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Dynamic Solvent Effects in Electrochemical Kinetics: Indications for a Switch of the Relevant Solvent Mode Pavel A. Zagrebin,† Richard Buchner,‡ Renat R. Nazmutdinov,§ and Galina A. Tsirlina*,† Department of Electrochemistry, Moscow State UniVersity, Leninskie Gory 1-str.3, 119991 Moscow, Russian Federation, Institute of Physical and Theoretical Chemistry, UniVersity of Regensburg, 93040 Regensburg, Germany, and Kazan State Technological UniVersity, K. Marx Str., 68, 420015 Kazan, Republic Tatarstan, Russian Federation ReceiVed: August 3, 2009; ReVised Manuscript ReceiVed: NoVember 15, 2009
The influence of solvent dielectric relaxation on the rate of electron transfer (ET) at an electrochemical interface is addressed using both experiment and model calculations. Water-ethylene glycol (EG) mixtures were chosen as the solvent because their optical permittivity remains practically constant over the entire composition range. This allows observation of the dynamic solvent effect with a very minor interference from the static solvent properties (being typically of opposite sign). Three groups of experimental results are presented to characterize the mixed-solvent system (dielectric spectra in the frequency range 0.1-89 GHz), the mercury/solvent interface (electrocapillary data), and the ET kinetics (dc polarography of peroxodisulphate reduction). To extract the true solvent influence on the electron transfer elementary step, the results from dc polarography are corrected for interfacial effects with the help of the electrocapillary data. An anomalous dependence of the ET rate on EG content (i.e., nonmonotonic dependence of the ET rate on macroscopic viscosity) can be inferred after all corrections. The interplay of different solvent modes is suggested to be responsible for the observed features of ET kinetics. A possible interpretation of the corrected ET rate in the framework of the Agmon-Hopfield formalism is proposed, where the dielectric spectra of the mixed solvent are modeled by a superposition of three Debye equations. The results demonstrate that the observed anomalous “viscosity effect” may be explained qualitatively by an increased contribution of the fast relaxation mode at high EG contents. Introduction While solvent dynamics generally has only limited influence on the dynamics of homogeneous self-exchange reactions,1 this seems not to be the case for electron transfer (ET) at electrodes, especially for nonaqueous solvents. The theory of electron transfer in polar media (see refs 2 and 3 and also refs therein) provides various models to describe solvent effects. Among these, the influence of solvent dynamics can be considered as the most challenging aspect of electron transfer (ET) reactions and is thought to control ET rates over extremely wide ranges of experimental conditions and operation modes. However, until now the predictive power of theory remains unsatisfactory, as it uses model parameters and quantities that cannot be accessed by direct and independent experiments. Moreover, for numerous studied redox systems, various required solvent properties, although available in principle by experiment, are still unknown or were only determined with insufficient accuracy. This problem is most crucial for solvent relaxation phenomena and their effect on ET rates, as the so-called dynamic solvent effect is the key determining numerous adiabatic ET reactions. The general approach2 assumes that the full dielectric spectrum, ε*(ν), of the solvent, covering a wide interval of frequencies, ν, (up to hundreds of gigahertz, if possible) should be used to compute the reaction rates. Simplified approaches are usually based on the assumption of a single solvent relaxation process exhibiting Debye behavior. Such an exponential relaxation is characterized by the static permittivity, ε, the high frequency †
Moscow State University. University of Regensburg. § Kazan State Technological University. ‡
permittivity, εopt (generally taken as the square of the optical refractive index), and the associated relaxation time, τ. Generally, these parameters are extracted from values of ε*(ν) at relatively low frequencies. However, the dielectric relaxation behavior of many pure solvents and most solvent mixtures is more complicated.4 Experimental studies of the dynamic solvent effect in a series of solvents can be roughly classified in terms of their degree of simplification. The most simple (and widely adopted) approach is to assume a straightforward relation of viscosity, η, and relaxation time on the basis of Debye’s model of dipole reorientation in liquids, and to consider the slope of reaction rate vs viscosity in a bilogarithmic plot as the “degree of adiabaticity”. This approach ignores the real relaxation behavior of the solvents, as well as the fact, that viscosity, static and optical permittivity change simultaneously. However the two latter properties are also important factors in ET kinetics, and the treatment of the observed influence of the solvent as a purely dynamic effect is basically erroneous. The next level is consideration of an additive combination of dynamic and static solvent effects, that is, contributions from both viscosity and permittivity,5,6 but still to assume for the solvent Debye behavior with a single relaxation time (recalculated from viscosity). A slight improvement is achieved when the values for dielectric relaxation time and the permittivity extracted from real spectra are used directly;7 i.e., viscosity is not involved. Usually, a thorough comparison of experimental data with model estimates, done in the frame of above-mentioned approaches, leads to the statement that the multistep relaxation of all real conventional solvents has to be addressed to achieve a more realistic and quantitative description. Unfortunately, it is
