Diffusion domain approximation

Under the diffusion domain approximation [1, 14, 17, 18], the hexagonal unit cell is approximated as a cylindrical cell of the same base area as shown in Figure 10.3. 

Previous studies have demonstrated that the results of simulations using the diffusion domain approximation show very good agreement with experiments [19]. 

As with the isolated microdisc simulations in Chapter 9, we here consider the simulation of the cyclic voltammetry of a simple fully reversible one-electron reduction. For an array, since each unit cell is identical, the concentrations of the electroactive species will necessarily be the same on either side of the cell boundary and there can be no flux of electroactive material across the boundary. After using the diffusion domain approximation, this boundary is at a distance r = Vd, therefore... 

Voronoi cells (b) diffusion domain approximation used to transform a Voronoi cell into a cylindrical cell of equivalent base area. 

In the same maimer as for the regularly distributed array, we can use the diffusion domain approximation to transform each Voronoi cell into a cyhndrical cell of the same base area, A , and of radius rd , as illustrated in Figure 10.6(b), thus reducing the problem of simulating each cell from... 

The correctness of the edge plane activity model for HOPG electrode kinetics is best justified by the close correlation of experiment and two-dimensional simulated voltammetry, using the diffusion domain approximation. 

The situation becomes more complicated for nanoelectrodes arrays since they typically have a total footprint in the order of microns, and hence even when adjacent diffusion fields fully overlap, behaviour akin to that at a single microelectrode is still observed. The extension of the diffusion domain approximation to nanoelectrode arrays was explored by Godino et al, who compared simulated voltammetry generated by 2D and 3D modelling with... 

Davies and Compton completed 2D simulations based on the diffusion domain approximation introduced by Amatore et al. and also found that 2R was not a sufficient guideline for separation distance in collections of regularly and irregularly spaced electrodes with radii <10 pm. " Davies and Compton suggested that minimum distance between neighboring electrodes to avoid diffusion layer overlap is not linearly related to electrode size. Rather, ideal separation distance is a function of diffusion coefficient of the electroactive species, and the scan rate of the CV. Simulations can thus be used to correctly predict ideal separation distance. For example, Davies and Compton estimated this distance for 100 nm radius electrodes to be -12 pm at 2 V s and 70 pm at 0.005 V s and for 10 nm radius electrodes to be 5 pm at 2 V s and 20 pm at 0.005 V s when the diffnsion coefficient of the electroactive species is 10 cm s . " ... 

These zones are approximated as being cylindrical, with the particle situated at the symmetry axis. If a random spatial distribution of microparticles is assumed, the respective diffusion domains (cylinders) are of different sizes, with a probability distribution function as follows [41] ... 

Figure 6.14 Coordinate system used to model the diffusion domain for a cylindrically approximated diffusion domain. The plane to be simulated is shaded [40].

The extra two intervals Xv+i — X and X +i n+i need not be expanding and are best set equal to Xn — Xn-u Alternatively, one might choose asymmetric backward-pointing difference approximation m"(4) for the point X -i and make Xn the bulk point, thus obviating the need for the extra two points. However, for methods where there is a Neumann (derivative) condition at the outer boundary, there is no such choice, and then Xn is the furthest point, and backwards derivative approximations must be applied on the boundary. This happens in such cases as thin layer eells, the diffusion domain discussed in connection with arrays of electrodes in Chap. 12, and a case the present authors have worked on, a flat polymer film containing an enzyme over a disk electrode [8]. 

Using these approximations the simulation task becomes essentially the same as already described for the simulation of a UMDF undisturbed by other electrodes. The pde to solve is given by (12.2). Normalisation is done as already described for a single UMDF, resulting in Fq. (12.17) for potential step or (12.27) for LSV conditions. The difference is that the UMDF is now in effect embedded in a walled area of base radius d, i.e. the radius of the diffusion domain, so that r ax = d and the boundary condition changes from... 

Lenhoff, J. Chromatogr., 384, 285 (1987)] or by direct numerical solution of the conservation and rate equations. For the special case of no-axial dispersion with external mass transfer and pore diffusion, an explicit time-domain solution, useful for the case of time-periodic injections, is also available [Carta, Chem. Eng. Sci, 43, 2877 (1988)]. In most cases, however, when N > 50, use of Eq. (16-161), or (16-172) and (16-174) with N 2Np calculated from Eq. (16-181) provides an approximation sufficiently accurate for most practical purposes. 

The low-pressure region displays the electroneutrality equation approximation [e ] = 2[Vx ]. Electrons predominate so that the material is an n-type semiconductor in this regime. In addition, the conductivity will increase as the partial pressure of the gaseous X2 component decreases. The number of nonmetal vacancies will increase as the partial pressure of the gaseous X2 component decreases, and the phase will display a metal-rich nonstoichiometry opposite to that in the high-pressure domain. Because there is a high concentration of anion vacancies, easy diffusion of anions is to be expected. 

Two numerical methods have been used for the solution of the spray equation. In the first method, i.e., the full spray equation method 543 544 the full distribution function / is found approximately by subdividing the domain of coordinates accessible to the droplets, including their physical positions, velocities, sizes, and temperatures, into computational cells and keeping a value of / in each cell. The computational cells are fixed in time as in an Eulerian fluid dynamics calculation, and derivatives off are approximated by taking finite differences of the cell values. This approach suffersfrom two principal drawbacks (a) large numerical diffusion and dispersion... 

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