Domain Equations for Tertiary Current Distributions Using the Nernst-Planck Equations and Electroneutrality
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:
where
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ci denotes the concentration of species i (SI unit: mol/ m3),
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Di the diffusion coefficient of species i (SI unit: m2/s),
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zi the charge number of the ionic species (dimensionless),
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um,i the ionic mobility of a species i (SI unit: mol·s/kg),
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F the Faraday constant (SI unit: C/mol), and
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l denotes the electric potential (SI unit: V).
The current balance includes the sum of the flux of all charged species, which yields the current density in the electrolyte:
The concentration gradients are not assumed to be negligible here, and so the contribution of ion diffusion to overall current density can be nonzero. (Compare with the Domain Equations for Primary and Secondary Current Distributions.)
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:
These formulations are also valid for the pore electrolyte in porous electrodes, except for the transport properties that have to be corrected for porosity and tortuosity. In such cases, the source or sink, Ql, denotes the charge transfer reactions in the porous electrode and/or the non-Faradaic source or sink due to double layer charge and discharge.
The charge transfer reaction can be parameterized by arbitrary functions of the concentrations of the species in the redox couple and the local electric and electrolyte potentials. The most common way to describe the reaction kinetics is to use a Butler-Volmer expression for the charge transfer current density, iloc (SI unit: A/m2), which in this case can contain concentration dependencies:
where i0 denotes the exchange current density (SI unit: A/m2), 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:
In the current balance in a porous electrode, the local current density multiplied by the specific surface area of an electrode gives a contribution to the source or sink, Ql, due to electrochemical reactions.