Constrained-chain model

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. 

In summary, the common feature of all constrained chain models is that they impose only limited constraints on chain fluctuations. [101] The constrained-junction fluctuation model restricts fluctuations of junctions and of the center of mass of network chains. The diffused constraint model restricts fluctuations of a single randomly chosen monomer for each network strand. Consequently, all these models can only represent the crossover between the phantom and afflne limits. [101] The phantom limit corresponds to a weak constraining case, while the affine limit corresponds to a very strong constraining potential. 

FIGURE 29.5. Mooney-Rivlin reduced stress plot showing comparison of experimental data with modified constrained chain model (MCC) predictions for dry (o) and swollen ( ) natural rubber networks [112,117]. Swelling agent n-Decane. continuous lines are theoretical curves calculated with paremeters /cT/l/o = 0.17MPa and kq =2.0. 

Erman, B. Mark, J. E., Stress-Strain Isotherms for Elastomers Cross-Linked in Solution. 2. Interpretation in Terms of the Constrained-Chain Model. Macromolecules 1992, 25,1917-1921. 

The constrained-junction model was formulated in order to explain the decrease of the elastic moduli of networks upon stretching. It was first introduced by Ronca and Allegra [39], and Flory [40]. The model assumes that the fluctuations of junctions are diminished below those of the phantom network because of the presence of entanglements and that stretching increases the range of fluctuations back to those of the phantom network. As indicated by the second part of Equation (26), the fluctuations in a phantom network are substantial. For a tetrafunctional network, the mean-square fluctuations of junctions amount to as much as half of the mean-square end-to-end vector of the network chains. The strength of the constraints on these fluctuations is measured by a parameter k, defined as... 

According to the models, the free energy of deformation and the stress in the network with constrained chains contains an additive contribution to that describing... 

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. 

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... 

It is well known that the elasticity of polymer networks with constrained chains in the rubbery state is proportional to the number of elastically active chains. The statistical (topological) model of epoxy-aromatic amine networks (see Sect. 2) allows to calculate the number of elastically active chains1 and finally the equilibrium modulus of elasticity Eca,c for a network of given topological structure 9 10). The following Equation 9) was used for the calculations of E, c ... 

The constraining potential represented by virtual chains must be set up so that the fluctuations of junction points are restricted, but the virtual chains must not store any stress. If the number of monomers in each virtual chain is independent of network deformation, these virtual chains would act as real chains and would store elastic energy when the network is deformed. A principal assumption of the constrained-junction model is that the constraining potential acting on junction points changes with network deformation. In the virtual chain representation of this con-... 

The constrained junction model has virtual chains (thin lines) connecting each network junction (circles) to the elastic background (at the crosses). 

The constrained-junction model relies on an additional parameter that determines the strength of the constraining potential, and can be thought of as the ratio of the number of monomers in real network strands and in wirtual chains NjnQ. If this ratio is small, the virtual-chain is relatively long... 

Experimental determination of the contributions above those predicted by the reference phantom network model has been controversial. Experiments of Oppermann and Rennar (1987) on endlinked poly(dimethylsiloxane) networks, represented by the dotted points in Figure 4.4, indicate that contributions from trapped entanglements are significant for low degrees of end-linking but are not important when the network chains are shorter. Experimental results of Erman and Wagner (1980) on randomly crosslinked poly(ethyl acrylate) networks fall on the solid line and indicate that the observed high deformation limit moduli are within the predictions of the constrained-junction model. 

The strength of the constraints is determined in this model by the prefactors of [R (s) - ft (s)] It is assumed that for the undeformed isotropic system the constraining potential is independent of the direction of the constrained chain. Consequently, the constraining potential has to be diagonal in the main axis system of the deformation tensor in the case of external deformation. 

Fig. 2. Entanglement model. Configuration of the constrained chain Allowed configuration of a constraining chain ------...

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