Describing Binary Mixture Diffusion in Carbon Nanotubes with the Maxwell−Stefan Equations. An Investigation Using Molecular Dynamics Simulations

Chemically routed interpore molecular diffusion in metal-organic framework thin films

Transport diffusivity of molecules in a porous solid is constricted by the rate at which molecules move from one pore to the other, along the concentration gradient, i.e. by following Fickian diffusion. In heterogeneous porous materials, i.e. in the presence of pores of different sizes and chemical environments, diffusion rate and directionality remain tricky to estimate and adjust. In such a porous system, we have realized that molecular diffusion direction can be orthogonal to the concentration gradient. To experimentally determine this complex diffusion rate dependency and get insight of the microscopic diffusion pathway, we have designed a model nanoporous structure, metal-organic framework (MOF). In this model two chemically and geometrically distinct pore windows are spatially oriented by an epitaxial, layer-by-layer growth method. The specific design of the nanoporous channels and quantitative mass uptake rate measurements have indicated that the mass uptake is governed by the interpore diffusion along the direction orthogonal to the concentration gradient. This revelation allows chemically carving the nanopores, and accelerating the interpore diffusion and kinetic diffusion selectivity.

Mutual and Thermal Diffusivities as well as Fluid-Phase Equilibria of Mixtures of 1-Hexanol and Carbon Dioxide

This work contributes to an improved understanding of the fluid-phase behavior and diffusion processes in mixtures of 1-hexanol and carbon dioxide (CO₂) at temperatures around the upper critical end point (UCEP) of the system. Raman spectroscopy and dynamic light scattering were used to determine the composition at saturation conditions as well as Fick and thermal diffusivities. An acceleration of the Fick diffusive process up to CO₂ mole fractions of about 0.2 was found, followed by a strong slowing-down approaching vapor-liquid-liquid equilibrium or critical conditions. The acceleration of the Fick diffusive process vanished at temperatures much higher than the UCEP. Experimental Fick diffusivity data were compared with predictions from equilibrium molecular dynamics simulations and excess Gibbs energy calculations using interaction parameters from the literature. Both theoretical methods were not able to predict that the thermodynamic factor is equal to zero at the spinodal composition, stressing the need for new methodologies under such conditions. Thus, new sets of temperature-dependent interaction parameters were developed for the nonrandom two-liquid model, which improve the prediction of the Fick diffusion coefficient considerably. The link between the Fick diffusion coefficient and the nonrandomness of the liquid phases is also discussed.

Transport diffusivity of propane and propylene inside SWNTs from equilibrium molecular dynamics simulations

The gas transport of two model gases (propane and propylene) inside the single-walled nanotubes (SWNTs) of various diameters was systematically investigated using the molecular dynamics (MD) simulations. The thermodynamic factor can be obtained directly from equilibrium MD simulations following the newly-minted method proposed by Schnell et al. (Chem. Phys. Lett., 2011, 504, 199-201). This process eliminates the need to implement the tedious and challenging Monte Carlo simulations for the adsorption isotherm, from which the thermodynamic factor is usually extracted. The satisfactory agreement between simulation and the literature is found for self-diffusivity, corrected diffusivity and transport diffusivity, as well as for the thermodynamic factor. The ideal selectivity for a propane-propylene mixture through SWNT membranes could be optimized through adjusting the concentration gradient. This method can be readily extended to the binary and multiple-component systems.

Atomistic simulations of CO2 and N2 within cage-type silica zeolites

The behavior of CO(2) and N(2), both as single components and as binary mixtures, in two cage-type silica zeolites was studied using atomistic simulations. The zeolites considered, ITQ-3 and paradigm cage-type zeolite ZK4 (the all-silica analog of LTA), were chosen so that the principles illustrated can be generalized to other adsorbent/adsorbate systems with similar topology and types of interactions. N(2) was chosen both because of the potential uses of N(2)/CO(2) separations and because it differs from CO(2) most significantly in the magnitude of its Coulombic interactions with zeolites. Despite similarities between N(2) and CO(2) diffusion in other materials, we show here that the diffusion of CO(2) within cage-type zeolites is dominated by an energy barrier to diffusion located at the entrance to the narrow channels connecting larger cages. This barrier originates in Coulombic interactions between zeolites and CO(2)'s quadrupole and results in well-defined orientations for the diffusing molecules. Furthermore, CO(2)'s favorable electrostatic interactions with the zeolite framework result in preferential binding in the windows between cages. N(2)'s behavior, in contrast, is more consistent with that of molecules previously studied. Our analysis suggests that CO(2)'s behavior might be common for adsorbates with quadrupoles that interact strongly with a material that has narrow windows between cages.

