Understanding Diffusion in Hierarchical Zeolites with House-of-Cards Nanosheets Peng Bai,† Emmanuel Haldoupis,‡ Paul J. Dauenhauer,† Michael Tsapatsis,† and J. Ilja Siepmann*,†,‡ †
Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue SE, Minneapolis, Minnesota 55455, United States ‡ Department of Chemistry and Chemical Theory Center, University of Minnesota, 207 Pleasant Street SE, Minneapolis, Minnesota 55455, United States S Supporting Information *
ABSTRACT: Introducing mesoporosity to conventional microporous sorbents or catalysts is often proposed as a solution to enhance their mass transport rates. Here, we show that diffusion in these hierarchical materials is more complex and exhibits non-monotonic dependence on sorbate loading. Our atomistic simulations of n-hexane in a model system containing microporous nanosheets and mesopore channels indicate that diffusivity can be smaller than in a conventional zeolite with the same micropore structure, and this observation holds true even if we confine the analysis to molecules completely inside the microporous nanosheets. Only at high sorbate loadings or elevated temperatures, when the mesopores begin to be sufficiently populated, does the overall diffusion in the hierarchical material exceed that in conventional microporous zeolites. Our model system is free of structural defects, such as pore blocking or surface disorder, that are typically invoked to explain slower-than-expected diffusion phenomena in experimental measurements. Examination of free energy profiles and visualization of molecular diffusion pathways demonstrates that the large free energy cost (mostly enthalpic in origin) for escaping from the microporous region into the mesopores leads to more tortuous diffusion paths and causes this unusual transport behavior in hierarchical nanoporous materials. This knowledge allows us to re-examine zero-length-column chromatography data and show that these experimental measurements are consistent with the simulation data when the crystallite size instead of the nanosheet thickness is used for the nominal diffusional length. KEYWORDS: diffusion, hierarchical materials, porous materials, zeolite, alkane, molecular simulation
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microporous regions have diffusivities much lower than would be expected on the basis of their structures.7−9 When analyzing hybrid systems with regions of distinct diffusion mechanisms, a two-region approach is often employed to model observed phenomena. In this approach, the overall diffusivity is simply taken to be the sum of contributions from the different regions. Very good agreement with measured data has been reported, where the Knudsen or kinetic gas expression is used for mesopore diffusion, and a separate, Arrhenius expression for the other region, such as the liquid layer on the mesopore surface of a Vycor glass10 or the micropores of a hierarchical porous material.11−14 On the basis of this type of analysis, the diffusivity in the microporous domains of two hierarchical zeolites was deduced to be orders of magnitude smaller than that in a conventional zeolite with the same pore structure.7,8
rystalline aluminosilicates known as zeolites are one of the most important classes of industrial catalysts used today. Zeolites possess uniform internal pores of molecular dimensions, allowing them to carry out a wide range of highly selective chemical transformations. The well-defined structural dimensions of these and related nanoporous materials also make them natural model systems that are widely studied to understand the effects of confinement on adsorption and diffusion.1 The sub-2 nm micropores and channels can selectively stabilize adsorbed species and transition states but at the same time can also present severe transport limitations for the reactant and product molecules. Introducing mesoporosity (2−50 nm) to the microporous materials, as a fast channel for diffusion, is being actively explored as a promising method to alleviate the mass transfer problem while preserving the catalytic activity/selectivity and hydrothermal stability of the original zeolites.2−4 It is tempting to assume that transport of molecules will take the path of least resistance,5,6 as in a network of electrical conductors with competing pathways of different resistances. However, recent studies using zero-length-column (ZLC) chromatography suggested that materials with very small length scales of © 2016 American Chemical Society
Received: April 29, 2016 Accepted: August 4, 2016 Published: August 4, 2016 7612
