J. Chem. SOC., Faraday Trans. I, 1989, 85(11), 3609-3621 Membrane Potential and Ion Transport in Inhomogeneous Ion-exchange Membranes Akon Higuchi" and Tsutomu Nakagawa Department of Industrial Chemistry, Faculty of Engineering, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki, Kanagawa 214, Japan The Nernst-Planck equation is modified for the transport of uni-univalent ions in inhomogeneous membranes, which contain continuous gradients of partition and activity coefficients of the ions and/or fixed charge density. Standard chemical-potential gradients in the membranes are employed in the equation as a new force. Numerical solutions of membrane potential and ion fluxes for 10 model membranes are obtained from simulation at the condition of zero current. Various potential and concentration profiles are observed, which depend on the partition coefficients and/or the fixed charge density in the membranes.The membrane potential, which arises between two aqueous solutions of an electrolyte separated by a charged membrane, has been the subject of theoretical and experimental studies. Theoretical treatment of the membrane potential and ion transport in a homogeneously charged membrane was developed by Teorell,' Meyer and Severs2 (TMS theory) and other investigator~.~-~~ There are also some fundamental ~ t u d i e s l ~ - ~ ~ for inhomogeneously charged membranes. Gierke and Hsu's'~ proposed fixed charge of Nafion was described as a cluster-channel network, and inhomogeneities in the fixed charge density in the membrane gave rise to a potential within the membrane which impeded co-ion transport relative to that of counterions.Selvey and Reissl' considered fluctuations in the fixed charge density due to ion clustering and solved the Nernst-Planck and Poisson equations using perturbation theory. Sonin & Grossman" studied membrane potential and ion transport in layered ion-exchange membranes, although their primary interest is the current-voltage characteristics across the membranes. 21 observed an unusual asymmetric potential, between two identical electrolyte solutions separated by composite mem- branes, having different charge density, or by asymmetrically charged membranes. The asymmetric potential is explained by the existence of the water phase between the mem- branes, which has lower concentration of ions than the external ~ o l u t i o n s .~ ~ The potential is generated by the permeation of ions into the membranes at the steady state, but not in the equilibrium. Takagi and Nakagaki22 reported interesting results of the facilitated and reverse transport of ions in inhomogeneous membranes, asymmetric with respect to the partition coefficients of ions and/or the fixed charge density. They tried to verify their results from their theory. Their equation, however, leads to permanent ion fluxes even for a situation in which two external solutions separated by the asymmetric membrane are identical and without consideration of a water phase in the membrane. It is obvious that this violates the first law of thermodynamics. However, their results are a product of the nature of the asymmetric potentials and should be regarded as transient in character.Nevertheless, the study has prompted us to try to develop an exact theory of membrane potential and ion transport in the inhomogeneous membranes. Liquori and BOtre,lg and other 36093610 In homogeneous lon-exchange Membranes we developed the theory for a multilamellar series array of ion- exchange membranes having independent partition coefficients and charge densities. The goal of this study is to propose a modified Nernst-Planck equation for inhomogeneous membranes that have continuous gradients in the partition coefficients of ions and/or fixed charge density. In our previous Model Membranes considered in this study have continuous gradients of ionic partition and activity coefficients and/or of the fixed charge density along the x axis (taken to be perpendicular to membrane surfaces).The membrane is defined to occupy the region 0 6 x 6 1. On either side the membrane is in contact with aqueous electrolyte solutions having concentrations C,, at x = 0 and C, at x = 1. In order to emphasize the essential points of an ideal system, the following assumptions were made. (a) All charges are considered to be point charges and ionic dimensions are neglected. (b) Gradients of the partition coefficients, activity coefficients, the fixed charge density, diffusion coefficients, ion concentration and electric potential only exist along the x axis in the membrane. (c) Hydrostatic pressure gradients and volume movements of the fluid are regarded as negligible. ( d ) Electroneutrality is observed in all parts of the membrane (current is not generated in the system).