Butler -Volmer-type equations

The free energy barrier for the flow of ionic charge across the oxide electrode-electrolyte interface has an electrical contribution and consequently the reaction rate can be formally described by Butler-Volmer-type equations [38]. The cation current density corresponding to the process... 

Table 29.3 Selection of Butler-Volmer-type equations from the models discussed.

Butler-Volmer type equations and exchange current density... 

The rate of anodic hydrogen oxidation is proportional to the current density of hydrogen oxidation (iH2)- As discussed in Section 11.3, this current density according to the electrochemical model follows a Butler-Volmer-type equation (Eqs. (10a) and (10b)) with a concentration-dependent exchange current... 

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. 

The Butler-Volmer Equation for an Elementary Step Charge-transfer l-V relations are given in LSE by the But-ler-Volmer type equation. For an elementary step in which a species with charge zq is transferred,... 

However, if the rate constant of one step in a reaction sequence is much smaller than those of all the others, so that that step controls the overall kinetics [the concept of the rate-determining step (rds)], while other steps are in the state of virtual equilibrium (i.e., the rates in each of them are practically equal in both directions), relation (30) simplifies into the usual Butler-Volmer type of equation... 

For a current density below the limiting one, a Butler-Volmer type of equation applies, taking into account the change in concentration of the depositing ions at the electrode surface for the cathodic direction, i.e.,... 

The oxidation reactions in equation (7.110) and equation (7.112) follow a Butler-Volmer-type kinetics ... 

If the kinetics of electron transfer does not obey the Butler-Volmer law, as when it follows a quadratic or quasi-quadratic law of the MHL type, convolution (Sections 1.3.2 and 1.4.3) offers the most convenient treatment of the kinetic data. A potential-dependent apparent rate constant, kap(E), may indeed be obtained derived from a dimensioned version of equation (2.10) ... 

Activation polarization arises from kinetics hindrances of the charge-transfer reaction taking place at the electrode/electrolyte interface. This type of kinetics is best understood using the absolute reaction rate theory or the transition state theory. In these treatments, the path followed by the reaction proceeds by a route involving an activated complex, where the rate-limiting step is the dissociation of the activated complex. The rate, current flow, i (/ = HA and lo = lolA, where A is the electrode surface area), of a charge-transfer-controlled battery reaction can be given by the Butler—Volmer equation as... 

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... 

We have the following unknown boundary values the two species nearsurface concentrations Cyo and Cb,o, the two species fluxes, respectively G and G n, the additional capacitive flux Gc, and the potential p, differing (for p > 0) from the nominal, desired potential pnom that was set, for example, in an LSV sweep or a potential step experiment. Five of the six required equations are common to all types of experiments, but the sixth (here, the first one given below) depends on the reaction. That might be a reversible reaction, in which case a form of the Nernst equation must he invoked, or a quasi-reversible reaction, in which case the Butler-Volmer equation is used (see Chap. 6 for these). Let us now assume an LSV sweep, the case of most interest in this context. The unknowns are all written as future values with apostrophes, because they must, in what follows below, be distinguished from their present counterparts, all known. 

As already shown in Fig. 1, a general feature of electrocatalysis is that the current passing through an electrode-electrolyte interface depends exponentially on overpotential, as described by the Butler-Volmer equation discussed in Sect. 2.4.1, so that logi versus r] U — C/rev) gives straight lines, termed Tafel plots (Fig. 1). On this basis, one would expect an exponential-type dependence of current on overpotential in Fig. 12 (curve labeled 7ac). Such a curve would correspond to pure activation control, that is, to infinitely fast mass-transport rates of reactants and products to and from the two electrodes. 

Guglielmi applied a Langmuir-type adsorption concept and a Butler-Volmer equation to formulate the final deposition process. But his final equation for the volume percent of co-deposited particles Xy is rather empirical. It can be written in the following form ... 

The irregular type of codeposition is very often characterized by simultaneous influence of cathodic potential and diffusion phenomena, i.e., it mainly occurs under the activation and/or mixed control of the electrodeposition processes. The rate of electrodeposition in such a case is expressed by the Butler-Volmer equation which is usually used for the kinetics of electrochemical processes [1,5] ... 

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... 

Table 29.2 Commonly used parameters for Butler-Volmer- and Tafel-type equations.

Often, the exponential dependence of the dark current at semiconductor-electrolyte contacts is interpreted as Tafel behavior [49], since the Tafel approximation of the Butler-Volmer equation [50] also shows an exponential increase of the current with applied potential. One should, however, be aware of the fundamental differences of the situation at the metal-electrolyte versus the semiconductor-electrolyte contact. In the former, applied potentials result in an energetic change of the activated complex [51] that resides between the metal surface and the outer Helmholtz plane. The supply of electrons from the Fermi level of the metal is not the limiting factor rather, the exponential behavior results from the Arrhenius-type voltage dependence of the reaction rate that contains the Gibbs free energy in the expraient It is therefore somewhat misleading to refer to Tafel behavior at semiconductor-electrolyte contacts. 

Regardless of fuel cell type, in a certain range of polarization voltages this rate is well approximated by the Butler-Volmer or Tafel equations. A better (non-Tafel) approximation of the reaction rate for DMFC anode is considered in Section 2.7. 

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