Models entanglement model

The cycle rank completely defines the connectivity of a network and is the only parameter that contributes to the elasticity of a network, as will be discussed further in the following section on elementary molecular theories. In several other studies, contributions from entanglements that are trapped during cross-linking are considered in addition to the chemical cross-links [23,24]. The trapped entanglement model is also discussed below. 

This model, whilst not entirely satisfactory, can give a reasonable prediction of the elasticity in the plateau zone. The approach does not readily lend itself to a prediction of the dynamic behaviour. There have been numerous attempts to predict the low shear viscosity using entanglement models. For example Graessley obtained... 

Figure 7.8 Escape of a chain from its entanglements (modeled as a tube) according to the...

Again the proposed entanglement model can be invoked to explain the ductility of compacted UHMW-PE reactor powders. The growing chains on the crystal surface can crystallize independently of each other and conse-... 

A value of c equal to 0.3, previously used to describe FT selectivity data on Ru catalysts (4), was also chosen here to describe the behavior of cobalt catalysts. This equation for hydrocarbon diffusion in melts reflects the strong influence of molecular size in reptation and entanglement models of transport in such systems (IJ6). Our model also requires the input of intrinsic values for jSn (given by the asymptotic j8r), jSo, j8r, and j8s, measured independently. After such parameters are specified, the model yields a non-Flory carbon number distribution of increasingly paraffinic hydrocarbons that agrees well with our experimental observations (Fig. 16). 

In the non-affine slip-tube model, entanglements are represented by slip-rings that are attached to the elastic... 

As discussed below in this chapter, the Doi-Edwards theory models entanglements as slip-links. Having taken the effects of chain slippage through the links into account, the theory gives ... 

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

The value v = —1 was obtained by a simple entanglement model by Thomas et al. The limiting case v = — 1, P = 1 describes a network with the modulus G = G(, + Gjj (see Eq. (42)), but with a phantom-like stress-strain relation P = 0 obviously yields a phantom network with G = 0. In the intermediate range 0 < p < 1, deviations from the phantom behaviour with G, P are observed. 

These varied aspects of the C2 stress suggest that it is associated with entangled chains in networks (Fig. 6) and specifically that it arises from restrictions on the conformations available to entangled chains, different from those operating at crosslink sites [7-9], Prager and Frisch [10] have pointed out that chains involved in model entanglements are governed by different statistics their conclusions are quite consistent with what is known of the C2 stress. 

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. 

TABLE 29.4. Parameters of the stress-strain isotherms for PDMS modei networks caicuiated from the entanglement model (Eq. (29.46)) [54]. 

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