Interplay between Solvent Effects of Different ... - ACS Publications

J. Phys. Chem. B 2009, 113, 10277–10284

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Interplay between Solvent Effects of Different Nature in Interfacial Bond Breaking Electron Transfer Renat R. Nazmutdinov,*,† Michael D. Bronshtein,† Galina A. Tsirlina,‡ and Nina V. Titova‡ Kazan State Technological UniVersity, K. Marx Str., 68, 420015 Kazan, Republic Tatarstan, Russian Federation, and Department of Electrochemistry, Moscow State UniVersity, Leninskie Gory 1-str.3, 119991, Moscow, Russian Federation ReceiVed: March 25, 2009; ReVised Manuscript ReceiVed: May 19, 2009

Solvent dynamics effects on electroreduction of peroxodisulphate anion on mercury electrode (a typical bond breaking electron transfer reaction) are explored in the framework of the Sumi-Marcus model. The reaction three-dimensional free energy surface is constructed using the Anderson model Hamiltonian. A new interpretation of short- and long-time survival times is presented as well. Since the reduction is assumed to proceed from aqueous sucrose and glucose solutions of different concentrations (which are used to vary the solution viscosity), unavoidable changes in the Pekar factor (static effect) are also taken into account. The results of model calculations are employed to interpret challenging experimental data on nonmonotonous constant rate vs solution viscosity dependence reported earlier (in part, appearance of an ascent plot). The influence of mixed solvent composition on the reaction rate and transfer coefficient is explained in terms of the saddle point avoidance in the vicinity of activationless discharge. Splitting of the reaction coordinates into slow (solvent) and fast (intramolecular) ones is argued to be crucial, as the most important reaction features cannot be described by means of more simplified models, even if both static and dynamic effects are addressed. I. Introduction A variety of electron transfer (ET) reactions in electrolyte solutions are accompanied with chemical bond break. When both ET and the bond break occur in a single elementary act, the reaction mechanism is usually treated as concerted. The bond breaking electron transfer (BBET) can be described in a simplified way in the framework of Saveánt’s model,1 which remains the most frequently used so far. A more rigorous theory to calculate the activation barrier of BBET reactions was developed by German and Kuznetsov.2 Solvent dynamics effects on BBET reactions in terms of stochastic theory were explored theoretically by Cukier et al.;3,4 useful analytical solutions were found for special limiting cases. The interplay between solvent dynamics and the electrode-reactant coupling effect in such reactions was addressed first in ref 5. The authors combined a spinless version of the Anderson Hamiltonian6 with Saveánt’s theory; the solvent dynamics was treated on the basis of the Sumi-Marcus (SM) model (frequently also referred to as the Agmon-Hopfield formalism in the literature).7,8 In ref 9, Ignaczak and Schmickler have reported the results of molecular dynamics simulations of adiabatic BBET processes using a twodimensional model potential5 and the stochastic collision model to address the solvent friction. However, despite considerable progress in this field, some aspects of the heterogeneous BBET kinetics remain unclear so far. In part, stochastic theory was never employed to model electrode reactions in the activationless kinetic regime. Usually, the dynamic solvent effect, i.e., the dependence of the reaction rate k on solution viscosity (or effective relaxation * Corresponding author. E-mail: [email protected]. Fax: +7 (843) 23657-68. † Kazan State Technological University. ‡ Moscow State University.

time τL), is examined in experiments, which forms a basis for a widely accepted semiqualitative technique of estimating the so-called “degree of adiabaticity” θ (θ ) -(d lg k)/(d lg τL); for strongly adiabatic ET reactions θ ) 1, while a zero value for θ is interpreted for a long time as the feature of diabatic ET). Experimental electrochemical studies of outersphere ET reactions10-17 (with a quite low intramolecular contribution to the activation barrier) confirmed zero or negative θ values. Moreover, the condition |θ| < 1 was always fulfilled, and the linearity of lg k vs lg τL plots in the absence of any corrections was reported for rather different experimental approaches, i.e., when the solvent effect was attained by changing either the nature of pure solvent or the composition of mixed solvent. ET reactions in all experiments mentioned above were observed in a narrow overpotential region. The state of the art in the field of molecular modeling of heterogeneous ET hardly gives any hope to describe a real electrochemical system at a quantitative level. At the same time, the most obvious experimental effects of qualitative nature (“sign inversion”) are of primary importance not only to elucidate their origin but also to understand the interplay of key theory parameters. Deviations of |θ| values from unity were already observed experimentally, and the interplay between dynamic and static solvent effects was discussed to interpret the experimental results (see, e.g., refs 11, 12, and 14). Note that these effects are of different sign for the overwhelming majority of possible sets of solvents: when τL increases (with decreasing the preexponential term), the optical dielectric permittivity εop increases as well. The latter results in the Pekar factor decrease, which entails decreasing the solvent reorganization energy and, therefore, lowering the activation barrier. We failed, however, to find in the literature any systematic observation of solvent effect turnover, i.e., the start of k increase with solvent viscosity

