The Simulation of Electrochemical Experiments

There are many different approaches to solving the mathematical equations that describe electrochemical experiments. Because electrochemical mechanisms that involve coupled chemical reactions can be quite complicated and varied, our primary requirement is that the method of solution be general. The method of simulation by explicit finite differences (EFD) lends itself to a quite general treatment. The EFD method can also be optimized to improve computational speed without compromising its generality. The straightforward manner in which the physical problem is translated into a numerical method also makes the EFD method attractive to study. More detailed treatments than that presented here can be found in the chapter by Maloy and the book by Britz.  

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Considering the above aspects, it is beneficial to establish a grid that mimics the concentration profile and provides a high number of points near the electrode surface but minimises the total number, which makes the simulation process more efficient. With this aim, unequally spaced grids or transformation of the spatial coordinates can be employed, the former being preferred in the simulation of electrochemical experiments [3]. 

Exponentially expanding grids are widely employed in the simulation of electrochemical experiments such that the distance between consecutive points of the grid expands exponentially according to the following definition ... 

The modelling of electrochemical experiments with these electrodes obliges us to deal with two-dimensional problems (Figure 1.4) and non-imiform surface fluxes which comphcate the numerical resolution of the problem. Procedures for the simulation of two-dimensional electrochemical problems will be given in Chapters 9 and 10. 

Rudolph M (2004) Digital simulations on unequally spaced grids. Part 3. Attaining exponential convergence for the discretisation error of the flux as a new strategy in digital simulations of electrochemical experiments. 1 Electroanal Chem 571 289-307... 

For this reason, the emphasis in this article is directed more towards the simulation of specific adsorption and, in particular, the recent encouraging comparison of electrochemical and UHV data for the interaction of bromine and chlorine with Ag 110 /7, 8/. A brief outline of the conclusions emerging from alkali-water coadsorption experiments is given to illustrate basic modes of ion-solvent interaction on metal surfaces and to discuss future directions of this research. 

The consecutive stages of a typical gas-phase-adsorption experiment, in which the simulation of an electrochemical interface is the aim, are schematically illustrated in Figure 1. The depicted sequence illustrates in particular the case when specific adsorption of an ion is expected from... 

The simulation of other electrochemical experiments will require different electrode boundary conditions. The simulation of potential-step Nernstian behavior will require that the ratio of reactant and product concentrations at the electrode surface be a fixed function of electrode potential. In the simulation of voltammetry, this ratio is no longer fixed it is a function of time. Chrono-potentiometry may be simulated by fixing the slope of the concentration profile in the vicinity of the electrode surface according to the magnitude of the constant current passed. These other techniques are discussed later a model for diffusion-limited semi-infinite linear diffusion is developed immediately. 

In the last section, we discussed the use of QC calculations to elucidate reaction mechanisms. First-principle atomistic calculations offer valuable information on how reactions happen by providing detailed PES for various reaction pathways. Potential energy surfaces can also be obtained as a function of electrode potential (for example see Refs. [16, 18, 33, 38]). However, these calculations do not provide information on the complex reaction kinetics that occur on timescales and lengthscales of electrochemical experiments. Mesoscale lattice models can be used to address this issue. For example, in Refs. [25, 51, 52] kinetic Monte Carlo (KMC) simulations were used to simulate voltammetry transients in the timescale of seconds to model Pt(l 11) and Pt(lOO) surfaces containing up to 256x256 atoms. These models can be developed based on insights obtained from first-principle QC calculations and experiments. Theory and/or experiments can be used to parameterize these models. For example, rate theories [22, 24, 53, 54] can be applied on detailed potential energy surfaces from accurate QC calculations to calculate electrochemical rate constants. On the other hand, approximate rate constants for some reactions can be obtained from experiments (for example see Refs. [25, 26]). This chapter describes the later approach. 

The simplest case corresponds to the one-electron transfer between the electrode and species that are chemically stable on the time scale of the experiments (Eq. (1.1)). However, electrochemical systems are frequently more complicated and the electroactive species take part in successive electron transfer reactions at the electrode (multistep processes) and/or in parallel chemical reactions in solution such as protonation, dimerisation, rearrangement, electron exchange, nucleophilic/electrophilic addition, disproportionation, etc., the product(s) of which may or may not be electroactive in the potential region under study. The simulation of these cases is described in Chapters 5 and 6. 

Extraction of quantitative chemical information from SECM requires a mathematical model of the interaction of the tip and substrate. Such modeling typically involves numerical solution of a reaction-diffusion equation with the boundary conditions appropriate to the interfacial kinetics. Simulation of SECM experiments is computationally much more demanding than for standard electrochemical experiments (discussed in Chapter 1.3). This is because diffusion in at least two dimensions must be considered and the discontinuity in the boundary condition between the tip metal and insulating sheath necessitates a fine mesh. 

The use of computers in both the management treatment of data of electrochemical experiments the digital simulation of voltammetric responses for a wide variety of mechanistic schemes has been well established in the last decade beyond. However early attempts involved the use of mainframe computers thus excluded widespread use in the interactive sense when experiments were in progress. The advent of cheap computers, in particular the IBM PC its compatibles, has transformed the situation in that dedicated interactive systems can be established cheaply with the speed of the latest products simulations matching previous sophistication also become feasible. 

Cyclic voltammetry is probably the electrochemical technique that is simulated most often, aiming at the analysis of electrode processes with respect to mechanism, kinetics, and thermodynamics of the reaction steps as well as transport properties of the molecules involved. The simulation of processes at (ultra)microelectrodes is also popular and highly important for the analysis of scanning electrochemical microscopy experiments [10]. 

Bieniasz LK (2002) Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 12. Patch-adaptive simulation of extimple transient experiments described by kinetic models defined over multiple space intervals in onedimensional space geometry. J Electroantd Chem 527 21-32. Corrigendum ibid. 565 141 (2004)... 

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