Solution Of Electrochemical Kinetic Equations Research Articles

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 16: Patch-adaptive strategy combined with the extended Numerov spatial discretisation

Electrochemical kinetic simulations by the patch-adaptive strategy outlined in the previous parts of this series of papers can be computationally expensive in the extreme cases of very thin boundary layers, moving fronts, or models defined over multiple spatial subintervals having disparate scales. The replacement of the second-order accurate conventional spatial discretisation (thus far used by the strategy) by the fourth-order accurate extended Numerov discretisation of Chawla [M.M. Chawla, J. Inst. Math. Appl. 22 (1978) 89], discussed in this work, allows one to reduce the computational costs. Numerical experiments with five representative examples of difficult-to-simulate kinetic models reveal that the reduction of the computational time, the number of spatial grid nodes needed, and spatial grid refinement levels, is particularly noticeable when a high spatial accuracy of the simulations is requested. At low accuracy demands the conventional spatial discretisation may be more efficient and more convenient to use. The patch-adaptive simulations by the extended Numerov scheme require spatial tolerance parameter TOLX values that are several times larger than the TOLX values that ensure a comparable accuracy by the conventional spatial discretisation.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 15: patch-adaptive simulation of example transient experiments described by Nernst–Planck–electroneutrality equations in one-dimensional space geometry

The patch-adaptive strategy for electrochemical kinetic simulations, introduced in Part 5 of the present series of papers, with extensions described in Part 14, is applied to five typical examples of time-dependent models of transient experiments represented by Nernst–Planck–electroneutrality equation systems in one-dimensional space geometry. The models describe: potential step chronoamperometry for a charge neutralisation reaction at a planar electrode in the absence of the supporting electrolyte; potential step chronoamperometry for a planar electrode coated with a polymer film under conditions of migration–diffusion of redox ions; potential step chronoamperometry of uncharged species at a hemispherical electrode in the presence of a low concentration of supporting electrolyte; cyclic voltammetry for an electrode reaction (initially at equilibrium) involving uncharged species, at a planar electrode in the absence of the supporting electrolyte; and current step chronopotentiometry for a liquid|liquid junction between two binary electrolytes. The calculations lead to analogous conclusions, with regard to the performance of the strategy, as in the case of the models represented by the Nernst–Planck–Poisson equations solved in Part 14. New observations reveal that highly non-linear boundary conditions present an additional challenge for the Newton iterative method utilised by the strategy. The ability of the strategy to solve non-local boundary conditions expressing current equality at distant boundaries is also demonstrated. However, formulating such boundary conditions correctly is not straightforward, so that their use is not recommended. Adaptive solutions suggest inaccuracies in the literature solutions for two example models.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 14: extension of the patch-adaptive strategy to time-dependent models involving migration–diffusion transport in one-dimensional space geometry, and its application to example transient experiments described by Nernst–Planck–Poisson equations

The finite-difference patch-adaptive strategy for electrochemical kinetic simulations, introduced in Part 5 and extended in Part 10 of this series of papers, is further extended to time-dependent models involving migration–diffusion transport in one-dimensional space geometry. The extensions include: spatial discretisation of generalised second spatial derivative expressions typical of migration–diffusion equations; allowance for the dependence of boundary conditions on displacement current; support for an a posteriori calculation of the displacement current as one of the model responses; optional calculation of steady-state initial conditions; the ability to use enhanced precision of floating point calculations. The extended strategy is used to simulate four examples of transient experiments, represented by Nernst–Planck–Poisson equation systems: coulostatic charge injection at an ideally polarised planar electrode; a voltage step for a thin layer asymmetric electrochemical cell; chronopotentiometry for an electrolyte|membrane|electrolyte system; and chronopotentiometry for a bipolar membrane. The strategy provides fairly reliable solutions, but its automatism and efficiency are less satisfactory compared to models without electric migration, owing to the need for model-dependent tuning of the method parameters, and increased computational cost necessary for the exact adaptive determination of the electric potential profiles.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 12. Patch-adaptive simulation of example transient experiments described by kinetic models defined over multiple space intervals in one-dimensional space geometry

