Diffusion Path Reversibility Confirms Symmetry of Surface Barriers

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Diffusion Path Reversibility Confirms Symmetry of Surface Barriers German Sastre, Jörg Kärger, and Douglas M. Ruthven J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b04528 • Publication Date (Web): 17 Jul 2019 Downloaded from pubs.acs.org on July 22, 2019

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The Journal of Physical Chemistry

Diffusion Path Reversibility Confirms Symmetry of Surface Barriers

German Sastre,a Jörg Kärger,b Douglas M. Ruthvenc *

a

Instituto de Tecnologia Quimica U.P.V.−C.S.I.C., Universidad Politecnica de Valencia, Avenida

Los Naranjos s/n, 46022 Valencia, Spain

b

Faculty of Physics and Earth Sciences, University of Leipzig, Linnéstraße 5, 04103 Leipzig,

Germany E-mail: [email protected]

c

Department of Chemical and Biological Engineering, University of Maine, Orono, Maine 04469,

United States E-mail: [email protected]

ACS Paragon Plus Environment

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ABSTRACT: The mass transfer resistance encountered by guest molecules entering or leaving a microporous host material (the “surface barrier”) is often comparable with or even greater than the internal diffusional resistance and therefore has a major effect on the overall rate of molecular uptake and release. However, direct measurement of surface resistance is difficult and this has impeded our understanding of this phenomenon. It has recently been suggested that surface resistance may be asymmetric - greater for a molecule leaving the pore than for one that is entering. To investigate this hypothesis we have carried out MD simulations with a model system, methane in silicalite with partial pore blocking at the external surface. The results confirm complete symmetry of the diffusion paths for adsorption and desorption and therefore of the surface resistance, in accordance with the Principle of Microscopic Reversibility.


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1. INTRODUCTION The advent of the Pulsed Field Gradient (PFG) technique of NMR1 enabled the first direct measurements of intra-crystalline diffusivities in zeolites2. In many cases, especially for small crystals, these diffusivities were found to exceed, by several orders of magnitude, the values estimated (assuming diffusion control) from molecular uptake and release rate measurements3,4. This discrepancy has been investigated over many years and, in many cases, can be explained by the intrusion of extra-crystalline resistances to mass and/or heat transfer. However, in some systems the discrepancy remains even when extra-crystalline resistances are eliminated, suggesting that, in these cases, the mass transfer rate is controlled by a transport resistance at the external surface of the individual zeolite crystallites. Speculation as to the nature of these “surface barriers”

5–7

has been followed by the development of experimental techniques and molecular

dynamics (MD) simulations8–13 to determine their magnitude. PFG NMR has proved to be particularly suitable for measurement of surface barriers since it is able to simultaneously determine (by “microscopic” measurements) the genuine intra-crystalline diffusivity and (by operating in the so-called “fast-tracer-desorption” mode5) the rate of tracer exchange within a bed of crystals14,15. With proper data analysis, information about the magnitude of surface barriers can also be derived from “macroscopic” measurements, including conventional uptake studies16,17, the Zero-Length-Column (ZLC) technique18–20, Frequency Response (FR) measurements21,22 and reaction yield analysis23. However, it is important to emphasize that in these techniques, as a prerequisite for discriminating between the influences of intra-crystalline diffusion and surface permeation data analysis is based on the assumption that, within the sample under study, the surface properties of the individual crystals are uniform. It was only with the introduction of the techniques of micro-imaging through interference microscopy (IFM) or IR


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microscopy (IRM)24 that transport resistances at individual crystal faces have become accessible to direct observation25,26. From such measurements it has been found that the surface permeance of even apparently identical crystals may vary substantially within the same sample. For example, for light hydrocarbons in zeolite SAPO-34, the permeance varied by more than two orders of magnitude27. The present stage of our knowledge leaves no doubt about the existence of surface barriers and their impact on mass transfer rates, especially in small microporous crystals which are widely used for miniaturization and performance enhancement28,29. Much effort has therefore been concentrated on the development of strategies for material synthesis and post-synthesis modification to produce microporous materials with enhanced surface permeabilities16,30–33. These generally focus on ways to shorten the diffusion pathways from the exterior into the intracrystalline pore space. In this context, the question arises as to the corresponding changes in the diffusion pathways leading out of the crystal. Although the principle of microscopic reversibility34– 36

