Butler-Volmer approach

Using a kinetic formalism, the flux corresponding to the transfer of an ion across a water-organic solvent interface can be written  

When at eqnilibrium the bulk concentrations are equal, the potential difference is eqnal to the formal potential for the ion-transfer reaction, and the global activation energy barrier is symmetrical [91]. As the potential is varied, the overall driving force zF (A (l)-A ( ) ) partially lowers the activation energy barrier and, as in the Butler-Volmer mechanism, the current is then given by 

Different approaches have been proposed to justify a Butler-Volmer behavior. In 1986, Gurevich and Kharkats proposed a stochastic approach to ion-transfer reactions [116-118], Here, the main equation is the Langevin equation  

The key problem to proceed is to know the potential F In a first approximation, Gurevich and Kharkats used the electrical potential and, using a harmonic model, they end up with an expression very similar to a Butler-Volmer equation. In 1995, Indenbom proposed taking into account the elastic properties of the interface to derive an expression of the activation energy barrier [119]. 

In 1997, Schmickler published an ion-transfer theory based on a lattice-gas model [120]. The potential of mean force experienced by a single ion in the vicinity of a liquid-liquid interface was then calculated. Typically, it shows a maximum and a minimum, which means that ion transfer is treated as an activated process involving the transfer over a barrier following a Butler-Volmer-type law. 

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The Butler-Volmer approach is not empirical, so it therefore helps to explain the observed Tafel behaviour. Here, we start by saying that the observed current always has two components, i.e. oxidative and reductive, with both components depending strongly on the applied potential, V. The oxidative (anodic) current is... 

We will now combine the Levich and Butler-Volmer approaches. The Levich relationship (equation (7.1)) is written in terms of the limiting current / jm, where limiting here means proportional to Canaiyte - in other words, the electrode reaction is so fast that the magnitude of the current is controlled only by the flux of analyte to the electrode solution interface, i.e. /um is mass>transport controlled. 

The kinetics of electrochemical O2 reduction on Pt has been studied extensively. °° There is a general consensus that it shows first order kineucs in O2. By following the Butler-Volmer approach, the rate expression for the ORR can be expressed by the relationship between kinetic current, i, and potential, E ... 

Another model that treats the ion transfer across ITIES as an activated transport process by applying classical transition state theory has been suggested by Girault and SchifFrin [55]. Also in this case the equation describing the ion flux across the interface is similar to that obtained by the Butler-Volmer approach. Naturally, the interpretation of the a values and k° is somewhat different in that case. Experimentally [85] and theoretically [79], the effect of solvent viscosity on the ion-transfer process has been confirmed indicating the importance of this parameter. 

Recently, several groups have taken cell-level macroscale models a step further to investigate the electrochemistry through the thickness of the electrodes using the mesoscale electrochemistry approach [19, 27, 31]. In these models, no assumptions are made about a reactive zone for the electrochemical reactions instead, the electrochemistry is modeled through the thickness of the electrodes based on a mesoscale electrochemistry approach (Section 26.2.4.2) in which the explicit charge-transfer reactions [27] or a modified Butler-Volmer approach [19, 31] are modeled. This extends the effects of the electrochemical reactions away from the electrolyte interface into the electrodes. In these cell-level models, the electrochemistry is coupled to the local species concentrations, pressures, and temperatures, and provides a more detailed view into the local conditions within the fuel cell and how these local conditions affect the overall SOFC performance. 

The expression for a given imposed current j is given by the Butler-Volmer equations, (11.2) and (11.3). These are essentially empirical equations [2,3] Bard and Faulkner refer to them as the Butler-Volmer approach . According to those equations the imposed current is given by... 

The effect of the phospholipids on the rate of ion transfer has been controversial over the last years. While the early studies found a retardation effect [6-8], more recent ones reported that the rate of ion transfer is either not retarded [9,10] or even enhanced due to the presence of the monolayer [11 14]. Furthermore, the theoretical efforts to explain this effect were unsatisfactory. The retardation observed in the early studies was explained in terms of the blocking of the interfacial area by the phospholipids, and therefore was related to the size of the transferring ion and the state of the monolayer [8,15]. The enhancement observed in the following years was attributed to electrical double layer effects, but a Frumkin-type correction to the Butler Volmer (BV) equation was found unsuitable to explain the observations [11,16]. Recently, Manzanares et al. showed that the enhancement can be described by an electrical double layer correction provided that an accurate picture of the electrical double layer structure is used [17]. This theoretical approach will be the subject of Section III.C. 

Analysis of the cyclic voltammetric responses is also possible if a kinetic law different from Butler-Volmers governs the electrode electron transfer. Derivation of the kinetic law from the cyclic voltammetric responses may benefit from a convolution approach similar to that described in the preceding section. 

However, as we saw in section 3.3 for platinum on YSZ, the fact that i—rj data fits a Butler—Volmer expression does not necessarily indicate that the electrode is limited by interfacial electrochemical kinetics. Supporting this point is a series of papers published by Svensson et al., who modeled the current—overpotential i—rj) characteristics of porous mixed-conducting electrodes. As shown in Figure 28a, these models take a similar mechanistic approach as the Adler model but consider additional physics (surface adsorption and transport) and forego time dependence (required to predict impedance) in order to solve for the full nonlinear i—rj characteristics at steady state. 