10.1021/jp902712g CCC: $40.75  2009 American Chemical Society Published on Web 07/06/2009

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due to the static solvent effect (θ < 0). A less significant effect of this sort has been reported in ref 18, but it is difficult to judge about the accuracy and reliability of only three k values tabulated in this brief communication. A nonmonotonous behavior of θ vs electrode overvoltage η was observed at the electroreduction of [PtCl4]2- complexes at a mercury electrode from sucrose-water-mixed solutions.19 This effect was explained in ref 20 on the basis of the Sumi-Marcus model; the saddle point avoidance was concluded to be the main reason of the maximum of θ(η) dependence. Very recently, DC polarography was applied to study the effect of sucrose and glucose concentration in aqueous medium on the rate of peroxodisulfate electroreduction on a mercury electrode.21 This BBET process takes place at extremely high overvoltages (up to 3 V), which makes it possible to elucidate the role of key parameters of theory in a wide range of the reaction free energy and to model the kinetics in the vicinity of activationless discharge. Glucose solutions provide, on the other hand, a possibility to widen the range of viscosity and dielectric permittivity, as at room temperature such systems do not demonstrate any glass formation at least up to 4.7 M. (Usually, any set of pure or mixed solvents at fixed temperature provides a possibility to increase the viscosity within the limits of 2 orders of magnitude. Even if effective solvent relaxation time is not proportional to solution viscosity in the framework of the simple Debye model, the widest possible interval for this parameter can hardly exceed 3 orders of magnitude.) A remarkable nonmonotonous dependence of the reduction rate on the syrup concentration was maintained after all corrections;21 this conspicuous qualitative effect is a challenge for ET theory. We made an attempt to develop a flexible and physically transparent model which describes static and dynamic solvent effects in electrochemical BBET reactions without details on the structure of the nearest reactant solvation shell. The Sumi-Marcus model is employed; i.e., the reaction coordinate is split into the slow (diffusive) solvent coordinate and the fast intramolecular degree of freedom. The paper is organized as follows. The pertinent details of model calculations are reported in section II. The computational results are discussed and compared with experimental data in section III. Some concluding remarks can be found in section IV. It should be stressed from the very beginning that the microscopic treatment of solvation in mixed water-carbohydrate solutions is out of the scope of the present study. II. Model and Computational Details We consider further the first (bond breaking) step of the peroxodisulphate electroreduction (S2O82- + e ) SO42- + SO4•-) which is apparently rate-controlling. A spinless version of the Anderson Hamiltonian6 is employed, in order to construct the reaction free energy surface (FES) E(q, r) along the solvent coordinate q and intramolecular degree of freedom r (the O-O bond length):

E(q, r) ) Ui(q, r) + yεa(q, r) +

εa2 + ∆2 ∆ ln 2π (ε - U )2 + ∆2 a c (1)

where εa is an effective electronic energy level describing a reactant, y is the occupation probability of an electron, ∆

Nazmutdinov et al. characterizes electrode-reactant coupling, and Uc is the bottom of the conduction band (≈12 eV 6). In eq 1, Ui(q, r) can be written in the form

Ui(q, r) ) λsq2 + U*i (r)

(2)

describes the intramolecular reorgawhere the potential U*(r) i nization of reactant. The occupation probability can be obtained from the following equation:

y)

{

εa(q, r) 1 arccot π ∆

}

(3)

