Electrochemical processes semi-infinite diffusion

For disk-type electrodes, usually with a radius of O.l-l.O cm the thickness of the diffusion layer that is depleted of reactant is much smaller than the electrode size so that mass transport can be described in terms of planar diffusion of the electroactive species from the bulk solution to the electrode surface as schematized in Figure 1.2a, where semi-infinite diffusion conditions apply. The thickness of the diffusion layer can be estimated as for a time electrolysis t and usually ranges between 0.01 and 0.1 mm (Bard et al., 2008). For an electrochemically reversible -electron transfer process in the absence of parallel chemical reactions, the variation of the faradaic current with time is then given by the Cottrell equation ... 

As described in the introduction, submicrometer disk electrodes are extremely useful to probe local chemical events at the surface of a variety of substrates. However, when an electrode is placed close to a surface, the diffusion layer may extend from the microelectrode to the surface. Under these conditions, the equations developed for semi-infinite linear diffusion are no longer appropriate because the boundary conditions are no longer correct [97]. If the substrate is an insulator, the measured current will be lower than under conditions of semi-infinite linear diffusion, because the microelectrode and substrate both block free diffusion to the electrode. This phenomena is referred to as shielding. On the other hand, if the substrate is a conductor, the current will be enhanced if the couple examined is chemically stable. For example, a species that is reduced at the microelectrode can be oxidized at the conductor and then return to the microelectrode, a process referred to as feedback. This will occur even if the conductor is not electrically connected to a potentiostat, because the potential of the conductor will be the same as that of the solution. Both shielding and feedback are sensitive to the diameter of the insulating material surrounding the microelectrode surface, because this will affect the size and shape of the diffusion layer. When these concepts are taken into account, the use of scanning electrochemical microscopy can provide quantitative results. For example, with the use of a 30-nm conical electrode, diffusion coefficients have been measured inside a polymer film that is itself only 200 nm thick [98]. 

By proper treatment of the linear potential sweep data, the voltammetric i-E (or i-t) curves can be transformed into forms, closely resembling the steady-state voltammetric curves, which are frequently more convenient for further data processing. This transformation makes use of the convolution principle, (A.1.21), and has been facilitated by the availability of digital computers for the processing and acquisition of data. The solution of the diffusion equation for semi-infinite linear diffusion conditions and for species O initially present at a concentration Cq yields, for any electrochemical technique, the following expression (see equations 6.2.4 to 6.2.6) ... 

Reinmuth has examined chronopotentiometric potential-time curves and proposed diagnostic criteria for their interpretation. His treatment applies to the very limited cases with conditions of semi-infinite linear diffusion to a plane electrode, where only one electrode process is possible and where both oxidized and reduced forms of the electroactive species are soluble in solution. This approach is further restricted in application, in many cases, to electrode processes whose rates are mass-transport controlled. Nicholson and Shain have examined in some detail the theory of stationary electrode polarography for single-scan and cyclic methods applied to reversible and irreversible systems. However, since in kinetic studies it is preferable to avoid diffusion control which obscures the reaction kinetics, such methods are not well suited for the general study of the mechanism of electrochemical organic oxidation. The relatively few studies which have attempted to analyze the mechanisms of electrochemical organic oxidation reactions will be discussed in detail in a following section. 

Semi-infinite linear diffusion conditions The rate of an electrochemical process depends not only on electrode kinetics but also on the transport of species to/from the bulk solution. Mass transport can occur by diffusion, convection or migration. Generally, in a spectroeiectrochemicai experiment, conditions are chosen in which migration and convection effects are negligible. The solution of diffusion equations, that is the discovery of an equation for the calculation of oxidized form [O] and reduced form [R] concentrations as functions of distance from electrode and time, requires boundary conditions to be assumed. Usually the electrochemical cell is so large relative to the length of the diffusion path that effects at walls of the cell are not felt at the electrode. For semiinfinite linear diffusion boundary conditions, one can assume that at large distances from the electrode the concentration reaches a constant value. 

This relationship holds for any electrochemical process that involves semi-infinite linear diffusion and is the basis for a variety of electrochemical methods (e.g., polarogra-phy, voltammetry, and controlled-potential electrolysis). Equation (50) is the basic relationship used for solid-electrode voltammetry with a preset initial potential on a plateau region of the cmrent-voltage curve. Its application requires that the electrode configuration be such that semi-infinite linear diffusion is the controlling condition for the mass-transfer process. 

As the first approximation, impedance of a porous electrode can always be considered as a series combination of two processes—a mass-transport resistance inside the pores and impedance of electrochemical reactions inside the pores. De Levie was the first to develop a transmission line model to describe the frequency dispersion in porous electrodes in the absence of internal diffusion limitations [66]. De Levie s model is based on the assumption that the pores are cylindrical, of uniform diameter 2r and semi-infinite length /, not intercoimected, and homogeneously filled with electrolyte. The electrode material is assumed to have no resistance. Under these conditions, a pore behaves like a imiform RC transmission line. If a sinusoidal excitation is applied, the transmission line behavior causes the amplitude of the signal to decrease with the distance from the opening of the pore, and concentration and potential gradients may develop inside the pore. These assumptions imply that only a fraction of the pore is effectively taking part in the double-layer charging process. The RpQi i- [ohm] resistance to current in a porous electrode structure with number of pores n, filled with solution with resistivity p, is ... 

---