A new experimental protocol for the evaluation of the flow activation volume is proposed. It is based on step-shear experiments carried out on polymer melts. The flow activation volume evaluated agrees with the volume of a tube confining the chain, and is the same for the polymer melt and its composites with as received and functionalized carbon nanotubes. The functional groups bonded to the carbon nanotubes surface facilitate the polymer melt flow, eliminating the solid-like behavior in the high temperature flow region. A model for the morphology of polymer melts with carbon nanotubes is discussed. Polymer-nanotube interaction energies are discussed and the relaxation time of these interactions is estimated. The link between the solid-like behavior at the flow region and the strong shear thinning observed for the carbon nanotube composites is explained analyzing the different response to shear flow of each of the three networks considered by a model for the morphology of these composites.