Diffusion capacitance

Two general approaches have been used in low-temperature studies. In the first, the uncompensated resistance, electrode capacitance, diffusion coefficient, and kinetic and thermodynamic parameters describing the electrode reaction are incorporated in a master model, which is treated (usually by some form of digital simulation) to calculate the expected voltammetric response for comparison with experiment [7,49]. 

There is a range of equations used describing the experimental data for the interactions of a substance as liquid and solid phases. They extend from simple empirical equations (sorption isotherms) to complicated mechanistic models based on surface complexation for the determination of electric potentials, e.g. constant-capacitance, diffuse-double layer and triple-layer model. 

Several models have been developed to describe reactions between aqueous ions and solid surfaces. These models tend to fall into two categories (1) empirical partitioning models, such as distribution coefficients and isotherms (e.g., Langmuir and Freundlich isotherms), and (2) surface-complexation models (e.g., constant-capacitance, diffuse-layer, or triple-layer model) that are analogous to solution complexation with corrections for the electrostatic effects at the solid-solution interface (Davis and Kent, 1990). These models have been described in numerous articles (Westall and Hohl, 1980 Morel, Yeasted, and Westall, 1981 James and Parks, 1982 Barrow, 1983 Westall, 1986 Davis and Kent, 1990 Dzombak and Morel, 1990). Travis and Etnier (1981) provided a comprehensive review of the partitioning and kinetic models typically used to define sorption of ions by soils. The reader is referred to the cited articles for details of the models. 

It is additionally noted that the changes in resistance/ capacitance/diffusivity and electrochemical or thermodynamic potential with lithium content were disregarded for the calculation of the polarization. This seems to be acceptable in the operating conditions adopted in this work due to small variation of lithium content during the discharging for 10 s. For further discussion on this topic, see [31]. 

Fig. 7. (a) Simple battery circuit diagram where represents the capacitance of the electrical double layer at the electrode—solution interface, W depicts the Warburg impedance for diffusion processes, and R is internal resistance and (b) the corresponding Argand diagram of the behavior of impedance with frequency, for an idealized battery system, where the characteristic behavior of A, ohmic B, activation and C, diffusion or concentration (Warburg... 

The model just presented describes what electrochemists call the diffuse part of the double layer and no account is made of the inner layer effects such as the plane of the closest approach. To have an idea what the impact of the effects predicted by this model on the measured capacitance could be, we assume the traditional inner and diffuse layer separation. However, we... 

Impedance Some of the errors arising from the use of linear polarisation resistance led to interest and development in a.c. systems.An early development used a fixed a.c. frequency and a commercial instrument was produced in the UK. Inaccuracies still occurred, however, and were due to the electrode impedance which is fequency dependent. Electrode reactions have a capacitance component, in addition to resistance, resulting in a requirement to measure the impedance. However, the total impedance comprises values for the reaction, solution, diffusion and capacitance. Measurements at different frequency are more reliable, particularly where high solution resistances occur. Simplifications for industrial monitoring have been developed consisting of two measurements, i.e. at a high (10 kHz) and low frequency (0-1 Hz). The high-frequency measurement can identify the... 

The capacitance of the double layer consists of combination of the capacitance of die compact layer in series with that of the diffuse layer. For two capacitors in series, the total capacitance is given by... 

Figure 1-13 displays the experimental dependence of the double-layer capacitance upon the applied potential and electrolyte concentration. As expected for the parallel-plate model, the capacitance is nearly independent of the potential or concentration over several hundreds of millivolts. Nevertheless, a sharp dip in the capacitance is observed (around —0.5 V i.e., the Ep/C) with dilute solutions, reflecting the contribution of the diffuse layer. Comparison of the double layer witii die parallel-plate capacitor is dius most appropriate at high electrolyte concentrations (i.e., when C CH). 

The heterogeneity of a metal surface is responsible for the curvature of the Parsons-Zobel (PZ) plot (1/Cvs. 1/Q, where Cis the experimental capacitance and Qthe diffuse layer capacitance calculated on the basis of... 

