The role of the amorphous fraction for the equilibrium shape of polymer single crystals

The equilibrium state of polymer single crystals is considered by explicitly taking into account the amorphous fraction formed by loops and tails of the chains using a statistical model introduced by Muthukumar (Philos. Trans. R. Soc. London, Ser. A 361, 539 (2003)). We show that under realistic conditions below the equilibrium melting temperature, tight loops and close re-entries are favored, and that the amorphous fraction can be mapped into an excess surface free energy. The model is extended to many-chain crystals where it is shown that the lamellar thickness increases with the number of chains in the crystal and extended-chain conformations are thermodynamically favored if the number of chains in the crystal is sufficiently large. The number of chains necessary to form an extended-chain crystal in thermodynamic equilibrium scales with the square of the degree of polymerization of the chains. We discuss the temperature behavior of the equilibrium crystal thickness in the under-cooled state.