Hydrogen bonding effects in adsorption of water-alcohol mixtures in zeolites and the consequences for the characteristics of the Maxwell-Stefan diffusivities

This work highlights a variety of peculiar characteristics of adsorption and diffusion of polar molecules such as water, methanol and ethanol in zeolites. These peculiarities are investigated with the aid of configurational-bias Monte Carlo (CBMC) simulations of adsorption isotherms, and molecular dynamics (MD) simulations of diffusivities in FAU, MFI, DDR, and LTA zeolites. Because of strong hydrogen bonding, significant clustering of the guest molecules occurs in all investigated structures. Because of molecular clustering, the inverse thermodynamic factor 1/Gamma(i) identical with (d[ln c(i)])/(d[ln f(i)]) exceeds unity for a range molar concentrations c(i) within the micropores. The degree of clustering is lowered as the temperature is increased. For the concentration ranges for which 1/Gamma(i) > 1, the Fick diffusivity, D(i), for unary diffusion is often lower than both the Maxwell-Stefan, D(i), and the self-diffusivity, D(i,self). For water-alcohol mixtures, the hydrogen bonding between water and alcohol molecules is much more predominant than for water-water, and alcohol-alcohol molecule pairs. Consequently, the adsorption of water-alcohol mixtures shows significant deviations from the predictions of the ideal adsorbed solution theory (IAST). The water-alcohol bonding also leaves its imprint on the mixture diffusion characteristics. The Maxwell-Stefan diffusivity, D(i), of either component in water-alcohol mixtures is lower than the corresponding values of the pure components; this behavior is distinctly different from that for mixtures of nonpolar guest molecules. The binary exchange coefficient D(12) for water-alcohol mixtures is also significantly lower than either self-exchange coefficients D(11) and D(22) of the constituent species. This implies that correlation effects are significantly stronger in water-alcohol mixtures than for the constituent species. Correlation effects are found to be significant for water-alcohol mixture diffusion in DDR and LTA zeolites, even though such effects are negligible for the pure constituents. The major conclusion to emerge from this investigation is that, unlike mixtures of nonpolar molecules, it is not possible to estimate water-alcohol mixture adsorption and diffusion characteristics on the basis of pure component data.

On the Maxwell-Stefan approach to diffusion: a general resolution in the transient regime for one-dimensional systems

The diffusion process in a multicomponent system can be formulated in a general form by the generalized Maxwell-Stefan equations. This formulation is able to describe the diffusion process in different systems, such as, for instance, bulk diffusion (in the gas, liquid, and solid phase) and diffusion in microporous materials (membranes, zeolites, nanotubes, etc.). The Maxwell-Stefan equations can be solved analytically (only in special cases) or by numerical approaches. Different numerical strategies have been previously presented, but the number of diffusing species is normally restricted, with only few exceptions, to three in bulk diffusion and to two in microporous systems, unless simplifications of the Maxwell-Stefan equations are considered. In the literature, a large effort has been devoted to the derivation of the analytic expression of the elements of the Fick-like diffusion matrix and therefore to the symbolic inversion of a square matrix with dimensions n x n (n being the number of independent components). This step, which can be easily performed for n = 2 and remains reasonable for n = 3, becomes rapidly very complex in problems with a large number of components. This paper addresses the problem of the numerical resolution of the Maxwell-Stefan equations in the transient regime for a one-dimensional system with a generic number of components, avoiding the definition of the analytic expression of the elements of the Fick-like diffusion matrix. To this aim, two approaches have been implemented in a computational code; the first is the simple finite difference second-order accurate in time Crank-Nicolson scheme for which the full mathematical derivation and the relevant final equations are reported. The second is based on the more accurate backward differentiation formulas, BDF, or Gear's method (Shampine, L. F. ; Gear, C. W. SIAM Rev. 1979, 21, 1.), as implemented in the Livermore solver for ordinary differential equations, LSODE (Hindmarsh, A. C. Serial Fortran Solvers for ODE Initial Value Problems, Technical Report; https://computation.llnl.gov/casc/odepack/odepack_ home.html (2006).). Both methods have been applied to a series of specific problems, such as bulk diffusion of acetone and methanol through stagnant air, uptake of two components on a microporous material in a model system, and permeation across a microporous membrane in model systems, both with the aim to validate the method and to add new information to the comprehension of the peculiar behavior of these systems. The approach is validated by comparison with different published results and with analytic expressions for the steady-state concentration profiles or fluxes in particular systems. The possibility to treat a generic number of components (the limitation being essentially the computational power) is also tested, and results are reported on the permeation of a five component mixture through a membrane in a model system. It is worth noticing that the algorithm here reported can be applied also to the Fick formulation of the diffusion problem with concentration-dependent diffusion coefficients.

