Electrochemical Reactions and the Difference Between a Primary and a Secondary Current Distribution
Both the primary and secondary current distribution models assume an electrolyte that conducts current according to Ohm's law, with a constant conductivity. The difference between a primary and a secondary current distribution lies in the description of the electrochemical charge transfer reaction at the interface between an electrolyte and an electrode. The charge transfer reactions can be defined as boundary conditions, or as sources or sinks in a domain for the case of porous electrodes.
The rate of the electrochemical reactions can be described by relating the reaction rate to the activation overpotential. For an electrode reaction, with index m, the activation overpotential, denoted ηm, is the following:
where Eeq,m denotes the equilibrium potential (also known as a reduction potential) for reaction m.
In the Primary Current Distribution interface, the electrochemical reactions are assumed to be fast enough that their kinetics have negligible influence on the cell voltage. Consequently, the activation overpotential, ηm, in this physics interface is equal to zero, and the difference in potential between the electrode and the electrolyte is always equal to the equilibrium potential, Eeq, m. This condition is implemented as the following constraint on boundaries between electrodes and electrolyte domains:
A good modeling practice is to estimate the influence of finite reaction kinetics on the cell voltage and to compare the associated activation losses to the ohmic losses in a cell to check the validity of the primary current distribution assumption.
In the Secondary Current Distribution interface, the electrochemical reactions are described as a function of the overpotential. The physics interface uses several relations for the charge transfer current density and the overpotential, such as Butler-Volmer and Tafel expressions. The most general expression is of Butler-Volmer type:
where iloc,m denotes the local charge transfer current density for reaction m, i0 the exchange current density, αa the anodic transfer coefficient, αc the cathodic charge transfer coefficient, F Faraday’s constant, and R the universal gas constant.
Both the exchange current density and the overpotential are typically concentration dependent. It is possible include the dependence of kinetics on concentration in the expression above. It also possible to use other kinetics expressions.
See Equilibrium Potentials and the Nernst Equation and Electrode Kinetics Expressions.
The sum of all electrode reaction currents is implemented as a current density condition on the boundary between an electrode and an electrolyte domain according to:
Both the Primary Current Distribution and Secondary Current Density Distribution interfaces allow for a domain definition for porous electrodes. For the Primary Current Distribution interface, the same constraint as above is applied.
In porous electrodes for the Secondary Current Distribution interface, the sum of all reaction currents appears as a source in the domain equations:
where Av,m is the surface area.
The only reason to add Porous Electrode descriptions to the Primary Current Distribution interface is to be able to expand a model to include charge transfer resistance.