[14]. A number of different local mechanisms have been proposed so far to explain
this phenomenon, but no consensus has emerged yet.
According to Payne, the three-dimensional structure network constructed by the
aggregation of carbon black filler significantly influences the dynamic viscoelastic
properties of carbon black filled rubbers. Kraus [15] proposed a model based on the
agglomeration/ deagglomeration kinetics of filler aggregates by assuming a Van der
waal’s type interaction between the particles. This model was further developed by
Huber and Vilgis [16–18] who relate G
0 and G
00 to the fractal dimension and the
connectivity of the network, and by Kluppel [19] who introduces the idea of cluster.
The fact that the temperature and the frequency dependence of the amplitude of the
Payne effect are not taken into consideration is certainly the most accepted criticism
of this approach. However, one of the major drawbacks comes from the evidence
brought by Funt [20] who shows from electrical conductivity measurements that the
Payne effect might occur although a continuous filler network does not exist
through the sample. As an alternative to the destruction and reformation of a filler
network, it has also been proposed that the dynamics of adsorption/desorption of the
polymer chains at the particle surface may be responsible for various linear and
nonlinear effects. Maier [21], Zhu and Sternstein [22, 23] have suggested that the
reduction of the storage modulus with the applied strain could be related to
polymer-filler interactions including the aspects of trapped topological
entanglements.
The interaction between the filler particles and the rubber matrix, which leads to
the adsorption of polymer chains on the particle surface, can be controlled by
varying the nature of the polymer-filler interface [24–29].
Maier and Goritz [30, 31] take into consideration the adsorption/ desorption
mechanism by considering the filler particles as multifunctional cross-link with
chains which are either loosely or strongly anchor to the surface. The molecular
interpretation of the Payne effect is then based on a variable network density when
the loosely tied chains are desorbed with the increase of the strain. A compromise is
also suggested, considering that the primary mechanism for the Payne effect
certainly involves the existence of cooperation between the breakdown and reformation of the filler network and the molecular disentanglement of the bound and
free rubber [20, 32–34]. Another explanation first proposed by Yatsuyanagi
et al. [35] considers the existence of a percolation network through the rigid
amorphous layer formed around the particles. Their interpretation of the Payne
effect equally relies on the competition between desorption and adsorption of this
rigid amorphous layer.
The importance of glassy layers in filled polymer has received considerable
attention very recently, when it was recognized that a glass transition gradient exist
near the surface and that the dynamics could be either enhanced or slowed down
according to the interaction of the chains with the surface. In a body of work,
[36–38] Montes et al. [38] among others have clearly shown in filled elastomers that
a maximum of reinforcement is obtained when this rigid or slow dynamics layer
forms a continuous path through the filler aggregates. In their work, [39] model the
Payne effect by considering that the stress is supported mainly by the cross section
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
195
this phenomenon, but no consensus has emerged yet.
According to Payne, the three-dimensional structure network constructed by the
aggregation of carbon black filler significantly influences the dynamic viscoelastic
properties of carbon black filled rubbers. Kraus [15] proposed a model based on the
agglomeration/ deagglomeration kinetics of filler aggregates by assuming a Van der
waal’s type interaction between the particles. This model was further developed by
Huber and Vilgis [16–18] who relate G
0 and G
00 to the fractal dimension and the
connectivity of the network, and by Kluppel [19] who introduces the idea of cluster.
The fact that the temperature and the frequency dependence of the amplitude of the
Payne effect are not taken into consideration is certainly the most accepted criticism
of this approach. However, one of the major drawbacks comes from the evidence
brought by Funt [20] who shows from electrical conductivity measurements that the
Payne effect might occur although a continuous filler network does not exist
through the sample. As an alternative to the destruction and reformation of a filler
network, it has also been proposed that the dynamics of adsorption/desorption of the
polymer chains at the particle surface may be responsible for various linear and
nonlinear effects. Maier [21], Zhu and Sternstein [22, 23] have suggested that the
reduction of the storage modulus with the applied strain could be related to
polymer-filler interactions including the aspects of trapped topological
entanglements.
The interaction between the filler particles and the rubber matrix, which leads to
the adsorption of polymer chains on the particle surface, can be controlled by
varying the nature of the polymer-filler interface [24–29].
Maier and Goritz [30, 31] take into consideration the adsorption/ desorption
mechanism by considering the filler particles as multifunctional cross-link with
chains which are either loosely or strongly anchor to the surface. The molecular
interpretation of the Payne effect is then based on a variable network density when
the loosely tied chains are desorbed with the increase of the strain. A compromise is
also suggested, considering that the primary mechanism for the Payne effect
certainly involves the existence of cooperation between the breakdown and reformation of the filler network and the molecular disentanglement of the bound and
free rubber [20, 32–34]. Another explanation first proposed by Yatsuyanagi
et al. [35] considers the existence of a percolation network through the rigid
amorphous layer formed around the particles. Their interpretation of the Payne
effect equally relies on the competition between desorption and adsorption of this
rigid amorphous layer.
The importance of glassy layers in filled polymer has received considerable
attention very recently, when it was recognized that a glass transition gradient exist
near the surface and that the dynamics could be either enhanced or slowed down
according to the interaction of the chains with the surface. In a body of work,
[36–38] Montes et al. [38] among others have clearly shown in filled elastomers that
a maximum of reinforcement is obtained when this rigid or slow dynamics layer
forms a continuous path through the filler aggregates. In their work, [39] model the
Payne effect by considering that the stress is supported mainly by the cross section
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
195