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# Stiffness

Modulus is one of the basic properties of composites and the goal of using particulate fillers is often to increase it (Bajaj et al. 1987). Modulus is not only the most frequently measured but also the most often modeled composite property. A large number of models exist which predict the composition dependence of stiffness or give at least some bounds for its value. The abundance of models is relatively easy to explain: modulus is determined at very small deformations, and thus the theory of linear viscoelasticity can be used in model equations. The large number of accessible data also helps both the development and the verification of models. Model equations developed for heterogeneous polymer systems can be classified in different ways (McGee and McGullough 1981, Dickie 1978). Ignoring completely empirical correlations, we distinguish four groups here:

• 1. Phenomenological equations which are similar to the spring and dashpot models used for the description of the viscoelastic properties of polymers (Dickie 1978).
• 2. Bounds. These are usually exact mathematical solutions which do not contain any or only very limited assumptions about the structure of the composite (Halpin and Kardos 1976).
• 3. Self-consistent models. The mechanical response of a composite structure is calculated in which the dispersed particle is assumed to be embedded into the continuous phase. A well-known model of this type, frequently used also for particulate-filled composites, is the Kerner equation (Dickie 1978). Although it was much criticized because of the incorrect elastic solution used (Christensen and Lo 1979), the model gained wide use and acceptance.
• 4. Semiempirical models. In spite of the effort of the self-consistent models to take into account the influence of microstructure, they very often fail to predict correctly the composition dependence of composite modulus; thus, additional, adjustable parameters are introduced in order to improve their performance. The most often applied equation of this type is the Nielsen (also called Lewis-Nielsen or modified Kerner) model (Nielsen 1974).

where G, Gm, and Gf are the shear moduli of the composite, the matrix, and the filler, respectively, vm is the Poisson’s ratio of the matrix, and is the filler content.

The equation contains two structure-related or adjustable parameters (A, ?). The two parameters, however, are not very well defined. A can be related to filler anisotropy, through the relation A = kE 1, where kE is Einstein’s coefficient, but the relation has not been thoroughly investigated and verified. ? depends on maximum packing fraction. ф™3* is related to anisotropy, but it is influenced also by the formation of an interphase which was not taken into consideration in the original treatment (Nielsen 1974). Its experimental determination is difficult. McGee and McCullough proposed a different form for ?, which is supposed to be based on a more rigorous treatment (McGee and McGullough 1981).

In spite of these uncertainties, the model is quite frequently used in all kinds of particulate-filled composites for the prediction of the composition dependence of modulus. In some cases, merely the existence of a good fit is established; in others, conclusions are drawn from the results about the structure of the composite. However, the attention must be called here to some problems of the application of these equations or any other theoretical model. The uncertainty of input parameters might bias the results considerably. Poisson’s ratios between 0.25 and 0.30, as well as moduli between 19.5 and 50 GPa, have been reported for CaCO3 (Chacko et al. 1982; Vollenberg 1987). Such changes in component properties may lead to differences in the final prediction which exceed the standard deviation of the measurement. Maximum packing fraction influences predicted moduli especially strongly, but its value is usually not known. A certain packing of the particles may be assumed, but this approach neglects the effect of particle size distribution and interactions. At the moment, the best solution is the fitting of the equation to the experimental data and the determination of A and ф f max . The model is very useful for the estimation of the amount of embedded filler in polymer/elastomer/filler composites, but otherwise its value is limited.

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