pec - American Chemical Society

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Voltammetry Can Reveal Differences between the Potential Energy Curve (pec) and Density of States (dos) Models for Heterogeneous Electron Transfer Liu-Bin Zhao State Key Laboratory of Physical Chemistry of Solid Surfaces and Department of Chemistry, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, 361005 Fujian, China

A. K. Mishra Institute of Mathematical Sciences, CIT Campus, Chennai 600113, India

David H. Waldeck* Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States S Supporting Information *

ABSTRACT: This work uses the potential energy curve (pec) and density of states (dos) methods to calculate the cyclic voltammogram for a redox adsorbate that undergoes a simple one-electron reversible redox reaction. It extends an earlier treatment for simulation of voltammograms, which used the simple Butler−Volmer kinetics and classic Marcus theory in the nonadiabatic limit, to calculate the voltammogram for an arbitrary electrode−reactant coupling strength. Voltammograms are calculated as a function of the electrode−reactant coupling strength and reorganization energy, and the dependence of the voltammetric peak potentials and normalized peak currents on sweep rate are calculated and compared for the pec and dos methods. Although the pec and dos approaches coincide in the nonadiabatic limit, they deviate very significantly for large electronic couplings, beyond the nonadiabatic limit. The most interesting finding is that the dos voltammetric wave splits into two separated peaks, under some conditions at large coupling strengths.

I. INTRODUCTION Electron transfer at interfaces is an area of great fundamental and practical importance, which plays a central role in many well-known and emerging technologies. Electrochemical electron transfer rates have been calculated by either the potential energy curve (pec) method1 or the density of states (dos) method.2 The pec method generates potential energy curves for an electron transfer reaction by using a generalized Anderson-Newns Hamiltonian,3−8 and then calculates the electron transfer rate by a transition state theory (TST) based description.9,10 An alternative theoretical treatment to electrochemical kinetics is the dos method, which is based on the overlap between electronic states of the electrode and those of the reactant in solution.11−14 The main idea in the dos method is that an electron transfer can take place from any occupied electronic state of the donor that is matched in energy with an unoccupied electronic state of the acceptor. In this treatment, an electron transmission probability is evaluated for each pair of states and the total electron transfer rate is © 2013 American Chemical Society

determined by a sum over all possible states. The reactant density of states appears naturally in this treatment and is similar to a modern Green’s function or correlation function based formalism for electron transfer.15−23 In the nonadiabatic limit, the pec and dos methods lead to identical results, but for strong electrode−reactant coupling, the two models predict distinctly different overpotential and reorganization energy dependences. The current work shows that this difference can become manifest in the cyclic voltammograms for an adsorbed, redox active layer on an electrode. Voltammetry is a primary method in electrochemical research that has many variations; however cyclic voltammetry is very widely used and highly informative. In this method, the voltage is swept in a cyclic fashion and the current response of the system is measured. The reversibility of electrode reactions, Received: July 18, 2013 Revised: September 10, 2013 Published: September 13, 2013 20746

dx.doi.org/10.1021/jp4071532 | J. Phys. Chem. C 2013, 117, 20746−20761

The Journal of Physical Chemistry C

Article

and from a reactant to the electrode, kox, as

possible intermediates, interfacial adsorption, as well as coupled chemical reactions can be evaluated from the shape of the current−voltage curve.24,25 An important application of cyclic voltammetry is the study of kinetics and the extraction of kinetic parameters for electrochemical reactions. This work uses fundamental theoretical models for a simple one-electron transfer reaction to calculate the shape of the current−voltage curves (voltammograms) in cyclic voltammetry as a function of the electronic coupling strength, the reorganization energy, and the voltage scan rate. An important problem in modern electrochemistry is the study of long-range electron transfer, specifically between a metal electrode and a redox-active monolayer self-assembled on the electrode surface.26−29 Voltammetry methods have been used to analyze redox kinetics in such systems by using either the classic Butler−Volmer model30 or the Marcus theory31,32 of electrochemical kinetics in the nonadiabatic limit.33−35 The standard electron-transfer rate constant can be derived from an analysis of the dependence of peak potential separation on the potential sweep rate.33−35 In previous studies, the voltammograms were simulated by using the nonadiabatic Marcus theory, and thus have not been calculated for moderate and strong electronic coupling conditions. In this paper, numerical simulations of voltammograms are performed by using pec and dos based rate constants that are calculated for arbitrary coupling strength, ranging from the weak coupling (nonadiabatic) to the strong coupling (adiabatic) limits. These simulations reveal significant differences in the voltammogram’s shape at large coupling strengths; most significantly the voltammogram displays two current peaks over a range of conditions. The remainder of the paper is separated into three sections. In section II, a synopsis of essential features of the electron transfer theory are provided; it includes a simple treatment for the nonadiabatic limit and presents the pec and dos methods which can be generalized to arbitrary coupling magnitudes |V|. Section III.a presents the dependence of the pec model’s potential energy curves and the reactant density of states on the electrode-reactant coupling and the reorganization energy. The rate constants for both nonadiabatic and adiabatic limits are presented in section III.b. Section III.c provides a comparison of voltammograms obtained from the pec and dos methods. The variation of the voltammogram with electrode-reactant coupling and reorganization energy, in particular the dependence of the voltammetric peak potentials, peak current, and peak shape on sweep rate, is investigated and discussed. Section III.d presents working curves of the voltammetric peak potential Epeak and the peak current Cpeak versus sweep rate. Section IV provides a summary and conclusions.

kox(η) =

(2)

(λ ± (ε − εF − eη))2 4λ

Ea(λ , η) =

(3)

The exponential term in eqs 1 and 2 can be viewed as the redox species density of states ρr(λ,η), which has a Gaussian distribution with a maximum at ε = εF + eη ∓ λ and a standard deviation of (2λ/β)−1/2. ⎛ (λ ± (ε − ε − eη))2 ⎞ β F ⎟ exp⎜ −β 4πλ 4λ ⎠ ⎝

ρr (λ , η) =

(4)

In the nonadiabatic limit, the activation energy and the redox species densities of states are typically taken to be independent of the electronic coupling V. Note that eqs 1 and 2 have been modified so that they extend to the adiabatic (large |V|) limit.1−3,5 The first change that must be made is to replace the pre-exponential factor (2π/ℏ)|V|2 by the product νnκn, in which νn is a nuclear frequency factor and kn is an electron transmission coefficient (see ref 1 for details). The second step is to account for the lowering of the activation energy and the broadening of reactant density of states by the reactant-electrode coupling V.1 II.b. Potential Energy Curve (pec) Model. Here we summarize features of the pec model, which are discussed in more detail in refs 1−5. A generalized Anderson−Newns Hamiltonian (spinless case) for the electron transfer system can be written as3−8

∑ εknk + ∑ (υkrck†cr + υrk cr†ck)

H = εrnr +

k

1 + 2

∑ ℏωi(pi 2 i

k

+ qi 2) + (nr − z) ∑ ℏωigiqi i

(5)

