Density Functional Theory for Adsorption of Gas ... - ACS Publications

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J. Phys. Chem. B 2010, 114, 2820–2827

Density Functional Theory for Adsorption of Gas Mixtures in Metal-Organic Frameworks Yu Liu,†,‡ Honglai Liu,†,* Ying Hu,† and Jianwen Jiang‡,* State Key Laboratory of Chemical Engineering and Department of Chemistry, East China UniVersity of Science and Technology, Shanghai 200237, China, and Department of Chemical and Biomolecular Engineering, National UniVersity of Singapore, Singapore 117576 ReceiVed: NoVember 3, 2009; ReVised Manuscript ReceiVed: January 22, 2010

In this work, a recently developed density functional theory in three-dimensional space was extended to the adsorption of gas mixtures. Weighted density approximations to the excess free energy with different weighting functions were adopted for both repulsive and attractive contributions. An equation of state for hard-sphere mixtures and a modified Benedict-Webb-Rubin equation for Lennard-Jones mixtures were used to estimate the excess free energy of a uniform fluid. The theory was applied to the adsorption of CO2/CH4 and CO2/N2 mixtures in two metal-organic frameworks: ZIF-8 and Zn2(BDC)2(ted). To validate the theoretical predictions, grand canonical Monte Carlo simulations were also conducted. The predicted adsorption and selectivity from DFT were found to agree well with the simulation results. CO2 has stronger adsorption than CH4 and N2, particularly in Zn2(BDC)2(ted). The selectivity of CO2 over CH4 or N2 increases with increasing pressure as attributed to the cooperative interactions of adsorbed CO2 molecules. The composition of the gas mixture exhibits a significant effect on adsorption but not on selectivity. 1. Introduction In a nanoconfined space, fluids have significantly different thermodynamic and structural properties from the bulk phase and exhibit rich surface phenomena such as adsorption, layering, and capillary condensation. Understanding the properties of confined fluids is not only of interest in physical and surface sciences, but also of practical importance for industrial applications. Toward this end, remarkable advances have been achieved on the basis of statistical mechanics for example, density functional theory (DFT) and integral equation theory. In particular, DFT turns out to be robust in the study of inhomogeneous fluids.1-3 In the past a few decades, a large number of various versions of DFT have been proposed.4-20 The central theme in DFT is to precisely estimate the excess free energy for the system of interest. In general, the excess free energy is decomposed into hard-core and attractive contributions with different approximations. For instance, the fundamental measure theory (FMT)21 and the weighted density approximation (WDA)22 have been used for the hard-core contribution. In the WDA, a weighting function is introduced to take into account the oscillatory density distribution. The Heaviside function is such an example that is easy to use in calculations and works fairly well for polymeric systems.23 A commonly used weighting function is the series density expansion proposed by Tarazona.24 The weighting function of Curtin and Ashcroft25 is rigorous and accurate; however, it is complicated to implement in calculations. Similarly to the hard-core contribution, different approximations are also used for the attractive contribution. One example is the mean-field approximation (MFA), which does not require information about a uniform fluid. However, the MFA is not very reliable at low temperatures.6,7 In contrast, the WDA is widely used with an appropriate equation of state for uniform fluids. * Corresponding authors. E-mail: [email protected] (H.L.), chejj@ nus.edu.sg (J.J.). † East China University of Science and Technology. ‡ National University of Singapore.

