Simulation, electrochemical

Evans found that molecular hydrogen was efficiently generated by the reaction of a simple diiron complex [CpFe(CO)2]2 (Fp2) with acetic acid (pA a = 22.3) in acetonitrile [202]. Electrochemical simulations revealed that Ep2, [CpEe(CO)2] (Fp ), and [CpFe(CO)2H] (FpH) were key intermediates in this catalytic mechanism (Scheme 61). Reduction of Fp2 produces both an Fp anion and an Fp radical, which is further reduced to give an Fp anion. The oxidation of the Fp anion by proton affords FpH. This protonation was found to be the rate-limiting step. The dimerization of the FpH generates Fp2 and H2. Alternatively, the FpH is reduced to afford the FpH anion, which is subsequently protonated... 

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

At the heart of impedance analysis is the concept of an equivalent circuit. We assume that any cell (and its constituent phases, planes and layers) can be approximated to an array of electrical components. This array is termed the equivalent circuit , with a knowledge of its make-up being an extremely powetfitl simulation technique. Basically, we mentally dissect the cell or sample into resistors and capacitors, and then arrange them in such a way that the impedance behaviour in the Nyquist plot is reproduced exactly (see Section 10.2 below on electrochemical simulation). 

We must first appreciate that electrochemical simulations are probably more useful to the research chemist than the quality control analyst an analyst involved in quality control is likely to repeat a large number of measurements on a known chemical system, each time asking how much , rather than needing to ponder complicated mechanistic questions. 

Although the theoretical basis of the technique is different from other electrochemical-simulation packages, in operation it is just as powerful as DigiSim, allowing parameters to be computed for a variety of mechanisms, electrode geometries and experiment types. More information and representative references to Condecon, can be found at its website ... 

One of the best web-based directories for gauging the status of electrochemical simulation is ... 

Electrochemical Simulation Package (ESP) is a free program which allows a PC to simulate virtually any mechanism by the following pulse techniques, i.e. cyclic voltammetry, square-wave voltammetry, chronoamperometry and sample DC polarography. The program can also be used in conjunction for fitting experimental data at solid and DME electrodes. It is the only package to explicitly claim to be bug-free . 

Britz, D., Digital Simulation in Electrochemistry, Springer-Verlag, Berlin, 1981 and MacDonald, D., Transient Techniques in Electrochemistry, Plenum Press, New York, 1977. Both of these books contain copious details concerning electrochemical simulations. Although these texts are extremely mathematical (as all simulation work has to be), the basic concepts are not too difficult to follow. The application notes to Condecon (see URL on page 301) are also a feast of detail. 

Based on such an electrochemically simulated system, further modification and characterization of the Chl-contalning interfacial layer on electrodes are expected to contribute to the discovery of useful information on the electron and energy transfer reactions involving Chi and other compounds of photosynthetic importance. 

See also http //www.basinc.com/products/ec/digisim/ and http //www.gamry.com/products/ digielch-electrochemical-simulation-software/... 

A better alternative approach is what will be called the Rudolph method [476], after the person who introduced it into electrochemical simulation. It was known before 1991 under various names, notably block-tridiagonal [280,412,470,471,528,570]. This comes from the fact that if one lumps the large matrix into a matrix of smaller matrices and vectors, the result is a tridiagonal system that is amenable to more efficient methods of solution. In the present context, we define some vectors... 

For the basics of this method, see Chap. 4. There it was mentioned that Bieniasz introduced this method to electrochemical simulation [100], preferring ROWDA3, a third-order variant that also has a smooth response. There exists a second-order variant with a smooth response, ROS2, due to Lang [347], which might be more appropriate if second-order spatial derivative approximations are to be used. Coefficients for some variants are given in Appendix A. The object here is to describe the way Rosenbrock methods are used in the present context. The Bieniasz paper [100] shows the way (but the standard symbols, as used in Chap. 4, are used here, rather than those used by Bieniasz). 

Here are a few brief references to recent or key works in which these methods have been described as used in electrochemical simulations. The interested reader is urged to look these up and follow the references contained in them to the seminal works and text books. Of necessity, much work is left uncited here. 

Another possible approach, apparently not taken by any electrochemical simulator, is to render the equation set into a DAE set. Instead of substituting for Co as described above, one replaces the first equation of (9.92) by the algebraic equation for the boundary condition, and uses one of the available DAE packages to solve the system. 

Yet another, quite different, approach to solving a system of odes, such as one obtains as an intermediate step when using, for example, MOL or OC, is the eigenvalue-eigenvector method. Its use for electrochemical simulations was described in two papers in 1989 and 1990 [255,332]. The method has some drawbacks, and does not appear to have seen much use since these two papers. It does have one unique feature there is no discretisation of time. A solution is generated by the algorithm, at any chosen time. So, although the method may at times be fairly inefficient, if one wants a current or concentrations at only one or a few time points, this could be faster than a time march with the usually small time intervals. 

In descriptions of this problem, the names of Randles [460] and Sevclk [505] are prominent. They both worked on the problem and reported their work in 1948. Randles was in fact the first to do electrochemical simulation, as he solved this system by explicit finite differences (and using a three-point current approximation), referring to Emmons [218]. Sevclk attempted to solve the system analytically, using two different methods. The second of these was by Laplace transformation, which today is the standard method. He arrived at (9.116) and then applied a series approximation for the current. Galus writes [257] that there was an error in a constant. Other analytical solutions were described (see Galus and Bard and Faulkner for references), all in the form of series, which themselves require quite some computation to evaluate. 

There are some special conditions in electrochemical simulations that have an effect on stability. 

Box method Chap. 9, Sect. 9.1. The original electrochemical simulation method. With boxes, most of the above techniques can be applied. There is an unresolved issue of whether this method is inherently better than the point method, or not. 

Of greatest interest here is the group of programs that to lesser or greater degree solve a variety of electrochemical simulation problems. There are quite a number of these, a few of them somewhat prominent and commercial, and some more or less private but accessible to others. 

The Rosenbrock method is described for odes in Chap. 4 and for electrochemical simulations, that is, DAEs, in Chap. 9. There are four variants, two of... 

The space/time over which the problem is formulated is covered with a mesh of points, often referred to as nodes . At each point, the derivatives in the material balance equation are approximated as differences of the concentrations at the given and surrounding points. This leads to a set of linear equations (based on a five-point stencil in two dimensions - each node is related to its four nearest neighbours) which can be solved to give the solution to the PDE. The methods are well suited to simulations in rectangular regions, which is often compatible with an electrochemical cell. These are by far the most popular methods for electrochemical simulations and will therefore be the focus of the remainder of this section. 

Electrochemical simulations of the concentration and scan-rate dependence of the voltammetry potentially provide the composition of the intermediates formed during the reaction cycle together with estimates of the rate and equilibrium constants. As shown in the preceding section spectroscopic information can greatly assist the elucidation of the molecular details of these reactions, however, reliable deduction of the structure is greatly enhanced by the incorporation of structural and computational information (Section 1.6). The rapid advance in computer power and implementation of density-functional theory allows a more quantitative approach for evaluation of proposed structures based on spectroscopic information and estimation of the relative energies of the proposed spe-cies. The recent computational study of the electrocatalytic reaction cycle proposed for illustrates the opportunities presented by the approach. 

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