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.     

Adaptive finite element simulation of chronoamperometry at microdisc electrodes

In this paper we shall introduce a transient finite element algorithm by considering the simplest problem of numerical simulation of the chronoamperometric current for an E reaction mechanism at a microdisc electrode. Such a numerical simulation is made difficult by the presence of a boundary singularity where the electrode meets the insulator (often known as the ‘edge-effect’) and by a time-singularity caused by the impulsive start of the experiment. We attempt to overcome these problems by using an adaptive finite element algorithm in which we derive an a posteriori bound on the error in the computed current. This is used to guide mesh refinement and adaptive time-stepping, resulting in a fully automated algorithm which is both accurate and efficient.     

How accurate is your two-dimensional numerical simulation? Part 1. An introduction

Numerical simulations of the diffusion processes at electrode surfaces are subject to three sources of error: those arising in the calculation of the concentration values, those arising in the numerical approximation of the flux at the electrode surface, and those arising from the integration of the flux over the electrode surface. In this paper we investigate the effects of each type of error on the accuracy of numerical simulation at the microdisc electrode by solving the steady state problem (for which the analytical solution is known). We are able to show that the major source of error is due to the boundary singularity at the electrode edge. By introducing a simple model problem, we demonstrate that the theoretical rates of convergence of the standard finite difference schemes can be attained in the absence of a boundary singularity, but these rates are destroyed by the presence of the singularity when solving for the electrode problem. Finally, we show that it is not possible to recover accuracy using n-point flux calculations or spline functions at the electrode edge.     

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.     

The boundary element method: applications to steady-state voltammetric simulations within domains extending to infinity

Application of the boundary element method (BEM) to the simulation of diffusion limited electrolysis reactions occurring within an infinite domain is outlined. This article focuses on the development of procedures that permit electrolysis simulations to be performed, where only an element mesh over the electrode region is required. This approach provides significant benefits over the traditional application of BEM simulations for electrochemical-based problems. In particular the reduction in mesh points to regions only over the electrode results in simpler grid generation procedures and significantly reduced computational times. The paper describes the theory and numerical details required to develop steady-state two-dimensional diffusional models for voltammetric simulations. The accuracy and versatility of the numerical procedures are tested by examining the current density at a range of well-defined electrode geometries.     

The excess current in cyclic voltammetry arising from the presence of an electrode edge

The popular inlaid disc electrode suffers from an edge effect that is usually, and sometimes unwarrantedly, ignored in analyzing transient voltammograms. This study addresses the role played by the edge in linear scan and cyclic voltammetries when the electron transfer is reversible or quasi-reversible. A simulation models the concentrations, current densities and currents in two circumstances—when the edge in important and when it is absent—simultaneously, and thereby the evolving edge current is quantified. Special attention is paid to the effect that the edge has on the heights and positions of the voltammetric peaks. It is demonstrated that disregarding the edge may lead to the bogus classification of a reversible electrode reaction as quasi-reversible.     

Analytical expression of non steady-state concentration for the CE mechanism at a planar electrode

The analytical solutions of the non-steady-state concentrations of species at a planar microelectrode are discussed. The analytical expression of the kinetics of CE mechanism under first or pseudo-first order conditions with equal diffusion coefficients at planar electrode under non-steady-state conditions are obtained by using Homotopy perturbation method. These simple new approximate expressions are valid for all values of time and possible values of rate constants. Analytical equations are given to describe the current when the homogeneous equilibrium position lies heavily in favour of the electroinactive species. Working surfaces are presented for the variation of limiting current with a homogeneous kinetic parameter and equilibrium constant. In this work we employ the Homotopy perturbation method to solve the boundary value problem. Furthermore, in this work the numerical simulation of the problem is also reported using Scilab program. The analytical results are found to be in excellent agreement with the numerical results.     

基于改进计时电流法精确模型的数值解仿真及实验验证

根据电极表面被测物浓度变化经验公式所建立的改进计时电流法的理论模型,只能定性分析响应电流峰值与被测物浓度关系,无法准确描述响应电流峰值与反应体系及激励电势中其它参数的关系.为解决该问题,利用能斯特方程与扩散定律,建立了描述电极表面被测物浓度与响应电流关系的积分方程,通过数值分析方法求解得到该积分方程数值解.进而利用数值仿真得到精确的电流与时间关系曲线,分析了电流峰值与试剂浓度、惯性时间常数、参比电极标准电势、激励电势初始值以及稳态值之间的关系.最后,利用改进计时电流法的电路装置对K3[Fe(CN)6]试剂进行测试,实验结果表明该数值模型比经验模型更接近真实情况,也验证了由该模型推导得到的各参数之间关系的正确性.     