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ACS Nano Such dramatic effects, which depend critically on the sample size and morphology,15−17 are often attributed to additional transport resistances between crystalline domains that manifest only in experimental techniques probing sufficient length scales. It is believed that structural imperfections such as twinning, prevalent in the popular MFI-type zeolites, and pore blocking can lead to enhanced surface barriers. For the single-file Zn(tbip) system, Heinke et al.18,19 estimated the fraction of open-pore entrances to be only 1 out of 2000. Similarly, Teixeira et al.9 estimated the fraction of open pores in MFI materials to be less than 1 out of 1000 in order to explain ZLC chromatography data. With regard to membrane permeation, Kocirik et al.20 suggested that the potential energy differences between gasphase, crystal surface and its interior can already give rise to a surface resistance. Arya et al.21 carried out molecular dynamics simulations for methane passing through 1-dimensional AlPO45 pores and presented a similar explanation for the surface barrier but based on free energy differences. Schuring22 derived a relationship between the entering probability of molecules adsorbed on the surface and their diffusion coefficient and equilibrium constant and proposed it as a way to decouple barriers arising from intrinsic energetic differences and those from structural defects. Similar energetic barriers have since been reported for other membrane systems as well.23−28 Although models based on reduced interface permeability were able to correlate experimental data,18,19,29 the exact origins of the extraordinary surface resistance necessary for these models remain unclear. Motivated by the interest to advance understanding of diffusion under confinement, especially in the context of a new class of hierarchical materials, we performed an atomistic simulation study of n-hexane in a hierarchical selfpillared pentasil (SPP) zeolite.30 Significant surface barriers were found in our model of a perfect crystalline structure, which leads to, under certain conditions, a reduction of diffusivity in the microporous domains and in the hierarchical material as a whole.
Figure 1. Calculated self-diffusion coefficients as a function of loading for n-hexane at T = 363 K. Data are shown for MFI, SPP, and only the microporous region of SPP. Note that Dmicro is given as a function of amount adsorbed in only the microporous region.
agreement with previous quasielastic neutron scattering (QENS) and pulsed field gradient (PFG) NMR studies,33,34 which reported values close to 10−9 m2/s at a loading of Q ≈ 0.7 mol/kg (1 molecule per intersection) with a relatively weak temperature dependence and an activation barrier of ∼5 kJ/ mol. We find that DSPP exceeds DMFI only at Q > 0.7 mol/kg. Interestingly, DSPP values exhibit a minimum at Q ≈ 0.4 mol/ kg. It should be emphasized that our SPP model system does not contain any surface disorder (beyond the rotational motion of the hydroxyl groups) or defects that would block entrance to the host framework. In fact, the calculations here treat the SPP zeolite as a single material, perfectly crystalline and with infinite extent, a bulk material possessing two types of channel systems. As loading decreases, the differences between self, collective, and transport diffusivities vanish, so this reduction in SPP is expected to hold for all three diffusivities. These observations are in stark contrast with the intuition that the large fraction of mesopore channels would necessarily lead to higher transport rates in the hierarchical material. We next characterize the anisotropy of the diffusion behavior in the two zeolites (see Figure 2). In MFI, the 1-D diffusivities along all three axes decrease with loadings, with DMFI‑b being the largest. Compared to the diffusion in zigzag channels, straight channels have a slightly lower activation barrier (see below) and allow for multiple correlated jumps due to the conservation of momentum. The diffusion along the c-direction can only occur through alternating movements in these two channel systems and is therefore the slowest. Similarly, in the low-loading regime, the three components of DSPP follow the same ranking as in MFI, but their magnitudes are smaller. For Q ≈ 0.1 mol/kg, the one-dimensional self-diffusion coefficients in both MFI and SPP zeolites yield Db ≈ 5Da ≈ 20Dc. DSPP‑a and DSPP‑b exhibit a minimum as loading increases before crossing over and exceeding the corresponding 1-D diffusivities for MFI (i.e., the presence of the mesopore channel also enhances diffusion in the directions perpendicular to its long axis). At Q > 0.8 mol/kg, DSPP‑c becomes the largest
