Understanding the Nature of Heterogeneous Electron Transfer in

Article Cite This: J. Phys. Chem. C 2019, 123, 14370−14381

pubs.acs.org/JPCC

Understanding the Nature of Heterogeneous Electron Transfer in Molecular and Ionic Solvents: Experiment, Theory, and Computations Victoria A. Nikitina,*,†,‡ Sergey A. Kislenko,§ and Renat R. Nazmutdinov∥ †

Skoltech Institute of Science and Technology, Nobel str. 3, 143026, Moscow, Russia M.V. Lomonosov Moscow State University, Leninskie Gory 1/3, 199991, Moscow, Russia § Joint Institute for High Temperatures of RAS, Izhorskaya 13/19, 125412, Moscow, Russia ∥ Kazan National Research Technological University, 420015, Kazan, Republic of Tatarstan, Russia

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‡

S Supporting Information *

ABSTRACT: We report a combined experimental and computational study on the heterogeneous electron transfer kinetics for a simple one electron transfer reaction (ferrocene/ ferrocenium Fc+/Fc couple) in a series of molecular solvents and ionic liquids. We focus on the diagnostics of the electron transfer regime (adiabatic vs nonadiabatic) and assess the parameters of the quantum mechanical electron transfer theory, which determine the observed tendencies in the solvent effect on the electron transfer rates. The applicability of the linear plots of the electron transfer rate constant vs longitudinal relaxation time (or solvent viscosity) for distinguishing between different ET kinetic regimes is analyzed. Classical molecular dynamics simulations were performed to calculate the potential of mean force for Fc and Fc+. The structure of the reaction layer derived from molecular dynamics is thoroughly investigated. The experimental dielectric spectra for the both type of solvents are used for quantum corrections of the outer-sphere reorganization energy as well as for estimations of the effective frequency factor in the limit of strong and weak electronic coupling. The electron transfer rate constants are calculated and discussed in the viewpoint of available experimental data. voltammetry results.13,14 Moreover, the rate constant values for Fc+/Fc couple in acetonitrile (AN) reported by different groups differ by 4 orders of magnitude,10,15 which demonstrates difficulties associated with the experimental determination of high rate constants. A Fc+/Fc redox couple is commonly regarded as a good model system for theoretical treatment due to the spherical shape of the reactant, a small intramolecular reorganization at the ET and reversibility of the process. The kinetic data for Fc+/Fc in a series of nonaqueous solvents reported in refs.16,17 have been considered as reliable and were frequently used in the analysis of solvent dynamic effects on the ET rates.2,11 The commonly adopted strategy in this analysis rests on the construction of log kapp vs log τL plots, i.e., the dependencies of the apparent ET rate constants on the solvent longitudinal relaxation times (τL). The linearity of the plot is considered to point to the adiabaticity of the Fc+/Fc reaction and predominant kinetic control by the solvent dynamics. Even

1. INTRODUCTION One of the major research goals in the field of heterogeneous electron transfer (ET) kinetics is the experimental verification of the quantum mechanical theory1 of ET in polar solvents. The pioneering works of Weaver have allowed one to gain insight into the complexity of even the simplest one electron transfer reactions, for which kinetics was demonstrated to be dictated by both static and dynamic solvent effects as well as by the electrode/reactant orbital overlap and reaction layer structure.2,3 A vast number of attempts to reach molecular level understanding of solvent effect on electrode processes was performed for heterogeneous as well as homogeneous ET for different reactants (transition and noble metal complexes with various ligands, metallocene derivatives, etc.).4−12 Unfortunately, not only model estimates of the rate constants are uncertain, but also the experimental data often exhibits a significant scatter due to the difficulties in estimating very high rate constants. For instance, some examples in literature illustrate a very weak solvent dependence of the ET rate constant for the ferrocenium/ferrocene (Fc+/Fc) redox couple in nonaqueous solvents, most likely due to the influence of uncompensated ohmic resistance on the fast scan cyclic © 2019 American Chemical Society

Received: February 5, 2019 Revised: May 5, 2019 Published: May 22, 2019 14370

DOI: 10.1021/acs.jpcc.9b01163 J. Phys. Chem. C 2019, 123, 14370−14381

Article

The Journal of Physical Chemistry C

methylimidazolium bis(trifluoromethanesulfonyl)imide [C2mim][NTf2] (Aldrich, >99%), and 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide [C4mim][NTf2] (Aldrich, 99%) were dried with activated 3 Å molecular sieves and kept in an argon-filled glovebox (MBRAUN, concentration of H2O < 0.1 ppm, concentration of O2 < 20 ppm). Ferrocene (Fluka, >99.5%) was used as received. Solution preparation and assembling of the cell was performed under argon atmosphere to avoid contamination with the traces of water and oxygen. Solutions of 20 mM Fc in ILs were prepared by directly mixing a weighted amount of Fc under argon atmosphere with the appropriate amount of IL. 2.2. Electrochemical Measurements. All electrochemical measurements were performed in a three-electrode glass cell with a single compartment (1 mL working volume). To allow measurements at high scan rates Au microelectrode (10 μm diameter, Metrohm Autolab) was employed as working electrode. Pt wires served as counter and quasi-reference (QRE) electrodes. Prior to measurements, the microelectrode was cleaned with freshly prepared H2SO4/H2O2 (4:1) mixture. The drift of the QRE potential never exceeded 1 mV during the experiment. In all the ILs, the current values at 1000 V·s−1 never exceeded 100 nA, so no correction for uncompensated ohmic resistance was necessary to obtain estimates of apparent rate constants. The potentiostat was a software controlled Autolab PGSTAT302N system (Eco Chemie BV, Netherlands). 2.3. Computational Details. MD Simulation of Au(111)/ Solvent Interfaces. We performed classical molecular dynamics simulation in the NVT ensemble using a rectangular box to estimate the Au(111)/solvent interface structure. The box contained the solvent volume of ca. 6 nm in width confined on both sides by Au(111) surfaces 3.52 × 3.55 nm2 in area. The Au(111) surfaces were constructed with five layers of gold atoms with fixed positions. Three-dimensional (3D) periodic boundary conditions were employed. To reduce the interaction between the simulation box and its images, the periodicity in the direction perpendicular to the surfaces was elongated to 25 nm. The GolP force field was used to describe the interaction of solvents with gold (111) surfaces.23 The force fields for AN, PC, EtOH, [C4mim][BF4], Fc, and Fc+ were taken from refs 21 and 24−28, respectively. The Ewald method with a real space cutoff of 1.5 nm was used to calculate the electrostatic interactions. The equations of motion were solved using the Verlet leapfrog integration algorithm with a time step of 1 fs. The cutoff radius of the van der Waals interaction was 1.5 nm. In the case of molecular solvents, simulations were performed at a temperature of 300 K; in the case of [C4mim][BF4], the temperature was 350 K to avoid glassy behavior. The potential of mean force (PMF) for Fc and Fc+ was estimated in the same manner as in our previous work.21 Briefly, the PMF was calculated by integrating the average force (acting on the solute Fc or Fc+) in the direction perpendicular to the surface. To estimate the average force, we ran a series of simulations (of 1 ns each) where the solute molecule is restrained at different distances from the surface. The simulations were performed for the fixed molecular orientation, in which cyclopentadienyl (Cp) rings were arranged perpendicular to the surface. The orientation of Fc (Fc+) and distance from the surface were fixed by harmonic potentials, acting on centroids of the Cp rings.

