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Effect of an ac Perturbation on the Electroosmotic Behavior of a Cation-Exchange Membrane. Influence of the Cation Nature
Journal of Colloid and Interface Science 240, 182–189 (2001) doi:10.1006/jcis.2001.7610, available online at http://www.idealibrary.com on
Effect of an ac Perturbation on the Electroosmotic Behavior of a Cation-Exchange Membrane. Influence of the Cation Nature V. M. Barrag´an1 and C. Ru´ız Bauz´a Departamento de F´ısica Aplicada I, Facultad de Ciencias F´ısicas, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain E-mail: [email protected]; [email protected] Received November 20, 2000; accepted April 6, 2001; published online June 26, 2001
The effect of an ac sinusoidal perturbation of known amplitude and frequency superimposed on the usual dc applied electric voltage difference on the electroosmotic flow through a typical cationexchange membrane has been studied using different monovalent electrolytes. As a general trend, the presence of the ac perturbation increases the value of the electroosmotic flow with respect to the value in the absence of ac perturbation. A dispersion of the electroosmotic permeability on the frequency of the applied ac signal has been found for the three studied electrolytes, observing that the electroosmotic permeability reaches maximum values for some characteristic values of the frequency. This behavior may be related to the different relaxation processes in heterogeneous mediums. °C 2001 Academic Press Key Words: electroosmosis; cation-exchange membrane; electroosmotic permeability; membrane relaxation phenomena.
INTRODUCTION
The electrokinetic phenomena originate as a consequence of the interaction between matter flux and electrical charge flux through porous media. Due to this fact, the “thermodynamics of irreversible processes” has been used satisfactorily for their study (1, 2). The transport equations for matter and electric charge through porous media obtained in that approach are JV = L 11 1P + L 12 1ϕ I = L 21 1P + L 22 1ϕ,
[1] [2]
where JV is the volume flow, I the electric current, 1P the pressure difference, and 1ϕ the electric potential difference. The coefficients L 11 and L 22 are, respectively, the hydraulic permeability and the electric conductance, whereas L 12 , which must be equal to L 21 according to the Onsager reciprocity relation, is the electrokinetic coefficient.
1
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From Eqs. [1] and [2], in the absence of pressure difference (1P = 0), one obtains the following relations: µ
¶ JV = L 12 1ϕ 1P=0 µ ¶ L 12 JV = = W. I 1P=0 L 22
[3] [4]
Equations [3] and [4] provide expressions for the electrokinetic coefficient, L 12 , and the electroosmotic permeability, W , in terms of the phenomenological coefficients. Hence, from measurements of the electroosmotic flow both L 12 and W can be determined. Usually, in the study of electroosmotic phenomena, static electric fields are used to obtain a dc current circulating through the membrane system (3–6). In the last years, however, studies of the electric behavior of different membrane/electrolyte systems under the influence of time-dependent electric fields have been carried out (7–13). In some papers, it was shown that a dispersion of the impedance with frequency of the alternating fields existed in some kind of system, revealing that the capacity and conductance of these systems depend on the frequency of the alternating field (10, 12, 13). On the other hand, the use of ac techniques has been shown as a useful instrument for studying electrochemical processes occurring within the interfacial regions between the membrane surface and the bulk electrolyte without disturbing the system. The functional properties of membranes and solid–aqueous systems, in general, are closely connected to the electrochemical processes that occur within the system, so the understanding of the dielectric behavior will be useful in improving membrane processes. In this sense, it is expected that the dispersion of the dielectric properties may influence the electrokinetic phenomena, and in particular on the electroosmotic transport. The aim of this paper is, continuing the work of Ref. (11), to check this idea and try to relate in a broader way the behavior found for the electroosmotic transport with the frequency of the ac perturbation to the different relaxation phenomena observed in similar systems from ac
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techniques. To this end, with use of different electrolytes, the influence of an ac sinusoidal perturbation of known amplitude and frequency, superimposed on the usual dc applied electric voltage difference, on the electroosmotic flow through a typical ion-exchange membrane have been studied. The frequency has been varied in the (10−2 −107 ) Hz range with the purpose of analyzing the possible influence of relaxation phenomena observed at low frequencies on the electroosmotic behavior of the membrane system. EXPERIMENTAL
Materials The membrane used in the present work was the commercial ionics cation-exchange membrane comprised of cross-linked sulfonated copolymers of vinyl compounds CR 61CZL412, whose principal characteristics (provided by the manufacturer) are as follows: selectivity, 94%; electrical resistance (in 0.1 mol/liter NaCl), 11 Ä cm2 ; and capacity, 2.7 meq/g. The membrane thickness was 540 ± 1 µm and the fractional void volume was 0.32. The materials employed in the experiments were 10−2 mol/ liter aqueous solutions of lithium, sodium, and potassium chloride. Pure proanalysis-grade chemicals and pure water (deionized, doubly distilled, doubly filtrated, and degassed) were used. Apparatus and Procedures The apparatus used in this research and the methodology employed were substantially similar to those described in earlier publications (12, 13). The membrane surface exposed to the flow was 0.79 cm2 . An Ag/AgCl electrode with a large active surface was introduced in each chamber on both sides of the membrane with injection of an electric current. The cell is provided with a chain-driven cell magnetic stirrer assembly, which permits stirring of both solutions. The temperature requirements were achieved by immersing the cell in a water thermostat that was maintained at the selected temperature with an accuracy of ±0.1◦ C. In addition, the complete unit was contained in a large ambient thermostat. All measurements were carried out under isothermal conditions at 30◦ C. To measure the current–voltage curves in each condition, the experimental device was modified to use the four-electrodes configuration. In this case, together with the electrodes used to inject the current, other Ag/AgCl electrodes were introduced in each chamber to measure the dc voltage. These electrodes consisted of two linear Ag wires of approximately 4-mm longitude and 1-mm diameter and they were also prepared by the usual method (14). The volume flow was determined by measuring the time displacement of the solution meniscus in the capillary tubes connected to each chamber when an electric potential difference was established through the membrane. Taking into account the volume change due to the electrochemical reactions at the Ag/AgCl electrodes, the volume change rate of the cathodic and anodic
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compartments, 1Vc and 1Va , respectively, is given by I ¯ (V Ag − V¯ AgCl ) F I 1Vc = −JV + (V¯ AgCl − V¯ Ag ), F
1Va = JV +
where F is the Faraday constant and V¯ Ag and V¯ AgCl are the partial molar volumes of Ag and AgCl, respectively. Taking into account that V¯ AgCl − V¯ Ag = 15.5 cm3 /mol, we have JV (ml/s) =
|1Va | + |1Vc | + (1.6 × 10−4 )I, 2
[5]
