Today big attention is expressed for nanocomposites. One of the basic characteristics of nanocomposites is the ultra small dimensions of filler particles (d<100nm) and a structure which is irregular fractal. Creating a theoretical basis to prognosticate the effective properties of nanocomposites is a topical important task nowadays.

In this work, the fractal structural model and the iteration averaging method based on the idea of a renormal group transformation is proposed to predict the thermal conduction of a filled polymer nanocomposite. The averaging on mesolevels is done with the help of a Voronoi polyhedron.

In general, the definition of effective properties can be carried out according to the following scheme: firstly, properties of different configurations are found at the initial stage, then they are averaged, and after that they are handed over to the next step.

We single out two kinds of sets of bond configurations: connecting sets of “conducting” bonds (CS), and non-connecting sets (NCS).

The formulae obtained on the basis of variation evaluations are used to calculate the conductivity of the connecting and non-connecting sets.

The conductivity of the connecting set at the k-th iteration step is defined as:

λ^(k)_con=x_k-1λ^(k-1)_con+(1-x_k-1)λ^(k-1)_non-

-[(x_k-1·(1-x_k-1)( λ^(k-1)_con-λ^(k-1)_non)²)/(x_k-1·λ^(k-1)_non+(1-x_k-1)λ^(k-1)_con+2λ^(k-1)_con)],

where λ⁽⁰⁾_con=λ₁, λ⁽⁰⁾_non=λ₂; x_k=Y(x_k-1), at this x₀=x - is the concentration of the component with a conductivity of λ₁.

The iteration procedure continues until the effective conductivity of the composite, λ, is equal to:

lim λ^(k)_con=lim λ^(k)_non=λ

 ^{k→∞              k→∞}                      

The obtained results can be further used to analyze the dependence of the conductivity of the composite on such characteristics as the fractal dimension, the percolation threshold, the radius of particle, the ratio of the interface layer thickness to the particle’s radius and the thermal conduction of the interface layer.

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1. Electrical conductivity of nanocomposite materials