In turn,

εa(q, r) ) Uf(q, r) - Ui(q, r)

(4)

The potential Uf(q, r) is defined as follows:

Uf(q, r) ) λs(q - 1)2 + U*f (r) - Fη

(5)

describes the intramolecular reorganization of where U*(r) f product and η is the electrode overpotential. and U*(r) [devised on The intramolecular potentials U*(r) i f the basis of density functional theory calculations for a peroxodisulphate anion and its reduced form (the first electron transfer step is assumed, with formation of sulfate radical and sulfate anion)] were taken from ref 23

U*i (r) ) D(1 - exp[-Rr])2

and

U*f (r) ) B exp[-βr] (6)

where D ) 1.86 eV, R ) 2.38 Å-1; B ) 3.709 eV, β ) 1.95 Å-1 (r is reckoned from the O-O equilibrium distance). These potentials provide a more realistic descripparameters of U*(r) i tion of the reaction under study as compared to a simplified Saveánt’s model1 (see discussion in ref 23). For heterogeneous ET reactions, the coupling parameter ∆ in eq 1 is difficult to estimate with sufficient accuracy. This is why we simply constructed free energy surfaces (FES) at several ∆ values ranging from 0.01 to 0.1 eV; it is easy to show that this interval corresponds to the adiabatic limit of ET. Varying the coupling parameter, one can significantly modify the FES shape in the vicinity of the barrier. If ∆ e 0.01 eV, the energy surface sections are of cusp-like form, while large ∆ values entail a smoothing of the activation barrier. A three-dimensional picture of the reaction free energy surface is portrayed in Figure 1. The activation barrier is defined by the saddle point of FES. To address the static solvent effect, we have recast the solvent reorganization energy in the form (to construct the dependence of the Pekar factor on syrup concentration, we used refractory index data, like in ref 21 for the same systems)

λs(τL) ) a + bx + cx2 + dx3 λs

(7)

where λs ) 0.6 eV (in ref 23, this value is estimated from the usual Marcus formula assuming a typical molecular size radius);

Interplay between Solvent Effects in Interfacial BBET

J. Phys. Chem. B, Vol. 113, No. 30, 2009 10279 very good approximation for a reaction proceeding at high overvoltages, in particular for the peroxodisulfate electroreduction.)

{

}

∂2 1 dU(q) ∂ + LˆP(q, τ) ) D P(q, τ) - kin(q) P(q, τ) 2 k ∂q BT dq ∂q (9) and P(q, τ) is the probability density to find a reactant in the initial state; D refers to the coefficient of diffusion along the solvent coordinate, D ) kBT/2λsτL; and U(q) is a section of the reaction FES. The sink term in eq 9, kin(q), is written as follows:

kin ) νin exp{-∆E*a (q)/kBT}

(10)

where νin is an effective frequency factor. The energy barrier along the intramolecular degree of freedom, ∆E*, a depends on the solvent coordinate q and is defined in the form Figure 1. Adiabatic free energy (E) surface along the solvent (q) and intramolecular (r) coordinates describing the electroreduction of S2O82constructed at η ) 2 V and ∆ ) 0.1 eV (a); FES sections U(qsaddle, r) along r built at three different ∆ values (b).

x ) lg τL; a ) 0.579; b ) -4.2 × 10-2; c ) -7.5 × 10-3; d ) -5.231 × 10-4. The dependence (eq 7) originates from the behavior of the Pekar factor, 1/εop - 1/εst (where εop and εst are optical and static solvent dielectric constants, respectively). The room temperature value of εop for the most concentrated sucrose and glucose solutions approaches 2.5; i.e., the Pekar factor decrease with carbohydrate concentration can attain one-third (bearing in mind that εop ) 1.8 for pure water). Note that the change of this quantity in other sets of pure or mixed solvents never exceeds dozens of percents. The dependence of solvent reorganization on the composition of various mixed solvents was thoroughly investigated for a number of homogeneous ET reactions in works.24-28 The authors put primary attention on the analysis of the Pekar factor as a function of the solution composition (this dependence can be both linear and nonlinear) and concluded that preferential solvation crucially affects the solvent reorganization. The nonlinear dependence of λs in aqueous solutions on the concentration of cosolvent discussed in ref 25 looks qualitatively similar to that suggested by us (see eq 7). The Sumi-Marcus model was employed recently to address interfacial electrochemical systems and, in part, to examine the mechanism of certain electrochemical reactions: electroreduction of EDTA complexes of transitional metals (Cr(III), Co(III))22 and aqua-chlorocomplexes of Pt(II).20 Various other applications of the SM model (Agmon-Hopfield formalism) were also intensively explored over the last two decades.29-43 Dealing with this model, we have to solve the equation

∂P(q, τ) ) LˆP(q, τ) ∂τ

(8)

where Lˆ is the Smoluchowski operator supplemented by a sink term. (Equations 8 and 9 presume that the reverse process along the intramolecular coordinate is neglected. This is apparently a

∆E*a (q) ) U(q, q*saddle(q)) - U(q, r ) 0)

(11)

where q*saddle(q) notes the saddle line on the three-dimensional free energy surface E(q, r) and q*saddle(q) is defined by a transcendent equation. The probability density P(q, τ) is described by the initial boundary problem

1 exp(-λsq2 /kBT) N*

P(q, 0) ) P*(q) )

(12)

and

∂P(q, τ) 1 ∂U(q) + P(q, τ)| q)a ) 0 ∂q kBT ∂q

(13)

where N* is the normalization coefficient (N* ) (λs/πkBT)1/2). Two different time scales characterizing different averaged survival times of the product in the initial state can be considered (see relevant discussion in ref 7):

τa )

∫0∞ ∫qq

R

L

P(q, τ) dq dτ

(14)

and

τb )

1 τa

∫0∞ ∫qq

R

L

τP(q, τ) dq dτ

(15)

where qL and qR are assumed q values at the left and right boundaries, respectively.20 Then, the ET rate constant (k) can be defined in two different ways, as 1/τa and 1/τb (for heterogeneous ET, k should be multiplied by the reaction volume to be compared with experimental values). We present here a new interpretation of both average times; some pertinent details are given below. Let us write the auxiliary expression resulting from eq 8:

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∂2P(q, τ) ) Lˆ2P(q, τ) ∂τ2

Nazmutdinov et al.

(16)

Then, we can multiply both of its parts by τ and perform integration over this variable: 2

∞ ∞ τ) dτ ) ∫0 τLˆ2P(q, τ) dτ ) Lˆ2 ∫0 τP(q, τ) dτ ∫0∞ τ ∂ P(q, 2 ∂τ

) Lˆ2P(1)(q) (17) where the new function P(1)(q) ) ∫0∞ τP(q, τ) dτ is introduced. It is easy to show using integration by parts and taking into account the initially boundary problem of eq 8:

Lˆ2P(1)(q) )

2

τ) ∂P(q, τ) dτ ) τ ∫0∞ τ ∂ P(q, ∂τ ∂τ2

|

∞

-

0

τ) dτ ) -P(q, τ)| ∞0 ) P*(q) ∫0∞ ∂P(q, ∂τ

(18)

where P*(q) is the normalized equilibrium distribution function (see eq 12). It is convenient to split operator eq 18 into two parts:

LˆP(1)(q) ) -P(0)(q)

(19)

LˆP(0)(q) ) -P*(q)

(20)

and

where P(0)(q) ) ∫∞0 P(q, τ) dτ (i.e., the non-normalized function P(0)(q) is constructed on the basis of solution of eq 8 with the initial boundary problem defined by eqs 12 and 13). Indeed,

LˆP(0)(q) )

τ) dτ ) ∫0∞ LˆP(q, τ) dτ ) ∫0∞ ∂P(q, ∂τ P(q, τ)| ∞0 ) -P*(q) (21)

In turn, P(1)(q) can be treated as the solution of eq 8 (integrated over τ) using P(0)(q) as the other initial boundary problem. Then, the survival times τa and τb (defined above by eqs 14 and 15) can be readily computed from operator eqs 19 and 20):

τa )

∫0∞ ∫qq

R

L

P(q, τ) dq dτ )

∫qq

R

L

P(0)(q) dq )

∫qq

-

R

L

Lˆ-1P*(q) dq (22)

mi

d2b ri 2

dτ

)b Fi - miγ

db ri +b F is dτ

1 τa

∫0∞ ∫qq

R

L

τP(q, τ) dq dτ ) -

∫qq

R

L

∫qq

R

L

P(1)(q) dq )

( )

P(0)(q) Lˆ-1 dq (23) τa

where 1/τa is a normalization coefficient which we have to use dealing with the function P(0)(q).

