Many chain crystals

In this work, I consider several aspects of the equiUbrium state of polymer single crystals using the model proposed by Muthukumar which will be extended to multi-chain crystals. In particular I will show that extended chain crystals are the equilibrium form for many chain crystals if sufficiently many chains are accessible and I will give a simple argument for their thermodynamic stability with reference to folded chain crystals. Furthermore, the role of finite flexibiUty of chains is discussed as well as the tilt of stems in extended chain crystals. 

The Pink model is found to exhibit a gel-fluid transition for lipids with sufficiently long chains, which is weakly first order. The transition disappears in bilayers of shorter lipids, but it leaves a signature in that one observes strong lateral density fluctuations in a narrow temperature region [200,201]. In later studies, the model has been extended in many ways in order to explore various aspects of gel-fluid transitions [202]. For example, Mouritsen et al. [203] have investigated the interplay between chain melting and chain crystallization by coupling a two-state Doniach model or a ten-state Pink model to a Potts model. (The use of Potts models as models for... 

Usually, synthetic polymers crystallize11 j15 from a melt or a solution in form of folded lamellae. Under specific circumstances it is sometimes also possible to obtain extended chain crystals which is the preferred arrangement in the crystallites of many natural polymers (cellulose, silk). Recently it has been found33 31 that in some cases another crystalline modification can be obtained, the so-called shish-kebabs, which are a sort of hybrid between folded lamellae and extended chain crystals. These shish-kebabs are obtained by shear-induced crystallization, a process in which the polymer crystallizes from solution under the influence of an elongated flow. 

Many copolymers are said to be internally plastidzed because of the flexibilization brought about by the presence of a second repeating unit in the polymer chain. In contrast. DOP and other liquid plastidzers are said to be external plastidzers. The presence of bulky pendant groups on the polymer increases segmental motion, and the flexibility of the polymer increases as the size of the pendant group increases. However, linear pendant groups with more than 10 carbon atoms reduce flexibility because of side chain crystallization. 

X-ray diffraction analysis can be used to characterize the crystallites themselves.15,68 First, it is possible to obtain information about the chain conformation adopted by the polymer chains in the crystallites. Many polymers crystallize in helical conformations. These can be defined by the following helical parameters. The designation pn tells how to rotate and simultaneously translate along an axis to generate the specified helix. First a point of reference is rotated around the proposed axis by nip times a complete rotation of In. Simultaneously, there should be a translation of nip of the crystallographic repeat distance, a quantity that is also obtained from diffraction data. Repetition of this scheme gives a helix having p repeat units in n turns of the helix. 

Very high molar mass polyethylene (M>2,000,000 g mol-1) crystallises without the, formation of a clear superstructure, sometimes referred to as the random lamellar structure [116]. The great many chain entanglements present in high molar mass polymers obstruct crystallisation and the crystals become small and their orientation less correlated with surrounding crystal lamellae. 

Extended chain crystals have been implicitly included in the treatment on lattice visualization of crystallized homopolymers. While crystals obtained by crystallization under extension may be imperfect compared to the classical examples of high-pressure crystallization [61], these examples are closer to many applications. The morphology of fibrils and microfibrils of PTFE has been analyzed by CM-AFM (Fig. 3.14). A second example discussed below refers to cold-drawn PET (Fig. 3.17), which is crystallized in extended chain crystals. 

Many inorganic crystals contain 1-dimensional chains of atoms that endow the material with interesting electronic properties. Using a computer, plot the electron density function ... 

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