where I is the dc current passing through the system. The last term in Eq. [5] is the corrective term that takes into account the volume change due to the chemical reactions in the Ag/AgCl electrodes. It has been checked that the electric current passing through the system was constant—up to the hundredth or tenth part of mA—during the measurement time. When the electric potential was applied, a steady current was reached after a few seconds. If an ac perturbation was superimposed on the dc signal, a steady effective dc current was also observed, this value being similar to the one obtained in the absence of ac perturbation under the same conditions. Only for frequencies below 1 Hz and for 10 Hz, a small oscillation of the dc current around a mean value was observed. This mean value was similar to the value in the absence of ac perturbation under the same conditions. In Fig. 1, the dependence of the dc current on time is shown in the absence and in the presence of ac perturbation for one of the studied cases. As can be observed, for the cited frequencies an oscillation is observed around the corresponding value in the absence of ac signal, shown in Fig. 1 as a continuous line. In these cases, this mean value was taken as an effective dc current, I ∗ , to calculate the correction of the reaction change volume to the volume flow in Eq. [5]. In this situation, from Eqs. [4] and [5], the effective apparent electroosmotic permeability, W ∗ , will be given by W ∗ (ml/C) =
|1Va∗ | + |1Vc∗ | + 1.6 × 10−4 , 2I ∗
[6]
where 1Vc∗ and 1Va∗ are, respectively, the volume change rate of the anodic and cathodic compartments in the presence of the ac perturbation. RESULTS AND DISCUSSION
The current–voltage curves of the system under the different studied experimental conditions have been measured by using the usual four-terminal configuration. From these curves, Fig. 2, the values of the limiting current density have been estimated by using a method described in (15). The values obtained were
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electric current circulating through the system was far away from the corresponding limiting values. The volume change rates in the cathodic and anodic compartments were estimated by measuring the displacement of the meniscus in the capillary tubes connected to each compartment as a function of time when the steady state was considered reached. Thus, the volume change rates were calculated from the slope of the straight lines obtained from the linear fits of the experimental data. It was observed that the ac perturbation did not affect this linear behavior, but only the value of the slope, depending on the ac perturbation frequency. In all cases, the measurements were reproducible within the experimental error range. The results obtained showed that a linear relation existed between the volume flow and the applied dc voltage under all the studied conditions. In this paper we have checked this behavior for three different 10−2 mol/liter electrolytes in the range 0– 8 V. The results are shown in Fig. 4, where the continuous lines represent the linear fits of the experimental data by using Eq. [3]. As can be observed, the value of the volume flow increases with decreasing cation atomic number, although the difference between the values corresponding to NaCl and KCl solutions are very small in comparison to their differences with respect to LiCl solution. This result is in agreement with the results
FIG. 1. Dependence of the dc current on time at different frequencies of the ac perturbation. The data correspond to the KCl electrolyte with a dc voltage of 6 V. The continuous line represents the data in the absence of ac perturbation. (s) 10−2 Hz; (+) 10−1 Hz; (m) 10 Hz.
9.8, 7.0, and 8.4 mA/cm2 , for LiCl, NaCl, and KCl solutions, respectively. It was important to determine these values to know the ohmic region of the current–voltage curves and to avoid the abrupt increase of the electroosmotic flow observed at current values close to the limiting current value (13). The behavior of the ohmic region of the (V, I ) curves in the presence of an ac perturbation has been studied. Figure 3 shows the current–voltage curves, in the 0- to 10-V range, for the three studied electrolytes and at different frequencies of the ac perturbation. The dotted lines correspond to the fit to the values in the absence of ac perturbation to a straight line. In all the cases, a linear dependence was found between the dc voltage applied and the dc current in the measured interval, and no significant influence of the frequency was observed. It means that, for the same dc voltage applied, the effective continuous component of the current is the same for all the frequencies, and so the effective ion transport is not affected by the ac signal in the studied frequency range. It was also checked that there were no changes in the described qualitative behavior when the same electrodes were used to measure the dc current as well as the dc voltage. The electroosmotic flow was estimated from Eq. [5]. In all the cases, the value of the applied dc voltage was such that the
FIG. 2. Typical current–voltage curves for the three studied electrolytes in the absence of ac perturbation. (s) LiCl; (n) NaCl; (h) KCl.
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this coefficient decreases when the atomic number of the cation increases, independent of the frequency of the ac perturbation. On the other hand, as a general trend, the values of L ∗12 increase, with respect to the corresponding L 12 value, for some values of the frequency. The increase depends on the ac perturbation frequency and on the electrolyte nature. The effective apparent electroosmotic permeability for the same dc voltage, 6 V, has been measured as a function of the frequency of the ac perturbation and at 10−2 mol/liter concentrations of the three electrolytes. The results obtained are shown in Fig. 6 for the different electrolytes used. The dotted horizontal lines correspond to the values of the apparent electroosmotic permeability in the absence of ac perturbation. The experimental results show that the presence of the ac perturbation seems to cause, as a general trend, the value of the electroosmotic flow to increase. For the three electrolytes, it is possible to observe that maximum values are reached at certain values of the frequency. These characteristic frequencies, w ∗ , depend on the electrolyte, but they appear in the same frequency ranges for the three studied electrolytes. One of them, w ∗1 , appears at low values of the frequency, in the 0.1- to 10-Hz range, observing a slight tendency to move toward lower values with increasing cation atomic number. A second maximum is observed at a characteristic frequency, w ∗2 , in the 100 Hz to 10 kHz
FIG. 3. Current–voltage curves in the linear range for the three studied electrolytes and at different ac perturbation frequencies. The dotted lines correspond to the fit to the values in the absence of ac perturbation to a straight line. (d) dc current; (s) 1 Hz; (j) 100 Hz; (h) 1 kHz; (m) 10 kHz; (n) 100 kHz; (r) 1 MHz; (♦) 10 MHz.
obtained in other papers about the influence of the cation nature on the electroosmotic transport (13, 16). To check the influence of ac perturbation on the validity of the linear phenomenological equation [3], for each electrolyte, the behavior of the effective volume flow, JV∗ , with the applied dc voltage has been studied at different frequencies of the ac perturbation and for a 10−2 mol/liter concentration. The results obtained were similar for the three studied electrolytes and they showed, in all the cases, that the presence of an ac perturbation did not influence the linear behavior range but it affected the value of the slope, depending on the frequency of the ac perturbation. The points corresponding to the same frequency have been fitted, as a function of the dc voltage, to a straight line, in the linear behavior range, to estimate the effective electrokinetic coefficient L ∗12 = (JV∗ /1ϕ)1P=0 at the different studied frequencies from the slopes of the fitted straight lines. The results obtained for the electrokinetic coefficient in the absence of ac perturbation, L 12 , and for the effective electrokinetic coefficient at the different frequencies, L ∗12 , are shown in Fig. 5 as a function of the frequency. The continuous lines correspond to the value of L 12 under the same conditions. As can be observed, the value of
FIG. 4. Volume flow as a function of the dc voltage for the three electrolytes in the absence of ac perturbation. Continuous lines represent the linear fits of the experimental data by using Eq. [3]. (s) LiCl; (d) NaCl; (×) KCl.