Simulation of the microdisc problem in spherical co-ordinates. Application to electrogenerated chemiluminescence

In this paper, we demonstrate a new approach for simulation of the 2D microdisc problem in spherical co-ordinates and apply it to the solution of the non-steady-state electrogenerated chemiluminescence (ECL) problem at a microdisc electrode.     

Theory of the steady-state current of a redox couple at interdigitated array electrodes of which pairs are insulated electrically by steps

The possibility of constructing an interdigitated array electrode (IDA) with a submicrometre gap is proposed in which adjacent microband electrodes are separated from each other by an insulated step. Then the IDA is an assembly of protrusive and hollow microband electrodes. The unit model of the IDA consisted of half of the lower (hollow) microband electrode, an insulated step and half of the upper (protrusive) microband electrode with a finite thickness on the step. The boundary value problem involving the two-dimensional Laplace equation is presented for redox cycling at the IDA under diffusion control and is solved numerically by a boundary element method. The steady-state current was computed as a function of the height of the step and the thickness of the upper electrode. It was larger than the current at the ordinary IDA, partly because the true electrode area was larger than the area of the ordinary IDA. The current varied linearly with the logarithm of the step height. It was expressed by a simple approximate equation in order to facilitate prediction of its numerical value.     

The electrochemistry of electrode edges and its relevance to partially blocked voltammetric electrodes

Analytical mathematics and digital simulation are used to predict the response, to a potential jump, of the junction between insulating and conducting regions of an electrode. The simulation is carried out differentially and employs other novel features. Concentrations in the vicinity of edges of positive and negative curvatures, as well as straight edges, are analyzed by the model and thereby the faradaic current densities and currents are predicted. It is shown that, in addition to the well-understood cottrellian current arising from the surface of the conducting electrode, currents are generated that are proportional to the length of the edge and to its curvature. These results are then applied to inlaid disks and to partially blocked electrodes. The possibility is explored of using the response to a potential step to gain information on the geometry of a partially blocked electrode.     

The boundary element method: chronoamperometric simulations at microelectrodes

The application of the boundary element method (BEM) as an efficient and powerful method for the analysis of the time-dependent electrochemical processes at multiple electrode configurations is presented. The paper describes the theory and numerical details required for developing one-, two- and three-dimensional transient diffusion models for chronoampermetric simulations under diffusion control. The benefits of the BEM approach are discussed, including the reduction in dimensionality brought about by the formulation procedure and complete elimination of the need for domain discretisation with the time-domain convolution approach. The versatility and efficiency of the numerical procedures are examined with respect to a number of electrode geometries. Results are presented for chronoamperometric simulations at microband, microhemishpere, microcylinder and microdisc electrodes compared to appropriate analytical theory. The three-dimensional simulations focus on the modelling of double-electrode configurations (microdisc and microhemisphere) operating in a generator–collector mode. The influence of electrode separation on the transient current response is presented and normalised working curves currently unavailable via analytical methods are provided.     

High-throughput dielectrophoretic manipulation of bioparticles within fluids through biocompatible three-dimensional microelectrode array

Dielectrophoresis (DEP) has been deemed as a potential and ideal solution for bioparticle manipulation. A 3-D carbon micro-electro-mechanical system (MEMS) fabricated from the latest developed carbon-MEMS approach has advantages of offering low-cost, biocompatible and high-throughput DEP manipulation for bioparticles. In this paper, a typical process for fabrication of various 3-D microelectrode configurations was demonstrated; accurate numerical analysis was presented on electric field gradient distribution and DEP force based on various microelectrode array configurations. The effects of electrode edge angle, electrode edge-to-edge spacing and electrode height on the electric field distributions were investigated, and optimal design considerations and rules were concluded through analysis of results. The outcomes demonstrate that the sharp edge electrode is more effective in DEP manipulation and both electrode edge-to-edge spacing and electrode height are critical design parameters for seeking optimal DEP manipulation. The gradient magnitude increases exponentially as the electrode spacing is reduced and the electric field extends significantly as the electrode height increases, both of which contribute to a higher throughput for DEP manipulation. These findings are consistent with experimental observations in the literature and will provide critical guidelines for optimal design of DEP devices with 3-D carbon-MEMS.     

Simulation of the microdisc problem in spherical co-ordinates. Application to electrochemiluminescence homogeneous analysis.

The chemical transformations of electrogenerated ion-radicals of a number of complex organic compounds may be accompanied by emission of photons. An electrogenerated chemiluminescence (ECL) quantum contains information both on the kinetics of the heterogeneous electrode processes and on the subsequent homogeneous chemical reactions in the solution. Application of ECL to solution analysis provides advantages in comparison to electrochemical methods. We demonstrate a new approach for numerical simulation of the microdisc problem in spherical co-ordinates and apply it to ECL homogeneous analysis under non-steady state electrolysis.     

