Network junction model for mechanical properties of filled vulcanizates

A nano-composite model of filler-rubber system is proposed. Contrary to the well-known Einstein-Guth-Smallwood hydrodynamic model, which was developed from low filler loading then increased to a practical level, the proposed model starts with the maximum filler concentration. That is, the filler network junction width is initially zero, then “diluted” to an average width in order to obtain the correct filler volume fraction. By assuming a simple cubic arrangement of the aggregates, the small strain behavior of the junction rubber is analyzed in extensional mode. Using only a few parameters from carbon black morphology, a formula for the filler loading effect on the Young’s modulus is obtained with results similar to those from the hydrodynamic models. However, this is achieved without resorting to the “occluded rubber” correction factor. At small strains, it is assumed that there is no slippage induced frictional energy loss on the filler surface. To account for the hysteresis, it is proposed instead to consider the friction between rubber molecular chains as a result of deformation and the boundary condition effect on the junction rubber. The model suggests that the hysteresis, tan δ, is inversely proportional to the junction width of the filler network.