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tration. In this paper, we have observed that the dispersion exists for all the electrolytes used. Although the behavior is similar for the three electrolytes, some differences could be observed, as was previously described. It is known that when an electric current circulates across a heterogeneous conductor, it gives rise to different polarization effects in different frequency domains (11, 17, 18). In biological systems and artificial membranes permeated by ionic solutions (11, 18), at very low frequencies, the dispersions are known as α-dispersions and they have their origin in diffusion polarization effects occurring in the electrolytes, unstirred layers, membrane interfaces, etc. At sufficiently high frequencies, the α-dispersions diminish, revealing β-dispersions. These dispersions arise when two contacting phases have different specific conductivities/permittivities and a free charge is formed to provide continuity of the electric current. Due to the diffusion, the charge is localized in some regions having non-zero thickness within each phase. This type of polarization is usually referred to as Maxwell polarization. At even higher frequencies these β-dispersions also diminish, revealing γ -dispersions, which arise from electric field reorientations of molecular dipoles. Thermal agitation causes a
FIG. 5. Effective electrokinetic coefficient, L ∗12 , as a function of the frequency of the ac perturbation. The dotted lines correspond to LiCl and KCl and the continuous line to NaCl in the absence of ac perturbation under the same conditions. (s) LiCl; (d) NaCl; (×) KCl.
frequency range. In this case, the position of the maximum seems to shift to the higher frequencies when the cation atomic number increases, although in the case of KCl this maximum becomes less appreciable or even undetectable. For a characteristic frequency of ≈100 kHz, a third maximum can be observed for the three electrolytes. At last, at frequencies higher than 1 MHz, still another maximum value appears around 3 MHz in the case of LiCl and NaCl solutions, but this last maximum is not observed in the case of the KCl solution. The influence of the amplitude of the ac perturbation on the effective apparent electroosmotic permeability has been studied, for a frequency of 1 kHz, and a dc voltage of 6 V. The obtained results are shown in Fig. 7 for the three electrolytes. The dotted lines correspond to the mean value. As can be observed, there was no significant influence of the amplitude on the effective apparent electroosmotic permeability. In a previous paper (11), we studied the influence of an ac perturbation in the electroosmotic flow of the same membrane under different experimental conditions with NaCl solutions of different concentrations. In that paper, we observed that the dispersion of the electroosmotic permeability with the frequency of the ac perturbation was similar for all the concentrations studied, although it became less appreciable at the highest NaCl concen-
FIG. 6. Effective electroosmotic permeability as a function of the frequency of the ac perturbation. The data correspond to a dc voltage of 6 V. (s) LiCl; (d) NaCl; (×) KCl. The dotted lines represent the electroosmotic permeability values in the absence of ac perturbation.
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tic diffusion times attributed to the thickness of the constituent layer. Thus, following the treatment of Zholkovski for multilayer membranes (19), for an 1 : 1 electrolyte solution, the characteristic frequency of a medium containing two charged species, 1 and 2, is given by w∗ =
1 2D1 D2 , d 2 D1 + D2
[7]
where D1 and D2 are the diffusion coefficients of the two charged species and d the thickness of the layer considered. At the lower frequencies, the α-dispersions have their origin in diffusion polarization effects occurring in the unstirred layers. The thickness, δ, of these unstirred layers can be estimated from the classical polarization theory (20) by the equation δ=
FIG. 7. Effective electroosmotic permeability as a function of the amplitude of the ac perturbation. The data correspond to a dc voltage of 6 V and an ac perturbation frequency of 1 KHz. (s) LiCl; (d) NaCl; (+) KCl. The dotted lines indicate the corresponding mean value.
relaxation of the polarization and this results in a dispersion of the dielectric permittivity with frequency. The observed dispersion of the electroosmotic permeability on the frequency may be closely related to the phenomena, which cause the dispersion of the permittivity of a membrane system. The typical characteristic frequencies seem to be very close to the value of the frequencies at which maximum values of the apparent electroosmotic permeability are observed. It leads us to think that the changes originating in the dielectric properties of the system could have a macroscopic effect in the electroosmotic behavior of the system. Thus, the different maxima observed in the behavior of the electroosmotic permeability vs the ac perturbation frequency would be closely related to the different mechanisms of polarization previously described. With the purpose of checking these ideas and giving them quantitative justification based on our experimental results, we can suppose the membrane system be considered as a multilayer electric conductor. The system includes at least the membrane itself and two adjacent electrolyte solutions. The system would be formed by “fictitious” layers corresponding to the different parts of the system with different properties, as an unstirred layer, Debye layer, membrane pore, etc., and the thickness of each layer would be a characteristic length of that part. The different relaxation times could be roughly evaluated as the characteris-
F Dc0 , 1t+ I L
[8]
where c0 is the concentration corresponding to the bulk solution, I L is the limiting current density, D is the salt diffusion coefficient, 1t+ = t¯+ − t+ is the difference between the cation transport numbers in the membrane and in the free solution, and F is the Faraday constant. Using the values of D and t+ found in the literature (21) for the different electrolytes at 30◦ C, the values of the limiting current density estimated from curves (V, I ) and taking into account that due to the highly selective character of the membrane used, it can be supposed that t¯+ ∼ = 1, the values of δ can be estimated from Eq. [8]. The results obtained are shown in Table 1, together with the values of 1t+ , I L , and D used for each electrolyte. From Eq. [7] and taking into account that in the polarization layer D1 = D2 = D, we can estimate the theoretical character∗1 istic frequency, wthe , of this layer by means of the expression ∗1 = wthe
D . δ2
[9]
The values obtained are also shown in Table 1 together with the experimental values for w ∗1 . As can be observed, the results are very close to the values of the frequencies at which a maximum is observed for the electroosmotic permeability in TABLE 1 Different Parameters Used in the Determination of the Theo∗1 retical Characteristic Frequency w the from Eq. [9], and the Corresponding Experimental Value w ∗1 , for the Different Electrolytes Used
I L (10−3 A/m2 ) 1t+ D (10−9 m2 /s) δ (10−5 m) ∗1 (Hz) wthe w ∗1 (Hz)
LiCl
NaCl
KCl
9.82 0.67 1.4 2.05 3.3 (1–10)
6.98 0.61 1.8 4.1 1.1 (0.1–1)
8.41 0.50 2.14 4.9 0.9 (0.1–1)
´ AND RU´IZ BAUZA ´ BARRAGAN
