Microscopic quantum mechanical Marcus theory for chemical reactions and its relationship to the Butler-Volmer equation used in redox flow battery analysis

The vanadium redox flow battery has been intensively examined since the 1970s. What is missing is a connection between the current-overpotential Butler-Volmer equation, which provides an extremely helpful starting point for analytical and numerical studies, and microscopic quantum mechanical behavior at the atomic level. Such a connection will allow further advancements beyond the macroscopic, though very useful and insightful, modeling already done in the literature. Here we show rigorously the connection between the Butler-Volmer transfer coefficients, and the Marcus Gibbs free energy quantum mechanical parameters, and develop the equation directly in terms of the quantum mechanical parameters.     

Electrochemical mapping reveals direct correlation between heterogeneous electron-transfer kinetics and local density of states in diamond electrodes

Conducting carbon materials: a multi-microscopy approach shows that local heterogeneous electron-transfer rates at conducting diamond electrodes correlate with the local density of electronic states. This model of electroactivity is of considerable value for the rational design of conducting diamond electrochemical technologies, and also provides key general insights on electrode structure controls in electrochemical kinetics.     

Kinetic studies on the single electron transfer reaction between 2, 2, 6, 6-tetramethylpiperidine oxoammonium ions and phenothiazines: the application of Marcus theory

Electron transfer reactions take place readily between 2, 2, 6, 6-tetramethylpiperidine oxoammonium ions (1a, 1b) and phenothiazines (2a—2g), giving corresponding nitroxides (3a, 3b) and phenothiazine radical cations (4a—4g). The rate constants for the electron self-exchange reactions between 1 and 3, as well as between 2 and 4, are determined by EPR and ~1H NMR line-broadening effect in acetonitrile. By application of the Marcus theory, the kinetics of the cross-exchange reactions between 1 and 2 is studied.     

Driving force dependence of intermolecular electron-transfer reactions of fullerenes

Pulse-radiolytic studies were performed to determine the rate constants of intermolecular electron transfer (k(et)) from fullerenes (C(60), C(76), and C(78)) to a series of arene radical cations in dichloromethane. The one-electron oxidation potentials of the employed arenes-corresponding to the one-electron reduction potentials of arene pi-radical cations-were determined in dichloromethane to evaluate the driving forces of electron-transfer oxidation of fullerenes with arene pi-radical cations. The driving force dependence of log k(et) shows a pronounced decrease towards the highly exothermic region, representing the first definitive confirmation of the existence of the Marcus inverted region in a truly intermolecular electron transfer. Electron-transfer reduction of fullerenes with anthracene radical anion was also examined by laser flash photolysis in benzonitrile. The anthracene radical anion was produced by photoinduced electron transfer from 10,10'-dimethyl-9,9',10,10'-tetrahydro-9,9'-biacridine [(AcrH)(2)] to the singlet excited state of anthracene in benzonitrile. The rate constants of electron transfer (k(et)) from anthracene radical anion to C(60), C(70), and a C(60) derivative were determined from the decay of anthracene radical anion in the presence of various concentrations of the fullerene. Importantly, a significant decrease in the k(et) value was observed at large driving forces (1.50 eV) as compared to the diffusion-limited value seen at smaller driving forces (0.96 eV). In conclusion, our study presents clear evidence for the Marcus inverted region in both the electron-transfer reduction and oxidation of fullerenes.     

Electron transfer in bis(hydrazines), a critical test for application of the Marcus model

Electron transfer in the cations of bis(hydrazines), bridged by six different π‐systems (compounds 1–6) is studied using ab initio and density functional theory (DFT) methods. Due to ionization from an antibonding combination of the lone‐pair orbitals of the nitrogens in one of the hydrazine units, conjugation is introduced in the N? N bond of that unit. This leads to a shortening of the N? N bond distance and an increase of the planarity around the nitrogens. Due to steric hindrance, this causes an increase of the angle, called φ, between the lone‐pair orbital on the nitrogen attached to the bridge and the p‐orbital on the adjacent bridge carbon for the ionized unit in the charge localized, relaxed state of the molecule. This angle controls the magnitude of the electronic coupling. In the fully delocalized symmetric transition state of the ion, however, this angle is low for both units, due to the fact that the conjugation introduced at the ionized hydrazine unit is now shared between both units. An extended π‐system is formed including the orbitals of the hydrazine units and the bridge, which leads to a large electronic coupling. The electronic coupling derived by optical methods, corresponding to the structure of the relaxed, asymmetric cation with a large φ for the ionized unit, appears to be much smaller. We believe this is due to an approximate cosine dependence on φ of the coupling. The calculations carried out support these conclusions. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 655–664, 2001     

