The rate of electrochemical electron-transfer reactions

Unified quantum theory of electrochemical kinetics by coupled ion-electron transfer

A general theory of coupled ion-electron transfer (CIET) is presented, which unifies Marcus kinetics of electron transfer (ET) with Butler-Volmer kinetics of ion transfer (IT). In the limit of large reorganization energy, the theory predicts normal Marcus kinetics of "electron-coupled ion transfer" (ECIT). In the limit of large ion transfer energies, the theory predicts Butler-Volmer kinetics of "ion-coupled electron transfer" (ICET), where the charge transfer coefficient and exchange current are connected to microscopic properties of the electrode/electrolyte interface. In the ICET regime, the reductive and oxidative branches of Tafel's law are predicted to hold over a wide range of overpotentials, bounded by the ion-transfer energies for oxidation and reduction, respectively. The probability distribution of transferring electron energies in CIET smoothly interpolates between a shifted Gaussian distribution for ECIT (as in the Gerischer-Marcus theory of ET) to an asymmetric, fat-tailed Meixner distribution centered at the Fermi level for ICET. The latter may help interpret asymmetric line shapes in x-ray photo-electron spectroscopy (XPS) and Auger electron spectroscopy (AES) for metal surfaces in terms of shake-up relaxation of the ionized atom and its image polaron by ICET. In the limit of large overpotentials, the theory predicts a transition to inverted Marcus ECIT, leading to a universal reaction-limited current for metal electrodes, dominated by barrierless quantum transitions. Uniformly valid, closed-form asymptotic approximations are derived that smoothly transition between the limiting rate expressions for ICET and ECIT for metal electrodes, using simple but accurate mathematical functions. The theory is applied to lithium intercalation in lithium iron phosphate (LFP) and found to provide a consistent description of the observed current dependence on overpotential, temperature and concentration. CIET theory thus provides a critical bridge between quantum electrochemistry and electrochemical engineering, which may find many other applications and extensions.

Theoretical study of nonadiabatic hydrogen atom scattering dynamics on metal surfaces using the hierarchical equations of motion method

Hydrogen atom scattering on metal surfaces is investigated based on a simplified Newns-Anderson model. Both the nuclear and electronic degrees of freedom are treated quantum mechanically. By partitioning all the surface electronic states as the bath, the hierarchical equations of motion method for the fermionic bath is employed to simulate the scattering dynamics. It is found that, with a reasonable set of parameters, the main features of the recent experimental studies of hydrogen atom scattering on metal surfaces can be reproduced. Vibrational states on the chemisorption state whose energies are close to the incident energy are found to play an important role, and the scattering process is dominated by a single-pass electronic transition forth and back between the diabatic physisorption and chemisorption states. Further study on the effects of the atom-surface coupling strength reveals that, upon increasing the atom-surface coupling strength, the scattering mechanism changes from typical nonadiabatic transitions to dynamics in the electronic friction regime.

Models of Electron Transfer at Different Electrode Materials

Electron transfer is the most important electrochemical process. In this review, we present elements of various aspects of electron transfer theory from the early work of Marcus and Hush to recent developments. The emphasis is on the role of the electronic, and to a lesser extent the geometrical, properties of the electrode. A variety of experimental works are discussed in light of these theoretical concepts. Because the field of electron transfer is so vast, this review is far from comprehensive; rather, we focus on systems that offer a special interest and illuminate aspects of the theory.

Theoretical Modeling of Electrochemical Proton-Coupled Electron Transfer

Proton-coupled electron transfer (PCET) plays an essential role in a wide range of electrocatalytic processes. A vast array of theoretical and computational methods have been developed to study electrochemical PCET. These methods can be used to calculate redox potentials and pK_a values for molecular electrocatalysts, proton-coupled redox potentials and bond dissociation free energies for PCET at metal and semiconductor interfaces, and reorganization energies associated with electrochemical PCET. Periodic density functional theory can also be used to compute PCET activation energies and perform molecular dynamics simulations of electrochemical interfaces. Various approaches for maintaining a constant electrode potential in electronic structure calculations and modeling complex interactions in the electric double layer (EDL) have been developed. Theoretical formulations for both homogeneous and heterogeneous electrochemical PCET spanning the adiabatic, nonadiabatic, and solvent-controlled regimes have been developed and provide analytical expressions for the rate constants and current densities as functions of applied potential. The quantum mechanical treatment of the proton and inclusion of excited vibronic states have been shown to be critical for describing experimental data, such as Tafel slopes and potential-dependent kinetic isotope effects. The calculated rate constants can be used as input to microkinetic models and voltammogram simulations to elucidate complex electrocatalytic processes.

Heat transport induced by electron transfer: A general temperature quantum calculation

Electron transfer dominates chemical processes in biological, inorganic, and material chemistry. Energetic aspects of such phenomena, in particular, the energy transfer associated with the electron transfer process, have received little attention in the past but are important in designing energy conversion devices. This paper generalizes our earlier work in this direction, which was based on the semiclassical Marcus theory of electron transfer. It provides, within a simple model, a unified framework that includes the deep (nuclear) tunneling limit of electron transfer and the associated heat transfer when the donor and acceptor sites are seated in environments characterized by different local temperatures. The electron transfer induced heat conduction is shown to go through a maximum at some intermediate average temperature where quantum effects are already appreciable, and it approaches zero when the average temperature is very high (the classical limit) or very low (deep tunneling).

