J. Phys. Chem. B 1997, 101, 2321-2326
2321
Transport of Ions across Bipolar Membranes. 2. Membrane Potential and Permeability Coefficient Ratio in CaCl2 Solutions Mitsuru Higa* and Akihiko Tanioka Department of Organic and Polymeric Materials, Tokyo Institute of Technology, Ookayama, Meguroku, Tokyo 152, Japan
Akira Kira The Institute of Physical and Chemical Research (RIKEN), Hirosawa 2-1, Wako-shi, Saitama 351-01, Japan ReceiVed: NoVember 12, 1996; In Final Form: January 20, 1997X
The simulations and the experimental data of ionic transport across a bipolar membrane in dialysis systems of a 2-1 type salt, where “2-1 type” means a combination of a bivalent cation and a univalent anion, show that the transport properties of a 2-1 type salt in the system have features different from those of a 1-1 type (univalent cation and univalent anion) salt in the following points: (1) the potential of a 2-1 type salt system has a constant value irrespective of the charge density ratio of the two charged layers when the membrane charge densities are much larger than the concentration of the salt solutions; (2) the potential-concentration curves have a peak under suitable conditions; (3) the permeability coefficient of the salt has a different value between two dialysis systems where the direction of the positive-negative layered structure of the membrane is reversed even if the mobility of the cation is the same as that of the anion.
Introduction A bipolar membrane has a sandwich-like structure with a positively charged layer, P, and a negatively charged layer, N. One of the most important properties of a bipolar membrane is valence-selectivity caused by the P-N layered structure. Bipolar membranes show selective permeability of univalent ions to high-valent ions in an electrodialysis system.1-4 This property has been applied for developing ion exchange membranes with high permselectivity for univalent ions. Another important property of a bipolar membrane is that ionic transport depends on the direction of the P-N layered structure to the ionic concentration gradient in a dialysis system. In this paper, a term “1-1 type” means a combination of a univalent cation and a univalent anion, and “2-1 type” of a bivalent cation and a univalent anion. In a dialysis system consisting of 1-1 type electrolyte solutions and a bipolar membrane whose charged layers have the same charge density, the membrane potential is positive if layer P contacts with the solution at the concentrated side of the dialysis system and negative if it does at the dilute side.5,6 Moreover, Higuchi et al.6,7 reported that permeability of a 1-1 type salt in a bipolar membrane depends on the direction of the P-N layered structure when the mobility of the cation has a value different from that of the anion. In a dialysis system of a salt where the valence of the cation differs from that of the anion, such as a 2-1 type salt, the direction of the P-N layered structure will affect the salt permeation even if the mobility of the cation is the same as that of the anion because theoretical calculations8 and experimental data9 show the permeability coefficient of a 2-1 type salt is larger in a negatively charged membrane but smaller in a positively charged membrane than that of a 1-1 type salt. There have been several reports5-7,10-19 on the analysis of ionic transport across a bipolar membrane in a system either of * To whom correspondence should be addressed. Present address: Department of Applied Chemistry and Chemical Engineering, Faculty of Engineering, Yamaguchi University, Tokiwadai 2557 Ube City, Yamaguchi 755, Japan. X Abstract published in AdVance ACS Abstracts, March 1, 1997.
