Stretching Individual Chains

Let us now do the following experiment. We take a single polymer at both ends and stretch it To do so, we have to do work because as the polymer is stretched its entropy is decreased or, in other words, its structural order increases. The number of possible configurations a polymer chain can assume is reduced when it is forced to stretch. Thus, to stretch a polymer, a force is required. This force has been calculated. For freely jointed chain, it is given by [1321, 1322] 

x is the distance between the two ends and F is the force appUed. U nfortunately, Eq. (11.8) cannot be written in an explicit form in which the force is given as a function of the distance. Only for very low and very high forces can the limits be expressed expUdtly. An expression for low forces is derived by writing the coth in a series for FIs C ksT [1321]  

We added a factor 2 on the right-hand side. Ab initio calculations and a comparison with experimental results showed that this leads to a more realistic description of highly stretched polymers [1323]. 

The segment elasticity is typically of the order of 10 Nm . To obtain an even better description of the stretching behavior at high forces, ab initio quantum chemical 

Kq takes into account an elastic stretching of the chain it corresponds to igks in the freely jointed chain model. Usually, Ko is assumed to be large and the last term is neglected. 

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The effect of solvent quality on the thickness of the layer L and free energy of the individual chains is demonstrated by the work of Milner et al. (1988) and Halperin (1988). The result for the chain configurations comprises an asymptotic solution of Eq. (31) for highly stretched chains, i.e., L n1/2l, via the WKB approximation. This yields the end segment probability as... 

Fig. 16 is casted into a simple but quantitative lattice algorithm. The basic idea is that individual chains successively increase their internal order (characterized by the degree of chain folding) during the crystallization process. The more the chain is ordered (the fewer folds it has) the lower is the surface area needed for this chain. The ultimate degree of order is represented by the completely stretched chain which only occupies a surface area proportional to the cross-section of one stem ao (area of a crystalUne unit cell), see Fig. 16. Let Ao be the area of the corresponding liquid chain, flatly adsorbed onto the surface. Then, M = Aq/ao N > 1 chains can occupy the same area Ao in the crystalUne state. By contrast, in a simple growth model [27] the area per particle remains constant and it is the original dilution of particles which is responsible for the various diffusion-controlled patterns [28,55].

With this, the work of stretching W = NkT As rc/nl) for an assembly of chains N per unit volume, where As rc/nl) of the individual chain is given by eq. (6.45), and the defining expression for the nominal stress t2 (per initial area)... 

Orientations in elongated mbbers are sometimes regular to the extent that there is local crystallization of individual chain segments (e.g., in natural rubber). X-ray diffraction patterns of such samples are very similar to those obtained from stretched fibers. The following synthetic polymers are of technical relevance as mbbers poly(acrylic ester)s, polybutadienes, polyisoprenes, polychloroprenes, butadiene/styrene copolymers, styrene/butadiene/styrene tri-block-copolymers (also hydrogenated), butadiene/acrylonitrile copolymers (also hydrogenated), ethylene/propylene co- and terpolymers (with non-conjugated dienes (e.g., ethylidene norbomene)), ethylene/vinyl acetate copolymers, ethyl-ene/methacrylic acid copolymers (ionomers), polyisobutylene (and copolymers with isoprene), chlorinated polyethylenes, chlorosulfonated polyethylenes, polyurethanes, silicones, poly(fluoro alkylene)s, poly(alkylene sulfide)s. 

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