provides a strong argument that any change in path length must apply equally in both directions,

collective effects such as hysteresis phenomena have been invoked to justify the possibility of systematic differences between ad- and desorption paths37. These differences were assumed to be reinforced by pore blocking, as a consequence of the differences in the “detours” that molecules within and outside the crystals are forced to take, leading to asymmetric adsorption and desorption paths22.

2. COMPUTATIONAL METHODS Today, molecular dynamic simulations offer the possibility of calculating all possible diffusion pathways for a molecule travelling between the ambient gas phase and the interior of a


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microporous crystal, in both directions. Following a proof-of-principle for the benzene – silicalite system38, we now report the results of an in-depth MD study of the diffusion pathways for the methane – silicalite system (Figure 1) with varying degrees of surface blocking. We follow the situation typical for uptake, release and tracer exchange experiments involving physical adsorption for which the principle of microscopic reversibility is firmly established. Three regions have been considered in our model (Figure 1): the reservoir, the external surface of the silicalite crystal, and the bulk crystal. Only the molecules that have penetrated deeper than 4 Å from the outer H atoms (belonging to silanol groups) at the external surface are considered to be inside the crystal. The rest of the molecules, that is, those which are located within the surface region plus the molecules in the reservoir (with which they are in rapid dynamic equilibrium) are considered to be outside the crystal. At any time during the MD run, it is possible to calculate the number of molecules inside/outside, giving the uptake or release curves for each simulation.

Figure 1. Model of a silicalite crystal employed for the calculations. The system comprises 8 unit


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cells (2×2×2) with 8 straight channels in the y direction. Blocking atoms are highlighted. The halflength within the crystal (between the reservoirs) is ℓ = 21.9 Å.

2.1. Structure of Silicalite. According to the procedure followed in a previous publication38, the so-called PARA-silicalite structure (P212121, #19, orthorhombic) was selected to model the silicalite crystal, based on a previous study39 in which the reproducibility of pore sizes was modelled using different force fields. This structure was solved taking into account the effect of p-xylene loading on the structure, so the structural model accounts for the deformation of the channels resulting from the presence of the guest molecules. A 2×2×2 unit cell (a = 2×20.121 Å, b = 2×19.82 Å, c = 2×13.438 Å) of silicalite (containing 8×96 SiO2 atoms) was used as the model. The cell length along [010] was enlarged to 79.46 Å and the resulting two surfaces perpendicular to [010] were terminated by silanol groups (≡SiOH). This model (Figure 1) contains 8 straight channels, each with two external surfaces, thus offering possible degrees of surface blocking between 0 and 16. The extended length (79.46 Å) along [010] corresponds to 43.8 Å of zeolite crystal (with silanol groups at both ends) and 35.66 Å of reservoir. The periodic nature of this crystal means that the two ends of the reservoir are joined and the periodic model is a repetition of [crystal-reservoir] units. The half-length within the crystal (between the reservoirs) is ℓ = 21.9 Å. In our study we have simulated first an ideal unobstructed crystal in which all channels are open to the ambient fluid at the external surface. We then examine the effect of surface resistance by inserting large blocking atoms at the entrances to a fraction of the channels. The simulation model is depicted in Figure 1 and explained, in more detail, in the Supporting Information (S1.1– S1.3) and in ref. 38. The intra-crystalline pore space is connected with the


