Unified Maxwell–Stefan description of binary mixture diffusion in micro- and meso-porous materials
Introduction
In the development and design of separation and reaction equipment involving micro- and meso-porous materials, the proper description of mixture diffusion is of vital importance (Delgado and Rodrigues, 2001; Farooq and Ruthven, 1991; Gavalas, 2008; Hansen et al., 2009; Higgins et al., 2009; Kärger and Ruthven, 1992; Keskin et al., 2009; Ruthven, 1984; van de Graaf et al., 1999; Wang et al., 1999; Wang and LeVan, 2007, Wang and LeVan, 2008). In the published literature the models for diffusion in these materials have adopted two distinctly different approaches, the need for which can be appreciated by considering the Lennard-Jones interaction potential (normalized with respect to the energy parameter, ) for methane and a silica pore wall; see Fig. 1a. The minimum in the potential energy for interaction with the wall surface occurs at a distance 0.39 nm from the wall, and for distances greater than about 0.6 nm from the pore wall the interaction potential is virtually zero. In meso-porous materials such as MCM-41, SBA-16, and Vycor glass that have pore sizes in the range 2–50 nm, there is a central core region where the influence of interactions of the molecules with the pore wall is either small or negligible. As illustration, Fig. 1b shows the radial distribution of the loading of methane as a function of the distance from the wall of a 3 nm pore; the core region is demarcated. The maximum in the concentration distribution occurs at the position corresponding to the minimum in the Lennard-Jones interaction potential. Meso-pore diffusion is governed by a combination of molecule–molecule and molecule–pore wall interactions. The Maxwell–Stefan (M–S) equations are commonly written for mixture diffusion as (Kerkhof, 1996; Krishna and van Baten, 2009a; Young and Todd, 2005)In Eq. (1) is the M–S diffusivities of species i, portraying the interaction between component i in the mixture with the surface, or wall, of the pore; it reflects a conglomerate of Knudsen and surface diffusion, along with the viscous flow contribution. It is noteworthy that Eq. (1) do not correspond to the dusty gas model (Mason and Malinauskas, 1983), that been a subject of intense criticism in the recent literature due to some inconsistencies and handling of the viscous flow contribution (Kerkhof, 1996; Young and Todd, 2005). The are exchange coefficients representing interaction between components i with component j. The Onsager reciprocal relations prescribe
The ci are the molar concentrations defined in terms of the pore volume, and the xi represent the component mole fractions
Molecular dynamics (MD) simulations for diffusion of a wide variety of binary (n=2) mixtures in cylindrical meso-pores of silica with various diameters have shown that the can be identified with the fluid phase diffusivity, , in the binary mixture at the same total molar loadings ct as within the pore (Krishna and van Baten, 2009a). The have the same values as for the pure component diffusion, evaluated at the total loading in the mixture, ct. These are very convenient results, and allow mixture diffusion characteristics to be estimated for engineering purposes from: (1) unary diffusion data within the same pore and (2) fluid phase mixture diffusivity at same loading ct.
In micro-porous materials such as zeolites, metal organic frameworks (MOFs), and covalent organic frameworks (COFs), that have typically pore sizes in the 0.35–2 nm range, the description of diffusion is significantly more complicated than within meso-pores. Within micro-pores the guest molecules are always within the influence of the force field exerted with the wall and we have to reckon with the motion of adsorbed molecules, and there is no “bulk” fluid region. In the published literature, the M–S equations for binary mixture diffusion in zeolites and MOFs are set up in a different manner (Chempath et al., 2004; Krishna and van Baten, 2008a, Krishna and van Baten, 2008b; Skoulidas et al., 2003)with fractional occupancies, used in place of the component mole fractions xi. The concentrations ci are commonly expressed either in terms of moles of component i per m3 of framework or per kg of framework; in the latter case the left member of Eq. (4) has to be multiplied by the framework density, , in order to yield fluxes Ni in the usual units of mol m−2 s−1. The , defined in Eq. (4), represent molecule–pore wall interactions; there are, however, fundamental differences with the corresponding , defined in Eq. (1) for meso-porous materials: there is no viscous contribution, and no “Knudsen” character to diffusion mechanism within micro-pores. Eq. (4) have evolved from a description of multicomponent surface diffusion (Krishna, 1990). Formally, however, we note that the definition of in Eq. (4) is consistent with that in Eq. (1), and there is no need to distinguish between the two sets; this explains the absence of a superscript * on the in Eq. (4).
The binary exchange coefficients defined in Eq. (4) reflect correlations in molecular jumps and the Onsager reciprocal relations require that satisfyThe estimation of is the key to the description of mixture diffusion characteristics; this parameter depends on a variety of factors: degree of confinement of the species within the pores, connectivity, and loading. In the published literature the following “empirical” interpolation formula:has been recommended for estimating using information on the self-exchange coefficients , obtainable from unary diffusion data on and self-diffusivities (Chempath et al., 2004; Krishna and van Baten, 2008a, Krishna and van Baten, 2008b; Skoulidas et al., 2003). The prediction of the demands a lot of input data, including the , that are accessible from molecular simulations, but not commonly from experiments.
