electrolyte solution interface

Electrochimica Acta 55 (2009) 68–77

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Electron transfer across a conducting nanowire (nanotube)/electrolyte solution interface Renat R. Nazmutdinov a,∗ , Michael D. Bronshtein a , Wolfgang Schmickler b a b

Kazan State Technological University, K. Marx Str., 68, 420015 Kazan, Republic Tatarstan, Russian Federation Theoretical Chemistry Department, University of Ulm, D-86069 Ulm, Germany

a r t i c l e

i n f o

Article history: Received 11 January 2009 Received in revised form 6 August 2009 Accepted 6 August 2009 Available online 13 August 2009 Keywords: Metal nanowire Carbon nanotube Heterogeneous electron transfer Hexacyanoferrate Electrical double layer Solvent reorganization energy

a b s t r a c t The reduction of anions (considering hexacyanoferrate (III) as example) at charged conducting cylinders of nano-size, i.e., metal wires or carbon tubes, in contact with electrolyte solution is explored over a wide range of overpotentials. Two key parameters contributing to the current (solvent reorganization energy and work terms) are addressed in the framework of a quantum mechanical theory of electron transfer. It is argued that double layer effects play a crucial role and entail a significant rise of the current density as compared with plain metal electrodes. The stationary diffusion to a nanocylinder was found to proceed much faster, which results in an additional enhancement of the current. Some qualitative effects of the electron transfer across a conducting nanocylinder are discussed (in part, the appearance of an inverted Arrhenius plot). © 2009 Elsevier Ltd. All rights reserved.

1. Introduction In the last decades electrochemists have learned to design and investigate electrode structures on a nanometer scale, thus laying the foundations for an emerging electrochemical nanotechnology. In this context, metal nanowires and carbon nanotubes in contact with electrolyte solutions are a particularly interesting class of electrochemical systems, both from a theoretical and a practical point of view. Recently such interfaces have become the subject of experimental studies on the basis of electrochemical scanning probe techniques [1–11]. Thus, the adsorption of anions and organic molecules from electrolyte solutions at copper and gold nanowires was investigated by Tao et al. [1–3]. Some data on the hydrogen adsorption–desorption on arrays of silver nanowires coated with Ag–Pd alloy sheaths were reported in Ref. [5]. Electrochemical deposition of Cu, Ag and Au at single-walled carbon nanotubes (SWNT) was explored by Lemay et al. [8]. Choi and Woo examined a bimetallic Pt–Ru nanowire anode in a direct methanol fuel cell [4]. The hydrogen evolution at Cu, Ag and Au nanocontacts was studied by Kiguchi et al. [6]; the first results on the electrochemical reduction of oxygen at a SWNT surface were reported as well [7].

∗ Corresponding author at: Kazan State Technological University, Inorganic Chemistry Department, K. Marx Str., 68, 420015 Kazan, Republic Tatarstan, Russian Federation. Tel.: +7 843 545 37 75; fax: +7 843 236 57 68. E-mail address: [email protected] (R.R. Nazmutdinov). 0013-4686/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2009.08.008

Krivenko et al. observed solvated electrons electrochemically generated from carbon nanotubes [9]. Using cyclic voltammetry the authors [3] studied redox properties of hemoglobin (Hb) immobilized onto the surface of CNT; the apparent heterogeneous electron transfer rate constant was estimated. It was concluded that the direct electron transfer rate of Hb greatly enhances after its immobilization onto the surface of CNT [10]. Thus, the electrochemistry of metal nanowires and carbon nanotubes turned into to a challenging and hot area of research, which calls for theoretical research at a molecular level. The electronic structure of conducting nanocylinders (CNC) was treated earlier on the basis of an electrostatic approach [12] and a jellium model [13]. Several attempts were made to describe the structure and electronic properties of gold nanowires [14,15], and of silver and carbon single wall nanotubes (SWNTs) [16,17] using quantum chemical calculations, mostly at the density functional theory (DFT) level. DFT was also employed to study the adsorption behaviour of oxygen and CO molecules at single gold chains [18], as well as some organic species attached to monoatomic Au wires [19,20]. Note that no solvent effects were considered in the studies mentioned above. Some properties of a charged metal nanowire/electrolyte solution interface as an electrochemical system were broadly addressed for the first time by Leiva et al. [21]. The authors explored the capacitance characteristics of the diffusive part of electrical double layer (EDL), surface stress and diffusion, and compared the model predictions with those well-known for macroscopic electrodes. Lemay

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et al. [22,23] employed a quantum mechanical theory of electron transfer (ET), in order to describe the influence of specific structure of the density of electronic states of carbon SWNTs on the reaction rate. In spite of these efforts, the modeling of charge transfer across the interface between a charged nanowire or nanotube and an electrolyte solution is still in its infancy. To the best of our knowledge, no systematic studies of the key kinetic parameters, such as reorganization energy, work terms, and electronic transmission coefficient that determine the rate constant for such nano-contacts have been performed so far; no real electrode reactions were modeled. The work terms (double layer corrections) in combination with modern quantum mechanical theories of electron transfer were not addressed either, although they play a crucial role, for example, in the electrochemical reduction of anions. Model predictions on some special electrochemical kinetic regimes occurring at large overvoltages (e.g., activationless discharge) for the case of charged nanowires (nanotubes) are practically absent as well. These open questions motivate out present work. To investigate reactions at these interfaces, we could simply consider a charged sphere as a reactant without assuming a specific redox system. Nevertheless, we have chosen, the reduction of a hexacyanoferrate (III) ([Fe(CN)6 ]3− + e = [Fe(CN)6 ]4− ) as a model system for the following reasons. First, since these complex anions bear a high charge, one can expect a significant influence of electrical double layer effects. The latter are responsible for a remarkable feature observed experimentally: the current decreases with the overvoltage growth and slightly increases again starting from a certain value of the electrode potential. This is the reason why this redox pair remained for many years a touchstone for Frumkin’s slow charge transfer theory [24,25]. Secondly, as electron is transferred to bonding molecular orbital of [Fe(CN)6 ]3− , the intramolecular reorganization of the hexacyaneferrate system is negligibly small; this facilitates a model treatment of this reaction on a quantum mechanical basis. Last, this reaction has been thoroughly examined by microelectrode techniques [26–31]. The current–voltage curves reported in these works demonstrate steady-state behaviour over a wide range of the effective radii of spherical or hemi-cylindrical microelectrodes. The experimental currents vs. potential dependencies reveal a sigmoidal shape, which noticeably differs from those obtained previously for mercury or other sp-metals [24,25]. An interesting finding is that a decrease of currents with increasing overvoltage is observed only in the presence of a supporting electrolyte solution. Of course, for some electrodes the behaviour of [Fe(CN)6 ]3−/4− redox pair is rather complicated and cannot be treated in terms of outer-sphere ET. For example, a Prussian blue-like deposit was observed by Galus et al. [32] when investigating this system at a SnO2 -covered glass electrode. It should be clarified, however, from the very beginning that the molecular modeling of specific features of the reduction of a hexacyanoferrate anion at various well defined solid electrodes is out of the scope of the present work. Further we will assume the absence of reactant–electrode specific interaction, which might result in a noticeable deformation of the nearest coordination sheath of [Fe(CN)6 ]3−/4− complex species and, therefore, to the inner-sphere ET. The electroreduction of a hexacyanoferrate was explored in the framework of a quantum-mechanical theory of charge transfer in Ref. [33] for gold electrodes and in Ref. [34] for mercury. The authors [34] modeled the current–voltage curves over a wide region of overvoltages, reproduced the experimentally observable “pit” [24] and maintained an activationless nature of the [Fe(CN)6 ]3− discharge. Due to the large charge of a hexacyanoferrate anion the formation of ionic pairs is highly probable even for dilute electrolyte solutions. In this work we consider a [Fe(CN)6 ]3−/4− ·Cs+ pair as a

