TRANSPORT OF IONS AND ION PAIRS IN LIQUIDION-EXCHANGE MEMBRANES
3871
Electrical Phenomena Associated with the Transport of Ions and Ion Pairs in Liquid Ion-Exchange Membranes. 11. Nonzero Current Properties's
by J. Sandblom,lb G. Eisenman, and J. L. Walker, Jr.'c Deportment ofPhy&logy, University of Chicago, Chicago,Illinois 606.27 (Received April 19,1967)
This paper continues the theoretical examination of the steady-state properties of liquid ion-exchange membranes in which sites and counterions are incompletely dissociated. The flux equations derived in the preceding paper are integrated for the case where only a single counterion species is present without the restriction of that paper to zero membrane current. The current-voltage relationship is deduced, and the steady-state fluxes and internal concentration profiles are related to the total membrane potential. On the basis of these relationships, the effects of association between sites and counterions are examined. A comparison is also made between the theoretically expected properties of liquid ionexchange membranes having complete dissociation us. strong association, as well as between the expectation for liquid us. solid ion exchangers. It is concluded that increasing association makes a mobile-site membrane progressively more difficult to distinguish from a fixedsite membrane on the basis of its purely electrical properties, but progressively easier to distinguish on the basis of a comparison of electrical and flux properties. Some biological implications of these results are also discussed.
In a previous paper, the effects of association on the properties of a mobile-site membrane were examined.2 The membrane under consideration was formally treated as a multicomponent system containing ionic as well as nonionic (ion pairs) species, assumed to be in chemical equilibrium at every point in the membrane but flowing independently of each other. Having chosen this approach as the basis for a quantitative treatment, the flux equations and boundary conditions for all the species were derived and the equations were solved under conditions of zero current. This paper constitutes the second part of the same treatment and deals primarily with the behavior of the system under nonzero current conditions. The equations are written and solved for the case of a single counterion in steady state. Expressions are obtained relating concentration profiles, fluxes, and resistance to the total membrane potential and from these expressions the various effects of association on the membrane properties are deduced. In the Case where only a single counterion species is present in the system' the set Of equations can be deduced from eq 1 to 16 in part I.2
(3) c1
= cs
(4) (5)
nd
(1) (a) This work was supported by National Science Foundation Grant ~13-4039and USPHs Grant GM 14404-01. It was assisted by USPHS General Research Support Grant FR-5367 and an NIH postdoctoral fellowship to J. L. Walker, Jr. (b) Institute of Physiology and Medical Biophysics, University of Uppsala, Uppsala, Sweden. (c) Department of Physiolow, University of Utah, (201lege of Medicine, Salt Lake City, Utah. (2) J. Sandblom, G. Eisenman, and J. L. Walker, Jr., J . Phya. Chem., 71,3862 (1967).
Volume 7 1 , Number 18 November 1967
J. SANDBLOM, G. EISENMAN, AND J. L. WALKER, JR.
3872
c(d) RT a’ + RT - In L- + - lnzlF cs(0) Z I P a”
V = A$
Js* = Js
+
J1, =
0
(8)
tain other cases, it has been found justifiable to approximate Q by a constant.a Equations 13 and 14 can be combined to eliminate c,(O)-and sS(O). This gives
(9)
2RT
(7)
Equations 8 and 9 are obtained from the condition of steady state, but all other expressions are valid in the nonsteady state as well. We shall solve this system of differential equations to obtain fluxes and concentration profiles in terms of total membrane potential and external solution con& tions, ~ i ~ i eqd 8 iby~u1~and eq 9 by usand adding the resulting expressions, we obtain
where tils is a constant defined as
which we shall call the interdiffusion mobility of the dissociated species. Inserting eq 1-3 into eq 10 and using the condition of electroneutrality (4),we get
3 Cls(x) + 2RTcS(z) = UlS
Uls
2RTrQ uls
+ 221Ful Id ( 1
-
$)
(16)
and finally, eq 4 and 5 are used t’o express eq 16 in terms of Cs(z>. After rearrangement, this gives an expression for the concentration a t any point 2, cs(x), in terms of the electric current and the various membrane parameters
(17) from which c,(x) can be solved in terms of the electric current. The quantity Q may then be evaluated by inserting the resulting expression in (15) and integrating, as will be discussed in the section, “Concentration Profiles.” I n order to relate the concentrations a t the membrane boundaries to the total membrane potential, eq 2 and 3 are added, divided by u,/zl, and integrated using the steady-state condition (9) for the flow of sites. This gives
This equation can be integrated directly from 0 to x to yield
-l a - -2RT[c,(x)
- c,(O)]
-
ZiFui 2RT
3 kd4 - s,(O)l Uls
(13)
Equation 12 is then integrated from 0 to d, and, taking into account eq 6, we get
