J. Phys. Chem. 1994, 98, 11791-11795
11791
Voltammetric Investigation of the Transport of Metal Cations in Polyelectrolyte Solutions Malgorzata Ciszkowskat and Janet G. Osteryoung* Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695-8204 Received: August 8, 1994@
The transport of singly and doubly charged cations of metals (T1+, Cd2+, and Pb2+) was studied in solutions of the polyelectrolyte poly(styrenesu1fonic acid), PSSA, by voltammetric measurements at a mercury film disk microelectrode in solutions of various ionic strengths. Metal cations present in concentrations much lower than the equivalent concentration of polyacid were used as probe counterions. The interactions between counterion and polyelectrolyte were evaluated by the ratio of diffusion coefficients in the solution with and without polyelectrolyte, DPSSAID~. These interactions were found to be much stronger for the doubly charged metal cations than for the singly charged counterions. The ratio of DPSSAID~ was 0.06 for lead and cadmium counterions and 0.35 for the thallium counterion. This ratio obtained for thallium cation in poly(styrenesulfonic acid) was very close to that found previously for hydrogen counterion in PSSA solution, 0.345. The transport of metal probe counterion was also studied in solutions of the sodium salt of poly(styrenesu1fonic acid). The dependence of the diffusion coefficient of metal cations on concentration of supporting electrolyte was studied over a wide range of ratios of metal cation concentration to polyelectrolyte concentration and over a wide range of concentrations of supporting electrolyte.
Introduction Transport of counterions in aqueous solutions of polyelectrolyte can be used as a model for the description of transport of simple ions in many kinds of heterogeneous media such as ion-exchange membranes, biological fluids and tissues, and other ion-exchanging systems.1-4 Strong, long-distance, electrostatic interactions between the polyion and the counterion suppress transport of the counterion. The most noticeable changes in the transport properties of simple counterions in polyelectrolyte solution are observed at very low ionic ~ t r e n g t h . ~ - ~ Various techniques have been used for the determination of diffusion coefficients of simple ions in solutions of polyelectrolytes of various ionic strength, including radioactive tracerg-15 and FT NMR.16-21 Recently, a simple voltammetric method employing a platinum disk microelectrode has been used to study the transport of hydrogen counterion in solutions of poly(styrenesulfonic acid), PSSA, without and with supporting e l e c t r ~ l y t e . ~The ~.~ transport ~ of hydrogen ion in PSSA solution with no supporting electrolyte was found to be 2.9 times slower than in the solution of simple acid, HC104. The use of a microelectrode allows the measurements in solutions of very low ionic strength. Additionally, the steady-state current at microelectrodes is proportional to the diffusion coefficient, and the measured signal is thus very sensitive to the changes in the diffusion coefficient value. Voltammetric methods have been used to investigate the transport of counterion in polyelectrolyte solutions before. However, they employed regular-size electrodes and, therefore, have been restricted to rather high levels of supporting e l e ~ t r o l y t e . ~ Classic ~ - ~ ~ polarography was used to study the transport of metal cations (n+, Cd*+)in polyelectrolyte solutions with no added supporting electrolyte 30 years but the concentration of ions in the solution was rather high because of leakage of ions from the salt bridge connecting the reference electrode with the solution, as was pointed out by the authors. The dependence of the normalized diffusion coefficient of Permanent address: Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland. @Abstractpublished in Advance ACS Abstracts, October 15, 1994.
0022-365419412098-11791$04.50/0
the singly charged counterion in polyelectrolyte solution on the concentration of a 1:1 supporting electrolyte has been described by the simple semiempirical equation22
DID, = (a
+ y ')l(y ' + 1)
where D and Do are the diffusion coefficients in the presence and the absence of the polyion, y ' is the ratio of the concentration of simple electrolyte to the equivalent concentration of the polyelectrolyte, and a is an empirical factor equivalent to the ratio of diffusion coefficient of the counterion with and without polyelectrolyte, DID,, both in the solution with no simple electrolyte. The experimental results for hydrogen counterion in solutions of poly(styrenesu1fonic acid) are in accordance with this d e p e n d e n ~ e . ~ ~ , ~ ~ The aim of this paper is to describe further development of the voltammetric approach for studying the transport of simple ions in polyelectrolytesolutions. This extension of the previous method employs probe ions which are electroactive and which are present at concentrations negligible in comparison with the equivalent concentration of polyion. The excellent sensitivity of the voltammetric measurement makes this possible. The experiment is successful with simple, easily reducible cations such as Pb2+, Cd2+, and T1+ because the reduction is carried out at a silver-based mercury film m i c r ~ e l e c t r o d e at , ~which ~~~~ these ions display a well-defined, easily measured, steady-state current on time scales of a few seconds.
