Electrode kinetics Butler-Volmer model

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. 

In Fig. 3.14a, the dimensionless limiting current 7j ne(t)/7j ne(tp) (where lp is the total duration of the potential step) at a planar electrode is plotted versus 1 / ft under the Butler-Volmer (solid line) and Marcus-Hush (dashed lines) treatments for a fully irreversible process with k° = 10 4 cm s 1, where the differences between both models are more apparent according to the above discussion. Regarding the BV model, a unique curve is predicted independently of the electrode kinetics with a slope unity and a null intercept. With respect to the MH model, for typical values of the reorganization energy (X = 0.5 — 1 eV, A 20 — 40 [4]), the variation of the limiting current with time compares well with that predicted by Butler-Volmer kinetics. On the other hand, for small X values (A < 20) and short times, differences between the BV and MH results are observed such that the current expected with the MH model is smaller. In addition, a nonlinear dependence of 7 1 e(fp) with 1 / /l i s predicted, and any attempt at linearization would result in poor correlation coefficient and a slope smaller than unity and non-null intercept. 

The Butler-Volmer formulation of electrode kinetics [16,17] is the oldest and least complicated model constructed to describe heterogeneous electron transfer. However, this is a macroscopic model which does not explicitly consider the individual steps described above. Consider the following reaction in which an oxidized species, Ox, e.g. a ferricenium center bound to an alkanethiol tether, [Fe(Cp)2]+, is converted to the reduced form, Red, e.g. [Fe(Cp)2], by adding a single electron ... 

In real (as opposed to model) electrochemical cells, the net current flowing will often be partly determined by the kinetics of electron transfer between electrode and the electroactive species in solution. This is called heterogeneous kinetics, as it refers to the interface instead of the bulk solution. The current in such cases is obtained from the Butler-Volmer expressions relating current to electrode potential [73,74,83,257,559]. We have at an electrode the process (2.18), with concentrations at the electrode/electrolyte interface cj q and cb,Oj respectively. We take as positive current that going into the electrode, i.e., electrons leaving it, which corresponds to the reaction (2.18) going from left to right, or a reduction. Positive or forward (reduction) current if is then related to the potential E by... 

Butler-Volmer equation — The Butler-Volmer or -> Erdey-Gruz-Volmer or Butler-Erdey-Gruz-Volmer equation is the fundamental equation of -> electrode kinetics that describes the exponential relationship between the -> current density and the -> electrode potential. Based on this model the -> equilibrium electrode potential (or the reversible electrode potential) can also be interpreted. 

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 reaction occurs at the electrode/electrolyte interface (sol-id/liquid interface at the surface of the particle). This reaction occurs as a source term in the equations for the macro scale. In the model equations, accounts for the electrochemical kinetics, (intercalation reaction from the electrolyte phase into the solid matrix and vice-versa). It is a modified form of the Butler-Volmer kinetics, and is given by the following expression ... 

Kawamoto (2) developed a two-dimensional model that is based on a double iterative boundary element method. The numerical method calculates the secondary current distribution and the current distribution within anisotropic resistive electrodes. However, the model assumes only the initial current distribution and does not take into account the effect of the growing deposit. Matlosz et al. (3) developed a theoretical model that predicts the current distribution in the presence of Butler-Volmer kinetics, the current distribution within a resistive electrode and the effect of the growing metal. Vallotton et al. (4) compared their numerical simulations with experimental data taken during lead electrodeposition on a Ni-P substrate and found limitations to the applicability of the model that were attributed to mass transfer effects. 

Based on Newman s well-known modelling approach [8 10] the impedance of a commercial cell is described. This approach combines concentrated solution theory, porous electrode theory and Butler-Volmer kinetics to form a set of coupled partial differential equations. 

One of the early mechanistic models for a PEM fuel cell was the pioneering work of Bemardi and Verbrugge [45, 46]. They developed a one-dimensional, steady state, isothermal model which described water transport, reactant species transport, as well as ohmic and activation overpotentials. Their model assumed a fully hydrated membrane at all times, and thus calculated the water input and removal requirements to maintain full hydration of the membrane. The model was based on the Stefan Maxwell equations to describe gas phase diffusion in the electrode regions, the Nemst-Planck equation to describe dissolved species fluxes in the membrane and catalyst layers, the Butler Volmer equation to describe electrode rate kinetics and Schlogl s equation for liquid water transport. 

The Warburg and Nernst impedances were derived under the assumption that the potential obeys the Nernst equation. The more realistic Randles model takes into account the kinetics of charge transfer as described by the Butler-Volmer equation. For the electrode reaction (5.147) this is written as... 

Fermeglia et al. [15] 2D cross flow Dynamic Reduced momen- tum balance Cell WGS (equil.) Overall Butler—Volmer kinetics Unclear Overdeter- mined electrode model... 

Eor very high exchange current densities (i.e., rapid reactions), a linearized form of Eq. 27 can often be used. For very slow reaction kinetics, either the anodic or cathodic term dominates the kinetics, and so the other term is often ignored, yielded what is known as a Tafel equation for the kinetics. Often, more complicated expressions than that of Eq. 27 are used. For example, if the elementary reaction steps are known, one can write down the individual steps and derive the concentration dependence of the exchange current density and the kinetic equation. Other examples include accounting for surface species adsorption or additional internal or external mass transfer to the reaction site [9]. All of these additional issues are beyond the scope of this chapter, and often an empirically based Butler-Volmer equation is used for modeling the charge transfer in porous electrodes. 

Figure 4-1 Model for the electrode process includes diffusion (Pick s laws), electrode kinetics (the Butler-Volmer equation), and chemical reaction kinetics.

Electrode Kinetic and Mass Transfer for Fuel Cell Reactions For the reaction occurring inside a porous three-dimensional catalyst layer, a thin-film flooded agglomerate model has been developed [149, 150] to describe the potential-current behavior as a function of reaction kinetics and reactant diffusion. For simplicity, if the kinetic parameters, such as flie exchange current density and diffusion limiting current density, can be defined as apparent parameters, the corresponding Butler-Volmer and mass diffusion relationships can be obtained [134]. For an H2/air (O2) fuel cell, considering bofli the electrode kinetic and the mass transfer, the i-rj relationships of the fuel cell electrode reactions within flie catalyst layer can be expressed as Equations 1.130 and 1.131, respectively, based on Equation 1.122. The i-rj relationship of the catalyzed cathode reaction wifliin the catalyst layer is... 

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