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).

Highlighting the Influence of Thermodynamic Coupling on Kinetic Separations with Microporous Crystalline Materials

The main focus of this article is on mixture separations that are driven by differences in intracrystalline diffusivities of guest molecules in microporous crystalline adsorbent materials. Such "kinetic" separations serve to over-ride, and reverse, the selectivities dictated by mixture adsorption equilibrium. The Maxwell-Stefan formulation for the description of intracrystalline fluxes shows that the flux of each species is coupled with that of the partner species. For n-component mixtures, the coupling is quantified by a n × n dimensional matrix of thermodynamic correction factors with elements Γ _ij ; these elements can be determined from the model used to describe the mixture adsorption equilibrium. If the thermodynamic coupling effects are essentially ignored, i.e., the Γ _ij is assumed to be equal to δ _ij , the Kronecker delta, the Maxwell-Stefan formulation degenerates to yield uncoupled flux relations. The significance of thermodynamic coupling is highlighted by detailed analysis of separations of five different mixtures: N₂/CH₄, CO₂/C₂H₆, O₂/N₂, C₃H₆/C₃H₈, and hexane isomers. In all cases, the productivity of the purified raffinate, containing the tardier species, is found to be significantly larger than that anticipated if the simplification Γ _ij = δ _ij is assumed. The reason for the strong influence of Γ _ij on transient breakthroughs is traceable to the phenomenon of uphill intracrystalline diffusion of more mobile species. The major conclusion to emerge from this study is that modeling of kinetic separations needs to properly account for the thermodynamic coupling effects.

Occupancy Dependency of Maxwell-Stefan Diffusivities in Ordered Crystalline Microporous Materials

Molecular dynamics simulation data for a variety of binary guest mixtures (H₂/CO₂, Ne/CO₂, CH₄/CO₂, CO₂/N₂, H₂/CH₄, H₂/Ar, CH₄/Ar, Ar/Kr, Ne/Ar, CH₄/C₂H₆, CH₄/C₃H₈, C₂H₆C₃H₈, CH₄/nC₄H₁₀, and CH₄/nC₅H₁₁) in zeolites (MFI, BEA, ISV, FAU (all-silica), NaY, NaX, LTA, CHA, DDR) and metal-organic frameworks (MOFs) (IRMOF-1, CuBTC, MgMOF-74) show that the Maxwell-Stefan (M-S) diffusivities, Đ ₁, Đ ₂, Đ ₁₂, are strongly dependent on the molar loadings. The main aim of this article is to develop a fundamental basis for describing the loading dependence of M-S diffusivities. Using the ideal adsorbed solution theory, a thermodynamically rigorous definition of the occupancy, θ, is derived; this serves as a convenient proxy for the spreading pressure, π, and provides the correct metric to describe the loading dependence of diffusivities. Configurational-bias Monte Carlo simulations of the unary adsorption isotherms are used for the calculation of the spreading pressure, π, and occupancy, θ. The M-S diffusivity, Đ _i , of either constituent in binary mixtures has the same value as that for unary diffusion, provided the comparison is made at the same θ. Furthermore, compared at the same value of θ, the M-S diffusivity Đ _i of any component in a mixture does not depend on it partner species. The Đ _i versus θ dependence is amenable to simple interpretation using lattice-models. The degree of correlations, defined by the ratio Đ ₁/Đ ₁₂, that characterizes mixture diffusion shows a linear increase with occupancy θ, implying that correlations become increasingly important as pore saturation conditions are approached.