to the exterior. Two of them were used as solution inlet and outlet by means of a peristaltic pump. Two Ag/AgCl electrodes were introduced in the other two orifices. One of them had a large active surface in order to inject the current, and the other was used to measure the electric potential difference values. An HP 6186C DC current source was used to maintain a constant electric current, and a Keithley 195 system DMM multimeter was used to measure the voltage. All the experiments were carried out under isothermal conditions (30°C). The temperature requirements were achieved by introducing the complete unit in a large ambient thermostat. The stirring of the solutions was assessed by circulating them by means of a peristaltic pump. RESULTS AND DISCUSSION
The curves (V, I) for the studied ion-exchange membranes have been studied under different experimental conditions. Figure 2 shows the curves (V, I) obtained, in the absence of stirring, for the cation-exchange membrane 61 CZL 386 at different KCl concentrations. Figures 3–7 show the curves (V,
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´ N AND RUI´Z-BAUZA ´ BARRAGA
FIG. 2. Curves (V, I) for membrane 61 CZL 386, in the absence of stirring, with different KCl concentrations. (E) 5 3 1023 mol/liter; (F) 1022 mol/liter; (1) 2 3 1022 mol/liter; (h) 5 3 1022 mol/liter. The lines correspond to Eq. [16].
FIG. 4. Curves (V, I) for membrane 61 AZL 183, at different stirring rates, with aqueous 1022 mol/liter KCl. (F) 0 ml/min; (E) 102 ml/min; (■) 195 ml/min; (h) 305 ml/min. The lines correspond to Eq. [16].
I) obtained for the different studied membranes at various stirring rates and with 1022 mol/liter concentration of KCl. As can be observed in each figure, the obtained curves (V, I) show the three usual characteristic regions defined in the introduction for this kind of membranes. In addition, a visual inspection of these figures permits us to establish that the
second region, corresponding to the “plateau,” occurs at greater current densities and becomes less appreciable when the concentration of the solution and the stirring rate increase. This behavior is similar for cationic and anionic membranes. From the data (V, I) obtained in the first region, the value of
FIG. 3. Curves (V, I) for membrane 61 CZL 386, at different stirring rates, with aqueous 1022 mol/liter KCl. (E) 0 ml/min; (F) 128 ml/min; (h) 208 ml/min. The lines correspond to Eq. [16].
FIG. 5. Curves (V, I) for membrane 204 SXZL 386, at different stirring rates, with aqueous 1022 mol/liter KCl. (F) 0 ml/min; (E) 129 ml/min; (!) 201 ml/min. The lines correspond to Eq. [16].
CURRENT–VOLTAGE CURVES
FIG. 6. Curves (V, I) for membrane 103 QZL 386, at different stirring rates, with aqueous 1022 mol/liter KCl. (ƒ) 0 ml/min; (Œ) 207 ml/min; () 236 ml/min; () 349 ml/min. The lines correspond to Eq. [16].
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FIG. 8. Values of IL obtained, in the absence of stirring, for the two studied cation-exchange membranes at different KCl concentrations. From the adjustment to Eq. [16]: (F) membrane 61 AZL 183; (E) membrane 61 CZL 386. From the usual methods: (3) membrane 61 AZL 183; (1) membrane 61 CZL 386.
the limiting current density can be obtained under every studied experimental situation by fitting these data to Eq. [16] by means of a minimization x2 method. The adjustment parameters are IL and R0, as was previously indicated. Some of the curves obtained are shown in Figs. 2–7 together with the corresponding experimental values (V, I).
The values of IL obtained from the fits for the two cationexchange membranes, in the absence of stirring and at the different studied concentrations, are shown as a function of log c0 in Fig. 8. Together with these values of the limiting current density, those obtained from the usual methods under the same
FIG. 7. Curves (V, I) for membrane 204 U 386, at different stirring rates, with aqueous 1022 mol/liter KCl. ({) 0 ml/min; (}) 96 ml/min; () 152ml/ min; () 373 ml/min. The lines correspond to Eq. [16].
FIG. 9. Values of R0 as a function of log c0 obtained, in the absence of stirring, for the two studied cation-exchange membranes. (F) membrane 61 AZL 183; (E) membrane 61 CZL 386.
