Structural implications of the elliptical form of small-angle reflections in oriented semicrystalline polymers

The intensity maxima of lamellar reflections in small-angle scattering patterns from uniaxially oriented polymers such as fibers do not generally fall on a straight layer line or on a circular arc. The shape of the reflection can be analyzed by measuring the periodicity L_φ of the lamellar planes as a function of the angle φ between the reflection and z. L_φ is measured parallel to the fiber axis z. The data give a straight line in a plot of L_φ² vs tan² φ. This shows that the reflection is elliptical and provides a basis for fitting small-angle patterns in an elliptical coordinate system using few parameters. Further analysis of patterns that extend to large values of φ shows that an elliptical fit is not merely a convenience; it is also the best fit. Many possible structures could cause the lamellar reflection to lie on a curve, but few models predict an ellipse. The simplest is affine deformation of the lamellar structure. Recognizing that a single lamellar population will give reflections of an elliptical form allows populations of lamellae to be distinguished even when the reflections overlap strongly, with long spacings L_M different by only 2-8%. This permits a greater understanding of the processes of fiber drawing and annealing.