Constrained junction fluctuation model

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. 

The affine and the phantom models derive the behavior of the network from the statistical properties of the individual molecules (single chain models). In the more advanced constrained junction fluctuation model the properties of these two classical models are bridged and interchain interactions are taken into account. We remark for completeness that other molecular models for rubber networks have been proposed [32,57,75-87], however, these are not nearly as widely used and remain the subject of much debate. Here we briefly summarize the basic concepts of the affine, phantom, constrained junction fluctuation, diffused constraint, tube and slip-tube models. 

The Constrained Junction Fluctuation Model. The affine and phantom models are two limiting cases on the network properties and real network behavior is not perfectly described by them (recall Fig. 29.2). Intermolecular entanglements and other steric constraints on the fluctuations of junctions have been postulated as contributing to the elastic free energy. One widely used model proposed to explain deviations from ideal elastic behavior is that of Ronca and Allegra [34] and Hory [36]. They introduced the assumption of constrained fluctuations and of affine deformation of fluctuation domains. 

FIGURE 29.4. Effect of constraints on the fluctuations of network junctions, (a) Phantom model and (b) constrained junction fluctuation model. Note that the domain boundaries (circles in the figures) are diffuse rather than rigid. The action of domain constraint is assumed to be a Gaussian function of the distance of the junction from B similar to the action of the phantom network being a Gaussian function of AR from the mean position A. 

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. 

The corresponding equation according to the Flory-Erman constrained junction fluctuation model is... 

The detailed calculations according to the constrained junction fluctuation model and other advanced models can only be performed numerically. The fitting of the stress-strain (or swelling) data to the Flory-Erman model, in principle, requires three parameters / ]ph, k and Here we briefly outline the steps of the fitting procedure [113,114] ... 

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... 

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... 

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-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... 

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 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-... 

In real polymer network the effects of excluded volume and chain entanglements should be taken into account. In 1977 Hory [26] formulated the constrained junction model of real networks. According to this theory fluctuations of junctiOTs are affected by chains interpenetration, and as the result the elastic free energy is a sum of the elastic free energy of the phantom network AAph (given by Eq. (5.78)) and the free energy of constraints AA ... 

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 [35, 36]. It was concluded that the suppression of the fluctuations was over-estimated in the theory, presumably because the entire effect of... 

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) ... 

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