Determination of rate constants for heterogeneous mass transfer kinetics in liquid chromatography

The heterogeneity of the stationary phase surface leads to undesired peak broadening and tailing in chromatography. In order to properly characterize the interactions between the solute molecules and the surface of the stationary phase, the determination of the equilibrium isotherm is usually necessary by overloading the column. In this study we show that measurements made in the linear range of the isotherm can also be utilized to estimate the heterogeneity of the stationary phase. When one studies the peak shape obtained after injecting a small (analytical) amount of sample, the peak shape parameters can be correlated with the kinetic rate constants. The peak of a mildly polar solute, phenol, was measured in reversed-phase HPLC under various conditions and kinetic information was obtained regarding the heterogeneity of interactions.     

Structural correlations in heterogeneous electron transfer at monolayer and multilayer graphene electrodes

As a new form of carbon, graphene is attracting intense interest as an electrode material with widespread applications. In the present study, the heterogeneous electron transfer (ET) activity of graphene is investigated using scanning electrochemical cell microscopy (SECCM), which allows electrochemical currents to be mapped at high spatial resolution across a surface for correlation with the corresponding structure and properties of the graphene surface. We establish that the rate of heterogeneous ET at graphene increases systematically with the number of graphene layers, and show that the stacking in multilayers also has a subtle influence on ET kinetics.     

Model system-bath Hamiltonian and nonadiabatic rate constants for proton-coupled electron transfer at electrode-solution interfaces

An extension of the Anderson-Newns-Schmickler model for electrochemical proton-coupled electron transfer (PCET) is presented. This model describes reactions in which electron transfer between a solute complex in solution and an electrode is coupled to proton transfer within the solute complex. The model Hamiltonian is derived in a basis of electron-proton vibronic states defined within a double adiabatic approximation for the electrons, transferring proton, and bath modes. The interaction term responsible for electronic transitions between the solute complex and the electrode depends on the proton donor-acceptor vibrational mode within the solute complex. This model Hamiltonian is used to derive the anodic and cathodic rate constants for nonadiabatic electrochemical PCET. The derivation is based on the master equations for the reduced density matrix of the electron-proton subsystem, which includes the electrons of the solute complex and the electrode, as well as the transferring proton. The rate constant expressions differ from analogous expressions for electrochemical electron transfer because of the summation over electron-proton vibronic states and the dependence of the couplings on the proton donor-acceptor vibrational motion. These differences lead to additional contributions to the total reorganization energy, an additional exponential temperature-dependent prefactor, and a temperature-dependent term in the effective activation energy that has different signs for the anodic and cathodic processes. This model can be generalized to describe both nonadiabatic and adiabatic electrochemical PCET reactions and provides the framework for the inclusion of additional effects, such as the breaking and forming of other chemical bonds.     

Overestimation of heterogeneous rate constants of hexacyanoferrate at nanometer-sized ultramicroelectrodes

Standard rate constants of hexacyanoferrate (Fe(III)/Fe(II)) at nanometer-scaled ultramicroelectrodes were evaluated from a plot of half-wave potentials of the steady-state voltammograms against logarithms of radii of ultramicroelectrodes. The smaller is the radius, the more positively shifts the anodic voltammogram and the more negatively does the cathodic wave. The shift is ascribed to the increase in the current density, which alters the diffusion-control into the charge-transfer control. However, the amount of the shift was smaller than that predicted from the conventional theory for the kinetic mass transport at ultramicroelectrodes. Consequently, the standard rate constant was overestimated by three times. A possible reason of the smaller shift is a non-negligible length of the mean free path of the diffusing redox particle in comparison with the electrode size. A stochastic diffusion model is proposed, in which a particle walks by random distances of which average is of the order of the electrode diameter. Monte Carlo simulation for this model showed that the concentration profiles were more uniform than the conventional profile, and hence the overpotential is observed to be apparently smaller.     

