Dynamic Monte Carlo simulations of binary self-diffusion in ZSM-5
Introduction
Zeolites are periodically ordered, microporous, crystalline aluminosilicates. Engineering applications of zeolites include catalysis, the active layer in membranes, and adsorption. In this paper, the diffusion of guest molecules in the zeolite pore space is considered. Molecules of different sizes and shapes adsorb and diffuse at distinct rates in zeolites, which forms the basis for the use of zeolites for the separation of mixtures. The use of zeolites in membranes is extremely useful due to the low energy requirements compared to distillation, which is a thermally-driven separation. The same mass transfer properties also affect performance in zeolite-hosted catalysis, where the overall rates of reactions are influenced by the size and shape dependence of the reactant and product diffusivities. Zeolites can also be selective adsorbents, which is useful for applications in gas sequestration: some molecules interact more strongly with the pore walls and adsorb in greater amounts, as e.g., polar molecules in cation-doped zeolites [1], [2], [3], [4], [5], [6]. Diffusion in zeolites has been the focus of many theoretical, simulation, and experimental studies, since diffusion can be the rate-limiting step in each of these applications [2], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].
Statistical mechanical simulation methods include molecular dynamics (MD), transition state theory (TST), and dynamic Monte Carlo (DMC) [21]. MD considers the dynamics of molecules moving in a potential energy landscape (PEL) defined on an atomistically detailed scale. Newtonian equations of motion are solved over a given time frame (typically tens of ns) to collect the trajectories and to determine the diffusivities. The use of TST is based on the observation that the PEL is typically very rough for diffusion of molecules in zeolites, so that diffusion is an activated process. Short-time MD trajectories or MC sampling are analyzed to determine the hopping rates [22], [23]. DMC, on the other hand, is a method to measure the diffusivities in a coarse-grained description of the zeolite pore lattice. While it lacks detail compared to MD and TST, it is a computationally efficient method that gives insight into the different parameters that influence diffusion in a straightforward way.
The self-diffusivity of a species i is calculated using Einstein’s equation,where DSi is the self-diffusivity of species i (i = 1 or 2), is the position vector at time t, is the position vector at time 0, and d is the embedding dimension of the lattice (2 or 3). This quantity DSi measures the motion of individual particles of species i in an equilibrium system, i.e., the macroscopic chemical potential gradient is zero. Another common quantity is the corrected-diffusivity, , which measures the motion of the center of mass of a swarm of particles and is discussed in detail later.
Experimental techniques to measure diffusion in zeolites include the well-described methods of pulsed field gradient nuclear magnetic resonance (PFG-NMR) [12], [24], [25] and quasi-elastic neutron scattering (QENS) [26]. These methods directly determine self- and corrected-diffusivities of light gases in zeolites, and have been successfully compared to diffusivities derived from MD [24], [25], [27], [28], [29], [30]. Experimental methods and MD complement each other, since atomistically detailed simulations provide physical insight in a practical manner, without expensive equipment and experimental complications, such as contaminants and internal grain boundaries.
Since the above methods are microscopic, there is also a need for theories that can be used in chemical engineering to predict membrane performance, and to guide and interpret experimental results. For this purpose, many research groups have used the Maxwell–Stefan (MS) approach to model diffusion in zeolite membranes and other microporous materials. The MS theory applies to multi-component fluids. Fluxes are related to gradients in chemical potential, which is thermodynamically more rigorous than Fick’s Law, a formulation based on gradients in concentration. The MS approach was adapted from bulk fluids to fluids in mesoporous materials in a form known as the dusty gas model (DGM) [31], [32]. However, the MS approach is not rigorous for microporous materials, where the pores are of atomic scale. It does not take into account the poorly connected and very constrained space of the zeolite pore lattice, where molecules typically cannot pass each other. Therefore, other analytical theories have also been considered, such as the effective medium approximation (EMA) [33], [34], [35], [36], [37], [38], [39] and the mean field theory (MFT) [40], [41], [42]. A combination of these two theories enables one to explicitly account for the zeolite pore lattice characteristics in calculations of diffusivities.