Electron transfer to sulfides and disulfides: intrinsic barriers and relationship between heterogeneous and homogeneous electron-transfer kinetics

The electron-acceptor properties of series of related sulfides and disulfides were investigated in N,N-dimethylformamide with homogeneous (redox catalysis) and/or heterogeneous (cyclic voltammetry and convolution analysis) electrochemical techniques. The electron-transfer rate constants were determined as a function of the reaction free energy and the corresponding intrinsic barriers were determined. The dependence of relevant thermodynamic and kinetic parameters on substituents was assessed. The kinetic data were also analyzed in relation to corresponding data pertaining to reduction of diaryl disulfides. All investigated reductions take place by stepwise dissociative electron transfer (DET) which causes cleavage of the C(alkyl)--S or S--S bond. A generalized picture of how the intrinsic electron-transfer barrier depends on molecular features, ring substituents, and the presence of spacers between the frangible bond and aromatic groups was established. The reduction mechanism was found to undergo a progressive (and now predictable) transition between common stepwise DET and DET proceeding through formation of loose radical anions. The intrinsic barriers were compared with available results for ET to several classes of dissociative- and nondissociative-type acceptors, and this led to verification that the heterogeneous and the homogeneous data correlate as predicted by the Hush theory.     

Intermolecular electron-self exchange kinetics measured by electron paramagnetic resonance-linebroadening effects: useful rate constants for the application of Marcus theory

The different methods to obtain rate constants of homogeneous intermolecular electron-self exchange reactions in solutions from dynamic linebroadening effects in CW-electron paramagnetic resonance (EPR) spectra are discussed in details. The limitations of the slow and fast exchange region are explained. The medium exchange region is also discussed. Experimental details are given. The exchange rates of the 2-methyl-1,4-naphtoquinone/2-methyl-1,4-napthoquinone radical anion and the 1-nitronaphthalene/1-nitronaphthalene radical anion redox couples are reported to 9.2 · 10⁸ M⁻¹ s⁻¹ (EtOH, 263 K) and 1.8 · 10⁸ M⁻¹ s^-1 (DMF, 266 K), respectively.     

Approximate method for calculation of electron transition probability for simple outer-sphere electrochemical reactions

The probability of an elementary act in an outer-sphere electrochemical electron transfer reaction is calculated with arbitrary values of the parameter of reactant-electrode electron interaction for diabatic freeenergy surfaces of the parabolic form. The dependence of effective transmission coefficient on the Landau-Zener parameter is found. Interpolation formulas are obtained that describe this dependence and allow calculating the electron transition probability using the results of quantum chemical calculations of the electronic matrix element as a function of distance.     

Photoinduced electron transfer between quinones and amines in micellar media: Tuning the Marcus inversion region

Electron transfer (ET) rates between quinone acceptors and amine donors in micellar media show Marcus inversion behavior on correlating with the free energy changes of the ET reactions. The onset of Marcus inversion in these systems is seen to be tuned by about 0.25 eV by changing the type of the micelle. The results are rationalized on the basis of two-dimensional ET theory where ET occurs along intramolecular coordinate with non-equilibrium configuration along solvation coordinate. Maximum ET rates are seen to vary by about one order of magnitude in different micelles, and are attributed to the micelle-dependent changes in the separations of the interacting quinone–amine pairs. Tunings of Marcus inversion and ET rates by changing micellar microenvironments have been observed and suggested to have useful implications in different applied areas.     