more often this correlation is constructed using the viscosity of the solvent, which is considered to be proportional to the relevant relaxation time.18,19 As the reorganization energy contributions are rarely included in these estimates (as these quantities are sometimes difficult to estimate reliably), deviations from the linearity of the log kapp vs log τL plots are often attributed to the difference in the reorganization energetics or to the reaction rate nonadiabatic control.10,11 For some of the solvent types, the correlation of the observable rate constant with the solvent frequency factor is less obvious. Namely, the kinetics of ET for metallocene/ metallocenium couple both in heterogeneous6 and homogeneous ET reactions is unexpectedly rapid.9 No solid conclusions on the reasons of this discrepancy were formulated, as no molecular level information on the reaction layer structure and electrode−reactant interactions were available so far. It was suggested that the relaxation modes at higher frequencies (which appear in addition to the predominant Debye relaxation) could contribute to the enhanced reaction rates in alcohols. Indeed, when higher frequency modes are used in the construction of the log kapp− log τL plots, the data points for nitrile, amide, and alcohol solvents are on the same linear trend.6,11 A more rigorous molecular level description of the ET step is required in order to understand the physical validity of this approach. The rationalization of anomalously fast kinetics in alcohols demands the specification of both effective frequency values, solvent reorganization energetics and the reaction layer structure.20 Room temperature ionic liquids (RTILs) represent another example of anomalously “fast” solvents. Recently, we addressed the rates of heterogeneous ET for Fc+/Fc in [C4mim][BF4] and compared it with the reaction rate in AN21,22 using molecular dynamics and quantum chemistry methods to assess the model parameters of the ET theory. In our analysis, unusually fast heterogeneous ET rates in this IL were attributed to high effective relaxation time. Further verification of the formalism employed prompts considering other molecular solvents and different ionic liquids. In this work, we compare the ET rates for Fc+/Fc couple at Au(111) surface for a series of molecular solvents (nitriles, alcohols, carbonates) and imidazolium-based ILs. The experimental kinetic data for the molecular solvents is adopted from refs 16 and 17 while the ET rate constants for Fc+/Fc redox process in ILs were estimated based on the electrochemical measurements reported in this work. We aim to implement a self-consistent test of the approaches for the estimation of key ET parameters. This makes it possible to argue on the accuracy, which can be achieved via the combination of dielectric continuum approximations and molecular modeling to compute the absolute reaction rates in ionic and molecular solvents. Our analysis rests significantly on the molecular level knowledge of the interface structure. We combine results of molecular dynamics (MD) simulations obtained by us previously (Au/ acetonitrile and Au/[C4mim][BF4] interfaces21) with those reported in the present work for the first time.

2. EXPERIMENTAL AND COMPUTATIONAL DETAILS 2.1. Materials and Solution Preparation. ILs 1-butyl-3methylimidazolium tetrafluoroborate [C4mim][BF4] (Aldrich, >99%), 1-butyl-3-methylimidazolium hexafluorophosphate [C4mim][PF6] (Merck, >99%), 1-ethyl-3-methylimidazolium tetrafluoroborate [C2mim][BF4] (Aldrich, >99%), 1-ethyl-314371


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Table 1. Apparent Rate Constant Values (kapp) for a Set of Molecular Solvents and ILs, Static and Optical Dielectric Constants (εs and εop) at 25°C,40 and Longitudinal Relaxation Time (τL), Calculated from Debye Relaxation Time τD (τL= τD·εop/εs) kapp, cm·s−1

εs (εopt)

τL, ps

electrolyte salt

electrode

20.7 (1.84)

0.3

TBAP

37.5 (1.80)

0.2

TBAP

28.2 (1.86) 24.5 (1.91)

0.3 0.4

TBAP TBAP

TPAP

Au(μ) Pt(μ) Pt(μ) HSChE Pt(μ) Au(μ) Au(μ) Pt(μ) Pt(μ) Pt(μ) Pt(μ) Pt(μ) Pt(μ) Pt(μ)

16 17 15 16 16 16 17 17 17 17 15 17 17

DMSO

0.3

46.7 (2.18)

TPAP

Pt(μ)

17

PC

0.2 0.15 0.2

65.0 (2.02)

2.7 12.3 31.0 0.4 (0.34) 1.3 (0.6) 2.4 (0.9) 2.7 (1.3) 1.0

LiClO4 LiClO4 LiClO4 TPAP

DMF

5.5 2.0 0.99 ± 0.22 3.9 3.0 0.95 0.7 2.34 0.82 0.47 0.35 ± 0.09 0.5 0.6

TPAP

Pt(μ) Au(μ) Pt(μ)

17 this work 17

0.02 0.05 0.05 0.015 0.0025 0.013 0.06 0.065−0.078 0.0075 0.21 ± 0.06 0.01−0.02 0.14 0.03 0.035−0045 0.006 0.09

12.2 (2.02)

189

13.6 (1.99) 11.8 (1.98)

23 236

12.3 (2.022)

22

11.52 (2.033)

85

Au(μ) Pt(μ) Au(μ) Au(μ) Pt Pt(μ) Au(μ) Pt(μ) SECM Pt(μ) Pt(μ) HSChE Pt Pt(μ) Au(μ) Pt(μ) SECM Pt(μ) Pt(μ)

this 18 this this 35 18 this 31 32 33 34 18 this 31 32 18

solventa AC AN EtCN PrCN MeOH EtOH 1-PrOH 1,2-DCE

THF [C4mim][BF4] [C2mim][BF4] [C4mim][PF6]

[C2mim][NT2]

[C4mim][NTf2]

32.7 24.5 20.4 9.0

(1.76) (1.85) (1.92) (2.03)

36.7 (2.04)

7.58 (1.97)

TPAP

ref

work work work

work

work

a

AC, acetone; AN, acetonitrile; EtCN, propionitrile; PrCN, butyronitrile; MeOH, methanol; EtOH, ethanol; 1-PrOH, 1-propanol; 1,2-DCE, 1,2dichloromethane; DMF, dimethylformamide; DMSO, dimethyl sulfoxide; PC, propylene carbonate; THF, tetrahydrofuran; TPAP, tetrapropylammonium perchlorate; TBAP, tetrabutylammonium perchlorate.

