Small‐angle x‐ray scattering from polyethylene

Abstract

AbstractA new theoretical model for the description of small‐angle x‐ray scattering from oriented polymer systems is developed and intensity functions are derived. The scattering system considered can be represented by a set of stacks of approximately parallel lamellae where the stack dimensions are assumed large in comparison with correlation distances within the stacks. Each point within the sample is envisioned as being surrounded by such a domain, so the theory is based on a concept of continuously varying local conditions. Fluctuations in the local lamellar distribution parameters are taken into account through a gross disorder parameter which specifies a spread in the local mean long periods. In addition, a long‐range disorder parameter has been included as a measure of the randomness of the actual amorphous layer spacings about the local mean lamellar spacing within each lamellar stack. Important characteristics of the diffraction curves such as the peak positions, heights, and widths; the number of observable orders; and the higher‐angle behavior allow one to obtain the mean lamellar spacing, the fluctuation parameters, and the effective widths of the refractive index perturbations associated with the amorphous regions. The number of lamellae that scatter x‐rays coherently can be predicted with ease. We show that geometric corrections are negligible for oriented samples. In addition, the Lorentz correction is shown to be unnecessary in all cases. We have measured diffraction patterns for polyethylene (PE) precipitated from solution using a Kratky system. Good agreement between theoretical and experimental results on PE 87 and PE 91 is achieved for three‐parameter fits. Finally, we have considered non‐Gaussian distributions of the local mean long periods, with excellent results being generated for a slightly positively skewed three‐Gaussian distribution.