(24)

where b r are the coordinates of atoms, mi are the atomic masses, b Fi are the intramolecular forces which are computed using the Hellman-Feinman theorem (the quantum chemical calculations were performed at the semiempirical PM3 level), γ is the friction Fsi notes a stochastic random force. coefficient (γ ) τL), and b The starting geometry of the intermediate species corresponded to the optimized geometry of peroxodisulphate anion, S2O82-. The run time of simulations did not exceed 0.8-0.9 ps, and a value of 1 fs was used as the time step. The timedependent total energy of S2O83- Etot(τ) was calculated for several τL values and fitted with the exponent:

Etot(τ) ) Etot(0) exp[-τ/τ*]

and

τb )

Note that both Lˆ and Lˆ-1 are negatively defined operators; i.e., their eigenvalues are negative (due to the sink term in eq 8). It is evident that all eigenvalues of -Lˆ and -Lˆ-1 should be positive. Hence, the main difference between τa and τb stems from a different initial problem for eq 8. The first definition, τa, presumes the thermally equilibrated distribution function (see eq 12) as the initial problem of eq 8. A nonequilibrium function (1/τa)P(0)(q) (see eqs 19 and 20) is assumed as the initial problem of eq 8 when calculating the second time, τb. In other words, τa can be considered as a short-scale survival time, while the second quantity, τb, is more suitable to describe the reaction kinetics in a longer time scale. It is evident that both survival times should be addressed in a comprehensive analysis of an arbitrary ET process in terms of eq 8, because both 1/τa and 1/τb might contribute to the observable reaction rate (they can coincide, however, for certain reactions). Further, we will present two types of rate constants (ka and kb, respectively). When calculating the rate constants on the basis of the SM model, the electrostatic effects (work terms) were neglected. We used an effective numerical scheme which provides stable solutions for a wide range of both the reaction free energy and solvent viscosity.20 All simulations were performed assuming room temperature (T ) 300 K). It can be seen from Figure 1b that at high electrode overvoltages (close to activationless discharge) and ∆ > kBT are very shallow or even disappear. the energy barriers ∆E*(q) a Then, the pre-exponential factor in eq 10 should depend on the solvent relaxation time. To estimate a possible Vin vs τL dependence, we performed a series of simulations for a S2O83anion (nonstable intermediate) using Langevin molecular dynamics as implemented in the Hyperchem 7.0 program suite.44 In this method, solvent effects are treated as a dissipative random force as follows (see, e.g., ref 45):

(25)

where Etot(0) is the total energy at τ ) 0 and τ* is the fitting parameter depending on τL. We treat τ* as a characteristic intramolecular relaxation time, which controls the descent of the system along a FES section. Then, the effective frequency factor can be recast as follows: νin(τL) ≈ 1/τ*(τL). Some results of the calculations are presented in Table 1. It can be seen that the Vin values change noticeably only in a narrow region of the solvent relaxation times.


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TABLE 1: Intramolecular Frequency Factor (Win) as a Function of the Solvent Relaxation Time (τL)a Calculated Using the Data of Langevin Molecular Dynamics Simulations τL/s Vin/s-1

10-13 1.04 × 1013

2.5 × 10-13 7.45 × 1012

5 × 10-13 5.87 × 1012

10-12 4.9 × 1012

10-11 4.73 × 1012

10-10 4.84 × 1012

a The τL values were chosen to represent a series of carbohydrate solutions of experimentally available concentrations (up to glass formation).

Figure 2. Model lg k vs lg τL dependencies constructed in the framework of the SM model for two different values of the coupling parameter ∆ (a value of 2.9 V was taken for overvoltage).

