Mass, scaling with size

Another important property of fractal aggregates is that their mass scales with their size raised to the fractal dimension power. For a naturally occurring aggregate, apparently random to the eye, size is difficult to define in terms of the typically ragged border and anisotropic shape. Thus, a convenient measure of size that can be precisely quantified is the radius of gyration, R defined as... 

What this means is that if we measure the fractal dimension describing how the area of a projected fractal scales with size, and if the dimension so obtained is less than two, then the value obtained is also the mass fractal dimension of the structure in three-dimensional space. We can express this sense of fractal scaling in images as... 

Why do small lizards and large lizards have different proportions Consider how various features of a lizard scale with size. The mass of a lizard is proportional to the volume of a lizard, which is proportional to the height cubed ... 

The summation term is the mass broken into size interval / from all size intervals between j and /, and S is the mass broken from size internal i. Thus for a given feed material the product size distribution after a given time in a mill may be deterrnined. In practice however, both S and b are dependent on particle size, material, and the machine utilized. It is also expected that specific rate of breakage should decrease with decreasing particle size, and this is found to be tme. Such an approach has been shown to give reasonably accurate predictions when all conditions are known however, in practical appHcations severe limitations are met owing to inadequate data and scale-up uncertainties. Hence it is stiH the usual practice to carry out tests on equipment to be sure of predictions. 

Let us make this point clearer by the following hypothetical experiment. At some initial time t0 a droplet of dye is put on the surface of a turbulent fluid (Fig. 22.9). At some later time t] the large-scale fluid motion has moved the dye patch to a new location which can be characterized by the position of the center of mass of the patch. In addition, the patch has grown in size because of the small (turbulent) eddies, more precisely, those eddies with sizes similar to or smaller than the patch size. 

The muon is about two hundred times heavier than the electron and its orbit lies 200 times closer to the nucleus. The nuclear structure effects scale with the mass of the orbiting particle as m3R2 (for the Lamb shift It is a characteristic value of the nuclear size) and as m R2 (for the hyperfine structure), while the linewidth is linear in m. That means, that from a purely atomic point of view the muonic atoms offer a way to measure the nuclear contribution with a higher accuracy than normal atoms. However, there are a number of problems with formation and thermalization of these atoms and with their collisions with the buffer gas. 

Diffusion-limited aggregation of particles results in a fractal object. Growth processes that are apparendy disordered also form fractal objects (30). Sol—gel particle growth has also been modeled using fractal concepts (3,20). The nature of fractals requires that they be invariant with scale, ie, the fractal must look similar regardless of the level of detail chosen. The second requirement for mass fractals is that their density decreases with size. Thus, the fractal model overcomes the problem of increasing density of the classical models of gelation, yet retains many of its desirable features. The mass of a fractal, Af, is related to the fractal dimension and its size or radius, R, by equationS ... 

In fact, if one measures the total number of bonds (sites) on the infinite cluster at the percolation threshold (pc) in a (large) box of linear size L, then this number or the mass of the infinite cluster will be seen to scale with L as where die (< d) is called the fractal dimension of the infinite cluster at the percolation threshold. Similar measurements for the backbone (excluding the dangling ends of the infinite cluster) give the backbone mass scaling as, de < die, where dfi is called the backbone (fractal) dimension. In fact, die can be very easily related to the embedding Euclidean dimension d of the cluster by... 

This volume corresponds to that of a cube with edges about 3.2 X 10 cm (0.32 nm) on a side. We conclude from this and other density measurements that the characteristic size of atoms and small molecules is about 10 cm, or about 0.1 nm. Avogadro s number provides the link between the length and mass scales of laboratory measurements and the masses and volumes of single atoms and molecules. 

---