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In this review, we consider a variety of aspects of polymer crystallization using a very simple lattice model. This model has three ingredients that give it the necessary flexibility to account for many features of polymer crystallization that have been observed experimentally. These ingredients are (1) a difference in attraction between neighboring (nonbonded) components, (2) attraction between parallel bonds, and (3) temperature-dependent flexibility due to the energy cost associated with kinks in the polymer chain. We consider this model using both dynamic Monte Carlo simulations and a simple mean-field theory. In particular, we focus on the interplay of polymer crystallization and liquid--liquid demixing in polymer solutions. In addition, we study the factors that are responsible for the characteristic crystal morphologies observed in a variety of homopolymer and statistical-copolymer crystals. Finally, we consider how the freezing of polymers in the bulk can be related to the crystallization of a single polymer chain.

The molecular mechanism of polymer crystallization is one of the most difficult problems and has defied innumerable efforts to understand the process over the last fifty years in spite of its great importance both from the academic and the industrial point of view. We have been studying this historical problem by use of the molecular dynamics simulation method. In this chapter of the book, we review our recent work on the crystal growth of polymers with special focus on polymer behavior at the crystal surface, either at crystal-vapor or crystal-melt interfaces. Our starting molecular model is a bead-spring chain, or a wormlike chain, made of methylene-like united atoms; the zigzag structure of polymethylene is here neglected in order to accelerate crystallization. We proceed with stepwise revisions of the model toward the realistic modeling of polymer crystallization from the dense melt. We start our discussion with the crystallization of polymers on a two-dimensional surface, which is a model of the chain strongly adsorbed on the growth surface. Then we treat the three-dimensional process of crystallization of a single chain from a vapor phase: the adsorption to and the ordering on the growth substrate. Lastly, polymer crystallization from the dense melt is investigated. We also report on fiber formation from a highly oriented amorphous state. Various important issues concerning the molecular mechanism of polymer crystallization are discussed in the light of findings from our direct molecular simulations.

We distinguish three main modes of crystallization for polymers with a relatively flexible main-chain, i.e., (i) usual lamellar crystallization occurring by cooling from the reference state (melt or solution) above the temperature T _0 down to T  >  T _ g ; (ii) crystallization from the glass; (iii) crystallization from a stable thermotropic mesophase. In all three cases we propose that structure development proceeds via high entropy pre-crystalline aggregates, which may influence features of the crystalline organization. Pre-crystalline structures characteristic of modes (i) and (ii) are identified with bundles, i.e., energy-driven hexagonal associations among chain segments. At T  <  T _0 the polymer solution is regarded as meta-stable, and in this state the bundle segments are essentially consecutive whereas in the melt and the glass bundles also comprise non-consecutive chain segments. The fold thickness L observed in lamellar crystallization, resulting from bundle aggregation and rearrangement, is basically controlled by the average fold length in the consecutive chain portions within bundles. For small values of Δ T (=  T _0 −  T _crystallization) we obtain L  ∝ 1/∆ T , in agreement with experimental data from polyethylene as well as with several simulation results; the proportionality factor appears to be the same for the solution and the melt. The bundle model appears to be consistent with indirect evidence such as segregation of short chains in the crystallization process and clustering of segments belonging to the same chain in the crystal. In mode (ii) it is plausible, at least in certain instances, that crystallization is preceded by bundle aggregation leading to phase separation. In the case of crystallization in mode (iii) we can identify the pre-crystalline high entropy state with the thermotropic mesophase itself. Such phases involve large domains of parallel, hexagonally packed, conformationally disordered chains, with a high propensity to fully extended macroconformations. They occur with polymers with a large persistence length of entropic (i.e., elastic) origin, mainly due to conformational disorder of the side groups. Folds and hairpins in these mesophases are energetically disfavored because adequate compensatory inter-stem attractions are missing. Finally, it is shown that crystallization of helical non-chiral polymers into crystalline modifications comprising isochiral helices only, may in certain cases be accounted for on the basis of hexagonal pre-crystalline intermediates like bundles and mesophases discussed in the present contribution.