10.1021/jp907479z 2010 American Chemical Society Published on Web 12/15/2009
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difficult to judge what is more crucial, this non-Debye behavior of solvent dynamics, or nonadditivity of dynamic and static solvent effects. Pioneering papers dealing with this problem7-9 have been published already 15-20 years ago but suffered from the lack of precise dielectric data. With the recent progress in the dielectric spectroscopy of liquids in the microwave region,10,11 more details of the solvent relaxation are accessible now. In a series of recent papers12-14 we revisited ET in aqueous carbohydrate solutions. Such mixtures were traditionally used to study the effect of solvent viscosity on ET reactions. Various approaches were used to thoroughly analyze the experimental data, with the main goal of finding an “informative” model ET reaction. With regard to clarification of the dynamic solvent effect, the principal requirement for the searched model reaction was the observation of features being in qualitative (not only quantitative) contradiction to the predictions of the simplified models. This is to ensure that the factors and phenomena ignored by the above approaches dominate the system under study. We were finally successful with the electrochemical reduction of peroxodisulfate on mercury,13 which exhibits a nonmonotonic dependence of the reaction rate on carbohydrate concentration. At high viscosities, a strong increase of the reaction rate was observed, which paralleled the rise of η. We proposed and adopted a procedure of data treatment that corrected the experimental rates for interfacial (adsorption-induced) effects. The latter also depend on cosolvent concentration since adsorption of the cosolvent can result in partial surface blocking and a change of the potential in the diffuse part of electrical double layer. Thus, the remaining “anomalous” increase of the corrected ET rate with viscosity in carbohydrate syrups does not result from interfacial effects. We argued that the anomalous increase of the ET rate at high carbohydrate concentrations results from a positive static solvent effect: the growth of the optical permittivity with syrup concentration induces the decrease of solvent reorganization energy. Thus, the resulting increase of ET rate exceeds the opposing dynamical effect associated with the growth of viscosity. However, other (less transparent) reasons for the observed anomalous behavior could not be fully excluded. Note that the peroxodisulfate reduction on mercury in aqueous media is well documented experimentally15,16 and was found to be close to the activationless regime.17 In what follows, we are looking for a similar “anomalous” rate increase in a mixed liquid of more simple molecular structure, namely water-EG mixtures. EG is miscible with water in any ratios, and the mixtures are able to dissolve various inorganic salts.18 At room temperature, viscosity can be varied in the range 0.9-17.0 mPa s-1 (and in an even wider range at lower temperatures) by changing the mixture composition from pure water to pure EG. Since the refractive index at optical frequencies is practically independent of the water/EG ratio, the solvent reorganization energy is constant. On the other hand, the dielectric spectra, and thus the effective relaxation time, change significantly with composition (see Figure 1). Thus, for this system dynamic solvent effects should not be masked by a (otherwise common) variation of the static solvent effect. By going from sucrose and glucose to the smaller EG molecules as the viscous additive, we also hope that the more simple molecular structure of this component will minimize the possible problem of an inhomogeneous solution structure that might be anticipated for the carbohydrates. Additionally, for EG the width of the potential window on the mercury electrode is even wider than that in water,19 which is a further advantage when studying
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Figure 1. Dielectric dispersion and loss spectra of EG-water mixtures for EG mole fractions, xEG, indicated in the figure; 20 °C.
ET reactions. The simplicity of the EG molecular structure may even allow modeling the solutions at a molecular level in the near future. These important advantages should be supplemented with precise dielectric relaxation spectra (DRS) for water-EG mixtures. Data for the static permittivity (εs) of pure EG over the temperature range of 243-453 K can be found in ref 20. Detailed static permittivity data for EG-water mixtures at different temperatures are also available.21 Dielectric spectra, ε*(ν), of EG covering frequencies between 10 MHz and 4 GHz were reported by Sengwa22 for 25, 35, 45, and 55 °C, by Jordan et al.23 for 10, 20, 30, and 40 °C, and by Salefran et al.24 for 25 °C. In ref 22, the spectra were fitted using a single Cole-Cole (CC) equation,22 whereas superposition of two Debye relaxations (D+D model) was suggested in refs 23 and 24. Recent experiments covering 30 MHz to 89 GHz at 25 °C indicated an additional high-frequency contribution, leading to the D+D+D model.25 Dielectric spectra of EG-water mixtures were reported in ref 26 up to 10 GHz between 0 and 40 °C, and described by a single Debye relaxation. However, keeping in mind the rather complex dynamics of pure EG,25 most of the available dielectric spectra are of limited use for ET studies because of their too narrow frequency coverage and the resulting too simple relaxation model. In this contribution we combine broad-band dielectric relaxation spectroscopy and electrochemical experiments to study ET rates and their dependence on solvent dynamics in water-EG mixtures over the entire composition range. The data are complemented with electrocapillary measurements to allow reliable correction for interfacial effects. Experimental Details Solvent mixtures were prepared by combining precisely measured volumes of 1,2-ethanediol (A-grade reagent) and water (purified with a Millipore line). Na2S2O8 was recrystallized twice from Milli-Q water. NaF and mercury was analytical grade from Merck. Mixtures with EG mole fractions 0.1 e xEG e 0.9 were investigated, with the salt solutions in these mixtures prepared volumetrically. The dielectric spectra, ε*(ν) ) ε′(ν) - iε′′(ν), ε′ being the relative permittivity and ε′′ the corresponding dielectric loss at frequency ν, were determined at 15, 20, 25, 30, and 35 ((0.02) °C in the frequency range 0.1 e ν/GHz e 89 using a time-domain reflectometer for ν e 6 GHz,27 a frequencydomain reflectometer covering 0.2-20 GHz,28 and a set of three
