Polarization-resistance equation

Starting with Eq 6.1, derive the polarization-resistance equation, Eq 6.25. 

This equation is of the general form i = KAEI(M + NAE), where K, M, and N are constants. Derivative dildAE of this function has the value KMliM + NAE) and in the particular case where A = 0 the derivative has the value KIM. Thus, when allowing for Eq. (13.17) we find for polarization resistance p, which for AE = 0 is equal to dE/di ... 

The calculation becomes more difficult when the polarization resistance RP is relatively small so that diffusion of the oxidized and reduced forms to and from the electrode becomes important. Solution of the partial differential equation for linear diffusion (2.5.3) with the boundary condition D(dcReJdx) = —D(d0x/dx) = A/sin cot for a steady-state periodic process and a small deviation of the potential from equilibrium is... 

The fact that the corrosion rate is inversely proportional to the polarization resistance is clearly seen by Eq. (3). Taking the logarithm of this equation, it is seen that log icon versus log Rp is linear with a slope of -1 and has the intercept log B ... 

This expression was originally used to estimate the polarization resistance for actively corroding metals whose impedance response was a well-behaved semicircular arc in the complex plane (73), but can be used in certain situations to estimate the equivalent resistance of conversion coated metal surfaces. Equation (9) is derived from the Kramers-Kronig transforms and subject to the conditions that limit their use (74-78). These conditions include... 

In case of the appearance of the blocking reaction layer on the surface of zirconia with polarization resistance Rf (see Figure 4.17, b), the operative impedance equation would read as follows ... 

When E > Ecorr, the first exponential term is greater than the second exponential term and Iex is positive. Plotted as E versus log Iex, Eq 6.5 plots as the upper solid curve in Fig. 6.2. For E < Ecorr, Iex is negative, and a plot of E versus log Iex plots as the lower solid curve in Fig. 6.2. These equations will be used in establishing relationships for the analysis of corrosion rates by the experimental techniques of Tafel-curve modeling and polarization resistance. 

Equation 6.25 may be rewritten in the following form, since the desired quantity in the polarization-resistance analysis is the corrosion current density ... 

Once Rp is determined, icorr is evaluated with the polarization-resistance (or Stem-Geary) equation, Eq 6.28. The two-electrode method is a relatively simple and fast method for evaluating when compared with the standard polarization-resistance and electrochemical-impedance methods. 

Use the approximate equation, Eq 6.31, to determine icorr from the polarization-resistance data in Fig. 6.12. 

From a mathematical point of view, this problem can be readily solved by introducing the concept of polarization resistance, R, [5] associated with the implicit function defined by the equation (11). For this purpose, it may be useful to remember that, after Bonhoeffer and Jena s definition [53], the differential polarization resistance is related to the first derivative of the function i(AF), calculated at the point AE=Q, by the relation... 

Denoting by F i, AE) the left-hand side member of equation (11), by Dini s theorem on implicit functions [54] the apparent polarization resistance, Rpa, is defined by the relation... 

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. 

Many different electrochemical and non-electrochemical techniques exist for the study of corrosion and many factors should be considered when selecting a technique. Corrosion rate can be determined by Tafel extrapolation from a potentiodynamic polarization curve. Corrosion rate can also be determined using the Stem-Geary equation from the polarization resistance derived from a linear polarization or an electrochemical impedance spectroscopy (EIS) experiment. Techniques have recently been developed to use electrochemical noise for the determination ofcorrosion rate. Suscephbility to localized corrosion is often assessed by the determination of a breakdown potenhal. Other techniques exist for the determinahon of localized corrosion propagahon rates. The various electrochemical techniques will be addressed in the next section, followed by a discussion of some nonelectrochemical techniques. 

As mentioned in Chapter 1.3, the slope, d /di, at the zero-current potential is a measure of the polarization resistance, Rp. The corrosion can be determined from the polarization resistance using the Stern-Geary equation, if the Tafel slopes are known ... 

Recall that both Tafel slopes are positive in this form of the equation. The Stern-Geary equation is the basis for the linear polarization method in which the polarization resistance is determined typically by scanning the potential from a value slightly below the corrosion potential to one slightly above the corrosion potential. It is an extremely easy technique that has been put to considerable use in corrosion monitoring. The polarization resistance can be determined by a simple two-point measurement at values above... 