69

model for an inner-sphere ionic associate. For the sake of simplicity no water molecules between the anion and cation was considered. As the Cs+ cation has the largest radius in the row Li+ < Na+ < K+ < Cs+ and therefore, the smallest hydration energy, an assumed structure of the associate looks quite reasonable and resembles qualitatively that found earlier in quantum chemical modeling of the S2 O8 2− ·Cs+ ionic pair [35]. In our work a metal nanowire or nanotube or a carbon nanotube are modeled as a classical perfectly conducting cylinder. The electric double layer (EDL) properties, such as differential capacity and potential distribution), and the currents were computed for both the charged nanocylinders and for a plain metal electrode, the latter serving as a reference system. To address the effect of reactant–electrode orbital overlap, we used results of DFT calculations. This paper is organized as follows. Details of model calculations are described in Section 2, and the results are reported in Section 3. Some concluding remarks are listed in Section 4. 2. Model calculations 2.1. Equation for the current density The diabatic nature of electron transfer at the [Fe(CN)6 ]3− electroreduction at a mercury electrode has been demonstrated first in Ref. [34]. The current density1 (j) of outer-sphere ET reactions in diabatic limit can be obtained by a double integration over electronic energy levels of a metal electrode (ε) and the reactant–electrode surface distance (x) [36]:





∞

j≈



(x) exp −Wi (x; )/kT dx x0



× exp

−



∞

dε(ε)f (ε) −∞

(s − ε + Wf (x; ) − Wi (x; ) − F) 4kTs

2

 ,

(1)

where f(ε) is the Fermi-Dirac distribution function, f (ε) = 1/1 + exp{ε/kT };  is the electronic transmission coefficient; (ε) is the density of electronic states; s is the solvent reorganization energy2 ;  is electrode overvoltage; x0 is the distance of closest approach to the electrode surface. In Eq. (1) Wi(f) is the work term of reactant (i) or product (f). Since we assume no specific interaction with the electrode surface, the work terms can be estimated as follows: Wi(f ) =



i(f )

→

qm ( r m ),

(2)

m →

i(f )

where qm are the atomic charges of reactant and product; ( r m ) is the potential of the electrical double layer.3 Eq. (1) contains two contributions working in different potential scales. The electro→

static potential ( r m ) is zero at PZC, which is a property of the electrode. The overvoltage  (free energy factor) is zero at the equilibrium potential, which is a reactant’s property. Both the factors, taken in their own relative scales, are independent on the nature of electrode or reactant in the framework of the model employed. The two relative scales have to be combined explicitly, however, by using the PZC and the equilibrium potential in one common, experimental potential scale, to calculate the current of a specific electrode reaction at a specific electrode.

1 A factor co Fωeff (co is the reactant concentration and ωeff is the effective frequency factor) in Eq. (1) was neglected for simplicity. 2 Although the solvent reorganization energy also depends on x, this effect is small and was ignored. 3 See a comprehensive discussion of the microscopic psi-prim effects in Ref. [37].

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Table 1 Atomic charges (a.u) of the complex reactants in oxidized and reduced (in parentheses) states.

3−/4−

[Fe(CN)6 ] [Fe(CN)6 ]3−/4− ·Cs+

Fe

C

N

Cs

−0.087 (−0.16) 0.06 (−0.12)

0.18–0.22 (0.23) 0.12–0.15 (+0.14)

−0.68 to −0.7 (−0.87) −0.61 to −0.64 (−0.76)

0.89 (0.86)

Note that the behaviour of (ε) for nano-size contacts can be rather complicated. It was found in Ref. [38] with the help of Raman spectroscopy that the density of electronic states (DOS) of a SWNT reveals the Van-Hove singularities. The DOS of a gold nanorod was imagined by scanning near-field optical microscope [40], as well as built on the basis of DFT calculations [39]. As follows from Refs. [39,40], the DOS of Au nanowires significantly differs from that for the metal polycrystalline. Although the influence of (ε) on the diabatic current (Eq. (1)) is challenging and deserves special attention [22,23], we did not address this effect. The simplest expression for (ε) well-known from the free electron gas theory was employed when calculating the current. To describe the geometry and electronic structure of [Fe(CN)6 ]3−/4− and [Fe(CN)6 ]3−/4− ·Cs+ , quantum chemical calculations were performed at the DFT level with the hybrid functional B3LYP as implemented in the Gaussian-03 program suite [41]. A basis set of DZ quality was employed to describe the valence electrons of Fe and Cs atoms, whereas the effect of inner electrons was included in a relativistic Effective Core Potential (ECP) developed by Hay and Wadt (LanL2) [42]. We used the standard basis set 6-31g(d, p) to describe the electrons of N and C atoms. The geometry of the complex was optimized for the oxidized and reduced states with symmetry restrictions. The optimal position of Cs cation in the ionic pair was found by scanning the Fe–Cs distance in different orientations keeping fixed the optimal geometry of [Fe(CN)6 ]3−/4− (the distortions in the geometry of complexes induced by the cation were found to be quite small). The solvent effects were addressed in the framework of Polarized Continuum Model (PCM).4 For the ionic pair the modeling of solvent environment is crucial, because in gas phase the nature of its acceptor molecular orbital cannot be properly described. The open shell systems were treated in terms of unrestricted formalism. The Fe–C bond lengths were found to be 0.1985–0.1997 nm ([Fe(CN)6 ]3− ) and 0.1981 nm ([Fe(CN)6 ]4− ); a value of 0.1176 nm was obtained for the C–N bond length. The equilibrium Fe–Cs distances are 0.398 nm and 0.381 nm for the oxidized and reduced states, respectively (see the optimized structure in Fig. 5b). We have employed the ChelpG method [43] to calculate the atomic charges. The results are listed in Table 1.

