Chain deformation

Correlations have been found between certain absorption patterns in the infrared and the concentrations of aromatic and paraffinic carbons given by the ndA/method (see article 3.1.3.). The absorptions at 1600 cm due to vibrations of valence electrons in carbon-carbon bonds in aromatic rings and at 720 cm (see the spectrum in Figure 3.8) due to paraffinic chain deformations are directly related to the aromatic and paraffinic carbon concentrations, respectively. )... 

The Warner function has all the desired asymptotical characteristics, i.e. a linear dependence of f(r) on r at small deformation and a finite length Nlp in the limit of infinite force (Fig. 3). In a non-deterministic flow such as a turbulent flow, it was found useful to model f(r) with an anharmonic oscillator law which permits us to account for the deviation of f(r) from linearity in the intermediate range of chain deformation [34] ... 

In an attempt to describe the behavior at large chain deformations, de Gennes [7] incorporated into the dumbbell model the FENE spring law along with a variable bead friction coefficient which increases linearly with the interbead distance ... 

The molecular models of rubber elasticity relate chain statistics and chain deformation to the deformation of the macroscopic material. The thermodynamic changes, including stress are derived from chain deformation. In this sense, the measurement of geometric changes is fundamental to the theory, constitutes a direct check of the model, and is an unambiguous measure of the mutual consistency of theory and experiment. 

We consider four possible relationships between sample deformation and chain deformation. 

The chain deformation parallel to the direction of stretch is much greater than that in the perpendicular direction. This is true for all models considered here. 

If restricted junction fluctuations are taken into account, the chain deformation is increased, and is more anisotropic. The effect of increasing k is much more evident in networks of low functionality, since fluctuations of junction points are of minor importance in networks of high functionality. 

If network unfolding takes place so that distances between junctions connecting the ends of a polymer chain deform less than that of a phantom network, molecular dimensions change less than by any other of the models considered. This is easily seen from the data presented for a not equal to zero. 

The experimental error was very large, with the apparent chain deformation greater than that expected for a phantom network, and closest to the curve anticipated where crosslink fluctuations are completely suppressed. 

C. C. Han, H. Yu and their colleagues (23) have presented some new SANS data on end-linked trifunctional isoprene networks. These are shown in Figure 10. Those materials of low molecular weight between crosslinks exhibit greater chain deformation consistent with the thesis that the junction points are fixed. This is the reverse of that found by Beltzung et al. for siloxane networks. 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

The data on isocyanate crosslinked polybutadiene (B2) is given in Table II, and for this the molecular weight (Mc) was calculated using eqn. (2) (Mc=l0,1OOF3). The expected value of Mc as obtained from GPC on uncrosslinked polymer should be somewhere in the range of Mn (5100) to Mw (7400). Comparison of GPC and swelling data yields that F3 is in the range of 0.51-0.73, which in turn determines the extent up to which the chains deform non-affinely. 

Taking into account the big sizes of polymeric chains deformation and their non-linear relation with the tension let us express the relative linear deformation clx/x, along /-direction of (/-dimensional space under the action of all main forces f, i = 1, d under the approximation of w—ball isotropy via the differential form [8]... 

Jakeways et al. [69] addressed only the crystalline chain deformation to explain PTT s elastic recovery. The macroscopic deformation must also simultaneously involve the partially irreversible amorphous chain deformation. The higher the applied strain, then the more dominant was the irreversible amorphous deformation with deviation from affine deformation. 

Although PBT fiber also has a plateau region in the stress-strain curve [4], the crystalline chains do not respond to external strain in the first few percent of deformation. They increased in length only when the strain is above 4% (see Figure 11.13). Therefore, initial macroscopic deformation involved viscous flow of the amorphous phase. Furthermore, PBT undergoes strain-induced crystal transformation at moderately low strains of 15-20% [75], The differences in their microscopic crystalline chain deformation explained why PTT has a better elastic recovery than PBT even though both have contracted chains and knees in their stress-strain curves [4, 69],... 

Despite the significant chain deformations predicted for the SSL, the domain (stretching) contribution per chain was found by Semenov (1985) to have the same scaling at for a Gaussian chain, namely,... 

