Predicting Noble Gas Separation Performance of Metal Organic

Article pubs.acs.org/JPCC

Predicting Noble Gas Separation Performance of Metal Organic Frameworks Using Theoretical Correlations Yeliz Gurdal and Seda Keskin* Department of Chemical and Biological Engineering, Koç University Rumelifeneri Yolu, Sariyer, 34450, Istanbul, Turkey S Supporting Information *

ABSTRACT: In this work, we examined the accuracy of theoretical correlations that predict the performance of metal organic frameworks (MOFs) in separation of noble gas mixtures using only the single-component adsorption and diffusion data. Single component adsorption isotherms and self-diffusivities of Xe, Kr, and Ar in several MOFs were computed by grand canonical Monte Carlo and equilibrium molecular dynamics simulations. These pure component data were then used to apply Ideal Adsorbed Solution Theory (IAST) and Krishna−Paschek (KP) correlation for estimating the adsorption isotherms and self-diffusivities of Xe/Kr and Xe/Ar mixtures at various compositions in several representative MOFs. Separation properties of MOFs such as adsorption selectivity, working capacity, diffusion selectivity, permeation selectivity, and gas permeability were evaluated using the predictions of theoretical correlations and compared with the data obtained from computationally demanding molecular simulations. Results showed that theoretical correlations that predict mixture properties based on single-component data make accurate estimates for the separation performance of many MOFs which will be very useful for materials screening purposes. Skoulidas et al.9 used GCMC simulations to report adsorption of Ar in MOF-2, MOF-3, MOF-5, and CuBTC and performed MD simulations to compute the self-diffusivity (corrected diffusivity) of Ar in MOF-2, MOF-3, MOF-5, and CuBTC (MOF-5) at room temperature. Greathouse et al.10 reported adsorption isotherms of Xe/Ar and Xe/Kr mixtures in IRMOF-1 using GCMC, computed the adsorption selectivity of IRMOF-1, and concluded that IRMOF-1 is a good adsorbent for Xe separation. Ryan et al.11 investigated adsorption of Xe/Kr mixtures in IRMOF-1, HKUST-1, UMCM-1, ZIF-8, MOF-505, NOTT-101, NOTT-108, and Pd-MOF and concluded that large pore materials are not desirable for efficient adsorption-based separation of Xe/Kr. Van Heest and coworkers12 screened a large number of MOFs for Ar/Kr, Kr/Xe, and Xe/Rn separations using molecular simulations. They chose 70 materials for further analysis and showed that GUPJEG01 (Mn(dcbp)) is the most promising MOF for Kr/Xe separations due to its extremely high adsorption selectivity. They also showed that Ideal Adsorbed Solution Theory (IAST) gives good agreement with the GCMC simulations except for materials in which pore blocking effects are important. Sikora et al.13 recently generated 137 000 hypothetical MOFs and screened them for adsorption-based separation of Xe/Kr mixtures using GCMC simulations at 273 K. They concluded that MOFs having pores just large enough to fit a single Xe atom and the ones having morphologies resembling tubes of

1. INTRODUCTION Noble gas separation plays a significant role in industry since these gases have specific properties such as the lack of chemical reactivity, very low conductivity, and very low boiling and melting points. For example, argon (Ar) is used in electric light bulbs, fluorescent lamps, and thermal insulation.1 Krypton (Kr) is the most common noble gas used in excimer lasers1 and testing of leaks in sealed containers.2 Xenon (Xe) is used in anesthetic,2 neutron counters, and ionization chambers for cosmic rays.1 Because of these large number of application areas, purification of Ar, Kr, and Xe should be economically practical for industrial usage. After cryogenic distillation of air, a mixture of 80% Kr and 20% Xe is obtained, and this has to be purified further with low cost.3 Nanoporous adsorbents such as zeolites and activated carbon have been used to separate Xe/Ar and Xe/Kr mixtures in the past.4 Metal organic frameworks (MOFs) are nanoporous materials that are comprised of self-assembly of metal ions that act as coordination centers and organic ligands that act as linkers between metal centers. MOFs have been widely studied for gas storage and gas separation because of their low densities (0.2− 1 g/cm3), high surface areas (500−6000 m2/g), high porosities, and good thermal and mechanical stabilities.5 A large number of molecular simulation studies investigated adsorption and diffusion of CO2, CH4, N2, and H2 in addition to separation of CO2/CH4, CH4/H2, CO2/N2, and CO2/H2 mixtures using grand canonical Monte Carlo (GCMC) and molecular dynamics (MD) simulations.6−8 Although a significant number of studies have focused on CO2 separation, only a few studies addressed separation of noble gas mixtures using MOFs. © 2013 American Chemical Society

Received: December 29, 2012 Revised: February 8, 2013 Published: February 12, 2013 5229