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waveguide interferometers for 13-89 GHz.29 Typical spectra are shown in Figure 1. To check the effect of the supporting electrolyte, NaF (necessary for the measurement of the ET rates), on the relaxation behavior of the solvent mixtures, several samples with 5-100 mM NaF added to the mixture with xEG ) 0.6 were also investigated. No significant change of ε*(ν) could be observed. The peroxodisulfate reduction rates were determined from dc polarography according to the procedure reported in refs 13, 14, and 17. The dropping period of the mercury electrode into the solution with added supporting electrolyte (NaF, 10 mM) at open circuit was 10-12 s. The raw data, which are mixedcurrent values, were corrected for adsorption effects according to the protocol introduced in ref 13. The required electrocapillary data were obtained with a Gouy setup and evaluated as described in ref 19. Experimental Results (i) Dielectric Relaxation. A typical set of the obtained dielectric spectra is displayed in Figure 1. As expected, the values for the frequency-dependent relative permittivity, ε′(ν), decrease with increasing EG content. Simultaneously, the dispersion region of ε′(ν) and the associated peak of the dielectric loss, ε′′(ν), shift to lower frequencies, indicating a slowing down of the overall dynamics. The increasing asymmetry of the spectra suggests a redistribution of the relative weights of the modes contributing to ε*(ν). To describe the experimental spectra, various conceivable models based on sums of n ) 1, ..., 4 Havriliak-Negami (HN) equations, or the simplified variants (the Cole-Davidson (CD), the Cole-Cole (CC), and the Debye (D) equation30), were tested by simultaneously fitting ε′(ν) and ε′′(ν) with a nonlinear leastsquares procedure.31 The quality of the fit was judged from the obtained value for the reduced error function, χr2, and by visual inspection for systematic deviations of the fit curve from the experimental data. It turned out that for all compositions investigated at 20 °C the superposition of three Debye equations (the D+D+D model)
ε*(ν) )
∆ε1 ∆ε2 ∆ε3 + + + ε∞ 1 + i2ντ1 1 + i2ντ2 1 + i2ντ3
(1) gives the best fit (see Table S1 of the Supporting Information). This applies also for almost all samples at the other studied temperatures (15, 25, 30, 35 °C), although fits with the CD+D model are occasionally of similar quality. In eq 1, ∆εi is the amplitude (relaxation strength) and τi the relaxation time of mode i. The obtained parameters for 20 °C are collected in Table 1; those for the other temperatures are given in Table S2 of the Supporting Information. Figure S1(a) of the Supporting Information compares the static permittivities obtained for all temperatures investigated in this study with the values determined by Uosaki et al.33 The agreement between both data sets is excellent. Figure 2a shows the dependence of the normalized amplitudes, n
gi ) ∆εi /
∑ ∆εj
(2)
j)1
on the mole fraction of EG. Corresponding relaxation times are displayed in Figure 2b. Whereas the mixture data for g1 and τ1 smoothly extrapolate to the values for pure EG, g2 and τ2 are clearly linked to the normalized amplitude (g ) 0.969) and
Figure 2. (a) Normalized amplitudes, gi, and (b) relaxation times, τi, of the relaxation processes i ) 1, ..., 3 obtained from fitting eq 1 to the dielectric spectra of EG-water mixtures at 20 °C.
TABLE 1: Amplitudes, ∆εi, and Relaxation Times, τi, of the Relaxation Processes 1-3 Obtained from Fitting Eq 1 to the Dielectric Spectra of EG-Water Mixtures at 20 °Ca xEG
εs
τ1, ps
∆ε1
τ2, ps
∆ε2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
80.08 72.71 67.18 61.95 57.6 53.84 50.55 48.4 46.03 43.69 41.78
24.33 35.89 47.26 60.65 73.72 91.84 106.31 122.54 136.14 150
32.06 40.8 43.69 43.08 37.53 35.30 36.01 34.00 32.75 34.48
9.43 12.69 15.36 17.97 20.37 31.87 34.46 31.77 33.98 33.59 20
74.19 34.14 20.20 12.23 8.73 9.85 9.44 6.86 6.64 5.74 2.8
τ3, ps ∆ε3
εinf
χr2
0.29 2.3 2.00 1.93 1.99 3.75 3.51 2.91 2.82 2.79 1.7
3.53 5.42 5.48 4.71 4.45 4.27 4.33 4.08 4.11 3.97 3.6
0.0376 0.0548 0.0324 0.0167 0.0410 0.0497 0.0361 0.0240 0.0310
2.36 1.09 0.70 1.32 1.34 2.19 1.48 1.45 1.28 1.23 0.9
a 0 and 100% from refs 25 and 32. Also tabulated are the static permittivity, εs, and the high-frequency permittivity limit, εinf, and the value of the reduced error function, χr2.
relaxation time (τ ) 9.43 ps) of the dominating mode for pure water, which is commonly associated with the cooperative relaxation of the H-bond network in this liquid.25 According to Figure 2a, the weight of the water-like mode strongly decreases up to xEG ≈ 0.3, whereas the relative magnitude of the slow mode strongly increases. At higher EG content, g1 and g2 remain essentially constant. The relaxation time of the slow mode, τ1, steadily increases from ∼24 ps at xEG ) 0.1 to 150 ps in pure EG. On the other hand, τ2 passes a flat, ill-defined maximum at xEG ≈ 0.6. However, it should be kept in mind that especially at low EG content the ratio τ1/τ2 is only ∼2-3, which makes the accurate separation of both modes difficult.
