**3. Predicting gas diffusivity using molecular dynamics simulations**

In the literature, MD simulations have been used to predict three different types of gas diffusivities in MOFs. These are transport diffusivity, corrected diffusivity and self diffusivity. The transport diffusivity, which is also known as Fickian diffusivity or chemical diffusivity, can be defined without approximation in terms of corrected diffusivity, *Do* and a

& Allendorf, 2008) In this force field, only nonbonded parameters were used to describe Zn−O interactions and the CVFF (consistent valence force field) was used with slight modifications to describe the benzene dicarboxylate linker. The magnitude of the diffusion constant was underestimated and this was attributed to the deficiencies in the CVFF portion

The literature summary presented so far indicates that the number of MD simulation studies with flexible MOFs and flexible force fields is very limited. More research will sure be helpful to understand the importance of lattice dynamics on diffusivity of gas molecules in MOFs. Studies to date indicated that the lattice dynamics are specifically important in computing diffusivity of large gas molecules (such as benzene) in MOFs having relatively narrow pores. Studies on flexible force fields also suggested that a force field developed for a specific MOF can be adapted to similar MOF structures (as in the case of IRMOFs) with slight modifications for doing comparative studies to provide a comprehensive

One major issue in carrying out MD simulations for MOFs is to assign partial charges to the MOF atoms that are required to calculate adsorbate-adsorbent interactions for some polar (including quadrupolar) adsorbates. Several MD studies computed the diffusivity of CO2 in MOFs' pores and to do this, partial charges must be assigned to MOF atoms. Recent studies showed that the effects of inclusion of framework charges are crucial at low loadings. If the charge-quadrupolar interactions are not taken into account in MD simulations then the diffusivities can be significantly overestimated.(Rankin et al., 2009) Force field-based classical MD simulations of MOFs typically treat electrostatic interactions between adsorbates and MOF atoms by assigning fixed point charges to each atom. In this context an important role for quantum mechanics (QM) calculations is to assign the point charges that can later be used in force field calculations. Unfortunately, multiple methods exist for partitioning the net electron density determined in a QM calculation(Keskin et al., 2009b) and none of these methods give an unambiguous definition of the resulting point charges. Keskin and coworkers reviewed the partial charges assigned to IRMOF-1 on the basis of QM calculations and showed that there is a significant variation in the charge values based on the method used.(Keskin et al., 2009b) This variation may have a significant impact on the outcome of classical force field calculations in examples where electrostatic interactions are important. Since QM calculations are time consuming and the charges obtained from these calculations are method sensitive, a strategy called connectivity based atom contribution method (CBAC) with which the partial charges of framework atoms can be estimated easily was proposed.(Xu&Zhong, 2010) A recent study on two different MOFs showed that CO2 adsorption isotherms and diffusivities computed using the charges from QM methods based on the ChelpG(Francl et al., 1996) DFT calculations are very similar to the ones

of the force field.

understanding of gas diffusion in flexible MOFs.

computed using charges from CBAC method.(Keskin, 2011a)

**3. Predicting gas diffusivity using molecular dynamics simulations** 

In the literature, MD simulations have been used to predict three different types of gas diffusivities in MOFs. These are transport diffusivity, corrected diffusivity and self diffusivity. The transport diffusivity, which is also known as Fickian diffusivity or chemical diffusivity, can be defined without approximation in terms of corrected diffusivity, *Do* and a thermodynamic correction factor, a partial derivative relating the adsorbate concentration, *c* and bulk phase fugacity, *f*

$$D\_t(\mathbf{c}) = D\_o(\mathbf{c}) \cdot \frac{\partial \ln f}{\partial \ln \mathbf{c}}.\tag{1}$$

The thermodynamic correction factor is fully defined once the single component adsorption isotherm is known. Well developed approaches exist for calculating the corrected diffusion coefficient from MD simulations.(Kärger&Ruthven, 1992; Keil et al., 2000; Skoulidas&Sholl, 2003; Skoulidas&Sholl, 2005) For systems with a single adsorbed component, the corrected diffusivity is equivalent to the Maxwell-Stefan diffusion coefficient.(Kapteijn et al., 2000; Ruthven, 1984; Sholl, 2006) The corrected diffusivity includes information on the collective motion of multiple adsorbed molecules that is relevant to net mass transport and can be calculated using the following expression:(Kärger&Ruthven, 1992; Keil et al., 2000)

$$D\_{o,i} = \lim\_{t \to \infty} \frac{1}{6Nt} \left\langle \left(\sum\_{l=1}^{N\_i} [r\_{il}(t) - r\_{il}(0)]\right)^2 \right\rangle \tag{2}$$

Here, *N* is the number of molecules, r*il*(*t*) is the three dimensional position vector of molecule *l* of species *i* at time *t* and the angular brackets denote that the ensemble average. Using MD simulations, one can record the trajectory of the gas molecules in the pores of MOFs and calculate the corrected diffusivity. A more microscopic measure of diffusion is the self diffusion coefficient which describes the motion of individual, tagged particles. In an isotropic three dimensional material, the self diffusivity is related to the mean squared displacement of tagged particles by the Einstein relation:

$$D\_{self,i} = \lim\_{t \to \infty} \frac{1}{6t} \left\langle \frac{1}{N\_t} \sum\_{l=1}^{N\_i} \left[ r\_{il}(t) - r\_{il}(0) \right]^2 \right\rangle \tag{3}$$

This definition of self diffusivity is applicable to both single component and multicomponent systems.(Sanborn&Snurr, 2000) In general, all three diffusion coefficients described here, transport, corrected and self diffusivities are the functions of concentration and they are only equal in the limit of dilute concentrations.(Sholl, 2006) In some extreme cases, the self and corrected diffusivities vary by orders of magnitude.(Ackerman et al., 2003; Skoulidas et al., 2002) This observation sometimes underscores the value of characterizing these two diffusivities independently. Applications such as modeling of membranes, pressure swing adsorption require the accurate description of net mass transfer and in these processes generally the transport diffusivity is of greatest interest.(Sholl, 2006)

Almost all applications of nanoporous materials in gas separations involve chemical mixtures; therefore it is important to describe the multi-component gas transport in nanopores. There are several mathematically equivalent formalisms such as Onsager, Fickian and Maxwell-Stefan to describe multi-component gas transport through nanoporous materials.(Krishna&van den Broeke, 1995; Wesselingh&Krishna, 2000) The Onsager formulation is based on irreversible thermodynamics and expresses the flux of each species in terms of chemical potentials. One can calculate the Onsager coefficient using MD simulations based on the method by Theodorou et al. (Theodorou et al., 1996):

$$L\_{ij} = \frac{1}{6Vk\_BT} \lim\_{t \to \infty} \frac{1}{t} \left\langle \sum\_{l=1}^{N\_i} [r\_{il}(t) - r\_{il}(0)] \cdot \sum\_{k=1}^{N\_j^i} \left[ r\_{jk}(t) - r\_{jk}(0) \right] \right\rangle \tag{4}$$

In this formulation, *V* is the subsystem volume, *kB* is the Boltzmann constant, *T* is temperature, r*il*(*t*) is the three-dimensional position vector of molecule *l* of species *i* at time *t*  and *Ni* is the number of molecules of species *i*. The Onsager coefficients and the matrix of Fickian coefficients are mathematically equivalent and they are related to each other without approximation by expressions involving derivatives of the mixture adsorption isotherm for the adsorbed species.(Skoulidas et al., 2003) The Onsager coefficients from MD simulations can be converted to Fickian diffusion coefficients using the followings:

$$D\_{ii} = \frac{k\_B T}{c\_i} \sum\_{j=1}^{N} L\_{ij} \left(\frac{\partial \ln f\_i}{\partial \ln c\_i}\right) \tag{5}$$

Recent Advances in Molecular Dynamics

structure of MOF-5.