In the Levich−Dogonadze (LD) theory, eqs 1 and 2 can be obtained from this Hamiltonian by first order perturbation. The first term describes the electronic state of the reactant, and the next term represents the electronic states in the metal. k and r label the electrode and the reactant electronic states, respectively. ε and n are the energy value and the number operator. Electron exchange between the metal and the reactant is incorporated in the third term; the amplitudes υ rk characterize the strength of the electronic coupling between

k red(η) β 2π 2 ∞ f ( ε ) ρ (ε ) |V | −∞ 4πλ ℏ ⎡ (λ + ε − ε + eη)2 ⎤ F ⎥d ε exp⎢ −β 4λ ⎦ ⎣

∫

where β = (kBT)−1, η is the overpotential, λ is the solvent reorganization energy, εF is the electrode Fermi energy, and |V|2 is the effective electronic coupling strength between the reactant and the electrode states. ρ(ε) is the electrode density of states, and f(ε) is the probability that a state of energy ε is occupied by an electron. The total rate constant contains the contributions from all energy states of the electrode. This expression treats the nuclear degrees of freedom classically.36 It is common to define an activation energy for the oxidation and reduction reactions that is a function of ε and depends parametrically on the reorganization energy λ and the overpotential η; namely,

II. THEORETICAL BACKGROUND II.a. Levich−Dogonadze Theory. In the nonadiabatic limit, the Fermi-Golden Rule expression may be used to write the rate constant for an electron transfer from an electrode to a reactant, kred, as

=

β 2π 2 ∞ (1 − f (ε))ρ(ε) |V | −∞ 4πλ ℏ ⎡ (λ + ε − ε − eη)2 ⎤ F ⎥d ε exp⎢ −β 4λ ⎦ ⎣

∫

(1) 20747



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the two states k and r, and c and c† are the creation and annihilation operators. The fourth term describes the nuclear bath of harmonic oscillators; the pi are momenta, the qi are coordinates, and the ωi are the oscillator frequencies. The coupling of the oscillator to the reactant is assumed to be linear and proportional to the reactant’s core charge z, as given by the last term in which gi describes the reactant-solvent linear coupling strength. Note that the multi-oscillator description of the solvent can be replaced by a single oscillator model using a coordinate transformation.5 q=

∑ giqi

where ηox cr is the overpotential at which the oxidation reaction becomes activationless. Using the energy barrier given above and approximating the electrode dos by its value at the Fermi level ρ(εF), the electron transfer rate constant for the case of multiple potential energy curves (multiple pec case) can be obtained from the activation energy determined by eqs 12−14 and a similar set of relations for ΔUred(ε,η), cf. ref 1. The integration over all energy states ε provides the total rate constant, pec k red (η) = νnκnρ(εF)

(15)

(6)

i

and

The pec treatment aims to find the potential energy curve(s) for the reaction and then to evaluate the rate constant along the reaction coordinate by transition state theory or a more elaborate model which accounts for dynamical effects.1 Using the wide-band approximation for the interaction energy, the potential energy u(q,ε) for an electron transfer between an electron state at the energy ε and a redox couple may be expressed as a function of the solvent polarization coordinate q u(q , ε) = q2 /4λ + (εr + q − ε)⟨nr(q)⟩ − zq Δ + ln[(εF − εr − q)2 + Δ2 ] + C 2π

koxpec(η) = νnκnρ(εF)

εr = λ(1 − 2z) + εF + η

II.c. Density of State (dos) Model. The density of states (dos) model computes the electron transfer rate from the transition probability for an electron to move from a donor (occupied) state to an acceptor (unoccupied) state. In order to extend the rate constant expression in eq 1 and 2 beyond the weak coupling limit, one must find the actual redox species density of states, so that one accounts for the broadening which arises from the interaction between the redox species and the electrode. The density of states within a classical treatment of the oscillator may be written as15−23 1 Re(w(z±)) ρr (ε , Δ, λ , ±η) = (17) 2 πP

(7)

(8)

where

Δ accounts for the interaction strength between the electrode’s electronic states and the redox species,

2

w(z±) = e−z± erfc( −iz±)

Δ = π ∑ |υrk |2 δ(ε − εk )

z± = ( −Q ± + iΔ)/2 P

= |V |2 π ∑ δ(ε − εk )

(19)

with

k

= |V | π ∑ ρ(εk ) 2

k

P = λ/β (9)

(20)

∞

∫−∞ f (ε)Re(w(z−))dε

(21)

and the rate constant for the oxidation reaction becomes

(10)

By introducing

dos kox (η) = νnκnρ(εF)

(11)

∞

∫−∞ (1 − f (ε))Re(w(z+))dε

(22)

In the limit that Δ → 0, eq 17 yields a Gaussian dos,

the energy barrier for the oxidation in different ranges of η is given by

lim ρr (ε , Δ, λ , η)

Δ→ 0

‡ ΔUox (ε , η < ηcrox ) = U (qmax , ε , η) − U (qmin 1 , ε , η)

=

(12)

≤ η ≤ (λ + ε)) = 0

Q ± = ε − εF − eη ± λ

dos k red (η) = νnκnρ(εF)

⎛ ε − εr − q ⎞ 1 1 ⎟ ⟨nr(q)⟩ = ρr (ε′, q)dε′ = + tan−1⎜ ⎝ ⎠ 2 π Δ −∞ ε

U (q , ε , η) = u(q , ε , η) − u(q = 0, ε , η)

and

See refs 1, 2, 20, and 21 for more details. Note that, in refs 1 and 2, ε was set to be −ε for an anodic process, but that convention is not used here. If one approximates the electrode’s density of states ρ(ε) by ρ(εF), as in the pec method, then the rate constant for the reduction reaction becomes

The electronic coupling per electrode electronic state |V|2 denotes an average of |υrk|2 over the total number of electrode electronic states. In the wide-band limit, Δ may be considered energy independent. The average occupation probability of the reactant orbital ⟨nr(q)⟩ equals

ηcrox

(18)

and

k

∫

∞

∫−∞ (1 − f (ε)) exp(−βΔUox‡ (ε , η))dε (16)

where C is a constant of integration, εF is the electrode Fermi energy, z is the reactant’s charge, and εr is the redox species orbital energy, which equals

‡ ΔUox (ε ,

∞

∫−∞ f (ε) exp(−βΔUred‡ (ε , η))dε

⎛ (λ ± (ε − ε − eη))2 ⎞ β F ⎟ exp⎜ −β 4πλ 4λ ⎠ ⎝

(23)

and the LD (or Marcus) result is recovered (see eq 4). As Δ increases, the form of ρr changes. For example, in the limit that Δ → λ, the redox species density of states converges to a Lorentzian form,

(13)

‡ ‡ ΔUox (ε , η′ > (λ + ε)) = ΔUox (ε , η = 2(λ + ε) − η′)