2.4. Calculation of Kinetic Parameters. The distance dependent rate constants for Fc+/Fc reaction in ILs and molecular solvents in adiabatic (ka) limit can be computed using the equation:21

k na(x) =

× exp[−(λ + Wf − Wi )2 /4λkBT )]

(2)

Note that zero electrode overpotential is assumed when writing eqs 1 and 2. The distance dependence in the master equations originates mainly from Wi and κe. In both ET limits, the observable rate constant k is obtained by integration over x. Then it is convenient to discuss k in terms of reaction volume δx,

1 ka(x) = exp( −σ ) exp( −Wi /kBT ) τeff × exp[−(λ + Wf − Wi )2 /4λkBT )]

ωeff κe exp( −σ ) exp( −Wi /kBT ) 2π

(1)

where λ is the solvent reorganization energy (the intermolecular reorganization is small for Fc+/Fc and can be neglected), Wi and Wf are work terms of reactant and product, respectively, τeff is the effective relaxation time of solvent, σ is the solvent tunneling factor, x is the electrode surface−reactant separation and T is ambient temperature. In the nonadiabatic limit, the simplest expression for the rate constant should include the classical polarization frequency ωeff and electronic transmission coefficient κe:

k=

∫x

x bulk 0

k(x)dx = k(x0)δx

(3)

where x0 is the distance of the closest approach and xbulk is a certain point in the solution bulk (usually ca. 12 Å). In what follows, we use the electronic transmission coefficients for a Fe+/Fe0 couple calculated in our previous work21 which were assumed to depend only slightly on the solvent nature. An approach developed therein is based on a 14372


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uncompensated ohmic resistance become more pronounced, which could explain the very low rate constants (∼10−3 cm· s−1) obtained for the Fc+/Fc reaction in some ILs.32,35 Comparison of the rate constants obtained in one and the same group for different ILs is more reliable, as it allows assessing a least the relative ET rates in different ILs, when the absolute rates are less reliable. For this reason, we determined the rate constants for Fc+/Fc ET at Au microelectrode in five imidazolium-based ILs: [C4 mim][BF4 ], [C2 mim][BF4 ], [C4mim][PF6], [C2mim][NT2], and [C4mim][NTf2], from high scan rate voltammetry data using Nicholson method.36 The rate constant for the reaction in PC was also estimated from voltammetry data, as no information was available for Au electrode. The Fc diffusion coefficients for the estimation of rate constants were adopted from refs 37−39, and the transfer coefficient value was set to 0.5 in all the solvents.17 Figure S1 shows sets of representative high scan rate cyclic voltammograms of Au microelectrode in 20 mM Fc solutions in [C4mim][PF6] and [C2mim][NTf2]. The results obtained agree well with the those reported in ref 18 from the largeamplitude Fourier transformed alternating current (FTAC) voltammetry measurements at the platinum microelectrode (Table 1). The kapp values obtained at the Pt and Au electrodes differ only slightly, which is typical for outersphere redox processes. The ET rate constant in PC at a Au electrode (0.15 cm·s−1) is also close to the value obtained in ref 17 at a Pt electrode. It was briefly mentioned in the Introduction that a linear dependence of the apparent rate constant on the solvent viscosity or on the longitudinal solvent relaxation time τL has long been regarded as a simple test of the reaction proceeding in adiabatic regime, i.e., under predominant solvent control. This approach does disregard the solvent dependence of outersphere reorganization energy λs, but the solvent dynamics influence is considered to be prevailing for some simple redox systems with low contributions of innersphere reorganization energy λin, which is just the case for Fc+/Fc redox process (λin is ca. 5 kJ·mol−140). Figure 1 shows the log kapp vs log τL plot (open symbols) for a set of molecular solvents and ionic liquids. A linear trend is observed for most molecular solvents (nitriles, AC, PC, 1,2-DCE, DMF, THF, and DMSO). Three

quantum mechanical model in the framework of perturbation theory; DFT calculations are employed to address wave functions of the reactant and the metal electrode. Reorganization energy values were estimated using classical continuum approach (the Marcus formula).29,30 Although we employed earlier molecular dynamics methods to compute reorganization energies in [C4mim][BF4] and AN,21 no experimental evidence exists in favor of continuum or MD estimates of λ so far. Time-dependent fluorescence measurements for dipolar solutes show that continuum-level estimates of the reorganization energies reproduce well at least the relative values for different solvents, if not the absolute ones.2 The λ values were corrected by a factor of ξ to cut off the contribution of highfrequency solvent quantum freedom degrees from the classical activation barrier (ξλ, where ξ < 1).1 The reduced part of solvent coordinates leads to the appearance of a tunneling factor exp(−σ) in the formula for the ET rate constant. Both ξ and σ quantities can be calculated from complex dielectric spectra of solvents extended to high-frequency (near IR) region. The effective relaxation time of solvent (τeff) and ωeff factor are also calculated using experimental dielectric spectra. Pertinent computational details can be found in ref 21. Finally, the work terms were calculated using the potentials of mean force (free energy profiles) as derived from the MD simulations.

3. RESULTS AND DISCUSSION 3.1. Apparent ET Rate Constants in Ionic and Molecular Solvents. Table 1 collects the ET apparent rate constants for Fc+/Fc couple in molecular and ionic solvents, which are considered in this work.15−18,31−35 Most commonly, Pt and Au polycrystalline microelectrodes are used to measure the ET rate constants by means of fast scan rate cyclic voltammetry, impedance spectroscopy, scanning electrochemical microscopy (SECM) or hydrodynamic methods (e.g., using high speed channel electrode, HSChE). The application of polycrystalline electrodes complicates the experimental data analysis, as no systematic information is available on the potentials of zero charge (pzc) in nonaqueous solvents for polycrystalline electrodes, which results in the impossibility to correct the rate constant for the double layer effects. In our analysis, we will use therefore the uncorrected kapp values, bearing in mind that our computed rate constants may differ from the experimental values not only due to uncertainties in model parameters, but also due to the uncertainties in the double layer structure at the equilibrium potential of the Fc+/ Fc redox process in a given solvent. We note that the kapp values from refs 15−17 in molecular solvents do not exhibit large scatter and the values obtained by different groups for them differ by a factor of 3 at most (Table 1). As nonaqueous solutions generally exhibit higher resistivity, the k app values can be subject to the influence of uncompensated ohmic resistance. For this reason, we use the highest kapp estimates reported. In the available series of solvents, the difference in rate constants reaches ca. 1 order of magnitude (from ∼5 cm·s−1 for acetone, AC, to ∼0.15 cm·s−1 for propylene carbonate, PC). Reliable determination of heterogeneous ET rate constants for fast processes in ILs is also difficult. The rate constants in ILs appear to be too fast to be measured by conventional impedance spectroscopy methods, so most typically fast scan rate cyclic voltammetry is used. As ILs are highly viscous and highly resistive solvents,18 the problems associated with the

