Cottrell system

Along with a group of models that have shown themselves useful, their particular normalisations will be presented. The first model, the Cottrell system, will also serve to introduce the concept of the Nernst diffusion layer. 

Potential step experiments are a popular way to look at electrochemical kinetics. The oldest known is the Cottrell system, where the potential stepped to is so far negative that the resulting current is limited by the transport of the active substance. If the step is not so far negative, one then has either... 

The solution of this is the one for the simple Cottrell system, multiplied by the decay factor... 

While the Cottrell system might be regarded as the simplest possible model with a Dirichlet boundary condition (that is, in which boundary concentrations are specified), the constant current case is the simplest possible for the Neumann boundary condition, in which a concentration gradient is specified at the boundary. This model can also be called the chronopotentiometric experiment since here, the current is given and it is the electrode potential that is measured against time. Mathematically this model is defined by the usual (2.33), here with the boundary conditions... 

One of the simplest examples of a homogeneous chemical reaction (her) is the Reinert-Berg system [464], in which an electroactive species is generated, for example by means of a light flash, and then reduced as a Cottrell system, while the species decays chemically with a first-order reaction. The reactions are then... 

For the Cottrell system, this is the distance, beyond which changes greater than 10-4 relative to the bulk concentration, are no longer observed. Figure 12.10 shows this line. In the range 0 < R < 1, it is simply the line... 

This, classical Cottrell, system has an analytical solution that can be used to test methods and programs. That solution is in terms of concentrations (see Chapter 2.2 in this volume for the solution). What one normally wants is the current i at a given time, and this is derived from the concentration profile by using Pick s first diffusion equation, relating flux to concentration gradients ... 

Normally, the equations as presented above, are rendered dimensionless (normalized) by expressing the variables as multiples of reference values. For the Cottrell system, the reference time value is the observation time, or duration of the experiment, t concentrations are referred to c and distance x to a suitable length scale 5. In view of the solution of Eqs. (1 and 2) (see Chapter 2.2, this volume and Ref. [2]), this scale is conveniently defined as 5 = /Dr, which means that the concentration profile will extend into the solution bulk by only a few 5-units. We then have the following... 

Here, the simple exphcit method is briefly gone through, in practical detail. The example taken is the one-dimensional Cottrell system, diffusion Eq. (8) and boundary conditions (9). In writing a simulation program, some practical decisions have to be made. Normally, the time interval for a complete simulation is 0 < T <1, and that leaves only the number of steps Np or its inverse 5T to set. Then there is the interval in X. This is related to the time interval by the important parameter X (see Eq. 25), and it might be better to decide on its value, which then provides that for H. Another quantity is the distance along X to which the simulation is to extend. Here, experience (and the solution of the Cottrell equation, see Chapter 2.2 in this volume)... 

The list starts with a few programs all simulating the potential jump experiment (Cottrell system), using different methods, such as EX (point-), EX (box-). A Pascal version of EX is shown, followed by examples of the RK2 and CN methods for the system. For CN, the CN subroutine shown in Sect. 9.3 is called. These are demonstration programs and print out the log error in G at T = 1 only. Mostly, 5- or 6-point G approximations are used. 

Potential step experiments are a popular way to look at electrochemical kinetics. The oldest known is the Cottrell system, where the potential stepped to is so far negative that the resulting current is limited by the transport of the active substance. If the step is not so far negative, one then has either Nernstian boundary conditions, or those for quasireversible or irreversible systems. All of these cases have been analytically solved. As well, there are two systems involving homogeneous chemical reactions, from flash photolysis experiments, for which there exist solutions to the potential step experiment, and these are also given they are valuable tests of any simulation method, especially the second-order kinetics case. 

To appreciate the problems with both CN and Laasonen, consider Fig. 8.1. This shows three curves, and we ignore the stippled one for the moment (but see Sect. 8.5.1.2). The plot shows simulations of the Cottrell system, using only 20 steps in the range 0 < T < 1 and a A value of 3 (equal spatial intervals). The vertical axis is the relative error e in the computed current, defined as... 

---