Effect of packing on the cluster nature of C nanotubes: An information entropy analysis

The possibility of the existence of single−wall carbon nanotubes (SWNTs) in organic solvents in the form of clusters is discussed. A theory is developed based on a bundlet model for clusters, which enables describing the distribution function of clusters by size. Comparison of the calculated values of solubility with experiments would permit obtaining energetic parameters characterizing the interaction of an SWNT with its surrounding, in a solid or solution. Fullerenes and SWNTs are unique objects, whose behaviour in many physical situations is characterized by remarkable peculiarities. Peculiarities in solutions show up first in that fullerenes and SWNTs represent the only soluble forms of carbon, what is related to the originality in the molecular structure of fullerenes and SWNTs. The fullerene molecule is a virtually uniform closed spherical or spheroidal surface, and an SWNT is a smooth cylindrical unit. Both structures give rise to the relatively weak interaction between the neighbouring molecules in a crystal and promote interaction of the molecules with those of a solvent. Another peculiarity in solutions is related to their trend to form clusters, consisting of a number of fullerene molecules or SWNTs. The energy of interaction of a fullerene molecule or SWNT with solvent molecules is proportional to the surface of the former molecule and roughly independent of the orientation of solvent molecules. All these phenomena have a unified explanation in the framework of the bundlet model of a cluster, in accordance with which the free energy of an SWNT involved in a cluster is combined from two components, viz. a volume one proportional to the number of molecules n in a cluster, and a surface one proportional to n1/2. Algorithms for classification are proposed based on the criteria information entropy and its production. Many classification algorithms are based on information entropy. When applying these procedures to sets of moderate size, an excessive number of results appear compatible with data, and this number suffers a combinatorial explosion. However, after the equipartition conjecture, one has a selection criterion between different variants resulting from classification between hierarchical trees. According to this conjecture, for a given charge or duty, the best configuration of a flowsheet is the one in which the entropy production is most uniformly distributed. Information entropy, cluster and principal component analyses agree.