J . Phys. Chem. 1987, 91, 3606-3612
3606
Spectrophotometric Measurements of Cation Transport in Nafion C. R. Chris Wang, Jerzy W. Strojek,? and Theodore Kuwana* Center for Bioanalytical Research, University of Kansas, Lawrence, Kansas 66046 (Received: October 27, 1986; In Final Form: February 13, 1987)
The construction and use of a novel Nafion-containing flow-through spectrophotometric cell are described. The cell allows the optical monitoring of ions transporting into and out of a Nafion membrane for the simultaneous determination of D, K , and kd values. Experimental measurements and digital simulations are used to test possible mechanisms for this diffusion-interaction process. The values of D for Co2+,Cu2+,and Cr3+,diffusing from pH 0 solution into Nafion, are as high as -3.2 X IO" cm2/s. The D values decrease at higher pH's.
Introduction Since Adams and Holmes' discovered the ion-exchange properties of phenol-formaldehyde resins in 1935, much work has been directed toward the production of ion-exchange membranes. Among these membranes is DuPont Nafion, which plays an important role as an electrolytic separator in electrochemical applications,2 particularly in the chlor-alkali industry. The indusas well as analyticalG8 properties of Nafion have been well documented. Recently, electroactive polymers have been viewed as candidates for developing useful macromolecular electronic devices in the f ~ t u r e . Storage ~ of electric charge and catalysis of electrochemical reactions have drawn the attention of electrochemists to electroactive polymeric film electrodes. Nafioncoated electrodes conduct current both by electron self-exchange and physical diffusion of electroactive counterions.10 They also show enhanced ~ e l e c t i v i t y and ~ * ~sensitivity, ~ which makes them possibly useful as electroanalytical sensors. Nafion, a cation-exchange membrane, consists of a hydrophobic matrix and hydrophilic ionic cluster regions. Much effort has been made to model the ionic cluster morphology of this material.12 The transport of cations through Nafion is, therefore, a basic concern when attempting to optimize its performance in current-conducting and separation systems. Experimentally, many kinetic parameters of cations associated with Nafion have been reported. For example, self-diffusion coefficients for the sodium ion and cesium ion have been measured by using radioactive tracers.I3 Apparent diffusion coefficients for R ~ ( b p y ) ~me~+, thylviologen (MV2+), and dopamine (DA') have been reported to be as low as ca. cm2/s8,14*15 by using the conto trolled-potential electrochemical method. The goal of the work reported herein was to apply a method which combined spectrophotometric measurements with digital simulations to investigate the process of cation transport through Nafion. The diffusion coefficient (D),distribution coefficient ( K ) , dissociation rate constant (kd), and order of reaction ( z ) for the transport of some chromophoric cations through Nafion were determined by this method. We have limited our work to a system controlled by the equilibria depicted in Figure 1. In this figure Nafion is represented by R-S03H, where R is the nonpolar backbone of the Nafion polymer. For this study the most important equilibrium constants are K3, the equilibrium of the metal ion-sulfonate complex, and K , and K z , the competitive equilibria between protons, or sodium ions, and metal ions at the polar side-chain groups of Nafion, respectively (see below). The parameters mentioned above can be calculated from measurements of metal ion transport into or out of the Nafion membrane. We decided to conduct these measurements by applying UV-visible spectrophotometric methods, and chose Co2+, Cu2+,and Cr3+as the sample cations. Briefly, the experimental 'Current address: Department of Chemistry, Silesian Technical University, 44100 Gliwice, Poland.