Despite substantial advances in the modeling of inhomogeneous fluids, DFT is mostly applied in simple confined geometries such as one-dimensional cylindrical and slit pores. In three-dimensional systems, DFT calculations are very demanding. Nevertheless, most practical systems are threedimensional, for example, adsorption in activated carbons, zeolites, and metal-organic frameworks (MOFs).26 Thus, threedimensional porous materials are regarded as promising candidate for gas adsorption and separation. Recently, Siderius and Gelb proposed a lattice-based DFT by combining Tarazona’s WDA for repulsion and the MFA for attraction; the predicted isotherm for H2 adsorption in MOF-5 was in good agreement with simulation at 298 K.27 Using the WDA for both attractive and repulsive contributions, we also developed a threedimensional DFT.28 Our theoretical predictions for H2 adsorption in two MOFs matched well with simulation and experimental results at both high and low temperatures. In addition, the MFA was found to be largely inaccurate at low temperatures.28 In this work, we extend the three-dimensional DFT to gas mixtures in MOFs. As a new class of organic-inorganic hybrid materials, MOFs consist of metal-based building blocks connected by organic linkers and have received considerable attention for gas storage and separation. The controllable length of the organic linkers and the variation of the metal oxides in MOFs allow for the tailoring of their pore size, volume, and functionality in a rational way.29 Whereas a large number of simulation studies have been conducted for the adsorption of gas mixtures in MOFs,30-39 no DFT study has been reported toward this end. In section 2, the DFT for mixtures in threedimensional space is introduced. Similarly to our previous work, the WDA is employed for both the repulsive and attractive contributions in the excess free energy. Section 3 briefly describes the simulation method employed to validate the DFT predictions. In section 4, we examine the adsorption of CO2/ CH4 (natural gas) and CO2/N2 (flue gas) mixtures in two MOFs, namely, ZIF-8 (ZIF ) zeolitic imidazolate framework) and Zn2(BDC)2(ted) (BDC ) benzene-1,4-dicarboxylate, ted )

10.1021/jp9104932  2010 American Chemical Society Published on Web 02/09/2010

DFT for Adsorption of Gas Mixtures in MOFs

J. Phys. Chem. B, Vol. 114, No. 8, 2010 2821

triethylenediamine). The adsorption isotherms, selectivities, and density distributions are predicted from DFT and compared with simulation. With CH4 as the major component, natural gas is an alternative substitute for environmentally unfriendly fossil fuels and impurities such as CO2 in natural gas need to be removed in order to improve the energy content. Flue gas primarily consists of CO2 and N2 emitted by power plants, and CO2 capture from flue gas is a pressing issue in environmental protection.

core and attractive potentials using the Weeks-ChandlerAndersen (WCA) perturbation theory

uLJ(r) ) uhc(r) + uattr(r) where uhc(r) represents the hard-core interaction

uhc(r) )

2. Theory For an open system with K types of components, the grand potential Ω can be expressed as a functional of density distributions Ω[F1(r),F2(r), ..., FK(r)]. For simplification, we define the density vector as G(r) ) [F1(r),F2(r), ..., FK(r)]. At equilibrium, the grand potential reaches its minimum, that is

δΩ[G(r)] )0 δFi(r)

(i ) 1, 2, ..., K)

K

Ω[G(r)] ) F[G(r)] -

∑

K

µiNi ) F -

i)1

∑ ∫ µiFi(r) dr i)1

(2) where µi is the chemical potential of component i and F is the free energy, which consists of three terms, namely, the idealgas, excess, and external-field contributions

F[G(r)] ) Fid[G(r)] + Fex[G(r)] + Fext[G(r)]

(3)

The ideal-gas term is

{

-ε,

uattr(r) ) 4ε[(σ/r) 0,

∑ ∫ Fi(r){ln[Fi(r)Λ3] - 1} dr

(4)

i)1

rh )

12

r < rm - (σ/r) ], rm < r < rc r > rc 6

1 + 0.2977T* σ 1 + 0.33163T* + 0.0010471T*2

(9)

(10)

where T* ) 1/βε is the reduced temperature. Equation 6 can then be used for LJ fluids. The hard-core contribution can be evaluated with the WDA as K

Fex hc )

∑ ∫ Fi(r) f hs(i)[G¯ hc(r)] dr

(11)

∑ ∫ Fi(r)

Viext(r)

(1) (K) (i) (r), ..., Fjhc (r)] and Fjhc (r) is the weighted where Gjhc(r) ) [Fjhc density for the hard-core contribution of component i

