Connecting theory and simulation with experiment for the study of diffusion in nanoporous solids

Energetic Characteristics of Hydrophobic Porous Materials as Candidates for Manufacturing of Nanorockets

A new propulsion mechanism for nano- and microrocket engines is hypothesized. It is based on the instantaneous expulsion from hydrophobic nanopores triggered by irradiation from electromagnetic microwaves, ultrasound, or sudden pressure release. A large energy output is needed for the propulsion of a nanoparticle, and the value can be determined experimentally and by means of atomistic simulations. As such, we measured the heat of intrusion of water into ITQ-29 (LTA) pure silica zeolite with cage structure of pores. The heat effect is exothermic and equal to -7.3 ± 0.8 J/g of zeolite. Similar values were reported for chabazite, ZIF-8, and grafted mesoporous silica EVA. All these materials have cage structures of pores. In contrast, silicalite-1 (MFI) zeolite with a channel structure of pores exhibits endothermic intrusion. Molecular dynamics simulations of pure silica zeolites with LTA, CHA, and MFI topologies at a broad range of water loadings show that water becomes thermodynamically stable in cage-shaped pores while it is unstable in channel-shaped pores. A large energy release is expected during water expulsion from channel-type pores.

Multiscale simulation of liquid chromatography: Effective diffusion in macro-mesoporous beds and the B-term of the plate height equation

We performed multiscale simulations of analyte sorption and diffusion in hierarchical porosity models of monolithic silica columns for reversed-phase liquid chromatography to investigate how the mean mesopore size of the chromatographic bed and the analyte-specific interaction with the chromatographic interface influence the analyte diffusivity at various length scales. The reproduced experimental conditions comprised the retention of six analyte compounds of low to moderate solute polarity on a silica-based, endcapped, C18 stationary phase with water‒acetonitrile and water-methanol mobile phases whose elution strength was varied via the volumetric solvent ratio. Detailed information about the analyte-specific interfacial dynamics received from molecular dynamics simulations was incorporated through appropriate linker schemes into Brownian dynamics diffusion simulations in three hierarchical porosity models received from physical reconstructions of silica monoliths with a mean macropore size of 1.23 µm and mean mesopore sizes of 12.3, 21.3, or 25.7 nm. The mean mesopore size was found to have a similar influence on the effective mesopore diffusivity as the analyte polarity and the mobile-phase elution strength, which together determine the analyte residence time on a column. A smaller mesopore size attenuated the increase of the effective mesopore diffusivity with increasing mobile-phase elution strength significantly. The effective bed diffusivity was limited by the analyte residence time rather than by morphological details of the mesopore space. The stronger an analyte was retained by the chromatographic interface inside the mesopores, the slower became the mass transfer between the pore space hierarchies and the lower was the effective bed diffusivity. The B-terms of the plate height equation were finally generated with the bed diffusivities and phase-based retention factors derived from the hierarchical porosity models using additional information about the stationary-phase limit obtained from the analysis of analyte-bonded phase contacts. The B-terms highlight analyte- and mobile phase-specific behavior relevant to isocratic and gradient elution conditions in chromatographic practice. In particular, U-shaped B-term curves are observed due to the dominating contribution of the retention factor and the bed diffusivity to the B-term at low and high elution strength of the mobile phase, respectively.

Correlation-Based Predictions of Gas Solute Diffusivity in Ionic Liquid Solvents Based on Solvent-Accessible Surface Area

In our prior study [Olowookere, F. V.; Turner, C. H. J. Phys. Chem. B 2023, 127(42), 9144-9154], we introduced a new scaling relationship to predict gas solute diffusion at challenging conditions, focusing on CO2 and SO2 diffusion in multivalent ionic liquid (IL) solvents. This work extends our initial exploratory study into a much broader array of systems, encompassing additional solutes (N2, CH4, C2H6, C3H8, C3H8O, and H2O) and a variety of different ionic liquid species ([Bzmim3]3+, [Bzmim4]4+, [BMIM]+, [EMIM]+, [HMIM]+, [NapO2]2-, [BzO3]3-, [BF4]-, [Tf2N]-, and [PF6]-). Our study demonstrates a remarkably robust logarithmic correlation between solute diffusion and solvent accessible surface area (SA) across 20 different additional systems. We perform comprehensive analyses of the underlying molecular phenomena responsible for this correlation, including solute lifetime distributions, void space dynamics, and Voronoi tessellation, in order to elucidate a stronger mechanistic understanding of this behavior. Our findings highlight a direct link between the solvent accessible SA and the size of the void domains. Overall, our scaling approach provides an efficient and reliable approach for predicting diffusion from analyses of short simulations at higher temperatures.

Confined fluid dynamics in a viscoelastic, amorphous, and microporous medium: Study of a kerogen by molecular simulations and the generalized Langevin equation

We combine the use of molecular dynamics simulations and the generalized Langevin equation to study the diffusion of a fluid adsorbed within kerogen, the main organic phase of shales. As a class of microporous and amorphous materials that can exhibit significant adsorption-induced swelling, the dynamics of the kerogen's microstructure is expected to play an important role in the confined fluid dynamics. This role is investigated by conducting all-atom simulations with or without solid dynamics. Whenever the dynamics coupling between the fluid and solid is accounted for, we show that the fluid dynamics displays some qualitative differences compared to bulk fluids, which can be modulated by the amount of adsorbed fluid owing to adsorption-induced swelling. We highlight that working with the memory kernel, the central time correlation function of the generalized Langevin equation, allows the fingerprint of the dynamics of the solid to appear on that of the fluid. Interestingly, we observe that the memory kernels of fluid diffusion in kerogen qualitatively behave as those of tagged particles in supercooled liquids. We emphasize the importance of reproducing the velocity-force correlation function to validate the memory kernel numerically obtained as confinement enhances the numerical instabilities. This route is interesting as it opens the way for modeling the impact of fluid concentration on the diffusion coefficient in such ultra-confining cases.

