Modelling the amorphous phase and the fold surface of a semicrystalline polymer—the Gambler's Ruin method

Abstract

A semicrystalline polymer with lamellar morphology consists of alternating amorphous and crystalline regions. If sufficiently long, each molecule in this system traverses both the crystalline and amorphous zones. The amorphous portion is comprised of portions of a molecule that form loops that re-enter the same lamella at some distance from the point of emergence, and bridges that form connections between two different crystal lamellae. (A tight fold is not considered to be a loop). The statistics of loops and bridges are shown to be identical to the classical Gambler's Ruin problem in mathematical statistics. This is a useful observation because the extensive existing literature on the Gambler's Ruin problem allows us immediately to transcribe results to the polymer system. Using this approach, the ratio of the number of loops to the number of bridges is determined to be M, the thickness of the amorphous zone in unit statistical steps. Also, the average number of steps comprising the amorphous run is determined to be 3 M+1 for a simple cubic lattice in three dimensions. This modelling leads to a calculation of the minimal fraction of crystal stems involved in tight folding in a semicrystalline polymer. For a simple cubic lattice this is found to be 2 3 . The effects of crystal structure and stiffness of the chain in the melt on this bound are discussed.