Unified Maxwell–Stefan description of binary mixture diffusion in micro- and meso-porous materials

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.

Insights into Sorption and Molecular Transport of Aqueous Glucose into Zeolite Nanopores

Liquid-phase heterogeneous catalysis using zeolites is important for biomass conversion to fuels and chemicals. There is a substantial body of work on gas-phase sorption in zeolites with different topologies; however, studies investigating the diffusion of complex molecules in liquid medium into zeolitic nanopores are scarce. Here, we present a molecular dynamics study to understand the sorption and diffusion of aqueous β-d-glucose into β-zeolite silicate at T = 395 K and P = 1 bar. Through 2-μs-long molecular dynamics trajectories, we reveal the role of the solvent, the kinetics of the pore filling, and the effect of the water model on these properties. We find that the glucose and water loading is a function of the initial glucose concentration. Although the glucose concentration increases monotonically with the initial glucose concentration, the water loading exhibits a nonmonotonic behavior. At the highest initial concentration (∼20 wt %), we find that the equilibrium loading of glucose is approximately five molecules per unit cell and displays a weak dependence on the water model. Glucose molecules follow a single-file diffusion in the nanopores due to confinement. The dynamics of glucose and water molecules slows significantly at the interface. The average residence time for glucose molecules is an order of magnitude larger than that in the bulk solution, while it is about twice as large for the water molecules. Our simulations reveal critical molecular details of the glucose molecule's local environment inside the zeolite pore relevant to catalytic conversion of biomass to valuable chemicals.

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.

Machine Learning-Based Upscaling of Finite-Size Molecular Dynamics Diffusion Simulations for Binary Fluids

Molecular diffusion coefficients calculated using molecular dynamics (MD) simulations suffer from finite-size (i.e., finite box size and finite particle number) effects. Results from finite-sized MD simulations can be upscaled to infinite simulation size by applying a correction factor. For self-diffusion of single-component fluids, this correction has been well-studied by many researchers including Yeh and Hummer (YH); for binary fluid mixtures, a modified YH correction was recently proposed for correcting MD-predicted Maxwell-Stephan (MS) diffusion rates. Here we use both empirical and machine learning methods to identify improvements to the finite-size correction factors for both self-diffusion and MS diffusion of binary Lennard-Jones (LJ) fluid mixtures. Using artificial neural networks (ANNs), the error in the corrected LJ fluid diffusion is reduced by an order of magnitude versus existing YH corrections, and the ANN models perform well for mixtures with large dissimilarities in size and interaction energies where the YH correction proves insufficient.

Thermodynamically Consistent Methodology for Estimation of Diffusivities of Mixtures of Guest Molecules in Microporous Materials

The Maxwell-Stefan (M-S) formulation, that is grounded in the theory of irreversible thermodynamics, is widely used for describing mixture diffusion in microporous crystalline materials such as zeolites and metal-organic frameworks (MOFs). Binary mixture diffusion is characterized by a set of three M-S diffusivities: Đ ₁, Đ ₂, and Đ ₁₂. The M-S diffusivities Đ ₁ and Đ ₂ characterize interactions of guest molecules with pore walls. The exchange coefficient Đ ₁₂ quantifies correlation effects that result in slowing-down of the more mobile species due to correlated molecular jumps with tardier partners. The primary objective of this article is to develop a methodology for estimating Đ ₁, Đ ₂, and Đ ₁₂ using input data for the constituent unary systems. The dependence of the unary diffusivities Đ ₁ and Đ ₂ on the pore occupancy, θ, is quantified using the quasi-chemical theory that accounts for repulsive, or attractive, forces experienced by a guest molecule with the nearest neighbors. For binary mixtures, the same occupancy dependence of Đ ₁ and Đ ₂ is assumed to hold; in this case, the occupancy, θ, is calculated using the ideal adsorbed solution theory. The exchange coefficient Đ ₁₂ is estimated from the data on unary self-diffusivities. The developed estimation methodology is validated using a large data set of M-S diffusivities determined from molecular dynamics simulations for a wide variety of binary mixtures (H₂/CO₂, Ne/CO₂, CH₄/CO₂, CO₂/N₂, H₂/CH₄, H₂/Ar, CH₄/Ar, Ne/Ar, CH₄/C₂H₆, CH₄/C₃H₈, and C₂H₆/C₃H₈) in zeolites (MFI, BEA, ISV, FAU, NaY, NaX, LTA, CHA, and DDR) and MOFs (IRMOF-1, CuBTC, and MgMOF-74).

