Physical Parameters

The general physical parameters need to be provided in echemfem.EchemSolver.physical_params, a dictionary.

The only key required in all cases is the following one:

Key:

"flow"

Type:

list of str

Description:

Each str represents a transport mechanism. Below is a list of tested options and their parameter requirements, and, if relevant, the added contribution to the flux of species \(k\), \(\mathbf N_k\).

  • "diffusion": Fickian diffusion

    • conc_params: "diffusion coefficient" defines \(D_k\)

    \[\mathbf N_k \mathrel{+}= -D_k \nabla c_k\]

  • "advection": Advection

    \[\mathbf N_k \mathrel{+}= c_k \mathbf u\]

  • "migration": Electromigration (where the mobility constant is given by the Nernst-Einstein relation)

    • conc_params: "diffusion coefficient", "z"

    • physical_params: "F", "R", "T"

    \[\mathbf N_k \mathrel{+}= -z_k F\frac{D_k}{RT} c_k \nabla \Phi_2.\]

  • "electroneutrality": Electroneutrality approximation. Implemented through a charge-conservation equation as described in Roy et al, 2022. This eliminates a concentration from the solution variables. By default, the last concentration in conc_params is eliminated. It can also be specified by passing "eliminated": True in a species’ conc_params entry.

    \[\sum_k z_k c_k = 0.\]

  • "electroneutrality full": Electroneutrality approximation, but implemented explicitly.

  • "porous": For flow in a porous media. Diffusion coefficients are replaced with the effective diffusion coefficients using the Bruggeman correlation. A Poisson equation is solved for the electronic potential \(\Phi_1\). If using Electrochemical Parameters, the reaction is volumetric and the provided reaction is multiplied by the specific surface area, \(a_v\).

    • physical_params:

      • "solid conductivity": defines the electronic conductivity in the bulk of the solid, denoted \(\sigma\)

      • "saturation": defines \(S\), the liquid saturation. By default \(S=1\).

      • "porosity": defines \(\epsilon\)

      • "specific surface area": defines \(a_v\).

    \[ \begin{align}\begin{aligned}D_k^\mathrm{eff} = (\epsilon S)^{1.5} D_k,\\-\nabla \cdot \left((1-\epsilon)^{1.5} \sigma \nabla \Phi_1 \right) = -a_v \sum_j i_j.\end{aligned}\end{align} \]

  • "poisson": Poisson equation for the ionic potential.

    • physical_params:

      • "vacuum permittivity": \(\epsilon_0\)

      • "relative permittivity": \(\epsilon_r\)

    \[-\nabla \cdot \left( \epsilon_0 \epsilon_r \nabla \Phi_2 \right) = F\sum_k z_k c_k,\]

    neglacting reaction terms. If the vacuum and relative permittivity are not defined, the ionic conductivity is used instead \(\kappa = \sum_k z_k^2 F^2 \frac{D_k}{RT} c_k\).

  • "finite size": Finite-size effect for ion interaction. Choosing "poisson", "diffusion", "migration", and "finite size" gives the Generalized Poisson-Nernst-Planck model (GMPNP).

    • conc_params: "solvated diameter", \(a_k\) of species \(k\)

    • physical_params: "Avogadro constant", \(N_A\)

    \[\mathbf N_k \mathrel{+}= -D_k c_k \left(\frac{N_A \sum_j a_j^3 \nabla c_j}{1- N_A \sum_j a_j^3 c_j}\right)\]

Below is a list of other keys that can appear in the dictionary

  • Key:

    "bulk reaction"

    Type:

    a function

    Description:

    Homogeneous/bulk reactions to be added to the right-hand side of mass conservation equations. These can instead be set using homog_params (Homogeneous Reaction Parameters).

    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:

    List of length equal to the number of species, each entry being the reaction term of the corresponding species. The order of the reaction terms must be the same as the order of the species in conc_params, which can also be found through self.i_c.

  • Key:

    "F"

    Type:

    float, firedrake expression

    Description:

    Faraday Constant

  • Key:

    "R"

    Type:

    float, firedrake expression

    Description:

    Molar Gas Constant

  • Key:

    "T"

    Type:

    float, firedrake expression

    Description:

    Absolute Temperature

  • Key:

    "porosity"

    Type:

    float, firedrake expression

    Description:

    Void fraction of the porous medium. Used when "porous" is in physical_params["flow"].

  • Key:

    "solid conductivity"

    Type:

    float, firedrake expression

    Description:

    Electronic conductivity of the bulk solid material. Used when "porous" is in physical_params["flow"].

  • Key:

    "specific surface area"

    Type:

    float, firedrake expression

    Description:

    Specific surface area of the porous medium. Used when "porous" is in physical_params["flow"], if Electrochemical Parameters are used.

  • Key:

    "saturation"

    Type:

    float, firedrake expression

    Description:

    Ratio of the void fraction occupied by liquid. Default value of 1.0. Used when "porous" is in physical_params["flow"].

  • Key:

    "vacuum permittivity"

    Type:

    float, firedrake expression

    Description:

    Used when "poisson" is in physical_params["flow"].

  • Key:

    "relative permittivity"

    Type:

    float, firedrake expression

    Description:

    Used when "poisson" is in physical_params["flow"].

  • Key:

    "Avogadro constant"

    Type:

    float, firedrake expression

    Description:

    Used when "finite size" is in physical_params["flow"].

  • Key:

    "U_app"

    Type:

    float, firedrake expression

    Description:

    Applied potential. Used for "applied", "liquid applied", and "robin" Boundary Conditions.

  • Key:

    "gap capacitance"

    Type:

    float, firedrake expression

    Description:

    Used for "robin" Boundary Conditions.

  • Key:

    "surface charge density"

    Type:

    float, firedrake expression

    Description:

    Used for "poisson neumann" Boundary Conditions.