Solution Interface

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Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

Electron Transfer across the Graphene Electrode/Solution Interface: Interplay between Different Kinetic Regimes Renat R. Nazmutdinov,*,† Michael D. Bronshtein,† and Elizabeth Santos*,‡,§ †

Kazan National Research Technological University, Karl Marx Street 68, 420015 Kazan, Russian Federation Institute of Theoretical Chemistry, University of Ulm, D-89069 Ulm, Germany § IFEG-CONICET, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina ‡

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S Supporting Information *

ABSTRACT: We investigate current−voltage curves for describing electron transfer across the basal plane of a graphene electrode, considering a wide range of electronic coupling. Emphasis is put on the specific electronic structure of graphene. The influence of defects, which induce mid-gap states, is systematically analyzed. Also, the effect of “hot” electrons is explored in the nonadiabatic regime. We have calculated electronic transmission coefficients taking the Fc/ Fc+ couple as a model system. The results are compared with those obtained for metal electrodes. The theoretical estimations are in qualitative agreement with available experimental data.

1. INTRODUCTION There are several excellent articles describing the fundamental physics of graphene such as its electronic and magnetic properties.1,2 Most of those concerning electrochemical aspects are oriented at specific applications such as energy storage or electroanalysis. The authors of a recent review about the electrochemistry of graphene3 claim, “We feel that a significant omission ... is a thorough understanding of how the properties of graphene manifest themselves in fundamental electrochemistry... we still do not have a complete understanding of the underlying physical chemistry of graphene (e.g., how electron transfer and interfacial capacitance change...)”, and they finish the chapter ascertaining that “... further experimental and theoretical developments are certainly needed to harness the full potential of this unique material in an electrochemical context.” This report is one of the motivations for our work. Our main goal is to understand how we can exploit the unusual physical properties of graphene in fundamental electrochemistry. Electron transfer reactions in an electrochemical environment are determined by the interplay between the interaction of the reactant with the solvent and with the electronic levels of the electrode surface.4 The first process is characterized by the reorganization energy λ. The strength of the interaction between the electronic states of the reactant and electrode is determined by the coupling parameter (“chemisorption function”) Δ(ε). This quantity reflects the electronic structure of the electrode material. There are three different regimes depending on the strength of the electrode−reactant electronic coupling: (1) weak interactions, where the reaction proceeds nonadiabatically and first-order perturbation theory can be © XXXX American Chemical Society

applied; (2) medium interactions, where the reaction is adiabatic but the energy of activation is not affected by the coupling; (3) strong interactions, where the reaction barrier is lowered (electrocatalysis). Nowadays, electron transfer reactions on metallic electrodes are well understood, and they mainly proceed adiabatically. On semiconductors and graphene, electron transfer reactions can also proceed nonadiabatically. The Gerischer framework5 is one of the most popular in explaining the experimental data of electron transfer at graphene electrodes. However, until now, it is controversial whether electron transfer reactions on carbonaceous electrodes are adiabatic or nonadiabatic.6,7 The electrochemistry of graphene is a new and quickly growing area of science; many experimental data have been collected so far (see, for example, refs 8−19). Li et al.9 estimated the electron transfer rate constant for the simple redox molecule ferrocenemethanol (FcMeOH) at the mechanically exfoliated graphene electrode to be ∼0.5 cm s−1 (more than 10 times faster than at the basal plane of bulk graphite), while they obtained a value of 0.042 cm s−1 at the chemical vapor-deposited (CVD) graphene electrode. The first value is of the same order of magnitude as for metals, indicating that the outer sphere reaction would proceed adiabatically while the second value seems to be too low for this mechanism. Velický et al.13 investigated the electron transfer rate for three redox mediators, ferricyanide, hexaammineruthenium, and hexachloroiridate, at mono- and Received: March 7, 2019 Revised: April 9, 2019 Published: April 11, 2019 A

