Growth and Fractal Structure of Ceramic Precursors
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dimensions provide an upper length scale. It is only meaningful to ask if an object is fractal over an appropriate length scale.
Figure 1. The Sierpinski Carpet, a regular (as opposed to random) fractal, clearly displays the property of self-similarity. Fractals have holes on all length scales, the largest being on the order of the object itself. The scaling properties of fractal structures, both surface structures and mass structures, are quantified by their fractal dimensions. A mass fractal dimension may be defined in the following way. The mass of an object may usually be expressed as a power law in its radius: M « RD For an ordinary (or Euclidian) object, the exponent D is equal to the dimension of space, e.g., in three dimensions the mass of a sphere scales as the radius to the third power. For a fractal object, the exponent D is not equal to the dimension of space, and in general is not an integer. However, D is analogous to the dimension in the equation for Euclidian objects and is called the fractal (from fractional) dimension. A surface fractal dimension may be defined in a similar manner. Mass fractal dimensions are always less than the dimension of the space in which the fractals exist. This means that as a fractal grows, its mass increases less rapidly than the volume it occupies. Therefore, the density of a fractal is not constant, but decreases with its size; as the fractal gets larger, the holes in it get
larger, faster, with the largest holes approaching the size of the object itself (Figure 2). Surface fractal dimensions, on the other hand, must lie in the range of one less than the dimension of space up to the dimension of space. For these objects, surface area is being added faster than volume. As a result, the surface is very folded and c o n v o l u t e d , the c o n v o l u t i o n s approaching the size of the object (Figure 3). The property of being highly decorated with holes or convolutions is referred to as "ramified." The expanding interest in fractal structures has been driven by their more and more frequent observation in nature, including ceramic precursor formation, and their occurrence in computer simulations of random growth processes. 3 A major goal has been to relate real structures to simulated processes, since real growth processes are very difficult to observe on an atomic or colloidal scale. This goal has been somewhat elusive. Many plausible models for random growth processes generate structures which have yet to be observed in nature, but in the absence of a model, an observed fractal dimension, by itself, provides little information about the physics and chemistry which generated the structure. Much of what is known about the type of random growth processes which generate fractal structures has been learned from computer simulations. Perhaps the best known example is the simulation of diffusion-limited aggregation conceived of by Witten and Sander" to describe colloid aggregation. Their simulation may be imagined as starting as a large chessboard with a pawn at its cent
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