Dynamics of adiabatic electron transfer reactions at metal electrodes

We have investigated the dynamics of adiabatic electron transfer reactions at metal electrodes using a Hamiltonian suggested by Schmickler (J. Electroanal. Chem., 204 (1986) 31). We show that in the adiabatic limit the problem reduces to that of dynamics of a single variable, the shift of the ionic orbital caused by its interaction with the solvent. This variable is identified as the reaction co-ordinate for the problem and we show that in certain limits, it obeys a non-linear Volterra type integral equation, with a stochastic inhomogeneous term. For an inhomogenous term with the autocorrelation function decaying exponentially, this may be converted into a differential equation for Brownian motion. This equation can be analysed to obtain the rate, through the associated Fokker-Planck equation. The rate so obtained, has a correction to the pre-exponential factor obtained by Schmickler. A possible extension to inner sphere reactions is also discussed.     

Method for determination of heterogeneous rate constants at carbon-fibre microelectrodes using multi-ac-voltammetry

A state-of-the-art FFT-instrument recently described in this journal (which measures the impedance at each dc potential applied to a working electrode) was shown to perform with excellent precision in a kinetic study at a carbon fibre microelectrode. A constant phase element (CPE) was used in a model in order to correct for double layer effects arising at the carbon fibre microelectrode (∼11 μm). This simple model in combination with the data collected by the FFT-instrument were then used in order to determine the heterogeneous rate constant for a well-defined red/ox couple. Good agreement with previously published values was obtained.     

A comparison between interfacial electron-transfer rate constants at metallic and graphite electrodes

The Fermi golden rule formalism has been used to derive the rate constant for interfacial electron transfer from a semimetallic electrode, such as highly ordered pyrolytic graphite (HOPG), to a redox couple in solution. A simple expression is presented that semiquantitatively relates the electron-transfer rate constant at a semimetallic electrode to that at a metallic electrode. The approach allows for the estimation of the value of the rate constant for interfacial charge transfer to nonadsorbing outer-sphere redox species at semimetallic electrodes. Rate constants for interfacial electron transfer for a variety of one-electron redox couples at semimetallic electrodes have been calculated relative to the rate constant of the ferrocenium/ferrocene redox couple at a gold electrode. Good agreement is found, in general, between the calculated and observed rate constants.     

Local electron transfer rate measurements on modified and unmodified glassy carbon electrodes

A natural and artificial distribution of electron transfer activity on glassy carbon electrodes can be observed and quantified by the use of scanning electrochemical microscopy (SECM). A large (sevenfold) spread in rate constant is found for randomly sampled sites on polished, untreated glassy carbon surfaces. Direct-mode oxidation with the SECM tip was used to produce small regions of oxidized carbon on a polished surface. A large increase in electron transfer rate for the Fe(II/III) ion is observed on the locally oxidized carbon surface in comparison to the unoxidized region. Rate constant measurements made along a line profiles the transition from unoxidized to oxidized surfaces. SECM images of defect sites show reaction–rate variations. Rate constants measured at several locations of the defective surface allows discrimination between the kinetic and topographic components of the SECM image. Dedicated to the 80th birthday of Keith B. Oldham     

Voltammetry at spatially heterogeneous electrodes

Recent advances are overviewed which enable simulation of the voltammetric behaviour of surfaces which respond in an electrochemically spatially heterogeneous fashion. By use of the concept of a “diffusion domain” computationally expensive three-dimensional simulations may be reduced to tractable two-dimensional equivalents. In this way the electrochemical response of partially blocked electrodes and microelectrode arrays may be predicted, and are found to be consistent with experimental data. It is, furthermore, possible to adapt the “blocked” electrode analysis to enable the voltammetric sizing of inert particles present on an electrode surface. Finally theory of this type predicts the voltammetric behaviour of electrochemically heterogeneous electrodes—for example composites whose different spatial zones display contrasting electrochemical behaviour toward the same redox couple.     

