Predictive Assessment of Surface Resistances in Zeolite Membranes Using Atomically Detailed Models

The use of collective variables and enhanced sampling in the simulations of existing and emerging microporous materials

Microporous materials, including zeolites, metal-organic frameworks, and cage compounds, offer diverse functionalities due to their unique dynamics and guest confinement properties. These materials play a significant role in separation, catalysis, and sensing, but their complexity hinders exploration using traditional atomistic simulations. This review explores collective variables (CVs) paired with enhanced sampling as a powerful approach to enable efficient investigation of key features in microporous materials. We highlight successful applications of CVs in studying adsorption, diffusion, phase transitions, and mechanical properties, demonstrating their crucial role in guiding material design and optimisation. The future of CVs lies in integration with techniques like machine learning, allowing for enhanced efficiency and accuracy. By tailoring CVs to specific materials and developing multi-scale approaches we can further unlock the intricacies of these fascinating materials. Simulations are a cornerstone in unravelling the complexities of microporous materials and are crucial for our future understanding.

Separation of SO2 and NO2 with the Zeolite Membrane: Molecular Simulation Insights into the Advantageous NO2 Dimerization Effect

NO₂ and SO₂, as valuable chemical feedstock, are worth being recycled from flue gases. The separation of NO₂ and SO₂ is a key process step to enable practical deployment. This work proposes SO₂ separation from NO₂ using chabazite zeolite (SSZ-13) membranes and provides insights into the feasibility and advantages of this process using molecular simulation. Grand canonical ensemble Monte Carlo and equilibrium molecular dynamics methods were respectively adopted to simulate the adsorption equilibria and diffusion of SO₂, NO₂, and N₂O₄ on SSZ-13 at varying Si/Al (1, 5, 11, 71, +∞), temperatures (248-348 K), and pressures (0-100 kPa). The adsorption capacity and affinity (SO₂ > N₂O₄ > NO₂) demonstrated strong competitive adsorption of SO₂ based on dual-site interactions and significant reduction in NO₂ adsorption due to dimerization in the ternary gas mixture. The simulated order of diffusivity (NO₂ > SO₂ > N₂O₄) on SSZ-13 demonstrated rapid transport of NO₂, strong temperature dependence of SO₂ diffusion, and the impermeability of SSZ-13 to N₂O₄. The membrane permeability of each component was simulated, rendering a SO₂/NO₂ membrane separation factor of 26.34 which is much higher than adsorption equilibrium (6.9) and kinetic (2.2) counterparts. The key role of NO₂-N₂O₄ dimerization in molecular sieving of SO₂ from NO₂ was addressed, providing a facile membrane separation strategy at room temperature.

Nonuniformity of Transport Coefficients in Ultrathin Nanoscale Membranes and Nanomaterials

The quest to reduce transport resistance in separations using nanomaterials has led to considerable interest in nanoscale adsorbents and ultrathin membranes. It is now established that interfacial resistance limits the performance of such nanosized materials; however, the origin of this resistance is uncertain. While it is associated with surface pore blockages and distortions in some materials, its existence even in ideal materials is largely putative. Here, we report equilibrium molecular dynamics (EMD) simulations with ideal zeolite-based nanosheets, indicating the transport resistance to be entirely distributed within the solid, without contribution from an interfacial effect. We demonstrate the presence of an internal entry region over which fluid decorrelation occurs, and in which the local transport coefficient inside the crystal is nonuniform and position-dependent, increasing to the uniform value in the bulk material at larger distances. Our EMD-based diffusivity profiles within the nanomaterial enable us to unequivocally determine the entry length, and reveal an internal excess resistance, frequently assumed to be an interfacial resistance, due to significant reduction of the internal transport coefficient in the entrance and exit regions. A decrease in the entry length with loading in PON zeolite nanosheets is seen. We demonstrate a reduction in external resistance in the external bulk chambers used in simulations, triggered by the interplay of incomplete decorrelation in the nanosheet and periodic boundary conditions imposed on the system comprising the nanosheet and surrounding bulk reservoirs when the nanosheet thickness is less than the entry length. Our analysis of the transport dynamics within the nanosheet demonstrates that, at least for ideal systems, decomposition of the inhomogeneous diffusivity-based internal resistance into an interfacial and a uniform transport coefficient-based internal contribution is not appropriate for finite-sized systems. Our results will enable the improved design of nanoscale membranes and materials for applications in separation and other processes.

