Fractal modelling of medium–high porosity SiC ceramics
Introduction
In characterising porous materials, a description of the geometry of the system of voids and their simulation with suitable models is a preliminary phase in the correlation of the physical and technical characteristics, particularly permeability. The use of Fractal Geometry has recently been established in this sector,[1], [2], [3] but the adoption of “simple models”, the most noted of which is effectively exemplified by Menger sponge4 presents two problems: the succession of the pore sizes which count only a few terms for every decade and the distribution of the volumes concentrated towards the class of larger pore diameters.
This study demonstrates how the use of a series of interconnected fractal units (Intermingled Fractal Units, IFU) can bring about an effective representation of the porosimetric characteristics of SiC based ceramics of medium to high porosity, produced using a variety of techniques.
The industrial production of SiC dates back more than 100 years. At first this was prevalently concentrated on exploiting its “diamond-like” hardness and successively the refractory characteristics of this ceramic compound.5 In recent years this material's many excellent characteristics have become the subject of much research and industrial exploitation. Stability at high temperature, and the resistance to aggressive fluids lead to the development of SiC with porosity of between 40% and 80%. This is now of particular interest in the production of filters, catalysts or heat accumulators.[6], [7], [8], [9], [10]
The experimental porosimetric data used in the following treatment are derived from the literature.[9], [10] They were obtained by the technique of mercury intrusion porosimetry (MIP).
Section snippets
Basic concepts of the application of fractal geometry in materials
In the last decades a geometry has been developed which refers to figures with fractional dimensions, so called fractals, term derived from the Latin fractus.4
“Roughly, dimension indicates how much space a set occupies near to each of its points, … it is a measure of the prominence of the irregularities of a set when viewed at very small scales, … the dimension reflects how rapidly the irregularities develop, …”11 According to Mandelbrot, the importance of this approach to irregular figures is
The intermingled fractal units (IFU) model
One of the most noted models for fractal porosity is the Menger sponge. Even though it expands infinitely, in practice it is evidently necessary to take into consideration just the dimensional range of the pores of the materials considered. In this way, if the voids distribution of a particular porous microstructure develops, or is experimentally determinable, between two limits (Λ is the superior) Menger sponge will be made up of a unit represented by a cube of side L0 = 3Λ and with a succession
Results and discussions
Pore size distributions with a bell-like form, and therefore with a unique peak value, characterise the samples of high porosity SiC discussed in reference.9 Data relative to three systems were processed, with pore size corresponding to the maximum volume distribution of 1, 4 and 10 μm (Figs. 4–6 of reference9), herein referred to as M1, M4 and M10. According to Pfeifer and Avnir procedure, the MIP experimental data are correlated with two distinct straight lines which intersect in
Conclusions
The use of a fractal model is capable of rigorously describing only those microstructures which can be effectively approximated by a model which predicts a regular growth of the distribution towards the extreme of the largest diameters. A pore size distribution with one or two peaks, frequently found in practice, cannot be representative of a true fractal because the volume–size of the pores is missing from the characteristics of scaling on the data.
The pore size distribution considered in this
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