where Vif is the resonance integral (one-electronic matrix element), which was computed on the basis of perturbation theory (see pertinent computational details in Refs. [34,46,47]; ω* is the effective polarization frequency of liquid water (≈1013 c−1 ) [36]. For simplicity, the calculations of Vif were performed assuming an uncharged mercury electrode surface which was taken as a reference point.5 The effect of a difference between the  values for a mercury electrode and monoatomic metal wires will be discussed in Section 3. The model (x) dependence was fitted in the form:

2.2. Solvent reorganization energy

 = k0 exp(−ˇx),

potential which can be recast in a more complicated form [45]: 2e0

(x, z, ϕ) = − 

∞  

m=−∞

∞

dkeim(ϕ−ϕ0 ) cos[k(z − z0 )]

0

Im (ka) Km (kx)Km (kx0 ), × Km (ka)

(4)

where Im and Km are the modified Bessel functions; a is the cylinder radius. ∗ is written as follows: Therefore, in this case an expression for Uim ∗ = Uim

1 1

(x0 , z0 , ϕ0 ) = − 2e0 

∞  

m=−∞

0

∞

dk

Im (ka) 2 [Km (kx0 )] . Km (ka)

(5)

The quantum modes of liquid water (which lead to a decreasing of the solvent reorganization energy) were addressed by using a factor 0.8 in Eq. (6) [36]. If the diameter of nanowires (nanotubes) has the same order of magnitude as the correlation length of some solvent modes, non-local electrostatics may be more adequate to describe the solvent dielectric response. For the sake of simplicity, however, we restricted ourselves only to local electrostatics. 2.3. Electronic transmission coefficient The electronic transmission coefficient () was treated in the framework of Landau–Zener theory [36]: ≈

 2 Vif   h ¯ ω∗

s kT

,

(6)

(7) ˇ = 22.2 nm−1 .

The solvent reorganization energy was estimated in the spirit of Marcus theory [44]:



s = e02 C

1 ∗ + Uim 2reff



,

(3)

where reff is the effective radius of reactant; the Pekar factor (C = 1/εopt − 1/εst , εopt and εst are optical and static dielectric con∗ is the image potential (normalized to stants of liquid water); Uim the electron charge e0 ) induced by reactant. ∗ = −1/4x. On the other hand, a For a plain metal electrode Uim point single charge near a conducting cylinder with coordinates (x0 , z0 , ϕ0 , where x is the radial coordinate) induces electrostatic

4 A value of 78 for static dielectric constant was assumed to describe aqueous electrolyte solutions.

where k0 = 6318.6 and Strictly speaking, one should multiply the  values by a factor of (εF )ıε* (where εF is the Fermi energy level of an electrode and ıε*6 is an effective energy level contributing to the current) to judge about the adiabaticity of electron transfer (see Ref. [34] for details). The electronic density of states (ε) was estimated

in the framework of free electron gas model7 : (ε) = 3/2εF ε/εF . According to our estimations ␬(εF )ıε*  1 both for [Fe(CN)6 ]3−

5 The jellium model was employed to describe the wave functions of a mercury electrode. The influence of a negative electrode charge on the electronic transmission coefficient (see Ref. [34] for details) was neglected. 6 A value of 0.4 eV for ıε* was obtained in Ref. [34] when modeling the reduction of a hexacyanoferate at a mercury electrode. 7 In general, (ε) should be normalized to volume, i.e. multiplied by a factor of n+ which is electronic density. The latter, however, was addressed when calculation the squared resonance integral |Vif |2 (see Eq. (6)).

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Fig. 1. Sketch of an assumed structure of the electrical double layer (EDL) at a metal electrode (a) and a conductive nanocylinder (b); 0 is the electrode potential reckoned from the electrolyte solution bulk; OHP is the potential at the outer Helmholtz plane (OHP); dH is the thickness of the EDL compact part; a is the radius of nanocylinder; ε1 and ε2 are the dielectric constants of the EDL compact and diffusive parts, respectively.

and for [Fe(CN)6 ]3− ·Cs+ , hence, the electron is transferred diabatically. The main reason explaining this result is the predominant contribution of 3d orbitals of the Fe atom to the acceptor molecular orbital (MO) of the complex reactant.8 The MO is localized mostly on the central atom and its overlap with the wave functions of the electrode is small. 2.4. Electrical double layer corrections In our work the electrical double layer at both a charged cylinder, and a plain electrode is described in terms of the classical Stern model [48] (Fig. 1). In this model the ions of electrolyte are treated as point charges, and a solvent is represented as a dielectric continuum. The electrostatic potential () inside the dense EDL part can be expressed analytically as a function of electrode charge density for the both types of electrodes:

⎧ ⎨ 0 − x (plain electrode) ε1  (x, ) = , ⎩ 0 − ln 1 + x (conducting cylinder) a

(8)

ε1

where 0 is the electrode potential with respect to the solution bulk; ε1 is the dielectric constant of the dense layer (we used a commonly adopted value of 5, which implies Coulombic screening due to the vibration modes of water molecules).9 Eq. (8) is fulfilled in the region 0 ≤ x ≤ dH , where dH is the dense layer thickness; a value of 0.3 nm (close to the radius of a water molecule) for dH was used in further calculations. The distribution of the ions in the diffusive part is described in terms of a non-linear version of the Poisson–Boltzmann equation (the Gouy–Chapman theory) [48]; we also assumed that the concentration of supporting electrolyte solution is noticeably larger than the reactant concentration. The basic Stern–Gouy–Chapman equations for a plane electrode are listed below. The dependence of electrode potential on the charge density ( ) can be written as follows:



dH RT +2 ln 0 = ε1 zF

+



4B2 + 2 2B



,

(9)



 The sum i ∈ Fe



= 2B sinh

zF(0 − dH /ε1 ) 2RT



.

(10)

The potential distribution in the diffusive region (x > dH ) is represented in the form of well-known equation: (x) =

4RT arctanh zF





tanh

zF / 4RT



exp



 d − x  H D

,

(11)

where D = RTε2 ε0 /2c(zF)2 and / is the potential at the outer Helmholtz plane (OHP). For the case of a charged conducting cylinder the following equation can be written to describe the potential distribution in the diffusive part of EDL: d2  1 d 2zFc + = sinh x dx ε2 dx2

 zF(x)  RT

.