Dual microband electrodes: current distributions and diffusion layer ‘titrations’. Implications for electroanalytical measurements

The simulation of transport to double microband electrodes in generator–collector mode is reported focusing especially on the ‘titration curve’ approach to electroanalysis in which a titrant is electrogenerated from a redox active precursor on the generator electrode and reacts homogeneously with the target analyte. The current on the detector electrode reflects the amount of titrant ‘surviving’ passage between the two electrodes. The form of the titration curve – plots of detector current as a function of generator current – is shown to be highly sensitive to the electrode kinetics of the redox couple driven at the generator electrode. Accordingly the naïve use of such methodology for analysis without accompanying simulation and kinetic analysis is fraught with danger. Use of the conformal mapping approach in combination with the ADI method for investigation of the ‘titration’ current distributions at the double band system gives fast and precise simulation of this and similar problems. Convergence analysis is described which allows for the automatic selection of the simulation grid size so as to obtain a chosen accuracy (for example 1%) of the current for all experimentally meaningful values of the geometrical and physico-chemical parameters of the system to be investigated.     

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.     

金属线电极分形生长的模拟

电沉积金属过程中,阴极沉积的金属边缘会出现包括枝晶生长在内的许多复杂形态,这会严重影响电沉积产品的质量和加工过程中的电流效率. 对枝晶分形生长的过程以及形貌进行研究,可以实现对沉淀物的可控生长. 本研究使用Python和Matlab软件相互结合,基于扩散限制凝聚（DLA）模型,建立平行线电极电沉积的模型. 通过分析不同粒子数、沉积概率、电极间距、运动步长、定向漂移条件下的分形生长的变化规律,以及模拟参数与实际电沉积因素对分形生长影响的内在联系,发现只要合理控制模拟的粒子数、沉积概率、线电极间距、运动步长、定向漂移概率参数即可与实际电化学体系的浓度和沉积时间、还原概率、两极间距、温度和电压、电极的相对位置和形状一一对应,从而模拟得到跟实际电沉积接近的分形图,最终可实现对分形生长的可控操作,对分形生长在工业电沉积等方面应用有很大的意义.     

Linear dependence and energy conservation in Gaussian wavepacket basis sets

We propose a method for dealing with the problem of linear dependence in quantum dynamics simulations employing over-complete Gaussian wavepacket (GWP) basis sets. In particular, by periodically projecting out redundant basis functions using the matching pursuit algorithm whilst simultaneously introducing GWPs which avoid linear dependence with the current basis set, we find that numerical conditioning of the equations-of-motion can be readily controlled. In applications to particle tunnelling in one- and two-dimensional potentials, this method allows us to reproduce the exact quantum-mechanical results with fewer GWP basis functions than similar calculations with non-adaptive basis sets, a result which we trace back to the improved energy conservation of our adaptive approach.     

Diffusion-controlled current at elliptically deformed microelectrodes

Diameters of invisible microelectrodes have been estimated from steady-state diffusion-controlled currents of a known concentration of redox species on the assumption of a disk form. However, geometry of the disk is often deformed by polishing the electrode surface obliquely against a polishing pad, by malleability of metal, and/or by distortion of metal wire. Then, the exposed surface is close to an ellipse with rough circumference. The diameter estimated from the steady-state current should be an average value among a major radius, a minor radius, and circumference length. In order to obtain a way of the average, we obtained here voltammetric steady-state currents at elliptic electrodes which were fabricated by polishing glass-coated platinum wire obliquely. Values of the diffusion-controlled currents at the elliptic electrode with smooth edge agreed with the theoretical values with 4% error. The steady-state current at a deformed electrode was approximately proportional to the square root of the area of the electrode rather than the length of the edge, as opposed to the conventional concept of the edge effect on the current. Even if electrode geometry is uncertain, the diameter evaluated from the steady-state current corresponds to the square root of the area.     

Diffusion-controlled current at elliptically deformed microelectrodes

Diameters of invisible microelectrodes have been estimated from steady-state diffusion-controlled currents of a known concentration of redox species on the assumption of a disk form. However, geometry of the disk is often deformed by polishing the electrode surface obliquely against a polishing pad, by malleability of metal, and/or by distortion of metal wire. Then, the exposed surface is close to an ellipse with rough circumference. The diameter estimated from the steady-state current should be an average value among a major radius, a minor radius, and circumference length. In order to obtain a way of the average, we obtained here voltammetric steady-state currents at elliptic electrodes which were fabricated by polishing glass-coated platinum wire obliquely. Values of the diffusion-controlled currents at the elliptic electrode with smooth edge agreed with the theoretical values with 4% error. The steady-state current at a deformed electrode was approximately proportional to the square root of the area of the electrode rather than the length of the edge, as opposed to the conventional concept of the edge effect on the current. Even if electrode geometry is uncertain, the diameter evaluated from the steady-state current corresponds to the square root of the area.