-Id - - -2RT 221Fui
FQ+ 2RT UlS
where the quantity Q is given by
+ d- [- - l ] i d c , 1
Q = E*
a1s uls
dx
(15)
I n general, Q is a function of the electric current, but it is Seen from eq 15 that in a number of limiting cases it is a constant, e.y., when zi1, = ulaor when the degree of dissociation is so small that the second term can be neglected in comparison with the first term. I n cerThe Journal of Physical Chemistry
After expressing cls(x) in terms of c,(x) t,hrough eq 4 and 5, the integral is solved, and when this is introduced into eq 7 , we obtain an expression for the total potential
V
RT zlF
= -In
a’ -
a”
2RT ulS 2RT c,(d) +In - + - __ X cs(0) zlF Ku, zlF
- CS(0)l
(19) The solution to the equations is essentially complete since eq 19 can be combined with eq 17 to yield explicit expressions for the concentration profile and electric current in terms of the total membrane potential and external solution conditions. Limiting Currents. Equations 17 and 19 indicate that the steady-state current-voltage relationship for a single counterion species depends on the mobilities of the undissociated as well as the dissociated species, the total concentration of sites, and the degree of dissocia[cs(d)
(3) J. L. Walker, Jr., G. Eisenman, and J. Sandblom, J . Phys. Chem., inpress.
TRANSPORT OF IONSAND IONPAIRS IN LIQUIDION-EXCHANGE MEMBRANES
tion. The left side of eq 17 contains only positive quantities, and by inspecting the right-hand side after setting the left side of eq 17 equal to zero for x = 0 and 2 = d ( i e . , c,(O) = 0 and c,(d) = 0, the respective concentrations a t 0 and d for strong applied voltages), it can be seen that the electric current is confined to an interval given by
Thus, we find that a finite limiting current, previously deduced as being characteristic of a mobile-site membrane in the limit of complete diss~ciation,~ is characteristic of mobile-site membranes quite generally regardless of their degree of dissociation. The end values of the interval in eq 20 define the values of the limiting currents I1 whose absolute values are given by
3873
membrane in which K = 4 X lo-’ mole/cm3, E* = mole/cma, and tZlS/u1,= 1/3.04. (These particular values were chosen because they correspond ta those characteristic of HC1 in isopropyl alcohol, a liquid system in which the present theory has been t e ~ t e d ) . ~Examining this figure, we notice that the concentration profiles are parabolic in form, in contrast to the linear profiles expected4 and observed5 in a completely dissociated mobile-site membrane. We also see that the concentration of sites becomes increasingly skewed as the applied field is increased, so that in the limit as 4 --+ 1 , the site Concentration is reduced to zero at one side of the membrane, as noted in deducing eq 20. Current-Voltage Relationship. An explicit relationship between current and voltage is obtained by expressing c8(0)and c8(d) in eq 19 in terms of eq 22. This gives
V
RT
= -In-
zlF
a’ a”
+ 2RT 2 9
Concentration ProJiles. In order to simplify the expressions, we shall introduce a quantity 4 defined as IdJ18
=
4RTFulu1,Q
Note that 4 --+ 1 as I -t 11. Substituting this definition in eq 17 and solving for c,(z), we get an explicit expression for the concentration at any x as a function of 4
However, in order to calculate the concentration profiles, an expression for Q must be obtained. This is done by inserting eq 22 in eq 15 and integrating, which yields
34KQ J from which Q can be calculated graphically or by successive approximation.8 Figure 1 illustrates the concentration profiles for the indicated values of calculated from eq 22 and 23 for a
The general shape of the current-voltage relationship for varying values of the dissociation constant K and constant total concentration is illustrated in Figure 2 for a membrane in which ?.ilS/Uls = 1/3.04. Figure 2 indicates that the voltage range over which the I-V relationship is nearly linear becomes increasingly extended with increasing association. This is because the increasing number of undissociated ion pairs, which are not acted on by electric forces, acts as a 4lS,as in Figure 2, the backflow of the neutral species has a larger effect than the depletion of free sites, and the limiting current increases with increasing association. The reverse will be the case when UI, < 41,. Mem$rane Resistance. The instantaneous resistance R, of the membrane, as measured using a step change in current or by a high-frequency alternating current is given by
R, = When the integration is performed, this gives
VOLTAGE (mV)
Figure 2. Theoretical current-voltage relationships for a membrane having the indicated degrees of dissociation ( K is expressed in moles/liter) and Ula = 3.04G1, corresponding to the measured values for HCl in ’,?-propanol. C* has been held constant a t 10-3 mole/l. Note the initial decrease in resistance and subsequent increase as K is varied from infinity to zero. Also note that the limiting current at high positive voltage increases monotonically as K is varied from infinity to zero.