Figure 1. Double logarithmic plots of the apparent rate constants kapp for Fc+/Fc couple in molecular and ionic solvents against longitudinal relaxation time τL (open symbols) and effective relaxation time τeff (filled symbols). 14373


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The Journal of Physical Chemistry C alcohols (MeOH, EtOH, and PrOH) as well as five ILs obviously are out of this trend, as the experimental rate constants appear to be higher than expected for the τL values, calculated from the slowest relaxations of these solvents with complex dielectric spectra. The reorganization energy factor does not allow for correcting the trend in Figure 1, as estimations based on continuum models result in similar values of reorganization energies for EtOH and PrCN. It was also found that the reorganization energies for dipolar solutes in AN and EtOH are very close.41 As PrCN demonstrates much faster reorientational dynamics, this should result in faster ET in the adiabatic limit. In contrast to these expectations, rate constants are very close in EtOH and PrCN solvents (Table 1). It was demonstrated for [C4mim][BF4] that the account of the faster solvent relaxations might explain the observed increase in the rate constant values.21 In order to rationalize experimental observations on the ET rate in alcohols, two major factors, which might lead to the increase in rate constants, can be considered: (i) the higher effective relaxation time, as compared to that calculated from the slowest solvent relaxation; (ii) specific structure of reaction layer, which provides lower barriers for reactant approach. In order to address solvent dynamic effect at a higher level of theory, the effective frequency values in the adiabatic ET limit can be computed using the formalism introduced in ref 42 and previously applied to describe ET in [C4mim][BF4] and AN.21 The parameters of the solvent correlation function M(τ) are summarized in Table S1, while Table 2 collects the computed

magnitude for [C4mim][BF4] and [C4mim][PF6] ILs and 1 order of magnitude for TFSI-based ILs and [C2mim][BF4] (Table 2). We can suppose that higher effective frequency factors of [C4mim][BF4] and [C4mim][PF6] compared to other ILs are primarily determined by the availability of the higher frequency dielectric frequency data for these solvents, while the τeff factor is overestimated in other ILs. However, without the availability of experimental information on the relaxation behavior in a wide frequency region, such considerations cannot be regarded as conclusive. The solvent tunneling factor σ was not found to be important, while the role of ξ (reducing the solvent reorganization energy) is more significant. For PC and AN solvents, the effective relaxation times are ca. 1 order of magnitude lower than the longitudinal relaxation times. Surprisingly, for alcohols these values are rather close (for instance, for EtOH τeff is 5.1 ps, while τL is 12.3 ps). If the data for ILs and selected molecular solvents are plotted against τeff (filled symbols in Figure 1), the observed linear trend does include the ILs ([C4mim][BF4] and [C4mim][PF6]), while the data points for alcohols are still out of this trend. These results support our previous suggestion on the high frequency factors being responsible for faster ET in ILs.21 The correlation for TFSI-based ILs is rather uncertain, which is presumably attributed to the overestimated τeff values. Indeed, the estimates collected in Table 1 for ILs are rather sensitive to the width of the available frequency data. For instance, when the calculations for [C4mim][BF4] are performed based on the data in ref 50 (1 MHz to 20 GHz frequency range), the values for ξ, σ, and ωeff become 0.93, 0.02, and 1.0 × 1013 s−1, respectively, which is hardly comparable to the estimates based on spectra in ref 48 (0.1 GHz to 3 THz). Resting on these estimates, we did not include in a further analysis the data for ILs for which the dielectric spectroscopy data in the sufficiently wide frequency range are not available. The similar analysis results in a different trend for alcohols. When the spectrum for methanol measured up to 10 THz is used for the calculations,51 the results do not show any significant differences from the data in Table 1, computed from the spectra in a narrower frequency range, which allows us to use the obtained parameter values in further calculations of the rate constant. As the account of the frequency factor in the adiabatic limit does not allow constructing a common trend line for ILs, molecular solvents and alcohols, the differences in other microscopic ET parameters should be addressed as well. In the next section we explore the effect of the work terms and the electrode/solvent interface structure on the ET kinetics in selected solvents: one of the solvents with very fast ET rates (AN), the molecular solvent where the ET rate is fairly low in spite of fast dynamics (PC), a typical alcohol (EtOH) and a well-characterized IL ([C4mim][BF4]). 3.2. Structure of Reaction Layers. The structure of the reaction layer determines the key parameters of ET in polar solvents: the distance of the reactant’s closest approach and the work terms for the reactant’s approach. For a quantitative or at least semiquantitative rate constant values estimation, the knowledge on the solvent structuring near the interface is essential, as organic solvents and especially ILs show a pronounced ordering near the metal phase.52 Simple electrostatic models cannot be used, therefore, to construct a molecular picture of ET.

Table 2. Kinetic Parameters Calculated for ILs and Molecular Solvents solvent [C4mim] [BF4]* [C4mim] [PF6]* [C2mim] [BF4] [C2mim] [TFSI] [C4mim] [TFSI] AN PC methanol ethanol propanol

