Simulations on âPowderâ Samples for Better Agreement with

Article Cite This: J. Phys. Chem. C 2019, 123, 16172−16178

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Simulations on “Powder” Samples for Better Agreement with Macroscopic Measurements Angela Mary Thomas and Yashonath Subramanian*

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Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India ABSTRACT: Different experimental measurement techniques give widely differing values for diffusivity (Ds) and activation energy (Ea) for the same sample. Values from molecular dynamics simulations agree with quasi-elastic neutron scattering measurements and not with those obtained from macroscopic techniques, such as uptake or zero length column chromatography. We report molecular dynamics simulations on different zeolite samples with intercrystalline space along 0-, 1-, 2-, and 3-dimensions. The results on nhexane in zeolite NaY samples with intercrystalline space along threedimensions exhibit values of Ds and Ea in excellent agreement with macroscopic techniques, demonstrating the importance of “powder” samples in simulations.

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would be the following: Which is the ‘correct’ value for use in modeling for industries? Kárger, Ruthven, and several others have suggested possible reasons for the different Ds values obtained from different experimental techniques.5−17 Uptake or chromatography techniques, which probe diffusion for longer time scales (and therefore, probably, longer length scales), usually yield lower values for self-diffusivity compared to techniques that measure Ds for shorter periods. This is also evident from Table 1. Several groups have reported investigations to get insights into the underlying reasons for these differences between diffusivities. Zhang et al. have investigated the effect of crystal size distribution on diffusion of light olefins in ZSM-5.13 Hashimoto and Yamashita measured contact-induced intercrystalline migration of aromatic molecules with the help of fluorescence microscopy.11 In spite of several such investigations, a complete understanding of the processes at the microscopic level is lacking. Recently, a molecular dynamics simulation of xenon in faujasite was reported on a purely intracrystalline system without any intercrystalline space as well as a system with intercrystalline space along only one of the axes.18 The values of Ds were lower for the latter, whereas the values of activation energies for diffusion, Ea, were higher. The basic idea is that the experimental sample is always a “powder” sample and never a single crystal. Is diffusion in such a powder sample very different from what one sees in a single crystal? If so, in what way? Also, we shall attempt to answer the following question: Can MD simulations ever provide estimates of Ds and Ea comparable to the experimentally measured values obtained from macroscopic measurement techniques, such as uptake or chromatography?

INTRODUCTION Micro- and mesoporous materials are extensively used in chemical and consumer industries. In particular, petroleum refineries employ these substances for purification, extraction, refining, and separation of multicomponent mixtures. A knowledge of diffusivity of various components of the mixture is sine qua non for all above applications. Various experimental techniques, such as uptake, zero length column chromatography (ZLC), pulsed field gradient-NMR (PFG-NMR), quasielastic neutron scattering (QENS), etc., are employed to measure the self-diffusivity (Ds) of guest molecules in these substances. Computational methods, such as molecular dynamics (MD), Brownian dynamics, and dissipative particle dynamics, have also been employed to compute diffusivities.1−4 Unfortunately, different experimental techniques yield very different values for diffusivity of guests for the same sample under the same conditions (temperature and pressure). In Table 1, we list Ds values for the same system (n-hexane in silicalite) measured using different experimental techniques. The values vary over seven orders of magnitude. This is a typical case and such behavior is seen across all guest species and different zeolitic systems. This makes it difficult for a chemical engineer to actually predict the behavior on an industrial scale using these Ds values. In fact, the question Table 1. Diffusivity Values of n-Hexane in Silicalite with a MFI Structure Measured Using Different Experimental and Computational Techniques30−34 method

time scale

temperature (K)

uptake ZLC PFG-NMR QENS MD

>s >s 10−100 ms ps to ns fs to ns

298 303 298 298 314

Ds (m2/s) 7.5 5.3 5.0 4.5 1.42

© 2019 American Chemical Society

× × × × ×

10−16 10−13 10−11 10−10 10−9

Received: March 19, 2019 Revised: May 27, 2019 Published: June 2, 2019 16172

DOI: 10.1021/acs.jpcc.9b02599 J. Phys. Chem. C 2019, 123, 16172−16178

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The Journal of Physical Chemistry C

Figure 1. Various types of zeolite NaY systems under study.