dx.doi.org/10.1021/jp312838v | J. Phys. Chem. C 2013, 117, 5229−5241

The Journal of Physical Chemistry C

Article

uniform width are ideal for Xe/Kr separation. Meek et al.14 studied polarizability effects on the adsorption of noble gases in IRMOF-1 and IRMOF-2. Their experimental and simulation results showed that increasing linker polarizability correlates with the increased adsorbate interaction and higher adsorption selectivity for Xe/Kr separation. Similar to the simulation studies, the number of experimental studies for noble gas adsorption is limited. Mueller et al.15 measured Xe, Kr, and Ar uptake of IRMOF-1, and Thallapally et al.16 measured Xe adsorption capacity of Ni/DOBDC (also known as CPO-Ni), IRMOF-1, and activated carbon. Their results showed that Ni/DOBDC has a higher Xe adsorption capacity compared to IRMOF-1 and activated carbon. Liu et al.17 recently measured adsorption of pure Xe, pure Kr, the Xe/Kr mixture, and the O2/N2/Ar/CO2/Xe/Kr mixture in HKUST-1 (also known as CuBTC), Ni/DOBDC, and activated carbon. They showed that Ni/DOBDC exhibits higher selectivity for pure Xe compared to pure Kr, and it is Xe selective in adsorption-based Xe/Kr separations. Dorcheh et al.18 investigated single-component Xe, Kr adsorption, and desorption in HKUST-1, MFU-4l, COF-102, and ZIF-8 and concluded that MFU-4l can be a good candidate for noble gas separation. Fernandez et al.19 examined Xe and Kr adsorption in FMOFCu and FMOFZn and suggested that sorption selectivity favors Kr at temperatures below 0 °C in FMOFCu. This result was attributed to the decreasing pore flexibility of FMOFCu below 0 °C which prevents Xe diffusion inside the channels. Previous studies have shown that characterizing the performance of MOF adsorbents and membranes using mixed gas feeds rather than single-component gases is crucial.20 Assessing the performance of MOF adsorbents (membranes) for noble gas separations requires the information of mixture adsorption (both mixture adsorption and diffusion). We recently used molecular simulations to predict the membrane-based separation performance of various MOFs for Xe/Kr and Xe/Ar mixtures.21 These calculations required a significant amount of computational time since performing mixture GCMC and mixture MD simulations is computationally demanding. Computing mixture diffusivities was the most time-consuming part of the membrane modeling due to the nature of the MD simulations. Theoretical correlations that can predict multicomponent diffusion coefficients from single-component diffusion data are extremely useful if these correlations are known to be accurate. This type of theoretical correlation was previously tested to predict transport diffusivities of H2/CH4 mixtures in CuBTC,22 self-diffusivities of H2/CH4 in ZIF-2,23 and COF-6,24 CH4/C4H10, and Xe/CH4 in silicate,25 and n-C4H10/i-C4H10 in IRMOF-1.26 In our recent work, we compared the results of molecular simulations with the experimentally available data for noble gas adsorption in MOFs and showed that there is a good agreement between molecular simulations and experiments.21 In this study, we tested the accuracy of theoretical correlations, which can replace computationally demanding molecular simulations, for predicting the performance of MOFs in the separation of noble gas mixtures. We used IAST and KP correlation to predict adsorption and diffusion of Xe/Kr and Xe/Ar mixtures based on the pure component adsorption and diffusion data of Xe, Kr, and Ar. Several separation properties of MOFs such as adsorption selectivity, working capacity, permeation selectivity, and gas permeability were predicted using theoretical correlations. We then compared these predictions with the

results of computationally demanding molecular simulations and discussed the accuracy of theoretical correlations.

2. COMPUTATIONAL DETAILS We studied IRMOF-1, ZIF-2, ZIF-3, ZIF-10, Zn(bdc)(ted)0.5, CPO-Ni, CPO-Co, bioMOF-11, and CuBTC in this work to represent MOFs having different metal sites, organic linkers, topologies, and pore sizes. Structural information of these MOFs such as unit cell parameters, pore sizes, and fractional pore volumes can be found in our early study.21 The atomic positions of all structures were obtained from their experimental XRD data (IRMOF-1,27 ZIF-2,28 ZIF-3,28 ZIF-10,29 Zn(bdc)(ted)0.5,30 CPO-Ni,31 CPO-Co,32 bioMOF-11,33 CuBTC34). We used rigid structures in our simulations because it has been recently shown that the results of molecular simulations using a rigid framework agree well with the ones using a flexible framework for noble gas adsorption in MOFs.21 2.1. Single-Component and Mixture Diffusion Coefficients from MD. Single-component self-diffusivity (Dsingle i,self ) which describes the motion of an individual tagged particle in an isotropic three-dimensional material and single-component corrected diffusivity (Đi) which describes the collective motion of multiple adsorbed molecules were computed as a function of adsorbed loading using MD simulations in canonical ensemble for Xe, Kr, and Ar. Mixture self-diffusivities (Dmixture i,self ) for Xe/Kr and Xe/Ar were computed at predefined adsorbed compositions of 50/50, 25/75, and 75/25 using NVT-MD simulations. The details of using MD simulations to compute singlecomponent and mixture diffusion coefficients have been described in the literature.35−37 The Dreiding38 force field was used in MD simulations to describe the MOF atoms because we previously showed that the results of molecular simulations using this force field agree well with the available experimental data for noble gas adsorption in MOFs.21 The Universal Force Field (UFF)39 was used for the parameters of the metal atoms in MOFs since they are not available in the Dreiding force field. Spherical Lennard-Jones (LJ) 12-6 potentials were used to model Xe (ε/kB = 221 K, σ = 4.01 Å), Kr (ε/kB = 166.4 K, σ = 3.63 Å), and Ar (ε/kB = 119.8 K, σ = 3.4 Å).11,40 The Lorentz−Berthelot mixing rules were employed to calculate the adsorbate−adsorbent and adsorbate−adsorbate LJ cross-interaction parameters. The intermolecular potentials were truncated at 13 Å, and the simulation box was increased up to 2 × 4 × 4 to contain enough particles at the lowest loadings considered. Periodic boundary conditions were applied in all simulations.41,42 Initial states with the appropriate loadings were created using GCMC simulations. We used a Nose-Hoover thermostat41 and performed 20 independent MD simulations for 12 ns for each loading we considered. Both single-component and mixture self-diffusivities of gases in the pores of MOFs were reported as average diffusivities using Di,self = (Di,x,self + Di,y,self + Di,z,self)/3 except CPOs which have one-dimensional pores. The diffusivities in CPOs were reported only in the z direction. 2.2. Predicting Mixture Diffusivities using KP Correlation. Once the single-component self- and corrected diffusivities were computed using MD simulations, they were fitted to continuous functions of fractional loading to apply KP correlation to predict the mixture self-diffusivities. Fractional loading of species i (θi) is defined as the ratio of adsorbed loading (Θi) to the saturation loading (Θi,sat) which are calculated from the GCMC simulations25 5230