(14) 20748



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Figure 1. (A) Plots of the potential energy curves U(q) versus q are shown for different values of Δ. (B) Plots of the redox species density of states ρr(ε) versus ε are shown for different values of Δ (λ = 0.8 eV, η = 0 V, εF = 0 eV, and T = 298 K). The minimum at the left (right) side of panel (A) corresponds to the equilibrium configuration of the reduced (oxidized) form of the reactant whose solvent averaged density of states are predicted by the left (right) side curve in panel (B).

lim ρr (ε , Δ, λ , η) ≈

Δ→ λ

some selected Δ values. In the limit of very high couplings, Δ ≥ 0.5 eV, the dos curves have a Lorentzian form. Figure 2 (left panel) shows how the potential energy curve U(q) changes with reorganization energy λ for three different values of Δ. In each case, the activation energy Ea increases with λ. For the weakest coupling strength Δ = 0.001 eV, which lies in the nonadiabatic limit, Ea is equal to λ/4 as predicted by Marcus theory for the energy neutral reaction (η = 0 V). As Δ increases, the electron transfer barrier significantly decreases, especially for small λ values. For the strongest coupling strength Δ = 0.5 eV, the activation energy Ea reduces to zero at λ = 0.8 eV. Figure 2 (right panel) shows how the density of states ρr(ε) changes with λ for three different values of Δ. In each case, the peak position of ρr(ε) is located at ε = ± λ for η = 0 V. In every case the ρr(ε) maximum decreases and its peak width broadens as λ increases. For the weakest coupling strength Δ = 0.001 eV, the density of states has a Gaussian form with a peak maximum of ρmax = (β/4πλ)1/2 and a full width at half-peak height of ω1/2 = 2.355(2λ/β)1/2. As Δ increases, the dos gradually changes from a Gaussian to a Lorentzian form (see bottom right panel). An important difference between the Gaussian and Lorentzian dos is that the former decays as exp(−ε2) whereas the latter vanishes in a power law fashion 1/ε2 for large |ε|; thus the Lorentzian dos has a longer tail in comparison to the Gaussian dos. Increasing Δ leads to broadening of the reactant dos and an increase in its overlap with the Fermi function. Table S1 in the Supporting Information shows the dependence of the activation energy Ea on λ and Δ from the pec method, and ρmax and ω1/2 on λ and Δ from the dos method. III.b. Electron-Transfer Rate Constant. In this subsection, we show how the electrochemical rate constant depends on the overpotential and the electronic coupling strength. Figure 3A pec plots the logarithm of the rate constant sum, (log(kpec ox + kred)) with respect to the overpotential η by the pec method for a range of Δ values in the weak coupling limit, 10−10 eV ≥ Δ ≥ 10−4 eV. The rate constant sum shows an initial increase as the overpotential deviates from zero, and then reaches a plateau. The multiple pec formalism gives a plateau because the system belongs to the normal overpotential region for one set of ε, even though it is in the inverted region for another set of ε. Thus, the overall rate constant does not display characteristics of the Marcus inverted region.

Δ 1 π (λ ± (ε − εF − eη))2 + Δ2 (24)

Later, we show that this behavior affects the form of the voltammograms.

III. RESULTS AND DISCUSSION III.a. PEC and DOS. In this subsection, we show how the shape of the potential energy curve and its activation barrier change with Δ and λ (see Figure 1A), and we show how the density of states for the reactant changes with Δ and λ (see Figure 1B). From the curves in Figure 1A, it is apparent that for very low Δ the reductant and oxidant curves cross at q = 0 and form a cusp-like barrier. In this limit of Δ ≤ 0.001 eV, the pec curves all have the same shape; that is, they overlap with each other and the activation energy agrees with the Marcus theory value of λ/4. As the coupling increases, the barrier is reduced and the maximum forms a smooth, rounded top. In the limit of very high couplings, Δ of 0.5 eV and higher, only one minimum is apparent. Table 1 reports the activation energies that are found for some selected Δ values. Table 1. Activation Energy Ea from the pec Model and Peak Density ρmax, and Half-Peak Width ω1/2 from the dos Model as a Function of Interaction Energy Δ, for λ = 0.8 eV, η = 0 V, and T = 298 K Δ/eV

Ea/eV

ρmax/eV−1

ω1/2/eV

0.001 0.01 0.1 0.2 0.3 0.4 0.5

0.198 0.183 0.103 0.054 0.023 0.006 0

1.960 1.893 1.382 1.037 0.817 0.668 0.562

0.48 0.50 0.60 0.72 0.88 1.02 1.20

Figure 1B shows density of states plots for the same Δ values that are used for the U(q) plots in Figure 1A. For very low Δ the dos curves have a Gaussian form and have a maximum of (β/4πλ)1/2 which agrees with Marcus theory. For values of Δ ≤ 0.001 eV, they overlap with each other. As Δ increases, the peak maximum decreases and the peak width broadens. The dos maxima and their half peak widths are listed in Table 1 for 20749



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Figure 2. Left: Potential energy curves, U(q) versus q, are plotted for different values of λ at three Δ values (top, Δ = 0.001 eV; middle, Δ = 0.1 eV; bottom, Δ = 0.5 eV, η = 0 V, εF = 0 eV, and T = 298 K). Right: Redox species density of states, ρr(ε) versus ε, are plotted for different values of λ at three Δ values. (top, Δ = 0.001 eV; middle, Δ = 0.1 eV; bottom, Δ = 0.5 eV, η = 0 V, εF = 0 eV, and T = 298 K). dos Figure 3B shows plots of the (log(kdos ox + kred ))versus η by the dos method for a range of Δ values in the weak coupling limit, 10−10 eV ≥ Δ ≥ 10−4 eV. The rate constant sums show an initial increase as η deviates from zero, and then saturates at a limiting value for high η values. This behavior follows from a progressively increasing overlap of the reactant dos with f(ε) as η increases. Once the area of the reactant dos profile overlaps completely with the electrode density of states, a further change in η does not affect the overlap and (kox + kred) attains its maximum value. Comparison of the rate constant sums in Figure 3A and B reveals that the pec and dos approaches give identical results in the weak coupling limit (10−10 eV ≥ Δ ≥ 10−4 eV), and that they compare well to Marcus (LD) theory values. Note that the pec and dos methods give significantly different results from the

Butler−Volmer kinetics, which predicts that the logarithm of the rate sum increases monotonously at large η. In the nonadiabatic limit, the pec and dos rate constant expressions reduce to Levich−Dogonadze formulas (see eqs 1 and 2), so that the rate constant is proportional to |V|2 or Δ, and the rate constant sum in these plots shows a linear relationship on the coupling strength Δ for small Δ. pec Figure 3C plots the (log(kpec ox + kred)) versus η by the pec method for a range of Δ values that extend into the strong coupling limit, 10−4 eV ≥ Δ ≥ 0.2 eV. As expected, the rate constant sum increases with an increase of Δ and it attains a saturation value at very large overpotential for a given Δ. The displacement between the curves at η = 0 V does not follow the same dependence as that shown in Figure 3A for weak coupling. The reason behind this difference is the nonlinear 20750



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Figure 3. In each panel, the logarithm of the rate constant sum is plotted as a function of overpotential for the parameters λ = 0.8 eV, εF = 0 eV, and pec −10 to 10−4 eV, obtained by the pec method. (B, top T = 298 K. (A, top left) Plots are shown of log(kpec ox + kred) versus η for Δ values ranging from 10 dos dos −10 −4 right) Plots are shown of log(kox + kred) versus η for Δ values ranging from 10 to 10 eV, obtained by the dos method. (C, bottom left) Plots are pec −4 to 0.2 eV, obtained by the pec method. (D, bottom right) Plots are shown of shown of (log(kpec ox + kred) versus η for Δ values ranging from 10 dos dos −4 log(kox + kred) versus η for Δ values ranging from 10 to 0.2 eV, obtained by the dos method. Note the scale change between panels (A) and (B), and panels (C) and (D).