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external gas phase (the reservoir) through 8 straight channels of silicalite. Blockage of some of these channels by atoms (blue balls in Figure 1) close to the crystal surface prohibits the passage of methane molecules (into and out of the reservoir) through the blocked channels. Surface permeability is varied by varying the number of blocking atoms. We considered three different possibilities regarding the blocking of channels: a) 0 blocking atoms (0% blocking); b) 8 blocking atoms (4 channels blocked at both ends, giving 50% blocking); c) 14 blocking atoms (6 channels blocked at both ends and 2 channels blocked only at the bottom end, giving 87.5% blocking). Blocking atoms are located 7 Å below the end of the crystal, hence 3 Å below the region called 'external surface' (Figure 1). Blocking atoms are fixed throughout the simulation and considered as large spheres in order to preclude methane molecules passing through. 2.2. Molecular Dynamics. In the simulations of uptake the reservoir is initially filled with a given number (40) of thermalized methane molecules while the zeolite crystal contains no methane. The simulation starts with a random configuration of methane molecules, all of them located in the reservoir, and from this configuration the system is allowed to evolve by itself (at constant volume, 298 K, and with a fixed number of particles), without any restraint or external force. As a consequence of the lower free energy of the methane-zeolite (intra-crystalline) with respect to the methane-methane (reservoir) systems, methane molecules tend to adsorb and diffuse inside the zeolite micropores, until the external pressure becomes low enough so as to reach the equilibrium state. DL_POLY 2.2040 was used for all the molecular dynamic simulations including full flexibility and periodic boundary conditions for all the atoms of the system, except the blocking atoms which remain in the same position at the centre of the channel (just inside the external surface). The


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temperature chosen is 298 K within the NVT ensemble. We employed the Verlet-leapfrog integration algorithm and the Evans thermostat, with a timestep of 1 fs. Each run comprised an equilibration stage of 5×104 steps followed by the necessary production stage so as to ensure sufficiently good statistics for the analysis, which, depending on the run, will amount to simulations times of 150 ns. The cut-off for the non-bonding forces was set to 9 Å, and the Ewald summation was employed for the Coulombic with a precision parameter set to 10-3. The configurations were saved every 100 time-steps (0.1 ps). Details of the force field employed are given as Supporting Information.

3. RESULTS AND DISCUSSION To illustrate and confirm the significance of our simulations, we start with the results on molecular uptake and fluctuation. Figure 2a (at short times) shows the transient uptake curve, approaching the final (equilibrium) state, for a crystal exposed at time zero to an ambient atmosphere containing methane molecules (the “reservoir”). Figure 2b, at a longer time scale, shows the fluctuations in Nt (the number of guest molecules in the crystal at time t) around the mean value N∞, which, from the long-time average is about 33 for all degrees of blocking. Ntotal (the total number of molecules) is 40, so the average number numbers of molecules remaining in the reservoir is 7 or Λ≈ 0.825, where Λ is the fraction of molecules finally adsorbed.


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Figure 2. Uptake of 40 methane molecules in silicalite system (Figure 1) with different degrees of surface blocking. Initial uptake up to 2 ns (a) and whole run during 150 ns (b). Exchange of adsorbed methane molecules during the equilibrium stage (c). The broken lines show the theoretical curves from Crank’s figure 4.6 according to a previous procedure38.

Following ref38, uptake and exchange dynamics are analyzed within the formalism of the statistical moments41,42 (see Supporting Information, sections S2.3 and S2.4 for more details), with the relative amount exchanged (Nt/N∞) resulting as the solution of the diffusion equation with the boundary condition for finite surface resistance, which may be approximated by: 𝑁𝑡 𝑁∞

−𝑡

(1)

= 1 − exp [(1−Λ)𝜏]

The time constant 𝜏 = 𝜏diff + 𝜏surf results as a simple superposition of the contributions from intra-crystalline diffusion and surface resistance, which are given by the relations: 𝜏diff = 𝑙 2 /(3𝐷)

;

𝜏surf = 𝑙/𝑘.