For the data in Fig. 6, the ohmic and polarization resistances can be determined to be about 0.3 and just under 100 cm, respectively. The value of Rp is slightly higher than that determined by linear polarization (Fig. 4) in a measurement that just preceded the EIS experiment on the same electrode. The double layer capacitance is seen to be 1/3000 SI cm = 333 pF cm . The polarization resistance determined by EIS can be used to determine the corrosion rate with the Stern-Geary equation, just as was described above for polarization resistance determined by linear polarization. EIS data provide no estimation of the Tafel slopes, which are required in the Stem-Geary equation. 

Linear polarization measurements are executed rapidly. The currents in linear polarization measurements are measured in the potential range between 10 and 20 mV from the equilibrium potential. The E-I dependence in this potential range follows a linear relationship. The slope of the plot, dE/ di, represents the polarization resistance. The corrosion current is calculated using the Stem-Geary equation for known values of the anodic and cathodic Tafel slopes. The ratio of the overpotential to the current represents the resistance in Ohm s law and is often termed the charge transfer resistance or the polarization resistance, Rp. 

A higher exchange current density impHes increased reaction rate, while lower exchange current density (higher polarization resistance) indicates sluggish corrosion kinetics. Equation (1.10) can be written in terms of charge transfer resistance as ... 

The linear polarization technique estimates instantaneous corrosion rates under various process conditions. The corrosion current, according to the Stem-Geary equation, is inversely proportional to polarization resistance, which allows the measured polarization resistance to be normalized directly into corrosion rates. Because the current follows the appHed overvoltage, the polarization resistance curve is plotted automatically. Because this technique accurately measures corrosion rates <0.1 mpy, it is of a great importance in water distribution systems and food industries that face problems with traces of impurities and contamination. It can be used to measure the corrosion rates in civil engineering structures that cannot be subjected to weight loss measurements. Usually, Hnear polarization measurements are executed in 10 min. As shown in Fig. 5.3, the current as a... 

Macdonald summarized the hmitations of EIS technique when used to measure the corrosion current (corrosion rates) of metals [79]. A high level of mathematics is required to analyze data and interpret properties of the corrosion system. Analysis of impedance data results in determination of the polarization resistance. However, it requires obtaining a large number of low-frequency data for an accurate estimate. It is necessary to extract the noise from the data obtained at low frequency ranges to obtain meaningful mechanistic information. To calculate the corrosion rate using the Stem-Geary equation, the Tafel method should be used to estimate the Tafel slopes as a function of time. Due to the variation of porosity of corrosion products on metals, the corrosion products (oxides and hydroxides) contributions to the overall impedance spectra are difficult to evaluate. 

Once Rp is known, the corrosion rate can be evaluated using the Stern-Geary equation. Polarization resistance and corrosion current are determined from the current measured close to the corrosion potential. Polarization resistance can be determined with minimum system perturbation with linear polarization resistance or by using EIS. Experimentally determined potential ranges that indicate expected iron corrosion intensity for different-measured corrosion potentials are shown in Fig. 12.3. 

The other main electrochemical method for determination of corrosion rates is the (linear) polarization resistance method (the LPR method). In a limited potential range around the corrosion potential (up to 20 mV) a linear relationship exists betw een the potential increment AE and the increment in external current Ale, as shown in Figure 9.2. It can be shown mathematically that the slope of the curve in this potential range is given by Stem-Geary s equation... 

Rp gives the technique its alternative name of polarization resistance . The change in potential must be kept to less than 20 mV or so for the equation to be valid and remain linear (hence the name linear polarization ). Also, there is an iR drop in the circuit. This is the voltage that exists because a current is flowing through concrete which has a finite electrical resistance. This is also referred to as the solution resistance . This can be measured by switching off the current at some point during the measurement process so that the potential without the iR drop is measured. The iR drop is discussed further in the section on cathodic protection. 

The measurement is made in one of two ways. Either steadily fixed levels of current are applied and the potential monitored (galvanostatic), or the current increased to achieve one or more target potentials (potentiostatic). In both cases allowances for the iR drop (solution resistance) may be made. A plot of change in current vs. change in potential gives a gradient of the polarization resistance Rp (equation 4.2) to calculate the steel section loss rate. 

From these equations the corresponding equations for Tafel lines, polarization resistance, and electrochemical reaction orders are obtained as was described in Section 6.1 for electron transfer. 

Rp is called the polarization resistance and can be determined from the slope of the linear portion of the i-E curve in the vicinity of the corrosion potential by using a polarization of only a few mV from Ecorr (this method is known as the polarization resistance method). From the measured Rp value and the known (or estimated) j8 values, the corrosion current can be calculated from Equation 15.9. The corrosion rate is then determined from Faraday s law [17] ... 

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