DOI: 10.1021/acs.jpcc.9b02164 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C multilayers of graphene and demonstrated that the basal plane of graphene is electrochemically active. They did not find a clear trend, since the response was strongly dominated by the local microscopic condition of the flake surface. The estimated rate constant values varied between 10−4 and 10−2 cm/s. Therefore, these processes would be considered as nonadiabatic. High-resolution electrochemical microscopy14 using [Ru(NH3)6]3+/2+ as a redox probe revealed complexities in graphite (HOPG) and exfoliated graphene as electrode materials, related to the local electronic and microscopic structure of these materials. The authors also turned to the nonadiabatic Gerischer framework to explain their results despite the extremely high value for the rate constant obtained on monoatomic steps (keff = 7 cm s−1). The first attempt to calculate the current−voltage dependencies for graphene and single-wall carbon nanotube (SWCNTs) was made by Lemay et al.18 Then, the specific graphene density of states (DOS) was addressed in model calculations of faradaic current performed by the other authors.14,16,17,19 Other challenging experimental data (current−voltage curves for different redox couples on the graphene surface) have been reported in the literature.12,16,17 The generation of “hot” electrons in a metal electrode by laser pulses can drastically increase the temperature of electrons, which can significantly accelerate the rate of interfacial redox processes proceeding in different kinetic regimes (see, for example, refs 20 and 21). The factor of “hot” electrons might be important for graphene also, both from the viewpoint of molecular electronics22 and for controlling electrochemical reactions.23 This gave us impetus to perform model calculations at two different temperatures of the electron gas of graphene and metal electrodes: Telec = 300 and 3000 K. One of the purposes of this work is to shed light on the controversy found in the literature about the electron transfer mechanism on graphene in an electrochemical environment. The behavior of graphene (both pristine and with defects) for electron transfer reactions will be compared with those of metals. We shall perform our calculations using idealized band shapes to understand the mechanism of the electron transfer reactions. We shall discuss our results by systematically analyzing the effects of different parameters, such as the electronic coupling between reactant and electrode, the applied potential, and the solvation energy. However, we shall show here the results for the reorganization energy of λ = 0.5 eV, which is close to the estimated value for both ferricyanide/ ferrocyanide24 and ferrocene/ferrocenium25 redox pairs. Results for a larger reorganization energy (λ = 1 eV) are shown in the Supporting Information. We shall follow a similar strategy as in our previous works on metallic and semiconductor materials.26−28

Figure 1. Normalized density of states (DOS) of all the model systems considered in this work: semielliptic model for a metal (red) and tight-binding model for both pristine graphene (green) and graphene with topological defects or impurities that induce mid-gap states on the density of states located near the Dirac point: −0.5 eV below the Fermi level (yellow), centered on the Fermi level (black), and +0.5 eV above the Fermi level (blue). The vertical line at zero energy indicates the position of the Fermi level.

impurities acts as doping and induces mid-gap states. We have accounted for such distortions in the structure of the DOS introducing a Lorenzian function at the Fermi level and at 0.5 eV below and above the Fermi level. All electronic bands +∞

ρ dε = 1) (for more details, were normalized to one (∫ −∞ elec see SI2). 2.2. Electronic Interactions between Reactant and Electrode. When a reactant A approaches the surface of the electrode from the bulk of the solution and an electron transfer reaction occurs, it is accompanied by changes in some intramolecular degrees of freedom and the reorganization of the solvent. According to Marcus−Hush theory,33,34 the fluctuations of the solvent shell around the reactant are characterized by the reorganization energy λ. The electronic energy of the reactant can be described by the energy level εA of the orbital of the reactant involved in the electron transfer. A key factor that describes the electronic interactions between the reactant and electrode is the coupling function Δ(ε), which depends on both the corresponding matrix elements (hopping integrals VAk = ⟨A|Hint|k⟩) and electronic density of states of the electrode, ρelec(ε) Δ(ε) = π ∑ |VAk|2 δ(ε − εk) ≈ π |Veff |2 ρelec (ε) k

(1)

We can neglect (a good approximation for most cases) the k dependence of the hopping integrals and replace it with an effective value |Veff|2. Then, the sum over k in eq 1 is reduced to the product of this constant and the density of states of the electrode (see details in SI3). As we have pointed out in the Introduction, depending on the strength of the electronic interactions between the reactant and electrode |Veff|2, an electrochemical reaction can proceed within a nonadiabatic or adiabatic regime. We shall consider first the nonadiabatic case.