Experimental Section All reagents except poly(styrenesu1fonic acid) were of reagent-grade purity and were used as received. Poly(styrenesulfonic acid) (MW 70 000, Polysciences, Inc.) contains approximately 5% aqueous sulfuric acid. It was removed in a 36-h dialysis with 10-fold excess volume of water using a MWCO 12- 14 OOO membrane (SpectraPor4, Spectrum Medical Industries). The water was changed 6 times. The concentration of PSSA reported in this work is the concentration of hydrogen ion in PSSA. This concentration was determined by conductometric titration using a YSI Model 31 conductivity bridge 0 1994 American Chemical Society
Ciszkowska and Osteryoung
11792 J. Phys. Chem., Vol. 98, No. 45, 1994 (Yellow Springs Instrument Co.) against standard sodium hydroxide solution. Ultrapure water (Milli-Q, Millipore Corp.) was employed in all rinses, dialyses, and preparations of solutions. Solutions were deoxygenated before voltammetric scans and blanketed with a stream of water-saturated argon. Electrochemical measurements were carried out with a threeelectrode system in a jacketed cell (25 “C) enclosed on an aluminum Faraday cage. A mercury film disk microelectrode of 15-pm radius, r, was used as the working electrode. Silver disk microelectrodes (Project Ltd., Warsaw, Poland) were used as substrates for the mercury films. The mercury film thickness, I, was 1 pm. Mercury was deposited at -0.5 V (vs SCE) from a solution of 5 mM Hg(I1) in 0.1 M HC104. The procedure for the preparation of this silver-based mercury film microelectrode has been described in detail.31 After mercury deposition, the electrode was washed carefully with a large amount of water. The surface of the mercury film was inspected with an inverted microscope (Leitz Wetzler, Germany) before using. A quasi-reference platinum electrode was used to prevent leakage of ions into the cell. The use of a platinum quasireference electrode in aqueous solutions with no electrolyte has been d e s ~ r i b e d . ~The ~ . ~counter ~ electrode was platinum. Staircase voltammograms were obtained by using a Model 273 potentiostat (EG&G PARC) connected with a Keithley Model 427 current amplifier and controlled by software via a PC 486 computer. Staircase voltammetry parameters were as follows: step height (AE), 5 mV; frequency 0, 1 Hz. Under these conditions, the limiting current for a 15-pm Hg film disk electrode should not exceed the steady-state value by more than 3%.33 The deposition of mercury on the Ag disk electrode was performed in a three-electrode system with Pt wire counter electrode and SCE reference electrode, with a Model 173 potentiostat connected to a Model 179 digital coulometer (EG&G PARC).
Results and Discussion Background Concentration of the Electrolyte. In the absence of added supporting electrolyte, there is some background concentration of adventitious electrolyte. The source of ions in solution can be the electrode which is transferred to the cell from the solution used to deposit the mercury film (even if the electrode is washed with a large amount of water after deposition of mercury). In order to determine the background concentration of electrolyte, we compared the reduction waves of hydrogen ion obtained in solutions without and with added supporting electrolyte. This procedure has been described in detail in ref 23. Experiments were performed for two concentrations of HC104,0.03 and 0.09 mM. The average background concentration of 1:1 electrolyte obtained by this procedure was lower than 10 pM. Thus, a solution without supporting electrolyte in this work is a solution with an electrolyte concentration of about 10 pM. Reduction of Thallium Cation in PSSA Solution. Voltammograms for the reduction of thallium cation in 20 mM PSSA solution without added supporting electrolyte are presented in Figure 1. The waves are very well defined, and the reproducibility of the measurements was better than 2%. Thallium cation is used as a probe counterion; that is, the concentration of metal cation is much lower than the concentration of polyacid. Under these conditions, when the concentration of hydrogen ion (from dissociation of PSSA, a strong acid) in the solution is 100 times higher than the highest concentration of thallium cation, the steady-state current, is, of reduction of thallium cation is controlled by diffusional transport and for a disk microelectrode
1 1 . 3 0 .
