Nonlinear Relaxation in Redox Processes in Ionic and Polar Liquids

Aug 20, 2008 - ... Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford, ... functions C(t) for fluctuations at equilibrium showing that ...
0 downloads 0 Views 1MB Size
Download PDF
14538

J. Phys. Chem. C 2008, 112, 14538–14544

Nonlinear Relaxation in Redox Processes in Ionic and Polar Liquids Ian Streeter,† R. M. Lynden-Bell,*,‡ and Richard G. Compton*,† Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford, United Kingdom OX1 3QZ, and Department of Chemistry, UniVersity of Cambridge, Lensfield Road, Cambridge, United Kingdom CB2 1EW ReceiVed: June 5, 2008; ReVised Manuscript ReceiVed: July 16, 2008

Simulations have been performed to investigate the relaxation dynamics of the solvent around probe ions in the ionic liquid dimethylimidazolium hexafluorophosphate and in the polar liquid acetonitrile. The time scale of the relaxation dynamics is found to be different in the two cases, although our earlier work showed that the Marcus free energy curves, solvent rearrangement energies, and redox activation free energies were very similar. We also observe differences between the nonequilibrium decay curves S(t) and the time correlation functions C(t) for fluctuations at equilibrium showing that the response is strongly nonlinear. Relaxation toward equilibrium is slower for processes in which the magnitude of the electrostatic interaction increases than when it decreases. We discuss whether this may affect the rate constants for electrochemical processes and how this could be observed. 1. Introduction Marcus theory has been used successfully to describe redox processes in solutions for many years.1,2 A redox reaction takes place when a solvent fluctuation occurs to give a favorable environment. According to Marcus, the solvent free energy F can be described as a function of a solvent order parameter, X. The functions Fox(X) and Fred(X) describing the fluctuations around the oxidized and reduced form are approximately parabolic, and the reaction occurs when the two curves cross. This point defines the activation free energy for the reaction. Marcus introduced the concept of the solvent reorganization energies, λred and λox. λred is the change between the solvent free energy in equilibrium with the reduced form, Fred(Xred), and the free energy the solvent would need to give the order parameter appropriate to equilibrium with the oxidized form, Fred(Xox); λox is the corresponding quantity for the oxidized form. In principle, the values of λ for the reduced and oxidized forms may differ, although if the solvent response is truly linear, these values are equal and the free energy curves are exactly parabolic. In earlier work,3,4 we performed a comparative study of the solvent free energy and solvation thermodynamics of probe ions in the same two liquids used in the current study. These were the polar liquid acetonitrile, MeCN, and the ionic liquid dimethylimidazolium hexafluorophosphate, [dmim][PF6]. We found that, in spite of the very different character of the two liquids, the solvation thermodynamics of a probe ion was similar. This is the result of the almost complete screening of the probe ion within a nanometer or so. However, as the mechanism of screening is primarily due to molecular orientation in the polar liquid and to relative positions of positive and negative solvent ions in the ionic liquid, the dynamics may differ. Here we present results for the time dependence of solvent relaxation after an instantaneous change in the charge of the probe ion. The results are presented in section 3 and discussed in section 4. The main * Corresponding authors: Fax +44 (0) 1865 275410; Tel +44 (0) 1865 275413; E-mail: [email protected], [email protected]. † Oxford University. ‡ University of Cambridge.