S1089-5647(96)03758-3 CCC: $14.00
1-1 type salt or of a salt whose cation has the same valence as the anion. P. Ramı´rez et al.20 studied current-voltage property of bipolar membranes in an electrodialysis system at various Na2SO4 concentrations. However, there have been no studies reported on the relationship between the ionic transport and the charged layer structure of a bipolar membrane for a salt where the cation has a different valence from the anion. This study aims to analyze the relationship between the ionic transport and the charged layer structure in order to reveal a mechanism of valence selectivity of bipolar membranes. Hence, we simulate both the total membrane potential and permeability coefficient of a 2-1 type salt across a bipolar membrane as a function of the salt concentration for selected values of the charge density ratio and the thickness ratio of the two charged layers. In order to examine the dependence of the transport properties on the direction of the P-N layered structure, these simulations are made for the two dialysis systems: the one where layer N contacts the solution at the concentrated side and the other at the dilute side. Both the potentials and permeability coefficients were measured for CaCl2 in the two dialysis systems, and the experimental results were compared with the simulations. Calculations Ionic transport in the dialysis system shown in Figure 1 is calculated by using the same method as in ref 5. The electroneutrality condition and the Donnan equilibrium give the following equation:
∑zi(K(j,k))z CRi (j) + zx(k)Cx(k) ) 0 i
(1)
where K(j,k) is the Donnan equilibrium constant at the interface between layers j and k. The flux of the ith ion in layer k, Ji(k), is given as
Ji(k) ) -
CRi (k) - CLi (k)βzi RTziωi ln β d βzi - 1
© 1997 American Chemical Society
(2)
2322 J. Phys. Chem. B, Vol. 101, No. 13, 1997
Higa et al. where ωc and ωa are the mobility of the cation and the anion, respectively. Experiments
Figure 1. Schematic diagram of a dialysis system with a bipolar membrane. The membrane consists of the two charged layers A and B whose valence and charge density are indicated as zx(A)Cx(A) and zx(B)Cx(B), respectively. Ci(d) ) rC0 and Ci(0) ) C0, where C0 and r are the salt concentration at the dilute solution (side o) and the concentration ratio between the two cells, respectively. CLi (k) and CRi (k) are the ionic concentration at the left and the right side of layer k (k ) A or B), respectively.
where zi and ωi are the valence and mobility of the ion. CRi (k) and CLi (k) are the ionic concentrations at the right-hand side and the left-hand side in layer k, respectively. β is the solution of the following equation:
(ωaCRa (k) + 4ωcCLc )β2 + ωa(CRa (k) - CLa (k))β (ωaCLa (k) + 4ωcCRc ) ) 0 (3) where subscripts a and c indicate the anion and cation, respectively. Equation 1 is a cubic equation in the system of CaCl2 and can have three real solutions. We solved them by using the Cardano method and checked the physical meaning of the solutions to leave only one real solution for every salt concentration. Both the total membrane potential and permeability coefficient of the salt in the systems were calculated by using this calculation method. In the calculation, for system I, zx(x) and Cx(x) are set by using the ratio of the charge densities, rx, as
zx(A)Cx(A) ) -5.0 × 10-2 mol dm-3
(4)
zx(B)Cx(B) ) rxzx(A)Cx(A)
(5)
and for system II as
zx(B)Cx(B) ) -5.0 × 10-2 mol dm-3
(6)
zx(A)Cx(A) ) rxzx(B)Cx(B)
(7)
where A and B indicate the charged layers shown in Figure 1. Hence, in the case of rx < 0, layer A is a negative layer and layer B a positive layer for system I and vice versa for system II. The permeability coefficient ratio, PCR, is defined as
PCR ) PII/PI
(8)
where PI and PII are the permeability coefficient of the salt in the systems I and II, respectively. In this simulation, rx is swept from -4.0 to 1.0. If rx < 0, the membrane is bipolar. The thickness ratio of the two charged layers, rd, is defined as
rd ) dA/dB
(9)
where dA and dB are the thickness of the charged layer A and B, respectively. The mobility ratio of the cation to the anion, rm, is defined as
rm ) ωc/ωa
(10)
