The Cottrell experiment

When initially experimenting with unequal intervals using Feldberg s function, it was found that a 2 leads to somewhat oscillatory behaviour, as does large Xq. The combination a = 2 and Xq = 3 seems about optimal and leads to a satisfactory convergence of the error plot. 

The value 6 for RK2 is higher than previously reported (Britz 1988) in the earlier work, a less accurate method of measuring cpu time was used. 

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We introduce here the diffusion-controlled potential-step experiment, hereafter called the Cottrell experiment [181]. Consider Fig. 2.3, showing a long thin tube representing an electrochemical cell, bounded at one end by an electrode and filled with electrolyte and an electroactive substance initially at concentration c (the bulk concentration). We place the electrode at x = 0 and the other, counter-electrode (not shown), at a large distance so that what happens there is of no consequence to us. We apply, at /. 0, a potential such... 

Fig. 2.4. Concentration profile changing with time for the Cottrell experiment...

In the Cottrell experiment, as described in the last section, we have a step to a very negative potential, so that the concentration at the electrode is kept at zero throughout. It is possible also to step to a less extreme potential. If the system is reversible, and we consider the two species A and B, reacting as in (2.18), then we have the Nernstian boundary condition as in (2.24). Using (2.29) and assigning the symbols CA and Cb, respectively, to the dimensionless concentrations of species A and B, we now have the new boundary conditions for the potential step,... 

In the world of numerical analysis, one distinguishes formally between three kinds of boundary conditions [283,528] the Dirichlet, Neumann (derivative) and Robin (mixed) conditions they are also sometimes called [283,350] the first, second and third kind, respectively. In electrochemistry, we normally have to do with derivative boundary conditions, except in the case of the Cottrell experiment, that is, a jump to a potential where the concentration is forced to zero at the electrode (or, formally, to a constant value different from the initial bulk value). This is pure Dirichlet only for a single species simulation because if other species are involved, the flux condition must be applied, and it involves derivatives. Therefore, in what follows below, we briefly treat the single species case, which includes the Cottrell (Dirichlet) condition as well as derivative conditions, and then the two-species case, which always, at least in part, has derivative conditions. In a later section in this chapter, a mathematical formalism is described that includes all possible boundary conditions for a single species and can be useful in some more fundamental investigations. 

Here the value of the boundary concentration is specified. A familiar example in the present context is the outer boundary, beyond the diffusion space, where the concentration usually remains at the initial bulk value during the whole period over which the simulation is carried out. This also applies to the case of the Reinert-Berg mechanism (page 20), in which the bulk concentration itself changes with time, but we know the bulk value at any time, because chemical reaction kinetics, uncomplicated by transport effects, is well understood. In such cases, we can set a given bulk concentration, albeit time-varying. Another familiar example arises from the Cottrell experiment, in which the concentration at the electrode, Co, is set to zero. This is a particular case of that concentration being set to a definite value, not necessarily zero. 

It remains to describe how to handle the boundary value Cq. Clearly, for the RL variant, there is no problem because the last concentration value calculated is C(, and Cq can then be computed from all the other C values, now known, according to the boundary condition. This leaves the LR problem. If the boundary concentration is determined as such (the Dirichlet condition, for example the Cottrell experiment), then this is simply applied. It is with derivative (Neumann) boundary conditions that there is a (small) problem. Here, we know an expression for the gradient G at the electrode. For simplicity, assume a two-point gradient approximation at time t + ST... 

Alternatively, one can simply work with the X-scaling as it is, and change the analytical solution correspondingly. For example, for the Cottrell experiment, with the usual normalisation, the gradient G at time T has the analytical solution I/VttT (see (2.44) on page 18) while, with the normalisation as used in OC (9.77), the analytical solution for the gradient at the electrode ((2.36, page 16) becomes... 

The above implies that / m increases indefinitely with T. However, this ignores the fact of a steady state at the UMDE at long times, so for long times, rm might be an overestimate. The choice of L in (12.70) is made on the basis of the Cottrell experiment at a planar shrouded electrode, and... 

This program simulates the Cottrell experiment, as discussed in Chap. 5. The output, upon running this for NT=100 and lambda=0.45, is... 

Unlike macroelectrodes which operate under transient, semi-infinite linear diffusion conditions at all times, UMEs can operate in three diffusion regimes as shown in the Figure for an inlaid disk UME following a potential step to a diffusion-limited potential (i.e., the Cottrell experiment). At short times, where the diffusion-layer thickness is small compared to the diameter of the inlaid disc (left), the current follows the - Cottrell equation and semi-infinite linear diffusion applies. At long times, where the diffusion-layer thickness is large compared to the diameter of the inlaid disk (right), hemispherical diffusion dominates and the current approaches a steady-state value. 

The cell in Figure 17.1.2a is designed for experiments involving semi-infinite linear diffusion of the electroactive species to the electrode surface (6). It is normally used for experiments in which one applies large-amplitude steps in order to carry out electrolysis in the diffusion-limited region, and one then records the change in absorbance, si, versus time. From an electrochemical standpoint, the result is the same as that of the Cottrell experiment described in Section 5.2.1. 