⎛ Θ ⎞ θi = ⎜⎜ i ⎟⎟ ⎝ Θi,sat ⎠

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2.4. Predicting Mixture Adsorption using IAST. After evaluating the single-component adsorption isotherms of each component up to 30 bar by GCMC, a dual-site Langmuir or dual-site Freundlich model was fitted to single-component adsorption isotherms of Xe, Kr, and Ar in all MOFs

(1)

In fitting of single-component self- and corrected diffusivities, the functions describing corrected diffusivities were constrained to give the observed self-diffusion coefficients at zero loading as required by the definition of diffusivities.43 In contrast to Keskin et al.,22 we did not constrain these functions at the saturation loading to vanish because nonzero diffusivities were reported from the MD simulations at loadings close to the saturation. At least ten data points were collected from the MD simulations for single-component self- and corrected diffusivities of each species between zero loading and saturation loading. In this way, the accuracy of the fitting was increased since no fitting was done to predict diffusivities when simulation data are not present. In an adsorbed mixture, two additional diffusion coefficients, the self-exchange (Đcorr ii ) and the binaryexchange (Đcorr ij ) diffusivities, define the correlation effects which may occur due to the topology of the adsorbent or momentum transfer between adsorbed molecules44 or the concerted motions of adsorbed molecule clusters.45,46 The self-exchange diffusivities (Đcorr ii ) of each species were calculated using the single-component self- and corrected diffusivities by the following expression43

Θi,Langmuir =

1 single Di,self (θ )

−

1 Đi(θ )

(2)

⎤Θ i/(Θ i+ Θ j) ⎡Θ Ðcorr(θ )⎤Θ j/(Θ i+ Θ j) Θj,satĐijcorr(θ ) = ⎡⎣Θj,sat Ðcorr ⎣ i,sat jj ⎦ ii (θ )⎦ (3)

Once the single-component self-exchange and binary-exchange diffusivities were calculated as described above, the KP correlation predicts the mixture self-diffusivities of each species 43 (Dmixture i,self ) from pure component diffusivities as follows 1 1 Đi

+

θi Điicorr

+

θj Đijcorr

eP f hP k + f g+P m + Pk

In these models, Θi is the adsorbed amount of molecule i (molecules/unit cell); P is the pressure (bar); and others are parameters of the models. The saturation loadings (Θi,sat) of the single components were evaluated at infinitely large pressures. Using fitted adsorption isotherms and applying IAST,48 adsorption isotherms of Xe/Kr and Xe/Ar mixtures were predicted. IAST assumes ideal gas behavior and does not consider the adsorbent heterogeneity; therefore, it works very well at low pressures and for materials with homogeneous adsorption sites. IAST has been widely used to predict mixture adsorption in MOFs such as C2H4/C2H6 adsorption in ZIF-8,49 CH4/N2 adsorption in MFI, MIL-47, IRMOF-12,50 and CO2/N2 adsorption in IRMOF-3, MOF-177, and UMCM-151 and found to make accurate predictions. Both IAST and KP correlation are known to fail in predicting mixture adsorption and diffusion data if strong heterogeneity exists for adsorbed mixtures. Predictions of these theories are expected to be less accurate when (a) there is a strong competition between adsorbates for the same adsorption site and (b) there is a significant difference in size and chemistry of the adsorbates.52 2.5. Predicting Noble Gas Separation Performances of MOFs Using Theoretical Correlations. The main objective of this study is to examine if one can evaluate the noble gas separation performance of MOFs using theoretical correlations without performing computationally demanding mixture simulations. Adsorption selectivity and working capacity (permeation selectivity and gas permeability) are the two important parameters that describe the potential of MOFs in adsorption-based (permeation-based) separation of noble gases. We calculated adsorption selectivity, permeation selectivity, working capacity, and gas permeability of MOFs for Xe/Kr and Xe/Ar mixtures using theoretical correlations, namely, IAST and KP, and compared our findings with the results of mixture simulations, GCMC and MD. Permeation selectivity (Spermeation(i/j)) can be approximated as the multiplication of adsorption selectivity (Sadsorption(i/j)) and diffusion selectivity (Sdiffusion(i/j)). More details of this approximation and its validation can be found elsewhere52

To use this expression for mixtures, we replaced the fractional single-component occupancy with the fractional total occupancy,22 θ = θi + θj = (Θi/Θi,sat) + (Θj/Θj,sat). The binary exchange 47 coefficients, Đcorr ij , were estimated by the following expression

mixture Di,self =

Θi,Freundlich =

(5)

θ

Điicorr(θ ) =

aP cP + b+P d+P

(4)

We predicted self-diffusivities of the Xe/Kr and Xe/Ar mixtures at adsorbed compositions of 50/50, 25/75, and 75/25 and compared these predictions with the results of mixture MD simulations. 2.3. Single-Component and Mixture Adsorption from GCMC. The GCMC simulations were carried out for predicting single-component and mixture adsorption isotherms. This is a well-developed method to compute the number of adsorbed molecules in a nanoporous material.41 The same potential models and force fields as described in Section 2.1 were used for GCMC simulations. The temperature and fugacity of the adsorbing gases were specified, and the number of adsorbed molecules was calculated at equilibrium. At the lowest fugacity of each system, simulations were started from an empty MOF matrix. Each subsequent simulation at higher fugacity was started from the final configuration of the previous run. Simulations consisted of a total of 3 × 107 trial configurations with the last half of the configurations used for data collection. A configuration is defined as an attempted translation or creation or deletion of the adsorbate molecules. For the mixture GCMC simulations, particle swap moves were also done.