Kinetics of the Reductions of (Ethylenediaminetetraacetato)cobaltate(III) by 4,4-Dipyridylamine Complexes of Pentacyanoferrate(II) and Pentaammineruthenium(II)

The reactions of Fe(CN)₅dpa^3? and Ru(NH₃)₅dpa²⁺ (dpa = 4,4′-dipyridylamine) with Co(edta)^? have been investigated kinetically. For Fe(CN)₅dpa^3? complex, a linear relationship was observed between the pseudo-First-order rate constants and the concentrations of Co(edta) which leads to a specific rate 0.876 ± 0.006 M^?1S^?1 at T = 25°C., μ = 0.10 M and pH = 8.0. For the Ru(NH₃)₅dpa²⁺ system, the plots k_obs vs [Co(edta)^?] become nonlinear at concentrations of Co(edta) greater than 0.01 M and the reaction is interpreted on the basis of a mechanism involving the formation of an ion pair between Ru(NH₃)₅dpa²⁺ and Co(edta)^? followed by electron transfer from Ru(II) to Co(III). The nonlinear least squares fit of the kinetic results shows that Q_ip = 10.6 ± 0.7 M^?1 and k_et = 93.9 ± 0.7 s^?1 at pH = 8.0,μ = 0.10 M and T = 25°C.     

Binding modes in metal ion complexes of quinones and semiquinone radical anions: electron-transfer reactivity.

9,10-Phenanthrenequinone (PQ) and 1,10-phenanthroline-5,6-dione (PTQ) form 1:1 and 2:1 complexes with metal ions (M (n+)=Sc (3+), Y (3+), Mg (2+), and Ca (2+)) in acetonitrile (MeCN), respectively. The binding constants of PQ--M (n+) complexes vary depending on either the Lewis acidity or ion radius of metal ions. The one-electron reduced species (PTQ(-)) forms 1:1 complexes with M (n+), and PQ(-) also forms 1:1 complexes with Sc(3+), Mg(2+), and Ca(2+), whereas PQ(-) forms 1:2 complexes with Y(3+) and La(3+), as indicated by electron spin resonance (ESR) measurements. On the other hand, semiquinone radical anions (Q(-) and NQ(-)) derived from p-benzoquinone (Q) and 1,4-naphthoquinone (NQ) form Sc(3+)-bridged pi-dimer radical anion complexes, Q(-)--(Sc(3+))(n)--Q and NQ(-)--(Sc(3+))(n)-NQ (n=2 and 3), respectively. The one-electron reduction potentials of quinones (PQ, PTQ, and Q) are largely positively shifted in the presence of M (n+). The rate constant of electron transfer from CoTPP (TPP(2-)=dianion of tetraphenylporphyrin) to PQ increases with increasing the concentration of Sc(3+) to reach a constant value, when all PQ molecules form the 1:1 complex with Sc(3+). Rates of electron transfer from 10,10'-dimethyl-9,9'-biacridine [(AcrH)(2)] to PTQ are also accelerated significantly by the presence of Sc(3+), Y(3+), and Mg(2+), exhibiting a first-order dependence with respect to concentrations of metal ions. In contrast to the case of o-quinones, unusually high kinetic orders are observed for rates of Sc(3+)-promoted electron transfer from tris(2-phenylpyridine)iridium(III) [Ir(ppy)(3)] to p-quinones (Q): second-order dependence on concentration of Q, and second- and third-order dependence on concentration of Sc(3+) due to formation of highly ordered radical anion complexes, Q()--(Sc(3+))(n)--Q (n=2 and 3).     

Determination of Electronic Recombination Free Energy in Zeolites: Effects of the Charge Balancing Cation and Confinement

The adsorption of DPH in M_6.6ZSM-5 (M=Na⁺, K⁺, Rb⁺, Cs⁺), RbFER and RbMOR channel zeolites takes place without chemical or structural modification. After photoexcitation of these systems, a radical cation–electron pair is observed and has a sufficiently long lifetime to be studied by diffuse reflectance UV-visible spectroscopy. The study of the recombination of this radical cation-electron pair was carried out at different temperatures and allowed the determination of the activation energy as a function of the nature of the charge-balancing cation but also of the confinement effect. It appears that the activation energy decreases progressively from Na⁺ to Cs⁺ but also when the confinement decreases. To go further, the free enthalpies have been calculated from the Marcus theory demonstrating experimentally that these systems are located in the inverted Marcus region.     