Samples. The bipolar membranes, BIP-1, BIP-2, and BIP-3 in the previous paper,5 which were made from positively charged PVA membranes and negatively charged ones, were used for the experiments. Measurement of the Potentials. The membrane surface potential of each membrane surface and the total membrane potential for CaCl2 solutions were measured with the same apparatus as described in the previous paper.5 The total membrane potential was measured for two dialysis systems: system I where the negatively charged layer of a bipolar membrane contacted with the solution of the concentrated side; system II where the membrane was turned over so that the negative layer contacted with the dilute side. Permeation Experiments. The permeation of CaCl2 was measured with the same apparatus as shown elsewhere.9 In this measurement, the membrane area was 7.0 cm2 and the volume of the dilute side and the concentrated side cell was 500 and 70 cm3, respectively. The concentration of the dilute side cell was measured by a conductivity meter [Horiba, Ltd. ES-12]. The permeability coefficient was evaluated from the initial slope of the time vs concentration curve. Results and Discussion Parts a and b of Figure 2 show the calculations of the total membrane potential for systems I and II, respectively, vs CaCl2 concentration at the dilute side in the systems, C0, at various charge density ratios of the layers, rx. In the case that the charge density of both charged layers is much smaller than the ionic concentration of the concentrated side cell, Cx(k) , rC0 (k ) A and B), the potentials in both systems for all the values of rx are equal to the diffusion potential in aqueous solutions of CaCl2: (ωca - ωcl)/(2ωca + ωcl)RT/F ln r ()-14.3 mV), where ωca and ωcl are the mobility of Ca2+ and Cl- ions in an aqueous solution. These values are 3.0 × 10-13 mol m2 J-1 s-1 and 8.2 × 10-13 mol m2 J-1 s-1, respectively.21 r is the concentration ratio in the solutions. This is due to the fact that the Donnan potentials are almost equal to zero under this condition. The calculation in both systems in the case of rx ) 1.0 shows the same potential-concentration curve as that of the systems with a negatively charged membrane calculated in terms of the Teorell-Meyer-Sievers theory.22,23 This fact supports the adequacy of our calculations in both systems because we set zx ) -1 in this paper so that the membrane of rx ) 1.0 indicates a negatively charged membrane. The potential-concentration curves on rx in the system of CaCl2 have the following features different from that in the system of KCl5 where the potential varied from -41.3 to 41.3 mV according to the value of rx. (1) In the case of Cx(k) . rC0, the potential of a bipolar membrane (rx < 0) in both systems is independent of the value of rx and has the constant value -(RT/F) ln r ()-41.3 mV). This value is the same as that of a positively charged membrane. The appendix indicates the reason why all bipolar membranes have the same constant value. (2) A peak appears in the potentialconcentration curve in CaCl2 solutions in the region 0 > rx > -0.2 in system I and in the region 0 > rx > -2.0 in system II. This peak appears because in these regions of rx, the Donnan potential at the negative-layer/salt-solution interface is larger than that at the positive-layer/salt-concentration interface when Cx(k) ≈ C0. The theoretical conditions for the appearance of the peak are Cx(A)/(rC0) > Cx(B)/C0 or |rx| < 1/r for system I and Cx(B)/C0 > Cx(A)/(rC0) or |rx| < r for system II. Hence,
Transport of Ions
J. Phys. Chem. B, Vol. 101, No. 13, 1997 2323
Figure 3. Simulations of the total membrane potential in system II vs C0. rd is the thickness ratio of the two layers.
Figure 2. Simulations of the total membrane potential as a function of the CaCl2 concentration at the dilute side, C0, where rx is the ratio of the charge densities in the two layers: (a) system I where the positive layer contacts the dilute side; (b) system II where the negative layer contacts the dilute side.
Figure 4. Simulations of the permeability coefficient ratio of system II to system I, PCR, vs C0 for rx ) -4.0. rd is the thickness ratio of the two layers.