procedure is as follows: after both sides of a Nafion sheet are contacted with a flowing solution containing the cation of interest (seeExperimental Section), a monochromatic light beam is passed through it. If the solution flow rate is high enough that it maintains a constant absorbance outside the Nafion in the spectral cell, then diffusion of metal ions into Nafion from the flowing solution causes a change in absorbance. Based on data from the l i t e r a t ~ r e ~ *and ' ~ " our measurements, we assumed the following: I. The transport of the cations through the Nafion is controlled by a series of processes, as shown in Figure 2, consisting of the self-diffusion of cations and chemical interaction between cations and ion-exchange sites in ionic cluster regions of Nafion. This is described by a D-C mechanism: (a) Diffusion of a cation, Mn+, into and out of the membrane is described by Fick's laws and controlled by values of the diffusion coefficients Dinand D,,,; (b) Chemical interactions of the cations, M"', with the ionexchange sites of nonprotonated (or protonated) Nafion are controlled by the heterogeneous reactions: M"+
+ n(R-SOy)
M"+ + n(R-S03H)
%
+ (R-S03),M
(R-S03),M
+ nH+
K3
(la)
K,
( I b)
where R-S03-, R-S03H, and (R-SO3),,M represent the ion-exchange sites of the Nafion in unprotonated, protonated, and metal ( 1 ) Adam, B. A.; Holmes, E. L. J . Soc. Chem. Ind.,London 1935,54, IT. (2) Perfuorinated Ionomer Membranes; Eisenberg, A., Yeager, H. L., Us.; American Chemical Society: Washington, DC, 1982; ACS Symp. Ser. No. 180, Chapters 14-19. (3) Grot, W. Chem. Ing. Tech. 1972, 44, 167. (4) Mauritz, K. A.; Hora, C. J. Polym. Prep. 1978, 324. (5) Manning, M. J.; Meisheimer, S. S. Ind. Eng. Chem. Fundam. 1983, 22, 311.
(6) Krishnan, M.; Zhang, X.; Bard, A. J. J . Am. Chem. SOC.1984, 106, 7376.
(7) Szentirmay, M. N.; Martin, C. R. J . Am. Chem. SOC.1984, 56, 1898. Szentirmay, M. N.; prim, N. E.; Martin, C. R. J. Phys. Chem. 1985,89, 3017. (8) Nagy, G.; Gerhardt, G. A.; Oke, A. F.; Rice, M. E.; Adams, R. N.; Moore, R. B.; Szentirmay, M. N.: Martin, C. R. J. Electroanal. Chem. 1985, 188, 58. (9) Chidsey, E. D.; Murray, R. W. Science 1986, 231, 25. (10) Martin, C. R.; Freiser, H. Anal. Chem. 1981, 53, 902. ( 1 1 ) Buttry, D. A,; Anson, F. C. J . Am. Chem. Soc. 1982, 104, 4824. (1 2) Perfluorinated Ionomer Membranes, Eisenberg, A., Yeager, H. L., Eds.; American Chemical Society: Washington, DC, 1982; ACS Symp. Ser. No. 180, Chapters 7-1 3. (1 3) Perfluorinured Ionomer Membranes; Eisenberg, A., Yeager, H. L., Eds.; American Chemical Society: Washington, DC, 1982; ACS Symp. Ser. No. 180, Chapter 4. (14) Martin, C. R.; Rubinstein, J.; Bard, A. J. J . Am. Chem. SOC.1982, 104, 4817. (15) Leddy, J.; Bard, A. J. J . Electroanal. Chem. 1985, 189, 203. (16) Adams, R. N. Electrochemistry at Solid Electrodes: Dekker: New York, 1969. (17) Crank, J.; Nicholson, P. Proc. Cambridge Philos. Sor. Math. Sci. 1947, 43, 50.
0022-36541871209 1-3606$01 SO10 0 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 13, 1987 3607
Cation Transport in Nafion
Figure 1. Main eauilibria during the process of cations exchange into Nafion.
-I * 'H
4
Solution
D In . )
c
Figure 3. Diagram of spectral cell. 1, metal holders containing optical window; 2, Pyrex plates with 2-mm-diameter holes; 3, O-rings; 4, gasket (DuPont Kalrez, 0.25-mm-thick sheet KO-5012 type); 5, Nafion sheet with a hole at the bottom to allow solution flow; 6, CTFE female h e r connector, Hamilton (32834 type).
Mn+
Dout n-l
n
If-----Figure 2. Schematic representation of the D-C process for cations M"' diffusing into Nafion from both sides. The Nafion membrane has been divided into n parts.