ξ

σ

ωeff, s−1

1/τeff, s−1

τeff, ps

0.70

0.38

2.45 × 1013

0.9 × 1012

1.1

0.918

0.105

2.07 × 1013

3.45 × 1012

0.3

0.98

0.002

3.6 × 1012

2.24 × 1011

4.5

0.98

0.002

3.35 × 1012

1.18 × 1011

8.5

0.95

0.0016

3.0 × 1012

0.95 × 1011

10.5

0.95 0.98 0.98 0.98 0.98

0.18 0.06 0.036 0.002 0.001

× × × × ×

0.01 0.3 0.3 5.1 12.4

2.9 1.74 1.29 9.5 1.93

× × × × ×

1013 1013 1013 1012 1013

7.9 3.11 3.67 1.96 0.809

1013 1012 1012 1011 1011

effective inverse relaxation time (1/τeff) values in the spirit of Zusman−Kramers theory. The dielectric spectroscopy data for alcohols were adopted from refs 43−45, and the data for AN and PC were from refs 46 and 47. For ILs employed in our study, the data in the terahertz frequency region are available only for [C4mim][BF4] and [C4mim][PF6] ILs.48 Necessary information on the relaxation dynamics in TFSI-based ILs (up to 20 GHz), as well as in [C2mim][BF4] (up to 89 GHz) was found in refs 49 and 50. Table 2 also compiles the nonadiabatic effective frequency values ωeff, tunneling (σ), and ξ factors. A weak correlation was found between the calculated τeff and τL values. A plot of the log τeff vs log τL is shown in Figure S2. The calculated τeff values are generally lower than τL values, and this difference is especially pronounced for ILs: 2 orders of 14374


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solvents under study. The pzc values for polycrystalline gold cannot be used, as the difference in pzc values for various crystal planes can reach as much as 0.5 V.54 The simulations suggest that AN and EtOH are adsorbed at the Au(111) surface with NC− and HO− fragments, which is in a good agreement with the available experimental and modeling data. For instance, spectroscopic studies indicate that AN is adsorbed at the gold surface with the N atom toward the interface.55,56 As can be seen from DFT calculations, the ethanol adsorption occurs preferentially on the top site of the Au(111) surface with the O atom directly adsorbed on the Au atom, and with the OH bond parallel to the electrode surface.57,58 We failed to find in literature any information on the orientation of PC at Au(111) surface. It follows from our data that the PC molecule should be adsorbed with the negative dipole end oriented toward the solution. In situ STM studies suggest that Au(111) surface undergoes potential-dependent surface reconstruction in PC solutions (analogously to aqueous acidic solutions), but the interaction of PC with the surface is relatively weak.59 Various potential dependent orientations of ethylene carbonate on Au(110) surface were deduced from SNIFTIRS experiments in aqueous solution of EC, but these results cannot be transferred to our system unambiguously. Typically imidazolium IL cations specifically adsorb at the gold surface.60 MD simulations revealed similar values of positive Δφdip for [C4mim][PF6] and [C4mim][NTf2] ILs61 and in both ILs the alkyl side chains were found to be oriented parallel to the interface while the imidazolium rings tended to be parallel to the interface in about 60% of the cases. PMF profiles for all the solvents reflect the structuring of the organic solvent near the uncharged metal surface (Figure 3); they can be regarded as the most important information extracted from the MD simulations. The shape of the profiles is rather similar: a sharp minimum is observed at ca. 4.5−5.0 Å, which is separated by a substantially high barrier from the second minimum at 8−9 Å. Relative concentration oscillations

Figure 2 shows electric potential profiles in AN, EtOH, PC, and [C 4 mim][BF 4 ], which were obtained from MD

Figure 2. Electric potential profile at Au(111)/solvent interface.

simulations. The potential drop across the Au(111)/solvent interface can be estimated as the difference between the electric potential values in the solution bulk (20 Å in Figure 2, where the oscillations due to the solvent structuring in the vicinity of the metal interface disappear) and at the metal surface. As under the conditions of simulations the metal surface is uncharged, the value obtained is directly related to the potential difference due to the oriented dipolar solvent molecules at the Au(111)/solution interface, Δφdip. The corresponding values of Δφdip amount to 0.2 V in AN, 0.2 V in EtOH, −0.3 V in PC, and 0.1 V in [C4mim][BF4]. The Δφdip value is related to the solution surface potential, modified due to the interaction with the metal, i.e., Δφdip = χS(M), which is positive, when the positive end of the dipole is oriented toward the solution side.53,54 It should be noted that the direct comparison of the obtained interfacial potential drops with the experimental data (potentials of zero charge, pzc) is problematic, because no data are available on the potential of zero charge values for Au(111) surface in the

Figure 3. PMF profiles for Fc+ and Fc at the Au(111)/solvent interfaces: (a) [C4mim][BF4], (b) AN, (c) EtOH, and (d) PC. 14375


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The Journal of Physical Chemistry C diminish at distances of ca. 15 Å. The tendencies in the energetics of the reactant’s approach for the neutral molecule and ferrocenium cation differ in this series of solvents. For AN and EtOH, the PMF values for Fc and Fc+ in the left minimum practically coincide, and the differences in the second minimum at 8−9 Å does not exceed 5 kJ·mol−1. The trend is different for [C4mim][BF4] and PC; in these two solvents energy values differ significantly for the charged and uncharged species. In PC, the PMF is more negative for Fc+ (by ca. −20 and −9 kJ·mol−1 at the distances of the first and the second minima, respectively), while the opposite trend is observed in IL, where the PMF for the Fc+ cation is more positive in both minima (by ca. 15 and 7 kJ·mol−1 at 4.5 and 8.5 Å, respectively). Stronger repulsion of a positively charged ion in IL is not surprising, if the specific adsorption of IL cations on gold is taken into account.61 Very negative PMF values in the left minimum in EtOH (−16 kJ·mol−1 for both Fc+ and Fc) and in PC (−27 kJ·mol−1 for Fc+) point to the possibility of specific adsorption of Fc and Fc+ at the surface. The layer of adsorbates could provide a significant hindrance for the reactant approach. Effectively, this could result in higher distances of the closest approach, x0, in both PC and EtOH. The work term values, Wi (Fc+) and Wf (Fc), estimated from the PMF curves were used in further rate constant calculations. 3.3. Reactant Specific Adsorption in EtOH and PC. A physically adequate description of the ET process should involve a possible scenario of the specific adsorption of reactant at the electrode. Solvent adsorption energies estimated from the MD data are considerably low for both solvents: ca. 3.5 and 5 kJ·mol−1 for EtOH and PC molecules, respectively (see Figure S3 for the Gibbs free energy profile). The corresponding barriers for desorption are also low (ca. 9 and ca. 10 kJ·mol−1 in EtOH and PC, respectively). Low adsorption energies can favor the specific adsorption of Fc+ and Fc from both solvents. Figure 4 shows the distribution probability for the angle between the surface normal and the vector in EtOH and PC molecules. The EtOH molecule is adsorbed via the OH group at Au(111), in agreement with previous simulations results.57,58 The situation is more complex for the orientation of PC at the surface. Two orientations were found to be the most probable: (i) “planar” orientation with PC ring nearly parallel to the metal surface (θ ∼ 80°) and (ii) “vertical” orientation (θ ∼ 150−160°) with the CO group oriented toward the solution. It can be suggested that the contribution of the vertical orientation results in the positive potential drop value at the Au(111)/PC interface, in contrast to the interfaces in EtOH and AN. The orientation effect is significantly weakened in the second layer and disappears in the third layer in both solvents. Figure S4 shows the snapshots of the first layers of PC and EtOH molecules at the Au(111) surface, where different orientations can be observed. The specific adsorption of Fc+ and Fc could provide rather high coverages at the Au(111) surfaces in PC and EtOH. The coverages for the bulk Fc+ and Fc concentrations equal to 10 mM were evaluated from the adsorption energies (calculated from PMF profiles) and Frumkin (for Fc+) and Langmuir (for Fc) adsorption isotherms (details on the calculation can be found in the Supporting Information, section IV). Two values of dielectric constant in the vicinity of the interface were adopted, εs = 5 and 10, but the results of the calculations turned out to depend only weakly on the εs values. For the 10