(12)

This equation was integrated numerically for a set of different values of the cylinder radius, the surface charge density and electrolyte concentration. The mathematical analysis of some asymptotic formulas for (x) was performed, to develop an efficient and reliable computational scheme (pertinent details are given in Appendix). Previously the equilibrium structure of diffusive electrical double layer at charged rods was thoroughly investigated in a series of works using the Poisson–Boltzmann theory [49–52], hypernetted chain/mean spherical approximation (HNC/MSA) [53–56], molecular dynamics [51,55], and Monte Carlo simulations [56]. Note that the EDL effects on voltammetry at spherical microelectrodes were addressed earlier in works [57–60]. The authors employed a more extended approach based on the Butler–Volmer kinetics and Nernst–Planck transport equation. 3. Results and discussion 3.1. Solvent reorganization energy and work terms

where B = 2FTcε2 ε0 ; c is the concentration of supporting electrolyte, which is assumed to consist of anions and cations of charge

8

numbers z and −z; ε2 is the dielectric constant of diffusive layer (78 for aqueous solutions); ε0 is the dielectric constant of vacuum. One has to solve, in general, a transcendental equation, to express the electrode charge density as a function of 0 :



ci2 /

ci2 (where ci are the weight coefficients of the acceptor

i

molecular orbital) could be considered as a straightforward criterion. According to our DFT calculations this sum amounts to 0.9. 9 Although the dielectric constants of the dense layer may be different for nanosize contacts and plain metal electrodes, this difference was neglected.

The distance dependent solvent reorganization energy (see Section 2.2) calculated for a plain metal electrode and two CNCs with different radii is shown in Fig. 2. A value of 0.4 nm was used for an effective radius of the hexacyanoferrate complex. Since the image terms for conducting cylinders are smaller than for a metal electrode surface, the corresponding reorganization energy is larger. The effect, however, is not significant; for example, the difference in s values at x = 0.4 nm amounts to ca 0.1 eV for a nanocylinder with the smallest radius.

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Fig. 2. Solvent reorganization energy of the [Fe(CN)6 ]3− reduction (s ) vs. electrode surface distance (x) calculated for a plain metal electrode (solid) and two conductive nanocylinders.

Fig. 4. Work terms (Wi ,) vs. electrode potential (0 ) calculated for [Fe(CN)6 ]3− at a plain metal electrode (solid) and two conductive nanocylinders (concentration of supporting electrolyte solution is 0.01 M); the Fe–OHP distance of the closest approach was assumed (see Fig. 5).

The assumed orientations of [Fe(CN)6 ]3− and its ionic pair are shown in Fig. 3. The edge nitrogen atoms of the complex touch the OHP, hence, this orientation provides the closest approach of reactant to the electrode surface without its penetration into the dense part of electrical double layer. The chosen orientation of [Fe(CN)6 ]3− ·Cs+ is less preferable from the viewpoint of electrostatics as compared with that when the cation is close to the OPH, but favours the best overlap between the molecular orbitals of hexacyanoferrate and the wave functions of electrode. The latter is crucial for the magnitude of electronic transmission coefficient (see Section 2.3). It can be seen from Fig. 4 that the repulsive work terms of the reactant calculated for the reference system largely exceed the Wi , values obtained for the CNCs (a qualitatively similar picture was found for the [Fe(CN)6 ]3− ·Cs+ ionic pair). This should entail, in turn, a noticeable increase of the reduction of anions going from a plain metal electrode to conducting nanocylinders. 3.2. Model current–voltage curves After the work terms and solvent reorganization energies were obtained, we calculated the currents according to Eq. (1). To avoid unnecessary details and keep some generality, we considered 0 = 0 (nhe) as the PZC and did not take into account a possible difference in this potential for a plain electrode and a CNC (see discussion at the end of this section). Note that our “model” PZC value is close (in sense of a large overvoltage region) to the experimental potentials of zero charge in the nhe scale for mercury and Au(110) electrodes. A value of 0.36 V (nhe) was used as the standard redox potential of [Fe(CN)6 ]3− /[Fe(CN)6 ]4− pair, in order to match the work terms with the overvoltage scale. Eqs. (9) and (10), as well as those presented in Appendix (A8, A9) were used to recalculate the surface charge scale to the potential scale.

Fig. 5. Model current–voltage curves describing the electroreduction of [Fe(CN)6 ]3− (a) and [Fe(CN)6 ]3− Cs+ (b) calculated for a mercury electrode (1) and two conductive nanocylinders: a = 0.4 nm (2) and a = 0.2 nm (3); concentration of supporting electrolyte solution is 0.01 M.

The model current–voltage curves shown in Fig. 5 reveal a common feature: the currents decrease at the overpotential rises (due to the increase of electrostatic repulsion between the anion and negatively charged electrode surface). No minima are observed, however, as compared with the j vs.  dependencies measured for the [Fe(CN)6 ]3− reduction at a mercury electrode. To address the possible origin of the voltammogram “pits”, one needs to include in the model some additional effects (for example, adsorption of cations, dependence of the electronic transmission coefficient on the electrode charge etc.; see a pertinent discussion in Ref. [34]). The dip of curves decreases going from [Fe(CN)6 ]3− complex to its

Fig. 3. Assumed orientations of [Fe(CN)6 ]3− (a) and [Fe(CN)6 ]3− ·Cs+ (b) relative to the OHP.

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Fig. 6. Ratio of model current–voltage curves (j and jn relate to a plain metal electrode and conducting nanocylinders, respectively; a = 0.2 nm – solid; a = 0.4 nm – dot) describing the Fe(CN)6 ]3− (a) and [Fe(CN)6 ]3− Cs+ (b) reduction; concentration of supporting electrolyte solution is 0.01 M.

ionic pair. The shape of the model current–voltage curves depends on both the CNC radius a, and the reactant charge. For example, [Fe(CN)6 ]3− ·Cs+ ionic pair yields at a = 0.2 nm the lg j() dependency of sigmoidal-like form; this warrants somewhat considering the ionic pairs. Another remarkable feature of the model current–voltage curves can be observed from Fig. 6. The ratio of currents vs. overvoltage is non-monotonous and displays a maximum at  = 0.9 – 1 V. This effect cannot be ascribed solely to the double layer corrections; it originates from the interplay between the work terms Wf − Wi in the quadratic part of the Eq. (1) for the current density. Fig. 7 is helpful to understand how sensitive is the acceleration effect to the radius of a conducting nanocylinder and concentration of supporting electrolyte solution. A considerable growth of the rate of [Fe(CN)6 ]3− reduction in diluted solutions (ca 1–3 orders in magnitude) is observed at a ≤ 1 nm, while starting from a > 2 nm the effect practically disappears. The dependence of jn /j ratio on the electrolyte concentration was found to be very strong as well (see Fig. 7). Electrochemists frequently prefer to discuss the current in terms of the reaction volume ıx (see Eq. (1)): j() = j(x = xo ; )ıx ().