ion-sulfonate complex forms, respectively. The equilibrium constants K3 and K , are described in Figure 1, and K , = k[/k,'. The concentration of R-S03H is determined by the equilibrium R-S03H
R-S03-
+ H+
KH
(2)
where K H is the equilibrium constant. 11. Some cations are bound to the polar groups of Nafion, while others are located in the hydrophilic volume of Nafion as "free" cations. Cations bound to the ion-exchange sites of the membrane, (R-S03),M, do not freely diffuse through the membrane. Only may diffuse in the membrane. the free cations, Mn+, 111. Diffusion of protons in Nafion is very fast and assumed to be negligible in consideration of the overall process (DH+>> DMn+).Additionally, as cations diffuse into Nafion, an excess of anions develops in the solution. This causes a potential drop between Nafion and the solution.18 We assume that, at our low pH experimental conditions, protons may quickly diffuse into or out of the Nafion and eliminate this potential drop. IV. There was no change in the A, of the three cations studied before and after their sorption in Nafion. Therefore, we assume that sample cations in aqueous solution and sample cations in and 6 . From Nafion, whether free or bound, have the same A, this, the Nafionsample cation complexes must be weakly linked; this may be confirmed by means of IR,I9 NMR,20 or other methods.2',22 On the basis of the above assumptions, a special spectral cell was fabricated for these measurements and computer programs were devised for digital simulation of the processes occurring in the cell. The Crank-Nicholson was used and modified to generate a program (CNC) which describes processes controlled (18) Feldberg, S . W., personal communication. (19) Quezado, S.; Kwak, J. C. T. Can. J. Chem. 1984,62, 958. (20) Komorowski, R. A,; Evans, T. R. Polym. Prep. 1984, 341. (21) Rubinstein, 1. J. Electroanal. Chem. 1985, 188, 227. (22) Pan, H. K.; Yarusso, D. J.; Knapp, G. S.; Pineri, M.; Meagher, A.; Coey, J. M. D.; Cooper, S. L. J. Chem. Phys. 1983, 79, 4736. (23) Britz, D. Digital Simulation in Elecrrochemistry; Lecture Notes in Chemistry, Vol. 23; Springer-Verlag: Berlin, 198 1.
by both diffusion and chemical interaction. In the C N C program, the heterogeneous chemical interaction of cations with ion-exchange sites inside the Nafion is described by k
Mi$M,
K
(3)
with parameters, K,kd, and z, where Mi and M , represent "free" cations and "bound" cations in Nafion, respectively. The distribution coefficient, K , is given by K = [M,]/[Mf] (also K = kd/kf). The dissociation rate constant is kd and z is the order of reaction, defined from the assumption that dissociation rate = d[Mf]/dt = kd[M,]'
- kf[Mf]'
(4)
where kd = k{[H+]" and kf = kC[R-SO,H]". The diffusion process of counterions inside the Nafion (for example, in the "washing" experiment-see Experimental Section) is described under boundary conditions as follows: [Mf] = 1 at t = 0, 1 I i I n; [Mf] = 0 at t > 0, i < 1; [M,] = 0 at t > 0, i > n. Counterion self-diffusion, as described by Fick's laws, as well as the interaction of counterions with ion-exchange sites are used separately and in sequence in our C N C simulation program. The concentration of counterions in Nafion is the sum of the concentrations of both bound and free ions. The flux of counterion, based on the previous assumption 11, is contributed only by the concentration gradient of the free cations, which are in equilibrium with the bound cations in the membrane within each very short time interval and distance. In addition, subroutines were written to store experimental data sets and compare them to the digital simulations.
Experimental Section Nafion membrane type 117 (E.W. 1loo), produced by E.I. DuPont de Nemours & Co., was used throughout. Solutions of different pH values (0-3), with or without sample metal ions, were prepared by using sulfate solutions only. The pH value of each solution was examined with a pH meter (Corning 130). Chromium sulfate (Allied Chemical), cupric sulfate and cobalt sulfate (J.T. Baker Chemical Co.), sodium sulfate (MC/B), and H2S04 (Mallinckrodt) were all reagent grade and were used without further purification. Buffers of pH 2, 4, and 6 (Micro Essential Laboratory) were used for the Nafion thickness measurements.
3608 The Journal of Physical Chemistry, Vol. 91, No. 13, 1987 W C S
A
E E
Wang et al.
0.226
(IC)
“ 0.223 c
1000
Q)
2 0.221
v,
g 0.218
500
c
2 0.216 c
0 0.213
v) v)
0)
E
C
x
.
0.211
Oe208 . 0.206
.
I
J
.
-2
-1
100
Cr (0.12 U ) Co (0.70M 50 -
.
0
.
.
I
2
*
3
.
4
*
c
5
6
PH Figure 4. Thickness of Nafion sheet as a function of pH under the following conditions: H2S04solutions with different pH values; (A) 1 M Na2S0, in H2S04solutions of different pH; (0) various buffer solutions (pH 2): tartaric acid, phosphoric acid, and potassium biphthalate; pH 4: potassium acid phthalate; pH 6 : sodium phthalate (dibasic) and potassium phosphate (monobasic). ( 0 )
t O.l6 0.14
pH.2
t
”.”..