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Despite the uncertainties in the numerical values of the obtained amplitudes and relaxation times, we can tentatively assign the low-frequency mode (∆ε1, τ1) to a process where the dynamics is dominated by EG molecules, although water molecules hydrating EG molecules must also be involved to explain the marked rise of g1 at low xEG. A similar behavior was observed for aqueous solutions of oligoethylene glycol surfactants34,35 and is reminiscent of water-alcohol and water-oligoethylene glycol mixtures,36,37 although a direct comparison is not possible due to the limited frequency range of the latter studies. The concentration dependences of g2 and τ2 indicate that the intermediate-frequency relaxation (∆ε2, τ2) is more water-like and probably dominated by the relaxation of H2O molecules mainly interacting with other H2O. However, a contribution from the intramolecular dynamics of EG cannot be excluded, since ∆ε2 does not vanish at xEG ) 1. The assumption of a fast relaxation (∆ε3, τ3), possibly associated with the relaxation of OH groups not involved in hydrogen bonding, is necessary for a satisfactory fit of the mixture spectra and compatible with the relaxation behavior of both pure water and pure EG, which also exhibit a fast mode.25 However, this relaxation just peaks at the high-frequency limit of this investigation (EG and all mixtures) or even above (water), so that the obtained values for τ3 are not reliable (see Figure 2b). Although the numerical data should not be overestimated, we can safely say that the relative weight of the fast relaxation is small and essentially constant, g3 ≈ 0.03. Due to the small separation of the slow and intermediate relaxations and the problems discussed just before, the values obtained for τ2 and τ3 at the other temperatures were too noisy for a detailed analysis of their temperature dependence. However, for τ1 the Eyring activation free energy, ∆G*,
τ ) τ0 +
(
h ∆G* exp kBT kB T
)
(3)
could be reliably determined (Figure S1(b) of the Supporting Information). As can be seen, ∆G* smoothly increases from 14 kJ mol-1 for xEG ) 0.2 to 17 kJ mol-1 for xEG ) 0.9. The data are always above the line connecting the ∆G* values of pure water, 9.9 kJ mol-1,38 and pure EG, 16.6 kJ mol-1,32 suggesting a marked positive excess activation free energy especially at low xEG. This behavior closely resembles n-alcohol-water mixtures and indicates major changes in the liquid structure.39 Despite the need for three modes to properly describe the spectra of EG-water mixtures, their dielectric behavior is considerably less complicated than that of sucrose and glucose solutions39,40 with their numerous closely superimposing relaxation events. (ii) Hg/Solution Interface in Water-EG Mixtures. Information concerning EG adsorption on the mercury/solution interface is rather fragmentary. Earlier studies on the absorption of aromatic compounds from water and EG solutions on mercury using differential capacity measurements demonstrate indirectly that the EG molecule tends to stronger chemisorption as compared to water.41,42 The Gibbs energy of adsorption from the gas phase was estimated to be around -50 to -60 kJ mol-1. Electrocapillary data for EG adsorption from diluted water solutions were described by a Langmuir isotherm, yielding an adsorption energy parameter of ca. -8 kJ mol-1.43 The doublelayer capacitance of alkali chloride solutions in EG-water mixtures was found to be lower as compared to the correspond-
ing aqueous solutions.44 However, this fact was discussed exclusively in the context of larger effective radii of the solvated ions. We applied the electrocapillary technique to obtain the systematic dependence of the surface tension, σ, on the potential, E, of the mercury electrode (aqueous SCE scale) and to estimate the Gibbs adsorption of EG for various mixture compositions. A smooth shift of the maximum of the (σ, E)-curves from ca. -0.5 to ca. -0.3 V (vs SCE) was observed with increasing xEG, as well as a simultaneous decrease of surface tension (Figure 3a). To estimate Gibbs adsorption, Γ, we applied the following thermodynamic relationship:
Γ)-
1 ∂σ RT ∂ ln a
(
)
(4)
E
where a is the activity of adsorbing species in solution. The activity coefficients for EG in water solutions are available only in the range 0 e xEG e 0.025 (these values are in the range of 1-1.1).45,46 Therefore, concentrations instead of activities were used for approximate adsorption calculations. Figure 3b demonstrates that adsorption of EG remains rather high over the entire potential range and always increases monotonically with EG content. The calculated EG adsorption for cEG ) 1.0 mol L-1 at the vicinity of zero charge potential is ca. 3 × 10-6 mol m-2, slightly higher than reported in ref 42 (∼2 × 10-6 mol m-2). Unfortunately, there are no other data for comparison. In relation to the problem of interfacial structure affecting ET in water-EG mixtures, the data in Figure 3a mean that if any electrode reaction is blocked by adsorbed EG (i.e., if the EG molecule operates like a spacer), the rate of this process is expected to decrease systematically with increasing EG content. Since, as we discuss below, the solvent effect is of opposite sign, the accuracy of our estimated Γ is not problematic. The electrocapillary data were also applied to determine electrode charge densities, q, at various potentials. The (q, E)curves obtained by differentiation of σ ) f(E) exhibit a systematic decrease of their slope with increasing xEG. To estimate the potential of the outer Helmholz plane, ψ1, the Gouy-Chapman equation47 was applied:
ψ1 )
(
q 2RT Arcsh F 2√2εε0RT
)
(5)