Table 2. Calculated Rate Constants as a Function of Interaction Energy (λ = 0.8 eV, T = 298 K)a Δ/eV −10

10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 0.1 0.2

k0pec/s−1 0.633 0.513 0.405 0.364 0.256 0.180 0.120 0.108 0.193 0.421 0.294

× × × × × × × × × × ×

−1 kmax pec /s −7

10 10−5 10−3 10−1 101 103 105 106 106 107 108

0.103 0.832 0.657 0.591 0.416 0.293 0.186 0.167 0.208 0.361 0.450

× × × × × × × × × × ×

0 log(kmax pec /kpec) −2

10 10−1 101 103 105 107 109 1010 1010 1010 1010

4.21 4.21 4.21 4.21 4.21 4.21 4.19 4.19 4.03 2.93 2.18

k0dos/s−1 0.632 0.512 0.404 0.364 0.257 0.192 0.183 0.767 0.688 0.671 0.131

× × × × × × × × × × ×

−1 kmax dos /s −7

10 10−5 10−3 10−1 101 103 105 106 107 108 109

0.103 0.831 0.657 0.590 0.415 0.292 0.177 0.159 0.160 0.156 0.152

× × × × × × × × × × ×

0 log(kmax dos /kdos) −2

10 10−1 101 103 105 107 109 1010 1010 1010 1010

4.21 4.21 4.21 4.21 4.21 4.18 3.98 3.32 2.37 1.37 1.06

a The data in columns 2−4 are the standard rate constant, the maximum rate constant, and their ratio calculated by the pec method. The data in columns 5−7 are the standard rate constant, the maximum rate constant, and their ratio calculated by the dos method.

variation of the pre-exponential term with respect to |V|2 or Δ in the strong coupling limit. Note also that the height of the plateaus for the rate constant sum at high η still increase slightly with Δ, even though the pre-exponential term becomes independent of Δ at large Δ values. The increase of kmax comes from the contribution of the integral term in eqs 15 and 16, because the electrode-reactant electronic coupling leads to a lowering of the reaction barrier. dos Figure 3D plots the (log(kdos ox + kred)) versus η by the dos method for a range of Δ values that extend into the strong coupling limit, 10−4 eV ≥ Δ ≥ 0.2 eV. It is found that the rate constant at η = 0 V increases with increasing Δ, corresponding

to the increase of the overlap between the electronic states of the reactant and the electrode. The rate constant maximum, however, reaches very similar plateau values at high η. This behavior shows that the pre-exponential term becomes nearly independent of Δ at large Δ values. Also note that the (log(kdos ox + kdos )) plots show a rounded shape for small η values, which is red more pronounced than that observed for the pec model (see Figure 3C). This feature arises because the dos tail becomes longer and smoother as Δ increases; that is, ρr increases more slowly in the small η region. For |η| > 1 V, the reactant dos reaches the maximum overlap with the Fermi function (see eqs 20751



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Figure 4. Parameters used are λ = 0.8 eV, εF = 0 eV, T = 298 K, and log[v/k0] = 0. (A, B) Plots of cyclic voltammograms are shown for the pec method (left) and the dos method (right) at different Δ values. The dashed lines are based on Marcus kinetics. (C, D) Plots of DELF versus η are shown for the pec method (left) and the dos method (right) at different Δ values. (E, F) Plots of RTE versus η are shown for the pec method (left) and the dos method (right) at different Δ values. For all of the panels, the black line is Δ = 10−3 eV, the red line is Δ = 10−2 eV, the blue line is Δ = 0.1 eV, and the magenta line is Δ = 0.2 eV; for the k0 values, see Table 2.

remains true for the pec method irrespective of the value of Δ, however, the Re(w(z±)) tail gets longer as Δ increases for the dos method. In the extreme Lorentzian limit, this tail decays in a power law fashion 1/ ε2 for large |ε| and leads to a better overlap of Re(w(z±)) with the Fermi function for larger Δ, as compared to the overlap of exp(−βΔU‡) with the Fermi function. Consequently, the dos standard rate constant is larger than the pec rate constant in the η = 0 V region. For large |η|, the pec maximum rate constant exceeds the dos rate constant for large Δ values. When the overpotential η is very large, the overlap of Re(w(z±)) with the Fermi function attains its maximum value, and the dos rate constant reaches the same limiting values. However, the overlap of exp(−βΔU‡) with the Fermi function still increases because of a lowering of the

21 and 22), and thus, all the curves coincide with each other for large Δ. Table 2 shows a comparison of the standard rate constant and maximum rate constant calculated by the pec and dos methods for selected Δ values at λ = 0.8 eV. For Δ ranging from 10−3 to 0.2 eV in Figure 3C and D, it is found that the pec rate constant is smaller than the dos rate for η = 0 V. When η = 0 V, the rate constant is determined largely by the overlap of exp(−βΔU‡) with the Fermi function in the pec method (eqs 15 and 16), or by the overlap of Re(w(z±)) with the Fermi function in the dos method (eqs 21 and 22). When Δ tends to zero, both exp(−βΔU‡) and Re(w(z±)) acquire a Gaussian shape, and the pec and dos methods give an identical result. In this limit, the integrand drops off as exp(−ε2). This behavior 20752



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activation energy with increasing Δ, so the pec maximum rate constant increases monotonously with Δ. The kmax in Table 2 is calculated for |η| = 2.0 V. For the pec method, kmax increases with increasing Δ. For the dos method, kmax first increases and then decreases; the latter decrease is caused by the broadening of reactant density of states for large Δ values. Lastly, note that the ratio of the maximum rate constant kmax to the standard rate constant k0 for the pec method is larger than that of the dos method for large Δ values. The dependence of the logarithm of the rate constant sum with respect to the overpotential η on the reorganization energy λ is shown in Figure S1 in the Supporting Information. It is dos found that the (log(kdos ox + kred)) plots show a rounded shape for large λ values because the reactant’s dos tail gets longer and smoother as λ increases. III.c. Voltammogram. The measured current in a voltammetry experiment is directly related to the reduction and oxidation rate.33−35,37 For cyclic voltammetry of redox adsorbates in which the potential is a continuous function of time, the rate of change of the molecular fraction of oxidized adsorbate f O is given by dfO dt

