Filled rubbers filler clusters

Figure 45a-c shows an adaptation of the developed model to uniaxial stress-strain data of a pre-conditioned S-SBR-sample filled with 40 phr N220. The fits are obtained for the third stretching cycles at various prestrains by referring to Eqs. (38), (44), and (47) with different but constant strain amplification factors X=Xmax for every pre-strain. For illustrating the fitting procedure, the adaptation is performed in three steps. Since the evaluation of the nominal stress contribution of the strained filler clusters by the integral in Eq. (47) requires the nominal stress aR>1 of the rubber matrix, this quantity is developed in the first step shown in Fig. 45a. It is obtained by demanding an intersection of the simulated curves according to Eqs. (38) and (44) with the measured ones at maximum strain of each strain cycle, where all fragile filler clusters are broken and hence the stress contribution of the strained filler clusters vanishes. The adapted polymer parameters are Gc=0.176 MPa and neITe= 100, independent of pre-strain. According to the considerations at the end of Sect. 5.2.2, the tube constraint modulus is kept fixed at the value Ge=0.2 MPa, which is determined by the plateau modulus Gn° 0.4 MPa [174, 175] of the uncross-linked S-SBR-melt (Ge=l/2GN°). The adapted amplification factors Xmax for the different pre-strains ( max=l> 1-5, 2, 2.5, 3) are listed in the insert of Fig. 45a.

It is demonstrated that the quasi-static stress-strain cycles of carbon black as well as silica filled rubbers can be well described in the scope of the theoretic model of stress softening and filler-induced hysteresis up to large strain. The obtained microscopic material parameter appear reasonable, providing information on the mean size and distribution width of filler clusters, the tensile strength of filler-filler bonds, and the polymer network chain density. In particular it is shown that the model fulfils a plausibility criterion important for FE applications. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. 

The CCA-model considers the filler network as a result of kinetically cluster-cluster-aggregation, where the size of the fractal network heterogeneity is given by a space-filling condition for the filler clusters [60,63,64,92]. We will summarize the basic assumptions of this approach and extend it by adding additional considerations as well as experimental results. Thereby, we will apply the CCA-model to rubber composites filled with carbon black as well as polymeric filler particles (microgels) of spherical shape and almost mono-disperse size distribution that allow for a better understanding of the mechanisms of rubber reinforcement. 

Equation (70) predicts a power law behavior G cp3-5 for the elastic modulus. Thereby, the exponent (3 + d ) / (3 - df) 3.5 reflects the characteristic structure of the fractal heterogeneity of the filler network, i.e., the CCA-clusters. The predicted power law behavior at higher filler concentrations is confirmed by the experimental results shown in Fig. 15, where the small strain storage modulus of a variety of carbon black filled rubbers is plotted against carbon black loading in a double logarithmic manner. It also agrees with older experimental data obtained by Payne [1] as shown in [63,64]. 

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-... 

The particles of carbon black are not discrete but are fused clusters of individual particles. The reinforcement conferred by the black is not influenced to any extent by the size of the unit but predominantly by the size of the particles within the unit. The primary particle typically has cross-sectional dimensions" of 5-100 nm. It is well established that the most appropriate way of describing the size of the primary particles is to express it as speciflc surface area/weight Particle size of itself has relatively little effect on the modulus. But tensile and tear strengths are affected by the particle size and both properties are normally enhanced as the surface area increases (i.e. surface area increases with decreasing particle size). The high surface area enhances the ability of the filler to wet the rubber and thus enhances the interaction at the rubber filler interface. It is the enhancement of the filler-rubber interface that provides the desired reinforcement in filled vulcanized rubber. 

The effect of carbon black on hysteresis depends primarily on the particle size of the filler and is related to breakdown and reformation of the agglomerations and the network, to slippage of polymer chains around the periphery of the filler clusters and the presence of occluded rubber. Figure 13 shows the difference of the temperature profiles of carbon black and silica filled rubber compoimds. 

SBS and SIS Thermoplastic Rubbers (Harlan, 1977 Chu, 1986) - Styrene-butadiene s rene and styrene-isoprene-s rene are thermoplastic rubber block copolymers. They were larst marketed commercially in 1965. The polymers have rubbery midblocks of butadiene or isoprene molecules and two plastic end blocks of styrene molecules. The polymers have the modulus and resilience of vulcanized butadiene and isoprene at room temperature and act as thermoplastics at higher temperatures. When SBS or SIS molecules are combined in the solid phase, a two-phase structure is formed by the clustering of the styrene endblocks. The plastic endblock regions are called domains which act as crosslinks between the ends of the rubber chains (butadiene or isoprene) locking them in place. The block copolymers act like a typical vulcanized rubber that is filled with dispersed reactive filler particles. 

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