J. Phys. Chem. 1995,99,5089-5093
5089
Transport of Ions across Bipolar Membranes. 1. Theoretical and Experimental Examination of the Membrane Potential of KCl Solutions Mitsuru Higa* and Akira Kira The Institute of Physical and Chemical Research (RIKEN), Hirosawa 2-1, Wako-shi, Saitama, 351-01 Japan Received: September 14, 1994; In Final Form: January 3, 1999
A calculation method was derived for ionic transport across a bipolar membrane in dialysis systems of mixed salt solutions containing multivalent ions. The calculation of the total membrane potential in the solutions containing KC1 alone by this method shows that the potential-salt concentration curve depends both on the direction of the arrangement of the membrane charged layers to the concentration gradient of the salt solutions and on the ratio of the charge densities of the two charged layers. The simulations based on experimentally determined parameters agree with the potential measurements for bipolar membranes produced so that the transport properties depend mainly on their charges.
Introduction Theoretical treatment of ionic transport in a homogeneously charged membrane was pioneered by Teorell' and Meyer and Sievers2 and has been developed by many Ionic transport in inhomogeneously charged membranes such as bipolar ones which consist of a layered structure of two oppositely-chargedlayers has recently received attention because these membranes show many interesting phenomena: permselectivity for monovalent ions,*-" water splitting,12-16 rectification properties,17-19 and electrical oscillations.20~21These phenomena are very inportant clues for developing highly selective membranes as well as for making clear the mechanism of ionic transport in biological systems. There are several methods for analysis of ionic transport across inhomogeneously charged m e m b r a n e ~ ; ~ Ohowever, -~~ these analyses, except for one by Nakagaki et al.,27928have been done in systems either of a single 1-1 type salt or of a salt whose cation has the same valence as the anion. In their analysis Nakagaki et al. neglected the Donnan potentials in the membrane in their derivation of the equation of ionic flux. Hence, their method is not applicable to bipolar membranes which have large Donnan potentials at the junction. Since the valence-selective transport of ions is one of the most important features of a bipolar membrane, it is invoked to establish a method to analyze the dependence of the transport properties of systems containing multivalent ions on its charges. No quantitative examinations of a bipolar membrane have been reported for the dependence of the transport properties on the charges, although there have been many experimental reports of transport properties. The reasons are probably as follows: (i) In order to examine the dependence, experiments should be performed by using the membrane whose transport properties depend mainly on its charges irrespective of its chemical and geometrical structures; however, biological membranes are too complex to examine the dependence, and artificial membranes made by pasting ion-exchange membranes using a binder depend on both the properties of the binder and the condition of the junction rather than on just the membrane charges. Almost all of the artificial membranes also have such high charge densities andor low water contents that their transport properties depend
* To whom correspondence should be addressed. Present address: Department of Organic and Polymeric Materials, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152 Japan. @Abstractpublished in Advance ACS Abstracts, March 15, 1995.
on the membrane structures rather than on the charges. (ii) Since there were no methods to measure the charge density of the two charged layers separately, the simulations could not be examined quantitatively by experiments. This study aims to derive a simulation method of ionic transport in a dialysis system consisting of a bipolar membrane and solutions of multivalent ions and to simulate the total membrane potential vs the ionic concentrations in the two dialysis systems where the negatively charged layer of the membrane in one case faced the concentrated side of the dialysis cell and in another case faced the dilute side. To compare the simulations with the membrane potential measurements, we prepared bipolar membranes whose transport properties depend mainly on their charges and measured their surface membrane potentials to obtain the charge densities of the two charged layers. The simulations calculated by using the charge densities were compared with the potential measurements.