Recent advances in simulating gas permeation through MOF membranes

In the last two decades, metal organic frameworks (MOFs) have gained increasing attention in membrane-based gas separations due to their tunable structural properties. Computational methods play a critical role in providing molecular-level information about the membrane properties and identifying the most promising MOF membranes for various gas separations. In this review, we discuss the current state-of-the-art in molecular modeling methods to simulate gas permeation through MOF membranes and review the recent advancements. We finally address current opportunities and challenges of simulating gas permeation through MOF membranes to guide the development of high-performance MOF membranes in the future.

Understanding CO2/CH4 Separation in Pristine and Defective 2D MOF CuBDC Nanosheets via Nonequilibrium Molecular Dynamics

The separation of CO₂/CH₄ gas mixtures is a key challenge for the energy sector and is essential for the efficient upgrading of natural gas and biogas. A new emerging field, that of metal-organic framework nanosheets (MONs), has shown the potential to outperform conventional separation methods and bulk metal-organic frameworks (MOFs). In this work, we model the CO₂/CH₄ separation in both defect-free and defective 2D CuBDC nanosheets and compare their performance with the bulk CuBDC MOF and experimental data. We report the results of external force nonequilibrium molecular dynamics (EF-NEMD) for pure components and binary mixtures. The EF-NEMD simulations reveal a pore blocking separation mechanism, in which the CO₂ molecules occupy adsorption sites and significantly restrict the diffusion of CH₄. The MON structure achieves a better selectivity of CO₂ over CH₄ compared to the bulk CuBDC MOF which is due to the mass transfer resistance of the methane molecules on the surface of the nanosheet. Our results show that it is essential to consider the real mixture in these systems rather than relying solely on pure component data and ideal selectivity. Furthermore, the separation is shown to be sensitive to the presence of missing linker defects in the nanosheets. Only 10% of missing linkers result in nonselective nanosheets. Hence, the importance of a defect-free synthetic method for CuBDC nanosheets is underlined.

MOF-Based Membranes for Gas Separations

Metal-organic frameworks (MOFs) represent the largest known class of porous crystalline materials ever synthesized. Their narrow pore windows and nearly unlimited structural and chemical features have made these materials of significant interest for membrane-based gas separations. In this comprehensive review, we discuss opportunities and challenges related to the formation of pure MOF films and mixed-matrix membranes (MMMs). Common and emerging separation applications are identified, and membrane transport theory for MOFs is described and contextualized relative to the governing principles that describe transport in polymers. Additionally, cross-cutting research opportunities using advanced metrologies and computational techniques are reviewed. To quantify membrane performance, we introduce a simple membrane performance score that has been tabulated for all of the literature data compiled in this review. These data are reported on upper bound plots, revealing classes of MOF materials that consistently demonstrate promising separation performance. Recommendations are provided with the intent of identifying the most promising materials and directions for the field in terms of fundamental science and eventual deployment of MOF materials for commercial membrane-based gas separations.

Separation Selectivity of CH4/CO2 Gas Mixtures in the ZIF-8 Membrane Explored by Dynamic Monte Carlo Simulations

Here we report a series of nonequilibrium dynamic Monte Carlo simulations combined with dual control volume (DCV-DMC) to explore the separation selectivity of CH₄/CO₂ gas mixtures in the ZIF-8 membrane with a thickness of up to about 20 nm. Meanwhile, an improved DCV-DMC approach coupled with the corresponding potential map (PM-DCV-DMC) is further developed to speed up the computational efficiency of conventional DCV-DMC simulations. Our simulation results provide the molecular-level density and selectivity profiles along the permeation direction of both CH₄ and CO₂ molecules in the ZIF-8 membrane, indicating that the parts near membrane surfaces at both ends play a key role in determining the separation selectivity. All densities initially show a sharp increase in the individual maximum within the first outermost unit cell at the feed side and follow a long fluctuating decrease process. Accordingly, the corresponding selectivity profiles initially display a long fluctuating increase in the individual maximum and follow a sharp decrease near the membrane surface at the permeation side. Furthermore, the effects of feed composition, temperature, and pressure on the relevant separation selectivity are also discussed in detail, where the temperature has a greater influence on the separation selectivity than the feed composition and pressure. More importantly, the predicted separation selectivities from our PM-DCV-DMC simulations are well consistent with previous experimental results.

