Clarify definition of PyBaMM's SPMe

[brosaplanella]

The definition of the SPMe remains a sticking point in PyBaMM. Even though the cited article for its definition is the Marquis et al (2019) paper, the actual implementation is a bit different (more accurate) as some terms have been "unlinearised"*. As part of improving the Creating Models notebooks (#3844) I wanted to write a BasicSPMe file and document the SPMe definition better. However, I have also noticed that the current approach falls somewhat in the middle: some terms are unlinearised and some are not. The marginal cost of doing the SPMe fully "unlinearised". All this discussion pertains to composite_conductivity.py

*by "unlinearised" I mean that some terms in the asymptotic expansion get later grouped again in their nonlinear form.

Current implementation

$$\phi_\mathrm{e,const} = - \overline{\Delta \phi}_\mathrm{n} + \overline{\phi}_\mathrm{n} - \left[ \chi(\overline{c}_\mathrm{e}) \frac{R T}{F} \frac{1}{L_\mathrm{n}} \int_0^{L_\mathrm{n}} \log \left(\frac{c_\mathrm{e,n}}{\overline c_\mathrm{e}} \right) \mathrm{d} x - i_\mathrm{app} L_\mathrm{n} \left(\frac{1}{3 \kappa_\mathrm{e,eff} (\overline{c}_\mathrm{e,n})} - \frac{1}{\kappa_\mathrm{e,eff} (\overline{c}_\mathrm{e,s})} \right)\right]$$ $$\phi_\mathrm{e,n} = \phi_\mathrm{e,const} + \chi(\overline{c}_\mathrm{e,n}) \frac{R T}{F} \log \left( \frac{c_\mathrm{e,n}}{\overline{c}_\mathrm{e}} \right) - \frac{i_\mathrm{app}}{\kappa_\mathrm{e,eff} (\overline{c}_\mathrm{e,n})} \frac{x_\mathrm{n}^2 - L_\mathrm{n}^2}{2 L_\mathrm{n}} - \frac{i_\mathrm{app} L_\mathrm{n}}{\kappa_\mathrm{e,eff} (\overline{c}_\mathrm{e,s})}$$ $$\phi_\mathrm{e,s} = \phi_\mathrm{e,const} + \chi(\overline{c}_\mathrm{e,s}) \frac{R T}{F} \log \left( \frac{c_\mathrm{e,s}}{\overline{c}_\mathrm{e}} \right) - \frac{i_\mathrm{app} x_\mathrm{s}}{\kappa_\mathrm{e,eff} (\overline{c}_\mathrm{e,s})}$$ $$\phi_\mathrm{e,p} = \phi_\mathrm{e,const} + \chi(\overline{c}_\mathrm{e,p}) \frac{R T}{F} \log \left( \frac{c_\mathrm{e,p}}{\overline{c}_\mathrm{e}} \right) - \frac{i_\mathrm{app} }{\kappa_\mathrm{e,eff} (\overline{c}_\mathrm{e,p})} \frac{x_\mathrm{p} (2 L_x - x_\mathrm{p}) + L_\mathrm{p}^2 - L_x^2}{2 L_\mathrm{p}}$$

Proposed implementation

$$\phi_\mathrm{e,const} = - \overline{\Delta \phi}_\mathrm{n} + \overline{\phi}_\mathrm{n} - \frac{R T}{F} \frac{1}{L_\mathrm{n}} \int_0^{L_\mathrm{n}} \int_0^x \chi(c_\mathrm{e,n}) \frac{\partial \log c_\mathrm{e,n}}{\partial s} \mathrm{d} s \mathrm{d} x + \frac{1}{L_\mathrm{n}} \int_0^{L_\mathrm{n}} \int_0^x \frac{i_\mathrm{e,n}}{\kappa_\mathrm{e,eff}(c_\mathrm{e,n})} \mathrm{d} s \mathrm{d} x$$ $$\phi_\mathrm{e,n} = \phi_\mathrm{e,const} + \frac{R T}{F} \int_0^x \chi(c_\mathrm{e,n}) \frac{\partial \log c_\mathrm{e,n}}{\partial s} \mathrm{d} s - \int_0^x \frac{i_\mathrm{e,n}}{\kappa_\mathrm{e,eff}(c_\mathrm{e,n})} \mathrm{d} s$$ $$\phi_\mathrm{e,s} = \phi_\mathrm{e,n} ( x = L_\mathrm{n}) + \frac{R T}{F} \int_{L_\mathrm{n}}^x \chi(c_\mathrm{e,s}) \frac{\partial \log c_\mathrm{e,s}}{\partial s} \mathrm{d} s - \int_{L_\mathrm{n}}^x \frac{i_\mathrm{e,s}}{\kappa_\mathrm{e,eff}(c_\mathrm{e,s})} \mathrm{d} s$$ $$\phi_\mathrm{e,p} = \phi_\mathrm{e,s} ( x = L_x - L_\mathrm{p}) + \frac{R T}{F} \int_{L_x - L_\mathrm{p}}^x \chi(c_\mathrm{e,p}) \frac{\partial \log c_\mathrm{e,p}}{\partial s} \mathrm{d} s - \int_{L_x - L_\mathrm{p}}^x \frac{i_\mathrm{e,p}}{\kappa_\mathrm{e,eff}(c_\mathrm{e,p})} \mathrm{d} s$$

I suggest moving to this version, which I believe is the most general one (I have dropped T dependence from parameters for simplicity, but will be accounted for). The version is taken from Brosa Planella & Widanage (2022), but the derivation is not explained with a lot of detail as the focus was the side reaction. My older paper Brosa Planella et al (2021) has a lot more detail but still assumes $\chi$ to be constant. This would make the integrated conductivity model redundant so we can get rid of it.

[valentinsulzer]

Sounds good to me

[rtimms]

Me too, thanks!