The advance in nanotechnologies requires increasingly more novel functional materials. As of now, materials with a special combination of properties (e.g., magnetic-transparent, conductive-transparent, catalytic-magnetic, etc.) are more and more required. Materials based on metal-polymeric composites will meet many present and future technological demands, since they possess the above combination of necessary properties, as well as the characteristic properties of metals as plasmon resonance, superparamagnetism, etc.

It is our opinion that at the moment there is no adequate theory which would describe such materials properly. Therefore, the theory which is able to explain and predict behavior of properties and characteristics metal-polymeric composites with chaotic and fractal structure should be crated.

Extensive investigations of composites with random structure have repealed changes of their properties with the structure and the concentration of the components. The problem of proper understanding of the behaviour of the composites requires correct description of their structure and correspondingly requires knowledge of the distribution function of the components.

A simple model of percolation on hierarchical lattices has been proposed and developed by the researches from the Chair of Higher Mathematics from Odessa National Polytechnic University. The main idea of this approach consists in step-by-step averaging of the physical parameters at different scale levels of the non-uniform media. The model is useful both to analyse conductivity and to study dielectric and elastic properties of some materials with fractal structures. It has been known that some of such complicated structures may exhibit rather surprising properties. Systems which exhibit such unusual properties have studied both by computer simulations and analytically. Materials, which have been manufactured only a few years ago (nanocomposites), as it is known, are important from the point of view of various potential technological applications.

A comprehensive description, more fundamental understanding and proper applications of various properties of the nanocomposite structures require a combined approach. It is necessary to exploit both the logical simplicity of the analytical description and the predictive power of the computer simulations. We also plan to investigate the dispersion dependencies of dielectric and viscoelastic properties for such systems because these properties can be investigated using similar theoretical methods and simulation techniques.

We present model of fractal chaotic structure of a non-uniform medium. The given model for metal-polymeric nanocomposite allows calculation of electrical and optical parameters was carried out, such as inductivity, effective conductivity, tangent of dielectric losses at various values of concentration of metal phase and frequency of exterior electromagnetic field.

   The generalized effective conductivity of an inhomogeneous medium with chaotic fractal structure is defined basing on the ideas of the renormalization group transformation method and the theory of fractals. Fractal sets obtained from rectangular lattices have been used to construct the structure of a composite with random distribution of components (phases). The calculation of the effective conductivity of a composite is compared with experimental data and with the calculations based on the effective medium approximation (Novikov V. V. “Physical properties of fractal structures” In the book ”Fractals, diffusion and relaxation in disordered complex systems” ed. by Stuart A. Rice, Guest editors: William T. Coffey and Yuri P. Kalmykov, p.203.(2006))...

   The elastic properties of an inhomogeneous medium with chaotic structure were derived within the framework of a fractal model using the iterative averaging approach. The predicted values of a critical index for the bulk elastic modulus and of the Poisson ratio in the vicinity of a percolation threshold were in fair agreement with the available experimental data for inhomogeneous composites...

   A model for the structure of filled polymeric composites has been developed. To simulate filler-filler and filler-matrix interaction on a mesolevel, the Voronoi polyhedron representation of smallest structural elements filler particles is used while fractal concepts are applied for the description of more coarse-grained structures. An iterative method based on the ideas of renormalization group transformations is presented to calculate viscoelastic properties such as the storage and loss modulus of the composite. The influence of frequency as well as the properties of filler and matrix on the effective viscoelastic properties of polystyrene melt filled with glass spheres has been elucidated in a wide concentration range. The calculations and the experiments are in good agreement. Moreover, model calculations coincide with results of percolation theory...

   This method allows defining the effective thermal conduction of filled polymeric composites with chaotic structure. Namely, the Voronoi polyhedron has been formed in order to define the thermal conduction of aggregated particles of the filler is built. The influence of the interface layer and dimensions of the filler particles on the effective thermal conduction of a composite has been analysed. Fractal sets are used to form the structure of a composite with random distribution of components (phases). The calculation of the thermal conduction of filled polymeric composites with the experimental data has been compared...

   The calculations of the effective electrical conduction of filled polymeric composites epoxy resin – copper, polyvinyl chloride – copper, epoxy resin – nickel, and polyvinyl chloride – nickel were carried out with the help of the iterative averaging method. The comparison of the calculation and experimental data shows their good agreement...

   It is explicitly assumed that the fractional derivative is related to the dimensionality of a temporal fractal ensemble (in a sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of a microstructure of inhomogeneous media exhibiting nonexponential dielectric relaxation is built by singling out groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. Different relaxation functions are derived assuming that the real (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) selfsimilarity level, the relaxation should be of a classical, Debye-like type, whatever the pattern of nonclassical relaxation at longer times...

   The iterative step-by step averaging method allows us to study Hall and Faraday effects in the composite over a large range of different parameters: concentrations, conductivities, Hall and Faraday parameters of components, and a magnetic field... The iterative averaging method allows us to obtain the specific dependencies of the effective Hall and Faraday properties on a scale (number of iteration steps). This dependence yields information about geometry prevailing at a given scale. The transformation to the regime of Euclidean geometry quasihomogeneous medium occurs on a characteristic scale where the logarithm of a property ceases to depend on a scale. Note that the scale, as well as dependencies of the effective Hall properties, is multiparametric dependency...

   Based on the fractal model of an inhomogeneous medium with a chaotic structure and the iteration method of averaging, frequency dependences of the dielectric properties of metal–insulator composites were determined. In the low-frequency limit, the considered methods of the investigation of two-component media were shown to permit one to obtain detailed information on the metal–insulator transition...

   There is important mechanism of the diffusion processes in the nanocomposites materials. The investigation methods of the irregular fractal may be apply to this materials. One of these methods is diffusion processes description with fractional differential equations. In compliance with Stable Random Walk Theory and Telegraph’s Equation with fractional derivate was express the diffusion process model. It takes into account inertness of the system and slow relaxation effect in the irregular medium. The Value of the exponent is computed for the fractional differential equation, which define the diffusion processes in the conducting irregular medium at the low magnetic field. This value is in agreement with the same values obtained by other researchers using different experimental methods...