The main objective of the present communication is to develop an alternate approach to modeling mixture diffusion in micro-pores using Eq. (1) with the ci defined in terms of the accessible pore volumes inside the zeolites, MOFs, and COFs. By comparing (1), (4) we find the inter-relation the two sets of exchange coefficientsFor micro-porous materials, the exchange coefficient defined by Eq. (1) cannot be directly identified with the corresponding fluid phase diffusivity because the molecule–molecule interactions are also significantly influenced by molecule–wall interactions. However, we shall demonstrate that the characteristics of are susceptible to a simpler physical interpretation than ; this is the major rationale for the alternative, unified, treatment developed in this paper. We aim to show that the for any guest–host structure combination is related to the fluid phase by a constant factor F defined asthat has a value smaller than unity. We also examine the variety of factors that influence the value of F, and suggest engineering estimation procedures for . We shall suggest interpolation procedures with a more transparent physical basis that aims to supplant the “empirical” Eq. (7).
To achieve our objectives we carried out MD simulations to determine the diffusivities , , and for the binary mixtures: neon (Ne)–argon (Ar), methane (C1)–Ar, C1–C2 (ethane), C1–C3 (propane), Ne–carbon dioxide (CO2), Ar–CO2, and C1–CO2 in seven different zeolites (MFI, AFI, BEA, FAU, LTA, CHA, and DDR), eight different MOFs (IRMOF-1, CuBTC, Zn(bdc)dabco, Co(bdc)dabco, MIL-47, Co-FA, Mn-FA and Zn(tbip)), three different COFs (COF-102, COF-103, and COF-108), and cylindrical silica pores with diameters dp ranging from 0.6 to 30 nm. Though the majority of simulations were with all-silica zeolites (Si/Al=∞), a few simulations were carried out for unary diffusion in zeolites with finite Si/Al ratios to investigate the influence of the presence of cations: NaX (106 Si; 86 Al; 86 Na++; Si/Al=1.23), NaY (144 Si; 48 Al; 48 Na+; Si/Al=3), LTA-5A (96 Si; 96 Al; 32 Na+; 32 Ca++; Si/Al=1), LTA-4A (96 Si; 96 Al; 96 Na+; Si/Al=1), and MFI (with 2Na+, 4Na+, 6Na+, and 8Na+).
Fig. 2, Fig. 3, Fig. 4 show the pore landscapes of the chosen zeolites, MOFs and COFs. The various structures are deliberately chosen to represent a wide variety of micro-pore topologies and connectivities: (a) one-dimensional channels (cylindrical silica pores, AFI, Zn(tbip), MIL-47, Co-FA, Mn-FA), (b) intersecting channels (MFI, BEA, Zn(bdc)dabco, Co(bdc)dabco), (c) cavities with large windows (FAU, NaX, NaY, IRMOF-1, CuBTC, COF-102, COF-103, and COF-108), and (d) cages separated by narrow windows (LTA, LTA-5A, LTA-4A, CHA, and DDR). The loadings within the pore space are varied to near saturation values. The accessible pore volumes of the various structures were determined using the helium probe insertion simulation technique described in the literature (Myers and Monson, 2002; Talu and Myers, 2001). The salient information, including the characteristic channel sizes, on the variety of structures investigated is listed in Table 1.
Additionally, we determined the fluid phase self-diffusivities of pure components along with the M–S diffusivity for fluid mixtures. The entire data base of simulation results is available in the Supplementary material accompanying this publication; this material includes details of the MD simulation methodology, description of the force fields used, and simulation data. A selection of the simulation results is discussed below with the aim of drawing a variety of generic conclusions.
Section snippets
Exchange coefficient for binary mixture diffusion
We start by underlining the differences in mixture diffusion characteristics of micro- and meso-pores. For this purpose we consider the dependence of for an equimolar C1–Ar mixture on the total concentration ct within cylindrical silica meso-pores of diameters dp=2, 3 and 4 nm; see Fig. 5a. The for binary C1–Ar fluid phase mixture diffusion, obtained from independent MD simulations, is also presented in square symbols. At molar loadings ct<4 kmol m−3 the decreases linearly with
M–S diffusivity
Though Eq. (1) are applied in the current paper to describe diffusion in both meso- and micro-pores, there are a number of important differences in the underlying physics that determine the , representing molecule–wall interactions. We now compare and contrast the characteristics of this parameter in micro- and meso-pores.
