Ion Pairing & Ionicity in Ionic Liquids (2D)

Ionic Liquid Controls

Temperature 330 K

Ion concentration 40 / species

Ion size / charge density 1.00×

Live Readouts

Cation (free) Anion (free) Ion currently paired

Ionicity (free fraction)

1.00

Paired ions

Capture radius

–

Pair lifetime

–

How it works

Every ion undergoes overdamped Langevin (Brownian) dynamics in 2D — soft-core Coulomb forces between all ion pairs drive attraction/repulsion, thermal noise drives diffusion:

dx = (F/γ)·dt + √(2·D_T·dt)·N(0,1)
D_T = k_BT/γ,   γ(T) = γ₀·exp[Ea(1/T − 1/T_ref)]

On top of the continuous forces, each ion carries an explicit bound/free state, tracked exactly like a real ion-pair lifetime measurement:

capture radius  r_cap(T)  = r0 · (T_ref/T)      [grows as T falls, Bjerrum-length-style]
pair lifetime   τ(T)      = τ0 · exp[Ea'(1/T − 1/T_ref)]   [grows as T falls]
release prob. per step    = 1 − exp(−dt/τ(T))
ionicity = free_ion_fraction = (free cations + free anions) / (total ions)

Temperature — controls both the capture radius (how close two opposite ions must drift before Coulomb attraction locks them into a pair) and the pair lifetime (how long a bound pair survives thermal jostling before separating again). Lower T means a larger capture radius and a longer lifetime, so more of the population sits paired at any instant.
Ion size / charge density — scales the capture radius directly, standing in for how compact/charge-dense the real cation and anion are (smaller, more charge-dense ions pair more readily, exactly as imidazolium vs. larger, delocalized-charge ions behave differently on a real Walden plot).
Ion concentration — more ions in the same box shortens average spacing, raising the encounter rate and hence the paired fraction, independent of temperature.

The bottom strip re-runs the same pairing kinetics as small independent ensembles held at fixed temperatures across the slider's range, and plots each ensemble's steady-state ionicity — a live, simulated Walden-style ionicity-vs-temperature curve built from the actual particle populations, not a formula plugged in after the fact.

Ionicity vs. Temperature (live, simulated)