the peak can be observed over a wider range of rx in system II than in system I. A membrane where rx > 0, except the case that rx ) 1.0 is not a bipolar membrane but an inhomogeneously charged membrane with negative charges. Hence, when Cx(k) . rC0 the potential has the same value as that of a negatively charged membrane, (RT/F) ln r. The potential of such membranes decreases with increasing C0 more gradually than that of a homogeneously charged membrane (rx ) 1.0). This phenomenon is similar to that of a membrane with asymmetric fixed charge distributions. The calculations in the membrane was first reported by M. Nakagaki and R. Takagi24,25 and developed by J. A. Manzares et al.13 The difference between the membrane with asymmetric fixed charge distributions and the membranes in this study is that the charge density in the former membrane changes its value continuously while the membrane in this study has discontinuous charge distribution between the two layers. Figure 3 shows the total membrane potential vs salt concentration calculated for different values of the thickness ratio, rd, in the case of rm < 1.0. The peak height increases with decreasing rd. Our simulations, whose results are not described in this paper indicate that in the case of rm > 1.0 the peak height also increases with decreasing rd. Hence, the increase of the peak height with decreasing rd is a typical phenomenon of a 2-1 type salt system. As mentioned in the Appendix, salt transport in a bipolar membrane depends mainly on the cation resistance in layer P, Rc(P). The decrease of rd always results
in decreasing Rc(P). Thus, the concentration gradient of the cation in layer P decreases and that in layer N increases with rd. Thus, the ionic concentration at the P-N interface side of layer P increases with rd; hence, the inner Donnan potential decreases with rd, and the maximum of the total membrane potential increases with rd. In the case of rd > 1.0, the thickness of layer N becomes smaller than that of layer P. Therefore, the change of rd affects less the transport phenomena in CaCl2 solutions than in the case of rd < 1.0. Figure 4 shows the permeability coefficient ratio of the two systems, PCR ) PII/PI for a few values of rd. When Cx(k) , rC0, PCR ) 1.0 because the effect of charged groups on the salt permeation is very low. When Cx(k) . rC0, PCR ) 1.0 because the concentration gradient of the cation in layer P in system I is the same as that in system II as mentioned in the Appendix. In the intermediate concentration region, PCR > 1.0, which means that PII > PI. The reason why this difference occurs is because of the fact that the cation concentration of layer P is smaller in system I than that in system II because layer P contacts the dilute side in system I and the concentrated side in system II. The peak height of PCR increases with rd. The reason is as follows. Rc(P) decreases with rd in both systems. However, the change of Rc(P) with rd in system I is steeper than that in system II because layer P contacts with the dilute side in system I and the concentrated side in system II. Other simulations, whose results are not described here, also revealed that the maximum value of the peak increases with
2324 J. Phys. Chem. B, Vol. 101, No. 13, 1997
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Figure 5. Simulations of PCR vs C0 for rx ) -1.0. rm is the mobility ratio of the cation over the anion.
Figure 7. Experimental data and the simulations of the total membrane potential (b, system I; O, system II): (a) for BIP-1; (b) for BIP-2; (c) for BIP-3.
Figure 6. Surface potential vs the CaCl2 concentration, Cc: (open symbols) the data at the negatively charged layer; (solid symbols) the data at the positively charged layer. The solid curves are calculated by fitting the data to the equation of the Donnan equilibrium.
