STUDIESOF
~ ~ ‘ E M B R A NPHENOMENA. E I. MEMBRANE POTENTIAL
Studies of Membrane Phenomena.
3981
I. Membrane Potential
by Yonosuke Kobatake, Noriaki Takeguchi, Yoshinori Toyoshima, and Hiroshi Fujita Department of Polymer Science, O3Uka University, Osaka, J a p a n
(Received J u n e 14, 1966)
An equation is derived, on the basis of the thermodynamics of irreversible processes, for the electric potential, Ap, which arises between two solutions of a uni-univalent electrolyte of different concentrations C1 and Cz that are separated by an (negatively) ionizable membrane. The most crucial in its derivation is to assume that the activities, a+ and a_, of ions in the membrane are represented by a+ = C- and a- = C-, where C- is the concentration of the negative ion species. An implication of this assumption is discussed a t some length. The equation so obtained for A p is substantially the same as the previous one of Kobatake. It contains three parameters to be evaluated from experiment. Methods for this purpose are described which utilize limiting behaviors of observed A p a t very high and low C2 for a fixed ratio of concentrations C2/C1. To check the theory, data are obtained with oxidized collodion membranes in aqueous KaC1, KCl, K I , Ei303, and HCI. It is demonstrated that these new data as well as typical ones of previous workers are fitted quite accurately by the equation derived. No such agreement with experiment is obtained in terms of the earlier theory of Teorell and of Meyer and Sievers, which corresponds to the case in which intramembrane ions behave ideally, ie., a+ = C+ and a- = C-.
Introduction A steady electromotive force (e.1n.f.) arises between solutions of an electrolyte of different concentrations at constant temperature and pressure when they are separated by a uniform membrane that contains fixed ionizable groups. This e.m.f., usually called the membrane potential, has been the subject of many theoretical and experimental studies,’-3 but it appears that no satisfactory theory has as yet been established. Various membrane phenomena have been successfully correlated in a quantitative manner by the thermodynamics of irreversible This type of treatment, however, does not provide information about the actual mechanism which produces observed membrane potentials. The earlier theory of TeorelP and of lleyer and Sieverss (the T.3f.S. theory) and its various refinements,’! which all had been based on a fixed charge membrane model, were criticized recently by Hills,’O who showed that these theories are inadequate to explain experimental results on membranes in which there is an incomplete ionic selectivity. Kobatake” integrated flow equations provided by the thermodynamics of irreversible processes to derive an equation for the membrane potential and found that the de-
rived equation agreed quite satisfactorily with typical experimental data available. However, in order to make the analytical integration possible he had to introduce an important assumption between the derivatives of the activities for the positive and negative ion species in the membrane. Furthermore, he used a capillary model for the membrane. (1) H. T . Clarke, Ed., “Ion Transport across Membranes,” Academic Press, New York, N. Y., 1954. (2) “Membrane Phenomena,” a General Discussion of the Faraday Society, University Press Ltd., Aberdeen, 1956. (3) R. Schlogl, “Stofftransport durch Membranen,” D. Steinkopff Verlag, Darmstadt, 1964. (4) J. Staverman, Trans. Faraday SOC.,48, 176 (1952). (5) G. Scatchard, J . Am. Chem. Soc., 75, 2883 (1953); Discussione Faraday SOC.,21, 30 (1956). (6) K. S. Spiegler, Trans. Faraday SOC.,54, 1408 (1958). (7) J. G. Kirkwood in “Ion Transport across Membranes,” Academic Press, New Tork, N. T., 1954, p. 119. (8) T . Teorell, Proc. SOC.Ezptl. Biol., 33, 282 (1935); Prog. Biophya. Biophys. Chem., 3, 305 (1953): 2.Elektrochem., 5 5 , 460 (1951). (9) K. H. Sleyer and J. F. Sievers, Helv. Chim. Acta, 19, 649, 665, 987 (1936). (10) G. J. Hills, P. W. AI. Jacobs, and N. Lakshminarayanaiah, Proc. Roy. Soc. (London), A262, 246, 257 (1961); N. Lakshminarayanaiah, J . Polymer Sci., A2, 4491 (1964). (11) T.Kobatnke, J . Chem. Phys., 28, 146, 442 (1958); Progr. Theor. Phys., Szcppl., 10, 226 (1959).
Volztme 60, Number 11
November 1965
Y. KOBATAKE, N. TAKEGUCHI, Y. TOYOSHIMA, AND H. FUJITA
3982
I n the present paper, we first describe a somewhat improved derivation of Kobatake’s equation for the membrane potential. Here we have no recourse to the capillary model and discuss at some length the implication of the assumption that was basic to the treatment of Kobatake.” Second, we present membrane potential data which we have redetermined with oxidized collodion membranes in five uni-univalent electrolytes covering wide ranges of concentration and show that these new data as well as typical ones of previous investigators are fitted accurately by the equation derived.
