Constrained-chain theory

Figure 2. Types of constraint in the molecular theories. In the earliest such constraint theoiy (uppermost portion of the figure) the total effects of the constraints were placed on the cross-links themselves. In the subsequent constrained-chains theory, they were placed at the mass centers of the network chains and, in the diffused-constraints theory, along the entire network chains. The lowermost portion of the figure shows how additional experimental information could suggest a more refined placement of the constraints.

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... 

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]. 

Kloczkowski, Mark, and Erman [95] compared the prediction of the diffused constraint model with the results of the Flory constrained-junction fluctuation theory [36] and the Erman-Monnerie constrained chain theory [94]. They found that the shapes of the [/ ] vs. a curves for all three theories were very similar. Rubinstein and Panyukov [101] reanalyzed the data of Pak and Flory [118] obtained for uniaxially deformed crosslinked PDMS samples. They concluded that the fit of the experimental data by the diffused... 

The Constrained-Chain Theory. This refinement of the constrained-junction model is based on reexamination of the constraint problem and evaluation of some neutron scattering estimates of actual junction fluctuations... 

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]... 

The models presented in the previous section are of an elementary nature in the sense that they ignore contributions from intermolecular effects (such as entanglements that are permanently trapped on formation of the network). Among the theories that take account of the contribution of entanglements are (1) the treatment of Beam and Edwards [19] in terms of topological invariants, (2) the slip-link model [20, 21], (3) the constrained-]unction and constrained-chain models [22-27], and (4) the trapped entanglement model [11,28]. The slip-link, constrained-junction, and constrained-chain models can be studied under a common format as can be seen from the discussion by Erman and Mark [7]. For illustrative purposes we present the constrained-junction model in some detail here. We then discuss the trapped entanglement models. 

Here k is a parameter which measures the strength of the constraints. For k = 0 we obtain the phantom network limit, and for infinitely strong constraints (k = oo) the affine limit is obtained. Erman and Monnerie [27] developed the constrained chain model, where constraints effect fluctuations of the centers of the mass of chains in the network. Kloczkowski, Mark, and Erman [28] proposed a diffused-constraint theory with continuous placement of constraints along the network chains. 

The constrained junction fluctuation theory was modified by Erman and Monnerie [94]. The fundamental difference between the modified and the original models is the adoption of the assumption that constraints affect the centers of mass of the chains rather than the junction points only. They considered two different cases (1) the fluctuations of aU points along the chains in the phantom network are independent of macroscopic strain (constrained chain scheme, CC) and (2) the fluctuations of the points in the phantom network are dependent on the macroscopic strain, only the junctions are invariant to strain (modified constrained chain... 

In fact, in the actual market environment, the supply chain node enterprises may face many kinds of real risks, under these circumstances, the decision-makers need to consider how to maximize supply chain benefits in terms of probabUily. Stochastic chance-constrained programming theory has a practical significance in dealing with such issues. Following is the brief introduction of stochastic chance-constrained programming theory. 

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... 

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... 

Inherent in the analytical SCF theory are (1) that the free ends of the chains can sample any position within the brush rather than be constrained to reside at the peripheral surface of the layer and, (2) that there is strong stretching, that is... 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

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. 

Approaches have been developed which modify the traditional double-layer theory by assuming that the dielectric constant depends upon position. This dependence is considered to occur because the dielectric constant is affected by the electric field [9-12]. However, the dielectric constant can also be changed by the presence of hairs on the surface. Indeed, hairy chains on the surface constrain the orientation of the water molecules nearby, and, as a result, in the region near the surface the dielectric constant becomes lower... 

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