Double layer charge distribution

The double layer charge distribution (eq 11) was employed in the derivation of the last equation. 

Wang, J. and Bard, A.J. (2001) Direct atomic force microscopic determination of surface charge at the gold/electrolyte interface - the inadequacy of classical GCS theory in describing the double-layer charge distribution. Journal of Physical Chemistry B, 105, 5217. 

The above discussion also applies to the thin double layer systems with surface conductance considered in the previous section. The surface conductance alters the double layer charge distribution, and so it gives rise to double layer distortion. 

To minimize absorption from the solution, optical thin layer cells have been designed. The working electrode has the shape of a disc, and is mounted closely behind an IR-transparent window. For experiments in aqueous solutions the intervening layer is about 0.2 to 2 ftm thick. Since the solution layer in front of the working electrode is thin, its resistance is high this increases the time required for double-layer charging - time constants of the order of a few milliseconds or longer are common - and may create problems with a nonuniform potential distribution. 

Basically, the impedance behavior of a porous electrode cannot be described by using only one RC circuit, corresponding to a single time constant RC. In fact, a porous electrode can be described as a succession of series/parallel RC components, when starting from the outer interface in contact with the bulk electrolyte solution, toward the inner distribution of pore channels and pore surfaces [4], This series of RC components leads to different time constant RC that can be seen as the electrical response of the double layer charging in the depth of the electrode. Armed with this evidence, De Levie [27] proposed in 1963 a (simplified) schematic model of a porous electrode (Figure 1.24a) and its related equivalent circuit deduced from the model (Figure 1.24b). 

A charged surface and the ions, which neutralize the surface, together create an electric double layer. The distribution of the ions can be evaluated from the Poisson-Boltzmann equation where the ions are treated as point particles and the primitive model is used. Further, all correlations between the ions are neglected, which means that the ions are interacting directly only with the colloids and through an external field given by the average distribution of the small ions. The distribution of the particles are assumed to follow Boltzmann s theorem [11]... 

For a porous electrode such as is found in a fuel cell, since the capacitance caused by double-layer charging is distributed along the length of the pores, the conventional double-layer capacitance is often replaced by a CPE. Then the equivalent circuit in Figure 4.15 can be modified to that shown in Figure 4.16a. 

The distribution of small ions between regions I and n is described as the expulsion of a certain amount of ions of the same sign as the colloid. This ion exclusion effect is described in the manner of Klaarenbeek [8], whose result is encapsulated in a quantity g that expresses the ratio of the co-ion deficit to the total double-layer charge. Let us write... 

The preceding method (Section 6.3.1) relies completely on the limited equilibration of the ion distribution in solution, due to the hindered diffusion. Additionally, the charging time of the double layer can be exploited for the local machining of surfaces. As mentioned in Section 6.2.2, the time constant for the double-layer charging is given by the product of solution resistance and double-layer capacity. In the above experiments employing the tip of an STM, which is a few nms distance to the surface, as local counter electrode this time constant is well below nanoseconds, even for diluted electrolytes. Nonetheless, for electrode separations in the... 

For L-p/l y>> 1 the limitations due to proton transport are practically absent and the impedance response forms a perfect semicircle in the Cole - Cole representation with M = R = 9tdiff/2, due to the parallel processes of charge-transfer and double-layer charging, which are distributed homogeneously within the layer. The frequency in the turning point of the semicircle is 2p, the approximation being the better the larger the ratio L-p/l. For Lp/l = 1 the error of this estimate is about 10%. 

FIGURE 2-1 Helmholtz model of the electrical double layer, (a) Distribution of counterions in the vicinity of the charged surface. (b) Variation of electrical potential with distance from the charged surface. 

Double-layer charging of the pores only (non-faradaic process) and inclusion of a pore-size distribution leads to complex plane impedance plots, as in Fig. n.5.7, i.e. at high frequencies, a straight line results in an angle of 45° to the real axis and, at lower frequencies, the slope suddenly increases but does not change to a vertical line [16]. 

Fig. IL5.7 Nyquist impedance plot due to a porous electrode with log-normal distribution of the pore sizes and with double-layer charging only...

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