Filler-rubber clusters

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. 

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]. 

There are a number of papers in the open literature explicitly reporting on the properties of boron cluster compounds for potential neutron capture applications.1 Such applications make full use of the 10B isotope and its relatively high thermal neutron capture cross section of 3.840 X 10 28 m2 (barns). Composites of natural rubber incorporating 10B-enriched boron carbide filler have been investigated by Gwaily et al. as thermal neutron radiation shields.29 Their studies show that thermal neutron attenuation properties increased with boron carbide content to a critical concentration, after which there was no further change. 

By analogy with the works which dealt with cellulose micro crystal-reinforced nanocomposite materials, microcrystals of starch [95] or chitin [96, 97] were used as a reinforcing phase in a polymer matrix. Poly(styrene-co-butyl acrylate) [95,96], poly(e-caprolactone) [96], and natural rubber [97] were reinforced, and again the formation of aggregates or clustering of the fillers within the matrices was considered to account for the improvement in the mechanical properties and thermal stability of the respective composites processed from suspensions in water or suitable organic solvents. 

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... 

Fig. 36 Schematic view of kinetically aggregated filler clusters in rubber below and above the gel point < the left side characterizes the local structure of carbon black clusters, build by primary particles and primary aggregates accordingly, every black disc on the right side represents a primary aggregate...

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. 

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... 

The application of this normalized, relative stress in Eq. (32) is essential for a constitutive formulation of cyclic cluster breakdown and re-aggregation during stress-strain cycles. It implies that the clusters are stretched in spatial directions with deu/dt>0, only, since AjII>0 holds due to the norm in Eq. (33). In the compression directions with ds /dt<0 re-aggregation of the filler particles takes place and the clusters are not deformed. An analytical model for the large strain non-linear behavior of the nominal stress oRjU(eu) of the rubber matrix will be considered in the next section. 

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. 

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 ... 

For filler reinforced rubbers, both contributions of the free energy density Eq. (35) have to be considered and the strain amplification factor X, given by Eq. (39) differs from one. The nominal stress contributions of the cluster deformation are determined by oAtfJ=dWA/dzA, where the sum over all stretching directions, that differ for the up- and down cycle, have to be considered. For uniaxial deformations E =e, E2=Ej= +E) m- one obtains a positive contribution to the total nominal stress in stretching direction for the up-cycle if Eqs. (29)-(36) are used ... 

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.

Figure 46a shows that a reasonable adaptation of the stress contribution of the clusters can be obtained, if the distribution function Eq. (55) with mean cluster size =25 and distribution width b= 0.8 is used. Obviously, the form of the distribution function is roughly the same as the one in Fig. 45b. The simulation curve of the first uniaxial stretching cycle now fits much better to the experimental data than in Fig. 45c. Furthermore, a fair simulation is also obtained for the equi-biaxial measurement data, implying that the plausibility criterion , discussed at the end of Sect. 5.2.2, is also fulfilled for the present model of filler reinforced rubbers. Note that for the simulation of the equi-biaxial stress-strain curve, Eq. (47) is used together with Eqs. (45) and (38). The strain amplification factor X(E) is evaluated by referring to Eqs. (53) and (54) with i= 2= and 3=(l+ ) 2-l.

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