2. THEORY 2.1. Electronic Structure Models Used for the Electrode Material. Figure 1 shows the idealized electronic structure of some selected electrode materials. In all cases, we shall refer the energy values to the Fermi level. Metals are modeled using a semielliptic band26,29 centered at zero and extending from −w = −6.3 eV to w = +6.3 eV (see details in SI1). The graphene density of states (DOS) was calculated in the framework of the tight binding model. It is well known30−32 that the presence of topological defects or

3. RESULTS AND DISCUSSION 3.1. Nonadiabatic Case. If the electronic interactions are weak, an electron transfer does not take place every time that B


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the electrode and reactant is given by the prefactor, which contains |Veff|2 and will be discussed later. Figure 2 shows the numerical calculations of J(ρelec,λ,η) as a function of the overpotential η, obtained using the density of

occupied (unoccupied) levels between electrode and reactant overlap. We have to consider into (from) which electronic level εk of the electrode the electron is actually transferred from (into) the reactant energy level. This changes its energy position around εA − 2λq due to the solvent shell fluctuations. Therefore, the expression for the energy of an electron transfer reaction takes the following form4 E(q , λ , ε) = λq2 + 2zλq + (εA − εk − 2λq)nA

(2)

Here, q is the normalized solvent coordinates, which describes the reaction path,4 z is the charge number (see SI4 and SI5), and nA is the electronic population of the reactant energy level. When an electron is transferred into the orbital of the reactant (reduction), it becomes occupied (nA = 1) and, in the final state, q is 0. For the reverse reaction (oxidation), it becomes unoccupied (nA = 0), and q = −1. For simplicity, we shall delete from now on the subscript k on the sub-index in εk. Because the coupling terms between the electrode and reactant are small, first-order perturbation theory can be employed, and the current−voltage dependency j(η) for the reduction of a species A+ + e− → A0 is described by the following expression35,36 jred = P

|Veff |2 ℏ

∫

Figure 2. Effect of the density of states ρelec of the electrode material on the current−overpotential dependencies for the nonadiabatic limit in the case of pristine graphene (green), metal (red), and graphene containing mid-gap electronic states at the Fermi level (black), at −0.5 eV below the Fermi level (yellow), and +0.5 eV above the Fermi level (blue). λ = 0.5 eV (see Figure S7 in SI6 for λ = 1.0 eV).

+∞

ρelec (ε)fFD (ε , Telec)Wox(ε , λ , η)dε

states for the different electrode materials ρelec of Figure 1. The absolute values of J(ρelec, λ, η) for graphene are significantly smaller than those for the metal electrode (about two orders of magnitude); this difference originates from the influence of electronic density of states ρelec solely, which are larger for the metal than for graphene. Another striking feature is that the usual near-activationless regime is attained at η of about 0.7 eV for the metal, while it is still not observed for graphene even at higher overpotentials. It can be explained by taking into account that, while the product f FDρelec for the metal is almost constant below the Fermi level, the product for the graphene increases monotonically when the electronic energy becomes more negative (see SI6 for more details). An interesting behavior is observed when the graphene contains electronic states in the mid-gap. At low overpotentials, the values of the integral are larger than those for pristine graphene. In the case of the states located around the Fermi level, these values are even similar to those of a metal. At large overpotentials, all curves converge to that of pristine graphene. In some cases, a maximum is observed, which could be interpreted as a vestige of the inverse region in Marcus theory. However, a clear explanation based on the Gerischer’s approach can be found. The contribution of the mid-gap states is stronger at low overpotentials because the overlap of the probability Wox(ε, λ, η) with these states is larger in this case (see Figure 3 and SI6). At larger overpotentials, this overlap occurs with the deeper electronic states, which are similar for all graphene materials. Therefore, a maximum appears at middle intervals. When the value of the reorganization energy λ is larger, these effects are similar, but they are shifted to higher overpotentials (see Figure S7 and discussion in SI6). The results of our model calculations for pristine graphene agree qualitatively with recent experimental findings.12,16,17 Thus, Ritzert et al.12 obtained for the reduction of [Fe(CN)6]3− from aqueous solutions at the graphene electrode a nearly linear ln k(η) overpotential dependence (where k is the

