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Development of a Density Functional Theory in Three-Dimensional Nanoconfined Space: H2 Storage in Metal-Organic Frameworks Yu Liu,†,‡ Honglai Liu,*,‡ Ying Hu,‡ and Jianwen Jiang*,† Department of Chemical and Biomolecular Engineering, National UniVersity of Singapore, Singapore 117576, and State Key Laboratory of Chemical Engineering and Department of Chemistry, East China UniVersity of Science and Technology, Shanghai 200237, China ReceiVed: May 25, 2009; ReVised Manuscript ReceiVed: August 2, 2009
A density functional theory (DFT) is developed in three-dimensional nanoconfined space and applied for H2 storage in metal-organic frameworks. Two different weighting functions based on the weighted density approximation (WDA) are adopted, respectively, for the repulsive and attractive contributions to the excess free energy. The Carnahan-Starling equation and a modified Benedicit-Webb-Rubin equation are used to calculate the excess free energy of uniform fluid. To compare with DFT predictions, grand canonical Monte Carlo simulations are carried out separately. For H2 adsorption in MOF-5 and ZIF-8, the isotherms predicted from the DFT agree well with simulation and experiment results, and the DFT is found to be superior to the mean-field-approximation (MFA)-based theory. The adsorption energies and isosteric heats predicted are also in accord with simulation results. From the predicted density contours, the DFT shows that the preferential adsorption sites are the corners of metal clusters in MOF-5 and the top of organic linkers in ZIF-8, consistent with simulation and experimental observations. 1. Introduction Density functional theory (DFT) is a versatile statisticalmechanics method to describe the thermodynamic and structural properties of inhomogeneous fluids.1,2 In the last two decades, DFT has been successfully applied in a wide variety of confined systems.3-16 The underlying issue in DFT is to calculate the excess free energy with a judiciously chosen density functional. Compared with the local density approximation (LDA), the weighted density approximation (WDA)14,17-19 and fundamental measure theory (FMT)20,21 are more accurate and commonly used. WDA is used to mimic the oscillatory density distribution in inhomogeneous fluids and various weighting functions can be introduced independently into WDA for the repulsive and attractive contributions. Several approaches based on WDA have been proposed for the repulsive contribution. For example, the simplest Heaviside function is easy in calculation and works fairly well for polymeric systems.22 A weighting function was established by Tarazona as a series expansion of density with coefficients determined by the Percus-Yevick approximation, and good agreement was found with simulation.23 In most DFTs, the attractive contribution to excess free energy is considered independently from the repulsive counterpart. Methods such as functional expansion,8 mean-field approximation (MFA),24 and WDA25 are usually used for the attractive contribution. The functional expansion approximation requires the direct correlation function of the corresponding uniform fluid. Original MFA is simple in formulation and fast in computation, and can be used without any information of uniform fluid; however, it fails in several situations, e.g., capillary condensation. WDA has been widely adopted, in which an equation of state (EOS) for uniform fluid is needed. * Corresponding author. E-mail:
[email protected] (J.J.);
[email protected] (H.L.). † National University of Singapore. ‡ East China University of Science and Technology.
Although numerous versions of DFT have been developed, DFT is only widely applied in simple confined geometries such as slit and cylindrical pores, and rarely used in three-dimensional systems. Unfortunately, most practical systems are threedimensonal, e.g., gas adsorption in activated carbons, zeolites, and metal-organic frameworks (MOFs). A lattice-based DFT was proposed to examine H2 adsorption in MOF-5 by combining Tarazona’s WDA for repulsion and MFA for attraction, in which a H2 molecule was mimicked as a spherical Lennard-Jones (LJ) particle.26 Nevertheless, it has been revealed that MFA is not very reliable for LJ fluid and a more sophisticated approximation is thus highly desired for attractive contribution.5,6 In this work, we develop a DFT in three-dimensional