J . Phys. Chem. 1994,98, 63526358
6352
Evaluation of Ion Transport Parameters in a Nafion Membrane from Ion-Exchange Measurements Zden4k Samec' and Antonin Trojhnek The J. Heyrovskj Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Dolej3kova 3, 182 23 Prague 8, Czech Republic
Eva SamcovA Department of Medical Chemistry and Toxicology. 3rd Medical Faculty, Charles University, Ruskh 87, 10000 Prague IO, Czech Republic Received: December 27, 1993; In Final Form: April 19, 1994' Coupled diffusion of two counterions in a fixed-site, permselective, ion-exchange membrane is treated by the method of moments. Theoretical expressions are given for the evaluation of ion diffusion coefficients from the ion-exchange measurements. They are used to analyze the transport properties of a Nafion membrane in the presence of proton and a univalent (Li+, Na+, K+, Rb+, or Cs+) or divalent (Ru(2,2'-bipyridine)32+) cation. Diffusion coefficients of alkali metal cations, as calculated from ion-exchange rates when the hydrogen ion is initially present in the membrane (the forward process) or in the solution (the reverse process), differ significantly. Moreover, only the latter parameters are comparable with the corresponding self-diffusion coefficients. These effects can be attributed to a different water content in Nafion prior to forward or reverse exchange measurements and to the cross effects arising from the simultaneous transport of the two counterions. While the rates of R~(2,2'-bipyridine)3~+ exchange into the as-received Nafion and the Nafion cast films are about the same, the present analysis yields a much lower value of the diffusion coefficient.
Introduction In the past two decades, considerable attention has been paid to transport properties of perfluorosulfonateion-exchangepolymers, such as Nafion (DuPont).'-8 Ion diffusion coefficients were calculated from ion transport rates as measured by the radioactive tracer method,' the rate of redox reactions of electroactive ions at polymer-coated electrodes2 (cf. also a reviewg), polymer conductivity$ or ion-exchange rates.5 While the radiotracer measurements yield the well-defined ion self-diffusioncoefficients, the evaluationof a single ion transport by other methods may not be a straightforward task owing to the necessity to account for the coupling of the ion and electrod or the two ion transport rates.7.8 The principal aim of this work was to evaluate the ion transport in a Nafion membrane from ion-exchange mesurements. This approach to membrane transport seems to be attractive for several reasons. First, the radiotracer method is applicable only to a limited number of ions exhibiting a measurable radioactivity. On the other hand, electrochemical methods are useful only when the ion can undergo an electron-transfer reaction at the electrode. Ion-exchangemeasurementscan be used for either ion and hence can serve for a comparison with the results obtained by one of the two methods above. Second, the measurement of ion-exchange rates represents a convenient way to analyze the coupled ion diffusion? which is a problem commonly encountered in electrochemical studies of membrane ion Only the numerical solution of the nonlinear transport problem involved9asb and the empirical formulas9a.chave been given so far. Here, we shall show that the approximate analytical solution can be derived, which enables to evaluate the diffusion coefficient of one ion, provided that the value for the other ion is known. Third, the water to exchange site mole ratio in a Nafion membrane depends on the nature of the ion.10 Hence, an ion exchange should be accompanied by the flow of water in or out of the membrane. In addition to its importance in the transport theory, the ratioof ion to water flow through theNafion membrane could be the limiting factor in electrochemical applications." *Abstract published in Advance ACS Abstracts, May IS, 1994.
