Diffusion nonplanar

C3.6.13 where large diffusion fluxes are indicated by —> and smaller diffusion fluxes by —+. For tire part of tire B front tliat protmdes into tire A region, fast diffusion of B leads to dispersal of B and suppresses tire autocatalytic reaction tliat requires two molecules of B. The front will have difficulty advancing here. In tire region where A protmdes into B, A will react leading to advancement of tire front. The net effect is to remove any initial nonplanarity and give rise to a planar front. 

Adsorbed additives also tend to undergo reduction during the electroless process, and become incorporated as impurities into deposits, most likely via a mechanism similar to that involved in ternary alloy deposition. In a manner similar to that discussed below in greater detail for dissolved 02, electroless deposition rates will be lower for features smaller than the stabilizer diffusion layer thickness. The edges of larger features, which experience higher stabilizer levels due to enhanced nonplanar... 

Here F is the Faraday constant C = concentration of dissolved O2, in air-saturated water C = 2.7 x 10-7 mol cm 3 (C will be appreciably less in relatively concentrated heated solutions) the diffusion coefficient D = 2 x 10-5 cm2/s t is the time (s) r is the radius (cm). Figure 16 shows various plots of zm(02) vs. log t for various values of the microdisk electrode radius r. For large values of r, the transport of O2 to the surface follows a linear type of profile for finite times in the absence of stirring. In the case of small values of r, however, steady-state type diffusion conditions apply at shorter times due to the nonplanar nature of the diffusion process involved. Thus, the partial current density for O2 reduction in electroless deposition will tend to be more governed by kinetic factors at small features, while it will tend to be determined by the diffusion layer thickness in the case of large features. 

The Cottrell equation states that the product it,/2 should be a constant K for a diffusion-controlled reaction at a planar electrode. Deviation from this constancy can be caused by a number of situations, including nonplanar diffusion, convection in the cell, slow charging of the electrode during the potential step, and coupled chemical reactions. For each of these cases, the variation of it1/2 when plotted against t is somewhat characteristic. 

Nonplanar electrodes such as spheres and cylinders exhibit an increase in it1/2 with increasing t. However, planar diffusion can be closely approximated... 

For a system with no kinetic or adsorption complications, the forward transition time x decreases while xr increases until finally x = xr in the limit, at steady state. (Because the convergence rate is slow, equality of x and xr is not commonly achieved experimentally before the onset of natural convection and nonplanar diffusion effects.) Quantitative treatments for single component systems, multicomponent systems, stepwise reactions, and systems involving chemical kinetics have been derived. The technique has not been used extensively. 

Carbon fibers have not been studied as extensively as GC or graphite, and in most cases the fiber is pretreated. Thus most of the electrochemical properties of fibers are discussed in the next section, Preparation. A few general points are useful here, dealing with size and resistance. Since the majority of carbon fibers are 5-15 pm in diameter, they will exhibit nonplanar diffusion under most conditions, whether they are used as disks or as cylinders. For example, VDt for a typical analyte (D = 5 x 10 6 cm2/s) equals 2.2 pm at 10 ms. This is a significant fraction of a typical fiber diameter, so diffusion will become nonplanar even at short times. Thus any experiment lasting more than a few milliseconds will deviate from a response predicted for planar diffusion. Note that the deviation depends on whether the fiber end is used as a disk electrode or an exposed fiber is used as a cylinder, but quantitative theories have been presented for both cases [48]. 

When nonplanar geometries are considered for the reaction scheme (3.II), the following diffusive-kinetic differential equations must be solved ... 

The ratios given in Eq. (4.66) are only dependent on the electrode shape and size but not on parameters related to the electrode reaction, like the number of transferred electrons, the initial concentration of oxidized species, or the diffusion coefficient D. For fixed time and size, the values of f or Qf2 are characteristic for a simple charge transfer (see Fig. 4.4 for the plot of Qf2 calculated at time (ti + T2) for planar, spherical, and disc electrodes) and, as a consequence, deviations from this value are indicative of the presence of lateral processes (chemical instabilities, adsorption, non-idealities, etc.) [4, 32]. Additionally, for nonplanar electrodes, these values allow to the estimation of the electrode radius when simple electrode processes are considered. 

For nonplanar electrodes there are no analytical expressions for the CV or SCV curves corresponding to non-reversible (or even totally irreversible) electrode processes, and numerical simulation methods are used routinely to solve diffusion differential equations. The difficulties in the analysis of the resulting responses are related to the fact that the reversibility degree for a given value of the charge transfer coefficient a depends on the rate constant, the scan rate (as in the case of Nemstian processes) and also on the electrode size. For example, for spherical electrodes the expression of the dimensionless rate constant is... 

In this section, the current-potential curves of multi-electron transfer electrode reactions (with special emphasis on the case of a two-electron transfer process or EE mechanism) are analyzed for CSCV and CV. As in the case of single and double pulse potential techniques (discussed in Sects. 3.3 and 4.4, respectively), the equidiffusivity of all electro-active species is assumed, which avoids the consideration of the influence of comproportionation/disproportionation kinetics on the current corresponding to reversible electron transfers. A general treatment is presented and particular situations corresponding to planar and nonplanar diffusion and microelectrodes are discussed later. 

The location of the response is not affected by the ratio between pulse times in the case of DNMPV. This effect can be seen in Fig. 7.4 in which the current potential curves calculated for different values of (t -zt) and planar electrodes have been plotted. This ratio only affects the magnitude of the current and not the peak potential, which coincides with the formal potential in all the cases. For nonplanar electrodes, this behavior also holds when the diffusion coefficients of oxidized and reduced species are equal [9]. 

In the case of Cyclic Square Wave Voltammetry (CSWV), the SWV curve obtained in the second scan is a mirror image to that of the first scan whatever the electrode geometry if the diffusion coefficients of species O and R are assumed as equal. In the contrary case, although the peak potentials of both scans are coincident, differences in the peak heights are observed for nonplanar electrodes. 

Only then is diffusion into the fiber possible. The tendency toward aggregation is therefore characteristic of substantive dyes, which also explains why coplanar dyes possess greater substantivity than nonplanar ones. 

Both of these methods may be applied to nonplanar electrodes if the results are obtained at electrolysis times sufficiently short that the diffusion layer remains thin in comparison to the radius of curvature of the nonplanar electrode surface. For example, the spherical hanging-mercurcy-drop electrode provides chronoamperometric data that deviate less than 1-2% from the linear-diffusion Cottrell equation out to times of about 1 s. With solid wire electrodes of cylindrical geometry, similar conclusions apply, but at short times surface roughness effects yields a real surface area that is larger than the geometric area. 

There are three basic distinct types of phenomena that may be responsible for intrinsic instabilities of premixed flames with one-step chemistry body-force effects, hydrodynamic effects and diffusive-thermal effects. Cellular flames—flames that spontaneously take on a nonplanar shape—often have structures affected most strongly by diffusive-thermal... 

Since the shape of a nonplanar tip is typically imperfect, one is not motivated to carry out extensive simulations for these complicated systems. All equations describing the approach curves under diffusion control are highly approximate, and no theory has been developed for finite kinetics at either tip or substrate electrode. 

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