2
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-
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/
,
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Figure 1. Staircase steady-state voltammograms for the reduction of 0.1 mM Tl+ in 2 mh4 PSSA. Concentration of LiC104 added (mM): (1) 0; (2) 1; (3) 3; (4) 300.
c
/
mw
Figure 2. Concentration calibration plots for the reduction of T1+ in (0)20 mM PSSA, no supporting electrolyte; (A) 20 mh4 PSSA, 2 M LiC104; (v) 20 mM HC104, 2 M LiC104; and (0) 20 mM HC104.
depends on the diffusion coefficient, D,through the relation34 is = 4nFCDr
(2)
where n is the number of electrons transferred, F is the Faraday constant, C is the concentration of electroactive substrate, and r is the radius of the microelectrode. That is, the polyacid acts as a supporting electrolyte for the reduction of T1+. If the concentration of polyacid is not higher than 30 times the concentration of thallium cation, the contribution of migration to the steady-state limiting current for thallium may be higher than 1%. General theoretical solutions have been obtained for the dependence of the steady-state limiting current on the concentration of supporting e l e ~ t r o l y t e . ~ Theory ~-~~ predicts that the ratio of limiting current (measured on the plateau of the wave) to diffusional current, illid, for the reduction of univalent cation, accompanied by univalent anion, depends on the concentration of 1:1 supporting electrolyte as35
il/id= 2
+ 2y - 2[y(1 + y)ll”
(3)
where y is the ratio of the concentration of electrolyte to the concentration of electroactive cation (here y = CH+/C~+). This theoretical dependence has been confirmed experimentally for the reduction of hydrogen ion in solutions of strong acidm and for the reduction of thallium cation.32 Under our experimental conditions, the lowest value of y was 10 (0.1 mM T1+, 1 mM PSSA). The ratio of illid equals 1.02 in this case, and the limiting current is only 2% higher than the diffusional current. Figure 2 presents calibration plots obtained for the reduction of thallium cation in four different solutions, 20 mM HC104 with 2 M LiC104, 20 mM PSSA with 2 M LiC104, 20 mM HC104, and 20 mM PSSA without supporting electrolyte. The concentration range of T1+ was from 0.05 to 0.25 mM. In
J. Phys. Chem., Vol. 98,No. 45, 1994 11793
Transport of Metal Cations in Polyelectrolyte Solutions 2.0 2
1 .o0.8-
* .-
I
1
*
*
*
*
*
0.5
_ i a'
0 . -04
-3
-2
-1
log
Figure 3. Dependence of limiting currents for the reduction of 0.15 mM Tl+on the normalized concentrationof supporting electrolyte ( y ' = CJCp) in (0)20 mM HC104 and (*) 20 mM PSSA.
solutions containing 2 M LiC104, the slopes of the calibration plots with PSSA and with HC104 are the same within 3% (curve 2 of Figure 2). The average value is 8.48 nA mM-'. This indicates that there are no interactions between thallium cation and polyelectrolytewhen the concentrationof simple electrolyte is high enough, and the transport of thallium cation is the same in the solution of simple acid and polyacid. In this case, the ratio of concentration of LiC104 to the concentration of PSSA was 100, and according to eq 1 and by using an a value of 0.345, obtained experimentally for hydrogen counterion in PSSA s o l ~ t i o n ,the ~ ~ ratio , ~ ~ of diffusion coefficient of monovalent counterion with polyelectrolyte to diffusion coefficient without polyelectrolyte, DPSSAID,,,should be 0.994. In the solution of 20 mM simple acid, HC104 (curve 3 of Figure 2), the slope of the calibration plot is 12.85 nA mM-l. The concentration of acid guarantees that there is no influence of migration on the limiting current,35 and changes in the diffusion coefficient of thallium cation due to the changes in activity are lower than 3%.41 If there is no simple electrolyte in the 20 mM PSSA solution (curve 1, Figure 2), the slope of the concentration calibration plot of the reduction of thallium cation is much lower, 4.56 nA mM-'. We can calculate the diffusion coefficient from the slope of calibration plots, according to eq 2. The diffusion coefficient of Tl+ obtained in the solution of 20 mM HC104 is 2.22 x cm2 s-l, which agrees very well with the value 2.10 x cm2 s-l reported for thallium cation at infinite dilution.42 The diffusion coefficient of thallium cation calculated from the slope of the calibration plot in 20 mM PSSA without supporting electrolyte is 7.78 x cm2 s-l. The ratio of diffusion coefficients with PSSA and with simple acid, both without supporting electrolyte, is 0.35. This value defined as a in eq 1 agrees very well with the a value obtained for hydrogen counterion in poly(styrenesu1fonic acid), 0.345.22,23This means that the transport of thallium ion in the solution of simple acid with no supporting electrolyte is 2.9 times faster than in the solution of PSSA. We studied the influence of various concentrations of the supporting electrolyte, LiC104, on the reduction current of thallium cation in PSSA solution. Figure 3 (curve 1) presents the results for 0.15 mM Tl+ in 20 mM PSSA solution. The normalized concentration of supporting electrolyte, CLicloJ CPSSA,is denoted as y ' . The currents increase when the concentration of LiC104 increases, and for very high concentrations of the supporting electrolyte (the highest concentration of LiC104 is 2 M), they decrease due to the changes in the activity of thallium cation and the viscosity of the solution. The lowest value of current was obtained in the absence of added supporting electrolyte. If we compare this dependence with the reduction currents of T1+ in the solution of 20 mM HC104 (curve 2), we