Theory Basic Equations. The system considered is composed of an ionizable membrane of uniform thickness which separates two bulk solutions of a uni-univalent electrolyte of concentrations C1 and CZ. For convenience, we set C1 < Cz. It is assumed that the system is isothermal and no pressure head is applied across the membrane. The ionizable groups are fixed on the polymer network which constitutes the given membrane and hence do not appear as the component in the flow equations. The flom of ions and water molecules occur in the direction of membrane thickness. We take the space coordinate x in this direction, its origin being placed on the membrane surface which is in contact with the solution of concentration C1. The value of x for the other membrane surface is denoted by L. Starting with the basic flow equations provided by the thermodynamics of irreversible processes,12 we may derive the following equations for the flux of the electrolyte component ( J 8 ) cand the electric current density ( I ) c , both relative to the frame of reference fixed to the membrane, when the system is in the steady state. (J&
=
- (I+C+ -
p RT d In a+ z-c-) d-( z+c+ -+ dx F dx
reference, Urnis the velocity of the local center of mass,
R is the molar gas constant, T is the absolute temperature of the system, and F is the Faraday constant. The evaluation of Urnis a formidably difficult problem. To solve this, the hydrodynamic equation which governs the viscous flow of an electrolyte solution through a complex polymer network must be derived and then integrated. Here we have recourse to a crude approximation, since the last term on the right-hand side of eq. 2 gives rise to only a small correction for the membrane potential, as will be seen below. We represent the viscous force acting on 1 cc. of solution in the membrane by - (l/k)Urn, where IC is a constant which is considered to depend on the viscosity of the solution and the structural details of the polymer network of which the membrane is composed. The same volume of solution undergoes an electric force which is represented by - F(C+ - C-)(dp/dx). I n the steady state, the sum of these two forces is zero, so that me have ~
Urn = -kF(C+ - C-)(dq/dx)
For convenience, me consider a membrane which is ionized negatively with a charge density 6 (in moles/ cc.). Then the requirement that the electric neutrality must be realized in any element of the membrane gives the relation
c+- e- = e
+ I-C-)
dp d In a+ - - RT(I,C+ -dx dx
Here p is the electric potential, C+ and C- are concentrations of positive and negative ions in moles per cubic centimeter of solution, a+ and a- are activities of positive and negative ions in moles per cubic centimeter of solution, I+ and 1- are molar mobilities of positive and negative ions defined in terms of the mass-fixed frame of The Journal of Phusical Chemistry
(4)
Since in the system considered here no electric field is applied externally across the membrane, no net electric charge is transported from one side of the membrane to the other. This means that ( I ) c must be zero at a cross section of the membrane. Substituting eq. 3 and 4 into eq. 2, putting (I)oequal to zero, and solving for dpldx, we obtain dv dx = -
- (RT F)x
Z+(C-
= -F(l+C+
(3)
+ B)(d In a+/dx) - LC-(d In a-ldx) ( I + + L ) C - + 1+e + kFO2
(3
To proceed further, the activities a+ and a- must be known as functions of C-. Assumptions for a+ and a_. It is one of the unsolved problems in the field of polyelectrolyte study to derive exact theoretical expressions for the activities of small ions in polyelectrolyte solutions. It is therefore understandable that Kobatake” had to assume (12) H. J. V. Tyrrell, “Diffusion and Heat Flow in Liquids,” Butterworth and Co. Ltd., London, 1961, p. 54; D. D. Fitts, “Nonequilibrium Thermodynamics,” McGraw-Hill Book Co., Inc., New York, N. Y . , 1962, p. 74.
