How Reliable Is the Ideal Adsorbed Solution Theory for the Estimation of Mixture Separation Selectivities in Microporous Crystalline Adsorbents?

Microporous crystalline adsorbents such as zeolites and metal-organic frameworks (MOFs) have potential use in a wide variety of separation applications. The adsorption selectivity S ads is a key metric that quantifies the efficacy of any microporous adsorbent in mixture separations. The Ideal Adsorbed Solution Theory (IAST) is commonly used for estimating the value of S ads, with unary isotherms of the constituent guests as data inputs. There are two basic tenets underlying the development of the IAST. The first tenet mandates a homogeneous distribution of adsorbates within the pore landscape. The second tenet requires the surface area occupied by a guest molecule in the mixture to be the same as that for the corresponding pure component. Configurational-bias Monte Carlo (CBMC) simulations are employed in this article to highlight several scenarios in which the IAST fails to provide a quantitatively correct description of mixture adsorption equilibrium due to a failure to conform to either of the two tenets underpinning the IAST. For CO2 capture with cation-exchanged zeolites and MOFs with open metal sites, there is congregation of CO2 around the cations and unsaturated metal atoms, resulting in failure of the IAST due to an inhomogeneous distribution of adsorbates in the pore space. Thermodynamic non-idealities also arise due to the preferential location of CO2 molecules at the window regions of 8-ring zeolites such as DDR and CHA or within pockets of MOR and AFX zeolites. Thermodynamic non-idealities are evidenced for water/alcohol mixtures due to molecular clustering engendered by hydrogen bonding. It is also demonstrated that thermodynamic non-idealities can be strong enough to cause selectivity reversals, which are not anticipated by the IAST.

Introduction

Microporous adsorbents such as zeolites and metal–organic frameworks (MOFs) offer energy-efficient alternatives to conventional separation technologies such as distillation. There has been a tremendous upsurge in research on the development of MOFs for a variety of applications such as CO2 capture and alkene/alkane, alkyne/alkene, and water/alcohol mixture separations. In industrial practice, there are two alternative configurations for utilizing the microporous materials: (i) as crystallites in fixed-bed devices that are operated in transient mode in pressure swing adsorption (PSA) technologies and (ii) as thin perm-selective layers in membrane constructs. A key metric that quantifies the separation performance of both fixed-bed adsorbers and membrane permeation units is the adsorption selectivity Sads. Intracrystalline diffusional influences serve to either enhance or diminish the separation efficacy dictated by mixture adsorption equilibrium. For n-component mixture adsorption, the selectivity of guest constituent i with respect to another guest constituent j in that mixture, Sads, , is defined bywhere q and q are the molar loadings of the constituents i and j in the adsorbed phase in equilibrium, respectively, with the bulk fluid phase mixture having partial fugacities f and f and mole fractions . For the estimation of the component loadings and selectivity Sads, , it is a common practice to use the Ideal Adsorbed Solution Theory (IAST)[1,2] that requires the unary isotherm data as inputs. The IAST approach has been used in a number of published works for evaluating and ranking microporous crystalline adsorbents for separating a wide variety of mixtures, including CO2/CH4,[3,4] CO2/N2,[3,5] CO2/H2,[6,7] SO2/CO2/N2,[8] C2H2/C2H4,[9−11] C2H2/CO2,[12] C2H4/C2H6,[13−17] C3H4/C3H6,[18−20] C3H6/C3H8,[16,21] Xe/Kr,[22,23] water/alcohol,[24−27] pentane isomers,[28] hexane isomers,[29−31] xylene isomers,[32−34] and ethylbenzene/styrene.[35,36]

Of these cited references, the validity of the use of the IAST for providing quantitatively accurate estimates of selectivities has been established by resorting to configurational-bias Monte Carlo (CBMC) simulations in the following limited number of cases: C2H2/C2H4 in ZUL-100 and ZUL-200,[11] hexane isomers in Fe2(BDP)3[30] and ZIF-77,[31] and xylene isomers in MAF-X8.[34]

Despite the widespread usage of the IAST, a limited number of investigations have found that IAST estimates of component loadings for mixture adsorption are not in quantitative agreement with experimental data. These studies include the adsorption of CO2/N2,[37] CO2/CH4,[38−41] CO2/C3H8,[42−44] CO2/C2H4,[45−47] CO2/H2S,[48] and H2S/C3H8[48] mixtures in cation-exchanged zeolites such as NaX (commonly known by its trade name 13X), LTA-5A, ZSM-5, and H-MOR.

The primary objective of this article is to investigate the reliability of IAST estimates of mixture adsorption equilibrium. We aim to highlight a variety of scenarios that would enable researchers to anticipate the possibility of the failure of the IAST to provide quantitative estimates of the component loadings in the adsorbed phase. To meet with the objectives, we resort to configurational-bias Monte Carlo (CBMC) simulations of the unary and mixture adsorption equilibrium for a wide variety of guest/host combinations. The CBMC simulations are performed using the methodology that is firmly established in the literature; details are provided in the Supporting Information accompanying this publication, which also includes (a) structural details of host materials, (b) CBMC data for unary and mixture adsorption, and (c) unary isotherm data fits.

