**5. Comparison of diffusivities from molecular dynamics with theories**

The prediction of transport properties in chemical mixtures from data taken from single component studies has been a long standing goal in describing mass transport in nanoporous materials. The validation of methods for this task can have great practical significance, but this type of validation can only be considered when high quality mixture diffusion data is available from MD. This section will present the validity of theoretical correlations by comparing their predictions for self diffusivities and Fickian diffusivities with the ones derived from MD. In the literature, Krishna-Paschek's (KP) correlation (Krishna&Paschek, 2002) and Skoulidas, Sholl and Krishna (SSK) correlation(Skoulidas et al., 2003) have been widely used to predict the self diffusivities and Fickian diffusivities of a binary gas mixture, respectively.

Keskin and Sholl studied MOF-5 as a membrane for separation of CO2/CH4 mixtures.(Keskin&Sholl, 2007) As discussed in previous sections, in order to study transport

thermostat.(Frenkel&Smit, 2002) Simulated data was in a good agreement with the experimentally measured data for both MILs. Experiments measured a diffusivity of 9×10-8 m2/s (1.65×10-7 m2/s) and simulations predicted 4.5×10-8 m2/s (1.5×10-7 m2/s) at a loading of 0.5 H2 molecules per unit cell of MIL-53(Cr) (MIL-47(V)). In a similar study, QENS measurements were combined with MD simulations in NVT ensemble using either Berendsen or Evans thermostat to determine the self diffusivity of H2 in the same MILs.(Salles et al., 2008) Two different force fields, spherical one site model(Frost et al., 2006) and explicit two atoms model(Yang&Zhong, 2005) were used in MD simulations of H2. Comparisons between QENS data and MD simulations clearly showed that the two force fields lead to very similar diffusivity values that produce the experimental value. This observation suggests that H2 diffusion is not significantly affected by the potential model. A combination of MD and QENS measurements were used to examine the diffusivity of water in MIL-53(Cr).(Salles et al., 2011) The breathing of this MOF upon water adsorption induces a structural transition between narrow pore (NP) and large pore (LP) forms. The self diffusivity of water was faster in LP form (8×10-10 m2/s) compared to the one in NP form (2.5×10-11 m2/s) since the confinement degree was much higher in NP structure. As an extension of this work, self, corrected and transport diffusivities of CO2 in MIL-47(V) were determined using MD and QENS.(Salles et al., 2010) While self and corrected diffusivities exhibited a decreasing profile with increased loading as expected, transport diffusivity presented an unexpected trend with a decrease at low loadings. This behavior was attributed to the unusual evolution of thermodynamic correction factor. This work was a good example of probing the transport

Two experiments studied diffusion of alkanes in MOFs: The diffusivity of n-butane, isobutane, 2-methylbutane and 2,2-dimethylpropane in CuBTC was investigated using infrared microscopy and MD simulations.(Chmelik et al., 2009) In another work, intracrystalline self diffusivities of propane, propene, n-butane, 1-butene, n-pentane and n-hexane in CuBTC were assessed using PFG-NMR and MD simulations.(Wehring et al., 2010) For the nalkanes, measured diffusivities within the experimental uncertainty agreed with the values from the MD simulations. The different trends observed in diffusivities of alkanes remained

**5. Comparison of diffusivities from molecular dynamics with theories** 

The prediction of transport properties in chemical mixtures from data taken from single component studies has been a long standing goal in describing mass transport in nanoporous materials. The validation of methods for this task can have great practical significance, but this type of validation can only be considered when high quality mixture diffusion data is available from MD. This section will present the validity of theoretical correlations by comparing their predictions for self diffusivities and Fickian diffusivities with the ones derived from MD. In the literature, Krishna-Paschek's (KP) correlation (Krishna&Paschek, 2002) and Skoulidas, Sholl and Krishna (SSK) correlation(Skoulidas et al., 2003) have been widely used to predict the self diffusivities and Fickian diffusivities of a

Keskin and Sholl studied MOF-5 as a membrane for separation of CO2/CH4 mixtures.(Keskin&Sholl, 2007) As discussed in previous sections, in order to study transport

diffusivity of gases in MOFs by combining MD and QENS.

as an unsolved issue.

binary gas mixture, respectively.

of gas mixtures in membranes mixture diffusivity data is required. However, at the time of that study there was no binary diffusion data available for MOF-5. Keskin and Sholl applied the SSK approach to quantify mixture diffusion of CO2/CH4 in MOF-5. This approach combines information from the loading dependence of the single component self diffusivities and corrected diffusivities (computed from MD simulations) with the binary adsorption isotherms (computed from GCMC simulations) to predict the loading and composition dependent matrix of binary diffusion coefficients. The SSK approach defines the mixture diffusivities for all loadings and compositions, an important feature of any description that will be used in examining a wide range of potential membrane operating conditions. Prior tests of this method by comparison with detailed atomic simulations of binary diffusion in silica zeolites and carbon nanotubes indicated that this approach is accurate for a wide variety of adsorbed mixtures.(Sholl, 2006)

A year later, Keskin and coworkers presented the validity of SSK approach in a MOF.(Keskin et al., 2008) They examined both KP and SSK approaches by comparing predictions of these methods with the results of MD simulations for mixture transport of H2/CH4 in CuBTC. In order to use SSK correlation, continuous functions describing the pure component self and corrected diffusivities were required. The self and corrected diffusivities of each species in H2/CH4 mixture were calculated by MD simulations. Based on these single component diffusivities, the SSK approach predicted the Fickian diffusivities. Mixture MD simulations in a Nosé-Hoover thermostat in the NVT ensemble calculated Onsager coefficients (Equation 4) for H2/CH4 mixture and these values were converted to Fickian diffusivities (Equations 5 and 6). The predictions of the SSK approach for the Fickian diffusivities were in good agreement with the direct MD simulations of binary diffusion, suggesting that this approach may be a powerful one for examining multicomponent diffusion in MOFs. Mixture self diffusivities were predicted using KP correlations based on single component self diffusivities, corrected diffusivities and fractional loadings. Comparison between KP predictions and mixture MD simulations were also found to be in a good agreement. The SSK approach was also used to obtain Fickian diffusivities of CH4/H2, N2/H2, N2/CH4, CO2/H2, CO2/N2 mixtures in MOF-5 and CH4/H2, CO2/CH4 in CuBTC.(Keskin&Sholl, 2009a; Keskin et al., 2009a)

Babarao and Jiang calculated self diffusivities of CH4 and CO2 in IRMOF-1 as a function of total loading based on the adsorption of an equimolar mixture using MD simulations and compared their results with the predictions of KP correlation.(Babarao&Jiang, 2008) Theory predictions were found to be in a fairly good agreement with MD simulations particularly for CH4 diffusivity in IRMOF-1 whereas the CO2 diffusivity was slightly overestimated by the theory. No certain reasoning was given for this overestimation. The predictions of KP correlations for mixture self diffusivities of CH4 and H2 were in reasonable agreement with the results of MD simulations for ZIF-68 and ZIF-70. (Liu et al., 2011)