decreasing rx or rm. The calculation of permeability coefficient in the system of a 1-1 type salt reported by Higuchi et al.6 shows that the difference between the two systems in less than 40%. However, PII in the present study is about 80% larger than PI in the case of rx ) -4, rd ) 0.1, and rm ) 0.3. Figure 5 shows PCR for a few values of rm. PII is about 30% greater than PI even if rm ) 1.0 while Higuchi et al.6 reported that the difference of the permeability coefficients is not observed in the calculations of a 1-1 type salt system when rm ) 1.0. This means that the ionic valence difference between the cation and the anion affects ionic selectivity of a bipolar membrane more than the ionic mobility difference does. PCR in a 2-1 type salt system slightly decreases with rm. Figure 6 shows the experimental data of the surface potential vs the concentration of the CaCl2 solution, Ce. As well as the results in the KCl solution,5 the surface potential of both charged-layer/solution interfaces is almost equal to zero for all
the membranes when Ce > 0.1 mol dm-3. This implies that the standard chemical potential of the ions in both layers is equal to those in the solution. For all the bipolar membranes, the potential-concentration curve of the positive layer in the system of CaCl2 is almost equal to that in the system of KCl1 because the counterion of the layer in the former is the same as that in the latter. The slope of the curve for the negative layers is half that of the KCl system at low salt concentration because the valence of the counterion in the former is twice the latter. The solid curves are the potentials calculated by fitting the data to the equation of the Donnan equilibrium. From this fitting, the charge density of the negatively charged layer and the positively charged layer was obtained. The charge densities in the CaCl2 system are equal to those obtained from those of the KCl system,5 which implies that the Donnan equilibrium in the three interfaces also holds ideally in the CaCl2 system. Figure 7 shows the experimental data and the simulations of the total membrane potential. In the dialysis system of CaCl2 as well as KCl, the potential of all the bipolar membranes in system I has different values from that in system II. Sample membrane BIP-3, which has the lowest value of rx, has a peak in system I as predicted by our calculations. The simulations calculated by using the charge densities obtained from the surface potential measurements are shown as solid curves in
Transport of Ions
J. Phys. Chem. B, Vol. 101, No. 13, 1997 2325 in this paper, revealed that the effect of both the charge density and the thickness of the negative layer on the salt transport is larger in a 1-2 type salt system than in 1-1 or 2-1 type salt systems. These results indicate that the transport properties of a bipolar membrane much depend on the charge density and the thickness of the charged layer, which has the opposite charge sites to a high-valent ion. Hence, the membrane potential and permeability of a high-valent ion across a bipolar membrane can be controlled by changing these values. This phenomenon gives very important clues for developing highly selective membranes as well as for understanding the mechanism of ionic transport in biological systems. Acknowledgment. Financial support from the Ministry of Education, Science and Culture and Uruma Trust for Research into Science and Humanity are gratefully acknowledged. Appendix
Figure 8. Experimental data and the simulations of PCR vs C0 for BIP-3.
Figure 7. They agree quantitatively with the experimental data in all the membranes and in both the systems. Figure 8 shows the experiments and the calculation of PCR vs CaCl2 concentration. In the experiments, PCR is larger than unity at all the concentrations and has a maximum at Ce ) 1.0 × 10-3 mol dm-3. The prediction of the concentration for the peak and the maximum value of the peak agree quantitatively with the experimental data. However, the peak of the data is not as sharp as that from the calculation. The deviation probably comes from the assumptions in the calculation, which are the same as those in ref 5: the boundary effects on the surfaces are negligible; ionic activity coefficients are unity