F¯ (i) hc(r) )

K

F [G(r)] )

(8)

i)1

where β ) 1/kBT, with kB as the Boltzmann constant and T as the absolute temperature, and Λ ) h/(2πmkBT)1/2 is the de Broglie wavelength. The external term can be expressed as

ext

∞, r < rh 0, r > rh

where rh is the hard-core diameter; rm ) 21/6σ is the minimum position of the LJ potential; rc is the cutoff distance, set as 3σ in this work; and σ and ε denote the diameter and well depth, respectively, of the LJ potential. In WCA theory, the hard-core diameter is estimated by iteratively solving a blip function, which is nontrivial and time-consuming. Therefore, approximations have usually been adopted such as rh ) 0.98σ;27 rh ) σ;40 and rh ) σ(RT/Tc + β)/(γT/Tc + δ) with R, γ, and δ fitted to Baker-Henderson (BH) theory and β fitted to simulation data.41 In this work, as a simple alternative, the hard-core diameter was calculated approximately by BH theory.42-44 Therefore, the hard-core diameter rh is expressed as45

K

βFid[G(r)] )

{

and uattr(r) is the attractive interaction

(1)

The key in DFT is to solve for the density distributions G(r) from eq 1. Following the definition of the grand potential, we have

(7)

dr

(5)

i)1

where Viext(r) denotes the external potential for component i at position r. The excess term cannot be derived exactly from statistical mechanics; however, it can be decomposed into hard-core and attractive contributions ex Fex[G(r)] ) Fex hc[G(r)] + Fattr[G(r)]

(6)

Equation 6 can be readily applied to hard-core square-well and hard-core Yukawa fluids, but not to soft Lennard-Jones (LJ) fluids. However, the LJ potential can be decomposed into hard-

∫ Fi(r′) w(i)hc(|r - r′|, F¯ (i)hc) dr′

(12)

(i) (r,F) as the hard-core weighting function. As a with whc compromise of computational speed and accuracy, the weighting function can be expanded in terms of density as proposed by Tarazona24

(i) (i) (i) 2 w(i) hc(r, F) ) w0(r, rh ) + w1(r, rh )Fi + w2(r, rh )Fi + · · · (13)

where rh(i) is the hard-core diameter of component i, which can be obtained from eq 10. By using the Percus-Yevick approximation and applying the normalization condition ∫w(i) hc(r,F) dr ) 1, the first three weighting functions on the right side of eq 13 can be expressed as

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w0(r, rh) )

Liu et al.

3 θ(rh - r) 4πrh3

(14)

The weighting function is given by28,40

w1(r, rh) )

{

(i) uattr (r)

(i) wattr (r) )

r < rh

0.475 - 0.648(r/rh) + 0.113(r/rh)2

0.288(rh /r) - 0.924 + 0.764(r/rh) - 0.187(r/rh)2 rh < r < 2rh r > 2rh 0

(15)

(i) (i) (i) (i) and f attr ) f LJ - f hs , where f LJ is the partial molecular excess free energy of component i for a LJ fluid46

5πrh3 w2(r, rh) ) [6 - 12(r/rh) + 5(r/rh)2]θ(rh - r) 144 (16) where θ(r) is a Heaviside step function. jhc(r)] is the partial molecular excess free energy In eq 11, f (i) hs[G of component i for a hard-sphere fluid46

(i) fhs )

( ) ex ∂Fhs

∂Ni

(23)

(i) (r) dr ∫ uattr

(i) fLJ )

( ) ex ∂FLJ ∂Ni

(24)

T,P,Fj*i

ex with f LJ as the excess free energy for a uniform fluid, which can be obtained from the modified Benedict-Webb-Rubin (MBWR) equation of state as49

ex FLJ ) Nεx

(17)

aiF*i + i i)1 8

∑

6

∑ biGi

(25)

i)1

T,P,Fj*i

where F* ) ΣFiσx3 is the reduced density and

ex f hs

where is the excess free energy for a uniform hard-sphere fluid and can be derived from an equation of state for hardsphere mixtures47,48

K

σx3 )

K

∑ ∑ XiXjσij3

(26)

i)1 j)1

[

ex βFhs 3BEη/F - E3 /F2 E3 /F2 + ) + N 1-η (1 - η)2

K

]

(E /F - 1) ln(1 - η) 3

2

(18)

K

∑

π Fσ3 6 i)1 i i

(19) Fi(r) )

B)

∑

K

Fiσi

i)1 K

,

E)

∑ Fi

∑

K

Fiσi2

i)1 K

∑ Fi

i)1

,

F)

∑

Fiσi3

i)1 K

i)1

∑ Fi i)1

(20) For the attractive contribution, the WDA is more accurate than the MFA as discussed earlier; therefore, the WDA was used. Similarly to its hard-core counterpart, the excess free energy can be expressed as K

ex Fattr )

(i) [G¯ attr(r)] dr ∑ ∫ Fi(r) fattr

(21)

i)1

with

({

∫ exp β

is the packing fraction; and the quantities B, E, and F are given by K

(27)

j)1

where Xi denotes the mole fraction of component i and εij and σ ij are cross parameters. With the expression for the excess free energy, the density distributions can be solved iteratively by combining eqs 1-5

where N is the total number of molecules

η)

K

∑ ∑ XiXjεijσij3

1 εx ) 3 σx i)1

µi - Viext(r) -

})

δFex[G(r)] dr δFi(r) (i ) 1, 2, ..., K) (28)

This DFT has been tested for H2 adsorption in MOFs, and the predictions are in good agreement with both simulation and experimental results.28 In this study, it was further applied to investigate the adsorption of CO2/CH4 and CO2/N2 mixtures in ZIF-8 and Zn2(BDC)2(ted) at room temperature. All adsorbate molecules were modeled as spherical LJ particles with potential parameters as listed in Table 1. The framework atoms were represented with the DREIDING force field.50 The cross interaction parameters were approximated by the LorentzBerthelot combining rules. In the calculations, the unit cell of ZIF-8 was divided into 36 × 36 × 36 grids, and that of Zn2(BDC)2(ted) was divided into 22 × 22 × 19 grids. The grid size was approximately 0.5 Å in both MOFs. The adsorbateadsorbent interaction energies Vext(r) were tabulated on the grids. In a highly repulsive region with Vext(r)/kB > 104 K, the density was assumed to be zero. The composite trapezoidal quadrature method was used for the numerical integration. 3. Simulation

(i) F¯ attr (r) )

(i) (|r - r′|) dr′ ∫ Fi(r′) wattr

(22)

To test the DFT predictions, grand canonical Monte Carlo (GCMC) simulations were carried out for the adsorption of CO2/



Figure 1. Atomic structures of (a) ZIF-8 and (b) Zn2(BDC)2(ted). Color code: Zn, green; O, red; N, blue; C, gray; H, white. Connolly surfaces are shown in purple (interior) and gray (exterior) and were generated using a probe of 1.0 Å radius.

TABLE 1: Lennard-Jones Potential Parameters interaction site CH4 N2 CO2 Zn C O N H

diameter σ (Å) Adsorbates 3.73 3.62 3.75 Adsorbent Atoms 4.04 3.47 3.03 3.26 2.85

