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

Abstract

The Maxwell–Stefan (M–S) formulation for binary mixture diffusion in micro-porous materials such as zeolites, metal organic frameworks (MOFs), and covalent organic frameworks (COFs), that have pore sizes typically smaller than 2 nm, is formulated in a manner that is consistent with corresponding description for meso-porous systems. The M–S equations are set up in terms of species concentrations, c_i, defined in terms of accessible pore volume space. Molecular dynamics simulations were carried out to determine the exchange coefficients ${Đ}_{12}$ for a large variety of binary mixtures in zeolites (MFI, AFI, BEA, FAU, LTA, CHA, and DDR), MOFs (CuBTC, IRMOF-1, Zn(bdc)dabco, Co(bdc)dabco, MIL-47, Co-FA, Mn-FA, and Zn(tbip)), COFs (COF-102, COF-103, and COF-108), and cylindrical silica pores of varying diameters. The exchange coefficients ${Đ}_{12}$ in all structures were found to be related by a constant factor, F, with the corresponding M–S diffusivity for binary fluid mixture, ${Đ}_{12,\mathit{fl}}$, at the same total mixture concentration, c_t, as within the pores. The factor F is primarily dictated by the degree of confinement of the guest molecules within the channels, defined as the ratio of the characteristic sizes of the guest molecules to that of the host channels. For meso-porous cylindrical silica pores: F=1, and ${Đ}_{12}={Đ}_{12,\mathit{fl}}$. For CuBTC, MIL-47, IRMOF-1, and COFs, that have structures with a high fractional open space and channel dimensions of 0.8–1.85 nm, the factor F is found to be in the range 0.55–0.85. For structures such as MFI, BEA, Co-FA, Mn-FA, and Zn(tbip) that have smaller fractional open space, and channels smaller than 0.6 nm, the factor F has values <0.2. The major conclusion of this study is that fluid mixture diffusivity ${Đ}_{12,\mathit{fl}}$ provides a good starting point for an engineering estimate of the exchange coefficient ${Đ}_{12}$ in 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, $\epsilon$) 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)

$$-\frac{c_i}{\mathit{RT}}\nabla {\mu }_i=\sum _{\underset{j\ne i}{j=1}}^n\frac{x_j{N}_i-x_i{N}_j}{{Đ}_{\mathit{ij}}}+\frac{N_i}{{Đ}_i};i=1,2,\dots ,n$$

In Eq. (1)${Đ}_i$ 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 ${Đ}_{\mathit{ij}}$ are exchange coefficients representing interaction between components i with component j. The Onsager reciprocal relations prescribe

$${Đ}_{\mathit{ij}}={Đ}_{\mathit{ji}}$$

The c_i are the molar concentrations defined in terms of the pore volume, and the x_i represent the component mole fractions

$$x_i=c_i/c_t;i=1,2,\dots ,n$$

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 ${Đ}_{12}$ can be identified with the fluid phase diffusivity, ${Đ}_{12,\mathit{fl}}$, in the binary mixture at the same total molar loadings c_t as within the pore (Krishna and van Baten, 2009a). The ${Đ}_i$ have the same values as for the pure component diffusion, evaluated at the total loading in the mixture, c_t. 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 c_t.

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)

$$-\frac{{\theta }_i}{\mathit{RT}}\nabla {\mu }_i=\sum _{\underset{j\ne i}{j=1}}^n\frac{c_j{N}_i-c_i{N}_j}{c_{i,\mathit{sat}}{c}_{j,\mathit{sat}}{Đ}_{\mathit{ij}}^{*}}+\frac{N_i}{c_{i,\mathit{sat}}{Đ}_i};i=1,2,\dots ,n$$

with fractional occupancies, ${\theta }_i$

$${\theta }_i\equiv c_i/c_{i,\mathit{sat}};i=1,2,\dots ,n$$

used in place of the component mole fractions x_i. The concentrations c_i 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, $\rho$, in order to yield fluxes N_i in the usual units of mol m⁻² s⁻¹. The ${Đ}_i$, defined in Eq. (4), represent molecule–pore wall interactions; there are, however, fundamental differences with the corresponding ${Đ}_i$, 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 ${Đ}_i$ 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 ${Đ}_i$ in Eq. (4).

The binary exchange coefficients ${Đ}_{\mathit{ij}}^{*}$ defined in Eq. (4) reflect correlations in molecular jumps and the Onsager reciprocal relations require that ${Đ}_{\mathit{ij}}^{*}$ satisfy

$$c_{j,\mathit{sat}}{Đ}_{\mathit{ij}}^{*}=c_{i,\mathit{sat}}{Đ}_{\mathit{ji}}^{*};i,j=1,2$$

The estimation of ${Đ}_{\mathit{ij}}^{*}$ 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:

$$c_{j,\mathit{sat}}{Đ}_{\mathit{ij}}^{*}=[c_{j,\mathit{sat}}{Đ}_{\mathit{ii}}^{*}{]}^{c_i/(c_i+c_j)}[c_{i,\mathit{sat}}{Đ}_{\mathit{jj}}^{*}{]}^{c_i/(c_i+c_j)}=c_{i,\mathit{sat}}{Đ}_{\mathit{ji}}^{*}$$

has been recommended for estimating ${Đ}_{\mathit{ij}}^{*}$ using information on the self-exchange coefficients ${Đ}_{\mathit{ii}}^{*}$, obtainable from unary diffusion data on ${Đ}_i$and self-diffusivities $D_{i,\mathit{self}}$ (Chempath et al., 2004; Krishna and van Baten, 2008a, Krishna and van Baten, 2008b; Skoulidas et al., 2003). The prediction of the ${Đ}_{\mathit{ij}}^{*}$ demands a lot of input data, including the $c_{i,\mathit{sat}}$, 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 c_i 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 coefficients

$$c_{j,\mathit{sat}}{Đ}_{\mathit{ij}}^{*}/c_t={Đ}_{\mathit{ij}}={Đ}_{\mathit{ji}}=c_{i,\mathit{sat}}{Đ}_{\mathit{ji}}^{*}/c_t;i,j=1,2,\dots ,n$$

For micro-porous materials, the exchange coefficient ${Đ}_{\mathit{ij}}$ defined by Eq. (1) cannot be directly identified with the corresponding fluid phase diffusivity ${Đ}_{\mathit{ij},\mathit{fl}}$ because the molecule–molecule interactions are also significantly influenced by molecule–wall interactions. However, we shall demonstrate that the characteristics of ${Đ}_{\mathit{ij}}$ are susceptible to a simpler physical interpretation than ${Đ}_{\mathit{ij}}^{*}$; this is the major rationale for the alternative, unified, treatment developed in this paper. We aim to show that the ${Đ}_{\mathit{ij}}$ for any guest–host structure combination is related to the fluid phase ${Đ}_{\mathit{ij},\mathit{fl}}$ by a constant factor F defined as

$$F\equiv {Đ}_{\mathit{ij}}/{Đ}_{\mathit{ij},\mathit{fl}}$$

that 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 ${Đ}_{\mathit{ij}}$. 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 ${Đ}_1$, ${Đ}_2$, and ${Đ}_{12}$ 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 d_p 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 ${Đ}_{\mathit{ii},\mathit{fl}}$ of pure components along with the M–S diffusivity ${Đ}_{12,\mathit{fl}}$ 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.