(13)

We found that for the reduction of hexacyanoferrate (III) anion the reaction volume is not too sensitive to the electrode size. Going from a plain metal electrode to a conducting cylinder with the smallest radius this quantity changes from 0.14–0.35 nm to 0.14–0.23 nm in the interval of overvoltages shown in Fig. 5. This

Fig. 7. Ratio of currents (j relates to a plain electrode, jn refers to a conductive nanocylinder) vs. the cylinder radius (a) calculated for two different supporting electrolyte concentrations (c); the reduction of [Fe(CN)6 ]3− at the overvoltage, which gives a maximal value of the jn /j ratio is assumed (see Fig. 6).

prediction remains the same for the ionic pair and practically does not depend on the supporting electrolyte concentration. We also made an attempt to explore the influence of temperature on the electroreduction currents. Besides varying of all kT terms in the equations for the current density and electrical double layer potential, the effect of temperature on the dielectric constant of liquid water in the diffusive layer (ε2 ) was taken into account as well (we used experimental data from Ref. [61]). At the same time a possible temperature dependence of the thickness of dense layer and the lower dielectric constant ε1 was ignored. For the reference system the Arrhenius plot has the expected shape (Fig. 8a), while the same plot built for a nanocylinder looks more puzzling and demonstrates an inverted temperature dependence (Fig. 8b). This conspicuous feature results mostly from an increase of the repulsive double layer potential with rising temperature. For CNCs with small radius this effect is stronger as compared with a plain electrode and cannot be compensated by increasing the currents due to the kT term in Eq. (1). As shows our analysis, the inverted Arrhenius plot disappears at a > 1 nm. So far we did not discuss a difference between the electronic transmission coefficient () values for plain metal electrodes and conducting nanocylinders (see Section 2.3). In diabatic limit this important quantity is proportional to the product |Vif (x = x0 )|2 (εF ), so that the resulting  value stems from a complex interplay of three different factors. First, the resonance integral Vif depends on the electronic density of an electrode. According to our preliminary periodical DFT calculations performed recently the electronic “tail” of monoatomic Cu, Ag, Au and Pt wires is slightly more spread as compared to the corresponding metal surfaces (the later were modeled as slabs) [62]. Secondly, the distance of the reactant closest approach x0 should be apparently smaller for the CNCs than for plain electrodes (mainly due to solvation contribution to the work terms). And last, the total density of electronic states at the Fermi level (εF ) was calculated to be noticeably smaller for the monoatomic metal wires [62]. The first and second contributions favour the reactant–electrode orbital overlap, i.e. the increasing of transmission coefficient, while the last factor does not. These competitive effects of different nature are expected to smooth possible difference between the  values for a metal nanowire and a plain electrode. The main problem which decreases significantly the accuracy of model  values is the uncertainty of reaction layer structure at a solid/solution interface (possible bridge-assisted ET, solvent screening, electrode charge effects [34,47] etc.). By this reason in the recent works on the model estimations of  for certain interfacial electrochemical systems the authors were focused mostly on some qualitative aspects of  behaviour [63–65]. For heterogeneous ET the resonance integral Vif as a function of x can be estimated from experiments where redox centers are separated from the electrode surface by self-assembled monolayers with variable thickness (see,

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R.R. Nazmutdinov et al. / Electrochimica Acta 55 (2009) 68–77

Fig. 8. Model ln j vs. 1/kT dependencies calculated for a mercury electrode (a) and a conducting nanocylinder, a = 0.2 nm (b); concentration of supporting electrolyte solution is 0.01 M.

e.g., Refs. [66,67]). These results predict a stronger coupling at the distances of closest approach (in comparison with our conclusions, see Eq. (7)), which leads to the significantly adiabatic character of ET.10 For example, the following Vif (x) dependence was found in Ref. [66] for ruthenium and ferrocene redox moieties attached to gold through alkanethiol monolayers, Vif (x) = V0 exp(−ˇx), where V0 = 0.76 eV and ˇ = 0.53 Å−1 . Adiabatic electron transfer at the [Fe(CN)6 ]3− reduction at Au and Pt electrodes was also argued in works [33,68], where the authors have reported the solvent friction effect on the reaction rate. The adiabatic ET at the reduction of hexacyanoferrate (III) anion cannot be, therefore, excluded for some electrochemical systems. That is why we made an attempt to address this kinetic regime for the reaction under study, in order to keep the general character of our approach. If the effect of resonance integral on the shape of reaction free energy surface is neglected, the current density (j) of outer-sphere ET reactions in adiabatic limit can be written in the form [36]:

 W (x ; ) + E∗  i 0 a

j ≈ exp −

kT

,

ference in the zero charge potentials (E =0 ) between electrodes of macro- and nano-size can be considered as one of the possible reasons. If the influence of solvent molecules on the behaviour of electronic tail of a metal is neglected, the following relation can be written for electrodes 1 and 2 (see Ref. [24]): 1 2 E =0 − E =0 ≈ We1 − We2 + 1solv − 2solv ,

(16)

where We is the “vacuum” work function of electron; solv is the potential drop induced by adsorption of solvent molecules at the electrode surface.

(14)

where the Franck–Condon barrier Ea∗ is defined as follows:

Ea∗ =

(s + Wf (x0 ; ) − Wi (x0 ; ) − F) 4s

2

(15)

(see Eq. (1) and notations therein). If s + Wf (x0 ;)–Wi (x0 ;) ≤ F, then Ea∗ = 0 (to avoid the inverted regime). Eqs. (14) and (15) are written for the distance of closest approach to the electrode surface; integration over x (the reaction volume factor) was ignored for simplicity. The results of model calculations performed for a plain electrode and conducting nanocylinders by using Eqs. (1) and (14) (a diluted electrolyte solution was assumed, c = 0.01 M) are shown in Fig. 9. It can be seen that the shape of current–voltage curves and their ratios look very similar going from diabatic to adiabatic limit. The ratios of model current–voltage curves calculated for a more concentrated supporting electrolyte solution are presented in Fig. 10. One can observe some differences in the “acceleration” effect for adiabatic and diabatic ET at large electrode overvoltages. These differences become, however, smaller when increasing the CNC radius. Thus, the adiabatic character of electron transfer for the electrochemical [Fe(CN)6 ]3− reduction does not change noticeably our main predictions made above and resulting basically from the electrical double layer effects. Note that the acceleration effects discussed above (see Figs. 5–7) are expected to be smaller in real electrochemical systems. A dif-

10 Note that our (x) dependence (see Eq. (7)) is steeper as compared with those reported in the literature [33,63,66,67].