0022-3654/94/2098-6352~04.5010
We have investigated the following ion-exchange processes between the Nafion membrane (m) and an electrolyte solution (SI,
H+(m) 2H+(m)
+ M+(s) F? H+(s) + M+(m)
+ Ru(bpy),'+(s)
(la)
e 2H+(s)
+ Ru(bpy)?(m)
(lb)
where M+ is an alkali metal cation and bpy = 2,2'-bipyridine. Kinetics of both forward (from left to right in eq 1) and the reverse process has been measured. Diffusion coefficients of protonic and some alkali metal cation'b.c in Nafion membrane are available from radioactive tracer measurements. The apparent diffusion coefficient of Ru(bpy)32+ ion was determined by membrane permeation chronopotentiometric measurements.*c
Theoretical Section According to the theory of the coupled diffusion? the concentrations ct of exchanged ions A and B in a permselective membrane are given by the nonlinear differential equation dCi/df
= d[D~~(dCi/dx)]/dx
(2)
is the ion charge, Di is the self-diffusion coefficient, t is time, and x is thecoordinate. Equations2 and 3 follow from the NernstPlanck equationsfor the ion fluxes JAand JB,the electroneutrality condition, ZACA + ZBCB = co (CO denotes the concentration of fixed univalent ion sites in the membrane), and the zero-current condition, ZAFJA+ Z#JB = 0 (F is the Faraday constant). The electroneutrality condition implies the absence of co-ions in the membrane (Donnan exclusion). Equation 2 was solved numerically for sphericalga and planar9b membrane geometries and for selected values of the ratio DAIDB. For a plane sheet membrane of a thickness d = 21, the following zi
0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, NO.25, 1994 6353
Ion Transport Parameters in a Nafion Membrane
and 24ab’
0.4
1
0
DAB(c) = DADB[DA (1 - C) 2
1
Figure 1. Initial rate of exchange d F / d r l P of an ion A (initially in the membrane) for an ion B (initially in the solution) vs the square root of the ratio of diffusion coefficients &IDA: ( 0 )numerical solution of eq 2;9b(A) calculated by eq 5; (solid line) a nonlinear least-squares fit to numerical data9bby an empirical function dF/dr1/2= 1.556( 1 - 0.8695
+ 0.626 exp(-2/y) - 0.74 exp(-3/y)].
initial and boundary conditions of the transport problem were assumed,
where the origin of the coordinate x has been placed in the center of the sheet. Thus, initially the membrane and solution are supposed to contain only ions A and B, respectively. However, due to a mobile equilibrium, which favors the exchange of B for A, the counterion ratio is reversed at the boundary x = -1 and x = 1. The result is the fractional attainment of equilibrium F(T) = m ( ~ ) / m ( = where ), m is the total amount of the ion B that has entered the membrane as a function of the dimensionless time T = DAt//’. The F vs plots show a very good linearity at low and medium values of F. The initial slopegbof these plots varies with the ratio D A / Das ~ displayed in Figure 1. For DA= DB,the initial slope 2 / d 2 = 1.128 follows from theanalytical solution.12 Then, the ion transport parameters could be inferred from ionexchange measurements by comparing the experimental and theoretical initial slopes dF/drl/’ 98.b or by using an empirical formula.9*lc In order to avoid an extrapolation or interpolation of numerical data, we have attempted to find an analyticalsolution also when DA # De. The transport problem described by eq 2 is known as the concentration-dependent diffusion, in the case of which the diffusion coefficient is a known function of the concentration of the diffusing particle.12bApproximate analytical solutions,which are usually valid in a certain region of the variable T , can be obtained for example by applying the method of moments.Izb Here we shall make use of Fujita’s treatment;*3cf. also ref 12b which can be adapted as follows. In the time region 0 C T C 71, where 71 is time at which the advancing front of the diffusion reaches the center of the sheet, F(T)is proportional to the square root of time with the slope A,
+a@’/’/12)~~/~
(9)
and
a = 30[(2-y)-’+2(1 -y)(2-y)-’ln(2/y)]
(11)
When the slope A is known from the experiment, 0 can be calculated from the equation 1/(4Az)]
+ 1/24’ = 0
(12)
which follows from eqs 5 and 7. After evaluating a from eq 7, the ratio y can be found from eq 9 or 11. This procedure was tested by calculating a,8, and the slope dF/ds112from eqs 9,7, and 5, respectively, for several values of the ratio y. The results are shown in Figure 1, and it is seen that the agreement with numerical datagbis very good. In view of the limitations mentioned above, the procedure is applicable when the ratio of diffusion coefficients y > 1. However, it is also apparent from Figure 1 that, for small y, the initial slope dF/ dT1/2reaches a limiting value; Le., the slope dF/dtl/’ is controlled only by DA. The plot in Figure 1 can be approximated by an exponential fit: u = dF/dd/’ = 1.556[1 - 0.8695 exp(-l/y) 0.626 exp(-2/y) - 0.74 exp(-3/y)]; cf. the solid line in Figure 1, which indicates that for y 324 must be fulfilled. Of two positive roots, the larger one agrees better with the test computations.12b Two specific cases are treated in this work experimentally. In the first one ZA = zg = 1, and eqs 3 and 6 can be written as
11
0.0
+ (a- l08)@+ 3/2 =O
(6)
Materials. A Nafion 117 membrane 0.007 in. thick in H form (Aldrich, Lot No. JX-08602KW) and the reagent grade hydrochloricacid (Fluka, puriss.p.a.) aswellasthealkalimetalchlorides (Fluka, MicroSelect) and Ru(bpy)3C1~6H~O (Aldrich) were used as received. A redistilled water was used throughout the work.