Other studies have also considered theories based upon the lattice description of the zeolites. One example is the lattice-gas cellular automaton (LGCA), where Demontis et al derived a diffusion theory specific to the ZK4 lattice [43], [44]. Van Tassel et al. derived a lattice diffusion theory for diffusion in NaA [45]. Their theory works exactly at lower loading and approximately at higher loading. Belova and Murch [46], [47] derived an analytical theory to describe diffusion in bimetallic alloys that shows nice agreement with their lattice simulations. Saravanan and Auerbach used mean field arguments to derive a theory to describe activated diffusion of benzene in NaY, which showed consistently strong agreement with Monte Carlo lattice simulations [41], [42].
For applications in catalysis and membranes, many zeolites contain atomic framework substitutions by introducing Al and a charge-balancing cation, such as Na+, [2], [48] in place of some of the Si atoms in the SiO2 lattice. This substitution creates different adsorption strengths within the stable basins, which are the minima of the PEL of the zeolite. This affects the behavior of guest molecules with different polarities, such that some guest molecules will spend much more time localized in the deeper basins, hindering diffusion. Hence, the concentration of cations in the zeolite affects the diffusion of molecules with different polarities, and this feature can be utilized to fine-tune the selectivity of a zeolite [21], [34], [49]. For example, when separating CO2 and N2 in a zeolite membrane, the Na+ cations will have stronger attractive quadrupolar interactions with CO2 than with N2. Therefore, CO2 will spend an increased time near the Na+ cations, which act as strong sites. In practice, different levels of Na+ can be used to enhance the CO2 adsorption relative to N2, though with lower diffusion of the CO2.
Shallow and deep stable basins are associated to weak and strong sites, respectively, with short and long average residence times, respectively, of the molecules present on them. This coexistence of different residence times is an example of static heterogeneity. With this terminology, static homogeneity means that all the sites have the same adsorption strength, hence a homogeneous PEL, with a single average residence time for each site. In contrast, dynamic heterogeneity occurs when a binary mixture of guest molecules coexists in the zeolite porous network, and one species is faster than the other one [34], [50]. This feature creates a strongly variable heterogeneous environment for each diffusing species.
Many experimental studies focus on the zeolite ZSM-5 (also known as MFI), since it is one of the most frequently used zeolites in practice, and it is relatively inexpensive to synthesize [51], [52], [53], [54], [55], [56]. Therefore, it is the example zeolite used in this work as well. ZSM-5 contains two types of channels – straight channels in the y-orientation and zigzag channels in the x–z plane – that periodically intersect, with a pore diameter of ∼5.5 Å. A schematic of the ZSM-5 unit cell for the pore network is shown in Fig. 1; note that this corresponds to half a crystallographic unit cell of ZSM-5. The level of the framework substitution is defined by the ratio of Si to Al atoms, Si/Al; Si/Al → ∞ means that there are no framework substitutions, corresponding to silicalite-1.
Previous studies indicated that the MS approach successfully predicts the self- and corrected-diffusivities for zeolites with a homogeneous PEL, such as found in silicalite-1 [15], [16], [57], [58], [59], but it becomes problematic for simple lattices and ZSM-5 when the Si/Al ratio is lower, i.e., for cases of high static heterogeneity [34], [50], [60]. Coppens and Iyengar [34], [50] demonstrated that the MS approach fails to properly predict the self-diffusivities when many strong sites are present. Sholl [60] tested the capability of the SSK theory, a version of the MS approach, on a two-dimensional square lattice with heterogeneity. Even for this simple structure, the SSK theory became inaccurate. In these situations, we would expect that methods that explicitly take into account the static heterogeneity, such as EMA, would perform better. The objective of this study is to compare the capability of various (semi-)analytical approaches; in particular, the MS approach, EMA, and MFT. On-lattice DMC simulations are used as a benchmarking measurement to judge the capability of the analytical theories to estimate the self-diffusivities on the ZSM-5 lattice.
DMC is a useful tool to investigate the capability of these analytical models, because DMC captures the main characteristics of activated diffusion in zeolites, including static and dynamic heterogeneity. Recent work by Collins et al. [61] improves on the DMC technique by using local coarse-graining over multiple sites by mean field approximation to considerably extend the length and time scales accessible. Results showed quantitative agreement at reasonable computational expense for several lattice case studies. Any theory of diffusion in zeolites should hold in the limit where DMC is valid. For the purpose of this study, DMC has a number of advantages compared to atomistically detailed MD simulations. DMC runs are faster than MD by several orders of magnitude, and avoid complicating details such as the atomistic structure of the zeolite lattice and of the guest molecules. MD is comparatively slow and more difficult to implement if the energy landscape is heterogeneous, because the integration steps are small, yet the trajectories should be long enough to cover a representative portion of the zeolite. In this study, simple point particles are considered, representing two species. These two species are identified by two distributions of residence times for two types of sites; namely, the weak and strong sites.
The next Section discusses the key concepts and assumptions behind DMC, the MS approach, EMA, and MFT. In Section 3, the results from DMC simulations are analyzed and compared to the analytical theories.
Section snippets
Dynamic Monte Carlo simulations (DMC)
DMC follows the particle movement on the ZSM-5 lattice over time, so that the diffusivities can directly be calculated. The method uses a coarse-grained, three-dimensional lattice representation with a random distribution of weak and strong adsorption sites. This ignores geometrical details at the atomic scale and the electronic structure of the guest molecules and the zeolite. While this coarse-graining represents an approximation, compared to MD, it preserves the main characteristics of the
Comparison between the MS approach and DMC
First, the ability of the MS approach is considered in predicting the self-diffusivity, as measured with DMC, for the case of binary diffusion on a statically homogeneous lattice, where f = 0. The average residence time on any site, before an attempted hop, is 100 times longer for the second, slow species, compared to the first, fast species (τs1 = τw1 = 1; τs2 = τw2 = 100). Fig. 2a, Fig. 2b present the self-diffusivities with respect to loading of the fast and slow species, respectively, in a binary
Conclusion
A number of general observations can be made for the effects of heterogeneity on binary diffusion on a ZSM-5 lattice. The dynamic heterogeneity does not compromise the accuracy of the MS approach to predict binary self-diffusivities, because the MS approach already includes the interactions between molecules of the different mobile species in its derivation. In stark contrast, the static heterogeneity strongly affects the accuracy of the MS approach to predict binary self-diffusivities, since
Acknowledgment
D.A. Newsome and M.O. Coppens gratefully acknowledge funding from the European Union via STREP project 014032, FUSION (Fundamental Studies of Transport in Inorganic Nanostructures).
References (67)
- et al.
Micropor. Mesopor. Mater.
(2006) - et al.
Micropor. Mesopor. Mater.
(2006) - et al.
Sep. Purif. Technol.
(2006) - et al.
J. Col. Int. Sci.
(2000) - et al.
J. Membr. Sci.
(2002) - et al.
Catal. Today
(2007) - et al.
Micropor. Mesopor. Mater.
(2007) - et al.
J. Membr. Sci.
(2008) - et al.
J. Membr. Sci.
(2007) - et al.
Micropor. Mesopor. Mater.
(2006)
Chem. Eng. Res. Des.
J. Membr. Sci.
Zeolites
Micropor. Mesopor. Mater.
Chem. Phys. Lett.
Chem. Eng. Sci.
Chem. Eng. Sci.
Chem. Eng. Sci.
Chem. Eng. Sci.
J. Phys. Chem. Solid
J. Phys. Chem. Solid
Chem. Eng. Sci.
Micropor. Mesopor. Mater.
J. Membr. Sci.
Chem. Eng. Sci.
Sep. Purif. Technol.
Chem. Eng. J.
Chem. Eng. J.
J. Membr. Sci.
Kor. J. Chem. Eng.
Ind. Eng. Chem. Res.
J. Phys. Chem.
Diffus. Zeolites Other Micropor. Mater.
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