crystallites having 2 × 2 × 4 unit cells of zeolite each. The simulation cell, thus, is of size 179.4144 × 179.4144 × 99.4144 Å3. (Figure 1c) (iv) 3d systemhere, intercrystalline regions are present along the crystallographic x-, y-, and z-directions after each crystallite of size 2 × 2 × 2 unit cells (comprising eight such crystallites). The simulation cell length is 179.4144 Å along all three directions. (Figure 1d) When the intercrystalline space was introduced, there was no addition of hydroxyl groups to satisfy the unsaturated silicon atoms at the boundaries. All simulations are performed using the LAMMPS package.20 The simulations are equilibrated for 5 ns, which is followed by a run of 20 ns during which positions, velocities, and forces are stored for later analysis. Zeolite atom coordinates are excluded from the MD integration throughout the simulations. Intramolecular interactions for the nC6 molecules are modeled using optimized potentials for liquid simulations (OPLS) parameters.21 Intermolecular nonbonded interactions between the zeolite atoms (O and Na) and the guest atoms are included using a Lennard-Jones potential, with self-parameters for zeolite atoms taken from Wender et al. and for guest atoms from the OPLS force field.21,22 A spherical cutoff of 12 Å is applied for these interactions. Periodic boundary conditions (PBCs) are imposed along the three directions. The total energy changes are less than 1 in 104 to ensure good energy conservation. Positions, velocities, and forces are stored at an interval of 1 ps for the calculation of properties, although here the latter two have not been used. As discussed above, the Ds values for different experimental techniques can vary over several orders of magnitude for the same hydrocarbon−zeolite system. It has been suggested by Kárger, Ruthven, and several others that this is due to

We report extensive molecular dynamics (MD) simulations of n-hexane in zeolite Y on crystalline and powder samples. The crystalline sample consists of (a) large single crystals. The powder simulations have been carried out with an intercrystalline region (b) in x-, (c) x- and y- as well as (d) x-, y-, and zdirections. The results are compared with each other and also with experimental measurements.

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METHODS Zeolite NaY with molecular formula Na56Si136Al56O384 is used as the host in the simulations. It belongs to the cubic space group, Fd3̅m, with a = 24.8536 Å. The atom coordinates in the unit cell provided by Fitch and co-workers are employed in the simulations.19 Each unit cell of this zeolite has eight α-cages or super-cages with a radius of ∼5.8 Å. These super-cages are connected tetrahedrally via 12-membered windows of free diameter ∼7.0 Å. There are in all 56 sodium ions as extraframework cations, which are distributed over the SI, SI′, and SII sites in each unit cell.19 The simulation cell is composed of 4 × 4 × 4 unit cells of zeolite Y; 128 n-hexane molecules (denoted as nC6 hereafter, 2 per unit cell) are introduced as guest molecules inside this zeolite host. Four different systems have been simulated in this study: (see Figure 1) (i) 0d systema purely intracrystalline system with no intercrystalline region. The simulation cell length along each direction is uniform, 99.4144 Å with 4 × 4 × 4 unit cells of zeolite Y. (Figure 1a) (ii) 1d systeman intercrystalline region of 40 Å is introduced in the x-direction after each of the two crystallites of size 2 × 4 × 4 unit cells. The resulting simulation cell dimensions are 179.4144 × 99.4144 × 99.4144 Å3. (see Figure 1b) (iii) 2d systemthe intercrystalline regions are present along both x- and y-axes between four 16173


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Figure 2. Density distribution, ρ, and potential of mean force (W) of the center of mass (COM) of n-hexane along the three crystallographic directions. The intercrystalline (indicated by IC) and crystalline (C) regions are shown with the vertical dashed lines indicating the boundaries. These are shown for the three systems investigated (1d, 2d, and 3d) along the (a) x-axis, (b) y-axis, and (c) z-axis at 200 K.

Figure 3. Density distribution, ρ, and potential of mean force (W) of the COM of n-hexane along the three crystallographic directions. The intercrystalline (indicated by IC) and crystalline (C) regions are shown with the vertical dashed lines indicating the boundaries. These are shown for the three systems investigated (1d, 2d, and 3d) along the (a) x-axis, (b) y-axis, and (c) z-axis at 800 K.

intercrystalline region at 200 K. The PMF shows a significant barrier in free energy in the intercrystalline regions. Near the interface between the crystal and the intercrystalline region, the density is negligible up to around 3 Å within the crystal but increases gradually as one goes deep into the crystal. But the density reaches values close to the maximum only around 12 Å from the boundary. This is not surprising, since the interactions between the nC6 and the crystal is maximum only when all interactions exist. This happens only when the interactions have decayed to zero, which occurs around 12 Å. In Figure 3, the density and the PMF are shown at 800 K. The ρ(x) values along the x-direction for 1d, 2d, and 3d are clearly different. Recall that all three systems studied, namely, 1d, 2d, and 3d, have intercrystalline space along the x-axis.

differences in the length and time scale over which the probe measures diffusivities.5−14 To examine whether this is true, we carried out long simulations for the four systems discussed above, and the results are presented below.