Direct evidence of nucleation during the induction period of nucleation from the melt is obtained for the first time by means of small angle X-ray scattering (SAXS). This confirmed that the induction period of crystallization from the melt corresponds to the process of nucleation, not to that of spinodal decomposition. This success is due to a significant increase in the scattering intensity ( I _x) from the nuclei (10^4 times as large as is normal), which was achieved by adding a nucleating agent (NA) to a “model polymer” of polyethylene (PE). I _x increased soon after quenching to the crystallization temperature ( T _c) and saturated after the induction time (τ_i). Lamellae start stacking later than the M _n. Power laws of the molecular weight ( M _n) dependence of the primary nucleation rate ( I ) and the growth rate ( V ) of PE, i.e., I or V  ∝  M _n^−H where H is a constant, were found for both morphologies of folded chain crystals (FCCs) and extended chain crystals (ECCs). As the power law was also confirmed on isotactic polypropylene (iPP), universality of the power law is suggested. It is to be noted that the power H increases significantly with increase of the degree of order of the crystal structure. The power law confirms that the topological nature of polymer chains, such as chain sliding diffusion and the chain entanglement within the interface between the nucleus and the melt or those within a nucleus, adopts a most important role in the nucleation and growth of polymers. This is theoretically explained by improving the “chain sliding diffusion theory” proposed by Hikosaka. Entanglement dependence of the nucleation rate I is qualitatively obtained for the first time by changing the number density of entanglement (ν_e) within the melt. An experimental formula of I as a function of ν_e was obtained on PE, I (ν_e) ∝ exp(−γν_e) where γis a constant.

This paper reviews the authors' investigation into polymer crystallization, especially involving a spinodal decomposition (SD) type phase separation due to the orientation fluctuation of stiff segments prior to crystal nucleation. Evidences for SD obtained from small-angle X-ray and neutron scattering (SAXS and SANS), depolarized light scattering (DPLS), Fourier-transform infrared spectroscopy (FT-IR) are discussed in detail in the case of the glass crystallization of poly(ethylene terephthalate) (PET) just above T _g. SD-like optical micrographs are also shown as a function of crystallization temperature for the melt crystallization of PET; their characteristic wavelengths Λ, which are of the order of μm above 120 °C, follow a van Aartsen equation derived from the Cahn–Hilliard theory for SD. By fitting the equation to the observed characteristic wavelengths the spinodal temperature T _s was determined to be T _s = 213 ± 5 °Cfor the PET melt, above which the SD pattern suddenly changed to the usual spherulite pattern. On the basis of a theory by Olmsted et al. [ 4 ], the general mechanisms of polymer crystallization are also discussed; the crystallization from the metastable melt causes the nucleation and growth (N&G) of dense (nematic) domains while that from the unstable melt causes SD into the dense (nematic) and less dense (isotropic) domains. Furthermore, the secondary phase separation of the SD-type phase separation into smectic and amorphous domains subsequently occurs inside the nematic domain for both these cases.

We summarize the salient conclusions derived from Langevin dynamics simulations of many flexible polymer molecules undergoing crystallization from solutions. These simulations reveal molecular mechanisms of nucleation and growth, and the accompanying free energy barriers, during the very early stages of crystallization. The simulation results are also analyzed by statistical mechanics theories. Major conclusions on the growth of density fluctuations in the primordial stage, birth of baby nuclei, which then mature into lamellae through a stage of smectic pearls, and spontaneous selection of finite equilibrium lamellar thickness are addressed. Furthermore, selection of shapes is addressed using a novel Monte Carlo algorithm for polymer crystallization in solutions. In addition, details of free energy landscape just in front of the growth front are summarized, based on Langevin dynamics simulations. The mechanism of growth is seen to be an adsorption process, in contrast to previous beliefs. Finally, the role of externally imposed flow on polymer crystallization is addressed by considering the molecular mechanisms behind the formation of shish-kebab morphology in extensional flows. The major conclusions from the reviewed simulation results are qualitatively different from the established models of polymer crystallization.