Electrode Kinetics Expressions
A number of different analytical expressions for iloc,m are available. In the following the index m is dropped. All parameters are understood to refer to a specific reaction.
The Equilibrium Potential and the Overpotential
The rate of the electrochemical reactions can be described by relating the reaction rate to the activation overpotential (or reduction potential). For an electrode reaction, the activation overpotential, denoted η, is the following:
where Eeq denotes the equilibrium potential.
Butler-Volmer
The most general expression is of Butler-Volmer type:
where αc denotes the cathodic charge transfer coefficient, αa the anodic charge transfer coefficient.
Linearized Butler-Volmer
The charge transfer reaction can be expressed by a linearized Butler-Volmer expression, which can be used for small overpotentials (η << RT/F) and is usually referred to as the low-field approximation. This approximation gives the following linearized equation:
where αc denotes the cathodic charge transfer coefficient, αa the anodic charge transfer coefficient, i0 the exchange current density, and η the overpotential.
Anodic and Cathodic Tafel Equations
By assuming either high anodic or cathodic overpotentials for a given current (that is, slow kinetics or low i0), one of the terms in the original Butler-Volmer potentials can be neglected.
The anodic Tafel equation is implemented as follows:
where Aa (SI unit: V) is the so-called Tafel slope. The cathodic Tafel expression is according to:
to account for the negative sign of cathodic charge transfer reactions. Ac is also required to be negative.
Concentration Dependent Kinetics
The following expression describes a charge transfer reaction according to the full Butler-Volmer equation, where the anodic and cathodic terms of the current density expression depend on the local concentrations of the electroactive species at the electrode surface:
where CR and CO are dimensionless expressions, describing the dependence on the reduced and oxidized species in the reaction.
Exchange Current Density and Rate Constants
The exchange current density typically depends on the local concentrations of the reacting species. For instance, for a one electron redox couple, defining CO = co / cref, CR = cr / cref, and i0=k0Fcref, the above expression can be rewritten as
where k0 (m/s) is the heterogeneous rate constant.
Limiting Current Density
The steady-state rate of electrode reactions never exceeds the rate at which reactants and products can be transported to and from the electrode surface. When explicitly including mass transport in a model, this dependence is typically described by a concentration dependent kinetics expression as described above.
By the assumption of a Nernst diffusion layer at the electrode surface, and a first order dependence between the charge transfer current and the local concentration of a reacting species, the following kinetics expression can be derived:
where iexpr (A/m2) is the current density expression in the absence of mass transport limitations for the species, and ilim (A/m2) is the limiting current density that corresponds to the maximum transport rate of the species. The derivation of this expression assumes either a strictly anodic or an cathodic reaction.