plained by the following considerat,ions. For the case of complete dissociation (K = a), the electric current at steady st3te is carried only by the counterions. This follows from the fact that in a completely disThe JOUTW~ qf Physical Chemistry
where, again, c,(d) and c,(O) may be expressed in terms of K , f i l , / ~ ~ ,4,, and Q through eq 22. The steady-state resistance Rois given by
Ro =
v - vo
____
I
and is obtained from eq 24 recalling that 4 = 1/1111. A comparison between the steady-state resistance Ro and the instantaneous resistance R, is most easily made in the limit of zero current (4 = 0). I n this case, eq 25 and 27 reduce, respectively, to
and
TRANSPORT OF IONS AND
LIQUID ION-EXCHANGE .hfEMBft4NES
+ KF2[[il,u,c, + + us)---K us
RO(+ = 0) =
I O N PAIRS IN
3875
C,
u1s
I
Q, = 1.0
IXd
ulSc82
(UI
08-
(28b) From these expressions it is seen that as the association increases ( K + 0), the instantaneous resistance approaches the steady-state resistance. In the other limit of complete dissociation ( K + a),the following relationship is obtained between the two resistances
R,(d
= 0 ) = (1 -
tJ X Ro(4 = 0)
+=09,\
04
(29)
~
~-
where t, is the transference number of the dissociated form of the sites, given by
t,
U S
+ us
= ___ u1
The relationship expressed by eq 29 was found and experimentally verified by Walker and E i ~ e n m a n . ~ Fluxes. Recalling that for the present single counterion case, J1* = I / F , the dependence of the total flux of the counterion species on the solution conditions and applied electric potential can be obtained from eq 24 and is shown in Figure 2. It has been mentioned that only the total fluxes are constant throughout the membrane, whereas the partial fluxes are subject to sources and sinks. This is seen by deriving explicit expressions for the partial fluxes J 1 , J,, and J1,. The partial flux J I , is obtained from eq 3-5
- -2RTu1, C , K
dc$ dx
(30)
from which dc,/dx and dcl,/dx may be expressed in terms of J1,. Inserting these expressions in eq 12, we get after rearrangement, recalling that J1*F = I
Recalling the dependence of c, on x in eq 22, eq 31 indicates the distance dependence of the partial flux J l , in the steady state for constant J1* (Le., constant current). This dependence is illustrated in Figure 3, which presents flux profiles of J1 as a fraction of the total flux J1* calculated from eq 31,22, and 8 using the same values of the parameters and currents as those given for Figure 1. Note that the ratio JI/Jl* is smaller than unit8yfor all currents less than the limiting one.
J!
0
02
04
06
08
---IO
X -
d
Figure 3. The partial flux J1of the counterion is plotted as a function of z in terms of the fraction of total flux (J,/Jl*) for 4 = 0, 0.5, 0.9, and 1.0. The dotted line represents the total flux J1* = I/F.
It is only when +
= 1 and the sites are depleted at one
end of the membrane that the counterions are seen to carry the total current. Notice that for small currents ( i e . , for low values of 4) the flux of counterion species is essentially constant for a given applied field, being independent of the position within the membrane. This is comparable to the situation for all applied currents in a fixed-charge membrane in the steady state. Moreover, notice that this flux (J1) is less than 40% of the total flux, demonstrating that circulation of sites contributes significantly to the flow of electric current even in the steady state despite the fact that the sites cannot cross the membranesolution interfaces. Indeed, notice that as I$ becomes increasingly large, it can be seen that the flux of the counterion becomes a function of distance, an effect which is surprisingly small until 4 is greater than 0.5. The slope of the curve is of course the rate of change of the flux of the counterion species with x, which determines the rate of transformation from the dissociated form to the undissociated form. JI*(namely the total flux of species 1 in all of its forms within the membrane) is exactly equal to the fluxes of species 1 entering and leaving the membrane. This can be seen to be constant throughout the membrane in the steady state, as required by eq 8 (see dotted line in Figure 3). The difference between curves J I * and J 1 for each applied field represents the flux of the site species J,, which is directly equal and opposite in this case t o the flux of the associated pairs (see eq 9). Di$usim Coeficients. For the present single counterion treatment, one interesting property of an associated Volume ‘71, ~Vumber12