O"
0 . 0 -4
-2 lOQ
0 3'
2
Figure 4. Dependence of normalized diffusion coefficientson the ratios of electrolyte (LiC104) to PSSA concentration ( y ' = CJCPSSA) for the reduction of thallium ion in (0)0.1 mM a+,2 mM PSSA; (A) 0.05 mM Tl+,5 mM PSSA; (V)0.1 mM 'IT, 5 mM PSSA; (0) 0.05 mM TP, 10 mM PSSA; and (0)0.1 mM Tl+,20 mM PSSA. (-) H+ cation in PSSA.
can see large differences between the current values in the solution without supporting electrolyte and the same currents with excess supporting electrolyte. In the following discussion, we present the results in terms of the dependence of D p s s ~ l D ~ on y '. Because steady-state current, is, is proportional to D (eq 2), this is equivalent to using curves such as curve 2 of Figure 3 as the normalizing factor for the results as in curve 1. The values of the ratio of the diffusion coefficients in the presence and the absence of polyelectrolyte, D ~ s s A I D (where ~ Do is the experimental diffusion coefficient of T1+ in the simple acid solution for the given concentration of the supporting electrolyte), for five different concentrations of thallium cation and PSSA were plotted vs the normalized concentration of supporting electrolyte, log y ', and are displayed in Figure 4. The solid line in Figure 4 is the dependence calculated according to eq 1 with value of a = 0.345 obtained experimentally for hydrogen counterion in PSSA solution. The experimental values for thallium cation are very close to the results obtained for hydrogen cation. However, there is a small shift between the dependencies for thallium and hydrogen cation. The average limiting value of D ~ ~ s AinI D solution ~ with no added electrolyte for five various T1+ and PSSA concentrations is 0.353, which agrees well with the ratio of the slopes of calibration plots, 0.350. This value of a is only 2% higher than the a value obtained for H+ counterion in PSSA solution. There are no visible differences in the diffusion coefficient of thallium cation for various concentrations of polyelectrolyte. For example, the dependencies look very similar for the case of 0.1 mM T1+ with 5 mM PSSA and 0.1 mM T1+ with 20 mM PSSA. Reduction of Lead or Cadmium Cation in PSSA Solution. Figure 5 presents voltammetric waves obtained for the reduction of lead cation in 20 mM PSSA solution without added supporting electrolyte. The reproducibility of the measurements was better than 2%. A white salt precipitates in solutions of Pb2+ and PSSA if the concentration of PSSA is lower than 10 mM, and the measurements of current for the reduction of lead ion in PSSA solution are possible only when the pH is low enough. (There is no precipitation in solutions of Cd2+ and PSSA over a wide concentration range.) For the conditions of Figure 5 , the concentration of hydrogen cation from the dissociation of PSSA is 100 times higher than the highest concentration of lead cation. In this case, the steady-state current is controlled by the diffusional transport of Pb2+ and depends on the diffusion coefficient through eq 2. If the concentration of hydrogen ion in the solution is not higher than 90 times the concentration of lead cation, the contribution of
Ciszkowska and Osteryoung
11794 J. Phys. Chem., Vol. 98,No. 45, 1994 1
.o
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5
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0 0 0 U
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Figure 6. Concentration calibration plots for the reduction of Pb2+in solutions of (0)20 mM PSSA, no supporthg electrolyte; (A) 20 mM PSSA, 2 M LiClO4; (V)20 mM HC104,2 M LiC104; and (0)20 mM
HC104. migration to the limiting current is expected to exceed 1%. Theory predicts that the ratio of limiting to diffusional current, illid, for two-electron reduction of doubly charged cation, accompanied by monovalent anion, depends on the concentration of the 1:1 supporting electrolyte through the relation35
ijid = 3
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*
-1
0
1
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Figure 7. Dependence of limiting currents for the reduction of 0.1
V
Figure 5. Staircase steady-state voltammograms for the reduction of 0.1 mM Pb2+in 20 mM PSSA solution. Concentration of LiC104 added (mM): (1) 0; (2) 20; (3) 50; (4)150;( 5 ) 650;(6) 2650.