IO
a 0
I
2
3
PH
Figure 6. Ratio of cation concentration in saturated Nafion (C,) to the cation concentration in solution used to attain saturation (C,) as a function of Cs and pH.
~
0
1
2
3
4
5
6
7
6
9
time (min)
Figure 5. Saturation curves of Cr3*(solution concentration 0.01 M) into Nafion as a function of time and pH. SYBRON/Barnstead NANOpure I1 water was used throughout. Spectrophotometric measurements were performed on a UVvisible recording spectrophotometer (Varian, Model DMS-90). Each experiment consisted of two parts: ( I ) saturation of the Nafion with metal ions and (2) removal of metal ions from Nafion, or ”washing out”. The home-made spectral cell is shown in Figure 3. It contains two solution regions (of total thickness r ) and a = Nafion region (of thickness d ) . Monochromatic light (A,, 589.5 nm for Cr”, 812 nm for Co2+,and 510 nm for Cu2+) is passed through the cell windows during the experiment. The measured absorbance is thus the sum of the absorbance caused by metal ions located in both solution and Nafion. Absorbance vs. time plots for both “saturation” and “washing” experiments were obtained with an X-Y recorder (Houston Instruments, Model 2000) connected to the output of the spectrophotometer. All of the experiments were carried out at 20 1 OC. The thickness of the Nafion membrane (d)was measured with a micrometer. After soaking the Nafion sheet in a solution of a given pH for about 30 min, the thickness of the Nafion sheet was recorded. These measurements were performed in three kinds of solutions: (1) sulfuric acid solutions with different pH values; (2) the same solution as (1) but with 1 M N a 2 S 0 4added; (3) several different non-sulfate buffer solutions. The total thickness of the solution ( r ) in the cell was calibrated
Figure 7. Three-dimensional plot of the relationship between C,, CS’l2, and pH for Cr3+. with K21rCI,, since this colored anion did not diffuse into Nafion within the 30 s calibration period. The K2IrCI6solution (pH 2) was introduced separately into both the spectral cell and a I-mm optical cell. By comparing the absorbance of the spectral cell with that of the optical cell, we calculated the total thickness of the solution regions of the spectral cell ( r ) to be 0.54 mm. Saturation experiments were carried out by first injecting 1-2 mL of a metal ion solution of known pH (0-3) from a syringe into the cell, which had been previously filled and equilibrated with sulfuric acid solution of the same pH. A flow of fresh metal ion solution was maintained through the cell at a constant rate (15 mL/min) by a tubing pump. The Reynold number N(Re) for the flow in the solution regions of the spectral cell was ca. 7830 in most cases (length = 2.9 cm, flow velocity = 30.86 cm/s, w / p = 1.18 cSt). Since the metal ion solution was constantly renewed, we assumed that the absorbance due to metal ions in solution regions of the spectral cell remained constant. Therefore, changes in measured absorbance were an indication of changes in the amount of metal ions in the Nafion. The absorbance vs. time was recorded immediately after the injection, until the absorbance became constant, at which time the Nafion had become saturated
Cation Transport in Nafion
The Journal of Physical Chemistry, Vol. 91. No. 13, 1987 3609 to
I
I \
_------- ----
PH 3 Figure 8. Normalized concentration profiles for Cu2+(solution concentration 1.1 M) at different pH and time t in Nafion membrane: (-) concentration profile for free cation Mf; (---) concentration profile for bound cation M,. to = initial time. x axis, distance within the Nafion membrane.
with metal ions. Data from absorbance vs. time graphs were read and stored on a floppy disk. Washing experiments followed a similar procedure, except that sulfuric acid solution was used to "wash" the metal ions out of Nafion which had been saturated with metal ions. About 1-2 mL of sulfuric acid solution with the same pH as that in the saturation experiment was first quickly injected from a syringe into the cell. Then the same sulfuric acid solution was pumped through the cell at a constant flow rate (15 mL/min). In this case we assumed that ions diffusing out of the Nafion into the washing solution were diluted and removed so rapidly that the absorbance of solution regions would remain essentially zero, and the measured absorbance was due only to ions within the Nafion. After sufficient absorbance vs. time data were collected for comparison with simulation results (usually after 10-20 min), the experiment was stopped and the cell was washed with 1 M sulfuric acid solution to quickly deplete the remaining metal ions from the Nafion membrane, as well as to reprotonate it.