S permeation

(i/j),sim

= Sadsorption

(i/j),GCMC

=

·Sdiffusion(i/j),MD

mixture (Θi,GCMC)/(Θj,GCMC) Di,self (Θi , Θj,),MD · mixture yi /yj Dj,self (Θi , Θj),MD

(6)

Here, the adsorption selectivity is the ratio of adsorbed loadings (Θi/Θj) normalized by the bulk-phase composition (yi/yj) of the gas mixture. The adsorbed loadings were obtained from the mixture GCMC simulations, and the bulk-phase composition of Xe/Kr and Xe/Ar mixtures was set to 20/80 to represent the industrial gas separation conditions. The diffusion selectivity was defined as the ratio of binary mixture self-diffusivities of mixture each species (Dmixture i,self /Dj,self ) and evaluated using the MD simulations at the corresponding adsorbed loadings obtained 5231



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Figure 1. Predictions of KP correlation (dotted lines) and the results of mixture MD simulations (symbols) for self-diffusion coefficients of (a) Xe/Kr:25/75, (b)Xe/Kr:50/50, and (c) Xe/Kr:75/25 mixtures in IRMOF-1. (d) Comparison of theory and simulations for mixture self-diffusivities at all loadings and compositions.

adsorption pressure of 10 bar and a desorption pressure of 1 bar at room temperature

from the GCMC simulations. The adsorption and permeation selectivity of MOFs for Xe/Kr and Xe/Ar separations can be estimated using only the pure component data and theoretical correlations as follows S permeation

(i/j),theory

= Sadsorption

(i/j),IAST

=

WC Xe,sim = qXe(10bar),GCMC − qXe(1bar),GCMC WC Xe,theory = qXe(10bar),IAST − qXe(1bar),IAST

·Sdiffusion(i/j),KP

mixture (Θi,IAST)/(Θj,IAST) Di,self (Θi , Θj,),KP · mixture yi /yj Dj,self (Θi , Θj),KP

(8)

The gas permeability through MOFs was calculated using the following expressions53

(7)

Pi,sim =

In these expressions, mixture adsorption amounts of each species were calculated from IAST based on pure component adsorption isotherms, and the mixture self-diffusivities were calculated using KP correlation based on single-component self-, corrected and exchange diffusivities. We compared the adsorption selectivities calculated from simulations (Sadsorption(i/j),GCMC) with the ones calculated from theory (Sadsorption(i/j),IAST) and permeation selectivities calculated from simulations (Spermeation(i/j),sim) with the ones calculated from the theory (Spermeation(i/j),theory) at 298 K and 10 bar. The Xe working capacities (WCXe) of MOFs were calculated as the difference of gas uptakes (q mol Xe/kg MOF) at an

mixture ϕ·Di,self,MD ·c i,GCMC

fi

Pi,theory =

mixture ϕ·Di,self,KP ·c i,IAST

fi (9)

In this expression, Pi is the permeability of species i (mol/(m s Pa)); ϕ is the fractional pore volume of the material; Dmixture is i,self the mixture diffusivity (m2/s); ci is the concentration of species i in the mixture at the upstream face of the membrane (mol/m3); and f i is the bulk-phase fugacity of the species i (Pa). We reported permeability values in Barrers since it is more commonly used in the membrane community. We finally computed ideal selectivity of MOFs for noble gas separations. The ideal selectivity is simply the ratio of adsorbed amounts of pure gases multiplied by the ratio of 5232



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Figure 2. Predictions of KP correlation (dotted lines) and the results of mixture MD simulations (symbols) for self-diffusion coefficients of (a) Xe/Kr:50/50 and (b) Xe/Kr:75/25 mixtures in ZIF-2. (c) Comparison of theory and simulations for mixture self-diffusivities at all loadings and compositions.

self-diffusivities. For a Xe/Kr separation, a MOF’s separation performance at 10 bar can be calculated as the following Sideal(Xe/Kr)sim. =

Similar calculations were carried out to estimate the ideal selectivity of MOFs for Xe/Ar separations.

single Θsingle Xe,GCMC,10bar DXe,self,MD(10bar) · single Θsingle Kr,GCMC,10bar DKr,self,MD(10bar)

3. RESULTS AND DISCUSSIONS We first examined the accuracy of KP correlation for predicting the mixture self-diffusivities of noble gases in MOFs by comparing the theoretical predictions with the results of mixture MD simulations. On the basis of this comparison, we categorized the MOFs into three groups: MOFs for which the predictions of KP correlation are in a very good agreement with the simulations (IRMOF-1, ZIF-3, and ZIF-10), MOFs for which the predictions of KP correlation are in a reasonable agreement with the simulations (ZIF-2, Zn(bdc)(ted)0.5, CPONi, and CPO-Co), and finally MOFs for which the predictions of KP correlation do not agree with the simulations (bioMOF11 and CuBTC). In every case, we showed the results for one representative MOF, and the results of all other MOFs for Xe/Kr and Xe/Ar mixtures are given in Figure S1 and Figure S2 of the Supporting Information, respectively. In all MOFs, the magnitude of the single-component selfdiffusivities has the following order: DXe,self < DKr,self < DAr,self.