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the lowest frequency interval. It leads one to think that this first maximum is due to the effect of the polarization concentration and originated in the same mechanism as the α-permittivity dispersions observed in this kind of system. At sufficiently high frequencies, β-dispersions appear to be due to the interfacial polarization effect in heterogeneous systems containing two contacting phases with different specific conductivities/permitivities. An ionic charge distribution, called an electrical double layer (22, 23), is always formed on a charge surface in an electrolyte. The electric charge is distributed between both phases and it is localized in a region with a thickness characterized by the Debye screening length, l D . Maxwell polarization is usually seen at frequencies of about 100 kHz. As can be observed in Fig. 6, for the three electrolytes a maximum value of the electroosmotic permeability is observed at approximately 100 kHz. For a high fixed charge membrane, D = D¯ 1 (diffusion coefficient of counterions inside the membrane) must be written in Eq. [7] and by analogy with Eq. [9], w∗ =
D¯ 1 , l 2D
[10]
where l D is, as was previously said, the thickness of this layer, which can be estimated by the corresponding Debye length given by the expression (23) r F2 2 −1 [11] c¯ z , lD = ε RT where c¯ is the electrolyte concentration inside the membrane, z its valence, ε its permittivity, T the absolute temperature, and R the gas constant. Knowing the fixed charge of the membrane, it is possible to estimate c¯ for each experimental situation. The values obtained are shown in Table 2. The value of the permittivity is ε = εr ε0 , where ε0 = 8.85 × 10−12 F−1 m is the permittivity of the vacuum and εr is the relative permittivity. From Eq. [11], and the values of the relative permittivity shown in Table 2 (24), it is possible to calculate the Debye length for each electrolyte at a concentration c¯ and temperature T . The results are also shown in Table 2. We have no information about the value of the diffusion coefficient of the counterion in the membrane phase. In this case, by using TABLE 2 Experimental Values w and the Different Parameters Used for Determining the Cation Diffusion Coefficient, D1 , in the Membrane from Eq. [10] for the Different Electrolytes ∗3
εr c¯ (103 mol/m3 ) l D (10−10 m) w∗3 (kHz) ¯ 1 (10−14 m2 /s) D
LiCl
NaCl
KCl
66 1.59 3.16 100 1.00
69 1.53 3.29 100 1.08
70 1.46 1.96 100 1.15
TABLE 3 Experimental Values w ∗2 and the Different Parameters Used for Determining the Equivalent Pore Size, d p , from Eq. [10] for the Different Electrolytes
w∗2 (Hz) d p (10−8 m)
LiCl
NaCl
KCl
100 1.0
1000 0.33
3000 0.19
Eq. [10] and supposing that the maximum observed is due to the ¯ 1 using Maxwell polarization, we can estimate the value of D ∗3 ¯ the experimental w . The results obtained for D1 together with the values of w ∗3 are shown in Table 2. As can be observed, the ¯ 1 increases when the cation atomic number increases. value of D In the case of LiCl and NaCl solutions, another maximum value is observed in the frequency interval characteristic of the α-dispersion. This maximum is almost inappreciable in the case of the KCl solutions. We think that they may be due to the existence of pores in the membrane structure. Kuang and Nelson (25) showed that a dielectric dispersion mechanism arised from membrane pores within artificial membranes. Theoretical analysis showed that a low-frequency dielectric dispersion would be produced in the same frequency range as the α-dispersion. If we consider the diffusion coefficient in the membrane phase previously estimated in Eq. [10] and the values of the experimental characteristic frequency, w∗2 , given in Table 3, we can use an analogue to Eq. [10] to estimate the equivalent radii, d P , of the pores. The results are shown in Table 3. As can be observed, the size of the pore decreases when the cation atomic number increases. If we consider the membrane as a simple array of parallel capillaries, it can be shown (26) that the equivalent pore radii can be estimated by means of the equation µ r=
8ησ L 11 L 22
¶1/2 ,
[12]
where σ is the specific conductivity of the solution, η its absolute viscosity, and L 11 and L 22 are defined by Eqs. [1] and [2]. In a previous paper (27), the coefficient and L 11 and L 22 were estimated at 30◦ C for the same membrane used in this work at different concentrations of KCl solutions. For a 10−2 mol/liter concentration, the values found were L 11 = 1.28 × 10−12 (m2 s)/kg, and L 22 = 200 Ä−1 m−2 . For a 10−2 mol/liter KCl solution the values of the especific conductance and the viscosity, at 30◦ C, are (21) 0.1563 Ä−1 m−2 and 8 × 10−4 kg/(m s), respectively. Using these data in Eq. [12], the equivalent pore radius estimated is 0.25 × 10−8 m. As can be observed, the agreement with the value of d P estimated from Eq. [10] for the KCl solution is quite satisfactory. At frequencies higher than 1 MHz, γ -dispersions arisen from electric field re-orientations of molecular dipoles are originated. The characteristic frequency depends on the molecular size in such a way that the higher the molecular size, the lower the characteristic frequency. In the case of LiCl and NaCl, a maximum
ac PERTURBATION AND ELECTROOSMOTIC BEHAVIOR
is observed at frequencies around 2 MHz, but no dispersion is observed for KCl in this interval. The characteristic frequency of these dispersions is 20 MHz for water. This frequency is related to the time that the molecules spend in re-orientating in the electric field. The presence of the electrolyte must affect this time since the water molecules are hydrating the ions in the solutions. From measurements of impedance spectroscopy, it is observed that, in the case of a slight external signal and a linear response, the impedance does not depend on the signal amplitude, but only depends on the frequency and on the morphology of the heterogeneous system. These results are in agreement with the results shown in Fig. 7, where no influence of the amplitude of the ac perturbation was observed on the values of the electroosmotic permeability. SUMMARY
1. The effect of an ac sinusoidal perturbation, of known angular frequency and small amplitude, superimposed on the usual applied dc electric voltage difference on the electroosmotic behavior of a cation-exchange membrane system has been studied for three 1:1 electrolytes. In all the studied cases, the experimental results show that the presence of the ac perturbation influences the electroosmotic permeability value. 2. As a general trend, the presence of the ac perturbation increases the value of the electroosmotic permeability with respect to the value in the absence of ac perturbation. 3. A dispersion of the electroosmotic permeability on the frequency of the ac signal applied has been found for the three electrolytes studied, observing that the electroosmotic permeability reaches maximum values for some characteristic values of the frequency. Although these characteristic values depend slightly on the cation nature, they appear in the same frequency domains for the three electrolytes. 4. This behavior has been related to the different relaxation processes in heterogeneous mediums. The maximum in the 0.1to 10-Hz interval has been related to the α-dispersion due to the concentration polarization, the maximum at ≈100 kHz with the β-dispersion due to the interfacial polarization, and the maximum around the MHz with the γ -dispersion due to the reorientations of molecular dipoles. The maximum found in the 100 Hz to 3 kHz range was related to the dispersion in the membrane pores. ACKNOWLEDGMENT Financial support from the University Complutense of Madrid under Project PR49/98-7756 is gratefully acknowledged.