Results and Discussion

The IAST and Its Prescriptions

In the Myers–Prausnitz development of the IAST,[1] the partial fugacities in the bulk fluid mixture are related to the mole fractions x in the adsorbed phase mixtureby the analogue of Raoult’s law for vapor–liquid equilibrium, i.e.,where P0 is the pressure for sorption of every component i, which yields the same spreading pressure π for each of the pure components, as that for the mixture:

In eq , A represents the surface area per kg of framework, and q0(f) is the pure component adsorption isotherm; the superscript 0 is used to emphasize that q0(f) relates the pure component loading to the bulk fluid fugacity. Since the surface area A is not directly accessible from experimental data, the surface potential,[40,43], with the unit mol kg–1, serves as a convenient and practical proxy for the spreading pressure π; the surface potential has also been termed the adsorption potential in several recent publications.[49−52]

For multicomponent mixture adsorption, each of the equalities on the right side of eq must be satisfied. These constraints may be solved using a suitable equation solver to yield the set of values of P10, P20, P30, ..., P0, all of which satisfy eq . The corresponding values of the integrals using P0 as upper limits of integration must yield the same value of the surface potential Φ for each component; this ensures that the obtained solution is the correct one.

The adsorbed phase mole fractions x are then determined from

The applicability of eq mandates that all of the adsorption sites within the microporous material are equally accessible to each of the guest molecules, implying a homogeneous distribution of guest adsorbates within the pore landscape, with no preferential locations of any guest species.

In view of eqs and 5, we rewrite eq as the ratio of the sorption pressures

Applying the restriction specified by eq , it follows that Sads,  is uniquely determined by the surface potential Φ. It is important to note that eq is valid irrespective of the total number of components in the mixture. In other words, the presence of component 3 in the ternary mixture has no direct influence on the adsorption selectivity Sads,12 = P20/P10 for the 1–2 pair, except for the fact that the surface potential Φ that satisfies eq is altered due to the presence of component 3.

A further key assumption of the IAST is that the adsorption enthalpies and surface areas of the adsorbed molecules do not change upon mixing with other guests. If the total mixture loading is q, the area covered by the adsorbed mixture is with the unit m2 (mole mixture)−1. Therefore, the assumption of no surface area change due to mixture adsorption translates as ; the total mixture loading is q = q1 + q2 + ... + q, which is calculated fromin which q10(P10), q20(P20), ..., q0(P0) are determined from the unary isotherm fits, using the sorption pressures for each component P10, P20, P30, ..., P0, that are available from the solutions to eq . The occurrence of molecular clustering and hydrogen bonding should be expected to invalidate the applicability of eq because the surface area occupied by a molecular cluster is different from that of each of the unclustered guest molecules in the adsorbed phase.

The ratio of the total mixture loading, q, to the saturation capacity of the mixture, qsat, mix, is the fractional pore occupancy, θ, which is relatable to Φ as follows (see the Supporting Information for the complete derivation)where the saturation capacity qsat, mix is calculated from the saturation capacities of the constituent guests

The surface potential Φ is therefore also interpretable as a proxy for the pore occupancy.

Armed with these concepts, let us compare the CBMC simulation data for mixture adsorption with the IAST predictions. Further details of the CBMC simulations (force fields used and host structures) and IAST (unary isotherm data fits) are provided in the Supporting Information.

Homogeneously Distributed Guests: Fulfilling the IAST Prescription

A quantitative procedure to verify the IAST precept of homogeneous distribution of guest adsorbates within the pore space is to perform CBMC simulations to determine the spatial locations of the guest molecules and to determine the intermolecular distances. As an illustration, we consider CO2/CH4 mixture adsorption in all-silica FAU zeolite with a total fugacity f = 500 kPa and y1 = 0.2 at 300 K. FAU zeolite has a large “open” structure that consists of cages with a volume of 786 Å3, separated by 12-ring windows with a size of 7.4 Å. By sampling a total of 105 simulation steps, the radial distribution of the separation distances between the molecular pairs CO2–CO2, CO2–CH4, and CH4–CH4 were determined. The data on the distances between the molecular pairs CO2–CO2, CO2–CH4, and CH4–CH4 are shown in Figure a; such plots are commonly termed radial distribution functions (RDFs). We note that the peaks occur at practically the same intermolecular distances. This indicates that there are no congregation or segregation effects and that the guest molecules are homogeneously distributed within the pore landscape, adequately fulfilling the precept of the IAST. Consequently, we should expect the IAST to provide a good quantitative description of CO2/CH4 mixture adsorption in all-silica FAU zeolite. As confirmation, Figure b presents CBMC data for CO2/CH4, and CO2/N2 adsorption selectivities for 50/50 CO2/CH4, 20/80 CO2/CH4, 15/85 CO2/N2, 20/80 CO2/N2, and 20/40/40 CO2/CH4/N2 mixtures in all-silica FAU. The CO2/CH4 and CO2/N2 selectivities are uniquely determined by the surface potential Φ, irrespective of the composition of the bulk fluid phase mixture and the presence of the third component. The IAST estimations, shown by the dashed lines, are in good agreement with the CBMC-simulated values of Sads.

Figure 1

(a) Radial distribution of guest pairs determined from CBMC simulations for the adsorption of CO2/CH4 mixtures in all-silica FAU zeolite at 300 K and total fugacity f = 500 kPa and y1 = 0.2. (b) CBMC data for adsorption selectivity for 50/50 CO2/CH4, 20/80 CO2/CH4, 15/85 CO2/N2, 20/80 CO2/N2, and 20/40/40 CO2/CH4/N2 mixtures in all-silica FAU. The x-axis represents the surface potential Φ. The dashed lines are the IAST estimations. All calculation details and input data are provided in the Supporting Information accompanying this publication.