Samec et al.
6354 The Journal of Physical Chemistry, Vol. 98, No. 25, 1994
I?
0.4 0.0
I
t
L
0
1
2
3
I 5
Figure 2. Ratio of the forward (uy) and the reverse (vr) rate d F / d ~ ' / ~ vs the square root of the ratio of diffusion coefficients ~ / D for A the exchange of an ion A (initially in the membrane) by an ion B (initially in the solution).
The nominal capacity of the Nafion 117 membrane corresponding to the equilvalent weight of 1100 is 9.09 X 10-4 mol equiv g-1. This value was confirmed by titration of the hydrogen ions released into a solution of 0.1 M KCl from the membrane, which was first equilibrated for 20 h in 0.1 M HCl. Apparatus. Praezitronic M 870pH meter (Germany) furnished with an analog output for a line recorder was used for continuous solution pH measurements in experiments involving hydrogen ion liberation from the membrane. When following the reverse process, i.e., displacement of an alkali metal cation by the hydrogen ion, the Varian SpectrAA 300 atomic absorption spectrophotometer served to monitor solution concentrations of the alkali metal cation. In experiments with Ru(bpy)3*+ complex cation the spectrophotometer Specord M42 was used for the determination of the residual complex concentration in the solution. These measurements were carried at 453 nm in a 2-mm cell. Procedure. To ensure the Donnan exclusion, the maximum electrolyte concentration we used in bathing membranes was 0.1 M. Prior to kinetic measurements involving alkali metal cations, the membrane was converted to H form or M form (M = Li, Na, K, Rb, Cs) by bathing in the 0.1 M solution of hydrochloric acid or appropriate alkali metal chloride, respectively, for at least 2 h. The membrane was then carefully washed in water, and after water droplets have been removed, it was immersed into 100 mL of the 0.1 M solution of the cation designated to substitute the ion present in the membrane. The solution was continuously agitated by a magnetic stirrer. Each measurement was repeated three times with a different membrane sample, all at the ambient temperature 20 f 2 O C . Fractional attainment of equilibrium F ( t ) was evaluated from the ion concentration determined as described below and the nominal membrane capacity, for a given volume of the solution and the weight of the membrane sample. Kinetic pH measurements of the hydrogen ion displacement by an alkali metal cation were carried out with H-form membrane pieces weighing from 0.1 to 0.2 g. The glass electrode was standardized by using a phthalate (pH 4) and phosphate (pH 7) buffers and calibrated with the standard HCl solutions. Figure 3 shows the changeof thesolution pH with time for the membrane immersed in the 0.1 M solution of LiCl or CsCl. After the injection of 150 Fmol of HCl in a blank experiment, the glass electrode shows a reasonably fast response, characterized by the response time less than about 1 s; cf. Figure 3. Measurements of the reverse process were performed with M-form membrane weighing from 0.025 to 0.035 g. The concentration of the released alkali metal cation was determined in 1-mL aliquots sampled in definite time intervals from the electrolyte solution. Compared to alkali metal ion-exchange reactions, the displacement of hydrogen ions by Ru(bpy)32+ was found to proceed
20
0
40
60
80
100
t / s
Figure 3. Solution pH vs time t after immersing an H-form Nafion membrane (0.15 g) into 100 mL of 0.1 M LiCl (solid line) or 0.1 M CsCl (dashed line) or after injecting 150pL of 0.1 M HCl into 100 mL of 0.1 M LiCl (dashed and dotted line). Stirring rate 150 rpm.
