the strength of the van der Waals forces between the particles. In the Merabia’s
vision, the thickness of the glassy layer decreases, diminishing as a consequence the
number of glassy bridges forming the network. For Maier and Goritz, the rate of the
chains desorption increases with temperature.
Application of the Model of Variable Network Density In the Maier and Goritz
model, [21, 30, 31] is assumed that the rubber molecules come in contact with the
filler surface and get adsorbed. After forming the first link, the neighboring segments have a high probability to attach to the next interaction position. These chains
form stable bonds to the filler surface, whereas the remaining chains coming
afterward form unstable bonds having very weak links to the particle. These can
be easily removed by a tensile force or by raising the temperature. According to this
model, the network density of a filled vulcanised elastomer (N ) comes from the
overall contribution of the chemical network density, N c , the chains network
density caused by stable bonds at the filler surface, N st , and the density of unstable
bonds between chains and filler, N l
N ¼ N c þ N st þ N l
ð6Þ
Storage modulus E
0 (γ) ¼ Nk B T where N is the network density, k B is the
Boltzmann constant, and T is the temperature in Kelvin. The strain dependent
modulus can be described by means of the theory of entropy and elasticity with
E
0
γ
ð Þ ¼ N c À N st þ N l γ
ð Þ
ð
Þ k B T
ð7Þ
By analogy with the Langmuir isotherm formation, it is assumed that the
adsorption/desorption process reaches a balance after a certain time. On the
assumption that the desorption rate is proportional to the strain amplitude, γ, the
dependence of the modulus on γ can be written as
E γ
ð Þ ¼ E
0
st þ E l = 1 þ cγ
ð
Þ
ð8Þ
with
E
0
st ¼ N e þ N st
ð
Þ k B T
ð9Þ
and
E
0
1 ¼ N k k B T
ð10Þ
The experimental curves were fitted to Eq. (8). Figure 16 shows the curve fits for
the storage modulus versus the strain amplitude. The fit parameters, E
0
st and E
0
l and
the values of 2/DOF and R 2 to assess the quality of fit are given in Table 1. Both E
0
l
and E
0
st increase with an increase in silica content. The values characterizing the
unstable part of the network are strongly influenced by the filler content, as can be
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
213
vision, the thickness of the glassy layer decreases, diminishing as a consequence the
number of glassy bridges forming the network. For Maier and Goritz, the rate of the
chains desorption increases with temperature.
Application of the Model of Variable Network Density In the Maier and Goritz
model, [21, 30, 31] is assumed that the rubber molecules come in contact with the
filler surface and get adsorbed. After forming the first link, the neighboring segments have a high probability to attach to the next interaction position. These chains
form stable bonds to the filler surface, whereas the remaining chains coming
afterward form unstable bonds having very weak links to the particle. These can
be easily removed by a tensile force or by raising the temperature. According to this
model, the network density of a filled vulcanised elastomer (N ) comes from the
overall contribution of the chemical network density, N c , the chains network
density caused by stable bonds at the filler surface, N st , and the density of unstable
bonds between chains and filler, N l
N ¼ N c þ N st þ N l
ð6Þ
Storage modulus E
0 (γ) ¼ Nk B T where N is the network density, k B is the
Boltzmann constant, and T is the temperature in Kelvin. The strain dependent
modulus can be described by means of the theory of entropy and elasticity with
E
0
γ
ð Þ ¼ N c À N st þ N l γ
ð Þ
ð
Þ k B T
ð7Þ
By analogy with the Langmuir isotherm formation, it is assumed that the
adsorption/desorption process reaches a balance after a certain time. On the
assumption that the desorption rate is proportional to the strain amplitude, γ, the
dependence of the modulus on γ can be written as
E γ
ð Þ ¼ E
0
st þ E l = 1 þ cγ
ð
Þ
ð8Þ
with
E
0
st ¼ N e þ N st
ð
Þ k B T
ð9Þ
and
E
0
1 ¼ N k k B T
ð10Þ
The experimental curves were fitted to Eq. (8). Figure 16 shows the curve fits for
the storage modulus versus the strain amplitude. The fit parameters, E
0
st and E
0
l and
the values of 2/DOF and R 2 to assess the quality of fit are given in Table 1. Both E
0
l
and E
0
st increase with an increase in silica content. The values characterizing the
unstable part of the network are strongly influenced by the filler content, as can be
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
213