Constant phase element behaviour

Zoltowski, P, On the electrical capacitance of interfaces exhibiting constant phase element behaviour. Journal of Electroanalytical Chemistry, 1998, 443 pp. 149-154... 

Kerner, Z, and T, Pajkossy, Impedance of rough capacitive electrodes the role of surface disorder. Journal of Electroanalytical Chemistry, 1998. 448 pp. 139-142 Ldng, G, and K.E, Heusler, Remarks of the energetics of interfaces exhibiting constant phase element behaviour. Journal of Electroanalytical Chemistry, 1998. 457 pp, 257-260 Liu, S.H, Eractal model for the ac response of a rough interface. Physical Review Letters, 1985, 55 pp, 529-532... 

In this book, an explanation of capacitive behaviour in similar and comparable systems is not directly possible with constant-phase elements because such a comparison is only possible if n values are equal, particularly in the study of surfaces covered with polymer coatings where a unification of the envisaged parameters is necessary. The impedances measured match with a relatively large amount of samples, of which the structure can be complex, showing many sources of non-idealities (e.g. variations in thickness of the membrane, pore size and pore density42 7). A good indication if such non-idealities occur can be found in the values of n. If they are not comparable, non-idealities occur. 

EIS data analysis is commonly carried out by fitting it to an equivalent electric circuit model. An equivalent circuit model is a combination of resistances, capacitances, and/or inductances, as well as a few specialized electrochemical elements (such as Warburg diffusion elements and constant phase elements), which produces the same response as the electrochemical system does when the same excitation signal is imposed. Equivalent circuit models can be partially or completely empirical. In the model, each circuit component comes from a physical process in the electrochemical cell and has a characteristic impedance behaviour. The shape of the model s impedance spectrum is controlled by the style of electrical elements in the model and the interconnections between them (series or parallel combinations). The size of each feature in the spectrum is controlled by the circuit elements parameters. 

Understanding the oxidation mechanism is important. Impedance spectroscopy was recently used to study methanol electrooxidation, and kinetic parameters can be deduced from impedance spectra. Figure 6.58 shows an equivalent circuit that was developed for methanol oxidation on a Pt electrode, but which is common for all electrochemical reactions. In this circuit, a constant phase element was used rather than a double-layer capacitance, since a CPE is more realistic than a simple capacitor in representing the capacitive behaviour. 

The compact interlacing of crystalline and amorphous regions within typical PEO-based polymer electrolytes could well account for the characteristic impedance plots obtained for these materials [20]. Such plots are characterized (Figure 1.4) by depressed semi-circles and tilted spikes, indicative of constant phase element (CPE) behaviour. A CPE can be thought of as a leaky capacitor such as could be formed by a thin imperfect dielectric (the crystalline lamella) separating the amorphous, inter-lamellar conducting regions. 

For example, whilst modelling high frequency noise in nanopore devices, Smeets el al. demonstrated the necessity of accounting for non-ideal capacitive behaviour and it has become common since for the generalised device capacitance to be replaced by either a parallel resistor-capacitor unit or a constant phase element [23]. Chien et al. used the circuit shown in Fig. 14 B, replacing C with a CPE, to determine pore resistance and diameter from Bode plots for Si pores. [42]. Pedone et al. used two parallel resistor-CPE units in series with a solution resistance for characterising the electrical properties of a pore-cavity-pore deviee [43]. 

Consider a small region in phase space, At, sufficiently small that the density is essentially constant within this region. The number of system points An = qAt. If the surface of the element is always determined by the same system points originally in the surface, the behaviour of the element can be followed as a function of time. None of the points within the surface may escape and there are no sources or sinks of system points. Consequently... 

Important selectivity enhancements are observed upon complexing with neutral ligands. The stability constants of the adsorbed complexes exceed the values in aqueous solution by two to three orders of magniture. Such observation may be relevant to the behaviour of transition elements in the environment in that stability constants of adsorbed organic matter complexes may differ from the values found for solution phase equilibria. Such effects are indeed observed for Cu (139) and Ca (132). 

The yields of aluminium and silicon are nearly constant for all the experiments, the yield of aluminium loeing slightly higher. The yield of incorporation of boron, instead, is very low when the synthesis gel is aluminium-rich (exp. -4). Boron and aluminium are incorporated at the same extent only when the available boron largely exceeds aluminium (exp. 5, 6). This behaviour can be accounted for by a competition between the Kinetics of incorporation of borosilicate and aluminosilicate units in the zeolite. No rate constants can be established without a knowledge of the partition coefficients of the elements between the liquid and solid phases of the synthesis gel. Anyway, the higher crystallization efficiency of the... 

Two distinct types of macroscopic theoretical model for the low strain mechanical behaviour of oriented solid polymers will be considered in this chapter. First, models which predict the changes in elastic constants with the development of orientation these will be referred to as orienting element models. Secondly, models which seek to explain the mechanical behaviour of both isotropic and oriented polymers in terms of a two phase material with separate components representing crystalline and amorphous fractions these we shall call composite structure models. 

The prepreg fibre bed is typically assumed to be an elastic porous medium with incompressible and inextensible fibres and fully saturated with the resin. The resin is assumed to flow in the pores between the fibres, and the fibre mass in the laminate remains constant during cure. The governing equations of the system must describe the behaviour of the composite constituents the fibre bed and the resin. Firstly, the equilibrium of forces on the representative element is considered. Secondly, the mass conservation for the representative element must be satisfied. For a porous medium saturated with a single phase fluid, the total stress tensor a,) is separated into two parts as (tensile stresses are considered positive) ... 

Table 6.3.4 shows that the back-extraction efficiencies of the studied elements remained almost constant for the three experiments. This behaviour was expected since the experimental conditions of this step were the same for the three runs only the initial amount of An and Nd in the metallic phase slightly differed from one experiment to the other. Pu and Am are quantitatively back-extracted (>99%) in a single stage while the... 

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