CONTRIBUTION OF ENTANGLEMENTS TO THE MECHANICAL-PROPERTIES OF CARBON-BLACK FILLED POLYMER NETWORKS

A rigorous molecular statistical model of filled polymer networks with quenched structural disorder coming from the conventional chemical cross-links between polymers and from an ensemble of rigid and highly dispersed multifunctional filler domains is presented. Specific surface and structure of the filler and its effect of ''hydrodynamic'' disturbance of intrinsic strain distribution are explicitly taken into account. The conformational constraints (entanglements, packing effects) of polymer chains in the mobile rubber phase are described by a mean-field-like tube model. The calculation of the elastic free energy follows a straightforward generalization of the non-Gibbsian statistical mechanics using the replica technique. Within the model proposed, the network is characterized by four typical length scales: the Kuhn's statistical segment length l(s) of the polymers; the root-mean-square end-to-end distance of the mobile network chains R(c); the square root b(PR) of the average area available to a couple site between mobile polymer phase and filler surface; and the lateral tube dimension d(o), which is equal to the mean spacing between two successive entanglements in the mobile rubber phase. Application of the theory to stress-strain experiments of unfilled and carbon black filled vulcanizates (styrene-butadiene copolymer rubbers, butadiene rubbers) yields the characteristic length scales R(c), b(PR), and d(o). The typical relations b(PR) almost-equal-to l(s) and 1(s) < d(o) < R(c) were found. The estimated values of the entanglement spacing d(o) agree with independent calculations for unfilled systems and direct (neutron scattering) measurements. It is further found that the contribution of entanglements to the equilibrium modulus of the elastomer is in the order of contributions coming from the polymer-polymer and polymer-filler junctions. The observation that the entanglement contribution to the elastic modulus decreases somewhat with increasing filler content (i.e., d(o) increases) is discussed within scaling considerations. It is concluded that the estimated coupling density between polymer phase and carbon black surface is related to the mean number of entanglements formed between the tightly adsorbed bound rubber and the bulk rubber. The problem of polymer adsorption on a carbon black surface is discussed within the concept of disorder-induced localization of polymers on disordered or fractal surfaces.