Bridging Microscopic Dynamics and Hydraulic Permeability in Mechanically-Deformed Nanoporous Materials

In the field of nanoconfined fluids, there are striking examples of deformation/transport coupling in which mechanical solicitation of the confining solid and dynamics of the confined fluid impact each other. While this intriguing behavior can be harnessed for applications (e.g., energy storage, phase separation, catalysis), the underlying mechanisms remain to be understood. Here, using molecular simulations, we investigate fluid flow in deformable nanoporous materials subjected to external mechanical stresses. We show that the pore mechanical properties significantly affect fluid flow as they lead to significant pore deformations and different fluid organization at the solid surface. Despite such mechanical effects, we show that the fluid thermodynamic properties (i.e., adsorption) can be linked consistently to Darcy's law for the permeability by invoking a pore size definition based on the concept of Gibbs' dividing surface. In particular, regardless of the solid stiffness and applied external stress, all data can be rationalized by accounting for the fluid viscosity and slippage at the solid surface (independently of a specific pore size definition). Using such a formalism, we establish that the intimate relation─derived using the linear response theory─between collective diffusivity and hydraulic permeability remains valid. This allows linking consistently microscopic dynamics experiments and macroscopic permeability experiments on fluid flow in deformable nanoporous materials.

Mind the Gap: The Role of Mass Transfer in Shaped Nanoporous Adsorbents for Carbon Dioxide Capture

Adsorptive separations by nanoporous materials are major industrial processes. The industrial importance of solid adsorbents is only expected to grow due to the increased focus on carbon dioxide capture technology and energy-efficient separations. To evaluate the performance of an adsorbent and design a separation process, the adsorption thermodynamics and kinetics must be known. However, although diffusion kinetics determine the maximum production rate in any adsorption-based separation, this aspect has received less attention due to the challenges associated with conducting diffusion measurements. These challenges are exacerbated in the study of shaped adsorbents due to the presence of porosity at different length scales. As a result, adsorbent selection typically relies mainly on adsorption properties at equilibrium, i.e., uptake capacity, selectivity and adsorption enthalpy. In this Perspective, based on an extensive literature review on mass transfer of CO2 in nanoporous adsorbents, we discuss the importance and limitations of measuring diffusion in nanoporous materials, from the powder form to the adsorption bed, considering the nature of the process, i.e., equilibrium-based or kinetic-based separations. By highlighting the lack of and discrepancies between published diffusivity data in the context of CO2 capture, we discuss future challenges and opportunities in studying mass transfer across scales in adsorption-based separations.

Cooperative Adsorption and Diffusion of Small Alcohols in Metal-Organic Framework ZIF-8 and Intrinsically Microporous Polymer PTMSP

Fundamental understanding of molecular interactions and transport within microporous materials displaying cooperative Type V adsorption is challenged by the unique features of this isotherm type. In order to capture a broad understanding of this uncommon, yet industrially relevant, behavior in microporous materials, this investigation examines the adsorption equilibria and kinetics of methanol and ethanol in both a metal-organic framework (MOF) material, ZIF-8, and a high free volume polymer of intrinsic microporosity, poly[1-(trimethylsilyl)-1-propyne] (PTMSP). A novel formulation that can capture the cooperative effects of small alcohols in its description of adsorption equilibria and kinetics is proposed. It is subsequently applied to successfully capture some previously uncharacterized or semiempirically characterized data for equilibria and the loading dependence of the diffusivity in both ZIF-8 and PTMSP, which are materials chosen for their industrial relevance. Finally, it is anticipated that the results of this study can fill the current void that exists in meaningful mechanistic and analytical descriptions of cooperative equilibrium and diffusion phenomena in microporous materials.

Universal superdiffusion of random walks in media with embedded fractal networks of low diffusivity

Diffusion in composite media with high contrasts between diffusion coefficients in fractal sets of inclusions and in their embedding matrices is modeled by lattice random walks (RWs) with probabilities p<1 of hops from fractal sites and 1 from matrix sites. Superdiffusion is predicted in time intervals that depend on p and with diffusion exponents that depend on the dimensions of matrix (E) and fractal (D_{F}) as ν=1/(2+D_{F}-E). This contrasts with the nonuniversal subdiffusion of RWs confined to fractal media. Simulations with four fractals show the anomaly at several time decades for p≲10^{-3} and the crossover to the asymptotic normal diffusion. These results show that superdiffusion can be observed in isotropic RWs with finite moments of hop length distributions and allow the estimation of the dimension of the inclusion set from the diffusion exponent. However, displacements within single trajectories have normal scaling, which shows transient ergodicity breaking.

Probing Self-Diffusion of Guest Molecules in a Covalent Organic Framework: Simulation and Experiment

Covalent organic frameworks (COFs) are a class of porous materials whose sorption properties have so far been studied primarily by physisorption. Quantifying the self-diffusion of guest molecules inside their nanometer-sized pores allows for a better understanding of confinement effects or transport limitations and is thus essential for various applications ranging from molecular separation to catalysis. Using a combination of pulsed field gradient nuclear magnetic resonance measurements and molecular dynamics simulations, we have studied the self-diffusion of acetonitrile and chloroform in the 1D pore channels of two imine-linked COFs (PI-3-COF) with different levels of crystallinity and porosity. The higher crystallinity and porosity sample exhibited anisotropic diffusion for MeCN parallel to the pore direction, with a diffusion coefficient of Dpar = 6.1(3) × 10-10 m2 s-1 at 300 K, indicating 1D transport and a 7.4-fold reduction in self-diffusion compared to the bulk liquid. This finding aligns with molecular dynamics simulations predicting 5.4-fold reduction, assuming an offset-stacked COF layer arrangement. In the low-porosity sample, more frequent diffusion barriers result in isotropic, yet significantly reduced diffusivities (DB = 1.4(1) × 10-11 m2 s-1). Diffusion coefficients for chloroform at 300 K in the pores of the high- (Dpar = 1.1(2) × 10-10 m2 s-1) and low-porosity (DB = 4.5(1) × 10-12 m2 s-1) samples reproduce these trends. Our multimodal study thus highlights the significant influence of real structure effects such as stacking faults and grain boundaries on the long-range diffusivity of molecular guest species while suggesting efficient intracrystalline transport at short diffusion times.

