Slip-link model

The elastic free energy of the constrained-junction model, similar to that of the slip-link model, is the sum of the phantom network free energy and that due to the constraints. Both the slip-link and the constrained-junction model free energies reduce to that of the phantom network model when the effect of entanglements diminishes to zero. One important difference between the two models, however, is that the constrained-junction model free energy equates to that of the affine network model in the limit of infinitely strong constraints, whereas the slip-link model free energy may exceed that for an affine deformation, as may be observed from Equation (41). 

Fig. 7.2 Mooney-Rivlin plot for the Slip-Link model...

Fig. 8.2 In the slip-link model, a hypothetical tensile force Feq = ikTja pulling at both chain ends is necessary to keep the polymer chain constrained by the slip-links otherwise, the polymer chain will soon shrink along the primitive path and leak out from the space between the slip-links or, so to speak, leak out of the tube.

The structural factors of the relaxation times t, tx, tb and tq and the hierarchy and universality among them as discussed above are viewed from a different angle in Appendix 9.D. Through dimensional analysis, it is shown that all of these are characteristics inherently contained in the extended slip-link model. 

Additional characteristic times are expected to arise from the introduction of the length scales a and L into the slip-link model as defined by Eq. (8.3). In addition to tq, the characteristic times in the slip-link model... 

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-... 

We distinguish three ways to simulate the configurational confinement of network strands. In the slip-link model each strand threads its way through a number of small rings. Such an approach has also been used successfully in calculating the viscoelastic properties of polymer melts The topological contributions are caused by the orientation of the subchains between the slip-links, i.e. the slip-link model may be called an alignment 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. 

The slip-link model incorporates the effects of entanglements along the chain contour into the elastic free energy. According to the mechanism of the slip link, sketched in Fig. 3, a link joins two different chains which may slide a distance a along the contour of the chains. The elastic free energy resulting from this model is... 

A different statistical-mechanical approach based on so called replica formalism was developed by Edwards and CO workers [29,30]. They studied the effect of topological entanglements between chains on the elastic free energy of the network and formulated the slip-link model. The elastic energy of constraints in the slip-link theory is... 

Vilgis and Erman that the constraint models and slip-link models have much in common, (iv) elucidating the effects of cross-link functionality and degree of cross linking, (v) exploring a variety of elastomeric polymers, particularly those having very different values of the plateau modulus, and (vi) generalizing rubber-elasticity models to include viscoelastic effects as well. 

Vilgis, T. A. Erman, B., Comparison of the Constrained Junction and the Slip-Link Models of Rubber Elasticity. Macromolecules 1993,26(24), 6657-6659. 

Figure 3.10 Slip-link model. (Reproduced with permission from Ball, Doi, Edwards and Warner, Polymer, 22, 1010 (1981))...

Despite the fact that there are plenty of papers on the tube theory and its modifications, there are not so many models in the sense of this chapter. We limited ourselves to the models easily available to us. There are four other models for entangled polymers in literature that are interesting to investigate in the same way as we did in this chapter. Only a lack of time and space had prevented us from doing so. They are the Twentanglements model, the transient forces model, the NAPLES model, and several slip-link models. In this sertion, we shall discuss them very briefly. 

Experiments of Rennar and Oppermann on end-linked ROMS networks indicate that contributions from trapped entanglements are significant for low degrees of end-linking but are not important when the network chains are shorter. The slip-link model of mbber elasticity recognizes the contributions from trapped entanglements. 

The constrained-junaion and the tube and slip-link models have been improved by several authors. " ... 

One of the most interesting alternative approaches is the slip-link model, which incorporates the effects of entanglements [40,41] along the network chains directly into the elastic free energy [42]. Still other approaches are the tube model [43] and the van der Waals model [44]. 

Janeschitz-Kriegl, H. (1982) Some pending problems in polymer melt rheology, as seen from the point of view of Doi slip-link model. Rheol. Acta., 21,388. 

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