STGDIESOF MEMBRANE PHENOMENA. I. MEMBRANE POTENTIAL
rather intuitively an unjustified relation between a+ and a-. His assumption is equivalent to setting such that
a+
=
C-; a- = C-
(6)
or y + = C-/(C-
+ 0); y- = 1
(7)
where y + and y- are the activity coefficients of posiOive and negative ions in the membrane. Alt'hough a t present, no theoretical justification exists for this arbitrary choice of the forms of a+ and a_, an interesting result is obtained when it, is applied to the membrane equilibrium problem. Consider a negatively ionizable polyelectrolyte gel or membrane immersed in a solution of a uni-univalent simple electrolyt'e. At equilibrium the following relation must be obeyed (C)oz = (a+)i(a-)i
(8)
where (C)(, is the equilibrium concentration of the electrolyte (in niole's/cc.) in the outer solution and ( . + ) i and are the act'ivities of positive and negative ions in the gel phase. I t has been assumed that the outer electrolyte solution behaves ideally; t'his assumption is not unreasonable in the treatment of polyelect,rolytes and may be taken out if desired. When the forms of a, aiid a- ns,wmed above are valid, it follows from eq. 8 that (C-ji
=
(C),
(9)
Thus the coiicentrat,iona of the negative ion species in the two phases become identical. This result is not obtained when the ideal Doniian equilibrium is established between the phases. In this case, the concentration of liegatjive ions in the gel phase, (C.-)i, is always lower than that in the outer solution, (C)o. I n passing, me note that tlie ideal Donnan equilibrium corresponds to the case in which (-y+)i = (y-)i = 1. Applying eq. 7 to eq. 8 and using eq. 9, we obt'ain the relation
(Cjo2/(C+)i(C-.)i = (C)o/[(C)o
+ 61
(10)
We have found that this simple relat,ion fits well typical data of previous worliers, notably, Sagasawa, et aZ.,13 Hills, et uZ.,'O and Gregor, et uZ.,l4 who determined the concentrations (C,) i and (C-j i in ion-exchange resins or membranes equilibrated with simple electrolyte solutions of given concentrations Recently, Katchalsky, et aZ.,l5 have point,ed out this similar fact. They also have demonstrated that equilibrium ion distributions between solutions of polyelectrolyte and simple salt separated by a semipermeable mem-
3983
brane can be represented remarkably well by eq. 10, unless the polyelectrolyte concentration is too low. This interesting consequence from the assumptions a+ = C- and a- = C- suggests that the assumed forms for the activities of small ions in polyelectrolyte gels or membranes may have a broad applicability. Presumably, they may be a much better approximation to the actual state in polyelectrolyte solutions or gels than the assumption of thermodynamic ideality} Le., a+ = C+ and a- = C-, which, as has been noted above, leads to the ideal Dolman distribution. However. it must be remembered that the agreement of eq. 10 with experiment does not always support the correctness of the underlying assumptions, eq. 6. The final justification is left for a theoretical study. For the time being, it would be most relevant to regard eq. 6 as a kind of working hypothesis and check with experiment the various consequences which would be obtained if this hypothesis were valid. I n what follows we proceed along this line of thought. Equation f o r Membrane Potential. With eq. 6 assumed for a+ and a_, eq. 5 becomes
9 = -(ET)dx
F
ac+ 1,8 + L ) C - + Z+8 + liF82]C- dx (1, - LjC-
[(Z+
(11) When the bulk solutions on both sides of the menibrane are vigorously stirred, no potential gradient is set up in them, so that the desired membrane potential Ap is obtained by integrating dp/dx over the thickness of the menibrane. Thus we have Ap =
-(?)lL
X
[(Z+
+ z+e dCdz + + SFP]C- dx
(6, - L)CL)C1+0
+
(12)
The flows of ions and water molecules in the membrane are sufficiently slow so that it is not unreasonable to assume that at the boundaries between the membrane and the outer electrolyte solutions thermodynamic equilibria are established. Then eq. 9 holds a t either niembrane surface, and n-e have C- = C1 a t z = 0 and C- = C2 at x = L as the boundary conditions for C- that are consistent with eq. 12. Performing the (13) K . Kanainaru, 11. Nagasawa, and K. Nakamura, Kogyo Kaguku Zasshi, 56, 436 (1953); M. Nagasawa and I. Kagawa, Discussions Faraday SOC., 21, 52 (1956). (14)H. P. Gregor, F. Gutoff, and J. I. Bregman, J. Colloid Sci.,6, 245 (1951); B. K. Sundheini, 11. H. Wasman, and H. P. Gregor, J. Phus. Chem., 57, 974 (1953). (15) A. Katchalsky, Z. Alesandrowicz, and 0. Kedem, personal communication.
