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 each 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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u is, the velocity vector (SI unit: m/s), and
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Ji denotes the molar flux relative to the convective transport(see Equation 3-2)
The material balances give one equation per unknown species concentration.
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.)
There is one more unknown dependent in the variable, the electrolyte potential, which requires an additional equation to close the system. 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:
Using the electroneutrality condition reduces the number of dependent concentration variables by one.
Further, by combining the electroneutrality condition with the sum of all species flux vectors and the species mass balance equation, multiplied by the individual species charges, results in expressions for the current vector and a current balance equation, respectively.
The expression for the current density in the electrolyte, il, reads:
whereas current balance in the electrolyte, used for solving for the electrolyte potential, then becomes:
where Ql can here be any source or sink. (Ql is typically nonzero for porous electrodes). 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 nonfaradaic 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. See The Butler–Volmer 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.