Stationary spherical electrode

Holub, K. and van Leeuwen, H. P. (1984). Influence of reactant adsorption on limiting currents in normal pulse polarography. 2. Theory for the stationary, spherical electrode, J. Electroanal. Chem., 162, 55-65. 

On a stationary spherical electrode, the reaction (1.1) can be mathematically represented by the following system of differential equations and boundary conditions ... 

The first applied potential is set at a value E at a stationary spherical electrode during the interval 0 < t < i. The diffusion mass transport of the electroactive species toward or from the electrode surface is described by the following differential equation system ... 

II.3.3 Simple Reactions on Stationary Spherical Electrodes and Microelectrodes... 

B) On a stationary spherical electrode, a simple redox reaction... 

The basic theory of SWV at a stationary spherical electrode is explained in the Appendix (see Eqs. IL3.A20 - IL3.A26). If the electrode reaction (Eq. IL3,1) is fast and reversible, the shape of the SWV response and its peak potential are independent of electrode geometry and size [46-48]. At a spherical electrode the dimensionless net peak current is linearly proportional to the inverse value of the dimensionless electrode radius y = [49, 50]. If nlsE - 5 mV and... 

In reference [18], the authors show the expression for the stationary current obtained at uniformly accessible electrodes in the case in which species R is not initially present in the solution (i.e., < ,) = 0). In the case of spherical electrodes, it can be written ... 

In this section, microdisc electrodes will be discussed since the disc is the most important geometry for microelectrodes (see Sect. 2.7). Note that discs are not uniformly accessible electrodes so the mass flux is not the same at different points of the electrode surface. For non-reversible processes, the applied potential controls the rate constant but not the surface concentrations, since these are defined by the local balance of electron transfer rates and mass transport rates at each point of the surface. This local balance is characteristic of a particular electrode geometry and will evolve along the voltammetric response. For this reason, it is difficult (if not impossible) to find analytical rigorous expressions for the current analogous to that presented above for spherical electrodes. To deal with this complex situation, different numerical or semi-analytical approaches have been followed [19-25]. The expression most employed for analyzing stationary responses at disc microelectrodes was derived by Oldham [20], and takes the following form when equal diffusion coefficients are assumed ... 

The variation of 7fphe with the potential for different values of the electrode radius obtained in CV with spherical electrodes, including the limiting case of a planar electrode (rs —> oo),canbeseeninFig.5.10a.Fromthesecurves, whichhave been calculated using Eq. (5.71) for a sweep rate v = 0.1 V s, it can be observed that the diffusion layer thickness decreases with the electrode radius and the constant value <5 = rs is reached when rs < 5 pm (i.e., a truly stationary I-E response is obtained as can be seen in Fig. 5.10b). 

Equation (6.42) clearly shows that the CV stationary responses of disc and spherical electrodes hold the same equivalence relationship as that observed for a simple charge transfer process ... 

In Fig. 7.1, the influence of the electrode radius rs on the dimensionless DSCVC i// Scv(7IA y curves (with A / = F AE /RT) is shown. Figure 7.1a corresponds to the planar contribution, Fig. 7.1b to the radial or stationary contribution, and the curves corresponding to spherical electrodes obtained from Eq. (7.19) are shown in Fig. 7.1c. From these curves, it can be seen that the peak potential does not coincide with the formal potential in the case of planar electrodes. In the case of spherical electrodes (Fig. 7.1c), the peaks of the first and second scans approach the formal potential as the electrode radius decreases, and Wdscvc responses of both scans become symmetrical. 

The electrode size is another important factor to be considered since it affects the magnitude of the diffusive transport, as shown in Fig. 7.14 for totally irreversible processes. At planar and spherical electrodes significant differences are found between double pulse and multipulse modes, with the discrepancy diminishing when the electrode radius decreases, since the system loses the memory of the previous pulses while approaching the stationary response. Thus, the relative difference in the peak current of a given double pulse technique and the corresponding multipulse variant is always smaller than 2 % when... 

