Statistical crystalline polymer chain

Balijepalli S and Rutledge G C (2000) Conformational statistics of polymer chains in the interphase of semi-crystalline polymers, Comput Theor Polym Sci 10 103-113. 

Ti numbers of statistical segments of the crystalline polymer chain... 

Structurally, plastomers straddle the property range between elastomers and plastics. Plastomers inherently contain some level of crystallinity due to the predominant monomer in a crystalline sequence within the polymer chains. The most common type of this residual crystallinity is ethylene (for ethylene-predominant plastomers or E-plastomers) or isotactic propylene in meso (or m) sequences (for propylene-predominant plastomers or P-plastomers). Uninterrupted sequences of these monomers crystallize into periodic strucmres, which form crystalline lamellae. Plastomers contain in addition at least one monomer, which interrupts this sequencing of crystalline mers. This may be a monomer too large to fit into the crystal lattice. An example is the incorporation of 1-octene into a polyethylene chain. The residual hexyl side chain provides a site for the dislocation of the periodic structure required for crystals to be formed. Another example would be the incorporation of a stereo error in the insertion of propylene. Thus, a propylene insertion with an r dyad leads similarly to a dislocation in the periodic structure required for the formation of an iPP crystal. In uniformly back-mixed polymerization processes, with a single discrete polymerization catalyst, the incorporation of these intermptions is statistical and controlled by the kinetics of the polymerization process. These statistics are known as reactivity ratios. 

A polymer stretched out to its full contour length is only one of the myriad conformations possible for a polymer at temperatures above Tg, or if the polymer is entirely crystalline. The chain length is expressed statistically as the RMS distance, which is only a fraction of the contour length. 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

This article deals with some topics of the statistical physics of liquid-crystalline phase in the solutions of stiff chain macromolecules. These topics include the problem of the phase diagram for the liquid-crystalline transition in die solutions of completely stiff macromolecules (rigid rods) conditions of formation of the liquid-crystalline phase in the solutions ofsemiflexible macromolecules possibility of the intramolecular liquid-crystalline ordering in semiflexible macromolecules structure of intramolecular liquid crystals and dependence of die properties of the liquid-crystalline phase on the microstructure of the polymer chain. 

Some examples of stiff-chain polymers able to form a liquid-crystalline phase in the solution are listed in Table l1. The ratio of the statistical segment length1 of a polymer chain, 1, to its width, d, (last column of Table 1) measures the degree of chain stiffness. For flexible macromolecules fid 1 stiff-chain macromolecules are those for which fid t> 1. 

These factors were mentioned soon after the first experimental works on liquid crystalline state of polymers were published. For example, Frenkel considered the effect of these factors by studying the change in the statistical flexibility parameter/ introduced by Flory in his analysis of flexible chain polymers Frenkel proposed the following equation to describe the effect of the type of the solvent on the flexibility of a polymer chain ... 

The path integral technique was first proposed by Feynmann (Feynmann Hibbs, 1965). The purpose of this technique was to deal with questions in quantum mechanics. It has been applied to the study of the statistical mechanics of polymer systems (Kreed, 1972 Doi Edwards, 1986) and liquid crystalline polymers as well (Jahnig, 1981 Warner et al, 1985 Wang Warner, 1986). The path integrals relate the configurations of a polymer chain to the paths of a particle when the particle is undergoing Brownian or diffusive motion. 

One of the advantages is that this non-homogeneous model connects the statistical properties of main chain liquid crystalline polymers to their molecular parameters. 

This chapter lays the groundwork for the various topics discussed in subsequent chapters. The mathematics associated with the statistics of the isolated chain are developed starting with the bonding and structure found in small molecules. Several models of chain structure are presented. Finally, the size distribution of polymer chains is introduced and their description in terms of mathematical equations derived, origin of rubber elasticity, the nature of polymer crystalline and polymeric heat capacities and the miscibility of polyblends. 

For the discussion of the results it has to be emphasized that the Kuhn-Griin relation is based on the application of the Lorenz-Lorentz-equation. It has already been discussed earlier that the assumption of a spherical internal field could not be valid for extended non-spherical statistical segments of polymer chains The strong influence of different internal fields on the sign of the stress optical coefficient has been described by Pietralla The same problem arises for liquid crystalline phases and has been discussed in detail for low molar mass LCs whereby the validity of the Lorenz -Lorentz equation in the isotropic state still has been assumed... 

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