AmericanChemicalSociety.com

We have been probing the flexibility of ideal zeolite frameworks. We have coined this term this year in order to better define the assumption underpinning a key concept that is emerging in our recent research. We have defined an ideal zeolite framework as a periodic assembly of perfect, rigid, vertex-sharing, tetrahedra that have force-free spherical joints at their vertices. Every tetrahedron is connected to exactly four other tetrahedra at their corners, is perfectly regular, and has no internal stress.

At a workshop on the mathematics of flexibility in Barbados, Jan 2009, the postdoctoral student, Vitaliy Kapko, and I met Simon Guest who is a structural engineer working on periodic trusses. His interests are the stability of deployable structures. He has proven that locally isostatic periodic structures have at least three infinitesimal deformation mechanisms, and can therefore never be fully rigid; they will always flex at least infinitesimally. Interpreted in terms of ideal zeolite frameworks, this important result means that they are guaranteed to have a small window of flexibility. Guest's insight, developed in a different field to ours, has provided us with an important new approach to the zeolite flexibility problem.

The insight can be broken down as follows. By locally isostatic we mean that each zeolite tetrahedron has 6 degrees of freedom, (3 translations x, y, z, and three rotations a, b, g), but has also four sets of x, y, z constraints on the vertices. Since each vertex is shared, these twelve constraints become 6 constraints per tetrahedron. Thus, each tetrahedron appears to be kinetically determined, with 6 degrees of freedom being perfectly balanced by 6 constraints. However, in a periodic system, there are also 6 unit cell degrees of freedom, the three cell edges and three cell angles. Thus, there will always be at least 6 more degrees of freedom than constraints, even in an infinite crystal. Three of these extra degrees of freedom correspond to trivial translations along the cell axes. However, the other three degrees of freedom must correspond to internal 'mechanisms,' where tetrahedra can rotate (librate) cooperatively thereby changing the volume. This is the origin of the flexibility window that we reported before. What we had not understood then is that such infinitesimal flexibility is innate to all tetrahedral frameworks.

However, not all locally isostatic frameworks are guaranteed a large flexibility window (range of densities). If the framework tetrahedra are stressed, then the window width is actually very small. Real zeolite frameworks have a large flexibility window, within which the tetrahedral have no stress. In addition, real zeolites prefer to adopt the lowest density structure within this window.

The insight obtained at the Barbados workshop has led to an exciting conjecture, hence our defintion of an ideal zeolite framework. In the ideal zeolite, there are no internal potentials (apart from the strong forces keeping the tetrahedra rigid). Thus, a model of a zeolite with force-free corner joints, will lack rigidity and just flop around on the table. If the framework is heated, the tetrahedra will start to move around due to thermal kinetic motion. The conjecture is that a heated ideal zeolite framework will undergo a free expansion (analogously to an ideal gas when a membrane separating the gas from a vacuum is broken). The zeolite will expand to its maximum volume kinetically. This corresponds to an entropic inflationary force that maximizes the entropy at maximum volume.

The exciting consequence of this idea is that zeolite frameworks do not need internal coulombic repulsive forces to inflate them to maximum volume. The frameworks can naturally expand to maximum volume through kinetics alone, unless they are at absolute zero temperature.

Using mathematical tools developed by the structural engineering community, such as singular-value decomposition, we are developing a computer program ZeNuSpEx (Zeolite Null Space Explorer) for probing zeolite flexibility. We have started to examine the flexibility characteristics of all the known zeolites. Some of the exceptions to our rule are providing us with valuable insights about the role of divalent cations in stabilizing some zeolite structures.

Figure 1 depicts the 6 folding mechanisms of the zeolite framework DFT. The central figure depicts the structure when all six mechanisms (shown on the outside) are randomly excited simultaneously. The framework is locked into its maximum volume state (center) because that is the only configuration where all six modes can coexist. To collapse, five of the modes must be inactive so that the sixth mode can evolve far enough along its path to permanently suppress the others. This is a statistically unlikely event, and so the zeolite remains expanded.

Our future work is to develop a molecular dynamics capability for the ZeNuSpEx program so that we can calculate the thermodynamic configurational entropy. We will also probe deeper into the MTN mystery reported previously. That this framework is inflexible at its maximum symmetry . However, because of degeneracy of constraints at this high symmetry, there are 39 degrees of freedom. It is possible that the configurational entropy gained at this high symmetry outweighs the internal energy cost associated with strained tetrahedra. This suggests that MTN materials will undergo a phase transition at low temperatures. Doug Dorset at ExxonMobil will be testing this hypothesis on his (rare) samples of ZSM-39 (MTN) at liquid nitrogen temperatures.

Finally, it is intended that once the ZeNuSpEx program is running reliably, we will apply it to the 8 million hypothetical frameworks in the database in order to identify the potentially flexible frameworks – as these are the frameworks that are most likely to be realized in the real world.

Figure 1. Center: the DFT framework at maximum symmetry (P4₂/mmc). The yellow tetrahedra are show red oxygen atoms at their shared vertices. The six surrounding structures show the six orthogonal folding mechanisms for DFT. When all six modes are randomly excited simultaneously, the framework spends essentially all of its time in the maximum volume (minimum density) state, center. This is the only state that supports all six modes simultaneously.