Theory of statistics of ties, loops, and tails in semicrystalline polymers

Polymer crystals grown from melts consist of alternating lamellar crystalline regions and amorphous regions. We study the statistics of ties: chains which bridge the adjacent lamellae, loops: chains which come out of one lamella and enter back into the same lamella before reaching the other lamellae, and tails: chains which end in an amorphous region. We develop a theory to calculate the probabilities of formation of ties, loops, and tails with consideration of finite chain length and cooperative incorporation of a chain into multiple lamellae. The results of our numerical calculations based on a field-theoretic formalism show that the fraction of ties increases with increasing chain length, and it decreases with increasing interlamellar separation. In the limiting case of an infinite chain confined between only two walls, we recover the classical results of the gambler's ruin model. We show that the density anomaly encountered in previous theories is avoided naturally in the present theory without forcing the majority of stems to form tight loops. The derived results on the probability of tie chains in the amorphous regions are pertinent to the mechanical properties of semicrystalline polymers.