Low-field approximation

In the high-overpotential case (cf. Section 7.2.3b.2), the first exponential term can be neglected for n 0, i.e., for net electronation, and the second exponential term for T) 0, i.e., for net deelectronation. In the low-field approximation, where both exponential terms in the Butler-Volmer equation can be linearized, Eq. (7.136) becomes... 

In order to obtain the low-field approximation, both exponential terms in (7.142)... 

Using an expression (low field approximation) for a membrane potential derived from this model calculate the membrane potential assuming that the difference of the reversible redox potentials of the reactions on either side of the membrane is 120 mV (i0)l = 10-4 A cm-2 (iQ)2 = 10-6 A cm-2 real area = 10 times the apparent area (0.2 cm2) and R = 104 ohms per sample of membrane. (Bockris)... 

Note that, in contrast to the low-field approximation, the Tafel equation is sometimes referred to as the high-field approximation since it is only valid for large values of overpotential. 

Just as the Butler-Vohner equation was shown to be linear near E " with a slope inversely proportional to j o (low-field approximation. Sect. 1.3.8), the net current in a mixed potential system, represented by Eq. (41), is Knear near corr. with a slope inversely proportional to icorr-Since polarization resistance measurements are made near the corrosion potential, the net currents are often low. Therefore, errors associated with ohmic potential drop are usually small. However, there are cases for which ohmic potential drop creates significant error in the Rp measurement. The measured R is the sum of the trae polarization resistance. 

Based on low-field approximation, a simple procedure for the evaluation of corrosion currents and corrosion rates was developed in 1938 by Wagner and Traud [22]. Stern and Geary [23] and Stern [24,25] developed an experimental procedure for measuring the corrosion rates known as the linear polarization technique. This technique wiU be discussed in detail in Chapter 5. 

Chapters 1 to 3 describe the theory of corrosion engineering and offer analyzed case studies and solved problems in the thermodynamics of corrosion processes, the relevance of electrochemical kinetics to corrosion, low field approximation theory, concentration polarization, the effects of polarization behavior on corrosion rate, the effect of mass transfer on electrode kinetics, and diffusion-limited corrosion rates. 

A low field approximate, rj < 0.002, the hyperbolic sine function, approximates to a linear function so that... 

The usual procedure for measuring the exchange current density ig is then to measure t] as a function of I and to plot In I vs. (Tafel plot). Such plots are shown in Figure 13.3 for Pt and Ag electrodes deposited on YSZ. From the slopes of the linear part of these plots ( ri > 200 mV, in which case Equation 13.15 are valid) one obtains the transfer coefficients and a. By extrapolating the linear part of the plot to r] = 0, one obtains io. One can then plot i vs. T] and use the low field approximation of the Butler-Volmer equation which is valid for < 10 mV, i.e.. 

In analogy to what has been found in LWR coolants at temperatures above 150°C and in SCW [71,72,77,78,83], the large number of defects in the oxide is not likely to support high electric field conditions during oxidation over the temperature range 500—700°C. Thus the low-field approximation of the generalized transport equation of Fromhold and Cook [84] was used for the flux of point defects of type j (j = I, O, and V, i.e., interstitial cations, oxygen, and cation vacancies) ... 

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