Electrochemical Parameters

The parameters of each charge-transfer reaction can be provided through echemfem.EchemSolver.echem_params, a list containing one dictionary for each charge-transfer reaction. Below is a list of different keys that can appear in each dictionary

• Key:

  "reaction"

  Type:

  a function

  Description:

  the current density from a charge-transfer reaction

  Args:

  u: solution state. The value of the different concentrations can be recovered through u([self.i_c["species name"]) within a echemfem.EchemSolver object.

  Returns:

  Current density \(i_k\).

• Key:

  "electrons"

  Type:

  float, int, firedrake expression

  Description:

  the number of electrons \(n_k\) being transferred in the reaction.

• Key:

  "stoichiometry"

  Type:

  dict

  Description:

  This entry defines the stoichiometry of the reaction. Each key in this dictionary should be the name of a species (as defined in the "name" key values of conc_params). The corresponding value is the stoichiometry coefficient \(s_{j,k}\) for the species in this reaction. It should be an integer: negative for a reactant, positive for a product.

• Key:

  "boundary"

  Type:

  str

  Description:

  In the non-porous case, this entry provides the name of the boundary where the charge-transfer reaction happens. This should correspond to a key in the dictionary self.boundary_markers containing the Boundary Conditions.

Here is an example of a reaction that can be implement via echem_params. CO reduction for ethanol:

\[2\mathrm{CO} + 7 \mathrm{H}_2 \mathrm{O} + 8 \mathrm{e}^- \rightarrow \mathrm{CH}_3\mathrm{CH}_2\mathrm{OH}_{(\mathrm{aq})} + 8 \mathrm{OH}^-\]

Here, we assume the dilute solution case where \(\mathrm{H}_2\mathrm{O}\) concentration is not tracked. In conc_params, we will have entries for the other species in this reactions, with "name" key values: "CO", "OH", and "C2H6O". Assuming we also defined a function reaction_C2H6O for the current density, we get the following echem_params entry:

{"reaction": reaction_C2H6O,
 "electrons": 8,
 "stoichiometry": {"CO": -2,
                   "C2H6O": 1,
                   "OH": 8},
 }

In the non-porous case, the reactions are surface reactions, which result in the following boundary conditions for the ionic current \(\mathbf{i}_2\), and mass flux \(\mathbf{N}_j\) of species \(j\)

\[ \begin{align}\begin{aligned}\mathbf{i}_2 \cdot \mathbf{n} = -\sum_k i_k,\\\mathbf{N}_j \cdot \mathbf n = -\sum_k \frac{s_{j,k} i_k}{n_k F},\end{aligned}\end{align} \]

where \(k\) represent each charge-transfer reaction in the system, and \(F\) is the Faraday constant, which must be passed through Physical Parameters.

In the porous case, i.e. when "porous" is passed through "flow" in Physical Parameters, the reactions are volumetric reactions, which are added to the right-hand sides of the equations. In the absence of homogeneous bulk reactions, we have

\[ \begin{align}\begin{aligned}\nabla \cdot \mathbf{i}_1 = -\nabla \cdot \mathbf{i}_2 = - a_v \sum_k i_k,\\\nabla \cdot \mathbf{N}_j = a_v \sum_k \frac{s_{j,k} i_k}{n_k F},\end{aligned}\end{align} \]

where \(a_v\) is passed as "specific surface area" through Physical Parameters.