Hydrodynamic voltammetry at channel electrodes: Part VII. Current transients at double channel electrodes

Expressions for the transient current at the downstream electrode in response to galvanostatic electrolysis at the upstream electrode in the channel flow cell were derived by applying double Laplace transformation when the electrode reaction at the upstream electrode is kinetically controlled. The ratio of the transient current to the steady state current or the transient collection efficiency was calculated as a function of electrode geometry and θ

, where U_m is the mean flow velocity in the channel cell, D the diffusion coefficient of the electroactive species, b the half height of the channel, x₁ the length of the upstream electrode and t the time elapsed since the beginning of the galvanostatic electrolysis at the upstream electrode. Curves for the transient collection efficiency can be applied to evaluating the amount of adsorption at the upstream electrode when metal at the electrode is anodically dissolved in solution. Digital simulation was carried out. Transient curves, obtained analytically, were in good agreement with those evaluated from the digital simulation. In order to allow one to draw transient curves readily, we derived a simple approximate equation.     

Hydrodynamic voltammetry at channel electrodes: Part III. Theory of kinetic currents

A theory of kinetic currents at a channel electrode is developed without any assumption on the magnitude of the rate constant of the preceding chemical reaction. An approximate equation with reasonable accuracy which relates the limiting current to the rate constant, flow velocity and the electrode geometries, is presented. The validity of the equation derived from the concept of the thin reaction layer is also discussed.     

Hydrodynamic voltammetry at channel electrodes: Part II. Theory of first-order kinetic collection efficiencies

The theory of first-order kinetic collection efficiencies at the double channel electrode is developed for the following two schemes: (I) A±n₁e→B (at the generator electrode), B→^kP (in solution), B±n₂e→Y (at the detector electrode), (II) A±n₁e→B, B→^kA, B±n₂e→Y. The exact expressions for the kinetic collection efficiencies are obtained as ascending and asymptotic series with respect to the kinetic parameter. Further, approximate formulae in exponential forms are given, which hold within an error of about 2% for conventional electrode geometry. Finally, the validity of the approximate procedure, which has been used previously to obtain the kinetic collection efficiencies for fast homogeneous reactions, is discussed in comparison with the present theory.     

Hydrodynamic voltammetry at channel electrodes: Part VIII. Theory of reversible voltammograms for chronoamperometry and linear sweep voltammetry

An expression for the reversible transient current at a channel flow electrode was derived when a potential with any time variation was applied to the electrode. First, the expression was adapted to chronoamperometry. As the electrolysis time elapses, the current density distribution varies from a Cottrellian uniform distribution to a non-uniformly steady-state distribution. Second, the expression for the reversible linear sweep voltammogram was derived. For small values of a dimensionless potential sweep rate, the voltammograms are in a sigmoidal form similar to the steady-state curve. As the values become large, the voltammograms have a peak which is often observed in quiescent solution. The dependence of the peak current and the potential at the half peak current on the dimensionless potential sweep rate was examined.     

Hydrodynamic voltammetry at channel electrodes: Part IV. Theory of chronopotentiometry for reversible electrode reactions

The theory of chronopotentiometric measurements at channel electrodes is developed for reversible electrode reactions, by rigorously solving the corresponding time-dependent boundary value problem, where the non-uniform accessibility of the channel electrode surface is taken into consideration. The theoretical equations of the transition time and the potential-time curve are derived as functions of (I_d/I), where I denotes the applied current intensity and I_d the limiting diffusion current obtained at steady-state. Finally, the time variation of the current density distribution at the electrode surface is given.     

Hydrodynamic voltammetry at an interdigitated electrode array in a flow channel: Part I. Numerical simulation

This paper describes the numerical simulation of convective diffusion at an interdigitated electrode array, consisting of multiple pairs of microelectrodes held at alternating applied potentials on one wall of a flow channel. The downstream microelectrode of each pair detects species generated at the upstream microelectrode. Concentration profiles in the channel, amperometric response, and signal-to-noise ratios for the detector electrodes are calculated. The simple backward implicit finite difference (BIFD) simulation approach is applicable for a wide range of channel conditions. The upper number of electrode pairs treatable is limited only by computational time. The agreement of the simulation with previous results for a single pair of electrodes under comparable conditions is very good. Substantial improvements in signal-to-noise ratio are predicted for the multi-electrode interdigitated electrode array relative to a single generator-detector pair of equal overall area. Electrode dimensions are discussed for optimum signal/noise ratio. Relative enhancement increases significantly with the number of generator-detector pairs.     