CH3 groups in IRMOF-18.

Simulations of Gas Diffusion in Metal Organic Frameworks 263

Skoulidas performed the first study of gas diffusion in a MOF material in the literature using equilibrium MD simulations and calculated the self, corrected and transport diffusivities of argon at 298 K as a function of pressure.(Skoulidas, 2004) Results showed that diffusion in CuBTC MOF is an activated process as in zeolites. The calculated diffusivities of Ar in CuBTC were similar to the diffusion in zeolites both in magnitude and concentration dependence. Sarkisov et al. used equilibrium MD simulations to calculate the self diffusivities of methane, n-pentane, n-hexane, n-heptane, cyclohexane and benzene in MOF-5 at 300 K at dilute loadings.(Sarkisov et al., 2004) They found that self diffusivities of n-alkanes in MOF-5 are comparable to those in the crystalline bipyridine system (0.1-3×10-8 m2/s), but they show a stronger dependence on chain length because of the more open

Skoulidas and Sholl(Skoulidas&Sholl, 2005) then used equilibrium MD simulations to probe the self, corrected and transport diffusivity of a number of gas species, Ar, CH4, CO2, N2 and H2 in MOF-5 as a function of pore loading at room temperature. They also calculated self, corrected and transport diffusivities of Ar in MOF-2, MOF-3, MOF-5, CuBTC and MFI to make a comparison among different MOFs and a zeolite. They concluded that diffusion of gas molecules in MOFs is mostly dominated by motions where the adsorbed species remain in close contact with the surfaces defined by the pore structure throughout their diffusion. At the time of their study, there was no experimental data for gas diffusion in MOFs. Therefore, Skoulidas and Sholl could not directly comment on the accuracy of the MD simulations, however, they pointed out that using similar molecular simulation methods for

Yang and Zhong performed constant temperature equilibrium MD simulations by a momentum scaling method to calculate the self diffusivity of H2 in isoreticular MOFs, IRMOF-1, IRMOF-8 and IRMOF-18 as a function of pressure.(Yang&Zhong, 2005) Their results showed that the diffusivity of H2 in IRMOFs is slightly larger than in zeolites due to the larger pore volume of IRMOFs. The self diffusivity of H2 in IRMOFs at 77 K at low pressures was around 1-3×10-8 m2/s whereas the self diffusivity of H2 in various zeolites were experimentally measured to be around 0.1-1×10-8 m2/s. The activation energy of H2 in IRMOFs was between 2-3 kJ/mol which is close to the values measured in zeolite NaX (4 kJ/mol(Bär et al., 1999)) and single walled carbon nanotubes (1.12 kJ/mol(Narehood et al., 2003)). This MD study examined the effects of framework topology on the diffusivity of H2. For example, H2 diffuses more rapidly in IRMOF-8 than that in IRMOF-1 because of the relatively larger pore sizes of the former. The diffusivity of H2 in IRMOF-18 is much slower than diffusion in IRMOF-1 and IRMOF-8 due to the steric hindrance effects of the pendant

IRMOFs can be further categorized as catenated and non-catenated structures. In catenation, two or more identical frameworks are intergrown at the expense of pore volume. Early studies showed that catenated MOFs can give better adsorption properties compared to their counterparts.(Ryan et al., 2008) The first study about the effects of catenation on the gas diffusion used equilibrium MD simulations in the canonical ensemble to investigate H2 diffusion.(Liu et al., 2008a) Nosé-Hoover chain thermostat as formulated by Martyna et al.(Martyna et al., 1996) was used to calculate room temperature self diffusivities of H2 in catenated and non-catenated MOFs. The results showed that H2 self diffusivity in the IRMOFs without catenation such as IRMOF-10, IRMOF-12, IRMOF-14, IRMOF-16 (30-90×10-

gas diffusion in zeolites they got excellent agreement with the experiments.

$$D\_{ij} = \frac{k\_B T}{c\_j} \sum\_{k=1}^{N} L\_{ik} \left( \frac{\partial \ln f\_k}{\partial \ln c\_j} \right) \tag{6}$$

In these equations, *T* is temperature, *ci* is the concentration of species *i*, *fj* is the fugacity of species *j*, *kB* is Boltzmann constant, *Lij* and *Lik* are the Onsager coefficients and *Dii* and *Dij* are the Fickian diffusivity coefficients. Using Onsager or Fickian diffusivities, one can calculate the flux (*J*) of a binary gas mixture through a membrane as follows:

$$
\begin{pmatrix} J\_1 \\ J\_2 \end{pmatrix} = - \begin{pmatrix} D\_{11} & D\_{12} \\ D\_{21} & D\_{22} \end{pmatrix} \cdot \begin{pmatrix} \nabla c\_1 \\ \nabla c\_2 \end{pmatrix} \tag{7}
$$

$$
\begin{pmatrix} J\_1 \\ J\_2 \end{pmatrix} = - \begin{pmatrix} L\_{11} & L\_{12} \\ L\_{21} & L\_{22} \end{pmatrix} \cdot \begin{pmatrix} \nabla \mu\_1 \\ \nabla \mu\_2 \end{pmatrix} \tag{8}
$$

$$
\stackrel{\rightarrow}{J} = -L\vec{\nabla}\,\stackrel{\rightarrow}{\mu} = -D\vec{\nabla}\,\stackrel{\rightarrow}{c}\tag{9}
$$

In these expressions, *<sup>i</sup> c* and *<sup>i</sup>* represent concentration gradient and chemical potential gradient of species *i* through the membrane, respectively. As Equation 9 suggests, gas fluxes in a MOF membrane can be calculated based on either of the formulations (Onsager or Fickian).

#### **3.1 Single component diffusion**

The transport rates of single component gas molecules inside the materials' pores are important in many potential applications of MOFs. For example, in equilibrium-based separations such as pressure swing adsorption, transport rates define limits on the cycle times that can be achieved. In these cases, molecular transport rates are mainly important if they are very slow. Since accurate characterization of molecular transport inside nanoporous materials using experiments is very challenging, most of the information that is currently available about single component gas diffusion in MOFs has obtained from MD simulations.

