Has DFN model in PyBaMM automatically modified the effective diffusivity and conductivity by Bruggeman Eq or not?

[XuboGU]

Hello!
I reduce the porosity of the separator, positive electrode, and negative electrode by up to 10% （use Mohtat2020.py parameter list) Theoretically，this change of porosity will adjust the Electrolyte diffusivity, Electrolyte conductivity [S.m-1]', Positive electrode diffusivity [m2.s-1], and Negative electrode diffusivity by the Bruggeman Equation. Since I didn't see the Bruggeman modification in the Mohtat2020.py, thus I'm wondering if the Bruggeman Equation will be modified in the dfn class or somewhere else or not?

I found the voltage curves did not change by a reduction of 10% of the porosity. The results are as follows:

param_umich_0 = get_parameter_values()
param_umich_0 = pybamm.ParameterValues(param_umich_0)

input_dict = {
    'Separator porosity': 0.38,
    'Negative electrode porosity': 0.3,
    'Positive electrode porosity': 0.3,
}          

sols = []
for discount in [0, 5, 10]:
    
    model2 = pybamm.lithium_ion.DFN()

    capacity = param_umich_0["Nominal cell capacity [A.h]"]
    param_umich_0.update({
        "Current function [A]": capacity * pybamm.InputParameter("C-rate")
    })

    # update param
    for key, value in input_dict.items():
        param_umich_0[key] = input_dict[key] * (1 - discount/100)
        print(param_umich_0[key])

    # update the mesh
    var = pybamm.standard_spatial_vars
    var_pts = {
    var.x_n: 10,
    var.x_s: 10,
    var.x_p: 10,
    var.r_n: 5,
    var.r_p: 5,
    }
    # define the simulation
    sim = pybamm.Simulation(
            model2,
            var_pts=var_pts,
            parameter_values=param_umich_0,
            # solver=pybamm.CasadiSolver(mode="fast")
        )

    sim.build(initial_soc=1) # start from 100% soc

    # fast solver
    solver=pybamm.CasadiSolver(max_step_decrease_count=5, 
                                rtol=1e-3,atol=1e-3,root_tol=1e-3,
                                return_solution_if_failed_early=True)
    # sim_CCCV = pybamm.Simulation(model2, experiment=experiment, parameter_values=param, solver=solver, var_pts=var_pts)

    Crate = 2  # 2C
    sol_ = sim.solve(t_eval=[0, 3600/Crate], inputs={"C-rate": Crate}, solver=solver)   # 
    ## collect the output
    sols.append(sol_)

for sol_ in sols:
    t_steps2 = sol_['Time [h]'].entries
    terminal_V2 = sol_["Terminal voltage [V]"].entries
    plt.plot(t_steps2, terminal_V2)
plt.xlabel('Time [h]')
plt.ylabel('Terminal voltage [V]')
plt.legend(['0%', '5%', '10%'])

Then I manually adjust the parameters by the Bruggeman Eq.:

    # modify by Bruggeman Eqution 
    param_umich_0.update({
        'Electrolyte diffusivity [m2.s-1]': 5.35 * 10 ** (-10) * param_umich_0['Separator porosity']**1.5,
        'Electrolyte conductivity [S.m-1]': 1.3 * param_umich_0['Separator porosity']**1.5,
        "Positive electrode diffusivity [m2.s-1]": 8 * 10 ** (-15) * param_umich_0['Positive electrode porosity']**1.5,
        "Negative electrode diffusivity [m2.s-1]": 5.0 * 10 ** (-15) * param_umich_0['Negative electrode porosity']**1.5,
    }) 

Thus, I would like to check if the effective diffusivity and conductivity are adjusted in DFN? Or maybe this 10% porosity change will not affect the terminal voltage?

In fact, I also do a lifetime simulation adding various sub-models, and the degradation curves are the same if I I did not manually adjust the parameters by Bruggeman Eq.

[TomTranter]

The issue I think is with the diffusivity and conductivity models in the Mohtat2020 parameter set which are all defined as constants e.g electrolyte_diffusivity_PeymanMPM :

def electrolyte_diffusivity_PeymanMPM(c_e, T):
    """
    Diffusivity of LiPF6 in EC:DMC as a function of ion concentration. The original data
    is from [1]. The fit from Dualfoil [2].

    References
    ----------
    .. [1] C Capiglia et al. 7Li and 19F diffusion coefficients and thermal
    properties of non-aqueous electrolyte solutions for rechargeable lithium batteries.
    Journal of power sources 81 (1999): 859-862.
    .. [2] http://www.cchem.berkeley.edu/jsngrp/fortran.html

    Parameters
    ----------
    c_e: :class:`pybamm.Symbol`
        Dimensional electrolyte concentration
    T: :class:`pybamm.Symbol`
        Dimensional temperature


    Returns
    -------
    :class:`pybamm.Symbol`
        Electrolyte diffusivity
    """

    D_c_e = 5.35 * 10 ** (-10)
    E_D_e = 37040
    arrhenius = np.exp(E_D_e / pybamm.constants.R * (1 / 298.15 - 1 / T))

    return D_c_e * arrhenius

I changed them to be same as Chen2020 and upped the "discount" you define and got some differences.

import pybamm
import matplotlib.pyplot as plt


def electrolyte_diffusivity_Nyman2008(c_e, T):
    """
    Diffusivity of LiPF6 in EC:EMC (3:7) as a function of ion concentration. The data
    comes from [1]

    References
    ----------
    .. [1] A. Nyman, M. Behm, and G. Lindbergh, "Electrochemical characterisation and
    modelling of the mass transport phenomena in LiPF6-EC-EMC electrolyte,"
    Electrochim. Acta, vol. 53, no. 22, pp. 6356–6365, 2008.