= −k red(t )fO + kox(t )(1 − fO )

step potential and are calculated by either the pec method or the dos method. The quantity f target is given by ftarget =

RT dfO RT Δf ≈ Fv dt F ΔE

0 finitial = f initial

⎛ ⎡ (k + k red) ⎤⎞ 0 + (ftarget − f initial )⎜1 − exp⎢ − ox ΔE ⎥⎟ ⎣ ⎦⎠ v ⎝ (29)

This f initial now appears as for the next ΔE increment in the applied potential. Equation 27 expresses the dimensionless current inorm as a product of the potential normalization factor (RT/F(ΔE)), a term that is the change in the fractional degree of oxidation DELF = f target − f 0initial, and a term that contains the rate constant sum and the scan rate, which ranges from zero at low overpotential to one at high overpotential RTE = 1 − exp[ − (kox + kred)ΔE/v]. The normalization factor is independent of the voltage scan rate and it does not affect the shape of voltammogram. DELF and RTE can change strongly, however, and they determine the peak position, the peak current, and the peak shape. Coupling Strength Dependence. Figure 4A and B shows plots of cyclic voltammograms for intermediate to strong couplings 10−3 eV ≥ Δ ≥ 0.2 eV, from the pec method and the dos method, respectively. The numerical value of the voltage scan rate (in V·s−1) is scaled by the standard rate constant (in s−1), and the plots are given as inorm versus overpotential. When Δ is small enough (0.001 and 0.01 eV), the pec method gives the same voltammogram shape as the Marcus (LD) theory, however, the predicted peak currents are smaller than the Butler−Volmer kinetics limit (see Figure S2 in the Supporting Information). As Δ increases, the peak current decreases, the peak width broadens, and the peak potential shifts slightly. The dos method gives distinctly different results from the pec method, however. The most interesting finding is that two peaks can appear in the voltammograms for Δ = 0.01 eV. One peak is located near 0 V, and the other is shifted to higher voltage. As Δ increases further, the peak near 0 V remains at the same height and the second peak gradually disappears. A comparison of the voltammograms from the pec method and the dos method shows that the peak currents from the pec method are larger than those from the dos method and the peak widths from the dos method are broader than those from the pec method. As noted earlier, the distinct features of the pec and dos voltammograms arise from the different shape of DELF and RTE. The steeper is the increase in the rate constant sum with overpotential, the faster will be the decay of DELF for η > 0 V; whereas a higher rate constant sum or a smaller scan rate leads to a larger RTE. Insight into the origin of the different behaviors for the pec and dos voltammograms can be found by analyzing the RTE and DELF dependence independently. Figure 4C and D shows plots of DELF versus η from the pec method and the dos f 0initial

(25)

(26)

where Δf is the change in the fractional degree of oxidation during a given time interval, and ΔE is the potential increment for each time interval. Δf can be obtained by solving eq 25 using a finite difference approximation, in which the rate constants kox and kred are assumed to have fixed values at each potential step, and the current response can be found by substitution of Δf into eq 26 at each time (potential) step. The normalized current inorm is given by inorm = =

(28)

where E is the applied electrode potential. Voltammograms were calculated iteratively, starting with a value of f 0initial for the first interval calculated by the Nernst equation at the initial potential and then constantly updating it by the actual value of Δf. After a potential step ΔE is applied, the starting fraction of oxidation gets updated as,

Note that the redox species is taken to be adsorbed on the surface so that mass transport need not be considered. The time dependence of the rate constant arises from the time dependence of the applied potential. In the numerical simulations, the voltammetric scan is considered to be a series of discrete small amplitude potential steps over fixed time intervals Δt whose duration depends on the scan rate v. The current at each interval is proportional to the amount of the electroactive material that is oxidized or reduced in response to each potential step. It is conventional to present current in dimensionless form (inorm) as the fractional degree of oxidation per unit of dimensionless potential. The dimensionless current inorm is therefore written as inorm =

1 1 + exp( −EF /RT )

⎛ RT /F ⎞ 0 ⎜ ⎟(f − f initial )(1 − exp[− (kox + k red)Δt ]) ⎝ ΔE ⎠ target ⎛ RT /F ⎞ 0 ⎜ ⎟(f − f initial )(1 − exp[− (kox + k red)ΔE /v]) ⎝ ΔE ⎠ target (27)

where Δt is the time interval over which the potential is applied Δt = ΔE/v, f 0initial is the fractional degree of oxidation in a given time interval before the potential step is applied, and f target is a “target” fractional degree of oxidation calculated by using the Nernst equation at the step potential. The kox and kred are the reduction and oxidation electron transfer rate constants at the 20753



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Figure 5. Parameters used are Δ = 0.01 eV, λ = 0.8 eV, εF = 0 eV, T = 298 K, and log[v/k0] = 0. (A, B) Plots of cyclic voltammograms are shown for the pec method (left) and the dos method (right). (C, D) Plots of f target (black line), f initial (red line), and DELF (blue line) versus η are shown for the pec method (left) and the dos method (right). (E, F) Plots of RTE versus η are shown for the pec method (left) and the dos method (right).

method, which leads to a broader voltammetric wave for the dos method. Figure 4E and F shows plots of RTE versus η from the pec method and the dos method, respectively. The maximum RTE value decreases as Δ increases. For Δ ≤ 0.001 eV, the RTE values range from 0 to 1 as the overpotential (η) increases, but for Δ = 0.2 eV, the RTE maximum is reduced to about 0.53 for the pec method and to about 0.06 for the dos method, because the ratio of (kox + kred)/v (v is scaled by the standard rate constant at η = 0; see Table 2) is smaller for larger Δ values. Meanwhile, for the same Δ value, the RTE values from the dos method are smaller than that from the pec method. Thus, the dos currents are smaller than the pec current, as shown in Figure 5A and B. The most interesting finding in the voltammograms is that two peaks appear from the dos method calculation for Δ = 0.01 eV. The voltammograms obtained by the pec and dos methods

method, respectively. As described above DELF quantifies the deviation of the oxidation fraction from its equilibrium value, DELF is zero for η values less than 0 V because the system is assumed to start at equilibrium. DELF increases sharply around η = 0 V as the system shifts away from its equilibrium. As the reaction rate increases at larger η, this deviation from equilibrium decreases; the DELF term decreases monotonously to zero. The decay of DELF to zero becomes softer for larger Δ values, however. The reason is that the ratio of rate constant sum (kox + kred) to the sweep rate (v) in eq 29 decreases with an increase in the Δ value. Note that the numerical value of the sweep rate (in V·s−1) is scaled by k0 (in s−1) and the ratio of kmax to k0 decreases with the increase of Δ; see Table 2. Thus, f initial increases more slowly and DELF decreases more slowly for larger Δ values. Note also that, for the same Δ value, the DELF tail from the dos method is longer than that from the pec 20754



