Simplified tube form factor for analysis of small-angle scattering data from carbon nanotube filled systems

This work presents an analysis method for small-angle scattering data utilizing a simplified tube (hollow cylinder) form factor. The simplified form factor captures the rod-like character of a tube at long length scales (one-dimensional), the sheet-like character of the tube wall at intermediate length scales (two-dimensional), and the surface characteristics of a tube at small length scales while suppressing the deep minima seen in the exact form factor. Ultra-small-angle X-ray scattering data from composites made with multi-walled carbon nanotubes and a bismaleimide resin are analyzed using the simplified form factor and compared with scanning electron micrographs. Although a hollow core is not evident via microscopy, a solid rod form factor does not fit the data. However, a tube form factor does fit the data and generates reasonable geometric parameters. At higher concentrations, evidence for aggregation is seen in the data. Aggregation is accommodated by including a fractal structure factor within the simplified approach, allowing facile analysis of data from aggregated (poorly dispersed) fillers.