A non-perturbative approach to simulate heterogeneous electron transfer dynamics: Effective mode treatment of the continuum electronic states

We propose a non-perturbative method to simulate heterogeneous electron transfer dynamics in systems described by a Newns-Anderson type of model. The coupling between the molecule and the continuum electronic states at the metal or semiconductor surface is represented using a set of effective modes, by employing an exponential expansion of the bath correlation functions. Depending on the nature of the problems, the nuclear degrees of freedom are either treated explicitly using wave functions and density operators or as dissipative modes using the techniques from the hierarchical equations of the motion method. Numerical examples are also presented for applications in problems including (1) photo-induced charge transfer at the molecule-semiconductor interfaces, (2) heterogeneous electron transfer at the molecule-metal interface, and (3) vibrational relaxation on a metal surface.

The dynamics of charge transfer with and without a barrier: A very simplified model of cyclic voltammetry

Using the Anderson-Holstein model, we investigate charge transfer dynamics between a molecule and a metal surface for two extreme cases. (i) With a large barrier, we show that the dynamics follow a single exponential decay as expected; (ii) without any barrier, we show that the dynamics are more complicated. On the one hand, if the metal-molecule coupling is small, single exponential dynamics persist. On the other hand, when the coupling between the metal and the molecule is large, the dynamics follow a biexponential decay. We analyze the dynamics using the Smoluchowski equation, develop a simple model, and explore the consequences of biexponential dynamics for a hypothetical cyclic voltammetry experiment.

Towards Understanding the Solvent-Dynamic Control of the Transport and Heterogeneous Electron-Transfer Processes in Ionic Liquids

The impact of temperature-induced changes in solvent dynamics on the diffusion coefficient and standard rate constant k⁰ for heterogeneous electron transfer (ET) of ethylferrocene (EFc) in 1-butyl-3-methylimidazolium hexafluorophosphate ([BMIM][PF₆ ]) is investigated. The results are analysed to understand the impact of solvent-dynamic control, solute-solvent interactions and solvent friction on the transport of redox probes and k⁰ . Concentration dependence of the diffusion coefficient of EFc in [BMIM][PF₆ ] is observed. This is attributed to the solute-induced enhancement of the structural organisation of the ionic liquid (IL), which is supported by the concentration-dependent UV/Vis absorption and photoluminescence responses of EFc/[BMIM][PF₆ ] solutions. Similar values of the activation energies for mass transport and ET and a linear relationship between the diffusion coefficient and the heterogeneous ET rate is observed. The ratio between the diffusion coefficient and the heterogeneous rate constant allows a characteristic length L_d , which is temperature-independent, to be introduced. The presented results clearly establish that mass transport and heterogeneous ET of redox probes are strongly correlated in ILs. It is proposed that the apparent kinetics of heterogeneous ET reactions in ILs can be explained in terms of their impact on thermal equilibration, energy dissipation and thermal excitation of redox-active probes.

Temperature-Driven Changeover in the Electron-Transfer Mechanism of a Thermophilic Plastocyanin

Electron-transfer kinetics of the thermophilic protein Plastocyanin from Phormidium laminosum adsorbed on 1,ω-alkanedithiol self-assembled monolayers (SAMs) deposited on gold have been investigated. The standard electron-transfer rate constant has been determined as a function of electrode-protein distance and solution viscosity over a broad temperature range (0-90 °C). For either thin or thick SAMs, the electron-transfer regime remains invariant with temperature, whereas for the 1,11-undecanethiol SAM of intermediate chain length, a kinetic regime changeover from a gated or friction-controlled mechanism at low temperature (0-30 °C) to a nonadiabatic mechanism above 40 °C is observed. To the best of our knowledge, this is the first time a thermal-induced transition between these two kinetic regimes is reported for a metalloprotein.

Fundamental signatures of short- and long-range electron transfer for the blue copper protein azurin at Au/SAM junctions

The blue copper protein from Pseudomonas aeruginosa, azurin, immobilized at gold electrodes through hydrophobic interaction with alkanethiol self-assembled monolayers (SAMs) of the general type [-S-(CH(2))(n)-CH(3)] (n = 4, 10, and 15) was employed to gain detailed insight into the physical mechanisms of short- and long-range biomolecular electron transfer (ET). Fast scan cyclic voltammetry and a Marcus equation analysis were used to determine unimolecular standard rate constants and reorganization free energies for variable n, temperature (2-55 degrees C), and pressure (5-150 MPa) conditions. A novel global fitting procedure was found to account for the reduced ET rate constant over almost five orders of magnitude (covering different n, temperature, and pressure) and revealed that electron exchange is a direct ET process and not conformationally gated. All the ET data, addressing SAMs with thickness variable over ca. 12 A, could be described by using a single reorganization energy (0.3 eV), however, the values for the enthalpies and volumes of activation were found to vary with n. These data and their comparison with theory show how to discriminate between the fundamental signatures of short- and long-range biomolecular ET that are theoretically anticipated for the adiabatic and nonadiabatic ET mechanisms, respectively.