Let us turn to mixture adsorption in MFI zeolite, a host structure in which the guest molecules are more strongly constrained. MFI (also called silicalite-1) has a topology consisting of a set of intersecting straight channels and zig-zag (or sinusoidal) channels with sizes of 5.4 Å × 5.5 Å and 5.4 Å × 5.6 Å. The IAST prescription demanding the homogeneous distribution of guest molecules within MFI zeolite is fulfilled only for light gaseous guest molecules such as H2, N2, CO2, CH4, C2H4, C2H6, C3H6, C3H8, and n-C4H10. These light gaseous guests can locate anywhere along the straight channels and zig-zag channels, and there are no perceptible isotherm inflections, as evidenced in the unary isotherms in Figure a. The saturation capacities follow the hierarchy H2 > CO2 > N2 ≈ CH4 > C2H4 ≈ C2H6 > C3H6 ≈ C3H8 > n-C4H10. Figure b shows computational snapshots for the adsorption of CO2 and CH4 within the intersecting channel topology of MFI zeolite. It is noticeable that neither guest species show any preferential location and there is no visual indication of segregated adsorption. The Coulombic interactions of CO2 with the negatively charged oxygen atoms in the zeolite framework are not strong enough to cause segregation between CO2 and CH4. We should therefore expect the mixture adsorption characteristics to be adequately well described by the IAST.

Figure 2

(a) CBMC simulations of unary isotherms for light gaseous molecules H2, N2, CO2, CH4, C2H4, C2H6, C3H6, C3H8, and n-C4H10 in MFI zeolite at 300 K. (b) Computational snapshots showing the location of CO2 and CH4 for binary mixture adsorption in MFI.

The IAST calculations for the adsorption selectivity Sads for five different binary mixtures CO2/CH4, CO2/H2, CO2/N2, CH4/N2, and C3H8/CH4 are compared with the corresponding Sads values determined from CBMC simulations in Figure a. For all five mixtures, the IAST estimations are in good agreement with the CBMC-simulated data, plotted as a function of the surface potential Φ. For CO2/CH4 and CO2/N2 mixtures, the Sads increases as pore saturation conditions are approached, i.e., Φ > 10 mol kg–1; θ > 0.5, because of entropy effects that favor the guest CO2 with the higher saturation capacity (cf. Figure a); the explanation of entropy effects is provided in the published literature.[33,53] For CO2/H2 and C3H8/CH4 mixtures, the Sads decreases as pore saturation conditions are approached because entropy effects favor the smaller guests H2 and CH4, respectively, that have significantly higher saturation capacities. For CH4/N2 mixtures, the Sads is practically independent of occupancy because the saturation capacities of CH4 and N2 are nearly the same, as evidenced in Figure a.

Figure 3

(a) CBMC simulations (indicated by symbols) of the adsorption selectivity Sads for five different binary mixtures: CO2/CH4, CO2/N2, CH4/N2, CO2/H2, and C3H8/CH4 in MFI zeolite at 300 K. The dashed lines are the IAST calculations for corresponding Sads values using the dual-site Langmuir–Freundlich fits of unary isotherms. (b) Comparison of CO2/CH4, CO2/N2, and CH4/N2 adsorption selectivities determined from binary mixtures, with the corresponding values in two different ternary mixtures: 5/15/80 CO2/CH4/N2 and 20/30/50 CO2/CH4/N2. The x-axes represent the surface potential Φ. All calculation details and input data are provided in the Supporting Information accompanying this publication.

Figure b presents a comparison of CO2/CH4, CO2/N2, and CH4/N2 adsorption selectivities determined from binary mixtures in MFI, with the corresponding values determined from CBMC simulations using two different ternary mixtures: 5/15/80 CO2/CH4/N2 and 20/30/50 CO2/CH4/N2. Each of the three selectivities shows a unique dependence on Φ, as prescribed by eqs and 6. In other words, the presence of component 3 in the ternary mixture has no direct influence on the adsorption selectivity for the 1–2 pair other than via Φ, as is anticipated on the basis of the development of the IAST.

Results analogous to those presented in Figures b and 3b, demonstrating the unique dependence of Sads on Φ, and the concomitant accuracy of IAST estimates are found for (i) CO2/CH4/N2 mixture adsorption in ISV that has intersecting channel structures of 6 Å (see Figure S28), (ii) CO2/CH4/N2 mixture adsorption in all-silica LTA zeolite that has cages separated by 4.11 Å × 4.47 Å 8-ring windows (see Figure S77), (iii) adsorption of ternary and quinary mixtures of hexane isomers in Mg2(dobdc), which has 1D hexagonal channels of 11 Å (see Figure S102a), and (iv) adsorption of ternary and quinary mixtures of hexane isomers in Co(BDP), which has 1D square channels of 10 Å (see Figure S102b). In all these cases, the IAST prescription is met because the guest molecules are homogeneously distributed within the pore landscape, allowing the guest species to compete equitably with one another.