surprise is that the response is nonlinear with differing relaxations for the changes q f q + 1 and q + 1 f q. This study of the behavior of a single probe ion in solution is relevant to electrochemical redox reactions. It has been shown that half-reactions may be treated separately in simulations and that the relative free energy of the oxidized and reduced states is tuned by varying the external electrical potential5 as in cyclic voltammetry.6 Cyclic voltammograms can be then be analyzed to yield rate constants for oxidation or reduction steps, which usually occur at different electrical potentials. In section 5 we discuss the probable relevance of the simulation results to determining these rate constants and whether the pre-exponential term may depend on the external voltage. Finally, in section 6 we discuss the circumstances in which any such dependence could be observed. 2. The Model System and Simulation Details The model system, which is the same as studied in our previous work,3,4 comprises a single ion with charge +qe dissolved in either of the solvents [dmim][PF6] and MeCN. The ion, I, is a similar size to a chloride ion. In our previous work the equilibrium properties of solutions of the ion with charge q ) +3, +2, ..., -2, -3 were studied. Here we take a number of independent samples from the previous runs at equilibrium and switch the charge by one electron. The resulting timedependent relaxation of the solvent is then monitored by measuring the instantaneous solvent contribution X to the vertical ionization potential (or energy gap). This quantity has been shown to be a good choice for an order parameter for the solvent reorganization.5 Full details of the intermolecular potentials are given in our previous paper.3 Simulations were carried out using a modified version of DL_POLY7 in a NVT ensemble using FCC (dodecahedral) periodic boundary conditions. The distance between an atom and its nearest image was 3.4 nm for all three solutions. Simulations with MeCN were carried out at 298 K, while those for the ionic liquid were carried out at 450 K (as these particular ionic liquids are not liquid at 298 K). A time step of 2 fs was used, and the temperature was controlled by a Nose´-Hoover

10.1021/jp804958p CCC: $40.75  2008 American Chemical Society Published on Web 08/20/2008

Redox Processes in Ionic and Polar Liquids

J. Phys. Chem. C, Vol. 112, No. 37, 2008 14539

thermostat with time constant of 0.5 ps. Some runs were carried out with a Berendsen thermostat for comparison, but no differences were observed. In addition, constant energy runs at equilibrium were performed to find the time correlation functions (C(t)) of fluctuations of X. The number of molecules in the simulation cells was 300 MeCN + 1 solute ion or 90 [dmim][PF6] ion pairs + 1 solute ion. The long-range Coulomb interactions were treated by the Ewald method with a uniform background charge to balance the charge on the dissolved ion. The instantaneous energy gap between the reduced and oxidized form is often used as the order parameter,8 and for an electrochemical half-reaction this is just equal to the vertical ionization energy.5 In such an electrochemical half-reaction the vertical ionization energy can be written as the sum of the gas phase ionization energy IEgas, a solvent contribution VIEsolv, and the electrochemical potential of the electron µ

VIE ) IEgas + VIEsolv + µ

(1)

The value of µ can be tuned by the external potential applied to the electrochemical cell. In classical simulations it is most convenient to use the solvent contribution VIEsolv as the order parameter X,8-10 and this is what is used in this paper. Units of kJ/mol are used for X and all energetic quantities. Nonequilibrium trajectories were run by taking configurations from the previous equilibrated runs and following the changes in the order parameter (X) and in energetic properties as a function of time following a switch in ion charge. In all cases the trajectories shown in the figures are averages of at least four starting points, while the particular example of +2 f +1 and +1 f +2 at least 12 trajectories were run for each of the four cases. The normalized solvent relaxation functions Sqq′ after the switching of the charge from q to q′ at t ) 0 are defined as

Sqq′ ) (X(0) - X(t))/(〈X〉q - 〈X〉q′)

Figure 1. Solvent relaxation in MeCN for 12 redox processes after charge switching. The bold dots show the initial equilibrium values corresponding to charges ranging from -3 (top) to +3 (bottom) as labeled by red figures. Red curves correspond to processes in which the electrostatic interaction increases and blue curves to ones where it decreases. The red curves reach equilibrium more slowly than the blue curves showing that the processes are nonlinear.

(2)

where the angular brackets denote ensemble averages in the presence of the charge specified by the subscript. 3. Simulation Results Figures 1 and 2 show the principal results of these numerical experiments. In these figures the solvent contribution to the instantaneous ionization potential is plotted as a function of time for six different oxidation and reduction processes in either acetonitrile or [dmim][PF6]. Figures 3 and 4 show the trajectories of the various relaxation processes in the X, ULJ space, where ULJ is the nonelectrostatic energy of interaction between the ion and the solvent which is dominated by short-range repulsive terms which are due to the nearest neighbors of the ion. Turning first to the time dependence of the approach to equilibrium, it is clear that for both the polar solvent and the ionic liquid there is a fast initial subpicosecond decay followed by a slower decay with a time constant of about 1-2 ps in acetonitrile and 10-15 ps in [dmim][PF6]. The fast decay accounts for 80%-90% of the total decay of S(t). There is a difference between the acetonitrile solution where the equilibrium values of X (shown by dots in the figure) are reached in 5 ps and the ionic liquid solutions where the decay is slower. This is also evident in the trajectory plots of Figures 3 and 4. In these plots it can be seen that the subpicosecond response is due to changes in the nearest-neighbor repulsion giving a “loop” in the trajectory plots. This immediate response in ULJ overshoots the equilibrium value (bold dot). The intermediate time