−∞

|V |2 = P eff (4πλkBT )−1/2 J(ρelec , λ , η) ℏ

(3)

where Wox(ε, λ, η) = (4πλkBT)−1/2 exp [ − (λ − ε − η)2/ (4λkBT)] and f FD(ε, Telec) = 1/[1 + exp (ε/kBTelec)]. f FD(ε,Telec) is the Fermi−Dirac distribution, that is, the probability to find an occupied state in the electrode. It is a great advantage of electrochemical systems that the driving force of interfacial reactions can be varied with the electrode potential producing shifts of the order of electron volts in the position of the electronic states participating in the reaction. Since in the simple model of electron transfer reactions of Marcus−Hush, the equilibrium conditions are reached when εA = −λ, we shall take the electrode overpotential reference in this way and write εA = −λ + η. T is the ambient temperature, and Telec is the temperature of the electrons in the solid electrode, which can be higher than T in the case of hot electrons, as we shall discuss below. P is an extra factor, which depends on the reactant work term (i.e., the Boltzmann factor that converts the bulk to the surface concentration) and on the reaction volume resulting from its integration over the distance (of the order of 10−10−10−9 m).4 According to Gerischer’s interpretation,5,37 Wox(ε,λ,η) is the normalized probability that an electron from state ε in the metal is transferred to an unoccupied (oxidized) state in the solution. It is a Gaussian function centered at ε = λ − η with a width of λ, which accounts for the fluctuations in the solvent shell, and therefore, it is independent of the electrode material. This must not be confused with the electronic density of states of the reactant. According to the first perturbation theory, which is valid in the limit |Veff| ≪ λ, these are sharply localized at a given energy: ρA = δ(ε − εA). A similar expression for the oxidation current is straightforward. Now, we shall focus on the integral J(ρelec,λ,η) of eq 3 to extract the pure effect of the electronic structure of the electrode on the current. The strength of the coupling between C


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Figure 3. (Left) Product of the Fermi−Dirac distribution with the density of states of the different electrodes (colors correspond to those in Figures 1 and 2). (Right) Gerischer’s probabilities for the reduction and oxidation species. Figure 4. Electronic transmission coefficient calculated for a Fc/Fc+ redox couple at a graphene surface for several orientations of Fc+.

reaction rate constant) in the η range from 0.2 to 0.7 V. The experimental data for the same redox couple at the graphene/ electrolyte interface were also presented by Narayanan et al. as the ln k versus η plot in the overpotential interval of 0.7 eV.16 The plot looks very similar to our current−voltage curve (compare Figure 3B in ref 16 with the green curve in Figure 2). Their experimental data cannot be fit with extant Marcus− Hush theory. Therefore, they assume a linear variation with energy of the DOS of graphene and consequently obtain an improved fit. However, the authors stress the necessity for a deeper theoretical investigation. Our model can explain these results in a consistent way. At last, a linear dependence on the overpotential of the rate constant corresponding to a Fc/Fc+ redox couple at the graphene surface in the narrow range (0.1 V ≤ η ≤ 0.4 V) was reported in a previous work.17 The experimental data clearly demonstrate that the current−voltage curves measured for the basal plane of graphene electrode are far from the activationless kinetic regime observed at metals at large overpotentials, which is in full agreement with our predictions. In the discussion above, we have ignored the prefactor to the integral in eq 3, which contains the coupling strength between the electronic states of the electrode and reactant, |Veff|. Next, we shall address these interactions using an approach involving the electronic transmission coefficient κe, which can be estimated on the basis of the Landau−Zener (LZ) theory25,35,36 κe ≈ 1 − exp −(2πγe(εj))

