Diffusion in nanoporous materials: fundamental principles, insights and challenges

Following a brief review of Fick's laws and the theory of diffusion in a homogeneous medium, we consider the application of the Fickian model to diffusion in nanoporous materials. If the pore system is statistically uniform the simple Fickian model is directly applicable. Inhomogeneities such as surface or internal barriers require some adjustments to the model but Fick's equations still provide a valid approach. Hierarchical pore systems present a more serious challenge. When there is rapid exchange between the different regions such systems conform to the simple Fickian model, with a diffusivity corresponding to the mean of the diffusivities in the different regions. In contrast, when the condition of rapid exchange is not fulfilled the simple Fickian model is not applicable and the situation becomes more complicated. Simple hierarchical pore structures such as the micropore/macropore system typically found in commercial adsorbents and catalysts can still be described by a dual resistance Fickian model but for more complex hierarchical pore structures Monte Carlo or MD simulations offer the only realistic approach. The measurement of self-diffusion by PFG NMR and by microimaging (notably by interference microscopy) is also reviewed and selected examples are presented to show the detailed information that can be extracted from such measurements, especially when accompanied by molecular simulations. Examples highlighting the relevance of a detailed knowledge of the various steps of mass transfer for a transport-optimized technological application of nanoporous materials, notably for molecular separations and mass conversions, are provided.