Fig. 9. Model current–voltage curves describing the electroreduction of [Fe(CN)6 ]3− in diabatic (a) and adiabatic (b) limit calculated for a plain electrode (solid) and conducting nanocylinders: a = 0.8 nm (dash), 0.4 nm (dot) and 0.2 nm (dash–dot); concentration of supporting electrolyte solution is 0.01 M.

R.R. Nazmutdinov et al. / Electrochimica Acta 55 (2009) 68–77

75

Because of the x in the denominator, the total current flowing through any cylinder with radius r and concentric with the cylinder axis is independent of the radius; this is a consequence of particle conservation. In particular, the current density on the surface of the cylinder is: cyl

cyl

jd (r) = nFD

C0 − Cs . r ln(ıN /r)

(19)

For given values of the bulk and surface concentrations C0 , Cs cyl cyl and Cs (and assuming that Cs ≈ Cs ) the enhancement of mass transport toward the CNC () is =

Fig. 10. Ratio of model current–voltage curves (j and jn relate to a plain metal electrode and conducting nanocylinders with two different radius a values, respectively) describing the Fe(CN)6 ]3− reduction in diabatic (solid) and adiabatic (dot) limits; concentration of supporting electrolyte solution is 0.1 M.

To illustrate a limiting case, we employ Eq. (16) for a comparative analysis of a mercury electrode vs. monoatomic silver wire. The electronic work function of liquid mercury is well-known (4.5 eV [61]); a calculated value of 5.41 eV was reported in Ref. [21] for a monoatomic Ag wire. The corresponding solv values (0.44 V (Hg) and 0.3 V (monoatomic Ag wire) were estimated earlier on the basis of molecular dynamics simulations [69,70]. Taking into account the above mentioned data, a significantly more positive E =0 value (ca 0.77 V larger) can be predicted for the silver wire as compared with a mercury electrode. Therefore, at a given overpotential value the repulsive work terms for the nanowire should exceed those for mercury. This effect reduces the acceleration of reduction of anions at CNCs but, of course, it diminishes with increasing of the radius of nanowire(-tube). 3.3. Analysis of mass transfer So far we have focused solely on electron transfer. Since mass transfer is also an important step of electrochemical processes, we consider briefly a few specific features of the diffusion of particles near a CNC as compared with a plain electrode. The case of timedependent diffusion without convection has been considered by Leiva et al. [21] on the basis of model calculations. In the recent experimental work by Lemay et al. [71] stationary concentration gradients were achieved and transport properties were probed on the scale of several nanometers using electrochemical measurements. Therefore, we here consider the stationary state with a constant thickness ıN of the diffusion layer, which can be achieved by letting the cylinder – or the cell – rotate with a constant angular velocity (see, e.g. Ref. [72]). Let C0 and Cs be the concentrations of the diffusing species in the bulk and at the surface of the nanocylinder, respectively, and r the effective radius of the cylinder (for example, a + dH ). The concentration C(x) as a function of the radial distance x from the wire is then: cyl

C(x) = Cs

cyl

+

C0 − Cs ln ln(ıN /r)

x r

cyl

For example, for a thickness of 103 nm of the diffusion layer [73] and a radius of 1 nm we obtain an enhancement factor of 145; qualitatively similar conclusions were made previously by Leiva et al. [21]. Note that a challenging non-linear dependence of the current on the concentration of electroactive specie was reported by the authors [71], which might question the simple continual approaches and calls for a more advanced microscopic consideration. 3.4. Comparison with experiment We have got apparent problems with comparison of our results with experiment, because experimental data on electrochemistry of anions in the vicinity of charged nanowires (nanotubes) are practically absent. Very recently a ferrocyanide/ferricyanide redox pair was probed at SWCNTs on edge plane pyrolytic graphite electrodes (EPPGE), as well as at SWCNTs decorated with Ni nanoparticles by using cyclic voltammetry and electrochemical impedance spectroscopy [74]. The authors [74] maintained that the charge transport proceeds ca 12 times faster going from the bare EPPGE to the EPPGE-SWCNT which agrees qualitatively with our main prediction. We also made an attempt to use some results obtained earlier using microelectrode technique. Morris et al. [27] have reported experimental data on the voltammetric response of the K4 [Fe(CN)6 ] oxidation at Pt band microelectrodes in 0.1 M KCl solution. It is curious to note that the ratio lg{jn (a = 2 nm)/j} extracted from these data11 amounts to 0.7, which is comparable with the effect observed for our system (see Fig. 7). As can be seen from Fig. 8, the dip of curves decreases at decreasing the cylinder radius and this agrees qualitatively with the experimental data reported recently by Chen and Kucernak [31]. The steady-state cyclic voltammograms obtained by the authors for the reduction of hexacyanoferrate (III) anion at nanometric-sized carbon electrodes also display a decrease of the dip of current–voltage curves going to the microelectrodes of smaller size. It should be noted that the shape of model curves depend on the reactant charge and the radius of cylinder (see Fig. 8). For the [Fe(CN)6 ]3− ·Cs+ ionic pair the lgj() dependencies slightly resemble sigmoidal voltammograms obtained in experiments with microelectrodes [27,31].

(17)

cyl

C0 − Cs

 ,

x ln

(20)

4. Conclusions ,

and the corresponding current density is jd (x) = nFD∇ x C(x) = nFD

ıN . r ln(ıN /r)

ıN r

(18)

where D is the diffusion coefficient, n is the number of electrons transferred and F is the Faraday constant.

In this work we explored the reduction of hexacyanoferrate (III) anion at charged conducting nanocylinders which serve as model systems for metal wires or carbon tubes. Main attention was focused on fundamental problems, because it is still difficult to predict some practical applications which probably might be expected

11

The microelectrodes were treated in Ref. [27] as hemicylinders.