potential ε/kB (K) 148.0 101.5 236.1 27.65 47.81 48.12 38.91 7.64

CH4 and CO2/N2 mixtures in ZIF-8 and Zn2(BDC)2(ted). ZIF-8 has a sodalite (SOD) zeolite-like topology with a space group of I4j3m.51 Each Zn metal in ZIF-8 is tetrahedrally coordinated by four N atoms of 2-methylimidazolate (mIM). Because of the long linkers rather than the bridging O atoms in zeolites, the pore size in ZIF-8 is almost twice as large as that in its zeolite counterpart. Nevertheless, the aperture size is small because of the blockage of the rings in mIM. The pores in ZIF-8 have a diameter of 11.6 Å, are connected by small apertures of 3.4 Å, and are accessible only through a narrow funnel in the six-membered ring of ZnN4 clusters. The rotation of methyl groups at the apertures could allow molecules to pass through, as evidenced by the adsorption of nitrogen and methanol and the release of solvent molecules.51 Zn2(BDC)2(ted) exhibits a paddle-wheel secondary building unit (SBU), namely, Zn2(COO)4.52,53 The SBU is connected with a BDC linker to form a two-dimensional net. There are intersecting channels of different sizes, namely, 7.5 × 7.5 Å and 4.8 × 3.2 Å. Figure 1 shows the Connolly surfaces of ZIF-8 and Zn2(BDC)2(ted). Eight (2 × 2 × 2) unit cells were used for ZIF-8 to represent the simulation box, whereas 64 (4 × 4 × 4) unit cells were used for Zn2(BDC)2(ted). To mimic infinitely large frameworks, periodic boundary conditions were used in three dimensions. The frameworks were assumed to be rigid in the simulations. The unit cell of each MOF was divided into grids where the potential energies were tabulated and then used by interpolation during simulations. Such a treatment accelerated simulations by 1-2 orders of magnitude. A spherical cutoff of 15 Å was used to evaluate the LJ

Figure 2. (a) Adsorption and (b) selectivity of a CO2/CH4 mixture (50:50 in the bulk) in ZIF-8 at 298 K. Lines, DFT; symbols, simulation.

Figure 3. Density distributions of a mixture of CH4 (Fbulk ) 0.0133 mol/L) and CO2 (Fbulk ) 0.0133 mol/L) in ZIF-8 at 0.6477 bar.

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Figure 4. Density distributions of CO2 for a CO2/CH4 mixture in ZIF-8 at 0.6477 bar. Density units: mol/L. Color code: Zn, green; O, red; N, blue; C, gray; H, white.

Figure 5. (a) Adsorption and (b) selectivity of a CO2/N2 mixture (15:85 in the bulk) in ZIF-8 at 298 K. Lines, DFT; symbols, simulation.

Figure 6. Density distributions of a mixture of N2 (Fbulk ) 0.0254 mol/L) and CO2 (Fbulk ) 0.00448 mol/L) in ZIF-8 at 0.7299 bar.

interactions. The chemical potentials in the GCMC simulations were converted into pressures by the MBWR equation of state. Four types of trial moves were conducted randomly, including displacement, regrowth, swap with reservoir, and identity exchange between adsorbates. The number of trial moves in a typical simulation was 2 × 107, in which the first 107 moves were used for equilibration and the second 107 moves were used for ensemble averages. 4. Results and Discussion Figure 2a shows the adsorption of CO2/CH4 mixture (50: 50 in the bulk) in ZIF-8 at 298 K. In the low-pressure range considered in this study, the amounts adsorbed for both CO2 and CH4 increase almost linearly with increasing pressure. Nevertheless, saturation is expected at high pressures. CO2 is more strongly adsorbed than CH4 because CO2 has a stronger interaction with the framework, as evidenced by the

potential parameters in Table 1. The predictions from DFT match well with simulation results despite slight overestimations at high pressures. Adsorption-based separation in a binary mixture of components i and j is usually characterized by the selectivity, defined as Si/j ) (xi/xj)(yj/yi), where xi and yi are the mole fractions of component i adsorbed and in the bulk phase, respectively. Figure 2b shows the selectivity of CO2 over CH4 in ZIF-8, and good agreement is observed between the DFT predictions and the simulation results. The selectivity increases slightly with increasing pressure, which is attributed to the cooperative attractions between CO2 molecules. The density distributions of confined fluids in a nanoscopic environment can be readily provided by DFT. Figure 3 shows the three-dimensional distributions of CH4 and CO2 in ZIF-8 at 0.6477 bar. The two adsorbates generally exhibit similar distributions. Near the framework surface, the density is low (0.83 mol/L). The highest density is on top of the mIM linkers



Figure 7. (a) Adsorption and (b) selectivity of a CO2/CH4 mixture (50:50 in the bulk) in Zn2(BDC)2(ted) at 298 K. Lines, DFT; symbols, simulation.

