Physics applied to electrochemistry: Tunneling reactions

Abstract

In this work, we explore how electrochemical tunneling reactions can be understood within the single-particle picture. That is, the formal approach in which band diagrams are typically utilized to understand electronic processes in solid-state materials and devices. This single-particle perspective is based upon the Gerischer–Hopfield description of electron transfer at solid–liquid interfaces. Both single and multiple electron tunneling reactions are discussed, as are related voltammetric concepts and trends. The impact of nuclear quantization on the Gerischer–Hopfield description is also addressed, as well as its compact representation of two probe electrochemical phenomena at low temperatures (often referred to as Franck–Condon blockade). In this manner, a perspective linking solid-state phenomena and tunneling electrochemical reactions is presented.

Faradaic Quantized Capacitance as an Ideal Pseudocapacitive Mechanism

“…The electrochemical potential of the electrode is shifted from its equilibrium location (μ eq ) by an applied bias V such that μ = μ eq + V . Hence, the rate of forward electron transfer to a nanoparticle in the N th charge configuration at a bias of V is and the corresponding rate for backward electron transfer is where M el is the electronic coupling between the reactant and electrode arising due to tunneling and D s is the density of states of the electrode substrate. ,− ,− The expressions can be worked out through Fermi’s golden rule or quantum transport methodssee refs ,,, and citations therein. Note that a positive applied bias ( V > 0) is considered to have lifted the electrode electrochemical potential relative to that of the solvent (see Figure c).…”

“…where M el is the electronic coupling between the reactant and electrode arising due to tunneling and D s is the density of states of the electrode substrate. 52,65−67,84−90 The expressions can be worked out through Fermi's golden rule or quantum transport methodssee refs 65,66,74,91 and citations therein. Note that a positive applied bias (V > 0) is considered to have lifted the electrode electrochemical potential relative to that of the solvent (see Figure 2c).…”

Benchmarking the ragone behaviour and power performance trends of pseudocapacitive batteries

“…Here we denote ε to be the single-particle energy [41]. The rate of interfacial electron transfer (k et ) is therefore determined by the interfacial potential drop (V int ) and the fraction of states in the reactant that are already filled as indicated in figure 2(a) [27,41].…”

“…Here we denote ε to be the single-particle energy [41]. The rate of interfacial electron transfer (k et ) is therefore determined by the interfacial potential drop (V int ) and the fraction of states in the reactant that are already filled as indicated in figure 2(a) [27,41]. The substrate density of states D s and interface electron transfer coupling M et are both taken to be constant [41]-h is Planck's constant.…”

“…The rate of interfacial electron transfer (k et ) is therefore determined by the interfacial potential drop (V int ) and the fraction of states in the reactant that are already filled as indicated in figure 2(a) [27,41]. The substrate density of states D s and interface electron transfer coupling M et are both taken to be constant [41]-h is Planck's constant. Within a reasonable approximation, when ET peaks are overlapping sufficiently well, D e can also be treated as a constant per equation ( 1) [27].…”