Modeling of the carbon black reinforcement mechanism in elastomers.

by Michel Gerspacher, Charles P. O'Farrell and H.H.Yang, Sid Richardson Carbon Co., and Claude Tricot, Ecole Polytechnique

The term reinforcement in tire technology can have numerous meanings, including tire treadwear resistance, heat dissipation, flex resistance, traction, etc. The object of this article is to present a cohesive model of the carbon black reinforcement mechanism which may be used to explain the filler's contribution on end use tire performances.

A tire is essentially an object undergoing dynamic deformation. This can be translated in terms of strain energy input which because of the viscoelastic nature of the elastomeric compounds will be partly stored as active energy and partly dissipated in the form of heat build-up. The stored energy (elastic) is essential to the proper functioning of the tire; however, too much stored energy may result in crack propagation within the polymeric matrix of the compound. To minimize crack propagation, more energy dissipation (heat) would be beneficial; however, this trade-off will certainly increase the rolling resistance of the tire, which is often not acceptable. Therefore, to characterize the role of carbon black in a rubber compound, measurements of its viscoelastic characteristics are mandatory.

Previous studies have shown (refs. 1-3) that strain dependency is much more important than the influence of frequency (up to 30 Hz) or temperature in distinguishing the reinforcing role of carbon black in polymeric compounds.

Viscoelastic behavior of filled compounds

The viscoelastic nature of elastomers has been the subject of a tremendous amount of work and theories, all of which are well documented in the literature (refs. 4 and 5).

An unfilled elastomer in the rubber plateau when undergoing a dynamic strain sweep, [gamma] = [[gamma].sub.o] sin wt. exhibits in general and in first approximation a linear behavior translated by a complex modulus:

a G*u = G'u + i[G".sub.f]u which is independent of the strain amplitude (figure 1).

The presence of carbon black in quasi linear viscoelastic elastomer changes its linear characteristic to non-linear behavior, clearly described by the dependence of the complex modulus on the strain amplitude:

b [G*.sub.f] ([gamma]) = G'f ([gamma]) + i[G".sub.f]f ([gamma])

Figure 1 clearly indicates that both G'f and [G".sub.f]f are strain dependent. G' is indicative of stored elastic energy; whereas, the dissipated energy is proportional to [G".sub.f] (refs. 4 and 5). It therefore appeared judicious to consider G'f = F1 ([gamma]) and [G".sub.f]f = F2 ([gamma]) as a parametric representation of a more general equation.

c [G".sub.f]f = F3(G'f)

The representation of equation c (figure 2) is referred to as < G-plot > by the authors in a previous paper (ref. 6) and allows for a more insightful understanding of the mechanism of carbon black reinforcement.

To explain the shape of the < G-plot >, which can be shown to be affined for all carbon black/elastomer composites (ref 7), a filler network is postulated. Other authors also use the filler-filler network concept to explain the strain dependency of filled rubber compounds (refs. 8 and 9).

Mathematical modeling

A mathematical model was derived (ref. 10) and can be summarized as follows:

* Two phenomena contribute to the complex modulus: the linear viscoelastic behavior of the polymer matrix and the non-linear elastic behavior of carbon black.

* Upon deformation, the polymer pulls the aggregate apart while the London-van der Waals force pushes them together.

* Because of the particular shape of the London-van der Waals force, two stable equilibrium points are possible for a pair of aggregates: they can be bound or unbound.

* The transition from the bound to unbound state is the cause of the decrease of G' when [gamma] increases.

* The high speed at which the transition from bounded to unbounded state occurs produces increased energy loss via increased friction in the polymeric continuum due to a high deformation rate. This is the cause of the peak in the plot of G" versus [gamma].

* An expression of viscoelastic constant of the idealized model of a pair of aggregates can be found. Its real and imaginary parts, respectively, are proportional to:

1 if [gamma] [greater than or equal to] [gamma]b

s' ([gamma], [gamma]b) [gamma]3b/[gamma]3 if [gamma] > [gamma]b

0 if [gamma] [greater than or equal to] [gamma]b

S"([gamma], [gamma]b) [gamma]2b/[gamma]2 if [gamma] > [gamma]b Where

- [gamma] is the amplitude of sinusoidal shear deformation

- [gamma]b is the critical deformation causing the pair to unbind.

* The composite is made of a collection of those elementary models, characterized by a different [gamma]b. The expression of the complex modulus for the material as thus becomes:

[Mathematical Expression Omitted] W([gamma]b) is a weighing function giving the contribution of aggregate pairs which break at [gamma] = [gamma]b.

-h is a constant proportional to the average width [delta[gamma] of the hysteresis cycle.

* With this expression, one can see why plots of G" for different polymers ate so similar (figure 2). The expression of s' and s" are universal, while W([gamma]b) is material-dependent. There is a link between G' and G" because they are derived from the convolution of the same function W([gamma]b). The link remains across different materials because s' ([gamma], [gamma]b) are material-independent.

This model was successfully evaluated for a wide range of filled compounds.

Evaluation of W([gamma]b)

It can be shown that:

d W([gamma]b) = d[G".sub.f]([gamma]b)/d([gamma]b)+2/[gamma]b [G".sub.f]([gamma]b) From experimental data it is therefore possible to evaluate that distribution function.

Various carbon blacks were mixed in D3191 ASTM model formulation in order to study the influence on W([gamma]) of:

* Carbon black structure at a given level of nitrogen adsorption (5 samples);

* Carbon black loading (3 samples);

* Carbon black type (6 samples).

From the measurements of G' and G" variations with increasing strain amplitude @ 30[degrees]C and 1 Hz the functions' were obtained and the results are summarized in figures 3-5. From these initial studies it is shown once again that the nitrogen adsorption data measuring a density of hi-energy surface sites on the carbon black rather than a specific surface area (ref. 11), is of primordial importance for low strain viscoelasticity. Indeed, the total amount of van der Waals bonding, directly related to the distribution function, as expected from the filler-filler network/model govern the physical properties of the filled rubber compounds. This is also confirmed by the fact that increasing the loading increases essentially the number of filler-filler bonds. On the other hand, the "structure" does not seem to drastically influence the results.

Future work

As a future work, the function is being evaluated as a function of the filler distribution in the polymeric continuum.

[Figures 1 to 5 ILLUSTRATION OMITTED]

References

[1.] A.E. Medalia, Rubber Chem. Tech., 1978, 51, 437. [2.] M. Gerspacher, C Lansinger, Paper #7, ACS Rubber Div. Meeting, May 1989. [3.] M. Gerspacher, J.-B. Donnet, Paper #42, ACS Rubber Div. Meeting, May 1989. [4.] J.D. Ferry, Viscoelasticity of polymers, 3rd ed, Wiley Sons. [5.] N. W. Tschoegl, The phenomenological theory of linear viscoelasticity, Springer Verlag. [6.] M. Gerspacher, C.P. O'Farrell, Kautschuk Gummi Kunststoffe, 45, 1992. [7.] M. Gerspacher, H.H. Yang, G.M. Starita, L'actualite Chimique, 106, March-April (1991). [8.] G. Y. Kraus, Appl Polymer Sci., Appl Polymer Symposium, 39, 75,(1984). [9.] L. C Burton, K. Hwang and T. Zhang, 1989, RCT 621838. [10.] A. van de Walle, C. Tricot, M. Gerspacher, Kautschuk Gummi Kunststoffe, to be published [11.] M. Gerspacher, CP. O'Farrell, W.A. Wampler, Rubber World, 26, June 1995.