(2)

l and D are, respectively, the crystal half-thickness (21.9 Å) and the coefficient of intracrystalline diffusion. The surface permeance k is defined as the factor of proportionality between the flux through the surface and the difference between intracrystalline concentration close to the surface and its value in equilibrium with the external gas phase. Λ is the fraction of molecules


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finally adsorbed. Figure 2a shows the first interval of the simulation period as molecular uptake vs time, together with the theoretical curves corresponding to the best fits of Eq.1 to the simulation data. Figure 2c shows the tracer exchange curves at t = 2 ns at which time equilibrium is closely approached (see Figure 2b) expressed as fractional uptake (Nt/N∞). All molecules initially within the crystal at time zero are distinguished from those in the reservoir by different labelling. The broken lines show the best fits of the data to Eq.1. Table 1 provides a summary of the diffusivities and surface permeances obtained from the best fits of Eqs 1-2 to the simulations shown in Figure 2. Further details are given in the Supporting Information (S2.4). Given the large scatter in the primary data, the simulations cannot be expected to yield more than order-of-magnitude estimates. However, these estimates are found to be in reasonably good agreement with the available experimental data.


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Table 1. Summary of Parameters Derived from Uptake and Tracer Exchange Curves Blockage uptake 0% 50% 87.5% tracer exchange 0% 50% 87.5%

D/10-9 (m2s-1)

k (ms-1)

0.825 275 0.825 783 0.825 2909

5.35 5.35 5.35

∞ 4.13 0.80

0.825 163 0.775 463 0.750 1766

9.0 9.0 9.0

∞ 7.0 1.3

Λ

𝜏 (ps)

At the relevant loading (< 20% saturation) one would expect uptake and tracer exchange to occur at similar rates. The diffusivities and permeances derived from the simulations are of the same order but, surprisingly, tracer exchange appears to be somewhat faster. More importantly, the selfdiffusivities are consistent with the values derived from PFG NMR measurements (≈8·10-9 m2s-1 43,44

) and from Quasi-Elastic Neutron Scattering (≈4·10-9 m2s-1

45

) and the values from MD

simulations (≈1.5·10-9 m2s-1 46, ≈1.3·10-8 m2s-1 47). To understand the observed decrease in permeance with increasing blocking we adopt the formalism of effective medium theory38,42,48. Assuming that surface permeation is possible only through holes of radius r with average spacing λsep, (λsep>>r) the surface permeance is given by: 𝑘 = 2𝐷𝑟/𝜆2sep

(3)

from which, for the present system, k may be shown38 to scale with (1 – fractional blocking). Increased blocking from 50% to 87.5% is thus expected to give rise to a decrease in permeance by a factor of 4. This is fairly close to the ratios of about 5 derived from the simulation data (Table 1).


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Our simulations have thus been shown to reflect the main features of molecular uptake and exchange in our model system reasonably well. It would probably be possible to improve the accuracy of the simulations, but at the cost of a substantial increase in computing times. This was not pursued as we preferred to focus on a detailed analysis of the diffusion pathways for molecules entering or leaving the crystal in order to answer the question that has been raised concerning possible asymmetry between adsorption and desorption, particularly in the presence of a surface barrier. Figure 3a illustrates the procedure used in our analysis. Pairs of boxes A and B are arbitrarily selected, with the only condition that A is within the crystal and B is part of the external surface. Any pathway among the simulated trajectories connecting A with B (B with A) is considered to be a desorption (adsorption) path. From the trajectory shown in Figure 3a, path going from ‘3’ to ‘4’ is a desorption path. Further details are given as Supporting Information (section S2.1). Path length distributions have been analyzed by considering (on a logarithmic scale) the mean of the path lengths within five equally spaced path length ranges. The complete data set may be found in the Supporting Information (section S2.2). The key message is provided by the graphical representation shown in Figure 3b. The path length distributions for different degrees of blocking are very similar with essentially the same mean values, and the absolute number (“Number” in Tables S1 and S2) of paths decreasing with increasing blockage. The distribution is broad (normalized std. dev. ≈1.5) but there is no perceptible difference in the numbers of the adsorption and desorption pathways or their length distribution. Finally, the free energy profile for methane molecules in the system with 0% blocking has been calculated (at 298K with a loading of 40 methane molecules) using the trajectories obtained from the molecular dynamics simulations. Free energy barriers of ca. 1 and 5 kJ/mol were found for


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adsorption and desorption near the external surface, while intra-crystalline free energy barriers of ca. 3 kJ/mol were found for diffusion throughout the straight channel of silicalite. Despite the difference in free energy between the gas and adsorbed phases the simulation results clearly show that adsorption and desorption are symmetric processes with the same distribution of path lengths. More details are given in the Supporting Information (Section S3).