Figure 2. Solvent relaxation in [dmim][PF6] for 12 redox processes after charge switching. The bold dots show the initial equilibrium values corresponding to charges ranging from -3 (top) to +3 (bottom) as labeled by red figures. Red curves correspond to processes in which the electrostatic interaction increases and blue curves to ones where it decreases. The red curves reach equilibrium more slowly than the blue curves showing that the processes are nonlinear. All the relaxation processes are slow compared to the corresponding processes in MeCN.

response consists of oscillations in the X, ULJ plane. These oscillations slowly reach the equilibrium well whose minimum is shown by a bold dot. In the case of the ionic liquid these oscillations do not reach the equilibrium value by the end of 10 ps. Thus, although the equilibrium values of Xq at equilibrium are similar in the two liquids (see Table 1), the dynamics is, indeed, different. Oscillations can also be seen in C(t) and S(t). A remarkable and unexpected observation is the difference between the decay curves S(t) for processes in which the magnitude of the charge is increased and those in which it becomes less. The upper part of Figure 5 shows the details of the time dependence of the decay to equilibrium S(t) for the processes in which the charge is switched from +1 to +2 and

14540 J. Phys. Chem. C, Vol. 112, No. 37, 2008

Streeter et al. TABLE 1: Thermodynamic Parameters for the Charged Ions in Acetonitrile and [dmim][PF6] in Units of kJ/mola property

MeCN

[dmim][PF6]

X (q ) 1) X (q ) 2) λ

-606 ( 3 -1166 ( 3 280 ( 20

-620 ( 5 -1143 ( 5 230 ( 20

a X is the solvent order parameter and λ the reorganization energy. Temperatures are 300 K for MeCN and 450 K for [dmim][PF6].

Figure 3. Trajectories in the X, ULJ plane for the first 10 ps of the relaxation in MeCN for the 12 redox processes shown in Figure. ULJ is the nonelectrostatic interaction energy between the probe ion and the solvent ions. The bold circles show the equilibrium points for ions of charges ranging from +3 (left-hand side) to -3 (right-hand side). Each trajectory is an average over four independent runs. The red curves correspond to switches from |q| to |q + 1| and the blue curves to switches from |q + 1| to |q|; note the difference in the red and blue trajectories.

Figure 4. Trajectories in the X, ULJ plane for the first 10 ps of the relaxation in the ionic liquid [dmim][PF6] for the 12 redox processes. ULJ is the nonelectrostatic interaction energy between the probe ion and the solvent ions. The bold circles show the equilibrium points for ions of charges ranging from +3 (left-hand side) to -3 (right-hand side). Each trajectory is an average over four independent runs. The red curves correspond to switches from |q| to |q + 1| and the blue curves to switches from |q + 1| to |q|. Note that the red and blue curves differ and that equilibrium values are not reached in 10 ps.

vice versa. Each of the decay curves in the upper portion of the figure is the average of 12 independent trajectories. The difference between the two relaxation processes is unexpected, as in a linear system the two responses should be identical to each other and to the time correlation function of fluctuations at equilibrium, C(t). The lower part of Figure 5 shows C(t) for probe ions with charge +1 and +2 in the two liquids. In both liquids there are initial damped oscillations which have a higher

Figure 5. Comparison of the short time behavior of solvent relaxation S(t) (above) with correlation functions for equilibrium fluctuations C(t) (below). In the upper part relaxation curves are shown for the oxidation process +1 f +2 (red) and reduction process +2 f +1 (blue) in the dipolar liquid MeCN and in the ionic liquid [dmim][PF6]. The lower part shows the time correlation functions at equilibrium for runs with q ) 1 (blue) and q ) 2 (red) in the same two liquids. The curves are displaced with the right-hand scale referring to [dmim][PF6] and the left-hand one to MeCN.