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Figure 6. Parameters used are Δ = 0.001 eV, εF = 0 eV, T = 298 K, and log[v/k0] = 0. (A, B) Plots of cyclic voltammograms are shown for the pec method (left) and the dos method (right) at different λ values. (C, D) Plots of DELF versus η are shown for the pec method (left) and the dos method (right) at different λ values. (E, F) Plots of RTE versus η are shown for the pec method (left) and the dos method (right) at different λ values. For all of the panels, the black line is λ = 0.2 eV, the red line is λ = 0.4 eV, the blue line is λ = 0.6 eV, the magenta line is λ = 0.8 eV, the olive line is λ = 1.0 eV, the navy line is λ = 1.4 eV, and the violet line is λ = 1.8 eV.

method increases more slowly with η than that found from the pec method. Because DELF is obtained by subtraction of f initial from f target, it displays a more gradual decrease for the dos method than for the pec method. Figure 5E and F shows plots of RTE versus η for the pec method and the dos method. The RTE term in the pec method changes from 0.10 (η = 0 V) to 1.00 (η = 2 V), whereas the RTE term from the dos method changes from 0.10 (η = 0 V) to 0.69 (η = 2 V). For the dos method, a flat region is evident at small |η|, because the rate constant (kox + kred) increases weakly around η = 0 V; see Figure 3D. Ultimately, this flatter region has its origin in the changes in shape of the dos with increasing Δ (Figure 1B).

for Δ = 0.01 eV are more carefully analyzed in Figure 5. The pec voltammogram in Figure 5A has a single oxidation peak with a potential of 0.155 V and a dimensionless peak current of 0.162. For the dos method (Figure 5B), two oxidation peaks appear. One peak occurs at 0.105 V and the other is at 0.380 V. The origin of this unusual behavior can be found by analyzing the contributions from the DELF and RTE terms. Figure 5C and D shows plots of f target (black line), f initial (red line), and DELF (blue line) versus η calculated by the two methods. f target is the “target” fractional degree of oxidation determined by the Nernst equation at the step potential. This term has the same value for the pec and dos methods. f initial is the fractional degree of oxidation in a given interval before the next potential step is applied. It is found that f initial from the dos 20755



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Figure 7. Parameters used are λ = 0.8 eV, εF = 0 eV, and T = 298 K. Panels A (Δ = 0.001 eV) and C (Δ = 0.1 eV) show the dependence of pec voltammograms on sweep rate. Panels B (Δ = 0.001 eV) and D (Δ = 0.1 eV) show the dependence of dos voltammograms on sweep rate.

The dimensionless current inorm is proportional to the product of DELF and RTE. For the dos method, the DELF term changes strongly around η = 0 V, whereas the RTE term increases more gradually around 0 V. So the first peak in the voltammogram mainly comes from the contribution of the DELF term. Because DELF reaches its maximum value and starts to decrease before the RTE term has started to increase, their product starts to decrease. The subsequent increase of the RTE term leads to an increase in the product of RTE and DELF, and a second peak in the voltammogram. For the pec method, however, the RTE term has nearly reached its maximum value as the DELF term has dropped and only one peak (albeit asymmetric) is found. From a more physical perspective, consider that the electrochemical reaction rate is proportional to the product of the molecular fraction (local reactant concentration) and the rate constant (see eq 25). The molecular fraction decreases from one to zero as the electrode potential moves to larger overpotential, and the rate constant increases from zero to the maximum value as the overpotential increases. For the pec method, the rate constant increases sharply around η = 0 V and causes a sharp decrease of molecular fraction. In this case, a single voltammetric peak results. However, for the dos method, the rate constant increases very weakly around η = 0 V for large Δ values (see Figure 3D), which leads to the molecular fraction decreasing weakly around η = 0 V. As the local concentration falls, the faradaic current decreases until the rate constant starts to increase more strongly at higher η, after which the current can rise. For a particular range of parameters, the voltammetric peak for a single electron transfer splits into two separated peaks. The first peak comes from the contribution of a large molecular fraction (large effective reactant concentration)

multiplied by a small rate constant, and the second peak comes from the contribution of a small molecular fraction multiplied by a large rate constant. Reorganization Energy Dependence. Figure 6A and B shows how the voltammograms depend on the reorganization energy for Δ = 0.001 eV, from the pec method and the dos method, respectively. In each case, the sweep rates are scaled by the standard rate constant. For the pec method, the peak current increases with λ in the range from 0.2 to 0.8 eV. For λ ≥ 0.8 eV, the peak current reaches its limiting value. The peak position shifts slightly with λ values for the pec method. In contrast, the peak current first increases and then decreases with increasing λ for the dos method. In addition, two peaks appear in the voltammogram for large λ values and the peak potential for the second peak shifts to higher η with increasing λ. It is also apparent that the dos peak width is much broader than the pec peak width. The calculated dependence of the voltammograms on the reorganization energy by the pec method agrees with the nonadiabatic Marcus (LD) treatment.33 Figure 6C and D shows plots of DELF versus η calculated by the pec and dos method, respectively. For the pec method, the DELF plots are insensitive to λ, whereas for the dos method the DELF tails become significantly longer with increasing λ. Figure 6E and F shows plots of RTE versus η from the pec method and the dos method, respectively. For λ values ranging from 0.2 to 0.6 eV, the pec and dos methods give almost identical results, with the RTE maximum increasing from 0.1 to 1.0. For λ = 0.8 eV and larger, the RTE plots from the pec method overlap with each other; however, the RTE plots from the dos method are distinctly different. The dos based RTE plots show a flat region about η = 0 and the width of the plateau increases with λ. From panels (D) and (F), one can find the reason why two peaks 20756



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Figure 8. Parameters used are λ = 0.8 eV, εF = 0 eV, and T = 298 K. (A) Dependence of dos voltammograms on sweep rate are shown for values of log[ν/k0] from −1 to 0 and Δ = 0.001 eV. (B) Dependence of dos voltammograms on sweep rate are shown for values of log[ν/k0] from 0 to 0.5 and Δ = 0.1 eV. (C) Dependence of DELF on sweep rate are shown for Δ = 0.001 eV from the dos method. (D) Dependence of DELF on sweep rate are shown for Δ = 0.1 eV from the dos method. (E) Dependence of RTE on sweep rate are shown for Δ = 0.001 eV from the dos method. (F) Dependence of RTE on sweep rate are shown for Δ = 0.1 eV from the dos method.