The κe values for the graphene were found to be noticeably smaller than those for the Au(111) surface. According to the calculated values for κe, the electron transfer across the graphene at distances larger than 5−6 Å proceeds in the nonadiabatic regime for all Fc+ orientations while, for gold in the same region, corresponds to the adiabatic limit. However, at short distances, they all converge to unity. The slope values for the κe versus x plots are ranged from 2.1 to 3.7 Å−1 for graphene and are nearly 1.2 Å−1 for the gold surface.25 As was shown earlier by Luque and Schmickler,7 starting from x ≥ 2 Å, the graphite surface reveals a stronger decay of the electronic density than the Au(111) surface. This finding elucidates the origin of a remarkable difference between the κe values for the graphene and gold surfaces. An interesting phenomenon is the generation of the socalled hot electrons by means of a laser pulse. Here, we assume that the heated electronic bath of the electrode is in a stationary state and obeys Fermi−Dirac statistics. A considerable increase of the current is observed in the overpotential range between 0 and 0.5 V (see Figure 5). Although the changes on the available electrons in the electrode due to the heating of the electrons are larger for the metal (see inset in Figure 5), the increase in the current seems to be comparatively larger for the graphene than for the metal.

(4)

where γe is the LZ factor, which is proportional to the |Veff|2 (for more details, see refs 25 and 36). We have chosen as an example the Fe/Fc+ redox couple and estimated its dependence on the separation distance of this reactant to the graphene surface following a similar strategy to our previous work on Au(111) electrodes.25 The electronic transmission coefficient was modeled using an exponential decay with the distance x κe = κo exp( −βx)

(5)

A dicircum coronene molecule (C96H24, see Figure S13A in SI8) was used to model the uncharged basal plane of graphene; details of quantum chemical calculations are also given in SI8. We investigated different orientations of the Fc molecule relative the graphene surface (see Figure S13B,C in SI8); two doubly degenerated acceptor energy levels of Fc were properly addressed. The results are plotted in Figure 4 and compared with those obtained previously for Fc/Fc+ at the Au(111) surface in the framework of the same model.25

Figure 5. Effect of the hot electrons on the current−overpotential curves for the nonadiabatic limit in the case of pristine graphene (green) and metal (red). The lines correspond to a temperature of the electron gas of Telec = 3000 K, while the symbols are the curves obtained by ambient temperature, Telec = 300 K. The reorganization energy of the solvent is λ = 0.5 eV. D


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The Journal of Physical Chemistry C 3.2. Adiabatic Case. Now, we focus on the electron transfer across the graphene/electrolyte interface in the regime of a strong electronic coupling |Veff|2 and compare the behavior with those obtained for a metal. In contrast to the nonadiabatic case, for adiabatic reactions, the reactant and electrode share their electrons. Therefore, a sharp level no longer characterizes the reactant’s electronic states. A broadening of the reactant energy level is produced, and the electronic states are described by the density of states ρ(ε). The width of this distribution is given by the coupling function Δ(ε) of eq 1. The expression for the density of states of the reactant is according to the Anderson−Newns model and our extended model26,27,29,38,39 ρA (ε , q , η) =

Δ(ε) 1 π [ε − εf (q , η) − Λ(ε)]2 + Δ(ε)2

We shall disregard the double layer and quantum capacitance effects. Indeed, if a graphene sheet is fixed at a metal electrode, the asymptotic behavior of Coulomb interaction of an external charge with the graphene surface is close to the image charge potential.40 Recently, Tao et al.41 observed a significant extension of the lattice of a charged graphene electrode. This can be addressed in our model by reducing the parameter t (see SI2). However, as test calculations have shown, the ET rate constant is practically not sensitive to this effect. An analytical expression for the integral in eq 9 does not always exist, but it can be calculated numerically. Figure 6