Theory Ionic transport in the dialysis system shown in Figure 1 will be simulated under the following assumptions: (a) the surfaces of both of the charged layers are always in a state of the Donnan equilibrium, with the same equilibrium constant for all ions; (b) the boundary effects on the surfaces are negligible; (c) all the electrolytes dissolve perfectly, and ionic activity coefficients are unity both in the aqueous solutions and in the layers A and B; (d) the standard chemical potentials in the two layers are equal to those in the solutions, respectively. The flux of the ith ion, J,, for the concentrations of the ith ion in the two solutions, C,(d) and C,(o); the charge density of the layers, C,(A) and C,(B); and the mobility of the ion, w,, are calculated by the following steps: 1. The Donnan equilibrium constant K o , ~at) the interface between layers j and k is a function of the ionic concentrations at these layers:35
where zj is the valence of the ith ion; F, R, and Tare the Faraday constant, the gas constant, and the absolute temperature, respectively. &donu,@ is the Donnan potential at the interface between layers j and k. The electroneutral condition at the left surface of layer k gives
0022-365419512099-5089$09.00/0 0 1995 American Chemical Society
Higa and Kira
5090 J. Phys. Chem., Vol. 99, No. 14, 1995
a criterion for ending the numerical computation: If the value s,
'=?(
Jj(B) - Ji(A) Jj(A)
(6)
then we stop the simulation and obtain
is smaller than the flux of ions
J j = J,(A)
%
(7)
Ji(B)
and the membrane potential, A#:
Salt solution
Salt solution
Bipolar Membrane
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 z,(A)C,(A) and z,(B)C,(B), respectively. Ci(d) and Ci(o)indicate the concentration of the ith ion at the concentrated solution (side d) and at the dilute solution (side o), respectively. This diagram also shows the potential appearing under the conditions z,(A) = -1, z,(B) = 1, A 4 is the total membrane potential; A#don(o,A), A&n(A,B), and A&n(B,d) are the Donnan potentials at the membrane interfaces; A&iff(A) and A#diff(B) are the diffusion potential difference in the charged layers.
Substitution of eq 1 into eq 2 gives
(3) Substitution of the solution of eq 3 with respect to Ko.,~) into eq 1 yields Ck(k). Hence, substitution of Cj(d) and Cj(0) into eq 3 yields Ck(A), the concentration of the ion at the left side surface of the layer A, and CY(B), that at the right side surface of the layer B, respectively. 2. At the initial stage, we assume that &A) = C ( A ) . 3. Ck(B) is also given by substituting CY(A) into eq 3. 4. Since layer k can be regarded as an homogeneously charged membrane, we calculate the flux of the ith ion in the layer, Jj(k), by using the ionic flux equation derived for a system consisting of the membrane and multivalent ions:36
where d is the membrane thickness; p = eXp(-FA#diff(k)/Z?T); A#diff(k) is the diffusion potential difference between the surfaces of layer k. /?is the solution of the following equation:
(B, 4- 4B2 4- 9B3)f 4- (2B14- 4B, 4- 9B3 - A,)p3 4-
+
+
(2B14- 4B2 - 2A, - 4A2)P2- (2A, 4A2 - B , 9A3)P (A, 4A, i- 9A3) = 0 (5)
+
where
4'
'@don(d,A)
+ '@diff(A) + '@don(A,B) + '@diff(B) + '@don(B,o)
If s
then CY(A) is set for the new value as
2
CF(A) = Cr(A)
+ Jj(B) - Jj(A)
(9)
where CY(A) is the previous value of the concentration, and steps 3-5 are repeated.
Experiment Samples. Aqueous solutions of a mixture of PVA [poly(vinyl alcohol), Wako Pure Chemical Industries Ltd., DP = 20001 and AP-2 [Kuraray Co. Ltd.] were cast for negativelycharged membranes, and those of PVA and PAAm [poly(allylamine), Nittobo Industries Inc.] were cast for positivelycharged membranes. The volume fraction of PVA to AP-2 or to PAAm was changed to control the charge densities. The AP-2 polymer contains sulfonic groups which provide negativelycharged sites. Bipolar membranes were made as follows. After swelling the charged PVA membranes in ion-exchanged water, we put one of the negatively-charged membranes on one of the positively-charged ones and then dried them at room temperature and annealed them at 160 "Cfor 20 min. This annealing process makes physical crosslinks of PVA polymers at the interface; thus, a monosheet bipolar membrane is produced without any binders. This bipolar membrane has a binderless junction of the two charged layers and a higher water content than ones made by binding ion-exchange membranes because PVA is a hydrophilic polymer. Consequently the assumptions in the theoretical calculation are met better by the use of these bipolar membranes. Thickness Measurement