frequency around the higher charged ion than around the q ) 1 ion. These can be attributed to the local vibrations of the first solvation shell. These damped oscillations are also seen in the red curves in the upper part of the figure. However, there is a significant difference between the behavior of the equilibrium

Redox Processes in Ionic and Polar Liquids

J. Phys. Chem. C, Vol. 112, No. 37, 2008 14541

time correlation functions C(t) and the time relaxation S(t) in both types of liquid, confirming the nonlinearity of the solvent response. One can interpret the initial fast decay and the oscillations as the result of rapid changes in the first solvation shell due to translational motion. In terms of a continuum description, the size of the cavity containing the probe ion changes. The oscillations are due to the balance between the repulsive forces between the probe ion and neighboring species and the electrostatic attractions. As the electrostatic forces are larger for q ) 2 ions than for q ) 1 ions, the oscillations are faster, the periods being 0.35 ps for q ) 2 and 0.5 ps for q ) 1. Oscillations with the same frequency are also seen in the equilibrium correlation functions for the Lennard-Jones energy of the probe ion (not shown). These oscillations are strongly damped with only 1 or 2 maxima and are followed by a slower decay with a time constant of about 5 ps in [dmim][PF6]. In MeCN the behavior of C(t) is qualitatively similar, but with a more rapid “slow decay”. 4. Discussion of Results A nonexponential solvent response has been observed in both dipolar solvents and ionic liquids by a number of workers, mainly in the simulation or observation of the response to the optical excitation of a dye molecule.11-13 Commonly it is found that there is a fast subpicosecond response, followed by a component decaying over ∼10 ps and possibly followed by even slower components. Differences between the time dependence of the solvent response function S(t) and the correlation of solvent fluctuations C(t) have also been observed previously.14,15 Kim and colleagues16 have made extensive molecular dynamics studies of electron transfer processes in ionic liquids. The two most relevant papers to this work are by Shim et al.11,17 in which they investigate the response to an internal charge separation in a probe dipolar molecule. They find that the relaxation functions Spfn(t) and Snfp(t) differ slightly for the relaxation to the nonpolar p f n form and to the polar form n f p, respectively, but the differences are smaller than we see in this work. The time correlation functions of the equilibrium fluctuations of the energy gaps between the two species Cp(t) and Cn(t) also differ, but again by comparatively small amounts. Maroncelli18 observed a larger difference in between S01(t) and S10(t) for of charge switching between q ) 0 and q ) +1 and vice versa in simulations of a spherical ion in model acetonitrile but found at very short (subpicosecond) times that S01(t) was similar to C0(t) and S10(t) was similar to C1(t). This type of argument led Geissler and Chandler19 to investigate the probability distributions of fluctuations in X as a function of time after the original charge switch. They showed that for a particular internal charge exchange process for a polar diatomic dissolved in TIP4P water these distributions were Gaussian at different times but that the widths were different, and they concluded that the solvent response was linear but nonstationary. Maroncelli, in a detailed simulation study of the solvation dynamics following a charge jump on an ion in acetonitrile,18 also looked at the distributions of X at different times (“solvation spectra”). Interestingly, he found that following the q ) 0 f 1 switch the distributions were much broader than after the reverse switch. His data were not good enough to distinguish between Gaussian and non-Gaussian distributions. Another possible cause of nonequilibrium effects is local heating or cooling after the charge switch.

Figure 6. Free energies of ions with charges q ) 1 (blue curve) and q ) 2 (red curves) as a function of the solvent order parameter X. The bold curves correspond to equilibrium (E ) E°f ). The dashed red curves correspond to offsets in the applied potential of (0.5 V while the thin red curves to offsets of 2.8 and -2.63 V, respectively. The points a, b, c, and d show the transition states for different applied potentials. O and R are the minimum free energies for oxidized and reduced states, respectively.

