Regular arrays of microdisc electrodes: simulation quantifies the fraction of 'dead' electrodes

Arrays of microdisc electrodes have found widespread use in electroanalysis. These are commonly produced lithographically and practical arrays may contain up to hundreds of individual disc electrodes (e.g. of gold, platinum, indium,...) to maximise sensitivity and minimise limits of detection. Typically, however, the lithographic fabrication process is imperfect resulting in a significant fraction (often tens of percent) of electrochemically inactive electrodes. We demonstrate that a 2-dimensional simulation based on the diffusion domain approximation in conjugation with simple experiments on the ferrocyanide redox couple in aqueous solutions can be used to rigorously 'count' the number of active electrodes in a non-destructive fashion. The agreement with an independent count in which active electrodes are identified via electro-plating with copper followed by ex situ microscopic examination is quantitatively excellent.     

Adaptive finite element simulation of currents at microelectrodes to a guaranteed accuracy. ECE and EC2E mechanisms at channel microband electrodes

We extend the work in the accompanying paper (K. Harriman et al., Electrochem. Commun. 2 (2000) 567) on the use of adaptive finite element methods to simulate the current for a steady state E reaction mechanism at a channel microband electrode to the more complex ECE mechanism, and the non-linear EC₂E mechanism. We again use the standard Galerkin approach for the diffusion dominated (low-flow) case, and the streamline diffusion finite element method (SDFEM) for convection-dominated (high-flow) case, and compare our results with previous numerical and analytical approximations. We give a general discussion on the implications of our results for numerical simulation at microelectrodes.     

Adaptive finite element simulation of currents at microelectrodes to a guaranteed accuracy. Theory

In our accompanying paper (K. Harriman et al., Electrochem. Commun. 2 (2000) 150) we demonstrated how, for the finite element method, an automatic mesh-adaptation algorithm can be derived, given an a posteriori error bound on the simulated current. In this paper we give the technical details of the background theory of the finite element method, and the derivation of the a posteriori error bound.     

Double potential step chronoamperometry at spherical electrodes and microelectrodes

A general and explicit analytical easy solution for double potential step chronoamperometry applicable to microelectrodes is presented. This solution considers that both electroactive species are initially present.The solution indicates that the smaller the electrode radius, the more notorious the influence of the diffusion coefficient ratio, so the use of microelectrodes is of great utility for determining the diffusion coefficients of the electroactive couple. The analytical solutions given until now can only be considered as partial solutions for this problem, since they lead to absurd results for electrode radii <10⁻² cm.This equation has also been applied to the study of amalgamation processes and it could also be applicable to the study of the ionic transfer at hanging and microspherical electrolyte drop electrodes (i.e., liquid–liquid interfaces). In these last cases, the use of this equation is limited by the validity of Koutecký’s approximation, which neglects the electrode finite volume.     

Effects of resistance and capacitance on the chronoamperometry of polymer-coated electrodes as modeled by a finite elements digital simulation

A finite elements digital simulation of chronoamperometry with polymer-coated electrodes is presented. The model takes account of mass transport by diffusion and migration, and the effects of the uncompensated resistance and capacitance of the system. A graphical method for the analysis of experimental data was introduced and applied to electrodes coated with poly[Ru(II)(bipy)₂(4-vpy)₂]⁺PF⁻₆.     

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.     

Cyclic voltammetry and chronoamperometry of lead ions at carbon fibre electrodes

Cyclic voltammetry of reversible redox couples. e.g., hexacyanoferrate (III)/(II), at carbon fibre electrodes yields sigmoidal anodic and cathodic scans that are almost superimposed. However, the cyclic voltammogram of lead (II) ions was markedly different. The voltammogram, and results from chronoamperometric measurements, can be interpreted on the basis of lead nucleation.     

Determination of the number of electrons by chronoamperometry at small electrodes

The number of electrons transferred, n, was determined by taking the ratio of the square of the slope to the intercept of Cottrell plots by chronoamperometry at a small disk electrode without knowing values of diffusion coefficients. The intercept is ascribed to the contribution of spherical diffusion. This technique is useful for estimating reaction mechanisms of unstable species of which lifetime is shorter than the period of bulk electrolysis. It takes advantage not only of getting the simultaneous acquisition of two independent variables (the slope and the intercept) in one chronoamperometric curve but also of evaluating the dependence of n for multi-step charge transfer reactions on electrode potentials. Ferrocene and N,N′-bis(4-methoxyphenyl)-N,N′-diphenyl-1,4-phenylenediamine were used for redox species at n = 1 and 2, respectively. Chronoamperometric currents were obtained at the platinum disk electrode 50 μm in radius for the time less than 3 s, responding to the various stepped potential values. Values of n were obtained at each potential even for sluggish charge transfer reactions.     

Adaptive finite element simulation of currents at microelectrodes to a guaranteed accuracy. An E reaction at a channel microband electrode

We extend our earlier work (K. Harriman et al., Electrochem. Commun. 2 (2000) 150) on adaptive finite element methods for disc electrodes to the case of an E reaction mechanism to the increasingly popular channel microband electrode configuration. We use the standard Galerkin finite element method for the diffusion-dominated (low-flow) case, and the streamline diffusion finite element method for the convection-dominated (high-flow) case. We demonstrate excellent agreement with previous approximate analytical results across the range of parameters of interest, on comparatively coarse meshes.     