    Parameters
    ----------
    c_e: :class:`pybamm.Symbol`
        Dimensional electrolyte concentration
    T: :class:`pybamm.Symbol`
        Dimensional temperature

    Returns
    -------
    :class:`pybamm.Symbol`
        Solid diffusivity
    """

    D_c_e = 8.794e-11 * (c_e / 1000) ** 2 - 3.972e-10 * (c_e / 1000) + 4.862e-10

    # Nyman et al. (2008) does not provide temperature dependence

    return D_c_e


def electrolyte_conductivity_Nyman2008(c_e, T):
    """
    Conductivity of LiPF6 in EC:EMC (3:7) as a function of ion concentration. The data
    comes from [1].

    References
    ----------
    .. [1] A. Nyman, M. Behm, and G. Lindbergh, "Electrochemical characterisation and
    modelling of the mass transport phenomena in LiPF6-EC-EMC electrolyte,"
    Electrochim. Acta, vol. 53, no. 22, pp. 6356–6365, 2008.

    Parameters
    ----------
    c_e: :class:`pybamm.Symbol`
        Dimensional electrolyte concentration
    T: :class:`pybamm.Symbol`
        Dimensional temperature

    Returns
    -------
    :class:`pybamm.Symbol`
        Solid diffusivity
    """

    sigma_e = (
        0.1297 * (c_e / 1000) ** 3 - 2.51 * (c_e / 1000) ** 1.5 + 3.329 * (c_e / 1000)
    )

    # Nyman et al. (2008) does not provide temperature dependence

    return sigma_e


param_umich_0 = pybamm.ParameterValues("Mohtat2020")
param_umich_0.update({
    "Electrolyte diffusivity [m2.s-1]": electrolyte_diffusivity_Nyman2008,
    "Electrolyte conductivity [S.m-1]": electrolyte_conductivity_Nyman2008,
    "Negative electrode diffusivity [m2.s-1]": 3.3e-14,
    "Positive electrode diffusivity [m2.s-1]": 4e-15,
    })
input_dict = {
    'Separator porosity': 0.38,
    'Negative electrode porosity': 0.3,
    'Positive electrode porosity': 0.3,
}          

sols = []
for discount in [0, 20, 30]:
    
    model2 = pybamm.lithium_ion.DFN({
    "particle": "Fickian diffusion",
    })

    capacity = param_umich_0["Nominal cell capacity [A.h]"]
    param_umich_0.update({
        "Current function [A]": capacity * pybamm.InputParameter("C-rate")
    })

    # update param
    for key, value in input_dict.items():
        param_umich_0[key] = input_dict[key] * (1 - discount/100)
        print(param_umich_0[key])

    # update the mesh
    var = pybamm.standard_spatial_vars
    var_pts = {
    var.x_n: 10,
    var.x_s: 10,
    var.x_p: 10,
    var.r_n: 5,
    var.r_p: 5,
    }
    # define the simulation
    sim = pybamm.Simulation(
            model2,
            var_pts=var_pts,
            parameter_values=param_umich_0,
            # solver=pybamm.CasadiSolver(mode="fast")
        )

    sim.build(initial_soc=1) # start from 100% soc

    # fast solver
    solver=pybamm.CasadiSolver(max_step_decrease_count=5, 
                                rtol=1e-3,atol=1e-3,root_tol=1e-3,
                                return_solution_if_failed_early=True)
    # sim_CCCV = pybamm.Simulation(model2, experiment=experiment, parameter_values=param, solver=solver, var_pts=var_pts)

    Crate = 2  # 2C
    sol_ = sim.solve(t_eval=[0, 3600/Crate], inputs={"C-rate": Crate}, solver=solver)   # 
    ## collect the output
    sols.append(sol_)

for sol_ in sols:
    t_steps2 = sol_['Time [h]'].entries
    terminal_V2 = sol_["Terminal voltage [V]"].entries
    plt.plot(t_steps2, terminal_V2)
    
plt.xlabel('Time [h]')
plt.ylabel('Terminal voltage [V]')
plt.legend(['0%', '20%', '30%'])

[XuboGU]

Thanks @TomTranter for your kind reply :)
Indeed, it seems like the concentration-independent diffusivity and conductivity models in the Mohtat2020 reduce the effect of porosity change.
Then as for the Bruggeman Eq., is it included in the DFN class (or somewhere else) to adjust the diffusivity and conductivity? If so, where I may find it? Thanks ahead for your further help :)

[TomTranter]

Have a look through the examples. If you open a notebook and load a model you can latexify the equations

[XuboGU]

Awesome function :) Checked and included. Thanks again, Tom :)