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REFERENCES 1. Prigogine, I., “Introduction to Thermodynamics of Irreversible Processes,” 3rd ed. Wiley, New York, 1968. 2. De Groot, S. R., “Thermodynamics of Irreversible Processes,” 4th ed. North Holland, Amsterdam, 1966. 3. Khedr, G., Schmitt, A., and Varoqui, R., J. Colloid Interface Sci. 66, 516 (1978). 4. Demish, H. U., and Pusch, W., J. Colloid Interface Sci. 76, 464 (1980). 5. Narebska, A., Kujawski, W., and Kotter, S., J. Membr. Sci. 30, 125 (1987). 6. Arulanandam, S., and Dongqing, L., J. Colloid Interface Sci. 225, 421 (2000). 7. Martinsen, Ø. G., Grimnes, S., and Karlsen, J., J. Colloid Interface Sci. 199, 107 (1998). 8. Plesner, I. W., Malmgren-Hansen, B., and Sørensen, T. S., J. Chem. Soc. Faraday Trans. 90, 2381 (1994). 9. Malmgren-Hansen, B., Sørensen, T. S., Jensen, J. B., and Hennenberg, M., J. Colloid Interface. Sci. 130, 359 (1989). 10. Holdik, H., Alcaraz, A., Ram´ırez, P., and Maf´e, S., J. Electroanal. Chem. 442, 13 (1998). 11. Sørensen, T. S., in “Surface Chemistry and Electrochemistry of Membranes” (T. S. Sørensen, Ed.), Chap. 19. Dekker, New York, 1999. 12. Alcaraz, A., Holdik, H., Rufling, T., Ram´ırez, P., and Maf´e, S., J. Membr. Sci. 150, 43 (1998). 13. Kuang, W., and Nelson, S. O., J. Colloid Interface Sci. 193, 242 (1997). 14. Barrag´an, V. M., and Ru´ız-Bauz´a, C., J. Colloid Interface Sci. 230, 359 (2000). 15. Ru´ız-Bauz´a, C., and Rueda-S´anchez, C., J. Non-Equilibr. Thermodyn. 15, 281 (1990). 16. Barrag´an, V. M., Ru´ız-Bauz´a, C., and Mengual, J. I., J. Membr. Sci. 95, 1 (1994). 17. Janz, G. J., in “Reference Electrodes,” Chap. 4. Academic Press, London, 1961. 18. Barrag´an, V. M., and Ru´ız-Bauz´a, C., J. Colloid Interface Sci. 205, 365 (1998). 19. Lakshminarayanaiah, N., J. Electrochem. Soc. 116, 338 (1969). 20. McCrum, N. G., Read, B. E., and Williams, G., “Anelastic and Dielectric Effects in Polymeric Solids.” Dover, New York, 1991. 21. Coster H. G. L., and Chilcott, T. C., in “Surface Chemistry and Electrochemistry of Membranes” (T. S. Sørensen, Ed.), Chap. 19. Dekker, New York, 1999. 22. Zholkovskij, E. K., in “Surface Chemistry and Electrochemistry of Membranes” (T. S. Sørensen, Ed.), Chap. 20. Dekker, New York, 1999. 23. Helfferich, F., “Ion-Exchange.” McGraw-Hill, New York, 1962. 24. Lobo, V. M. M., “Electrolyte Solutions: Literature Data on Thermodynamics and Transport Properties.” Coimbra Editora, Coimbra, Portugal, 1975. 25. Lakshminarayanaiah, N., “Transport Phenomena in Membranes.” Academic Press, New York, 1969. 26. Bagotzky, V. S., “Fundamentals of Electrochemistry.” Plenum, New York, 1993. 27. Bockris, J., and Reddy, A., “Modern Electrochemistry,” Vol. 1. Plenum/ Rosseta, New York, 1973. 28. Gonz´alez-Caballero, F., and De las Nieves, F. J., J. Membr. Sci. 16, 225 (1983). 29. Barrag´an, V. M., and Ru´ız-Bauz´a, C., J. Non-Equilibr. Thermodyn. 22, 374 (1997).
Effect of an ac Perturbation on the Electroosmotic Behavior of a Cation-Exchange Membrane. Influence of the Cation Nature V. M. Barrag´an1 and C. Ru´ız Bauz´a Departamento de F´ısica Aplicada I, Facultad de Ciencias F´ısicas, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain E-mail: [email protected]; [email protected] Received November 20, 2000; accepted April 6, 2001; published online June 26, 2001
The effect of an ac sinusoidal perturbation of known amplitude and frequency superimposed on the usual dc applied electric voltage difference on the electroosmotic flow through a typical cationexchange membrane has been studied using different monovalent electrolytes. As a general trend, the presence of the ac perturbation increases the value of the electroosmotic flow with respect to the value in the absence of ac perturbation. A dispersion of the electroosmotic permeability on the frequency of the applied ac signal has been found for the three studied electrolytes, observing that the electroosmotic permeability reaches maximum values for some characteristic values of the frequency. This behavior may be related to the different relaxation processes in heterogeneous mediums. °C 2001 Academic Press Key Words: electroosmosis; cation-exchange membrane; electroosmotic permeability; membrane relaxation phenomena.
INTRODUCTION
The electrokinetic phenomena originate as a consequence of the interaction between matter flux and electrical charge flux through porous media. Due to this fact, the “thermodynamics of irreversible processes” has been used satisfactorily for their study (1, 2). The transport equations for matter and electric charge through porous media obtained in that approach are JV = L 11 1P + L 12 1ϕ I = L 21 1P + L 22 1ϕ,
[1] [2]
where JV is the volume flow, I the electric current, 1P the pressure difference, and 1ϕ the electric potential difference. The coefficients L 11 and L 22 are, respectively, the hydraulic permeability and the electric conductance, whereas L 12 , which must be equal to L 21 according to the Onsager reciprocity relation, is the electrokinetic coefficient.
1
To whom correspondence should be addressed. Fax: +34 91 394 51 91.
0021-9797/01 $35.00
C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
From Eqs. [1] and [2], in the absence of pressure difference (1P = 0), one obtains the following relations: µ
¶ JV = L 12 1ϕ 1P=0 µ ¶ L 12 JV = = W. I 1P=0 L 22
[3] [4]
Equations [3] and [4] provide expressions for the electrokinetic coefficient, L 12 , and the electroosmotic permeability, W , in terms of the phenomenological coefficients. Hence, from measurements of the electroosmotic flow both L 12 and W can be determined. Usually, in the study of electroosmotic phenomena, static electric fields are used to obtain a dc current circulating through the membrane system (3–6). In the last years, however, studies of the electric behavior of different membrane/electrolyte systems under the influence of time-dependent electric fields have been carried out (7–13). In some papers, it was shown that a dispersion of the impedance with frequency of the alternating fields existed in some kind of system, revealing that the capacity and conductance of these systems depend on the frequency of the alternating field (10, 12, 13). On the other hand, the use of ac techniques has been shown as a useful instrument for studying electrochemical processes occurring within the interfacial regions between the membrane surface and the bulk electrolyte without disturbing the system. The functional properties of membranes and solid–aqueous systems, in general, are closely connected to the electrochemical processes that occur within the system, so the understanding of the dielectric behavior will be useful in improving membrane processes. In this sense, it is expected that the dispersion of the dielectric properties may influence the electrokinetic phenomena, and in particular on the electroosmotic transport. The aim of this paper is, continuing the work of Ref. (11), to check this idea and try to relate in a broader way the behavior found for the electroosmotic transport with the frequency of the ac perturbation to the different relaxation phenomena observed in similar systems from ac
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techniques. To this end, with use of different electrolytes, the influence of an ac sinusoidal perturbation of known amplitude and frequency, superimposed on the usual dc applied electric voltage difference, on the electroosmotic flow through a typical ion-exchange membrane have been studied. The frequency has been varied in the (10−2 −107 ) Hz range with the purpose of analyzing the possible influence of relaxation phenomena observed at low frequencies on the electroosmotic behavior of the membrane system. EXPERIMENTAL