RESULTS AND DISCUSSION Construction of the Hierarchical SPP Zeolite. Our model SPP zeolite is composed of single-unit-cell thick (2 nm) microporous nanosheets stacked together perpendicularly to form mesopore channels with a side length of 6 nm. The microporous nanosheets have the MFI structure,31 which is characterized by intersecting channels of 10-oxygen rings, both with a diameter of ∼5.5 Å. The straight channels align with the crystallographic b-direction, while the zigzag channels run along the a-direction in the ac plane. The model SPP system is constructed by cutting along the pentasil chains of a 4 × 4 × 3 MFI super cell and terminating all dangling O atoms with H atoms (i.e., as hydroxyl groups), resulting in a structure with mesopore volume fraction of about 0.55. Previously, it was found that this model SPP system allows one to capture accurately the unusual adsorption behavior of alkanes.32 Comparison of Conventional and Hierarchical Zeolites. We calculate self-diffusion coefficients, D, using equilibrium molecular dynamics (MD) simulations via the Einstein relation (see the Methods). The data for n-hexane in hierarchical SPP and conventional MFI zeolites at T = 363 K are compared in Figure 1 (numerical data are provided in the Supporting Information). Computed DMFI values range from 0.14 to 3.5 × 10−9 m2/s and show an exponential decay with increasing pore filling. These simulation data are in excellent 7613
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−33 kJ/mol. The PMFs continue to rise and reach values near zero about 15 Å away from the nanosheet surfaces. The third type of PMF profiles correspond to motion at the nanosheet surfaces. The partially open intersections at the surfaces are elevated in free energy to about −27 kJ/mol, and the corner sites have intermediate values around −32 kJ/mol. However, these PMFs are very rugged, and surface sites are separated by barriers of about 18 kJ/mol. For comparison, the zero-coverage enthalpy of adsorption in both MFI and SPP materials is −65 kJ/mol,32 which is a factor of 5/3 larger in magnitude than the value of the free energy minimum. The larger free energy costs for partial desorption and diffusion along the surface lead to a more tortuous diffusional path in the microporous region of SPP that results in DSPP < DMFI for Q < 0.7 mol/kg. As illustrated in Figure 4, at the lowest loading (Q = 0.097 mol/kg), diffusion proceeds almost entirely within the microporous region but with the sorbate molecule exploring nearly every side channel leading to a pore mouth. On only a few occasions does the sorbate molecule slither along the mesopore surface to reach a neighboring pore entrance or travel across the mesopore. At Q = 0.097 mol/kg, we find an average success rate of less than 10−4 for jumps leading to desorption to the mesopore surface or mesopore interior. This low probability is consistent with the PMF profiles indicating that the diffusion barrier within the micropore interior is smaller by ∼25 and ∼33 kJ/mol than for jumps along the mesopore surface or to the mesopore interior. Thus, it is clear that diffusion in SPP-type hierarchical materials at low loading is dominated by micropore diffusion and the appropriate diffusional length is the dimension of the crystalline particle and not the nanosheet thickness. Single-molecule trajectories at intermediate and high loadings are also shown in Figure 4. At Q = 0.388 mol/kg, the molecule still mostly diffuses within the microporous region. However, the fraction of time spent in the mesopore region is about three times larger than observed for Q = 0.097 mol/kg and motion within the mesoporous region makes a significant contribution to the overall diffusion. While the sorbate resides in the mesopore, it either creeps along the mesopore surface or, less frequently, flies across the mesopore interior to another surface where it immediately adsorbs again, i.e., its behavior in the mesopore does not conform with the Knudsen mechanism. At the highest loading, the single-
Figure 2. Calculated self-diffusion coefficients along the a (left, zigzag channel), b (middle, straight channel), and c (right, mesopore long axis) crystallographic directions as a function of loading for n-hexane in zeolites MFI (magenta circles) and SPP (blue triangles) at T = 363 K.
