Affine deformation model

To illustrate the usefulness of birefringence measurements in orientation studies, we now briefly discuss two simple models of orientation leading to different expressions of the second moment of the orientation function the affine deformation model for rubbers and the pseudo-affine model more frequently used for semi-crystalline polymers. 

Figure 3.8 shows a diagram of the deformation. The sample initially possesses dimensions Xq, yo, zq, and when deformed it takes on dimensions X, y, z. According to the affine deformation model, the end-to-end distance vector of the chains of the strained network will change its components in... 

It is actually possible within the framework of the statistical theory of elasticity to deduce an expression similar to Eq, (3.33) that considers the experimentally observed decrease in modulus. This is done by using a model different from the affine deformation model, known as the phantom network model. In the phantom network the nodes fluctuate around mean... 

Comparison of Eqs. (3.36a) and (3.33) indicates that the value of modulus G obtained from the affine deformation model is two times the value corresponding to the phantom network. This would mean that the latter model is more applicable in the region of moderate deformations and the affine model is more suitable in the region of low deformations. 

Several models have been proposed that attempt to describe the development of orientation in drawn polymers. The pseudo-affine deformation model... 

One of the best-known of such schemes is the AFFINE deformation model for rubbers. The rubber is considered to be a network of flexible chains, and the macroscopic strain is imagined to be transmitted to the network such that lines joining the network junction points rotate and translate exactly as lines joining corresponding points marked on the bulk material. If we assume that the flexible chains consist of rotatable segments called random links , and that some statistical model can describe the configurational situation, it is then possible to obtain explicit expressions which relate the segmental orientation to the macroscopic deformation. 

Fig. 10. Degree of orientation given as versus the effective draw ratio for the range 8 < L < 80. The drawn curve presents the affine deformation model (see text). The experimental data have been derived from the modulus values given in Fig. 9 [230]...

Fig. 9 presents the modulus of PpPTA fibers as a function of the applied draw ratio for air-gap spinning at a very low winding tension. This figure demonstrates the importance of the spin-stretch stage in the spinning process. It wiU be shown here that these results are in agreement with the affine deformation model [230]. 

Figure 10 presents the curve for the affine deformation model calculated with Eq. (11) in terms of log (sin cp) versus logX for the range 8 < X < 80. The experimental values for the overall order parameter (P2> were derived from from the modulus values presented in Fig. 9. Division of these values by the molecular...

Miscibility of segmented rigid-rod polyimide (PI), viz., biphenyl dianhydride perfluoromethylbenzidine (BPDA-PFMB), and flexible polyether imide (PEI) molecular composites was established by differential scanning calorimetry. The composite films of BPDA-PFMB/PEI were drawn at elevated temperatures above their glass transitions. Tensile moduli of the films were evaluated as a function of composition and draw ratio. Molecular orientations of polyimide were determined by birefringence and wide-angle X-ray diffraction. The crystal orientation behavior of the 80/20 BPDA-PFMB/PEI was analyzed in the framework of the affine deformation model. 

Purvis and Bower [13] also examined drawn specimens of amorphous poly(methyl methacrylate), using four peaks in the Raman spectrum. They were not able to distinguish between two plausible structural models, one essentially linear and the other helical, but they were able to show that their results are consistent with the affine deformational model. 

Figure 3.10 shows schematically the difference between the affine network model and the phantom network model. The affine deformation model assumes that the junction points (i.e. the crosslinks) have a specified fixed position defined by the specimen deformation ratio L/Lq, where L is the length of the specimen after loading and Lq is the length of the unstressed specimen). The chains between the junction points are, however, free to take any of the great many possible conformations. The junction points of the phantom network are allowed to fluctuate about their mean values (shown in Fig. 3.10 by the points marked with an A) and the chains between the crosslinks to take any of the great many possible conformations. 

As discussed briefly in the introduction the elastic and relaxational properties of polymer networks are also expected to be influenced significantly by the presence of entanglements. The classical theories, the phantom network modeP and the affine deformation model, describe the two extreme points of view. In the first, at least in its original form, the network strands and the crosslinks are not subject to any constraint besides connectivity and functionality. The other extreme considers the crosslinks to be fixed in space and deform affinely under deformation. A number of modifications of these theories have been proposed in which the junction fluctuations are partially suppressed. All of these models however consider the network strands as entropic springs. The entropic force, as... 

The classical models of rubber elasticity reduce the elastic properties to a study of entropic springs. In the phantom model the EV interaction only gives the volume conservation, while in the affine deformation model it also is responsible for the affine position transformation of the crosslinks. Otherwise the entropic forces and the excluded volume interactions and consequently the entanglements, as they are a result of the EV interaction, completely decouple from the chain elasticity. The elasticity is entirely determined by the strand entropy. It is obvious that this is a... 

The behavior of PBT displayed in Figure 7b is quite different the dependence of L on the external macrodeformation can be followed in a much broader deformation interval (up to j > 18%) and a very well expressed linear relationship is observed between the macro- (e) and microdeformation (L). The respective line coincides with that of the affine deformation model, and the deformation is completely reversible up to = 12%. The similarity of the PET and PBT scattering patterns on the one hand, and the differences in their deformation behavior on the other hand (Figure 7a and b, respectively), i.e., their macrodeformation limits, the reversibility of deformation and its character (affine or non-affine) lead to a preliminary conclusion regarding the reason for the differences observed. They can be attributed to the different chemical composition — replacement of the ethylene glycol (EG) moieties of PET by the longer and more flexible tetramethylene glycol (TMG) units of PBT. 

The relationship between macro- and microdeformation on the nano-level is reflected in the relationship L s), For PET (Figure 7a), there is no well-defined relationship. For the other materials, one observes a linear relation (Figures 7b-d and 9) that obeys the affine deformation model, except for the blend (Figure 7c). At low elongations, this increase is completely reversible, i.e., the L values measured in the corresponding deformation range in the absence of stress, are very close to the initial value Lq (best seen in Figure 9). 

Figure 6.10 presents the curve for the affine deformation model calculated with eqn (6.11) in terms of log(sin ( > versus log/I for the range 8lo = 8, the observed data plotted against the effective draw ratio were found to coincide with the theoretical curve of the affine deformation. This pre-draw value is in agreement with the one derived from the dimensions of the entrance cone of the spinneret used in these experiments. Note that an incorrect value for P2 has a considerable effect on the experimental values derived for large draw ratios, thereby causing a change of slope of the curve determined by the experimental data in Fig. 6.10. The curve for the affine deformation has also been drawn in Fig. 6.9.

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