Congregation of Charged Guests around Cations

Let us consider CO2/CH4 mixture adsorption in cation-exchanged NaX zeolite that has the same pore topology as FAU zeolite; per unit cell NaX zeolite has 106 Si, 86 Al, and 86 Na+ with Si/Al = 1.23. Figure a presents the RDF data determined from CBMC simulations. If we compare the first peaks, it is noteworthy that the CO2–CO2 and CO2–Na+ pairs are close together, indicating that the major proportion of CO2 congregates around the cations. A further point to note is that the CO2–CH4 separation distance is significantly larger than the CO2–CO2 and CH4–CH4 separation distances. This implies that the CH4 molecules face less severe competitive adsorption with CO2 than is anticipated by the IAST. Consequently, as seen in Figure b, the values of Sads estimated by the IAST are significantly higher, by about a factor of two, than those determined by CBMC. Also shown in Figure b are the CBMC data for CO2/CH4 mixture adsorption in NaY zeolite (138 Si, 54 Al, 54 Na+, and Si/Al = 2.56); the IAST estimates are also in excess of the CBMC data, but the departures are less than that experienced with NaX because congregation effects are reduced due to the presence of fewer cations in NaY. Of course, in the total absence of cations, the IAST estimates are in excellent agreement with CBMC data, as already witnessed in Figure b.

Figure 4

(a) Radial distribution of guest pairs determined from CBMC simulations for the adsorption of CO2/CH4 mixtures in NaX zeolite at 300 K and total fugacity f = 1 MPa, and y1 = 0.01. (b) Comparison CO2/CH4 adsorption selectivities determined from CBMC simulations for NaY (138 Si, 54 Al, 54 Na+, and Si/Al = 2.56) and NaX (106 Si, 86 Al, 86 Na+, and Si/Al = 1.23) zeolites at 300 K. The CBMC-simulated values (indicated by symbols) are compared with RAST (continuous solid lines) and IAST (dashed lines) estimates. All calculation details and input data are provided in the Supporting Information accompanying this publication.

The inhomogeneous distribution of adsorbates is a common feature of mixtures of charged and neutral guests in cation-exchanged zeolites. Figure a shows the RDF data for CO2/C3H8 mixture adsorption in NaX zeolite. The CO2–C3H8 separation distance is significantly higher than between the CO2–CO2 and CO2–Na+ pairs, indicating that C3H8 experiences reduced competition with CO2 partners. The consequences of this reduced competition is reflected by the CBMC data for CO2(1)/C3H8(2) mixture adsorption in three different CBMC campaigns: (i) equimolar mixtures, y1 = y2 = 0.5, with varying f = f1 + f2, (ii) f = 1 MPa with varying y1, and (iii) f = 50 kPa with varying y1. The assumption of an ideal adsorbed mixture anticipates all three data sets to follow a unique Sads – Φ dependence, as shown by the dashed line in Figure b. However, the CBMC data (indicated by symbols) show that the CO2(1)/C3H8(2) adsorption selectivity Sads does not follow a unique dependence on Φ.

Figure 5

(a) Radial distribution of guest pairs determined from CBMC simulations for the adsorption of CO2/C3H8 mixtures in NaX zeolite at 300 K and total fugacity f = 1 MPa and y1 = 0.5. (b) Adsorption selectivity Sads for CO2(1)/C3H8(2) mixture adsorption in NaX zeolite at 300 K for three different CBMC campaigns, plotted as a function of the surface potential Φ: (i) constant composition y1 = 0.5 with varying f = f1 + f2, (ii) f = 1 MPa with varying composition y1, and (iii) f = 50 kPa with varying y1. The CBMC-simulated values (indicated by symbols) are compared with RAST (continuous solid lines) and IAST (dashed lines) estimates. (c) Activity coefficients for CO2(1) and C3H8(2), determined from campaign (i). (d) Activity coefficients for CO2(1) and C3H8(2) determined from campaign (ii). The continuous solid lines in panels (b) and (c) are RAST/Margules model calculations. All calculation details and input data are provided in the Supporting Information accompanying this publication.

To quantify non-ideality effects and departures from the IAST, we need to abandon Raoult’s law assumption in eq and introduce activity coefficients γ

Figure c presents the activity coefficients calculated from the CBMC data for campaign (i) for equimolar mixtures of CO2 and C3H8 with varying f. As Φ → 0, both activity coefficients tend to unity γ → 1; this corresponds with the Henry regime of adsorption. In other words, at vanishing small values of pore occupancy, non-ideality effects can be ignored, as should be expected. With increasing pore occupancy, the activity coefficient of C3H8 steadily decreases below unity, whereas the activity coefficient of CO2 remains close to unity over the entire range of Φ values.

Figure d presents the activity coefficients calculated from the CBMC data for campaign (ii) with f = 1 MPa and varying bulk fluid mixture composition; in this campaign, the variation of Φ is minimal and falls in the range 24 < Φ < 30 mol kg–1. Both activity coefficients are strongly dependent on the composition of the adsorbed phase mixture, x1, and satisfy the requirement x → 1; γ → 1.

Following the approaches of Myers, Talu, and Sieperstein,[43,48,54] we model the excess Gibbs free energy for binary mixture adsorption as follows

For the calculation of the total mixture loading q = q1 + q2, we need to replace eq by

The excess reciprocal loading for the mixture can be related to the partial derivative of the Gibbs free energy with respect to the surface potential at constant composition

For quantitative modeling of the data on activity coefficients, a variety of models such as regular solution,[43] Wilson,[50,51,55] NRTL,[56] SPD,[48] and Margules[52,57] have been used. For example, the Margules model takes the following form

In eq , C is a constant with the unit kg mol–1. The introduction of (1 – exp ( – CΦ)) imparts the correct limiting behaviors for the activity coefficients in the Henry regime: Φ → 0; γ → 1. As pore saturation conditions are approached, this correction factor tends to unity: (1 – exp( –CΦ)) → 1.

