Calculations of the energy of mixing carbon nanotubes with polymers

A method for calculating the energy of mixing carbon nanotubes (CNTs) with polymers is presented. The formation of the nanocomposite is analyzed in terms of a simple path in which the nanotubes are exfoliated from a bundle and dispersed in a distorted polymer with cylindrical cavities to accommodate the nanotubes. From this perspective, the energy of mixing is the difference between the energy required to exfoliate the nanotubes from a bundle and the energy needed to extract the nanotubes from the polymer matrix relative to the relaxed polymer without any nanotubes. These energy components are evaluated by performing molecular mechanics calculations on individual, localized models representing the polymer, nanotube bundles, and polymer/CNT agglomerates. This method is applied to polystyrene/CNT composites and the factors that determine their thermodynamic stability are identified. To a first approximation, the interaction energies (per unit surface area of the nanotubes) are independent of the lengths and chiral indices, but dependent on the diameters of the component nanotubes. By the application of this method, we show that the energy of mixing CNTs with PS is endothermic until the diameters of the component nanotubes exceed about 2.2 nm; at diameters greater than this value the energy of mixing becomes exothermic. This may explain why it is so difficult to obtain good dispersion of single-walled CNTs (SWCNTs) in PS, since they rarely grow to have diameters greater than about 1.4 nm. On the other hand, since the diameters of multi-walled CNTs typically exceed 10 nm, we would expect them to disperse much better than SWCNTs in polystyrene.