November 1967
3876
J. SANDBLOM, G. EISENMAN, AND J. L. WALKER, JR.
mobile site membrane which contrasts with the situation in a fixed-site membrane is that the diffusion coefficient measured by tracer flux (Dtr) differs from that measured from the high-frequency electrical resistance (De,). Since the labeled and nonlabeled isotopes have the same properties, there will not be any potential gradient or site gradient a t zero current regardless of the concentrations in the solution phases. The total tracer flux J1*t’ can therefore be written directly in terms of concentration gradients of the tracer in the membrane. When the tracer is added to one of the external soluis measured on tions and the rate of appearance JI*~‘ the other side, we obtain
__
Ddiff
Del
-
Ula ___ UI
+ us
The discrepancy between diffusion Coefficient measured electrically and that measured by tracer diffusion is therefore inversely proportional to the square root of K for strong ast&ation ( K 0).
-
Discussion Physical Systems to Which the Present Analysis I s Applicable. The present analysis is restricted to mem-
branes in which (a) univalent counterions and sites are assumed to behave ideally except insofar as they can associate through a simple law of mass action to form neutral ion pairs, and in which (b) the sites are assumed to be unable to cross the menibrane-solution interface, dcltr dclstr J1*“ = -RTul - - RTul, - (c) the concentration of co-ions is assumed to be negdx dx ligible, and (d) the mobilities of species are assumed to be constant. (e) In addition, we have considered only the situations in which the diffusion of counterions is membrane controlled (ie., we have assumed that negligible concentration polarization occurs in the solutions adjacent to the membrane and have neglected where 8 is the specific activity (curies/g), M1the molecuinterfacial effects). Xembranes for which these aslar weight of the counterion, and clt’ the concentration sumptions are expected to be approximately valid are of tracer in the membrane which is measured in curies those containing the usual monofunctional liquid ion c1 is the (uniform) concentration of counterion exchanged dissolved in water-immiscible solvents over as defined before and is measured in moles cm-3. J1*tr some as yet undefined range of dielectric constant. is measured in curies sec-’ The measurement of Notice that for solvents of very low dielectric constant, tracer flux can be compared with the measurement of the assumption (a) may no longer be correct since aselectric conductance performed a t zero current and highsociation is likely to be complicated by the formation of frequency alternating current. “triple ions” and other higher order aggregates.7 InThis conductance is obtained from eq 28a, recalling deed, typical liquid ion exchangers such as diisooctyl hythat for a single counterion el = cs drogen phosphate are known to form associated complexes in low dielectric constant solvents whose degree F2(U1 Uh1 Gm(+ = 0) = (33) of aggregation is a function of the size and charge of the d counterions? The present system may also be thought Dividing eq 32 and 33, we get of as a model for certain types of “carrier transport” mechanisms postulated for biological membranes.9 The conditions under which the present treatment is expected to apply to membranes nirtde from the usual liquid ion exchangers are examined in more detail below; however, it is worth noting here that a simple u1 uls (34) physical system has been devised and studied by ~1 us ~1 us K Walker, Eisenman, and Sandblom3 to test the principal expectations of the present theory. This system, Equation 34 is seen to define a ratio between a diffusion coefficient measured by diffusion (Ddift = RT[ul (6) C. F. Coleman, C. A. Blake, Jr., and K. I3 Brown, Talanta, 9, 297 (C~/K)UI~]) and an electrically measured diffusion co(1962). efficient (Del = RT(u1 u,)). (7) R. A. Fuoss and hl. Krauss, J . Am. Chem. Soc., 55, 476 (1933); For the limit of complete dissociation (K + w ) , this R. A. Fuoss, ibid., 57, 488 (1935); Trans. Faraday Soc., 32, 594 (1936). ratio is given by the transference number of the counter(8) G. J. Jan2 and S . S. Danyluk, Chem. Reu , 60, 209 (1960); cf. pp ion (see eq 34). On the other hand, for strong associ228-230. ation ( K 0 ) , c,* = Kc,, + KE* and eq 34 reduces to (9) W. Wilbrandt and T. Rosenberg, Pharmarol. Rev., 13, 109 (1961).