Materials The membrane used in the present work was the commercial ionics cation-exchange membrane comprised of cross-linked sulfonated copolymers of vinyl compounds CR 61CZL412, whose principal characteristics (provided by the manufacturer) are as follows: selectivity, 94%; electrical resistance (in 0.1 mol/liter NaCl), 11 Ä cm2 ; and capacity, 2.7 meq/g. The membrane thickness was 540 ± 1 µm and the fractional void volume was 0.32. The materials employed in the experiments were 10−2 mol/ liter aqueous solutions of lithium, sodium, and potassium chloride. Pure proanalysis-grade chemicals and pure water (deionized, doubly distilled, doubly filtrated, and degassed) were used. Apparatus and Procedures The apparatus used in this research and the methodology employed were substantially similar to those described in earlier publications (12, 13). The membrane surface exposed to the flow was 0.79 cm2 . An Ag/AgCl electrode with a large active surface was introduced in each chamber on both sides of the membrane with injection of an electric current. The cell is provided with a chain-driven cell magnetic stirrer assembly, which permits stirring of both solutions. The temperature requirements were achieved by immersing the cell in a water thermostat that was maintained at the selected temperature with an accuracy of ±0.1◦ C. In addition, the complete unit was contained in a large ambient thermostat. All measurements were carried out under isothermal conditions at 30◦ C. To measure the current–voltage curves in each condition, the experimental device was modified to use the four-electrodes configuration. In this case, together with the electrodes used to inject the current, other Ag/AgCl electrodes were introduced in each chamber to measure the dc voltage. These electrodes consisted of two linear Ag wires of approximately 4-mm longitude and 1-mm diameter and they were also prepared by the usual method (14). The volume flow was determined by measuring the time displacement of the solution meniscus in the capillary tubes connected to each chamber when an electric potential difference was established through the membrane. Taking into account the volume change due to the electrochemical reactions at the Ag/AgCl electrodes, the volume change rate of the cathodic and anodic
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compartments, 1Vc and 1Va , respectively, is given by I ¯ (V Ag − V¯ AgCl ) F I 1Vc = −JV + (V¯ AgCl − V¯ Ag ), F
1Va = JV +
where F is the Faraday constant and V¯ Ag and V¯ AgCl are the partial molar volumes of Ag and AgCl, respectively. Taking into account that V¯ AgCl − V¯ Ag = 15.5 cm3 /mol, we have JV (ml/s) =
|1Va | + |1Vc | + (1.6 × 10−4 )I, 2
[5]
where I is the dc current passing through the system. The last term in Eq. [5] is the corrective term that takes into account the volume change due to the chemical reactions in the Ag/AgCl electrodes. It has been checked that the electric current passing through the system was constant—up to the hundredth or tenth part of mA—during the measurement time. When the electric potential was applied, a steady current was reached after a few seconds. If an ac perturbation was superimposed on the dc signal, a steady effective dc current was also observed, this value being similar to the one obtained in the absence of ac perturbation under the same conditions. Only for frequencies below 1 Hz and for 10 Hz, a small oscillation of the dc current around a mean value was observed. This mean value was similar to the value in the absence of ac perturbation under the same conditions. In Fig. 1, the dependence of the dc current on time is shown in the absence and in the presence of ac perturbation for one of the studied cases. As can be observed, for the cited frequencies an oscillation is observed around the corresponding value in the absence of ac signal, shown in Fig. 1 as a continuous line. In these cases, this mean value was taken as an effective dc current, I ∗ , to calculate the correction of the reaction change volume to the volume flow in Eq. [5]. In this situation, from Eqs. [4] and [5], the effective apparent electroosmotic permeability, W ∗ , will be given by W ∗ (ml/C) =
|1Va∗ | + |1Vc∗ | + 1.6 × 10−4 , 2I ∗
[6]
where 1Vc∗ and 1Va∗ are, respectively, the volume change rate of the anodic and cathodic compartments in the presence of the ac perturbation. RESULTS AND DISCUSSION
The current–voltage curves of the system under the different studied experimental conditions have been measured by using the usual four-terminal configuration. From these curves, Fig. 2, the values of the limiting current density have been estimated by using a method described in (15). The values obtained were
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electric current circulating through the system was far away from the corresponding limiting values. The volume change rates in the cathodic and anodic compartments were estimated by measuring the displacement of the meniscus in the capillary tubes connected to each compartment as a function of time when the steady state was considered reached. Thus, the volume change rates were calculated from the slope of the straight lines obtained from the linear fits of the experimental data. It was observed that the ac perturbation did not affect this linear behavior, but only the value of the slope, depending on the ac perturbation frequency. In all cases, the measurements were reproducible within the experimental error range. The results obtained showed that a linear relation existed between the volume flow and the applied dc voltage under all the studied conditions. In this paper we have checked this behavior for three different 10−2 mol/liter electrolytes in the range 0– 8 V. The results are shown in Fig. 4, where the continuous lines represent the linear fits of the experimental data by using Eq. [3]. As can be observed, the value of the volume flow increases with decreasing cation atomic number, although the difference between the values corresponding to NaCl and KCl solutions are very small in comparison to their differences with respect to LiCl solution. This result is in agreement with the results
FIG. 1. Dependence of the dc current on time at different frequencies of the ac perturbation. The data correspond to the KCl electrolyte with a dc voltage of 6 V. The continuous line represents the data in the absence of ac perturbation. (s) 10−2 Hz; (+) 10−1 Hz; (m) 10 Hz.
9.8, 7.0, and 8.4 mA/cm2 , for LiCl, NaCl, and KCl solutions, respectively. It was important to determine these values to know the ohmic region of the current–voltage curves and to avoid the abrupt increase of the electroosmotic flow observed at current values close to the limiting current value (13). The behavior of the ohmic region of the (V, I ) curves in the presence of an ac perturbation has been studied. Figure 3 shows the current–voltage curves, in the 0- to 10-V range, for the three studied electrolytes and at different frequencies of the ac perturbation. The dotted lines correspond to the fit to the values in the absence of ac perturbation to a straight line. In all the cases, a linear dependence was found between the dc voltage applied and the dc current in the measured interval, and no significant influence of the frequency was observed. It means that, for the same dc voltage applied, the effective continuous component of the current is the same for all the frequencies, and so the effective ion transport is not affected by the ac signal in the studied frequency range. It was also checked that there were no changes in the described qualitative behavior when the same electrodes were used to measure the dc current as well as the dc voltage. The electroosmotic flow was estimated from Eq. [5]. In all the cases, the value of the applied dc voltage was such that the
FIG. 2. Typical current–voltage curves for the three studied electrolytes in the absence of ac perturbation. (s) LiCl; (n) NaCl; (h) KCl.
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185
this coefficient decreases when the atomic number of the cation increases, independent of the frequency of the ac perturbation. On the other hand, as a general trend, the values of L ∗12 increase, with respect to the corresponding L 12 value, for some values of the frequency. The increase depends on the ac perturbation frequency and on the electrolyte nature. The effective apparent electroosmotic permeability for the same dc voltage, 6 V, has been measured as a function of the frequency of the ac perturbation and at 10−2 mol/liter concentrations of the three electrolytes. The results obtained are shown in Fig. 6 for the different electrolytes used. The dotted horizontal lines correspond to the values of the apparent electroosmotic permeability in the absence of ac perturbation. The experimental results show that the presence of the ac perturbation seems to cause, as a general trend, the value of the electroosmotic flow to increase. For the three electrolytes, it is possible to observe that maximum values are reached at certain values of the frequency. These characteristic frequencies, w ∗ , depend on the electrolyte, but they appear in the same frequency ranges for the three studied electrolytes. One of them, w ∗1 , appears at low values of the frequency, in the 0.1- to 10-Hz range, observing a slight tendency to move toward lower values with increasing cation atomic number. A second maximum is observed at a characteristic frequency, w ∗2 , in the 100 Hz to 10 kHz
FIG. 3. Current–voltage curves in the linear range for the three studied electrolytes and at different ac perturbation frequencies. The dotted lines correspond to the fit to the values in the absence of ac perturbation to a straight line. (d) dc current; (s) 1 Hz; (j) 100 Hz; (h) 1 kHz; (m) 10 kHz; (n) 100 kHz; (r) 1 MHz; (♦) 10 MHz.
