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Electron Transfer Across the Graphene Electrode/Solution Interface: Interplay Between Different Kinetic Regimes Renat R. Nazmutdinov, Michael Davydovich Bronshtein, and Elizabeth Santos J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 11 Apr 2019 Downloaded from http://pubs.acs.org on April 11, 2019

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The Journal of Physical Chemistry

Electron Transfer Across the Graphene Electrode/Solution Interface: Interplay Between Different Kinetic Regimes Renat R. Nazmutdinov*1, Michael D. Bronshtein1 and Elizabeth Santos*2 1

Kazan National Research Technological University, K. Marx Str. 68, 420015 Kazan, Russian Federation 2 Institute of Theoretical Chemistry, University of Ulm, D-89069 Ulm, Germany and IFEG-CONICET, UNC, Argentina

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 non-adiabatic 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 properties1,2. 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: "... 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 surface4. The first process is characterized by the reorganization energy λ. The strength of the interaction between the electronic states of the reactant

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and of the 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 - the reaction proceeds non-adiabatically and first order perturbation theory can be applied; (2) medium interactions - the reaction is adiabatic but the energy of activation is not affected by the coupling; (3) strong interactions - 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 non-adiabatically. The Gerischer framework5 is one of the most popular to explain experimental data of electron transfer at graphene electrodes. However, until now, it is controversial, whether electron transfer reactions on carbonaceous electrodes are adiabatic or non-adiabatic6,7. 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 ten 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 graphene electrode (CVD). 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. Velicky et al.13 investigated the electron transfer rate for three redox mediators, ferricyanide, hexaammineruthenium, and hexachloroiridate at mono- and 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 to 10-2 cm/s. Therefore, these processes would be considered as non-adiabatic. 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 (k eff = 7 cm s-1). The first attempt to


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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 authors14,16,17,19 . Other challenging experimental data (current – voltage curves for different redox couple on the graphene surface) have been reported in the literature12,16,17. 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, 21. The factor of “hot” electrons might be important for graphene as well, both from the viewpoint of molecular electronics22, and for controlling electrochemical reactions23. This gave us impetus to perform model calculations at two different temperatures of the electron gas of graphene and metal electrodes, T elec = 300 K 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 behaviour of graphene (both pristine and with defects) for electron transfer reactions will be compared with that of metals. We shall perform our calculations using idealized band shapes in order to understand the mechanism of the electron transfer reactions. We shall discuss our results systematically analysing 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 ferri-/ferrocyanide24 and ferrocene/ferrocenium25 redox pairs. Results for a larger reorganization energy (λ= 1 eV) are shown in the Supporting Information (SI). We shall follow a similar strategy as in our previous works on metalic and semiconductor materials26-28.

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 modelled



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 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 de Fermi level. All electronic bands were normalized to one (

). (For more details see SI2)

Figure 1. Normalized density of states (DOS) of all the model systems considered in this work: semielliptic model for a metal (red); tight-binding model for both pristine graphene (green) and graphene with topological defects or impurities that induce midgap 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. 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 of changes on some intramolecular degrees of freedom and the reorganization of the solvent. According to Marcus-Hush theory33-34 the fluctuations of the solvent shell around the reactant are


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characterized by the reorganization energy λ. The electronic energy of the reactant can be described by the energy level ε A of the orbital of the reactant involved in the electron transfer. A key factor that describes the electronic interactions between the reactant and the electrode is the coupling function ∆(ε), which depends on both the corresponding matrix elements (hopping integrals V Ak =), and the electronic density of states of the electrode, ρ elec (ε): (1)

We can neglect (a good approximation for most cases) the k dependence of the hopping integrals and replace it by an effective value |V eff |2. Then the sum over k in equation (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 the electrode |V eff |2, an electrochemical reaction can proceed within a non-adiabatic or an adiabatic regime. We shall consider first the non-adiabatic case. 3. RESULT AND DISCUSSION 3.1 Non-adiabatic case If the electronic interactions are weak, an electron transfer does not take place every time that occupied (unoccupied) levels between electrode and reactant overlap. We have to consider into (from) which electronic level ε k of the electrode the electron is actually transferred from (into) the reactant energy level. This changes its energy position around ε A -2λq due to the solvent shell fluctuations. Therefore the expression for the energy of an electron transfer reactions takes the following form4: (2) Here, q is the normalized solvent coordinates, which describes the reaction path4, z the charge number (see SI4-SI5) and n A is the electronic population of the reactant energy level. When an electron is transferred into the orbital of the reactant (reduction), it becomes occupied (n A = 1) and in the final state q is 0. For the reverse reaction (oxidation) it becomes unoccupied (n A = 0) and q=-1. For simplicity, we shall delete from now on the subscript k on the sub index in ε k .



