Theory constrained-junction

As already described, the upper three portions of Figure 2 summarize the differences in the way the constraints are applied in the constrained-junction theory, constrained-chain theory, and the diffused-constraints theory, respectively [4], Additional comparisons between theory and experiment for a variety of elastomeric properties should be very helpful [20], Also, neutron-scattering measurements conducted on series of networks having different values of the junction functionality , which is the number of chains emanating from a junction (cross-link), would be extremely useful in suggesting how to position the constraints along a chain in refining such models, since should have a pronounced effect on the... 

These observations can be qualitatively explained in terms of the constrained-junction theory. If a network is cross-linked in solution and the solvent then removed, the chains collapse in such a way that there is reduced overlap in their configurational domains. It is primarily in this regard, namely reduced chain-junction entangling, that solution-cross-linked samples have simpler topologies, and these diminished constraints give correspondingly simpler elastomeric behavior. 

Elastic term (Oeias) the earliest elasticity mode based on Gaussian chain distributions (Flory—Rehner and Flory theories) improved elasticity model based on constrained junction theory an elastic expression that accounts for the limits of elongation Prausnitz et al. [36] Saito et al. [42]... 

In Eq. (29.23) W 0) is the distribution of constraints among different points along the network chain and 0 = i/n is the position of the /th segment of the chain as a fraction of the contour length between two crosslinks. If the distribution is uniform, then W 0) = 1 inside the integrand of Eq. (29.23). In the case when constraints are assumed to affect only fluctuations of junctions (as in the constrained-junction theory), 0 is limited to 0 = 0 or 0 = 1 only. [95] It is important to note that this theory does not reduce identically to the constrained-chain theory, because the latter characterizes the deformation-dependent fluctuations of the centers of mass of the chains and not the deformation-independent fluctuations of the midpoints [95]. 

This article presents uniaxial extension measurements on cis-1,4-polybutadiene networks of known junction functionality. The observed values of the reduced force from uniaxial extension measurements conform to the constrained junction theory of Flory. The reduced force intercept at 1/cX = 0 is fully comprehensible in terms of the cycle rank of the network, and can be calculated from chemical considerations. This holds even though the polybutadiene melt has a high plateau modulus. Therefore, discrete topological entanglements do not contribute perceptibly to the equilibrium modulus of polybutadiene networks. 

Experimental results indicate that the response to deformation of a network generally falls between the affine and phantom limits [31-34]. At low deformations, chain-junction entangling suppresses the fluctuations of the junctions and the deformation is relatively close to the affine limit. This is illustrated in Fig. 1.8, which shows schematically some of the results of the constrained-junction theory based on this qualitative idea [32-34]. In the case of the two limits, the affine deformation and the non-affine deformation in the phantom-network limit, the reduced stress should be independent of a. Because of junction fluctuations, the value for the... 

It is also possible to interpret the upturns in modulus in these isotherms using analytic expressions, for example the Fixman-Alben modification [121] of the Gaussian distribution function, combined with the constrained-junction theory and reasonable values of the constraint parameter k [122]. 

Experimental results on networks of natural rubber in shear deformation [134] are not well accounted for by the simple molecular theory of rubber-like elasticity. The constrained-junction theory, however, was found to give excellent agreement with experiment. Shear measurements have also been reported for some unimodal and... 

Hooper et al [12] give mathematical details concerning the use of constrained-junction theory for and Donnan equilibria for jCjQp. Here we are concerned with the... 

Fig. 3. Reduced stress as a fiinction of reciprocal extension ratio. The upper and lower horizontal lines represent results from affine and phantom network models, respectively. Circles show representative data from experiments, and the curves are from the constrained junction theory.

Some measurements in biaxial extension have involved the direct stretching of a sample sheet in two perpendicular directions within its plane, by two independently variable amoimts. In the equibiaxial case, the deformation is equivalent to compression. A good account of such experimented results (142) has been given by the simple molecular theory, with improvements at lower extensions upon use of the constrained-junction theory (5,13). 

The constrained-junction theory successfully describes most of the features of numerous investigations that have been made on stress-strain relationships involving a variety of t5rpes of deformations (1-3,13,220,249-252). Specifically, the decrease in modulus f ] with the increase in elongation is viewed as the deformations becoming more nonaffine as the stretching of the network chains... 

The reduced birefringence shows some properties that parallel the reduced stress or modulus. In other respects, however, it can be quite different, showing for example significant increases with the increase in degree of swelling. Nonetheless, experimental results and those from the constrained-junction theory seem to be in at least fair agreement (201,202). 

The predictions of the constrained junction theory are consistent with all the experimental data, and the range of k is in agreement with values previously reported. According to the interpenetration concept, k should be a decreasing function of the cross-link density. Effectively, K regularly decreases with increasing amount of dicumyl peroxide, as expected. 

Figure 15 Plot of equilibrium reduced forces vs. reciprocal of the extension ratio. Experimental points were obtained by extrapolation of relaxation measurements to infinite time. Continuous curves are predictions of the constrained junction theory of rubber elasticity calculated with parameters listed in Table 1 (reproduced by permission of Butterworth from J. P. Queslel, P. Thirion and L. Monnerie, Polymer, 1986, 27, 1869)...

The simple statistical mechanical models, i.e. affine and phantom, or the phenomenological approach cannot reproduce the maximum in the plot of Aln(ai/a ) vs. n6,28i jjjjg maximum is predicted, however, by the constrained junction theory but, as shown in Figure 19, agreement with experiment is only qualitative.This has to lead to questioning of the separability of mixing and elastic contributions in equation (145). This postulate may be valid only in the limit of swelling equilibrium. ... 

Characterization of elastomeric networks by swelling equilibrium measurements may take advantage of the applicability of the constrained junction theory already demonstrated for mechanical testing. Use of the interpenetration concept (equation 124 or 125), and of topological expressions for (equations 47 or 60) cause equation (156) to depend only on and x for a perfect network, or on M, M , and x for a randomly cross-linked network. An example of such an application was given by Erman and Baysal. Two cross-linked polystyrene networks were... 

Table 2 Characterization of Polystyrene Networks by Swelling Equilibrium Measurements in Toluene, and Application of the Constrained Junction Theory ...

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