(6)

where εf(q,η) = −λ − 2λq − η is the acceptor energy level of the reactant. Similar to the nonadiabatic case, the electronic states of the reactant fluctuates due to the nonequilibrium response of the solvent, and its position depends on the solvent configuration. Therefore, its occupation changes with these fluctuations, and its position center εA shift according to the solvent reorganization energy λ through the effective normalized solvent coordinate q4,33,34 (for more details, see SI4 and SI5). When a potential is applied to the interface, the center also moves but in a controlled way: a positive potential to the equilibrium shifts the states of the reactant to more negative energies, and when they cross the Fermi level, electrons can be transferred to the electrode, a cathodic current flows, and the reactant is reduced (the opposite occurs for the oxidation reaction). The chemisorption function Λ(ε) produces an additional shift of the electronic states of the reactant and, in the case of very strong interactions, can also result in a splitting of peaks. This shift and the broadening are interrelated through the Hilbert transform29 (see also SI3) Λ (ε) =

1 P π

∫ εΔ−(ε′ε)′ dε′

Figure 6. One-dimensional free energy curves describing an adiabatic electrochemical reaction (A+1 + e− = A0) at equilibrium (η = 0; λ = 0.5 eV = −ε0 ; |Veff|2= 0.5 eV2) occurring on the different electrodes materials as described in Figure 1. Marcus approach (cyan curves) and the electronic contributions (dashed lines) are also shown for comparison (see Figure S8 in SI7 for λ = 1.0 eV).

shows free energy curves versus the solvent coordinate q for the different materials for specific parameters. The behavior according to Marcus−Hush,4,33,34 where the electronic interactions are absent, is also shown. In the latter case, we obtain two different parabolas for nA = 1 and nA = 0.* At the equilibrium potential (η = 0), the energy of the oxidized and reduced species is expressed as −λ,4,33,34 and the activation barrier for both oxidation and reduction reactions is λ/4.4,33,34 The electronic interaction with the electrode surface reduces the energy. This effect is larger for the metal than for the graphene materials for the same coupling constant |Veff|2 because of the higher density of electronic states of the metal. The curves are symmetric for the metal and pristine graphene, the barrier is located at q = −0.5, and the occupation at the transition state is nA = 0.5. When mid-gap states are present in the electronic structure of graphene, the shape of the energy curves depends on the position of these states relative to the Fermi level. When the center of these states is located at the Fermi level, the curves are also symmetric like for the metal and pristine graphene, and the activation barrier is lower than for pristine graphene but still larger than for the metal. When the center of the midgap states is localized below the Fermi level, the decrease in the energy of the oxidized state is larger than that of the reduced species, and the position of the barrier is shifted toward more positive values of the solvent coordinate (qbarrier > −0.5, and the occupation is nA > 0.5). The opposite effect is observed when the center of the mid-gap states is localized above the

(7)

The chemisorption function Δ(ε) can be obtained by introducing the electronic structures of the different materials in eq 1. Then, the chemisorption function Λ(ε) can be calculated from eq 7 and subsequently the density of states of the reactant using eq 6 for different coupling constants |Veff|2, solvent reorganization energies λ, and overpotentials η. Now, the electronic contribution to the total energy of the system is given by Eelec(q , η) =

EF

∫−∞ ερA (ε , q , η)dε

(8)

In practice, the bottom of the conductivity band (ca. −12 eV) is taken as the lower integration limit. Adding the terms containing the energy of solvent reorganization and interaction between the reactant and solvent, the total energy* results to Etot(λ , q , η) = Esolv (λ , q) + Eelec(λ , q , η) = λq2 + 2zqλ +

EF

∫−∞ ερA (ε , q , η)dε

(9)

This expression is valid for a given distance of the reactant to the electrode, since both |Veff|2 and λ change when the reactant approaches the surface. E


