How to use custom Li plating kinetics in half-cell

[jijn]

I am using the PyBaMM half-cell model and need to implement a custom kinetic expression for lithium plating. I am unable to reformulate my desired kinetic form to match the expression in Xu2019 paper (Eq. 13, I believe this is the expressions used in PyBaMM?).

The Desired Kinetic Form

I want to use the following form for the plating current density, $j$:

$$j = 2 j_0 \sinh \frac{F}{RT} \frac{\eta}{2}$$

where the overpotential, $\eta$, is defined as:

$$\eta = \phi_s - \phi_e - E - \frac{RT}{F} \ln \frac{c_e}{c_s}$$

Definitions of Variables

• $\phi_s$ : Electrostatic potential of the negative electrode (Li)
• $\phi_e$ : Electrolyte potential at the Li-electrolyte interface
• $E$ : Reference potential
• $c_e$ : Electrolyte concentration at the interface
• $c_s$ : Electrode surface concentration (constant in my case)
• $j_0$ : Exchange current density (a constant in my case)

My Question

I can modify the exchange current density. But I don't know where can I modify the form of the kinetics. Where in PyBaMM should I implement this custom kinetic form?

Example Code

This is the code I have for now.

import pybamm

model = pybamm.lithium_ion.DFN(options = {"working electrode" : "positive"})
param = pybamm.ParameterValues("Xu2019")
param.update({
    "Negative electrode OCP [V]": 0.0,
    'Negative electrode charge transfer coefficient': 0.5,
    "Exchange-current density for lithium metal electrode [A.m-2]": 30,
})
Iapp = 1.0*10*param['Electrode height [m]']*param['Electrode width [m]'] # 1 mA/cm^2
exp = pybamm.Experiment(
    [f"Discharge at {Iapp}A until 3.5 V", f"Charge at {Iapp}A until 4.2 V"]
)
sim = pybamm.Simulation(
    model,
    parameter_values=param,
    experiment=exp,
)
sim.solve(
    solver = pybamm.CasadiSolver(mode="safe"),
)
sol = sim.solution

I would appreciate any help and recommendations to achieve this custom kinetic form.