The time required to establish such stationary states will depend on the diffusion coefficients of the ions in the solution and the size of the electrodes. For a small spherical electrode of radius ro the time for establishment of a quasi-stationary state will be of the order of f ro/ir D. For ro 0.1 cm and D 10 cm /sec, t is about 100 sec, BO that for large electrodes the times can become quite long. For ions, the diffusion of O and R is not independent of the speed of negative ions in the solution because the condition of electroneutrality requires a coupled diffusion mechanism. 

Equation (5.18), in contrast to (5.16), contains a term that does not depend on time - a stationary component of the limiting diffusion current. Therefore, poten-tiostatic electrolysis on a spherical electrode differs from that on a planar electrode and ends up with a stationary regime. The process rate does not change over time in this mode (Fig. 5.10). 

The theoretical treatment of mass transfer in LSV and CV assumes that only diffusion is operative. Supporting electrolyte concentrations of the order of 0.1 M are generally used at substrate concentrations of the order of 10-3 M, which should preclude the necessity of considering mass transfer by migration. Here, it is assumed that planar stationary electrodes are used under circumstances where diffusion can be considered to be semi-infinite linear diffusion. Other types of electrode may give rise to spherical, cyclindrical or rectangular diffusion and these cases have been treated. 

The voltammetric behavior of the first-order catalytic process in DDPV for different values of the kinetic parameter Zi(= ( 1 + V) Ti) at spherical and disc electrodes with radius ranging from 1 to 100 pm can be seen in Fig. 4.25. For this mechanism, the criterion for the attainment of a kinetic steady state is %2 > 1-5 (Eq. 4.232) [73-75]. In both transient and stationary cases, the response is peakshaped and increases with j2. h is important to highlight that the DDPV response loses its sensitivity toward the kinetics of the chemical step as the electrode size decreases (compare the curves in Fig. 4.25a, c). For the smallest electrode (rd rs 1 pm, Fig. 4.25c), only small differences in the peak current can be observed in all the range of constants considered. Thus, the rate constants that can... 

In the case of disc electrodes, a similar behavior to that observed for spherical ones is obtained (although in this case the diffusion layer thickness is an average magnitude), whereas for band or cylinder electrodes, the diffusion layer thickness is always potential dependent, and no constant limit is achieved, even for very small values of the electrode characteristic dimension, confirming the impossibility of these electrodes achieving a true stationary response [5, 8, 16, 29, 30]. 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

Equation (7.116) for the SW charge-potential response of surface-bound molecules presents a potential dependence identical to that obtained for the stationary SW current of a one-electron electrochemical reaction between solution soluble molecules at electrodes of any geometry (see for example Eqs. (7.35) and (7.36) for spherical and disc electrodes, respectively). Therefore, the expression of the peak parameters will be analogous to those obtained in the case of solution soluble molecules (see Eq. (7.37)) ... 

The dimensionless SWV curve is identical to the stationary SWV curve obtained for a fast electrochemical reaction with solution soluble molecules given by Eqs. (7.35) or (7.36) for spherical or disc electrodes, respectively, and also to that obtained for a reversible surface electrode process (Eq. (7.116)). Therefore, the peak potential coincides with the formal potential and the half-peak width is given by Eq. (7.32). The peak height is amplified by kc ... 

Sample of solid suspension in the supernatant, collected after the attainment of adsorption equilibrium, was transferred to the thermostated microelectrophoresis celt. The velocity U for at least 10 particles was measured at the two stationary levels and the average value taken the polarity of electrodes was reversed after each velocity measurement. For a spherical particle, the following equation is satisfied ... 

The influence of the nonlinearity of diffusion on the observed complex plane plots is shown in Fig. 13. Spherical mass transfer causes the formation of a depressed semicircle at low frequencies instead of the linear behavior observed for linear semi-infinite diffusion. For very small electrodes (ultramicroelectrodes) or low frequencies, the mass-transfer impedances become negligible and the dc current becomes stationary. On the Bode phase-angle graph, a maximum is observed at low frequencies. 

Electrophoresis refers to the movement of a charged surface relative to a stationary liquid. For instances, under an applied electric field, a negatively charged particle will migrate towards the positive electrode (anode). For spherical particles the velocity is given as [1]... 

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