Electrochemical kinetics and dimensional considerations at the nanoscale

It is shown that the consideration of the density of states variation in nanoscale electrochemical systems yields modulations in the rate constant and concomitant electrical currents. The proposed models extend the utility of Marcus-Hush-Chidsey (MHC) kinetics to a larger class of materials and could be used as a test of dimensional character. The implications of the study are of much significance to an understanding and modulation of charge transfer nanostructured electrodes.

A critical understanding of the thermodynamics and kinetics inherent to electrochemical reactions is necessary for scientific insights into charge transfer [1] as well as in applications ranging from biochemical reactions [2] to charge storage in capacitors and batteries [3]. While the foundational attributes have almost always been reckoned in terms of one-electron based charge transfer [4,5], much of the theoretical and experimental analysis has only obliquely referred to the considerations of dimensionality. Consequently, three-dimensional electrode characteristics and classical thermodynamics have been implicitly assumed in heterogeneous electron transfer kinetics, encompassing the widely used Butler-Volmer (BV) formulations and the subsequent Marcus [6,7] -Hush [8] interpretations. In this regard, Arrhenius based activation theory, leading to the BV approaches, has been used for over a century, and extensively documented in standard electrochemistry textbooks [4,9]. In the BV case, the rate constant (K BV ), considering that for the forward reaction rate (K F ) and for the backward reaction (K B ), is: solvent and the redox species vis-à-vis the electrochemical reactions and the electrolyte (through the macroscopic dielectric constant). We first briefly review the salient features of the MH kinetics approach and its extension by Chidsey [11]. Consequently, we consider typical [6,7] free energy (G) -reaction coordinate (q) curves: Fig. 1. Typically, the reaction coordinate has been broadly interpreted [12], and may refer to the distance [13], in a multi-dimensional extensive variable sense (e.g., the change of bond length, electrical charge, etc.), between the oxidized (O) and reduced (R) species in an electrochemical redox reaction, of the type: O + e -! R. While the progressive lowering of the minimum energy of the R parabola (e.g., through increasing the η) always decreases the free energy of reaction ΔG o , the free energy of activation ΔG a initially decreases, reaching zero when the R parabola passes through the minimum of the O parabola, and subsequently increases, due to a shift of the R free energy curves to the left hand side of the O parabola: Fig. 1(a). The concomitant increase and decrease of the electrochemical reaction rate constant K MH , i.e., as represented in Eqn. (2), with v as the attempt frequency, reaches a maximum when ΔG a = 0. (2) Such a non-intuitive increase and subsequent decrease of the reaction rate with increasing driving force (i.e., η) constitutes the essence of the inverted region, particular to the Marcus-Hush theory. Such a notion on the maximum of a rate constant has been experimentally confirmed [14], e.g., in intramolecular reactions, concerning molecules with bridged donoracceptor units [15]. It may also be derived that [14], , with λ as the reorganization energy -which is related to the energy required for both the internal (e.g., due to the bond configuration changes) and the external (e.g., in the rearrangement of the solvation shell, surrounding electrolyte, etc.) configurational changes. Subsequently, it is evident that a zero ΔG a would imply that the peak of the K MH is at a value of λ ∼ -ΔG o .
However, such a theory seemed to be incompatible with the notion of long distance interfacial electron transfer where the rate constant decreases exponentially with increased donor-acceptor separation distances [16] as considered through the seminal work of Chidsey [11]. Additionally, the experimental observation, in certain metal electrode based electrochemical ensembles, of the saturation of the electrochemical current with increasing η, prompted the consideration of a continuum of energy level states. The consequently derived rate constant K MHC , considering energy level occupancy through the Fermi-Dirac distribution f FD , and the explicit introduction of a constant metallic electronic density of states (DOS) (= ρ), was of the form [11]:  First, we reinterpret the classical free energy -reaction coordinate curves depicted in Fig.   1(a), in the context of lower dimensional structures. The initial decrease in ΔG a followed by a subsequent increase, can be related by analogy to the availability and subsequent lack in the number of energy levels (related to the DOS) accessible for electron transfer. Such a modulation is apparent in the DOS of one-dimensional nanostructures [17], with increasing carrier concentration and change of the E F , and may be induced through appropriate η. We have then observed that such non-constant DOS yields novel electrical current -voltage response in related electrochemical systems. Fig. 1(b) indicates the correspondence for lower dimensional systems where the decreasing DOS at higher energy may be taken analogous to the increasing λ. Indeed, saturation of the electrical current/rate constant curves may be indicative of the limit of a finite DOS. Miller, et al. [15], Chidsey [11], and Bai, et al, [19].
