Some pitfalls in the use of the Knudsen equation in modelling diffusion in nanoporous materials

Adsorption of Diffusing Tracers, Apparent Tortuosity, and Application to Mesoporous Silica

The apparent tortuosity due to adsorption of diffusing tracers in a porous material is determined by a scaling approach and is used to analyze recent data on LiCl and alkane diffusion in mesoporous silica. The slope of the adsorption isotherm at small loadings is written as β = qA/qG, where qA is the adsorption-desorption ratio and qG = ϵ/(as) - 1 is a geometrical factor depending on the range a of the tracer-wall interaction, the porosity ϵ, and the specific surface area s. The adsorption leads to a decrease of effective diffusion coefficient, which is quantified by multiplying the geometrical tortuosity factor τgeom by an apparent tortuosity factor τapp. In wide pores or when the adsorption barrier is high, τapp = β + 1, as obtained in previous works, but in narrow pores there is an additional contribution from frequent adsorption-desorption transitions. These results are obtained in media with parallel pores of constant cross sections, where the ratio between the effective pore width ϵ/s and the actual width is ≈0.25. Applications to mesoporous silica samples are justified by the small deviations from this ideal ratio. In the analysis of alkane self-diffusion data, the fractions of adsorbed molecules predicted in a recent theoretical work are used to estimate τgeom of the silica samples, which is ≫1 only in the sample with the narrowest pores (nominal 3 nm). The application of the model to Li+ ion diffusion leads to similar values of τgeom and to a difference of energy barriers of desorption and adsorption for those ions of ∼0.06 eV. Comparatively, alkane self-diffusion provides the correct order of magnitude of τgeom, with adsorption playing a less important role, whereas adsorption effects on Li+ diffusion are much more important.

Comprehensive Molecular Dynamics Study of Oxygen Diffusion in Carbon Mesopores: Insights for Designing Fuel-Cell Catalyst Supports

Mesoporous carbon is often used as a support for platinum catalysts in polymer electrolyte fuel-cell catalyst layers. Mesopores in the carbon support improve the performance of fuel cells by inhibiting the adsorption of ionomer onto the catalyst particles. However, the mesopores may impair mass transport. Hence, understanding molecular behaviors in the pores is essential to optimizing the mesopore structures. Specifically, it is crucial to understand the oxygen transport in the high-current region. In this study, the diffusion coefficients of oxygen molecules in carbon mesopores were calculated for various pore lengths, pore diameters, filling rates, and water contents in the ionomer via molecular dynamics simulations. The results show that oxygen diffusion slows by 2 orders of magnitude because of pore occlusion, and it slows down by an additional 1 or 2 orders of magnitude if ionomers are present in the pores. The occlusion can be theoretically predicted by considering the surface free energy. This theory provides some insight into mesoporous carbon designs; for instance, the theory suggests that narrow pores should be shortened to prevent occlusion. Slow diffusion in the presence of ionomers was attributed to the localization of oxygen at the dense ionomer-carbon interface. Thus, to improve oxygen transport properties, carbon surfaces and ionomer structures may be designed in such a manner as to prevent densification at the interface.

Diffusion Anisotropy of Gaseous Helium-3 in Ordered Aerogels at Low Temperatures

We report on the first observation of diffusion anisotropy of gaseous helium-3 entrapped in ordered aerogels at 4.2 K. The origins of ³He diffusion anisotropy in aerogels of different porosity are discussed. The correlations between gas diffusion coefficient and basic parameters of aerogels, such as porosity, fiber diameter, and fiber's degree of alignment, are inspected using simple diffusion simulations within the framework of classical diffusion model in both oriented and chaotic aerogels under conditions of diffuse (Knudsen diffusion) and specular reflections of atoms from the walls. The failure of the two-phase and Knudsen diffusion models at low temperature in isotropic and anisotropic aerogels is observed. The effect of a wall attractive potential on the gas dynamics is suspected to play a crucial role in the gas diffusion and its anisotropy. The rough theoretical estimates of that effect at low temperatures in aerogel space confirm this assumption. The observed peculiar diffusion is universal and is expected to occur with other probe gases at higher temperatures.

