Geometric capacitance

There is also a term representing the impedance of the second electrode in the cell and a term representing the geometrical capacitance of the whole cell. These latter two can, however, be minimised by proper choice of cell geometry, but we cannot eliminate the first two in any practical measurement, with the result that our final equivalent circuit for the cell looks like ... 

The geometrical capacitance Cg is of the order of 10 10 F cm 2 for the majority of films studied, whose thickness is 1 pm by order the corresponding impedance would be an order higher than the film resistance Rs (with exception of very low-doped films) and can be neglected. We then obtain a simpler three-element circuit (Fig. 10b) often called the Randles circuit [65], An essential assumption is that all elements of the circuits in Figs. 10a and b are frequency-independent. 

The time (t) dependences of the charging current (Ic) of a cell where the capacitance of the electrode investigated is much smaller than that of the reference electrode (e.g., saturated calomel electrode) and the counter electrode (e.g., high surface area of platinum electrode) as well as the geometrical capacitance of the cell are neglected, therefore the total capacitance of the cell is equal to the capacitance of the electrode (Q) under study, and - ohmic (solution) resistance is Rs> - are as follows. 

Figure 6.2 (a) Schematic representation of equivaient circuit for an ion conductor put between a pair of nonbiocking eiectrode, and (b) the corresponding Nyquist plot. The high-frequency equivaient circuit fa -1 j /s composed of the buik resistance of the sampie fRb and geometrical capacitance fCgj, white the iow-frequency equivaient circuit (a-2) is composed of the resistance to charge transfer across the sampie/eiectrode interface fRct and double-layer capacitance (Ca)-... 

Polarisation effects at electrodes become most prominent when the material of a specimen shows some appreciable bulk conductivity. Characteristically, there is an apparent increase in the relative permittivity at low frequencies. The anomaly originates in a high-impedance layer on the electrode surface. This may be caused by imperfect contact between the metal electrode and the specimen, aggravated by the accumulation of the products of electrolysis, etc. At low frequencies there is sufficient time for any slight conduction through the specimen to transfer the entire applied field across the very thin electrode layers, and the result is an enormous increase in the measured capacitance. For a purely capacitive impedance Ce at the electrodes, in.series with the specimen proper (geometrical capacitance C0), Johnson and Cole (1951) showed that the apparent relative permittivity takes the approximate form ... 

But the geometrical capacitance per unit area between them for a separation x is given by... 

In this case R is the extrapolated dc-resistance of the material, C the capacitance, and o) the angular frequency. The use of these fit functions reduces the complex impedance function describing the electrical properties of each material to one value for the resistance and the capacitance, respectively. Although the capacitance can be assigned to the geometric capacitance of the IDC, the resistance values under the respective measuring conditions are taken for the determination of the material sensitivity. 

Figure 12.3 Mott-Schottky plot for a thin (80 nm) film of n-CdS on tin oxide-coated conducting glass, showing the transition from Mott-Schottky behaviour to the geometric capacitance limit when the space-charge region extends to the substrate. Electrolyte 0.1 mol dm" Na2S, pH 13. Adapted from Ozsan et al. (1996).

Figure 26.6 Model of geometric gate capacitances for (a) back-gated and (b) electrochemically gated SWCNT-FETs. In the case of back-gating, the geometric capacitance is determined by the capacitance of the Si02 dielectric. During electrochemical gating the electro-...

Initial measurements on valinomycin-doped membranes (13, 14) showed that lipid bilayers, which provide cells with an effective permeability barrier, are equilibrium objects by this criterion. Within an accuracy of several percent, the experimentally obtained values of the spectral density of voltage noise showed agreement with those calculated from relation 1. Figure 1 illustrates this agreement for three valinomycin concentrations. For the valinomycin-K+ system chosen for these experiments and for the frequency range used in measurements, the dispersion in membrane impedance was caused only by geometrical capacitance of the bilayer the characteristic times of the transport process itself were too small to influence impedance in this range. 

Figure 1. The agreement of the spectral density of voltage fluctuations from valinomycin-modified phospholipid bilayers at equilibrium conditions (13, 14) with the Nyquist relation 1. An aqueous 0.01-M KCl solution at 33 °C was used in the experiments. Bilayer direct current resistances and valinomycin solution concentrations were 0.52-Mfl and 1.5 X 10 8 M (l), 0.19 Mfl and 5 X 10 8 M (2), and 0.055 Mi2 and 1.5 X 10 7 M (3). Solid lines are drawn in accordance with relation 1 for the impedance of a parallel resistance-capacitance (RC) circuit using foregoing resistance values and a value of membrane geometrical capacitance.

The leftmost lumped subcircuit (Rcb,p1 bQ) contains the geometrical capacitance, the bulk ionic conductance, and the conduction of ions parallel in the grain boundaries. The next (Rgb,sQgb) contains the resistance and capacitance across grain boundaries, and where g is a constant phase element [37], often with n A reflecting a dispersion of... 

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