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TABLE 2 Values of IL and R0 Obtained for the Cation-Exchange Membrane 61 AZL 183 v (ml/min)
R0 (KV cm2)
IL (mA/cm2)
ITL (mA/cm2)
TABLE 3 Values of IL and R0 Obtained for the Cation-Exchange Membrane 61 CZL 386 v (ml/min)
1023 mol/liter 0 168 192 288
5.46 5.27 5.30
— 6 0.06 6 0.05 6 0.06
R0 (KV cm2)
IL (mA/cm2)
ITL (mA/cm2)
5 3 1023 mol/liter — 0.50 6 0.14 0.65 6 0.01 0.77 6 0.21
0.26 0.53 0.59 0.68
0 159 190
2.57 6 0.05 2.74 6 0.01 2.66 6 0.04
0.51 6 0.04 0.76 6 0.01 0.85 6 0.12
0.55 0.75 0.86
1022 mol/liter 23
5 3 10 0 105 200 286
1.37 1.35 1.58 1.36
mol/liter
6 0.01 6 0.03 6 0.01 6 0.02
0.99 6 0.02 1.26 6 0.5 1.47 6 0.06 1.68 6 0.27
1.07 1.26 1.39 1.60
1022 mol/liter 0 102 195 305
0.894 0.885 0.882 0.888
6 0.006 6 0.015 6 0.006 6 0.003
1.45 6 0.02 2.04 6 0.32 2.61 6 0.05 3.27 6 0.05
1.64 2.10 2.60 3.40
5 3 1022 mol/liter 0 100 196 312
0.20 0.204 0.203 0.200
6 0.01 6 0.003 6 0.001 6 0.003
6.53 6 0.02 9.75 6 0.32 15.57 6 0.05 17.85 6 0.05
6.62 10.0 14.3 20.0
1021 mol/liter 0 99 198 206 305
0.085 6 0.001 0.0954 6 0.0008 0.1004 6 0.0001 0.0975 6 0.0005 0.0969 6 0.0002
11.5 6 20.1 6 28.1 6 29.7 6 30.5 6
0.5 0.5 0.3 0.9 0.3
12.0 18.0 24.2 25.2 31.5
experimental conditions are shown; observe that the two sets of values are very similar. For membrane 61 AZL 183, the slope obtained for the curve which represents log IL against log c0 is 1.2, a result which is in agreement with the behavior IL } c5/4 0 predicted by the hydrodynamics theory (5, 6). In Fig. 8, a change of slope can also be observed occurring at concentrations higher than 5 3 1022 mol/liter, obtaining in this case 0.8, a result that can be explained by taking into account the electrogravitational effects. Thus, in this range, the hypothesis of the hydrodynamics theory is not applicable, in agreement with the results obtained by other authors. The values of R0 obtained in the absence of stirring are shown as a function of concentration logarithm in Fig. 9 for the former membranes. As can be expected, the value of R0 decreases when the concentration of the external solution increases. Tables 2 and 3 show the values of IL and R0 obtained from the fit to Eq. [16] for the cation-exchange membranes 61 AZL
0 128 208
1.47 6 0.01 1.40 6 0.01 1.35 6 0.01
1.90 6 0.26 2.12 6 0.02 2.65 6 0.17
1.90 2.10 2.30
183 and 61 CZL 386 at different stirring rates for every concentration studied. In Table 4 the values of IL and R0 obtained for the three anion-exchange membranes studied are shown, with a KCl 1022 mol/liter solution and at various stirring rates. In these tables, the value of the limiting current density obtained from the usual methods is also shown. As can be observed, a great similitude exists between the values obtained from both methods. It can be observed that in all the studied cases the obtained limiting current density increases, for a given concentration, when the stirring rate increases; that is a natural requirement since it is expected that the stirring decreases the thickness of the layers and, so, favors the transport process. On the contrary,
TABLE 4 Values of IL and R0 Obtained for the Anion-Exchange Membranes with a 1022 mol/liter Concentration of KCl v (ml/min)
R0 (KV cm2)
IL (mA/cm2)
ITL (mA/cm2)
204 SXZL 386 0 129 201
1.241 6 0.006 1.241 6 0.005 1.186 6 0.003
0.95 6 0.02 2.16 6 0.07 2.94 6 0.02
0.9 2.2 2.9
103 QZL 386 0 207 236 345
1.141 6 0.003 1.118 6 0.005 1.091 6 0.003 1.174 6 0.003
1.17 6 0.02 3.06 6 0.14 3.50 6 0.11 4.58 6 0.04
1.2 2.9 3.6 4.2
1.16 6 0.04 1.50 6 0.01 1.71 6 0.01 2.59 6 0.01
1.2 1.5 1.6 2.5
204 U 386 0 102 195 305
1.074 6 0.006 1.148 6 0.004 1.136 6 0.002 1.096 6 0.002
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CURRENT–VOLTAGE CURVES
d ~v ! 5
d0 , av 1 1
[17]
where d0 is the thickness of the layer in the absence of stirring. Expression [17] has been previously used to justify the dependence of the electroosmotic permeability of this kind of membrane on the rate of stirring, obtaining satisfactory results (7), and to consider the phenomena of temperature polarization in experiments on thermoosmosis and membrane distillation (8). Taking into account Eqs. [2] and [17], it is obtained that I L 5 I L0 1 b v ,
[18]
I L0 5 ~FDc 0 !/~Dt id 0 !
