Asymptotic Coagulation-Fragmentation Equations

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 describing the distribution function of clusters by size. The phenomena have a unified explanation in 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: a volume one, proportional to the number of molecules n in a cluster, and a surface one, proportional to n^1/2. The model yields an activation barrier and predicts that pores with a radius below a certain critical value are unstable, while those above this radius will grow indefinitely until the membrane ruptures. During the latter stage of the fusion process, the dynamics were governed by the displacement of the volume of liquid around the fusion site. Based on a simple kinetic model micellization of rod-like aggregates occurs in three separated stages. A convenient relation is obtained for <n> at transient stage; at equilibrium another relation determines binding energy alpha. A relation with surface dilatational viscosity is obtained. The model predicts that pores with a radius below a certain critical value are unstable.