Constant Potential Conditions

Under the assumption of constant potential, the boundary conditions for the problem 

However, the mathematical difficulties here are even higher than in the case of a single diffuse layer (Chapter 3), so, as it was done there, diverse simplifying assumptions were introduced, resulting in a number of different theories. 

1 Linear Approximation The Hogg-Healy-Fuerstenau Theory 

The simplest approximation is of the Debye-Hiickel type, first proposed by Hogg, Healy, and Fuerstenau (1966), assuming low potentials (ZiFy/ RT) and linearizing Equation 6.24, leading to (see Section 3.2.1) 

For two spheres of radii and R2 at a distance d, the following expression is obtained using the Derjaguin approximation  

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Controlled potential methods have been successfully applied to ion-selective electrodes. The term voltammetric ion-selective electrode (VISE) was suggested by Cammann [60], Senda and coworkers called electrodes placed under constant potential conditions amperometric ion-selective electrodes (AISE) [61, 62], Similarly to controlled current methods potentiostatic techniques help to overcome two major drawbacks of classic potentiometry. First, ISEs have a logarithmic response function, which makes them less sensitive to the small change in activity of the detected analyte. Second, an increased charge of the detected ions leads to the reduction of the response slope and, therefore, to the loss of sensitivity, especially in the case of large polyionic molecules. Due to the underlying response mechanism voltammetric ISEs yield a linear response function that is not as sensitive to the charge of the ion. 

For large distances (i>Ac) both boundary conditions lead to identical forces. At small distances the constant charge condition leads to more repulsive forces than the constant potential condition (Fig. 6.10). 

Jacobsen-Katsuki-type chiral (salen)cobalt(II) complex 267 (10 mol%) was used by Dunach and coworkers to catalyze electrochemical radical 5-exo cyclizations of ort/m-bromophenyl allyl ethers 293a,b to dihydrobenzofurans 295a,b (and 296a,b) [332]. Constant current or constant potential conditions... 

Making use of Eq. (3.91) it is easy to note that if Ym and Xd are known, there no need to assume constant charge or constant potential conditions. As a matter of fact, self consistent values of >o and Q0 can be generated at each integration of Eq. (3.90) carried out with the given Ym and Xd. Thus, the response of Y0 and Q0 to the overlap of the two diffuse electric layers can be obtained without any additional approximations except those inherent to the PB approximation itself. 

The different situations that can arise between charged surfaces acting either under constant charge or constant potential conditions may be summarized thus ... 

The cell shown in Figure 1 is usually used for the reaction carried out under constant current conditions. For carrying out the reaction under a constant potential condition a cell equipped with three electrodes Is required. 

Which boundary condition is more realistic depends on the materials used. In addition, the electrolyte and the speed of the approach might have an influence. Prica and coworkers measured force curves between 2irconia, which showed a constant charge behavior [84]. Also the force between two surfaces coated with densely packed carboxylic groups followed constant charge conditions [94]. In other cases, constant potential conditions are more appropriate. Most cases, however, lay between the two extremes. Then often, a charge regulation model is applied [156]. 

At the line electrodes, constant potential conditions apply ... 

FIGURE 6.13 Schematic representation of the relative interaction strength of the different contributions (a), and the resulting total DLVO interaction (b), at different ionic strengths for two identical spheres at constant potential conditions with Ci = fl2 = 10 nm, = aa ... 

Although ABTS can be generated either potentiostatically or galvanostatically, the operation of the cell in the galvanostatic mode has some advantages over the operation under constant-potential conditions. When a constant current (i, pA) is imposed on the... 

The interpretation of experiments on supercapacitors usually relies on the use of equivalent electric circuits. ° The MD simulations allow to test the relevance of such macroscopic models. The most commonly used model in the case of nanoporous electrodes is that of a transmission line, in which the charge penetrates progressively into the electrode. This model is based on an infinite succession of slices connected in parallel through the constant potential condition. Each of them consists of a capacitor, which accounts for the possibility to store the charge by adsorption of the ions at the surface of the pores, and of a resistance which describes the transport of the electrolyte inside the pores. 

Let us turn from planar, parallel surface to spheres. The interaction between spheres can be calculated in different ways, (a) We can start with expressions (4.63-4.66) and apply the Derjaguin approximation [416,427-430). This leads to a good approximation for Ri, R2 3> X.d and short distance, (b) Or, we superimpose the potentials around spheres [430-433]. This is a good approximation for small spheres and large distances, (c) Ohshima solved the linearized Poisson-Boltzmann equation in two dimensions analytically for constant potential conditions [434]. As a leading term, he obtained... 

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