For micro-pore diffusion, previous work (Chempath et al., 2004; Krishna and van Baten, 2005b, Krishna and van Baten, 2008a, Krishna and van Baten, 2008b; Skoulidas et al.,
Self-exchange coefficient
Let us apply Eq. (1) to equimolar diffusion () in a system consisting of two species, tagged and un-tagged, that are identical with respect to diffusional properties:Eq. (11) defines the self-diffusivity within a poreand so we derive the expression
The in Eq. (13) is the self-exchange coefficient within the pore and can be evaluated from MD simulations of both , and . The is related to
Degree of correlations Đi/Đii
The encapsulate the influence of correlation effects in unary diffusion. The larger the value of the M–S diffusivity with respect to self-exchange the stronger are the consequences of correlation effects, and we may consider the ratio as a measure of the degree of correlations.
For meso-pores, i.e. dp>2 nm, the factor Fi=1, and consequently the ratio progressively increases with increasing pore diameter; see Fig. 13a. This implies that correlation effects are stronger in
Estimation of for micro-porous structures
There are two ways to estimate the for micro-porous structures. In the first approach we proceed via the fluid phase . The for fluid mixtures can be estimated from unary self-diffusivities using the Darken (1948) relations (Krishna and van Baten, 2005a):We also note, in passing, that the formula (16) has been misprinted in Krishna and van Baten (2009a). The are more accessible, both experimentally (Bidlack and Anderson, 1964; Helbaek et
Self-diffusivities in mixtures
The M–S equations (1) can be applied to derive the following expression for the self-diffusivities in n-component mixtures inside micro- or meso-pores:
Invoking the Darken or Vignes interpolation schemes, allows the estimation of the from unary diffusion data. Fig. 16 presents a comparison of MD simulated values of the self-diffusivities in a variety of a binary mixtures in different micro-porous hosts with the predictions
Estimation of the matrix [] for binary mixture diffusion
For binary mixtures the M–S equations (1) can re-written to evaluate the fluxes Ni explicitlywhere the elements of of the matrix [] are directly accessible from MD simulations. From Eq. (1) we derive
Some representative comparisons of the MD simulated values of with estimations using MD simulated unary diffusion data on and at the mixture loading ct, along with the Vignes interpolation formula (21) are shown in Fig. 17
Conclusions
The M–S equations (1) provide an unified description of mixture diffusion in both micro- and meso-porous materials. The unified approach uses loadings ci, expressed in terms of accessible pore volume inside the porous structures.
The major conclusions of the present study are summarized below.
- (1)
For mixture diffusion inside cylindrical silica meso-pores, dp > 2 nm, the binary exchange coefficient , is found to be equal to the corresponding value in the binary fluid mixture, , over the
Notation
ci concentration of species i, mol m−3 ct total concentration in mixture, mol m−3 dp pore diameter, m self-diffusivity of species i within pore, m2 s−1 self-exchange coefficient, m2 s−1 self-diffusivity of species i in fluid phase, m2 s−1 M–S diffusivity for molecule–wall interaction, m2 s−1 zero-loading M–S diffusivity for molecule–wall interaction, m2 s−1 M–S exchange coefficient defined by Eq. (1), m2 s−1 M–S exchange coefficient defined by Eq. (4), m2 s−1 M–S diffusivity in
Acknowledgment
RK acknowledges the grant of a TOP subsidy from the Netherlands Foundation for Fundamental Research (NWO-CW) for intensification of reactors.
References (95)
- et al.
Inflection in the loading dependence of the Maxwell–Stefan diffusivity of iso-butane in MFI zeolite
Chemical Physics Letters
(2008) - et al.
Adsorption and diffusion of alkanes in CuBTC crystals investigated using infrared microscopy and molecular simulations
Microporous and Mesoporous Materials
(2009) - et al.
Molecular simulation of adsorption sites of light gases in the metal–organic framework IRMOF-1
Fluid Phase Equilibria
(2007) - et al.
Numerical-simulation of a kinetically controlled pressure swing adsorption bulk separation process based on a diffusion-model
Chemical Engineering Science
(1991) - et al.
Influence of exchangeable cations on the diffusion of neutral diffusants in zeolites of type LTA-an MD study
Chemical Physics Letters
(1995) - et al.
Gas adsorption on silicalite
Journal of Colloid and Interface Science
(1994) - et al.
PFG NMR self-diffusion of small hydrocarbons in high silica DDR, CHA and LTA structures
Microporous and Mesoporous Materials
(2008) - et al.
PFG NMR self-diffusion of propylene in ITQ-29, CaA and NaCaA: window size and cation effects
Microporous and Mesoporous Materials
(2007) - et al.
