Surface form potential when using double layer capacitance

[Yanisiqi]

Hello everyone:
When I want to include double layer capacitance in DFN model, pybamm guildeline tells me to set surface potential option to be differential: model = pybamm.lithium_ion.DFN(options={'surface form': 'differential'}).
Then I notice that electrolyte current density i_e is calculated in the following way (full_surface_form_conductivity.py):
i_e = conductivity * (
param.chiRT_over_Fc(c_e, T) * pybamm.grad(c_e)
+ pybamm.grad(delta_phi)
+ i_boundary_cc / sigma_eff
)
However, in normal DFN model, i_e is calculated in this way:
i_e = (param.kappa_e(c_e, T) * tor) * (
param.chiRT_over_Fc(c_e, T) * pybamm.grad(c_e) - pybamm.grad(phi_e)
)
These two equations are quite different. I wonder why double-layer capacitance modelling need to use a different i_e expression compared with normal DFN model? Can someone give some explanations or relevant papers? Thank you.

Best regards

[valentinsulzer]

We have to rearrange in order to get a single equation in terms of delta_phi (which is phi_s - phi_e). We can do this by combining the equations for i_e and i_s using the fact that i_e + i_s = I. See equations 47 and 48a in https://iopscience.iop.org/article/10.1149/2.0301910jes/pdf

[Yanisiqi]

Thanks