The model more generally accepted for metal/electrolyte interfaces envisages the electrical double layer as split into two parts the inner layer and the diffuse layer, which can be represented by two capacitances in series.1,3-7,10,15,32 Thus, the total differential capacitance C is equal to... 

Measurements based on the Gouy-Chapman-Stem theory to determine the diffuse double-layer capacitance 10, 24,72, 74... 

Frumkin was the first to give a qualitative consideration of the electrochemical properties of pc electrodes.10,20 70 He noted that the charge fixed value of the potential E and this may change the form of the capacitance curve near the diffuse-layer capacitance minimum. Important results were obtained in a pioneering paper by Valette and Hamelin.67 They compared experimental capacitance curves for a pc-Ag electrode and its three basic faces. They found that the capacitance of a pc-Ag electrode can be obtained by the superposition of the corresponding Cj, E curves for individual faces exposed at the pc surface, i.e. 

A weighted sum of C,E curves for the faces was found to be similar to the C,E curve for a pc electrode. According to Valette and Hamelin,67 all main Ag faces [(111), (100), and (110)] are exposed on the surface, their fractions 0j on the surface being 0.31, 0.23, and 0.46, respectively. These authors demonstrated that the diffuse-layer capacitance minimum potential E a of a pc-Ag electrode was only slightly less negative (30 mV) than the pzc of the Ag(110) face, i.e., for the face with the more negative value of EamQ. The diffuse-layer capacitance minimum for pc-Ag was wider and less deep than for the Ag faces. 

Mathematical simulation of C, E curves shows that the shape of the diffuse-layer capacitance minimum depends on the difference of Eamo in individual faces and their fractions, as well as on the shape of partial Cj, E curves (Fig. 9). 

The results of experimental capacitance studies at two plane model pc-Bi electrodes were in agreement with these conclusions.2 266 Thus it has been shown that the potential of the diffuse-layer capacitance minimum for a pc electrode does not correspond to the zero charge potential of the whole surface, i.e., Zfipj Oat E n-... 

In the second group of models, the pc surface consists only of very small crystallites with a linear parameter y, whose sizes are comparable with the electrical double-layer parameters, i.e., with the effective Debye screening length in the bulk of the diffuse layer near the face j.262,263 In the case of such electrodes, inner layers at different monocrystalline areas are considered to be independent, but the diffuse layer is common for the entire surface of a pc electrode and depends on the average charge density <7pc = R ZjOjOj [Fig. 10(b)]. The capacitance Cj al is obtained by the equation... 

The division of the interface into an inner layer and a diffuse layer has been a matter of discussion in view of the molecular dimensions of the inner layer.122-126,279-285 However, the contribution of a constant capacitance is an experimental fact. Furthermore, molecular theories for electrolytes near a charged hard wall282 as well as phenomenological nonlocal electrostatic theories283 predict such a component without artificial introduction of any inner layers. This turns out to be an effect of the short-range structure of the solvent.279-285... 

The electrical double-layer structure at Ga/DMF, In(Ga)/DMF, and Tl(Ga)/DMF interfaces upon the addition of various amounts of NaC104 as a surface-inactive electrolyte has been investigated by differential capacitance, as well as by the streaming electrode method.358 The capacitance of all the systems was found to be independent of the ac frequency, v. The potential of the diffuse layer minimum was independent of... 

Borisova et al. A diffuse-layer minimum was found in the C, E curves, but the capacitance dispersion was appreciable and the value of C at Emib for Cd was higher than that calculated using the C, E curve for Hg 10,220,221 jt was notect10 that one of the reasons for this was the roughness of the pc-Cd surface. Therefore, Cd, Pb, and T1 electrodes were remelted in an inert atmosphere to give solidified drop electrodes. The capacitance dispersion was somewhat lower, but the difference between calculated and experimental capacitance was still substantial.221... 

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