Role of entanglement in crystalline polymers 1. Basic theory

When stacked lamellar crystals are formed in melts, entanglements are trapped and condensed in amorphous domains (a-domains); this makes the free energy of the a-domains increase and inhibits crystallization, thus, determines the crystalline structure of the system. Based on the local-knot (LK) model of entanglement proposed by Iwata and Edwards, the entanglement state of the system is described by three parameters, condensation ratio χ, trapping ratio ξ and average number ν̄ of LKs trapped per stem in the a-domains. It is shown that crystallinity wc, average lamellar thickness L̄c and average a-domain thickness L̄a are written in terms of χ, ξ and ν̄ alone; this means that structure of stacked lamellar crystals is determined by entanglement. Particularly, the microscopic structure is determined by how LKs are partitioned in the stems of the a-domains. The equilibrium amorphous ratio in the limit M→∞ (which is called ‘limiting equilibrium amorphous ratio ŵa∞’) is a universal function of a reduced degree of supercooling, τ=(NcΔhm/kBTm0)ΔT/T, where Δhm is the enthalpy of fusion and Nc is the critical chain length of the entanglement transition; this means that ŵa∞(τ) is independent of polymer species, thermal history or morphological properties of the system. Based on this result, a method is proposed to determine trapping ratio ξ experimentally. Magnitudes as well as T- and M-dependence of wc, l̄c and l̄a predicted by the theory agree reasonably with experiments. It is shown that the topological free energy of entanglement accumulated in the a-domains plays an important role in the melting phenomena: For example, folding surface energy σe changes largely from that estimated by the usual Thompson–Gibbs equation, and the abnormal increase of σe with increasing M in stacked lamellar crystals, the phenomenon found by Schultz and Manderkern, is explained by the accumulated topological energy. Mechanism of trapping LKs in the crystallization process is discussed in detail.