( e ) Anion and cation fluxes are equal in any part of the membrane. (f) The total membrane potential Aq5 is given as the sum of the Donnan potential at the two interfaces between the membrane and the external solution, A#Don, and potential generated inside the membrane, (g) Equilibrium conditions are maintained at the surfaces between the external solution phase and the membrane surfaces. (h) The external solution is a uni-univalent salt solution. (i) Activity coefficients of ions in the external solution are unity. ( j ) The diffusion and partition coefficients of ions and the fixed charge density in the membrane are constant or dependent on ion concentration, Theory General Equation Assumptions (d) and (f) lead to A# = '#Don +'$in C-(x) = C+(x) + wC,(x) where C+(x) and C-(x) are the concentration of cation and anion at x in the membrane, cu has a value of + 1 or - 1 for a positively or negatively charged membrane and C,(x) is the fixed charge density at x in the membrane.Electrochemical potentials of cation and anion at x in the membrane, ~ + ~ ( x ) and pTm(x), are given by (3) (4) where pYrn(x) and ,uU",(x) are the standard chemical potentials of cation and anion at x in the membrane, y+ and y- are the activity coefficients of cation and anion at x in the membrane, qhrn(x) is the electric potential at x in the membrane, F is the Faraday constant, R is the gas constant and T is the absolute temperature.p+rn(x) = ~Yrn(x) + RTln Y+ C+(X) + f'$rn(x) P-~(X) = p?,(x) + RTln y- CJx) - Fq5,(x)A . Higuchi and T. Nakagawa 361 1 Donnan Potential When the membrane is assumed to be at equilibrium in the external solution with concentration C+soln and CPsoln, the electrochemical potential is given as ( 5 ) (6) Where pLo+soln and pZsoln are the standard chemical potentials of cation and anion, respectively, in the external solution, and Qsoln is the electric potential in the solution. We define (7) (8) po+soln + C+soln + F$soln = po+m(x) + RTln Y+ C+(x) + F # m POsoln + RTln C-soln-F#soln = p:m(X) + RTln y- C - ( X ) - F # ~ . pu",m(x> -p;soln = - RTln K+ pO,(x) -pOsoln = - RTln K - . Qm - #soln = ( - RT/F) In [Y+ C+(X)/K+ C+solnl = ( - RT/F) In [ K - C-soln/~- C-(x)].Then Qm - $soln is (9) (10) From this we obtain C + s o l n C-soln(K+ K-/Y+ Y-1 = C+(X> C-(X>* With K+(x) = K+/Y+ and K-(x) = rc-/y-, and C+soln CPsoln = C : or C:, we obtain Finally, by combining eqn (2), (1 1) and (1 2), the concentration at the surface is given as Flux Equation The fluxes of cations and anions, J+ and L, respectively, are given by J+ = - D+(x) C+(x) dpu,,(x)/L dx = - D+(x) C+(x) [RTd In y+ C+(x)/dx + Fd#,/dx + dpo+,(x)/dx]/l (1 6) J- = - D-(x) C-(x) [RTd In y- C-(x)/dx - Fd#,/dx + dpu",(x)/dx]/l (1 7) where D+(x) and D-(x) are the mobilities of the cations and anions at x in the membrane and L is membrane thickness. Since the fixed charge density and the partition coefficients of ions give different values at different places in the inhomogeneous membranes, the standard chemical potential will not be constant but will be a function of position x in the membrane.dp;,(x)/dx and dpO,(x)/dx cannot therefore, be regarded as zero for the imhomogeneous membranes, while eqn (1 6) and (1 7) reduce to the conventional Nernst-Planck equation for homogeneous membranes neglecting dp;,(x)/dx and dpu" (x) / dx.3612 Inhomogeneous Ion-exchange Membranes Eqn (22) and (23) are applicable in cases where the diffusion coefficients, the partition coefficients and the fixed charge density are the function of not only the position x, but the concentration from the derivation processes of the equations. The permeability coefficient, P, is finally calculated from P = J+ L/ABS(Co - Cl).