The Steady State Characteristics of Multicomponent Diffusion in Micro- and Mesoporous Media for Adsorbable and Nonadsorbable Species

The study addresses one of the fundamental issues in the mathematical modeling and quantitative process analysis of complex multicomponent diffusion in meso- and microporous materials. The model presented here incorporates combined molecular diffusion, Knudsen diffusion, viscous flow, and surface diffusion. A methodology and algorithm for the determination of steady states of such complex diffusive processes are proposed. The adopted form of the surface diffusion model does not require the calculation of the thermodynamic factor matrix. The method was verified by comparing the profiles of the state variables with those obtained from the dynamic model for sufficiently long diffusion times. The application of the method is illustrated for two diffusion processes involving three components. In the first one, all components are subject to adsorption. In the other, one gaseous component is an inert and is therefore not adsorbed and does not participate in surface diffusion. It is shown that the presence of inerts as well as their number does not impede the application of the proposed algorithm for the determination of steady states.

The Determination of Pore Shape and Interfacial Barrier of Entry for Light Gases Transport in Amorphous TEOS-Derived Silica: A Finite Element Method

The interfacial barrier of entry for light gas transport in a nanopore was a crucial factor to determine the separation efficiency in membrane technologies. To examine this effect, amorphous silica was prepared by sol-gel process, and its characterization results revealed that the commonly used cylindrical pore shape failed to represent the adsorption behavior of gases, but instead the pore shape had to be represented by a slit pore model. A finite element method (FEM) was developed to analyze the interfacial resistance by integrating a Lennard-Jones (LJ) potential over the layer area. It was found that the strong repulsion/attraction at the pore interface could be paired with the motion energy of guest molecules to predict the ideal selectivity between gases, thereby providing a solution to preliminarily screen the separation performance among a host of membrane candidates.

Understanding the Loading Dependence of Adsorbate Diffusivities in Hierarchical Metal-Organic Frameworks

Using atomistic simulations, we studied the diffusion of n-hexane in a series of isoreticular hierarchical metal-organic frameworks (MOFs) NU-100x. Nonmonotonic diffusivity-loading relationships that depend on the pore sizes were observed, which can be explained by the spatial distribution of adsorbates at different loadings. For one of the MOFs in the series, NU-1000-M, the diffusivity-loading relationship is almost identical to the previously reported results of n-hexane diffusion in the hierarchical self-pillared pentasil (SPP) zeolite. Detailed analysis revealed that the similarity results from their similar micropore and window sizes, which was confirmed by free-energy mapping. The effects of temperature and adsorbate chain length on the diffusion were also studied, which supported our conclusion that the diffusivity in hierarchical nanoporous materials is primarily controlled by the sizes of the micropores and the connecting windows, particularly at relatively low loadings.

Statistical method for modeling Knudsen diffusion in nanopores

This paper presents a statistical method for the calculation of gaseous flux and diffusion coefficients through a Knudsen-regime cylindrical nanopore. A general integral formula for the flux is derived in terms of collision frequency, molecular density, and a scattering path length probability distribution. Under appropriate steady-state assumptions, the general formula simplifies to Fick's first law, from which an expression for the diffusion coefficient is derived. The model is shown to be dimensionally consistent with the Einstein relation. The conditions for agreement with Fick's second law are investigated. Using a model probability distribution the model leads to an expression for the diffusion coefficient for a pore of finite length. This result is shown to compare favorably with a classic formula from the literature.