Modified carbon-containing electrodes in stripping voltammetry of metals. Part II. Composite and microelectrodes

The second part of the review, which covers modified carbon-containing electrodes, describes composite and microelectrodes. Electrodes made of commercial and laboratory carbon-containing composite materials are discussed. Impregnated and thick-film electrodes and microelectrodes made of carbon fibers form a separate group. Various modifiers and methods of electrode modification are presented. Prospects for the future development of solid-state modified electrodes are considered.     

Voltammetry at microcylinder electrodes: Part I. Linear sweep voltammetry

An expression for reversible linear sweep voltammograms at stationary microcylinder electrodes is presented. From numerical calculations theoretical voltammograms are obtained for various values of the dimensionless parameter, p=(nFa²v/RTD)^{$\text{1}\text{2}$}, where a is the radius of the electrode, v the potential sweep rate and D the diffusion coefficient. The peak current and the peak potential are evaluated from the voltammograms as functions of p and are expressed by approximate equations with high precision. In order to examine the validity of the equations, an experimental study was made at platinum wire micro-electrodes (a=10?10) μm). The experimental voltammograms were in good agreement with the ones predicted theoretically for various values of the sweep rates and for several different radii of the electrodes.     

Hydrodynamic voltammetry with channel microband electrodes: Resolution of ECE and DISP1 processes

Gold microband channel electrodes (30 μ m in length) are used to study the reduction of ortho- bromonitrobenzene in dimethylformamide solution. An ECE mechanism is shown to operate and a rate constant of 250 s⁻¹ for the chemical step is deduced. The ability to distinguish between ECE and DISP1 processes with the experimental protocol adopted is noted.     

Potentiostatic voltammetry at spherical electrodes and microelectrodes in the presence of product

An explicit analytical solution for the complete anodic–cathodic I–E–t response of a reversible electrode process in presence/absence of amalgamation under potentiostatic conditions is presented. To obtain this solution we have taken into account the electrode curvature, using in the case of amalgamation, Koutecký’s approximation (i.e. the finite electrode volume has not been considered). Explicit expressions for the concentration profiles and the surface concentrations have been also deduced. All the results are applicable to electrodes of any radius including ultramicroelectrodes when both species are soluble in the electrolytic solution for any value of the ratio D_O/D_R. When amalgamation takes place, analytical results were compared with the numerical ones deduced by using the rigorous condition of null net flux in the electrode centre, pointing out that Koutecký’s approximation remains valid even for electrode radius and time values at which the diffusion layer reaches the electrode centre. For high electrode sphericities the I–t curves present a cross-linking for applied potential values higher than the equilibrium potential which is the cause of the appearance of a “peak” near to the anodic limit of the I–E curve.     

Voltammetry at microcylinder electrodes: Part IV. Normal and differential pulse voltammetry

Expressions for reversible normal pulse (NP) and differential pulse(DP) voltammograms at microcylinder electrodes, which are powerful and convenient tools in in vivo measurements, were derived in connection with the effects of successive pulse electrolysis. These expressions were computed numerically, varying the pulse interval, τ, and the pulse width, δ. Consequently, it was found that the use of τ>10δ for NP and τ>5δ for DP did not distort the well-known shapes of the voltammograms. By choosing τ=10δ for NP voltammetry or τ=5δ for DP voltammetry, the analysis time was greatly reduced. Methods of analysing reversible NP and DP waves are presented. NP and DP measurements were carried out at platinum and carbon fiber microcylinder electrodes for various values of τ and δ. The minimum values of δ for undistorted voltammograms found experimentally were τ = 5δ for NP voltammetry and τ = 2δ for DP voltammetry.     

