Filler clusters

It is important to note here that the presence of rigid filler clusters, with bonds in the virgin, unbroken state of the sample, gives rise to hydrodynamic reinforcement of the mbber matrix. This must be specified by the strain amplification factor X, which relates the external strain of the... 

The flocculation results presented in Figure 22.2 give strong evidence that kinetically aggregated filler clusters or networks are formed in elastomer composites, as shown in Figure 22.5. 

For a consideration of filler-network breakdown at increasing strain, the failure properties of filler-filler bonds and filler clusters have to be evaluated in dependence of cluster size. This allows for a micromechanical description of tender but fragile filler clusters in the stress field of a strained mbber matrix. A schematic view of the mechanical equivalence between a CCA-filler cluster and a series of soft and hard springs is presented in Figure 22.9. The two springs with force constants... 

FIGURE 22.5 Schematic view of kinetically aggregated filler clusters in mbber below and above the gel point <1>. The left side characterizes the local stmcture of carbon black clusters, built by primary particles and primary aggregates. (Every black disk in the center figure [ and on the right-hand side

primary aggregate.) (From Kliippel, M. and Heinrich, G., Kautschuk, Gummi, Kunststojfe, 58, 217, 2005. With permission.)... 

The second addend in Equation 22.21 considers the energy stored in the substantially strained fragile but soft filler clusters ... 

Here, Ga is the elastic modulus and Sa/j, is the strain of the fragile filler clusters in spatial direction /i =x,y,z). The dependence of these quantities on cluster size and external strain is specified in [24]. 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

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... 

An explanation of the observed relaxation transition of the permittivity in carbon black filled composites above the percolation threshold is again provided by percolation theory. Two different polarization mechanisms can be considered (i) polarization of the filler clusters that are assumed to be located in a non polar medium, and (ii) polarization of the polymer matrix between conducting filler clusters. Both concepts predict a critical behavior of the characteristic frequency R similar to Eq. (18). In case (i) it holds that R= , since both transitions are related to the diffusion behavior of the charge carriers on fractal clusters and are controlled by the correlation length of the clusters. Hence, R corresponds to the anomalous diffusion transition, i.e., the cross-over frequency of the conductivity as observed in Fig. 30a. In case (ii), also referred to as random resistor-capacitor model, the polarization transition is affected by the polarization behavior of the polymer matrix and it holds that [128, 136,137]... 

So far, we have considered the elasticity of filler networks in elastomers and its reinforcing action at small strain amplitudes, where no fracture of filler-filler bonds appears. With increasing strain, a successive breakdown of the filler network takes place and the elastic modulus decreases rapidly if a critical strain amplitude is exceeded (Fig. 42). For a theoretical description of this behavior, the ultimate properties and fracture mechanics of CCA-filler clusters in elastomers have to be evaluated. This will be a basic tool for a quantitative understanding of stress softening phenomena and the role of fillers in internal friction of reinforced rubbers. 

Fig. 41 Schematic view demonstrating the mechanical equivalence between a filler cluster and a series of soft and stiff molecular springs, representing bending-twisting- and tension deformation of filler-filler bonds, respectively...

For analyzing the fracture behavior of filler clusters in strained rubbers, it is necessary to estimate the strain of the clusters in dependence of the external strain of the samples. In the case of small strains, considered above, both strain amplitudes in spatial direction n are equal (t A F i). because the stress is transmitted directly between neighboring clusters of the filler network. For strain amplitudes larger than about 1%, this is no longer the case, since a gel-sol transition of the filler network takes place with increasing strain [57, 154] and the stress of the filler clusters is transmitted by the rubber matrix. At larger strains, the local strain eAtfl of a filler cluster in a strained rubber matrix can be determined with respect to the external strain if a stress equilibrium between the strained cluster and the rubber matrix is assumed ea GpX =6rm( u)) With Eq. (29) this implies... 

This allows for an evaluation of the stress contribution of the stretched filler clusters if the size distribution of the clusters in the system is known. Note that for small deformations, where holds, the situation is differ-... 

Fig. 43 Schematic view of the decomposition of filler clusters in rigid and fragile units for pre-conditioned samples. The right side shows the cluster size distribution with the pre-strain dependent boundary size min...

Hydrodynamic reinforcement of the rubber matrix by the fraction of hard, rigid filler clusters with strong filler-filler bonds that have not been broken during previous deformations. 

Cyclic breakdown and re-aggregation of the residual fraction of fragile, soft filler clusters with weaker filler-filler bonds. 

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 ... 

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.

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