 1 1 1 1 lim ( ) (0) ( ) (0) <sup>6</sup> *Ni Nj*

In this formulation, *V* is the subsystem volume, *kB* is the Boltzmann constant, *T* is temperature, r*il*(*t*) is the three-dimensional position vector of molecule *l* of species *i* at time *t*  and *Ni* is the number of molecules of species *i*. The Onsager coefficients and the matrix of Fickian coefficients are mathematically equivalent and they are related to each other without approximation by expressions involving derivatives of the mixture adsorption isotherm for the adsorbed species.(Skoulidas et al., 2003) The Onsager coefficients from MD simulations

1

1

In these equations, *T* is temperature, *ci* is the concentration of species *i*, *fj* is the fugacity of species *j*, *kB* is Boltzmann constant, *Lij* and *Lik* are the Onsager coefficients and *Dii* and *Dij* are the Fickian diffusivity coefficients. Using Onsager or Fickian diffusivities, one can calculate

> 1 11 12 1 2 2 21 22 *J DD c J c D D*

1 11 12 1 2 2 21 22

*J L Dc* 

gradient of species *i* through the membrane, respectively. As Equation 9 suggests, gas fluxes in a MOF membrane can be calculated based on either of the formulations (Onsager or

The transport rates of single component gas molecules inside the materials' pores are important in many potential applications of MOFs. For example, in equilibrium-based separations such as pressure swing adsorption, transport rates define limits on the cycle times that can be achieved. In these cases, molecular transport rates are mainly important if they are very slow. Since accurate characterization of molecular transport inside nanoporous materials using experiments is very challenging, most of the information that is currently available about single component gas diffusion in MOFs has obtained from MD simulations.

*J LL J L L*

*N B k*

*k T <sup>f</sup> D L c c* 

*i i j*

*j j k*

*N B i*

*k T <sup>f</sup> D L c c* 

*ii ij*

*ij ik*

ln ln

ln ln

(5)

(6)

(9)

*<sup>i</sup>* represent concentration gradient and chemical potential

(8)

(7)

(4)

*ij il il jk jk <sup>t</sup> <sup>B</sup> l k L rt r r t r Vk T t*

can be converted to Fickian diffusion coefficients using the followings:

the flux (*J*) of a binary gas mixture through a membrane as follows:

In these expressions, *<sup>i</sup> c* and

**3.1 Single component diffusion** 

Fickian).

Skoulidas performed the first study of gas diffusion in a MOF material in the literature using equilibrium MD simulations and calculated the self, corrected and transport diffusivities of argon at 298 K as a function of pressure.(Skoulidas, 2004) Results showed that diffusion in CuBTC MOF is an activated process as in zeolites. The calculated diffusivities of Ar in CuBTC were similar to the diffusion in zeolites both in magnitude and concentration dependence. Sarkisov et al. used equilibrium MD simulations to calculate the self diffusivities of methane, n-pentane, n-hexane, n-heptane, cyclohexane and benzene in MOF-5 at 300 K at dilute loadings.(Sarkisov et al., 2004) They found that self diffusivities of n-alkanes in MOF-5 are comparable to those in the crystalline bipyridine system (0.1-3×10-8 m2/s), but they show a stronger dependence on chain length because of the more open structure of MOF-5.

Skoulidas and Sholl(Skoulidas&Sholl, 2005) then used equilibrium MD simulations to probe the self, corrected and transport diffusivity of a number of gas species, Ar, CH4, CO2, N2 and H2 in MOF-5 as a function of pore loading at room temperature. They also calculated self, corrected and transport diffusivities of Ar in MOF-2, MOF-3, MOF-5, CuBTC and MFI to make a comparison among different MOFs and a zeolite. They concluded that diffusion of gas molecules in MOFs is mostly dominated by motions where the adsorbed species remain in close contact with the surfaces defined by the pore structure throughout their diffusion. At the time of their study, there was no experimental data for gas diffusion in MOFs. Therefore, Skoulidas and Sholl could not directly comment on the accuracy of the MD simulations, however, they pointed out that using similar molecular simulation methods for gas diffusion in zeolites they got excellent agreement with the experiments.

Yang and Zhong performed constant temperature equilibrium MD simulations by a momentum scaling method to calculate the self diffusivity of H2 in isoreticular MOFs, IRMOF-1, IRMOF-8 and IRMOF-18 as a function of pressure.(Yang&Zhong, 2005) Their results showed that the diffusivity of H2 in IRMOFs is slightly larger than in zeolites due to the larger pore volume of IRMOFs. The self diffusivity of H2 in IRMOFs at 77 K at low pressures was around 1-3×10-8 m2/s whereas the self diffusivity of H2 in various zeolites were experimentally measured to be around 0.1-1×10-8 m2/s. The activation energy of H2 in IRMOFs was between 2-3 kJ/mol which is close to the values measured in zeolite NaX (4 kJ/mol(Bär et al., 1999)) and single walled carbon nanotubes (1.12 kJ/mol(Narehood et al., 2003)). This MD study examined the effects of framework topology on the diffusivity of H2. For example, H2 diffuses more rapidly in IRMOF-8 than that in IRMOF-1 because of the relatively larger pore sizes of the former. The diffusivity of H2 in IRMOF-18 is much slower than diffusion in IRMOF-1 and IRMOF-8 due to the steric hindrance effects of the pendant CH3 groups in IRMOF-18.

IRMOFs can be further categorized as catenated and non-catenated structures. In catenation, two or more identical frameworks are intergrown at the expense of pore volume. Early studies showed that catenated MOFs can give better adsorption properties compared to their counterparts.(Ryan et al., 2008) The first study about the effects of catenation on the gas diffusion used equilibrium MD simulations in the canonical ensemble to investigate H2 diffusion.(Liu et al., 2008a) Nosé-Hoover chain thermostat as formulated by Martyna et al.(Martyna et al., 1996) was used to calculate room temperature self diffusivities of H2 in catenated and non-catenated MOFs. The results showed that H2 self diffusivity in the IRMOFs without catenation such as IRMOF-10, IRMOF-12, IRMOF-14, IRMOF-16 (30-90×10-

Recent Advances in Molecular Dynamics

mass transport.

loading in MOFs having large pore volumes.