Adaptive finite element methods in electrochemistry

In this article, we review some of our previous work that considers the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the edge effect. Our approach to overcoming this problem has involved the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm, allowing calculation of the quantity of interest (the current) to within a prescribed tolerance. We illustrate the generic applicability of the approach by considering a broad range of steady-state applications of the technique.     

Chronoamperometric current at finite disk electrodes

The chronoamperometric current at a stationary finite disk electrode is studied using both analytical and digital simulation techniques. The exact long-time expansion of the current is obtained and its short-time behavior is considered. Digital simulation of the current using an explicit hopscotch algorithm is presented. In contrast to the usual explicit difference method, the ‘hopscotch’ algorithm is unconditionally stable, and thus, it is particularly suited for studying electrochemical problems at intermediate and long times. A simple analytic expressions for the current, which is accurate to 0.6% for all times, is presented.     

超微盘电极上双电位阶跃计时电流和计时库仑法理论及验证

本文报道超微盘电极上双电位阶跃计时电流和计时库仑法理论,用自制的微机多功能电分析仪及亚铁氰化钾、氯化钾体系进行验证,实验结果与理论相符合.     

Anodic stripping voltammetry at in situ bismuth-plated carbon and gold microdisc electrodes in variable electrolyte content unstirred solutions

Carbon and gold microdisc electrodes (30 and 10 μm, respectively) have been tested as substrates for in situ bismuth film plating from unstirred solutions of variable acetate buffer content and were subsequently used in the anodic stripping voltammetry determination of Pb(II) and Cd(II) ions. The effects of Bi(III) concentration, analyte accumulation time, stirring as well as supporting electrolyte content have been studied. Under optimal conditions good voltammetric responses were obtained by means of square wave anodic stripping voltammetry in unstirred analyte solutions of 5 × 10⁻⁸ to 10⁻⁶ M, even in the absence of added buffer. In an indicative application, Pb(II) ion levels were determined in tap water using bismuth-plated carbon microdisc electrodes.     

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.     

Steady state voltammetry at non-uniformly accessible electrodes: a study of Tafel plots for microdisc and tubular flow electrodes in the reversible and irreversible limits of electron transfer

Steady state concentration profiles are obtained via numerical solution for an electron transfer at the microdisc and tubular flow electrodes. The non-uniform flux profiles over the electrode surfaces and their effect on the voltammetric waveshape are discussed. An approach is suggested for the most appropriate way to estimate the charge transfer coefficient from an experimental voltammogram of an irreversible electron transfer.     

Adaptative finite element simulation of currents at microelectrodes to a guaranteed accuracy. Application to a simple model problem

In this series of papers we consider the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the ‘edge-effect’. Our approach to overcoming this problem involves the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm. This allows us to calculate the current to within a prescribed tolerance. We begin by demonstrating the power of the method for a simple model problem — an E reaction mechanism at a microdisc electrode — for which the analytical solution is known. In this paper we give the background to the problem, and show how an a posteriori error bound can be used to drive an adaptive mesh-generation algorithm. We then use the algorithm to solve our model problem and obtain very accurate results on comparatively coarse meshes in minimal computing time. We give the technical details of the background theory and the derivation of the error bound in the accompanying paper.     

Computational electrochemistry: finite element simulation of a disk electrode with ultrasonic acoustic streaming

Computer simulations using the finite element method (FEM) are used to predict the correlation between the transport limiting current (I(lim)) and parameters such as diffusion coefficient, source to electrode separation, source power, and medium viscosity for a sonicated disk electrode in "face on" mode. The fluid dynamics and diffusion layer are modeled directly using FEM and predict that the electrode is uniformly accessible, I(lim) is proportional to the diffusion coefficient to the 2/3 power and I(lim) is proportional to the square root of the source power. Curves are also calculated relating I(lim) to the source to electrode separation and liquid viscosity.     

Finite element simulation of the amperometric response of recessed and protruding microband electrodes in flow channels

The convective mass transfer for recessed and protruding microband electrodes, two geometries found in practical devices, is determined by the finite element method. The problem is solved by a two-step method, first solving the Navier-Stokes equation to compute the real hydrodynamic flow, then solving the convective diffusion equation to calculate the electrochemical response of the systems. In this work, we focus on the chronoamperometric response of recessed and protruding microband electrodes, emphasizing the role of the edge effects and convection streamlines, in particular the role of stagnant recirculating eddies due to these particular geometries.     

Steady state simulation of electrode processes with a new error bounded adaptive finite element algorithm

In a series of papers, Harriman et al. [1], [2], [3], [4], [5] have presented a reliable means of simulating steady state currents with adaptive finite element. They have demonstrated that multi-step, even non-linear, mechanisms, alongside convection, can be incorporated. However, there is considerable complication in the mathematical approach taken, and it would seem to be limited as it stands to certain types of electrode reaction models – notably those without heterogeneous kinetics or transient effects. In this paper we discuss alternative approaches to error estimation and adaptivity, and present a simpler formulation, capable of simulating systems with heterogeneous kinetics; transient simulations also appear more attainable. We introduce, apparently for the first time in electrochemistry, the use of gradient recovery methods [6] to both error estimation and accurate current calculations. The result is an algorithm with considerably more potential for generalisation, closer to the ideal of an entirely flexible automatic simulation program, capable of dealing with any mechanism or electrode geometry. In tests we find our method to perform more efficiently than that cited above, producing accurate results with simpler meshes in less time.