Diffusive transport through mesoporous silica membranes
Microporous and Mesoporous Materials
(2009) A modified Maxwell–Stefan model for transport through inert membranes: the binary friction model
Chemical Engineering Journal
(1996)
Multicomponent surface diffusion of adsorbed species—a description based on the generalized Maxwell–Stefan equations
Chemical Engineering Science
Investigation of slowing-down and speeding-up effects in binary mixture permeation across SAPO-34 and MFI membranes
Separation and Purification Technology
Onsager coefficients for binary mixture diffusion in nanopores
Chemical Engineering Science
Insights into diffusion of gases in zeolites gained from molecular dynamics simulations
Microporous and Mesoporous Materials
Segregation effects in adsorption of CO2 containing mixtures and their consequences for separation selectivities in cage-type zeolites
Separation and Purification Technology
An investigation of the characteristics of Maxwell–Stefan diffusivities of binary mixtures in silica nanopores
Chemical Engineering Science
Surface diffusion, atomic jump rates and thermodynamics
Surface Science
A semi-empirical approach for predicting the performance of mixed matrix membranes containing selective flakes
Journal of Membrane Science
Separation and permeation characteristics of a DD3R zeolite membrane
Journal of Membrane Science
Natural gas purification with a DDR zeolite membrane; permeation modelling with Maxwell–Stefan equations
Studies in Surface Science and Catalysis
Structural transformation and high pressure methane adsorption of Co2(1,4-bdc)2dabco
Microporous and Mesoporous Materials
Maxwell–Stefan theory for macropore molecular-diffusion-controlled fixed-bed adsorption
Chemical Engineering Science
Modelling of multi-component gas flows in capillaries and porous solids
International Journal of Heat and Mass Transfer
Selective adsorption and separation of xylene isomers and ethylbenzene with the microporous vanadium(IV) terephthalate MIL-47
Angewandte Chemie—International Edition
Molecular dynamics simulation of benzene diffusion in MOF-5: importance of lattice dynamics
Angewandte Chemie—International Edition
Kinetic separation of hexane isomers by fixed-bed adsorption with a microporous metal–organic framework
Journal of Physical Chemistry B
Inclusion complexes of faujasite with paraffins and permanent gases
Proceedings of the Royal Society of London Series A
A breathing hybrid organic-inorganic solid with very large pores and high magnetic characteristics
Angewandte Chemie—International Edition
Molecular understanding of diffusion in confinement
Physical Review Letters
Understanding diffusion in nanoporous materials
Physical Review Letters
Molecular transport in nanopores
Journal of Chemical Physics
Mutual diffusion in the system hexane–hexadecane
Journal of Physical Chemistry
Nonequilibrium MD simulations of diffusion of binary mixtures containing short n-alkanes in faujasite
Journal of Physical Chemistry B
A chemically functionalizable nanoporous material [Cu3(TMA)2(H2O)3]n
Science
Diffusion, mobility and their interrelation through free energy in binary metallic systems
Transactions of the American Institute of Mining and Metallurgical Engineers
A Maxwell–Stefan model of bidisperse pore pressurization for Langmuir adsorption of gas mixtures
Industrial & Engineering Chemistry Research
Understanding the window effect in zeolite catalysis
Angewandte Chemie—International Edition
Separation and molecular-level segregation of complex alkane mixtures in metal–organic frameworks
Journal of the American Chemical Society
Computer simulation of incommensurate diffusion in zeolites: Understanding window effects
Journal of Physical Chemistry B
Exceptional negative thermal expansion in isoreticular metal–organic frameworks
Angewandte Chemie—International Edition
Microporous manganese formate: a simple metal–organic porous material with high framework stability and highly selective gas sorption properties
Journal of the American Chemical Society
Fluid transport properties by equilibrium molecular dynamics. II. Multicomponent systems
Journal of Chemical Physics
Designed synthesis of 3D covalent organic frameworks
Science
Hybrid porous solids: past, present, future
Chemical Society Reviews
Self diffusion and binary Maxwell–Stefan diffusion in simple fluids with the Green–Kubo method
International Journal of Thermophysics
Pore-filling-dependent selectivity effects in the vapor-phase separation of xylene isomers on the metal–organic framework MIL-47
Journal of the American Chemical Society
Cited by (114)
The shape selectivity of zeolites in isobutane alkylation: An investigation using CBMC and MD simulations
2021, Chemical Engineering ScienceMass produced NaA zeolite membranes for pervaporative recycling of spent N-Methyl-2-Pyrrolidone in the manufacturing process for lithium-ion battery
2019, Separation and Purification TechnologyThe theory of diffusion in a binary mixture of molecules coadsorbed on a two-dimensional lattice
2019, Separation and Purification TechnologyCitation Excerpt :Another interesting effect is the uphill diffusion and overshooting in the adsorption of binary mixtures, which was predicted theoretically [7–10]. A large number of theoretical studies of the diffusion have been carried out using the numerical techniques [11–18]. There are, also, some monographs on the subject [19–23].