Simulation of H2/CH4 mixture permeation through MOF membranes using non-equilibrium molecular dynamics

Grand canonical Monte Carlo (GCMC) simulations are widely used with equilibrium molecular dynamics (EMD) to predict gas adsorption and diffusion in single-crystals of metal-organic frameworks (MOFs). Adsorption and diffusion data obtained from these simulations are then combined to predict gas permeabilities and selectivities of MOF membranes. This GCMC + EMD approach is highly useful to screen a large number of MOFs for a target membrane-based gas separation process. External field non-equilibrium molecular dynamics (NEMD) simulations, on the other hand, can directly compute gas permeation by providing an accurate representation of MOF membranes but they are computationally demanding and require long simulation times. In this work, we performed NEMD simulations to investigate H₂/CH₄ separation performances of MOF membranes. Both single-component and binary mixture permeabilities of H₂ and CH₄ were computed using the NEMD approach and results were compared with the predictions of the GCMC + EMD approach and experimental measurements reported in the literature. Our results showed that there is a good agreement between NEMD simulations and experiments for the permeability and selectivity of MOF membranes. NEMD simulations provided the direct observation of the mass transfer resistances on the pore mouth of MOF membranes, which is neglected in the GCMC + EMD approach. Our results suggested that once the very large numbers of MOF materials were screened using the GCMC + EMD approach, more detailed NEMD calculations can be performed for the best membrane candidates to unlock the actual gas transport mechanism before the experimental fabrication of MOF membranes.

Interfacial barriers to gas transport in zeolites: distinguishing internal and external resistances

The gas separation performance of ultrathin membranes is dictated by the interfacial barriers that exist on the solid side of the interface.

Computational study of pressure-driven methane transport in hierarchical nanostructured porous carbons

Using the reflecting particle method together with a perturbation-relaxation loop developed in our previous work, we studied pressure-driven methane transport in hierarchical nanostructured porous carbons (HNPCs) containing both mesopores and micropores in non-equilibrium molecular dynamics simulations. The surface morphology of the mesopore wall was systematically varied by tuning interaction strength between carbon atoms and the template in a mimetic nanocasting process. Effects of temperature and mesopore size on methane transport in HNPCs were also studied. Our study shows that increased mesopore wall surface roughness changes the character of the gas-wall interaction from specular to diffuse, while the gas-gas interaction is diminished due to the decrease of adsorption density. Effects of the mesopore wall surface morphology are the most significant at low temperatures and in small channels. Our systematic study provides a better understanding of the transport mechanisms of light gases through carbon nanotube composite membranes in experiments.

Thermodynamic Resistance to Matter Flow at The Interface of a Porous Membrane

Nanoporous materials are important in industrial separation, but their application is subject to strong interfacial barriers to the entry and transport of fluids. At certain conditions the fluid inside and outside the nanoporous material can be viewed as a two-phase system, with an interface between them, which poses an excess resistance to matter flow. We show that there exist two kinds of phenomena which influence the interfacial resistance: hydrodynamic effects and thermodynamic effects, which are independent of each other. Here, we investigate the role of the thermodynamic effects in carbon nanotubes (CNTs) and slit pores and compare the associated thermodynmic resistance with that due to hydrodynamic effects traditionally modeled by the established Sampson expression. Using CH4 and CO2 as model fluids, we show that the thermodynamic resistance is especially important for moderate to high pressures, at which the fluid within the CNT or slit pore is in the condensed state. Further, we show that at such pressures the thermodynamic resistance becomes comparable with the internal resistance to fluid transport at length scales typical of membranes used in fuel cells, and of importance in membrane-based separation, and nanofluidics in general.

