Multiphase modeling of pressure-dependent hydrogen diffusivity in fractal porous structures of acrylonitrile butadiene rubber-carbon black composites with different fillers

The hydrogen diffusivities of acrylonitrile butadiene rubber (NBR) composites with different types and contents of carbon black (CB) fillers were investigated in the exposure pressure range of 0.5–10 MPa using a volumetric analysis system. These measured diffusivities exhibited distinct pressure-dependent behavior. The diffusivities of unfilled NBR and NBR composites containing medium-thermal CB filler decreased from 39.7 $\times$ 10⁻¹¹ m²/s to 9.4 $\times$ 10⁻¹¹ m²/s as the pressure increased. On the other hand, the pressure-dependent diffusivities of NBR composites containing high-abrasion furnace, fast-extrusion furnace and semireinforcing furnace CB fillers revealed left-biased unimodal shaped curves, with peak values ranging from 19.7 $\times$ 10⁻¹¹ m²/s to 5.6 $\times$ 10⁻¹¹ m²/s. To model this observed behavior, the diffusion resistance theory with a heterogeneous NBR–CB composite and a fractal porous structure for H₂ was introduced. The theoretical parallel diffusion resistance model was found to coexist independently as the surface, Knudsen, and bulk diffusion phases. This theoretical multiphase modeling was applied to the measured diffusivities, determining diffusion resistance parameters for each phase. The obtained individual parametric characteristics for each diffusion were interpreted by considering the CB filler content and volume fraction of the filler. As a result, the diffusivities calculated by multiphase diffusion modeling were in quite agreement with the measured diffusivities for all investigated specimens, where the determined squared correlation coefficient (R²) by fitting process was in the range from 0.69 to 0.97.