Figure 8. Density distributions of a mixture of CH4 (Fbulk ) 0.0133 mol/L) and CO2 (Fbulk ) 0.0133 mol/L) in Zn2(BDC)2(ted) at 0.6477 bar.

Figure 9. Density distributions of CO2 for a CO2/CH4 mixture in Zn2(BDC)2(ted) at 0.6477 bar. Density units: mol/L. Color code: Zn, green; O, red; N, blue; C, gray; H, white.

in the six-membered rings. This reveals that the organic linkers are the most favorable adsorption sites in ZIF-8, confirmed by a recent experimental and computational study.54 This behavior is in remarkable contrast to most MOFs, in which the regions near metal clusters are the strongest. Apparently, CO2 has a significantly higher density (13.28 mol/L) on top of the mIM linkers because of stronger adsorption compared to CH4 (3.32 mol/L). Figure 4 shows the two-dimensional density distributions of CO2 for the CO2/ CH4 mixture at 0.6477 bar. Because of the symmetry in the ZIF-8 framework, the distributions in the XY, XZ, and YZ planes are identical. It is clearly demonstrated that the regions near the mIM linkers are the preferential sites, followed by the corners around the metal clusters. Figure 5a shows the adsorption of a CO2/N2 mixture (15:85 in the bulk) in ZIF-8 at 298 K. Even though the mole fraction of N2 in the bulk phase is 5.67 times larger than that of CO2, a

greater amount of the latter is adsorbed. Similarly to Figure 2a, the adsorbed amounts of CO2 and N2 increase linearly with increasing pressure because the adsorption is far from saturation. The adsorption and selectivity predicted by DFT are in accord with simulation. As shown in Figure 5b, the selectivity of the CO2/N2 mixture increases slightly from 10 to 11 over the range of pressure considered. The value of selectivity here is larger than that for the CO2/CH4 mixture. This is because, among the three adsorbates, CO2 has the strongest adsorption, followed by CH4 and finally N2. Consequently, the CO2/N2 separation factor is greater than the CO2/CH4 separation factor. The density distributions of the CO2/N2 mixture in ZIF-8 are shown in Figure 6 and are similar to those of the CO2/CH4 mixture in Figure 3. The most preferential adsorption sites are also on top of the mIM linkers. Figure 7a shows the adsorption of a CO2/CH4 mixture (50: 50 in the bulk) in another MOF, Zn2(BDC)2(ted). The DFT

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Figure 10. (a) Adsorption and (b) selectivity of a CO2/N2 mixture (15:85 in the bulk) in Zn2(BDC)2(ted) at 298 K. Lines, DFT; symbols, simulation.

Figure 11. Density distributions of a mixture of N2 (Fbulk ) 0.0254 mol/L) and CO2 (Fbulk ) 0.00448 mol/L) in Zn2(BDC)2(ted) at 0.7299 bar.

Figure 12. (a) Adsorption and (b) selectivity of CO2/CH4 mixtures with different CO2 compositions in ZIF-8 at 1 bar. Lines, DFT; symbols, simulation.

predictions and simulation results are again in good agreement. The adsorbed amount here is higher than in ZIF-8 at the same pressure. This implies that CO2 is more strongly adsorbed in Zn2(BDC)2(ted) than in ZIF-8. However, the behaviors of the selectivity of the CO2/CH4 mixture in the two MOFs are similar, as observed in Figures 2b and 7b, increasing slightly from 4 to 5 over the pressure range studied. The density distributions of the CO2/CH4 mixture in Zn2(BDC)2(ted) at 0.6477 bar are shown in Figure 8.