Marcus theory of outer-sphere heterogeneous electron transfer reactions: dependence of the standard electrochemical rate constant on the hydrodynamic radius from high precision measurements of the oxidation of anthracene and its derivatives in nonaqueous solvents using the high-speed channel electrode

The validity of Marcus theory for outer-sphere heterogeneous electron transfer for the electro-oxidation of a range of anthracene derivatives in alkyl cyanide solvents is investigated. The precision measurement of these fast electron transfers (k(0) >or= 1 cm s(-1)) is achieved by use of the high-speed channel electrode and, where necessary, fast-scan cyclic voltammetry. First, the solvent effect on the rate of electron transfer is studied by considering the first oxidation wave of 9,10-diphenylanthracene in the alkyl cyanide solvents: acetonitrile, propionitrile, butyronitrile, and valeronitrile. Second, the variation of k(0) for a series of substituted anthracenes is investigated by analyzing the voltammetric response of the one-electron oxidations of 9-phenylanthracene, 9,10-dichloroanthracene, 9-chloroanthracene, 9,10-dicyanoanthracene, 9-cyanoanthracene, 9-nitroanthracene, 9,10-diphenylanthracene, and anthracene in acetonitrile. It is shown that the rate of electron transfer of a single compound in different alkyl cyanides is determined by the longitudinal dielectric relaxation properties of the solvent, while differences in rate between the substituted anthracenes in acetonitrile can be quantitatively rationalized by considering their relative hydrodynamic radii. This makes possible the accurate prediction of electron-transfer rates for a molecule by interpolation of rate constants known for related molecules.     

Unusually highly efficient,singlet state,visible light photoinitiators based on styrylbenzimidazolium phenyltributylborate photoredox pairs for vinyl monomers free radical polymerization

In this article, hemicyanine dye–borate complexes, for example, 1,3‐dimethyl‐2‐[4‐(N,N‐dialkylamino)styryl]benzimidazolium phenyl‐tri‐n‐butylborates, were employed as the novel, very effective photoinitiators operating in the visible light region. The influence of the sensitizers and electron donor structure on the photopolymerization kinetics of multiacrylate monomer was investigated by photo‐DSC. The maximum photopolymerization quantum yield measured for 2‐ethyl‐2‐(hydroxymethyl)‐1,3‐propanediol triacrylate (TMPTA) was about 67 for sample of thickness of about 1 mm under 100 mW/cm² laser irradiation. It was found that the polymerization rate and the final conversion degree were depended on the dye structure. Moreover, the photoinitiating systems described gave a double bond conversion higher than the photoinitiator possessing as chromophore RBAX (Rose Bengal derivative), the common triplet state initiator. Additionally, the rate of photopolymerization depends on ΔG_el of electron transfer between borate anion and styrylbenzimidazolium cation. This latter value was estimated for a series of styrylbenzimidazolium borate salts. The relationship between the rate of polymerization and the free energy of activation for electron transfer reaction gives the dependence predicted by the classical theory of electron transfer. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 4119–4129, 2009     

Positional Effect of the Triphenylamine Group on the Optical and Charge‐Transfer Properties of Thiophene‐Based Hole‐Transporting Materials

Hybrid organic‐inorganic perovskite solar cells (PSCs) have shown significant potential for use in the energy field. Typically, hole‐transporting materials (HTMs) play an important role in affecting the power conversion efficiency (PCE) of PSCs. A deep understanding of the structure‐property relationship plays a vital role in developing efficient HTMs. Herein, the relationship between the structure and properties of two small organic HTMs H2,5 and H3,4 were systematically investigated in terms of the electronic and optical properties, the hole‐transporting behavior by using density functional theory (DFT) and Marcus electron transfer theory. The results demonstrated that the high power conversion efficiency of the H2,5‐ based PSC was caused by strong interactions with the perovskite material on the interface and an enhanced hole mobility in H2,5 compared with H3,4 . The strong interaction derives from the short bond length of O atom of HTM and Pb atom of perovskite material, and the highly hole mobility derives from the quasi‐planar conjugated conformation and tight packing model of neighboring molecules in H2,5 . In addition, we found that the planar structure enhances the intermolecular interaction between HTM and perovskite materials compared with the ′V′‐shaped molecule. Importantly, we also note that the HOMO level of the isolated molecule is not always proportional to the open‐circuit voltages of PSCs since the HOMO level might move toward a higher level when the interaction between HTM and interface of perovskite was included. The work gives essential information for rational designing efficient HTMs.