Results and Discussion Measurements of the thickness (6)of the Nafion membrane used in our experiments show the effect of electrostatic repulsion between ion-exchange sites on the thickness of the membrane. These results are presented in Figure 4. In sulfuric acid solutions of varying pH, the Nafion membrane thickness decreases as the solution pH becomes less than 1. The thickness decreases ca. 0.9% as the pH changes from 1 to 0 and ca. 6.6% as the pH goes to -1. This "shrinkage" can be understood by considering the electrostatic repulsion between like-charged fixed ion-exchange sites. More ion-exchange sites are in an unprotonated form when the solution pH is increased, thus increasing the electrostatic repulsion. This, together with increased hydration at ion-exchange
sites (water uptake), leads to the expansion of the Nafion membrane; thus a greater thickness is measured. As the solution pH becomes greater than ca. 2, the thickness of the membrane reaches a maximum and is independent of pH. This is consistent with the observationz4 that the internal pH of the Nafion membrane does not change much with the variation of the external pH when it is greater than ca. 2. However, for Nafion membranes in sulfuric acid solutions with 1 M Na2S0, added, the thickness of the membrane is independent of pH. Each solution examined had the same relatively high concentration of Na'. The strong neutralization of ion-exchange sites by Na+ overwhelms the change in electrostatic repulsion and hydration of ion-exchange sites. Therefore, the pH independence is shown. The measurements of thickness in different buffer solutions gave irregular results because the affinities of the two counterions (Na'; K') to the ion-exchange sites are different,z5 and the amount of counterion (Na'; K') neutralizing the ionexchange sites in each buffer solution is irregular. Since the investigated D-C transport process is dependent upon four parameters, a working curve could not be calculated as had been done by Strojek et and others.z7 Therefore, the following procedure was used in order to obtain a match between experimental results and simulated curves. First, the experimental data (from a given file) are analyzed by using a subroutine SQUARE, which checks whether the first few data points fit the relation Abs = at1/z.z8At the beginning of each experiment, the process occurs under semiinfinite diffusion control until the initial concentration (24) Ceynowa, J. Polymer 1982, 203. (25) Hsu,W . Y.; Gierke, T. D. J . Membr. Sci. 1983, 13, 307. (26) Strojek, J. W.; Kuwana, T.; Feldberg, S. W. J. Am. Chem. SOC.1968, 90, 1353. (27) Blount, H . N.; Kuwana, T. J . Electroanul. Chem. 1970, 2, 464.
3610
Wang et al.
The Journal of Physical Chemistry, Vol. 91, No. 13, 1987
‘i! cu M=e D= 2 . 9 3 E(-6) Hz 9 . 2 8 ; kd- 8.9313 ; z=l
Cu
:D
uH=l 3.17 E(-6)
X= 9.625 ; kd=9.9199 ; z= 1.6
‘hx
Figure 9. Examples of matching CNC simulation curves (lines) with experimental results (points) for washing Cu2* from saturated Nafion. The Nafion was presaturated with 1.1 M CuS04 solution at different pH values.