Adsorbate molecules are primarily located in the 7.5 × 7.5 Å channels, which appear to be located at the four corners of a unit cell and are along the Z axis. The lowest density of 0.83 mol/L is close to the framework surface for both CO2 and CH4. However, CO2 has the highest density of 24.91 mol/L in the centers of the channels, which is much higher than the 3.32 mol/L value for CH4. Comparing Figures 3 and 8, one can see that, at the same pressure (0.6477 bar), CO2 is more densely packed in Zn2(BDC)2(ted) than in ZIF-

DFT for Adsorption of Gas Mixtures in MOFs 8. This is consistent with the above discussion that CO2 exhibits stronger adsorption in Zn2(BDC)2(ted). Figure 9 shows the two-dimensional density distributions of CO2. As observed from the XY plane, the CO2 molecules are preferentially adsorbed in the 7.5 × 7.5 Å channels. In addition, adsorption also occurs in the 4.8 × 3.2 Å channels but to a lesser extent. Figure 10a shows the adsorption of the CO2/N2 mixture (15: 85 in the bulk) in Zn2(BDC)2(ted). Again, the DFT predictions match well with the simulation results. The adsorption of CO2/ N2 in Zn2(BDC)2(ted) is larger than that in ZIF-8. However, the selectivity shown in Figure 10b is similar to that in ZIF-8, implying that the two MOFs have similar separation capacities for both CO2/CH4 and CO2/N2 mixtures. The density distributions of the CO2/N2 mixture in Zn2(BDC)2(ted) are shown in Figure 11 and largely resemble those shown in Figure 8. Finally, the adsorption and selectivity of CO2/CH4 mixtures with different CO2 concentrations (mole fractions) were examined in ZIF-8 at 298 K and 1 bar. As shown in Figure 12a, with increasing CO2 content, the CO2 adsorption increases sharply, whereas the CH4 adsorption decreases slowly. When the composition is 20% CO2, the same amounts of CO2 and CH4 are adsorbed. Whereas the adsorption is substantially affected by the composition, the selectivity is not. As shown in Figure 12b, the selectivity is a weak function of CO2 composition. 5. Conclusions We have extended a recently developed three-dimensional density functional theory from pure components to mixtures. In the proprosed DFT, the excess free energy is decomposed into repulsive and attractive contributions, for which weighted density approximations are employed with different weighting functions. The equations of state for hard-sphere and LennardJones mixtures are used to estimate the excess free energy of uniform fluids. The theory was applied to investigate the adsorption of CO2/CH4 and CO2/N2 mixtures in ZIF-8 and Zn2(BDC)2(ted). The theoretically predicted adsorption and selectivity are in good agreement with simulation data. In the pressure range under study, the adsorption of the gas mixtures increases linearly with increasing pressure, whereas the selectivity rises marginally. Among the three adsorbates, CO2 has the strongest adsorption, followed by CH4 and then N2. Consequently, CO2/N2 exhibits a larger selectivity than CO2/CH4. With increasing CO2 content, the adsorption of CO2/CH4 mixture in ZIF-8 changes substantially; however, the selectivity does not. Although the adsorption in Zn2(BDC)2(ted) is greater than that in ZIF-8, the two MOFs appear to have similar separation capacities for CO2/CH4 and CO2/N2 mixtures. Acknowledgment. This work was supported by the National University of Singapore (R279-000-198-112/133 and R279-000243-123), the Singapore National Research Foundation on the Competitive Research Programme (R279-000-261-281), the National Natural Science Foundation of China (No. 20736002), the National High Technology Research and Development Program of China (No. 2008AA062302), the Program for Changjiang Scholars and Innovative Research Team in University of China (No. IRT0721), and the 111 Project of China (No. B08021).

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