Salt permeability maxima in charged poly (methyl methacrylate) stereocomplex membranes

Salt permeability maxima in charged poly (methyl methacrylate) stereocomplex membranes

Journal of Membrane Science, 32 (198’7) 69-82 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands 69 SALT PERMEABILITY MAXIMA I...

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Journal of Membrane Science, 32 (198’7) 69-82 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

69

SALT PERMEABILITY MAXIMA IN CHARGED POLY (METHYL METHACRYLATE) STEREOCOMPLEX MEMBRANES TADASHI NAKANISHI, MASAO KUNIOKA, TAKAO DAI, MITSURU SATOH and JIRO KOMIYAMA Department of Polymer Science, Tokyo Institute of Technology, Meguru- ku Ookayamu, Tokyo 152 (Japan) (Received August 18,1986; accepted in revised form December l&l986

)

Summary The permeabilities of NaBr and four tetraalkylammonium bromides ( TAABr) were measured for negatively charged (anionic), positively charged (cationic) and neutral poly (methyl methacrylate) (PMMA) stereocomplex membranes in aqueous systems. For anionic membranes, the dependence of salt permeability on the upstream salt concentration varies according to the cation size; e.g., the permeability of NaBr increases to approach a leveled-off value with the increase of the upstream salt concentration, whereas the permeability of tetra-n-butylammonium bromide shows a maximum. The analysis of these results in terms of the Teorell Meyer-Sievers theory reveals that the permeability maximum can be attributed to imbalanced hindrances to the diffu sion of the large and small pair ions within these membranes.

Introduction The object of this study was to obtain information about salt transport through charged and uncharged PMMA membranes. Toray Industries, Inc, has been producing water-swollen PMMA hollow fiber membranes by utilizing complex formation between isotactic and syndiotactic PMMAs for hemodialysis membranes. These stereocomplex membranes possess stoichiometric and thermoreversible crosslinking between the isotactic and syndiotactic polymer components. The membrane permeabilities to water and solutes are easily controlled by the ratio of isotactic to syndiotactic polymers in the membrane preparation solution. Preparation in sheet form was first reported by Sakai and Tanzawa [ 11. Kobayashi et al. [ 21 discussed the pore structure of the PMMA hollow fiber membrane by measuring the water permeability and the differential scanning calorimetric behavior. Kunitomo et al. [ 3,4] proposed a design for an anionic PMMA membrane for rapid hemodialysis under regulated acid-base balance of dialyzed blood through the permselectivity of the charged membrane [ 51. In this study, the permeability coefficients of NaBr and four tetraalkylamQ316-7386/87/$03.50

0 1987 Elsevier Science Publishers B.V

70

monium

bromides

( TAABr ) , namely,

tetramethylammonium

bromide

(Me,NBr ) , tetraetliylammonium bromide (Et,NBr 1, tetra-n-propylammonium bromide (Pr,NBr ) and tetra-n-butylammonium bromide (Bu,NBr ) were measured for anionic, cationic and neutral PMMA membranes over a salt concentration range of O-O.2 mol-dm-3. The results were quantitatively discussed on the basis of the Teorell-Meyer-Sievers theory ( TMS theory) [ 6-81. This theory takes into account the Donnan exclusion of salt at the solution-membrane interfaces, the mobilities of the component ions of salt and the diffusion potential generated in the membrane by the difference in the mobilities. The salts studied here are composed of an anion and cations of increasingly large size. This variation must affect the three factors mentioned above, hence the permeabilities should show some specific features according to the cation size. Experimental

Materials The membranes used in this study were kindly supplied by Basic Research Laboratories, Toray Industries, Inc. They are classified into the following three groups [9]: (1) MS membranes: uniform and symmetric stereocomplex membranes composed of isotactic PMMA and syndiotactic PMMA-based copolymer containing a few mol% sodium p-styrenesulfonate. The content of charged groups was ca. 0.3 eq/kg dry membrane [ 91. ( 2) MQ membrane: a uniform and symmetric stereocomplex membrane composed of the same base substrate as MS membranes. The syndiotactic part of the stereocomplex contains a few mol% 2-methacryloyloxyethyltriethylammonium chloride as quaternary ammonium groups. The content of charged groups was ca. 0.15 eq/kg dry membrane [ 91. (3) DIA membrane: a uniform and symmetric PMMA stereocomplex membrane. This membrane is uncharged. All three types of membranes were prepared for hemodialysis experiments with the aim of blocking the permeation of albumin and other large solutes. MS membranes were supplied with different water contents, 0.72 and 0.61 in weight fraction. The thickness of the water-swollen membranes was measured by means of an electric micrometer (Tokyo Seimitsu Co.). The water content of the membranes was determined by the weight difference between the water-swollen and the dry membranes. Each membrane sample equilibrated with distilled water was blotted with filter paper and weighed. Then, the sample was dried under reduced pressure at 120 ? 5’ C until constant weight was reached, and weighed again. These characteristics of the membranes are shown in Table 1. The num-

