Dmitry I. Kopelevich
and Hsueh-Chia
Chang
*Department of Chemical Engineering, University of Notre Dame,*
*Notre Dame, IN 46556*

We are investigating possible driving forces for diffusion of molecules in zeolites. Diffusion is typically associated with a noise-driven process. In the case of diffusion in zeolites, the noise is usually attributed to lattice vibration. It has been found, however, that in some systems (e.g., methane in silicalite) lattice vibration has a weak effect on the diffusion process. In such systems, transition state theory (TST), assuming large lattice-sorbate coupling (high friction and noise), overestimates diffusivity by orders of magnitude [1]. To the contrary, molecular dynamics (MD) simulations for these systems yield quite accurate estimates of diffusion coefficients even if the simulations neglect lattice vibrations (see, e.g. [2]).

This suggests that for at least some diffusive processes, inertial effects are important while external noise is not. In such systems, we assume the absence of energy exchange between the sorbate molecule and the thermal bath on a characteristic time scale of diffusion and define a diffusion coefficient as a Maxwell-Boltzmann average of diffusivities at fixed energies:

.

Here D(E) is a coefficient of deterministic diffusion known as Arnold diffusion. The driving force for this diffusion is the coupling of different degrees of freedom of a Hamiltonian. For example, the azimuthal degree of freedom couples with and drives the longitudinal one due to channel corrugation in a zeolite pore. We combine Mel'nikov analysis [3] with the theory of low-noise diffusion [4] and estimate the rate of Arnold diffusion for a simple case of a spherical molecule in a single-pore zeolite with an axisymmetric channel. We obtain a power-law scaling of the diffusion coefficient with temperature

.

This theoretical finding is in good agreement with
our MD simulations for methane in AlPO_{4}-5
which show that

.

We also define a quantitative criterion for the importance
of lattice vibrations. We determine that, for small spherical molecules,
the effects of lattice vibrations are negligible and Arnold dynamics is
dominant, while for larger molecules (e.g., benzene, long paraffins) noise
due to lattice vibrations becomes significant.

[1] D. I. Kopelevich and H.-C. Chang, preprint.

[2] P. Demontis, G. B. Suffritti, and P. Mura, Chem.
Phys. Lett. 191 (1992) 553.

[3] B. V. Chirikov, Phys. Rep. 52 (1979) 263.

[4] W. Young, A. Pumir, and Y. Pomeau, Phys. Fluids
A 1 (1989) 462.