Helium-3 gas self-diffusion in a nematically ordered aerogel at low temperatures: enhanced role of adsorption

We performed ³He gas diffusion measurements for the first time in a highly porous ordered Al₂O₃ aerogel sample at a temperature of 4.2 K using a nuclear magnetic resonance field gradient technique. A strong influence of ³He adsorption in the aerogel on self-diffusion is observed. The classical consideration of adsorptive gas diffusion in mesopores leads to anomalously high tortuosity factors. The application of a more sophisticated model than the simple combination of empirical two-phase diffusion and the Knudsen gas diffusion models is required to explain our results. Anisotropic properties of the aerogel are not reflected in the observed gas diffusion even at low gas densities where the anisotropic Knudsen regime of diffusion is expected. The observed gas densification indicates the influence of the aerogel attractive potential on the molecular dynamics, which probably explains the reduced diffusion process. Perhaps this behavior is common for any adsorptive gases in nanopores.

Characterizing Structural Complexity in Disordered Carbons: From the Slit Pore to Atomistic Models

The reliable characterization of nanoporous carbons is critical to the design and optimization of their numerous applications; however, the vast majority of carbons in industrial use are highly disordered, with complex structures whose understanding has long challenged researchers. The idealized slit pore model represents the most commonly used approximation to a carbon nanopore; nevertheless, it has been only partially successful in predicting adsorption isotherms and fails significantly in predicting transport properties because of its inability to capture structural disorder and its effect on fluid accessibility. Atomistic modeling of the structure has much potential for overcoming this limitation, and among such approaches, hybrid reverse Monte Carlo simulation has emerged as the most attractive. This method reconstructs the structure of a carbon based on the fitting of its experimentally measured pair distribution function and appropriate properties such as porosity while minimizing the energy. The method is shown to be best implemented using a multistage strategy, with the first stage used to attain a deep minimum of the energy and subsequent stages to refine the structure based on the fitting of specific properties. Methods to determine the accessibility of gases based on the atomistic structure are outlined, and it is shown that energy barriers are very sensitive to small differences in the sizes of constrictions and pore entries. The ability to accurately predict macroscopic transport coefficients of adsorbates in nanoporous carbons appears to be the greatest limitation of such models. Overcoming this will require the fitting of properties more sensitive to long-range disorder than the currently used pair distribution and the use of a suitable multiscaling strategy, which is suggested as a future direction for advancing atomistic models. The inclusion of heteroatoms in the structure is also an important area requiring further attention, particularly in the development of computationally efficient force fields incorporating their interactions.

Applying Alkyl-Chain Surface Functionalizations in Mesoporous Inorganic Structures: Their Impact on Gas Flow and Selectivity Depending on Temperature

Porous inorganic capillary membranes are prepared to serve as model structures for the experimental investigation of the gas transport in functionalized mesopores. The porous structures possess a mean pore diameter of 23 nm which is slightly reduced to 20 nm after immobilizing C₁₆-alkyl chains on the surface. Gas permeation measurements are performed at temperatures ranging from 0 to 80 °C using Ar, N₂, and CO₂. Nonfunctionalized structures feature a gas transport according to Knudsen diffusion with regard to gas flow and selectivity. After C₁₆-functionalization, the gas flow is reduced by a factor of 10, and the ideal selectivities deviate from the Knudsen theory. CO₂ adsorption measurements show a decrease in total amount of adsorbed gas and isosteric heat of adsorption. It is hypothesized that the immobilized C₁₆-chains sterically influence the gas transport behavior without a contribution from adsorption effects. The reduced gas flow derives from an additional surface resistance caused by the C₁₆-chains spacially limiting the adsorption and desorption directions for gas molecules propagating through the structure, resulting in longer diffusion paths. In agreement, the gas flow is found to correlate with the molecular diameter of the gas species (CO₂ < Ar < N₂) increasing the resistance for larger molecules. This affects the ideal selectivities with the relation [Formula: see text]. The influence on selectivity increases with increasing temperature which leads to the conclusion that the temperature induced movement of the C₁₆-chains is responsible for the stronger interaction between gas molecules and surface functional groups.