71 TABLE 1 Characteristics of membranes Membrane MS-12 MS-61 MQ-65 DIA

(pm)

Water content” (in wt. fraction)

122 283 125 120

0.72 0.61 0.65 0.68

Thickness”, d

“For water-swollen membranes.

bers in the membrane codes indicate the percentage water contents of the membranes. NaBr was purchased from Kanto Chemical Co., Inc. Me,NBr, Et,NBr, Pr,NBr and Bu,NBr were purchased from Tokyo Chemical Industry Co., Ltd. All the salts were of extra pure grade, and were used without further purification Measurements of permeability coefficients The coefficient was measured at 25 +-0.05”C with a two-compartment glass cell shown in Fig. 1. The effective area of the membrane was 7.09 cm’. The upstream and the downstream compartments of the cell were filled with 250 cm3 salt solution and 50 cm3 distilled water, respectively. The solutions in both compartments were stirred effectively by magnetic stirring bars in order to

Fig. 1. Permeation cell: a, magnetic stirring bar; b, clamp; c, conductivity cell; d, membrane,

72

minimize stagnant layers on the surfaces of the membrane. Permeabilities of the salts through the membranes were determined by measuring the conductivity of the downstream solutions. The conductivities of the downstream solutions were recorded against time. The conductivity changed linearly with time within 3 min from the start of each test. Permeability coefficients, P t cm2set-l), were calculated from the slopes of the linear portions of the recorded curves using the following equation: P=J*d/(

c, -C,)

(1)

where J, ( mol-cm-2-sec-1) is the salt flux and d (cm ) is the membrane thickness. C, and Cd ( mol-cm-3) are the salt concentrations of the upstream and the downstream solutions, respectively. At the end of each test, Cd did never exceed 3% of C,,. Therefore, Cd may be taken as zero m the calculation of the permeability coefficient. Sorption isotherms Sorption isotherms were determined at 25 + 0.05 ’ C by the desorption method. After 0.15-0.25 g of the water-swollen membranes had been soaked in salt solutions of various concentrations for 24 hr, the membranes were blotted quickly with filter paper and soaked in 50 cm3 of distilled water for 24 hrs The amounts of salt desorbed were determined by conductometry. Results

Salt permeabilities Figures 2 and 3 show the plots of the permeability coefficients, P, of NaBr and TAABr against the upstream salt concentration, C,, for anionic MS-72 and MS-61 membranes, respectively. With the MS-72 membrane, the permeability coefficients of NaBr and Me4NBr increase monotonically to approach leveled-off values with the increase of the upstream salt concentration. Each of the permeability curves of the Pr,NBr and Bu,NBr exhibits a maximum at about C,,=0.02 mol-dm-3 and a decrease to a leveled-off value in the range over this concentration. The permeability coefficients at the highest upstream salt concentration decrease with increasing size of the cations. With the MS61 membrane, the patterns of the concentration dependences of the permeability coefficients are similar to those for MS-72 membrane; a permeability maximum was also observed at C, = 0.03 mol-dm-” for Bu,NBr. Figure 4 shows the concentration dependence of the permeability coefficients of NaBr and TAABr for the cationic MQ-65 membrane. The permeability coefficients of all salts increase monotonically to approach leveled-off values with the increase of the upstream salt concentration. No permeability maxima were observed in this case. Figure 5 shows the permeability coefficients of NaBr and TAABr for the

73

1

0.1 Co /mol.dmm3

1

0.2

Fig. 2. Salt permeabilities of MS-72 membrane plotted against upstream salt concentration: 0, N&r; 0, Me,NBr; A, Et,NBr; 0, PqNBr; V, Bu,NBr.

