Semi-flexible polymers

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 conformational states of semi-flexible chains can be represented by the disorder parameter/, defined as 

Why can the lattice model calculate the mixing free energy of polymer solutions  

Why are two species of non-polar polymer chains not easy to mix with each other  

Why is the shear flow easy to induce phase separation in the multi-component polymer systems  

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J. S. Pedersen, M. Laso, P. Schurtenberger. Monte Carlo study of excluded volume effects in worm-like micelles and semi-flexible polymers. Phys Rev E 54 R5917-R5920, 1996. 

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 discussion of the influence of the interphase need not be limited to just linear polyethylenes. Interphases of several nm have been reported in polyesters and poly-hydroxy alkanoates. One major difference between the interphase of a flexible polymer like polyethylene and semi-flexible polymers like PET, PEN and PBT is the absence of regular chain folding in the latter materials. The interphase in these semi-flexible polymers is often defined as the rigid amorphous phase (or rigid amorphous fraction, RAF) existing between the crystalline and amorphous phases. The presence of the interphase is more easily discerned in these semi-flexible polymers containing phenylene groups, such as polyesters. 

The other and most popular way is to copolymerize the flexible segment into the polymer main chains. De Gennes (1975) predicted that incorporation of both a rigid and a flexible segment in the repeating unit should afford semi-flexible polymers exhibiting thermotropic liquid crystallinity,... 

Interestingly, it has been shown that the collapse transition of semi-flexible polymers leads to substantial and cooperative local ordering in the compact state [43,67,68]. 

In this section, we consider the case of solutions of rigid or semi-flexible polymers which display one or several liquid crystalline phases in a given range of concentration. The main control parameter is not flie temperature as is the case for thermotropic LCPs but rather the concentration of polymer in the solvent. There are many different kinds of lyotropic LCPs. Some are synthetic like Kevlar which has become a very important structural material with mechanical properties comparable to those of steel. Some are natural like the Tobacco Mosaic Virus and like DNA which shows a nematic and a hexagonal phase. Some are mineral like the vanadium pentoxide ribbons. In the next section, we shall first describe the lyotropic system which is probably best known, namely the tobacco mosaic virus. 

Berry, G. C., Crossover behavior in the viscosity of semi-flexible polymers intermolecular interactions as a function of concentration and molecular weight, J. Rheol., 40, 1129-1154 (1996). 

Up to this point we have confined ourselves to ideally flexible chains. Thus, the theories developed on the models of such chains (for example, the spring-bead chain) should no longer be adequate for polymers whose chemical stmcture suggests considerable stiffness of the chain backbone. Many cheiin models may be used to formulate a theory of stiff or semi-flexible polymers in solution, but the most frequently adopted is the wormlike chain mentioned in Section 1.3 of Chapter 1 it is sometimes called the KP chain. This physical model was introduced long ago by Kratky and Porod [1] to represent cellulosic polymers. However, significant progress in the study of its dilute solution properties, static and dynamic, has occurred in the last two decades. 

The present chapter aims to describe some typical contributions from recent studies on stiff polymers in dilute solution. We will be mainly interested in (1) applicability of the wormlike chain model to actual polymers, (ii) validity of the hydrodynamic theories [2-4] recently developed for this model, and (iii) the onset of the excluded-volume effect on the dimensions of semi-flexible polymers. Yamakawa [5, 6] has generalized the wormlike chain model to one that he named the helical wormlike chain. In a series of papers he and his collaborators have made a great many efforts to formulate its static and dynamic properties in dilute solution. In fact, the theoretical information obtained is now comparable in both breadth and depth to that of the wotmlike chain (see Ref. [6] for an overview). Unfortunately, however, most of the derived expressions are too complex to be of use for quantitative anal) sis and interpretation of experimental data. Thus, we only have a few to be considered with reference to the practical aspects of the helical wormlike chain, and have to be content with mentioning the definition and some basic features of this novel model. 

The most important of recent theoretical studies on semi-flexible polymers is probably the formulation of Yamakawa and Fuji [2,3] for the steady transport coefficients of the wormlike cylinder. This hydrodynamic model, depicted in Figure 5-2, is a smooth cylinder whose centroid obeys the statistics of wormlike chains. In the figure, r denotes the normal radius vector drawn from a contour... 

Many other semi-flexible polymers are known which can be reasonably well modeled on wormlike chains. The Ml values obtained for most of such polymers are close to those calculated from crystallographic data, suggesting that stiff polymers in dilute solution maintain their crystalline conformations at least locally. 

Bohdanecky found that published data on many semi-flexible polymers fit eq 3.5. Figure 5-8 shows this with the data [40] for poly (tere-phthalamide-p-benzohydrazide) (PPAH) in dimethyl sulfoxide (DMSO). The linear relation as observed here permits unequivocal estimation of I and S. If we have one more relation among q, Ml, and d, it becomes possible to determine these three unknowns. Molecular weight dependence data of (5 ) or / may be utilized as such an additional relation. Bohdanecky [39] proposed the use of the relation... 

Unquestionably, Yamakawa and collaborators have made a substantial contribution to the understanding of transport behavior of semi-flexible polymers in dilute solution. However, their theories still leave something to be desired, as revealed by the recent careful experiments mentioned above. Their formulation is essentially the combination of the the Kirkwood-Riseman hydrodynamics and the statistics of wormlike chains. As mentioned in Chapter 2, this hydrodynamics fails to be good for flexible chains, but we have seen that it seems to work well for stiff chains. The reason is that the Kirkwood-Riseman formalism gives the exact solution in the limit of rigid rods. 

For the sake of reference we summarize reported values of q and Ml for some typical semi-flexible polymers in the Appendix. 

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-... 

In Table 1 we present the Zp values determined in THF and two different THE/DME mixtures. These values, on the order of 1 rm, are comparable to those reported by Discher and coworkers [38,70] and by Bates and coworkers [71,72] for PEO-PI cylindrical micelles with a core diameter of 20 nm in water. Here PEG denotes poly(ethylene oxide). Bates and coworkers deduced their values of Zp from small-angle neutron scattering experiments, whereas Discher and coworkers determined the Zp values using fluorescence microscopy. The fact that the Zp values that we determined from viscometry are comparable to those of the PEO-PI cyUndrical micelles with similar core diameters again suggests the validity of the YFY theory in treating the nanofiber viscosity data. This study demonstrates that block copolymer nanofibers have dilute solution properties similar to those of semi-flexible polymer chains. 

Another property of semi-flexible polymer chains is the formation of nematic phases in concentrated solution. According to the theories of On-sager [73] and Flory [74], polymer chains with Zp/dh > 6 should form a liquid crystalline phase above a critical concentration. We were able to show the presence of such a liquid crystalline phase by polarized optical microscopy for PS-PCEMA nanofibers dissolved in bromoform at concentrations above 25 wt. % [75] Furthermore, we observed that these hquid crystalline phases disappeared when these solutions were heated to a temperature above a well-defined liquid-crystalline-to-disorder transition temperature, Tij. Such observations suggest nanofibers have concentrated solution properties similar to those of semi-flexible polymers. 

The static flexibility of a semi-flexible polymer chain is related not only to the potential energy difference Ae, but also to the temperature T. The following quantities are often used to characterize the conformational states of semi-flexible polymer chains. 

Kusner, I. and Srebnik, S. 2006. Conformational behavior of semi-flexible polymers confined to a cylindrical surface. Chemical Physics letters 430 84—88. 

MONTE CARLO SIMULATIONS OF SEMI-FLEXIBLE POLYMERS... 

Monte Carlo simulations of semi-flexible polymers 171... 

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