III. Results and Discussion The results of model calculations were found to be sensitive to the coupling parameter ∆, describing the shape of the energy barrier. The lg k vs lg τL dependencies computed for two different ∆ values at certain overvoltages are shown in Figure 2. It can be seen that for ∆ ) 0.01 eV the curves demonstrate either descent (kb) or plateau (ka) when increasing the solvent relaxation time. At larger ∆ values (more smooth energy barrier), a plateau region (kb) and even a minimum (ka) can be attained. That is why a value of 0.1 eV for the coupling parameter was employed in further calculations, in order to reinforce the manifestation of a possible effect. We used values of 2.8 and 2.9 V for peroxodisulfate reduction overvoltage, which characterize the region of “pit” in the current-voltage curves.22 We present in Figure 3a the corrected experimental lg i vs lg τ*L plots taken from ref 21 and corresponding to 50 mM Na2SO4 as a supporting electrolyte. The data treatment therein includes a thorough consideration of the formation of ionic pairs in the solution bulk, partial surface blockage with adsorbed cosolvent molecules, and electrode-reactant repulsion. The main aim of this procedure is the separation of theoretically predicted solvent effects (both static and dynamic) from various effects of the solvent-dependent structure of the reaction layer. The asterisk denotes the difference of plotted relaxation time values from the bulk values for a given syrup concentration. This correction of time scale reflects the increased concentration of carbohydrates in the reaction layer, c*; the c* values were recalculated from surface excess values of sucrose and glucose (obtained from the electrocapillary measurements). For each c*, τL* was calculated from the corresponding macroscopic viscosity as the longitudinal relaxation time in the framework of the Debye model of liquid. This approximation, being obviously rough, practically does not affect the main experimental finding, nonmonotonous behavior of the plot in Figure 3a. The points for sucrose and glucose solutions as well as the points for two supporting electrolyte concentrations in sucrose solutions coincide, while uncorrected experimental points for these solutions differ strongly (an order of magnitude or higher21). This confirms somewhat a reasonable character of such corrections. The results of model calculations presented in Figure 3b clearly demonstrate the most important features observed in experiment: a minimum on lg k vs lg τL dependencies and their

Figure 3. Experimental data on the relaxation time dependencies of peroxodisulfate reduction rate taken from ref 21 (a); model dependencies of the rate constants on the solvent relaxation time constructed in the framework of the SM model for two overvoltage values (b).

asymmetry; i.e., the descent of curves takes place in a narrower interval of the effective relaxation times as compared with the ascent plot. The predictions depend on the type of survival time involved in the calculations: the τb values yield a steeper descent (ca. 1 order) of the model curves; it is predicted, however, for a wider interval than in experiment. The origin of this conspicuous behavior can be explained in terms of the saddle point avoidance, which results, in turn, from the diffusive character of the slow solvent coordinate q. A value of the solvent coordinate q˜ related to the maximum of product kin(q) P*(q) may be rendered as the “reaction window” (see also a pertinent discussion in ref 20). The intramolecular coordinate r˜ which corresponds to q˜ is found from the equation for the saddle line of the reaction free energy surface. As follows from our analysis, the “reaction window” is shifted to the left (i.e., toward the initial state) from the saddle point along the solvent coordinate, when the solution viscosity increases. This entails some increase of effective energy barriers across the intramolecular coordinate as compared with that in the saddle point. In such a kinetic regime, the solvent static effect becomes more sluggish, while the solvent dynamics leads to lowering of the rate constant. Starting from a certain region of the solvent

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Nazmutdinov et al.

Figure 5. Slopes of curves plotted in Figure 3b presented in terms of the apparent θ values; the error bars were estimated from the corrected experimental data in Figure 3a (see also ref 21).

Figure 4. Transfer coefficients vs solvent relaxation time estimated from the experimental data21 (a) and computed using the SM model (b).

relaxation times, the “reaction window”, however, moves no longer, and the static effect contributes more noticeably to increasing the ET rate with the solution viscosity. We also modeled two other experimentally determined quantities, the transfer coefficient R, which presents the dependence of the reaction rate on overvoltage, and θ (see section I). To confirm the model hypothesis, we look, therefore, for its ability to describe simultaneously the k, R, and θ dependencies on the relaxation time. The experimental values of R obtained from corrected current-voltage curves are plotted in Figure 4a. The accuracy of transfer coefficient extraction from experiment usually amounts to 0.05 (even for usual aqueous solutions); this value is expected to be even larger (ca. 0.1) when corrections for mixed solvents induce additional accuracy problems. The difference in values for sucrose and glucose in Figure 4a is close to 0.1 or lower. However, the growth of transfer coefficient with relaxation time surely exceeds the experimental accuracy and can be considered as a reliable feature. Our predictions on the behavior of transfer coefficient are exhibited in Figure 4b. This quantity was calculated as follows:

R).

kBT ∂ ln k eo ∂η

(26)

Using both types of survival times leads to ascent plots on the R(τL) function, when the solvent viscosity increases. The shape of Ra(τL) curves highly resembles the qualitative behavior of experimental dependencies: a plateau-like plot after a steep ascent. The derivative θ ) (d lg k)/(d lg τL) calculated as a function of τL is shown in Figure 5 and compared with the estimates of this quantity from experimental data21 (see bars in Figure 5; the problem of accuracy of θ determination is discussed in ref 21). It can be argued that the model curves reproduce qualitatively the experimentally observed tendency: decreasing θ values with the growth of solvent relaxation time. Even a quantitative agreement with experiment is reached for certain τL intervals. Experimental data on the solvent effect are often discussed in terms of a simplified phenomenological relationship for the current density i (the latter is proportional to the rate constant of electrochemical reaction; see, e.g., ref 45):

i ) Const

( )

∆Ea 1 exp kBT τLθ

(27)

where ∆Ea is the activation energy, Const ) (ωeff/2π)1-θ, ωeff is the effective polarization frequency (≈1013 s-1 for liquid water),45 and θ is a phenomenological parameter which ranges between 0 and 1. Equation 27 presumes a constant slope of the lg k vs lg τL plot. This is valid only for a narrow region of the solvent relaxation time, while, for a wide lg τL interval, such plots demonstrate a more complicated behavior;20,22 i.e., in general, θ is a function of τL. A traditional treatment of experimental data in terms of eq 27 neglects any possible dependence of Const on solvent nature and rests on the hypothesis about additivity of two solvent contributions of opposite sign to the resulting effect: dynamic (presented by 1/τθL) and static (presented by the ∆Ea term which depends on the Pekar factor). The slope of lg k vs lg τL dependence corresponds to the apparent θ value in eq


Figure 6. Model lg k vs lg τL plots calculated using eq 27 for three different θ values.

27, if experimental data are treated in a simplified manner (assuming that Const and ∆Ea in eq 27 are solvent-independent). In order to compare two different approaches, we built lg k vs lg τL plots using phenomenological eq 27. The activation energy, however, is estimated from calculations accounting for both the molecular features of the barrier and the static solvent effect (see eqs 1-7). The results of calculations of the rate constants at three different θ values are shown in Figure 6. At θ ) 1 and 0.5, the model dependencies are descending and practically linear. Although for lower θ significant deviations from linearity were found (as well as a slight hint of nonmonotonous behavior), this takes place only in the widest possible interval of effective relaxation time available for real solvents. To the best of our knowledge, experiments of other authors were always arranged in more narrow τL intervals (usually ca. 1 order of magnitude). Thus, Figure 6 clearly illustrates a general impossibility to explain the experimental data presented in Figure 3a in the framework of phenomenological eq 27; i.e., the effect of the 1/τLθ factor on the rate constant is stronger than the decrease of activation barrier with τL. IV. Concluding Remarks The viscosity influence on electrochemical reduction of a peroxodisulphate anion from sucrose and glucose aqueous solutions discloses some qualitatively interesting and challenging features observed experimentally.21 These effects prompt elucidation of them in the framework of modern charge transfer theories. The following main conclusions on the reaction mechanism have been made using the results of model calculations: (1) at least two anisotropic reaction coordinates (slow diffusive and fast) should be assumed to address properly the solvent dynamics, which can result in the saddle point avoidance; (2) some possibility to observe a nonmonotonous dependence of ET rate on the solvent relaxation time really exists only in the vicinity of activationless discharge; i.e., the activation barrier should be smooth; (3) the static solvent effect (decrease of the Pekar factor with the solution viscosity growth) plays an essential role, which can be misunderstood using simplified “additive” approaches. We have shown that the observed experimental features stem from the interplay between these three important features. The simulations discussed above do not require any detailed knowledge about work terms. This makes our analysis of the viscosity effects rather attractive, since reliable calculation of

