Charge Transfer Overpotential Butler-Volmer Equation

6 CHARGE TRANSFER OVERPOTENTIAL BUTLER-VOLMER EQUATION 

John Alfred Valentine Butler (1899-1977) was an English physical chemist, who greatly contributed to theoretical electrochemistry. Particularly, he contributed to developing a relationship between electrochemical kinetics and thermodynamics. He is best known for his contribution to the development of the famous Butler-Volmer equation. 

Inzelt, and R Scholz, Electrochemical Dictionary, Springer, Berlin, Germany, 2008. 

Max Volmer (1885-1965) was a German physical chemist, who made important contributions to electrochemistry, in particular on electrode kinetics. He codeveloped the Butler-Volmer equation. 

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Charge transfer resistance — At low - overpotentials (q RT/nF) none of the -> partial current densities is negligible (see also activation overpotential, - charge-transfer overpotential, -> Butler-Volmer equation). 

The activation overpotential Tiac,w is due to slow charge transfer reactions at the electrode-electrolyte interface and is related to current via the Butler-Volmer equation (4.7). A slow chemical reaction (e.g. adsorption, desorption, spillover) preceding or following the charge-transfer step can also contribute to the development of activation overpotential. 

Butler27 and Volmer28 advanced Tafel s equation by relating overpotentials to activation barriers. The quantitative relationship between current and overpotential is called the Butler-Volmer equation (eqn (32)), and is valid for electrochemical reactions that are rate limited by charge transfer. 

This general equation covers charge transfer at electrified interfaces under conditions both of zero excess field, low excess fields, and high excess fields, and of the corresponding overpotentials. Thus the Butler-Volmer equation spans a large range of potentials. At equilibrium, it settles down into the Nernst equation. Near equilibrium it reduces to a linear / vs. T) (Ohm slaw for interfaces), whereas, if T) > (RT/fiF) (i.e., one is 50 mV or more from equilibrium at room temperature), it becomes an exponential /vs. T) relation, the logarithmic version ofwhich is called Tafel s equation. 

It is an experimental fact that whenever mass transfer limitations are excluded, the rate of charge transfer for a given electrochemical reaction varies exponentially with the so-called overpotential rj, which is the potential difference between the equilibrium potential F0 and the actual electrode potential E (t) = E — Ed). Since for the electrode reaction Eq. (1) there exists a forward and back reaction, both of which are changed by the applied overpotential in exponential fashion but in an opposite sense, one obtains as the effective total current density the difference between anodic and cathodic partial current densities according to the generalized Butler-Volmer equation ... 

For a given overpotential, the effective current density depends on the magnitude of the charge-transfer coefficient a as well as on the exchange current density iu. If the overpotential is high enough—that is, if either —(a Frj/RT) or (a Fr/)/RT > 1—then one of the partial current densities in the Butler-Volmer equation overrules the other ... 

If the electrocrystallization is controlled by formation of two- or three-dimensional isolated nuclei, the current—overpotential relationship has a stronger dependence on 17 than predicted by the Butler—Volmer equation for charge transfer control [151]... 

Charge transport is modeled by Ohm s law (Equation (3.10)) and the charge conservation equation (Equation (3.68)), while the current density distribution at the electrode/electrolyte interface is modeled through the Butler-Volmer equation (Equation (3.102)). It should be noted that, contrarily to Section 3.7, Equation (3.102) is here derived from Equation (3.37) rather than Equation (3.39), because the former allows for a better agreement between experimental and simulated results. Equations (3.40)-(3.42) are used to model, the exchange current density, the activation overpotential, and the ideal potential drop at the electrode/electrolyte interface, respectively. Heat transfer is modeled through Equation (3.6), and the appropriate heat terms for each domain. 

Fig. 9.21 Simulink electrochemical model with R-C model capacitance behavior. The BV Fen block is the quasi-steady Butler-Volmer overpotential equation giving current through the Ret charge transfer resistor as a function of charge transfer overpotential, r).

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... 

In the limit of small overpotentials (rj 0), the Butler-Volmer equation can be linearized to yield the charge transfer resistance (Rct) ... 

In the case of a redox system out of equilibrium, the current density can be expressed as a function of the charge transfer overpotential Pc, according to the Butler-Volmer equation ... 

Aetivation overpotential arises from the kinetics of charge transfer reaction across the eleetrode-eleetrolyte interface. In other words, a portion of the electrode potential is lost in driving the eleetron transfer reaction. Activation overpotential is directly related to the nature of the eleetroehemical reactions and represents the magnitude of activation energy, when the reaction propagates at the rate demanded by the current. The activation overpotential can be divided into the anode and cathode overpotentials. The anode and cathode activation overpotentials are calculated from Butler-Volmer equation (3.33 and 3.34). 

This is the most common form of the Butler-Volmer equation. It is applicable to electrode reactions whose rate is entirely limited by charge transfer at the interface. This process is sometimes called activation control, the corresponding overpotential is then called the activation overpotential. 

Equation 6 is referred to as the Butler-Volmer equation. Normally, for significant overpotentials, either one or the other of the two terms is dominant, so that the current-density exponentially increases with r, i.e. In i is proportional to 3tiF/RT in the case, for example, of appreciable positive t] values. Here the significance of Tafel s b coefficient (Equation 1) is seen b = dn/d In i = RT/3F for a simple, single-electron charge-transfer process. 

When a load is applied to the fuel cell and the current flows through the circuit, the active overpotential from the kinetics of charge transfer reactions, the ohmic overpotential from component resistances, and the mass transfer overpotential from the limited rate of mass transfer will rise. The Butler-Volmer Equation (Equation 23.8) describes a relation between the overpotential and the eurrent density on an... 

The cell activation overpotential is the sum of the activation overpotentials at the anode and cathode and is caused by slow electrocatalytic (charge transfer) reactions at both electrodes. The anodic and cathodic activation overpotentials, 33 3 and r 3c c respectively, can be related to the current density i via the Butler-Volmer equation ... 

The charge transfer overpotential is described by the Butler-Volmer equation. Two parameters of the Butler-Vohner equation are the exchange current density (/o) the symmetry factor p. The physical meaning of these parameters should be clearly understood. 

In general, the charge transfer and diffusion overpotential add to each other according to T = t ct + i1d/ arid the measured current density is limited by both effects. At small values, X is determined by the Butler-Volmer equation due to a dominating charge transfer reaction and becomes diffusion limited at larger overpotentials as shown in Figure 1.22 by the dashed line. 

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