appear in the dos voltammogram. The first peak arises from the sharp increase of the DELF term, and the second peak comes from the sharp increase of the RTE term that occurs at high overpotential. Figure S3 in the Supporting Information shows plots of the dependence of the voltammograms on the reorganization energy from the pec method and the dos method for Δ = 0.1 eV. Sweep Rate Dependence. Figure 7 shows pec and dos voltammograms calculated for several values of log[v/k0] at Δ = 0.001 eV and Δ = 0.1 eV (Figure S4 in the Supporting Information shows pec and dos voltammograms calculated for the same values of v). For a chosen coupling strength, we see that as log[v/k0] is increased, the voltammetric peak potential moves to larger overpotentials, but in addition the waveforms

become progressively broader and the peak currents smaller. At log[v/k0] = −2, the voltammogram has an almost reversible shape, whereas at log[v/k0] = 2, the voltammetric wave becomes very irreversible. With the increase of Δ, the peak potential shifts become larger and the peak current smaller for a given log[v/k0]; note that k0 increases as Δ increases and thus v is larger for a given log[v/k0]. The peak shifts are discussed more fully below, see Figure 9. For the same log[v/k0] value, the dos peak potential shifts are larger, however the peak currents are smaller than the corresponding pec ones. From the scan rate dependence of the voltammograms obtained by the dos method, a sudden jump of the largest current, peak potential was found for log[ν/k0] from −1 to 0 in Figure 7B and for log[v/k0] from 0 to 0.5 in Figure 7D. This 20757



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Figure 9. Parameters used are λ = 0.8 eV, εF = 0 eV, and T = 298 K. (A, B) Working curves of peak potentials versus log[v/k0] are shown for the pec method (left) and the dos method (right) with 10−3 eV ≥ Δ ≥ 0.2 eV. (C, D) Working curves of peak currents versus log[v/k0] are shown for the pec method (left) and the dos method (right) with 10−3 eV ≥ Δ ≥ 0.2 eV. The dashed lines are based on Marcus kinetics, and the dotted lines are based on Butler−Volmer kinetics.

behavior is associated with the “two peaks phenomenon” found in the dos voltammograms. Figure 8 shows dos voltammograms with the same parameters used in Figure 7 but with smaller log[v/ko] ranges. As shown in Figure 8A (Δ = 0.001 eV), the peak potentials gradually shift to higher overpotential and the peak currents decrease for log[v/k0] from −1 to −0.4. As the log[v/k0] increases from −0.4 to −0.2, the peak potential shows a sharp shift and the voltammetric wave splits into two separate peaks. A further increase in the sweep rate causes the gradual disappearance of the first peak. The sudden jump of peak potential is also observed as log[v/ k0] increases from 0 to 0.5 for Δ = 0.1 eV as shown in Figure 8B. In Figure 8B, we clearly see two voltammetric peaks. The height of the first peak gradually decreases as log[v/k0] increases. When the log[v/ko] reaches a value of 0.2, the height of the second peak exceeds the first peak and this peak becomes the primary peak in the voltammograms. With a further increase of sweep rate, the first peak gradually disappears. The voltammetric peak splitting occurs in the transition region where the voltammetric peak changes from being controlled by the local reactant population (DELF control) to being controlled by the rate constant (RTE control). The distinct features of the voltammograms can be understood by analyzing DELF and RTE independently as shown in Figure 8. Note that the voltammetric wave splitting is predicted from the theoretical simulation by the dos method, however, this phenomenon has not been reported experimentally. According to the calculations, the voltammetric peak splitting appears in the case that the electrode-reactant electronic coupling strength is large enough (adiabatic limit). For an electron transfer

system at Δ = 0.001 eV and λ = 0.8 eV, the standard rate constant k0 is 7.7 × 105 s−1 by the dos method. The modeling predicts that the peak splitting can be observed if the voltage scan rate is larger than 3.9 × 105 V s−1. Current state-of-the-art ultrafast cyclic voltammetry is a few megavolts per second, and it has been used to measure electrochemical charge transfer rates of 106−107 s−1.38 Thus, the phenomenon should be accessible to experiment. If the coupling strength increases to 0.1 eV, the standard rate constant k0 increases to 6.7 × 107 s−1, and the peak splitting would be observed if the voltage scan rate is larger than 1.3 × 108 V s−1. In order to observe the peak splitting phenomenon, a fast scan voltammetry technique and a strong coupling electron transfer system are needed. III.d. Working Curve. Figure 9 shows the working curves of voltammetric peak potentials and normalized peak currents versus sweep rate at various electrode-reactant coupling strengths for the pec and dos methods. In general, the peak potentials shift to higher overpotential and the peak currents decrease with the increase of the voltage scan rate. In the weak coupling limit (Δ < 0.001 eV), the pec and dos methods give similar results to Marcus (LD) treatments (see Figure S5 in the Supporting Information).33,34 The pec and dos methods give significantly different results from the Butler−Volmer kinetics, in which the peak potentials scale linearly with the logarithm of the rate and the peak currents are independent of log[v/k0] for large η. For the pec method, the peak potential shift increases and the peak current decreases as Δ increases at the same log[v/k0] value. The dos based working curves give quite different results. In particular, the working curves of peak potential in Figure 9B show a sharp jump and the working 20758



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Figure 10. Parameters used are Δ = 0.001 eV, εF = 0 eV, and T = 298 K. (A, B) Working curves of peak potentials versus log[v/k0] are shown for the pec method (left) and the dos method (right) at different λ values. (C, D) Working curves of peak currents versus log[v/k0] are shown for the pec method (left) and the dos method (right) at different λ values.

energies for Δ = 0.1 eV are shown in Figure S6 in the Supporting Information. As shown in Figures 9 and 10, the peak splitting behavior is predicted by the dos method for large coupling strength Δ and reorganization energy λ values. Increasing both Δ and λ gives rise to a broadening of the reactant’s density of states and thus leads to the dos rate constant increasing very slowly around the equilibrium electrode potential (see Figure 3 and Supporting Information Figure S1). The rounded shape of the dos rate constant versus η is the origin of the change in shapes of DELF and RTE, which strongly influence the waveform of voltammograms (see Figures 4 and 6). Ultimately, the voltammetric peak obtained by the dos method is controlled by the DELF term for small scan rate and controlled by RTE term for large scan rate. In the transition region, log[v/k0] ranging from 0 to 1, the peak splitting phenomenon occurs.

curves of peak current do not show a monotonous decrease with sweep rate. For the peak current working curves as shown in Figure 9D, there is an abnormal region, in which the peak current increases with the sweep rate. The abnormal region gradually disappears with the increase of Δ. These interesting findings are associated with the voltammogram’s peak splitting into two separated peaks for large sweep rate. The working curves from Figure 9B and D coincide with the dependence of voltammograms on sweep rate as shown in Figure 7. Figure 10 shows the working curves of voltammetric peak potentials and normalized peak currents versus sweep rate at various reorganization energies for Δ = 0.001 eV. For the pec method, the differences between peak potentials and peak currents are minor (at a given log[v/k0]) for different reorganization energies, until a very large peak overpotential is reached. At very large log[v/k0] value, the peak potentials and the peak currents increase with the reorganization energies. For the dos method, a sharp jump is apparent in the working curves of peak potential, and larger reorganization energies give rise to greater peak potential shifts. The critical sweep rate, at which the peak potential jump occurs, increases with the reorganization energy. The working curves of peak currents from the dos method do not decrease linearly with the scan rate for large λ values. The peak current first decreases corresponding to a gradual disappearance of the first peak, and then increases corresponding to the appearance of the second peak. Note that the extent of the peak current jump increases with λ. The working curves of voltammetric peak potentials and normalized peak currents versus sweep rate at various reorganization