Simulations of Gas Diffusion in Metal Organic Frameworks 265

Zeolites are known with the Al(Si)O2 unit formula, whereas ZIFs are recognized by M(Im)2 where M is the transition metal (zinc, cobalt, copper, etc.) and Im is the imidazolate-type linker. Recent MD simulations focused on gas diffusion in ZIFs. For example, self and corrected diffusivities of CO2, CH4 and H2 were simulated using equilibrium MD in ZIF-68 and ZIF-70.(Rankin et al., 2009) That study underlined the importance of including chargequadrupole interactions on the diffusivity of CO2. Simulation results clearly revealed that addition of charge-quadrupole interaction terms results in almost one order of magnitude drop in the self and transport diffusivities of CO2 at low loadings. At high loadings diffusivities calculated from MD simulations with or without charge-quadrupole interaction terms converge towards the same values. The diffusivities of CO2, CH4 and H2 in ZIF-68 were found to be lower than the ones in ZIF-70 since ZIF-68 has narrower pores hence provides stronger confinement of the adsorbate molecules in the pores. Self diffusivity of CO2 in ZIF-68 and ZIF-69 was also computed by MD simulations.(Liu et al., 2009) The diffusion of CO2 in ZIF-68 and ZIF-69 was found to be nearly an order of magnitude slower than that in IRMOF-10 and IRMOF-14. This was attributed to the smaller pores of ZIFs and their structural characteristic that causes larger steric hindrance. Pantatosaki and coworkers computed H2 self diffusion in ZIF-8 using both LJ and FH potentials at 77 and 300 K.(Pantatosaki et al., 2010) The diffusivity predictions showed that quantum mechanical description of H2 at ambient temperatures is unimportant whereas MD simulations showed a marked difference between the values obtained from the classical and quantum mechanical description at 77 K. A recent MD study computed self diffusivities of H2, CO2, CH4 and N2 in ZIF-2, ZIF-4, ZIF-5, ZIF-8 and ZIF-9.(Battisti et al., 2011) Results showed that gases except H2 do not diffuse appreciably in ZIF-5 at least within the time interval of the MD calculations which makes ZIF-5 promising in H2 separations as a molecular sieve.

Self diffusivities of H2, CH4 and CO2 in bioMOF-11 were computed from canonical ensemble MD simulations at 298 K.(Atci et al., 2011) BioMOFs are another subclass of MOFs that have been recently discovered. They incorporate simple biomolecules and biocompatible metal cations in their structures as linkers and metals.(An et al., 2009a; An et al., 2009b) Gas diffusion in bioMOFs was found to be similar to IRMOFs in terms of magnitude and loading dependence. As can be seen from the literature reviewed so far, most of the MD studies on MOFs computed self diffusivity of gases rather than corrected diffusivities since the calculation of the latter is computationally demanding. Keskin computed both single component self and corrected diffusivities of CH4 and H2 as a function of fugacity and pore loading in CPO-27-Ni.(Keskin, 2010a) The diffusivity of H2 (4×10-3 cm2/s) was faster than CH4 (6×10-4 cm2/s) as expected. Single component corrected diffusivities were found to be higher than the self diffusivities, since corrected diffusivity by definition includes information on the collective motion of multiple adsorbed molecules that is relevant to net

Figure 3 represents the self diffusivity of CO2 computed from MD simulations in the widely studied MOFs at room temperature. Gas diffusion in MOFs having large pores (IRMOF-1, CuBTC, Zn(bdc)(ted)0.5) is higher than the one in MOFs having narrow pores (Cu(hfipbb)(H2hfipbb)0.5, MMIF). The CO2 self diffusivity decreases with increased adsorbed loading in bioMOF-11, Cu(hfipbb)(H2hfipbb)0.5 and MMIF since CO2 reaches saturation in these MOFs due to their small pore volumes. The diffusivities in large pore MOFs do not change significantly with increased loadings since CO2 is further away from the saturation

8 m2/s) are two or three times of those (10-20×10-8 m2/s) in their corresponding catenated counterparts IRMOF-9, IRMOF-11, IRMOF-13, IRMOF-15. This implied that the motion of H2 molecules in these MOFs is restricted by their catenated structures.

Lee and coworkers also investigated the diffusion of H2 in catenated MOFs, IRMOF-9, IRMOF-11 and IRMOF-13 at 77 K.(Lee et al., 2009) Diffusivities reported by Liu et al. were larger than the ones reported by Lee et al. by one order of magnitude since the former group performed the MD simulations at room temperature. The results of two studies were consistent; the diffusion rate of H2 is dramatically reduced by the catenation of IRMOFs due to the interpenetrated chains of the catenated structures and/or by tighter binding of the H2 molecules in catenated structures. Equilibrium MD simulation studies showed that the effect of catenation on CH4 diffusivity is much larger than that on H2 diffusivity at room temperature.(Xue et al., 2009) Xue and coworkers discussed that the motion of both CH4 and H2 is restricted by the catenated structures of IRMOF-11 and IRMOF-13 while the stronger interactions between CH4 and atoms of the catenated frameworks lead to stronger confinement effects than that of H2 in these IRMOFs.

Liu and coworkers(Liu et al., 2008b) investigated the influence of quantum effects on H2 diffusivity using MD simulations. They used both the classical and the Feynman-Hibbs (FH)(Feynman&Hibbs, 1965) effective Buch potentials with UFF in their MD simulations to calculate self, corrected and transport diffusivities of H2 in a MOF called Zn(bdc)(ted)0.5 at 77 K. The inclusion of quantum effects increased the self diffusivity of H2 at zero loading which was explained by the decrease in the diffusion energy barrier due to a non-uniform smearing of solid-fluid potential within the FH formalism. At higher loadings, inclusion of quantum effects decreased H2 diffusivity which was attributed to the steric hindrance in narrow pores due to the increase in the effective size parameter for the solid-fluid and fluidfluid interactions. In contrast to self diffusivity, transport diffusivity is not strongly influenced by the quantum effects at 77 K.

In order to compare the diffusivities of gases in MOFs with those in zeolites, MD simulations were performed to calculate self, corrected and transport diffusivities of CH4 and CO2 in silicalite, IRMOF-1 and C168 schwarzite.(Babarao&Jiang, 2008) The simulations were carried out in a canonical ensemble with a Nosé-Hoover thermostat and the equations of motion were integrated using a sixth order Gear predictor-corrector algorithm. (Allen&Tildesley, 1987) Both self and corrected diffusivities of CH4 and CO2 were found to be larger in IRMOF-1 (Dself-CH4:4-5×10-8 m2/s, Dself-CO2:2-3×10-8 m2/s) compared to the diffusivities in MFI and C168. This was attributed to the large pore volume of the IRMOF-1. This work also showed that in the limit of infinite dilution the diffusivities at various temperatures exhibit a good Arrhenius relationship. In another MD study, NVT ensemble with Berendsen(Frenkel&Smit, 2002) thermostat was used to examine the self diffusivity of CH4 in alkoxy functionalized IRMOF-1.(Jhon et al., 2007) As expected, CH4 diffusion was hindered due to the constriction of the pores as the length of the alkoxy chains increases. Comparison of the results with the early MD studies of Sarkisov et al.(Sarkisov et al., 2004) revealed good agreement whereas there is an unexplained small discrepancy between the results of Jhon et al. and Skoulidas et al.(Skoulidas&Sholl, 2005) at higher loadings.

