Electrochemical kinetics and dimensional considerations at the nanoscale: the influence of the density of states

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Electrochemical kinetics and dimensional considerations at the nanoscale: the influence of the density of states H. Yamada, Department of Electrical Engineering, University of California, San Diego, La Jolla, CA 92093, USA R. Narayanan, Department of Nanoengineering, University of California, San Diego, La Jolla, CA, 92093, USA P. R. Bandaru, Department of Electrical Engineering, University of California, San Diego, La Jolla, CA 92093, USA; Department of Nanoengineering, University of California, San Diego, La Jolla, CA, 92093, USA; Program in Materials Science, Department of Mechanical Engineering, University of California, San Diego, La Jolla, CA 92093, USA Address all correspondence to P. R. Bandaru at [email protected] (Received 14 July 2017; accepted 29 August 2017)

Abstract The theories to describe the rate at which electrochemical reactions proceed do not consider explicitly the dimensionality or the occupancy of the energy levels of nanostructured electrodes. It is shown here that the density of states variation in nanoscale electrochemical systems yield novel modulations in the rate constant and concomitant electrical currents. The proposed models extend the utility of presently used Marcus– Hush–Chidsey kinetics to a larger class of materials and could be used as a test of dimensional character. The new models are applied to explain the experimental variation of the electrochemical rate constant of single-layer graphene.

Introduction 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[3,4] and batteries.[5] While the foundational attributes have almost always been reckoned in terms of one-electron-based charge transfer,[6,7] much of the theoretical and experimental analysis has only obliquely referred to the considerations of dimensionality. Consequently, three-dimensional (3D) 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[8,9]–Hush[10] 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.[6] In the BV case, the rate constant (KBV), considering that for the forward reaction rate (KF) and for the backward reaction (KB), is K BV = KF + KB     aeh (1 − a)eh + K o exp − . = K o exp kB T kB T

(1)

In Eq. (1), α is the electron transfer coefficient and η refers to the overpotential (=V − Vo), with V as the applied voltage and Vo as the standard redox potential. The e is the elementary unit

of electronic charge, kB is the Boltzmann constant, and T is the temperature. While simple to use, in principle, such an approach does not yield substantial insight into the type and involvement of