Rubber elasticity filler effects

The above equations gave reasonably reliable M value of SBS. Another approach to modeling the elastic behavior of SBS triblock copolymer has been developed [202]. The first one, the simple model, is obtained by a modification of classical rubber elasticity theory to account for the filler effect of the domain. The major objection was the simple application of mbber elasticity theory to block copolymers without considering the effect of the domain on the distribution function of the mbber matrix chain. In the derivation of classical equation of rabber elasticity, it is assumed that the chain has Gaussian distribution function. The use of this distribution function considers that aU spaces are accessible to a given chain. However, that is not the case of TPEs because the domain also takes up space in block copolymers. 

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 ... 

In the hair that is swollen by the 8 M LiBr/BC diluted system, there is a globular HS protein that contains a large number of SS bonds. In such a heterogeneously crosslinked system, the ordinary rubber elasticity theory cannot be applied. Hence, a two-phase structure of swollen keratin networks is assumed. This structure consists of the matrix (domain phase) that is a tightly crosslinked and mechanically stable globular HS protein, and continuous networks (rubbery phase) that are made of low crosslink density LS protein chains. The domain phase was hypothesized to provide the filler effect in rubber networks [56]. Equation (2) is the relationship between equilibrium stress F and elongation ratio of rubber phase a ... 

So far the micro-mechanical origin of the Mullins effect is not totally understood [26, 36, 61]. Beside the action of the entropy elastic polymer network that is quite well understood on a molecular-statistical basis [24, 62], the impact of filler particles on stress-strain properties is of high importance. On the one hand the addition of hard filler particles leads to a stiffening of the rubber matrix that can be described by a hydrodynamic strain amplification factor [22, 63-65]. On the other, the constraints introduced into the system by filler-polymer bonds result in a decreased network entropy. Accordingly, the free energy that equals the negative entropy times the temperature increases linear with the effective number of network junctions [64-67]. A further effect is obtained from the formation of filler clusters or a... 

Equation (70) is a scaling invariant relation for the concentration-dependency of the elastic modulus of highly filled rubbers, i.e., the relation is independent of filler particle size. The invariant relation results from the special invariant form of the space-filling condition at Eq. (67) together with the scaling invariance of Eqs. (68) and (69), where the particle size d enters as a normalization factor for the cluster size only. This scaling invariance disappears if the action of the immobilized rubber layer is considered. The effect of a hard, glassy layer of immobilized polymer on the elastic modulus of CCA-clusters can be de-... 

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