2.5 Reinforcing Factors of Particulate Nanofiller
29
believed to decompose during the mechanical mixing, while the mechanical shear
force cannot reduce the primary aggregate to the primary particles. On the change
of aggregation level during the processing of CB compounds, refer to the comment
in 2.6.3.
Up to this point, primary carbon particle is represented by a circle (see Fig. 2.3).
The circle is an idealized and simplified representation. Approximation by a perfect
circle enables us to make scientific consideration and any calculations much easier.
Later in Fig. 2.7, we use this modeling, but remember that the perfect circle is simply
hypothetical. In reality, the surface of primary CB particle may have many irregular
hole or void, and a treatment as surface fractal is a must for the advanced study of
the surface structure in details [48].
2.5.4 Hydrodynamic Volume Effect by Filler Mixing
At the last paragraph of 2.5.1, hydrodynamic volume effect is mentioned as well
as bound rubber and structuring of nanofiller. Payne introduced it as a most basic
factor of rubber reinforcement [85]. He discussed his finding (now known as the
Payne effect) and showed Fig. 2.6 for his explanation. In the figure, he shows that the
shear modulus of CB-loaded rubber is composed of three components in comparison
with the pure gum stock: Hydrodynamic volume effect, strong links between CB and
rubber (bound rubber), and structuring of CB.
This figure, quite often with some alterations, has been reprinted in many papers on
rubber reinforcement. Payne used this figure to account for the dynamic mechanical
behaviors under relatively small strains. However, lots of papers used the figure in
their discussion of uni-axial tensile strength observed at the ultimate strain (usually
from a few to several hundred percent strains). These usages may not be an abuse,
but an overuse. In spite of these overuses, discussions on rubber reinforcement have
been highly activated by the Payne’s well-written chapter and his insight on the
Payne effect. This is the reason why we refrain from declaring the past usages to be
an abuse.
Interestingly, the origin of the hydrodynamic volume effect is Einstein’s paper
published in 1905 [86], which is one of the five famous papers by Einstein published
in 1905 [87–89]. After this paper, Einstein continued his study and proposed the
Einstein’s viscosity equation [90–93];
η = η 0 (1 + 2.5ϕ)
(2.1)
where η is viscosity of liquid in which rigid particles are dispersed, η 0 is viscosity of
the pure liquid, and ϕ is volume ratio of the dispersed rigid sphere particle. Einstein
himself explained this equation as follows [93]:
If very small rigid spheres are suspended in a liquid, the coefficient of internal friction is
thereby increased by a fraction which is equal to 2.5 times the total volume of the spheres
suspended in a unit volume, provided that total volume is very small.
29
believed to decompose during the mechanical mixing, while the mechanical shear
force cannot reduce the primary aggregate to the primary particles. On the change
of aggregation level during the processing of CB compounds, refer to the comment
in 2.6.3.
Up to this point, primary carbon particle is represented by a circle (see Fig. 2.3).
The circle is an idealized and simplified representation. Approximation by a perfect
circle enables us to make scientific consideration and any calculations much easier.
Later in Fig. 2.7, we use this modeling, but remember that the perfect circle is simply
hypothetical. In reality, the surface of primary CB particle may have many irregular
hole or void, and a treatment as surface fractal is a must for the advanced study of
the surface structure in details [48].
2.5.4 Hydrodynamic Volume Effect by Filler Mixing
At the last paragraph of 2.5.1, hydrodynamic volume effect is mentioned as well
as bound rubber and structuring of nanofiller. Payne introduced it as a most basic
factor of rubber reinforcement [85]. He discussed his finding (now known as the
Payne effect) and showed Fig. 2.6 for his explanation. In the figure, he shows that the
shear modulus of CB-loaded rubber is composed of three components in comparison
with the pure gum stock: Hydrodynamic volume effect, strong links between CB and
rubber (bound rubber), and structuring of CB.
This figure, quite often with some alterations, has been reprinted in many papers on
rubber reinforcement. Payne used this figure to account for the dynamic mechanical
behaviors under relatively small strains. However, lots of papers used the figure in
their discussion of uni-axial tensile strength observed at the ultimate strain (usually
from a few to several hundred percent strains). These usages may not be an abuse,
but an overuse. In spite of these overuses, discussions on rubber reinforcement have
been highly activated by the Payne’s well-written chapter and his insight on the
Payne effect. This is the reason why we refrain from declaring the past usages to be
an abuse.
Interestingly, the origin of the hydrodynamic volume effect is Einstein’s paper
published in 1905 [86], which is one of the five famous papers by Einstein published
in 1905 [87–89]. After this paper, Einstein continued his study and proposed the
Einstein’s viscosity equation [90–93];
η = η 0 (1 + 2.5ϕ)
(2.1)
where η is viscosity of liquid in which rigid particles are dispersed, η 0 is viscosity of
the pure liquid, and ϕ is volume ratio of the dispersed rigid sphere particle. Einstein
himself explained this equation as follows [93]:
If very small rigid spheres are suspended in a liquid, the coefficient of internal friction is
thereby increased by a fraction which is equal to 2.5 times the total volume of the spheres
suspended in a unit volume, provided that total volume is very small.