of the Charged Layers. The membranes were soaked in an acid dye solution (Suminal Fast Blue R, Sumitomo Chemical Co.) for 24 h. Since the dye colored only the positively-charged layer dark blue, the thickness of the charged regions can be measured with a microscope. The Definition of the Charge Densities of Bipolar Membranes. The charge densities of the two charged layers can be obtained from the membrane surface potential measured by the apparatus shown in a previous paper.39 The total membrane potential of a bipolar membrane A# is the sum of the five potentials as shown in Figure 1 and eq 8. When the two salt solutions in the dialysis system have the same concentration, A#, A#diff(A) and A#diff(B) are equal to zero. Thus, '@don(A,B)
and the superscripts, z+ and z- mean z-valent cation and anion, respectively. 5 . In the steady state, the ionic flux at the layer A, Ji(A), should be equal to that at the layer B, Ji(B). This condition is
(8)
- -('@don(d,A)
+ '@don(B,o))
(10)
In the case where the membrane surface at the left side contacts with the salt bridge, which contains concentrated KCl solution, instead of being in contact with the salt solution, A#diff(A) and A#diff(B) and A#diff(d,A) can be ignored; hence, the measured
J. Phys. Chem., Vol. 99, No. 14, 1995 5091
Transport of Ions across Bipolar Membranes potential is given by
50
40
Similarly, A~&(B,,) is also determined. The Total Membrane Potentials in System I and System 11. The total membrane potential was measured for two opposite systems, I and 11. In system I, the negatively-charged layer of a bipolar membrane faced the solution at the concentrated side (side d). In system 11, the membrane was turned over so that the negative layer faced the dilute side.
30
>
2
The total membrane potential at various KCI concentrations was simulated as a function of the ratio of the charge densities, r,, for system I, setting z,(x) and C,(x) as
10
0.0
L1
--- --_. \.
ai 8
Results and Discussion
20
--.---___ --- /-. ).*..
..................'... , ...... . ............_.-. . ,=:.--. ".=.7 n
-I
,
-10
(13) and for system I1 setting them as Co /moldm'3 50
where zC , , = -1 x 1.0 x mol dm-3. In this simulation r, is swept from -20 to unity, and the membrane where r, < 0 is regarded as a bipolar one. Figure 2 parts a and b shows the calculations of the total membrane potential of systems I and 11, respectively, vs the ionic concentration at the dilute side, C,. Where C,(A) C, and C,(B) >> C,. The simulations in both systems where r, = 1 show the same profile as that in the systems with negatively-charged membranes calculated from the Teorell-Meyer-Sievers (TMS) This fact supports the adequacy of our calculations because r, equal to unity indicates an homogeneous negatively-charged membrane. The potential in the plateau region at low C, decreases with decreasing r,. The value in the plateau region of system I decreases more steeply with decreasing r, than that of system II, because in system I the charge density of the layer contact with the dilute solution is swept. When both r, = -0.2 in system I and r, = -5 in system 11, the potentials are almost equal to zero irrespective of the concentration. This potential profile is apparently the same as that of a neutral membrane, since the Donnan potentials of the two membrane/solution interfaces have the same magnitude with the opposite sign under each of the above conditions. When r, = -1 the membrane has the same charge densities at the two charged layers and the potential in the two systems makes a symmetrical profile with respect to AC#J = 0. Figure 3 shows the experimental data of the surface potential lis the concentration of KC1 solution, C,. At a high value of C,,the surface potential of both surfaces of all the membranes became unity, which implies that assumption d is valid for the present system and that the surface potentials are approximately equal to the Donnan potentials. The higher charge densities the charged layers have, the more the potential-concentration curves shift to the high-concentration side. The solid curves are the potentials calculated by fitting the data to the Donnan
40
30
..-. .--,
-- 3 q ~ .4
,
-_.__ ..-,
10.3
*
lo'*
10 .IO
100
10' i
Co /moldm'3
Figure 2. The simulations of the total membrane potential vs the KCl concentration at the dilute side, C,, in the dialysis system by using the ratio of the charge densities in the two regions, r,, as a parameter: (a) system I where the negative layer faces the concentrated solution; (b) system I1 where the positive layer faces the concentrated solution.