0.5
1
-I
P
0 0
0.0 0
10
20
30
Figure 4. Fractional attainment of equilibrium F vs the square root of
time 2: (0)Li+ion exchange into an H-form Nafion and ( 0 )the reverse exchange. Bars indicatethe standard deviationfrom three measurements. with much lower rate. The necessity to carry out long-term measurements with low-volume samples excluded pH measurement as a monitoring method. Instead, visible spectrophotometry was used, taking advantage of the high extinction coefficient of the ruthenium complex. Two pieces of an H-form Nafion weighing 0.022 g each were dropped into two flasks containing 100 mL of 2 X 10-4 or 5 X 10-4 M Ru(bpy)3Cl~solutions. The aliquots of 1 mL were sampled to determine the change in the complex concentration at time periods ranging from 1 to 100 h. A similar procedure was used to follow the reverse process, which however was so slow that even after 30 days from placing a Ru(b~y)~Z+-saturated membrane into 3 mL of 0.1 M HCl there were no detectable amounts of the ruthenium complex found in the solution.
Results and Discussion Stoichiometry and Thermodynamics of the Ion Exchange. Kinetics of the forward and reverse ion exchange is illustrated in Figures 4,5, and 6 for Li+, Cs+, and Ru(bpy)32+, which is the sequence of ions with the steeply increasing selectivity coefficient for Nafion.5b.10 For alkali metal cations (Figures 4 and 5), the fractional attainment of equilibrium F(r) reaches the maximum value of 1.0 f 0.1, in accordance with the assumption of a 1:l
Ion Transport Parameters in a Nafion Membrane
The Journal of Physical Chemistry, Vol. 98, No. 25, 1994 6355
1.o
f * h
LL
?If
0.5
%
0.a
#
10 t'/2
,
20
30
s'/2
Figure 5. Fractional attainment of equilibrium F vs the square root of time t: (0)Cs+ion exchange into an H-form Nafion and ( 0 )the reverse exchange. Ban indicatethe standarddeviationfrom three measurements.
0.6
t
Oa2 O 0.0 1
0
1000 t1/2
,
2000
I 3000
4000
*1/2
Figure 6. Fractional attainment of equilibrium F vs the square root of time t for Ru(bpy)l2+ ion exchange into an H-form Nafion membrane. Ban indicate the standard deviation from three measurements.
stoichiometry. On the other hand, the hydrogen ion exchange by R~(bpy)3~+ ion (Figure 6) seems to reach a limit, which is somewhat less than unity (assuming 1:2 stoichiometry). This result suggests that there may exist regions in Nafion, into which this ion is not partitioned. In electrochemical ~tudies,k.c.~i the nonhomogeneous distribution of Ru(bpy)12+ has been assumed to be the reason why, in contrast to other ions, the apparent diffusion coefficient was independent of the ion concentration. Long-time equilibrium required in the forward process (about 100 days) is compatible with the observed behavior of the thin Nafion films.k.sb A Nafion film about 2 pm thick was shown to be fully loaded with Ru(bpy)32+ ion after about 15 minek In another study?b a 1-pm Nafion film was found to be fully loaded only after a week. Since the rate dF/dN2is inversely proportional to the membrane thickness (cf. eq 5), for a membrane 200 pm thick these times should be multiplied by a factor of 104, yielding indeed some 100 days for accumulation. A very low rate of Ru(bpy)3*+ ion eluting from Nafion is apparently due to an unfavorable ratio of the counterion ooncentrations at the membrane boundary; vide infra. From the thermodynamic point of view, ion-exchangereaction 1 is characterized by the equilibrium constant K#,I0
where n = 1,2, u1 is the ion activity, and xi is the equivalent ionic
fraction in the membrane. Other author@ have introduced the selectivity coefficient K#' = (K#)". For alkali metal cations, the values of the equilibrium constant K# are equal to'& 0.579, 1.22, 3.97, 6.26, and 9.11 for Li+, Na+, K+,Rb+, and Cs+, respectively, at 25 OC and for a Nafion 120 membrane. The selectivity coefficient KN,M = 5.7 X 106 was reportedsb for the R~(bpy)~~+/N ion a +exchange. When multiplied by the value of 1.22 for Na+ referred to above, it yields K H ~ =' 7 X 106 for the equilibrium Ru(bpy)12+/H+ion exchange. These data imply that the boundary conditions given by eq 4 are realistic for ion exchange (la) and the forward exchange (lb). Indeed, because the maximum concentration of proton released from the membrane in the forward ion exchange was less than 2 mM, the ionic fractions on the membrane side of the boundary should fulfill the inequality XM/XH > 29 for alkali metal cations and X M / X H ~> 8.7 X lo8 for the R ~ ( b p y ) , ~ion. + Analogously, the maximum concentrationof the alkali metal cation released during the reverse process was less than 0.27 mM, and hence XH/XM > 41. On the other hand, the reverse