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.

On De Gennes narrowing of fluids confined at the molecular scale in nanoporous materials

Beyond well-documented confinement and surface effects arising from the large internal surface and severely confining porosity of nanoporous hosts, the transport of nanoconfined fluids remains puzzling in many aspects. With striking examples such as memory, i.e., non-viscous effects, intermittent dynamics, and surface barriers, the dynamics of fluids in nanoconfinement challenge classical formalisms (e.g., random walk, viscous/advective transport)-especially for molecular pore sizes. In this context, while molecular frameworks such as intermittent Brownian motion, free volume theory, and surface diffusion are available to describe the self-diffusion of a molecularly confined fluid, a microscopic theory for collective diffusion (i.e., permeability), which characterizes the flow induced by a thermodynamic gradient, is lacking. Here, to fill this knowledge gap, we invoke the concept of "De Gennes narrowing," which relates the wavevector-dependent collective diffusivity D0(q) to the fluid structure factor S(q). First, using molecular simulation for a simple yet representative fluid confined in a prototypical solid (zeolite), we unravel an essential coupling between the wavevector-dependent collective diffusivity and the structural ordering imposed on the fluid by the crystalline nanoporous host. Second, despite this complex interplay with marked Bragg peaks in the fluid structure, the fluid collective dynamics is shown to be accurately described through De Gennes narrowing. Moreover, in contrast to the bulk fluid, the departure from De Gennes narrowing for the confined fluid in the macroscopic limit remains small as the fluid/solid interactions in severe confinement screen collective effects and, hence, weaken the wavevector dependence of collective transport.

Poly(2-vinylpyridine)/MCM-41 Composites with Micropores and Switchable Mesopores for the Removal of Cr(VI)

Porous structure design and reversible regulation of pore size during adsorption-desorption are crucial to the removal of pollutants in water such as Cr(VI). In this paper, micropores and switchable mesopores were constructed on MCM-41 to further improve adsorption-desorption performance of Cr(VI) via the confinement effect of micropores and opening and closing of mesopores. 2-Vinylpyridine was introduced and polymerized into the pores and on the pore mouth of MCM41 modified by C═C group (AM41) under the irradiation of ultraviolet light. The obtained samples (PM41) possessed mesopores (2.73 nm) and micropores (1.36 nm), where mesopores could open or close under different pH and micropores showed the confinement effect because their pore size is close to Cr(VI) diameter (0.87 nm). Compared with MCM-41, the introduction of poly(2-vinylpyridine) enhanced obviously its adsorptive ability and it trapped most of the Cr(VI) (99%) in solution, 12 times higher than that of the parent sample. The change of pore size is favorable to the cycle performance, and after 3 times recycling, the removal rate of Cr(VI) by PM41-20 remained above 88%. Langmuir isotherm showed a better data correlation than the Freundlich model. Cr(VI) in solution was removed by electrostatic interaction between the pyridine group and Cr(VI) and the confinement effect from micropores.

Mixture Adsorption in Nanoporous Zeolite and at Its External Surface: In-Pore and Surface Selectivity

Using toluene, ethylene, and water as gas compounds with different representative molecular interactions, we perform atom-scale simulations for their mixtures to investigate the selectivity of the core nanoporosity and external surface in a prototypical zeolite. As expected, the overall behavior suggests that increasing the pressure of a given component promotes the desorption of the coadsorbing species. However, for water-toluene mixtures, we identify that the pseudohydrogen bonding between water and toluene leads to beneficial coadsorption as toluene adsorption in the low-pressure range promotes water adsorption. Moreover, when the zeolite is completely filled with water, toluene adsorption does not occur due to steric repulsion, and ethylene shows oversolubility as the amount of ethylene per water molecule is significantly larger than in bulk water. The underlying oversolubility mechanism is found to be due to localized ethylene adsorption in the density minima arising from the layering of water in nanoconfinement. Despite these specific effects, the relatively weak coadsorption effects in the zeolite nanoporosity, which are found to be reasonably captured using the ideal adsorbed solution theory, arise from the fact that adsorption of these gases having different molecular sizes occurs in distinct pore regions (channel type, channel intersection). Finally, in contrast to confinement in the nanoporosity, mixture adsorption at the external surface does not show coadsorption effects as it mostly follows the Henry regime. These results show that selectivity is mostly governed by the confinement effects as the external surface leads to a selectivity loss.

Storage and Diffusion of Carbon Dioxide in the Metal Organic Framework MOF-5─A Semi-empirical Molecular Dynamics Study

Metal-organic frameworks (MOFs) have attracted increasing attention due to their high porosity for exceptional gas storage applications. MOF-5 belongs to the family of isoreticular MOFs (IRMOFs) and consists of Zn4O6+ clusters linked by 1,4-benzenedicarboxylate. Due to the large number of atoms in the unit cell, molecular dynamics simulation based on density functional theory has proved to be too demanding, while force field models are often inadequate to model complex host-guest interactions. To overcome this limitation, an alternative semi-empirical approach using a set of approximations and extensive parametrization of interactions called density functional tight binding (DFTB) was applied in this work to study CO2 in the MOF-5 host. Calculations of pristine MOF-5 yield very good agreement with experimental data in terms of X-ray diffraction patterns as well as mechanical properties, such as the negative thermal expansion coefficient and the bulk modulus. In addition, different loadings of CO2 were introduced, and the associated self-diffusion coefficients and activation energies were investigated. The results show very good agreement with those of other experimental and theoretical investigations. This study provides detailed insights into the capability of semi-empirical DFTB-based molecular dynamics simulations of these challenging guest@host systems. Based on the comparison of the guest-guest pair distributions observed inside the MOF host and the corresponding gas-phase reference, a liquid-like structure of CO2 can be deduced upon storage in the host material.