Multiscale adsorption and transport in hierarchical porous materials

This review presents the state-of-the-art of multiscale adsorption and transport in hierarchical porous materials.

Diffusive dynamics of nanoparticles in ultra-confined media

Differential dynamic microscopy (DDM) was used to investigate the diffusive dynamics of nanoparticles of diameter 200-400 nm that were strongly confined in a periodic square array of cylindrical nanoposts. The minimum distance between posts was 1.3-5 times the diameter of the nanoparticles. The image structure functions obtained from the DDM analysis were isotropic and could be fit by a stretched exponential function. The relaxation time scaled diffusively across the range of wave vectors studied, and the corresponding scalar diffusivities decreased monotonically with increased confinement. The decrease in diffusivity could be described by models for hindered diffusion that accounted for steric restrictions and hydrodynamic interactions. The stretching exponent decreased linearly as the nanoparticles were increasingly confined by the posts. Together, these results are consistent with a picture in which strongly confined nanoparticles experience a heterogeneous spatial environment arising from hydrodynamics and volume exclusion on time scales comparable to cage escape, leading to multiple relaxation processes and Fickian but non-Gaussian diffusive dynamics.

Heterogeneous Diffusion of Alkanes in the Hierarchical Metal-Organic Framework NU-1000

The metal-organic framework (MOF) NU-1000 is a hierarchical material that comprises both micropores and mesopores in its crystalline structure. Because the pore structure is perfectly defined, NU-1000 is an interesting material for improving our understanding of diffusion in hierarchically structured materials. Here, we present molecular dynamics simulations aimed at probing the transport properties of n-alkanes in NU-1000 and introduce methods from the microrheology literature for analyzing the mean-squared displacements and their spatial heterogeneity. Adsorption occurs initially in the smaller channels, and diffusion at low loading is limited by interaction between adsorbate and framework atoms. The larger channels provide a region of low density where molecules are able to diffuse at higher rates predominantly along the channel axes. The disparate size of the channels gives rise to heterogeneity in the diffusivity of the guest molecules, whereas the asymmetry of the channels leads to anisotropic diffusion. Together, the channels form a network of "highways" and "side streets" that provide enhanced diffusion in one dimension.

Silylated mesoporous silica membranes on polymeric hollow fiber supports: synthesis and permeation properties

We report the synthesis and organic/water separation properties of mesoporous silica membranes, supported on low-cost and scalable polymeric (polyamide-imide) hollow fibers, and modified by trimethylsilylation with hexamethyldisilazane. Thin (∼1 μm) defect-free membranes are prepared, with high room-temperature gas permeances (e.g., 20,000 GPU for N2). The membrane morphology is characterized by multiple techniques, including SEM, TEM, XRD, and FT-ATR spectroscopy. Silylation leads to capping of the surface silanol groups in the mesopores with trimethylsilyl groups, and does not affect the integrity of the mesoporous silica structure and the underlying hollow fiber. The silylated membranes are evaluated for pervaporative separation of ethanol (EtOH), methylethyl ketone (MEK), ethyl acetate (EA), iso-butanol (i-BuOH), and n-butanol (n-BuOH) from their dilute (5 wt %) aqueous solutions. The membranes show separation factors in the range of 4-90 and high organic fluxes in the range of 0.18-2.15 kg m(-2) h(-1) at 303 K. The intrinsic selectivities (organic/water permeability ratios) of the silylated membranes at 303 K are 0.33 (EtOH/water), 0.5 (MEK/water), 0.25 (EA/water), 1.25 (i-BuOH/water), and 1.67 (n-BuOH/water) respectively, in comparison to 0.05, 0.015, 0.005, 0.08, and 0.14 for the unmodified membranes. The silylated membranes allow upgradation of water/organics feeds to permeate streams with considerably higher organics content. The selective and high-flux separation is attributed to both the organophilic nature of the modified mesopores and the large effective pore size. Comparison with other organics/water separation membranes reveals that the present membranes show promise due to high flux, use of scalable and low-cost supports, and good separation factors that can be further enhanced by tailoring the mesopore silylation chemistry.