[19]
where
b 5 I L0 a . FIG. 10. Values of IL as a function of stirring rate for membrane 61 AZL 183 at different KCl concentrations. (1) 1023 mol/liter; (E) 5 3 1023 mol/liter; (F) 1022 mol/liter; (Œ) 5 3 1022 mol/liter; (ƒ) 1021 mol/liter.
the value of R0 for every concentration is, as could be expected, practically independent of the stirring rate. The values of IL obtained for the cation-exchange membrane 61 AZL 183 have been shown versus stirring rate for the different concentrations used, Fig. 10. The observation of the experimental data of this figure induces one to think of a possible linear relation between the limiting current density and the stirring rate. For this reason, the data have been fitted to straight lines, obtaining in all cases a satisfactory result. The straight lines found are shown in Fig. 10 together with the values of IL. The finding of this linear behavior can be justified taking into account that, for a determined membrane separating a solution of given nature and concentration, the value of the limiting current density is only a function, according to Eq. [2], of the thickness of the polarization layers. These thicknesses only depend on the stirring rate, so that, the stirring process causes a decrease of the layer thickness. This last fact can be taking into account in a quantitative way by supposing the following relation between the thickness of the layers for a given stirring rate, d(v), and the stirring rate, v,
[20]
Equation [18] expresses a linear relation between the limiting current density and the stirring rate, just as it is obtained in the experiments. The fit of the values of IL against stirring rate to Eq. [18] permits us to determine the parameters IL0 and b for each situation. From Eqs. [19] and [20] the values of a and the thickness of the polarization layer in the absence of stirring, d0, can be estimated. The results obtained for all the used membranes with a 1022 mol/liter KCl concentration are shown in Table 5. Table 6 shows the results obtained with the cationexchange membrane 61 AZL 183 and different KCl concentrations. As can be observed in Tables 5 and 6, the thickness of the polarization layers in the absence of stirring is of the same order of magnitude as those obtained by other authors [9]. Likewise, as can be observed in Table 4, this thickness is the greater the greater the concentration of the external solution in contact with the membrane is. It can be also deduced, from the results shown in Tables 5 and 6, that the values of a and b depend, for the same concentration, on the membrane used, and for a given membrane, depend on the value of the concentration, with a trend to increase with the increase of concentration. This result is in agreement with those obtained in [7], where the value of a was estimated from the study of the influence of the polarization layers on the electroosmotic transport for the two cation-exchange membranes used in this work.
