Sweep diffusion, described

The transient method characterized by linearly changing potential with time is called potential-sweep (potential-scan) voltammetry (cf. Section 5.5.2). In this case the transport process is described by equations of linear diffusion with the potential function... 

A complete comprehension of Single Pulse electrochemical techniques is fundamental for the study of more complex techniques that will be analyzed in the following chapters. Hence, the concept of half-wave potential, for example, will be defined here and then characterized in all electrochemical techniques [1, 3, 8]. Moreover, when very small electrodes are used, a stationary current-potential response is reached. This is independent of the conditions of the system prior to each potential step and even of the way the current-potential was obtained (i.e., by means of a controlled potential technique or a controlled current one) [9, 10]. So, the stationary solutions deduced in this chapter for the current-potential curves for single potential step techniques are applicable to any multipotential step or sweep technique such as Staircase Voltammetry or Cyclic Voltammetry. Moreover, many of the functional dependences shown in this chapter for different diffusion fields are maintained in the following chapters when multipulse techniques are described if the superposition principle can be applied. 

For LSV, the diffusion equation for a UMDE is a little different from that for a potential jump. The LSV case can be considered as one of a group of possible cases, in which the characteristic time is defined independently of the disk radius. In general, let that characteristic time be r. For LSV, as described on page 26, it is the time taken by the potential to sweep over one... 

The development of ultramicroelectrodes with characteristic physical dimensions below 25 pm has allowed the implementation of faster transients in recent years, as discussed in Section 2.4. For CA and DPSC this means that a smaller step time x can be employed, while there is no advantage to a larger t. Rather, steady-state currents are attained here, owing to the contribution from spherical diffusion for the small electrodes. However, by combination of the use of ultramicroelectrodes and microelectrodes, the useful time window of the techniques is widened considerably. Compared to scanning techniques such as linear sweep voltammetry and cyclic voltammetry, described in the following, the step techniques have the advantage that the responses are independent of heterogeneous kinetics if the potential is properly adjusted. The result is that fewer parameters need to be adjusted for the determination of rate constants. 

The electrode surface was assumed to contain N electroactive metal or metal oxide centers, respectively, which can be not only uniformly but also (mimicking more realistic experimental conditions) randomly distributed an example is the results of atomic force microscopy (AFM) studies on microparticle electrodes [53]. Here, the diffusion domain approach (as described in Section 6.3.2.2.1) has been employed that is, the electrode surface is assumed to be an arrangement of independent diffusion domains of radius Fq. If all particles are of the same radius, rj, but are distributed in a random manner, then a distribution of diffusion domains with different domain radii, ro, follows. The local position-dependent coverage is given by T. The electroactive microparticle flat disks of the radius rj are located in the center of the respective diffusion domain cylinder. The simulated (linear sweep voltammetric) reaction follows a one-electron transfer, and species B is stripped from the electrode surface into the solution, forming A, or ... 

The theoretical treatments for the different voltammetric methods (e.g., polarography, linear sweep voltammetry, and chronopotentiometry) and the various kinetic cases generally follow the procedures described previously. The appropriate partial differential equations (usually the diffusion equations modified to take account of the coupled reactions producing or consuming the species of interest) are solved with the requisite initial and boundary conditions. For example, consider the EfCi reaction scheme ... 

Since the forward reaction for a potential step to the limiting current region is unperturbed by the irreversible following reaction, no kinetic information can be obtained from the po-larographic diffusion current or the limiting chronoamperometric i-t curve. Some kinetic information is contained in the rising portion of the i-E wave and the shift of 1/2 with Wx- Since this behavior is similar to that found in linear potential sweep methods, these results will not be described separately. The reaction rate constant k can be obtained by reversal techniques (see Section 5.7) (32, 33). A convenient approach is the potential step method, where at = 0 the potential is stepped to a potential where Cq(x = 0) = 0, and at t = T it is stepped to a potential where Cr(x = 0) = 0. The equation for the ratio of (measured at time j.) to (measured at time Figure 5.7.3) is... 

Third, Morgan etal.[ teach the use of a plurality of passages (fibres, tubes, or other conduits) embedded in the retentate end tube sheet that allow fluid communication between the retentate header and the shell. The pressure difference between the header and shell drives a portion of the retentate product back into the shell. The sweep flow rate is determined by the number and size of the passages and cannot be regulated externally. This internal sweep design is used extensively. Durban et al. [12] describe a modification in which a diffuser is used to distribute the sweep more uniformly in the shell. A channel or conduit extends from the retentate header through the tube sheet into the fibre bundle. The channel end in the header is left as an open orifice while the channel end in the fibre bundle is capped by a porous diffuser. 

Now, consider the reduction of solution species O at a planar electrode surface during a voltammetry experiment in which the potential is swept with time in a linear manner. This is, of course, equivalent to the first sweep of a cyclic voltammogram, and the faradaic current that we observe is ideally limited by the diffusion of O to the electrode surface. In this case the observed peak current i is typically described [1-4] by the Randles-Sevcik equation—Eq. (5). 

In a linear potential sweep experiment performed on a RDE, the potential of the working electrode is scanned from a potential where no reaction occurs to a potential that causes a reaction to occur. A limiting current is achieved when the overpotential is high enough so that the reaction rate is determined by the mass transport rate of the reactant at a given electrode rotation rate. The surface concentration of the reactant drops to zero, and a steady mass transport profile is attained as C/L, where L is the diffusion layer thickness. At a fixed electrode rotation rate, L does not change, and thus C/L does not change. Therefore, a steady-state diffusion-controlled current is achieved, described by the Levich equation ... 

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