Figure 3. (left) Schematic representation of a typical path; (right) Path length distributions for different degrees of blocking.

4. CONCLUSIONS Molecular dynamics simulations thus confirm what might have been guessed already from the time invariance in the solutions of the underlying differential equations but which, to the best of our knowledge, has never been previously demonstrated explicitly by MD simulations. In


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particular, it has been shown that symmetry in adsorption and desorption paths is maintained with increasing blockage. It follows that pore blocking at the surface cannot give rise to asymmetry of adsorption/desorption. Adsorption and desorption paths are always equally affected by surface blockages, in accordance with the Principle of Microscopic Reversibilty36. The investigation of this issue for more complex models of host-guest interaction in the crystal surface, including the identification of any conditions leading to a breakdown of the observed adsorption-desorption symmetry, would be a worthwhile goal for future molecular dynamics simulation studies.

ASSOCIATED CONTENT Supporting Information Force fields; Definitions of adsorption and desorption paths; Tracer exchange plots; Method of moments to calculate time constants; calculation of free energy landscape. Notes The authors declare no competing financial interests. Acknowledgements MINECO of Spain is thanked for funding through “Severo Ochoa” (SEV-2016-0683) and RTI2018-101784-B-I00 projects. G.S. thanks the CTI-CSIC and ASIC-UPV for the use of computational facilities. J. K. and D. M. R. acknowledge financial support by the Alexander von Humboldt Foundation and the Fonds der Chemischen Industrie.


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(39) Bermudez, D. G.; Sastre, G. Calculation of pore diameters in zeolites. Theor. Chem. Acc. 2017, 136, 116. (40) Smith W.; Forester T. DL POLY 2.0: A general-purpose parallel molecular dynamics simulation package”; J. Mol. Graph. 1996, 14, 136 (41) Barrer R. M. Zeolites and Clay Minerals as Sorbents and Molecular Sieves; Academic Press: London, 1978. (42) Kärger J.; Ruthven, D. M.; Theodorou, D. N. Diffusion in Nanoporous Materials p.41; Wiley - VCH: Weinheim, 2012. (43) Heink, W.; Kärger, J.; Pfeifer, H.; Salverda, P.; Datema, K. P.; Nowak, A. K. HighTemperature Pulsed Field Gradient Nuclear Magnetic Resonance Self-Diffusion Measurements of n-Alkanes in MFI-Type Zeolites. J.C.S. Faraday Trans. 1992, 88, 3505– 3509. (44) Caro, J.; Bülow, M.; Schirmer, W.; Kärger, J.; Heink, W.; Pfeifer, H.; Zhdanov, S. P. Microdynamics of Methane, Ethane and Propane in ZSM-5 Type Zeolites. J. Chem. Soc. Faraday Trans. I 1985, 81, 2541–2550. (45) Jobic, H. Diffusion of Linear and Branched Alkanes in ZSM-5. A Quasi-Elastic Neutron Scattering Study. J. Mol. Catal. A: Chem. 2000, 158, 135–142. (46) June, R. L.; Bell, A. T.; Theodorou, D. N. A Molecular Dynamics Study of Methane and Xenon in Silicalite. J. Phys. Chem. 1990, 94, 8232–8240. (47) Goodbody, S. J.; Watanabe, K.; MacGowan, D.; Walton, J. P. B.; Quirke, N. Molecular Simulation of Methane and Butane in Silicalite. J. Chem. Soc. Faraday Trans. 1991, 87, 1951–1958.


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(48) Dudko, O. K.; Berezhkovskii, A. M.; Weiss, G. H. Diffusion in the presence of periodically spaced permeable membranes. J. Chem. Phys. 2004, 121, 11283–11288.


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Definition of 'adsorption' (B-->A) and 'desorption' (A-->B) paths through trajectories obtained from molecular dynamics. 166x106mm (96 x 96 DPI)


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Diffusion Path Reversibility Confirms Symmetry of Surface Barriers

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