of cations in the middle section of the membrane starts to change. The initial concentration is at a maximum for washing experiments and zero for saturation. The value of coefficient a allows estimation of D. Next, the data are analyzed by using the subroutine LOG, which checks whether the last few points close to the end of the data set match the relation log (Abs’) = btSz9 If they do, coefficient b allows prediction of kd. Values of K and z were obtained by considering the maximum absorbance value and the shape of the experimental curve. In addition, z values other than 1 for simulation were used only when a fit could not be made between experimental data and simulated curve using z = 1. After initial values for D,K,kd, and z were obtained, several trials were run. Each trial consisted of a comparison between the C N C simulation curve and experimental data. The comparison was done by subroutine COMP. After several trials, correspondence between experimental and simulated results was attained so that final values of D, K, kd, and z could be assigned. Results of Cation Saturation in Nafion. The saturation concentration of the metal ions in Nafion (C,) at a certain condition ( 2 8 ) In the ’washing” experiment, when the transport of counterions is under semiinfinite diffusion control, the absorbance can be expressed as Abs(t)
Ab@ - 4 ~ ( D / n ) ~ / ~ P f ’ / ~
where Abso is the initial (maximum) absorbance and c is the molar absorptivity. The initial concentration of sample cation in the Nafion can be used to express C for estimation of D. Therefore Abs = Abso - Abs(t) = ~ C ( D / T ) ’ ~=~oC ~Y ~~/ ~/ ~ a =~ c ( D / T ) ~ / ~ C
( 2 9 ) As the transport of counterions in the “washing” experiment is controlled by the first-order dissociation reaction, the relationship between absorbance and time can be expressed as
log (Abs’) = log (Abso/Abs(t)) = (kd/2.303)r b = kd/2.303
(pH and concentration of metal ion solution) was determined in two ways: (1) According to Beer’s law:
c~= AbsN(max)/(cd)
(5)
where d is the thickness of Nafion (in mm) and AbsN(max) is the maximum absorbance of metal ions in Nafion. By measuring the absorbance of the metal ion solution in a 1-mm optical cell the maximum (Abs,) and in the spectral cell (Abss) at A,, absorbance of metal ions in Nafion is equal to AbsN(max) = Abs,(max) - (Abso.0.54) (6)
(2) If the reading AbsN(ma.) is taken directly from the initial point of the absorbance vs. time curve in the “washing” experiment, CN can be calculated from eq 5 . When the Nafion membrane is equilibrated with a sulfuric acid solution at a given pH value, the extent of protonation of the ion-exchange sites is controlled by equilibrium 2. Once the metal ion solution is introduced into the cell, the metal ions diffuse from the solution regions of the cell into the Nafion. This “saturation” process, as mentioned earlier, is of the D-C type. The increase of free metal ion concentration in the Nafion causes the equilibrium l a or l b to shift to the right until the metal ion concentration in the Nafion reaches an equilibrium saturation. Figure 5 presents an exampb of Cr3+saturation experimental results for different pH values. The observed increases in absorbance at higher pH indicate that the distribution coefficient ( K ) is larger for cations in Nafion at higher pH. This means that more cations are bound to the ion-exchange sites of Nafion in a higher pH environment. Also, the saturation time is longer at higher pH. Nafion saturation measurements for each of the three cations studied were conducted by varying both the cation concentration (in a range from saturation in water to 0.01 M) and the pH (0-3). Analysis of the data shows that the maximum (equilibrium) cation concentration in Nafion, CN,can be described by the relationship: C, = m + nCS’J2 (7)
The Journal of Physical Chemistry, Vol. 91. No. 13, 1987 3611
Cation Transport in Nafion
*og .
TABLE I: Simulation Results of D , K, k,, and z for C$+, Co'+, and CuZt, Transporting out of Nafion 117 (E.W. 1100) pH 0 1 2 3
Cr"
Co2+
0 1
2 CuZt
0 I 2 3
D, cm2/s
K
kd, I / S
2
X IO" X 10" X X
0.91 1.09 3.57 4.76
1.17 1.28 2.83 3.32
3.55 X 10" 2.03 X 6.04 X
0.27 0.59 1.43
1.47 X 6.02 X 6.23 X IO4
1
2.93 3.17 6.73 7.86
0.28 0.63 1.43 2.70
3.23 X 1.09 X 4.61 X 2.17 X lo4
1 1.6 1 1
3.36 1.01 5.98 3.58
X 10" X 10" X X lo-'
X X lo-' X lo4 X lo4
-
A
I
1.6 1
where the value of n depends on the type of cation and on the solution pH. This value was especially large for Cr3+. Data from saturation measurements are presented in Figure 6. The concentration of each of the cations studied in Nafion increases a t higher solution pH until it reaches a maximum between pH 2 and pH 3. Additionally, the ratio of C,/C, will be fairly large when the concentration of cations in solution is low. This illustrates one property of Nafion membranes-the ability to concentrate cations from solution. It is clear that the saturation concentration of a cation in Nafion is affected by both its concentration in solution (C,) and the solution pH. Figure 7 presents an example of the three-dimensional relation for Cr3+. The data for the concentration in Nafion are taken from Figure 6. From the data, it is also evident that the saturation process is relatively fast and that its rate is determined by diffusion when the solution p H is near or less than zero. This is especially true for low valence cations such as Cu2+or Co2+. For these cations, the Nafion becomes 95% saturated after only a few minutes of exposure to the cation solution. When the solution pH has a value of 1 or higher, the process becomes limited by the kinetic step (interaction between ion-exchange sites and cations). This is especially evident for highly charged cations such as Cr3+. In this case, it is necessary to have the Nafion in contact with the solution for several hours in order to get 95% saturation. Results for Cations "Washing" from Saturated Nafion. The measurements for cations diffusing out of Nafion were carried out by filling the solution regions of the cell with (1) a solution containing only a known amount of sulfuric acid, or (2) a solution containing known amounts of sodium sulfate and sulfuric acid. Under these conditions, metal ions diffuse out of the Nafion. This process, which we called "washing", is also of the D-C type as assumed in the Introduction. In both cases, self-diffusion removes metal ions from the Nafion membrane, which shifts equilibria l a and l b to the left. The diffusion of protons or sodium ions into the membrane also pushes the equilibrium to the left, according to reaction 1b or nNa+
+ (R-S03),M * n(R-S03Na) + Mn+
a
..
