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380817 Transport into Zeolite Nanosheets: Test of Diffusion Equations

Ultrathin porous materials, such as zeolite nanosheets, are prominent candidates for performing catalysis, drug supply, and separation processes [1,2] in a highly efficient manner due to exceptionally short transport paths. Predictive design of such processes requires applying diffusion equations that were derived on the assumption of a macroscopic structureless continuum to nanoscale, nanostructured host systems. In this context, it is important to emphasize that numerous measurements [3] and simulation studies [4] have highlighted the general validity of conventional Fickian diffusion to describe guest molecule transport into nanoporous materials. In the experiments, crystals are however well in the micrometer range and the spatial resolution for direct observation of concentration profiles is limited to 500 nanometers [3]. Thus, testing the applicability of diffusion equations to guest transport into nanometer sized objects such as zeolite nanosheets can to date be done in a most direct manner by molecular simulation approaches only. This represents the core of the present contribution, which investigates methane transport into nanosheets of two different zeolite structures under industrially relevant non-steady-state conditions by means of simulations. The central question is: Does the reliability of describing guest transport into nanoporous materials end at the nanoscale?

Computationally demanding transient molecular dynamics (TrMD) simulations are performed, paralleling conditions found in cyclic conducted processes and diffusion experiments with constant sorbate supply: initially empty nanosheets are filled over time with guest molecules. In contrast to conventional equilibrium molecular dynamics (EMD) where the number of atoms in the simulation remains constant, this number is allowed to change in TrMD by performing spatially restricted Grand-Canonical Monte Carlo trials (insertion and deletion). In doing so, an infinite fluid reservoir can be mimicked. We determine guest molecule concentration profiles over the nanosheet as averages from 40 independent TrMD simulations because single-simulation profiles are very noisy. The thus obtained data are fitted to standard analytical solutions of Fick’s diffusion equations [5]. As we investigate two different boundary conditions two sets of fitted transport coefficients are obtained: first, a single apparent transport diffusion coefficient, *D*_{T,app}, that lumps together interface (fluid–zeolite) and intracrystalline transport; second, a transport diffusivities, *D*_{T}, and surface permeabilities, *α*, where the latter is an explicit measure of the interface transport rate. The two zeolite structures chosen are siliceous AFI and LTA which feature smooth channels (AFI) and bulky cages connected by narrow windows (LTA), respectively. Nanosheets build from unit cells of these structures are investigated for various sizes.

Fitted concentration profiles using the apparent diffusivity give rise to large discrepancies from the TrMD data, particularly at early uptake times and for the smallest nanosheets studied. The resulting apparent diffusion coefficients increase as the nanosheet thickness becomes larger for both zeolites. On the other hand, the concentration profiles with the second boundary condition (so-called surface evaporation boundary condition) fit the data perfectly and the majority of transport coefficients are constant over sheet thickness. Hence, we conclude as a general finding that the one-diffusivity model is too simple to accurately describe guest transport into nanosheets. Moreover, the transport coefficients agree well with conventional EMD predictions. The exception to the rule is that TrMD simulations with the smallest AFI sheet (2.6 nm) yield a diffusivity distinctly exceeding the former values and the EMD prediction. This represents a size limitation to the applicability of Fick’s laws because transport coefficients must not vary with primary geometric parameters. Because we observe the effect only for the smooth AFI pores we conjecture that the molecular explanation is found in a memory effect [6]. Molecules entering the initially empty AFI pores cannot equilibrate in the first cage, perform a cascade of jumps across several cages (“multijumps”), and, thus, violate random walk theory on the length scale of a single zeolite cage. While these so-called kinetic jump correlations have been known before in the context of EMD simulations [7] they have been mostly viewed as artifacts of a bad choice of simulation setup [8]. It is thus important to realize that this is not true, that these jump correlations are physical, and, most importantly, that they can cause deviations from predictions of transport times into zeolite nanosheets and, if the mode is changed, discrepancies from true transport resistances of membranes relying on ultrathin zeolite films.

**References**

[1] J. Caro, M. Noack: *Zeolite membranes–recent developments and progress*; Microporous Mesoporous Mater. **115**, 215–233 (2008).

[2] P. S. Wheatley, A. R. Butler, M. S. Crane, S. Fox, B. Xiao, A. G. Rossi, I. L. Megson, R. E. Morris: *NO-releasing zeolites and their antithrombotic properties*; J. Am. Chem. Soc. **128**, 502−509 (2006).

[3] C. Chmelik, J. Kärger: *In situ study on molecular diffusion phenomena in nanoporous catalytic solids*; Chem. Soc. Rev. **39**, 4864–4884 (2010).

[4] R. Krishna, *Diffusion in porous crystalline materials*; Chem. Soc. Rev. **41**, 3099–3118 (2012).

[5] J. Crank: *The Mathematics of Diffusion*; Oxford University Press, UK (1975).

[6] N. E. R. Zimmermann, T. J. Zabel, F. J. Keil: *Transport into Nanosheets: Diffusion Equations Put to Test*; J. Phys. Chem. C **117**, 7384−7390 (2013).

[7] D. Dubbeldam, E. Beerdsen, T. J. H. Vlugt, B. Smit: *Molecular simulation of loading-dependent diffusion in nanoporous materials using extended dynamically corrected transition state theory*; J. Chem. Phys. **122**, 224712 (2005).

[8] N. E. R. Zimmermann, S. Jakobtorweihen, E. Beerdsen, B. Smit, F. J. Keil: *In-Depth Study of the Influence of Host-Framework Flexibility on the Diffusion of Small Gas Molecules in One-Dimensional Zeolitic Pore Systems*; J. Phys. Chem. C **111**, 17370–17381 (2007).

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