The patch-adaptive strategy for electrochemical kinetic simulations, with extensions described in Part 10, is applied to three representative examples of time-dependent kinetic models defined over multiple space intervals in one-dimensional space geometry. The models describe: cyclic voltammetry for a diffusion-controlled electron transfer reaction at a liquid ∣ liquid interface; linear potential sweep voltammetry for a diffusion-controlled charge transfer at a planar amalgam film electrode; and potential step chronoamperometry for a free monoenzymatic amperometric biosensor. The strategy provides accurate, efficient and nearly automatic solutions for these examples, which confirms the overall adequacy of the extensions of the strategy, associated with the handling of multiple space intervals. However, limiting cases of the models, characterised by disparate lengths of the spatial intervals, appear difficult to solve. A possible occurrence of an error in the literature simulation results for the biosensor example is also revealed.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 11. Patch-adaptive simulation of example transient experiments described by kinetic models involving simultaneously distributed and localised unknowns, in one-dimensional space geometry

The patch-adaptive strategy for electrochemical kinetic simulations, with extensions described in Part 10, is applied to three representative examples of time-dependent kinetic models in one-dimensional space geometry, characterised by the simultaneous presence of spatially distributed and localised unknowns. The models describe: potential step chronoamperometry for a diffusion-controlled charge transfer reaction accompanied by the adsorption of reactants at the electrode; cyclic voltammetry for an electrochemical deposition–precipitation mechanism; and double potential step chronoamperometry for an oscillatory system involving a diffusion-controlled charge transfer reaction having a negative polarisation resistance, in the presence of double layer charging, and ohmic potential drops. The strategy provides accurate, efficient and nearly automatic solutions for these examples, which confirms the overall adequacy of the extensions of the strategy, associated with the handling of localised unknowns. In the third example, the adaptive simulations are very detailed and informative. They confirm the occurrence of numerical artefacts in the literature results of fixed grid simulations for this example.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 10. Extension of the patch-adaptive strategy to kinetic models involving spatially localised unknowns at the boundaries, multiple space intervals, and non-local boundary conditions, in one-dimensional space geometry

The finite-difference patch-adaptive strategy for electrochemical kinetic simulations, described in Part 5 of this series of papers, is extended to time-dependent models in one-dimensional space geometry, involving spatially localised unknowns at the boundaries, multiple space intervals, and non-local boundary conditions. These extensions are of interest, among others, for the modelling studies of: electrochemical adsorption and/or electrocatalytic reactions, liquid ∣ liquid interfaces, amalgam electrodes, electrochemical biosensors, and pattern formation at electrodes. The following new features are included in the strategy. Grid objects, used by the spatial grid adaptation algorithm, are decomposed into interior and boundary subgrid objects, keeping discrete values of the spatially distributed and localised unknowns, respectively. Linear algebraic equation systems, characterised by quasi-block-tridiagonal matrices having variable block dimensions, and isolated block-pentadiagonal rows, are solved by a specially developed variant of the Thomas algorithm, previously published by the present author. Relative error control is applied to localised unknowns. The calculation of consistent boundary solutions takes into account temporal derivatives present in boundary conditions. Accurate a posteriori calculation of temporal derivatives of the localised unknowns is enabled.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Advantage of time step adaptation, using example of current spikes in linear potential sweep voltammograms for the EqrevEqrev-DISP reaction mechanism

Electrochemical kinetic transients may exhibit brief, but rapid temporal variations, the location of which along the time axis, and duration, are difficult or impossible to predict. For example, very narrow current spikes may occur on linear potential sweep voltammograms for the E qrev E qrev-DISP reaction mechanism. Traditional fixed time grids require a dense grid spacing over large time intervals in such cases, which renders digital simulations computationally expensive. In contrast, the adaptive time step selection, included into the previously described finite-difference patch-adaptive strategy (PAS), enables a detailed temporal resolution of the current spikes, simultaneously with an efficient simulation of large-scale temporal variations. Using this technique, the phenomenon of the current spikes is investigated.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Patch-adaptive simulation of moving fronts in non-linear diffusion models of the switching of conductive polymers