exchange of Ru(bpy)02+by the hydrogen ions from the solution is not favored thermodynamically. Based on the theoretical capacity, the concentration of the released ruthenium complex cation in the solution should not exceed 1 mM and X H ~ / X M > 1.4 X 10-6, which is too low to ensure a measurable exchange. RateDetermining Step. Besides the ion exchange being favorablethermodynamically,eq 4 assumesthat the ion-exchange kinetics is controlled by the membrane ion transport; i.e., the contributions from the interfacial or the solution ion transport are negligible. There has been no evidence that the interfacial ion transport across a Nafionlelectrolyte solution interface could be the ratedeterminingstep. The absenceof the kineticregion on impedance plots is typical for the transfer of alkali-metal cations across a Nafion membrane/solution interface." Although the impedance analysis of Ru(bpy)p2+ion transport at frequencies higher than about 100 Hz reveals a slow interfacial ion transfer with an apparent heterogeneous rate constant of 1.2 X 10-4 cm s-I,l4 on the time scale of the present experiments this step can hardly play any role. Owing to the large difference between the ion diffusion coefficients in solution and in a Nafion membrane,Ia.b+s*as well as to the convective contribution due to stirring, the solution ion transport should not be a significant factor, too. We examined the role of the solution transport by measuring the Li+/H+ and Cs+/H+ ion exchange at several stirring rates of 70, 100, 150, and 200 rpm and by measuring the Ru(bpy)s2+/H+ion exchange at various ionconcentrationsin the bathing solution. Both effects were quite negligible. Diffusion Coefficients of Alkali Metal Cations. Table 1 summarizes the initial slopes dF/dtl/2 for the ion-exchange processes studied. On going from Li+ to Cs+,the slope for the forward process increases and that for the reverse process decreases. According to the theory of the coupled diffusion, the observed ratio v& > 1 should result from the superior proton mobility in Nafion; cf. Figure 2. In such a case, the succession of the initial slopes should correspond to the succession of diffusion coefficients for alkali metal cations, irrespective of the actual value of the proton diffusion coefficient; cf. Figure 1. Since the assumption that 7 = D H / D M>> 1 appears to be plausible, the ion diffusion coefficients can be evaluated with the help of eq 12 for the forward exchange and eq 13 for the reverse exchange. In the former case, the value of the proton diffusion coefficientin Nafion must be known. This parameter is difficult to acquire by the radiotracer method, because the hydrogen isotope can be partitioned among various hydrogen species present in the membrane. An indirect evaluation yields DH = 3.5 X 10-6 cm2 s-l at 20 OC.lC The nominal thickness of a dry Nafion 117 membrane is 175 pm. In a swollen state, its thickness was found
Samec et al.
6356 The Journal of Physical Chemistry, Vol. 98, No. 25, 1994 TABLE 1: Illitial Slopes dF/dP/* and Diffusion Coefficients D of Cations in a Nafmn Membrane Evaluated from the Forward and Reverse (in Parenthesea) Ion-Exchange Measurements at the Ambient Temperature, 22 f 2 OC 107D/cm29-I 102(dF/dt1/2)/ ion 8-w this work'J previous work wate+ Li+
Na+
K+ Rb+
cs+
9.1 (9.2) 14.2 (8.6)
1.7 (3.7) 6.8 (3.0)
14.1 15.5 16.1 (4.0)
6.8 9.3 10.6 (0.59)
0.022
Ru(bpy)32+
2.8 X 1od
9.44c 11.2d 9.83e 0.52c 1.70" 1.88e (2-3) X l ( r f
103 133 196 207 205
Assuming 41= 3.5 X l o d cm2s-l,le b From limiting ionic mobilities OC.25 C Self-diffusion coefficients at 25 OC.lb Self-diffusion coefficients at 25 O C after two years in a as-received dry Nafion 12O.Ic e Self-diffusion coefficients at 25 OC in freshly rcccived sample of Nafion 120.lcf From membrane permeation chronoamperometricmeasurements." a
at 25
to be in the range 190-210 pm depending on the cation Therefore, the value of 1 = 100 pm was used in calculations. Diffusion coefficients are given in Table 1. It is seen that the diffusion coefficients derived from the forward exchange measurements follow the same sequence as in an aqueous solution, Le., Li+ < Na+ < K+ < Rb+ < Cs+, which is opposite to the sequence of the self-diffusion coefficients in a Nafion membrane, H+le > Na+ Ib > Cs+ lb. While diffusion coefficients of alkali metal cations in H-form Nafion tend to be higher than the correspondingself-diffusion coefficients, those derived from the reverse ion exchange follow the opposite trend. Although in the latter case the values of the diffusion coefficients of Na+ and Cs+ are approximately 3 times lower, their ratio is about the same, DNalDCa
z=z
5.