J. Phys. Chem. B, Vol. 113, No. 30, 2009 10283 the work terms is a rather complicated problem. Our qualitative conclusions seem to be of quite general nature and might be useful to elucidate the mechanism of other adiabatic interfacial ET reactions. The computational approach described in section II offers an efficient way to model the solvent dynamics effects in multifarious ET processes proceeding in viscous media (for a wide range of key parameters). From the viewpoint of its further development, it would be worthwhile to combine the Agmon-Hopfield formalism with “two-dimensional” molecular dynamics simulations (see ref 9). A continuum model employed in this work to describe the solvent reorganization in mixed solutions is, of course, simplified. The structure of aqueous glucose, sucrose, and fructose solutions was studied by means of X-ray diffraction, nearinfrared, and Raman spectroscopy (see e.g. refs 46-48). The experimental results point to a significant role of short-range order of solute molecules and bound water molecules in concentrated carbohydrate solutions. These local effects are very important to model properly selective solvation (solvent reorganization) in such media and will be addressed by us in the future using quantum chemistry and molecular dynamics methods. If ions remain solvated with water selectively in highly concentrated syrups, one can expect that the permittivitiy in the closest reactant surroundings differs from that in the solution bulk. Moreover, when the nature of cosolvent predominating in the solvation shell is changed, the effective size of the reactant should be changed as well. Both effects might be important for more reliable estimates of solvent reorganization energy and its dependence on solvent composition. The setting up of new experiments to examine the viscosity effects on interfacial ET kinetics should be aimed first of all at quantitative separation of contributions (1)-(3) to the reaction mechanism (see the first paragraph of Section IV). To overcome the problem resulting from the complex interplay between the reactant electronic structure, static and dynamic solvent properties, one should look for a mixed solvent with as weak as possible Pekar factor dependence on solvent composition. For example, experimental results on the kinetics of electrochemical reactions in water-ethyleneglycol mixtures are very promising as a new challenge for theory.49 Acknowledgment. In memory of our colleague and friend Alexander M. Kuznetsov. Michael Probst’s participation in quantum chemical calculations is also appreciated, as well as a kind help of Dmitrii V. Glukhov and Tamara T. Zinkicheva. This work was supported in common by the RFBR project 0803-00769-a and RFBR-FWF project 09-03-91001-a. References and Notes (1) Saveańt, J.-M. J. Am. Chem. Soc. 1987, 109, 6788. (2) Herman, E. D.; Kuznetsov, A. M. J. Phys. Chem. 1994, 98, 1189. (3) Spirina, O. B.; Cukier, R. I. J. Chem. Phys. 1996, 104, 539. (4) Zhu, J.; Spirina, O. B.; Cukier, R. I. J. Chem. Phys. 1994, 100, 8109. (5) Koper, M. T. M.; Voth, G. A. Chem. Phys. Lett. 1997, 101, 3168. (6) Schmickler, W. Chem. Phys. Lett. 1995, 237, 152. (7) Sumi, H.; Marcus, R. J. Chem. Phys. 1986, 84, 4894. (8) Nadler, W.; Marcus, R. J. Chem. Phys. 1987, 86, 3906. (9) Ignaczak, A.; Schmickler, W. Chem. Phys. 2002, 278, 147. (10) Zhang, X.; Leddy, J.; Bard, A. J. J. Am. Chem. Soc. 1985, 107, 3719. (11) Zhang, X.; Yang, H.; Bard, A. J. J. Am. Chem. Soc. 1987, 109, 1916. (12) Hecht, M.; Fawcett, W. R. J. Phys. Chem. 1996, 100, 14248. (13) Khoshtariya, D. E.; Dolidze, T. D.; Krulic, D.; Fatouros, N.; Devilliers, D. J. Phys. Chem. B 1998, 102, 7800. (14) Moressi, M. B.; Zoń, M. A.; Fernańdez, H. Electrochim. Acta 2000, 45, 1669.

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