component, as c is the direction of the mesopore channels that is not interrupted by microporous layers. Free Energy Profiles and Single-Molecule Trajectories. Local free energy profiles can provide quantitative insights into the dynamic behavior of adsorbed molecules. Here, we focus on the potentials of mean force (PMF) along the zigzag and straight channels. We observe three types of PMF profiles for the hierarchical SPP zeolite (see Figure 3). For channels that are completely enclosed within the nanosheets, the PMFs exhibit similar variations as in the conventional MFI zeolite. The minima with a PMF value of −39 kJ/mol are found at the location of the channel intersections. Away from the intersections, the PMFs along the straight channel (located at x = 30.0 Å) and the zigzag channel (located at y = 24.9 Å) fluctuate around −34 and −32 kJ/mol; i.e., the diffusive barrier is slightly larger along the zigzag channel and the value of ∼5 kJ/mol for diffusion along the straight channel agrees with the experimental data for MFI.33,34 The PMFs away from the junctions of stacked MFI nanosheets also exhibit pronounced minima of about −39 kJ/mol at the sheet centers, corresponding to the channel intersections. Another pair of adsorption sites exist within the straight and zigzag channels and directly connect to the pore openings with PMF values of
Figure 3. Potentials of mean force as a function of x (left) and y (right) coordinates for n-hexane in the hierarchical SPP zeolite at zeroloading and T = 363 K. The lines are colored according to the other coordinate, y (left) and x (right), as indicated in the middle panel. 7614
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Figure 4. Illustration of the diffusive pathway taken by a specific n-hexane molecule in the SPP zeolite at Q = 0.097 (left), 0.388 (middle), and 1.55 mol/kg (right) at T = 363 K. The black, cyan, and magenta colors indicate parts of the trajectory with the sorbate in the interior channels of the microporous region, in channels leading to a pore mouth, and in the mesoporous region, respectively. These specific molecules are picked because their fraction of time spent in the microporous region is close to the corresponding ensemble averaged xmicro.
estimate of Dmeso from the simulation data indicates that the kinetic gas model and the Knudsen model overpredict Dmeso by factors of ∼20000 and ∼20, respectively. As mentioned above, the creep along mesopore walls and the capture on pore walls after motion across the mesopore that are observed in the single-molecule trajectories are incompatible with the kinetic gas and Knudsen models.43 The simulations demonstrate that the framework−sorbate interactions strongly modify the behavior of n-hexane molecules in the mesopores and that the MFI nanosheets are not mere spectators of the dynamic behavior in the mesopore channels surrounding them. Temperature and Loading Effects. The effects of temperature on DSPP and DMFI are illustrated in Figure 5. For Q < 0.8 mol/kg, the data for MFI exhibit only a very weak
molecule trajectory indicates (i) a few small connected sections where the sorbate moves within the micropore region, (ii) numerous locations where the sorbate transverses a nanosheet to move from one mesopore to the next, (iii) motion along the mesopore walls, and (iv) flights across the mesopore interior. Micropore and Mesopore Contributions to Diffusion. Diffusion in hierarchical materials is sometimes described in terms of a weighted linear combination based on contributions of the microporous and mesoporous regions:10−14 DSPP = xmicroDmicro + (1 − xmicro)Dmeso, where xmicro is the average fraction of molecules in the microporous region. The boundary between micro- and mesoporous domains is defined here by the planes consisting of the surface hydroxyl oxygen atoms, and an n-hexane molecule is considered to be in the MFI nanosheets when its center-of-mass is inside the boundary. At T = 363 K, xmicro ranges from 99.86 to 68.11% for the lowest and highest loadings considered here. Dmicro is only reported for Q < 0.4 mol/kg where residence times in the microporous region are sufficiently long (see the Methods and Figure 4). Figure 1 shows that Dmicro is smaller than DMFI by factors of 2.7 ± 0.8 and 6.3 ± 1.6 at Q = 0.10 and 0.37 mol/kg, respectively. Clearly, the assumption that Dmicro ≈ DMFI does not hold for the SPP material. Diffusivity in mesopores is often estimated as gas