both in the solutions and in the two charged layers; the ionic mobility in the two layers has the same value in an aqueous solution. P. Ramı´rez et al.15 reported the effect of diffusion double layer on the transport properties of a bipolar membrane. M. Higa and A. Kira26 reported that the ionic mobility in a water-swollen membrane depends on the membrane potential. Another effect that should be taken into account is the possible existence of a neutral layer between the two charged layers. Although these factors were not taken into account, the calculation demonstrates qualitatively the relationship between the charge selectivity of the bipolar membranes and the P-N layered structure. Conclusions We simulated both the total membrane potential and permeability coefficient of a 2-1 type salt across a bipolar membrane as a function of the salt concentration, varying the charge density ratio, rx, the thickness ratio, rd, of the two charged layers, and the mobility ratio of the cation over the anion, rm. Both the membrane potential and the permeability coefficient depend on the direction of the P-N layered structure in the system. The membrane potential vs the salt concentration curve in a 2-1 type salt system has features different from that in a 1-1 type salt system: (1) the potential of a 2-1 type salt system has the constant value -(RT/F) ln r for all values of rx when Cx(k) . rC0, while that of a 1-1 type salt system changes with rx; (2) the potential-concentration curves have a peak when Cx(k) ≈ C0 under the condition that the Donnan potential at the negativelayer/solution interface is larger than that at the positive-layer/ solution interface; (3) the height of the potential peak increases with decreasing rd; (4) even if rm ) 1.0 the permeability coefficient ratio of system II over system I has a peak when Cx(k) ≈ C0; (5) the permeability coefficient ratio increases with decreasing rd. Our simulations, whose results are not described
In a dialysis system consisting of a bipolar membrane and 2-1 type salt solutions, the flux of the two ions in the two charged layers has the same value compared with each other:
JAc ) JAa ) JBc ) JBa
(A18)
and ionic flux in a charged layer is proportional to its concentration in the layer. Hence, in the case of Cx(k) . C0, the salt transport across the bipolar membrane depends on the concentration and the concentration gradient of the bivalent cation in the positively charged layer because the ion in the layer has the lowest concentration in the membrane. We obtain the ionic concentration profile in the bipolar membrane and examine the dependence of the profile on the transport properties of the bipolar membrane. In a charged layer of a bipolar membrane, when Cx(k) . C0, the following relations for the diffusion potential and the concentration of the cation in the two charged layers are assumed in the two dialysis systems as
∆φdiff ≈ 0
(A1)
CRc (A) ≈ CLc (A)
(A2)
CLc (B) ) pkCRc (B) (k ) I or II)
(A3)
where pk is a constant. In system I, the Donnan equilibrium constant of each interface is
K(d,A) )
K(B,0) )
K(A,B) )
( ) ( ) CLc (A) C0
1/2
CRc (B) rC0
1/2
CLc (B)
1/2
( )
(A4)
(A5)
(A6)
CRc (A)
Hence, the total Donnan potential is given as
∆φIdon ) -
(
)
RT K(d,A)K(A,B) ln F K(B,0)
1/2
≈-
RT ln(pIr) (A7) 2F
Similarly, in system II,
∆φIIdon ≈ -
()
RT pII ln 2F r
(A8)
2326 J. Phys. Chem. B, Vol. 101, No. 13, 1997
Higa et al.
TABLE 1: Calculations of Ionic Concentrations in the Dialysis Systems system
ion
I
Ca2+
II
Ci(d)
5.0 × Cl- 1.0 × 10-4 Ca2+ 1.0 × 10-5 Cl- 2.0 × 10-5 10-5
CLi (A)
CRi (A)
CLi (B)
CRi (B)
2.5 × 4.5 × 10-6 2.5 × 10-2 4.0 × 10-7
2.5 × 4.5 × 10-6 2.5 × 10-2 4.0 × 10-7
2.0 × 5.0 × 10-2 1.6 × 10-12 5.0 × 10-2
1.6 × 5.0 × 10-2 2.0 × 10-10 5.0 × 10-2
10-2
10-2
10-10
10-12
The following relations for the concentrations of the anion are also obtained:
CRa (A) ) qkCLa (A) (k ) I or II)
(A9)
CLa (B) ≈ CRa (B)
(A10)
Hence, the total Donnan potential in the two systems is given as
∆φIdon
(
)
RT K(d,A)K(A,B) ) - ln F K(B,0) ∆φIIdon ≈
-1
()
RT r ≈ ln F qI