The finite-difference patch-adaptive strategy (PAS) for electrochemical kinetic simulations, previously described by the author, is extended and applied to two examples of non-linear diffusion in one-dimensional space geometry, characterised by moving fronts, and related to the modelling of redox switching of conductive polymers. The extension consists of an appropriate spatial discretisation of the second derivative diffusion term involving concentration-dependent diffusion coefficient, and of an improved selection of starting approximations for the Newton iterations within the extrapolation time-stepping scheme. Adaptive solutions obtained are reliable, accurate, efficient, and superior to fixed-grid solutions.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations

The patch-adaptive strategy described in Part 5 proves to provide correct, fairly efficient, and nearly automatic solutions to six representative example electrochemical kinetic models in one-dimensional space geometry, that exhibit difficult-to-resolve local spatial and temporal solution structures at the electrodes. The models describe: the square-wave controlled potential transient for an uncomplicated heterogeneous charge transfer reaction, the potential step method and linear potential sweep voltammetry for the standard catalytic reaction mechanism with a pseudo-first-order homogeneous reaction, linear potential sweep voltammetry for an EC reaction mechanism involving a follow-up homogeneous dimerisation, steady-state voltammetry at a rotating disc electrode, and steady-state voltammetry at a spherical microelectrode. These examples are characterised by extremely thin reaction layers and other instances of local spatial regions of considerable variations of the solution, including transient effects caused by temporal discontinuities of boundary conditions. The strategy creates dynamically spatio-temporal grids well adapted to such solution structures, without any a priori knowledge about their location. The accuracy of the numerical solutions proves well controlled by means of error tolerance parameters. Numerical difficulties observed with the previously described adaptive moving grid technique do not occur in the present strategy. However, further work is needed to improve the performance of the strategy in the cases of thin reaction layers associated with fast second-order homogeneous reactions, and boundary layers associated with hydrodynamic electroanalytical methods, for which the computational effort appears too large.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations

Abstract Local, difficult-to-solve spatio-temporal structures such as extremely thin reaction layers at the electrodes, thin moving reaction fronts, fast transient variations, isolated temporal discontinuities in boundary conditions, edge effects, etc. are characteristic of the initial boundary value and boundary value problems occurring in electrochemical kinetic modelling. Further progress in the simulation methodology is hindered by the lack of automatic solution techniques for such problems. Based on a critical consideration of various adaptive grid strategies for partial differential equations reported in the scientific literature, a new finite-difference adaptive grid strategy has been formulated, that is especially designed for electrochemical kinetic simulations, and is currently limited to models in one-dimensional space geometry. The intention is to enable an automatic solution of the governing partial/ordinary differential equations to a prescribed accuracy, without any a priori knowledge about the spatio-temporal location of the emerging solution structures. In view of the importance of the solution (concentration) gradients for the electrochemical theory, simultaneous control of the spatial errors of the solutions and their spatial gradients is included in the strategy. Spatial grid adaptation is based on a local uniform grid refinement, using overlapping grid patches. Temporal step selection uses a recent control theoretic algorithm, combined with a simple method of detecting temporal discontinuities of the boundary conditions. A third-order accurate, implicit and L-stable Rosenbrock time-stepping scheme ROWDA3 is used to enable efficient and non-oscillatory temporal integration in the time intervals where a regular solution occurs. A second-order accurate Lawson–Morris–Gourlay extrapolation scheme is used to enable a reliable time-stepping at temporal discontinuities.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations

The patch-adaptive strategy described in Part 5 proves to provide correct, fairly efficient, and nearly automatic solutions to three representative example electrochemical kinetic models in one-dimensional space geometry, that exhibit difficult-to-resolve moving reaction fronts in the electrolyte, away from the electrode. The models describe: double potential step experiments for a simple mechanism of electrochemically generated luminescence, linear potential sweep voltammetry for an EE-DISP mechanism with Nernstian charge transfers and a reversible homogeneous disproportionation reaction, and linear potential sweep voltammetry for an RRC-initiated electropolymerization reaction scheme. The strategy dynamically creates spatio-temporal grids that adaptively concentrate in the regions of the reaction fronts, without any a priori knowledge about their location. Numerical difficulties, such as electric current oscillations observed with the previously described adaptive moving grid technique in the case of moving reaction fronts, do not occur in the present strategy. However, further work is needed to improve the performance of the strategy in the cases of reaction fronts associated with fast and irreversible second-order homogeneous reactions, for which the computational costs are too large. Improvements are also desirable in the case of the simulation of a rapidly moving reaction front occurring in the electropolymerization model, for which the strategy tends to provide incorrect front positions.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 4: The adaptive moving-grid solution of one-dimensional fast homogeneous reaction-diffusion problems with extremely thin reaction zones away from the electrodes

The adaptive moving-grid strategy suggested in Parts 1–3 for the solution of electrochemical kinetic partial differential equations in one space dimension has been subject to further evaluation and tuning. The strategy has been applied to two kinetic examples of fast homogeneous reaction-diffusion systems involving extremely thin moving reaction zones away from the electrodes. One example, the simulation of the double-potential step transient for a simple mechanism of electrogenerated chemiluminescence, has been considered in much detail. As the second example the simulation of the linear potential scan voltammetry for a double-electron transfer with “nuances” has been briefly discussed. The adaptive grid strategy provides effective, satisfactorily accurate and complete solutions to these difficult problems, which would not be possible to obtain by traditional fixed-grid finite difference methods, except at much higher computational cost or with radical simplifications.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 3. An adaptive moving grid—adaptive time step strategy for problems with discontinuous boundary conditions at the electrodes

The finite-difference adaptive moving grid strategy suggested in Parts 1 and 2 for the solution of electrochemical kinetic one-dimensional partial differential equations has been further extended to allow temporal grid adaptation as well as time-stepping across discontinuities in boundary conditions. This extension aims at the development of a general simulation algorithm, capable of a largely automatic solution of a variety of kinetic problems. The validity of the strategy developed has been tested on three examples of the modelling of electrochemical transients: square-wave-controlled potential transient in pure diffusion conditions, potential-step transient for the catalytic electrode reaction mechanism with a fast homogeneous reaction and linear potential scan voltammetric transient for an electrode reaction accompanied by a fast follow-up homogeneous dimerization reaction.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 2. An improved finite-difference adaptive moving grid technique for one-dimensional fast homogeneous reaction-diffusion problems with reaction layers at the electrodes

The adaptive moving grid strategy suggested in Part 1 for the solution of electrochemical kinetic problems in one-dimensional geometry has been examined thoroughly in calculations and developed further, with special emphasis laid on the resolution of extremely thin reaction layers at the electrodes. Numerical difficulties observed previously have been identified and eliminated. The use of a more accurate finite-difference approximation to second spatial derivatives in the regridding stage and padding of the monitor functions has led to a more reliable and nearly automatic spatial grid adaptation algorithm. Some improvement has also been achieved by using an expanding simulation layer. The improved algorithm has been applied to the example of linear potential scan voltammetric for the EC mechanism involving a fast homogeneous dimerization reaction.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 1. Introductory exploration of the finite-difference adaptive moving grid solution of the one-dimensional fast homogeneous reaction-diffusion problem with a reaction layer

An effective finite difference solution of extremely fast homogeneous reaction-diffusion problems in electrochemical kinetics might well be achieved by using dynamically adaptive grid techniques, instead of considering limiting cases resulting from steady- state or equilibrium assumptions and traditional calculations on fixed grids. A simple adaptive moving-grid strategy has been applied to the example of linear potential scan voltammetry and a catalytic mechanism in one-dimensional geometry. Although not entirely satisfactory at present, the strategy permits homogeneous rate constants greater by as much as 20 orders of magnitude than the maximum possible values in corresponding fixed-grid calculations using the same number of space grid points at not much greater computational cost.