equilibrium equation (eq 3). From this fitting, the charge density of the negatively-charged layer, C,-,and that of the positivelycharged layer, C,+, were obtained as listed in Table 1. The charge density increases with increasing volume fraction of the polyelectrolytes. Figure 4 shows the experimental data and the simulations of the total membrane potential. In all the bipolar membranes the potential profile in system I differs from that in system 11. In the dialysis system the transport properties of a bipolar membrane depend on the direction of the arrangement of the
5092 J. Phys. Chem., Vol. 99, No. 14, 1995
Higa and Kira
TABLE 1: Characteristics of the Bipolar Membranes: The Polyanion Content, Cpa; the Polycation Content, C,; the Water Content, H;the Charge Density of the Negative-Charged Layer and the Positively-Charged One, Cx+, Respectively; the Ratio of the Two Charged Lasers, rr: the Thickness of the Negative and the Positive Lavers, d- and d+. ResDectivelv samples
C,, C,
H
C,-
wt % (in dry membrane) wt % (in dry membrane) vol % x lo-* mol dm-3 I \
BIP-1
BIP-2
BIP-3
13.2 4.5 65 5.0
1.o 4.5 59
13.2 0.6 55 6.0
1
0.8 I
e BIP-1 i+i
30
0 BIP-1 1-1 A 0
BIP-2 I-' BIP-3 It BIP-3 1-1
- (a)
BIP- 1
I
BIP-2 20
> g
10
-10 -20 -30 -
*o
4
CJ /moldm-3
Figure 3. The surface potentials vs the KCl concentration: open symbols, the data at the negatively-charged layer; solid symbols, the data at the positively-charged layer. The solid curves are the calculation by fitting the data to eq 3. membrane charged layers to the driving force of ions. A similar phenomenon is observed in an electrodialysis system using the membrane as a rectifier. For the BIP-1 membrane where C,Cx+,the absolute value of the potential in system I is almost the same as that of system I1 but the sign of the potential is opposite. For the BIP-2 membrane the potential profile in system I is almost the same as that in a dialysis system with a positively-charged membrane, and the profile in system I1 is almost the same as that in the system with a neutral membrane. Since Cx+is larger than Cx- in the BIP-2 membrane, in system I the Donnan potential at the positive layeddilute solution interface is much larger than that at the negative layer/ concentrated solution, and in system 11 the potential at the negative layeddilute solution interface is almost the same in magnitude but opposite in sign to that at the positive layer/ concentrated solution one. The potential of the BIP-3membrane has positive values in both systems, because Cx- is much larger than Cx+ and the negative charges hardly affect the Donnan potential. The solid curves in Figure 4 represent simulations by using the charge densities obtained from the surface potential measurements, which agree quantitatively with the experimental data in all the membranes.
Conclusions A simulation method for ionic transport across a bipolar membrane in a dialysis system containing multivalent ions was
0
I
40
30
-
I I
I I
BIP-3
(C)
Co /moI.dm-2
Figure 4. The experimental data and the simulations of the total membrane potential. Solid circles, system I; open circles, system 11. (a) For BIP-1; (b) for BIP-2; (c) for BIP-3.
derived on the basis of the theory for ionic flux across an homogeneously charged membrane in the system. The total membrane potential in the dialysis system of KC1 solutions was simulated. The simulations show that the potential-concentration curve has different profiles, depending on the direction of the arrangement of the membrane charged layers to the concentration gradient in the system. The profile also depends on the ratio of the charge densities of the two charged layers. The experiments for the examination of the simulation have the following advantages: (1) the charge density, which is a crucial variable in the simulation, was determined from the surface potential at each surface of the bipolar membranes and
Transport of Ions across Bipolar Membranes
(2) the potential measurements were performed for the membranes which have an ideal junction of the two charged layers and also have little dependence of the transport property on their geometrical and chemical structures. Although in this study the proposed calculation method is not examined for the system containing multivalent ions, we believe that this method is probably valid for the system, because the calculations of the theory on which the proposed method is based agree quantitatively with measurements of both the potential and ionic permeability coefficient in mixed salt systems.@ In the dialysis system containing multivalent ions, we are measuring the dependence of the membrane potential and ionic permeability coefficient on the direction of the membrane charged layers to the concentration gradient of ions in the system. Acknowledgment. This work was supported in part by the system of the Basic Scienc~eProgram from the Agency of Science and Technology of Japan and was supported in part by Special Grant for Promotion of Research from The Institute of Physical and Chemical Research. References and Notes (1) (2) (3) (4) 498.
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