Quasi-one-dimensional system of molecules inside carbon nanotubes: Exact solution for the lattice gas model and its application to fullerene-filled nanotubes

The statistical mechanics of a system of molecules encapsulated inside a carbon nanotube was analyzed using a one-dimensional lattice gas model. Both open and closed tubes were studied and both the grand-canonical and the canonical partition functions were evaluated exactly. The formulas for the frequency of occurrence of clusters of various sizes were also derived. The model was applied to a (10,10) nanotube containing ${\mathrm{C}}_{60}$ molecules. The calculations gave detailed information on the clustering of molecules in both open and closed nanotubes. Analysis of the open system yields information on the conditions under which the nanotubes can be filled when they are in equilibrium with an external gas. The results show that an open nanotube can be filled very efficiently at room temperature provided there are enough external fullerenes in the gas phase. For a closed system at room temperature, we found a high degree of clustering that decreases with increasing temperature. Because of the strong interaction between fullerenes, the system is far from its random state even at the highest temperatures studied. For both cases, linear equations of state were determined as well. Obtained results are in accord with the experimental observations.