When doing so, we considered NaF and Na2S2O8 solutions as electrolytes tending to purely electrostatic adsorption. For the studied region of electrode charge densities, the ψ1 values (Figure 3c) indicate a negligible dependence on EG content. This is a favorable finding for further data analysis because electrostatic interfacial effects, even if only taken into account in a rather approximate manner, thus cannot seriously affect the trends observed for the dependence of the ET rate on solvent composition at fixed potential. (iii) Electron Transfer Kinetics. For heterogeneous ET in electrochemical systems, the reaction rate is easily determined by measuring current density (or current I, if the electrode surface area remains the same in comparative experiments). This rate depends on the electrode potential E: the stronger the deviation of this value from equilibrium potential, the higher is the reaction free energy. Rate constants of electrochemical reactions reported in the literature correspond to reaction rates at fixed (typically, but not necessarily equilibrium) potential. The only systematic experimental study of heterogeneous ET in water-EG mixtures we found in the literature is a series of papers by Hecht and Fawcett,48,49 who studied the reduction of
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J. Phys. Chem. B, Vol. 114, No. 1, 2010 315 with solvent composition. On the other hand, the data on europium(III) reduction50 should be mentioned. Here, a slight increase of the reaction rate constant was observed when going from pure water to 18 and 36 vol % aqueous EG, instead of the expected rate decrease with viscosity. Unfortunately, the absence of experimental details prohibits judging the accuracy and reliability of this result. In any case, the range of EG content in ref 50 is too narrow for consideration of the viscosity effect. For deeper understanding of the solvent effect in the water-EG system we prefer to deal with the dependence of current on potential in a potential range as wide as possible. In particular, this approach allows one (under favorable circumstances) to ignore the shift of equilibrium potential with solvent composition. The latter shift is a fundamental problem having no exact solution, as it requires the knowledge of the potential drop at the interface between two solvents. However, if we obtain different current vs xEG dependences for different potentials, we can be sure that the change of equilibrium potential is not a major factor of the observed solvent effect. The present experimental dc polarogramms of peroxodisulfate reduction on mercury in mixed water-EG solvents (Figure 4a) demonstrate a significant decrease of the observed current for the mixtures as compared to water, accompanied by a systematic shift of the current “pit” (minimum in the polarograms) with xEG toward less negative potentials. The “pit” is a typical feature of the electroreduction of anions at negatively charged metals in dilute solutions, when the reactant-electrode repulsion is not screened by a supporting electrolyte. Just this feature makes peroxodisulfate such an attractive model reactant: due to repulsion its reduction escapes mass transport limitations over a wide potential interval. The increase of the current (right branch of the pit) with increasingly negative potential is explained by the interplay of repulsion and an increasing reaction free energy.15-17 The dependence of the current on xEG at a given potential is nonmonotonic. Similar features were also observed previously for the same reaction in carbohydrate syrups.10 For highly viscous solutions, at xEG > 0.4, the observed reaction rate increases with xEG, especially at the right branch of the polarogram (current growth). In a certain xEG interval there is no correlation of the current with the monotonically increasing slow relaxation time, τ1, but clear correlations with the shorter relaxation times, τ1 and τ3 (Figure 2b). The currents given in Figure 5a are of mixed (ET and mass transfer) nature. At potentials more negative than -1.6 to -1.7 V the limiting diffusion current, Id, is reached for all solutions under study. According to the Ilkovic equation51
Id ) 6.29 × 10-3nFD1/2m2/3τ1/6c Figure 3. (a) Electrocapillary data (surface tension, σ, of the mercury electrode as a function of its potential, E) for solutions of 0.01 M NaF in EG-water mixtures at 20 °C. (b) Excess surface concentration, Γ, of adsorbed EG and (c) the potential, Ψ1, at the outer Helmholz plane (OHP) as a function of EG mole fraction, xEG, calculated from (σ, E) curves according to eqs 4 and 5 for three electrode charge densities indicated in the figure.
V(III) ethylenediamine tetraacetate. These authors discovered a disappearance of the inhibitory viscosity effect on the rate constant at intermediate EG concentrations and attributed this to an increased contribution of reactants containing innersphere EG molecules (instead of water as inner-sphere ligand). For the problem we try to address here the system studied in refs 48 and 49 looks too complex, as in reality this system possesses two or even three reacting species, and their ratio is changed
(6)
this value is proportional to the square root of the diffusion coefficient, D, of the reactant. In eq 6, n is the number of electrons taking part in the reaction, m the mass flow rate of mercury, τ the drop lifetime, and c is the reagent concentration. Figure 4b shows the dependence of D (normalized to the value for peroxodisulfate in pure water, D(w) ) 1.03 10-9 cm2 s-1) on reduced viscosity η/η(w). Despite the limited accuracy of Id, deviations from linearity are obvious. In relation to this finding, it is important to know what deviations were found for the dependence of electric conductivity (or the diffusion coefficients) on inverse viscosity for other electrolyte solutions in pure EG or EG-water mixtures: For Co(NO2)(NH3)5SO4 and Fe(phen)3SO4 solutions,52 as well as for alkali metal chlorides, positive derivations from linearity were observed in pure EG when viscosity was controlled by temperature.53 On the other
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Figure 5. (a) Corrected Tafel plots and (b) dependence of the corrected ET rate on solvent composition.