Computational Study of Pressure-Driven Gas Transport in Nanostructured Carbons: An Alternative Approach

We demonstrated a computationally efficient method in nonequilibrium molecular dynamics (NEMD) simulations to study pressure-driven gas transport in porous media. The reflecting particle method (RPM)14 was used to establish a steady-state gas flow along the transport channel, and the gas density in the feed chamber was properly adjusted to allow a constant pressure drop under various conditions by using a perturbation-relaxation loop developed here. This method was validated for methane flow through carbon nanotubes over a wide range of temperatures, giving results comparable to those of the commonly used dual control volume grand canonical molecular dynamics (DCV-GCMD) method but at least 20 times more efficient, even though the transport condition tested is favorable for the latter. This made it possible to perform systematic studies on the effects of temperature, pressure, and channel size on the transport behaviors. Our study shows that adsorption density varies significantly with temperature, which dramatically influences the transport mechanisms, especially in small channels at low temperatures and under high pressures. This newly developed NEMD method can be readily extended to study gas transport through channels with more complex surface morphology.

Correlating surface permeability with intracrystalline diffusivity in nanoporous solids

The rates of uptake and release of guest molecules in nanoporous solids are often strongly influenced or even controlled by transport resistances at the external surface ("surface barriers") rather than by intraparticle diffusion, which was assumed to be rate controlling in many of the earlier kinetic studies. By correlating the surface resistance with the intracrystalline diffusivity, we develop here a microkinetic model which closely reproduces the experimentally observed results for short-chain alkanes in Zn(tbip), a member of the novel metal-organic framework family of nanoporous materials. It seems likely that this mechanism, which is shown to provide a rational explanation of the commonly observed discrepancies between "macro" and "micro" measurements of intracrystalline diffusion, may be fairly general.

Self-diffusion of water and simple alcohols in single-walled aluminosilicate nanotubes

Understanding transport phenomena of fluids through nanotubes (NTs) is of great interest in order to enable potential application of NTs as separation devices, encapsulation media for molecule storage and delivery, and sensors. Single-walled metal oxide NTs are interesting materials because they present a well-defined solid-state structure, precisely tunable diameter and length, as well as a hydrophilic and functionalizable interior for tuning transport and adsorption selectivity. Here, we study the transport properties of hydrogen-bonding liquids (water, methanol, and ethanol) through a single-walled aluminosilicate NT to investigate the influence of liquid-surface and liquid-liquid interactions and the effects of competitive transport of different chemical species using molecular dynamics (MD) simulations. The self-diffusivities (D(s)) for all the three species decrease with increasing loading and are comparable to bulk liquid diffusivities at low molecular loadings. We show that the hydrogen-bond network associated with water makes its diffusion behavior different from methanol and ethanol. Mixtures of water and methanol show segregation in the NT, with water located closer to the tube wall and the alcohol molecules localized near the center of the NT. D(s) values of water in an analogous aluminogermanate NT are larger than those in the aluminosilicate NT due to a larger pore diameter.

Temperature control algorithms in dual control volume grand canonical molecular dynamics simulations of hydrogen diffusion in palladium

The effectiveness of five temperature control algorithms for dual control volume grand canonical molecular dynamics is investigated in the study of hydrogen atom diffusion in a palladium bulk. The five algorithms, namely, Gaussian, generalized Gaussian moment thermostat (GGMT), velocity scaling, Nosé-Hoover (NH), and its enhanced version Nosé-Hoover chain (NHC) are examined in both equilibrium and nonequilibrium simulation studies. Our numerical results show that Gaussian yields the most inaccurate solutions for the hydrogen-palladium system due to the high friction coefficient generated from the large velocity fluctuation of hydrogen, while NHC, NH, and GGMT produce the most accurate temperature and density profiles in both equilibrium and nonequilibrium cases with their feedback control mechanisms. However, this feedback control also overestimates the self-diffusion coefficients in equilibrium systems and the diffusion coefficient in nonequilibrium systems. Velocity scaling thermostat produces slight inhomogeneities in the temperature and density profiles, but due to the dissipated heat accumulated in the control volumes it still yields accurate self-diffusion coefficients that are in good agreement with the experimental data at a wide range of temperatures while others tend to deviate.