Nanoconfined Water in Pillared Zeolites Probed by 1H Nuclear Magnetic Resonance

Here, we report the results of our 1H nuclear magnetic resonance study of the dynamics of water molecules confined in zeolites (mordenite and ZSM-5 structures) with hierarchical porosity (micropores in zeolite lamella and mesopores formed by amorphous SiO2 in the inter-lamellar space). 1H nuclear magnetic resonance (NMR) spectra show that water experiences complex behavior within the temperature range from 173 to 298 K. The temperature dependence of 1H spin-lattice relaxation evidences the presence of three processes with different activation energies: freezing (about 30 kJ/mol), fast rotation (about 10 kJ/mol), and translational motion of water molecules (23.6 and 26.0 kJ/mol for pillared mordenite and ZSM-5, respectively). For translational motion, the activation energy is markedly lower than for water in mesoporous silica or zeolites with similar mesopore size but with disordered secondary porosity. This indicates that the process of water diffusion in zeolites with hierarchical porosity is governed not only by the presence of mesopores, but also by the mutual arrangement of meso- and micropores. The translational motion of water molecules is determined mainly by zeolite micropores.

The Electrochemical Actuation Performances of Nanoporous Ternary AlCoCu Alloy with a Unique Nanosheet Structure

Compared to traditional actuators (such as piezoelectric ceramics), metal actuators possess the advantages of a low energy consumption, large strain amplitude, and high strain energy density. However, most of the existing metal actuators with an excellent comprehensive performance are composed of precious metals, which are limited by high costs and have almost no possibility for large-scale production in the future. This study focuses on non-precious metal materials and exploits a one-step chemical dealloying method to prepare bulk nanoporous (NP) CoCuAl actuators (NP-CCA) from Al70Co20Cu10 alloy. The microstructure and actuation properties of the NP-CCA were analyzed in detail. The dense continuous nanoscale pores provide an excellent network connectivity for a large strain response, enabling the NP-CCA to achieve a strain amplitude of up to 1.19% (more than eight and two times that of NP-Pt and NP-Ag, respectively), comparable to precious metal actuators. In addition, the NP-CCA possesses a high strain energy density, which is prominent in many precious metal actuation materials (such as NP-Au, NP-Ag, and NP-Pt).

Predicting Gaseous Solute Diffusion in Viscous Multivalent Ionic Liquid Solvents

Calculating solute diffusion in dense, viscous solvents can be particularly challenging in molecular dynamics simulations due to the long time scales involved. Here, a new scaling approach is developed for predicting solute diffusion based on analyses of CO2 and SO2 diffusion in two different multivalent ionic liquid solvents. Various scaling approaches are initially evaluated, including single and separate thermostats for the solute and solvent, as well as the application of the Arrhenius relationship and the Speedy-Angell power law. A very strong logarithmic correlation is established between the solvent-accessible surface area and solute diffusion. This relationship, reflecting Danckwerts' surface renewal theory and the Vrentas-Duda free volume model, presents a valuable method for estimating diffusion behavior from short simulation trajectories at elevated temperatures. The approach may be beneficial for enhancing predictive modeling in similar challenging systems and should be more broadly evaluated.

Molecular Diffusion in Nanoreactors' Pore Channel System: Measurement Techniques, Structural Regulation, and Catalytic Effects

Nanoreactors, as a new class of materials with highly enriched and ordered pore channel structures, can achieve special catalytic effects by precisely identifying and controlling the molecular diffusion behavior within the ordered pore channel system. Nanoreactors-driven molecular diffusion within the ordered pore channels can be highly dependent on the local microenvironment in the nanoreactors' pore channel system. Although the diffusion process of molecules within the ordered pore channels of nanoreactors is crucial for the regulation of catalytic behaviors, it has not yet been as clearly elucidated as it deserves to be in this study. In this review, fundamental theory and measurement techniques for molecular diffusion in the pore channel system of nanoreactors are presented, structural regulation strategies of pore channel parameters for controlling molecular diffusion are discussed, and the effects of molecular diffusion in the pore channel system on catalytic reactivity and selectivity are further analyzed. This article attempts to further develop the underlying theory of molecular diffusion within the theoretical framework of nanoreactor-driven catalysis, and the proposed perspectives may contribute to the rational design of advanced catalytic materials and the precise control of complex catalytic kinetics.

Machine Learning Predictions of Simulated Self-Diffusion Coefficients for Bulk and Confined Pure Liquids

Diffusion properties of bulk fluids have been predicted using empirical expressions and machine learning (ML) models, suggesting that predictions of diffusion also should be possible for fluids in confined environments. The ability to quickly and accurately predict diffusion in porous materials would enable new discoveries and spur development in relevant technologies such as separations, catalysis, batteries, and subsurface applications. In this work, we apply artificial neural network (ANN) models to predict the simulated self-diffusion coefficients of real liquids in both bulk and pore environments. The training data sets were generated from molecular dynamics (MD) simulations of Lennard-Jones particles representing a diverse set of 14 molecules ranging from ammonia to dodecane over a range of liquid pressures and temperatures. Planar, cylindrical, and hexagonal pore models consisted of walls composed of carbon atoms. Our simple model for these liquids was primarily used to generate ANN training data, but the simulated self-diffusion coefficients of bulk liquids show excellent agreement with experimental diffusion coefficients. ANN models based on simple descriptors accurately reproduced the MD diffusion data for both bulk and confined liquids, including the trend of increased mobility in large pores relative to the corresponding bulk liquid.