transport in either the molecular (kinetic gas) or Knudsen regimes. The first assumes that the mean free path between molecular collisions is shorter than the container size, and we calculate it by separate MD simulations in the canonical ensemble of pure gas-phase nhexane at the same density as found within SPP’s mesopore channels. In Knudsen diffusion, molecule−wall collisions dominate, and therefore, its value depends on pore width, dpore. For cylindrical pores with smooth walls, the Knudsen diffusion is given by DKnudsen = dpore(8kBT/πMA)1/2/3, where MA is the molecular mass of the diffusing species. However, this model is too simplistic to describe the diffusion in actual mesopores, and the prefactor is sometimes adjusted to account for tortuosity of the mesopore network, different mesopore cross-section, and/or ruggedness of the walls.7,8,10−14,35−43 Applying the two-zone diffusion description to obtain an
Figure 5. Temperature dependence of the calculated self-diffusion coefficients as a function of loading for n-hexane in SPP (filled symbols connected by dashed lines) and MFI (open symbols connected by dotted lines). For clarity, the MFI data are shown only at T = 363 and 543 K. 7615
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host−guest interactions. When thermodynamics dictates that sorbate molecules are essentially absent from the mesopore channels, then the potentially high mobility afforded by these transport “highways” does not play a role and the more tortuous diffusion pathway in the microporous region can instead lead to a reduction in the diffusivity. Mehlhorn et al.12 showed that increased tortuosity can also be achieved by saturating the mesopores with large coadsorbing molecules that cannot enter the micropores, thus creating “forbidden” regions of space and leading to decreased diffusion rates in hierarchical materials. At high loading and/or temperature, a significant fraction of sorbate molecules populates the mesoporous region, and diffusion in a hierarchical material by far exceeds that of the corresponding microporous material. The faster diffusivity in the mesoporous region and the short pathways to reach adsorption/catalytic sites are the reasons that hierarchical materials show faster uptake and improved catalyst utilization.30,45−47 Our results may also prove useful in designing processes for hierarchical sorbents with optimized operating conditions (e.g., loading and temperature).
temperature dependence in agreement with QENS and PFG NMR measurements.33,34 For the highest loading (Q = 1.37 mol/kg), raising the temperature from 363 to 543 K leads to an increase of DMFI by only a factor of 2.6 ± 0.4. In contrast, the diffusion data for the SPP zeolite show a more pronounced temperature dependence with increases by factors of 12 ± 3, 46 ± 10, and 3.3 ± 0.4 at Q = 0.10, 0.39, and 1.55 mol/kg, respectively, as the temperature is increased from 363 to 543 K. Furthermore, the minimum in DSPP found near 0.4 mol/kg at T = 363 K shifts to lower loading at 423 K and vanishes at the two higher temperatures. In addition, DSPP exceeds DMFI at all Q values for T = 483 and 543 K. The reason for the pronounced temperature dependence in the hierarchical material is that, at a given loading, xmicro decreases with increasing temperature32 and molecules spend more time in the mesoporous region where they can diffuse faster. The temperature dependence of DSPP therefore shows an unusually large activation energy (e.g., ΔEa ≈ 28 kJ/mol for Q = 0.194 mol/kg) that falls in between the activation free energy for micropore diffusion of ∼5 kJ/mol and the magnitude of the PMF minimum of −39 kJ/mol. Large ΔEa can also be inferred from the experimental data of Mehlhorn et al.12,13 and Chang et al.7 and are also consistent with the temperature dependence of surface resistance reported previously for membrane systems.21,23,27 As loading in SPP is increased, the PMF profiles become more compressed;32 that is, the free energy differences between channel intersections, nanosheet surfaces, and mesopore interior are reduced. As surface regions and mesopores start filling, the exchange between micro- and mesopores becomes more facile and the channels leading to pore mouths are no longer “sinks” of long residence times. Similarly, as temperature is increased, the entropic cost of confinement in the micropores and to a lesser extent at the nanosheet surfaces increases and more molecules are found