Co /molsdm-3 Fig. 3. Salt permeabilities of MS-61 membrane plotted against upstream salt concentration: a, NaBr; 0, Me,NBr; A, Et,NBr; 0, Pr,NBr; V , Bu,NBr.

neutral DIA membrane. The permeability coefficients of each salt are almost independent of the upstream salt concentration. The permeabilities of TAABr decrease with increasing alkyl chain length of the quaternary ammonium ions. DIA membrane gave relatively high permeabilities for ail salts, e.g., the perme-

74 5a

A

0 V

I

I

0.1

0.2

Co /mol.dm-3 Fig. 4. Salt permeabilities of MB-65 membrane plotted against upstream salt concentration NaBr; 0, Me,NBr; A, Et,NBr; 0, Pr,NBr; V, Bu,NBr.

000 4-

00

0

0

0.

l,

0

0

l

A : AA (?3-

AA

2 b 7 n 2 _oao~o vvvvv

A

A

0 0

V V

1 -

0'

0

1

1

0.1

0.2

Co/mol.dm-3 Fig. 5. Salt permeabilities of DIA membrane plotted against upstream salt concentration: 0, NaBr; 0, Me,NBr; A, Et,NBr; 0, Pr,NBr; V, Bu,NBr.

75

0.3 V

0 s

OO

I 0.1 C /mol.dme3

0.2

Fig. 6. Sorption isotherms for MS-72 membrane: 0, NaBr; 0, Me,NBr; A. Et,NBr; Cl, Pr,NBr; V , Bu,NBr.

of NaBr in DIA membrane was found to be about 27% of the corresponding value for a layer of water. ability

coefficient

Sorption isotherms Figure 6 shows as an example the sorption isotherms of NaBr and TAABr for the MS-72 membrane. The sorption isotherms of these salts were found to be of partition type for all membranes. The partition coefficients, K( = C/C, where C is the internal salt concentration based on the volume of water, and C is the external salt concentration), obtained from the slopes of the isotherms, are listed in Table 2. The coefficients of the salts for MS-72, MQ-65 and DIA membranes are 0.8~1.1, except for Bu,NBr. This suggests that the sorbed salts are dissolved mainly into the water in the membranes. The coefficients for TABLE 2 Partition coefficients” Membrane

NaBr

Me,NBr

Et,NBr

Pr,NBr

Bu,NBr

MS-72 MS-61 MB-65 DIA

0.90 0.69 0.86 0.91

0.94 0.62 0.80 0.97

0.94 0.63 1.04 0.99

0.97 0.83 1.15 1.14

1.23 1.44 1.68 1.56

“Calculated by dividing solute concentration in the membrane (mol soluteivolume of water in membrane) by the concentration in external solution.

76

Bu,NBr are larger than those for the other salts by a factor of about 1.5. This salt has affinity to the membrane substrate [lo]. Discussion

In the preceding section, various types of the concentration dependences of the salt permeabilities were shown. The permeability coefficients for anionic MS and cationic MQ membranes are strongly dependent on the upstream salt concentration, while the coefficients for the neutral DIA membrane are independent of the upstream salt concentration. Usually, in the low salt concentration range not exceeding the charge density of the membranes, salt permeability of charged membranes increases monotonically with the increase of the external salt concentration because the electrostatic salt exclusion from the membranes is relaxed at high salt concentration. This is indeed observed for anionic MS-72 and MS-61 membranes with NaBr and Me,NBr. However, the permeability curves of Pr,NBr and Bu,NBr gave rather unexpected maxima. This suggests that the diffusivities of the salts composed of large cations and small anions are affected by these membranes in some unique manner. In this study, the experimental permeabilities are analyzed on the basis of the TMS theory, from which the mobilities of the ions in the membranes are evaluated. The TMS, fixed charge model applied to synthetic charged membranes explains well the permselectivity of salt. This theory is based on the following assumptions [ 111: (a) The membrane behaves as a strongly ionized exchanger with fixed charges. (b) Donnan equilibria are instantaneously established at the interfaces between the external solutions and the membrane. (c) Flux of each ion through the membrane is subject to the Nernst-Planck equation. Under conditions of electroneutrality and zero electrical current, one may integrate the Nernst-Planck equation to obtain the salt flux expression in terms of the interfacial salt concentrations in the membrane. These concentrations are related to the external concentrations by the Donnan equilibria, so the salt flux can be expressed as a function of the external salt concentrations. The permeability coefficient, P, of a uni-univalent salt is given as

o+o_KRT P=(w++w-H50-rd)