The Anatomy of Amorphous, Heterogeneous Catalyst Pellets

This review focuses on disordered, or amorphous, porous heterogeneous catalysts, especially those in the forms of pellets and monoliths. It considers the structural characterisation and representation of the void space of these porous media. It discusses the latest developments in the determination of key void space descriptors, such as porosity, pore size, and tortuosity. In particular, it discusses the contributions that can be made by various imaging modalities in both direct and indirect characterisations and their limitations. The second part of the review considers the various types of representations of the void space of porous catalysts. It was found that these come in three main types, which are dependent on the level of idealisation of the representation and the final purpose of the model. It was found that the limitations on the resolution and field of view for direct imaging methods mean that hybrid methods, combined with indirect porosimetry methods that can bridge the many length scales of structural heterogeneity and provide more statistically representative parameters, deliver the best basis for model construction for understanding mass transport in highly heterogeneous media.

Scaling laws for single-file diffusion of adhesive particles

Single-file diffusion refers to the Brownian motion in narrow channels where particles cannot pass each other. In such processes, the diffusion of a tagged particle is typically normal at short times and becomes subdiffusive at long times. For hard-sphere interparticle interaction, the time-dependent mean squared displacement of a tracer is well understood. Here we develop a scaling theory for adhesive particles. It provides a full description of the time-dependent diffusive behavior with a scaling function that depends on an effective strength of adhesive interaction. Particle clustering induced by the adhesive interaction slows down the diffusion at short times, while it enhances subdiffusion at long times. The enhancement effect can be quantified in measurements irrespective of how tagged particles are injected into the system. Combined effects of pore structure and particle adhesiveness should speed up translocation of molecules through narrow pores.

Features of Self-Diffusion of Tridecane Molecules in a Porous Medium of Kaolinite Used as a Model of a Chemically Inert Membrane

The present work focused on the experimental study of the specific features of self-diffusion of tridecane molecules in macroporous kaolinite, which is used as a raw material for the production of chemically inert membranes. The measurements of self-diffusion coefficients by pulsed magnetic field gradient nuclear magnetic resonance (PMFG NMR) revealed an increased translational mobility of tridecane molecules in kaolinite with incomplete filling of the pore space. This effect was accompanied by a sharp change in the slope of the Arrhenius plot of the self-diffusion coefficients of tridecane molecules in kaolinite. An analysis of the diffusion spin echo decay in the tridecane-kaolinite system revealed a discrepancy between the experimental data and the theoretical predictions, considering the effect of the geometry of porous space on molecular mobility. It was shown that the experimental results could be interpreted in terms of a model of two phases of tridecane molecules in the pores of kaolinite, in the gaseous and adsorbed state, coexisting under the fast-exchange conditions. Within the framework of the model, the activation energies of self-diffusion were calculated, which agreed satisfactorily with the experimental data. Additionally, the effects of the internal magnetic field gradients arising in a porous medium loaded with a gas or liquid on the data of the PFG NMR measurements were calculated. It was shown that the effect of magnetic field inhomogeneities on the measured self-diffusion coefficients of tridecane in kaolinite is small and could be neglected.

Insights and Heuristics for Predicting Diffusion Rates of Chemical Warfare Agents in Zirconium Metal-Organic Frameworks

Designing nanoporous catalysts to destroy chemical warfare agents (CWAs) and environmental contaminants requires consideration of both intrinsic catalytic activity and the mass transfer of molecules in and out of the pores. Polar adsorbates such as CWAs experience a heterogeneous environment in many metal-organic frameworks (MOFs) due to the arrangement of the metal nodes and organic linkers of the MOF. However, quantitative relationships between the pore architecture and the resulting diffusion properties of polar molecules have not been established. We used molecular dynamics simulations to calculate the diffusion coefficients of the CWA simulant dimethyl methyl phosphonate (DMMP) in a diverse set of 776 MOFs with Zr₆ nodes. We developed a 4-parameter machine learning model to predict DMMP diffusivities in Zr₆ MOFs and found the model to be transferable to the CWA sarin. We then developed a simplified heuristic based on the machine learning model that the node-node distance and accessible surface area should be maximized to find MOFs with rapid CWA diffusion.

Molecular modeling of thin-film nanocomposite membranes for reverse osmosis water desalination

The scarcity of freshwater resources is a major global challenge causedby population and economic growth. Water desalination using a reverse osmosis (RO) membrane is a promising technology to supply potable water from seawater and brackish water. The advancement of RO desalination highly depends on new membrane materials. Currently, the RO technology mainly relies on polyamide thin-film composite (TFC) membranes, which suffer from several drawbacks (e.g., low water permeability, permeability-selectivity tradeoff, and low fouling resistance) that hamper their real-world applications. Nanoscale fillers with specific characteristics can be used to improve the properties of TFC membranes. Embedding nanofillers into TFC membranes using interfacial polymerization allows the creation of thin-film nanocomposite (TFNC) membranes, and has become an emerging strategy in the fabrication of high-performance membranes for advanced RO water desalination. To achieve optimal design, it is indispensable to search for reliable methods that can provide fast and accurate predictions of the structural and transport properties of the TFNC membranes. However, molecular understanding of permeability-selectivity characteristics of nanofillers remains limited, partially due to the challenges in experimentally exploring microscopic behaviors of water and salt ions in confinement. Molecular modeling and simulations can fill this gap by generating molecular-level insights into the effects of nanofillers' characteristics (e.g., shape, size, surface chemistry, and density) on water permeability and ion selectivity. In this review, we summarize molecular simulations of a diverse range of nanofillers including nanotubes (carbon nanotubes, boron nitride nanotubes, and aquaporin-mimicking nanochannels) and nanosheets (graphene, graphene oxide, boron nitride sheets, molybdenum disulfide, metal and covalent organic frameworks) for water desalination applications. These simulations reveal that water permeability and salt rejection, as the major factors determining the desalination performance of TFNC membranes, significantly depend on the size, topology, density, and chemical modifications of the nanofillers. Identifying their influences and the physicochemical processes behind, via molecular modeling, is expected to yield important insights for the fabrication and optimization of the next generation high-performance TFNC membranes for RO water desalination.

Autonomous circular heat engine

A dynamical model of a highly efficient heat engine is proposed, where an applied temperature difference maintains the motion of particles around the circuit consisting of two asymmetric narrow channels, in one of which the current flows against the applied thermodynamic forces. Numerical simulations and linear-response analysis suggest that, in the absence of frictional losses, the Carnot efficiency can be achieved in the thermodynamic limit.

