PID Diffusion Front & Arrhenius Diagnostics (2D)

Accelerated Stress Test (IEC 62804)

System voltage bias -1000 V

Module temperature 60 °C

Relative humidity 85 %

Test speed ×12

Live Readouts

Elapsed stress time

0.0 h

Leakage current

0 µA

Front depth d₅₀

0.0 mm

Effective D_eff

— m²/s

Avg. shunt resistance R_sh

3000 Ω

Module power loss

0.0 %

How it works (2D method)

This is the 2D-native counterpart to the 3D PID module simulator. Instead of orbiting a cell grid, it simulates the ion migration itself: each sodium ion performs a free 1D random walk (Euler–Maruyama integration of Brownian motion, dW = √(2·D·dt)·N(0,1)) starting from the frame edge. By the reflection principle, the fraction of a large ensemble of such walkers whose absolute displacement exceeds a depth d at time t is exactly:

P(|W(t)| > d) = erfc( d / (2·√(D·t)) ) = C(d,t)/C0

D(T)   = D0 · exp(−Ea / (kB·T))     [Arrhenius, Ea = 0.8 eV]
D_eff  = D(T) · (|V|/Vref) · (RH/100)^1.5

— the identical closed-form drift-diffusion law used by the 3D simulator, but reached here by literally stepping a stochastic ensemble instead of placing ions from the formula directly. The upper strip shows the live particle swarm's depth positions; the concentration curve above it is the same ensemble's empirical survival profile, converging to the erfc curve as more steps accumulate.

The Arrhenius panel plots ln(D) against 1000/T(K) — the straight-line diagnostic used to extract activation energy from real PID accelerated-test data — with the current operating point marked, plus the separate voltage and humidity multipliers that scale the Arrhenius baseline up to D_eff.

The bottom strip plots the front depth d₅₀ (where concentration crosses half its boundary value) against √(elapsed hours). Fickian diffusion predicts this is exactly linear — the dashed reference line is the theoretical slope 0.954·√D for the current settings, so you can watch the live trace track it.

Voltage bias — scales D_eff directly; more negative accelerates every trace.
Temperature — moves the operating point along the Arrhenius line; every +10 °C roughly doubles D.
Humidity — raises both the humidity multiplier and the boundary ion concentration C0.
Diffusivity changes made mid-run are integrated correctly: the engine accumulates ∫D dt rather than multiplying the current D by total elapsed time.

Na⁺ ion (random walk) Concentration front C(d,t) Arrhenius / linearization traces