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The Journal of Physical Chemistry C Fermi level (qbarrier < −0.5, and the occupation is nA < 0.5). In both cases, the barriers are similar, lower than for pristine graphene, but larger than for the metal and the system with mid-gap states located at the Fermi level. These differences can be explained by the different contributions along the reaction coordinate q of the electronic energy Eelec, which components are also included in Figure 6. Although the absolute value of the barrier is similar for the systems with mid-gap states above and below the Fermi level, the activation energy for the oxidation and the reduction reactions are now different (see Figure 6). Therefore, different to the case when Marcus−Hush applies, near the surface, the equilibrium condition is no longer for εA = −λ. For larger λ, the potential energy curves are shifted to more negative energies, and the barriers are larger (see Figure S8 in SI7). These effects can be better understood by analyzing the density of states of the reactant. Figure 7 shows the density of states of the reacting species A at three different positions on the reaction coordinate q for the different electrode materials. Near the equilibrium of the solvent configuration for the reduced species (A0, q = 0), the center of the density of states is located below the Fermi level at εf = −λ, and its occupation is unity. Near the equilibrium of the solvent configuration for the oxidized species (A+, q = −1), the center of the density of states is located above the Fermi level at εf = +λ, and it is unoccupied. At the transition state, the center just crosses the Fermi level (εf = 0) at about q = −0.5. In the case of the metal, the electronic states of the electrode ρelec are almost constant in the energy range where the electron transfer reaction takes place, the shape of the electronic states of the reactant ρA is similar, and its width Δ is also almost constant. Therefore, the lowering of activation energy originates basically from the smoothing of the adiabatic free energy curves in the vicinity of the barrier. In the case of pristine graphene, the electronic states of the electrode ρelec increases above and below the Fermi level. Therefore, the density of states of the reactant ρA shows a very sharp peak for the transition state at the Fermi level, since at that energy, ρelec is almost zero, but these are broader for the oxidized and reduced states. Therefore, the barrier decreases due to the change of the curvature in the wells corresponding to initial and final states, while the barrier itself keeps a cusplike shape. Because for the same value of the coupling constant |Veff|2, the ρelec values for the metal are larger than for pristine graphene in the energy range where the reaction occurs, and the decrease of the barrier for the metal electrode is more significant than for graphene. The analysis is somewhat more complicated when mid-gap states are present in the electronic structure of graphene. If they are located at the Fermi level, at the transition state, the ρA shows a broader peak than for the case of pristine graphene. Depending on the strength of |Veff|2, this peak can split. Thus, the reactivity becomes closer to that of the metal, since due to the broadening, the electronic energy becomes lower. An interesting behavior is observed when the mid-gap states are localized below or above the Fermi level. In the first case when the center of the mid-gap state coincides with the center of the electronic states corresponding to the reduced species, the energy of the reduced species is strongly decreased than that of the oxidized species. In the second case, when the center of the mid-gap state coincides with the center of the

Figure 7. Density of states ρA for the reacting species at three different points of the reaction path (A+1 + e− = A0) for the different electrode materials (same colors as in Figure 1) (η = 0; λ = 0.5 eV = −ε0; |Veff|2 = 0.5 eV2). The densities of states of the electrode are also shown (violet lines).

electronic states corresponding to the oxidized species, the energy of the oxidized species is strongly decreased. In the first case, a larger broadening of the distribution of electronic states related to the reduced species (ρA) becomes evident, while in the second case, the larger broadening is observed for the oxidized species. This is the cause of the asymmetric behavior. In conclusion, the results for the free energy curve of Figure 6 can be clearly explained in terms of the electronic structure of the electrode materials. F


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The Journal of Physical Chemistry C In the case of a larger reorganization energy λ, the center of the density of states for the reduced species lies deeper, and that for the oxidized species is higher in energy. Then, the effect on the density of states due to the mid-gap states is weaker (see Figure S9 in SI7). Obviously, this effect becomes more remarkable when the coupling constant |Veff|2 is also larger (see SI7). Figure 8 shows

Figure 9. Activation energy versus overpotential for the oxidation reaction (A0 = A+1 + e−) at the different electrode materials λ = 0.5 eV = −ε0; |Veff|2 = 0.3 eV2. Graphene containing mid-gap electronic states at the Fermi level (black square), at −0.5 eV below the Fermi level (yellow square), and 0.5 eV above the Fermi level (blue square).

maximum like for the nonadiabatic regime is never observed (compare Figures 2 and 9). It is because the system is always in electronic equilibrium in the latter case. At large overpotentials, the electron transfer process occurs immediately without activation energy, since the initial state is unstable.

