Electrode polarization linearity

After complete formation of each successive monolayer of atoms, the next layer should start to form. This requires two-dimensional nucleation by the union of several adatoms in a position 1. Like three-dimensional nucleation, two-dimensional nucleation requires some excess energy (i.e., elevated electrode polarization). Introducing the concept of excess linear energy p of the one-dimensional face (of length L) of the nucleus, we can derive an expression for the work of formation of such a nucleus (analogous to that used in Section 14.2.2). When the step of two-dimensional nucleation is rate determining, the polarization equation becomes, instead of (14.39),... 

Bias-dependent measurements were performed in order to check to what extent the mechanism depends on the electrical operation conditions. Fig. 43 shows double-logarithmic plots of the electrode polarization resistance (determined from the arc in the impedance spectrum) versus the microelectrode diameter observed at a cathodic bias of —300 mV and at an anodic bias of +300 mV respectively. In the cathodic case the electrode polarization resistance again scales with the inverse of the electrode area, whereas in the anodic case it scales with the inverse of the microelectrode diameter. These findings are supported by I-V measurements on LSM microelectrodes with diameters ranging from 30-80 pm the differential resistance is proportional to the inverse microelectrode area in the cathodic regime and comes close to an inverse linear relationship with the three-phase boundary (3PB) length in the anodic regime [161]. 

The Electrical Conductivity System consists of a frequency generator (the frequency is usually between 1,000 and 5,000 Hz) that provides an AC potential across the cell. As already discussed, an AC potential must be used to avoid electrode polarization. The sensor is usually placed in one arm of a Wheatstone bridge as shown in figure 3. The out-of-balance signal is then rectified with a precision rectifier and the DC signal either passed to nonlinear amplifier or a computer data acquisition system. The non- linear amplifier modifies the signal so that the output is linearly related to ion concentration. Alternatively, if the output from the precision rectifier is passed directly to the... 

For a small electrode polarization (denoted as AE, which can be no more than 20 mV) that produces a current of /, the exponential terms in Equation (26.162) can be linearized, resulting in the following current-voltage expression ... 

All of these techniques monitor the response of the electrode to stimulation by a potential change. The magnitude of the potential stimulation and current response decreases in the order potentiodynamic polarization, linear polarization, EIS, ENA. Each of the methods provides information, and there are trade-offs involved in the decision of which is the best to use. 

There are sources of nonlinearity both in electrode polarization and tissue impedance (Schwan 1992). Onaral and Schwan (1982) studied electrode polarization impedance and found that the limit voltage of linearity was about 100 mV and frequency independent. The corresponding hmit current is of course impedance dependent and therefore frequency dependent, and may be about 5 pA/cm at a frequency of 1 Hz. 

Furthermore, we note the existence of a linear part appearing at low frequencies, which is caused by electrode polarization. 

The electrodes are arranged to constitute one arm of a Wheatstone Bridge. When ions are present in the detector cell, the electrical resistance changes and the out-of-balance signal is passed to a suitable amplifier. The amplifier output is either digitized, and the binary number sent to a computer for storage, or is passed directly to a potentiometric recorder. The detector actually measures the electrical impedance between the electrodes which, by suitable non-linear amplification, provides an output that is linearly related to solute concentration. To avoid electrode polarization, an AC voltage (about 10 kHz) is applied across the electrodes to measure the cell impedance. Typical specifications for the conductivity detector are as follows ... 

Electrode behavior in the nonlinear region may be studied by electrode polarization impedance Z = R + jX measured as a function of sinusoidal amplitude. The limit current of linearity iL may, for instance, be defined as the amplitude when the values of R or X deviate more than 10% from low current density values. Often 1l is increasing with frequency proportional to f (Schwan s law of nonlinearity) (Onaral and Schwan, 1982 McAdams and Jossinet, 1991a, 1994). m is the constant phase factor (as defined in this book) under the assumption that it is obeying Fricke s law and is frequency independent (Section 9.2.5). When the measuring current is kept limit current of linearity will usually be lower for X than for R. 

Geddes et al. (1975a) measured the current/voltage characteristics with a 5 ms duration heavily damped sinusoidal defibrillator pulse. Standard defibrillator electrodes 3.5" in diameter (60 cm ) were used with current pulses up to 80 A. The electrodes were face-to-face at 1 cm distance with the space filled with a 8.4 Q-cm electrode paste. It was found that the impedance of both electrodes (defined as the ratio of peak voltage to peak current) at such current levels was only a fraction of 1 Q. With the usual thorax tissue impedance of about 50 Q, little energy therefore is lost in electrode polarization processes. The 0.01 Hz impedance of the same electrode pair with small linear AC current levels was found to be about 2 kQ. This shows the extreme degree of nonlinearity. 

Fig. 8.15 Simulated complex plane plots of impedance response for ideally polarized disk electrode (a) linear plot showing effect of dispersion at frequencies AT > 1 as deviation from vertical line (b) plot in logarithmic scale for imaginary impedances (From Ref [358], Reproduced with permission of Electrochemical Society)...

In addition, ionic conduction causes a charge qiit) to pass through the material, which rises from zero and then becomes linearly dependent on time to give a constant current I(t) = dgi/d. In contrast to qp(t), which becomes constant at long times (since the dipolar system reaches thermodynamic equilibrium in the presence of the steady field), qiit) increases without limit in the absence of interfacial and electrode polarization effects, giving a continual dissipation of energy via the conduction process. The fact that the steady conduction current takes time to get established means that the ac real conductivity a ico) is constant at low frequencies and becomes /-dependent at high frequencies. 

Resistance to current flow also occurs as a consequence of solid corrosion product buildup on the metal surface. This phenomenon is most pronounced in environments containing H2S. Iron sulfide is a semiconductor whose conducting properties depend on the nature of the environment. It had been observed [39] that the anodic and cathodic polarization curves on iron sulfide covered electrodes are linear rather than exponential. In this case, the current flow is entirely controlled by the charge transfer across the interphase (not interface) consisting of FeS. The polarization admittance (1/Rp) becomes... 

Electrode polarization was considered by Tobias in terms of adopting a linear approximation to the Tafel equation. The influence of polarization, which is in essence a bubble-independent resistance, acts in series with the ohmic resistance, improving the uniformity of current distribution. 

Fricke (1932) developed the following linear range relations to describe the components of alternating-current electrode polarization ... 

Schwan, H. P. and Maczuk, J. G., 1965, Electrode polarization impedance limits of linearity, Proc. I8th ACEMB, Philadelphia, p. 24. 

The signal source pulse I (t) was simply a rectangular pulse of 1-msec duration. Pulse amplitude could be varied from 0 to 10 V peak without overloading the amplifier system or affecting the linear range of the electrode polarization impedance. In each of the photographic records which follow. 

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