Chronoamperometry and cyclic voltammetry at conical electrodes, microelectrodes, and electrode arrays: theory

The finite difference method is used to simulate chronoamperometry, linear sweep voltammetry, and cyclic voltammetry at conical electrodes and microelectrodes. Techniques for the numerical simulation of these processes at microdiscs are adapted and extended to accurately model diffusion to the electroactive cone surface. Simulated results are analyzed, and trends are rationalized in terms of the cone apex angle, alpha. The diffusion domain approximation is used to extend the theory to regular and random arrays of conical electrodes.     

Voltammetry at partially covered electrodes: Part II. Linear potential sweep and cyclic voltammetry

Effect of inhomogeneity of the electrode surface on the linear potential sweep and cyclic voltammograms is investigated theoretically and experimentally using model electrodes partially covered with photoresist layer. Good agreement between the theoretical and experimental results is obtained.     

Computational electrochemistry: Lattice Boltzmann simulations of voltammetry at microelectrodes

This paper describes the development of a two-dimensional lattice Boltzmann method for the simulation of mass transport at a range of microelectrode geometries using potential step and linear sweep voltammetric techniques. The simulations of the current response for microband and micro hemicylinder geometries were compared with previously published analytical solutions. Excellent agreement between the numerical models and the analytical solutions was observed for a series of experimental parameters. To illustrate the flexibility of the lattice Boltzmann method the simulation of the current response for a range of electrode geometries, distorted from microband electrode geometry to micro hemicylinder electrode, is also described.     

The dual reciprocity method: simulation of potential step voltammetry at microelectrodes

The dual reciprocity method (DRM) is presented as an efficient and powerful method for the analysis of electrochemical processes occurring under diffusional transport control. This article outlines the theory and numerical details required to apply the DRM for modelling chronoamperometry at microband and microcylinder electrodes. The results obtained are found to be in good agreement with analytical approximations for the two geometries. The benefits of the DRM approach are discussed, including the reduction in dimensionality brought about by the formulation procedure which permits a two dimensional diffusion problem to be modelled using a one dimensional boundary mesh. The flexibility of the method for tackling geometries of complex topography are noted.     

Voltammetry at microcylinder electrodes: Part III. Chronopotentiometry

Expressions for chronopotentiometry at stationary microcylinder electrodes are presented. The transition time, τ, is expressed as a function of a single parameter λ ( = nFc^*_RD / ia, where a is the radius of the electrode, D is the diffusion coefficient, i is the current density, n is the number of transferred electrons, F is the Faraday constant and c^*_R is the bulk concentration. When the values of λ are small, the transition-time constant, i√τ / c^*_R, depends linearly on λ, with the intercept predicted from the Sand equation. Conversely, when values of λ increase, the transition time also increases exponentially. Therefore transition necessarily occurs no matter how small a current flows. An approximate equation for the transition time is presented, from which one can evaluate the diffusion coefficient. Equations for the potential-time curve and the quarter-wave potential are also obtained. The equations were tested experimentally using carbon fiber electrodes (a = 4.1 μ m) and platinum wire electrodes (a = 10-100μ m). The transition times obtained experimentally were in good agreement with those predicted theoretically for various values of the applied current, for several different radii of the electrodes.     

Voltammetry at partially covered electrodes: Part I. Chronopotentiometry and chronoamperometry at model electrodes

Equations for chronopotentiometry and chronoamperometry at partially covered electrodes have been derived using a model of hexagonal array of cylidrical spaces terminated, at the electrode surface, by concentric active and inactive regions. The boundary value problem was shown to be analogous to that for a charge transfer preceded by a chemical reaction. Experiments with the reduction of ferricyanide on gold model electrodes partially covered with photoresist layer showed excellent agreement with the theory. Application of the equations to estimation of coverage and size of active sites distributed on a electrode surface is discussed.     

Voltammetry at partially covered electrodes: Part III. Faradaic impedance measurements at model electrodes

Faradaic, impedances at model electrodes partially covered with a photoresist layer have been studied theoretically and experimentally. Equations for the faradaic impedance are derived based on the theoretical model and approach described in Part I of this series of papers. Experimental data for the hexacyanoferrate system at various model electrodes give excellent agreement, with theoretical predictions for the diffusion impedance behavior, and the applicability of the derived equations to the estimation of the degree of coverage and the size of the active regions is confirmed. Furthermore, the application of such model electrodes to the kinetic study of electrode reactions with high heterogeneous charge transfer rates is suggested.