Junction fluctuations

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]. 

B. The mean vector connecting the chain ends is deformed affinely. The crosslink junctions fluctuate according to the theory of Brownian motion (2, 5, 7) ... 

C. Fluctuations are partially damped by chain interferences. This leads to a result intermediate between A. and B. To be specific, a model proposed by Flory (16) is adopted here. In this model, the inhibitory influence on junction fluctuations is assumed to be affine in the strain. One finds... 

The parameters of neutron scattering theory of polymer networks are A, the macroscopic stretching of the sample, or linear degree of swelling, f, the network functionality, K. which accounts for restricted junction fluctuations and a, a measure of the degree to which chain extension parallels the macroscopic sample deformation. The functionality is known from knowledge of the chemistry of network formation, and A is measured. Both K and a must be extracted from experiments. 

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. 

The effect of restricted junction fluctuations on S(x) is to change the scattering function monotonically from that exhibited by a phantom network to that of the fixed junction model. Network unfolding produces the reverse trend, the change of S(x) with x is even less than that exhibited by a phantom network. Figure 6 illustrates how the scattering function is modified by these two opposing influences. 

Figure 6. A plot of S(x)/SD(x) versus f for a network in which junction fluctuations are partially inhibited (k 0) and where molecular deformations are less than...

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. 

Therefore, Flory s theory concludes that as the functionality of a network increases, the constraint contribution, fc, should decrease and eventually vanish. Furthermore, in the extreme limit in which junction fluctuations are totally suppressed, the Flory theory reduces to the affine network model ... 

When the network junctions are entirely immobilized by the surrounding chains, h equals zero. Then the junctions in a deformed specimen are displaced in proportion to the macroscopic strain, i.e., the deformation is affine. Alternatively, h equals unity when junction fluctuations are not impeded, the defining characteristic of a phantom network (16, 17). The parameter h was introduced (13) to allow empirically for different degrees of fluctuations. For undiluted networks at small deformations, h should usually be small, though not necessarily zero. 

Ronca and Allegra (12) and Flory ( 1, 2) assume explicitly in their new rubber elasticity theory that trapped entanglements make no contribution to the equilibrium elastic modulus. It is proposed that chain entangling merely serves to suppress junction fluctuations at small deformations, thereby making the network deform affinely at small deformations. This means that the limiting value of the front factor is one for complete suppression of junction fluctuations. 

Figure 5.11 Dependence of the reduced equilibrium shear modulus, Ge/wg// 7" on the molar ratio of [OH]/[NCO] groups, ah, for poly(oxypropylene)triol (Niax LG 56)-4,4 -diisocyanatodiphenylmethane system (—-) limits of the Flory-Erman junction fluctuation rubber elasticity theory. The dependence has been reconstructed from data of ref. [78]...

Figure 9. Reduced equilibrium modulus of polyurethane networks from POP trlols and MDI in dependence on the sol fraction. networks from POP triol Mjj - 708, o networks from POP triol Mjj = 2630. C-) calculated dependence using Flory junction fluctuation theory for the value of the front factor A indicated. (Reproduced from Ref. 57. Copyright 1982 American Chemical Society.)...

Thus, this consideration shows that the thermoelasticity of the majority of the new models is considerably more complex than that of the phantom networks. However, the new models contain temperature-dependent parameters which are difficult to relate to molecular characteristics of a real rubber-elastic body. It is necessary to note that recent analysis by Gottlieb and Gaylord 63> has demonstrated that only the Gaylord tube model and the Flory constrained junction fluctuation model agree well with the experimental data on the uniaxial stress-strain response. On the other hand, their analysis has shown that all of the existing molecular theories cannot satisfactorily describe swelling behaviour with a physically reasonable set of parameters. The thermoelastic behaviour of the new models has not yet been analysed. 

This refinement of the constrained-junction model is based on re-examination of the constraint problem and evaluation of some neutron-scattering estimates of actual junction fluctuations [158, 159]. It was concluded that the suppression of the fluctuations was over-estimated in the theory, presumably because the entire effect of the inter-chain interactions was arbitrarily placed on the junctions. The theory was therefore revised to make it more realistic by placing the effects of the constraints along the network-chain contours, specifically at their mass centers [4, 160, 161]. This is illustrated in the second portion of Figure 2. Relocating the constraints in this more realistic way provided improved agreement between theory and experiment. 

The assemblage of chains is constructed to represent the affine network model of rubber elasticity in which all network junction positions are subject to the same affine transformation that characterizes the macroscopic deformation. In the affine network model, junction fluctuations are not permitted so the model is simply equivalent to a set of chains whose end-to-end vectors are subject to the same affine transformation. All atoms are subject to nonbonded interactions in the absence of these interactions, the stress response of this model is the same as that of the ideal affine network. 

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],... 

Limit of full suppression of junction fluctuation (fixed phantom network, affine... 

The mean field approach can be applied in different stages of elaboration. In the first stage, models are introduced that contain additional free parameters which are not determined by microscopic theory. Using these models, the influence of the constraints on network properties has been calculated and discussed. Box models slip-link models constraining springs constrained junction fluctuation and different tube models are predominantly used. The main charac-... 

Tube models, and their simpler versions, the box models, are distinguished by the advantage that the constraints act equally on each segment of the chains. This assumption corresponds much more directly to the physically well settled idea of the mean field approach than an introduction of discrete constraints as in the models of constrained junction fluctuation and of constraining springs in slip-link... 

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