Crystal stability and mechanical properties of drawn polymers: A statistical‐mechanical approach

Abstract

AbstractThe partition function is formulated by the generating function method for a stacked lamellar model of alternating crystalline and amorphous layers. The random‐walk problem of enumerating statistical weights for conformations of amorphous chains confined by two parallel walls is solved for the body‐centered cubic lattice as a generalization of the one‐wall model treated by Roe. The mean lengths of loop, tie, and cilia chains and the free energy of the system are calculated for the random‐reentry and‐bridge model as a function of the distance h of separation between crystal layers and as a function of the number N of loop chains in a crystal block as the basic structural element of the system. The mean length of amorphous chains decreases at a given thickness l of the crystal layers with decreasing h or with increasing N. The free energy of the system exhibits no minimum with respect to N, showing that the folded‐chain crystal is thermodynamically stable, especially for relatively small l. Additionally, it is shown that another requirement for stabilizing relatively small crystals (small l) is the formation of an aggregate structure of crystals, whereas a large single‐crystal (large l) is relatively stable, irrespective of h and N. Furthermore, a theoretical model is developed to calculate the force and elastic modulus of a highly deformed stacked system, assuming that the only change in the configurations of amorphous chains within the interlamellar regions is due to deformation, except for scission of tie chains having fewer segments than are needed to span the interlamellar distance of the deformed system. It becomes evident that taut tie chains are effective in increasing the modulus of the stacked system.