The interconnection of efficiency and the degree of aggregation of nanofiller in polymer nanocomposites

The rule of mixtures was the first theoretical model, used for a description of the elastic modulus of multicomponent systems, including those of polymer nanocomposites. However, the use of nominal magnitudes of the characteristics of system components in such an approach led to overestimated values of their elastic modulus. Therefore, various modified versions of the rule of mixtures are currently used for this purpose, which significantly complicate its application and do not indicate the physical factors leading to overestimated theoretical results. In this study, a modified rule of mixtures was proposed, taking into account the decrease in the effective (actual) elastic modulus of the nanofiller in a polymer matrix of the nanocomposite compared to the nominal value determined only by the aggregation of the nanofiller. It is known that the aggregation process is the main negative factor reducing the final properties of nanomaterials, while other factors (for example, the interfacial adhesion level, nanofiller orientation, etc.) depend on the degree of aggregation. The physical sence of the aggregation process is a decrease in the relative proportion of nanofiller-polymer matrix interfacial regions, i.e., the effectiveness of a nanofiller as a reinforcing element of a nanocomposite is determined by its ability to generate high-modulus interfacial regions. The rule of mixtures modified in this way correctly describes the dependence of the elastic modulus of the nanocomposite on the content of the nanofiller, regardless of the type of the latter (carbon nanotubes, graphene, etc.). Therefore, the nanofiller efficiency indicator can serve as a complex parameter that is characteristic of the nanocomposite quality.