Hierarchical Nanomanufacturing: From Shaped Zeolite Nanoparticles to High-Performance Separation Membranes

Despite more than a decade of intense research on the high-resolution selectivity of thin zeolite films as alternatives to energy-intensive industrial separations, membranes consisting of intergrown, oriented zeolite crystals have fallen short of gaining wide commercial application. Factors including poor performance, high cost, and difficulties in scale up have contributed to this, and have also stunted their application in other niche markets. Until recently, rational design of these materials was limited because of the elusive mechanism of zeolite growth, and forced more empirical approaches. New understanding of zeolite growth along with recent advances in the molecular engineering of crystal microstructure and morphology, assembly of crystal monolayers, and synthesis of ordered films constitute a strong foundation for meeting stringent industrial demands in the future. Together with new processing capabilities, such a foundation should make it possible to synthesize commercially viable zeolite membranes through hierarchical approaches. Such advances open exciting prospects beyond the realm of separations for assembly of novel and complex functional materials including molecular sensors, mechanically stable dielectrics, and novel reaction-diffusion devices.

Molecular Dynamics Simulations of Mass Transfer Resistance in Grain Boundaries of Twinned Zeolite Membranes

The net mass transfer resistance for gas molecules permeating through zeolite membranes includes contributions from intracrystalline diffusion and contributions from interfacial effects. These interfacial effects can arise either from gas-zeolite interfaces or from interfaces that exist within zeolite crystals due to grain boundaries. We present the first atomically detailed simulations that examine interfacial mass transfer resistance due to internal grain boundaries in zeolites that are relevant for membrane applications. Our calculations examine twinned silicalite crystals in crystallographic configurations that have been identified in previous experiments. We used the dual control volume grand canonical molecular dynamics method to simulate the permeance of CH(4) and CF(4) through thin twinned silicalite crystals. The magnitudes of the grain boundary resistances are quite substantial, at least for the thin crystals that are accessible in our simulations.

Transport diffusion of gases is rapid in flexible carbon nanotubes

Molecular dynamics simulations of rigid, defect-free single-walled carbon nanotubes have previously suggested that the transport diffusivity of gases adsorbed in these materials can be orders of magnitude higher than any other nanoporous material (A. I. Skoulidas et al., Phys. Rev. Lett. 2002, 89, 185901). These simulations must overestimate the molecular diffusion coefficients because they neglect energy exchange between the diffusing molecules and the nanotube. Recently, Jakobtorweihen et al. have reported careful simulations of molecular self-diffusion that allow nanotube flexibility (Phys. Rev. Lett. 2005, 95, 044501). We have used the efficient thermostat developed by Jakobtorweihen et al. to examine the influence of nanotube flexibility on the transport diffusion of CH4 in (20,0) and (15,0) nanotubes. The inclusion of nanotube flexibility reduces the transport diffusion relative to the rigid nanotube by roughly an order of magnitude close to zero pressure, but at pressures above about 1 bar the transport diffusivities for flexible and rigid nanotubes are very similar, differing by less than a factor or two on average. Hence, the transport diffusivities are still extremely large compared to other known materials when flexibility is taken into account.

Influences of interfacial resistances on gas transport through carbon nanotube membranes

Carbon nanotubes have significant promise as gas separation membranes. Gas permeation through nanopores involves mass transfer resistances from molecules entering and leaving pores (so-called surface resistances) and diffusion within the pores. We use molecular simulations to give the first estimates of surface resistances for gas transport through nanotubes. For CH4 transport through (20,0) carbon nanotubes at 300 K, surface resistances are small for nanotubes 5-10 mum in length but can be significant for shorter nanotubes.

Understanding macroscopic diffusion of adsorbed molecules in crystalline nanoporous materials via atomistic simulations

The diffusion rates of molecules inside nanoporous materials lie at the heart of many large-scale industrial applications of these materials. Quantitatively describing this diffusion, particularly diffusion of chemical mixtures in situations leading to net mass transport, remains challenging. Molecular dynamics (MD) simulations can play an important complementary role to experiments in this area. This Account describes applications of MD to diffusion in nanoporous materials with a particular focus on macroscopic diffusion, that is, diffusion involving mass transport. These methods have made useful contributions to developing mixing theories for predicting multicomponent diffusion from single-component data and to screening new classes of materials for practical applications.