Combining eqs –14, we derive

The parameters A12, A21, and C can be fitted to match the CBMC data on activity coefficients; the fitting procedure is detailed in the Supporting Information accompanying this publication. The continuous solid lines in Figure c,d are calculations following the Real Adsorbed Solution Theory (RAST) with fitted Margules parameters A12 = – 3.082, A21 = – 2.170, and C = 0.038 kg mol–1.

With the introduction of activity coefficients, eq needs to be replaced by the more generalized expression for the adsorption selectivity for the i–j pair

Equations and 16 imply that the Sads,  depends on both Φ and the composition of the adsorbed mixture; this point is underscored in the RAST calculations (indicated by the continuous solid lines) of the selectivity for the three campaigns in Figure b. An important consequence of this complex dependence is the occurrence of selectivity reversal phenomena. Figure presents CBMC data on the CO2/C3H8 and CO2/n-C4H10 selectivities for cation-exchanged zeolites NaX and LTA-4A; in the simulations, the total mixture fugacity f is maintained at a fixed value. With increasing proportion of CO2 in the bulk fluid mixture, selectivity reversals in favor of the alkane occur; all such reversals are not anticipated by the IAST (see Figures S69, S75, S76, and S86). Experimental evidence of such selectivity reversals, attributable to congregation of CO2 around cations, has been reported for CO2/C3H8[42−44] and CO2/C2H4[45] in cation-exchanged zeolites.

Figure 6

CBMC simulation data for CO2/alkane selectivities determined from three different CBMC campaigns: (i) CO2(1)/C3H8(2) mixture adsorption in NaX with f = 50 kPa and varying y1, (ii) CO2(1)/C3H8(2) mixture adsorption in LTA-4A with f = 1 MPa and varying y1, and (iii) CO2(1)/n-C4H10(2) mixture adsorption in LTA-4A with f = 500 kPa and varying y1. The CO2/alkane selectivity values in each case are plotted against the mole fraction in the bulk fluid mixture, y1. All calculation details and input data are provided in the Supporting Information accompanying this publication.

For the adsorption of the CO2-bearing mixture in Mg2(dobdc), the preponderance of CO2 around the unsaturated Mg2+ sites causes quantitative failure of the IAST; see CBMC data in Figures S95 and S96.

Other examples of the failure of the IAST, along with alternative approaches to RAST modeling of non-idealities, are available in the literature.[2,58−62]

Preferential Location of Guests at Channel Intersections of MFI Zeolite

Due to configurational considerations, branched alkanes prefer to locate at the channel intersections of MFI zeolite because of the extra “leg room” that is available here. An extra “push” is required to locate these molecules within the channel interiors. This extra push results in an inflection in the pure component isotherms at a loading of four molecules per unit cell because per unit cell of MFI, there are four channel intersection sites;[63−66] see Figure a. Cyclic hydrocarbons, such as cyclohexane, benzene, and ethylbenzene, also prefer to locate at the intersections; the unary isotherm for benzene also exhibits a strong inflection at a loading of four molecules per unit cell (cf. Figure a). For C3H6(1)/benzene(2) mixture adsorption in MFI, the computational snapshots in Figure b clearly show that the aromatics are exclusively located at the channel intersections, whereas the linear propene can locate anywhere along either the straight or zig-zag channels. Figure c plots the CBMC data for adsorption selectivity Sads of C2H4(1)/benzene(2) and C3H6(1)/benzene(2) mixtures as functions of Φ. For both mixtures, the IAST (indicated by the dashed lines) significantly overestimates the Sads value in favor of benzene, except for the limiting case of low pore occupancy Φ → 0, θ → 0. The IAST calculation assumes that alkene molecules (C2H4 or C3H6) compete with all of the benzene molecules, making no allowance for segregation and preferential adsorption of benzene at the intersections. Due to segregation effects, the competition faced by alkene molecules within the channels is lower than that in the entire pore space. In other words, the IAST anticipates a stiffer competition between benzene and alkenes as it assumes a uniform distribution of adsorbates; consequently, the separation selectivity is overestimated. Due to the preferential location of benzene at the intersections, some alkene molecules are farther removed from benzene and suffer diminished competition.

Figure 7

(a) CBMC simulations of unary isotherms for branched alkanes and benzene in MFI zeolite at 300 K. (b) Computational snapshots showing the location of guest molecules for C3H6(1)/benzene(2) mixture adsorption in MFI zeolite at 300 K. (c) Adsorption selectivity Sads for benzene/C2H4 and benzene/C3H6 mixtures in MFI zeolite, plotted as a function of the surface potential Φ. The CBMC-simulated values (indicated by symbols) are compared with RAST (continuous solid lines) and IAST (dashed lines) estimates. All calculation details and input data are provided in the Supporting Information accompanying this publication.

A further point to be noted is that the benzene/alkene selectivity reduces significantly with increasing values of Φ; this reduction in Sads is a direct consequence of entropy effects that favor alkene because of significantly higher saturation capacity. The CBMC data for C3H6/benzene mixtures and entropy effects are strong enough to cause selectivity reversals in favor of propene, for Φ > 5 mol kg–1, corresponding to θ > 0.93. Such selectivity reversal is not quantitatively matched by the IAST; the use of the RAST is necessary for a good quantitative description of Sads – Φ characteristics.