C

(1+4c3’~-

(l+4gP2 _ZWln

(1+4~3”2+zw (1+453”2+zw

1

(2) x ’

whereW=(w+-cc,_)/(cc,++o_),r=CK/X,andZ=lforanionic,Z=-l for cationic membranes. ws are the molar mobilities, X is the concentration of fixed charge groups and subscripts + and - refer to cation and anion, respectively. R is the gas constant and T is the absolute temperature.

Using the salt partition coefficients, K, determined from the sorption isotherms, we fitted eqn. ( 2) to the experimental permeability data and evaluated the parameters o +, co_ and X. In the fittings, the tortuosity factor of the membrane was assumed to be identical for both cations and anions as w+=Bc0T w_

(3)

=e co”

(4)

where 8 is the tortuosity factor, OS are the apparent molar mobilities of the ions, and w “s are the molar mobilities based on the actual diffusion path. We first employed the molar mobilities in water as w ’ + and ok’ _ in eqns. ( 3) and (4)) and then carried out curve fittings of eqn. ( 2 1 to the experimental data for the permeabilities of charged MS-72 MS-61 and MQ-65 membranes by adjusting the two parameters, 8 and X. For anionic MS-72 and MS-61 membranes, the concentration dependences of the permeabilities of NaBr, MelNBr and Et,NBr were accounted for by the theory. However, the permeability maxima observed for Pr*NBr and Bu,NBr could not be reproduced. These facts imply that the true mobilities, w a + and w ’ _-) of the ions within the membranes differ from the corresponding mobilities in water. With the downstream salt concentration, Cd, zero, eqn. (2) may be rewritten for anionic membranes as P=

(2”;;yjT +

(1+4cz)1/2--1-W

2 1/2+W ln(1+4:2W

0 [

I

(5)

Differentiation of eqn. ( 5 ) gives

ap/ag, =

o+c!_KRT (w+ +0-K:

(l[

w)[(1+4t;)1’2-l] (1+45:)l’2+

w

i ‘I2 + W -~ l+W 1 (6)

+ w ln (I+. 4lK

When Wz 0 (i.e., the mobility of the cation is larger than or equal to that of the anion), dP/d<, is positive, that is, the theoretical permeability increases with the increase of the upstream salt concentration. When W < 0, one maximum appears in the permeability curve. For selected values of W, Table 3 shows the reduced salt concentrations, &,, giving the permeability maxima and the permeability coefficients divided by the hypothetical values of the permeability coefficients at Co= co. As the value of W approaches - 1, the maximum permeability coefficient becomes progressively larger and the value of C, giving the maximum approaches to zero. The distinct permeability maxima observed for anionic MS-72 and MS-61 membranes with Bu,NBr cannot be reproduced with W= - 0.60 estimated from the mobilities of the component ions in water.

78 TABLE 3 Salt peme&fiity

P,,J

W” -0.30 -0.50 -0.70

-0.90 -0.95 - 0.97 -0.99

predicted by the TMS theory

maxima

25.3 4.42

1.55 0.539 0.344 0.256 0.143

for aniomcmembranes

lim P(C”jC G-m

1.006 1.05

1.21 1.89 2.61 3.33 5.72

“W=(w+-w_)/(W++W_).Foro+andw_,seetext. “A’zG% G beingthe external concentration giving the permeability maximum. For K and X, “The maximum permeability coefficient divided by the coefficient at C, = co.

