Abstract
A model for strain-dependent dynamic properties of filler loadedrubber systems ( Payne effect ) has been derived based on theLinks-Nodes-Blobs (L-N-B) model of percolation theory. It isthe first time that a L-N-B model is applied in the study ofdynamic properties of filled rubbers. A blob in the L-N-B modelcan be a primary aggregate or a cluster formed by coagulation ofprimary aggregates, occluded rubber and bound rubber. The linkscorrespond to tenuous filler bondings between dense filleraggregates. The chains made of links and blobs are called as L-N-B chains. The connected points among L-N-B chains are callednodes. During deformation, the blobs are assumed entirely rigidso that they are not deformed. However, the links deform undertension, bending and/or torsion. The links even tend to breakoff when some failure strain is attained.Important parameters in the model include (1) the average forceconstant of a L-N-B chain, (2) the apparent yield strainamplitude, (3) the storage modulus at large strain and the lossmodulus at large strain, (4) the critical length of the L-N-Bmodel, (5) the density distribution function of the number ofsingly connected bonds, and (6) the density distributionfunction that accounts for the break-down of secondaryaggregate attributed to non-linear rubber phase deformation.The average force constant corresponds to the angulardeformation element ( primarily attributed to bound rubber) inseries with the tension element (controlled by van der Waalsforce between fillers ).Simulation results indicate that the apparent yield strainamplitude and the density distribution function control thebreak-down and recombination of the filler network. Tworecombination mechanisms are adopted in this study. Result ofsimulations from the extreme ends recombination mechanismmatches the experimental data better than that from the zerostrain recombination mechanism. With the combination of van derWaals force model, these parameters can be described in terms ofbasic physical material properties of filler and rubber matrix.The temperature dependence of dynamic properties of filledrubbers is successfully described in terms of the seriescombination of the angular deformation element in series withthe tension element. Moreover, normalized storage modulus canbe expressed as a master curve. The parameter of the mastercurve is almost a constant for different carbon black filledsystems. The distributions of blob and link in a L-N-B modelare similar for carbon black filled rubber systems. Besides theeffect of fillers (carbon black vs. silica), the effect ofmatrix ( filler-in-oil vs. filler-in-rubber) and the effect ofmixing time are discussed too. The limitationof L-N-B model is also discussed. The result of simulationssuggests that, at very small strain region, deformation insideof aggregates such as slippage between primary particle surfaceand deformation of occluded rubber contribute some degree ofhysteresis loss.