(10)

Early studies have shown that ideal selectivity can be significantly different than the mixture selectivity.20 A revised version of the ideal selectivity that considers the composition effect of the mixture was recently suggested.54 This revised version calculates ideal selectivity by considering the singlecomponent adsorption and diffusion of each component at their partial pressures in the mixture. For example, the ideal selectivity of a MOF for a Xe/Kr:20/80 bulk mixture at 10 bar can be computed as follows

Sideal(Xe/Kr),sim.with composition effect

⎛ (Θsingle Xe,GCMC,2bar) ⎞ ⎜ single ⎟ single ⎝ Θ Kr,GCMC,8bar ⎠ DXe,self,MD(2bar) = · single yXe DKr,self,MD(8bar) y Kr

(11) 5233



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Figure 3. Predictions of KP correlation (dotted lines) and the results of mixture MD simulations (symbols) for self-diffusion coefficients of (a) Xe/Kr:50/50 and (b) Xe/Kr:75/25 mixtures in bioMOF-11. (c) Comparison of theory and simulations for mixture self-diffusivities at all loadings and compositions.

mixture self-diffusivities of Xe and Kr are almost independent from the adsorbed loading since the material’s free volume is high and both gases are away from saturation. Figure 1d compares the predictions of KP correlation with the results of mixture MD simulations at all compositions and suggests that KP correlation accurately predicts the self-diffusivities of Xe/Kr mixtures at various conditions. The same discussion is valid for Xe/Ar mixtures in IRMOF-1, and the results are shown in Figure S2 (Supporting Information). As we mentioned in the beginning of discussion, KP also makes accurate predictions for self-diffusivities of Xe/Kr and Xe/Ar mixtures in ZIF-3 and ZIF-10 using pure component diffusivities of Xe, Kr, and Ar. These MOFs also possess larger cavity diameters and higher free volumes compared to the other MOFs we considered in this work. Similar to IRMOF-1, the self-diffusivity values for Xe/Kr and Xe/Ar mixtures do not significantly change with the loadings in ZIF-3 and ZIF-10. In Figure 2, we showed the results of ZIF-2, a representative MOF for which there is a fair agreement between theory and simulations for self-diffusivities of noble gas mixtures. The KP correlation captures the diffusion behavior of equimolar mixtures well but makes less accurate predictions if one component is dominated in the adsorbed mixture. Figure 2a shows that the self-diffusivities of Xe and Kr are almost

The adsorbate with the highest interaction energy parameter (Xe) has a stronger adsorption tendency and therefore lower mobility, whereas the adsorbate with the lowest interaction energy parameter (Ar) has a higher tendency to diffuse. The diffusion order also agrees with the molecular weight of the gases, and the lighter adsorbate (Ar) shows the higher diffusivity. As expected from the single-component diffusion behavior, Kr (Ar) diffuses faster than Xe in Xe/Kr (Xe/Ar) mixtures in all MOFs we studied. Figure 1 shows the self-diffusivities of the Xe/Kr mixture as a function of total adsorbed loading at three different adsorbed compositions in IRMOF-1. The predictions of KP correlation agree well with the results of mixture MD simulations at all loadings. This good agreement between theory and simulations can be explained by considering the mixture correlation effects: IRMOF-1 is the most porous material among the ones we considered in this work with a large cavity diameter of ∼15 Å and total pore volume of 74%. Due to this high free volume, the adsorbates act as they are in the single-component case. For example, the presence of the slow Xe (fast Kr) atoms does not significantly decrease (increase) the diffusion of fast Kr (slow Xe) atoms. This shows that the correlation effects between different adsorbates are negligible, which leads to high accuracy of KP. Another observation from Figure 1a−c is that the 5234



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Figure 4. Predictions of KP correlation (dotted lines) and the results of mixture MD simulations (symbols) for self-diffusion coefficients of (a) Xe/Kr:50/50 and (b) Xe/Kr:75/25 mixtures in CuBTC. (c) Comparison of theory and simulations for mixture self-diffusivities at all loadings and compositions.

independent from the adsorbed loadings in an equimolar adsorbed mixture, and the KP predictions agree well with the simulations at this composition. On the other hand, Figure 2b shows that the diffusivity of Kr decreases sharply when the concentration of slowly diffusing Xe atoms increases in the mixture. It is well-known that more mobile components (in our case Kr) are much affected from the correlation effects compared to slow components.43 Since the slower component (Xe) does not vacate the adsorption site quickly, a sharp decrease is observed in Kr diffusivity. Figure 2c shows the comparison between theory predictions and simulation results for the mixture self-diffusivities of Xe/Kr at 50/50 and 75/25 adsorbed compositions in ZIF-2. The KP correlation captures self-diffusivities at 50/50 loading but overestimates both Xe and Kr diffusivities in the Xe/Kr:75/25 mixture. Here it is important to explain why we did not study the mixture of Xe/Kr:25/75 in ZIF-2. The industrial gas mixture has a bulk composition of Xe/Kr:20/80 (Xe/Ar:20/80) which corresponds to adsorbed compositions of Xe > 50 in all MOFs (see Table S1 and Table S2, Supporting Information). The minimum adsorbed Xe composition was observed in IRMOF-1. Therefore, except IRMOF-1, we did not show the results for Xe/Kr:25/75 and Xe/Ar:25/75 throughout this paper.