For precisely analogous reasons, adsorption of C3H8/iso-C4H10, n-C4H10/iso-C4H10, and n-hexane/2-methylpentane mixtures in MFI zeolite shows significant deviations from IAST estimates of component loadings and selectivities; see Figures S19–S25 of the Supporting Information.

Preferential Location of CO2 at Window Regions of Cage-Type Zeolites

For the separation of CO2 from gaseous mixtures with CH4, cage-type zeolites such as CHA, DDR, LTA, and ERI are of practical interest;[67−69] these materials consist of cages separated by narrow windows in the 3.3–4.5 Å range. For adsorption of CO2/CH4 mixtures, CBMC simulations[67] show that the window regions of cage-type zeolites have a significantly higher proportion of CO2 than within the cages; see computational snapshots in Figure for (a) CHA and (b) DDR zeolites.

Figure 8

Computational snapshots for CO2(1)/CH4(2) mixture adsorption in (a) CHA and (b) DDR zeolites at 300 K.

In Figure a, the CBMC-simulated values of the adsorption selectivity Sads for CO2/CH4 mixture adsorption in CHA, determined from three different CBMC campaigns, are plotted as a function of Φ. For all three sets of CBMC data, the IAST calculations overestimate the values of Sads because the competition faced by CH4, which locates predominantly within the cages, is less severe than anticipated because of the preferential location of CO2 in the window regions. The deviations of IAST estimates from CBMC-simulated values increase with increasing pore occupancies.

Figure 9

(a) CBMC data for adsorption selectivity Sads for CO2(1)/CH4(2) mixture adsorption in CHA zeolite, determined for three different campaigns: (i) constant composition y1 = 0.5 with varying f = f1 + f2, (ii) constant composition y1 = 0.15 with varying f, and (iii) f = 1 MPa with varying y1. (b) CBMC data for adsorption selectivity Sads for CO2(1)/CH4(2) mixture adsorption in DDR, determined for two different campaigns: (i) constant composition y1 = 0.5 with varying f and (ii) f = 1 MPa with varying composition y1. The x-axes represent the surface potential Φ. The CBMC-simulated values (indicated by symbols) are compared with RAST (continuous solid lines) and IAST (dashed lines) estimates. All calculation details and input data are provided in the Supporting Information accompanying this publication.

Precisely analogous results are obtained for CO2/CH4 mixture adsorption in DDR, determined from two different CBMC campaigns; see Figure b. The CBMC-simulated Sads values for the two sets of campaigns are not uniquely related to Φ, as is anticipated by the IAST; the non-unique Sads – Φ characteristics are quantitatively captured by the RAST. As pore saturation conditions are approached, the IAST predictions of selectivities become increasingly optimistic.

Preferential Location of CO2 within Pockets of AFX and MOR

Earlier works have shown that AFX zeolite is particularly effective for CO2 capture applications.[70−72]Figure a shows snapshots for adsorption of the binary mixture of CO2 and CH4. In one unit cell of AFX, there are four 490 Å3-sized cages, connected to four small pockets each of 98 Å3. The 8-ring windows separating two cages are 3.44 Å × 3.88 Å in size. Guests such as CH4, N2, or H2 are preferentially located within the cages. The competition experienced by CH4, N2, or H2 from coadsorption with CO2 should be expected to be significantly lowered because the window regions and the small pockets are preferred locations for CO2.[67,70,71,73] Consequently, the IAST should be expected to overestimate the CO2/CH4 selectivity. The CBMC data for CO2/CH4 mixture adsorption in AFX confirms this expectation; see Figure a. Figure a also shows that the IAST overestimates the CO2/CH4 selectivity values in MOR zeolite because CO2 gets firmly ensconced in the side pockets (cf. snapshots in Figure b), far removed from the CH4 partners that preferentially reside in the main 12-ring 1D channels.

Figure 10

(a) Snapshots showing the location of guest molecules for CO2(1)/CH4(2) mixture adsorption in AFX zeolite at 300 K. (b) Snapshots showing the location of guest molecules for CO2(1)/C3H8(2) mixture adsorption in MOR zeolite at 300 K.

Figure 11

(a) CBMC simulation data on the adsorption selectivity Sads for equimolar f1 = f2 CO2(1)/CH4(2) mixture adsorption in AFX and MOR zeolites at 300 K, plotted as a function of the surface potential Φ. (b) CBMC simulation data on the adsorption selectivity for CO2(1)/C3H8(2) mixture adsorption in MOR zeolite at 300 K. The total fluid phase fugacity is f = 1 MPa, and the composition y1 is varied. In panels (a) and (b), the CBMC-simulated values (indicated by symbols) are compared with RAST (continuous solid lines) and IAST (dashed lines) estimates. All calculation details and input data are provided in the Supporting Information accompanying this publication.