5. Possible Implications for Electrochemistry The rates for the oxidation and reduction processes in an electrochemical redox reaction can be determined from steady state and cyclic voltammetry. Figure 6 shows Marcus free energy plots for the oxidized and reduced states for a number of different external potentials. The reduction reaction kred

e + I(q+1)+ 98 Iq+

(3)

occurs from the free energy curves on the left of the figure to the free energy curve on the right. Changing the potential shifts the relative positions of the free energy curves of the oxidized and reduced states by amounts ∆ ) F(E - Ef°), where E and Ef° are the applied potential and the formal potential for the redox couple, respectively, and F is the Faraday constant, so that ∆ is a molar energy. The transition states between oxidized and reduced forms are indicated by lower case letters. It can be seen from this figure that the activation free energy depends on the applied potential through ∆. In Marcus theory this dependence is quadratic, while the Butler-Volmer formalism gives a linear dependence. Equilibrium requires that the ratio of the rates of the oxidation reaction to the reduction reaction is given by

kox /kred ) exp(∆/RT)

(4)

The individual rate constants can be written

kred ) A exp[-G∗red /RT]

(5)

14542 J. Phys. Chem. C, Vol. 112, No. 37, 2008

kox ) A exp[-G∗ox /RT]

Streeter et al.

(6)

where the activation free energy G/ox for the oxidation reaction is the free energy difference between the crossing point of the curves (a, b, c, or d on the figure depen) and point R, which depends / of the value of ∆. Equation 4 shows that G/ox - Gred ) -∆. Normally it is assumed that preexponential factor A, which depends on solvent friction, is independent of the applied potential. However, our simulations suggest that this may not be true, as will be explained in section 5. We have seen that the process for decay from point a on the free energy curve for the reduced species to the equilibrium point R is faster than the decay from point R on the free energy curve for the oxidized species to equilibrium at the minimum. This difference in solvent friction is likely to manifest itself in a change of the preexponential term. The effects of solvent friction on chemical rate constants have been studied for many years following the work of Kramers.2,20 Three regimes can be distinguished. If the solvent friction is low, thermal equilibrium is not achieved in the reaction well, and the rate of reaction is less than that predicted by transition state theory and is proportional to the friction. If the solvent friction is large, then the path through the transition state is impeded and again the rate is reduced compared to the transition state prediction. In this regime the rate is inversely proportional to the solvent friction. In an intermediate regime the rate is that predicted by transition state theory. Several workers have investigated the effects of solvent friction on redox processes in solution.21-25 It is generally found that the appropriate regime for electrochemical redox processes in solution is the first one, with low friction and diabatic or weakly adiabatic crossing of the two free energy curves. The expression for the rate constant at equilibrium (∆ ) 0) given by Hynes22 is

k-1 ) kTST-1 + 2kD-1

(7)

where kTST is the transition state theory rate and kD is the rate of of relaxation in the well

kD )

1 G* τ RTπ

1/2

( )

exp[-G*/RT]

(8)

In this expression, G* is the activation free energy per mole at equilibrium and τ is an effective relaxation time for relaxation in the wells. In a Debye dielectric fluid with a single relaxation process this time is just equal to the longitudinal relaxation time. In real liquids, both polar and ionic, solvent relaxation is more complex with fast processes, intermediate processes, and slow processes. Hynes22 showed that the shorter time processes dominate. This suggests that the high viscosity and slow selfdiffusion observed in ionic liquids are probably irrelevant to the solvent friction but that the processes on time scales up to 30 ps which can be observed in simulations are relevant. However, theoretical treatments2 assume that the solvent response is linear and so cannot be applied directly to discuss the friction in this model, as we have found its relaxation to be nonlinear. Nevertheless, qualitative arguments can be given for the dependence of the pre-exponential part of the rate on the applied potential. Referring to Figure 6, we have found that the relaxation from point a to point R is faster than the relaxation from point R to the bottom of the oxidized curve. This suggests

that the more negative the offset (the nearer the transition state is to the oxidized form), the faster the relaxation and the larger the value of the preexponential factor A(∆). The conclusion of this section is that the pre-exponential factor in the electrochemical redox rate constants is likely to depend on the applied potential at which the process is observed. As the oxidation and reduction waves in cyclic voltammetry occur at different applied potentials, the value of the preexponential part of the rate constant may differ. 6. Could This Dependence on the Applied Potential Be Observed? The analysis of the kinetics of electrode processes is conventionally conducted within the formalism of Butler-Volmer kinetics,6,26,27 which views the electron transfer as an activated process. For the simple process

e-(metal) + Ox(solution) a Red(solution)