It was, therefore, decided to include the molar mobilities of the tetraalkylammonium ions as adjustable parameters in the curve fittings. For the permeability curves having maxima in anionic MS-72 and MS-61 membranes, the fittings of eqn. (2) were carried out again by adjusting the three parameters, o” +, 0 and X, employing the w ’ _ values in water. The parameters obtained by these curve fittings are shown in Table 4. The theoretical permeability curves obtained with these parameters are shown as TABLE 4 Parameters for the interpretation of the permeability coefficients for MS-72 and MS-61 membranes

MembraneSolute

w ’ +e

Ob

X’ (mol-drn3)

Wd

(lop9 cm’-mol-JJ’-set-‘) MS-72

MS-61

NaBr Me4NBr Et,NBr Pr,NBr Bu,NBr

5.4 4.8 1.5 0.68 0.35

(5.4)” (4.8) (3.5) (2.5) (2.1)

0.28 0.28 0.47 0.53 0.55

0.03 0.03 0.05 0.04 0.03

-0.22 -0.27 -0.70 -0.85 -0.92

(-0.54) C- 0.60)

NaBr MelNBr Et.,NBr

5.4 4.8 3.5

(4.8) (5.4) (3.5)

0.35 0.30 0.30

0.07 0.07 0.055

-0.22 -0.27 -0.41

(-0.22) (-0.27) ( -0.41)

Pr,NBr 1.8 (2.5) BuJVBr 0.44 (2.1)

0.24 0.05 0.37 0.08

“Mobilityof cationbasedon the actual diffusion path. ?Tortuosity. ‘Concentration of fixed charge group. dW=(~+--W_)/(~++w_).Forw+

andw_,seetext.

“Thevaluesin parentheses represent thecorresponding values in free solution.

(-0.22)’

( - 0.27) ( -0.41)

- 0.64 ( - 0.54) -0.90 ( -0.60)

79 TABLE 5 Parameters for the interpretation of the permeability coefficients for the MQ-65 membrane” Solute

e

WD+

(1O-9 cm*-mol-J-‘-set-‘)

NaBr Me,NBr Et,NBr Pr,NBr Bu,NBr

5.4 4.8 2.2 1.0 0.54

(5.4) (4.8) (3.5) (2.5) (2.1)

0.32 0.36 0.36 0.36 0.36

X (mol-dn0.011 0.011 0.011 0.011 0.011

W ‘)

-

-0.22 -0.27 -0.59 -0.78 -0.88

(-0.22) (-0.27) (-0.41) ( -0.54) (-0.60)

“See footnotes for Table 4 for the symbols.

solid curves in Figs. 2 and 3. For the MS-72 membrane, the permeability curves of NaBr and Me,NBr are well reproduced by the TMS theory when the aqueous mobilities of the cations are used. However, the curves of EtJNBr, Pr4NBr and Bu,NBr can be reproduced only employing relatively small o ’ + values; the values evaluated for the MS-72 membrane as percentages of the aqueous mobilities are 43% for Et,NBr, 27% for Pr,NBr and 1’iF for Bu,NBr. The radius of the effective pores of MS membranes is supposed to be more than double the radius (0.49 nm) of the tetra-n-butylammonium ion, since the membranes were prepared as hemodialysis membranes with the aim of blocking the permeation of albumin (effective radius: 1.8 nm ) ; however, the diffusivity of the tetra-n-butylammonium ion is markedly reduced in the MS-72 membrane. The reduction in cation diffusivity in the membrane was also taken into consideration in interpreting the permeability curves of Pr,NBr and Bu,NBr for the MS-61 membrane. The tortuosity factor, 0, of the MS-61 membrane was almost constant, regardless of the kind of the salt. However, I!? of the MS-72 membrane increased with increasing cation size; for instance, 0 = 0.28 for NaBr and 0 = 0.55 for Bu,NBr. A possible reason for this inconsistency may be the presence of dead spaces, which trap small Ions only. The permeability curves of NaBr and TAABr for the cationic MQ-65 membrane show no maxima. The permeabilities increase to approach leveled-off values with increasing upstream salt concentration. This is as expected from the TMS theory, from which dP/a&, for the cationic membrane is given by eqn. (6) when W is replaced by - W. The parameters, cc)r + _H and X, evaluated for the MQ-65 membrane, are listed in Table 5. These parameters are obtained by curve fittings assuming that 8 and X are independent oft he kind of the salt. NaBr is an exception, in that a slightly lower 0 value has to be assumed. Note that these values may contain large errors because of the characterless form of the curves. w ’ +s decreased with increasing cation size. This reduction in cation mobility is in line with that found for anionic MS membranes. For cationic membranes also, maxima in the permeabihty curves of salts with o + ,> w _ are expected from the theory. We measured the permeability