Molecular insight into adsorption and diffusion of alkane isomer mixtures in metal-organic frameworks

Adsorption and diffusion of alkane isomer mixtures (C(4) and C(5)) are investigated in catenated and noncatenated metal-organic frameworks (IRMOF-13, IRMOF-14, PCN-6, and PCN-6') using molecular simulations. Competitive adsorption between isomers is observed, particularly at high pressures, at which a linear isomer shows a larger extent of adsorption due to configurational entropy. An inflection is found in the isotherm as a consequence of sequential adsorption in multiple favorable sites. Compared with the noncatenated counterparts, IRMOF-13 and PCN-6 have a greater loading at low pressures because of the constricted pores and stronger affinity with adsorbate. However, the reverse is true at high pressures due to the smaller pore volume. Catenated MOFs exhibit larger adsorption selectivity for alkane mixtures than the noncatenated counterparts. Adsorption selectivity in the four MOFs is comparable to that in carbon nanotube and silicalite, though adsorption capacity is lower in the latter. It is found that diffusivity of alkane in MOFs decreases with the degree of branching because a slender isomer diffuses faster. With the presence of constricted pores, diffusivity in catenated MOFs is smaller than that in noncatenated counterparts. In IRMOF-14 and IRMOF-13 diffusivity decreases monotonically, while it initially increases and then decreases in PCN-6'. The diffusion selectivity in catenated IRMOF-13 and PCN-6 is larger than that in noncatenated IRMOF-14 and PCN-6'. This work provides insightful microscopic mechanisms for the adsorption and diffusion of alkane isomers in MOFs and reveals that both adsorption and diffusion selectivities can be enhanced by catenation.

Efficient methods for screening of metal organic framework membranes for gas separations using atomically detailed models

Metal organic frameworks (MOFs) define a diverse class of nanoporous materials having potential applications in adsorption-based and membrane-based gas separations. We have previously used atomically detailed models to predict the performance of MOFs for membrane-based separations of gases, but these calculations require considerable computational resources and time. Here, we introduce an efficient approximate method for screening MOFs based on atomistic models that will accelerate the modeling of membrane applications. The validity of this approximate method is examined by comparison with detailed calculations for CH4/H2, CO2/CH4, and CO2/H2 mixtures at room temperature permeating through IRMOF-1 and CuBTC membranes. These results allow us to hypothesize a connection between two computationally efficient correlations predicting mixture adsorption and mixture self-diffusion properties and the validity of our approximate screening method. We then apply our model to six additional MOFs, IRMOF-8, -9, -10, and -14, Zn(bdc)(ted)0.5, and COF-102, to examine the effect of chemical diversity and interpenetration on the performance of metal organic framework membranes for light gas separations.

Testing the accuracy of correlations for multicomponent mass transport of adsorbed gases in metal-organic frameworks: diffusion of H2/CH4 mixtures in CuBTC

Mass transport of chemical mixtures in nanoporous materials is important in applications such as membrane separations, but measuring diffusion of mixtures experimentally is challenging. Methods that can predict multicomponent diffusion coefficients from single-component data can be extremely useful if these methods are known to be accurate. We present the first test of a method of this kind for molecules adsorbed in a metal-organic framework (MOF). Specifically, we examine the method proposed by Skoulidas, Sholl, and Krishna (SSK) ( Langmuir, 2003, 19, 7977) by comparing predictions made with this method to molecular simulations of mixture transport of H 2/CH 4 mixtures in CuBTC. These calculations provide the first direct information on mixture transport of any species in a MOF. The predictions of the SSK approach are in good agreement with our direct simulations of binary diffusion, suggesting that this approach may be a powerful one for examining multicomponent diffusion in MOFs. We also use our molecular simulation data to test the ideal adsorbed solution theory method for predicting binary adsorption isotherms and a method for predicting mixture self-diffusion coefficients.