IV. CONCLUSIONS A comparative study of the pec and dos based descriptions of the electrochemical rate constant and the corresponding prediction for the cyclic voltammogram has been made. These two formalisms were used to examine how the voltammograms vary with the electrode-reactant coupling strength and reorganization energy and to determine the dependence of the voltammetric peak potentials and normalized peak currents on the voltage sweep rate. Although the pec and dos approaches coincide in the nonadiabatic limit, they deviate very significantly for large electronic couplings, beyond the nonadiabatic limit. When the sweep rate is scaled by the standard rate constant, the pec 20759



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(6) Mishra, A. K.; Schmickler, W. Potential energy surface for an electron transfer reaction mediated by a metal adlayer. J. Chem. Phys. 2004, 121, 1020−1028. (7) Anderson, P. W. Localized Magnetic States in Metals. Phys. Rev. 1961, 124, 41−53. (8) Newns, D. M. Self-Consistent Model of Hydrogen Chemisorption. Phys. Rev. 1969, 178, 1123−1135. (9) Miller, R. J. D.; McLendon, G. L.; Nozik, A. J.; Schmickler, W.; Willing, F. Surface Electron Transfer Process; VCH Publishers, Inc.: Weinheim, Germany, 1995. (10) Kuznetsov, A. E. Charge Transfer in Physics, Chemistry and Biology; Gordon and Beach: Reading, MA, 1995. (11) Gerischer, H. Kinetics of Oxidation-Reduction Reactions on Metals and Semiconductors. I. General Remarks on the Electron Transition between a Solid Body and a Reduction-Oxidation Electrolyte. Z. Phys. Chem. NF 1960, 26, 233−247. (12) Gerischer, H. Kinetics of Oxidation-Reduction Reactions on Metals and Semiconductors. II. Metal Electrodes. Z. Phys. Chem. NF 1960, 26, 325−338. (13) Gerischer, H. Kinetics of Reduction-Oxidation Reaction on Metals and Semiconductors. III. Semiconductor Electrodes. Z. Phys. Chem. NF 1961, 27, 48−79. (14) Miller, C. J. In Physical Electrochemistry, Principles, Methods and Applications; Rubinstein, I., Ed.; Marcel Dekker: New York, 1995; p 27. (15) Mishra, A. K.; Rangarajan, S. K. Theory of Electron-Transfer Processes via Chemisorbed Intermediates. J. Phys. Chem. 1987, 91, 3417−3425. (16) Mishra, A. K.; Rangarajan, S. K. Electron Transfer through FilmCovered Surfaces. Case of Monolayer Submonolayer - A Coherent Potential Approximation Formalism. J. Phys. Chem. 1987, 91, 3425− 3430. (17) Mishra, A. K.; Bhattacharjee, B.; Rangarajan, S. K. Electron Transfer through Monolayers: Dependence of Current on Potential and Coverage. J. Eletroanal. Chem. 1991, 318, 387−393. (18) Mishra, A. K.; Bhattacharjee, B.; Rangarajan, S. K. Theory of Electron Transfer Processes via Chemisorbed Intermediates: Part II. Current-Potential Characteristics. J. Eletroanal. Chem. 1992, 331, 801− 813. (19) Mishra, A. K.; Rangarajan, S. K. Scanning Tunnelling Microscopy Currents in the Presence of A Chemisorbate. THEOCHEM 1996, 361, 101−109. (20) Mohr, J.-H.; Schmickler, W. Exactly Solvable Quantum Model for Electrochemical Electron-Transfer Reactions. Phys. Rev. Lett. 2000, 84, 1051−1054. (21) Mohr, J.; Schmickler, W.; Badiali, J. P. A Model for Electrochemical Electron Transfer with Strong Electronic Coupling. Chem. Phys. 2006, 324, 140−147. (22) Galperin, M.; Ratner, M. A.; Nitzan, A. Molecular Transport Junctions: Vibrational Effects. J. Phys.: Condens. Matter 2007, 19, 103201(81pp). (23) Cruz, A. V. B.; Mishra, A. K. Electron Transfer Reaction through An Adsorbed Layer. J. Electroanal. Chem. 2011, 659, 50−57. (24) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications; Wiley: New York, 2000. (25) Zoski, C. G. Handbook of Electrochemistry; Elsevier Science: Oxford, 2007. (26) Darwish, N.; Eggers, P. K.; Ciampi, S.; Tong, Y.; Ye, S.; PaddonRow, M. N.; Gooding, J. J. Probing the Effect of the Solution Environment around Redox-Active Moieties Using Rigid Anthraquinone Terminated Molecular Rulers. J. Am. Chem. Soc. 2012, 134, 18401−18409. (27) Pramanik, D.; Sengupta, K.; Mukherjee, S.; Dey, S. G.; Dey, A. Self-Assembled Monolayers of Aβ peptides on Au Electrodes: An Artificial Platform for Probing the Reactivity of Redox Active Metals and Cofactors Relevant to Alzheimer’s Disease. J. Am. Chem. Soc. 2012, 134, 12180−12189. (28) Salvatore, P.; Karlsen, K. K.; Hansen, A. G.; Zhang, J.; Nichols, R. J.; Ulstrup, J. Polycation Induced Potential Dependent Structural

method predicts a lowering of the peak current and a broadening of the peak width with an increase in coupling strength Δ and an increase of peak current and narrowing of peak width with an increase in reorganization energy λ. The dos method, however, gives significantly different results from the pec method. The most interesting finding is that the dos voltammetric wave splits into two separated peaks for large coupling strength and large reorganization energy. The voltammetric peak splitting behavior is manifest in a transition region where the voltammetric peak changes from being controlled by the local concentration to being controlled by the rate constant. The working curves of peak potentials and peak currents versus voltage sweep rate are calculated by the pec and the dos method. The pec results are quite similar to the classic Marcus (LD) theory, while the dos method gives totally different results and even the peak splitting. To the best of our knowledge, such a phenomenon has not been reported experimentally; however it should be possible to observe for the right system (high enough Δ) by using modern fast scan voltammetry methods.38 Experimental observation of the peak splitting likely will require an electron transfer system for which the redox moiety is adsorbed to the electrode and diffusion is not important, as assumed in the model. Studies of this sort on a specific electron transfer system could be useful for distinguishing between the two (pec and dos) mechanistic models for describing the voltammetric behavior for an electron transfer system in the adiabatic limit.

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ASSOCIATED CONTENT

* Supporting Information S

Additional numerical results for (a) the dependence of the pec and dos parameters on λ and Δ, (b) rate constant data as a function of η, (c) voltammograms for the four models at Δ = 0.1 eV, (d) a comparison of voltammograms for Δ = 0.1 and 0.001 eV, and (e) working curves for the dependence of peak potential on sweep rate. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS D.H.W. acknowledges support from the NSF (CHE-1057981). REFERENCES

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