In a solid polymer electrolyte (e.g. LiX salt dissolved in poly(ethylene oxide), PEO), there is no free liquid solvent. A Li⁺ ion is coordinated by several ether-oxygen lone pairs on the chain and moves only when local segmental motion — the backbone wiggling above the glass transition — briefly opens a path to a neighboring coordination site, letting the ion hop and the chain "reptate" the vacancy shut behind it. Crystalline lamellae are far too rigid for this motion, so conduction is confined almost entirely to the amorphous fraction.
Because it is gated by segmental relaxation rather than a fixed activation barrier, the conductivity does not follow simple Arrhenius behavior — it follows the Vogel–Tammann–Fulcher (VTF) law used throughout the polymer-electrolyte literature:
σ(T) = A · φ_am · f(c) · T^(-1/2) · exp[ -B / (T - T₀) ]
φ_am = amorphous volume fraction (1 − crystallinity)
f(c) = mobile-carrier factor: rises with salt concentration c,
then falls as ion pairs / aggregates lock up carriers
T₀ = Vogel (ideal glass-transition) temperature, ≈ Tg − 50 K
B = pseudo-activation term for the free-volume hop
- Temperature — raises the segmental wiggle amplitude and hop-attempt rate (more free volume opens up more often).
- EO:Li ratio — sets how many Li⁺ ions are dissolved; too dilute means few carriers, too concentrated means ion pairing removes mobile carriers, giving a conductivity optimum at intermediate salt loading.
- Crystallinity — grows the rigid slab (center band) that ions cannot cross, cutting the amorphous pathway.
- Chain flexibility — a stand-in for plasticizer/comb-branch content that raises segmental mobility independent of temperature.
Real-world relevance: this mechanism is exactly why room-temperature PEO-based polymer electrolytes still lag liquid electrolytes in lithium-ion and lithium-metal batteries, and why battery makers add plasticizers, use comb-shaped polymers, or blend in ceramic fillers to boost the amorphous, mobile fraction.