TABLE 5 Results Obtained for All the Studied Membranes with a 1022 mol/liter Concentration of KCl Membrane
61 AZL 183
61 CZL 386
204 SXZL 386
103 QZL 386
204 U 386
IL0 (mA/cm2) a (1023 min/ml) b (1023 min A/cm5) d0 (mm)
1.443 6 0.005 4.15 6 0.03 5.99 6 0.02 287
1.84 6 0.18 1.9 6 0.9 3.4 6 1.3 217
0.95 6 0.03 10.4 6 0.5 9.9 6 0.2 435
1.17 6 0.01 8.4 6 0.2 9.78 6 0.08 353
1.12 6 0.01 3.5 6 0.1 3.93 6 0.05 369
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TABLE 6 Results Obtained for the Cation-Exchange Membrane 61 AZL 183 at Different Concentrations of KCl c0 (mol/liter)
1023
5 3 1023
1022
5 3 1021
IL0 (mA/cm2) a (1023 min/ml) b (1023 min A/cm5) d0 (mm)
0.26 6 0.04 7 62 1.8 6 0.3 162
0.990 6 0.005 2.43 6 0.02 2.40 6 0.01 210
1.443 6 0.005 4.15 6 0.03 5.99 6 0.02 287
6.53 6 0.11 5.6 6 0.2 3.63 6 0.05 313
21
10
11.1 6 0.6 8.1 6 0.7 8.9 6 0.3 365
The limiting current, IL, is associated with the absence of a transport mechanism in the limiting diffusion layer that makes it possible for the electric current to increase over IL. However, as was said in the introduction, the limiting current is not so limiting in practice, and there continues to exist an additional transport of charge which causes electric currents over the “limiting” value. These are the so-called overlimiting currents, the origin of which is still not well understood. The existence of overlimiting currents has been appointed to very different causes in the literature, such as water splitting (10, 11), electroosmotic transport (12), loss of selectivity of the membrane (13), interactions between the membrane matrix and its electroactive part (14), the variation of the thickness of the diffusion-limiting layer with current (15), deviations of the hypotheses of local electroneutrality and local equilibrium (1, 16). In relation to the overlimiting current, it is interesting to emphasize the differences existing between the behaviors of the cationic and the anionic membranes, differences which are usually associated with the different chemical groups fixed in the membrane matrix. In anion-exchange membranes, the main mechanism of transport seems to be the water splitting, favored by
the electric fields developed in the membrane–solution interface and catalyzed by the membrane fixed groups. In cation-exchange membranes, this splitting is scarcely found and another mechanism appears, of a seemingly convective nature. To analyze this difference in the overlimiting behavior, the overpotential V*AB has been shown against the reduced current, Ir 5 I/IL, for a cation-exchange membrane, Fig. 11, and for an anion-exchange membrane, Fig. 12, with a 1022 mol/liter concentration of KCl and the different studied stirring rates. In both cases, it can be observed that the overpotential is practically zero up to values of the current close to the limiting current density, Ir > 1. At this value occurs a sudden step in V*AB and afterwards, there is an increase in V*AB at higher values of Ir . Although this behavior is similar for both kinds of membranes, the observed step is higher for the cation-exchange membrane than for the anionexchange membrane, which may indicate that the transport mechanism starts before for the latter membrane. On the other hand, relating to the influence of the stirring rate, it is observed that an increase in the stirring rate seems to increase the overpotential in the cation-exchange membrane,
FIG. 11. Overpotential V*AB as a function of the reduced current density Ir for the cation-exchange membrane 61 CZL 386 with aqueous 1022 mol/liter KCl. (F) 0 ml/min; (E) 128 ml/min; (1) 208 ml/min.
FIG. 12. Overpotential V*AB as a function of the reduced current density Ir for the anion-exchange membrane 103 QZL 386 with aqueous 1022 mol/liter KCl. (F) 0 ml/min; (E) 207 ml/min; (1) 349 ml/min.
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CURRENT–VOLTAGE CURVES
whereas the opposite behavior seems to occur in the anionexchange membrane. SUMMARY
1. The current–voltage curves obtained show the three usual characteristic regions: the first region of approximately ohmic behavior, the second region corresponding to the “plateau,” followed by the third region of rapid current increase. 2. A new method has been developed for determining the limiting current density, IL, from the adjustment of the data corresponding to the first region of the current–voltage curves. 3. The values obtained for IL under the different experimental conditions considered are close to those obtained from the usual methods, ITL, under the same conditions, increasing when the solution concentration increases. 4. The values of the limiting current density depend linearly on the stirring rate of the solutions. The fit of these values to straight lines allows for a determination of— among other parameters—the thickness of the polarization layers in the absence of stirring.
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