1 1.4 1 1
0 0
.OW
0
2
I
PH
L
3.0
t
/
Io o + .-
.01
0
2
I
'
PH
3
.)
'N
t
Y
(8)
When sodium sulfate solution was used to wash the metal ions from Nafion, the results were almost the same as those in which sulfuric acid solution of the same concentration was used. Figure 8 presents an example of the simulated concentration profiles in the membrane. The normalized concentration profiles for Cu2+ (both bound and free cations within the membrane) at different time t (in the order to, t , , t2, ..., etc.) and different pH during the washing processes are shown. The dashed lines show the changes in concentration profiles for bound cations, which are unable to move and are in equilibrium with the free cations. The solid lines show the changes in concentration profiles for the free cations, which are able to diffuse. Note that the membrane is saturated at to. The concentrations of both bound and free cations, outside the membrane, are equal to zero during the "washing" experiment. Figure 9 shows examples of data collected when Cu2+ was washed from Nafion with acid solutions of pH 0 to 3 (points),
3
0
2
I
'
3
PH
Figure 10. Simulation results of (a) kd,(b) K,and (c) D at different pH for Cr3+ (-), Co2+ (---), and Cu2+ (E.W. 1100).
(e-),
transporting out of Nafion 117
and the results of C N C simulation (lines). The CNC curves match well with the experimental points. The transport parameters are also shown. Our experimental results strongly indicate that the transport process of sample cations through Nafion is driven by hydronium ions. For example, in Figure 9, as the Nafion membrane, which has been saturated with the sample cation at a higher pH (Le., pH 3) is washed by sulfuric acid solution (pH 0), protons dominate the competitive equilibria between metal cations and protons at the ion-exchange sites (see Figure 1). Thus, all the sample cations quickly diffuse out of the membrane. However, if a dilute sulfuric acid solution of pH 3 is used for washing, sample cations diffuse
3612
J . Phys. Chem. 1987, 91, 3612-3616
out of the membrane very slowly because, in this case, the protons have lost their dominance in the competitive reactions. Similarly in Figure 6, more sample cations can diffuse into the membrane from the sample solution with higher pH and vice versa. The results of the washing experiments for three different cations and comparisons of the data with the C N C simulation are presented in Table I and Figure 10. In Figure loa, the dissociation rate constants for three cations are shown to follow the order Cr3+ < Co2+ C Cu2+. This implies that the Cr3+ions are bound more tightly at ion-exchange sites than are Co2+and Cu2+. The dissociation rate constant decreases at higher solution pH. This is consistent with the assumption that kd = kd'[H+In. However, the decrease of kd does not exactly follow the same order of [H']" change if k i is pH-independent. Figure 10b shows that the distribution coefficient K becomes larger when solution pH is increased. The distribution coefficients for Cr3+ in Nafion are larger than those for Co2+ and Cu2+. As expected, the order of reaction is 1.0 in most cases (see Table I). However, the z values for solutions at pH 1 are different from those obtained at other pH's. While no attempt has been made to interpret this observation in this paper, it is interesting to note that the pK, value of Nafion (E.W. = 1100) is believed to be near 1. The simulated results of diffusion coefficients for three cations at different pH are shown in Figure 1Oc. Generally speaking, the tortuosity of the paths available for diffusion and the interaction of counterions with fixed ion-exchange sites retard counterion transport in a membrane. It has been proposed that a Nafion membrane can be classified into three regions:I3 fluorocarbon region (A), interfacial region (B), and cluster ion region (C). Large-sized ions, especially those with some hydrophobic properties such as DA+ and Ru(bpy),2+, may interact with region B. Ions with larger charge density and hydration energy, such as Co2+, Cu2+, and Cr3+, would prefer region C. The environment for cationic transport in region B and in region C is different; however, the transport of counterions in region C can be expected to be more rapid than in region B. Our simulated values of D for the sample cations (ca. 3 X 10" cm2/s at pH 0), which are only slightly lower than those in bulk