(24) Note that at steady state P should not depend on x. Computational Method To obtain the membrane potential and the flux analytically we must integrate eqn (22) and (23), which is, at present, difficult. However, a numerical solution for the flux and the integration of eqn (22) can be achieved using a computer. If we know the flux, concentration profiles in the membrane can be built up stepwise, via eqn (23), from dC+(x)/dx = {J+ L / [ D + ( X ) RTI + [m2 - K:(x)/K+(x)l C+(x)>/(m, - 1). (25) Computational procedures are shown as follows. (1) Set the control data on the situation to be studied and define the membrane model [the definition of the functions of C,(x), K+(x), K-(x), D+(x) and D-(x), and read T, L, u, Co and C,]. (2) Set Ax which is small compared with unity (0 < Ax $ 1).(3) Calculate C+(O) and C+(l) from eqn (13). (4) Input J+ as an initial value, and set n = 1.A . Higuchi and T. Nakagawa 3613 Table 1. Model membranes for the calculation. D+(x) = 5.382 x lo-’ cm2 rnol s-’ J-l, D-(x) = 8.201 x lo-’ cm2 mol s-l J-l, T=298Kandc;o=-l membranes C,(x)/equiv. dm-3 K(x) model I a model I b model IIa model I1 b model IIIa model IIIb model IV model V model VIa model VIb 0.0 1 0.0 1 0.0 199 - 0.0198~ 0.0001 +0.0198~ 0.0 1 0.0 1 0.005 + 0.03~-0.03~* 0.015-0.03~+0.03~~ 0.002+0.016x 0.018-0.016~ 1 0.5 1 1 1.8-1.6~ 0.2 + 1 . 6 ~ 0.25 + 1 . 5 ~ + 1 .5x2 0.1 +0.8x 0.75 - 1 . 5 ~ + 1 .5x2 0.9 - 0 . 8 ~ ( 5 ) Calculate dC+[(n - 1) Ax]/dx from eqn (25). (6) Calculate C+(nAx) from eqn (26): C+(nAx) = C+[(n - 1) Ax] + AxdC+[(n - 1) Ax]/dx.(26) (7) Set n = n+ 1 and repeat ( 5 ) and (6) until n = l/Ax. (8) If C+(nAx) at n = l/Ax > C+(l), new 4. is generated under the condition that (9) If C+(nAx) at n = l / A x < C+(l), new J+ is generated under the condition that (10) If C+(nAx) at n = l/Ax is approximately equal to C+(l), J+ and C+(x) are (1 1) Integrate eqn (22) from Simpson’s equation and calculate Aq5in(A$in = Ji Qm dx). (12) Calculate ADon and A$ from eqn (1) and (15). (13) Print out all results. J+(new) = J+(old)a (1 < a at C, > C,, 0 < a < 1 at C, < C,). Go to ( 5 ) with n = 1. J+(new) = J+(old)b(O < b < 1 at Co > C,, 1 < b at C, < C,). Go to ( 5 ) with n = 1. obtained from the above procedures. Go to (1 1). Results and Discussion Model Membranes The method proposed above was used to calculate the membrane potential and the flux for some model membranes.In order to emphasize the essential points, the calculations performed in this study were confined to the following conditions : co = - 1, T = 298 K, dD+(x)/dx = 0, dD-(x)/dx = 0, K+(x) = K-(x), Ji K+(x) dx = 0.5 or 1.0, j’: C,(x) dx = 0.01 equiv. dm-3 and Ax = 0.001. The values of D+(x) and D-(x) for sodium ion and chloride ion in bulk water, were chosen to be 5.382 and 8.201 x cm2 mol s-’ J - I . C,(x) and K+(x) for the 10 models addressed in this study are summarized in table 1 . Calculations were performed using a 16-bit personal computer (PC-980lVX, NEC Corp.) with N88 B~sIc(86) language on MS-DOS ver. 3.10 (Microsoft Corp.).Membrane Potential Fig. 1 shows membrane potentials of models Ia, IIa and IIb. Model I a represents a homogeneous cation-exchange membrane with C,(x) = 0.01 equiv. dm-3 and K(x) = 1. Models I1 a and I1 b are inhomogeneous membranes having asymmetric charge density, and model IIa is identical to model IIb with values of Co and C, interchanged. The three models give different membrane potentials at C, > low3 mol dm-3, although the models have the same values of Ji C,(x) dx and JiK(x) dx.3614 120 80 40 -40 In homogeneous Ion-exchange Membranes -80 -120 . 1 L 1 o - ~ 1 o - ~ lo-* lo-' 1 C, /mol dm-3 Fig. 1. Membrane potentials calculated for models I (-), IIa (----), IIb (--) and TMS fitting curve (..