(9)

this leads to the following expressions for the cathodic and anodic rate constants:

(

kred ) k◦red exp -

R∆ RT

)

(10)

( β∆ RT )

kox ) k◦ox exp

(11)

where ∆ ) F(E - Ef°) was defined in section 5 and R and β are the transfer coefficients where R + β ) 1. The parameters k°red and k°ox are usually assumed to be independent of ∆, in which case their values must be equal (kred ° ) kox ° ) k°), because at equilibrium the principle of microscopic reversibility requires the rate of the forward and backward electrode reactions to be equal. The parameter k° is then the rate of the reaction at equilibrium (∆ ) 0):

ko ) A exp( -G*(∆ ) 0)/RT)

(12)

These equations have been very successful in describing voltammetric data, which can be analyzed to give the parameters k°, R, and Ef°. In there case where the parameters kox ° and kred ° are not equal, we shall show that it is not possible to determine their values independently unless there is also an independent measurement of Ef°. For an electrochemically irreversible heterogeneous electron transfer with Butler-Volmer kinetics, the rate of electron transfer is described in terms of the flux of electroactive species at the electrode, J, by eq 13 for the reduction and eq 14 for the oxidation:

( (

Jred ) -k◦red exp -

Jox ) -k◦ox exp

)

RF(E - E◦f ) [Ox] RT

)

βF(E - E◦f ) [Red] RT

(13)

(14)

where the rate constants kox ° and kred ° are not necessarily equal in value. From inspection of eqs 13 and 14, it can be seen that it is possible to change the value of the rate constants without

Redox Processes in Ionic and Polar Liquids

J. Phys. Chem. C, Vol. 112, No. 37, 2008 14543

Figure 7. Steady state voltammetry at a hemispherical electrode calculated using eq 17. The solution contains equal amounts of the oxidized and reduced species. The three current-potential curves plotted coincide, confirming that the curves can be reproduced with different choices of parameters (see text).

having any effect on the overall rate of reaction, providing Ef° is also changed by an appropriate amount as compensation. Furthermore, if kred ° and kox ° have different values, it is possible to replace them with a single rate constant, k*, and to adjust the formal electrode potential to a value E/f such that the overall rates of reaction, and their potential dependence, are left unchanged. The values k* and Ef/ can be found from

k* ) (k◦red)β(k◦ox)R

E∗f ) E◦f +

( )

k◦red RT ln ◦ F kox

(15)

(16)

The implication is that it is impossible to infer the individual rate constants, kred ° and kox °, for any electrochemically irreversible electron transfer with Butler-Volmer kinetics. Rather, it is only possible to infer the composite rate constant, k*. This is illustrated below for both steady state voltammetry at a hemispherical microelectrode and for cyclic voltammetry at a planar macroelectrode. For an irreversible electron transfer at a hemispherical electrode, the steady state current is given by

i ilim

)

1 1+

D kr

(17)

where ilim is the mass transport limiting current, D is the diffusion coefficient, r is the radius of the hemisphere, and k is the rate constant for the electron transfer (including the exponential terms from eqs 13 and 14). This expression can be found by considering the rate of diffusion at the electrode surface and the rate of the electron transfer.28 Figure 7 shows an example of steady state voltammetry at a hemispherical electrode when both the reduced and oxidized species are present in bulk solution. Three current-potential curves are plotted: one is for an irreversible oxidation with Ef° ) 0, kox ° r/D ) 5 × 10-3, R ) 0.4; the second is for an irreversible reduction with Ef° ) 0, kred ° r/D ) 5 × 10-4, R ) 0.4. The third curve, shown by white circles, exactly overlays the other two curves and uses Ef/ ) -59.1 mV, k* ) 1.256 × 10-3, and R ) 0.4. The

Figure 8. Cyclic voltammogram. The two waves shown coincide, confirming that the curves can be reproduced with different choices of parameters (see text).