solution (ca. 6 X 10" m2/s), reflect only the tortuosity restrictions that cations encounter during self-diffusionthrough the membrane. The experimental results (see Table I and Figure 1Oc) demonstrate that the values of D are high even when the transport of sample cations in the membrane is low (in the case of pH 3). A pH dependence is also found for values of D which decrease at higher pH values. Referring to the results for a cluster morphology calculation for Nafion (E.W. 1200) neutralized with group IA (group 1)30 cations,25we can explain our experimental results for the pH dependence of the diffusion coefficient. Since more sample cations are bound to the ion-exchange sites (increase in K ) as the pH increases, the cluster diameter will decrease in association with the increase of elastic deformation. Thus an increase in tortuosity of the paths is possible, which would decrease the diffusion coefficient for counterion transport in Nafion. However, Knudsen's theory of solid-gas catalysis may be applied to this problem. It states that D = kr', where r'is the radius of the pores of porous solids, and k is a constant for a certain diffusing molecule at a given temperature. The dependence of D on pH can be explained under the assumptions that r'is very small compared to the mean free path of the cation in solution and that the ionic cluster diameter of the Nafion (or the diameter of intercluster channels25) decreases at higher pH. More work is necessary to confirm this.
Acknowledgment. We gratefully acknowledge financial support of this work by a grant from the National Science Foundation. Helpful suggestions by Steve Feldberg of Brookhaven National Laboratories are appreciated. Registry No. Co, 7440-48-4; Cu, 7440-50-8; Cr, 7440-47-3; Nafion, 39464-59-0.
(30) in this paper the periodic group notation (in parenthesis) is in accord with recent actions by IUPAC and ACS nomenclature committees. A and B notation is eliminated because of wide confusion. Groups IA and 1IA become groups I and 2. The d-transition elements comprise groups 3 through 12, and the p-block elements comprise groups 13 through 18. (Note that the former Roman number designation is preserved in the last digit of the new numbering: e.g., 111 3 and 13.)
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Theoretical Prediction of Translatlonal Diffusion Coefficients of Small Rigid Molecules from Their Molecular Geometry Pedro J. Espinosa and Jose Garcia de la Torre* Departamento de Quimica Fhica, Facultad de Ciencias Quimicas y Matembticas, Uniuersidad de Murcia, 30001 Murcia, Spain (Received: November 4, 1986)
Translational diffusion coefficients of small, rigid solute molecules in various nonaqueous solvents have been calculated from molecular geometries. The calculations start from a molecular model built from bond lengths and angles. Atoms are replaced in the model by spherical frictional elements with either the covalent or the van der Waal radii. The frictional coefficients of the elements contain a numerical factor,f, which varies fromf= 1 for stick boundary conditions tof= 2/3 for slip conditions. A Kirkwood-Riseman treatment with the Oseen hydrodynamic interaction tensor, which is well-known in polymer hydrodynamics, is employed in the calculations. When covalent radii are assigned to the spherical elements, we find that for a number of solute/solvent systems the experimental value of the translational diffusion coefficient is bracketed by the theoretical predictions corresponding tof = 1 andf = 2/,. This indicates an hydrodynamic behavior intermediate between the two limiting cases. Regardingfas an adjustable parameter, we find thatf= 0.79 yields theoretical values that deviate only a few percent from the experimental data. A similar study was attempted using atomic van der Waals radii that represent better the volume of the diffusing entities. It was found that the calculations are strongly affected by singularities due to use of Oseen tensors for unequal, overlapping elements. Introduction The translational diffusion coefficient, D,, of the solute in an infinitely dilute solution can be described well, in many instances, by a Stokes-Einstein equation of the type 0022-3654/87/2091-3612$01.50/0
D, = kT/c?rqop (1) where k T is Boltzmann factor, T~ is the viscosity of the pure solvent, p is some length characterizing the size of the solute, and c is a numerical factor that depends on the shape of the solute
0 1987 American Chemical Society
