Tafel curve modeling

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. 

Tafel Curve Modeling (Ref 4, 5). Equation 6.5 provides the form of the experimental polarization curve when the anodic and cathodic reactions follow Tafel behavior. The equation accounts for the curvature near Ecorr and Icorr, which is observed experimentally. Physically, the curvature is a consequence of both the anodic and cathodic reactions having measurable effects on Iex at potentials near Ecorr. Tafel-curve modeling uses experimental data taken within approximately 25 mV of Ecorr where the corrosion process is less disturbed by induced corro-... 

As previously stated, once Rp is determined, calculation of icorr requires knowledge of the Tafel constants. These constants can be determined from experimental anodic and cathodic polarization curves, or by Tafel-curve modeling, forthe material and solution of interest as discussed earlier. In the absence of these values, an approximation is sometimes used. 

In contrast to the EIS method, the Tafel-extrapolation, Tafel-curve-modeling and polarization-resistance methods are conducted under essentially dc conditions. In these cases, in generating the appropriate Eexp versus log iex or iex curve, the potentiodynamic potential scan rate is very slow, or the time between potentiostatic potential steps is very long. The common practice is a potential scan rate of 600 mV/h or an equivalent step rate of 50 mV every 5 min. Underthese conditions, it is assumed that a steady-state, extemal-current-density results at every discrete potential. Consequently, every element in the electrical circuit is purely resistive in nature, and therefore, the applied potential and resultant extemal-current-density are exactly in phase. Since the impedance (normalized with respect to specimen area) is dEexp/diex, under these conditions, the impedance, Z, at Ecorr is given by (see Eq 6.29) ... 

If the potential-current (E-i) characteristics of the individual reactions were measured, the reactions could be readily modeled as electrochemical reactions with the battery at open circuit as indicated by the processes in Figure 10. If dynamic electrode potential-current relationships were determined, the electrode is expected to show the classic Tafel slope behaviors as the exchange current of the anodic-cathodic equilibrium is shifted into either direction. From the Tafel curves a value for the Eq and Iq of the electrode could be defined. 

Applying the Tafel equation with Uq, we obtain the polarization curves for Pt and PtsNi (Fig. 3.10). The experimental polarization curves fall off at the transport limiting current since the model only deals with the surface catalysis, this part of the polarization curve is not included in the theoretical curves. Looking at the low current limit, the model actually predicts the relative activity semiquantitatively. We call it semiquantitative since the absolute value for the prefactor on Pt is really a fitting parameter. 

Spiro [27] has derived quantitative expressions for the catalytic effect of electron conducting catalysts on oxidation-reduction reactions in solution in which the catalyst assumes the Emp imposed on it by the interacting redox couples. When both partial reaction polarization curves in the region of Emp exhibit Tafel type kinetics, he determined that the catalytic rate of reaction will be proportional to the concentrations of the two reactants raised to fractional powers in many simple cases, the power is one. On the other hand, if the polarization curve of one of the reactants shows diffusion-controlled kinetics, the catalytic rate of reaction will be proportional to the concentration of that reactant alone. Electroless metal deposition systems, at least those that appear to obey the MPT model, may be considered to be a special case of the general class of heterogeneously catalyzed reactions treated by Spiro. 

We see that this simple model well describes the characteristic S-shaped profiles of local polarization curves far from the channel inlet, detected in experiments [197, 200]. Physically, these maxima result from the effect of oxygen starvation . Qualitatively, Eq. (154) shows that local current is a product of two factors. The preexponential factor a describes the growth of local current with the increase in rj due to exponential dependence of the rate constant of ORR on rj (Tafel law). The second (exponential) term in Eq. (154) describes oxygen consumption upstream from the given point z. 

The Tafel equation can be used to model the activation losses in polarization curve, in fact assuming that these are the only losses in a fuel cell, the cell voltage is given by ... 

This simple model of the inhibitor action which is based essentially on the potential independence of the inhibitor adsorption is, however, often not applicable. Kaesche (15) indicates that the corrosion inhibition of pure iron in sulfuric or perchloric acid by phenyl-thiourea strongly affects the slopes of the polarization curves, leaving the corrosion potentials almost unchanged Fig.7. In fact, the polarization curves for the inhibited situation do not exhibit real Tafel behavior. This behavior finds a partial explanation in the fact that the mechanism of the hydrogen evolution appears to be changed in the presence of... 

Omran et al. have proposed a 3D, single phase steady-state model of a liquid feed DMFC [181]. Their model is implemented into the commercial computational fluid dynamics (CFD) software package FLUENT . The continuity, momentum, and species conservation equations are coupled with mathematical descriptions of the electrochemical kinetics in the anode and cathode channel and MEA. For electrochemical kinetics, the Tafel equation is used at both the anode and cathode sides. Results are validated against DMFC experimental data with reasonable agreement and used to explore the effects of cell temperature, channel depth, and channel width on polarization curve, power density and crossover rate. The results show that the power density peak and crossover increase as the operational temperature increases. It is also shown that the increasing of the channel width improves the cell performance at a methanol concentration below 1 M. 

Based on an analytical model for PFFCs, Kulikovsky et al. presented a two-step procedure to evaluate the parameters Tafel slope, exchange current density, and cell resistance from two sets of polarization curves for an HT-PFFC [36]. The method was validated with experimental data. Shamardina et al. described an analytical model taht accounts for the crossover of gases through the membrane [37]. The model is pseudo-two dimensional and describes mainly the effects across the MFA. Temperature and pressure variations in the cell were not considered. From these analytical studies, it follows that the crossover effect has a considerable influence only at a low stoichiometry of oxygen. 

The majority of the models discussed have in common that the electrochemical reaction at the electrodes is described by a Butler-Volmer- or Tafel-type equation. Therefore, all of these models share a very similar set of fundamental parameters, which are summarized in Table 29.2. These can be compared in order to check for model consistency. Nevertheless, aU of these models are based on slightly different assumptions. The parameters that are obtained by curve-fitting procedures are... 

Fig. 3.9 -- At high overpotentials the activation energy in one direction is predicted to fall to zero, as shown in the figure. At higher overpotentials the model of intersecting energy curves predicts that the rate will fall again as the point of intersection moves up the left hand side of upper energy curve. This gives rise to a maximum in the Tafel plots )see Fig. 3.8).

Both macrohomogeneous and flooded agglomerate models predict doubling of the Tafel slope in the proton- and oxygen-limiting regimes. Thus, polarization curves cannot be used to distinguish between these two models. The decisive experiment would be to measure the distribution of... 

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