There are several factors which can be responsible for the difference between transport behavior of Nafion in self-diffusion and ion-exchange measurements. One factor can be the pretreatment of the membrane having an effect on its morphology. Another factor is the difference in water content in Nafion prior to forward and reverse exchange measurements due to the presence of more hydrophilic proton or a more hydrophobic alkali metal cation, respectively. Finally, the cross effects associated with mutual interactions of exchanged ions or ions and solvent should be considered. It has been known that the properties of the as-received Nafion membrane and the membrane pretreated by drying,5asolvent exchange,la-b,5aJObor heating10b.c can differ. In particular, when Nafion is pretreated by boiling in water, an expanded form of the membrane is produced with an increased water to exchange site mole ratio.10b From this point of view, the present results should be comparable with the self-diffusion data reportedlbvc for a nonexpanded form of Nafion. On the other hand, the morphology of the present batch of Nafion may not be exactly the same as in other studies. A change in technology has been supposed to be the reason why the mobility of Na+ in Nafion samples of a more recent manufacture did not change, whereas that of Cs+ increased more than three times.Ic The water to exchange site mole ratio in Nafion depends on the nature of the ion and amounts to 16.7, 14.3, 11.9, 8.8, 7.7, and 6.6 for H+, Li+, Na+, K+, Rb+, and Cs+, respective1y.lh.b In accordancewith the free-volumeapproach to diffusion,ISa change in water content can have an effect on the diffusion coefficient D in a polymer,16
D = Doexp[-bV,/( 1 - v,)] where DO is the aqueous diffusion coefficient, V, is the volume fraction of polymer in the water-swollen material, and b is an
1
2
3
4
5
6
VJCl -VJ
Figure 7. Logarithm of the ratio of the ion diffusion coefficients in the membrane and in the solution vs polymer fraction function for M-or H-form Nafion 117: Li+ (A), Na+ (v),K+ ( O ) , Rb+ (A),and Cs+(0). Also displayed is the value for H+(e) (solution value D = 9.33 X cm2s - I ) ~ ~ calculated from the coefficient of the self-diffusion,le which has been the basis for the analysis of the forward ion-exchange measurements.
ion-dependent constant. According to eq 15, the ion diffusion coefficient should increase with an increasingwater content. This effect was demonstrated for ion transport in various poly(styrenesulfonate) membranes" and also in Nafion, the samples of which with varied water contents were prepared by using Na+ and Cs+ mixed ionic forms or by heating the sample in water.lc However, important differences were seen, which were ascribed to the ion cluster morphology of Nafion.lC*I8Apart from the inverse ratio of self-diffusion coefficients of Na+ and Cs+,the effect of water content in Nafion is much less pronounced than in poly(styrenesu1fonate) membranes, in particular for Cs+.lCThe lower self-diffusion coefficient of Cs+ compared to that of Na+ in Nafion was proposed to be due to a different, more tortuous, diffusion path.Ic Unlike the Na+ ion, Cs+is probably partitioned mainly into the interfacial zone of Nafion, containing polymer sidechain material,a smaller amount of water, and some sulfonate exchangesites which have not been involved in forming clusters.1C In ion-exchangemeasurements, the initial and final values of the water to exchange site ratio differ, and the membrane water content should vary. However, the initial water flow associated with the ion exchange can be compensated by the electrosmotic flow in the opposite direction; vide infra. Therefore, the diffusion coefficients derived from the initial ion-exchange rates probably correlate with the initial water content. As in the self-diffusion study,lctheeffectofthewater content should beseen by comparing the diffusion coefficients of Na+ or Cs+ions in the mixed proton and alkali metal ion forms, when either proton or the alkali metal cation prevails. The ratio of the diffusion coefficients DIDO in eq 15 was calculated from the data in Table 1. Values of Vp/( 1 - V,) were calculated using water to exchange site ratioslhub referred to above, the molar concentration of exchange sites in the swollen polymer corresponding to the equivalent weight of 1100, and the partial molar volume of water of 17 mL mol-', as previously described.lc The results are displayed in Figure 7. Surprisingly, the data for Na+ or Cs+ fall on a single straight line indicatinga significant effect of the water content,quite analogous for both ions. When the data obtained for Li+, Na+, and Cs+in M-form Nafion alone are inspected, the picture is the same. On the other hand, the data obtained for ions in H-form Nafion group together, except for Li+. Thus, in contrast to the selfdiffusion measurements, no difference in the mechanism of the transport between Na+ and Cs+ ions is seen, and a single value of the parameter b characterizesthe transport of various univariant ions. Rates of the forward ion exchange for Na+ and Cs+in Table 1 are also at variance with an earlier result for the as-received