nanospace by using WDA for both repulsive and attractive contributions but with different weighting functions. To demonstrate its applicability, H2 adsorption in two MOFs is examined by the developed DFT. H2 is an ideal energy carrier and pollution-free fuel for future sustainable economy. The onboard utilization of H2 requires a safe and high-capacity system for H2 storage. As a new class of hybrid inorganic-organic materials, MOFs are considered as promising candidates for H2 storage. The readily controllable organic linkers and the variation of metal oxides in MOFs allow for tailoring their pore size, volume, and functionality in a rational manner.27 Following this section, the theoretical framework of the DFT is introduced in section 2. Section 3 describes Monte Carlo simulations for H2 adsorption in MOFs, which were performed to compare with DFT predictions. In section 4, the predicted results (adsorption isotherms, density distributions, adsorption energies, and isosteric heats) are presented and compared with simulation and experimental data. Finally, concluding remarks are summarized in section 5. 2. Theory The grand potential of an open system is a function of density distribution F(r)
10.1021/jp904872f CCC: $40.75 2009 American Chemical Society Published on Web 08/19/2009
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J. Phys. Chem. B, Vol. 113, No. 36, 2009 12327
Ω[F(r)] ) F[F(r)] - µN ) Fint[F(r)] -
∫ [Vext(r) - µ]F(r) dr
(1)
where fhs(F) is the excess free energy density for uniform hardsphere fluid, which is calculated by the Carnahan-Starling (CS) equation of state29
where Fint[F(r)] denotes the intrinsic Helmholtz free energy, µ is the chemical potential, and Vext(r) represents the external potential. At equilibrium, Ω is subject to
δΩ[F(r)] )0 δF(r)
(2)
βfhs(F) )
Fint[F(r)] ) F [F(r)] + F [F(r)] ex
(1)
(9)
η ) π F σ3/6 is the packing fraction. Fjhc(r) is the weighted density for hard-core contribution
F¯ hc(r) )
The intrinsic Helmholtz free energy can be split into ideal-gas free energy Fid[F(r)] and excess free energy Fex[F(r)] id
4η - 3η2 (1 - η)2
∫ F(r′)whc(|r - r′|, F¯ ) dr′
(10)
where whc(r, F) is the weighting function. Since the simplest Heaviside function is not very accurate for LJ fluid but accurate weighting functions are computationally expensive, Tarazona’s weighting function23 is used here
The ideal gas term is known exactly
βFid[F(r)] )
∫ {ln[F(r)Λ3] - 1}F(r) dr
whc(r, F) ) w0(r) + w1(r)F + w2(r)F2 + ... The weighting function wi(r) is given by19
where β ) 1/kBT; kB is the Boltzmann constant, and T is the absolute temperature. For LJ fluid, the potential can be split into hard-core and attractive terms
uLJ(r) ) uhc(r) + uattr(r)
(3)
where
{
∞ r < rh uhc(r) ) 0 r > rh
(11)
(2)
w0(r) )
(12)
w1(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) 0
2
rh < r < 2rh r > 2rh
(13)
(4) w2(r) )
and
3 θ(rh - r) 4πrh3
5πrh3 [6 - 12(r/rh) + 5(r/rh)2]θ(rh - r) 144
(14)
{
-ε
uattr(r) ) 4ε[(σ/r) 0
12
r < rm - (σ/r) ] rm < r < rc r > rc 6
(5)
In eqs 3-5, σ and ε are the diameter and well-depth of LJ potential, rh represents the hard-core diameter, rm ) 21/6σ is the minimum position of LJ potential, and rc is the cutoff distance. The hard-core diameter rh can be approximated by28
rh )
1 + 0.2977T* σ 1 + 0.33163T* + 0.0010471T*2
(6)
where T* ) 1/βε is the reduced temperature. Therefore, the excess free energy can be decomposed into hard-core and attractive contributions ex Fex[F(r)] ) Fex hc[F(r)] + Fattr[F(r)]
where θ(r) is the Heaviside step function. Similar to the hard-core contribution, WDA is also used for the attractive contribution but with a different weighting function ex Fattr [F(r)] )
fattr(F) ) fLJ(F) - fhs(F)
∫ F(r)fhs[F¯ hc(r)] dr
(8)
(16)
where fLJ(F) is the excess free energy density for uniform LJ fluid. On the basis of simulation data, Johnson et al. have developed a modified Benedicit-Webb-Rubin (MBWR) equation of state for LJ fluid30
aiF*i + i i)1 8
fLJ(F)/ε )
Fex hc[F(r)] )
(15)
where Fjattr(r) is the weighted density for attractive contribution and fattr(F) is the excess free energy density of the corresponding uniform fluid and can be calculated for LJ fluid
(7)
Using WDA, the hard-core contribution is expressed as
∫ F(r)fattr[F¯ attr(r)] dr
∑
6
∑ biGi
(17)
i)1
where F* ) Fσ3, coefficients ai and bi are functions of temperature only, and Gi is a function of F*.
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Liu et al.
