Resistance/capacitance values

Figure 11. Experimental and predicted differential conductance plots of the double-island device of Figure 10(b). (a) Differential conductance measured at 4.2 K four peaks are found per gate period. Above the threshold for the Coulomb blockade, the current can be described as linear with small oscillations superposed, which give the peaks in dljdVj s- The linear component corresponds to a resistance of 20 GQ. (b) Electrical modeling of the device. The silicon substrate acts as a common gate electrode for both islands, (c) Monte Carlo simulation of a stability plot for the double-island device at 4.2 K with capacitance values obtained from finite-element modeling Cq = 0.84aF (island-gate capacitance). Cm = 3.7aF (inter-island capacitance). Cl = 4.9 aF (lead-island capacitance) the left, middle and right tunnel junction resistances were, respectively, set to 0.1, 10 and 10 GQ to reproduce the experimental data. (Reprinted with permission from Ref [28], 2006, American Institute of Physics.)...

It is much less clear how the adsorption leads to such a dramatic change as a potential decay of several hundred volts, occurring within milliseconds. This short time is difficult to associate with film thinning, as assumed in the adsorption mechanism of pit initiation. It is not only that the mechanism of dissolution changes so much that the current efficiency falls from virtually 100% to virtually zero, but also that the resistance of the oxide decreases by orders of magnitude. The control of the process is, to a great extent, taken over by the events at the O/S interface, judging from the capacitance values measured,115 which approach those typical of the electrochemical double layer (cf. Fig. 22). 

A typical ceramic sample contains contributions from the bulk, the grain boundaries, and the electrode. Each of these is characterized by a semicircular arc with a maximum at RCu> = 1, where the values of resistance, capacitance, and frequency refer directly to the bulk, grain boundaries, or electrodes (Fig. 6.7c). The separation of resistance due to the bulk from that of the grain boundaries is thus easily achieved using impedance spectroscopy. 

Figure 7. (A, top) Simple battery circuit diagram, where Cdl represents the capacitance of the electrical double layer at the electrode—solution interface (cf. discussion of supercapacitors below), W depicts the Warburg impedance for diffusion processes, Rj is the internal resistance, and Zanode and Zcathode are the impedances of the electrode reactions. These are sometimes represented as a series resistance capacitance network with values derived from the Argand diagram. This reaction capacitance can be 10 times the size of the double-layer capacitance. The reaction resistance component of Z is related to the exchange current for the kinetics of the reaction. (B, bottom) Corresponding Argand diagram of the behavior of impedance with frequency, f, for an idealized battery system, where the characteristic behaviors of ohmic, activation, and diffusion or concentration polarizations are depicted.

Figures 11.3 and 11.4 show the frequency spectra of 2600F, 0.5mQ BCAP0010 DLC capacitance and series resistance for three different polarization voltages. It is interesting to observe the low-capacitance value when there is no voltage polarization. This phenomenon should be studied as a function of the electrode thickness. If ionic depletion was the cause, a thick electrode should display a more pronounce effect.

The capacitance and the series resistance have values which are not constant over the frequency spectrum. The performances may be determined with an impedance spectrum analyzer [70], To take into account the voltage, the temperature, and the frequency dependencies, a simple equivalent electrical circuit has been developed (Figure 11.10). It is a combination of de Levie frequency model and Zubieta voltage model with the addition of a function to consider the temperature dependency. 

Electrical properties of membranes. Biological membranes serve as barriers to the passage of ions and polar molecules, a fact that is reflected in their high electrical resistance and capacitance. The electrical resistance is usually 10 ohms cm, while the capacitance is 0.5-1.5 microfarad (pF) cm . The corresponding values for artificial membranes are 10 ohms cm and 0.6 - 0.9 pF cm . The lower resistance of biological membranes must result from the presence of proteins and other ion-carrying substances or of pores in the membranes. The capacitance values for the two types of membrane are very close to those expected for a bilayer with a thickness of 2.5 nm and a dielectric constant of 2. 4 The electrical potential gradient is steep. 

Remember 9.3 While all resistance contributions to an electrical circuit can be observed at low frequencies, the inability to measure at sufficiently high frequencies may make it impossible to obtain all capacitance values. 

