The Slip Link Model

Edwards and co-workers have proposed a different molecular model that subsumes the considerations discussed above, which are based on phantom, affine and constraint junctions to describe the deformation of a molecular network. Edwards and his colleagues have 

It can be seen that the first term in Equation (4.34) is the free energy of a phantom network, equivalent to the first term in Equation (4.32) of the constrained junction model. The second term can be thought of as equivalent to the second term in Equation (4.32), but because there is no fixed limit to the ratio of slip links Ns to permanent cross-links Nc, the maximum value of the free energy can be greater than that of the affine network that would replace V2 Nc in the first term by Nc. 

Edwards and Vilgis [16] subsequently extended this theory to include the limitation of finite extensibility for the network that they regarded as arising when the network chains 

---

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.

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... 

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

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. 

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]. 

Figure 4.13 The slip-links model of Doi and Edwards, which hypothesizes that the primitive chain is defined by a line segment between slip-links and that it passes through small rings, where the slip-links are separated by a distance a.

Figure 9.22 Illustration of the slip link model, in which an entanglement between two chains is represented by a slip linkthat is created when the end of chain /fluctuates outward randomly entangling partner chain j. if either chain / or fluctuates inward, the slip link is destroyed. From Shanbhag and Larson [62].

Figure 9.23 Normalized dielectric constant o - s ( ), where Sq is the zero-frequency dielectric constant, and dielectric loss constant ( ) at 40 °C for a 6-arm polyisoprene star with = 459,000. Symbols are data ofWatanabeef a/. [66], and the lines are predictions of the slip link model. The parameters of the model = 4650 and Tq=42 s,are used for all calculations with the...

The success of the slip link model for symmetric and asymmetric star polymers inspires its application to more complex architectures, such as H polymers. The mechanisms of relaxation and branch point motion were established in studies of symmetric and asymmetric star polymers, as were the parameters (Mf,, and Tg) that allow the simulations to be compared... 

Edwards and colleagues formulated the contributions from entanglements by two different models, (i) the tube model and (ii) the slip-link model. In the tube model (217), the topological contributions are applied to every monomer of the network chain, confining its fluctuations to a tube. This potential is independent of network deformation. In a more recent work (218), the strength of tube constraints was made proportional to deformation. This model is referred to as the Nonaffine Tube Model . 

---