Network models affine

IR dichroism has also been particularly helpful in this regard. Of predominant interest is the orientation factor S=( 1/2)(3—1) (see Chapter 8), which can be obtained experimentally from the ratio of absorbances of a chosen peak parallel and perpendicular to the direction in which an elastomer is stretched [5,249]. One representation of such results is the effect of network chain length on the reduced orientation factor [S]=S/(72—2 1), where X is the elongation. A comparison is made among typical theoretical results in which the affine model assumes the chain dimensions to change linearly with the imposed macroscopic strain, and the phantom model allows for junction fluctuations that make the relationship nonlinear. The experimental results were found to be close to the phantom relationship. Combined techniques, such as Fourier-transform infrared (FTIR) spectroscopy combined with rheometry (see Chapter 8), are also of increasing interest [250]. 

Note that X =X if a=o, which represents no change from the affine models, and X =1 where network unfolding takes place upon deformation without chain extension. If a is non-zero, network unfolding can be introduced into the previous equations by subr-stitutlng X for X in Eqs. 18, 19 or 20. The most general result of the models discussed above is obtained by replacing Xz by Xz in Eq. 20. Appropriate choices of k and a lead to the results of Eq. 18 or 19 as well. 

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. 

In the affine model of network deformation, the cross-links are viewed as firmly embedded in the elastomeric matrix, and thus as moving linearly with the imposed macroscopic strain [1-4, 20]. In the alternative phantom model, the chains are treated as having zero cross-sectional areas, with the ability to move through one another as phantoms [2-4]. The cross-links in this model undergo consider-... 

A treatment of the classic phantom network model is contained in Ref. [6], page 252-256. However, the statement on p. 256 that this model leads to the same stress-strain relation as the affine model is incorrect because it neglects the effect of junction fluctuations on the predicted shear modulus. See Graessley [17],... 

Fig.2 compares the predictions of the pseudo-affine model and the affine model with different values of the nvunber n of links per chain. It is clearly seen that the pseudo-affine scheme gives a much more rapid initial orientation than the affine network model. 

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. 

Consider an affine model of an incompressible network with polydisperse strands between crosslinks. Prove that the stress a in this network due to... 

A pseudo-affine model predicts the variation of 2 with the deformation of a semi-crystalline polymer. It assumes that the distribution of crystal c axes is the same as the distribution of network chain end-to-end vectors r, in a rubber that has undergone the same macroscopic strain. Figure 3.12 showed the affine deformation of an r vector with that of a rubber block. 

In a stretched rubber, the molecules elongate, and the r vectors move towards the tensile axis. Fience the variation of Pi with extension ratio will differ from the pseudo-affine model. For moderate strains the increase of Pi with extension ratio is linear, but at high extensions the approximation used in Eq. (3.12), that both q and q are large, breaks down. Treloar (1975) described models which consider the limited number of links in the network chains. Figure 3.33 shows that the orientation function abruptly approaches 1 as the extension ratio of the rubber exceeds v. Although the model is successful for rubbers, it fails for the amorphous phase in polypropylene (Fig. 3.32), presumably because the crystals deform and reduce the strain in the amorphous phase. 

In the affine model of the network it is assumed all junction points are imbedded in the network, and each Cartesian component of the chain end-to-end vector transforms linearly with macroscopic deformation... 

Importantly, the model spans the behavior between the phantom and affine models. When k = oo and = 0 we recover the affine network behavior. In this case the junction fluctuations are completely suppressed, i.e., o = 0. When K = 0, i.e., the junctions are free to fluctuate, we recover the phantom network model. 

A recent model, based on the Donnan equilibrium concept, but with a different approach, is the elastic polyelectrolyte network (EPN) model. In the EPN model (Orsetti, Andrade, and Molina 2010), the HS particles are considered as divided in to two fractions an inner fraction gf, which behaves as a gel in Donnan equilibrium with the bulk solution, and an external fraction 1-g/, which is in equilibrium with the bulk (Figure 13.8) thus, it is assumed that a fraction gf of the total number of sites resides inside the gel phase, whereas a fraction f-g/ resides outside the gel given an affinity spectrum, it applies to the external sites in equilibrium with the bulk solutions, with activity a for ion i, whereas for the internal medium, it is applied the internal activity aP, in Donnan equilibrium with the bulk ... 

The earliest and the simplest model of mbber elasticity is the affine model which assumes that the junction points in the network transform affinely with maaoscopic deformation, that is,... 

The classical affinity model assumes that the doss-links are immobile with respect to the whole network. The fluctuations of the positions of cross-links induced by thermal motion are taken into account in the phantom model proposed by James and Guth. It suggests that the fluctuations of a given CTOss-link proceed independently of the presence of subchains linked to it, and during such fluctuations the subchains can pass freely through each other like phantoms. The classic phantom theory predicts the shear modulus G as ... 

The classical rubber elasticity model considers, however, that the crosslink points are particular, such that the cut-off occurs by these points in real space. The corresponding calculations for a chain obliged to pass by several crosslinks are recalled in Ref The calculation for the junction affine model was accomplished by Ull-mann for R and by Bastide for the entire form factor for the case of the phantom network model, this was achieved by Edwards and Warner using the replica method. 

Fig. 14. Comparison of data for long chains in the plateau regime with the calculated form factor for a labeled chain (M, = 2,6 10 ) crosslinked in a rubber of mesh Me = 20000. Solid line below, isotropic above, junction affine model dashed line, phantom network. Data set VI (Table 1) 0> = 10 mn , t " = 40 mn O, isotropic...

These networks (99) were extended up to a = 1.6 and characterized by SANS. Owing to large experimental error, no definitive conclusion could be reached, although the data fit the junction affine model better than either the chain affine model or the phantom network model (Section 9.10.5.2). 

From the observed changes in the parallel and pernendicular components of the radius of gyration relative to macroscopic extension ratio, after appropriate correction for the dangling chain contributions, the chain extensive deformation is found to follow a behavior intermediate between the junction affine model and the phantom network model which allows unrestricted fluctuations of network junctions. On the other hand, the chain contractive deformation follows closely the chain affine model, indicating an asymmetry between extensive and contractive chain deformation. In either case, the deformation behavior is found to be the same for the two molecular weights. 

For the network structure, let us consider regular networks. When such networks deform, the front factor of Eq. (20) differs by two orders of magnitude. This depends on whether it is assumed that the crosslink point moves proportionately to the deformation of the whole body, although they are fixed spatially (affine model) [38], or whether the networks are fixed to the frame while the internal crosslink points can move freely (ghost model) [39]. In any case, if Eq. (20) is rewritten as... 

Taking this value for CmocL in account reduces the difference with the standard model somewhat. The actual value of the modulus then can reasonably well be described by the affine network model for the LJ system. Including Cmod = 1.5 one gets G% 0.027. Using p = 0.37cr (the density of the elastically active part) one gets within the affine model an effective strand length of Neff 13-14, while the cluster analysis yields Ng/f 11. The soft core model is somewhere in between the affine and the phantom model. 

The two models postulate an affine displacement of the positions occupied by the cross-links of the network resulting from a deformation, but differ about the movements undergone by these cross-links. For the Flory-Rehner affine model, cross-links move proportionally to the macroscopic deformation and remain in a given position of space at constant deformation. In the James-Guth phantom model, cross-links are assumed to freely move or fluctuate around an average position corresponding to the affine deformation. The amplitude of such fluctuations is independent of the deformation but depends on the valence of the cross-links and the length of elastic chains ... 

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