coincidence of these curves confirms the conclusion that the results can be reproduced using different parameters. Cyclic voltammetry under conditions of planar Fickian diffusion to a macroelectrode is simulated by finite difference methods.29 Figure 8 shows some sample voltammetry for an irreversible electron transfer when only the reduced species is present in bulk solution. Two waves are shown: one uses Ef° ) 0, kox ° ) 2 × 10-5 cm s-1, kred ° ) 1 × 10-4 cm s-1, R ) 0.5; the second, shown with white circles, uses Ef/ ) 41.33 mV, k* ) 4.4721 × 10-5 cm s-1, and R ) 0.5. For both waves, D ) 10-5 cm 2 s-1 for both species, scan rate ν ) 0.1 V s-1, [Red] ) 10-6 mol cm-3, electrode area A ) 1 cm2. Again, the two curves calculated with different parameters are identical, showing that the results are consistent with many sets of parameters. Although the activation energy in Butler-Volmer kinetics (eqs 10 and 11) is assumed to be linear in the applied potential and hence in ∆ ) F(E - Ef°), the Marcus prediction is a quadratic dependence. It can readily be shown that this gives rate constants

(

∆2 ∆ exp 4λRT 2RT

(

∆2 ∆ exp 4λRT 2RT

kred ) k◦M red exp -

kox ) k◦M ox exp -

) (

)

(18)

) (

)

(19)

where λ is the solvent reorganization energy. This has the same form as the Butler-Volmer equations with R ) 0.5, but with k° replaced by

(

k° ) k◦M ox exp -

∆2 4λRT

)

(20)

If there were no ∆ dependence in the pre-exponential factor A (eqs 5 and 6), then k°M would just be the rate constant at equilibrium (∆ ) 0). Our simulations, however, suggest that both the nonlinear dependence of the activation free energy on ∆ and the dependence of solvent friction on ∆ may cause a ∆ dependence of the value of k° in the Butler-Volmer analysis. It is evident from the above discussion that conventional voltammetric analysis using Butler-Volmer kinetics does not permit the validity of the assumptions of a constant value of A or a linear dependence of the activation energy on ∆ to be deduced. At first sight it appears that the principle of microscopic

14544 J. Phys. Chem. C, Vol. 112, No. 37, 2008 reversibility is flouted by the use of eqs 13 and 14 with different values of the pre-exponential factor for the oxidation and reduction processes. However, although this principle would apply to the forward and reverse reactions at any particular value of the potential, for irreversible electrochemical processes the forward and reverse reactions of eq 9 are measured at different potentials. Thus, a dependence of A and k° on ∆ could be determined, but only if there is an accurate independent determination of the formal electrode potential Ef°. 7. Discussion and Conclusions In this paper we have shown that the solvent relaxation following a charge switch of a model probe ion is nonlinear and differs for increasing and decreasing changes in charge in models of both acetonitrile, a typical polar solvent, and [dmim][PF6], an ionic liquid. In all cases the relaxation consists of a fast subpicosecond process, followed by a slower picosecond to 10 ps process. The difference in the slower of these processes in the ionic liquid and the polar liquid is probably the source of the difference in observed electrochemical rate constants.30 Both types of process are much faster than the time scale for self-diffusion in ionic liquids. Theories of solvent friction effects on chemical and electrochemical reactions are usually based on the generalized Langevin equation.2 This equation is derived assuming a linear response of the solvent and does not apply when the response is nonlinear. Thus, it is not possible for us to draw quantitative conclusions about electrochemical kinetics from our simulations. Further the electronic crossing regime (adiabatic, weakly adiabatic, or diabatic) depends on the details of the experimental system considered. We suggest that the observed nonlinearity of the relaxation after “up” and “down” charge switching in our model system shows that the effective friction depends on the position of the crossing point of the Marcus curves. Hence, the pre-exponential part of the electrochemical rate constants may vary with the applied potential and so differ for oxidation and reduction processes in, for example, a cyclic voltammagram. We have shown that the standard analyis of steady state and cyclic voltammetry does not yield unambiguous values for the pre-exponential parts of the forward and reverse rate constants without an independent measurement of the electrode potential at equilibrium, which makes it difficult to observe kinetic asymmetry. Other possible sources of kinetic asymmetry depend on the details of the solvent structure near the electrode31,32 which will vary with the electrode potnetial and also with the solvent.