As mentioned above, the mean-field approximation (MFA) is fast in computation and can be used as well for the attractive contribution. With MFA, the excess free energy is expressed by ex Fattr [F(r)] )
1 2
A uattr(|r - r′|)F(r)F(r′) dr dr′ (18)
However, the MFA has been found to be not very reliable for LJ fluid,5,6 as also demonstrated below. Unless otherwise specified, the DFT theoretical results presented here are based on WDA rather than MFA. The weighted density Fjattr(r) in eq 15 for the attractive contribution can be expressed as
∫ F(r′)wattr(|r - r′|) dr′
F¯ attr(r) )
(17)
where wattr(r) is the weighting function and differs from that in the hard-core contribution. The weighting function for attractive contribution can be approximated as
wattr(r) ∝ uattr(r)
uattr(r)
(19)
∫ uattr(r) dr
Finally, the density distribution F(r) can be calculated by iteration from
F(r) )
({
∫ exp β
µ - Vext(r) -
δFex[F(r)] δF(r)
})
dr
(20)
The developed DFT in three-dimensional space was applied for H2 adsorption in two MOFs, MOF-5 and ZIF-8. A more detailed description for MOF-5 and ZIF-8 is provided in the next section. H2 was modeled as a spherical LJ particle with potential parameters σH2 ) 2.96 Å and εH2/kB ) 34.2 K.31 The framework atoms in MOFs were represented by the DREIDING force field.32 The interactions between H2 and MOF atoms were accounted for by the LJ potential, and the cross LJ parameters were obtained by Lorentz-Berthelot combining rules. In the numerical calculations, the unit cell of MOF-5 was divided into 44 × 44 × 44 grids, while ZIF-8, into 34 × 34 × 34 grids. The interaction energies Vext(r) between H2 and MOF on grids were tabulated. For a highly repulsive region with Vext(r)/kB > 104 K, the density was assumed to be zero. The initial density in the other region was simply obtained by ideal-gas approximation. For efficient iteration, eq 22 was modified as
F(n+1)(r) )
[
1 κF(n)(r) + 1+κ
where n denotes the iteration step and κ is a parameter that does not affect the final results but its use accelerates convergence. We found that a small κ should be used at low pressures, whereas a large κ, at high pressures. In our calculations, κ increased from 0 to 49 with increasing pressure. The iteration was considered to be converged upon the density difference between two consecutive steps being less than 10-9
(18)
Considering the normalization condition, ∫wattr(r) dr ) 1, wattr(r) is then
wattr(r) )
Figure 1. Atomic structures of (a) MOF-5 and (b) ZIF-8. The metal clusters are represented by tetrahedra. Color code: O, red; N, blue; C, gray; H, white. Connolly surfaces are shown in purple (interior) and gray (exterior) generated using a probe of 1.0 Å in radius.
({
∫ exp β
µ - Vext(r) -
δFex[F(n)(r)] δF(n)(r)
}) ] dr
(21)
|F(n)(r) - F(n-1)(r)| < 10-9
(22)
3. Simulation Figure 1 shows the atomic structures of MOF-5 and ZIF-8, respectively. MOF-5 is an isoreticular MOF and also known as IRMOF-1.33 It has a lattice constant of 25.832 Å, a crystal density of 0.593 g/cm3, and a formula of Zn4O(BDC)3, where BDC is 1,4-benzenedicarboxylate. Each Zn4O tetrahedron is edge-bridged by six carboxylate linkers, which results in an octahedral Zn4O(O2C-)6 building unit. There are straight pores in MOF-5 with sizes of approximately 15 and 12 Å. ZIF-8 has an SOD zeolite-like topology with a cubic space group I4j3m.34 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 the zeolite counterpart. Nevertheless, the aperture size is small due to the blockage of the rings in mIM. The pore in ZIF-8 has a diameter of 11.6 Å connected through a small aperture of 3.4 Å and is only accessible through a narrow funnel in the six-membered ring of ZnN4 clusters. In Figure 1, the Connolly surfaces in MOF-5 and ZIF-8 are shown in purple (interior) and gray (exterior). Grand canonical Monte Carlo (GCMC) simulations were carried out for H2 adsorption in MOF-5 and ZIF-8. In GCMC simulation, the chemical potential of sorbate in adsorbed and bulk phases is identical and can be easily converted into pressure. In our study, the MBWR equation of state was used for