Figure 8. Dependence of the activation energy on the coupling constant for the different electrode materials of Figure 1 for the adiabatic case. The full symbols correspond to the reduction reaction (A+1 + e− = A0), and the open symbols correspond to the oxidation reaction (A0 = A+1 + e−) (same colors as in Figure 1). η = 0; λ = 0.5 eV = −ε0.

4. CONCLUSIONS In the framework of a quantum mechanical theory of electron transfer and resting on simple analytical models, we found intriguing features in the electron transfer kinetics for the graphene/electrolyte interface as compared with a metal electrode. These features originate basically from the particular shape of the density of states for graphene, as well as from a weaker electronic coupling. Some of our findings for nonadiabatic electron transfer are in a qualitative agreement with available experimental data. We have also explored the electron transfer process when mid-gap states are present and analyzed the effects according to their positions relative to the Fermi level. It is well known that these states appear when graphene shows topological defects or is doped with foreign atoms. In this case, an interesting effect appears: a maximum is observed in the current−overpotentials curves for the nonadiabatic regime. Since no electronic equilibrium conditions govern the reaction process, the probability of electron transfer is larger at the potentials where the overlap with the density of states of the electrode is larger also. We believe that our predictions can be partially extended to the electrochemical reactivity of graphite and carbon nanotubes. Another point that deserves attention is modeling of the electronic structure of the charged graphene (in particular, its electronic density profile) and its role in redox processes. Because of the rather flexible and robust model approach we employed, it would be tempting to investigate in future other challenging issues of the electrochemistry of graphene.

the dependence of the activation energy on the coupling constant for both oxidation and reduction reactions at η = 0 for the different electronic structures of the investigated systems. The activation energy decays with increasing interaction, and the effects are stronger for the metal than for the graphene systems. It is also clear that, for the systems containing mid-gap states below and above the Fermi level, the electrochemical equilibrium is no longer at εA = −λ due to the nonsymmetrical effect of the electronic energy on the oxidized and reduced states. The curves for the reduction in the case of mid-gap states below the Fermi level coincide with those of the oxidation in the case of mid-gap states above the Fermi level and vice versa. At larger λ, this asymmetry is less evident (see Figure S10 in SI7). 3.3. Overpotential Effects in the Adiabatic Regime. Figure 9 shows the potential dependence of the activation energy for the different electrode materials in the adiabatic limit behavior. The activationless regime for strong electronic coupling is reached noticeably earlier for the metal as compared with the graphene electrode. The presence of mid-gap states reduced considerably the barrier in comparison with pristine graphene. The asymmetry between the systems containing mid-gap states below and above the Fermi level becomes evident. Near η = 0, the barrier is slightly lower when the mid-gap states are localized below the Fermi level, as they are above it. The most favorable system is when these mid-gap states are centered at the Fermi level. When the reorganization energy is larger, the activationless regime is attained at larger overpotentials for all electrode materials, while for larger coupling constant |Veff|2, it is reached at lower overpotentials (see Figure S11 in SI7). In the case of the reduction reaction, the behavior is complementary (curves not shown). It is important to stress that, in this case, a current

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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.9b02164. Details of calculations of the metal DOS; details of calculations of the graphene DOS; chemisorption functions for the different electronic structures; interactions with the solvent; density of states of the reactant; electron transfer reactions: nonadiabatic case; electron G


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transfer reactions: adiabatic case; details of quantum chemical calculations (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (R.R.N.). *E-mail: [email protected] (E.S.). ORCID

Elizabeth Santos: 0000-0002-9417-433X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are indebted to W. Schmickler for helpful discussions. E.S. thanks CONICET for the continuous support. This work was supported by the RSF (project no. 17-13-01274).

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ADDITIONAL NOTE For a sharp electronic level (weak coupling), this equation reduces to the Marcus−Hush model: *

E(q , λ , ε) = λq2 + 2zλq +

EF

∫−∞ εδ(ε − (εA − 2λq))dε

≈ λq2 + 2zλq + (εA − 2λq)nA

.

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REFERENCES

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Solution Interface

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