Direct electron transfer of PQQ-glucose dehydrogenase at modified carbon nanotubes electrodes

In order to establish efficient enzyme-electrode-contacts for the pyrroloquinoline quinone dependent glucose dehydrogenase (PQQ-GDH) different immobilisation strategies are investigated. Multi-walled carbon nanotubes (MWCNT) on gold electrodes are modified by chemical treatment and by (poly)-aniline derivatives. The electropolymerisation of methoxy-m-anilinesulfonic acid and m-aminobenzoic acid on the MWCNTs allows the covalent coupling of the PQQ-GDH. Such a poly-[ASA-ABA]/MWCNT/Au electrode can achieve current densities of up to 500 μA/cm² at a potential of 100 mV vs. Ag/AgCl. Furthermore investigations with small amounts of free PQQ indicate direct electron transfer between enzyme and electrode.     

Mediator-modified electrodes for catalytic NADH oxidation: high rate constants at interesting overpotentials

Carbon paste electrodes were modified with a nitrofluorenone derivative, 2,4,7-trinitro-9-fluorenone, adsorbed on zirconium phosphate (ZP). After electrochemical reduction of the fluorenone derivative, it turns into a very efficient mediator for electrocatalytic NADH oxidation, with a formal potential of about +250 mV vs. Ag/AgCl. The electrochemistry and the electrocatalytic properties of the mediator were investigated with cyclic voltammetry and rotating disk electrode methodology. The second order rate constant with NADH was evaluated and found to be higher than 10(6) M(-1) s(-1), thus approaching true diffusion controlled currents for NADH oxidation.     

Theory of photoinduced heterogeneous electron transfer

We consider electron injection into the conduction band of a semiconductor, from an electronically excited state of a dye molecule, adsorbed on its surface. For arbitrary width of the conduction band, the survival probability of the excited state can be calculated using a Green's-function approach. We show that the existence of a split-off state can play an important role in the total injection probability. In the wide band limit, the survival probability decays exponentially, but for finite band widths it does not. We further investigate the effect of vibrations on the process. A Green's operator technique may be used to solve this too exactly. We show that the problem may be reduced to a non-Hermitian eigenvalue problem for the vibrational states alone. Exact results can be obtained for arbitrary bandwidth and for a few vibrational degrees of freedom. In the wide band limit, the dynamics is particularly simple and we find that (1) the survival probability of the excited state is unchanged by the inclusion of vibrational motion, but (2) each vibrational state now has a finite lifetime. Numerical results are presented for the effects of reorganization energy, energy of the injecting level, and the variation of the matrix element for the electron injection, on the survival probability of the electron in the excited state. As an illustration of the approach, we also present results of numerical calculation of the absorption spectrum of perylene adsorbed on TiO(2) and compare it with experimental results.     

Estimation of electron transfer rate constants by static (optical and electrochemical) measurements

A kinetic study of the reduction of Co(NH₃)₅(DMSO)³⁺ (DMSO=dimethylsulfoxide) by Fe(CN)₆⁴⁻ has been carried out in several water–cosolvent mixtures at 298.2 K. The cosolvents used were ethylene glycol, acetonitrile, methanol and glucose. The free energies of activation for this thermal electron-transfer reaction have been calculated from a combination of spectroscopic and electrochemical data and compared with those obtained from the kinetic study. Quantitative agreement is found between both series of data. This shows the possibility of estimating activation free energies for electron transfer reactions from these (static) measurements.     