Figure 4. (a) dc polarograms, current I as a function of electrode potential, E, of peroxodisulfate electroreduction in various EG-water mixtures containing 2 mM Na2S2O8 and 10 mM NaF. (b) Normalized diffusion coefficient, D/D(w), of peroxodisulfate, recalculated from the limiting currents Id, as a function of reduced viscosity (taken from ref 15). (c) ET rates corrected from mass transfer contribution as IET ) IdI/(Id - I).
hand, for magnesium m-benzenedisulfonate solutions at 25 °C a close-to-linear relation was found,54 whereas for ZnSO4 a small negative derivation appeared at high EG concentrations.55 A possible reason for these deviations is the solvent-dependent size of reacting species (specific salvation by different solvent components at different xEG). Another reason can be a heterogeneous structure of the liquid, providing some channels for ion transport. In any case these effects are less pronounced in water-EG as compared to water-carbohydrate mixtures and can thus be neglected here. To get the true ET rates, we first recalculated the polarographic data (Figure 4a) to eliminate mass transport contributions using the simple relation IET ) IdI/(Id - I). The resulting true
ET values (Figure 4c) exhibit a clearly nonmonotonic dependence on solvent composition over a wide potential range. We attempted to introduce corrections for possible surface blocking by adsorbed EG using Gibbs adsorption values. This procedure, proposed in ref 14, assumes that ET occurs exclusively at the surface fragments free from adsorbed EG. Correspondingly, the corrected value should be IET/(1 - Γ/Γmax). The exact value of surface coverage Γ/Γmax depends on Γmax, which reflects monolayer adsorption. For this reason correction depends on the assumed EG orientations in the adlayer, but for all possible values for Γmax it results in a sharp increase of the corrected ET rate with xEG. As a consequence, we escape any correction for surface blocking in the following analysis because we prefer to consider underestimated, not overestimated, values for anomalous current growth with viscosity. We can assume that, in contrast to large carbohydrate molecules, the molecules of EG cosolvent have a size similar to water molecules and can hardly affect the distance of closest approach for any reactant. Just this distance is the most important parameter for possible blocking. To correct the ET rates for electrostatic repulsion, the traditional corrected Tafel plots15,17 were constructed (Figure 5a) using the above-discussed ψ1 potentials (Figure 3c). The disappearance of the “pit” in the corrected Tafel plots demonstrates that this correction is reasonable. This correction is surely approximate because of some ambiguity for the potential in the plane where the reactant is really localized (it can deviate slightly from ψ1). However, we should stress again that, irrespective of the details of this correction, the dependence of corrected ET rate on the solvent composition at fixed potential is not affected by this approximation. The final dependences of the rate of the elementary ET act are plotted in Figure 5b for several values of the ψ1-corrected potential.
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Qualitatively, the observed dependences (Figure 5b) of the corrected current (reaction rate) on xEG, with their asymmetric minima, look similar to corresponding plots for the peroxodisulfate reduction in carbohydrate solutions.13 However, for water-EG mixtures it appears to be possible to extract the corrected rates over a much wider potential region because of the more straightforward correction of interfacial effects and the absence of cosolvent reduction at the ascending branch of the polarograms. Moreover, the anomalous current increase with rising viscosity is more pronounced than in carbohydrate syrups. Finally, we should stress that for the water-EG system we were able to observe a clear tendency: the minimum shape and position depend on the electrode potential, which cannot be explained by a conventional dependence of the equilibrium potential on solvent composition. The systematic increase of the slope of the polarization curves, presented in Figure 5a, may become important in the future for a deeper understanding of the effect. In terms of the transfer coefficient, R, which is a basic characteristic of the ET elementary act,56 the viscosity increase leads to a pronounced increase of R for the system under study. The values of R observed for aqueous solution are 0.03-0.08, increasing with increasingly negative potential; for xEG ) 0.7, the corresponding interval of R is 0.06-0.28. Basically, this looks similar to the behavior of R for the same model reaction in carbohydrate syrups57 determined for a more narrow potential range (without proper possibility to judge about the potential dependence of R). Basically, so low R values result from very high overvoltage, corresponding to the vicinity of activationless region. Modeling the Observed Effect The effects of solvent dynamics on the electroreduction of peroxodisulphate anion at the mercury electrode were explored recently for aqueous sucrose and glucose solutions.57 The following main conclusions have been made on the reaction mechanism resting on the results of model calculations: (1) at least two anisotropic reaction coordinates, a slow diffusive solvent coordinate and a fast intramolecular (reactant) coordinate, should be introduced to address properly the solvent dynamics within the Sumi-Marcus model;3 (2) the nonmonotonic dependence of the ET rate on the solvent relaxation time can only be observed in the vicinity of activationless discharge; and (3) the static solvent effect plays an essential role. However, it is important to stress that, in contrast to recent results for aqueous sucrose and glucose solutions,57 the Sumi-Marcus model is not able to predict the ascending plots observed for the present system (i.e., the increase of IET with increasing xEG for xEG >0.3 in Figure 5b). The main reason is the less pronounced change of the Pekar factor, (1/εopt - 1/ε), with solvent composition for water-EG mixtures as compared to sucrose and glucose solutions. Indeed, the values for the Pekar factor fall into the interval 0.55-0.48 when going from pure water to 70% EG solution (see Table S2 of the Supporting Information), so that the static solvent effect should be rather constant. This prompts us to look for other possible physical reasons that could lead to a reinforcement of the static solvent effect. One possible scenario is discussed below. Let us consider the correlation function describing a fluctuating reactant energy level in solution that can be defined as follows:
M(τ) ) 2kBTλs
Q(τ) Q(0)
(7)
where λs is the solvent (outer-sphere) reorganization energy.58-61 The function Q(τ) is written in the form
Q(τ) )
i 2π
[
1 1 ∫-∞∞ exp(-iωτ) ε(ω) εopt
]
dω ω
(8)
where ε(ω) is the complex dielectric spectrum at angular frequency ω ) 2πν. For a solvent exhibiting dielectric relaxation described by the D+D+D model the correlation function M(τ) can be exactly expanded into the sum: 3
M(τ) ) 2kBTλs
∑ δi exp(-τ/τ*)i
(9)
i)1
3 where the δi are the contributions from ith solvent mode (Σi)1 δi ) 1, not to be confused with gi used above!) to the solvent reorganization energy and τ*i are the characteristic correlation times (which are not identical to the dielectric relaxation times τ). To perform this expansion, we employed the inverse Laplace transform technique described in detail in ref 12. The results are shown in Figure S2 of the Supporting Information and reveal differing behavior for the three solvent modes. The component associated with the longest correlation time, τ*1 , does practically not contribute to the solvent reorganization energy over the entire composition range. The contribution of the fast mode, characterized by τ*3 , decreases with the EG concentration, while the behavior of intermediate component, with correlation time τ*2 , is inverse. The most important consequence of expansion (9) is that we can consider three independent solvent coordinates b q ) (q1, q2, q3) to address the solvent contribution to the multidimensional reaction free energy surface E. In the weak coupling limit (small reactant-electrode orbital overlap) the latter can be recast in the form
Ei(q b,r) ) Ui(q b) + U*(r) i Ef(q b,r) ) Uf(q b) + U*(r) f
(10)
where r is the intramolecular coordinate (the O-O bond length); subscripts “i” and “f” refer to the initial and final states, and U*(r) are intramolecular potentials respectively, and U*(r) in f 57 describes bond rupture). The “solvent” parts of eqs 10, (U*(r) f b) and Uf(q b), take the form Ui(q 3
Ui(q b) )
∑ δjλsqj2 j)1
3
Uf(q b) )
∑ δjλs(qj - 1)2
(11)
j)1
Thus, the product δjλs can be treated as a part of the solvent reorganization energy associated with the jth solvent mode. Then we can write the master equation in terms the Agmon-Hopfield formalism:
∂P(q b,τ) ) ∂τ
3
∑ Dj j)1
{
}
b) ∂ ∂2 1 ∂U(q + P(q b,τ) 2 k T ∂q ∂qj ∂qj B j kin(q b) P(q b,τ)
(12)
which is an extended version of the Sumi-Marcus model. In eq 12 P(q b,τ) is the probability density to find the reactant in its initial state; Dj refers to the coefficient of diffusion along the jth solvent coordinate, Dj ) kBT/2δjλsτL, and U(q) is a section of the reaction free-energy surface. The sink term in eq 12, kin(q b), is written as follows:
kin ) νin exp{-∆E*(q a b)/kBT} where νin is an effective frequency factor.57
(13)
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Zagrebin et al.
The energy barrier along the intramolecular degree of freedom, ∆E*, a depends in general on all solvent coordinates (q1, q2, q3) and is defined as
b, q*saddle(q b)) - U(q b,r ) r0) ∆E*(q a b) ) U(q
(14)
where q*saddle(q), defined by a transcendent equation, denotes the saddle line on the multidimensional free energy surface, E(q,r) is the equilibrium O-O bond length in S2O82-. Two time scales characterizing different averaged survival times of the product in the initial state can be considered (see relevant discussion in ref 57).
τa )
∫0∞ ∫bqbq
R
L
P(q b, τ) dq b dτ
(15)
and
1 τb ) τa
∫0 ∫bq ∞
b qR L
τP(q b,τ) dq b dτ
(16)
where b qR and b qL are assumed to be the values of q1, q2, and q3 at the left and right boundaries of the reaction coordinates, respectively.62 Then the ET rate constants, ka and kb, can be alternatively defined as 1/τa and 1/τb. The accurate numerical solution of eq 12 is complicated and challenging; such work is currently in progress and will be reported separately. Instead, we suggest here a simple approach to the qualitative analysis of this problem solely. The main idea is to split eq 12 into three independent equations corresponding to the three solvent modes q1, q2, and q3:
{
}
∂Pj(qj,τ) ∂2 1 ∂U(qj) ∂ + P (q ,τ) ) Dj 2 ∂τ kBT ∂qj ∂qj j j ∂qj kin(qj) Pj(qj,τ)
(17)
where j ) 1-3 and Pj(qj,τ) is the “partial” probability density to find a reactant in its initial state. The intersects of U(qj) of the reaction free energy surface for initial (in) and final (f) states are written in the form:
Ui(qj) ) δjλsqj2 Uf(qj) ) δjλs(qj - 1)2
(18)
Equations 17 can be readily solved on the basis of an effective computational scheme suggested in ref 62, leading to an assumed expression for the observable rate constant, k˜, which is convenient for further qualitative analysis: (1) (2) (3) k˜ ) c1(xEG)ka(b) + c2(xEG)ka(b) + c3(xEG)ka(b)
(19)
where the cj(xEG) are some functions on the EG concentration. Using the present experimental data, we calculated three “partial” rate constants as a function of the mixed solvent for different values of potential and temperature. Since we did not find noticeable differences between the survival times τa and τb, only the k(i) a values are discussed below. As can be seen from Figure 6, the “partial” rate constants associated with the three solvent modes show quite different composition dependence. The k(i) a values associated with the fast solvent mode noticeably increase with the EG concentration (due to the significant static solvent effect; see Figure S2, Supporting Information), whereas the “intermediate” rate constant decreases. The “slow” rate constant reveals only a slight decrease when going to pure EG. The results obtained for other values of overvoltage and temperature are qualitatively similar to those shown in Figure
Figure 6. Partial rate constants corresponding to three solvent modes calculated according to eq 17 for 20 °C and the electrode overpotential E0 - E of 2.65 V.
