SEI growth gives a square-root law
The solid-electrolyte interphase (SEI) forms on graphite and consumes cyclable lithium. Growth is limited by electrons/solvent diffusing through the existing layer, so dδ/dt = D/δ, which integrates to δ ~ √t. Calendar loss is therefore Qcal = k(T,SOC)·√t: fast at first, then slowing down.
Arrhenius temperature and SOC
k = k0 exp[Ea/R (1/298 − 1/T)] exp[b (SOC − 0.5)], with Ea = 50 kJ/mol, so +10 °C roughly doubles the rate (check above). Storing at high charge also accelerates SEI growth.
Cycle ageing
Qcyc = B(T, C, DoD, SOC)·FECz with z = 0.75 (sub-linear, SEI-dominated). FEC = full equivalent cycles = Ah throughput / capacity. Conditions can change in time, so the model advances Q1/z by B1/zΔFEC, which is exact for piecewise-constant conditions.
Lithium plating
When the graphite potential falls below 0 V vs Li/Li+ (cold, fast, near-full charge), lithium metal deposits instead of intercalating: dQpl = kp P1.5 dFEC, with risk index P from temperature, C-rate and top SOC. Plating is partly irreversible and can form dendrites (the silver spikes in the particle view). Pre-heating before charging removes most of it.
Resistance and range
R/R0 = 1 + 3 Qcal + 2 Qcyc + 4 Qpl. Range = usable energy / consumption, reduced by resistive losses that grow with R. Reference: 75 kWh pack, 0.165 kWh/km at 20 °C.
Honest simplifications
This is a semi-empirical teaching model, not a cell-specific fit: the activation energies, SOC sensitivities, plating coefficient and resistance factors are illustrative values chosen to give typical orders of magnitude (about 20 % loss after 8 years at 25 °C / 1500 FEC). Real cells differ by chemistry (NMC, LFP, NCA), have a late “knee” from accelerating plating or particle cracking, and capacity may be partly recoverable. Components are added linearly. The Blender cutaway and SEI thickness are exaggerated for visibility.