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RESULTS AND DISCUSSION In Figure 2a, the plot of density distribution, ρ, and potential of mean force (PMF, W = −ln RTρ) at 200 K are plotted along the crystallographic x-axis by averaging over y- and zdirections. Similar plots are shown along y- and z-directions averaged over the other two directions in Figure 2b,c. It is seen that there is only a slight difference between different curves belonging to the 1d, 2d, and 3d systems. The density is higher in the intracrystalline region and essentially zero in the 16174


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Figure 4. Mean square displacement of COM of nC6 in zeolite Y without intercrystalline space (0d) and with intercrystalline space along the x-axis (1d), along the x- and y-axes (2d) and along the x-, y-, and z-axes (3d) at temperatures (a) 200 K, (b) 400 K, and (c) 800 K calculated with the trajectories from 20 ns runs.

Table 2. Diffusivities of nC6 in Zeolite Y without Intercrystalline Space (0d) and with Intercrystalline Space along the x- (1d), along the x- and y- (2d), and along the x-, y-, and z-Axes (3d) from the Slopes of MSDs Obtained over the Time Range of 0−6 nsa Ds (×10−9 m2/s) T (K)

0d

1d

2d

3d

200 400 800

0.88(0.05) 9.23(0.25) 16.14(0.34)

0.61(0.08) 7.95(0.32) 4.37 × 102(30.1)

0.22(0.07) 12.48(1.79) 2.28 × 103(106.8)

0.09(0.03) 36.17(4.77) 7.72 × 103(439.6)

a

Simulations are done for 20 ns at 200, 400, and 800 K. Activation energies are calculated for the total diffusion from Arrhenius plots.

through the intercrystalline region (see W(x) in Figure 2). As a result, when the intercrystalline space is present along several directions, n-hexane is more and more confined, leading to reduced displacement. This explains why when we go from 1d to 3d, there is a reduction in the slope of the mean square displacements (MSDs). At a higher temperature (800 K), nhexane samples the intercrystalline region in addition to intracrystalline region, leading to a significant increase in density in the intercrystalline regions. Motion in the intercrystalline region is more facile than in the intracrystalline region, and the availability of the intercrystalline region increases with increased dimensionality. This leads to higher diffusivity with the increase in dimensionality of the system at 800 K. One of the consequences of this is the pronounced increase in the diffusivity with temperature in the higher dimensional systems (2d and 3d). This, as we shall see, leads to what is seen in the macroscopic measurements (where the guest molecules sample both the intracrystalline and intercrystalline space): a higher activation energy. Furthermore, it will be seen that the macroscopic measurements invariably report lower diffusivity (see Table 1). In fact, the longer the sampling duration of the experimental technique, the lower will be the diffusivity (at lower temperatures). This arises from the fact that the diffusing molecules typically sample the intracrystalline region and they, therefore, remain confined to the crystallites and are unable to leave these

Thus, even though all three systems are identical along the xdirection, the W(x) value is maximum in the intracrystalline space for 1d and minimum for 3d. In contrast, in the intercrystalline space, the population density of the 1d system is the least, whereas that of the 3d system is the highest. Similarly, in the y-direction, the population density of nC6 is the highest in the intercrystalline space in the 3d as compared to the 2d system. These differences are not seen at lower temperatures because the intercrystalline space in all three different powder samples studied is hardly populated. Instead, nC6 populates almost exclusively the intracrystalline space at lower temperatures. The density is almost uniform without variation within the intercrystalline space, whereas within the intracrystalline space, there are significant variations or oscillations. These may be attributed to variations in the interactions within the crystal. We have computed the time evolution of the mean square displacement (MSD) of the center of mass (COM) of nC6 over 6 ns for all four systems at three different temperatures, and these are shown in Figure 4. At 200 K, the distance traversed over 6 ns decreases with increase in dimensionality. In contrast, at 800 K, the distance traversed increases with increase in dimensionality. To understand these contrasting trends, we need to look at the earlier plots of density and PMF. At 200 K, there is little or no sampling of the intercrystalline region, since there is a large free energy barrier for passage 16175


