Semi-flexible polymer molecules

With Mw higher than 3xl05, the experimental points could be fitted by a smooth convex curve. The slope of the convex curve was about 1.54 for Mw between 5xl05 and lxlO6 and about 1.15 for Mw between 3xl06 and 3xl07. This change in slope implies that the molecule is rodlike at lower MW and approaches a spherical coil as Mw increases, which is the characteristic behavior of semi-flexible polymers. 

The rearrangement of more than one inner site in a flexible or semi-flexible chain molecule can be conveniently performed if the geometric constraints that guarantee chain closure are taken into account every time a site is repositioned [50] (see Fig. 1). Performance can be enhanced by favoring low-energy trial positions for each growing site (extended continuum configurational bias, or ECCB method). Since this method can be applied to inner sites of arbitrary functionality, it has been used to study linear, branched, and crosslinked polymers [50-52]. 

Chain stiffness prevents the monomer units of a polymer molecule from having contact with one another. Thus, we may expect that there exists for a semi-flexible polymer a certain contour length Lc below which the excluded-volume effect on the chain dimensions disappears (in a statisitical sense). This prediction is borne out by the data of Figure 5-3, which show that (5 )/M follows the curve for an unperturbed wormlike chain until Mw reaches 3 x 10 (L 4 X 10 ). Obviously, Lc ought to be larger for a stiffer polymer, i.e., one with larger persistence length q. 

As one-dimensional objects, cylindrical micelles and polymer nanotubes have many features in common with semi-flexible polymer chains, but on a different size scale. Nanotubes tend to be longer, thicker, and more rigid than individual polymer molecules, but both are characterized by a distribution of end-to-end lengths, a radius of gyration, and a persistence length. Figure 9 compares the structures of a poly(n-hexyl isocyanate) or PHIC chain, a PS-... 

Fetters LJ, Lohse DJ, Graessley WW (1999) Chain dimension and entanglement spacings in dense macromolecnlar systems. J Polym Sci, Polym Phys Ed 37 1023-1033 Fischer EW, Bakai A, Patkowski AW, Steffen W, Reinhardt L (2002) Hettaophase fluctuations in supercooled liquids and polymers. J Non-Cryst Solids 307-310 584-601 Flory PJ (1956) Statistical thcamodynamics of semi-flexible chain molecules. Proc R Soc London A234 60-73... 

Flory PJ (1941) Thermodynamics of high polymer solutions. J Chem Phys 9(8) 660 Flory PJ (1942) Thermodynamics of high polymer solutions. J Chem Phys 10(1) 51-61 Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, Ithaca Flory PJ (1955) Theory of crystallization in copolymers. Trans Faraday Soc 51 848-857 Flory PJ (1956) Statistical thermodynamics of semi-flexible chain molecules. Proc R Soc Lond A Math Phys Sci 234(1196) 60-73... 

The rheological behaviour of polymeric solutions is strongly influenced by the conformation of the polymer. In principle one has to deal with three different conformations, namely (1) random coil polymers (2) semi-flexible rod-like macromolecules and (2) rigid rods. It is easily understood that the hydrody-namically effective volume increases in the sequence mentioned, i.e. molecules with an equal degree of polymerisation exhibit drastically larger viscosities in a rod-like conformation than as statistical coil molecules. An experimental parameter, easily determined, for the conformation of a polymer is the exponent a of the Mark-Houwink relationship [25,26]. In the case of coiled polymers a is between 0.5 and 0.9,semi-flexible rods exhibit values between 1 and 1.3, whereas for an ideal rod the intrinsic viscosity is found to be proportional to M2. 

The conformation parameter a (=A/Af, where Af is A of a hypothetical chain with free internal rotation) for cellulose and its derivatives lies between 2.8-7.5 2 119,120) and the characteristic ratio ( = A2Mb//2, where Ax is the asymptotic value of A at infinite molecular weight, Mb is the mean molecular weight per skeletal bond, and / the mean bond length) is in the range 19-115. These unexpectedly large values of a and Cffi suggest that the molecules of cellulose and its derivatives behave as semi-flexible or even inflexible chains. For inflexible polymers, analysis of dilute solution properties by the pearl necklace model becomes theoretically inadequate. Thus, the applicability of this model to cellulose and its derivatives in solution should be carefully examined. 

In 1956, Hory introduced the semi-flexibility into the classical lattice statistical thermodynamic theory of polymer solutions (Flory 1956). From the classical lattice statistics of flexible polymers, we have derived the total number of ways to arrange polymer chains in a lattice space, as given by (8.15). The first two terms on the right-hand side of that equation are the combinational entropy between polymers and solvent molecules, and the last three terms belong to polymer conformational entropy. Thus the contribution of polymer conformation in the total partition function is... 

The regularity of the polymer backbone is the key factor isotactic polypropylene crystallizes forming a rigid stable solid, whereas atactic polypropylene does not and forms a rubbery elastic solid. For flexible polymers, the structure of the solid is dictated by the symmetry of the polymer backbone. For the formation of a semi-crystalline solid it is necessary for there to be either an element of symmetry in the repeat unit chemical structure or strong interactions to aid the packing of the molecule and initiate the alignment that is required for the crystal growth process. 

Models for Stiff-Chain Polymers.— Flory briefly reviewed some of the consequences of separating the configurational partition function for long chain molecules and their solutions into inter- and intra-molecular parts. In particular, he pointed out that this separation, and hence the partition function derived therefrom, are valid only for sufficiently flexible chains, or when the polymer concentration is sufiiciently low. He indicates how this can be rectified in statistical mechanical models for semi-rigid and rod-like polymer molecules. For the latter case this is pursued in considerable detail in a very recent series of papers by Flory and co-workers. ... 

Linear polymers in the semi-crystalline state are metastable nanostructured systems with the complicated morphology, which are divided into nano-, submicro-, or microphases with crystalline, amorphous, and intermediate (mesophase and other) molecular packing. These different phases are connected in the flexible-chain polymers, such as PE, POM, poly(ethylene terephthalate) (PET), and many others, via strong covalent coupling between crystallites and disordered regions since the typical polymer molecules of l-lOOpm in contour length participate in several nanophases. Due to the multilevel structure, polymers with rather high levels of crystallinity may show up unique dynamics and properties which vary with the thermal and mechanical histories of materials. This has been confirmed by different techniques (DMA, DSC, NMR, DRS, and others) in numerous studies. 

Flory et al. considered only the shape anisotropy of the rod-like molecules, they derived expressions for the phase equilibrium in mixtures of low molar mass mesogens with rod-like polymers, semi-rigid polymers, and flexible coil polymers ... 

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