this conversion. One unit cell was used to represent the simulation box for MOF-5 and eight (2 × 2 × 2) unit cells for ZIF-8. The periodic boundary conditions were exerted in all three dimensions to mimic infinitely large frameworks. Our study for H2 adsorption was focused on the low-energy equilibrium configurations; therefore, the MOF structures were assumed to be rigid in the GCMC simulations. A flexible adsorbent allowing local vibration would not give significantly different adsorption properties for light gases such as H2. Consequently, the unit cell of MOF was divided into threedimensional grids with the potential energy tabulated in advance and then used by interpolation during simulations. Such a
H2 Storage in Metal-Organic Frameworks
Figure 2. Isotherms of H2 adsorption in MOF-5 at 77 and 298 K. Solid lines, WDA-based DFT; dashed lines, MFA-based DFT; triangles, MC; circles, experiment.35
treatment accelerated simulations by approximately 2 orders of magnitude. A spherical cutoff of 12.5 Å was used to evaluate the LJ interactions. Three types of trial moves were conducted randomly in the GCMC simulations, namely, displacement, regrowth, and swap with reservoir. 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 for ensemble averages. The block transformation technique was used to estimate statistical uncertainties. Unless otherwise mentioned, the uncertainties were smaller than the symbol sizes in the figures presented below. 4. Results and Discussion Figure 2 shows the isotherms of H2 adsorption in MOF-5 at 77 and 298 K. The isotherms belong to type I (Langmuirian),
J. Phys. Chem. B, Vol. 113, No. 36, 2009 12329 the signature of adsorption in microporous adsorbents. At low temperatures, the excess free energy contribution in DFT is more dominant and the capability of DFT can be tested more rigorously. Here, adsorption at a low temperature of 77 K is considered. The DFT predictions at 77 K match well with both simulation and experimental results,35 despite slight underestimation at high pressures. The deviations imply that more accurate weighting functions might be needed to improve the theory. At a high temperature of 298 K, the agreement between the DFT, simulation, and experiment is perfect over the entire pressure range under study; the isotherm increases linearly with pressure because the adsorption is far away from saturation in the pressure range examined. If the MFA is used for the attractive contribution, the theory largely fails at 77 K though performs well at 298 K. This reveals that the WDA adopted in our DFT is superior to the MFA. One salient feature in DFT is that structural information of inhomogeneous fluid can be readily obtained. Figure 3 shows the density distributions of H2 adsorption in MOF-5 at 298 K and three different pressures. At 34.73 bar, the isosurface with a constant density of 1.66 mol/L is proximal to the Connolly surface of MOF-5, implying that H2 molecules are adsorbed closely to the framework atoms. Interestingly, several discrete regions with a higher density of 11.62 mol/L are observed in the corners of Zn4O metal clusters. This reveals that the corner regions surrounding the metal clusters are the preferential adsorption sites, consistent with simulation studies for gas adsorption in MOF-5.36-38 Nevertheless, the corner regions are small and get saturated rapidly. At 118.64 bar, the corners are almost completely covered and adsorption starts to occur near the BDC linkers. At 296.53 bar, the regions near organic linkers are fully occupied and further adsorbed molecules fill in the straight pores. The highest density of 58.11 mol/L is observed near the corners of metal clusters. The sequence of adsorption as a function of pressure clearly demonstrates the strengths of various adsorption sites.
Figure 3. Density distributions of H2 adsorption in MOF-5 at 298 K and three different pressures. Isosurface: purple, 1.66 mol/L; green, 11.62 mol/L; red, 58.11 mol/L.
Figure 4. (a) Adsorption energy and (b) isosteric heat of H2 adsorption in MOF-5 at 298 K. Lines, DFT; triangles, MC.