Quantitating Diffusion Enhancement in Pore Hierarchies

A microporous continuum traversed by a set of mutually perpendicular channels is considered to be a model for a hierarchically porous system of the mesoporous zeolite type. Transient profiles of molecular uptake as determined by kinetic Monte Carlo (kMC) simulation are found to be in excellent agreement with the result attained by the application of the two-region model (the Kärger model) of molecular diffusion. In particular, it is found that, in the two limiting cases referred to as fast exchange and slow exchange, there exist two simple analytical expressions for the rate of molecular uptake and hence for the quantification of transport enhancement in comparison with the purely microporous adsorbent. In the general case, transport enhancement is simply recognized by the reciprocal addition of the expressions in the two limiting cases.

Confinement Effects on Distribution and Transport of Neutral Solutes in a Small Hydrophobic Nanopore

Molecular dynamics simulations are used to study confinement effects in small cylindrical silica pores with extended hydrophobic surface functionalization as realized, for example, in reversed-phase liquid chromatography (RPLC) columns. In particular, we use a 6 nm cylindrical and a 10 nm slit pore bearing the same C₁₈ stationary phase to compare the conditions inside the smaller-than-average pores within an RPLC column to column-averaged properties. Two small, neutral, apolar to moderately polar solutes are used to assess the consequences of spatial confinement for typical RPLC analytes with water (W)-acetonitrile (ACN) mobile phases at W/ACN ratios between 70/30 and 10/90 (v/v). The simulated data show that true bulk liquid behavior, as observed over an extended center region in the 10 nm slit pore, is not recovered within the 6 nm cylindrical pore. Instead, the ACN-enriched solvent layer around the C₁₈ chain ends (the ACN ditch), a general feature of hydrophobic interfaces equilibrated with aqueous-organic liquids, extends over the entire pore lumen of the small cylindrical pore. This renders the entire pore a highly hydrophobic environment, where, contrary to column-averaged behavior, neither the local nor the pore-averaged sorption and diffusion of analytes scales directly with the W/ACN ratio of the mobile phase. Additionally, the solute polarity-related discrimination between analytes is enhanced. The consequences of local ACN ditch overlap in RPLC columns are reminiscent of ion transport in porous media with charged surfaces, where electrical double-layer overlap occurring locally in smaller pores leads to discrimination between co- and counterionic species.

Relationship Between the Mobility of Aggregates and Fluid Penetration Depth Across a Range of Fractal Dimensions Using Stokesian Dynamics

The hydrodynamic behavior of fractal aggregates plays an important role in various applications in industry and the environment, and has been a topic of interest over the past several decades. Despite this, crucial aspects such as the relationship of the mobility radius, R_m, with respect to the fractal dimension, d_f, and the fluid penetration depth, δ, have largely remained unexplored. Herein, we examine these aspects across a wide range of d_f's through a Stokesian dynamics approach. It takes into account all orders of monomer-monomer interactions to construct the resistance matrix for the entire cluster, which is assumed to be rigid. Statistical fractals created using algorithms such as diffusion limited aggregation (DLA), cluster-cluster aggregation (CCA), tunable Monte Carlo algorithm, and a deterministic Vicsek fractal, with d_f varying from 1.76 to 3, and the number of monomers ranging from 20 to 10 240 are considered. While confirming the expected asymptotic cluster-size independence of the hydrodynamic ratio, β = R_m/R_g (where R_g is the radius of gyration of the cluster), this study reveals a monotonically increasing trend for β with increasing d_f. The decay of the fluid velocity within the aggregate is quantified via the concept of penetration depth (δ). Analysis shows that the dimensionless penetration depth (δ* = δ/R_g) approaches asymptotic constancy with respect to cluster size in contrast to a weak dependency of the form δ* ∼ (R_g/a)^{-(d_f - 1)/2}, predicted by the mean-field theory (a being the monomer radius). Furthermore, the penetration depth is found to decrease rapidly, in an exponential manner, with increasing β. This establishes a quantitative relationship between the resistance experienced by the cluster and the degree of penetration of fluid into it. The implications of these results are further discussed.

An atomistic view on the uptake of aromatic compounds by cyclodextrin immobilized on mesoporous silica

The effect of immobilized $$\upbeta$$ β -cyclodextrin (bCD) molecules inside a mesoporous silica support on the uptake of benzene and p-nitrophenol from aqueous solution was investigated using all-atom molecular dynamics (MD) simulations. The calculated adsorption isotherms are discussed with respect to the free energies of binding for a 1:1 complex of bCD and the aromatic guest molecule. The adsorption capacity of the bCD-containing material significantly exceeds the amount corresponding to a 1:1 binding scenario, in agreement with experimental observations. Beside the formation of 1:2 and, to a lesser extent, 1:3 host:guest complexes, also host–host interactions on the surface as well as more unspecific host–guest interactions govern the adsorption process. The demonstrated feasibility of classical all-atom MD simulations to calculate liquid phase adsorption isotherms paves the way to a molecular interpretation of experimental data that are too complex to be described by empirical models.

Mass transfer of toluene in a series of metal-organic frameworks: molecular clusters inside the nanopores cause slow and step-like release

The mass transfer of the guest molecules in the pores is fundamental for the application of nanoporous materials like metal-organic frameworks, MOFs. In the present work, we explore the uptake and release of toluene in a series of Zr-based MOFs with different pore sizes. We find that intermolecular guest-guest interaction, sterically controlled by the pore size, has a substantial impact on the release kinetics. While the adsorption is rather fast, the desorption process is many orders of magnitude slower. Depending on the pore size, molecular clusters form, here (most likely) toluene dimers, which are rather stable and their break-up is rate-limiting during the desorption process. This results in a step-like desorption kinetics, deviating from the plain Fickian-diffusion-controlled release. Temperature-dependent experiments show that the minimum and maximum of the release rates are obtained at the same toluene loadings, independent of the temperature. Moreover, the activation energy for the release coincides with the binding energy of a toluene dimer. The work shows the importance of intermolecular guest-guest interaction, controlled by the MOF-nanoconfinement, for the uptake and release from nanoporous materials.

