The thermodynamic stability of clathrate hydrate. II. Simultaneous occupation of larger and smaller cages

The thermodynamic stability of a clathrate hydrate encaging methane or xenon has been investigated by examining the free energy of formation. The total free energy is divided into several contributions: the interaction between water and guest molecules, the entropic contribution arising from the combinations of cage occupancy, and also the free energy arising from the guest molecular motions inside cages. Our method is based on the generalized van der Waals and Platteeuw theory used for the study of the hydrate encaging propane and is free from some of the fundamental assumptions in the original theory. This enables us to evaluate separately the factors which contribute significantly to the thermodynamic stability of the hydrate, and to address a question as to what extent the original theory is applicable to the prediction of the phase diagrams. While the potential energy curve of the guest molecule with surrounding water molecules in a smaller cage has a single minimum and the molecular motion can be accurately approximated to a collection of harmonic oscillators strongly coupled with the host molecules, the guest molecular motion in a larger cage does not couple with the host. To show evidence that the fixed lattice approximation is sufficient to describe molecular motions inside the larger cage, two kinds of molecular dynamics simulations have been carried out. In one simulation, both host water and guest molecules move according to the classical equations of motion. In another simulation only guest molecules are allowed to move, interacting with fixed host molecules. We perform two kinds of analyses on those motions. In the first analysis, the velocity autocorrelation functions are calculated from molecular dynamics simulations at several temperatures and the power spectra are obtained by the Fourier transform of the correlation functions. In the second, a so-called normal mode analysis is performed and the densities of state for intermolecular vibrations are obtained. The densities of state (corresponding to 0 K) are compared with the power spectra. It is revealed that the fixed lattice approximation can be applied when describing the molecular motions of methane and xenon in larger cages. The free energy for the accommodation of an extremely large CF₄ or small argon guest is also examined.