Current-potential curves Butler-Volmer equation

Therefore, the current density depends on the exchange current density ( o), transfer coefficient ( p), overpotential r ), and temperature (r). Fig. 7 represents typical current-overpotential curves based on Eq. (39). The net current is the result of the combined effects of the forward (anodic) and reverse (cathodic) currents. Although the Butler-Volmer equation for an electrochemical reaction in PEMFC is valid over the full potential range, simpler approximate equations may often be used for limited conditions. Thus, for the common value dp = 1/2, Eq. (39) becomes... 

Figure 5.2 Current-potential curves according to the Butler-Volmer equation.

Fig. 7.3 Current-potential curve according to the Butler-Volmer equation for a = 0.5. Dashed lines represent the partial currents. (After ref. [2])...

A theoretical current-potential curve (/7/q vs. fj) is given in Fig. 7.3 for r] = 0.5. It should be emphasized here that Eq. (7.11) is only valid in this simple form if the current is really kinetically controlled, i.e. if diffusion of the redox species toward the electrode surface is sufficiently fast. According to the Butler-Volmer equation (Eq. 7.11) the current increases exponentially with potential in both directions. In this aspect charge transfer processes at metal electrodes differ completely from those at semiconductors. When the overpotential is sufficiently large, erj/kT 1. one of the exponential terms in Eq. (7.11) can be neglected compared to the other. In this case we have either... 

We will not illustrate the polarization curves because a more generalized treatment and a more complete expression of the current vs. potential profile comprising the entire mass transfer-modified Butler-Volmer equation are available in the literature [7], We only discuss here the most determining features of the curves that are deduced from our detailed comments presented above ... 

The current-voltage curves corresponding to these processes are depicted in Fig. 5.1. As the net current across the metal/solution interface is zero the potential Ep assumed by the particle under stationary conditions is given by the point of intersection of the two i(E) curves. At this potential the anodic and cathodic currents are equal and their value corresponds to iR. The latter defines the overall reaction rate. Both the mixed potential Ep and the reaction current iR may be evaluated from electrokinetic theory. Application of the Butler-Volmer equation to reaction (5.2) gives for the reaction rate V the expression... 

These equations are straight lines in a plot of E versus log i, such as that in Fig. 2. This plot of versus log i is called an Evans diagram. Actually, the Butler-Volmer equation is described better by the curve in Fig. 5. The net current at the reversible potential is zero because the forward and reverse current, each equal to the exchange current density, balance each other. The log of the current density approaches negative infinity at the reversible potential at which the net current density goes to zero, and the polarization curve points down at the reversible potential when plotted on semilogarithmic axes. 

As already mentioned previously, the derivation of the Butler-Volmer equation, especially the introduction of the transfer factor a, is mostly based on an empirical approach. On the other hand, the model of a transition state (Figures 7.1 and 7.2) looks similar to the free energy profile derived for adiabatic reactions, that is, for processes where a strong interaction between the electrode and redox species exists (compare with Section 6.3.3). However, it should also be possible to apply the basic Marcus theory (Section 6.1) or the quantum mechanical theory for weak interactions (see Section 6.3.2) to the derivation of a current-potential curve. Ac-... 

Figures 9 and 13 show current potential curves for reversible and steady state processes. The curves were calculated from the Nernst and Butler-Volmer equations, respectively. If an experimental curve is adequately described by the Nernst equation, this is a reversible reaction. However, it is not convenient to check the current potential curve itself for reversibility, it is much easier to replot it according to equation 3.43. The potential is plotted on the abscissa and log on the ordinate. If a straight line is...

To further illustrate the points above, hypothetical current density-potential curves for reactions with different reaction rates (exchange current densities) and/or standard electrode potentials were calculated and plotted (Figure 3.5). These curves were calculated based on the Erdey-Graz-Volmer (Butler-Volmer) equation [Eq. (3.2)] describing the relation of current density and potential when the rate of reaction (current) is controlled solely by the rate of the electrochemical charge transfer process (activation kinetics) and the effect of mass transfer limitation on the current is not considered ... 

Other techniques that are commonly applied while using RDE system are electrochemical impedance spectroscopy (EIS) and cyclic voltammetry (CV). Using EIS, one can extrapolate the exchange current density of an electrode. This is achieved by completing the EIS spectrum close to the rest potential of the electrode. Fig. 11 shows the EIS curves developed at two overpotentials and the low frequency intercept at close to rest potential or equilibrium potential helps quantify the exchange current density. Let us apply simple mathematical treatment to attain the Jq (exchange current density). Butler-Volmer equation can be simplified and written as... 

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