The deviations between KP correlation and MD simulations can be also discussed in terms of binary-exchange diffusivities. According to eq 3, the ratio of binary exchange diffusivities (Đij/Đji) is equal to the ratio of saturation loadings of pure components (Θi,sat/Θj,sat) at all mixture compositions. For example, the single-component saturation loadings of Xe and Kr in ZIF-2 were calculated as 25.3 and 28.6 molecules/unit cell at room temperature, respectively. Since these values are close to each other, the binary-exchange diffusivities, Đij and Đji, are also similar. For equimolar mixtures, getting similar binaryexchange diffusivity values is reasonable, which is also supported by the accurate predictions of KP. The KP correlation assumes that the rate of filling a Xe vacancy by a Kr atom is almost equal to the rate of filling a Kr vacancy by a Xe atom. However, this is not a good assumption for mixtures where one species is dominant in the mixture such as Xe/Kr:75/25. Therefore, less accurate predictions are expected from the KP correlations for this composition. Our results also showed that the KP predictions are generally better for Xe/Kr mixtures compared to Xe/Ar mixtures, and this can be attributed to the fast diffusion of Ar which is more affected by the correlation effects as we discussed above. 5235



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Figure 5. Ratio of single-component self-diffusivity to the mixture self-diffusivity for (a) Kr in Xe/Kr:50/50, (b) Kr in Xe/Kr:75/25, (c) Xe in Xe/Kr:50/50, and (d) Xe in Xe/Kr:75/25 in all MOFs.

ZIF-3, and ZIF-10 and the decreasing diffusivity trend observed in ZIF-2, Zn(bdc)(ted)0.5, CPO-Ni, CPO-Co, and bioMOF-11 (see Figures S1 and S2, Supporting Information). The loading dependency of self-diffusivity in CuBTC can be explained by the Type III diffusion behavior proposed by Karger et al.55 They suggested that some adsorption sites of the material have strong adsorption energies, and these sites attract the molecules at the lowest loadings and prevent their diffusion. As the loading increases, these strong binding sites become occupied, and other less-strong adsorption sites are filled by adsorbates which lead to an increase in the self-diffusivity. The selfdiffusivity becomes constant as the loading further increases because every strong adsorption site is filled at high loadings. This distinct self-diffusion behavior in CuBTC indicates that CuBTC has strong adsorption sites. Snurr and co-workers11 also showed that Xe preferentially occupies octahedral pockets of CuBTC and prevents Kr from adsorbing to these pockets which also proves that octahedral pockets are strong adsorption sites of CuBTC. Due to the existence of these sites and competitive adsorption, less accurate predictions are expected from the KP. Figure 4 shows that KP captures the increasing trend of mixture diffusivities at low loadings qualitatively but does not make very accurate predictions quantitatively. This is an expected result because it is well-known that KP correlation

In Figure 3, we presented the results of a MOF, bioMOF-11, for which the agreement between theory and simulations is weak. The mixture self-diffusivities of both components show sharp decrease with increasing loading for all adsorbed compositions of Xe/Kr mixtures instead of slightly decreasing behavior proposed by KP correlation. The failure of KP correlation can be attributed to several reasons: The cavities of bioMOF-11 are 5.8 Å in diameter, and its apertures between cavities are 5.2 Å in diameter, which indicates that the pores of this material are too narrow for mutual passage of Xe and Kr. Furthermore, the pore volume of bioMOF-11 (0.45 cm3/g)32 is low compared to other MOFs, and saturation loadings of Xe, Kr, and Ar are very low (see Table S1 and S2, Supporting Information). Therefore, as the loading increases, steric hindrance effects become dominant, and the self-diffusivities of both gases decrease sharply. Since KP does not include steric hindrance and topology effects, it overestimates the selfdiffusivities of both components. We also presented the self-diffusivities of Xe/Kr mixtures in CuBTC because the diffusivity trends of gases in CuBTC are different from the ones in other MOFs. Figure 4 shows that self-diffusivities of Xe and Kr in CuBTC increase with loading and then approach a plateau at higher loadings. This is in contrast to the loading-independent diffusion in IRMOF-1, 5236



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Figure 6. Ratio of single-component self-diffusivity to the mixture self-diffusivity of (a) Ar in Xe/Ar:50/50, (b) Ar in Xe/Ar:75/25, (c) Xe in Xe/Ar:50/50, and (d) Xe in Xe/Ar:75/25 in all MOFs.

single-component diffusivities, and the KP correlation makes accurate estimates for mixture diffusivities using singlecomponent data. Figure 5 represents that this ratio is close to one for equimolar adsorbed composition of Xe/Kr mixtures in all MOFs except bioMOF-11, and we already showed that KP correlation makes accurate estimates for all MOFs except bioMOF-11 if the composition is equimolar. Deviation from one was observed for Xe/Kr:75/25 mixtures and becomes more observable for bioMOF-11, ZIF-2, and Zn(bdc)(ted)0.5, the materials for which KP makes less accurate predictions for nondiagonal compositions. mixture We also examined the effect of composition on Dsingle i,self /Di,self using the data shown in Figure 5. For example, the maximum mixture for Kr in bioMOF-11 is 3.94 at the value of Dsingle i,self /Di,self Xe/Kr:50/50 mixture. This value increases up to 5.06 for Xe/Kr:75/25 because the mixture self-diffusivity of Kr decreases as the number of slow Xe atoms increases in the mixture for Xe mixture. Similarly, the maximum value of Dsingle i,self /Di,self in bioMOF-11 is 5.19 at Xe/Kr:75/25, and this value decreases to 3.87 for the Xe/Kr:50/50 mixture. As the amount of Kr increases in the mixture, Kr atoms fasten the slow Xe atoms; the self-diffusivity of Xe atoms increases; and the value of mixture single mixture values Dsingle i,self /Di,self decreases. Figure 6 shows Di,self /Di,self for Xe/Ar mixtures. Similar to the Xe/Kr case, the deviation of the ratio from one is less for equimolar gas mixtures.