Use of Boundary-Driven Nonequilibrium Molecular Dynamics for Determining Transport Diffusivities of Multicomponent Mixtures in Nanoporous Materials

The boundary-driven molecular modeling strategy to evaluate mass transport coefficients of fluids in nanoconfined media is revisited and expanded to multicomponent mixtures. The method requires setting up a simulation with bulk fluid reservoirs upstream and downstream of a porous media. A fluid flow is induced by applying an external force at the periodic boundary between the upstream and downstream reservoirs. The relationship between the resulting flow and the density gradient of the adsorbed fluid at the entrance/exit of the porous media provides for a direct path for the calculation of the transport diffusivities. It is shown how the transport diffusivities found this way relate to the collective, Onsager, and self-diffusion coefficients, typically used in other contexts to describe fluid transport in porous media. Examples are provided by calculating the diffusion coefficients of a Lennard-Jones (LJ) fluid and mixtures of differently sized LJ particles in slit pores, a realistic model of methane in carbon-based slit pores, and binary mixtures of methane with hypothetical counterparts having different attractions to the solid. The method is seen to be robust and particularly suited for the study of study of transport of dense fluids and liquids in nanoconfined media.

MOF-enabled confinement and related effects for chemical catalyst presentation and utilization

A defining characteristic of nearly all catalytically functional MOFs is uniform, molecular-scale porosity. MOF pores, linkers and nodes that define them, help regulate reactant and product transport, catalyst siting, catalyst accessibility, catalyst stability, catalyst activity, co-catalyst proximity, composition of the chemical environment at and beyond the catalytic active site, chemical intermediate and transition-state conformations, thermodynamic affinity of molecular guests for MOF interior sites, framework charge and density of charge-compensating ions, pore hydrophobicity/hydrophilicity, pore and channel rigidity vs. flexibility, and other features and properties. Collectively and individually, these properties help define overall catalyst functional behaviour. This review focuses on how porous, catalyst-containing MOFs capitalize on molecular-scale confinement, containment, isolation, environment modulation, energy delivery, and mobility to accomplish desired chemical transformations with potentially superior selectivity or other efficacy, especially in comparison to catalysts in homogeneous solution environments.

Green Synthesis of Zeolitic Imidazolate Frameworks: A Review of Their Characterization and Industrial and Medical Applications

Metal organic frameworks (MOF) are a class of hybrid networks of supramolecular solid materials comprising a large number of inorganic and organic linkers, all bound to metal ions in a well-organized fashion. Zeolitic imidazolate frameworks (ZIFs) are a sub-group of MOFs with imidazole as an organic linker to metals; it is rich in carbon, nitrogen, and transition metals. ZIFs combine the classical zeolite characteristics of thermal and chemical stability with pore-size tunability and the rich topological diversity of MOFs. Due to the energy crisis and the existence of organic solvents that lead to environmental hazards, considerable research efforts have been devoted to devising clean and sustainable synthesis routes for ZIFs to reduce the environmental impact of their preparation. Green chemistry is the key to sustainable development, as it will lead to new solutions to existing problems. Moreover, it will present opportunities for new processes and products and, at its heart, is scientific and technological innovation. The green chemistry approach seeks to redesign the materials that make up the basis of our society and our economy, including the materials that generate, store, and transport our energy, in ways that are benign for humans and the environment and that possess intrinsic sustainability. This study covers the principles of green chemistry as used in designing strategies for synthesizing greener, less toxic ZIFs the consume less energy to produce. First, the necessity of green methods in today's society, their replacement of the usual non-green methods and their benefits are discussed; then, various methods for the green synthesis of ZIF compounds, such as hydrothermally, ionothermally, and by the electrospray technique, are considered. These methods use the least harmful and toxic substances, especially concerning organic solvents, and are also more economical. When a compound is synthesized by a green method, a question arises as to whether these compounds can replace the same compounds as synthesized by non-green methods. For example, is the thermal stability of these compounds (which is one of the most important features of ZIFs) preserved? Therefore, after studying the methods of identifying these compounds, in the last part, there is an in-depth discussion on the various applications of these green-synthesized compounds.

Enhanced Adsorption and Mass Transfer of Hierarchically Porous Zr-MOF Nanoarchitectures toward Toxic Chemical Removal

Zirconium-based metal-organic frameworks (Zr-MOFs) have shown tremendous prospects as highly efficient adsorbents against toxic chemicals under ambient conditions. Here, we report for the first time the enhanced toxic chemical adsorption and mass transfer properties of hierarchically porous Zr-MOF nanoarchitectures. A general and scalable sol-gel-based strategy combined with facile ambient pressure drying (APD) was utilized to construct MOF-808, MOF-808-NH₂, and UiO-66-NH₂ xerogel monoliths, denoted as G808, G808-NH₂, and G66-NH₂, respectively. The resulting Zr-MOF xerogels demonstrated 3D porous networks assembled by nanocrystal aggregates, with substantially higher mesoporosities than the precipitate analogues. Microbreakthrough tests on powders and tube breakthrough experiments on engineered granules were conducted at different relative humidities to comprehensively evaluate the NO₂ adsorption capabilities. The Zr-MOF xerogels showed considerably better NO₂ removal abilities than the precipitates, whether intrinsically or under simulated respirator canister/protection filter environment conditions. Multiple physicochemical characterizations were conducted to illuminate the NO₂ filtration mechanisms. Analysis on adsorption kinetics and mass transfer patterns in Zr-MOF xerogels was further performed to visualize the underlying structure-activity relationship using the gravimetric uptake and zero length column methods with cyclohexane and acetaldehyde as probes. The results revealed that the synergy of hierarchical porosities and nanosized crystals could effectively expedite the intracrystalline diffusion for the G66-NH₂ xerogel as well as alleviate the surface resistance for the G808-NH₂ xerogel, which led to accelerated overall adsorption uptake and thus enhanced performance toward toxic chemical removal.