76

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for certain anionic redox pairs in the presence of an array of charged nano-tubes(wires). The key parameters contributing to the current were addressed in the framework of quantum mechanical theory. It is argued that the double layer effects play the most important role and entail a significant growth of the current density as compared to plain metal electrodes. We have predicted some qualitatively interesting features of the electron transfer across a CNC: non-monotonous behaviour of the ratio of currents (Fig. 6) and the inverted Arrhenius plot (Fig. 8b). Therefore, the charged nanowires (nanotubes) in contact with an electrolyte solution may offer new possibilities to control electrochemical reactions. The effects mentioned above provide a new insight into the charge transfer at the nanoscale and certainly call for experimental verification. Bearing in mind a fast progress in this research area we believe that pertinent experimental studies will be performed in the nearest future. Although some simplifications made in this work are apparently crude (the solvent is treated as continuum and the ions of supporting electrolyte solution are represented as point charges), the model approach is physically transparent; being rather flexible it can be readily extended to other more complicated redox pairs (for example, to bond breaking electron transfer). In particular, we have performed calculations for another model system, the electroreduction of a peroxodisulphate anion. The results demonstrate clearly a “pit” on the current–voltage curves and its gradual disappearance going to charged nanocylinders with small radii. The reduction of cations proceeding in a wide overvoltage region (e.g., Eu(III)) can be considered as another potentially interesting system. Opposite to anions, “inhibition” effects may be predicted for such redox pairs at a negatively charged CNC (in comparison with plain electrodes). The modeling of redox processes at charged semi-conducting carbon nanotubes should be mentioned as a challenging area of future research. It was already mentioned in Section 3.2 that reliable calculations of the electronic transmission coefficient remain an important problem for both nano-size electrodes and plain metal surfaces. It would also be tempting to combine DFT calculations with molecular dynamics, in order to get a deeper insight into the nature of specific interactions of reactants with the charged nanoelectrodes, as well as into environmental effects at a proper molecular level. In case of a nanowire with a diameter comparable to the size of molecules and ions, it seems to be of crucial importance to address the molecular structure of water. Some recent results of classical molecular dynamics simulations of a monoatomic silver nanowire in contact with electrolyte solution look very promising [70].

Acknowledgements This work was supported in part by the Deutsche Forschungsgemeinschaft and Russian Foundation for Basis Research (project No. 08-03-00769-a). We thank Dr. Tamara Zinkicheva, Dr. Dmitrii Glukhov, Dr. Sergei Vassiliev, and Dr. Qijin Chi for useful remarks and technical assistance.

Appendix A. It is easy to show that |(x)| < |* (x)|, where * is the potential of the EDL diffusive part for a plain electrode. Note that zF(x)/RT < 0.005 at x > D (3 + ln c* ), ∗

where c =

+ +



4B2 + 2 − 2B 4B2 + 2 + 2B



exp dH /D



and B is defined in Eq. (9) (see the article text).

(A1)

One can write for an assumed value for the solution error O(x) ≈ 0.005 (i.e. ≈0.5%):



sinh zF(x)/RT



= (1 + O(x))zF(x)/RT

(A2)

Therefore, at x > D (3 + ln c* ) zF ˜ 0 (x) = mK RT

 x  D

(1 + O(x)),

(A3)

˜ is a special parameter (see below). where m To find (x) at x ∈ [a + dH , D (3 + ln c* )], one needs to solve numerically the Cauchy problem: 2zFc 1 d d2  = + sinh x dx ε2 dx2

 zF(x) 

,

(A4)

zF ˜ 0 (3 + ln c ∗ ), (D (3 + ln c ∗ )) = mK RT

(A5)

and



zF ∂(x)  RT ∂x 

˜ = −m x=D

(3+ln c ∗ )

RT

1 K1 (3 + ln c ∗ ), D

(A6)

It should be stressed that the integration must be fulfilled going from x > D (3 + ln c* ) to x = a + dH , i.e. from the solution bulk to the electrode surface, otherwise the computational procedure is not stable and the solution tends to infinity. According to our estimation:



˜ ∈ m

0,

zF RT

∗ (D (3 + ln c ∗ )) K0 (3 + ln c ∗ )



.

(A7)

˜ value. Then Eqs. This condition is used to choose an initial m (A4)–(A6) are integrated numerically by using a dichotomy procedure to improve the initial estimate. Two possible boundary problems are considered. For a given value of the surface charge density on a nanocylinder ( ) we have



a d  . =  ε2 ε0 (dH + a) dx dH +a

(A8)

On the other hand, for a given value of the surface potential of a nanocylinder (0 ): −



ε2 (dH + a) d  ln{(dH + a)/a} + (dH + a) = 0 .  ε1 dx dH +a

(A9)

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

C.Z. Li, H.X. He, A. Bogozi, J.S. Bunch, N.J. Tao, Appl. Phys. Lett. 76 (2000) 1333. H.X. He, C. Shu, C.Z. Li, N.J. Tao, J. Electroanal. Chem. 522 (2002) 26. B. Xu, H. He, S. Boussaad, N.J. Tao, Electrochim. Acta 48 (2003) 3085. W.C. Choi, S.T. Woo, J. Power Sources 124 (2003) 420. Y. Sun, Z. Tao, J. Chen, T. Herricks, Y. Xia, J. Am. Chem. Soc. 126 (2004) 5940. M. Kiguchi, T. Konishu, S. Miura, K. Murakoshi, Nanotechnology 18 (2007) 4240. F. Wang, S. Su, J. Electroanal. Chem. 580 (2005) 68. B.M. Quinn, C. Dekker, S.G. Lemay, J. Am. Chem. Soc. 127 (2005) 6146. A.G. Krivenko, N.S. Komarova, N.P. Piven, Electrochem. Commun. 9 (2007) 2364. C. Cai, J. Chen, Anal. Biochem. 325 (2004) 285. M. Yang, F. Qu, Y. Li, Y. He, G. Shen, R. Yu, Biosens. Bioelectron. 23 (2007) 414. M. Krˇcmar, W.M. Saslow, A. Zangwill, J. Appl. Phys. 93 (2003) 3495. C. Yannouleas, U. Landman, J. Phys. Chem. B 101 (1997) 5780. A. Hasegawa, K. Yoshizawa, K. Hirao, Chem. Phys. Lett. 345 (2001) 367. H. Häkkinen, R.N. Barnett, A.G. Scherbakov, U. Landman, J. Phys. Chem. B 104 (2000) 9063. C.T. White, J.W. Mintmire, J. Phys. Chem. B 109 (2005) 52. S.L. Elisondo, J.W. Mintmire, Phys. Rev. B 73 (2006) 1. S.R. Bahn, N. Lopez, J.K. Nørskov, K.W. Jacobsen, Phys. Rev. B 66 (2002) 081405. H. Häkkinen, R.N. Barnett, U. Landman, J. Phys. Chem. B 103 (1999) 8814. P. Vélez, S.A. Dassie, E.P.M. Leiva, Phys. Rev. Lett. 95 (2005) 045503. E.P.M. Leiva, C.G. Sánchez, P. Vélez, W. Schmickler, Phys. Rev. B 74 (2006) 035422. I. Heller, J. Kong, H.A. Heering, K.A. Williams, S.G. Lemay, C. Dekker, Nanoletters 5 (2005) 137. I. Heller, J. Kong, K.A. Williams, C. Dekker, S.G. Lemay, J. Am. Chem. Soc. 128 (2006) 7353. A.N. Frumkin, Potentials of Zero Charge, Moscow, Nauka, 1979.