Zeolite imidazolate frameworks (ZIFs) are a subclass of MOFs with their tetrahedral networks that resemble those of zeolites with transition metals linked by imidazolate ligands.(Banerjee et al., 2008; Banerjee et al., 2009; Hayashi et al., 2007; Park et al., 2006)

8 m2/s) are two or three times of those (10-20×10-8 m2/s) in their corresponding catenated counterparts IRMOF-9, IRMOF-11, IRMOF-13, IRMOF-15. This implied that the motion of

Lee and coworkers also investigated the diffusion of H2 in catenated MOFs, IRMOF-9, IRMOF-11 and IRMOF-13 at 77 K.(Lee et al., 2009) Diffusivities reported by Liu et al. were larger than the ones reported by Lee et al. by one order of magnitude since the former group performed the MD simulations at room temperature. The results of two studies were consistent; the diffusion rate of H2 is dramatically reduced by the catenation of IRMOFs due to the interpenetrated chains of the catenated structures and/or by tighter binding of the H2 molecules in catenated structures. Equilibrium MD simulation studies showed that the effect of catenation on CH4 diffusivity is much larger than that on H2 diffusivity at room temperature.(Xue et al., 2009) Xue and coworkers discussed that the motion of both CH4 and H2 is restricted by the catenated structures of IRMOF-11 and IRMOF-13 while the stronger interactions between CH4 and atoms of the catenated frameworks lead to stronger

Liu and coworkers(Liu et al., 2008b) investigated the influence of quantum effects on H2 diffusivity using MD simulations. They used both the classical and the Feynman-Hibbs (FH)(Feynman&Hibbs, 1965) effective Buch potentials with UFF in their MD simulations to calculate self, corrected and transport diffusivities of H2 in a MOF called Zn(bdc)(ted)0.5 at 77 K. The inclusion of quantum effects increased the self diffusivity of H2 at zero loading which was explained by the decrease in the diffusion energy barrier due to a non-uniform smearing of solid-fluid potential within the FH formalism. At higher loadings, inclusion of quantum effects decreased H2 diffusivity which was attributed to the steric hindrance in narrow pores due to the increase in the effective size parameter for the solid-fluid and fluidfluid interactions. In contrast to self diffusivity, transport diffusivity is not strongly

In order to compare the diffusivities of gases in MOFs with those in zeolites, MD simulations were performed to calculate self, corrected and transport diffusivities of CH4 and CO2 in silicalite, IRMOF-1 and C168 schwarzite.(Babarao&Jiang, 2008) The simulations were carried out in a canonical ensemble with a Nosé-Hoover thermostat and the equations of motion were integrated using a sixth order Gear predictor-corrector algorithm. (Allen&Tildesley, 1987) Both self and corrected diffusivities of CH4 and CO2 were found to be larger in IRMOF-1 (Dself-CH4:4-5×10-8 m2/s, Dself-CO2:2-3×10-8 m2/s) compared to the diffusivities in MFI and C168. This was attributed to the large pore volume of the IRMOF-1. This work also showed that in the limit of infinite dilution the diffusivities at various temperatures exhibit a good Arrhenius relationship. In another MD study, NVT ensemble with Berendsen(Frenkel&Smit, 2002) thermostat was used to examine the self diffusivity of CH4 in alkoxy functionalized IRMOF-1.(Jhon et al., 2007) As expected, CH4 diffusion was hindered due to the constriction of the pores as the length of the alkoxy chains increases. Comparison of the results with the early MD studies of Sarkisov et al.(Sarkisov et al., 2004) revealed good agreement whereas there is an unexplained small discrepancy between the

results of Jhon et al. and Skoulidas et al.(Skoulidas&Sholl, 2005) at higher loadings.

Zeolite imidazolate frameworks (ZIFs) are a subclass of MOFs with their tetrahedral networks that resemble those of zeolites with transition metals linked by imidazolate ligands.(Banerjee et al., 2008; Banerjee et al., 2009; Hayashi et al., 2007; Park et al., 2006)

H2 molecules in these MOFs is restricted by their catenated structures.

confinement effects than that of H2 in these IRMOFs.

influenced by the quantum effects at 77 K.

Zeolites are known with the Al(Si)O2 unit formula, whereas ZIFs are recognized by M(Im)2 where M is the transition metal (zinc, cobalt, copper, etc.) and Im is the imidazolate-type linker. Recent MD simulations focused on gas diffusion in ZIFs. For example, self and corrected diffusivities of CO2, CH4 and H2 were simulated using equilibrium MD in ZIF-68 and ZIF-70.(Rankin et al., 2009) That study underlined the importance of including chargequadrupole interactions on the diffusivity of CO2. Simulation results clearly revealed that addition of charge-quadrupole interaction terms results in almost one order of magnitude drop in the self and transport diffusivities of CO2 at low loadings. At high loadings diffusivities calculated from MD simulations with or without charge-quadrupole interaction terms converge towards the same values. The diffusivities of CO2, CH4 and H2 in ZIF-68 were found to be lower than the ones in ZIF-70 since ZIF-68 has narrower pores hence provides stronger confinement of the adsorbate molecules in the pores. Self diffusivity of CO2 in ZIF-68 and ZIF-69 was also computed by MD simulations.(Liu et al., 2009) The diffusion of CO2 in ZIF-68 and ZIF-69 was found to be nearly an order of magnitude slower than that in IRMOF-10 and IRMOF-14. This was attributed to the smaller pores of ZIFs and their structural characteristic that causes larger steric hindrance. Pantatosaki and coworkers computed H2 self diffusion in ZIF-8 using both LJ and FH potentials at 77 and 300 K.(Pantatosaki et al., 2010) The diffusivity predictions showed that quantum mechanical description of H2 at ambient temperatures is unimportant whereas MD simulations showed a marked difference between the values obtained from the classical and quantum mechanical description at 77 K. A recent MD study computed self diffusivities of H2, CO2, CH4 and N2 in ZIF-2, ZIF-4, ZIF-5, ZIF-8 and ZIF-9.(Battisti et al., 2011) Results showed that gases except H2 do not diffuse appreciably in ZIF-5 at least within the time interval of the MD calculations which makes ZIF-5 promising in H2 separations as a molecular sieve.

Self diffusivities of H2, CH4 and CO2 in bioMOF-11 were computed from canonical ensemble MD simulations at 298 K.(Atci et al., 2011) BioMOFs are another subclass of MOFs that have been recently discovered. They incorporate simple biomolecules and biocompatible metal cations in their structures as linkers and metals.(An et al., 2009a; An et al., 2009b) Gas diffusion in bioMOFs was found to be similar to IRMOFs in terms of magnitude and loading dependence. As can be seen from the literature reviewed so far, most of the MD studies on MOFs computed self diffusivity of gases rather than corrected diffusivities since the calculation of the latter is computationally demanding. Keskin computed both single component self and corrected diffusivities of CH4 and H2 as a function of fugacity and pore loading in CPO-27-Ni.(Keskin, 2010a) The diffusivity of H2 (4×10-3 cm2/s) was faster than CH4 (6×10-4 cm2/s) as expected. Single component corrected diffusivities were found to be higher than the self diffusivities, since corrected diffusivity by definition includes information on the collective motion of multiple adsorbed molecules that is relevant to net mass transport.

