Entanglement network tube model

An important role in the present model is played by the strongly non-linear elastic response of the rubber matrix that transmits the stress between the filler clusters. We refer here to an extended tube model of rubber elasticity, which is based on the following fundamental assumptions. The network chains in a highly entangled polymer network are heavily restricted in their fluctuations due to packing effects. This restriction is described by virtual tubes around the network chains that hinder the fluctuation. When the network elongates, these tubes deform non-affinely with a deformation exponent v=l/2. The tube radius in spatial direction p of the main axis system depends on the deformation ratio as follows ... 

Therefore, it is well established that topological entanglements dominate and control the modulus of polymer networks with long network strands. The Edwards tube model explains the non-zero intercept in plots of network modulus against number density of strands (see Figs 7.11 and 7.12). The modulus of networks with very long strands between crosslinks approaches the plateau modulus of the linear polymer melt. The modulus of the entangled polymer network can be approximated as a simple sum. 

The deformation dependence of the stress in the Edwards tube model is the same as in the classical models [Eqs (7.32) and (7.33)] because each entanglement effectively acts as another crosslink junction in the network. Therefore, the Edwards tube model is unable to explain the stress softening at intermediate deformations, demonstrated in Fig. 7.8. The reason for the classical functional form of the stress strain dependence is that the confining potential is assumed to be independent of deformation. 

The results in this section were all derived for unentangled networks. The Edwards tube model for entangled networks gives identical results with N replaced by N, the number of Kuhn monomers in an entanglement strand in the preparation state, because both entanglement strands and network strands are assumed to deform affinely in the Edwards tube model. If the Edwards tube model were correct, the universal relations [Eqs (7.91) and (7.92)] would still apply for entangled networks, since they are independent of N. However, the non-affine tube models predict that entangled networks will swell considerably more than the Edwards tube model predicts. 

Part I summarizes the main ideas of de Gennes, Doi and Edwards about tube models and reptation in entangled polymer systems. Attention has been limited to properties for which predictions can be made without invoking the independent alignment approximation macromolecular diffusion, linear viscoelasticity in the plateau and terminal regions, stress relaxation following a step strain from rest of arbitrary magnitude, and equilibrium elasticity in networks. 

In the Edwards tube model [80] the topological potential is applied to every monomer of the chain restricting its fluctuations to a confining tube with the diameter a K bNl where is the degree of polymerization be-— In [B,( ) + 1] In [f X ) + l]]dfl (29.23) tween network entanglements. [101] In the model this... 

The role of positional fluctuations in polymer networks is central to some theories of elasticity, and has been investigated with an MC method based on a modified bond-fluctuation model (265). The simple model used in the simulations gave results close to those calculated from theory for a Bethe lattice (also known as a Cayley tree). More extensive results bearing on the role of fluctuations in polymer networks have been reported by Grest and co-workers (225). They find that entanglements limit fluctuations, giving behavior similar to the description provided by the tube model. 

In equation 76a the reptation time A,rep is the time for the chain to escape from the tube (orientation relaxation occurs from the end to the center of the chain). Gn is the entanglement plateau modulus (this value is slightly different from that implied from rubber elasticity of an entangled network) and f pit) is a normalized relaxation modulus for the reptation process. In this time regime, equation 76a implies that the modulus is separable into a time fimction and a modulus function. This becomes important in discussing the nonlinear response, which is done, in more detail, below. Some other viscoelastic functions from the DE tube model of reptation are... 

The mutual steric restrictions of entangled chains at deformation are accotmted for in a tube model considering the reptation motion of network subchains. This approach was proposed by Edwards and Vilgis ° and Heinrich et al. Later the tube model was further developed (see References 42 and 43). 

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]. 

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