R.R. Nazmutdinov et al. / Electrochimica Acta 55 (2009) 68–77 [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48]

J. Nerut, P. Möller, E. Lust, Electrochim. Acta 49 (2004) 1597. L.M. Peter, W. Dürr, P. Binda, H. Gerisher, J. Electroanal. Chem. 71 (1976) 31. R.B. Morris, D.J. Franta, H.S. White, J. Phys. Chem. 91 (1987) 3559. M.B. Rooney, D.C. Coomber, A.M. Bond, Anal. Chem. 72 (2000) 3486. C. Lee, F.C. Anson, J. Electroanal. Chem. 323 (1992) 381. C. Beriet, D. Pletcher, J. Electroanal. Chem. 361 (1993) 93. S. Chen, A. Kucernak, J. Phys. Chem. B 106 (2002) 9396. J. Kawiak, P.J. Kulesza, Z. Galus, J. Electroanal. Chem. 226 (1987) 305. D.E. Khoshtariya, T.D. Dolidze, L.D. Zusman, D.H. Waldek, J. Phys. Chem. A 105 (2001) 1818. R.R. Nazmutdinov, D.V. Glukhov, G.A. Tsirlina, O.A. Petrii, Russ. J. Electrochem. 39 (2003) 105. R.R. Nazmutdinov, D.V. Glukhov, G.A. Tsirlina, O.A. Petrii, J. Electroanal. Chem. 582 (2005) 118. A.M. Kuznetsov, Charge Transfer in Physics, Chemistry and Biology, Gordon, 1995. G.A. Tsirlina, O.A. Petrii, R.R. Nazmutdinov, D.V. Glukhov, Russ. J. Electrochem. 38 (2002) 132. M.S. Dresselhaus, G. Dresselhaus, A. Jorio, A.G. Souza Filho, M.A. Pimenta, R. Saito, Acc. Chem. Res. 35 (2002) 1070. B. Wang, S. Yin, G. Wang, A. Buldum, J. Zhao, Phys. Rev. Lett. 86 (2001) 2046. K. Imura, T. Nagahara, H. Okamoto, J. Chem. Phys. 122 (2005) 154701. M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria et al., Gaussian 03, Revision B.04. P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 270. C.M. Breneman, K.W. Wiberg, J. Comp. Chem. 11 (1990) 361. R.A. Marcus, J. Chem. Phys. 38 (1963) 1858. M. Zamkov, H.S. Chakraborty, A. Habib, N. Woody, U. Thumm, P. Richard, Phys. Rev. B 70 (2004) 115419. R.R. Nazmutdinov, W. Schmickler, A.M. Kuznetsov, Chem. Phys. 310 (2005) 257. R.R. Nazmutdinov, D.V. Glukhov, O.A. Petrii, G.A. Tsirlina, G.N. Botukhova, J. Electroanal. Chem. 552 (2003) 261. P. Delahay, Double Layer and Electrode Kinetics, Interscience, New York, 1965.

[49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66]

[67] [68] [69] [70] [71] [72] [73] [74]

77

M. Guéron, G. Weisbuch, Biopolymers 19 (1980) 353. M. Le Bret, B.H. Zimm, Biopolymers 23 (1984) 287. M. Deserno, C. Holm, S. May, Macromolecules 33 (2000) 199. K. Bohinc, A. Igliˇc, T. Slivnik, V. Kralj-Igliˇc, Bioelectrochemistry 57 (2002) 73. E. Gonzales, M. Lozada-Cassou, D. Henderson, J. Chem. Phys. 83 (1985) 361. V. Vlachy, D.A. McQuarrie, J. Chem. Phys. 83 (1985) 1928. M. Deserno, F. Jimínez-Ángeles, C. Holm, M. Lozada-Cassou, J. Phys. Chem. B 105 (2001) 10983. F. Jimínez-Ángeles, G. Odriozola, M. Lozada-Cassou, J. Chem. Phys. 124 (2006) 134902. D. Norton, H.S. White, S.W. Feldberg, J. Phys. Chem. 94 (1990) 6772. W.D. Murphy, J.A. Manzanares, S. Mafé, H. Reiss, J. Phys. Chem. 96 (1992) 9983. C.P. Smith, H.S. White, Anal. Chem. 65 (1993) 3343. X. Yang, G. Zhang, Nanotechnology 18 (2007) 335201. Handbook of Chemistry and Physics, CRC Press, Inc., Boca Raton, FL, 1989. R.R Nazmutdinov et al., in preparation. C.-P. Hsu, R.A. Marcus, J. Chem. Phys. 106 (1997) 584. S. Gosavi, R.A. Marcus, J. Phys. Chem. 104 (2000) 2067. R.R. Nazmutdinov, N.V. Roznyatovskaya, D.V. Glukhov, I.R. Manyurov, V.M. Mazin, G.A. Tsirlina, M. Probst, Inorg. Chem. 47 (2008) 6659. J.F. Smalley, H.O. Finklea, C.E.D. Chidsey, M.R. Linford, S.E. Creager, J.P. Ferraris, K. Chalfant, T. Zawodzinsk, S.W. Feldberg, M.D. Newton, J. Am. Chem. Soc. 125 (2003) 2004. M.D. Newton, J.F. Smalley, Phys. Chem. Chem. Phys. 9 (2007) 555. D.E. Khoshtariya, T.D. Dolidze, D. Krulic, N. Fatouros, D. Devilliers, J. Phys. Chem. B 102 (1998) 7800. J. Böcker, R.R. Nazmutdinov, E. Spohr, K. Heinzinger, Surf. Sci. 335 (1995) 372. S. Frank, C. Hartnig, A. Gro␤, W. Schmickler, Chem. Phys. Chem. 9 (2008) 1371. D. Krapf, B.M. Quinn, M.-Y. Wu, H.W. Zandbergen, C. Dekker, S.G. Lemay, Nanoletters 6 (2006) 2531. D.R. Gabe, J. Appl. Electrochem. 4 (1974) 91. C.H. Haman, W. Vielstich, Electrochemie II, Verlag Chemie, Weinheim, 1981. A.S. Adekunle, K.I. Ozoemena, Electrochim. Acta 53 (2008) 5774.