BeveridgeHydrogen permeation through a metal membrane
Hydrogen enters one face of a thin metal sheet and is detected leaving the other. How fast it gets through depends on the lattice, on traps that hold it, and on grain boundaries that act as fast paths. Change the conditions and rerun the experiment; every curve is checked against its exact solution.
Experiment settings
Defaults: polycrystalline nickel, typical literature values.
| Model | Effective D (m²/s) | Time lag, exact | Time lag, simulated | Max curve error | Check |
|---|
Status. Each simulated curve is checked against the exact solution of the same equations (pass: largest difference below 1% of the steady flux). These curves have not yet been compared with the figures in the papers; that comparison is added once the papers' exact settings are entered.
What is being computed
1 · Lattice diffusion
Fick's second law across the membrane, ∂c/∂t = D ∂²c/∂x², with a fixed concentration on the entry face and zero on the exit face. Lattice diffusivity D_L = D₀ exp(−Q/RT); for nickel D₀ = 6.9×10⁻⁷ m²/s, Q = 39.5 kJ/mol.
2 · Trapping
Traps in local equilibrium with the lattice slow the hydrogen down: D_eff = D_L / (1 + (N_T/N_L) exp(E_b/RT)). Example: N_T/N_L = 10⁻⁵ and E_b = 30 kJ/mol at 298 K give a factor 1 + 1.8 ≈ 2.8 slower.
3 · Grain-boundary fast paths
A share f of the material is grain boundary, f = 3δ/d with boundary width δ = 0.5 nm and grain size d. The effective diffusivity mixes both paths: D = (1 − f) D_eff + f D_GB.
Flux leaving the membrane, as a share of its final value: J/J_ss = 1 + 2 Σ (−1)ⁿ exp(−n²π² D t / L²). Time lag (the classic measurement): t_lag = L² / 6D. Example: L = 0.5 mm and D = 1×10⁻¹³ m²/s give t_lag = (5×10⁻⁴)² / (6×10⁻¹³) ≈ 4.2×10⁵ s ≈ 116 h. The simulation solves the same equation step by step (finite differences, 120 points across the membrane) and must reproduce the exact curve.
Open the simulation code
Papers behind this experiment
Research by Dr. Sathiskumar JOTHI on hydrogen transport in polycrystalline metals. All publications.
| Meso-microstructural computational simulation of the hydrogen permeation test to calculate intergranular, grain boundary and effective diffusivities Journal of Alloys and Compounds, 2015 |
| Influence of grain boundary misorientation on hydrogen embrittlement in bi-crystal nickel International Journal of Hydrogen Energy, 2014 |
| Modelling the influence of microstructural morphology and triple junctions on hydrogen transport in nanopolycrystalline nickel Composites Part B: Engineering, 2015 |
| Multi-phase modelling of intergranular hydrogen segregation/trapping for hydrogen embrittlement International Journal of Hydrogen Energy, 2015 |