80

F&. 7. plot of the apparent diffusion coefficient in DIA membrane against the diffusion coefficient in water for each of the salts indicated. of NaC1, NaBr, NaI and NaClO, for cationic MQ-72 and MQ-64 membranes. However, the permeability coefficients of these salts showed only a monotonic increase approaching leveled-off values with increasing upstream salt concentration. These results are reasonable since the sizes of the anions of the latter salts, iodide and perchlorate ion, are still sufficiently small to manifest the aforementioned asymmetry for the permeability curves. Figure 7 plots the apparent diffusion coefficient, Dapparent ( =P/K) , in the DIA membrane against the diffusion coefficient, D,, in water for each of the salts. If we assume that the reduction of the diffusion coefficients of the two small salts, NaBr and Me,NBr, is caused only by the tortuosity of the membrane, 8 is estimated to be 0.27-0.30. As shown in the figure, Dapparentvalues of Pr,NBr and Bu,NBr deviate downward from the line for 0 = 0.30. With the adjustable mobility ratio, o + /w _, or the parameter, W, in the membrane, we fitted eqn. ( 2) to the experimental permeabilities for DIA membrane under coefficients

TABLE 6 Parameters for the interpretation of the permeability coefficients for the DIA membrane” Solute

aa+

(lOmgcm2-mol-J-‘-set

NaBr Me,NBr E&NBr PrlNBr BulNBr

5.4 4.8 2.9 1.4 0.73

(5.4) (4.8) (3.5) (2.5) (2.1)

“See footnotes for Table 4 for the symbols.

e

W

0.27 0.30 0.30 0.30 0.30

-0.22 - 0.27 -0.48 -0.72 - 0.84

‘) ( -0.22) ( -0.27) (-0.41) (-0.54) ( -0.60)

81

the condition that 8 is constant and X= 0. The Wvalues determined this way are shown in Table 6. We find that the values for Pr,NBr and Bu,NBr are again smaller than those in water, in reasonable consistency with the results found for the MS-72 membrane. Concluding

remarks

In this study, we measured the salt permeabilities of anionic, cationic and neutral membranes composed of PMMA stereocomplex. All the membranes gave high permeabilities for NaBr and Me,NBr, whereas they suppressed the permeabilities of Pr,NBr and Bu,NBr rather strongly. For the permeation of Pr,NBr and Bu,NBr through the anionic membranes, a distinct maximum was observed in each curve of the concentration dependence of the permeability. TMS theory predicts that such a maximum appears at w _ <*CL for anionic membranes and at o + % o _ for cationic membranes. A large imbalance in the mobilities between the two cations and the anion is effect,ed in these membranes. For conventional charged membrane systems, however, these extreme conditions can hardly be attained. This is the reason why such an unusual concentration dependence has not been reported previously, The present results indicate that charged membranes can be characterized with respect to the effective diffusivity of the permeating entities. This may help in the interpretation of membrane structure in some detail. Acknowledgment

We are grateful to Drs. H. Tanzawa and T. Kunitomo of Toray Industries, Inc. for the generous donation of the membranes.

References Y. Sakai and H. Tanxawa, Poly (methyl methacrylate) membranes,
82 5

6 7 8 9 10 11

T. Hanai, H. Kataoka and T. Kunitomo, Permselectivity and membrane potential of charged membranes - An introduction of a new parameter to represent permselectivity, Abstracts of the 7th Annual Meeting of the Membrane Society of Japan, 1985, pa 70. T. Teorell, An attempt to formulate a quantitative theory of membrane permeability, Proc. Sot. Exp. Biol. Med., 33 (1935) 282-285. K.H. Meyer and J.-F. Sievers, La permeabiliti des membranes I Thborie de la permkabilite ionique, Helv. Chim. Acta, 19 (1936) 649-664. K.H. Meyer and J.-F. Sievers, La permeabiliti des membranes. II Essais avec des membranes s&ctives artificielles, Helv. Chim. Acta, 19 (1936) 665-677 Jpn. Kokai Tokkyo Koho, Japanese Patents 59/166158,59/166159,1984. D. Ketisz and F. de K&&y, The influence of counter-ions on the properties of a polyethylene phosphonic acid membrane, Isr. J. Chem., 6 (1968) 103-l 14 T. Teorell, Transport processes and electrical phenomena m iomc membranes, Prog. Biophys. Biophys. Chem., 3 (1953) 305-369.