Figure 3 represents the self diffusivity of CO2 computed from MD simulations in the widely studied MOFs at room temperature. Gas diffusion in MOFs having large pores (IRMOF-1, CuBTC, Zn(bdc)(ted)0.5) is higher than the one in MOFs having narrow pores (Cu(hfipbb)(H2hfipbb)0.5, MMIF). The CO2 self diffusivity decreases with increased adsorbed loading in bioMOF-11, Cu(hfipbb)(H2hfipbb)0.5 and MMIF since CO2 reaches saturation in these MOFs due to their small pore volumes. The diffusivities in large pore MOFs do not change significantly with increased loadings since CO2 is further away from the saturation loading in MOFs having large pore volumes.

Recent Advances in Molecular Dynamics

of magnitude faster than CH4 in these MOFs.

mixtures is harder than studying a single species.

membrane using:(Keskin&Sholl, 2009b)

**3.2 Mixture diffusion** 

Simulations of Gas Diffusion in Metal Organic Frameworks 267

where *υ* is the pre-exponential factor (1012-1013 s-1), *Etrans* is the transition energy barrier computed using DFT calculations for a flexible MOF, *kB* is the Boltzmann constant, *T* is

where *a* is the cage-to-cage distance along the pore. Keskin studied diffusion of CH4 and CO2 in a microporous metal-imidazolate framework and similarly observed that CH4 diffusion in this MOF is not accessible by MD due to a very large energy barrier (95 kJ/mol) that exists for the adsorbate to move through the pore.(Keskin, 2011b) It is important to note that both Watanabe et al. and Keskin concluded that Cu(hfipbb)(H2hfipbb)0.5 and MMIF are very promising materials for separation of CO2 from CH4 since CO2 diffuses several orders

In most practical applications, gases exist as mixtures rather than single components. For example, in membrane-based separations, at least two gas components exist. The relative transport rates of these components inside the material of interest are crucial in determining the overall performance of a material. Therefore, understanding mixture diffusion in MOFs is essential to design these materials as separation devices. In this section, MD simulations which predicted multi-component mixture diffusion in MOFs will be reviewed. These studies have mostly focused on self diffusivities of CO2/CH4, CO2/H2, CO2/N2, CO2/CO and CH4/H2 mixtures. The diffusion selectivities of MOFs for these gas mixtures have been computed using MD to understand the potential of MOFs in kinetic-based separations. It is important to highlight the fact that number of mixture MD simulations is limited compared to the number of single component MD simulations since characterizing diffusivity of gas

Keskin and coworkers provided the first gas mixture diffusivity data in a MOF material using MD simulations.(Keskin et al., 2008) They computed self and Fickian diffusivities of CH4/H2 mixtures at various compositions in CuBTC. Theoretical correlations that can estimate mixture self and Fickian diffusivities based on single component data were tested in that work and the predictions of the correlations were compared with the results of direct MD simulations. This will be discussed in detail in Section 5. Keskin and Sholl later showed that if MD (GCMC) simulations are used to compute the mixture diffusivities (adsorbed amounts of each species) in a MOF, one can simply estimate the selectivity of this MOF as a

1, 1 2 1 2

*D qq q q D qq y y*

*self*

*self*

2, 1 2 1 2 (,) / (,) /

*perm diff sorp*

where α*perm,1/2*, α*diff,1/2* and α*sorp,1/2* represent permeation selectivity, diffusion selectivity and sorption selectivity of species 1 over species 2, respectively. In this approximate expression, the diffusion selectivity is defined as the ratio of self diffusivities in a binary mixture (*Di,self(qi)*) and the sorption selectivity is described as the ratio of adsorbed molar loadings, *qi*. This expression predicts a membrane's selectivity at a specified feed pressure and composition based on a single mixture GCMC simulation and an MD simulation performed

,1/2 ,1/2 ,1/2

  (12)

<sup>2</sup> *D ka* 1/2 (11)

temperature. These calculations resulted in a one-dimensional diffusivity,

As the literature cited so far indicated most MD studies focused on diffusion of small gas molecules such as Ar, CH4, CO2, N2, H2 in MOFs. Limited number of studies investigated diffusion of larger gases. The self diffusivities of hexahydro-1,3,5-trinitro-1,3,5-triazine (RDX) were generated by MD in IRMOF-1, IRMOF-3, IRMOF-10.(Xiong et al., 2010) The trend for the self diffusivities of RDX in MOFs followed the pore sizes, highest in IRMOF-10 and lowest in IRMOF-3. The self diffusivity of ethane, n-butane, n-hexane and cyclohexane in a MOF with the organic linker tetrakis[4-(carboxyphenyl)oxamethyl]methane was studied using MD simulations.(Sun et al., 2011) For linear alkanes, the diffusivities decreased dramatically with increased chain length. The specific MOF studied in this work exhibited high selectivity towards n-hexane as a result of kinetics.

adsorbed loading (molecules/unit cell MOF)

In some cases, the adsorbate molecules cannot move in the MOF pores at a rate that can be measured by MD simulations. This case is generally observed when the kinetic diameter of the gas molecule is very similar in size to the pore diameter of the MOF. For example, initial MD simulations of adsorbed CH4 in a rigid Cu(hfipbb)(H2hfipbb)0.5 indicated that CH4 can not move between adjacent cages on the nanosecond time scales accessible using MD due to the large energy barrier.(Watanabe et al., 2009) The authors used a simple transition state expression to estimate the diffusivity of CH4 in Cu(hfipbb)(H2hfipbb)0.5 by assuming that the CH4 hopping rate between cages is,

$$k = \nu \cdot \exp\left(-E\_{trans} \;/\; k\_B T\right) \tag{10}$$

where *υ* is the pre-exponential factor (1012-1013 s-1), *Etrans* is the transition energy barrier computed using DFT calculations for a flexible MOF, *kB* is the Boltzmann constant, *T* is temperature. These calculations resulted in a one-dimensional diffusivity,

$$D = 1 \;/\; 2 \cdot k \cdot a^2 \tag{11}$$

where *a* is the cage-to-cage distance along the pore. Keskin studied diffusion of CH4 and CO2 in a microporous metal-imidazolate framework and similarly observed that CH4 diffusion in this MOF is not accessible by MD due to a very large energy barrier (95 kJ/mol) that exists for the adsorbate to move through the pore.(Keskin, 2011b) It is important to note that both Watanabe et al. and Keskin concluded that Cu(hfipbb)(H2hfipbb)0.5 and MMIF are very promising materials for separation of CO2 from CH4 since CO2 diffuses several orders of magnitude faster than CH4 in these MOFs.