Ion Transport Parameters in a Nafion Membrane H-form Nafion 12LS8 Times necessary to reach 90% exchange for Na+ and Cs+were found to be 0.03 and 0.67 h, respectively, of which only the former value is comparable with present data. However, such a behavior is consistent with the results of the self-diffusion measurements.lc Indeed, since the diffusion coefficient of Cs+was found to be independent of the water content,lc a higher value of the water to exchange site mole ratio in an H-form membrane would hardly cause a change in the rate the Cs+ ion transport. Therefore, the sample of the H-form Nafion used in this study seems to offer a faster diffusion path for hydrophobic ions like Cs+.There is an evidence that the presence of the hydrogen ion in Nafion membrane could lead to an enhanced penetration of water molecules into interfacial region or channels connecting ion clusters, thereby facilitatingthe ion transport. Earlier infrared spectroscopic studies have indicated the existence of two water environments in a Na-form Nafion, which are presumably associated with ion cluster and interfacial or fluorocarbon regions.lg On theother hand, IR spectra weremuch less indicative of such separation in an H-form membrane.*O More recent 1 ’ 0 NMR study of the as-received H-form Nafion has not confirmed the existence of two distinct water regions, though it has been admitted that this phenomenon may not bedetectable byNMR.21 Transport properties of an expanded H-form Nafion have led the authorslf to a conclusion that a cluster network morphology’s does not provide an adequate representation of the transport and that all the pore diameters inside the membrane are about 60 A. An interesting feature of the coupled ion diffusion, which can be manifested in the differencebetween the diffusion coefficients from the forward and reverse ion-exchange measurements, is a hysteresis in water This effect is caused by the electroosmosis,22which tends to preserve the initial water content of the membrane. Owing to the positive surface excess charge on the solvent inside the membrane, which compensates the negative fixed charge, the electroosmotic flow will be opposite to the gradient of the electrical potential. According to the theory of the coupled diffusion? the membrane transport of the counterion with the lower diffusion coefficient is accelerated and that with the higher diffusion coefficient is retarded. Acceleration or deceleration is brought about by the gradient of the electrical potential, which has the same.or oppositesign as the concentration gradient, respectively; cf. the Nernst-Planck equation
where cp is the electrical potential. Thus, when an alkali metal cation diffuses into an H-form Nafion, the electrical potential gradient is negative and the electroosmotic water flow is positive in the direction toward the membrane interior. At the same time, water should flow out of membrane due to a lower water content in a M-form Nafion. A similar compensation in the water flow is predicted for the reverse ion exchange. The cross effect arising from the simultaneous transport of the two counterions in the membrane can be treated within the framework of nonequilibrium thermodynami~s.~3 As outlined in the Appendix, eqs 2 and 3 apply, but the coefficient of the coupled diffusion DABis given in terms of the phenomenologicalparameters ZB = 1. lij, e.g., for Z A
In order to solve eq 2 or to integrate in eq 6,1, must be expressed as functions of ion concentrations. While the assumption that the diagonal coefficients are proportional to corresponding ion concentration is plausible (see Appendix), an analogous relationship for the cross-term coefficient IAB has no justification.24 Nevertheless, somewhat lower values of diffusion coefficients inferred from reverse ion-exchange measurements suggest that
The Journal of Physical Chemistry, Vol. 98, No. 25, I994 6357