Volume G9, Kitmber i i
h'ovember 1965
Y. KOBATAKE, N. TAKEGUCHI, Y. TOYOSHIMA, AND H. FUJITA
3984
integration on the right-hand side of eq. 12 with these conditions into account, there results the final expression for the membrane potential
Cz -
Acp = -(-F)[;1nC RT 1
where
and these parameters have been assumed to be independent of salt concentration. Linaiting F o m s of Eq. 13. As will be shown below, we have compared eq. 13 with the data from a series of measurements in which values of A y on a given membrane-electrolyte system were determined as a function OS C:! with the concentration ratio y = Cz/ C1 (>1) heing fixed at a given value. We derive two limiting forms of eq. 13 useful for the analysis of this type of experimental data. (a) When Cp becomes sufficiently small with y fixed, eq. 13 may be expanded to give
(1
+
- 2a)(4)
+ O[
which has the form expected from experimental information. Equation 19 indicates that the intercept for a plot of l/t- against l/Cz a t a fixed y allows the value of a to be determined. If this value of a is inserted in the relation obtained from the initial slope for lay,\ us. CZ,the desired value for e can be determined. Once a and P are known in the manner described above, the 0 may also be evaluated from the initial slope for l / t - us. ~ / C Z . Providing our membrane potential equation is correct, the two values of 6 obtained in this way from the opposite limits should agree with one another. Expression for (JJ,. It is a simple matter to eliminate dy/dJ: from eq. 1 and 2, using ( I ) c = 0 and eq. 3 for U,. The resulting expression for (Js),can be put in a form which contains only C- as variable, by making use of the neutrality condition, eq. 4, and the assumptions for a, and a- made in eq. 6. By integration from x = 0 to x = L of the equation so obtained for (JS),we obtain the following expression for the steady-state flux of the electrolyte component
where Do is the diffusion coefficient of the electrolyte component defined by (:)2]
(16)
-(-)
Do= 2RT F
1
where A& is the absolute value of a reduced membrane potential defined by Apr = F A p / R T
(17)
Equation 16 indicates that the value of P and a relation between a and 6 may be obtained by evaluating the intercept and the initial slope of a plot for (A++\ against C2 obtained with y fixed. (b) It is well known experimentally that at a fixed y the inverse of an apparent transference number t- for the co-ion species in a negatively charged membrane is proportional to the inverse of the concentration C2 in the region of high salt concentration. Here t- is defined by the relation jA(p,j = (1 - 2t-) In y
(18)
Substituting for Acp from eq. 13 and expanding the resulting expression for l/t- in powers of l/Cz gives 1 -
t-
1 ___ (1 - a )
+
The Journal of Physical Chemistry
1+1-
I,
+ E-
Experimental tests of eq. 20 will be reported in a forthcoming paper of this series.
Experimental Section Oxidized collodion membranes, which are negatively ionizable in aqueous media, were prepared according to the method reported by Sollner,16and those which had been found appropriate by preliminary experiments were chosen for the measurements. Except in the measurements with HC1, a given membrane was repeatedly used for different electrolytes, but no significant change occurred in measured e.1n.f. when the experimental conditions were identical. Five uni-univalent electrolytes were examined, which were NaC1, KC1, KI, KN03, and HC1. Before use, XaC1, KC1, and KNOs were purified by repeated recrystallization. Both KI and HC1 of analytical grade were used as de(16) K. Sollner and H. P. Gregor, J . Phys. Chem., 50, 470 (1946); 51, 299 (1947); J . Phys. CoZZoid Chem., 54,325,330 (1950); K. Sollner, Ann. N . Y . Acad. Sei., 57, 177 (1953); “Ion Transport across Membranes,” Academic Press, New York, N . Y.,1954,p. 144.
STUDIES OF MEMBRANE PHENOMENA. I. MEMBRANE POTENTIAL
3985
-15
Stirrer SoltBridge I
SdtBridge Stirrer I
I
-1 0
I
-7, Gasket
Membrane
Figure 1. Schematic diagram of the cell used for the measurement of membrane potentials.
"'i.
Figure 3. Typical data of the membrane potential as a function of log Ct at constant y (=C,/C, = 2 ) : 8,KC1; 0, KI with membrane 2; Q, NaCl; , DHC1 with membrane 3; e, KNOI with membrane 4.
IO
-7
5
Figure 2. Effect of the rate of stirring on membrane potential. I
livered. The water used as solvent was prepared by treating distilled water with both cation and anion exchangers. Carbon dioxide dissolved in it was not degassed. Figure 1 shows a schematic diagram of the apparatus used for measuring membrane potentials. The e m f . which arose between the bulk solutions was conducted by saturated IiCl bridges and calomel electrodes and measured by a vibrating reed electrometer (Takeda Riken Co. TR-84 B Type) or by an electronic potentiometer of a Du Bridge type. The caps of the salt bridges were carefully ground so that the test solutions might not be contaminated due to leakage of KC1. Since we have no definite information about the liquid junction potential between the salt bridge arid the electrolyte solution, the measured e.m.f. were not corrected for this potential. The bulk solutions were stirred by a pair of plastic fans at a speed of 33 c.p.s. As illustrated in Figure 2, the measured potential became independent of stirrer speed when it was
0
50
0 IO
100
Z for various electrolytes a t Figure 4. l / t - us. ~ / C plots Notations and membranes are the same as in Figure 3.
=
2.
faster than about 10 C.P.S. On each system five determinations of e.m.f. were made. The results agreed within 0.2 mv. and their average was taken as the desired result. All measurements were made in an air oven at 25". For a given membrane-electrolyte system the data mere taken as a function of CZwith the concentration ratio y (=C,/C,) fixed to desired values. The values of y studied were 1 E (e