#### **3.2 Mixture diffusion**

266 Molecular Dynamics – Theoretical Developments and Applications in Nanotechnology and Energy

As the literature cited so far indicated most MD studies focused on diffusion of small gas molecules such as Ar, CH4, CO2, N2, H2 in MOFs. Limited number of studies investigated diffusion of larger gases. The self diffusivities of hexahydro-1,3,5-trinitro-1,3,5-triazine (RDX) were generated by MD in IRMOF-1, IRMOF-3, IRMOF-10.(Xiong et al., 2010) The trend for the self diffusivities of RDX in MOFs followed the pore sizes, highest in IRMOF-10 and lowest in IRMOF-3. The self diffusivity of ethane, n-butane, n-hexane and cyclohexane in a MOF with the organic linker tetrakis[4-(carboxyphenyl)oxamethyl]methane was studied using MD simulations.(Sun et al., 2011) For linear alkanes, the diffusivities decreased dramatically with increased chain length. The specific MOF studied in this work exhibited

0 5 10 15 20

In some cases, the adsorbate molecules cannot move in the MOF pores at a rate that can be measured by MD simulations. This case is generally observed when the kinetic diameter of the gas molecule is very similar in size to the pore diameter of the MOF. For example, initial MD simulations of adsorbed CH4 in a rigid Cu(hfipbb)(H2hfipbb)0.5 indicated that CH4 can not move between adjacent cages on the nanosecond time scales accessible using MD due to the large energy barrier.(Watanabe et al., 2009) The authors used a simple transition state expression to estimate the diffusivity of CH4 in Cu(hfipbb)(H2hfipbb)0.5 by assuming that the

exp( / ) *trans B k E kT*

Fig. 3. The self diffusivity of CO2 in IRMOF-1, CuBTC, Zn(bdc)(ted)0.5, bioMOF-11, Cu(hfipbb)(H2hfipbb)0.5 and MMIF at 298 K computed from MD simulations.(Atci et al., 2011; Erucar&Keskin, 2011;Keskin, 2011b; Keskin&Sholl, 2009b;Watanabe et al., 2009)

adsorbed loading (molecules/unit cell MOF)

 IRMOF-1 CuBTC

MMIF

 Zn(bdc)(ted)0.5 BioMOF-11 Cu(hfipbb)(H2

hfipbb)0.5

(10)

high selectivity towards n-hexane as a result of kinetics.

1E-7

CH4 hopping rate between cages is,

1E-6

1E-5

D

self CO2 (cm2/s)

1E-4

1E-3

In most practical applications, gases exist as mixtures rather than single components. For example, in membrane-based separations, at least two gas components exist. The relative transport rates of these components inside the material of interest are crucial in determining the overall performance of a material. Therefore, understanding mixture diffusion in MOFs is essential to design these materials as separation devices. In this section, MD simulations which predicted multi-component mixture diffusion in MOFs will be reviewed. These studies have mostly focused on self diffusivities of CO2/CH4, CO2/H2, CO2/N2, CO2/CO and CH4/H2 mixtures. The diffusion selectivities of MOFs for these gas mixtures have been computed using MD to understand the potential of MOFs in kinetic-based separations. It is important to highlight the fact that number of mixture MD simulations is limited compared to the number of single component MD simulations since characterizing diffusivity of gas mixtures is harder than studying a single species.

Keskin and coworkers provided the first gas mixture diffusivity data in a MOF material using MD simulations.(Keskin et al., 2008) They computed self and Fickian diffusivities of CH4/H2 mixtures at various compositions in CuBTC. Theoretical correlations that can estimate mixture self and Fickian diffusivities based on single component data were tested in that work and the predictions of the correlations were compared with the results of direct MD simulations. This will be discussed in detail in Section 5. Keskin and Sholl later showed that if MD (GCMC) simulations are used to compute the mixture diffusivities (adsorbed amounts of each species) in a MOF, one can simply estimate the selectivity of this MOF as a membrane using:(Keskin&Sholl, 2009b)

$$\alpha\_{perm, 1/2} = \frac{\mathcal{D}\_{1, \text{self}}(q\_1, q\_2)}{\mathcal{D}\_{2, \text{self}}(q\_1, q\_2)} \cdot \frac{q\_1 / \, q\_2}{y\_1 / \, y\_2} = \alpha\_{\text{diff}} \,\_{1/2} \cdot \alpha\_{\text{sup}, 1/2} \tag{12}$$

where α*perm,1/2*, α*diff,1/2* and α*sorp,1/2* represent permeation selectivity, diffusion selectivity and sorption selectivity of species 1 over species 2, respectively. In this approximate expression, the diffusion selectivity is defined as the ratio of self diffusivities in a binary mixture (*Di,self(qi)*) and the sorption selectivity is described as the ratio of adsorbed molar loadings, *qi*. This expression predicts a membrane's selectivity at a specified feed pressure and composition based on a single mixture GCMC simulation and an MD simulation performed

Recent Advances in Molecular Dynamics

faster ones through the channel of Zn(tbip).

in the synthesized MOF structure.

Simulations of Gas Diffusion in Metal Organic Frameworks 269

indicated that diffusion of CH4 is increased with increasing concentration of H2 in the CH4/H2 mixture, while the diffusivity of H2 decreases with increasing CH4 concentration. In contrast, the diffusivity of CH4 was essentially independent of the concentration of CO2 in the CO2/CH4 mixture, while CO2 diffusivity decreases with increased CH4 loading, even though the diffusivity of CH4 is substantially larger than that of CO2. This unusual behavior was explained in terms of differences in adsorption site preferences due to chargequadrupole interactions. Another recent MD study examined the self diffusivities of equimolar CO2/ethane, CH4/ethane and CO2/methanol mixtures in Zn(tbip) including the flexibility effects.(Seehamart et al., 2011) Similar to previous observations, faster diffusing molecules accelerate the slower diffusing molecules whereas the slower ones slow down the

**4. Comparison of diffusivities from molecular dynamics with experiments** 

Measuring diffusivities of gas molecules in nanoporous materials is a challenging process, therefore experimentally measured diffusion data for gases in the pores of MOFs is still very limited. Stallmach and co-workers(Stallmach et al., 2006) carried out the first experimental study in the literature for diffusivity of hydrocarbons in MOF-5. They measured diffusion of methane, ethane, n-hexane, benzene by pulsed field gradient-nuclear magnetic resonance (PFG-NMR) which is a well-established technique for intra-crystalline diffusion studies in nanoporous materials. Diffusion of methane and ethane in MOF-5 was found to be faster than in NaX which was attributed to the larger pores of the former. This study supplied the first experimental data points for gas diffusion in MOFs for direct comparison between experiments and MD simulations. The measured diffusivity of n-hexane, 3.2-4.1×10-9 m2/s, was found to be in a good agreement with the value of 2.2×10-9 m2/s predicted by earlier MD simulations(Sarkisov et al., 2004) for a slightly higher loading. However, the self diffusivity of CH4 measured by PFG-NMR was about one order of magnitude higher than the value of 3.1×10-8 m2/s reported in MD simulations.(Sarkisov et al., 2004; Skoulidas&Sholl, 2005) Stallmach and coworkers attributed this discrepancy to the imperfections that may exist in the MOF structure and loadings used in MD simulations which were lower than the ones considered in the experiments. Zhao and coworkers(Zhao et al., 2009) measured diffusivity of CO2 in MOF-5 and reported a value (8×10-13 m2/s) which is several orders of magnitude smaller than the one obtained by the MD simulations (4×10-9 m2/s)(Skoulidas&Sholl, 2005) and also significantly smaller than the diffusivity of larger adsorbates such as n-hexane, benzene measured by other groups. This large difference between experiments and simulations can be again attributed to the imperfections

