General remarks and scaling

Although understanding the behavior of polymers on heterogeneous surfaces is a general problem in theoretical physics it provides deep insight into the problem of reinforcement and contributions. It is well accepted that the filler particles form large clusters which diffuse throughout the mixture to provide the most significant reinforcement effect on large macroscopic scales [179, 181, 197, 198]. Consequently these clusters form large surfaces inside the elastomer and allow significant polymer–filler contacts. Figure 9.1 shows a typical particle aggregate, with its hierarchy of length scales. The aggregate consists of individual particles, each with an irregular rough surface. As the particles form larger aggregates the irregular surfaces become very large. Moreover, the aggregates themselves form large clusters when the filler concentrations are high enough. Therefore we can expect major contributions to the reinforcement from the interaction between the polymer matrix and the irregular, rough surfaces.

However, the filler particles do not have homogeneous surfaces, but are strongly disordered. The disorder can be categorized in two extreme cases. In the first, the filler particles are spatially disordered. The second extreme case arises from the irregularity of the interactions. Imagine the surface to be spatially flat, but with the interaction energy varying randomly at each point on the surface. Such surfaces show non-trivial effects on the surrounding polymers as well. Both cases are driven by typical “disorder effects,” which we will study in Section 9.2. Indeed several studies [135, 156, 199] suggest a strongly heterogeneous surface. Gerspacher and coworkers provided some data which even suggest fractal surface properties for several carbon blacks [135, 199].