does not work accurately for energetically heterogeneous materials.56,57 It is important to note that KP was also reported to be inaccurate for predicting mixture diffusion of CO2/H2 in CuBTC,52 and IAST was reported to be less accurate for Xe/Kr and CO2/H2 adsorption in CuBTC.11,58 One conclusion from the results we had so far is that KP correlation makes accurate predictions especially at low loadings although it may not be accurate at high loadings. At zero loading, the single-component self-diffusivity is equal to the corrected diffusivity since there is no adsorbate−adsorbate interaction. At low loadings, the mixture self-diffusivity approaches to single-component self- and corrected diffusivities mixture Dsingle i,self (θ → 0) ≅ Đi(θ → 0) ≅ Di,self (θi → 0, θj → 0) (see eqs 2 and 4). Therefore, KP makes accurate predictions at low loadings for mixture self-diffusivities by using the singlecomponent self- and corrected diffusivities. As the loading increases, adsorbate−adsorbate interactions become important, and predictions of KP get less accurate. If the mixture selfdiffusivities show a similar trend to the single-component selfdiffusivities, a high accuracy is expected from the KP correlation. To examine this, we computed the ratio of pure component self-diffusivity to the mixture self-diffusivity (Dsingle i,self / ) for each component in Xe/Kr and Xe/Ar as a function Dmixture i,self of total loading using MD simulations. If this ratio is around one that means mixture self-diffusivities do not deviate from the 5237



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IAST. The results showed that IAST predictions for adsorption of Xe/Kr and Xe/Ar mixtures are good for all materials. Small deviations were observed in predicting working capacity due to the accumulation of the errors in adsorbed concentration values at 1 and 10 bar. The largest deviation between the mixture GCMC simulations and IAST was observed for the Xe/Ar adsorption selectivity of bioMOF-11. IAST overestimates Ar adsorption and underestimates Xe adsorption in bioMOF-11; therefore, its selectivity prediction is less than the one calculated from mixture GCMC. This figure shows that using IAST instead of computationally demanding mixture GCMC simulations can give accurate answers for evaluating the adsorption-based separation performance of MOFs. Previous research showed that NaX zeolite has an adsorption selectivity of ∼6 for Xe over Kr,11,59 and NaA zeolite has a Xe selectivity of 4−4.5 (20−25) from an equimolar Xe/Kr (Xe/Ar) mixture at 300 K between 1 and 10 bar.60 Except IRMOF-1 and ZIF-10, all the MOFs considered in this work show higher Xe selectivity than NaA and NaX zeolites. Once mixture self-diffusivities are predicted using KP and mixture adsorption isotherms are predicted using IAST, the permeation selectivity and permeability of MOFs for membrane-based separation of Xe/Kr and Xe/Ar mixtures can be estimated using eqs 7 and 9, respectively. Figure 8

If we compare Figures 5 and 6, we can see that deviation of the ratio for Ar is higher than the one for Kr in the mixtures where adsorbed composition of Xe is 75%. For example, at a total loading of 18 molecules/unit cell for Zn(bdc)(ted)0.5, the value of ratio is 2.53 and 3.68 for Kr and Ar, respectively. This observation also supports the idea that the fast diffusing component (Ar) is much more affected from the correlation effects. These results suggest that one can predict whether the KP correlation will give accurate estimates or not by computing mixture using MD simulations at a few loadings prior to Dsingle i,self /Di,self mixture is doing extensive calculations. If the value of Dsingle i,self /Di,self around 1, then the KP correlation can be used to accurately predict the mixture self-diffusivities of the components at a wide range of loadings for that material. It is important to note that the adsorbed loadings considered in Figures 5 and 6 are different for each MOF. Since saturation loadings of IRMOF-1, ZIF-10, and CuBTC are higher than the other MOFs, pure component and mixture-self-diffusivities were examined up to higher loadings for these MOFs compared to others. After testing the accuracy of a theoretical correlation for predicting mixture self-diffusivities, we now turn to the theoretical method that can predict the mixture adsorption based on single-component adsorption data. Figure 7 shows

Figure 7. Adsorption selectivity and working capacity of MOFs calculated from GCMC simulations and predicted by IAST for (a) Xe/Kr and (b) Xe/Ar separations at 10 bar, 298 K.

Figure 8. Permeation selectivity and gas permeability of MOFs calculated from simulations (GCMC and MD) and predicted by theory (IAST and KP) for (a) Xe/Kr and (b) Xe/Ar separations at 10 bar, 298 K.

adsorption selectivity and working capacity of MOFs calculated at 10 bar and 298 K using mixture GCMC simulations and

compares the Xe permeability and permeation selectivity calculated from mixture simulations (GCMC and MD) with 5238



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We finally compared the ideal selectivity of MOFs for Xe/Kr and Xe/Ar separations with the mixture selectivities calculated from theoretical correlations and simulations. Table S.3 and Table S.4 (Supporting Information) show that ideal selectivity calculated using single-component data is significantly different than the mixture selectivity for Xe/Kr and Xe/Ar separations. The ideal selectivity that considers the composition effect is slightly higher, but its predictions are not close to the mixture selectivity since the competition between adsorbates is not considered. For example, both theory and simulations suggest that ZIF-2 is a good candidate for separation of Xe/Ar mixtures due to its high Xe selectivity (predicted as 10.3 by theory and computed as 10.2 by simulations) compared to other MOFs. However, ideal selectivity assumes that ZIF-2 is only weakly Ar selective (1.89) under the same conditions. This result underlines the importance of evaluating a material’s separation performance based on mixed gas feeds rather than the single gas feeds. We can conclude that if the single-component adsorption and diffusion data are available, using KP and IAST to estimate the mixture selectivity is much more accurate than calculating the ideal selectivity to evaluate the performance of a material in separation processes.

