Understanding  molecular transport inside zeolite pores is important for optimal designs of zeolites as sieves, sorbents and catalysts. Diffusion in zeolites has peculiar dependence on temperature, loading, and molecular structure which does not appear in more conventional non-equilibrium bulk gas diffusion and Knudsen diffusion due to  collisions and other near-equilibrium noise-driven activated transport in most condensed media with atomic  scale periodicity. Such differences are due to the lattice  medium around the sorbate molecule, which is periodic over a large  scale but not so locally,   and the peculiar interaction between the  two. In contrast to discrete bombardment by a homogeneously  distributed medium in gas diffusion, the sorbate molecule in a  zeolite is under constant influence of its highly anisotropic local lattice medium. Other than the static lattice/sorbate interaction  that defines the potential landscape the sorbate must traverse,  thermal lattice vibration  has also been speculated as the main driving force behind the transport of a single sorbate molecule  in the dilute limit. Due  to the rigidity of the lattice and the elongated geometry of the  pore within relative large crystal structures, this noise field due
to sorbate/lattice interaction is expected to be spatially  inhomogeneous and escapes any mean-field or periodic coarse field  approximation.

    In existing theoretical studies, the sorbate and lattice are usually  assumed to be near equilibrium such that sorbate inertia (kinetic energy) is  negligible and the classical transition state theory (TST) can be used, which yields  the  Arrhenius scaling of diffusivity with respect to temperature that often  disagrees with data. On the other hand, most of molecular dynamics (MD)  simulations often either neglect the lattice vibration or model it by one of many thermostating techniques which inevitably introduces a somewhat arbitrary model for the noise. For smaller molecules, our review of the literature has  shown that the simulated transport rate is quite often independent  of the noise model -- a highly surprising result for a transport  process that is thought to be noise-driven [Kopelevich, D.I. and Chang, H.-C., "Nonequilibrium Diffusion in Zeolites due to Deterministic Hamiltonian Chaos", Phys. Rev. Lett., 83, p. 1590 (1999)] Such dilemma attests to our lack of understanding of the  basic mechanism behind zeolite transport.

    In this study, we quantitatively estimate  lattice - sorbate interaction as a noise source and  scrutinize its role on transport through zeolite. We develop a model for transport of sorbates in zeolites using linear approximation for lattice vibration and for the interaction between lattice vibration and sorbate. This  small-amplitude expansion is justified by the rigidity of the  lattice. Our normal mode vibration analysis includes a small wave  number asymptotic expansion of the eigenvalue problem such that  acoustic phonon modes with wavelength  longer than the unit  cell can be obtained. The estimated  vibration spectra are in agreement with measured optical mode  vibration spectra from infrared and Raman scattering. We then derive a Langevin equation for this transport and show that the dominant contribution to the thermal noise is due to long lattice  phonon modes. This noise is not homogeneous and we obtain its  explicit dependence on sorbate position in the potential landscape. We estimate the magnitude of the noise for different sorbate molecules and find that it is negligible for small molecules and becomes more important as the size of molecules increases. This explains the  insensitivity to noise model in the aforementioned molecular  dynamics simulations. Therefore, TST cannot be used for diffusion of small molecules in zeolites.
In such systems, there is no energy exchange between the sorbate molecule and the thermal bath on a time scale of diffusion.

    The driving mechanism for small molecules must hence have a  deterministic origin independent of noise. Our examination of the  deterministic Newton's equations for sorbate reveals a new  non-equilibrium Arnold diffusion mechanism due to coupling between different degrees of freedom of a sorbate at high inertia. We consider motion of a small spherical molecule in a single pore zeolite. Due to corrugation of the zeolite pore, azimuthal motion of the molecule interacts chaotically with the motion in the longitudinal direction. We use a random-phase approximation for the resulting Arnold diffusion and a boundary-layer analysis of the quasi-steady Fokker-Planck equation, to estimate the effective diffusivity to scale as T^1/2 for an axisymmetric, single-pore zeolite in contrast to the Arrhenius temperature dependence of high-barrier, noise-driven transport of near-equilibrium diffusion. This theoretical finding is in good agreement with our MD simulations for methane in a single-pore zeolite ALPO 4-5.

   In the future, we will develop a quantitative criterion distinguishing between Arnold and noise-driven diffusion. The importance of inertia in the former  non-equilibrium mechanism is suppressed for long molecules. Averaging of  our anisotropic noise field in the direction normal to the steepest  descent path along the potential landscape will allow us to formulate a  near-equilibrium transport theory for our position-specific noise  fields. We will use this approach to study noise-driven diffusion of long-chain alkanes in zeolite silicalite.

    We will also tackle the intriguing problem of transport of  bulky molecules through narrow bottlenecks in zeolite pores (such as benzene in  silicalite). In such cases, the local lattice dynamics are at the  same time scale as the sorbate dynamics and the lattice vibration  amplitude is large. Consequently, the lattice dynamics cannot be  delegated as noise and must be included in parallel to the sorbate  dynamics. The resulting nonlinear interaction can produce nonlinear  resonance that may also explain some peculiar aspects of zeolite  transport.
 
 
 

Projection of a methane trajectory with E= -5.4 kJ/mol in a rigidAlPO4-5 single- pore zeolite in the v-z phaze space.
 
 
 
 
 
 
 
 

Streamlines of the potential Uz for the uncouplet longitudinal degree of freedom.
 
 
 
 
 
 

Iterations of Poincare map for the dynamical system.
 
 
 
 
 
 

   Probability distribution calculated from the Poincare map is almost uniform.
 
 
 
 
 
 

    We hope to eventually develop a hybrid MD-continuum scheme for hydrodynamics with significant molecular contributions - dewetting, jet breakup, polymeric flow etc.
 
 

Relevant publications: [120], [128],[130],[133],[134],[149],[151],[152],[155]