6; the “humps” on the model curves become smoother for other temperatures. The way we suggested to treat eq 12 is obviously crude, and we cannot expect receiving quantitative estimates for the resulting (observable) rate constants. The aim was to demonstrate that some inherent properties of EG-solvent mixtures could result in a nontrivial solvent effect. The conclusion is allowed, however, that at low EG concentrations the slow and intermediate solvent relaxation processes dominate the ET rate, whereas for solutions with high EG content the contribution from the fast mode becomes more significant. The interplay between the “partial” model constants, ka(j), corresponding to three solvent modes (distinctly shown in Figure S2, Supporting Information), might lead to the observed anomalous dependence of the rate of S2O82- electroreduction on the composition of water-EG mixtures. Concluding Remarks The presented experimental results confirm the existence of an anomalous viscosity effect on the ET rate of the electrochemical reduction of S2O82- in EG-water mixtures. In contrast to a similar effect reported in refs 48 and 49, the present case cannot result from a change of the molecular structure of the reactant with changing composition of the mixed solvent. Our semiquantitative interpretation assigns this effect to the change of the solvent relaxation mode, which provides the major contribution to the ET rate constant. We can argue, therefore, that a static solvent effect can originate both from the noticeable change of the Pekar factor and from the behavior of different solvent modes. The latter effect probably takes place in carbohydrate-water systems as well, but it is certainly damped due to smeared “non-Debye” behavior of the dielectric response.40 It follows from model calculations in ref 57 that some details of the free energy surface specific for bond break reactions are crucial to observe the nonmonotonic dependence of reaction rate on solution viscosity. For the peroxodisulfate model reaction, the main feature is a smooth barrier in the vicinity of activationless discharge. According to our additional analysis, the same features of free energy surfaces play also an important role for the dynamic effect in water-EG mixtures. Therefore, most likely the anomalous viscosity effect on the ET rate should be associated with bond rupture. For example, a quite monotonic ET rate vs viscosity dependence was found for the Cr(EDTA)- reduction in these mixtures.63 This redox couple also demonstrates a noticeable intramolecular reorganiza-
Dynamic Solvent Effects in Electrochemical Kinetics tion that can be described, however, in terms of the linear response theory.12 For the present system it appears that the anomalous viscosity effect on the ET rate is associated with bond rupture. However, there are still open questions that will hopefully be solved by ongoing work on pertinent theoretical model and associated experimental investigations. Further theoretical efforts are necessary to base our qualitative explanation of the experimental effect on more solid ground. At present, our analysis is based on the traditional representation of the static solvent effect in terms of a continuum solvent model. However, a possible reason for a positive static solvent effect may arise from the specific solvent structure. For example, in glasses the reaction rates in water-EG mixtures can be higher than in the liquid because some solvent modes are frozen (see experimental observations in ref 64). To judge whether we can assign the observed current increase to some “excluded volume” in concentrated EG solutions, we checked the available conductivity data but found no support for such an alternative hypothesis. However, further systematic studies of electrolyte conductivity in EG-water mixtures would nevertheless be promising, as the salts studied so far were rather exotic. As for a more quantitative treatment of eq 12, we consider molecular dynamics simulations in the four-dimensional reaction free energy space (with four different friction coefficients, corresponding to the solvent and intramolecular degrees of freedom) as the most promising approach. Recently, this method was successfully employed for a more simple two-dimensional case.65 New insight into the dynamic solvent effect on the electroreduction of peroxodisulphate should also be gained from molecular modeling of the S2O82- solvation in water-EG mixtures. Such simulations based on the QM/MM method and CPMD are currently in progress. To increase the strength of experimental verification, we are going to address the solventdependence of the transfer coefficient as well. The influence of solvent dynamics is seen in electrode ET reactions more frequently as compared with the homogeneous ones.1 The main reason seems to be a possibility to control the ET activation barrier when changing the electrode potential (the solvent dynamics effects appear only when the energy barrier is small enough; otherwise the ET kinetics is described pretty well in terms of the transition state theory). Acknowledgment. We are greatly indebted to Michael D. Bronshtein for fruitful discussions. We thank A. Stoppa for the VNA measurements and G. Hefter and W. Kunz for the provision of laboratory facilities. This work was supported in part by the RFBR-FWF project 09-03-91001-a and by the Deutscher Akademischer Austauschdienst (travel grant for P.A.Z.). Supporting Information Available: Figures S1 and S2 of static permittivity, activation free energy, and relative contributions to reorganization energy. Tables S1-S3 of reduced error function; amplitudes, relaxation times, and permittivities; and refractive index, dielectric constants, and Pekar factors. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Swaddle, Th. W. Chem. ReV. 2005, 105, 2573. (2) Kuznetsov, A. M.; Ulstrup, J. Electron Transfer in Chemistry and Biology; Wiley: Chichester-N.Y.-Weinheim-Brisbane-Singapore-Toronto, 1999. (3) Sumi, H.; Marcus, R. A. J. Chem. Phys. 1986, 84, 4894. (4) Barthel, J.; Buchner, R. Pure Appl. Chem. 1991, 63, 1473. (5) Zang, X.; Yang, H.; Bard, A. J. J. Am. Chem. Soc. 1987, 109, 1916.
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