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The Journal of Physical Chemistry C crystallites to sample the intercrystalline regions, leading to lower diffusivity of the diffusing or adsorbed molecules. This is because experimental measurements are made at lower temperatures, where the guest molecules sample largely the intracrystalline region and less frequently the intercrystalline region. In Table 2, the overall self-diffusivities, Ds, at three different temperatures for all four systems are listed. It is seen that at 200 K diffusivities are in the order Ds (0d) > Ds (1d) > Ds (2d) > Ds (3d). At 800 K, the order is reversed, namely, Ds (0d) < Ds (1d) < Ds (2d) < Ds (3d). From Table 2, it is seen that the diffusivity varies only by one order of magnitude between the 0d (0.88 × 10−9 m2/s) and 3d (0.09 × 10−9 m2/s) systems at 200 K. At 800 K, it is seen that the diffusivity between the 0d and 3d systems differs by several (between 2 and 3) orders of magnitude: 0d (16.14 × 10−9 m2/s) and 3d (7720.0 × 10−9 m2/s). These findings may be compared with the PFG-NMR investigation by Kárger and Spindler, who have made measurement of mean square displacements (MSDs) as a function of observation times.23 Thus, they could obtain diffusivities for short times, which correspond to the intracrystalline diffusivity or Ds for the 0d system of the present study, as well as MSDs from long observation times corresponding to motion both within the crystallites and the intercrystalline regions. They observed that while the Ds is higher at lower temperatures for intracrystalline motion as compared to that for intercrystalline motion, at higher temperatures, the Ds value is higher for intercrystalline motion as compared to that for intracrystalline motion. In other words, the present study confirms their finding and there is a one-toone correspondence between the experimental findings and the findings from our study here. The reasons for the rather large value of Ds at higher temperatures is the relatively large increase in the population in the intercrystalline space. Indeed, this has been pointed out in several experimental studies by Kárger and co-workers, who studied single-component adsorption, and Rittig et al., who investigated multicomponent mixtures.23−25 They proposed that the effective diffusivity, Deff, can be written as Deff = pinter Dinter

Figure 5. Activation energies obtained from MD simulations on the 0d and 3d systems obtained from self-diffusivity data between 200 and 800 K. The activation energy for the 3d system is comparable to that obtained from macroscopic measurement on powder samples.

Table 3. Activation Energies, Ea, Calculated for the Total Diffusion Using Arrhenius Relations Ea (kJ/mol) system

0d

1d

2d

3d

nC6 in NaY (this study) nC6 in NaY (gas chromatography)28 nC6 in NaX (PFG-NMR)27 nC6 in NaX (PFG-NMR)29 nC6 in MgA (uptake)35 nC6 in 5A (PFG, SE)36 nC6 in FER (ZLC)37

6.6

23.7

34.6

38.9

experiments 30.9 14.6 ± 2.5 40−75 42−50 28.1−27.6 32

nC6 in NaX27 (see Table 3). The PFG-NMR probe for milliseconds and the n-hexane molecules are likely to sample both intra- and intercrystalline regions but predominantly the intracrystalline region. Thus, the activation energy appears to be between 6.6 and 23.7 kJ/mol. Bobok et al. have reported Ds values at different temperatures with the help of gas chromatography from which we have obtained an Ea value of 30.9 kJ/mol for nC6 in NaY.28 Also listed are the values of Ea for nC6 in three other systems: MgA (uptake), 5A (PFG, SE), and FER (ZLC). Both these techniques probe for seconds to hours and, therefore, their values sample adequately the intercrystalline region even at low temperatures giving higher values for Ea. The Ea values for nC6 in these zeolites are in the range between 27 and 50 kJ/mol. Kárger and Volkmer found that the Ea value lies between 40 and 75 kJ/mol for n-hexane in NaX. Our estimate here, 38.9 kJ/mol, is in good agreement with their measurements.29 Thus, the simulations on the 3d systems appear to yield values of activation energy that are in the right range and appear to yield values close to those obtained from macroscopic measurements.

(1)

where pinter is the population in the intercrystalline region and Dinter is the diffusivity of the guest molecule in the intercrystalline region. The temperature dependence of Ds is of interest as they are obtainable from experimental data as well. Arrhenius plots of ln Ds vs 1/T were plotted, and activation energy Ea was derived for each system (see Figure 5). These are listed in Table 3. We have also listed Ea from different experimental measurements. The value of Ea for the 0d system consisting of purely intracrystalline motion is 6.6 kJ/mol. With the introduction of an intercrystalline region along the x-direction, the Ea value obtained from the total MSD increases to 23.7 kJ/mol. For the 2d system, one obtains an activation energy of 34.6 kJ/mol, which increases to 38.9 kJ/mol for the 3d system. Also listed in Table 3 are the values of activation energies from various experimental techniques obtained from the literature. Usually, QENS measurements, which probe for a short duration (ps to ns), will give values of activation energies that are close to values derived from MD.26 However, in the present case, there are no QENS measurements for n-hexane in zeolite NaY. They have reported a value of 14.6 kJ/mol for