Kinetic, Thermodynamic, and Mechanistic Patterns for Free (Unbound) Cytochromec at Au/SAM Junctions: Impact of Electronic Coupling, Hydrostatic Pressure, and Stabilizing/Denaturing Additives

Combined kinetic (electrochemical) and thermodynamic (calorimetric) investigations were performed for an unbound (intact native-like) cytochrome c (CytC) freely diffusing to and from gold electrodes modified by hydroxyl-terminated self-assembled monolayer films (SAMs), under a unique broad range of experimental conditions. Our approach included: 1) fine-tuning of the charge-transfer (CT) distance by using the extended set of Au-deposited hydroxyl-terminated alkanethiol SAMs [-S-(CH(2))(n)-OH] of variable thickness (n=2, 3, 4, 6, 11); 2) application of a high-pressure (up to 150 MPa) kinetic strategy toward the representative Au/SAM/CytC assemblies (n=3, 4, 6); 3) complementary electrochemical and microcalorimetric studies on the impact of some stabilizing and denaturing additives. We report for the first time a mechanistic changeover detected for "free" CytC by three independent kinetic methods, manifested through 1) the abrupt change in the dependence of the shape of the electron exchange standard rate constant (k(o)) versus the SAM thickness (resulting in a variation of estimated actual CT range within ca. 15 to 25 A including ca. 11 A of an "effective" heme-to-omega-hydroxyl distance). The corresponding values of the electronic coupling matrix element vary within the range from ca. 3 to 0.02 cm(-1); 2) the change in activation volume from +6.7 (n=3), to approximately 0 (n=4), and -5.5 (n=6) cm(3) mol(-1) (disclosing at n=3 a direct pressure effect on the protein's internal viscosity); 3) a "full" Kramers-type viscosity dependence for k(o) at n=2 and 3 (demonstrating control of an intraglobular friction through the external dynamic properties), and its gradual transformation to the viscosity independent (nonadiabatic) regime at n=6 and 11. Multilateral cross-testing of "free" CytC in a native-like, glucose-stabilized and urea-destabilized (molten-globule-like) states revealed novel intrinsic links between local/global structural and functional characteristics. Importantly, our results on the high-pressure and solution-viscosity effects, together with matching literature data, strongly support the concept of "dynamic slaving", which implies that fluctuations involving "small" solution components control the proteins' intrinsic dynamics and function in a highly cooperative manner as far as CT processes under adiabatic conditions are concerned.

ULTRAFAST ELECTRON TRANSFER AT THE MOLECULE-SEMICONDUCTOR NANOPARTICLE INTERFACE

Electron transfer across the molecule-semiconductor interface is a fundamental process that is relevant to many applications of nanoparticles, such as dye-sensitized solar cells and molecular electronics. This review summarizes recent progress in understanding electron transfer dynamics from molecular adsorbates to semiconductor nanoparticles. Photoexcitation of molecular adsorbates to their excited states is followed by electron injection into semiconductor nanoparticles. The products of electron injection (oxidized adsorbate and electrons in semiconductor) are monitored by their electronic and vibrational spectra, allowing direct measurement of injection rate. The dependence of injection rate on the properties of semiconductor nanoparticle, molecular adsorbate, intervening bridging and anchoring group, and interfacial environment are discussed and compared with Marcus theory of interfacial electron transfer.

Comment on “Comment on ‘Constant temperature molecular dynamics simulations by means of a stochastic collision model. II. The harmonic oscillator’ [J. Chem. Phys. , 3732 (1996)]” [J. Chem. Phys. , 1646 (1997)]

An argument brought forward by Sholl and Fichthorn against the stochastic collision-based constant temperature algorithm for molecular dynamics simulations developed by Kast et al. is refuted. It is demonstrated that the large temperature fluctuations noted by Sholl and Fichthorn are due to improperly chosen initial conditions within their formulation of the algorithm. With the original form or by suitable initialization of their variant no deficient behavior is observed.

Laser pulse control of ultrafast heterogeneous electron transfer: A computational study

Laser pulse control of the photoinduced 90 fs charge injection from perylene into the conduction band of TiO2 is studied theoretically. The approach accounts for the electronic-ground state of the dye, the first excited state, the ionized state formed after charge injection, and the continuum of the electronic states in the conduction band, all defined vs a single reaction coordinate. To address different control tasks optimal control theory is combined with a full quantum dynamical description of the electron-vibrational motion accompanying the charge injection process. First it is proved in which way the charge injection time can be changed by tailored laser pulses. In a second step a pump-dump scheme from the perylene ground state to the first excited electronic state and back to the ground state is discussed. Because of the strong coupling of the excited perylene state to the band continuum of TiO2 this control task is more suited to an experimental test than the direct control of the charge injection.