in the mesoporous region. Comparison to Chromatographic Measurements. With the understanding that diffusion in SPP-type hierarchical materials at low loading is dominated by motion in the microporous region and that the dimension of the crystalline particle is the appropriate nominal diffusional length, we can resolve the puzzling situation observed in the ZLC chromatography experiments that also probe diffusion in the zero-loading limit.7 Using 75 and 176 nm as the nominal diffusional length for the SPP and MFI particles used in the experiments,7 yields DMFI/DSPP = 1.4 for cyclohexane at T = 363 K that is in remarkable agreement with the ratio of 2.7 ± 0.8 observed in the simulations for n-hexane, whereas using the nanosheet thickness as the nominal diffusion length would yield DMFI/ DSPP = 12000.7 This comparison suggests that diffusion of cyclohexane in SPP also follows a tortuous pathway in the microporous region. The simulation data obtained for a perfect hierarchical system with infinitely long mesopore channels demonstrate that a lack of enhanced diffusivity requires neither (physical) pore blockage nor a lack of mesopore connectivity.44 The decrease in DMFI/DSPP with increasing temperature is also consistent with the ZLC data.7
METHODS The n-hexane sorbate molecules are modeled using the united-atom version of the TraPPE force field48 and consist of six connected CHx segments with fixed bond lengths but flexible bending and dihedral angles. The inorganic SiO2 framework is treated assuming a rigid interior but with flexible surface hydroxyl groups (only for the SPP material), and framework−sorbate interactions are described by the TraPPE-zeo force field.49 The molecular dynamics calculations were performed using the GROMACS package, version 5.0.4,50,51 starting from equilibrium configurations obtained by previous Monte Carlo simulations32 with initial velocities drawn randomly from the Maxwell−Boltzmann distribution. The simulations were equilibrated for 1 ns, followed by production periods of 0.5−1 μs. The uncertainties reported are 95% confidence intervals estimated by slicing the trajectories into 0.1 μs segments. The velocity Verlet algorithm52 was used for integrating the equations of motion, with a time step of 1 fs. The temperature was maintained using a Nose−Hoover thermostat53,54 with τt = 100 fs. Fixed bond lengths were realized using the LINCS algorithm.55 All pair interactions were switched to zero over 11.5 Å ≤ rij ≤ 12 Å. Diffusion coefficients were calculated using the Einstein relation.56 For the calculations of DMFI and DSPP, the entire production periods were used to obtain the mean-square-displacements (MSDs) as a function of time. Logarithmic plots of MSD versus time were used to find the linear region with a slope of unity to be used for the calculation of the diffusion coefficients, and these regions extended from 1 to >30 ns. For the calculation of Dmicro, only the parts during which the diffusing molecules reside in the microporous region were used (where the boundary between micro- and mesoporous domains is defined here by the planes consisting of the surface hydroxyl oxygen atoms). Due to transfers between the regions, the linear MSD region was found to extend only to (4 nm)2 (i.e., the square of twice the thickness of the nanosheets). To obtain PMFs along the zigzag and straight channels, we computed local free energies (i.e., excess chemical potentials) for evenly divided rectangular cuboids (extended along the z-axis) throughout the simulation box using the Widom insertion method and configurational-bias Monte Carlo algorithm.57,58 The PMF is expressed as a function of the position of the second carbon atom, as the more intuitive choice of using the center of mass has been shown to underestimate free energy barriers.59
CONCLUSIONS Our simulations demonstrate the possibility of slower diffusion in a model SPP zeolite that contains a large mesopore fraction but no physical pore blockage. The phenomenon is, however, by no means limited to this specific system but is instead expected for any hierarchical material with sufficiently favorable 7616
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ASSOCIATED CONTENT S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.6b02856. Numerical values of the simulation data are provided in tabular form (PDF)
AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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