obtained in other papers about the influence of the cation nature on the electroosmotic transport (13, 16). To check the influence of ac perturbation on the validity of the linear phenomenological equation [3], for each electrolyte, the behavior of the effective volume flow, JV∗ , with the applied dc voltage has been studied at different frequencies of the ac perturbation and for a 10−2 mol/liter concentration. The results obtained were similar for the three studied electrolytes and they showed, in all the cases, that the presence of an ac perturbation did not influence the linear behavior range but it affected the value of the slope, depending on the frequency of the ac perturbation. The points corresponding to the same frequency have been fitted, as a function of the dc voltage, to a straight line, in the linear behavior range, to estimate the effective electrokinetic coefficient L ∗12 = (JV∗ /1ϕ)1P=0 at the different studied frequencies from the slopes of the fitted straight lines. The results obtained for the electrokinetic coefficient in the absence of ac perturbation, L 12 , and for the effective electrokinetic coefficient at the different frequencies, L ∗12 , are shown in Fig. 5 as a function of the frequency. The continuous lines correspond to the value of L 12 under the same conditions. As can be observed, the value of
FIG. 4. Volume flow as a function of the dc voltage for the three electrolytes in the absence of ac perturbation. Continuous lines represent the linear fits of the experimental data by using Eq. [3]. (s) LiCl; (d) NaCl; (×) KCl.
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tration. In this paper, we have observed that the dispersion exists for all the electrolytes used. Although the behavior is similar for the three electrolytes, some differences could be observed, as was previously described. It is known that when an electric current circulates across a heterogeneous conductor, it gives rise to different polarization effects in different frequency domains (11, 17, 18). In biological systems and artificial membranes permeated by ionic solutions (11, 18), at very low frequencies, the dispersions are known as α-dispersions and they have their origin in diffusion polarization effects occurring in the electrolytes, unstirred layers, membrane interfaces, etc. At sufficiently high frequencies, the α-dispersions diminish, revealing β-dispersions. These dispersions arise when two contacting phases have different specific conductivities/permittivities and a free charge is formed to provide continuity of the electric current. Due to the diffusion, the charge is localized in some regions having non-zero thickness within each phase. This type of polarization is usually referred to as Maxwell polarization. At even higher frequencies these β-dispersions also diminish, revealing γ -dispersions, which arise from electric field reorientations of molecular dipoles. Thermal agitation causes a
FIG. 5. Effective electrokinetic coefficient, L ∗12 , as a function of the frequency of the ac perturbation. The dotted lines correspond to LiCl and KCl and the continuous line to NaCl in the absence of ac perturbation under the same conditions. (s) LiCl; (d) NaCl; (×) KCl.
frequency range. In this case, the position of the maximum seems to shift to the higher frequencies when the cation atomic number increases, although in the case of KCl this maximum becomes less appreciable or even undetectable. For a characteristic frequency of ≈100 kHz, a third maximum can be observed for the three electrolytes. At last, at frequencies higher than 1 MHz, still another maximum value appears around 3 MHz in the case of LiCl and NaCl solutions, but this last maximum is not observed in the case of the KCl solution. The influence of the amplitude of the ac perturbation on the effective apparent electroosmotic permeability has been studied, for a frequency of 1 kHz, and a dc voltage of 6 V. The obtained results are shown in Fig. 7 for the three electrolytes. The dotted lines correspond to the mean value. As can be observed, there was no significant influence of the amplitude on the effective apparent electroosmotic permeability. In a previous paper (11), we studied the influence of an ac perturbation in the electroosmotic flow of the same membrane under different experimental conditions with NaCl solutions of different concentrations. In that paper, we observed that the dispersion of the electroosmotic permeability with the frequency of the ac perturbation was similar for all the concentrations studied, although it became less appreciable at the highest NaCl concen-
FIG. 6. Effective electroosmotic permeability as a function of the frequency of the ac perturbation. The data correspond to a dc voltage of 6 V. (s) LiCl; (d) NaCl; (×) KCl. The dotted lines represent the electroosmotic permeability values in the absence of ac perturbation.
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ac PERTURBATION AND ELECTROOSMOTIC BEHAVIOR
tic diffusion times attributed to the thickness of the constituent layer. Thus, following the treatment of Zholkovski for multilayer membranes (19), for an 1 : 1 electrolyte solution, the characteristic frequency of a medium containing two charged species, 1 and 2, is given by w∗ =
1 2D1 D2 , d 2 D1 + D2
[7]
where D1 and D2 are the diffusion coefficients of the two charged species and d the thickness of the layer considered. At the lower frequencies, the α-dispersions have their origin in diffusion polarization effects occurring in the unstirred layers. The thickness, δ, of these unstirred layers can be estimated from the classical polarization theory (20) by the equation δ=
FIG. 7. Effective electroosmotic permeability as a function of the amplitude of the ac perturbation. The data correspond to a dc voltage of 6 V and an ac perturbation frequency of 1 KHz. (s) LiCl; (d) NaCl; (+) KCl. The dotted lines indicate the corresponding mean value.
relaxation of the polarization and this results in a dispersion of the dielectric permittivity with frequency. The observed dispersion of the electroosmotic permeability on the frequency may be closely related to the phenomena, which cause the dispersion of the permittivity of a membrane system. The typical characteristic frequencies seem to be very close to the value of the frequencies at which maximum values of the apparent electroosmotic permeability are observed. It leads us to think that the changes originating in the dielectric properties of the system could have a macroscopic effect in the electroosmotic behavior of the system. Thus, the different maxima observed in the behavior of the electroosmotic permeability vs the ac perturbation frequency would be closely related to the different mechanisms of polarization previously described. With the purpose of checking these ideas and giving them quantitative justification based on our experimental results, we can suppose the membrane system be considered as a multilayer electric conductor. The system includes at least the membrane itself and two adjacent electrolyte solutions. The system would be formed by “fictitious” layers corresponding to the different parts of the system with different properties, as an unstirred layer, Debye layer, membrane pore, etc., and the thickness of each layer would be a characteristic length of that part. The different relaxation times could be roughly evaluated as the characteris-
F Dc0 , 1t+ I L
[8]
where c0 is the concentration corresponding to the bulk solution, I L is the limiting current density, D is the salt diffusion coefficient, 1t+ = t¯+ − t+ is the difference between the cation transport numbers in the membrane and in the free solution, and F is the Faraday constant. Using the values of D and t+ found in the literature (21) for the different electrolytes at 30◦ C, the values of the limiting current density estimated from curves (V, I ) and taking into account that due to the highly selective character of the membrane used, it can be supposed that t¯+ ∼ = 1, the values of δ can be estimated from Eq. [8]. The results obtained are shown in Table 1, together with the values of 1t+ , I L , and D used for each electrolyte. From Eq. [7] and taking into account that in the polarization layer D1 = D2 = D, we can estimate the theoretical character∗1 istic frequency, wthe , of this layer by means of the expression ∗1 = wthe
D . δ2
[9]
