The design and development of many separation and catalytic process technologies require a proper quantitative description of diffusion of mixtures of guest molecules within porous crystalline materials. This tutorial review presents a unified, phenomenological description of diffusion inside meso- and micro-porous structures. In meso-porous materials, with pore sizes 2 nm < dp < 50 nm, there is a central core region where the influence of interactions of the molecules with the pore wall is either small or negligible; meso-pore diffusion is governed by a combination of molecule–molecule and molecule–pore wall interactions. Within micro-pores, with dp < 2 nm, the guest molecules are always under the influence of the force field exerted with the wall and we have to reckon with the motion of adsorbed molecules, and there is no “bulk” fluid region. The characteristics and physical significance of the self-, Maxwell–Stefan, and Fick diffusivities are explained with the aid of data obtained either from experiments or molecular dynamics simulations, for a wide variety of structures with different pore sizes and topology. The influence of adsorption thermodynamics, molecular clustering, and segregation on both magnitudes and concentration dependences of the diffusivities is highlighted. In mixture diffusion, correlations in molecular hops have the effect of slowing-down the more mobile species. The need for proper modeling of correlation effects using the Maxwell–Stefan formulation is stressed with the aid of examples of membrane separations and catalytic reactors.

Diffusion in porous crystalline materials

Adding mesopore networks in microporous materials using the principles of hierarchical structure design is recognized as a promising route for eliminating their transport limitations and, therefore, for improving their value in technological applications. Depending on the routes of physico-chemical procedures or post-synthesis treatments used, very different geometries of the intentionally-added transport mesopores can be obtained. Understanding the structure-dynamics relationships in these complex materials with multiple porosities under different thermodynamical conditions remains a challenging task. In this review, we summarize the results obtained so far on experimental and theoretical studies of diffusion in micro-mesoporous materials. By considering four common classes of bi-porous materials, which are differing by the inter-connectivities of their sup-spaces as one of the most important parameter determining the transport rates, we discuss their generic transport properties and correlate the results delivered by the equilibrium and non-equilibrium techniques of diffusion measurements.

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Transport properties of hierarchical micro–mesoporous materials

The sorption isotherms and self-diffusivities of CO2 and N2 in silicalite have been calculated via grand canonical Monte Carlo and equilibrium molecular dynamics simulations over a wide range of occupancies, using various force fields proposed in the literature. Predictions for the sorption thermodynamics are in very favorable agreement with the experiment, especially when detailed point-charge models are used to represent the interaction of the quadrupole moments of the sorbate molecules with the lattice field and with each other. They indicate that the zeolite cannot be in its para (P212121) form under the conditions of the measurements. Permeabilities corresponding to a perfectly crystalline membrane have been estimated for CO2 and N2, as well as for methane, examined in past simulation work, from the predicted sorption isotherms and low-occupancy self-diffusivities by invoking the Darken equation. The ratios of pure component permeabilities obtained in this way agree very well with actual macroscopic ...

Prediction of Permeation Properties of CO2 and N2 through Silicalite via Molecular Simulations

Adsorptive separation of light olefins from paraffins

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.

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Diffusion in nanoporous materials: fundamental principles, insights and challenges

The coefficients of intracrystalline self-diffusion of the n-alkanes from propane to n-alkanes from propane to n-hexane adsorbed in zeolite ZSM-5 are studied by means of the pulsed field gradient (PFG) NMR method with field gradient amplitudes up to 25 T m–1 over a temperature range from –20 to +380 °C. The diffusivities are found to decrease monotonically with increasing chain lengths. They are in satisfactory agreement with the results of quasielastic neutron scattering and MD simulations as well as with non-equilibrium measurements by the frequency response and single-step adsorption technique. However, in comparison with zero-length-column (ZLC) desorption and permeation studies, these data, though exhibiting the same trends of chain length dependence, are ca. two orders of magnitude larger. As expected, the diffusivities are intermediate between those for zeolite NaX and NaCaA, exhibiting, however, the smallest activation energies. This indicates that the differences in the potential energies of the adsorbate molecules on their intracrystalline trajectories are in general smaller than in zeolites NaCaA and NaX.

High-temperature pulsed field gradient nuclear magnetic resonance self-diffusion measurements of n-alkanes in MFl-type zeolites

Pulsed-field gradient (PFG) NMR is applied to the study of self-diffusion of the n-alkanes from methane to n-heptane, adsorbed in zeolite NaCaA. Using magnetic field gradients up to 10 T m–1 over a temperature range from – 100 to 350 °C, diffusivities between 10–13 and 10–8 m2 s–1 can be covered. For methane and ethane in NaCaA the diffusivity is found to increase by up to two orders of magnitude with increasing concentration. In contrast to the case for zeolites NaX and HZSM-5, there is a dramatic jump in the chain-length dependence of the diffusivities between ethane and propane, corresponding to the fact that the critical molecular diameter of propane and the longer n-alkanes is close to the diameter of the free apertures between adjacent supercages. With further increasing chain length there is a monotonic decrease in the diffusivities. The values obtained are in satisfactory agreement with the results of uptake measurements and neutron scattering experiments.

Andreas K. Nowak

Papers

High-temperature pulsed field gradient nuclear magnetic resonance self-diffusion measurements of n-alkanes in MFl-type zeolites

Self-diffusion measurements of n-alkanes in zeolite NaCaA by pulsed-field gradient nuclear magnetic resonance