The domain equations can be derived by starting with The Nernst-Planck Equations. The material balance equation for the species
i in the electrolyte is given by the continuity equation, with a flux given by the Nernst-Planck equation:
In the equations above, il denotes the current density vector in the electrolyte. The current balance in the electrolyte then becomes:
where Ql can here be any source or sink. (
Ql is typically nonzero for porous electrodes). The current balance and the material balances give one equation per unknown species concentration. However, there is still one more unknown, the
electrolyte potential, which requires an additional equation. This equation is the electroneutrality condition, which follows from dimensional analysis of Gauss’s law. In a typical electrolyte solution, it is accurate over lengths greater than a few nanometers:
where i0 denotes the exchange current density (SI unit: A/m
2),
αa the anodic charge transfer coefficient (dimensionless),
αc the cathodic charge transfer coefficient (dimensionless),
γi,a the anodic reaction order for species
i (dimensionless), and
γi,c is the cathodic reaction order for species
i (dimensionless). The overpotential,
η, is defined as in preceding sections, according to the following equation: