A statistical mechanics study of ring size, ring shape, and the relation to pores found in zeolites

In this work, we survey the size and flatness of rings that occur in known zeolites and also in a set of hypothetical structures. The results suggest that 16-membered rings, while rarely observed in zeolites, are not unique and should be thermodynamically accessible. Conversely, the results also show that rings of a given flatness, or planarity, become exponentially less likely as ring size increases. We compare the geometry of rings in known zeolites with the geometry of unconstrained rings as determined from Monte Carlo simulation. The rings that occur in zeolites are flatter than unconstrained rings due to the constraints imposed by the crystal. The thermodynamic factors that determine the flatness of rings in crystals is investigated by using a reverse umbrella sampling technique. Interestingly, the energy required to bring rings from an unconstrained state to the crystalline flat state is roughly 5 kJ/mol Si, which is similar to the range of stabilities observed for zeolites and also to the range of interaction energies between zeolites and structure-directing agents.