Davis K. Cope

Diffusion current at a band electrode by an integral equation method

Abstract The diffusion current at a finite band electrode of length L and width W in a cell of finite width (W) but infinite length and depth is obtained by an integral equation method which offers a promising alternative to finite difference methods. A Laplace transform with respect to time T is applied to the original diffusion problem, resulting in a boundary value problem for the transformed current I(S). This new problem appears typically as an integral equation for I(S), which is then obtained and inverted numerically to yield the current I(T). The diffusion current at a hemicyclindrical electrode is also obtained as a check of the method, and a comparison of the currents obtained at these two electrode geometries is presented. Approximate formulae are provided which permit calculation of current (to within ca. 1%) at a band electrode at short (0 1 2 1 2 > 4.9 L) times.

Transient behavior at planar microelectrodes: Diffusion current at the disk electrode by the integral equation method

Abstract The transient diffusion-limited current at the ring electrode is calculated by the integral equation method over significant time intervals (to steady state or near steady state) for the full range of significant ring sizes. We also report on the development of generic programs for the integral equation method which will aid in the implementation of this method by others working in this area. The results of this paper were obtained by the application of these programs to the ring electrode kernel.

Transient behavior at planar microelectrodes: A comparison of diffusion current at ring, band and disk electrodes

Abstract The integral equation method is an efficient, accurate numerical approach for computing transient current at planar microelectrodes. This method has been used to compute the diffusion-limited current at disk, band, and ring electrodes. In this paper we present a critical assessment of the influence of geometry on the behavior of the transient current. The disk and band geometries are shown to be limiting forms of the ring geometry. A normalization is introduced which permits differences in transient behavior at the various geometries to be ascribed to purely geometric factors.

Transient behavior at planar microelectrodes: Diffusion current at a band electrode by an integral equation method. Part II

We have previously formulated an integral equation method for the calculation of current in response to a potential step at a microelectrode and have applied it to obtain the diffusion-limited current at a band electrode. For systematic application to other electrode boundary conditions and cell geometries, generic programs for solution of relevant integral equations and inversion of resulting Laplace transforms have been developed. In this paper, we report on initial uses of these new programs for the band electrode case, including calculation of diffusion-limited current over a range 40 times greater than our earlier report and comparison with experiment.

Representation of reversible current for arbitrary electrode geometries and arbitrary potential modulation

Abstract For simple electron transfer O+ n e − ⇄ R with equal diffusion coefficients ( D o = D R ), it is shown that reversible current can be expressed in terms of diffusion-limited current for arbitrary cell/electrode geometries and arbitrary time-dependent potentials. This result provides a better understanding of a formula due to Aoki and co-workers and suggests new experimental approaches which simplify interpretation of data obtained at microelectrodes.

Calculation of convective diffusion current at a strip electrode in a rectangular flow channel: Inviscid flow

Abstract The determination of analytes in flowing streams has become a major area of research in electroanalytica chemistry, due in large part to the increasing popularity of amperometric and coulometric detection in liquid chromatography and flow injection analysis. Experimental results from our laboratory suggest that the replacement of the traditional macroelectrode (ca. 2–3 mm in size) by a collection of microelectrodes (each less than ca. 50 μm in size) in the flow channel of an amperometric detector can result in a signal enhancement leading to improved limits of detection. A theoretical analysis of convective diffusion current at a single microelectrode provides insight into the origins of this signal enhancement. In this paper we calculate the convective diffusion current as a function of flow rate and electrode size at a single strip electrode placed on one boundary of a rectangular flow channel, the one assumption in the calculation being that of inviscid flow. The electrode sizes investigated extend into the microelectrode region and the flow rates range from coulometric behavior (slow flows) to amperometric behavior (fast flows). Significantly, these calculations include the contribution of lateral diffusion (the edge effect) to the current, a contribution which becomes more appreciable as electrode size is reduced and which is responsible in part for the signal enhancement obtained with microelectrode ensembles. Further enhancement of current due to depletion layer recharge is explored by a simple analysis of a sequence (array) of strip electrodes.

Transient behavior at planar microelectrodes: Diffusion current at the ring electrode. A comparison of experiment with theory

Abstract In the preceding paper we described the transient behavior of diffusion current in response to a potential step at ring microelectrodes, as predicted from numerical calculations using the recently introduced integral equation method. This method is applicable to rings of arbitrary thickness and diameter and describes accurately the current over several orders of magnitude in time. In this report we describe results of transient measurements of diffusion current at ring electrodes of various dimensions and compare the experimentally observed transients with those predicted from the integral equation method.

Diffusion current at the tubular band electrode by the integral equation method

Abstract The integral equation method is used to compute the transient diffusion current at the tubular band electrode, a geometry consisting of a band electrode inlaid concentrically around the inside wall of an infinitely long insulating tube. When the ratio μ of diameter-to-width of the tubular band electrode is large (at least 100) the transient current closely resembles that obtained at the planar band electrode, with deviation becoming apparent only at very long time. For μ > 2 the edge effect enhancement of the diffusion current more than compensates for the depletion effect and the limiting current exceeds the Cottrell current at longer times, the net enhancement being proportional to μ. For μ t −1 2 dependence (i.e. a second Cottrell region), corresponding to complete depletion of the electroactive species within the solution space encompassed by the electrode and the resulting rectilinear diffusion of the species from each direction within the tube.

Simplifications in the theory of step experiments at microelectrodes

Abstract Theoretical studies of the chronoamperometry of the reaction O+ n e− ⇄ R often consider (1) the diffusion-limited reaction, (2) the totally irreversible reaction, and (3) the reversible reaction (governed by the Nernst equation), and these can all be considered as special limiting cases of (4) quasi-reversible reactions, electron transfer kinetics governed by the general current-potential characteristic. Cases (1) and (2) simplify to one-component diffusion systems for the concentration of O; cases (3) and (4) are necessarily two-component diffusion systems involving the concentrations of both O and R. We show that, when O and R have equal diffusion coefficients, the solution for the reversible reaction (3) can be expressed in terms of the solution for the diffusion-limited reaction (1) and that the solution for the quasi-reversible reaction (4) can be expressed in terms of the solution for the totally irreversible reaction (2). These results hold for arbitrary electrode geometries.

Calculation of convective-diffusion current at multiple strip electrodes in a rectangular flow channel implications for electrochemical detection

Abstract If a single strip channel electrode (of mm dimension, for example) is subdivided into a number of smaller strip electrodes (perhaps microelectrodes of μm or smaller dimension) separated from one another by insulating regions of the channel boundary, the convective-diffusion current density at the array of smaller electrodes is enhanced relative to that at the larger single electrode. A spectral method is used to examine theoretically the origins of this current enhancement at microelectrode arrays. Assuming viscous flow but ignoring edge diffusion, we first assess the contribution to the enhancement from depletion layer recharge which takes place as analyte flows over the insulating regions between electrodes of the array. Then, extending results of calculations previously reported for inviscid flow, we examine the importance of longitudinal edge diffusion at small electrodes and its contribution to the current under convective conditions. The implications of these results are then discussed from the point of view of designing electrochemical detectors having higher signal-to-noise ratios for improved detection in flowing streams.