the values predicted from theoretical correlations (KP and IAST) for Xe/Kr and Xe/Ar mixtures. Since permeation selectivity is the product of adsorption selectivity and diffusion selectivity, the differences between theory and simulations are due to the deviation of IAST and/or KP. Figure 7 showed that adsorption selectivities predicted from IAST are in good agreement with the ones directly calculated from mixture GCMC; therefore, differences observed in permeation selectivity in Figure 8 must be due to the deviation of KP. For example, the Xe/Kr:20/80 bulk mixture in bioMOF-11 corresponds to an adsorbed mixture of Xe/Kr:73/27. We showed that KP overestimates both Xe and Kr self-diffusivities compared to MD at an adsorbed composition of Xe/Kr:75/25 in Figure 3, and overestimation for Kr was higher than Xe. Due to this reason, permeation selectivity calculated from theory is less than the value calculated from simulations. One interesting case from Figure 8 is the Xe/Ar separation performance of ZIF-2. In this case, theory overestimates Xe permeability of ZIF-2 in Xe/Ar separations but predicts the Xe selectivity well compared to the simulations. This is due to the overestimation of individual diffusion coefficients by the KP correlation. The KP correlation overestimates the mixture mixture self-diffusivities of Xe and Ar (Dmixture Xe,self and DAr,self ) in the same amount and therefore predicts almost the same mixture diffusion selectivity (Dmixture Xe,self /DAr,self ) with the mixture MD simulations. Since the estimates of IAST agree well with the GCMC results, the Xe permeation selectivity calculated by the simulations and predicted by the theories are almost the same, although the Xe permeability was overestimated (see Table S2, Supporting Information). In summary, Figure 8 indicates that using theoretical correlations to evaluate the performance of MOFs for permeation-based separation of noble gas mixtures for prescreening purposes is appropriate. The predictions of theory agree with the results of simulations about the materials’ performance in separation applications. For example, both theory and simulations suggest that CPOs are good candidates in adsorption-based and membrane-based separation of noble gases due to their high gas permeability and good selectivity. The example that we discussed above for ZIF-2 indicates that diffusion selectivity derived from theory and the one obtained from simulations can be very similar, but this does not guarantee that the individual diffusion coefficients are accurately predicted by the KP correlation. Because of this reason, it is better to compare the adsorbed amounts and self-diffusivities of each component obtained from theory and simulations to make a better judgment about the accuracy of theoretical correlations. With this aim, we compared the results of simulations and predictions of theories for the adsorbed concentrations and self-diffusivities of each component in Xe/Kr and Xe/Ar mixtures at 10 bar, 298 K in Figure S3 (Supporting Information). The adsorbed amounts predicted by IAST perfectly agree with the ones calculated from GCMC in Figure S3a-b (Supporting Information). On the other hand, deviations between KP correlation and MD simulations are observed for bioMOF-11 and Zn(bdc)(ted)0.5 in Figure S3c-d (Supporting Information). For instance, KP correlation underestimates (overestimates) both Xe and Kr mixture self-diffusivities in Zn(bdc)(ted)0.5 (bioMOF-11) by nearly the same ratio; therefore, Xe permeation selectivities obtained by simulations and predicted by theory are similar to each other for this MOF even though the self-diffusivities are not.

4. CONCLUSION In this work, we computed adsorption and diffusion of Xe/Kr and Xe/Ar mixtures in several MOFs by using both computationally demanding mixture simulations (GCMC and MD) and theoretical correlations (IAST and KP). Our results showed that IAST predicts the mixture adsorption well for both Xe/Kr and Xe/Ar in all MOFs, but the KP predictions may deviate from MD results in some cases. The KP correlation that predicts the mixture diffusivities based on single-component diffusivities is expected to be more accurate for equimolar mixtures, and it becomes less accurate if one of the components is dominant in the adsorbed phase. The agreement between theory and simulations is best at zero loading since correlation effects are negligible at low loadings. These effects are dominant at high adsorbate loadings and in materials that have heterogeneous adsorption sites. Since the KP correlation does not take all these effects into account, it is expected to make less accurate predictions under these conditions. Finally, it is important to note that the presence of heterogeneous adsorption sites cannot be the only reason for the failure of KP correlation. Any kind of heterogeneity such as existence of different pore sizes, shapes, loading, and composition of the adsorbates also play important roles in the accuracy of theoretical correlations. Several separation properties of MOFs such as adsorption selectivity, working capacity, diffusion selectivity, permeation selectivity, and gas permeability can be accurately predicted by applying theoretical correlations. By using these correlations, it is possible to rapidly examine a large range of potential operating conditions (pressure, temperature, composition) for chemical mixtures as soon as information on each species in the MOF of interest is known. If methods equivalent to the correlations we have explored were not available, the performance of MOFs for chemical mixtures could only be tested by accumulating detailed mixture diffusion and adsorption data at every state point relevant to the macroscopic process being examined using computationally demanding simulations. 5239



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ASSOCIATED CONTENT

S Supporting Information *

The predictions of KP correlation and the results of mixture MD simulations for self-diffusion coefficients of Xe/Kr and Xe/Ar mixtures in all MOFs at various compositions, comparison of adsorbed compositions obtained from IAST and mixture GCMC simulations for Xe/Kr and Xe/Ar mixtures in all MOFs, selectivities and permeabilities of MOFs calculated from theory and simulation for Xe/Kr and Xe/Ar mixtures. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support provided by The Scientific and Technological Research Council of Turkey (TUBITAK) National Scholarship Programme for MSc Students (BIDEB-2228) is gratefully acknowledged. We thank to Ms. Lutfiye Hallioglu for valuable discussions.

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REFERENCES

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