-....... ) with K(x) = 1, C,(x) = 0.0307 equiv. dmP3 and D+/D- = 0.0325.C , = mol dm-3. We can judge whether a given membrane is inhomogeneous by observing two separate membrane potentials under the same conditions; for example, if Co is varied from lop3 to 1 mol dm-3 with C, = mol dm-3 and likewise C, is varied with C, = 10-3 mol dm-3, TMS theory predicts the two potentials to be identical. Model IIb is also concluded to be the inhomogeneous membrane from the fact that the membrane potential observed in model I1 b can not be explained by TMS theory. The best fit of the membrane potential from eqn (1) of ref. (13) (TMS theory) using a non- linear least-squares method gives a negative value of the fixed charge density ( - 7.80 x 10-3eq/l). The fixed charge density and ion-mobility ratio (D+/D-) can also be estimated in the TMS theory from maximum membrane potential and C, at A+ex (C0-e,).13 Membrane potential calculated by this procedure with A#ex = 28.84 mV, CO-ex = 5.725 x mol dm-3, C,(x) = 0.0307 equiv.dm-3 and D+/D- = 0.0325 is also shown in fig. 1. The figure suggests that TMS theory explains the membrane potential at Co < 0.03 mol dmP3, but the discrepancy of the membrane potential between TMS theory and model I1 b gradually increases with the increase of C, at Co 2 0.03 mol dm-3. This is due to an inflection point at C, ca. 0.1 mol dmF3 observed in the membrane potential of model IIb. It is known that the membrane potential for a bipolar membrane, which consists of juxtaposed cation and anion exchange membranes, also shows the inflection point in some cases.23 Since the bipolar membranes can also be regarded as the inhomogeneous membrane, the inflection point found in the membrane potential should be a characteristic aspect for the inhomogeneous membrane. Fig.2 shows the membrane potentials of models IIIb, IV, V, VIa and VIb. The membrane potential of model IIIa, which is not shown in the figure, is estimated to be ca. identical to that of model V at Co < 0.5 mol dm-3. Models IIIa and I11 b are the inhomogeneous membranes with respect to the partition coefficients and not to the charge density, and model I11 a is identical to model I11 b with the values of C, and C, interchanged. Models IV or V are symmetrical for the charge density and the partition coefficients about the line of x = 0.5 in the membranes. Models VIa and VIb are constructed on the conditions that K(x) = H(x) and 1 -H(x) is linearly proportional toA .Higuchi and T. Nakagawa 361 5 120 80 40 $ 0 \ a d - 4 0 -80 -120 /----- -/*- .A- /----- . I 1 lo-& 1 o - ~ lo-* lo-’ 1 c,, /mol dni3 Fig. 2. Membrane potentials calculated for models 111 b (-), IV (----), V (---), VIa (-a*-) and VIb (.......... ). C, = mol dmV3. CJx) where H(x) is water content at x in the membrane. The model VIa is identical to model VIb with the values of Co and C, interchanged. It is found in fig. 2 that the membrane potentials of models IIIa and VIa also show different values estimated from the potentials of models IIIb and VIb, although membrane properties between models IIIa and IIIb, or models VIa and VIb are identical except in the direction of ion fluxes.Models I a, I1 a, I1 b, I11 a and I11 b have the properties of Ji K(x) dx = 1 and Ji CJx) dx = 0.01 equiv. dm-3, and models I b, IV, V, VI a and VI b have the conditions of JiK(x) dx = 0.5 and Ji CJx) dx = 0.01 equiv. dmP3. The models, however, give different potentials according to the membrane model. It is found that the macroscopic properties of the membrane such as Ji K(x) dx and Ji C,(x) dx cannot be estimated solely from the results of the inhomogeneous membrane potential, whereas the properties can be determined if the membrane is homogeneous. This is because the profiles of the partition coefficients and charge density in the membrane significantly influence the membrane potentials, as found in fig. 1 and 2 . Permeability Difference Another index which characterizes the given membrane as an inhomogeneous one is the variation of the ion flux with direction.Fig. 3 shows the ratios of Pb to Pa for models 11, I11 and VI where Pa and Pb are the permeability coefficients of models Y, and yb ( Y = 11, I11 or VI). Since models IV and V are identical owing to the symmetry about the line of x = 0.5, permeability differences with direction are not observed although models IV and V describe inhomogeneous membranes. Maxima were observed in the plots of Pb/Pa against C, for models 11, I11 and VI. p b / p a for model VI was found to be higher than that for models I1 and III. Similar maxima were reported for the bipolar membranes in a previous It was suggested that the permeability difference due to the direction of flux is caused by the diffusion potential difference resulting from the flux direction in the bipolar rnembrane~.