Potential Dependent Activation Barriers

Voltammetry is an important tool for evaluating electrochemical and electro-catalytic processes. In a voltammetric experiment, the potential of a working electrode is varied with time relative to a reference electrode. The current of the working electrode is measured and reported as a function of potential. If the potential is swept linearly with time, peaks or waves are observed, which can be attributed to the various electrochemical processes possible in the system. For comparison with experiment, DFT calculated energetics can be used to predict voltammetry results in much the same way microkinetic models are used to predict catalytic kinetics. In the sections above, we have discussed DFT methods to calculate elementary reaction or adsorption free energies as a function of electrode potential. These free energy differences can be used to calculate potential dependent equilibrium constants. Section 3.2.5 will present a method to calculate potential dependent activation barriers. With these values for all possible elementary reaction steps, we could use microkinetic modeling to simulate voltammetry and compare with experiment. 

When A = BH4, the reaction represents the activation barrier for the initial surface oxidation reaction within the overall mechanism of borohydride oxidation, which is the reaction occurring at the anode of a direct borohydride fuel cells. Figure 3.12 illustrates the initial, transition, and final states of BH4 dissociation. The DFT computed activation barrier for the reaction BH4 - -H + BH3 is 0.37 eV." The potential dependent activation barrier becomes... 

With an assumed value of 3, a potential dependent activation barrier is determined. This potential dependent barrier was used within a microkinetic model for the overall borohydride oxidation reaction, which allowed for simulation of a borohydride oxidation LSV." A value of near to 0.5 was found to provide a simulated voltammogram that matched experiment, though the same value was used for all reaction steps as an approximation. 

Figure 18 shows the dependence of the activation barrier for film nucleation on the electrode potential. The activation barrier, which at the equilibrium film-formation potential E, depends only on the surface tension and electric field, is seen to decrease with increasing anodic potential, and an overpotential of a few tenths of a volt is required for the activation energy to decrease to the order of kBT. However, for some metals such as iron,30,31 in the passivation process metal dissolution takes place simultaneously with film formation, and kinetic factors such as the rate of metal dissolution and the accumulation of ions in the diffusion layer of the electrolyte on the metal surface have to be taken into account, requiring a more refined treatment. 

The principal details of the proton transfer in Eq. (9.6) will become apparent after an analysis of the calculations [32-42] that determine the dependence of the form of potentials and activation barriers on the distance X.X ... 

When an atom or molecule approaches a surface, it feels an attractive force. The interaction potential between the atom or molecule and the surface, which depends on the distance between the molecule and the surface and on the lateral position above the surface, detemiines the strength of this force. The incoming molecule feels this potential, and upon adsorption becomes trapped near the minimum m the well. Often the molecule has to overcome an activation barrier, before adsorption can occur. 

Figure 18. Dependence of activation barrier A f for the nucleation of a thin oxide film on the metal surface as a function of electrode potential. Ey is the equilibrium potential of anodic oxide formation.7 The solid line represents the value of A against and the dotted line corresponds to the critical potential for the film formation. AE = 0.2 V, Cd= -1Fm-2, am = 0.411 m 2, a -0.01 J m-2,...

If we move the chemisorbed molecule closer to the surface, it will feel a strong repulsion and the energy rises. However, if the molecule can respond by changing its electron structure in the interaction with the surface, it may dissociate into two chemisorbed atoms. Again the potential is much more complicated than drawn in Fig. 6.34, since it depends very much on the orientation of the molecule with respect to the atoms in the surface. For a diatomic molecule, we expect the molecule in the transition state for dissociation to bind parallel to the surface. The barriers between the physisorption, associative and dissociative chemisorption are activation barriers for the reaction from gas phase molecule to dissociated atoms and all subsequent reactions. It is important to be able to determine and predict the behavior of these barriers since they have a key impact on if and how and at what rate the reaction proceeds. 

The interaction of hydrogen (deuterium) molecules with a transition metal surface c an be conveniently described in terms of a Lennard--Jones potential energy diagram (Pig. 1). It cxxislsts of a shallcw molecular precursor well followed by a deep atomic chemisorption potential. Depending on their relative depths and positions the wells m or may not be separated by an activation energy barrier E as schematically Indicated by the dotted cur e in Fig. 1. 

The height of the potential barrier is lower than that for nonadiabatic reactions and depends on the interaction between the acceptor and the metal. However, at not too large values of the effective eiectrochemical Landau-Zener parameter the difference in the activation barriers is insignihcant. Taking into account the fact that the effective eiectron transmission coefficient is 1 here, one concludes that the rate of the adiabatic outer-sphere electron transfer reaction is practically independent of the electronic properties of the metal electrode. 

From the given Hamiltonian, adiabatic potential energy surfaces for the reaction can be calculated numerically [Santos and Schmickler 2007a, b, c Santos and Schmickler 2006] they depend on the solvent coordinate q and the bond distance r, measured with respect to its equilibrium value. A typical example is shown in Fig. 2.16a (Plate 2.4) it refers to a reduction reaction at the equilibrium potential in the absence of a J-band (A = 0). The stable molecule correspond to the valley centered at g = 0, r = 0, and the two separated ions correspond to the trough seen for larger r and centered at q = 2. The two regions are separated by an activation barrier, which the system has to overcome. 

The data in Figure 5 can now be considered in light of the conduction model developed above. As stated previously, conduction in reduced poly-I behaves like an activated process. There are two sources that potentially could be responsible for this behavior. The first is the Boltzmann type concentration dependence of the 1+ and 1- states discussed above. The number of charge carriers is expected to decrease approximately exponentially with T. The second is the activation barrier to self-exchange between 1+ and 0 sites and 0 and 1- sites. For low concentration of charge carriers both processes are expected to contribute to the measured resistance. 

Hydrogen evolution, the other reaction studied, is a classical reaction for electrochemical kinetic studies. It was this reaction that led Tafel (24) to formulate his semi-logarithmic relation between potential and current which is named for him and that later resulted in the derivation of the equation that today is called "Butler-Volmer-equation" (25,26). The influence of the electrode potential is considered to modify the activation barrier for the charge transfer step of the reaction at the interface. This results in an exponential dependence of the reaction rate on the electrode potential, the extent of which is given by the transfer coefficient, a. 

This result is quite in contrast to the common expectation that the electrode potential changes the activation barrier at the interface which would result in a temperature independent transfer coefficient a. Following Agar s discussion (30), such a behavior indicates a potential dependence of the entropy of activation rather than the enthalpy of activation. Such "anomalous" behavior in which the transfer coefficient depends on the temperature seems to be rather common as recently reviewed by Conway (31). 

The latter discussion confirms the results of the potential dependence of the current in that the activation barrier for the hydrogen evolution reaction is, at least on copper and silver, not affected by the electrode potential. This behavior is, on the other hand, connected with the observation of straight lines in a Tafel plot. It would be premature to come up with a comprehensive model that would explain this behavior more experimental work is necessary to substantiate and quantify the effects for a larger variety of systems and reactions. A few aspects, however, should be pointed out. 

The single particle auto-correlation time tc in Eq. 9 can, of course, exhibit also a non-critical temperature dependence. Consider a set of independent hydrogen bonds with symmetric double well potentials and a barrier a between the wells. In this case the motion is thermally activated and tc shows an Arrhenius behaviour ... 

The formal rate constants kfh and kb h are potential dependent. At any given potential, in order for the transition from the oxidized form to occur, it will be necessary to pass over an activation free-energy barrier, AGf, as illustrated by the Morse curves in Figure 2.14. The rate of reaction will be proportional to exp (-AGf /RT). 

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