The values obtained are also shown in Table 1 together with the experimental values for w ∗1 . As can be observed, the results are very close to the values of the frequencies at which a maximum is observed for the electroosmotic permeability in TABLE 1 Different Parameters Used in the Determination of the Theo∗1 retical Characteristic Frequency w the from Eq. [9], and the Corresponding Experimental Value w ∗1 , for the Different Electrolytes Used
I L (10−3 A/m2 ) 1t+ D (10−9 m2 /s) δ (10−5 m) ∗1 (Hz) wthe w ∗1 (Hz)
LiCl
NaCl
KCl
9.82 0.67 1.4 2.05 3.3 (1–10)
6.98 0.61 1.8 4.1 1.1 (0.1–1)
8.41 0.50 2.14 4.9 0.9 (0.1–1)
´ AND RU´IZ BAUZA ´ BARRAGAN
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the lowest frequency interval. It leads one to think that this first maximum is due to the effect of the polarization concentration and originated in the same mechanism as the α-permittivity dispersions observed in this kind of system. At sufficiently high frequencies, β-dispersions appear to be due to the interfacial polarization effect in heterogeneous systems containing two contacting phases with different specific conductivities/permitivities. An ionic charge distribution, called an electrical double layer (22, 23), is always formed on a charge surface in an electrolyte. The electric charge is distributed between both phases and it is localized in a region with a thickness characterized by the Debye screening length, l D . Maxwell polarization is usually seen at frequencies of about 100 kHz. As can be observed in Fig. 6, for the three electrolytes a maximum value of the electroosmotic permeability is observed at approximately 100 kHz. For a high fixed charge membrane, D = D¯ 1 (diffusion coefficient of counterions inside the membrane) must be written in Eq. [7] and by analogy with Eq. [9], w∗ =
D¯ 1 , l 2D
[10]
where l D is, as was previously said, the thickness of this layer, which can be estimated by the corresponding Debye length given by the expression (23) r F2 2 −1 [11] c¯ z , lD = ε RT where c¯ is the electrolyte concentration inside the membrane, z its valence, ε its permittivity, T the absolute temperature, and R the gas constant. Knowing the fixed charge of the membrane, it is possible to estimate c¯ for each experimental situation. The values obtained are shown in Table 2. The value of the permittivity is ε = εr ε0 , where ε0 = 8.85 × 10−12 F−1 m is the permittivity of the vacuum and εr is the relative permittivity. From Eq. [11], and the values of the relative permittivity shown in Table 2 (24), it is possible to calculate the Debye length for each electrolyte at a concentration c¯ and temperature T . The results are also shown in Table 2. We have no information about the value of the diffusion coefficient of the counterion in the membrane phase. In this case, by using TABLE 2 Experimental Values w and the Different Parameters Used for Determining the Cation Diffusion Coefficient, D1 , in the Membrane from Eq. [10] for the Different Electrolytes ∗3
εr c¯ (103 mol/m3 ) l D (10−10 m) w∗3 (kHz) ¯ 1 (10−14 m2 /s) D
LiCl
NaCl
KCl
66 1.59 3.16 100 1.00
69 1.53 3.29 100 1.08
70 1.46 1.96 100 1.15
TABLE 3 Experimental Values w ∗2 and the Different Parameters Used for Determining the Equivalent Pore Size, d p , from Eq. [10] for the Different Electrolytes
w∗2 (Hz) d p (10−8 m)
LiCl
NaCl
KCl
100 1.0
1000 0.33
3000 0.19
Eq. [10] and supposing that the maximum observed is due to the ¯ 1 using Maxwell polarization, we can estimate the value of D ∗3 ¯ the experimental w . The results obtained for D1 together with the values of w ∗3 are shown in Table 2. As can be observed, the ¯ 1 increases when the cation atomic number increases. value of D In the case of LiCl and NaCl solutions, another maximum value is observed in the frequency interval characteristic of the α-dispersion. This maximum is almost inappreciable in the case of the KCl solutions. We think that they may be due to the existence of pores in the membrane structure. Kuang and Nelson (25) showed that a dielectric dispersion mechanism arised from membrane pores within artificial membranes. Theoretical analysis showed that a low-frequency dielectric dispersion would be produced in the same frequency range as the α-dispersion. If we consider the diffusion coefficient in the membrane phase previously estimated in Eq. [10] and the values of the experimental characteristic frequency, w∗2 , given in Table 3, we can use an analogue to Eq. [10] to estimate the equivalent radii, d P , of the pores. The results are shown in Table 3. As can be observed, the size of the pore decreases when the cation atomic number increases. If we consider the membrane as a simple array of parallel capillaries, it can be shown (26) that the equivalent pore radii can be estimated by means of the equation µ r=
8ησ L 11 L 22
¶1/2 ,
[12]
where σ is the specific conductivity of the solution, η its absolute viscosity, and L 11 and L 22 are defined by Eqs. [1] and [2]. In a previous paper (27), the coefficient and L 11 and L 22 were estimated at 30◦ C for the same membrane used in this work at different concentrations of KCl solutions. For a 10−2 mol/liter concentration, the values found were L 11 = 1.28 × 10−12 (m2 s)/kg, and L 22 = 200 Ä−1 m−2 . For a 10−2 mol/liter KCl solution the values of the especific conductance and the viscosity, at 30◦ C, are (21) 0.1563 Ä−1 m−2 and 8 × 10−4 kg/(m s), respectively. Using these data in Eq. [12], the equivalent pore radius estimated is 0.25 × 10−8 m. As can be observed, the agreement with the value of d P estimated from Eq. [10] for the KCl solution is quite satisfactory. At frequencies higher than 1 MHz, γ -dispersions arisen from electric field re-orientations of molecular dipoles are originated. The characteristic frequency depends on the molecular size in such a way that the higher the molecular size, the lower the characteristic frequency. In the case of LiCl and NaCl, a maximum
ac PERTURBATION AND ELECTROOSMOTIC BEHAVIOR
is observed at frequencies around 2 MHz, but no dispersion is observed for KCl in this interval. The characteristic frequency of these dispersions is 20 MHz for water. This frequency is related to the time that the molecules spend in re-orientating in the electric field. The presence of the electrolyte must affect this time since the water molecules are hydrating the ions in the solutions. From measurements of impedance spectroscopy, it is observed that, in the case of a slight external signal and a linear response, the impedance does not depend on the signal amplitude, but only depends on the frequency and on the morphology of the heterogeneous system. These results are in agreement with the results shown in Fig. 7, where no influence of the amplitude of the ac perturbation was observed on the values of the electroosmotic permeability. SUMMARY
1. The effect of an ac sinusoidal perturbation, of known angular frequency and small amplitude, superimposed on the usual applied dc electric voltage difference on the electroosmotic behavior of a cation-exchange membrane system has been studied for three 1:1 electrolytes. In all the studied cases, the experimental results show that the presence of the ac perturbation influences the electroosmotic permeability value. 2. As a general trend, the presence of the ac perturbation increases the value of the electroosmotic permeability with respect to the value in the absence of ac perturbation. 3. A dispersion of the electroosmotic permeability on the frequency of the ac signal applied has been found for the three electrolytes studied, observing that the electroosmotic permeability reaches maximum values for some characteristic values of the frequency. Although these characteristic values depend slightly on the cation nature, they appear in the same frequency domains for the three electrolytes. 4. This behavior has been related to the different relaxation processes in heterogeneous mediums. The maximum in the 0.1to 10-Hz interval has been related to the α-dispersion due to the concentration polarization, the maximum at ≈100 kHz with the β-dispersion due to the interfacial polarization, and the maximum around the MHz with the γ -dispersion due to the reorientations of molecular dipoles. The maximum found in the 100 Hz to 3 kHz range was related to the dispersion in the membrane pores. ACKNOWLEDGMENT Financial support from the University Complutense of Madrid under Project PR49/98-7756 is gratefully acknowledged.
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