The first experimental exploration of the H2 self diffusivity in MOFs was performed by quasielastic neutron scattering (QENS) measurements.(Salles et al., 2008) The QENS technique has proved to be very powerful to extract the loading dependence of the diffusivities for a wide range of adsorbates including H2 diffusivity in zeolites.(Jobic et al., 1999) Combining QENS technique with molecular simulations has been successful in the past to characterize the diffusion mechanism of various adsorbates in nanoporous materials.(Jobic&Theodorou, 2007) The self diffusivities of H2 in MOFs, MIL-47(V) and MIL-53(Cr) were extracted from QENS measurements and compared with the ones predicted by MD simulations performed in the NVT ensemble using the Evans isokinetic

at the loadings determined from this GCMC calculation. Keskin and Sholl computed self diffusivities for CH4/H2, CO2/CH4 and CO2/H2 mixtures in several MOFs, IRMOF-1, IRMOF-8, IRMOF-9, IRMOF-10, IRMOF-14, COF-102, Zn(bdc)(ted)0.5 using MD simulations and based on these diffusivities they estimated the membrane selectivity of these MOFs.

Mixture self diffusivities of CH4/H2 in CPO-27-Ni and CPO-27-Co were computed at 298 K for a wide pressure range and selectivity of these MOFs in CH4/H2 separations were predicted.(Keskin, 2010a) MD calculations were carried out to evaluate diffusion selectivities and permeation selectivities of ZIF-3 and ZIF-10 for CH4/H2, CO2/CH4 and CO2/H2 mixtures.(Keskin, 2011a) Figure 4 shows the diffusion selectivities for these mixtures in the pores of ZIF-3 and ZIF-10 as a function of pressure at room temperature. Using the same approach, Krishna and van Baten computed diffusion selectivities and permeation selectivities for equimolar CO2/CH4 and CO2/H2 mixtures as a function of total pore loading in CPO-27-Zn, CPO-27-Mg, IRMOF-1.(Krishna&van Baten, 2011) Atci and coworkers evaluated the mixture self diffusivities of CH4/H2, CO2/CH4 and CO2/H2 in bioMOF-11 at the adsorbed loadings calculated from GCMC simulations.(Atci et al., 2011)

Fig. 4. The diffusion selectivities of ZIF-3 (closed symbols) and ZIF-10 (open symbols) computed from MD simulations. The first species on the label indicates the selected one. The compositions of the bulk gas mixtures are CH4/H2:10/90, CO2/CH4:10/90 and CO2/H2:1/99.(Keskin, 2011a)

The self diffusivities of adsorbed CH4/H2 mixtures were examined at different compositions in Zn(bdc)(ted)0.5.(Keskin, 2010b) The self diffusivities of CH4 (H2) in the CH4/H2 mixture were larger (smaller) than pure component CH4 (H2) self diffusivity at the same loading. This observation is natural since the fast diffusing H2 molecules in the mixture speeds up the slowly diffusing CH4 molecules. Self diffusivities of CO2/CH4 and CH4/H2 mixtures were computed in ZIF-68 and ZIF-70 using NVT-MD simulations.(Liu et al., 2011) Results

at the loadings determined from this GCMC calculation. Keskin and Sholl computed self diffusivities for CH4/H2, CO2/CH4 and CO2/H2 mixtures in several MOFs, IRMOF-1, IRMOF-8, IRMOF-9, IRMOF-10, IRMOF-14, COF-102, Zn(bdc)(ted)0.5 using MD simulations and based on these diffusivities they estimated the membrane selectivity of these MOFs.

Mixture self diffusivities of CH4/H2 in CPO-27-Ni and CPO-27-Co were computed at 298 K for a wide pressure range and selectivity of these MOFs in CH4/H2 separations were predicted.(Keskin, 2010a) MD calculations were carried out to evaluate diffusion selectivities and permeation selectivities of ZIF-3 and ZIF-10 for CH4/H2, CO2/CH4 and CO2/H2 mixtures.(Keskin, 2011a) Figure 4 shows the diffusion selectivities for these mixtures in the pores of ZIF-3 and ZIF-10 as a function of pressure at room temperature. Using the same approach, Krishna and van Baten computed diffusion selectivities and permeation selectivities for equimolar CO2/CH4 and CO2/H2 mixtures as a function of total pore loading in CPO-27-Zn, CPO-27-Mg, IRMOF-1.(Krishna&van Baten, 2011) Atci and coworkers evaluated the mixture self diffusivities of CH4/H2, CO2/CH4 and CO2/H2 in bioMOF-11 at the adsorbed loadings calculated from GCMC simulations.(Atci et al., 2011)

> H2 /CO2

 H2 /CH4

CH4

/CO2

0 10 20 30 40 50

Fig. 4. The diffusion selectivities of ZIF-3 (closed symbols) and ZIF-10 (open symbols) computed from MD simulations. The first species on the label indicates the selected one. The

compositions of the bulk gas mixtures are CH4/H2:10/90, CO2/CH4:10/90 and

Fugacity (bar)

The self diffusivities of adsorbed CH4/H2 mixtures were examined at different compositions in Zn(bdc)(ted)0.5.(Keskin, 2010b) The self diffusivities of CH4 (H2) in the CH4/H2 mixture were larger (smaller) than pure component CH4 (H2) self diffusivity at the same loading. This observation is natural since the fast diffusing H2 molecules in the mixture speeds up the slowly diffusing CH4 molecules. Self diffusivities of CO2/CH4 and CH4/H2 mixtures were computed in ZIF-68 and ZIF-70 using NVT-MD simulations.(Liu et al., 2011) Results

1

CO2/H2:1/99.(Keskin, 2011a)

10

Diffusion selectivity

100

indicated that diffusion of CH4 is increased with increasing concentration of H2 in the CH4/H2 mixture, while the diffusivity of H2 decreases with increasing CH4 concentration. In contrast, the diffusivity of CH4 was essentially independent of the concentration of CO2 in the CO2/CH4 mixture, while CO2 diffusivity decreases with increased CH4 loading, even though the diffusivity of CH4 is substantially larger than that of CO2. This unusual behavior was explained in terms of differences in adsorption site preferences due to chargequadrupole interactions. Another recent MD study examined the self diffusivities of equimolar CO2/ethane, CH4/ethane and CO2/methanol mixtures in Zn(tbip) including the flexibility effects.(Seehamart et al., 2011) Similar to previous observations, faster diffusing molecules accelerate the slower diffusing molecules whereas the slower ones slow down the faster ones through the channel of Zn(tbip).