The circuit equivalent taking into account parasitic effects consists of a contact resistance (Rs), a lead inductance Ls in series as well as a stray capacitance C() and the resistance of the substrate material between the leads 1/G0 parallel to the impedance of the sample. These parasitic effects, resulting from the compact arrangement of the electrodes can be eliminated by an offset adjustment. Capacitive parasitic effects are acquired by compensation measurements. The electrode array comprises some IDCs, which have almost negligible parasitic pathway effects, like position 5-5 (Fig. 11.1), where the conductor pathways are perpendicular to each other.22 These positions show the smallest capacities, which result only from the IDC and can be taken to determine the parasitic capacitive values of the other positions.24 Inductive parasitic effects Ls are acquired by data fitting and elimination from the sample impedance. Resistances of the conductor path (Rs generally <10 mQ) and conductivity of the substrate material (1/G0, >20mQ, exceeds measurement limit of the impedance analyser) are unaccounted. 

The capacitances corresponding to the low frequency maximum are of the order of 10-400 pF cm and decrease in magnitude with increasingly cathodic potential. Resistances corresponding to these maxima are of the order of 1000 ohms and decrease with increasing cathodic potential. Since most of the capacitance values are higher than those characteristic of the double layer, and vary with potential, they may be attributed to surface states. 

The average capacitance and specific resistivity of the barrier aluminium oxide films are determined to be 430. .. 470nF/cm and 1.3. .. 2.4 10 " Qcm, respectively. By using the anodisation factor of 1.2 nm/V for the films formed at low formation voltage, dielectric constants of 5.8. .. 6.4 are calculated from the measured capacitance values. The comparatively low dielectric constant is in agreement with the formation of an amorphous anodic aluminium oxide film as discussed above rather than a crystalline structure for which a higher dielec-... 

The cell membrane resistance is generally much greater than the reactance of the membrane and is ignored. Likewise the capacitance of the cell cytoplasm can be ignored when its reactance is compared to the cell cytoplasm resistance. The values of the electrical components in the circuit are as follows ... 

Table 9.1 provides the values of membrane resistance (/ ), capacitance (Cm), and thickness d) of artificial BLMs and natural cell membranes [11,18]. The resistance of artificial membranes is much higher than that of biological membranes. This results from the presence of translocators such as peptides and proteins in the cell membranes. The resistance of artificial membranes can however be reduced to the levels of natural cell membranes when ion translocators are inserted. Specific capacitance (C ) is the primary criterion to distinguish between solventless BLMs and black lipid films. Table 9.1 exhibits that the specific capacitance of the solventless BLMs (about 0.9 /itF cm ) approaches the values measured for natural cell membranes, and is almost twice the magnitude observed for black lipid membranes. These values of specific capacitance can be used to estimate the hydrocarbon thickness, d, of membranes using the equation... 

This operation determines the values of R and C that, in series, behave as the cell does at the measurement frequency. The impedance is measured as a function of the frequency of the ac source. The technique where the cell or electrode impedance is plotted V5. frequency is called electrochemical impedance spectroscopy (EIS). In modem practice, the impedance is usually measured with lock-in amplifiers or frequency-response analyzers, which are faster and more convenient than impedance bridges. Such approaches are introduced in Section 10.8. The job of theory is to interpret the equivalent resistance and capacitance values in terms of interfacial phenomena. The mean potential of the working electrode (the dc potential ) is simply the equilibrium potential determined by the ratio of oxidized and reduced forms of the couple. Measurements can be made at other potentials by preparing additional solutions with different concentration ratios. The faradaic impedance method, including EIS, is capable of high precision and is frequently used for the evaluation of heterogeneous charge-transfer parameters and for studies of double-layer structure. 

At a given frequency, the equivalent circuit of the cell can be taken as in Figure 10.1.14, but we measure its impedance as a resistance value R and the capacitance value Cb in series [or equivalently as Zrc = im l/coC ]. One approach to obtaining... 

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.

Capacitance values of the PtO, Cox, from the best curve fit is lower for the membrane-coated electrode than for the bare PtO electrode. Also the resistance, Rox, is higher for the membrane-coated electrode than for the... 

Based on this consideration and on the certainty that this technique is more effective than the current interruption method, the codes SOFTCOR-AC-GS [71] and SOFTCOR-AC-GE [72] have been developed for determining the value of the resistance R, through sruface impedance measurements over the interval [7, 13] kHz when the capacitance values are rather small. 

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