~~ The permeability difference was, however, observed even in3616 4 0 20 Inhomogeneous Ion-exchange Membranes .I 1.4 1.3 s 1.2 1.1 1.0 CL" 10-3 1 o-2 10" 1 C , /mol dm-3 Fig. 3. Permeability ratios of Pb to Pa calculated for models I1 (-), I11 (----) and VI (---). C, = mol dm-3. -0 3 -20 -40 -60 > 3 a d h - 8 0 t -100 0 0.2 0.4 0.6 0.8 1.0 Fig. 4. Membrane potential profiles calculated for models I a (-), I1 a (----), I1 b (---), I11 a (_.._) and IIIb (.......... ). the case of D+(x) = D-(x) = 5.382 x lo-' cm2 mol J-l s-' for model VI (e.g. Pa = 2.546 x cm2 s-l and Pb = 3.535 x lo-* cm2 s-l on the conditions of Co = 0.06 mol dm-3 and C, = 0.001 mol dm-3).It is found that the potential generated inside the membrane such as the standard chemical potential gradient also plays an important role for the permeability difference in the inhomogeneous membranes.A . Higuchi and T, Nakagawa 3617 60 40 20 - 0. 3 -20 - 4 0 -60 > h v Y a -a -80 -100 0' \ 1 I I . 0 0.2 0.4 0.6 0.8 1.0 Fig. 5. Membrane potential profiles calculated for models I b (-), IV (----), V (--.-), VIa (-. .-) and VI b (. . . . . . . . . . ). C, = 0.01 rnol dm-3 and C, = mol dm-3. Table 2. Patterns of C,(x), K(x), C+(x), CJx) and A$(x) in model membranesa membranes C,(x) K(x) C+jx) CJx) A$(x) model I a model Ib model IIa model IIb model IIIa model IIIb model IV model V model VIa move1 VIb I L I 7 L 7 u n I f a -+, Constant; 7, increase from Co side to C, side; I , decrease from C, side to C, side; n, convex upward from C, side to C, side; u, convex downward from C,, side to C, side.Potential Profiles Membrane potential profiles for 10 model membranes are calculated by integrating eqn (22) over distance, x, in the membranes. The results at C, = 0.01 mol dm-3 and C, = 0.001 mol dm-3 are depicted in fig. 4 and 5. The potentials of models IV and V show convex profiles in the membrane where the profiles of the partition coefficients and the charge density also show convex function in the membrane (see table l), and are completely different profiles from those estimated from TMS theory.3618 Inhomogeneous Ion-exchange Membranes 2.8 2.4 0.8 0.4 0 0 0.2 0.4 0.6 0.8 1.0 Fig. 6.Concentration profiles of cation calculated for models Ia (-), I1 a (----), I1 b (-*-), IIIa (--..-) and IIIb (..-...-... ). C, = 0.01 mol dm-3 and C, = mol dm-3. 0 0.2 0.4 0.6 0.8 1.0 Fig. 7. Concentration profiles of cation calculated for models I b (-), 1V (----), V (-.--), VI a (_.._) and VIb (.......... ). C,, = 0.01 mol dm-3 and C, = mol dm-3.A . Higuchi and T. Nakagawa 3619 1.4 1.2 CO m -1.0 'E 0.8 k 2 0.6 U - \ I L, 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1.0 Fig. 8. Concentration profiles of anion calculated for models I a (-), I1 a (----), I1 b (-.--), X I11 a (-a*-) and I11 b (-.......-. ). C,, = 0.01 mol dm-3 and C, = mol drnp3. The inhomogeneous membranes in this study suggest that the membrane potential increases with decreasing fixed charge density or with increasing partition coefficients along the distance, x, and the potential decreases with increasing fixed charge density or with decreasing partition coefficient along the x axis (see table 2).The potential profiles are, therefore, directly related to the functions of the fixed charge density and the partition coefficients, and may be valuable properties to estimate the functions of CJx) and K(x) if the potential profiles can be directly obtained from the experiments. Concentration Profiles Concentration profiles of cation and anion for 10 model membranes are calculated by eqn (2) and (26). The concentration profiles at Co = 0.01 mol dmP3 and C, = 0.001 mol dm-3 are shown in fig. 6-9. Several patterns of concentration profiles are observed in the figures.Convex profiles were observed for C+(x) in models IIIb, IV and V, or C-(x) in models I11 b and V. It was also found that the concentration in the membrane