Fast Method for Calculating Spatially Resolved Heterogeneous Electron-Transfer Kinetics and Its Application to Graphene with Defects

Abstract

Establishing the relationship between the electrochemical activity of a surface and its chemical structure is extremely important for the development of new functional materials for electrochemical energy conversion systems. Here, we present a fast method, which combines a theoretical model and density functional theory calculations, for the prediction of nonadiabatic electron-transfer kinetics at nanoscale surfaces with spatial resolution. We propose two approaches for the calculation of electronic coupling, which characterizes the interaction strength between electronic states of a redox-active molecule and surface electronic states and depends on the position of the molecule above the surface. The first one is based on the linear approximation between the electronic coupling and overlap integral and takes into account the molecular wave function explicitly. The second one uses Tersoff–Hamann and Chen approximations that are based on the model assumption about the molecular orbital structure and allow ultrafast electron-transfer kinetic calculations only from the surface wave function. The proposed method was applied for the electron-transfer kinetic investigation of graphene with defects. We have shown that defects can act as electrocatalytic sites, selectively increasing the electron-transfer rate in a different range of standard redox potentials depending on the defect type.

Modulation of the kinetics of outer-sphere electron transfer at graphene by a metal substrate

“…The standard rate constant of electron transfer from the surface to a molecule located at a distance x from the surface can be obtained by integrating over all electronic states ε of the surface: 41,54 where f ( ε ) and ρ ( ε ) are the Fermi distribution and the density of electronic states in the electrode, λ is the solvent reorganization energy, ω is the effective fluctuation frequency of the solvent, ε F is the Fermi energy, κ ( ε , x ) is the electronic transmission coefficient, which characterizes the probability of electron transfer in the transition state from the ε level to a molecule in a solution located at a distance x . Eqn (8) shows that the rate of nonadiabatic ET increases with an increase in the density of electronic states ρ ( ε ) near the Fermi level and the electronic transmission coefficient κ ( ε , x ).…”

“…To calculate the matrix element, the Tersoff–Hamann approximation was used, 55 which was tested in our previous work in relation to the electrochemical problem: 41 | H i , k ( ε , r⃑ )| 2 ∼ | ψ i , k | 2 …”

“…In this case, graphene is charged and the Fermi level shift at equilibrium depends on the redox potential, which leads to redoxdependent kinetics of the outer-sphere nonadiabatic ET on the graphene surface. 38,41,42,49 Thus, for a complete analysis of the metal underlayer effect, we should predict the alignment of electronic levels at the metal/graphene/redox electrolyte interface. In this work, we used the model proposed in our previous study.…”

“…The standard rate constant of electron transfer from the surface to a molecule located at a distance x from the surface can be obtained by integrating over all electronic states e of the surface: 41,54…”

“…22,24 The theoretical description of the ET at graphene defects was proposed by us earlier in ref. 41–44.…”