This is a genuine 2D kinetic Monte Carlo (KMC) lattice-gas simulation, not a flattened view of a 3D chain bundle. Every coordination site sits on a fixed grid: columns are positions along the amorphous polymer backbone (periodic, unbounded — the real bulk conduction direction), rows are which of a small stack of chains a site belongs to, and a rigid band of rows through the middle is crystalline. Li⁺ ions occupy at most one site each (hard-core exclusion, like real ions that cannot overlap) and hop stochastically to an empty neighbor with a rate gated by the same segmental-relaxation physics as the closed-form theory:
Γ_success(T,ratio,flex,site) = attemptRate(T,flex) · hopProb(site,T,ratio)
hopProb ∝ amorphFactor(site) · pairing(ratio) · exp[ B/(T-T₀ref) − B/(T-T₀) ]
Then, independently of that rate, the engine MEASURES:
D = ⟨Δx²⟩ / (2t) (Einstein relation, along-chain axis)
σ_meas = n·q²·D / (k_B·T) (Nernst–Einstein relation)
...and compares σ_meas against the closed-form VTF law used in the original model:
σ_VTF(T) = A·φ_am·f(ratio)·T^(−1/2)·exp[ −B/(T−T₀) ]
These are two independently computed numbers: the VTF line is a closed-form formula evaluated directly from the sliders, while the measured value comes from actually tracking every ion's real lattice-hop trajectory and applying the Einstein/Nernst–Einstein relations used throughout diffusion physics and molecular-dynamics literature. They will not match exactly — hard-core exclusion between ions ("traffic jamming") measurably slows real tracer diffusion below the naive single-particle estimate, an effect the closed-form formula cannot see — but both should move in the same direction under every control.
- Temperature — raises both the attempt rate and the VTF-gated success probability, so measured D and predicted σ should rise together.
- EO:Li ratio — sets the ion count (more salt near the 18:1 optimum, then ion-pairing suppresses mobile carriers on both sides of it) and changes the pairing factor in the hop probability.
- Crystallinity — widens the rigid row-band; ions already inside can rarely leave (and vice versa), collapsing the measured hop rate the same way the amorphous-fraction term collapses σ_VTF.
- Chain flexibility — scales the attempt-rate prefactor for both computations equally.
Real-world relevance: this same divide — a clean analytical transport law versus a particle-based simulation that captures crowding/correlation effects — is exactly why battery researchers run both molecular-dynamics/KMC studies and fit VTF curves to their conductivity data, and why real polymer electrolytes rarely match a fitted VTF line perfectly once ion concentration gets high enough for real ion-ion interactions to matter.