Single chain crystal

In the following I consider only the case p > 1, which is the physical relevant solution for single chain crystals. The scaling variable can be related to the average loop length in the amorphous fraction... 

The only part which is left to prove is the stability of the extended chain form with respect to the folded chain form. This can be easily inferred from the following Gedankenexperiment as sketched in Fig. 2.7. Let us consider a single chain crystal formed by a huge chain of length N in thermodynamic... 

Fig. 2 Particle diameter distribution of i-PS single-chain crystals...

Selected-area electron diffraction was carried out with the Philips EM-420 for the larger crystal which may contain several tens of chains and for simultaneous analysis of a large number of single-chain crystals. Micro-beam electron diffraction was performed on isolated single-chain crystal. For the latter, the electron beam was concentrated to a spot with a diameter of 30-40 nm, to cover no more than a single-chain ciystal. 

Periodic strips were found on the top surface of single-chain crystal with an average periodic length of 0.52 nm between strips, by using scanning tunnelling microscopy. Based on the hexagonal shape of... 

The partition function and free energy of the single-chain crystal are calculated. From the calculation of free energy, the equilibrium thickness of the single-chain crystal and melting temperature corresponding to the equilibrium state of the minimum free energy can be obtained. 

Polymer crystals most commonly take the form of folded-chain lamellae. Figure 3 sketches single polymer crystals grown from dilute solution and illustrates two possible modes of chain re-entry. Similar stmctures exist in bulk-crystallized polymers, although the lamellae are usually thicker. Individual lamellae are held together by tie molecules that pass irregularly between lamellae. This explains why it is difficult to obtain a completely crystalline polymer. Tie molecules and material in the folds at the lamellae surfaces cannot readily fit into a lattice. 

Fig. 63. Molecular arrangement in (a, c) plane of a mixed ethylene-chlorine binary crystal illustrating (a) radical pair formation, (b) single chain growth and (c) chain growth in the vicinity of product line. Molecules labelled 1-4 are ethylene (C2H4), chlorine, chloroethyl radical (C2H4CI) and anti 1,2-dichloroethane (C2H4CI2), respectively.

The single crystal of a polymer is a lamellar structure with a thin plateletlike form, and the chain runs perpendicular to the lamella. The crystal is thinner than the polymer chain length. The chain folds back and forth on the top and bottom surfaces. Since the fold costs extra energy, this folded chain crystal (FCC) should be metastable with respect to the thermodynamically more stable extended chain crystal (ECC) without folds. 

The most widely used method for preparing extended-chain crystals involves solid-phase polymerization when the monomer exists as a single crystal. The polymerization of single crystals of the monomer permits the preparation of a polymer material with a maximum orientation a polymeric single crystal composed of fully extended macromolecules. Such polymer crystals are needle-shaped22. ... 

It should be noted that the concept extended-chain crystal (ECC) does not mean a crystal, in which not a single molecule can form a told. ECC can contain some molecules with folded conformation but the folds play a role of defects... 

In what follows, we use simple mean-field theories to predict polymer phase diagrams and then use numerical simulations to study the kinetics of polymer crystallization behaviors and the morphologies of the resulting polymer crystals. More specifically, in the molecular driving forces for the crystallization of statistical copolymers, the distinction of comonomer sequences from monomer sequences can be represented by the absence (presence) of parallel attractions. We also devote considerable attention to the study of the free-energy landscape of single-chain homopolymer crystallites. For readers interested in the computational techniques that we used, we provide a detailed description in the Appendix. ... 

Thus far, we have been discussing the crystallization of a multichain system. However, under suitable conditions, crystallization can even occur in a single-chain system. Using a combination of biased sampling, multihistogram techniques, and parallel tempering [ 125], we can directly compute the... 

Fig. 16 Height of the equilibrium free-energy barrier for melting and crystallization vs. the chain length in single-chain systems. The circles are the simulation results, and the solid line is calculated from Eq. 20 with fitting parameter a = 15 p [127]... 

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