From CA related experiments and I=I o exp (-K MHC t), such characteristics may be considered equivalent to electrical current I-η plots. The figure also indicates a re-plotting of experimental data previously obtained [15]. It is to be noted that while the BV kinetics indicates a linear variation with η, the MH model exhibits a peak as a function of the η. It is also relevant to note that the experimental curves were also fit through employing Poisson statistics [18]  while avoiding the steeper drop-off of the MH curve, and was considered [15] the best fit to certain chronoamperometric data.
We now consider the influence of a variable DOS, on the K variation with η. The number of electrons available for the redox reaction from the electrode: n, would be: where E c is the energy at the bottom of the conduction band.
We concomitantly introduce a new DOS based reaction rate constant: K MHC-DOS , considering the influence of the energy levels, through: The integration may again be either over the negative interval (-∞, 0] or the positive interval: [0, ∞), as previously discussed. In a limiting case corresponding to Eqn. (3), the DOS would be a constant (e.g., ρ), reverting to the original Chidsey formulation [11]. In the subsequent treatment, the E c was taken as reference energy and set to zero. Such a formulation involving the energy variation of the DOS [17] as a function of the dimensionality, D (e.g., DOS 3D ~ constant or ~ allows for a variable height of the step function, depicted on the left hand side of Fig. 2. The resulting K MHC-DOS -η curves, as a function of the dimensionality dependent DOS are indicated in Fig. 4. In addition to the parabolic Energy-k vector dispersion, we have also incorporated a linear E-k dispersion as seems to be necessary to describe the characteristics of graphene and related 2D materials [20,21]. function of the electrode dimensionality (i.e., a = ½ for a three-dimensional semiconductor; a = 0 or 1, for a two-dimensional system, a = -½ for a one-dimensional system) and is Deltafunction like for zero-dimensional systems, such as quantum dots. The case of a = ½ involves a bandgap, which causes the K η=0 to be smaller than that for the other cases. Generally, a reduction of the K corresponds to a decreasing DOS with energy.
The respective influences of the dimensionality and the dispersion are clearly evident.
While the traditional MHC based formulations assumed a constant DOS, particular to bulk-like/three-dimensional (3D) metallic electrodes, the energy variation of the DOS in lower dimensional systems yields rich and involved behavior. For instance, the behavior of a twodimensional (2D) material with parabolic energy dispersion, e.g., involving a quantum well, is seen to differ compared to one with linear energy dispersion, e.g., graphene. In the latter case, an increasing DOS with electron kinetic energy is responsible for the observed variation. The situation for a one-dimensional (1D) material, e.g., a carbon nanotube (CNT), constituted electrode -with parabolic energy dispersion along the long-axis and quantization along the two perpendicular directions, with a decreasing DOS vis-à-vis energy, corresponds to inversion in the K-η curves at a sufficiently large η, as posited in the original MH formulations. In onedimensional systems, the initial increase of the DOS upon the E F reaching the band edge and the subsequent E -1/2 induced decrease yields a corresponding modulation of the K and the electrical currents.
We then predict the occurrence of oscillations in the K/K η=0 -η curves in one-dimensional nanostructures as a function of chirality in Fig. 5. As is well known [22], the specific nature of wrapping of a constituent graphene sheet, through the chirality index [m, n], dictates whether the resulting CNT is metallic/semiconducting. We depict the corresponding DOS for a (i) semiconducting [10,0] nanotube, and a (ii) metallic [9,0]   to an R state (as in O + e -! R), cf., Fig. 1, can be considered analogous to an energy level width. When the ΔE is larger (/smaller) compared to the λ, the interaction of the electrode energy levels (and relevant electron exchange/redox interactions) with respect to the electrolyte would be more (/less) sharply defined, and yield an oscillatory (/smooth) K-η variation. A small λ implies that the nuclear reconfiguration and the coordinating solvent interactions [23] accompanying the redox reaction is negligible. At a large enough λ/ΔE, a continuous electronic distribution/DOS may be assumed, yielding smooth MHC kinetics, with an increase of the K up to η ~ λ/e, and subsequent plateauing of the K-η curves. The discussed K-η variation as related to the λ/ΔΕ ratio is indicated in Fig. 6. CNT actually causes K/K η=0 to decrease with increasing λ, in contrast to the [9,0] CNT.
As it was recently indicated that a λ of ~ 0.2 eV seemed to be effective for modeling MHC based charge transfer kinetics at LiFePO 4 battery electrode interfaces [19], such modulations could be experimentally probed. Additionally, the K/K η=0 increases with λ for a [9,0] CNT, as was previously indicated [10], but shows the opposite variation in a [10,0] CNT.
The bandgap in the semiconducting [10,0] CNT causes the K η=0 value to be smaller than that for the metallic [9,0] CNT; such an effects is stronger for smaller λ, cf. Fig. 2.