increases from x = 0 to x = I for C+(x) in models I1 a and VIa, or C-(x) in model VI b, although these profiles are opposite to the flux direction. Table 2 summarizes the patterns of C,(x), K(x), C+(x), C-(x) and A$(x) in the membranes. There seems to be some relationship between C+(x) and the function of the fixed charge density in the membranes, while any relationship between CJx) and Cz(x) or K(x) is not observed in table 2. Evidently, the relationship between C+(x) and Cz(x) should be observed if the fixed charge density is in a high-concentration region, or C, and C, are in low- concentration regions. This is because the counter-ion concentration, C+(x) in this case, is mainly determined by C,(x) from eqn (2). The flow direction for the 10 membranes of this study having various concentration profiles was observed to be from high concentration to low concentration, although some concentration profiles were against the flow direction.3620 Inhomogeneous Ion-exchange Membranes 1.4 1.2 CO -1.0 'E ,.,' 0.8 k 0.6 d 0.4 0.2 0 U M \ h 0 0.2 0.4 0.6 0.8 1.0 Fig.9. Concentration profiles of anion calculated for models I b (-), IV (----), V (----), VI a (_.._) and VIb (.......... ). C, = 0.01 mol dm-3 and C, = mol dm-3. Comments on Reverse Transport Takagi and Nakagaki22 studied theoretically and found experimentally ' the reverse transport ' of NaCl through inhomogeneous membranes such as NaOH- treated collodion-collodion, where this reverse transport refers to solutes permeating in an opposite direction to the concentration gradient between the two external solutions of the membranes.Their experimental results can be related to the phenomena of the asymmetric p~tentiall'-~l and should be regarded as of temporary nature, since for such reverse transport not to require any energy is obviously against the first law of thermodynamics. One reason why their theory led to such a misleading result is that they applied the conventional Nernst-Planck equation to the inhomogeneous membranes, and the standard chemical potential gradient in the membranes, which is considered in this study, was not considered in their theory. They also demonstrated reverse transport even for a membrane having K(x) = constant and asymmetric charge distribution.In this case the modified Nernst-Planck equation reduces to the conventional Nernst- Planck equation because of dK(x)/dx = 0. Therefore, in this case, we cannot attribute the reverse transport solely to a disregard of the standard chemical potential gradient. There was also the assumption presented below according to in order to solve flux equation analytically, uiz. C+(x) = c1 - w1 C+(O) + w C+( 1) C-(x) = [ 1 - 6(x)] C ( 0 ) + 6(x) c-( 1) (27) (28) where d(x) is a continuous function of x that does not depend on the concentration, and is the same for C+(x) and C-(x). By combining eqn (27) and (28), we obtainA . Higuc hi and T. Nakaga w a 362 1 On the other hand, the differential of eqn (2) is represented by eqn (30).dC+(x)/dx = dC-(x)/dx - dC,(x)/dx. (30) There is no gradient of fixed charge density in eqn (29), whereas there is such a gradient in eqn (30). Eqn (29) and (30) reduce to the same equation in the special case of dC,(x)/dx = 0 : dC+(x)/dx = dC-(x)/dx. (31) In our study, the membrane potential and ion fluxes were calculated by the simulation method, and eqn (27) and (28) were not used. It is concluded that eqn (27) and (28), the validity of which is not clear, lead to the misleading equations in their theory. Although we found that Takagi and Nakagaki22 treated their equation inadequately for the transport of ions in the inhomogeneous membranes, it should be noted that it was their studies which stimulated and prompted us to develop the present investigation. References 1 T. Teorell, Proc. SOC. Exptl. Biol. Med., 1935, 33, 282. 2 K. H. Meyer and J. F. Sievers, Hehi. Chim. Acta, 1936, 19, 649. 3 K. S. Spiegler, J. Electrochem. Soc., 1953, 100, 303. 4 R. Schlogl, Z . Elektrochem., 1952, 56, 644. 5 G. Schmid, Z. Elektrochem., 1950, 54, 424; G. Schmid, Z. Phys. Chem., N.F., 1954, 1, 305. 6 M. Nagasawa and 1. Kagawa, Discuss. 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