+
-
r
+ +-- +
+
+
-
The Journal of Physical Chemietry
TRANSPORT OF IONS AND ION PAIRSIN LIQUIDION-EXCHANGE MEMBRANES
in which all of the present assumptions are satisfied, consists of a solution of hydrochloric acid in 2-propanol bounded by two chloridized silver plates. I n this system, chloride ions can enter or leave the aqueous phase, but hydrogen ions can o d y redistribute within it. We have therefore only one species of counterion-the chloride ion, while the hydrogen ion corresponds to the mobile site. The mobile associated species is the HC1 molecule, whose dissociation is governed by a simple law of mass action. After conventional measurements had been made of the dissociation constant for HCl and of the individual mobilities of H + and C1-, the mobility of the undissociated HC1 species was measured from the value of the limiting current density using eq 21 of the present theory. With the value of this parameter known, the steadystate current-voltage relationship is completely specified for all HC1 concentrations and so are the concentration profiles, electric potential profiles, and instantaneous conductances for the steady state of any applied current. Measurements were therefore made of all of these properties and were found to be in quantitative agreement with the expectations of the present paper. The Conditions under Which Assumptions b and e Are Satisfied Simultaneously. Assumption b that the sites are completely trapped within the membrane is an idealization whose validity is a function of the solubility of both the membrane phase and of the sites in the external solution phases. Since the site species must have a finite solubility in the external solutions which can be considered as infinite reservoirs, some fraction of the sites must be transported across the membrane under nonequilibrium conditions. Thus, if an electric field is applied to the equilibrium situation of a membrane separating two identical solutions, accumulation of sites will occur in the solution a t one of the membrane-solution interfaces, together with a corresponding depletion of sites in the solution a t the other interface as the system passes to the stationary state. Consequently, the number of sites contained in the membrane will change during the process. If,however, the ionic redistribution within the membrane is rapid compared to the rate of loss or gain of sites, the sites will transiently act as if they were trapped. Therefore, the physical situation which might be realized in a liquid ion exchanger corresponding to assumption b is that the transport of sites must be rate limited by the membrane boundaries (e.g., film controlled)lo whereas to satisfy assumption e the transport of permeant species must be rated limited by the membrane phase (e.g., membrane controlled) .lo Since the activation energies for counterion diffusion across the membrane-solution interfaces may be much larger than within the mem-
3877
brane phase,“ it is necessary to have a sufficiently thick system in order to ensure that this effect is negligible, while at the same time being certain that the membrane is sufficiently thin to maintain rate-limiting surfaces for the diffusion of sites. The above conditions can be formulated quantitatively for the situation where the limiting part of the membrane boundaries may be attributed to an unstirred layer of thickness 6 in the external solutions adjacent to the membrane surfaces. Confining ourselves to the case of a single counterion species and completely dissociated sites arid counterions, the time const,ant of establishing interior profiles, TD, is given by
(35) where d is the membrane thickness and D, = RT&. In order to calculate the time constant rSfor the loss of sites, we shall consider the case when the electric current is zero and the concentration of sites in the external solution is zero initially. If the ionic redistribution within the membrane is rapid compared to the rate of loss of sites, the concentration profile within the membrane is essentially uniform. The rate a t which the sites disappear must therefore be given by
where S is the distribution coefficient of the sites, S = [C* (m)/E,* (f) 3, and Dt is the diffusion coefficient (RTG,,) in the film. The time constant r 9 calculated from eq 36 is
(37) and the condition for trapped sites may therefore be expressed as rS >> T D
or
6D, 2 s-->>dDt r2 A quantitative formulation can also be given to the condition (e) that the counterions are membrane controlled. Helfferich’o (eq 8-8) has derived such an expression by considering self-diff usion in the stationary state, and under these conditions his treatment is also (10) F. Helfferich, “Ion Exchangers,” McGraw-Hill Book Go., Inc., New York, N. Y.,1962. (11) J, T.navies, J. Phys. Chem.,54, 185 (1950).
Volume 71, Number 13 November 1967
J. SANDBLOM, G. EISENMAN, AND J. L. WALKER, JR.
3878
valid for a dissociated liquid ion-exchange membrane. The expression he obtains is
z*
6 Ulm
c d
Uif
