Fractal dimension of fractals tensor product ferroelectric ceramic materials frontiers

Abstract

BaTiO3-ceramics is a porous material made out of grains and pores. Such a structure is best explained by fractal nature approach. There is a strong porosity influence on microelectronic properties, so a pore itself acts as a microcapacitor with traces of inductivity. A cluster of pores than makes an integral impedance with an equivalent circuit scheme. Fractal nature theory approach helps modeling such microimpedance. The experimental data are acquired using SEM micro-graphs of the selected specimens. Further method is based on fractal analytic procedures based on extraction of fractal dimension and involving this as a key parameter. Such an approach, in combination with methods developed by the authors completes the extensive method for determination of relationship between microelectronic properties of sintered barium-titanate ceramics and technological parameters used in the sintering process. The issue of additives is also considered due the fact that dopants have considerable influence on electronic characteristics. Fractal analyzes and fractal dimension materials characterization are very important as a new tool with in the materials, electronic ceramics, and concretely ferroelectric applications. In this paper we applied fractal Cantor set and also developed fractal curvature tensor product. This enlighten new frontiers in electroceramics materials surface characterization.

KEYWORDS:

• Fractals
• tensors product
• ferroelectric-ceramics

Funding

This research is a part of the Project “Directed synthesis, structure and properties of multifunctional materials” (172057). The authors gratefully acknowledge the financial support of Serbian Ministry of Education, Science and Technological Development for this work.

Figure 1. Cantor ternary set - fractal.

Figure 2. Horizontal view of Cantor 2D dust (left) and vertical (right).

Figure 3. Cantor 2D dust in process of infinitesimal division (three-dimensional view).