32
2 Filler and Rubber Reinforcement
collaboration with Mark very well, began formulating for the first time the ‘phantom network theory’ [104–106]. This model is in contrast to ‘affine’ model by Flory
et al. [107–109], and the junction points of a network can fluctuate freely without
being hindered by the neighboring chains. The phantom model was mathematically
further sophisticated by another James [110, 111]. The active discussions between
the two groups seem to have resulted in the acceptance of phantom network model
by Flory’s group later [112–115]. Yet, even the most sophisticated phantom network
model [116, 117] seems still on the way to its maturity [103, 113–121].
Return to the extension of Einstein’s viscosity equation. Guth, who had managed
to present Eq. (2.2) in the USA [99], had an idea of the equivalent equation of the
mechanical modulus in his mind. Almost simultaneously, H. Smallwood at U. S.
Rubber Co. had started calculating the modulus of rubber loaded with rigid sphere
particles. Supposedly taking Einstein’s papers [90–93] as a model, he managed to
derive the Smallwood equation [122];
E = E 0 (1 + 2.5ϕ)
(2.3)
where E is the Young modulus for the filled rubber, E 0 is that for the pure gum (without any filler), and ϕ is the volume fraction of spherical rigid filler. This Smallwood
equation is formally identical with the Einstein equation, Eq. (2.1). However, the
similarity is superficial: Smallwood conducted mathematical derivations by studying Einstein’s calculation with care. He delivered a lecture at Division of Polymer
Physics, American Physical Society meeting on June 24, 1944, which was published
later in Journal of Applied Physics [122].
At the same meeting, Guth also gave a lecture and presented the Guth equation;
E = E 0
1 + 2.5 ϕ + 14.1 ϕ
2
(2.4)
which is again formally identical with the Guth–Gold equation, Eq. (2.3) on viscosity.
This lecture was published, too [123]. The two lectures were given at the same
meeting, and the difference of the publication year (1944 and 1945) did not matter at
all. Both Smallwood and Guth became aware of the similarity between the viscosity
of the suspension of rigid sphere in liquid and the rubber mixed with particulate filler,
independently at the same time. On this occasion, Guth also presented the modified
Guth equation for a non-sphere filler whose shape factor was f;
E = E 0
1 + 0.67 f ϕ + 1.62 f
2
ϕ
2
(2.5)
Guth passed away on July 5, 1990, and in the memorial article of him [124], his
achievement is written as follows:
Guth generalized the viscosity theory of suspension, first developed by Einstein in his PhD
thesis, and proved the theory’s isomorphism to that of a ‘solid suspension’, like carbon black
in rubber.
2 Filler and Rubber Reinforcement
collaboration with Mark very well, began formulating for the first time the ‘phantom network theory’ [104–106]. This model is in contrast to ‘affine’ model by Flory
et al. [107–109], and the junction points of a network can fluctuate freely without
being hindered by the neighboring chains. The phantom model was mathematically
further sophisticated by another James [110, 111]. The active discussions between
the two groups seem to have resulted in the acceptance of phantom network model
by Flory’s group later [112–115]. Yet, even the most sophisticated phantom network
model [116, 117] seems still on the way to its maturity [103, 113–121].
Return to the extension of Einstein’s viscosity equation. Guth, who had managed
to present Eq. (2.2) in the USA [99], had an idea of the equivalent equation of the
mechanical modulus in his mind. Almost simultaneously, H. Smallwood at U. S.
Rubber Co. had started calculating the modulus of rubber loaded with rigid sphere
particles. Supposedly taking Einstein’s papers [90–93] as a model, he managed to
derive the Smallwood equation [122];
E = E 0 (1 + 2.5ϕ)
(2.3)
where E is the Young modulus for the filled rubber, E 0 is that for the pure gum (without any filler), and ϕ is the volume fraction of spherical rigid filler. This Smallwood
equation is formally identical with the Einstein equation, Eq. (2.1). However, the
similarity is superficial: Smallwood conducted mathematical derivations by studying Einstein’s calculation with care. He delivered a lecture at Division of Polymer
Physics, American Physical Society meeting on June 24, 1944, which was published
later in Journal of Applied Physics [122].
At the same meeting, Guth also gave a lecture and presented the Guth equation;
E = E 0
1 + 2.5 ϕ + 14.1 ϕ
2
(2.4)
which is again formally identical with the Guth–Gold equation, Eq. (2.3) on viscosity.
This lecture was published, too [123]. The two lectures were given at the same
meeting, and the difference of the publication year (1944 and 1945) did not matter at
all. Both Smallwood and Guth became aware of the similarity between the viscosity
of the suspension of rigid sphere in liquid and the rubber mixed with particulate filler,
independently at the same time. On this occasion, Guth also presented the modified
Guth equation for a non-sphere filler whose shape factor was f;
E = E 0
1 + 0.67 f ϕ + 1.62 f
2
ϕ
2
(2.5)
Guth passed away on July 5, 1990, and in the memorial article of him [124], his
achievement is written as follows:
Guth generalized the viscosity theory of suspension, first developed by Einstein in his PhD
thesis, and proved the theory’s isomorphism to that of a ‘solid suspension’, like carbon black
in rubber.