3.4.2 Activation Overpotential

The general electrochemical reaction is, according to [ 6],

where $a_j$ is the stoichiometric coefficient of species $j$, $A_j$ is the chemical species, and $n$ is the number of electrons.

The reaction rate is:

where $\phi$ is the voltage, $k_a$ and $p_i$ are the rate constant and the reaction order for the anodic direction, $k_c$ and $q_i$ are the rate constant and the reaction order for the cathodic direction, $\alpha_{\rm a}$ is the anodic transfer coefficient, $\alpha_{\rm c}$ is the cathodic transfer coefficient, and $n$ is the number of electrons that are released. At equilibrium, the forward and the backward reaction rates are the same, therefore:

where $i_0$ is the exchange current density.

The reaction rate (i.e., current) can be written in terms of the exchange current density $i_0$ to obtain the Butler-Volmer formulation [ 6]:

The activation overpotential is the energy lost due to the slowness of electrochemical reactions at the anode and the cathode electrodes.

Using this relation, the Butler-Volmer equation can be written as:

where

with $i_{\rm0, ref}$ being the exchange current density at the reference condition, $(\chi_j)$ is the mole fraction and $\gamma_{j}$ is the concentration exponent for species $j$. More specifically, at the anode side, you have:

Likewise, at the cathode side, you have:

Given values for $\alpha_{\rm a}$ and $\alpha_{\rm c}$. the full version of the Butler-Volmer equation can be solved using the Newton method, therefore finding the activation overpotential at the anode ( $\eta_{\rm act, a}$) and the cathode ( $\eta_{\rm act, c}$).