Boundary conditions potential sweeps

One of the main uses of digital simulation - for some workers, the only application - is for linear sweep (LSV) or cyclic voltammetry (CV). This is more demanding than simulation of step methods, for which the simulation usually spans one observation time unit, whereas in LSV or CV, the characteristic time r used to normalise time with is the time taken to sweep through one dimensionless potential unit (see Sect. 2.4.3) and typically, a sweep traverses around 24 of these units and a cyclic voltammogram twice that many. Thus, the explicit method is not very suitable, requiring rather many steps per unit, but will serve as a simple introduction. Also, the groundwork for the handling of boundary conditions for multispecies simulations is laid here. 

In Fig. 8D, we compared the potential applied to the interface during linear potential sweep with and without an uncompensated solution resistance. Clearly, the error is a maximum at the peak, where the current has its highest value. Just before and during the peak, the effective sweep rate imposed on the interphase is much less than that applied by the instrument. The assumption that v = constant, which has been used as one of the boundary conditions for solving the diffusion equation, does not apply. In this sense the experiment is no longer conducted "correctly". 

Note that equations need not be written for species Y, since its concentration does not affect the current or the potential. If reaction (12.2.2) were reversible, however, the concentration of species Y would appear in the equation for 5Cr(x, t) dt, and an equation for 5Cy(, t) dt and initial and boundary conditions for Y would have to be supplied (see entry 3 in Table 12.2.1). Generally, then, the equations for the theoretical treatment are deduced in a straightforward manner from the diffusion equation and the appropriate homogeneous reaction rate equations. In Table 12.2.1, equations for several different reaction schemes and the appropriate boundary conditions for potential-step, potential-sweep, and current-step techniques are given. 

K. The scan rate v determines the time of the experiment tc, since tc = (Vf- V )/v, where V is the final potential and is the initial potential of the sweep. The other boundary conditions for the situation shown in Fig. 6.6 are... 

Thus far we have only considered potential step experiments, and these only in the potential region where the electron transfer step occurs at a diffusion limited rate. The major part of any simulation for any technique is the same as that already outlined, but the boundary conditions, and hence the finite difference expressions for the first box, do vary. As an example of a different type of experiment we will now consider sweep voltammetry. Here the boundary conditions are a function of time, and for a reversible system can be derived from the Nernst equation... 

The bulk concentration (Co and Cr) of each species is set to the one used during the experiment and the adsorbed species A is not present in this subdomain. At the tip boundary, z=0,0 < r < a, the inward flux condition following Butler-Volmer kinetics (Equations 16.13 and 16.14) is written and in the case of a linear potential sweep. Equation 16.15 is added ... 

The boundary conditions for solving the diffusion equation for linear potential sweep are really the same as those written for the potential step experiment, as discussed in Section 14.2. because in both cases the potential is the externally controlled parameter. As before, we can distinguish between the reversible case, in which it is assumed that the concentrations at the surface are determined by the potential via... 

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