A scattering function for correlated lamellae

Melt crystallized polymers display an emergent, multi-hierarchical, ordered structure made up of stacked lamellar single crystals that form fibrous or other meso structures which, in turn, form macroscopic crystallites. A dominant feature of small-angle scattering from these complex assemblies is a correlation peak associated with the stacking period. A new model-based function is proposed for small-angle scattering data from such correlated lamellar multi-hierarchical structures. Generally, routine use of scattering data has been limited to a 1-d analysis to determine the long period from Lorentz corrected data (I(q)q(2) versus q). Fourier transforms of the data are sometimes used to determine the 1-d pairwise correlation function for the electron-density distribution which has been further analyzed in terms of the structure of these materials. A simple 1-d fitting model limited to infinite width 2-d sheets was introduced by Hermans (1944; Hosemann, 1950) [1,2] in the 1940s with some success. A new approach, the Unified Born-Green Function (UBG), is proposed that uses the Unified Function as adapted to correlated lamellar structures and incorporates a Born-Green description of one-dimensional correlations. The UBG fit allows quantification of the average lamellar aspect ratio, the local degree of crystallinity within a stack, quantification of the stacking versus non-stacking amorphous, and two types of disorder in addition to the stacking period and lamellar thickness. UBG can account for higher levels of structure such as crystalline domains in block copolymers and convoluted lamellar structure. The UBG fit is compared to the Hermans (1944; Hosemann, 1950) [1,2] and a hybrid-Hermans function. Fits to data sets from a wide range of polyethylene are shown ranging from molecular weight standard samples that are isothermally crystallized, to commercial HDPE quenched from the melt and a metallocene blown film sample. Several other examples from the literature are explored. It is shown that the Unified fit allows for new understanding of the impact of thermal and mechanical history, chain structure, fillers, nucleating agents, and additives on the crystalline structure and the resulting physical properties. Limitations of the UBG approach are noted.