Streeter et al. Lastly, from a pragmatic point of view, we emphasize that a good or even perfect fit of Butler-Volmer kinetics to experimental voltammetric data does not provide an indication of the constancy of the pre-exponential terms in eqs 5 and 6, unless Ef° can be independently determined. Rather, as implied by eqs 15 and 16, quite different values for oxidation and reduction can apply. References and Notes (1) Marcus, R. Annu. ReV. Phys. Chem. 1964, 15, 155. (2) Nitzan, A. Chemical Dynamics in Condensed Phases; Oxford University Press: Oxford, 2006. (3) Lynden-Bell, R. M. J. Phys. Chem. B 2007, 111, 10800. (4) Lynden-Bell, R. M. Electrochem. Commun. 2007, 9, 1857. (5) Blumberger, J.; Sprik, M. In Computer Simulations in Condensed Matter: from Materials to Chemical Biology; Springer: Berlin, 2006; Vol. 704, pp 481-506. (6) Compton, R. G.; Banks, C. E. Understanding Voltammetry; World Scientific: Singapore, 2007. (7) Smith, W.; Forester, T. R. The DL_POLY manual, Daresbury Laboratory, 1996; http://www.cse.scitech.ac.uk/ccg/software/DL_POLY/. (8) Warshel, A. J. Phys. Chem. 1982, 86, 2218. (9) Hartnig, C.; Koper, M. T. M. J. Chem. Phys. 2001, 115, 8540. (10) Yelle, R. B.; Ichiye, T. J. Phys. Chem. 1997, 101, 4127. (11) Shim, Y.; Kim, H. J. J. Phys. Chem. B 2007, 111, 4510. (12) Kobrak, M. N. J. Chem. Phys. 2006, 125, 064502. (13) Arzhantsev, S.; Jin, H.; Baker, G. A.; Maroncelli, M. J. Phys. Chem. B 2007, 111, 4978. (14) Skaf, M. S.; Ladanyi, B. M. J. Phys. Chem. 1996, 100, 18258. (15) Maroncelli, M.; Flemin, G. R. J. Chem. Phys. 1988, 89, 5044. (16) Shim, Y.; Jeong, D.; Manjari, S.; Choi, M. Y.; Kim, H. J. Acc. Chem. Res. 2007, 40, 1130. (17) Shim, Y.; Choi, M. Y.; Kim, H. J. J. Chem. Phys. 2005, 122, 044510. (18) Maroncelli, M. J. Chem. Phys. 1991, 94, 2084. (19) Giessler, P. L.; Chandler, D. J. Chem. Phys. 2000, 113, 9759. (20) Kramers, H. A. Physica 1940, 7, 284. (21) Weaver, M. J.; McManis, G. E. Acc. Chem. Res. 1990, 23, 294. (22) Hynes, J. T. J. Chem. Phys. 1986, 90, 3701. (23) Zusman, L. D. Chem. Phys. 1980, 49, 295. (24) Calef, D. F.; Wolynes, P. G. J. Phys. Chem. 1983, 87, 3387. (25) Ignaczak, A.; Schmickler, W. Electrochim. Acta 2007, 52, 5621. (26) Butler, J. A. V. Trans. Faraday Soc. 1924, 19, 734. (27) Erdey-Gruz, T.; Volmer, M. E. Z. Phys. Chem. 1930, 150A, 203. (28) Albery, W. J. Electrode Kinetics; Clarendon Press: Oxford, 1975. (29) The mass transport equations of species Ox and Red are discretized by the implicit finite difference method using a geometrically expanding simulation grid. The Butler-Volmer boundary condition is applied at the electrode surface, and a bulk solution boundary condition is applied at a distance 6(Dt)1/2. The matrix equations are solved by a generalized form of the Thomas algorithm. (30) Fietkau, N.; Clegg, A. D.; Evans, R. G.; Villa´gran, C.; Hardacre, C.; Compton, R. G. ChemPhysChem 2006, 7, 1041. (31) Pinilla, C.; Del Po´polo, M. G.; Lynden-Bell, R. M.; Kohanoff, J. J. Phys. Chem. B 2005, 109, 17922. (32) Pinilla, C.; Del Po´polo, M. G.; Kohanoff, J.; Lynden-Bell, R. M. J. Phys. Chem. B 2007, 111, 4877.

JP804958P