Diffusion of methane in supercritical carbon dioxide across the Widom line

Diffusion of methane diluted in supercritical carbon dioxide is studied by experiment and molecular simulation in the temperature range from 292.55 to 332.85 K along the isobars 9.0, 12.5 and 14.7 MPa. Measurements of the Fick diffusion coefficient are carried out with the Taylor dispersion technique. Molecular dynamics simulation and the Green-Kubo formalism are employed to obtain Fick, Maxwell-Stefan and intradiffusion coefficients as well as shear viscosity. The obtained diffusion coefficients are on the order of 10-8 m2/s. The composition, temperature and density dependence of diffusion is analyzed. The Fick diffusion coefficient of methane in carbon dioxide shows an anomaly in the near-critical region. This behavior can be attributed to the crossing of the so-called Widom line, where the supercritical fluid goes through a transition between liquid-like and gas-like states. Further, several classical equations are tested on their ability to predict this behavior and it is found that equations that explicitly include the density are better suited to predict the sharp variation of the diffusion coefficient near the critical region predicted by molecular simulation.

Perfluoropolymer/Molecular Sieve Mixed-Matrix Membranes

Despite the outstanding chemical, thermal and transport properties of amorphous and glassy perfluorinated polymers, only few works exist on the preparation and transport properties of perfluoropolymer/molecular sieves mixed-matrix membranes (MMMs), probably because of their poor compatibility. In this review, the compatibilization of ceramic molecular sieves with perfluorinated matrices is considered first, examining the effect of the surface treatment on the gas transport properties of the filler. Then the preparation of the defect-free hybrid membranes and their gas separation capabilities are described. Finally, recent modelling of the gas transport properties of the perfluoropolymer MMMs is reviewed. The systematic use of molecular sieves of different size and shape, either permeable or impermeable, and the calculation of the bulk transport properties of the molecular sieves-i.e., the unrestricted diffusion and permeability-allow to understand the nature of the physical phenomena at work in the MMMs, that is the larger the perfluoropolymer fractional free volume at the interface, and restricted diffusion at the molecular sieves. This knowledge led to the formulation of a new four-phase approach for the modelling of gas transport. The four-phase approach was implemented in the frame of the Maxwell model and also for the finite element simulation. The four-phase approach is a convenient representation of the transport in MMMs when more than one single interfacial effect is present.

Finite-Size Effects of Binary Mutual Diffusion Coefficients from Molecular Dynamics

Molecular dynamics simulations were performed for the prediction of the finite-size effects of Maxwell-Stefan diffusion coefficients of molecular mixtures and a wide variety of binary Lennard-Jones systems. A strong dependency of computed diffusivities on the system size was observed. Computed diffusivities were found to increase with the number of molecules. We propose a correction for the extrapolation of Maxwell-Stefan diffusion coefficients to the thermodynamic limit, based on the study by Yeh and Hummer ( J. Phys. Chem. B , 2004 , 108 , 15873 - 15879 ). The proposed correction is a function of the viscosity of the system, the size of the simulation box, and the thermodynamic factor, which is a measure for the nonideality of the mixture. Verification is carried out for more than 200 distinct binary Lennard-Jones systems, as well as 9 binary systems of methanol, water, ethanol, acetone, methylamine, and carbon tetrachloride. Significant deviations between finite-size Maxwell-Stefan diffusivities and the corresponding diffusivities at the thermodynamic limit were found for mixtures close to demixing. In these cases, the finite-size correction can be even larger than the simulated (finite-size) Maxwell-Stefan diffusivity. Our results show that considering these finite-size effects is crucial and that the suggested correction allows for reliable computations.

