Full-pattern analysis of two-dimensional small-angle scattering data from oriented polymers using elliptical coordinates

Two-dimensional small-angle scattering (SAS) data from oriented polymers are parameterized by profile fitting the intensity distribution to a product of two orthogonal functions. Elliptical cylindrical coordinates were found to best describe the observed small-angle scattering data. Each of the essential features of the small-angle X-ray/neutron scattering from uniaxially oriented polymers—the equatorial streak, lamellar reflections and interfibrillar interference peaks—is described completely by a single function in the elliptical coordinates. The parameters of the fit are used to describe the fibrillar and the lamellar structures. The analysis is illustrated with data from nylon 6 fibres, and the results are compared with those from a previous analysis of the same data as a series of one-dimensional scans. The method has enabled us to follow the changes in the weak equatorial scattering attributed to fluid-like organizations of the fibrils. The elliptical coordinates can be used to describe the wide variety of small-angle patterns that have been reported in the literature. The applicability of the elliptical coordinate system to SAS data is shown to be a natural consequence of the scattering object being elongated along the flow or the draw direction. Ellipticity could be used as a quantitative measure of the shape and the orientation distribution of the scattering object.