Figure 4. Probability density distributions for the angle θ between the normal to the Au(111) surface and the vector in the EtOH (a) and PC (b) molecules (insets) in the first, second, and third solvent layers.

mM concentration of the reactants in the reaction layer in PC, the coverage of the Fc+ is in the range 0.1−0.15, while for the neutral Fc molecule it is 0.21−0.23. The solvation shell of the adsorbed Fc+ involves six PC molecules (Figure S4). The coverages are noticeably lower in EtOH: θ(Fc+) = 0.015− 0.018, θ(Fc) = 0.035. High coverage of the reactant is likely to impede the ET in PC, while the effect for EtOH might be minor. In what follows, we consider the specific adsorption of Fc+ and Fc in PC when calculating the rate constants. It is also interesting to note that the rate constants in PC are very close at Au and Pt electrodes, which could point to the similar energetics of Fc+/Fc and PC adsorption at these metals. 3.4. Estimation of the Rate Constants. First, it is useful to reconstruct the log kapp vs log τeff plot with account of the work terms. As the equation for the rate constant includes the exponential term exp(−Wi/kBT), we plot the kapp value against the τeff exp(Wi/kBT) factor, where Wi is the work term for Fc+ cation at the distance of the reactant’s closest approach (4.5−5 Å). In order to get a semiquantitative description of the ET process in the three groups of solvents, we use the work term computed for EtOH for all three alcohols and the work term in [C4mim][BF4] for [C4mim][PF6]. Although this approximation is obviously crude, it provides a sufficient insight into the role of work terms in ET processes in the solvents under consideration. As in other ILs the effective frequency values are less reliable, we will refrain from rate constants estimations for these solvents. The correction for the work terms makes it possible to construct a common trend line for AN, alcohols, and ILs (Figure 5). It can be judged from the figure that the fast ET 14376


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second minimum in the PMF profile). All the λ values were corrected for the quantum solvent modes (i.e., reduced by the factor of ξ, Table 2). The transmission coefficient value was set to 1 in the adiabatic regime, while κe values in the nonadiabatic limit were set to 0.01.21 Given the close values of the ωeff for all the solvents under consideration ((1−2) × 1013 s−1), this assumption is unlikely to affect our results significantly.

( ) in eqs 1 and 2 presumes

Using the Boltzmann factor exp

−Wi kBT

steady-state conditions between the reactant species in the two PMF minima separated by a barrier. However, a nonequilibrium scenario can be considered as well. Then the resulting rate constant can be recast as follows: k = kDkET/(kD + kET), where kD describes the diffusive overcoming the PMF barrier and kET addresses the ET rate in terms of eqs 1 and 2

Figure 5. Double logarithmic plot of the apparent rate constants kapp for Fc+/Fc couple in molecular and ionic solvents against the τeff exp(−Wi/kBT) factor. The arrow marks the shift of the data point for PC after the correction for the reactant’s specific adsorption.

( ) factor. However, as shown by our

but without the exp

−Wi kBT

estimations, this scenario does not change qualitative conclusions made before. Finally, we have estimated the rate constants ka and kna according to eqs 1 and 2 with account of the distance dependencies of work terms, reorganization energies and x bulk transmission coefficient. Then the integrals ∫ k(x)dx (see

rates in alcohols are determined just by the energetics of approach and not by the non-Debye dielectric relaxation behavior. However, in such a treatment, the value for PC is obviously off the trend, as Wi is very negative in PC and the theory predicts a much higher rate constant in this solvent. However, as we emphasized in the previous sections, the specific adsorption could alter the work terms, and as the coverage is thought to be sufficiently high, this could explain the observed discrepancy between the prognosis and the experiment. If the closest approach distance (x0) of Fc+ and Fc is increased from 4.5 to 7−8 Å (where the reaction is thought to proceed still under mixed control), the work term factor decreases to ±5 kJ·mol−1, which allows to bring the point for PC into the linear trend (arrow in Figure 5 denotes the change in the coordinates of point for PC when correcting for higher x0 values). In the above discussion, we disregarded the solvent dependence of the reorganization energy λ to illustrate the major factors, which influence the linearity of the common plot of the apparent rate constant vs solvent relaxation time or sometimes the viscosity of the solvent.18 It is still necessary to include this factor in the ET description and to calculate the absolute ET rates in the solvents considered in order to understand the extent to which the theoretical expressions allow to explain the experimental tendencies. Table 3 collects the solvent reorganization energy (λ) values calculated in the adiabatic limit (λa; x0 = 4.5−5.0 Å, the position of the first minimum in the PMF profile) and in the nonadiabatic limit (λna; x0 = 8.0−8.5 Å, the position of the

x0

eq 3) were calculated for x0 values in the adiabatic and nonadiabatic limits as well. The corresponding reaction volumes δxa and δxna were estimated from the integrals. Table 3 collects the estimated values, while Figure 6 shows the plot of the calculated against the experimental rate constants. The rate constant for PC was assessed both with and without the account for the reactant’s specific adsorption and the corresponding increase of the distance of closest approach in the former case. The calculated adiabatic rate constants tend to be 1−2 orders of magnitude higher than the respective experimental values. This discrepancy might originate from the underestimation of the reorganization energy values, which can be substantially higher compared to the predictions of the undoubtedly oversimplified continuum model. Another reason can be related to the higher work terms at the equilibrium potential of the Fc+/Fc reaction, which is different from the potential of zero charge, involved in our model treatment. This factor should be especially remarkable for ILs, where various potential-dependent interfacial structural rearrangements are reported,62 and the difference between the calculated and experimental rate constants is indeed the highest for the two ILs considered. Higher work terms can also be caused by the adsorption of background electrolyte, which is always present