Ultrafast electron dynamics at ice-metal interfaces: competition between heterogeneous electron transfer and solvation

Microscopic insight into heterogeneous electron transfer requires an understanding of the participating donor and acceptor states and of their respective interaction. In the regime of strong electronic coupling, two limits have been discussed where either the states overlap directly or the states are separated by a potential barrier. In both situations, the transfer probability is determined by the magnitude of the wave function overlap, whereby in the case of the potential barrier, its width and height are rate limiting. In our study, we observe a dynamical crossover between these two regimes by investigating the electron-transfer dynamics of localized, solvated electrons at ice-metal interfaces. Employing femtosecond time-resolved two-photon photoelectron spectroscopy, we analyze the population dynamics of excess electrons in the ice layer, which experience the competing processes of transfer to the metal electrode and energetic stabilization in the ice by molecular reorientation. Comparing the dynamics of D(2)O on Cu(111) and Ru(001), we observe an early regime at t < 300 fs, where the transfer time is determined by wave-function overlap with the metal and a second regime (t > 300 fs), where the transfer proceeds nearly independent of the substrate. The assignment of these two regimes to the established mechanisms of electron transfer is backed by an empirical model calculation that reproduces the experimental data in an excellent manner.     

Free energy correlation of rate constants for electron transfer between organic systems in aqueous solutions

Recent experimental data concerning the rate constants for electron transfer reactions of organic systems in aqueous solutions and their equilibrium constants is examined for possible correlation. The data is correlated quite well by the Marcus theory, if a reorganization parameter, λ, of 18 kcal/mole is used. Assuming that the only contribution to λ is the free energy of rearrangement of the water molecules, an effective radius of 5A?for the reacting entities is estimated. For the zero free energy change reaction, i.e., electron exchange between a radical ion and its parent molecule, a rate constant of about 5 × 10⁷ M^?1 s^?1 is predicted.     

Polymer films on electrodes: Part XVIII. Determination of heterogeneous electron transfer kinetics at poly(vinylferrocene) and nafion/Ru(bpy)2+3 polymer-modified electrodes by convolution voltammetry

Convolution voltammetry was used to evaluate the rates of heterogeneous charge transfer to ferrocene groups in poly(vinylferrocene) and to Ru(bpy)²⁺₃ in Nafion-modified electrodes under semi-infinite conditions. This technique allows correction for uncompensated resistance and double layer capacitance, as well as detrmination of the diffusion coefficient, D, transfer coefficient, α, and half-wave potential, E_1/2, from a single cyclic voltammogram. Vinylferrocene in solution and a bound copolymer of vinylferrocene and styrene in a ratio of 58:42 were also examined. For the polymer films, the heterogeneous charge transfer rate constants, k°, are 10^?4 ≥ k° ≥ 10^?5 cm/s; these values are about two order of magnitude smaller than those for the similar species in homogeneous solution. The values of k°/D^1/2, however, are comparable to those in soluton; 10 > (k°/D^1/2) > 0.1 s^?1/2.     

Determination of rate constants of exciplex formation

Theoretical and Experimental Chemistry -     

Photoinduced electron transfer reactions at methylene blue adsorbed nafion and clay coated electrodes

Nafion and montmorillonite clay adsorbed methylene blue coated onto platinum electrode were prepared. These dye modified electrodes were used as photoelectrodes in a photogalvanic cell in the presence of Fe²⁺ ions. The photoelectrochemical investigations showed that the dye coated electrodes behaved as cathode upon irradiation whereas the plain platinum electrode dipped in a homogeneous solution containing methylene blue and Fe²⁺ ions behaved as anode. It is suggested that the intermediate complex formed between the photoreduced methylene blue and ferric ion lead to the reductive reaction at the coated electrode.     

Determination of the activation rate constants of alkyl halide initiators for atom transfer radical polymerization

The second-order activation rate constants k_A for low-molar-mass alkyl halides catalyzed by cuprous halide complexes Cu(I)X/2L (X = Cl or Br; L = 4,4′-diheptyl-2,2′-bipyridine) were determined by the nitroxide capping method along with ¹H NMR. The k_A for 1-phenylethyl bromide, a typical initiator for atom transfer radical polymerization (ATRP), with the Cu(I)Br complex was found to be close to the known value of the k_A for a polystyryl bromide, being large enough for the initiation to be completed at an early stage of polymerization. It was also found that k_A strongly depends on the kind of halogen and the steric factor of the alkyl halide in question.