Because the early computations in a finite difference approach are inaccurate, a computation of the current for the first iteration is usually not of importance. Recall that in the Cottrell experiment, / oo as / 0. 

Figure B.2.1 is a FORTRAN listing of an actual simulation. The treated problem is the Cottrell experiment, which was solved analytically in Section 5.2.1. An electroreactant A is uniformly distributed initially, but a potential step is applied at = 0 to force the surface concentration of A to zero by converting it faradaically to species B.

B.2 Using a spreadsheet, set up a simulation of the Cottrell experiment and work through the first 10 iterations. Use = 50 and Dm = 0.40. Calculate Z(k) for each iteration and compare it to Zcott( )-Calculate the x values corresponding to the first 12 boxes, and plot the concentration profiles /a and... 

An estimation of the thickness of the diffusion layer can be obtained through the Nernst diffusion layer, 6 (m), defined as shown in Figure 1.3. For the Cottrell experiment under linear diffusion conditions the 6 value for a species j is given by... 

Fig. 1.3. Variation with time of the concentration profiles of species A and B in the Cottrell experiment (see text) at a planar macroelectrode. The Nernst diffusion layer for ti (<5i) is indicated on the graph, is the bulk concentration of species A. Da = Db.

Fig. 4.1. Concentration profile of species A in the Cottrell experiment at T = Tmax-The solid line corresponds to the profile obtained with a uniform grid with h = 0.0001 the circle points indicate the position of the spatial nodes when an expanding grid is used with h = 0.0001 and ujx = 1-1. AT = 0.01. The value AT/VTinax is chosen as x-axis since this gives an estimation of the distance relative to the linear diffusion layer thickness X/ /Tmax = xly D t max ...

The distribution of the points in the grid can be adjusted by means of the parameters h = ho) and cox- For example, the distribution of nodes corresponding to ujx = 1.1 and h = 0.0001 in the Cottrell experiment are shown in Figure 4.1. The expanding grid enables us to have a very dense grid (Xi — Xo = 0.0001) next to the electrode surface and, at the same time, to cover all the simulation space from X = 0 to 6VTmax with only 92 points. 

Fig. 4.2. Variation of the current response with time in the Cottrell experiment. The circle points indicate the position of the timesteps when an expanding grid is used with AT = 0.01 and ujt = 105.

In a potential step experiment such as the Cottrell experiment, we must know the starting conditions, that is, the concentrations for the whole cell since we are dealing (for the moment) in only one dimension, this means concentrations along the space variable x. As the experiment proceeds, the concentrations at a range of distances from the electrode will change, but at the electrode x = 0), it will be fixed at zero, and at some sufficient distance from the electrode, it will retain the value c, the initial bulk concentration. All this is expressed formally as a set of boundary conditions ... 

For the point at the electrode, there are technically three classes of conditions. The Dirichlet condition is that in which the concentration there has a certain value (perhaps time variant) the Cottrell experiment... 

To simulate the Cottrell experiment, outputting the current at doubling intervals. Current G is calculated as 3-pt approx, use STUFF implicit none... 

Thus, after j iterations, when T = jST, the current is presented as belonging to (j — O.S)ST. The only argument for this is that it seems to work, providing more accurate current values for systems like the Cottrell experiment. However, there is no formal justification for the trick. This is a fudge and should not be used. The trick is also applied in the case of backward differentiation formula (BDF) (see Sect. 1.3.9) and there it is fully justified, as will be seen later. 

The reader may notice a small anomaly the second DO-loop starts with variable Cl equal, once again, to Cq - why should this not be augmented by its own Indeed it should, normally, but exactly how depends on the experiment simulated and will be gone into below. Fig. 5.2 would hold if Cq is constant throughout, as in the Cottrell experiment. 

In many (most ) simulations of practical value, there is no external supply of c values, as there is for the Cottrell experiment, where Cq =... 

The correct discrete expression, Eq. 6.12, has rarely been used (for example, by Hanafey et al, 1978) - most simulators using the box method prefer to use Eq. 3.9. If we recognise CN as an accurate technique when used to simulate the Cottrell experiment, then it confirms the 4/3 factor. This implies that, when using Eq. 3.9, one is (unintentionally) applying a fudge factor of 0.75. It so happens that this makes the result better for the explicit box method but this must again be error cancelling. 

There is a third argument which can yield the time offset of ijST simulate the Cottrell experiment and, after n time steps, compute the current (or G). Then, using the exact solution, Eq. 6.14 and thus Eq. 6.17, calculate the time corresponding to this G value. I. Ruzic supplied the following results (box method), for X - 0.45 (and n=2) ... 

Here, we simulate a system where a chemical reaction takes place in solution. An electroactive species is produced in solution, homogeneously distributed at bulk concentration Cj, by a radiation flash. The species decays with a first order chemical reaction. A potential is applied to the electrode such that the diffusion limited current is measured this corresponds to the Cottrell experiment on a decaying species. Reinert and Berg (1962) have solved this system analytically and found the ratio r of current at time t to the (diffusion) current i, in the absence of the chemical reaction ... 

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