Table 3. Calculated Rate Constants in the Adiabatic (ka, cm·s−1) and Nonadiabatic (kna, cm·s−1) Limits and Corresponding Reaction Volumes (δxa, δxna, Å) and Solvent Reorganization Energies (λa, λna, kJ·mol−1) solvent

kapp

ka

δxa

kna

δxna

λa

λna

[C4mim][BF4] [C4mim][PF6] AN PC MeOH EtOH 1-PrOH

0.02 0.013 3 0.15 2.34 0.82 0.47

9.2 6.8 360 0.2 (1010)a 470 47 30

0.5 0.5 0.2 0.9 (0.2)a 0.2 0.2 0.2

2.4 0.3 0.07 0.4 0.07 0.1 0.4

3.5 3.2 10 6 2.5 2.5 1.5

45.0 60.1 81.3 71.0 82.0 76.3 72.0

55.7 74.3 96.8 85.5 101.5 92.0 89.0

a

Values computed without account of the specific adsorption of Fc+ and Fc in PC. 14377


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Figure 6. Plot of the calculated rate constants in the adiabatic and nonadiabatic limits (kcalc) against the experimental rate constants (kapp) for Fc+/Fc redox process in ILs and molecular solvents. kcalc values are reduced by the factor of 10 for better representation. Filled and empty symbols correspond to adiabatic and nonadiabatic ET, respectively.

MeOH, which is hardly physically meaningful. However, when the correction for the Fc+ and Fc specific adsorption is introduced, the ka value decreases to a moderate value of 0.2, which is sufficiently close to the experimental value. Interestingly, the difference between the ka and kna values vanishes, which implies the mixed reaction rate control as in case of [C4mim][BF4]. This is a particularly important result, which shows that the initial statement on the adiabaticity of ET in PC, formulated based on the linearity of log kapp vs log τL(eff) dependency for a series of solvents, is misleading and does not warrant the physical validity of the assumptions involved. The results for alcohols allude that ET in these solvents proceeds under the adiabatic control, as for these solvents the ka values are several orders of magnitude higher than the kna values (Figure 6). The computed rates for alcohols reproduce the experimental tendencies quite well, showing significant differences between the rates in MeOH and EtOH and lower difference in rates in EtOH and PrOH. This implies the general validity of our conclusions on the factors, which primarily determine the high ET rates for alcohols, the plausible energetics of reactants approach.

at high concentrations in molecular solvents. We have also to keep in mind that the kinetic parameters were computed using the experimental dielectric spectra obtained for bulk electrolyte solutions ignoring local effects induced by the electrode surface and the reactant itself; this challenging issue can be addressed by intensive MD simulations. Deviations of 2 orders of magnitude are observed in AN and MeOH solvents, which demonstrate the fastest ET kinetics in the series. Here the difference between the experimental and computed values can be partially determined by the insufficient accuracy of the determination of the high rate constant and the difficulties of the adequate compensation of the solution ohmic resistance; the actual rate constants in these solvents can be somewhat higher than those reported. However, even given the large differences of the computed and experimental rate constants, a comparative analysis of the ET patterns in different solvents can still be performed, which undoubtedly provides more insight into the influence of various factors on the charge transfer rate than the mere numerical agreement of the values. If the results for the two ILs are considered, the adiabatic rate constants and reaction volumes are very close for these two solvents, which agrees with the experimental observations, with the rate constant for [C4mim][BF4] being slightly higher. As in our analysis we used the PMF profile computed only for one IL, one may suggest that the work terms are primarily determined by the nature of the cation, while the effect of the anion on the energetics of the reactant’s approach is not particularly strong. The ka and kna values are rather close for [C4mim][BF4] (9.2 and 2.4 cm·s−1), while ka is 20 times higher in [C4mim][PF6] that kna. These findings imply that while the ET in [C4mim][BF4] is likely to proceed under intermediate or mixed control, the ET in [C4mim][PF6] can be treated as adiabatic. The rate constant ka for AN is sufficiently high, and the difference between the adiabatic and the nonadiabatic cases reaches 4 orders of magnitude. The same results were obtained previously using MD values of reorganization energy, which indicates that for simple solvents qualitative trends are well reproduced using the continuum level estimates of λ values. For PC, initial estimates of ka result in very high values (ca. 1000 cm·s−1), which are higher than the values in AN and

4. CONCLUSIONS In this work, we present a semiquantitative description for one of the simplest ET reactions, the Fc+/Fc redox process in a series of molecular and ionic solvents, based on the combination of molecular modeling and continuum-level estimates in the framework of quantum chemical theory of charge transfer in polar media. Our analysis rests on the consideration of literature values of rate constants in molecular solvents as well as new experimental information on the Fc+/ Fc ET rates in imidazolium-based ionic liquids. The reported molecular-level information on the reaction layer structure adds original calculations on the Au/EtOH and Au/PC interphases to the data reported in our previous work,21 which allows to attain some generality in our conclusions on the factors, which control the ET rates in polar solvents. This gives an opportunity to revisit the classical test for the reaction adiabaticity, which implies the construction of plots of experimental rate constants against the solvent relaxation times, estimated from various spectroscopic data or, even more commonly, from solvent viscosity. The latter approach was shown to be particularly misleading for solvents with a complex dielectric behavior. For instance, the most viscous IL in our series, [C4mim][PF6] shows rather fast effective relaxation (relaxation time is close to the values in PC and MeOH). The ET rate in [C4mim][PF6] is correspondingly higher than it would be predicted from the solvent’s viscosity. A molecular-level description of the reaction layer structure in both molecular and ionic solvents was also shown to be essential for the diagnostics of the ET regime and the estimation rate constants. The ET rate in PC was found to be dramatically affected by the specific adsorption of the reactants, which effectively resulted in the decrease in the rate constant by several orders of magnitude and “merging” of the rates estimated in adiabatic and nonadiabatic limits. These results illustrate that even for the relatively simple solvents the linearity of the log kapp vs log τ plot can result from the mutual compensation of various factors, which can be isolated only when a careful model kinetic analysis is performed. The corrections for the computed work terms for ILs and alcohols resulted in opposite trends: the repulsive interactions in ILs determine the decrease of the ET rate, which would otherwise 14378