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Because the coupling terms between 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:

(3)

where: f FD (ε,T elec ) is the Fermi-Dirac distribution, i.e. 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 ε  = -λ, we shall take the electrode overpotential reference in this way and write ε  = -λ+η. T is the ambient temperature and T elec 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 interpretation5,37, W ox (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 centred at ε=λ−η with a width of λ,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 first perturbation theory, which is valid in the limit |V eff | -0.5 and the occupation is n A > 0.5). The opposite effect is observed when the center of the mid-gap states is localized above the Fermi level (q barrier < -0.5 and the occupation is n A < 0.5). In both cases the barriers are similar, lower than for pristine graphene, but larger than for the metal and for 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 E elec , which components are also included in the Fig.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 Fig. 6.). Therefore, different to the case when Marcus – Hush applies, near the surface the equlibrium condition is no longer for ε A = -λ.

Figure 6. One-dimensional free energy curves describing an adiabatic electrochemical reaction (A+1 + e- = A0) at equilibrium (ηλ0.5ε  ; V eff 20.5 eV2) occurring on the different electrodes materials as described in Fig.1.  approach (cyan curves), and the electronic contributions (dashed lines) are also shown for comparison. (see Fig.S8 in SI7 for λ = 1.0 eV) For larger λ, the potential energy curves are shifted to more negative energies and the barriers are larger (see Fig S8 in SI7).


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These effects can be better understood analysing 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 ε f = -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 ε f = +λ, and it is unoccupied. At the transition state, the center just crosses the Fermi level (ε f =0) at about q= -0.5. In the case of the metal, the electronic states of the electrode ρ elec are almost constant in the energy range where the electron transfer reaction takes place, the shape of the electronic states of the reactant ρ A is similar and its width ∆ 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 ρ elec increases above and below the Fermi level. Therefore, the density of states of the reactant ρ A shows a very sharp peak for the transition state at the Fermi level, since at that energy

ρ elec 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 cusp-like shape. Because for the same value of the coupling constant V eff 2 the ρ elec for the metal are larger than for pristine graphene in the energy range where the reaction occurs, 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 ρ A shows a broader peak than for the case of pristine graphene. Depending on the strength of V eff 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

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of

the

electronic

states

corresponding to the reduced species, the energy of the reduced species is stronger 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 electronic states corresponding to the oxidized species, the energy of the oxidized species is stronger decreased. In the first case, a larger broadening of the distribution of electronic states related to the reduced species (ρ A ) 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 Fig. 6 can be clearly explained ain terms of the electronic structure of the electrode materials. In the case of a larger reorganization energy λ, the center of the density of states for the reduced species lies deeper and that of the oxidized species higher in energy. Then the effect on the density of states due to the mid-gap sates is weaker (see Fig. S9 in SI7).

Figure 7. Density of states ρ A 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λ0.5ε  ; V eff 20.5 eV2). The density of states of the electrode are also shown (violet lines).

Obviously, this effect becomes more remarkable when the coupling constant |V eff |2 is also larger (see SI7). Figure 8 shows the dependence of the activation energy on the


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coupling constant for both oxidation and reduction reaction at η=0 for the different electronic structures of the investigated systems.

The activation energy decays with increasing interaction, the effects are stronger for the metal than for the graphene systems. Here is also clear that for the systems containing mid-gap states below and above the Fermi level the electrochemical equilibrium is no longer at ε A = -λ due to the non symmetrical 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 Fig.S10 in SI7).

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 to the oxidation reaction (A0= A+1 + e-) (same colors as in Figure 1). ηλ0.5ε  .

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.

Figure 9. Activation energy versus overpotential for the oxidation reaction (A0=A+1 + e-) at the different electrode materials. λ0.5ε  ; V eff 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). 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 the barrier is slightly lower when the mid-gap states are localized below the Fermi level as they are above it. The most favourable 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 V eff 2 it is reached at lower overpotentials (see Fig.S11in SI7). In the case of the reduction reaction, the behaviour is complementary (curves not shown). It is important to stress that in this case never a current-maximum like for the non-adiabatic regime never is observed (compare Fig. 2 and Fig. 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.


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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 non-adiabtic 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 as well. 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 the electronic structure of 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.

ASSOCIATED CONTENT

Supporting Information S1. Details of calculations of the metal DOS. S2. Details of calculations of the graphene DOS. S3. Chemisorption functions for the different electronic structures. S4. Interactions with the solvent. S5. Density of states of the reactant. S6. Electron transfer reactions: non-adiabatic case. S7. Electron transfer reactions: adiabatic case. S8. Details of quantum chemical calculations. This material is available free of charge via the Internet at http://pubs.acs.org



AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected] [email protected]>

Notes

The authors declare no competing financial interest.

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

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