AB should have a positive value, in agreement with data for concentrated solutions of strong electrolytes.23b Diffusion Coefficientsof Ru(bpy)o*+. Diffusion coefficient of the R u ( b p ~ ) 3 ~ion, + 2.8 X 10-13 cm2 s-l, was evaluated from the forward ion-exchangemeasurements by using the procedure based on eqs 7.11, and 12. The initial slope dF/dtll* of 2.2 X 1 V s-V2 in Table 1 corresponds to F related to the nominal capacity of the Nafion membrane. Since the actual limit appears to be about 60% of the nominal value, the slope was divided by the factor of 0.6. Owing to 1:2 stoichiometry, the ratio of the diffusion coefficients y was calculated from eq 11. The diffusion coefficient of this ion was evaluated also from chronoamperometricmeasurementsof the Ru(bpy)3” permeation through the Nafion cast film,&the morphology of which however can differ from the morphology of the as-received Nafion membrane. In the permeation experiment, R~(bpy)3~+ was assumed to diffuse into a homogeneous membrane without considering the other counterion.2C An equivalent model would involve the exchange of two counterions with an equal diffusion coefficient. By using the same model, the ion-exchange rate reported in this work provides a comparablevalue of the diffusion coefficient. Namely, the initial slope dF/dr112 = ( 2 / d 2 ) ( D / P)l12 = 3.7 X 1V s-Il2 corresponds to D = 1.1 X 10-11 cm2 s-I, which is about the value reported.2c Obviously, this model is not realistic, and the value of the diffusion coefficient of Ru(bpy)32+ ion in the as-received Nafion is much lower. Conclusions Theoretical expressions were derived for the evaluation the ion diffusion coefficient in a permselective membrane from ionexchange measurements. The exchangeof most alkali metal cation for proton in a Nafion membrane was found to be faster than the reverse process. Such a behavior is an inherent feature of the coupled ion diffusion, which is manifested here thanks to a higher proton mobility. However, the quantitative data analysis suggests that the difference in the forward and reverse ion-exchange rates is too big. In fact, the forward or the reverse rates yield a higher or a lower diffusioncoefficientof the alkali metal cation, respectively, than the corresponding coefficient of the self-diffusion. A lower ion mobility can be ascribed to the cross effects arising from the simultaneous transport of two counterions. A higher mobility is probably due to a change in transport properties of Nafion in the presence of proton, which supports a more extensive penetration of water molecule into hydrophobic regions of Nafion. While the rates of R ~ ( b p y ) ~ 2exchange + into the as-received Nafion and the Nafion cast films are comparable, the diffusion coefficients evaluated from ion-exchange measurements with the help of theory of the coupled ion diffusion are considerably lower. Thus, the role of the coupled ion diffusion in a Nafion membrane has been demonstrated also in this case. On the other hand, it should be noted that this feature of the simultaneous transport of two ions in a permselective membrane is tied up by the zerocurrent condition. When this condition is relaxed, e.g., in an electrochemical experiment, a different diffusion-migration problem is to be solved.
Appendix Followingthe treatment applied to transport processes in binary electrolyte systems,23bthe two Nernst-Planck equations for ions A and B in a permselective membrane must be replaced by two phenomenological relationships (cf. also ref 24)
-JB
= ~ B A ( ~ P A / ~ X+) l B B ( a ~ B / a x )
where lij’S are phenomenologicalcoefficients,~ i )electrochemical s
6358 The Journal of Physical Chemistry, Vol. 98, No. 25, 1994
potentials, and Jl's the relative ion fluxes,
where c, is the solvent concentration; Ji* and Js* are the ion and solvent fluxes, which are referred to a common, arbitrary frame.23b The number of independent coefficients It, is reduced when using the Onsanger reciprocal relation, IAB = IBA. Provided that the zero-current and electroneutrality conditionsapply as above, and variations in ion activity coefficients are neglected, the coupled ion transport in the solvent fixed frame (J,' = 0) can be described again by eqs 2 and 3 with
Since diagonal phenomonological parameters can be related to the self-diffusion coefficients DtO through 1, = DAocA/RT and = D B O C B / R Tin, ~a ~limit ~ / A B = ~ B A= 0 and DA (RT/CA)I, = DAO and DB = (RT/CB)lBB= DBO.
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