Large Cathodic Current We have seen from Figure 6.7 that for the large negative values of overpotential r], the partial cathodic current density i approaches i, i i. For these conditions the Butler-Volmer equation (6.45) can be simplified. Analysis of Eq. (6.45) shows that when rj becomes more negative, the first exponential term in the equation (corresponding to the anodic partial current) decreases, whereas the second exponential term (corresponding to the cathodic partial reaction) increases. Thus, under these conditions. 

The Butler-Volmer (BV) approximation is the simplest approach to model and capture the essential features of the empirical Tafel equation. It considers an electrochemical half-cell reaction as an activated process, with the forward and backward reaction rates following an Arrhenius type law according to... 

The approach curves are recorded at different potentials applied to the titanium number sample. The obtained effective rate constants are fitted to the Butler-Volmer equation. 

In this section, both approaches will be compared in chronoamperometry under limiting current conditions at spherical electrodes and microelectrodes. As is well known, for spherical electrodes and taking into account the Butler-Volmer model, the value of the diffusion-controlled reduction current at large overpotentials, e B is given by the following expression (see Eq. (2.147) of Sect. 2.5.2) ... 

The two models most commonly applied to the heterogeneous electron transfer kinetics are the Butler-Volmer model, which is primarily a macroscopic approach... 

Lorenz and Sali610 also propose an alternative kinetic approach to the determination of / this assumes a potential dependence of the adsorption and desorption rates of the pet process of Eq. (1) which is entirely analogous to that of the Butler-Volmer equation ... 

In spite of the above justification for the kinetic approach to the estimate of l, this has a number of drawbacks. First of all, there is no point in using a kinetic approach to determine a thermodynamic equilibrium quantity such as l. The justification of the validity ofEqs. (42) and (45) by the resulting equilibrium condition of Eq. (46) is far from rigorous, just as is the justification of the empirical Butler-Volmer equation by the thermodynamic Nernst equation. Moreover, the kinetic expressions of Eq. (41) involve a number of arbitrary assumptions. Thus, considering the adsorption step of Eq. (38a) in quasi-equilibrium under kinetic conditions cannot be taken for granted a heterogeneous chemical step, such as a deformation of the solvation shell of the... 

This mechanism is denoted as an EC mechanism (Testa and Reinmuth, 1961 Bott, 1997). Thus homogeneous kinetic terms may be combined with the expressions for diffusion and convection [i.e. a modified version of (18)] to give the temporal variation of the concentration of a species in an electrode reaction mechanism. In order to model the voltammetric response associated with this mechanism, a knowledge of , a, ko and k is required, or deduced from a theoretical-experimental comparison, and the set of concentrationtime equations for species A, B and C must be solved subject to the constraints of the Butler-Volmer equation and the experimental design. Considerable simplification of the theory is achieved if the kinetics for the forward and reverse processes associated with the E step are fast, which is a good approximation for many organic reactions. Section 7 describes the approaches used to solve the equations associated with electrode reaction mechanisms, thus enabling theoretical simulation of voltammetric responses to be achieved. 

The transition state approach leads in a natural way to the Butler-Volmer equation, but is relatively weak in its predictive properties regarding the exchange current, ( 0, which is proportional to the frequency factor kr(., i and to exp(—AG J). The latter is quite closely related to the enthalpy and standard entropy of formation of the adsorbed reduction product or intermediate, and this is one main reason for the very intense modern efforts to develop predictive theoretical tools for the ab initio computation of adsorption energies at... 

In this equation, and represent the surface concentrations of the oxidized and reduced forms of the electroactive species, respectively k° is the standard rate constant for the heterogeneous electron transfer process at the standard potential (cm/sec) and oc is the symmetry factor, a parameter characterizing the symmetry of the energy barrier that has to be surpassed during charge transfer. In Equation (1.2), E represents the applied potential and E° is the formal electrode potential, usually close to the standard electrode potential. The difference E-E° represents the overvoltage, a measure of the extra energy imparted to the electrode beyond the equilibrium potential for the reaction. Note that the Butler-Volmer equation reduces to the Nernst equation when the current is equal to zero (i.e., under equilibrium conditions) and when the reaction is very fast (i.e., when k° tends to approach oo). The latter is the condition of reversibility (Oldham and Myland, 1994 Rolison, 1995). 

As already mentioned above, 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 (Figs. 7.1 and 7.2) looks similar to the free energy profile derived for adiabatic reactions, i.e. for processes where a strong interaction between 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. According to these models the activation energy is given by (see Eq. 6.10)... 

One can derive the Butler-Volmer kinetic expressions by an alternative method based on electrochemical potentials (8, 10, 12, 19-21). Such an approach can be more convenient for more complicated cases, such as requiring the inclusion of double-layer effects or sequences of reactions in a mechanism. The first edition develops it in detail. ... 

Wave-slope plot. Totally irreversible steady-state voltammograms give linear plots of E vs. log [(/d — /)//] in accord with (5.5.48). The slope provides a and the intercept at E yields l if DqIvq is known. This approach involves the assumption that Butler-Volmer kinetics apply. For a totally irreversible wave based on early transients, the wave-slope plot is predicted to be slightly curved consequently it does not have quantitative utility. 

In deriving (6.7.14) and (6.7.17), we assumed that Butler-Volmer kinetics apply, as expressed in the i-E characteristic, (3.3.11). Indeed, this assumption (or the adoption of some other model) is necessary before equations can be derived for most electrochemical approaches. However, with the convolutive technique, this assumption is not essential, for the rate law can be written in the general form (27),... 

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