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Figure 5. Isotherms of H2 adsorption in ZIF-8 at 125, 200, and 300 K. Solid lines, WDA-based DFT; dashed lines, MFA-based DFT; triangles, MC; circles, experiment.41
To precisely characterize an adsorption mechanism, isosteric heat rather than adsorption isotherm is usually used, as the former is more sensitive to the change of adsorption energy. In our work, the isosteric heat Qst was calculated from
Qst ) RT -
( ) dUa dNa
(23)
N,T
where R is the ideal gas constant, Ua is the adsorption energy, and Na is the number of adsorbed molecules. Figure 4a shows the adsorption energy as a function of H2 loading in MOF-5 at
298 K. The DFT gives results in accord with simulation, particularly at low loadings. With increasing loading, the adsorption energy increases (negatively) almost linearly. As a consequence, Qst shown in Figure 4b is nearly a constant (4.7 kJ/mol) and changes marginally with loading. The theoretically predicted Qst agrees well with simulated results over the entire range of loading, and slightly higher than the experimental value (4-4.3 kJ/mol).39 Figure 5 shows the isotherms of H2 adsorption in ZIF-8 at 125, 200, and 300 K. At 200 and 300 K, the DFT predictions agree well with simulation and experimental data. In addition, a linear trend is found for the extent of adsorption as a function of pressure. This is similar to the isotherm of H2 in MOF-5 at 298 K, because saturation is not reached yet over the pressure range studied. At 125 K, the DFT predictions and simulation results are consistent with each other; however, both overestimate experimental data. The models of H2 and ZIF-8 used in the DFT and simulation are identical; therefore, the good agreement between the DFT and simulation at all three temperatures implies that the DFT is accurate. Nevertheless, the models may not exactly mimic the real sample. The difference between model and real case appears negligible at a high temperature (300 K) but not at a low temperature. From Figures 2 and 5, it is found that H2 adsorption in MOF-5 is greater than in ZIF-8 under similar conditions. This is because MOF-5 is lighter (0.59 g/cm3) compared with ZIF-8 (0.94 g/cm3) and MOF-5 has a larger porosity (0.82) than ZIF-8 (0.52). Though the predictions from the MFA-based theory appear to match fairly well with experimental data, they are not in good agreement with simulation results compared to those from the WDA-based DFT. Again, this suggests that the WDA-based DFT outperforms the MFA-based theory.
Figure 6. Density distributions of H2 adsorption in ZIF-8 at 200 K and three different pressures. Isosurface: purple, 4.98 mol/L; green, 19.92 mol/L; red, 119.52 mol/L.
Figure 7. (a) Adsorption energy and (b) isosteric heat of H2 adsorption in ZIF-8 at 200 K. Lines, DFT; triangles, MC.
H2 Storage in Metal-Organic Frameworks Figure 6 shows the density distributions of H2 adsorption in ZIF-8 at 200 K. At all three pressures, the low-density isosurface is close to the Connolly surface of the ZIF-8 framework, which is similar to Figure 3 for H2 adsorption in MOF-5. At 37.5 bar, a higher density of 19.92 mol/L is observed on the tops of methylimidazolate linkers in the six-membered rings of ZnN4. This indicates that the tops of organic linkers are the most favorable adsorption sites in ZIF-8, as confirmed by a recent experimental and computational study.40 Such behavior is in remarkable contrast to MOF-5 and other MOFs, in which the regions near metal clusters are the strongest. At 103.2 bar, the apertures are occupied, through which the pores are accessible. At 227.1 bar, the centers of six-membered rings are more populated and the highest density remains on the top of the methylimidazolate linkers. The adsorption energy and isosteric heat are shown in Figure 7 for H2 adsorption in ZIF-8 at 200 K. The trend is generally similar to that in Figure 4 for H2 adsorption in MOF-5. Good agreement is found between the DFT and simulation, particularly at low loadings. As a function of loading, the adsorption energy increases almost linearly. At infinite dilution, Qst predicted by the DFT and simulation is approximately 5.3 kJ/mol and a bit larger than the experimental value (∼4.5 kJ/mol).41 As the loading rises, Qst increases slightly, as attributed to the increased cooperative attraction between adsorbed H2 molecules. 5. Conclusions We have developed a DFT in three-dimensional space, in which the repulsive and attractive contributions to the excess free energy were estimated by the weighted functional approximation with two different weighting functions. The Carnahan-Starling and MBWR equations were used to calculate the excess free energy of uniform LJ fluid. The application of the DFT was illustrated for H2 adsorption in two MOFs, MOF-5 and ZIF-8. The adsorption isotherms, density distributions, adsorption energies, and isosteric heats predicted from the DFT agree well with simulation and experimental results if available, particularly at high temperatures. The deviations at low temperatures suggest that more accurate weighting functions are needed for further improvement. The DFT performs better than the mean-field-approximation-based theory. Consistent with simulation and experimental observations, the DFT shows that the favorable adsorption sites in MOF-5 are near the corners of metal clusters but in ZIF-8 are on the tops of organic linkers. The strengths of different adsorption sites were identified from the DFT and reflected by the sequence of adsorption with increasing pressure. It was found that the developed DFT is well suited for H2 storage in MOFs. In future work, we will further apply the DFT to investigate the adsorption of other gases or gas mixtures in a variety of MOFs and other nanoporous materials. Acknowledgment. This work was supported by the National University of Singapore (R279-000-243-123), the National Natural Science Foundation of China (No. 20736002), the
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