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(2) Weaver, M. J. Dynamical Solvent Effects on Activated ElectronTransfer Reactions: Principles, Pitfalls, and Progress. Chem. Rev. 1992, 92 (3), 463−480. (3) Weaver, M. J. Comprehensive Chemical Kinetics; Compton, R. G., Ed.; Elsevier Science Publishers B.V.: The Netherlands, 1987; Vol. 27, pp 1−60. (4) Hupp, J. T.; Liu, H. Y.; Farmer, J. K.; Gennett, T.; Weaver, M. J. The Prediction of Electrochemical Reactivities from Contemporary Theory: Some Comparisons with Experiment. J. Electroanal. Chem. Interfacial Electrochem. 1984, 168 (1), 313−334. (5) Gennett, T.; Milner, D. F.; Weaver, M. J. Role of Solvent Reorganization Dynamlcs in Electron-Transfer Processes. TheoryExperiment Comparlsons for Electrochemical and Homogeneous Electron Exchange Involving Metallocene Redox Couples. J. Phys. Chem. 1985, 89, 2787−2794. (6) McManis, G. E.; Golovin, M. N.; Weaver, M. J. Role of Solvent Reorganization Dynamics in Electron-Transfer Processes. Anomalous Kinetic Behavior in Alcohol Solvents. J. Phys. Chem. 1986, 90, 6563− 6570. (7) Weaver, M. J.; Phelps, D. K.; Nielson, R. M.; Golovin, M. N.; McManis, G. E. Solvent Dynamic Effects in Electron Transfer: Electrochemical Versus Self-Exchange Kinetics of Tris(Hexafluoroacetylacetonato)Ruthenium(Iii/Ii) and Comparison with Other Probe Reactants. J. Phys. Chem. 1990, 94 (7), 2949−2954. (8) Pyati, R.; Murray, R. W. Solvent Dynamics Effects on Heterogeneous Electron Transfer Rate Constants of Cobalt Tris(Bipyridine). J. Am. Chem. Soc. 1996, 118 (7), 1743−1749. (9) Yang, E. S.; Chan, M.-S.; Wahl, A. C. Electron Exchange between Ferrocene and Ferrocenium Ion. Effects of Solvent and of Ring Substitution on the Rate. J. Phys. Chem. 1980, 84 (23), 3094−3099. (10) Fawcett, W. R.; Opallo, M. The Kinetics of Heterogeneous Electron Transfer Reaction in Polar Solvents. Angew. Chem., Int. Ed. Engl. 1994, 33 (21), 2131−2143. (11) Fawcett, W. R.; Opallo, M. On the Differences in the Magnitude of the Observed Solvent Effect in the Kinetics of Simple Heterogeneous Electron Transfer Reactions. J. Electroanal. Chem. 1993, 349 (1), 273−284. (12) Zagrebin, P. A.; Buchner, R.; Nazmutdinov, R. R.; Tsirlina, G. A. Dynamic Solvent Effects in Electrochemical Kinetics: Indications for a Switch of the Relevant Solvent Mode. J. Phys. Chem. B 2010, 114 (1), 311−320. (13) Kadish, K. M.; Ding, J. Q.; Malinski, T. Resistance of Nonaqueous Solvent Systems Containing Tetraalkylammonium Salts. Evaluation of Heterogeneous Electron Transfer Rate Constants for the Ferrocene/Ferrocenium Couple. Anal. Chem. 1984, 56 (9), 1741−1744. (14) Abbott, A. P.; Miaw, C. L.; Rusling, J. F. Correlations between Solvent Polarity Scales and Electron Transfer Kinetics and an Application to Micellar Media. J. Electroanal. Chem. 1992, 327 (1), 31−46. (15) Clegg, A. D.; Rees, N. V.; Klymenko, O. V.; Coles, B. A.; Compton, R. G. Marcus Theory of Outer-Sphere Heterogeneous Electron Transfer Reactions: High Precision Steady-State Measurements of the Standard Electrochemical Rate Constant for Ferrocene Derivatives in Alkyl Cyanide Solvents. J. Electroanal. Chem. 2005, 580 (1), 78−86. (16) Safford, L. K.; Weaver, M. J. The Evaluation of Rate Constants for Rapid Electrode Reactions Using Microelectrode Voltammetry: Virtues of Measurements at Lower Temperatures. J. Electroanal. Chem. 1992, 331 (1), 857−876. (17) Baranski, A. S.; Winkler, K.; Fawcett, W. R. New Experimental Evidence Concerning the Magnitude of the Activation Parameters for Fast Heterogeneous Electron Transfer Reactions. J. Electroanal. Chem. Interfacial Electrochem. 1991, 313 (1), 367−375. (18) Bentley, C. L.; Li, J.; Bond, A. M.; Zhang, J. Mass-Transport and Heterogeneous Electron-Transfer Kinetics Associated with the Ferrocene/Ferrocenium Process in Ionic Liquids. J. Phys. Chem. C 2016, 120 (30), 16516−16525.

be very high due to high frequency factors, while the attractive interactions result in faster rates for alcohols, in complete agreement with the experimental data. We are aware of the fact that in our treatment of heterogeneous ET processes in polar solvents a vast number of inevitable oversimplifications and approximations were made. For instance, we employed the simple Marcus model to calculate reorganization energies, the work terms were constructed for the uncharged metal surface, and we neglected the effect of the supporting electrolyte on the rates in molecular solvents. However, qualitative and even semiquantitative agreement was achieved for the systems under consideration, which allowed us to suggest factors that control the ET rate and to test the validity of the theoretical assumptions. At this point, further specification of the model parameters and more accurate heterogeneous ET description is limited by the lack of high-quality experimental data (potentials of zero charge at single crystal electrodes in nonaqueous solvents and ionic liquids, dielectric spectroscopy data in the terahertz frequency region, accurate and reproducible rate constant values in a large series of solvents). We hope that further experimental and computational efforts in the field of ET modeling will prompt a more general and persuasive conclusion resting on the predictive power of the quantum mechanical ET theory.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.9b01163.

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Cyclic voltammograms at high scan rates; parameters of the solvent correlation function; additional MD data; calculation of surface coverage values in PC (PDF)

AUTHOR INFORMATION

Corresponding Author

*Tel: +7(495)9391321. E-mail: [email protected]. ORCID

Victoria A. Nikitina: 0000-0002-0491-3371 Sergey A. Kislenko: 0000-0003-3579-5221 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are indebted to G.A. Tsirlina for useful comments. R.R.N.’s work in subsections 3.1 and 3.4 was supported by the RSF (Project No. 17-13-01274); S.A.K.’s work in subsections 3.2 and 3.3 was supported by the state assignment of JIHT RAS (Theme No. GR AAAA-A-16-116051810080-3). The research was carried out using supercomputers at Joint Supercomputer Center of the Russian Academy of Sciences (JSCC RAS) and the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University. The authors acknowledge the Supercomputer Centre of JIHT RAS for providing computing time.

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REFERENCES

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