Diffusion and separation of CO2 and CH4 in silicalite, C168 schwarzite, and IRMOF-1: a comparative study from molecular dynamics simulation

Recently we have investigated the storage and adsorption selectivity of CO(2) and CH(4) in three different classes of nanoporous materialssilicalite, IRMOF-1, and C(168) schwarzite through Monte Carlo simulation (Babarao, R.; Hu, Z.; Jiang, J. Langmuir, 2007, 23, 659). In this work, the self-, corrected, and transport diffusivities of CO(2) and CH(4) in these materials are examined using molecular dynamics simulation. The activation energies at infinite dilution are evaluated from the Arrhenius fits to the diffusivities at various temperatures. As loading increases, the self-diffusivities in the three frameworks decrease as a result of the steric hindrance; the corrected diffusivities remain nearly constant or decrease approximately linearly depending on the adsorbate and framework; and the transport diffusivities generally increase except for CO(2) in IRMOF-1. The correlation effects are identified to reduce from MFI, C(168) to IRMOF-1, in accordance with the porosity increasing in the three frameworks. Predictions of self-, corrected, and transport diffusivities for pure CO(2) and CH(4) from the Maxwell-Stefan formulation match the simulation results well. In a CO(2)/CH(4) mixture, the self-diffusivities decreases with loading, and good agreement is found between simulated and predicted results. On the basis of the adsorption and self-diffusivity in the mixture, the permselectivity is found to be marginal in IRMOF-1, slightly enhanced in MFI, and greatest in C(168) schwarzite. Although IRMOF-1 has the largest storage capacity for CH(4) and CO(2), its selectivity is not satisfactory.

Diffusion of chain molecules and mixtures in carbon nanotubes: The effect of host lattice flexibility and theory of diffusion in the Knudsen regime

A novel algorithm for modeling the influence of the host lattice flexibility in molecular dynamics simulations is extended to chain-like molecules and mixtures. This technique, based on a Lowe-Andersen thermostat, maintains the advantages of both simplicity and efficiency. The same diffusivities and other properties of the flexible framework system are reproduced. Advantageously, the computationally demanding flexible host lattice simulations can be avoided. Using this methodology we study the influence of flexibility on diffusion of n-alkanes inside single-walled carbon nanotubes. Furthermore, results are shown for diffusion of two mixtures (methane-helium and ethane-butane). Using these results we investigate the accuracy of theories describing diffusion in the Knudsen regime. For the dynamics in carbon nanotubes the Knudsen diffusivities are much too low. The Smoluchowski model gives better results. Interestingly, the extended Smoluchowski model can reproduce our simulation results obtained with a rigid host lattice. We modify this model to also treat collisions with a flexible interface correctly. As the tangential momentum accommodation coefficient is needed for the theoretical models, we introduce a simple concept to calculate it.

Testing predictions of macroscopic binary diffusion coefficients using lattice models with site heterogeneity

Quantitatively predicting mass transport rates for chemical mixtures in porous materials is important in applications of materials such as adsorbents, membranes, and catalysts. Because directly assessing mixture transport experimentally is challenging, theoretical models that can predict mixture diffusion coefficients using only single-component information would have many uses. One such model was proposed by Skoulidas, Sholl, and Krishna (Langmuir, 2003, 19, 7977), and applications of this model to a variety of chemical mixtures in nanoporous materials have yielded promising results. In this paper, the accuracy of this model for predicting mixture diffusion coefficients in materials that exhibit a heterogeneous distribution of local binding energies is examined. To examine this issue, single-component and binary mixture diffusion coefficients are computed using kinetic Monte Carlo for a two-dimensional lattice model over a wide range of lattice occupancies and compositions. The approach suggested by Skoulidas, Sholl, and Krishna is found to be accurate in situations where the spatial distribution of binding site energies is relatively homogeneous, but is considerably less accurate for strongly heterogeneous energy distributions.