ResponZIF Structures: Zeolitic Imidazolate Frameworks as Stimuli-Responsive Materials

Zeolitic imidazolate frameworks (ZIFs) have long been recognized as a prominent subset of the metal-organic framework (MOF) family, in part because of their ease of synthesis and good thermal and chemical stability, alongside attractive properties for diverse potential applications. Prototypical ZIFs like ZIF-8 have become embodiments of the significant promise held by porous coordination polymers as next-generation designer materials. At the same time, their intriguing property of experiencing significant structural changes upon the application of external stimuli such as temperature, mechanical pressure, guest adsorption, or electromagnetic fields, among others, has placed this family of MOFs squarely under the umbrella of stimuli-responsive materials. In this review, we provide an overview of the current understanding of the triggered structural and electronic responses observed in ZIFs (linker and bond dynamics, crystalline and amorphous phase changes, luminescence, etc.). We then describe the state-of-the-art experimental and computational methodology capable of shedding light on these complex phenomena, followed by a comprehensive summary of the stimuli-responsive nature of four prototypical ZIFs: ZIF-8, ZIF-7, ZIF-4, and ZIF-zni. We further expose the relevant challenges for the characterization and fundamental understanding of responsive ZIFs, including how to take advantage of their flexible properties for new application avenues.

Mixed-dimensional membranes: chemistry and structure-property relationships

Tremendous progress in two-dimensional (2D) nanomaterial chemistry affords abundant opportunities for the sustainable development of membranes and membrane processes. In this review, we propose the concept of mixed dimensional membranes (MDMs), which are fabricated through the integration of 2D materials with nanomaterials of different dimensionality and chemistry. Complementing mixed matrix membranes or hybrid membranes, MDMs stimulate different conceptual thinking about designing advanced membranes from the angle of the dimensions of the building blocks as well as the final structures, including the nanochannels and the bulk structures. In this review, we survey MDMs (denoted nD/2D, where n is 0, 1 or 3) in terms of the dimensions of membrane-forming nanomaterials, as well as their fabrication methods. Subsequently, we highlight three kinds of nanochannels, which are 1D nanochannels within 1D/2D membranes, 2D nanochannels within 0D/2D membranes, and 3D nanochannels within 3D/2D membranes. Strategies to tune the physical and chemical microenvironments of the nanochannels as well as the bulk structures based on the size, type, structure and chemical character of nanomaterials are discussed. Some representative applications of MDMs are illustrated for gas molecular separations, liquid molecular separations, ionic separations and oil/water separation. Finally, current challenges and a future perspective on MDMs are presented.

Nanoconfinement and mass transport in metal-organic frameworks

The ubiquity of metal-organic frameworks in recent scientific literature underscores their highly versatile nature. MOFs have been developed for use in a wide array of applications, including: sensors, catalysis, separations, drug delivery, and electrochemical processes. Often overlooked in the discussion of MOF-based materials is the mass transport of guest molecules within the pores and channels. Given the wide distribution of pore sizes, linker functionalization, and crystal sizes, molecular diffusion within MOFs can be highly dependent on the MOF-guest system. In this review, we discuss the major factors that govern the mass transport of molecules through MOFs at both the intracrystalline and intercrystalline scale; provide an overview of the experimental and computational methods used to measure guest diffusivity within MOFs; and highlight the relevance of mass transfer in the applications of MOFs in electrochemical systems, separations, and heterogeneous catalysis.

A Short Review on the Utilization of Incense Sticks Ash as an Emerging and Overlooked Material for the Synthesis of Zeolites

The traditional hydrothermal synthesis methods are mainly performed under batch operation, which generally takes few days to weeks to yield a zeolite with the desired properties and structure. The zeolites are the backbone of the petrochemical and wastewater industries due to their importance. The commercial methods for zeolite synthesis are expensive, laborious and energy intensive. Among waste products, incense sticks ash is a compound of aluminosilicates and could act as a potential candidate for the synthesis of zeolites for daily needs in these industries. Incense sticks ash is the byproduct of religious places and houses and is rich in Ca, Mg, Al and Si. As a result, incense sticks ash can be proven to be a potential candidate for the formation of calcium-rich zeolites. The formation of zeolites from incense sticks ash is an economical, reliable and eco-friendly method. The application of incense sticks ash for zeolite synthesis can also minimize the problem related to its disposal in the water bodies, which will also minimize the solid waste in countries where it is considered sacred and generated in tons every day.

Molecular Insight into the Slow Dynamics of C4 Hydrocarbons in the Zeolitic-Imidazole Framework (ZIF-8)

The family of zeolitic-imidazole framework (ZIF) materials is currently considered for the challenging separation of C4 hydrocarbons. However, yet, the microscopic diffusion mechanism for these hydrocarbons in these narrow gate porous materials remains elusive by conventional methods due to its very slow nature. Experimental (solid-state ²H nuclear magnetic resonance-NMR) and computational (molecular dynamics-MD) approaches were applied together to derive slow diffusional dynamics of n-butane and 1-butene in ZIF-8. By means of the ²H NMR technique, we evidenced the presence of two adsorption sites for the guests localized inside the cages of ZIF-8 and in the vicinity of the gates. We characterized the molecular mobility at each site and revealed that the translational intercage diffusion is realized by a slow directional motion associated with the gate-crossing. MD simulations provide an in-depth analysis of the diffusion and fully support the proposed dynamics picture for both n-butane and 1-butene. These calculations enable the derivation of the diffusivity and barriers for the long-range diffusion of both hydrocarbons in ZIF-8 and unraveled the microscopic diffusion mechanism implying intracage and intercage motions. We show that this NMR approach combined with modeling is a valuable tool to probe the molecular mobility for slow diffusing species in ordered cagelike porous frameworks.