A number of different analytical expressions for the current density perpendicular to the electrode surface, iloc,m, are available. In the following paragraphs, the index
m is dropped. All parameters are understood to refer to a specific reaction.
where Eeq denotes the equilibrium potential.
where αc (unitless) denotes the cathodic charge transfer coefficient,
αa (unitless) the anodic charge transfer coefficient, and
i0 (SI unit: A/m
2) is the exchange current density.
The exchange current density is generally concentration dependent. For certain conditions, it is possible to derive analytical expressions for i0. Assuming a concentration dependent equilibrium potential defined by the Nernst equation (see previous section
Equilibrium Potentials and the Nernst Equation), the kinetics to follow the law of mass action, and the condition that
αa + αc = n, then the exchange current density becomes as follows:
where i0,
ref is the exchange current density (SI unit: A/m
2) at the reference state. The above expression can be derived from the mass action law, which gives the following expression for the local current density:
where the overpotential ηref (SI unit: V) is measured using relative to a reference state, which yields:
where γi are generic exponential coefficients. For this case,
αa and
αc may be independently defined. By rearranging the Butler-Volmer expression using
ηref similarly to what was done for the mass action law case above, the
γi coefficients relate to the anodic,
ξa,i, or cathodic,
ξc,i, reaction orders according to
For instance, for a one electron redox couple of concentrations co and c
r, with the same reference concentration
cref for both species, and
i0, ref = k0Fc
ref, the mass action law expression above can be rewritten as
where k0 (m/s) is the heterogeneous rate constant.
Here CR and
CO are dimensionless expressions, describing the dependence on the reduced and oxidized species in the reaction. Note that if
CO ≠ CR when
η = 0, this kinetics expression results in
iloc ≠ 0, thus violating the concept of equilibrium. This may result in thermodynamical inconsistencies, for instance when coupling an electrochemical model to heat transfer.
where Aa (SI unit: V) is the so-called Tafel slope.
Aa relates to the corresponding transfer coefficient as follows
where the sign accounts for the negative cathodic charge transfer current. Here, Ac is required to be negative and relates to the transfer coefficient according to
where iexpr (A/m
2) is the current density expression in the absence of mass transport limitations for the species, and
ilim (A/m
2) is the limiting current density that corresponds to the maximum transport rate of the species. The derivation of this expression assumes high overpotentials so that either the anodic or an cathodic term in the Butler-Volmer equation may be neglected.
where ν and
k are positive numbers and the desired behavior is that the rate
r and the concentration
c should equal zero in the converged solution at infinite time. However, if
c, due to numerical fluctuations in the solver process, becomes negative during iterating, issues may arise.
First consider the case when ν equals 1 (or any odd positive integer). Negative values of
c will then cause the rate to become positive, resulting in a “self stabilizing” situation where
c will be approaching 0 with time.
A second case to consider is when ν is an even integer larger than 1. The rate then will become increasingly negative for negative values of
c, resulting in an “exploding” solution, iterating
c towards minus infinity. The standard solution for these cases, which also works for non-integer
ν's larger than 1, is to change the expression
c in the rate term to max(
c,eps), where eps is a small number. This will avoid the “exploding” behavior, but result in poor convergence rate for negative
c values since the Jacobian of the rate with respect to
c then becomes zero for negative
c's.
The third case is when ν is a non-integer between 0 and 1. Note now that Jacobian with respect to
c of the rate expression now contains
cν−1, which will approach minus infinity when
c approaches zero from the positive side. This may result in poor convergence, and the max() wrapping will not improve convergence in this case.