Interest in thermotropic liquid crystals has focussed mainly on macroscopic properties studies relating these properties to the microscopic molecular order are new. Lyotropic liquid crystals, e.g. lipid-water systems, however, are better known from a microscopic point of view. We detail the descriptions of chain flexibility that were obtained from recent DMR experiments on deuterated soap molecules. Models were developed, and most chain deformations appear to result from intramolecular isomeric rotations that are compatible with intermodular steric hindrance. The characteristic times of chain motions can be estimated from earlier proton resonance experiments. There is a possibility of collective motions in the bilayer. The biological relevance of these findings is considered briefly. Recent similar DMR studies of thermotropic liquid crystals also suggest some molecular flexibility. 

Discussion. We can now propose a coarse description of the paraffinic medium in a lamellar lyotropic mesophase (potassium laurate-water). Fast translational diffusion, with D 10"6 at 90 °C, occurs while the chain conformation changes. The characteristic times of the chain deformations are distributed up to 3.10"6 sec at 90 °C. Presence of the soap-water interface and of neighboring molecules limits the number of conformations accessible to the chains. These findings confirm the concept of the paraffinic medium as an anisotropic liquid. One must also compare the frequencies of the slowest deformation mode (106 Hz) and of the local diffusive jump (109 Hz). When one molecule wants to slip by the side of another, the way has to be free. If the swinging motions of the molecules, or their slowest deformation modes, were uncorrelated, the molecules would have to wait about 10"6 sec between two diffusive jumps. The rapid diffusion could then be understood if the slow motions were collective motions in the lamellae. In this respect, the slow motions could depend on the macroscopic structure (lamellar or cylindrical, for example)... 

Localised SDZs are usually constituted of a set of micro-SDZs in which the main-chain deformation is quite significant and homogeneous. In contrast, in diffuse SDZs there is a gradual change of the main-chain deformation both along the length and the width of the SDZ. Furthermore, diffuse SDZs are considerably larger than localised SDZs [24],... 

Fig. 14.9 Snapshots of a system of twenty 100 carbon atom long polyethylene chains deformed at 300 K. The initial slab at the top rapidly deforms with the applied stress in the x dimension of the slab, roughly doubling in the first 500 ps to / — 2.64 (second image from the top) then the rate of deformation is slower and doubles again in 1500ps to X — 5.15 (third image from the top). Beyond this point the cell deforms even more slowly to reach a final deformation of X = 6.28 (bottom image). In absolute values, the initial cell of dimensions 1.88 x 5.32 x 5.32 nm deforms to 11.8 x 2.23 x 1.96nm. [Reprinted by permission from M. C. Levine, N. Waheed, and G. C. Rutledge, Molecular Dynamics Simulation of Orientation and Crystallization of Polyethylene during Uniaxial Extension, Polymer, 44, 1771-1779, (2003).]...

We again stress that this relationship has rigorous physical meaning only if the distribution of local kink orientations has cylindrical symmetry about the director nevertheless, it may provide a semi-quantitative relationship between the REV-8 lineshape and the concentration of chain deformations if the process is at least of a highly random nature. If the ratio p changes with temperature according to a Boltzmann factor, we expect the quantity... 

Drawn isotactic polypropylene (iPP) fibres have been studied by using deuterated n-decane as a 2H NMR probe of the chain deformation ratio in the amorphous regions. It is observed that the slope P=A/(X2-X 1) (defined in the limit of low deformations) indeed depends on the annealing temperature [82]. Thus, annealing above the melting temperature Tm of the crystallites allows chains to relax to some extent. Then, the local deformation ratio in the amorphous phase Xt becomes lower than the macroscopic one X and depends on the annealing temperature, i.e., on the amount of chain relaxation. Therefore, such systems have strongly non-affine deformation at the chain scale. 

Modulus. Jackson et al. (1) calculate the contribution of amorphous material to the shear modulus of a semicrystalline polymer by assuming that only tie chains (chains whose ends are attached to different crystallites) contribute to the modulus and that these chains follow Gaussian statistics. They assume that the chains deform affinely. The predicted modulus values are lower than the observed values. The... 

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