The main objectives of this study are to investigate the role of chain entanglements and aggregate interaction and to estimate their contributions in terms of tube-like configurational constraints in carbon black filled networks at the thermodynamical equilibrium. The first part of present study is based on the Heinrich-Vilgis(H-V) model of filled rubbers in which contributions of crosslink and constraint to the stress-strain behavior can be separated by generalizing and combining recent theoretical results resting on "the non-Gibbsian statistical mechanics" using the replica technique and the configurational tube approach. The role of filler particles in rubber networks can be described by their "hydrodynamic effect" together with its action as an ensemble of highly dispersed "multifunctional domains". The conformational constraints (entanglements, packing effects) of polymer chains in the mobile rubber phase may be described by a mean-field-like tube model. In the second part, the model is applied to static and dynamic mechanical properties and fracture mechanics of carbon black filled rubbers with different types of network structures and polymers. In this case, "the occluded rubber effect" is, additionally, taken into account. It is considered for the polymer to penetrate into the void space of the individual carbon aggregate, partially shielding it from deformation. The introduction of a coupling between mobile rubber phase and active filler domains to explain the enhancement of the vulcanizate properties is based on the experimental experience that the network modulus is higher than that predicted from the pure hydrodynamic interaction model. The coupling with the additional reinforcement is recognized to be caused by entanglement formed between tightly adsorbed bound rubber on the filler surface and the bulk rubber far removed from the surface.
Application of the H-V model to stress-strain experiments yields the network parameters which characterize network structures of unfilled and carbon black filled vulcanizates(natural rubber(SMR-20), styrene-butadiene copolymer(SBR-1712)) ; such as $M_C$ the number-average molecular weight of chain between networks, $v_{f,eff}$ the number density of the coupling sites between mobile polymer phase and filler surface, $v_{c,eff}$, $v_{e,eff}$ the elastically effective network chain density and constraint density due to chain entanglements, respectively, $d_o$ the lateral tube dimension which is equal to the mean spacing between two successive entanglements in the mobile rubber phase, $R_c$ the root-mean-square end-to-end distance of the mobile network chains, $A_{PF}$ the average area available to a couple site between mobile polymer phase and filler surface. As a result, it is founded that the contribution of entanglements to the equilibrium modulus of the elastomer is in the order of contributions coming from the polymer-polymer, polymer-filler, and filler-filler junctions. Moreover, the elastic modulus increases with increasing filler content (i.e., $d_o$ and $M_c$ decrease). It is thus concluded that the estimated coupling density between polymer phase and carbon black surface is closely related to the mean number of entanglements formed between the tightly adsorbed bound rubber and bulk rubber. Finally, it can be further proved that the H-V model can be applied to the characterizations at microscopic and/or molecular level of the network parameters influencing on the mechanical properties of filled rubber networks.
