Electrode Kinetics Expressions

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.

The rate of the electrochemical reactions can be described by relating the reaction rate to the activation overpotential. For an electrode reaction, the activation overpotential, denoted η (V), is the following:

where Eeq denotes the equilibrium potential.

A common expression for the current density as a function of the activation overpotential, in modeling of electrochemical systems, is the Butler-Volmer equation:

where αc (unitless) denotes the cathodic charge transfer coefficient, αa (unitless) the anodic charge transfer coefficient, and i0 (SI unit: A/m2) is the exchange current density.

It should be noted that, although used extensively for all sorts of reactions in the electrochemical community, the Butler-Volmer equation was originally derived for single-electron transfer reactions (for instance implying that the reaction does not involve the breaking or creation of a chemical bond).

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/m2) 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:

This latter form of the Butler-Volmer equation, where the concentration overpotential and the exchange current density are related to the same reference state, is less error prone and preferable in a modeling context.

The law of mass action is not always the most practical way for treating complex reactions involving multiple electron steps. For certain multi-electron reactions, where one electron transfer step is rate limiting, it is possible to derive a lumped Butler-Volmer expressions using the following relation for the exchange current density (see Ref. 1):

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

In electroanalysis, one commonly defines electrode kinetics in terms of rate constants rather than exchange current densities.

For instance, for a one electron redox couple of concentrations co and cr, with the same reference concentration cref for both species, and i0, ref = k0Fcref, the mass action law expression above can be rewritten as

where k0 (m/s) is the heterogeneous rate constant.

This expression type gives more freedom for the user to define concentration dependent Butler-Volmer types of expressions, where the anodic and cathodic terms of the current density expression, typically depending on the local concentrations of the electroactive species at the electrode surface, may be individually defined:

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.

The charge transfer reaction can be expressed by a linearized Butler-Volmer expression, which can be used for small overpotentials (η << RT/F) and is usually referred to as the low-field approximation. This approximation gives the following linearized equation:

By assuming either high anodic or cathodic overpotentials for a given current (that is, slow kinetics or low i0), one of the terms in the original Butler-Volmer potentials can be neglected.

The anodic Tafel equation is implemented as follows:

where Aa (SI unit: V) is the so-called Tafel slope. Aa relates to the corresponding transfer coefficient as follows

The cathodic Tafel expression is defined according to:

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

The steady-state rate of electrode reactions can never exceed the rate at which reactants and products can be transported to and from the electrode surface. When explicitly including mass transport in a model, this dependence is typically described in concentration dependence of the equilibrium potential and the exchange current density as described above.

When not explicitly including mass transfer in the domain equations one can still include the effect of transport limitations by the assumption of a Nernst diffusion layer at the electrode surface, and a first order dependence between the charge transfer current and the local concentration of a reacting species, resulting in the following kinetics expression:

where iexpr (A/m2) is the current density expression in the absence of mass transport limitations for the species, and ilim (A/m2) 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.

Consider a concentration-based kinetic expression

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.