Stereological Models Simulation of BaTiO3 Ceramics Grains

Abstract

The Stereological configuration of the position of BaTiO3-grains and the influence of grain displacement of contact surfaces is studied in this paper. We add two important conjectures to the recent authors' result that the fractal dimension of the contours of BaTiO3-ceramics grains is ≈ 1.0677699. The function that represents spacial distribution of the intergrains' contacts is not a totally random function but a function that contains an amount of “order” which makes the fractal dimension lower than Ds < 3. Next is the supported conjecture that the intergrain contact surface has a global shape of minimal surface, while the local shape is fractal which increases the size of the smooth surface and also increases the microcapacity generated at the contact. To simplify calculations, grain shape is approximated by polyhedron, sphere, ellipsoid and a real grain's shape, obtained by the SEM method. Relevant mathematical relations are established that give an approximate size of the intergrains contact area as the function of grains displacement inside the pattern used. The offered method consists of two components: 1) Fractal analysis, which gives the fractal dimension of the grain on the basis of microphotography, 2) Adding the fractal grains surface on Euclidean models—polyhedron, sphere, ellipsoid and the real grain's shape.