Influence of Structural Heterogeneity on Diffusion of CH4 and CO2 in Silicon Carbide-Derived Nanoporous Carbon

We investigate the influence of structural heterogeneity on the transport properties of simple gases in a Hybrid Reverse Monte Carlo (HRMC) constructed model of silicon carbide-derived carbon (SiC-DC). The energy landscape of the system is determined based on free energy analysis of the atomistic model. The overall energy barriers of the system for different gases are computed along with important properties, such as Henry constant and differential enthalpy of adsorption at infinite dilution, and indicate hydrophobicity of the SiC-DC structure and its affinity for CO₂ and CH₄ adsorption. We also study the effect of molecular geometry, pore structure and energy heterogeneity considering different hopping scenarios for diffusion of CO₂ and CH₄ through ultramicropores using the Nudged Elastic Band (NEB) method. It is shown that the energy barrier of a hopping molecule is very sensitive to the shape of the pore entry. We provide evidence for the influence of structural heterogeneity on self-diffusivity of methane and carbon dioxide using molecular dynamics simulation, based on a maximum in the variation of self-diffusivity with loading. A comparison of the MD simulation results with self-diffusivities from quasi-elastic neutron scattering (QENS) measurements and, with macroscopic uptake-based low-density transport coefficients, reveals the existence of internal barriers not captured in MD simulation and QENS experiments. Nevertheless, the simulation and macroscopic uptake-based diffusion coefficients agree within a factor of 2-3, indicating that our HRMC model structure captures most of the important energy barriers affecting the transport of CH₄ in the nanostructure of SiC-DC.

Diffusion Study by IR Micro-Imaging of Molecular Uptake and Release on Mesoporous Zeolites of Structure Type CHA and LTA

The presence of mesopores in the interior of microporous particles may significantly improve their transport properties. Complementing previous macroscopic transient sorption experiments and pulsed field gradient NMR self-diffusion studies with such materials, the present study is dedicated to an in-depth study of molecular uptake and release on the individual particles of mesoporous zeolitic specimens, notably with samples of the narrow-pore structure types, CHA and LTA. The investigations are focused on determining the time constants and functional dependences of uptake and release. They include a systematic variation of the architecture of the mesopores and of the guest molecules under study as well as a comparison of transient uptake with blocked and un-blocked mesopores. In addition to accelerating intracrystalline mass transfer, transport enhancement by mesopores is found to be, possibly, also caused by a reduction of transport resistances on the particle surfaces.

Diffusion in hierarchical mesoporous materials: applicability and generalization of the fast-exchange diffusion model

Transport properties of cyclohexane confined to a silica material with an ordered, bimodal pore structure have been studied by means of pulsed field gradient nuclear magnetic resonance. A particular organization of the well-defined pore structure, composed of a collection of spatially ordered, spherical mesopores interconnected via narrow worm-like pores, allowed for a quantitative analysis of the diffusion process in a medium with spatially ordered distribution of the fluid density for a broad range of the gas-liquid equilibria. The measured diffusion data were interpreted in terms of effective diffusivities, which were determined within a microscopic model considering long-range molecular trajectories constructed by assembling the alternating pieces of displacement in the two constituting pore spaces. It has further been found that for the system under study, in particular, and for mesoporous materials with multiple porosities, in general, this generalized model simplifies to the conventional fast-exchange model used in the literature. Thus, not only was justification of the applicability of the fast-exchange model to a diversity of mesoporous materials provided, but the diffusion parameters entering the fast-exchange model were also exactly defined. The equation resulting in this way was found to nicely reproduce the experimentally determined diffusivities, establishing a methodology for targeted fine-tuning of transport properties of fluids in hierarchical materials with multiple porosities.