The segregation between CO2 and its partners in MOR also results in selectivity reversals. Figure b shows CBMC data for CO2(1)/C3H8(2) mixture adsorption in all-silica MOR zeolite for a campaign in which the total fluid phase fugacity f = 40 kPa and the bulk fluid phase mixture composition y1 = f1/f is varied. For y1 < 0.6, Sads > 1, and the selectivity is in favor of CO2. The CBMC simulations show that the adsorption selectivity Sads is increasingly lowered below unity, i.e., in favor of alkane, with increasing proportion of CO2 in the bulk gas phase. The IAST anticipates Sads to be virtually independent of y1 and does not anticipate the selectivity reversal phenomena. Experimental evidence is available for such selectivity reversals, which require the use of the RAST for quantification.[48,50]

Hydrogen Bonding in Water/Alcohol Mixtures

For water/alcohol mixture adsorption in zeolites and MOFs, the manifestation of hydrogen bonding between water and alcohol molecules can be demonstrated by sampling the spatial locations of the guest molecules to determine the O····H distances of various pairs of molecular distances. For water(1)/ethanol(2) mixture adsorption in DDR zeolite at 300 K, the RDFs of O····H distances for water–water, water–ethanol, and ethanol–ethanol pairs are shown in Figure . We note that the first peaks in the RDFs occur at a distance less than 2 Å, which is characteristic of hydrogen bonding.[74,75] The heights of the first peaks are a direct reflection of the degree of hydrogen bonding between the molecular pairs. The degree of H-bonding between water–ethanol pairs is significantly larger, by about an order of magnitude, than for water–water and ethanol–ethanol pairs.

Figure 12

RDF of O····H distances for molecular pairs of water(1)/ethanol(2) mixture adsorption in DDR zeolite at 300 K. The partial fugacities of components 1 and 2 are f1 = 2.5 kPa and f2 = 7.5 kPa. The magnitudes of the first peaks are a direct reflection of the degree of hydrogen bonding between the molecular pairs.

Figure a presents CBMC data on the ethanol/water selectivity in DDR for mixtures in which the partial fugacities are maintained equal for both guests, i.e., f1 = f2. With increasing values of the surface potential, the selectivity increasingly favors water adsorption due to its smaller size. For Φ ≈ 10 mol kg–1, corresponding to a pore occupancy θ ≈ 0.9, the mixture adsorption is water-selective. Although the IAST calculations (dashed lines) correctly anticipate the selectivity reversal phenomenon, the quantitative agreement of IAST estimates with CBMC data is poor. For Φ < 10 mol kg–1, the IAST overestimates Sads due to enhanced water uptake resulting from molecular clustering. A further, distinct consequence of molecular clustering effects induced by hydrogen bonding is that the effective size difference between the guest molecules is reduced. Consequently, entropy effects are moderated by clustering phenomena. The IAST that does not account for clustering anticipates an exaggerated influence of entropy effects. In other words, for Φ > 10 mol kg–1, cluster formation tends to moderate entropy effects, and the IAST anticipates stronger water selectivity than found in CBMC simulations.

Figure 13

(a, b) CBMC simulation data for the ethanol/water selectivity Sads for water(1)/ethanol(2) mixture adsorption in DDR at 300 K for two different campaigns. (a) In this campaign, the total fugacity f is varied, maintaining equal partial fugacities, f1 = f2, in the bulk fluid phase mixture. (b) In the second campaign, the total bulk fluid phase fugacity f = f1+ f2 = 10 kPa; the water composition in the bulk fluid mixture, y1, is varied from 0 to 1. The CBMC-simulated values (indicated by symbols) are compared with RAST (continuous solid lines) and IAST (dashed lines) estimates. (c, d) RAST calculations of the activity coefficients, using fitted Margules parameters, for the two campaigns shown in panels (a) and (b). All calculation details and input data are provided in the Supporting Information accompanying this publication.

Figure b presents CBMC data for ethanol/water selectivity in DDR zeolite for a campaign in which the bulk fluid composition is varied while holding the total bulk mixture fugacity constant at f = 10 kPa. The CBMC data show that for water-rich mixtures, y1 > 0.5, the adsorption is ethanol-selective; this is desired of adsorbents, say, in recovery of bioethanol from fermentation broths. However, for feed mixtures that are richer in ethanol, y1 < 0.5, the adsorption is water-selective; this is a desirable feature, say, for use of DDR in membrane constructs for water-selective pervaporation processes.[76] The IAST (dashed line) anticipates ethanol-selective adsorption over the entire range of compositions y1.

The combined set of component loadings in the two CBMC campaigns was used to determine the set of Margules parameters A12 = –5.325, A21 = –1.665, and C = 1.868 kg mol–1 to quantify the non-idealities. Figure c presents the RAST calculations of the activity coefficients for equimolar water/ethanol mixtures with varying f. As the surface potential Φ → 0, both activity coefficients tend to unity γ1 → 1, γ2 → 1, as is expected in the Henry regime. The water activity coefficient exhibits a deep minimum for 0.01 < Φ < 10 mol kg–1; under these conditions, there is significant enhancement in the water ingress that is caused by hydrogen bonding. With increasing pore occupancy, there is a monotonous decrease in the activity coefficient of ethanol below unity.

Figure c presents the activity coefficients for the campaign in which f = 10 kPa and the bulk fluid mixture composition is varied. In this campaign, the variation of the surface potential is minimal and Φ ≈ 7 mol kg–1. Both activity coefficients are strongly dependent on the composition of the adsorbed phase mixture, x1, and satisfy the requirement x → 1; γ → 1. The use of the RAST is essential for quantitative modeling the selectivity reversals observed in Figure a,b.

