Fast Charging Without Lithium Plating: SPMe Safe C-Rate

The full canonical SPMe, Marquis et al. (2019), dimensional Eq. (48), attached pp. 32–33, is solved with the tuned LG M50 parameters from Chen et al. (2020), Table IX, attached p. 20, as implemented in PyBaMM 25.10.2 Chen2020. The measured Table VII diffusivities and concentrations are not substituted for that tuned set.

For each particle, $\partial_t c_s=r^{-2}\partial_r(r^2D_s\partial_r c_s)$, with zero centre flux and prescribed Faradaic surface flux. Across the three cell regions, $\epsilon\partial_t c_e=\partial_x(\epsilon^{1.5}D_{e,0}\partial_xc_e)+(1-t^+)a j/F$, with concentration/flux continuity and blocked collector boundaries. Charge is positive in the user interface; the paper's discharge-positive current is its negative.

Spherical shell volumes and harmonic interface diffusion resistances form conservative finite-volume operators: 28 radial cells per particle and 18/6/18 through-cell cells. A mass-weighted symmetric eigendecomposition retains every discrete diffusion mode. Each constant-current segment is propagated by the exact matrix exponential, including the conserved zero mode. This solves the complete spatially discretized transient equations without a particle-profile approximation or a steady electrolyte substitution.

The exchange density is $j_0=m(T)\sqrt{c_s(c_{s,\max}-c_s)}\langle\sqrt{c_e}\rangle$; symmetric Butler–Volmer is inverted analytically as $\eta=2RT/F\,\mathrm{asinh}(j/(2j_0))$. Chen's 35.0 and 17.8 kJ/mol charge-transfer activation energies supply $m(T)=m_{ref}\exp[(E_a/R)(1/T_{ref}-1/T)]$ (attached p. 17). The canonical electrolyte diffusivity and conductivity are evaluated at 1000 mol/m³, as required by Eq. (48).

The electrode-averaged voltage includes reaction, electrolyte concentration, electrolyte Ohmic and solid Ohmic terms. Local anode solid and electrolyte potentials are reconstructed at every anode cell centre and both boundaries; the minimum of $\phi_s-\phi_e$, rather than the averaged anode potential or the arbitrarily grounded collector potential, is compared with 0 V versus Li/Li⁺ and the entered margin.

A 12-iteration bisection tests 121 nonuniform time samples over each candidate segment. Final CC curves use 221 samples and each of four stepped bands uses 161. The stepped search carries both particle fields and the electrolyte field between bands and prevents current increases.

• A fresh, initially relaxed LG M50 cell with uniform particle concentrations and 1000 mol/m³ electrolyte; fixed, spatially uniform cell temperature.
• Nominal SOC counts 5 Ah from the tuned full-charge concentrations (29866 and 17038 mol/m³). The resulting empty-end stoichiometries follow lithium conservation on this capacity scale; they are not the measured Table VII endpoints.
• Constant tuned solid diffusivities, fixed particle radii/porosities, ideal electrolyte activity, transference number 0.2594 and Bruggeman exponent 1.5. The anode's graphite–SiOₓ mixture is represented by one effective particle, as in this parameter set.
• Temperature changes charge-transfer kinetics and thermal voltage. Solid diffusion and electrolyte transport retain the 25 °C reference values because this source set does not supply their temperature activation energies.
• No side reactions, SEI film resistance, hysteresis, thermal feedback, mechanical stress, loss of lithium or loss of active material.

“Feasible” means the resolved SPMe trajectory stays above the requested local anode potential margin, below the entered terminal voltage, within particle filling 0.0001–0.9999 and within electrolyte concentration 50–2500 mol/m³. These concentration guards are computational operating bounds chosen here, not experimentally established validity limits. Rates are searched from 0.005C to the entered ceiling; the bracket width is at most (ceiling − 0.005)/4096 C. No result claims a maximum above that ceiling.

The canonical SPMe is an asymptotic porous-electrode model. Large electrolyte deviations from 1000 mol/m³, steep OCP regions, high currents, or strongly nonuniform reactions reduce its accuracy. Its linearized electrolyte potential and reference transport coefficients are intrinsic to the cited canonical model. Temperature extrapolation below Chen's 25–60 °C kinetic measurements is particularly uncertain, and the omitted temperature dependence of transport can substantially overestimate cold-cell capability. A positive potential margin does not account for parameter uncertainty, local defects or plating nucleation.

The finite grid and finite event samples can miss thin initial diffusion layers or brief potential dips. All finite-volume modes are retained; only spatial and time-sampling resolution is finite. Particle surface filling uses quadratic extrapolation. The radial picture uses PCHIP through the solved radial cells and the electrolyte picture uses PCHIP between the 42 solved through-cell points; animation evaluates the stored segment state at the selected time, rather than replaying the charge search.

Independent 60-cell-per-particle, 60/20/60 electrolyte PyBaMM benchmarks configure the canonical Eq. (48) transport and averaged-exchange kinetics explicitly; they are not unmodified PyBaMM composite-SPMe runs. These checks verify the implementation, not real-cell fast-charge accuracy. The four-band profile is sequentially feasible and need not be globally time optimal. Check actual cell temperature, aging, cooling and manufacturer charge limits separately. The trial trace intentionally continues past the voltage threshold to diagnose the proposed CC window; discard its predictions after a concentration guard is crossed.