Diffusion of Supercritical Fluids through Single-Layer Nanoporous Solids: Theory and Molecular Simulations

With the advent of graphene material, membranes based on single-layer nanoporous solids appear as promising devices for fluid separation, be it liquid or gaseous mixtures. The design of such architectured porous materials would greatly benefit from accurate models that can predict their transport and separation properties. More specifically, there is no universal understanding of how parameters such as temperature, fluid loading conditions, or the ratio of the pore size to the fluid molecular diameter influence the permeation process. In this study, we address the problem of pure supercritical fluids diffusing through simplified models of single-layer porous materials. Basically, we investigate a toy model that consists of a single-layer lattice of Lennard-Jones interaction sites with a slit gap of controllable width. We performed extensive equilibrium and biased molecular dynamics simulations to document the physical mechanisms involved at the molecular scale. We propose a general constitutive equation for the diffusional transport coefficient derived from classical statistical mechanics and kinetic theory, which can be further simplified in the ideal gas limit. This transport coefficient relates the molecular flux to the fluid density jump across the single-layer membrane. It is found to be proportional to the accessible surface porosity of the single-layer porous solid and to a thermodynamic factor accounting for the inhomogeneity of the fluid close to the pore entrance. Both quantities directly depend on the potential of mean force that results from molecular interactions between solid and fluid atoms. Comparisons with the simulations data show that the kinetic model captures how narrowing the pore size below the fluid molecular diameter lowers dramatically the value of the transport coefficient. Furthermore, we demonstrate that our general constitutive equation allows for a consistent interpretation of the intricate effects of temperature and fluid loading conditions on the permeation process.

Tracing the origins of transient overshoots for binary mixture diffusion in microporous crystalline materials

Due to thermodynamic coupling overshoots in the loading of the more mobile species are observed during transient uptake.

Flow of methane in shale nanopores at low and high pressure by molecular dynamics simulations

Flow in shale nanopores may be vastly different from that in the conventional permeable media. In large pores and fractures, flow is governed by viscosity and pressure-driven. Convection describes the process. Pores in some shale media are in nanometer range. At this scale, continuum flow mechanism may not apply. Knudsen diffusion and hydrodynamic expressions such as the Hagen-Poiseuille equation and their modifications have been used to compute flow in nanopores. Both approaches may have drawbacks and can significantly underestimate molecular flux in nanopores. In this work, we use the dual control volume-grand canonical molecular dynamics simulations to investigate methane flow in carbon nanopores at low and high pressure conditions. Our simulations reveal that methane flow in a slit pore width of 1-4 nm can be more than one order of magnitude greater than that from Knudsen diffusion at low pressure and the Hagen-Poiseuille equation at high pressure. Knudsen diffusion and Hagen-Poiseuille equations do not account for surface adsorption and mobility of the adsorbed molecules, and inhomogeneous fluid density distributions. Mobility of molecules in the adsorbed layers significantly increases molecular flux. Molecular velocity profiles in nanopores deviate significantly from the Navier-Stokes hydrodynamic predictions. Our molecular simulation results are in agreement with the enhanced flow measurements in carbon nanotubes.

Development Trends in Porous Adsorbents for Carbon Capture

Accumulation of greenhouse gases especially CO2 in the atmosphere leading to global warming with undesirable climate changes has been a serious global concern. Major power generation in the world is from coal based power plants. Carbon capture through pre- and post- combustion technologies with various technical options like adsorption, absorption, membrane separations, and chemical looping combustion with and without oxygen uncoupling have received considerable attention of researchers, environmentalists and the stake holders. Carbon capture from flue gases can be achieved with micro and meso porous adsorbents. This review covers carbonaceous (organic and metal organic frameworks) and noncarbonaceous (inorganic) porous adsorbents for CO2 adsorption at different process conditions and pore sizes. Focus is also given to noncarbonaceous micro and meso porous adsorbents in chemical looping combustion involving insitu CO2 capture at high temperature (>400 °C). Adsorption mechanisms, material characteristics, and synthesis methods are discussed. Attention is given to isosteric heats and characterization techniques. The options to enhance the techno-economic viability of carbon capture techniques by integrating with CO2 utilization to produce industrially important chemicals like ammonia and urea are analyzed. From the reader's perspective, for different classes of materials, each section has been summarized in the form of tables or figures to get a quick glance of the developments.