Mass transport of O2 and N2 in nanoporous carbon (C168 schwarzite) using a quantum mechanical force field and molecular dynamics simulations

A hierarchical approach is used to calculate the single-component fluxes of N2 and O2 in nanoporous carbon molecular sieves (represented by C168 schwarzite) over a wide range of pressures and pressure drops. The self- and corrected diffusivities are calculated using equilibrium molecular dynamics simulations with force fields for the gas-carbon interactions obtained from quantum mechanical calculations. These results are combined with previously reported adsorption isotherms of N2 and O2 in C168 to obtain transport diffusivities and, by use of the Fick's equation of mass transport, to obtain single-component fluxes across the membrane. The diffusion coefficients and fluxes are also calculated using an empirical potential, which has been obtained by fitting low coverage adsorption data of N2 and O2 on a planar graphite sheet. By analyzing the diffusivities calculated with the ab initio potential in the limit of infinite dilution over the temperature range from 80 to 450 K, it is observed that the N2/O2 separation is energetically driven and a high selectivity of O2 over N2 can be obtained at low temperatures. However, with the empirical potential both the energetic and entropic contributions to selectivity were found to be close to unity. Similarly, by calculating single-component fluxes and ideal selectivities at 300 K and finite pressures it is found that the ab initio potential better explains the large O2/N2 selectivities of similarly sized molecules that have been observed experimentally. An interesting reversal in ideal selectivity is observed by adjusting the pressure at the two ends of the membrane. As a consequence, we predict that a highly selective kinetic separation in favor of either nitrogen or oxygen could be obtained with the same membrane depending on the operating conditions.

The mechanism of water diffusion in narrow carbon nanotubes

Carbon nanotubes show exceptional physical properties that render them promising candidates as building blocks for nanostructured materials. Many ambitious applications, ranging from gene therapy to membrane separations, require the delivery of fluids, in particular aqueous solutions, through the interior of carbon nanotubes. To foster these and other applications, it is necessary to understand the thermodynamic and transport properties of water confined within long narrow carbon nanotubes. Previous theoretical work considered either short carbon nanotubes or short periods of time. By conducting molecular dynamics simulations in the microcanonical ensemble for water confined in infinitely long carbon nanotubes of diameter 1.08 nm, we show here that confined water molecules diffuse through a fast ballistic motion mechanism for up to 500 ps at room temperature. By comparing the results obtained for the diffusion of water to those obtained for the diffusion of a reference Lennard-Jones fluid, we prove here that long-lasting hydrogen bonds are responsible for the ballistic diffusion of water clusters in narrow carbon nanotubes, as opposed to spatial mismatches between pore-fluid and fluid-fluid attractive interactions which, as shown previously by others, are responsible for the concerted motion of simple fluids in molecular sieves. Additionally we prove here for the first time that, despite the narrow diameter of the carbon nanotubes considered which may suggest the existence of single-file diffusion, when the trajectories of confined water are studied at time scales in excess of 500 ps, a Fickian-type diffusion mechanism prevails. Our results are important for designing nano fluidic apparatuses to develop, for example, novel drug-delivery devices.

Adsorption and diffusion of carbon dioxide and nitrogen through single-walled carbon nanotube membranes

We have used atomically detailed simulations to examine the adsorption and transport diffusion of CO2 and N2 in single-walled carbon nanotubes at room temperature as a function of nanotube diameter. Linear and spherical models for CO2 are compared, showing that representing this species as spherical has only a slight impact in the computed diffusion coefficients. Our results support previous predictions that transport diffusivities of molecules inside carbon nanotubes are extremely rapid when compared with other porous materials. By examining carbon nanotubes as large as the (40,40) nanotube, we are able to compare the transport rates predicted by our calculations with recent experimental measurements. The predicted transport rates are in reasonable agreement with experimental observations.