Q
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-1.2
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e
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+ 2y - 2 [ y ( 2 +
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where y is the ratio of the concentration of electrolyte to the concentration of doubly charged, electroactive cation (here y = CH+/cw+). This dependence has been c o n f i i e d experimentally for the reduction of lead and cadmium cations.32 Under our experimental conditions, the lowest values of y were 10 and 20 (e.g., 0.1 mM CdZ+with 1 mM PSSA and 0.05 mM Cd2+with 1 mM PSSA). The ratio of illid equals 1.09 and 1.04 for y values of 10 and 20, respectively. Therefore, the limiting currents obtained for y values lower than 20 were corrected according to eq 4. Concentration calibration plots obtained for the reduction of lead cation in four different solutions, 20 mM HC104 with 2 M LiC104, 20 mM PSSA with 2 M LiC104, 20 mM HC104, and 20 mM PSSA without supporting electrolyte, are presented in Figure 6. The concentration range of lead cation was from 0.05 to 0.2 mM. The slopes of the calibration plots in 2 M LiC104 solutions with and without polyelectrolyte are the same within 2% (curve 2 of Figure 6). The average value of the slope is 8.62 nA mM-'. Similar values of the slopes indicate no interactions between doubly charged counterion and polyion in the solution of high ionic strength. The slope of the concentration calibration plot obtained in the solution of simple acid, 20 mM HC104, is 11.19 nA mM-' (curve 3). A much lower value, 0.67 nA mM-', was obtained in the solution of 20 mM PSSA with no supporting electrolyte (curve 1 of Figure 6).
mM Pb2+on the normalized concentration of supporting electrolyte ( y ' = CJCH+)in (0)20 mM HClOd and (*) 20 mM PSSA.
The diffusion coefficient of lead cation, calculated from the slope of the calibration plot in the solution of simple acid, is 9.66 x cm2 s-l. This value agrees very well with the value of 9.5 x cm2 s - l reported for lead cation at infinite The diffusion coefficient of lead cation in 20 mM PSSA solution without supporting electrolyte, calculated from the slope of the appropriate calibration plot, is 5.8 x cm2 s-l. The ratio of diffusion coefficient with and without PSSA, D ~ S S A I in D ~solution , without supporting electrolyte is 0.06. That is, the transport of Pb2+ ions in simple acid solution is 16.7 times faster than in the solution of PSSA. This ratio, defined as a in eq 1, is only 17% of the a value for monovalent thallium cation. The influence of various concentrations of supporting electrolyte, LiC104, on the reduction current of lead cation in PSSA solutions is presented in Figure 7 (curve 1). The normalized concentration of supporting electrolyte, CL~CQICPSSA, is denoted as y '. The current increases when the concentration of supporting electrolyte increases, and for very high concentrations of the supporting electrolyte (2 M LiC104) the current decreases. Compare the currents for the reduction of thallium ion in solutions of PSSA with the currents obtained in 20 mM HC104 (curve 2 of Figure 7). There is a striking difference without supporting electrolyte, whereas the currents are the same with excess supporting electrolyte. The shape of this dependence for lead counterion is very similar to that for thallium cation (Figure 3), but the suppression in currents in the PSSA solution with no supporting electrolyte is much more marked. The ratio of diffusion coefficients in the presence and the absence of polyelectrolyte, DPSSAID~ (Do is the experimental diffusion coefficient of Pb2+ in the simple acid solution), for various concentrations of lead and cadmium counterions, and various concentrations of PSSA, was