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CONCLUSIONS The results on powder samples here provide several interesting insights into the diffusion in inhomogeneous systems. The variation of diffusivity within an inhomogeneous system over a certain temperature range is far wider than that in the homogeneous system (0d system). At low temperatures, the guest species preferentially populates the intracrystalline 16176


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chromatographic measurements are carried out. Such samples typically contain Avogadro number of molecules (∼1024). This is similar to the use of rather small sizes of the system (∼103− 106 atoms) on which simulation is carried out as compared to much larger number of atoms in experimental samples. The use of periodic boundary conditions (PBC) along the three perpendicular directions ensures the powder nature of the sample and excellent agreement in terms of the various properties computed. The present study suggests that the presence of the intercrystalline region by which the singlecrystal sample becomes a powder sample might be a significant advance in the computational approach or strategy akin to that achieved by the use of PBC. These results augur well for the future of computational approaches. It will be interesting to study the effect of the presence of intercrystalline regions on other properties, such as adsorption isotherms. We plan to carry out studies to compute other properties.

regions, whereas at sufficiently high temperatures, the intercrystalline regions are preferentially populated. Before we discuss some of the interesting results obtained in the present study, we like to emphasize the following. In Table 2, we see the diffusivities obtained from our simulations for the 0d to 3d systems listed. The 0d system has only an intracrystalline region, and the diffusivity computed for this system actually corresponds to short-time measurements by techniques such as normal MD on non-powder samples or QENS. Diffusivity obtained for the 3d systems really corresponds to long probe times by techniques such as uptake or ZLC. We have seen that (see Table 1) the diffusivity of the guest molecule, n-hexane, is lower when the experimental measurement technique probes the sample for longer periods. Thus, we see that uptake measurements, which can take seconds to hours to measure, give very low diffusivity values (of the order of 10−16). Other techniques such as ZLC or PFG-NMR have diffusivities that are of the order of 10−13 and 10−11, respectively. This trendwhere the longer the probe time, the lower the diffusivityis so well known that it is taken for granted. The present study, in contrast, suggests that the diffusivity obtained from a technique such as uptake can yield values that are greater than the value obtained from a technique such as PFG-NMR (see Table 2) at somewhat higher temperatures. This is evident from Table 2: both at 400 and 800 K, the values for the 3d system are greater than the values for the 0d system. As the former would correspond to macroscopic measurement techniques such as uptake or chromatography and the latter to QENS, it proves that this is the case at higher temperatures. The present investigation provides activation energies in good agreement with Ea values obtained from macroscopic measurement techniques. The present study uses very small crystallites (≈50 Å), whereas the crystal sizes used in experimental samples might be of the order of micrometers or 10 000 Å. This could lead to differences in the results, but we do not understand how this could alter the results as yet. Similarly, the magnitude or size of the intercrystalline region employed in the simulation is likely to be quite different from those found in the samples used in the experiments. We have no measure of the intercrystalline region in the experiment. Use of more appropriate sizes for the crystal and the intercrystalline region might lead to better agreement between the simulation and experimental Ds as well as Ea values. The results obtained here raise new questions. (a) How do the dynamics and the activation energy barriers depend on the crystal size? (b) What will be the effect of variation of the intercrystalline region on these properties? (c) What is the influence of a small crystallite that is not an integral number of unit cells? Note that previous studies by Liu and co-workers have actually quantified the influence of surface barriers on mass transfer.16 They have also investigated how channels of different sizes play different roles in diffusion and reaction. However, the introduction of intercrystalline regions within the system mimicking the powder sample appears to already provide a value for Ea that is quite close to the experimentally derived value even with modest system size and simulation duration. The dramatic improvement and better agreement with macroscopic measurements of the Ea values from the simulations suggest that it may not be necessary to use the actual sizes of the crystallites present in the actual laboratory sample of zeolites on which experiments such as uptake or

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Angela Mary Thomas: 0000-0002-3922-5237 Yashonath Subramanian: 0000-0003-1646-374X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Department of Science and Technology, New Delhi, and Nano-mission, DST, New Delhi, for support. The authors also thank the reviewers for their useful comments giving experimental references that agree with the results reported here, which has improved the discussion.

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REFERENCES

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