Precisely analogous results are obtained for water/alcohol mixture adsorption in CHA; see Figures S36–S38. The adsorption of alcohol-rich feed mixtures in CHA is water-selective; therefore, CHA membranes are used for the purification of alcohols by membrane pervaporation because diffusion through 3.8 Å × 4.2 Å 8-ring windows of CHA also favors water.[29,75,77−79]

Segregated Adsorption Due to the Selective Size Exclusion of Guest Molecules

One scenario in which it is evident that the mandate of homogeneous distribution of adsorbates is not fulfilled is the one in which one of the guest molecules is effectively excluded from the pore space on the basis of molecular size. We discuss below three examples of mixture separations exploiting size exclusion; in all these cases, CBMC simulations of mixture adsorption are not feasible.

For reducing the nitrogen content of natural gas, consisting predominantly of CH4, one practical solution is to choose materials such as Ba-ETS-4 (ETS = Engelhard titano-silicate; ETS-4 is also named as CTS-1 = contracted titano-silicate-1) with pore size ≈ 3.7 Å so as to effectively exclude the spherical CH4 molecule (3.7 Å) while allowing entry for the pencil-like nitrogen molecule (4.4 Å × 3.3 Å).[80−84] The experimental data of Bhadra[85] for the binary mixture adsorption equilibrium of CH4/N2 mixtures in Ba-ETS-4 demonstrate the failure of the IAST due to the segregated nature of adsorption.[86]

For C3H6/C3H8 mixture separations, a potent strategy is to employ NbOFFIVE-1-Ni (KAUST-7)[87] or Co-gallate,[88] which almost completely excludes the saturated alkane from the pores. For C2H4/C2H6 separations, near total exclusion of C2H6 is achieved by use of an ultramicroporous metal–organic framework UTSA-280 [Ca(C4O4)(H2O)], which possesses rigid one-dimensional channels.[14] The 1D channels are of a similar size to C2H4 molecules (all of atoms of which lie on the same plane) but, owing to the size, shape, and rigidity of the pores, practically exclude the C2H6. The applicability of the IAST to describe the mixture adsorption equilibrium for the aforementioned MOFs for alkene/alkane separations is clearly open to question.

Conclusions

The derivation of the IAST is based on two tenets: (i) homogeneous distribution of guest adsorbates in the pore space, allowing for equitable competition for the occupation of adsorption sites, and (ii) the surface area occupied by a guest molecule in the mixture that is essentially the same as for unary adsorption, implying no occurrence of clustering with partners. An important implication of the IAST is that the adsorption selectivity for the i–j pair, Sads, , is uniquely determined by the surface potential Φ, irrespective of the mixture composition and the presence of additional partners in the mixture. CBMC simulations of mixture adsorption in a wide variety of host materials have been used to investigate and highlight scenarios in which the IAST tenets are violated.

For the adsorption of CO2-bearing mixtures, an inhomogeneous distribution of adsorbates is engendered due to congregation of CO2 around the extra-framework cations in zeolites and exposed “open” charged metal sites of MOFs. Due to the inhomogeneous distribution of adsorbates, the partner molecules endure a reduced degree of competition with CO2 than is presumed in the IAST. Consequently, the IAST generally tends to anticipate a higher selectivity of CO2 with respect to partner species. The IAST also fails to anticipate reversals in the selectivity of CO2-bearing mixtures of varying composition.

For the adsorption of CO2-bearing mixtures in cage-type zeolites such as CHA and DDR, the CO2 molecules prefer to perch at the window regions; partner molecules such as CH4 prefer to locate within the cages and enjoy reduced competition with partner CO2 molecules. The IAST estimates of Sads are overly optimistic. The preferential location of CO2 within the side pockets of zeolites MOR and AFX leads to quantitative failure of the IAST for analogous regions. In severe cases, such as for CO2/C3H8 adsorption in MOR, the IAST fails to anticipate selectivity reversals; such failure has been confirmed by experiments.[48]

The IAST mandate of homogeneous distribution of guest adsorbates is clearly violated for MOFs and zeolites that rely on the principle of size exclusion to enable separations.

For separations of linear and branched alkanes using MFI zeolite, thermodynamic non-ideality effects arise due to the preferential location of the branched alkanes at the channel intersections that offer more “leg room”. Aromatic molecules such as benzene also prefer to locate at the intersections, and consequently, the IAST estimates of component loadings and selectivities of adsorption of benzene/alkene and benzene/alkane mixtures are not of acceptable accuracy.

For water/ethanol adsorption, molecular clustering occurs due to strong hydrogen bonding between water and ethanol. The IAST fails to provide quantitative predictions of selectivities for two separate reasons depending on the value of the surface potential Φ and pore occupancy θ. At relatively low values of Φ, water/ethanol clusters tend to increase the uptake of water, far in excess of the values anticipated by the IAST. Consequently, the IAST overestimates the ethanol/water selectivity. For large values of Φ, close to pore saturation, the occurrence of water/ethanol clusters has the effect of moderating entropy effects that normally favor the smaller water molecule with the higher saturation capacity. The IAST overestimates entropy effects and anticipates a higher degree of water selectivity than found in the CBMC simulations. The IAST does not also anticipate reversals that favor water in ethanol-rich mixtures.

For quantification of non-ideality effects, activity coefficients γ need to be introduced as shown in eq . While the γ can be backed out from CBMC data on mixture adsorption, there are no reliable procedures for estimating these a priori. Streb and Mazzotti[40,41] discuss a procedure for the estimation of the RAST model parameters from cyclic experiments for CO2/CH4 mixture adsorption in 13X zeolite.