Entanglement network constraints

The mathematical treatment that arises from the dynamic dilution hypothesis is remarkably simple - and very effective in the cases of star polymers and of path length fluctuation contributions to constraint release in Hnear polymers. The physics is equally appealing all relaxed segments on a timescale rare treated in just the same way they do not contribute to the entanglement network as far as the unrelaxed material is concerned. If the volume fraction of unrelaxed chain material is 0, then on this timescale the entanglement molecular weight is renormalised to Mg/0 or, equivalently, the tube diameter to However, such a... 

In Figure 1.33 the dependence of the front-factor A value on molecular weight is shown. As follows from the adduced plot, the cluster fluctuations constraint is systematically changed from 1.0 at = 0 to 0.5 at = 3600 g/mole. In other words, if the amorphous phase of a semi-crystalline polymer represents a single cluster (supercluster), fluctuations are completely suppressed, and its behaviour corresponds to the affine model [110]. The value = M = 3600 g/mole corresponds to polyethylene melt [20], where constraints imposed by crystallites are absent, and in this case entanglement network behaviour for polyethylenes corresponds to the phantom alternative [113]. 

With regard to entanglement networks some network model with a value of the front factor and an assumption about constraints of fluctuations is indispensable. 

Equations 22.3-22.14 represent the simplest formulation of filled phantom polymer networks. Clearly, specific features of the fractal filler structures of carbon black, etc., are totally neglected. However, the model uses chain variables R(i) directly. It assumes the chains are Gaussian the cross-links and filler particles are placed in position randomly and instantaneously and are thereafter permanent. Additionally, constraints arising from entanglements and packing effects can be introduced using the mean field approach of harmonic tube constraints [15]. 

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... 

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). 

These observations can be qualitatively explained in terms of the constrained-junction theory. If a network is cross-linked in solution and the solvent then removed, the chains collapse in such a way that there is reduced overlap in their configurational domains. It is primarily in this regard, namely reduced chain-junction entangling, that solution-cross-linked samples have simpler topologies, and these diminished constraints give correspondingly simpler elastomeric behavior. 

The mean average molecular mass of the network chains is determined for the elastomer matrix outside the adsorption layer. Contributions to the network structure fi om different types of junctions (chemical junctions, adsorption junctions, and topological hindrances due to confining of chains in the restricted geometry (entropy constraints or elastomer-filler entanglements) are estimated. The major contributions to the total network density are provided by the topological hindrances near the filler surface and by the adsorption junctions. The apparent number of the elementary chain units between the topological hindrances is estimated to be approximately 40-80 elementary chain units. 

The number of monomers in virtual chains is assumed to change with deformation according to Eq. (7.60), similar to the constrained-junction and diffused-constraints models. If one virtual chain is attached to every entanglement strand of monomers, it contains of order virtual monomers in the undeformed state of the network. The number of monomers in each virtual chain changes as the network is deformed... 

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