Fractal Models and the Structure of Materials
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Fractal Models and the Structure of Materials Dale W. Schaefer Introduction
• Science often advances through the introdction of new ideas which simplify the understanding of complex problems. Materials science is no exception to this rule. Concepts such as nucleation in crystal growth and spinodal decomposition, for example, have played essential roles in the modern understanding of the structure of materials. More recently, fractal geometry has emerged as an essential idea for understanding the kinetic growth of disordered materials. This review will introduce the concept of fractal geometry and demonstrate its application to the understanding of the structure of materials. Fractal geometry is a natural concept used to describe random or disordered objects ranging from branched polymers to the earth's surface.1 Disordered materials seldom display translational or rotational symmetry so conventional crystallographic classification is of no value. These materials, however, often display "dilation symmetry," which means they look geometrically selfsimilar under transformation of scale such as changing the magnification of a microscope. Surprisingly, most kinetic growth processes produce objects with self-similar fractal properties. It is now becoming clear that the origin of dilation symmetry is found in disorderly kinetic growth processes present in the formation of these materials. Fractal concepts have been used to treat numerous problems ranging from fracture to electrochemical response in batteries. However, many of these applications are speculative with the real value of introducing fractal ideas yet to be demonstrated. Accordingly, this review concentrates strictly on the structure of materials. This emphasis arises because, being a geometric concept, application of fractal analysis to structural issues is natural. 22
The goal of this article is to demonstrate the applicability of fractal models, not only as a scheme to classify structures, but as a tool to identify the essential factors which determine structure. To see the importance of growth processes, consider the generation of glasses via the sol-gel process (see Figure 1). Typically one begins with a soluble silicate such as tetraethylorthosilicate [Si(OC2H5)4, TEOS] and polymerizes in alcohol solution by the addition of water. The initial polymerization is a chemical growth process which can lead to a variety of structures from highly ramified branched polymers to compact colloidal particles (A, B, and C in Figure 1). In the later stages of growth, gelation can take place either by continued chemical branching (D) or by physical aggregation of primary colloidal particles (E). A solid material is then generated by drying and sintering. Materials generated by various paths on the diagram can have drastically different properties although they are chemically identical.
Fractal Structures
Before examining the correspondence between fractal models and growth, it is necessary to define fractal structures and show how they are measured in real experiment
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