Metal-organic framework based mixed matrix membranes: a solution for highly efficient CO2 capture?

The field of metal-organic framework based mixed matrix membranes (M(4)s) is critically reviewed, with special emphasis on their application in CO2 capture during energy generation. After introducing the most relevant parameters affecting membrane performance, we define targets in terms of selectivity and productivity based on existing literature on process design for pre- and post-combustion CO2 capture. Subsequently, the state of the art in M(4)s is reviewed against these targets. Because final application of these membranes will only be possible if thin separation layers can be produced, the latest advances in the manufacture of M(4) hollow fibers are discussed. Finally, the recent efforts in understanding the separation performance of these complex composite materials and future research directions are outlined.

Zeolite membranes – a review and comparison with MOFs

The latest developments in zeolite and MOF membranes are reviewed, with an emphasis on synthesis techniques. Industrial applications, hydrothermal stability, polymer-supported and mixed matrix membranes are some of the aspects discussed.

Influence of adsorption thermodynamics on guest diffusivities in nanoporous crystalline materials

Published experimental data, underpinned by molecular simulations, are used to highlight the strong influence of adsorption thermodynamics on diffusivities of guest molecules inside ordered nanoporous crystalline materials such as zeolites, metal-organic frameworks (MOFs), and zeolitic imidazolate frameworks (ZIFs). For cage-type structures (e.g. LTA, CHA, DDR, and ZIF-8), the variation of the free energy barrier for inter-cage hopping across the narrow windows, -δFi, provides a rationalization of the observed strong influence of pore concentrations, ci, on diffusivities. In open structures with large pore volumes (e.g. FAU, IRMOF-1, CuBTC) and within channels (MFI, BEA, MgMOF-74, MIL-47, MIL-53), the pore concentration (ci) dependence of the self- (Di,self), Maxwell-Stefan (Đi), and Fick (Di) diffusivities are often strongly dictated by the inverse thermodynamic correction factor, 1/Γi≡∂ln ci/∂ln pi; the magnitudes of the diffusivities are dictated by the binding energies for adsorption. For many guest-host combinations Đi-ci dependence is directly related to the 1/Γivs. ci variation. When molecular clustering occurs, we get 1/Γi > 1, causing unusual Đivs. ci dependencies. The match, or mis-match, between the periodicity of the pore landscape and the conformations of adsorbed chain molecules often leads to non-monotonic variation of diffusivities with chain lengths.

Diffusion of methane and carbon dioxide in carbon molecular sieve membranes by multinuclear pulsed field gradient NMR

Carbon molecular sieve (CMS) membranes are promising materials for energy efficient separations of light gases. In this work, we report a detailed microscopic study of carbon dioxide and methane self-diffusion in three CMS membrane derived from 6FDA/BPDA(1:1)-DAM and Matrimid polymers. In addition to diffusion of one-component sorbates, diffusion of a carbon dioxide/methane mixture was investigated. Self-diffusion studies were performed by the multinuclear (i.e., (1)H and (13)C) pulsed field gradient (PFG) NMR technique which combines the advantages of high field (17.6 T) NMR and high magnetic field gradients (up to 30 T/m). Diffusion measurements were carried out at different temperatures and for a broad range of the root-mean-square displacements of gas molecules inside the membranes. The diffusion data obtained from PFG NMR are compared with the corresponding results of membrane permeation measurements reported previously for the same membrane types. The observed differences between the transport diffusivities and self-diffusion coefficients of carbon dioxide and methane are discussed.