RT ln(rqII) F
(A11) (A12)
In order to obtain pI, qI, pII, and qII, the ionic concentration in each side of each charged layer was calculated in terms of the method cited above and shown in Table 1. The following relations are given from the calculations and eqs A11 and A12:
pI ≈ r-3
(A13)
pII ≈ r3
(A14)
qI ) qII ≈ 1
(A15)
Equations A13, A14, and A15 indicate that the concentration gradient of the co-ion in layer P is much larger than that of the counterion so that the flux of the co-ion in layer P is equal to the flux of the counterion. In other words, the concentration drop in the system occurs mainly in the layer having much higher resistance for salt transport than in the other, which is similar to an electric circuit. The calculations also indicate that ∆φdiff ≈ 0 in the two charged layers and that CRc (A) ≈ CLc (A) and CLa (B) ≈ CRa (B), which means the adequacy of the assumption in eqs A1, A2, and A10. The potentials obtained by substituting the value of the four constants into eqs A7, A8, A11, or A12 have the constant value -(RT/F)ln r. Another explanation why the membrane potential of all the bipolar membranes has a constant value when Cx(k) . C0 is the following. The membrane potential is written as k k k + ∆φdon(A,B) + ∆φdon(B,0) (A16) ∆φkdon ) ∆φdon(d,A) k We divide the inner Donnan potential ∆φdon(A,B) as
Ci(o)
∆φdiff(A)
∆φdiff(B)
1.0 × -7.1 × -1.4 × 2.0 × 10-5 5.0 × 10-5 7.1 × 10-8 1.4 × 10-7 1.0 × 10-4 10-5
10-8
10-7
∆φdon(d,A) ∆φdon(A,B) ∆φdon(B,o) -79
239
-200
-100
301
-159
k k ∆φdon(A,B) ) ∆φkdon(N) + ∆φdon(P)
(A17)
k k and ∆φdon(P) are the Donnan potential in the where ∆φdon(N) interface between the charged layers caused by the negatively charged and the positively charged sites, respectively. When Cx(k) . C0, the calculations show that CLa (A) ≈ CRa (A) and k k CLa (A) ≈ CRa (A). Hence, ∆φdon(d,A) ≈ - ∆φdon(N) . Therefore, the total Donnan potential under the condition is given as
k k ∆φkdon ) ∆φdon(P) + ∆φdon(B,0)
This means that the potential of all the bipolar membranes under the condition has the same value as that of a positively charged membrane, -(RT/F)ln r, because the Donnan potential at the two surfaces of the negatively charged layer cancels each other. References and Notes (1) Glueckauf, E.; Kitt, G. P. J. Appl. Chem. 1956, 6, 511. (2) Tanaka, Y.; Seno, M. J. Membr. Sci. 1981, 8, 115. (3) Mizutani, Y. J. Membr. Sci. 1990, 54, 233. (4) Urairi, M.; Tsuru, T.; Nakao, S.; Kimura, S. J. Membr. Sci. 1992, 70, 153. (5) Higa, M.; Kira, A. J. Phys. Chem. 1995, 99, 5089. (6) Higuchi, A.; Nakagawa, T. J. Membr. Sci. 1987, 32, 267. (7) Higuchi, A.; Nakagawa, T. J. Chem. Soc., Faraday Trans. 1 1989, 85, 3609. (8) Higa, M.; Tanioka, A.; Miyasaka, K. J. Membr. Sci. 1990, 49, 145. (9) Higa, M.; Tanioka, A.; Miyasaka, K. J. Membr. Sci. 1991, 64, 255. (10) Bassignana, I. C.; Reiss, H. J. Phys. Chem. 1983, 87, 136. (11) Higuchi, A.; Nakagawa, T. J. Chem. Soc., Faraday Trans. 1 1991, 87, 2723. (12) Mafe´, S.; Manzanares, J. A.; Pamı´rez, P. Phys. ReV. A 1990, 42, 6245. (13) Manzanares, J. A.; Mafe´, S.; Pellicer, J. J. Phys. Chem. 1991, 95, 5620. (14) Sokirko, A. V.; Ramı´rez, P.; Manzanares, J. A.; Mafe´, S. Ber. Bunsen-Ges. Phys. Chem. 1993, 97, 1040. (15) Ramı´rez, P.; Mafe´, S.; Manzanares, J. A.; Pellicer, J. J. Electroanal. Chem. 1996, 404, 187. (16) Sonin, A. A.; Grossman, G. J. Phys. Chem. 1972, 76, 3996. (17) Lakshminarayanaiah, N. Transport Phenomena in Membranes; Academic Press: New York and London, 1969. (18) Coster, H. G. L.; George, E. P.; Simons, R. Biophys. J. 1969, 9, 666. (19) Simons, R. Electrochim. Acta 1985, 30, 275. (20) Ramı´rez, P.; Rapp, H.-J.; Mafe´, S.; Bauer, B. J. Electroanal. Chem. 1994, 375, 101. (21) International critical tables; McGraw-Hill: New York, 1949. (22) Teorell, T. Proc. Soc. Exp. Biol. Med. 1935, 33, 282. (23) Meyer, K. H.; Sievers, J.-F. HelV. Chim. Acta 1936, 19, 649. (24) Nakagaki, M.; Takagi, R. Chem. Pharm. Bull. 1984, 32, 3812. (25) Takagi, R.; Nakagaki, M. J. Membr. Sci. 1986, 27, 285. (26) Higa, M.; Kira, A. J. Phys. Chem. 1992, 96, 9518.