plotted vs the normalized LiC104 concentration, log y ', and is presented in Figure 8. The average limiting value of Dpss~lD,in solution with no added electrolyte for various Pb2+ or Cd2+ and PSSA concentrations is 0.062, which agrees well with the ratio of the slopes of the calibration plots, 0.060. Note that the simple normalizations, for example, a a / z i and ci ziti, do not serve to bring the results such as those of Figure 8 into correspondence with those for a singly charged counterion. Reduction of Thallium Cation in the Solution of PSSA Sodium Salt. Instead of poly(styrenesu1fonic acid), we used its sodium salt as a polyelectrolyte. As in the case of PSSA, very well-defined waves of reduction of thallium cation in PSSNa solution were obtained. The dependence of the normalized diffusion coefficient of T1+ in sodium poly(styrenesulfonate) solution on the normalized concentration of supporting electrolyte, y ', is presented in Figure 9. The diffusion coefficient of T1+ in polyelectrolyte solution was normalized
-
-
Transport of Metal Cations in Polyelectrolyte Solutions
J. Phys. Chem., Vol. 98, No. 45, 1994 11795 I
1.21
0.81
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9 P
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v A
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.Ot
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0
1
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Figure 9. Dependence of normalized diffusion coefficient, Dps.@,, of thallium cation on the normalized concentration of supporting electrolyte ( y ’ = CJCPSSA), in 0.1 mM “W with (0)20 mM PSSNa and (A) 20 mM PSSA. (-) H+ cation in PSSA.
vs the experimental diffusion coefficient obtained in the solution of simple salt, NaC1. The experimental results for thallium counterion in PSSNa solution are very close to the results obtained for hydrogen ion in PSSA solution (solid line in Figure 9). There is a small shift between the dependencies on y ’ for Tl+ counterion in PSSA and PSSNa solutions, but the limiting values (with no or with excess supporting electrolyte) are the same. Conclusions. The results reported here show that voltammetry of a probe ion at a microelectrode is an effective technique for studying ionic interactions in polyelectrolyte solutions. Voltammetry with mercury film microelectrodes allows measurements of the reduction of amalgam-forming metal cations in solutions of various ionic strength. The current signal is very sensitive to the changes in diffusion coefficient value, and the diffusion coefficient can be determined with very good precision (2-3%). The sodium form of this anionic polyelectrolyte can be used as well as the acid form. The interactions between the acid or sodium form of poly(styrenesulfonate), and singly charged metal cation, Tlf, in the solution without supporting electrolyte were found to be very similar to these for hydrogen cation. Transport of thallium cation in solutions of PSSA is only one-third as fast as in the solution of simple acid; the ratio of D p s s ~ / D is~0.35. Stronger interactions between cation and polyelectrolyte were observed for divalent cations, Pb*+ and CdZ+. The ratio of the diffusion coefficient in the polyelectrolyte solution to that without either supporting electrolyte was 0.06 (D~SSA = Dd16.7). The focus in this paper is on presenting an inexpensive, accurate, and precise technique for probing charge interactions in complex media. The voltammetric method described here
presents the possibility for characterizing these interactions in detail not heretofore practical. Data arising from this approach will contribute to an empirical definition of the phenomena and may stimulate progress in the theoretical description of these and similar systems.
Acknowledgment. This work was supported in part by the National Science Foundation under Grant CHE9208987. References and Notes (1) Polyelectrolytes: science and technology; Hara, M., Ed.; Marcel Dekker: New York, 1993. (2) McLaughlin, S. A. Current Topics in Membrane Transport; Bronner, T., Kleinzeller, T., Eds.; Academic: New York, 1977; Vol. 9. (3) Record, M. T., Jr.; Mazur, S. J.; Melancon, P.; Roe, J. H.; Shaner, S. L.; Unger, L. Annu. Rev. Biochem. 1981, 50, 997. (4) Manning, G. S. Annu. Rev. Phys. Chem. 1972, 23, 117. (5) Marcus, R. A. J. Chem. Phys. 1955, 23, 1057. (6) Lifson, S.; Jackson, J. L. J . Ch” Phys. 1962, 36, 2410. (7) Kwak, J. C. T.; O’Brien, M. C.; MacLean, D. A. J. Phys. Chem. 1975, 79, 2381. (8) Szymczak, J.; Holyk, P.; Ander, P. J. Phys. Chem. 1975, 79, 269. (9) Ander, P.In Warer-Soluble Polymers; ACS Symposium Series 467; Shalaby, S. W., McCormick, C. L., Butler, G. B., Eds.; American Chemical Society: Washington, DC,1991. (10) Boyd, G. E. J. Phys. Chem. 1974, 78, 735. (11) Kowblansky, M.; Ander, P. J . Phys. Chem. 1976, 80, 297. (12) Ueda, T.; Kobatake, Y. J. Phys. Chem. 1973, 77, 2995. (13) Ander, P.; Kardan, M. Macromolecules 1984, 17, 2436. (14) Ander, P.;Kardan, M. Macromolecules 1984, 17, 2431. (15) Henningson, C. T.; Karluk, D.; Ander, P. Macromolecules 1987, 20, 1286. (16) Stejskal, E. 0.;Tanner, J. E. J. Chem. Phys. 1965, 42, 288. (17) Rose, D. M.; Bean, M. L.; Record, M. T., Jr.; Bryant, R. G. Proc. Natl. Acad. Sei. USA. 1980, 77, 6289. (18 ) Ndsson, L. G.; Nordenskiold, L.; Stilbs, P.; Braunlin, W. H. J . Phys. Chem. 1985, 89, 3385; J. Phys. Chem. 1987, 91, 6210. (19) Bratko, D.; Stilbs, P.; Bester, M. Macromol. Chem., Rapid Commun. 1985, 6, 163. (20) Qian, C.; Asdjodi, M. R.; Spencer, H. G.; Savitsky, G. B. Macromolecules 1989, 22, 995. (21) Stilbs, P.;Lindman, B. J . Magn. Reson. 1982, 48, 132. (22) Moms, S. E.; Ciszkowska, M.; Osteryoung, J. G. J . Phys. Chem. 1993, 97, 10453. (23) Ciszkowska, M.; Osteryoung, J. G. J . Phys. Chem. 1994,98,3194. (24) Carter, M. T.; Rodriguez, M.; Bard, A. J. J . Am. Chem. SOC.1989, 111, 8901. (25) Van Leeuwen, H. P.; Cleven, R.; Buffle, J. Pure Appl. Chem. 1989, 61, 255. (26) Jiang, R.; Anson, F. C. J . Phys. Chem. 1992, 96, 452. (27) Tanford, C. J. Am. Chem. Soc. 1951,73,2066;1952,74,211; 1952, 74, 6036. (28) Lapanje, S.; Oman, S. Macromol. Chem. 1962, 53, 46. (29) Lapanje, S. Biopolymers 1964, 2, 585. (30) Lapanje, S.Biopolymers 1966, 4, 85. (31) Ciszkowska, M.; Donten, M.; Stojek, Z . Anal. Chem., in press. (32) Ciszkowska, M.; Osteryoung, J. G. Anal. Chem., submitted. (33) Sinru, L.; Osteryoung, J. G.; O’Dea, J. J.; Osteryoung, R. A. Anal. Chem. 1988, 60, 1135. (34) Wightman, R. M. Anal. Chem. 1981, 53, 1125A. (35) Myland, J. C.; Oldham, K. B. J . Electroanal. Chem. 1993, 347, 49. (36) Amatore, C.; Fosset, B.; Bartelt, J.; Deakin, M. R.; Wightman, R. M. J . Electroanal. Chem. 1988, 256, 255. (37) Baker, D. R.; Verbmgge, M. W.; Newman, J. J. Electroanal. Chem. 1991, 314, 23. (38) Cooper, J. B.; Bond, A. M.; Oldham, K. B. J . Electroanal. Chem. 1992, 331, 877. (39) Oldham, K. B. J . Electroanal. Chem. 1992, 337, 91. (40) Ciszkowska, M.; Stojek, Z.; Moms, S. E.; Osteryoung, J. G. Anal. Chem. 1992,64,2372. (41) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Plenum: New York, 1970; Vol. 1. (42) Heyrovsky, J.; Kuta, J. Principles OfPolarography; Academic: New York, 1966.
